NORTHWESTERN UNIVERSITY LIBRARY Manuscript Theses Unpublished theses submitted for the Master's and Doctor's degrees and deposited in the Northwestern University Library are open for inspection, but are to be used only with due regard to the rights of the authors. Bibliographical references may be noted, but passages may be copied only with the permission of the authors, and proper credit must be given in subsequent written or published work. Extensive copying for publication of the thesis in whole or in part requires also the consent of the Dean of the Graduate School of Northwestern University, This thesis by ..................... has been used by the following persons, whose signatures attest their acceptance of the above restrictions, A library which borrows this thesis for use by its patrons i is expected to obtain the signature of each user. Demonstrative geomet as a means of improving critical thinking. NAME AND ADDRESS DATE Northwestern University Library Manuscript Theses Unpublished theses submitted Tor the M a s t e r ’s and Doctor's degrees and deposited in the Northwestern University Library are open Tor inspection, but are to be used only with due regard to the rig its of the authors. Bibliographical references may be noted, but passages may be copied only with the permission of the authors, and proper credit must be given in subsequent written or published work. Extensive copying or publication of the thesis in whole or in part requires also the consent of the beau of the Graduate School of Northwestor:i University. Thi s thesis ly has been used by the following persons, whose signatures attest their acc^pt&nco of the above restrictions. A library which borrows this thesis for use by its patrons is expected to Secure the signature of each user. NAME AND ADDRESS / y< -A . , --*LAc:' f C I -*> DATE y •' //ff : wp\Vvv~" 48**'* f, » '"'0 * ZZtr? TJtC / /?*f ^.zrt '*S NORTHWESTERN UNIVERSITY DEMONSTRATIVE GEOMETRY AS A MEANS NOR IMPROVING CRITICAL THINKING A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY DEPARTMENT OF EDUCATION BY RICHARD EDWARD GADSKE EVANSTON, ILLINOIS JUNE, 1940 ProQuest Num ber: 10101427 All rights reserved INFORMATION TO ALL USERS The q ua lity o f this re p ro d u ctio n is d e p e n d e n t upon th e q u a lity o f th e c o p y subm itted. In th e unlikely e v e n t th a t th e a uthor did n o t send a c o m p le te m anuscript a n d th e re are missing pages, these will be no te d . Also, if m aterial had to b e re m o ved, a n o te will in d ic a te th e deletion . uest. ProQuest 10101427 Published by ProQuest LLC (2016). C opyright o f th e Dissertation is held by th e Author. All rights reserved. This work is p ro te c te d against unauthorized co p yin g under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 TABLE OF CONTENTS Page Chapter I - Introduction . . . . » .................................. 1 Nature of the Problem........................................ . 1 The Nature of Critical Thinking................................ 3 Need for Improving Critical Thinking Abilities of High School P u p i l s ........................................... 15 Implications of Critical Thinking for Mathematics Teachers . . 20 1. Objectives for M a t h e m a t i c s ..........................20 2♦ How Critical Thinking May Be D e v e l o p e d .............. 25 Related Studies............. . ............................... 30 Chapter Summary............................................... 36 Chapter II - Experimental Situations............................... 38 The Experimental and Control Groups......................... 38 Selection of Teachers..................... 39 1# Selection of Experimental Group Teachers . . . . . . 39 2. Selection of Control Group Teachers.................. 40 Description of School S i t u a t i o n s ............................. 40 Characteristics of Pupils Involved in the S t u d y ................ 48 1. Intelligence.............................. 49 2* Geometry P r o g n o s i s ................................. 55 3. Critical Thinking Abilities Prior to the Study of Geometry ....................................... 60 Supplementary Information Concerning Experimental and Control Group Pupils ....................................... 62 Chapter Summary........................................ 68 Chapter III - Description of P r o c e d u r e s ........................... 70 Control Group Procedures ........................... • 70 Assumptions Underlying the Experimental Teaching Procedures . . 70 Description of Experimental Group Procedures . ............. 1# Procedures Common to the Six Experimental Group T e a c h e r s ............................. 2* Examples of Teaching Procedures in Terms of the Ten Elements of Critical Thinking ..................... 74 74 78 Observer1s Reports Relative to Teaching P r o c e d u r e s ........... 109 Chapter Summary.............................................. 120 Chapter IV - Geometric Content Introduction Geometric 1. 2* 3. 4. 5* ..... ......................... . 122 ..................................... 122 Content Developed Through Cooperative Effort . . . . 122 Undefined T e r m s .................................... 123 Defined T e r m s ...................................... 124 Assumptions................................ 127 Facts or Theorems............... 131 Fundamental Construction Problems................... 145 Geometric Content Developed Through IndividualEffort . . . . . 147 1. P r o j e c t s .......................................... 147 ............. 153 2. Voluntary Mathematical Contributions Chapter Summary ........... 163 Chapter V - Non-Mathematical Conten t .............................. 164 Introduction................... 164 Non-Mathematical Content Developed ThroughGroupEffort. . . . 164 1. Group Project on Capital Punishment......... 165 2* Group Project on Compulsory Education............... 172 3. Other Examples of Group P r o j e c t s ................... 178 4. Analyses of Instruments for P r o p a g a n d a ............. 184 Non-Mathematical Content Developed ThroughIndividualEffort 186 1* Individual Projects .......................... 187 2. Voluntary Contributions............................ 198 Chapter Chapter VI - Summary.............. 217 E v a l u a t i o n ......................................... 219 Introduction................................................ 219 Bases Upon Which Outcomes Are to Be Evaluated................. 221 Written Tests and Inventory Questionnaire ................... 222 1. The Columbia Research Bureau Plane Geometry Achievement T e s t ....................... 223 2. Other Mathematical Achievement..................... 228 3, Nature of Proof Test, Number 5.3 . ............ 229 4, Test in Critical Thinking...........................233 5. Periodic Exercises and Tests ..................... 237 6, Inventory Questionnaire ....................... 240 Chapter Summary............................................. 249 Chapter VII - Evaluation (Continued)............................... Introduction jj . ............................................ 253 253 j! Direct Observations............................. 253 1. Reports from 165 Classroom Teachers................* 253 2. Heports by an Experienced O b s e r v e r ................... 2S0 j! 3. Heports by Parents end Classroom Visitors............. 261 !; Anecdotal Records.............................................. 262 !| ti l! ij ji '! ; Analyses of Pupil Diaries....................................... 269 Analyses of Written W o r k ..................................... 282 1. Excerpts from T h e m e s ................................ 283 2. Excerpts from Written Reading Reports................. 285 ! Individual and Group Projects................................... 293 ij ! i Analyses of Reading Interests . . . . . . . .................... 289 j! Chapter Summary................................................ 293 i; !iChapter VIII - Evaluation (Continued)............................. 296 Introduction Ij ! !j ij ii ji j . . . . . . . ................................... 296 Evaluation by P u p i l s ...........................................296 1. An Example Where Two Evaluations Were Submitted . . . . 298 2# An Example Where Three Evaluations Were Submitted . . . 299 3* An Example Where Six Evaluations Were Submitted. . . . 301 4. Self-Evaluations by a Pupil Transferred to an Experimental Group at the End of the First Semester . . 307 Pupil Reactions to Their Course inGeometry . .................. 313 Chapter Summary................................................ 319 |iChapter IX - General Summary of the Study and Co n c l u s i o n s........320 General Summary of the S t u d y .................................... 320 I; !I ! Conclusions . ................................................ 329 ii ij ij Selected Bibliography........... i!Appendix A - Analysis of Situations for Elements of Critical Thinking 332 339 j! ;|Appendix B ................. 364 ;! I. Complete Tabulationof Tests Results andReports by j Observers for Each Pupil in Experimental and Control Groups 365 j' II. Summary of Responses to the Initial Form of the Inventory j! Questionnaire Administered to Both Grhups at the Beginning l ! of the School Y e a r ...........................................384 I! III. Summary of Responses to the Follow-up Form of the Inventory Questionnaire Administered to Both Groups at the End of the School Year .......................................... 387 Appendix C .......................................................... 390 1. Inventory Questionnaire (Preliminary F o r m ) ................... 391 2. Otis Self-Administering Tests of Mental A b i l i t y ............. 397 3, Orleans Geometry Prognosis T e s t .............................398 4* Progressive Education Association Nature of Proof Test 5.3 . . 399 5# Columbia Research Bureau Plane Geometry T e s t ................. 406 6. Critical Thinking T e s t .................................... . 407 7* Inventory Questionnaire (Pinal or Follow-up F o r m ) ........... 413 8. Examples of Periodical Tests ....................... Appendix D - Analyses of Literature on the Teaching of Mathematics for O b j e c t i v e s 415 . 427 Appendix E - Examples of Self-Evaluations by Pupils of Experimental Groups.................................... 442 Appendix F - Enrichment Materials Related to Mathematics for High School P u p i l s .................................... 469 1. References of a Historical and Cultural N a t u r e ............... 470 2* References Relating to Social and Practical Uses of Mathematics and to Other School Subjects.....................474 3# References Relatingto the Concept of N u m b e r ................. 477 4, References Relatingto 5* References Relatingto the Concept of Function............... 479 6. References Relatingto the Concept of P r o o f ................. 480 the Concept of Measu r e m e n t........... 478 7. References Relating to Mathematical I n s t r u m e n t s............. 481 8. References Relating to Recreations and Extra-Curricular Activities . . . . . . . . . ............................... 482 LIST OF TABLES I. jj II. Page Teachers, Schools, and Pupils Involved in the Study . . . . I.Q. Comparisons of Experimental end ControlGroups .... ! IV. I.Q. Ranges and Means for Teacher Group 49 53 . jj III. I.Q. Ranges and Means for School G r o u p .......... 39 .............. 53 ij i| V. j| jj jj VI. Distribution of I.Q. *s by Quartiles for Each Teacher Group .......................................... 54 Geometry Prognosis Scores Comparisons of Experimental and Control G r o u p s ....................................... 56 VII* Ranges and Means of Geometry Prognosis Scores for School G r o u p ............. 58 VIII Ranges and Means of Geometry Prognosis Scores Per jj Teacher G r o u p ............................................. 59 ij IX. i! !i!| ijX. jj ijXI. ! I!XII. I! jj jj jiXIII ij j! ||XIV. |j ii jXV. jj jj i' XBI. ij Distribution of Geometry Prognosis Scores by Quartiles for Each Teacher G r o u p ................................... 60 Distribution and Classification of Individual Projects Distribution of Voluntary Contributions . . 147 ................. 198 i ! Comparison of Experimental and Control Groups in Terms of I.Q. *s, Geometry Prognosis Scores, and Geometry Achievement S c o r e s .................................... 323 Decile Distribution of Scores on the Geometry Prognosis and Achievement T e s t s .................................... 225 Showing the Expectation of Success in Geometry from Standings in the Geometry Prognosis Test . . . . . . . . . 226 Tabulation of Means, Probable Error of Means, and Ranges of Responses to Each Question on the Critical Thinking T e s t .................................................... 234 Summary of Responses to Part II of the Follow-Up Form of the Inventory Questionnaire...............................315 ACKNOWLEDGMENTS I take this opportunity to acknowledge with gratitude my profound indebtedness to all who have assisted either directly or indirectly in the development of this study, because I do recognize that no research is the exclusive work of one individual. This is particularly true of the present study, which has enlisted the cooperation of nearly ninehundred people. I am particularly indebted to Dr s. William G. Brink, Chairman, S. A. Hamrin, J. M. Hughes, and L. W. Webb, members of my advisory committee, and to other members of the Northwestern University faculty for their kind advice and patient assistance; to Dr. Harold P. Fawcett of Ohio State University High School for many very helpful suggestions; to R. S. Peterson of New Trier Township High School for reading the entire manu script and offering constructive criticisms. Furthermore, I am indebted! (l) to Genelle Bell of the A. 0. Jones High School in Beeville, Texas; Ethel M. Evans of New Trier Township High School in Winnetka, Illinois; and to Gladys Brand, Dora Frazee, and Helen F. Luecki all of Roosevelt Senior High School, Chicago, Illinois for their patience and diligence in putting the Experimental procedures into prac tice in their respective schools; and (2; to Harold E. Qhlson of the Niles Township Community High School in Niles Center, Illinois; E. T. Foster and Anne L. Sandberg, both of Senn Senior High School in Chicago, Illinois; and to Iva Eae Bevins, Mary Jane Hartmann, and Robert S. Woodruff, all of Oak Park and River Forest Township High School for faithfully administering the various Comparative tests to their respective pupils. I am also indebted to the administrators and particularly the department heads of mathematics in these schools for permitting the various teachers to cooperate in the experiment. I am also very much indebted to C. A. Michelman of Northwestern University for visiting the various schools and observing the methods of teaching. I am further indebted to him and to Dean Flagg, the assistant principal of the Niles Township Community High School for their scoring and checking of the Critical Thinking Test constructed for this study. (The former acted as a neutral scorer, since he was not involved in the experiment, while the latter represented the Control Groups, and the writer of course represented the Experimental Groups.) My indebtedness should also be extended to the 165 classroom teachers who acted as observers for critical thinking behavior and who contributed many valuable anecdotal records regarding the pupils in the Experimental Groups. Finally, I feel that my greatest obligation is to the pupils them selves for the fine spirit of cooperation and enthusiasm they have shown without which this study would perhaps never have been realized. R.E. G. A free, virtuous, and enlightened people must know well the great principles and causes on which their happiness depends, — James Monroe CHAPTER I INTRODUCTION Nature of the Problem j Through the use of demonstrative geometry as a means, this study attempts to describe and to evaluate teaching procedures directed toward j i the improvement of critical thinking abilities of young people. In order to determine the relative effectiveness of the teaching procedures, Experimental and Control groups were established in several public high | schools. The Experimental procedures were applied by six teachers in three public high schools, while the Control procedures were applied by six different teachers in three other public high schools* Specifically stated, the problem is to compare Experimental and Control teaching procedures in demonstrative geometry for the purpose of determining their relative effectiveness in developing critical thinking abilities of high school pupils. 2 In order to determine the relative value of the two teaching pro cedures, it becomes the purpose of this study to seek answers to the following questions? (1) Can demonstrative geometry be used by mathematics teachers as a medium or a means to develop more effective critical thinking abilities among high school pupils? (2) Does the usual course in demonstrative geometry accomplish this purpose? (3) How do pupils in geometry classes where critical thinking is stressed compare in this ability with pupils in geometry classes where this objective is not emphasized? (4) How do pupils in geometry classes where critical thinking is emphasized compare in knowledge of the usual subject matter of geometry with pupils in geometry classes where this objective is not emphasized? (5) How do pupils react or respond to situations where improvement of critical thinking is the major teaching objective as against situations where this objective is not emphasized? In order to bring about an effective solution of the problem it was found necessary to: (1) clarify the meaning of critical thinking in the way in which it is to be used and emphasized in this study; (2) analyze the behavior of pupils in several different situations in order to determine common elements of critical thinking these elements then to be used as a frame of reference for the study; (3) point out the need for this kind of thinking and its impli cations for mathematics teachers; and then to: (1) select schools, teachers, and pupils for the study; (2) establish Experimental and Control groups; (3) develop Experimental teaching procedures; (4) devise ways and means for securing and recording data; (5) evaluate the outcomes. 3 The Hature of Critical Thinking There is a pattern of thought in our contemporary .American mind, which for the want of a name may he referred to as “critical thinking’1* Educational literature is replete with the use of this term and it is used with a variety of connotations* Controversy frequently arises over the question of what “thinking11 really is. Keyser* points out that, "Psychologists have not yet heen ahle to tell us what thinking is. But most knowledge is knowledge of the undefined, as knowledge of love, for example, or knowledge of life; and as for thinking, though we cannot de fine it we know enough ahout it to know that there are many kinds". Some psychologists, perhaps, would take issue with Keyser hy pointing out that the end products of thinking are so numerous and so dominant in our current civilization that we can readily define thinking through human behavior. 2 Dewey tells us, "The various ways in which men do think can be told be described in their general features . . . andcan The origin of thinking issome perplexity, confusion or doubt." *2 Childs says, "Thinking is the active seeking of an adequate stimulus, the remaking of the problematic situation into a situation with meanings so clarified that the behavior can go forward . . • Thinking is inquiring. Logic is a study of the conditions which lead to success and failure in conducting inquiry. AL1 thinking originates in some specific affair. is a piecemeal, not a wholesale, activity. It starts in a situation It of 1 Keyser, Cassius J., "Thinking ibout Thinking", Hew York: E. P. Dutton and Company, 1926, p.l. 2 Dewey, John, "How We Think". Hew York? D. 0. Heath and Company, 1933, pp. 3-15. 3 Childs, John L., "Education and the Philosophy of Experimental!sm". Hew York! The Century Company, 1931, pp. 77-79. ambiguity; its business is to get rid of that ambiguity." Dewey^ defines thinking as "that operation in which present facts suggest other facts (or truths) in such a way as to induce belief in what is suggested on the ground of real relation in the things themselves, a relation between what suggests and what is suggested." To point out further that there are many kinds of thinking, Keyser® gives three general types! (l) Organic thinking - a type that involves organic behavior, that is, a response to stimulus; (2) Empirical thinking - a type that involves trial and error, and is experimental in nature; and (3) Postulational thinking - which is the "If - Then kind, distinguished from all others by its characteristic form! if so-and-so, then so-and-so"* He makes the further distinction* "(l) If thinking be autonomous (meaning postulational) or empirical, it is human, (2) if it be human, it is autonomous or else empirical, (3) if it be subhuman, whether done by animals or by human beings, it is neither autonomous nor empirical; and (4) if it be neither of these, it is subhuman." Bobinson® distinguishes four types of thinking! (l) the reverie, or a free association of ideas; (2) Making practical decisions - weighing a decision, in other words "homely decisions forced upon us by every day needs"; (3) Rationalization - is stimulated when anyone questions our beliefs and opinions; and (4) Creative thinking - "the kind of thought that has raised man from his pristine, subsavage ignorance and squalor to the degree of knowledge and comfort which he now possesses". 4 Dewey, John, "How We Think", op. cit* p. 12. 5 Keyser, Cassius J., "Thinking .About Thinking", op. cit. pp. 6-45. 6 Eobinson, James Harvey, "The Mind in the Making". Hew York; Harper and Brothers Publishers, 1921, pp. 38-49. 5 j Dewey likewise identifies thinking in terms of four types! (l) Thinking as a stream of consciousness - day dreaming; (2) Thinking of an imaginative sort - restricted to things not directly perceived, an orderly chain of ideas; (3) Thinking as practically synonymous with belief - little or no evidence for thinking that one does, however, it does involve a controlling purpose or end; and (4) Heflective thinking, which he defines as "active, persistent, and careful consideration of any belief or supposed form of knowledge in the light of the grounds that support it and the further conclusions to which it tends." Because of the wide variety of meanings ascribed to the term "thinking" it is apparent that its scope is too broad for the purposes of this study. Furthermore, the present study is concerned with a type of thinking that requires personal examination, questioning, scrutiny and inquiry, - a type that aims at valid conclusions. Therefore, if some persons wish to call this type of thinking "reflective", or "postulational" or "creative", they may be fully justified in doing so. There still remains the task of describing, clarifying, and defining the type of thinking that is to be emphasized in this study. In the beginning, therefore, let it be assumed that there are many kinds of thinking. Specifically described by the adjective "critical" there may also be many kinds and varieties of critical thinking, and it in its turn may be sub-divided into several varieties in the different areas of human knowledge. Educational literature is replete with the use of this term, yet what one person may term critical thinking another may not so describe. The term very much involves meanings (semantics). Stuart Chase** 7 Dewey, John, "How We Think", op. cit. pp. 1-9. 8 Chase, Stuart, "The Tyranny of Words", Harpers Magazine. Hov*1937,p.563. 6 points out that, "Three investigators - Korzybski, Ogden, and Bichards agree broadly that the two besetting sins of language are the identi fication of words with things■ and the misuse of abstract words". Because there are many men of many minds and many men who do not agree, few would select precisely the same examples of critical thinking. However, in order to clarify the major objective underlying this study, namely to improve critical thinking abilities among high school pupils, it was decided to state some specific situations wherein "critical thinking" was in evidence. The following three situations will illustrate more clearly the writer*s conception of at least three levels of critical thinking. Other situations are presented in Appendix A. Situation Uo. 1 - Mary read that singeing the stems of cut flowers before putting them into water will make them keep longer. She thought she would conduct an experiment to prove this statement. She did this and concluded that her experiment proved that singeing the stems of cut flowers makes them stay fresh longer. She was so elated over the results of her experiment that she had to tell her science teacher all about it. The teacher, of course, wanted to know how the experiment was performed so Mary described it as follows: "Last Sunday evening I cut some roses and placed them in a vase of water. Monday morning I cut some sweet peas and after singeing the stems I put them into some cold water from the refrigerator. I then set the vase containing the sweet peas in the library beside the vase of roses. Wednesday morning I noticed that the roses were very wilted although the sweet peas still looked fresh. The sweet peas did not begin to show signs of wilting until Thursday evening. Since the sweet peas remained fresh for a day longer than the roses, I proved that singeing the stems of cut flowers before putting them into water will make them keep longer." To Mary*s disappointment the teacher very graciously informed her that the results of her experiment proved very little, if anything. Mary was thinking critically only when she decided to verify the statement by means of an experiment of her own. Had she made a critical analysis of her 7 method of procedure she would have exercised a higher degree of critical thinking. Situation Ho.2 - While reading a newspaper one evening, W.C.'s attention was directed to an overcoat advertisement. It read as follows: 'Its fabric (referring to the Angora Knit-tex) is based on the soft, silken hair of the Angora goat which lives in regions where the days are hot and the nights cold, therefore, the protective hair of this animal which is used in this coat is adaptable to varying degrees of temperature1* W.C. recognized that this statement was the same type of deductive reasoning that he had encountered in proving some of the theorems in his geometry class* He brought the advertisement to class the next day and pointed out why he doubted that certain living qualities of the Angora pelt, subjected to varying degrees of temperature, were actually retained in the Knit-tex material after the animal was killed. ?.M. remarked that W.C. was making an assumption, because there was no evidence presented for or against the statement. W.C* insisted, however, that facts in support of the conclusion in this advertisement were lacking and furthermore the advertiser was trying to sell his product regardless of the facts that were necessary to support his conclusion. Had W.C. or any other member of the class searched for more pertinent facts in this situation, consulted experts, looked up references relative to the fur industry and made a detailed analysis of Knit-tex, its meaning, the nature of the material, the manufacturing process, and other information, such inquiry would have represented a higher level of critical thinking* Situation No- 3 - In discussing the inductive method of reasoning in a geometry class, the teacher led his pupils from the properties of a point to that of a fourth dimensional figure. It was inferred that by this method one could readily determine the properties of a fifth, sixth, and n-th dimensional figure. It seems that this portion of the discussion aroused the interest of every member of the class* Several pupils insisted that one could not describe properties of a fourth dimensional figure if he could not see it, let alone a fifth, sixth, and so on up the scale. H.B. remarked that she understood her older brother, who was studying chemistry at that time, to say that properties of some of the elements in chemistry were known long before the element was discovered* S.S. remarked that this was also true in several instances in astronomy and physics* (Ehe teacher, following the discussion, referred the class to Slosson's "Easy Lessons in Einstein", and remarked that 8 this hook contained some information ahout the 11tesseraet" (a fourth dimensional solid)* W.M. burst into the teacher's room after school and remarked* "I just looked up the tesseract and found that it had the exact number of points, lines, squares, and cubes that we figured out inductively it would have. Is it possible to work out a formula for the properties of an i£-th dimensional figure in the same way?" The teacher replied that to the extent of his knowledge he had not encountered such a formula, but that through the inductive method one should be able to make such a generali zation. W.M* also remarked that the cubes in the tesseract did not look like cubes. The teacher asked him to reason that out for himself, and added, "If you can answer these questions and perhaps devise some ingenious construction of a tesseract on a three dimensional screen, such that it would have the appearance to the eye, of eight perfect cubes then your name would undoubt edly go down in history as one of the world's renowned contri butors to knowledge". Many elements of critical thinking entered into this situation when the pupils questioned the terms used, the assumptions behind the inductive development of properties of geometric figures of n- dimens ions, the desire for more facts, the motive behind the teacher's presentation, W.M.'s inquisitiveness, the conclusions reached and their tentativeness, their consequence to the individual and to the group. W.M. was thinking critically in this situation and if he continues to do so he may turn out some highly creative work. As one looks over the situations presented herein, and as one thinks about others, one is readily convinced that human experience is saturated with conflicts and perplexities of every kind and description and that the mental processes involve many kinds of thinking. The way in which one may respond to such situations, in either event, implies thinking. This thinking may be loose as in the singeing experiment performed by Mary, or it may become very highly critical and creative as exemplified by Lobachewsky when he challenged one of Euclid's assump tions and through his process of critical thought created a geometry that has revolutionized the thinking of mankind. 9 The question might he raised? Way not call this type of mental activity just plain “thinking0 and let it go at that? Precisely stated, it is “thinking0, hut on the other hand the term “thinking0 is too hroad and too inclusive in its scope to describe the type of behavior with which this study is vitally concerned. The adjective critical is a derivative of the word critic, which refers to a person or one who expresses a judg ment on any matter with respect to its value, truth, or beauty. The adjective “critical0, therefore, suggests a very special phase of thinking. Thus, critical thinking becomes a process of becoming aware and criticising the thinking that has already taken place* In other words, it is a process of thinking about thinking from the point of view of a critic. In this sense, however, it does not imply an attitude of destruction and tearing asunder that which has taken mankind untold energy and effort to build up. In fact it implies an appreciation for the past as well as the present thinking that has taken place plus one additional feature, namely that of questioning and examining existing values, not only for their inconsistencies or weaknesses, but also for the purpose of bringing about improvements that will benefit mankind. In other words, it implies a process of mental activity instead of mental passivity and becomes inherent in human behavior when a person is analyzing his own thinking as well as the thinking of others through the media of self-scrutiny, questioning, discrimination, search, and research with respect to any situation that may be of interest or of vital concern to him. Since critical thinking was described as a process of thinking about thinking, that is analyzing and criticizing thinking, it becomes necessary to study and to analyze situations where critical thinking has taken place in order to determine the common elements of this process of 10 thinking* That this he done is very essential to the success or failure with regard to the outcomes of this study, - and as Dewey^ points out "Common elements are the basis of so-called 1transfer1. That is, the carrying over of skill and understanding from one experience to another is dependent upon the existence of like elements in both experiences . . . A young child whose acquaintance with quadrupeds is limited to a dog will tend to call any four-footed animal of a similar size 'doggie1. Similar qualities are always the bridge over which the mind passes in going from a former experience to a new one. Now thinking is a process of grasping in a conscious way the common elements. It thus adds greatly to the availability of common elements for purposes of transfer. Unless these elements are seized and held by the mind (as they are in a rudimentary way by the symbol 'dog'), any transfer occurs only blindly, by sheer accident." It is with this view in mind that the following two situations were selected as examples of analyses for elements of critical thinking. (Other examples of such analyses may be found in Appendix A.) Situation No. 1 - A group of pupils was presented with this statement: "Many people in the United States have recently stated that they would never bear arras in any way. Others feel that it is their duty to support their government at all times whether in peace or in war." They gave a wide variety of responses which had in them several elements of critical thinking. Some of the elements and responses follow: (1) The element of seeking motives became apparent when R.B. asked the question* "Who made this statement? What is he driving at?" (2) The element of seeking the meaning of terms was pointed out by several pupils, for example: "How many neo-ole? How recently? How many would never bear arms in any war? Does he mean financial or -physical support? It seems he ought to explain what he means by duty. When are two nations at war? Are Ghina and Japan at war? Does government mean all of the citizens of a nation, or the persons in power or in control?** 9 Dewey, John, "How We Think", op. cit. p.67. 11 (3) The element of detecting assumptions “became apparent when D.W. remarked, "The group representing the former statement assumed that all wars were unnecessary, while the latter assumed that war may be necessary and that loyalty to one's country should be the cardinal virtue of every citizen." (4) The element of seeking more facts was brought out when W.G. said, "We ought to have more facts relative to both points of view." Several facts were presented as follows* V.M. remarked, "The pastor of our church delivered a sermon recently in which the 'Eye Report* of the munitions investigation was mentioned. He told us how the American munitions interests made it possible for Germans to kill our soldiers with American-made products and for German soldiers to be killed with German-made products." E.G. added, "Certain interests utilized the press to spread propaganda in order to stir the emotions of the people for war. This was achieved in the last World War and it seems that it is being tried today. l*or some reason or other, however, the people have not fallen for this propaganda as yet." B.C. remarked, "Your government is made up only of people that you put into office; therefore if they decide it is best for the country to declare war, then you should support the idea or be more careful as to whom you elect for the office." R.M. said, "I would agree with B.C., if the persons who decide that war is best for their country would, themselves, be the first to do the fighting." (5) The element of testing facts for pertinency was brought out by S.R. when he said, "We can get many facts in favor of war and many facts opposed to war. What we are trying to decide here is whether or not it is one's duty to support one's government at all times and to agree upon what we mean by support." (6) The element of discriminating between facts and assumptions was brought out when Y.M. said that B.C. was merely expressing an opinion or rather making an assumption and not stating an established fact. (7) The element of bias or prejudice entered into this situation several times. D.R. said, "Our fore-fathers fought for democracy so it is our duty to preserve it for future generations. Any one who wouldn't fight for the American flag under any circumstances is unpatriotic." V.M. replied, "What do you mean by patriotic or unpatriotic, or by fight? You are not thinking critically when you make such statements, because S.R. has already pointed out how we failed in our fight for democracy as a result of the last war. Besides, you should realize that European dictatorships have imperiled democracy more than ever. X think you are biased because you are considering only the facts in favor of your own point of view." (8) The element of testing conclusions for consistency is apparent in this situation. In either event the basic assumptions under lying peace or war ought to determine the nature of the 12 conclusions reached# For example, D. Me* said, "If no one would hear arms we would not have wars.11 This conclusion is consistent with his assumption# On the other hand S.R. remarked that, "If we assume that some wars are just, then under such circumstances it becomes the duty of every ablebodied citizen to support his government in such a crisis.1* (9) The element of recognizing the need for formulating only tentative conclusions was brought out by R# Me. when he said, '•It is difficult to conceive of a situation wherein no one would bear arms. On the other hand it is also probable that a supposedly just war may be found to be unjust. Therefore, any conclusions that we reach must be only temporary, because changing conditions may bring out facts necessitating changes in our assumptions." (10) The element of testing conclusions for their individual and social consequences was implied in the previous item. However, R.B. remarked, "If it is the duty of every citizen to serve his country to the best of his knowledge, then going to war is not to the best knowledge of some people. Besides, those that declare war should be the first to go# * • Furthermore, it must be remembered how war destroys many men that could make great contributions to the general welfare of society# Finally, the outcomes of past wars have seriously handicapped human welfare and human progress#" Situation Ho. 2 - Recently, a certain class in geometry, having had no previous experience or knowledge of the proof concerning the theorem which states that the sum of the interior angles of a triangle is equal to 180°, proved this theorem by a method unknown to the writer prior to this situation. This is an excellent example of critical thinking in a geometry class, because the group was permitted to reason about the situation in their own way, to challenge and question each other at will, to formulate any assumptions upon which the entire group would agree, and to test for pertinency any facts that were presented* Thus through mutual agreement the group arrived at the conclusion that, "If certain definitions and assumptions are accepted, the sum of the interior angles of a triangle is equal to 180°." The following analysis of this situation reveals several elements of critical thinking. (l) The teacher's motive was to teach the pupils some geometry and at the same time provide an oppor tunity for critical thinking. Several pupils recognized this motive. Then too, there were pupil motives involved, for example, motives appearing in 13 the form of a desire to remove a perplexity and to gain recog nition or reward for effort. (The latter motive, however, has little educational value if it becomes extrinsic in its nature.) (2) The meanings of various terms, such as vertex, transversal, alternate interior angles, supplementary angles, point, line, angle, right angle, straight angle, parallel lines, perpendicular lines, and opposite angles, were questioned by the different members of the class until mutual agreement ensued. (3) The following assumptions were questioned during this discussion: "A quantity may be substituted for its equal in any expression. A line may be extended, or it may be limited at any point. Only one perpendicular may be drawn to a line from an external point, or only one perpendicular may be drawn to a line at a point in the line." In general, the assumptions corresponded very closely to the Euclidean postulates. (4) The group brought out a number of related facts, some of which were as follows: "If two or more parallel lines are cut by a third line, the corresponding angles are equal. If two or more parallel lines are cut by a third line, the alternate interior angles are equal. If the interior angles on the same side of a crossing line (transversal) are supplementary, the lines are parallel. Lines perpendicular to the same line ©re parallel to each other." (5) The group tested the above facts for pertinency to the situation and agreed that only the second and fourth facts applied* (6) During the discussion the observer noted a tendency on the part of the group to discriminate between facts and assumptions. Eor example, "Lines perpendicular to the same line ©re parallel" was regarded as an assumption by some and a fact by others. The group, as a result of this phase of the discussion, agreed to classify a statement as a fact only after it was proved. Since this statement had been proved earlier, the group agreed to classify it with the facts. Two other statements were questioned in the same way, namely "90° in a right angle" and "When two straight lines intersect, the opposite angles are equal." (7) Some pupils became somewhat stubborn and refused to relinqtiish their position without Offering reasons or evidence in support of their belief. Eor example, M.S. insisted that "Lines per pendicular to the same line are parallel" is an assumption, yet she refused to give adequate reasons that would satisfy the other members of the group. In other words either she needed to evaluate herself for bias or she should have been able to convince at least some of the members in the group that her position was a valid one. (8) There was also the element of formulating a generalization, or reaching a conclusion and then testing this conclusion for consistency with the assumptions underlying it. Some of the 14 questions raised at this point are as follows? "Does the conclusion follow logically from the premises or the assumptions that were made? Could the same conclusion he reached using different assumptions? Different facts?" (9) Another element seamed to he prevalent in the discussion, namely, the recognition of the tentativeness of conclusions* This element grew out of the previous one in that the pupils became sensitive to the dependence of conclusions upon their assumptions. In other words, the conclusions were true only if the assumptions were true, (10) The conclusions reached in any area of thought imply individual as well as social consequences. In this situation the questions were asked? "What are the limitations of Euclidean geometry? Why did Einstein use Riemaiu^s geometry in establishing his theory of relativity? What is the essential difference between the Euclidean, elliptic, and hyper?>olic geometries with respect to the conclusion reached in this situation?" Answers to these questions were somewhat beyond the maturation level of this group; however, the idea that consequences are involved became apparent. In the light of the preceding analyses as well as those to be found in Appendix JL, certain elements of critical thinking seem to be common. One may generalize these findings by stating that the behavior of a pupil who exenqplifies a high level of critical thinking will be marked by the following characteristics? (1) He will try to detect motives behind any situation of concern to him. (2) He will question the meaning of terms in the situation and seek satisfactory definitions or descriptions of them. (Hereafter by the situation is meant pny situation of concern to the -pupil.) (3) He will detect and question underlying assumptions in the situation (the stated as well as the unstated ones). (4) He will search for more facts pertaining to the situation. (5) He will test these facts for pertinency to the situation. (6) He will endeavor to discriminate between facts and assumptions in the situation. (7) He will evaluate himself and others for bias or prejudice in the situation. 15 (8) He will test conclusions for consistency with underlying assumptions in the situation. (9) He will recognize the importance of formulating only tentative conclusions in the situation. (10) He will evaluate conclusions in the situation in terms of individual and social consequences. It is of particular importance to note at this point that the above ten elements of critical thinking will serve henceforth as a frame of reference for this study. In other words, any question pertaining to the study will he answered and evaluated in terms of these ten elements. Heed for Improving Critical Thinking Abilities of High School Pupils The need for developing more effective critical thinking abilities of high school pupils is generally recognized. This need has been brought to the educational foreground as a result of one of m a ^ s great periods of transition. Modern life, during the past several decades, has become extremely conplex and greatly enhanced through the medium of invention and discovery. Greatly improved methods of transportation make it possible to travel extensively and within a relatively short period of time. Highly improved ways and means of communication facilitate the exchange of ideas. Modern production and distribution is fostered by widespread advertising. In the light of these penetrating and far-reaching social forces, the citizenry in a democratic society are challenged in their thinking to the limits of their capacities. A society that calls itself democratic, therefore, necessitates the education of individuals who are capable of self-direction and a high degree of critical thinking. If young people ©re to participate intelli gently in such a society, it is of utmost importance that their behavior be based upon the best thinking they are capable of doing. This is in 16 keeping with Kilpatrick*s10point of view when he says, "Our rapid social change requires thinking, and not mere habit, to deal with it. . . unequal change has produced cultural lags which upset our former cultural balance, so that we now need further changes in order to make culture work, part with part, as an effectual whole together* These things require that our citizens shall learn to criticise and judge our various institutions so 11 as to bring the lagging parts up abreast of the rest.11 Brink expresses a similar point of view when he says, “In a democratic form of society such as that in which we live, it is a peculiar responsibility of the school to prepare boys and girls for intelligent citizenship. It must help them not only to become aware of the problems of their times, but also to think through these problems, to discover underlying causes, to judge the sound ness of opinions expressed over the radio, in the press, and on the lecture platform* Without an intelligent citizenry, democracy is imperiled.11 This need for improving critical thinking abilities among high school pupils is further stressed by C o e ^ when he makes the statement that, “A youth is not well-educated until he has had practice in the critical valuation of the institutions and the ways of the society of which he is a part. He must have practice in making his own judgments; his teachers cannot possibly do this for him, though they can spur him to judge his own judgments. . . The only conceivable way in which the student can get ready for independent judgment then is by practicing independent judging now and securing correction for his errors before they become too costly.0 10 Kilpatrick, William Heard, “Remaking the Curriculum*1, Hew York! Hewson and Company, 1936, p* 34* 11 Brink, William G., “Directing Study Activities in Secondary Schools11, Hew York? Doubleday, Doran and Company, Inc., 1937, p*3. 12 Coe, George A., “What Ails Our Youth*1. Hew YorkJ Charles Scribner and Sons, 1924, p.50. 17 This does not necessarily shift all of the responsibility from the adult to the child, but it does imply that in our current society youth must become critically-minded and sufficiently self-directive so as to stand ontts own feet, Stewart 13 expresses a similar viewpoint: “Youth cannot sit back and wait for the adult world to solve its problems. must help plan and fashion its own future. It Tor the young people of today live in a different world from that in which their parents grew up." 14 Bertrand Russell expresses a similar point of view: “I should encourage a habit of intelligent controversy among older boys and girls, and I should place no obstacles in their way even if they questioned what I regarded as important truths. I should make it my object to teach thinking, not orthodoxy, or even heterodoxy ... to knowledge shall exist of any sort or kind. In my school, no obstacle I shall seek virtue by the right training of passions and instincts, hot by lying and deceit. In the virtue that I desire, the pursuit of knowledge, without fear and without limitation, is an essential element, in the absence of which the rest has little value." Past schooling has sought to give pupils a knowledge of many subjects and facts, some of which are highly scientific, nevertheless this education has failed to produce in young people a critical attitude towards certain beliefs, or towards "certain authorities which claim their adherence", Everett^ says, "Business has made use of many agencies - among them the schools, the cinema, the radio, and the press - in 13 Stewart, Maxwell S., "Youth in the World Today", Public Affairs Pamphlets. Hew York: Public Affairs Committee, Inc.,No.22,1938,p.37. 14 Russell, Bertrand, "Education and the Good Life". Hew York: Boni and Liveright, Inc., 1926, pp. 287-289. 15 Everett, Samuel, "Democracy Paces the Future". Hew York: Columbia University Press, 1935, p. 135. 18 propagandizing the great mass of the American people. Educational agencies have taught people to read, hut not to discriminate and to do critical thinking. “ The Institute for Propaganda Analysis16 elaborates upon this idea somewhat further by pointing out that, “Young people should come to know such elementary facts as the following* propaganda plays a role of enormous importance in the lives of all of us* There are many of these propagandas; our nation and the world teem with them and with their con flict* groups* They come from organized groups or representatives of organized They touch every aspect of our lives. political, religious* Chiefly, they are commercial, They often determine the brand Jof toothpaste we use, the kind of clothes we wear, the school we attend, what we think and do about President Hoosevelt*s New Deal measures, the election of a candidate for mayor of our city, or war in China or Spain* We are fed propagandas by our political parties, our schools, our churches, our clubs, our news papers and magazines, our radios and motion pictures, and even by our textbooks.* The responsibility for developing more effective critical thinking abilities among our young people is being neglected at the present time. TO Dwwey points out that, “Our schooling does not educate, if by education be meant a trained habit of discriminating inquiry and discriminating belief, the ability to look beneath a floating surface to detect the conditions that fix the contour of the surface, and the forces which create its waves and drifts. We dupe ourselves and others because we have not that inward protection against sensation, excitement, credulity, 16 Institute for Propaganda Analysis* “Propaganda, How to Recognize It and Deal with It“, New York, 1938, pp. 15-16. 17 Dewey, John, Characters and Events. New York: Henry Holt and Company, 1929. Vol. II, pp. 779-781. 19 and conventionally stereotyped opinion which is found only in a trained mind* This fact determines what passes as an educational system • * . What will happen if teachers "become sufficiently courageous and emanci pated to insist that education means the creation of a discriminating mind, a mind that prefers not to dupe itself or to he the dupe of others?11 The authoritarian nature of past schooling tended to stifle critical thinking and in its stead propagated and imposed certain fixed patterns of thought and belief upon immature minds. The teacher was a task master whose word was law; textbooks were infallible; pupils were led to verbalize, imitate and memorize in accordance with the whims and fancies of the teacher. Observation of teaching practices in many situations still impresses one with the machine-like manipulative and routine aspects of 18 current instruction. Brink again points out that, "The inadequacies of the traditional school are apparent. Youth has not been truly educated, but merely trained to accept without question the dogmas of religion, philosophy, science, politics, and convention, to repeat parrot-fashion the precepts of parents, teachers, and preachers, and to recite glibly the rules and facts found in a few carefully prescribed textbooks." 19 Today in one of his recent books, Dewey suggests the dangers of not only the "old" but also the "new" education when he says, "Unless the problem of intellectual organization can be worked out on the ground of experience, reaction is sure to occur toward externally imposed methods of organization. There are signs of this reaction already in evidence. We are told that our schools, old and new, are failing in the main task. They do not develop, it is said, the capacity for critical discrimination 18 Brink, William Gr., op. cit., pp. 3-4. 19 Dewey, John, Experience and Education. Few York! Macmillan Company, 1938, p. 107. 20 and the ability to reason."2^ finally, Homer P. Hainey21 writesi "It has been emphasized from the beginning of .American democracy that a democratic form of government must be built upon a system of universal education - an education which would enable the masses to be intelligent about all the problems involved in a democratic society." One has only to look around him to see that modern .American society is replete with situations that call for the type of thinking suggested herein. If we continue to cherish such ideals as equality of opportunity, freedom of speech, universal suffrage, respect for the rights of others, social justice and a classless society, then critical thinking ought to become one of the major emphases of our public system of education. Implications of Critical Thinking for Mathematics Teachers Having considered the need for developing critical thinking abilities among yoking people, the question now arises; "What are the implications of this type of thinking for mathematics teachers and particularly for the teaching of demonstrative geometry?" In order to answer this question it appears advisable! (l) to examine the extent to which objectives in teaching demonstrative geometry emphasize the development of critical thinking, and (2) to point out how these abilities may be developed# 1. Objectives for Mathematics A close scrutiny of sources reveals a wide variety of objectives# Space does not permit detailed treatment of the voluminous literature that has been published on this subject. Por a complete and thorough 20 Note: Dewey interprets education as the scientific method by means of which man studies the world, acquires cumulatively knowledge of meanings gynfl values in order to use these as data for critical study and intelligent living. 21 Hainey, Homer P., How Pare Auerican Youth. Hew York: D. .Appletort-Century Company, 1937. p# 48# 21 historical treatment with regard to significant changes and trends in the teaching of geometry the reader is referred to the work of Stamper^. Eor a more recent historical treatise on this same iubject the reader is referred to the work of Shibli23. Besides the two references just mentioned the following sources for teaching objectives in mathematics were investigated in terms of the elements of critical thinking developed earlier in this chapter. These 24 sources are as follows5 (l) Books on the teaching of mathematics by David E. Smith, J. W. A. Young, Arthur Schultze, Ernest B. Breslich, <J. 0. Hassler and R. R. Smith, William L. Schaaf, Raleigh Schorling, David E. Smith and William D. Reeve, and the Eifth Yearbook of the Rational Council of Teachers of Mathematics; (2) Reports of various committees on the teaching of mathematics, namely the Rational Committee of Eifteen on Geometry Syllabus - 1912, Rational Committee on Mathematical Requirements 1923, Eirst Committee on Geometry - 1929, Second Committee on Geometry 1930, Third Committee on Geometry - 1932, Tentative Report of the Math ematics Committee of the Progressive Education Association - 1938, and a Preliminary Report by the Joint Commission of the Mathematical Association of America and the Rational Council of Teachers of Mathematics - 1938; and the studies by Christofferson and by Eawcett. The books and reports first mentioned reveal considerable disparity among the aims or objectives in teaching mathematics. This disparity is due, at least in part, to the existing confusion underlying the purposes for teaching mathematics. Eundamentally, it is due to a lack of agreement 22 Stamper, Alvin W., The History of the Teaching of Elementary Geometry. Rew York: Bureau of Publications, Teachers College, Columbia University,1909. 23 Shibli, J., Recent Developments in the Teaching of Geometry. State College; Pennsylvania State College, 1932. 24 References for each of the following are recorded in the bibliography. 22 upon a common psychology and philosophy of education. The earlier teaching objectives of secondary-school mathematics are reflected in the classification found in the 1923 report of the National Committee on Mathematical Beorganization , which is as follows! I. Practical Aims 1. The immediate and undisputed utility of the fundamental processes of arithmetic. 2. An understanding of the language of Algebra. 3. A study of the fundamental laws of algebra. 4. The ability to understand and interpret correctly graphic representations. 5. Pamiliarity with the geometric forms common in nature, industry, and life; mensuration of these forms; development of space perception; exercise of spatial imagination. II. Disciplinary Aims 1. The acquisition in precise form of the ideas or concepts in terms of which the quantitative thinking of the world is done. 2. Development of the ability to think clearly in terras of such ideas and concepts. This involves training in analysis of a complex situation, recognition of logical relations, and generalizations. 3. Acquisition of mental habits and attitudes. 4. The idea of relationship and dependence. III. Cultural Aims 1. Acquisition of appreciation of beauty in geometrical forms. 2. Ideals of perfection as to a logical structure, precision of statement and thought, logical reasoning,discrimination between true and false. 3. Appreciation of the power of mathematics. In contrast to the preceding report the more recent emphases are broader in scope, including such terms as '’Social Sensitivity", "Critical Thinking", "Self-direction", "Cooperation", "Creativeness", and others. In other words, teaching objectives are being directed toward greater social consciousness, which is a more modern outlook regarding the function of general education. This emphasis is particularly evident in the preliminary 25 National Committee on Mathematical Requirements, The Beorganization of Mathematics in Secondary Education. Boston! Houghton Mifflin Company, 1923, pp. 6-12. 23 report of the joint commission selected from members of the Mathematical Association of America and the National Council of Teachers of Mathematics,*^ Very early in the report it is stated that, “Educational objectives in the last analysis, will center around three permanent factors, namely Nature, Society, and the Child,“ further evidence of this trend is found in the 27 tentative report by the Mathematics Committee Association. of the Progressive Education It says, “We regard the objectives of general education as emerging from consideration of the needs of the individual as an active member of society.11 28 Brink very aptly summarizes the most conspicuous trends in the teaching of mathematics as follows: 1, Critical revaluation of the objectives for teaching mathematics has resulted in less emphasis upon the development of formal skills and upon disciplinary values, and more upon the acquisition of useful knowledges and upon functional values, 2, There has been a trend toward pushing down the simpler and more practical aspects of algebra and geometry to the lower levels of school work and toward the scientific grade placement of materials. This has led to the development of general or unified courses in mathematics, particularly on the junior high school level, 3, The subject matter of mathematics courses has been made more practical and lifelike through the (emission of much obsolete material, by the introduction of reading materials which contribute to the attainment of informational and cultural objectives, and by the use of problems which will aid the pupil in dealing with the quantitative situations of everyday life*. 26 The Mathematical Association of America, Incorporated, and the National Council of Teachers of Mathematics, - A Preliminary Beport by the Joint Commission, The Place of Mathematics. in_ Secondary Education. Ann Arbor, Michigan, Edwards Brothers, Incorporated, 1938, p.10. 27 Commission on the Secondary School Curriculum of the Progressive Education Association, “Mathematics in General Education”, Tentative Beport of the Mathematics Committee: Progressive Education Association. June 1938, p. 8 (Introduction). 28 Brink, William G., Directing Study Activities in Secondary Schools, op. cit., pp. 616-517. 24 4* Attempts are "being made to provide more adequately for individual differences "by organizing courses in such a way as to meet the needs of various students, and by basing drill activities upon pupils1 deficiencies as revealed by diagnostic tests, 5* Traditional logical or topical methods of teaching are giving way to the project and unit methods of teaching, 6, The teaching of mathematics is being vitalized by the use of a wide variety of supplementary materials, visual aids, and concrete materials, and by utilization of field trips, club and library activities* It was pointed out earlier that the preceding sources were to be investigated in terms of their contribution toward the development of critical thinking abilities. The general aims and the more recent trends in the teaching of mathematics have resulted in numerous teaching objec tives, which if restated in terms of pupil behavior may definitely contribute toward improvement of critical thinking abilities of young people. The more pertinent teaching objectives in this respect were selected from the previously mentioned sources and were restated in terms of pupil behhvior. The selected teaching objectives and their restatement in terms of the type of behavior being emphasized in this study, are as follows? (see Appendix D for others.) 1) Recognizing the place of undefined terms11 and “Developing appreciation of the place and function of definitions and postulates in the proof of any conclusion11 could be interpreted to mean that a pupil*s behavior would be marked by these characteristics when he questions the meaning of terms in any situation of interest or of concern to him, thereby seeking satisfactory definitions or descriptions of them. 2) “Realizing the importance of assumptions11 could mean that the behavior of a pupil who realizes the importance of assumptions would be marked by his ability to detect and to question underlying assumptions in a situation, - the stated as well as the unstated ones, because as Bell89 points out, “without assumptions there is no proof11 and “no demonstration proves more than is contained in the assumptions11. 29 Bell, Eric T., “The Meaning of Mathematics11, New York? Eleventh Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, 1936, p. 138. 25 3) 11Gathering and organizing data” could he interpreted to mean that such behavior would he marked by effort on the part of a pupil to seek as many facts as possible and to test them for pertinency to the situation so that the organization can be made upon a more intelligent basis. 4) "Appreciating logical interrelationships between truths11 could be interpreted to mean that the pupil is testing facts for pertinency to the situation, endeavoring to discriminate between facts and assumptions, and evaluating conclusions for consistency with the assumptions. 5) "Discriminating between that which has been assumed and that which has been demonstrated" could be interpreted in terms of pupil behavior by observing whether or not he is trying to discriminate between fact and assumption. 6) "Cultivating the habit of self-scrutiny" could mean that the behavior would be marked by a distinct effort on the part of a pupil to evaluate himself for bias or prejudice in the situation. ?) "Establishing and judging claims of proof", "Drawing intelligent con clusions", and "Generalizing conceptions" could mean a variety of behavior characteristics on the part of a pupil. JTor example, he would test conclusions reached for their consistency with assumptions, and furthermore recognize and accept these conclusions as tentative. 8) "Realizing the significance of provisional conclusions" could be interpreted to mean that the behavior of a pupil would be marked by his acceptance of conclusions as being tentative and evaluating them in terms of their consequences. Other meanings could be ascribed to the preceding objectives and they could be restated in several ways. However, the main emphasis at this point is to indicate that a restatement of some of the existing teaching objectives in terms of pupil behavior is possible and when re stated in this way they are found to correspond very closely to at least eight of the ten elements of critical thinking developed earlier in this chapter. 2. How Critical thinking Ma.v Be Developed There are undoubtedly some educators who are honestly skeptical that the preceding aims may be realized through a study of demonstrative geometry, and perhaps they are justified in their position. However, if such aims could not be realized, the psychological problem of "transfer" would lead one to a contradiction, since as Webb30 points out "transfer of training is concerned with the question as to whether or not the learning of material A, say mathematics, aids, hinders, or has no effect in the learning of subsequent materials, say physics or chemistry. This theory assumes nothing about faculties of the mind but is concerned how the organism meets situations B, C, D, and so forth, after having had an experience in situation A. Educationally it is concerned with finding out how the organism can benefit to the largest degree from its experience in situation A when it finds itself face to face with situations B, C, D, and so forth." This statement brings to the forefront precisely the problem that all subject-matter teachers must face. It becomes extremely important for the teacher to know that the amount of transfer is dependent upon the method and the teaching involved* 31 Grata , in his summary of this problem, writes? "It is safe to conclude, therefore, that from the stand point of the teacher and the school in general, the solution of the problem of transfer of training is to train for transfer." He points out further I Thorndike's own modified position? "Studies of the transfer of training j also have shown that the methods used in guiding the pupils1 learning activities have marked effect upon the degree of transfer. The more clearly the crucial elements or fact or principle in a situation is brought to the Ipupilte attention the more readily the same element or fact or principle may be identified in another situation." Hence, if a study of demonstrative geometry is to bring about Ibehavior that has characteristics of critical thinking then mathematics |teachers must modify their methods so as to teach for transfer into this |30 Webb, L.W., "The Transfer of Training", Chapter XIII of Skinner, Ghas. E., Educational Psychology. Hew York? Prentice Hall, Inc.,1936, p. 327. j31 Grata, Pedro T., "Transfer of Training and Educational Pseudo-Science", i The Mathematics Teacher. Volume XXVIII, Ho. 5, May 1935, p. 269. 27 mode of thinking. 3P Vera Sanford says, f,If we "believe that training in critical thinking comes through conscious practice in critical thinking, then we must scan the curriculum to find something to think about.” In other words we must seek common elements of critical thinking and then teach young people in such a way that this thinking carries over into non-mathematical situations. We must give pupils practice in thinking in the non-mathematical situations as well as in the mathematical ones in terms of the common elements involved. Wheeler 33 supports this view with the following statement! “Wo transfer will occur unless the material is learned in connection with the field to which transfer is desired. Isolated ideas and subjects do not integrate. Learning is not bond-forming. It is an orderly and organized process of differentiating general grasps of situations with respect to experience. !The details emerge organized, as they differentiate from previous knowledge, in the face of new situa tions, not repeated ones.*1 Current teaching practices of demonstrative geometry are being seriously questioned. Many instances can be cited which are comparable to that of a metropolitan newspaper editor who remarked somewhat as follows! "I cannot think of a single tough spot in my existence in which Euclid reached down to lend me a helping hand.11 Undoubtedly, this editor could find a great deal of support for this statement. How many of our boys and girls of today will say the same thing of their experience with this 32 Sanford, Vera, ”Why Teach Geometry”, The Mathematics Teacher. Volume XXVIII, May 1935, p. 294. 33 Wheeler, R.H., ttThe Hew Psychology of Learning”, New York! Tenth Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, 1935, p. 239. 28 subject? Eeeve 34 cites a number of criticisms of mathematics in his article, "Attacks on Mathematics and How to Meet Them". Fawcett*^ points out that "Actual classroom practice indicates that the major emphasis is placed on a body of theorems to be learned rather than on the method by which these theorems are established, The pupil feels that these theorems are important in themselves and in his earnest effort to ,knowt them he resorts to memorization." Young likewise points out that, "The mere memorisation of a demonstration in geometry has about the same educational value as the memorizing of a page from the city directory. And yet it must be admitted that a very large number of our pupils do study mathematics in just this way# fault lies with the teaching." There can be no doubt that the While scores of other criticisms of teaching mathematics could be brought to bear upon this problem, never theless it might suffice at this point to state that the opportunities for teaching pupils worthwhile outcomes through a study of demonstrative geometry have been widely misused. After all it is not the fault of mathematics that certain objectives are not attained, but rather it is the fault of teachers in presenting and utilizing the subject-matter at hand. Finally, the writer has had the pleasant experience of teaching high school pupils for the past twelve years, and nowhere has he ■ ■P , ■ ■! — ! I ■ ■ ■> ,. . - 1 ■ ■ ■■ ■■ ■■■ ■ ' 1 ■ 1 ' ■■ ■ 1 mm 34 Beeve, William "Attacks on Mathematics and How to Meet Them", Hew York; Eleventh Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, 1936, pp. 1-21. 35 Fawcett, Harold P., "The Nature of Proof", New York: Thirteenth Yearbook of the National Council of Teachers of Mathematics, Bureau of Publi cations, Teachers College, Columbia University, 1938, p.l. 36 Young, John Wesley, "Lectures on Fundamental Concents of Algebra and geometry". New York; The MacMillan Company, 1925, pp. 4-5. 29 encountered another subject less void of emotional M a s or prejudice than demonstrative geometry* It seems that here pupil and teacher alike can arrive at clear-cut conclusions, yet recognizing that with intelligent control of one's emotions one should likewise arrive at clear-cut con clusions in other areas of experience* Some critics have questioned this position hy insisting that other subjects or even courses in formal logic, when used as a means, can accomplish this purpose equally as well. There is no experimental evidence to indicate that these other courses or even training in formal logic can achieve or have achieved this end better or even equally as well. The one cardinal fact still remains, that we can use geometry instead of a course in pure logic because of the pupil's familiarity with the figures used in the former subject. It is these figures that are made the basis for discussion and from these figures, with clearly defined objectives and their common elements, we can try to teach for transfer into other fields of knowledge. Fortunately, some teachers of mathematics are beginning to recognize more and more their opportunities to guide the thinking of young people from mathematical to non-mathematical situations. They are becoming more and more aware of the fact that real problem solving is not merely the process of working examples or exercises out of a textbook, but a process that calls for activity involving analysis, seeking relationships, seeking meanings, detecting assumptions, discriminating between facts and assumptions* seeking facts and discovering data, testing the facts and data for relevancy, scrutinizing self for bias and prejudice, reaching valid conclusions, formulating intelligent generalizations, formulating hypotheses, testing conclusions and studying their consequences to the individual and to society. These same teachers are encouraging and stimulating their pupils 30 to find and to solve problems that they encounter in their daily experience rather than relying solely upon artificial problems found in textbooks* problems which are generally superficial and void of meaningful social implication. It must be understood at this point* however, that the writer does not assume it is the sole responsibility of mathematics teachers to develop this critical thinking ability, but he does contend that this is an important responsibility of mathematics teachers and that they possess a unique opportunity for developing this objective. Related Studies Research efforts in the teaching of geometry have been directed along the line of subject-matter revision, elimination of content in some cases and addition in others, rearrangement of logical sequences, and the like. The patterns were generally fixed logically by adult minds and placed in textbooks for pupils to follow. Little attention has been given to changes in the nature of the content which are necessary if the habits of thought, which it is hoped may be developed through a study of this subject, are to transfer to non-geometric situations* One of the early though faint beginnings in this direction may be attributed to an experimental study by Elsie Parker*^, Her fundamental assumption was that transfer of training from one field of experience to another is possible under certain favorable conditions, and in order to realize the benefits therefrom one must teach pupils the conscious use of a technique of thinking. To test this hypothesis, she set up a controlled 37 Parker, Elsie, ’*Teaching Pupils the Conscious Use of a Technique of Thinking1*, The Mathematics Teacher. Volume XVII, Ho. 4, 1924, pp. 191-201. 31 experiment and sought to answer the question! "Can pupils of geometry he taught to prove theorems more economically and effectively when trained to use consciously a technique of logical thinking; and furthermore, does such training, more than the usual method, increase the pupil's ability to analyze and see relationships in other non-geometrical situations?" She points out that, "The traditional method of instruction has been to let the pupil discover for himself a method of reasoning which he there after uses without in many cases being aware of the fact that he is using that mode of procedure*" The experimental group was led to study consciously the logical thought processes essential to proving theorems, while the control group sought to learn a body of theorems with very little attention given to the thought processes involved in their proof* While she recognizes the limitations of her experiment, nevertheless she does make some claims which are brought out in the following report of her conclusions* "Original theorems in geometry were given to the classes before and after training in the conscious use of a technique of thinking. While the classes had the same number of correct proofs on the first one, on the second one after training, the experimental group had one and one-third as many complete and perfect proofs as the control, while, counting also the proofs which were on the right track, but not complete, the ratio was 12 to 21 in favor of the experimental group. As in the previous experiment, it was found that those trained in the new method exhibited more perseverance in endeavoring to find a proof. In this case, one and a half times as many of the control group quit before the end of the time allowed as in the experimental group •. * These data would seem to offer conclusive evidence, in so far as one experiment can be considered to do so, that when pupils 32 are taught to use consciously a technique of logical thinking, they try more varied methods of attack, reject erroneous suggestions more readily, and without "becoming discouraged maintain an attitude of suspended judgment until the method has "been shown to be correct.11 The data on the reasoning tests would seem to indicate that such training in logical thinking with the materials of geometry tends to carry over these methods of attack and these attitudes to other problem situa tions not concerned with geometry. Whether or not her experiment provides conclusive evidence that this conscious use of a technique of thinking carries over into nonr* geometric situations, is still problematical# The fact remains that her last statement lacks supporting evidence and is therefore a mere conjecture because no effort appears to have been made to test or to evaluate carry over, if any into actual non-geometric situations. -Another interesting experiment somewhat along this same line was performed by Winona Perry 38 • and two controlled divisions. This experiment involved one experimental In the experimental division major emphasis was placed in the development of a technique in reasoning which stressed the Hif - then11 type of thinking and also the analytical method. The controlled divisions sought to learn the theorems and were definitely guided by their textbook. In one of these controlled divisions, proposi tions as found in the textbook were emphasized as being of primary impor tance, while in the other division the major emphasis was placed on proving 11originals11. The class in each of the controlled divisions was conducted largely by the question and answer technique and no attention was given to any particular method of thinking. 38 Perry, Winona, A Study in the Psychology of Learning in Geometry. Hew York! Bureau of Publications, Teachers College, Columbia University, 1925. 33 Non-mathematical subject-matter was not included in the experiment, yet Miss Perry reports finding fchht,”In the experimental group the ability to solve problems non-mathematical in character was markedly improved, following the period of training in the solution of exercises in geometry* This increased ability was most noticeable as resulting from those tests more nearly similar to the type of reasoning emphasized in demonstrative geometry in form and in content.’1 The same criticism as applied to the former experiment can again be applied to this one, that non-mathematical materials were not used in either experiment and yet the reports indicate that the experimental techniques facilitated the solution of non-mathematical problems. That this may have been true to some extent is not being doubted provided there were common elements of these techniques in thinking present in both the geometry and the non-mathematical situations referred to* This leads us to another experiment, perhaps one of the most significant studies ever attempted in the field of educational method* Its implications for all teachers, and particularly for mathematics teachers, are quite profound and far-reaching. The study being referred to is an experiment on the ’’Nature of Proof” performed by Harold P* Pawcett^ at the Ohio State University High School. He assumes in the beginning that ”(l) a senior high school pupil has reasoned accurately before he begins the study of demonstrative geometry; (2) a senior high school pupil should have the opportunity to reason about the subject-matter of geometry in his own way; (3) the logical processes which should guide the development of the work should be those of the pupil and not those of the teacher; and (4) opportunity should be provided for the application of the postulational 39 Pawcett, Harold P., Thirteenth Yearbook, op. cit. 146 pp. method to non-mathematical material."^ The purpose of his study, using his own words, is 11to describe classroom procedures by which geometric proof may be used as a means for cultivating critical and reflective thought, and to evaluate the effect 41 of such experiences on the thinking of the pupils.11 He points out that the general problem he is attacking consists of three related problems! (l) the problem of leading the pupil to under stand the nature of deductive proof through the study of geometric sit uations; (2) the problem of generalizing this experience so that effective transfer will result; and (3) the problem of evaluating the resulting change in the behavior of the student. Eurthermore, he assumes that a pupil understands the “nature of deductive proof” when he understands that (1) the place and significance of undefined concepts in proving any con clusion; (2) the necessity for clearly defined terms and their effect on the conclusion; (3) the necessity for assumptions or unproved propositions; and (4) no demonstration proves anything that is not implied by the assump tions and that the “real value of this sort of training to any pupil is determined by its effect on his behavior". The following is an outline summary of general procedures with the experimental group in Fawcetts study. No specific textbook was assigned. Each pupil developed his own text and was given the opportunity to develop it in his own way. Undefined terms were selected by the pupils. No attempt was made to reduce the number of dndefined terms to a minimum. terms needing definition were selected by the pupils and the definitions The 35 were an outgrowth of the work rather than the "basis of it* were then made "by the pupils. The definitions Certain loose and ambiguous statements were refined and improved "by criticisms and suggestions until they were accepted "by all of the pupils in the class. Propositions which seemed obvious to the pupils were accepted as assumptions and these assumptions were made by the pupils and were recognized by them as the product of their own thinking. Ho attempt was made to reduce the number of assumptions to a minimum. Every encouragement was given toward detection of implicit or tacit assumptions and the pupils learned to recognize this as being important. Ho statement of anything to be proved was given the pupil. erties of a figure were assumed properties. Certain prop Ho generalized statement was made before the pupil had an opportunity to think about the implications of the particular properties assumed. (Through the assumptions made, the attention of all pupils in the class was directed toward the discovery of a few theorems which seemed important to the teacher. Assumptions leading to theorems that were relatively unimportant were suggested in mimeographed material which was available to all of the pupils but not required of any. Matters of common concern, such as the selection of undefined terms, the making of definitions, the statement of assumptions and the generalizing of an implication, were topics for general discussion, while periods of supervised study provided for individual guidance. The major emphasis was not on the theorems proved but rather on the method of proof. This method was generalized and applied to non-mathematical situations. Prom the results of his evaluation, (Fawcett made the following gener ali zat ions 5 (l) Mathematical method illustrated by a small number of theorems yields a control of the subject matter of geometry at least equal to that obtained from the usual formal course* (2) By following the procedures outlined in his study he found that it is possible to improve the reflective thinking of high school pupils. (3) He points out that this improvement in the pupils’ ability for reflective thinking is general in character and transfers to a variejiy of situations. (4) Finally, he concludes that the usual formal course in demonstrative geometry does not improve the reflective thinking of the pupils. While this related study is a significant one for mathematics teachers, as previously mentioned, it has nevertheless several short comings. In the first place the study was not conducted in a normal public high school situation, and in the second place it dealt with a small number of pupils, whereas the usual number of pupils per teacher or the teaching load in the average public high school is considerably heavier. Specifically, there were only twnety-five pupils in Fawcett’s experimental group and twenty-five in a comparable group which was taught formal demonstrative geometry by another teacher. Furthermore, the median I.CJ. for Fawcett’s class was 115 while that of the other group was 109. Chanter Summary The problem is to determine whether or not demonstrative geometry may be taught in such a way as to improve the critical thinking abilities of young people. As a purpose of the study five questions were raised concerning the problem, and answers to these are to be sought. In order to work within a more definite frame-work, it was found necessary to clarify what is meant by the term ’’critical thinking” as it is to be used in this study and to determine the common elements of this 37 type of thinking. Critical thinking, therefore, was defined as a process of thinking about thinking from the point of view of a critic. By means of an analysis of situations wherein this form of thinking had taken place, ten common elements were found. These ten elements are to serve as a frame of reference for this study. The need for improving critical thinking abilities of young people was then pointed out and this need was supported by statements from leading philosophers, educators, organizations like the Institute for Propaganda -Analysis, and other leaders concerned with youth problems. This being a study in the teaching of mathematics, naturally led to a consideration of implications critical thinking may have for teachers of mathematics. Most of the literature on the teaching of mathematics swas examined for teaching objectives. These objectives that were in any way pertinent to this study were selected and restated in terms of pupil behavior. It wan then pointed out how the psychological problem of ^transfer11 could be applied to different situations so that critical thinking abilities might be improved. In this connection, however, attention was called to the fact that it is not the sole responsibility of mathematics teachers to develop these critical thinking abilities, but that it is an important responsibility for them and that they possess unlimited opportunities for developing this objective, Pinally, several related studies were mentioned and their findings discussed. One in particular, namely Pawcett*s study con cerning the HMature of Proof11 merited special mention because of its far-reaching implications for mathematics teachers and for the education of young people. CHAPTER II EXPERIMENTAL SITUATIONS The Experimental and Control Groups The data for this experiment were secured from approximately 700 high school sophomores whose average chronological age was fifteen. This group of sophomores represented a cross-section from six different public high schools! two large urban schools, two large suburban schools, and two small high schools. Three of the schools were paired with the other three on the basis of their location and size, such as urban with urban, for the purpose of making comparisons. The pupils selected from these paired schools will be referred to as Experimental Groups and Control Groups. Six teachers were selected from the three schools producing Experimental Groups and likewise six teachers were selected from the other three schools which produced the Control Groups. In order to keep the Experimental and Control situations as normal as possible, no attempt was made to pair the pupils or the teachers of the two major groups# Each of the two major groups was made up of approximately 350 pupils, which should be sufficiently adequate for purposes of making statistical com parisons of differences, if any were to be found. Table I summarizes data concerning teachers, schools, and pupils involved in the study. Table I reveals similarities between the Experimental and Control groups with respect to type of school and size of enrollments. However, the number of classes and number of pupils per teacher show considerable variation. In the cases of Teacher Groups C and D, as well as J and K 39 TABLE I._______ TEACHERS, SCHOOLS, AND PUPILS INVOLVED IN TEE STUDY Teacher A B and D E 3T School No. No* No. No. No* V I I III III O^pe of School Small Large Large Large Large uj^ent GeoZtoy Classes H" l s f EXPERIMENTAL GROUPS 300 Rural Urban 4000 4000 Urban 2700 Suburban 2700 Suburban 4 3 2 2 2 99 72 58 56 48 Total 333 CONTROL GROUPS a H i and K L No. No. No. No. No. VI II II IV IV Small Large Large Large Large Suburban Urban Urban Suburban Suburban 375 4100 4100 3900 3900 39 112 43 49 83 2 4 2 2 3 Total 326 the pupils were under the direction of the former during the first semester and the latter teacher during the second semester. Although there were over 700 pupils involved at the "beginning of the experiment, the final number was reduced to 659. This reduction was due to several factors, namely, changes in a pupil*s program of studies, withdrawal from school, transferring to another class or another school, and the like* Selection of Teachers 1. Selection of Experimental Group Teachers As previously mentioned, five teachers "besides the writer comprised 42 what will hereafter he referred to as the six Experimental Group teachers . Three of this group taught geometry in a large urban high school, two in a 42 The six Experimental Group teachers were students in an integrated course, designed toward the improvement of teaching procedures in secondary schools, at Northwestern University during the 1937 summer session. A section of this course, namely the Teaching of Mathematics in Secondary Schools, was directed by Dr. Harold P* Fawcett of Ohio State University High School. It was here that the writer found five teachers with a common interest toward the development of critical thinking abilities among young people. 40 large suburban high school, and one in a small high school located in another state. The teaching experience of this group ranged from five to fifteen years, This group of teachers was primarily selected on the "basis of a common interest in the problem of improving critical thinking abilities among high school pupils. Furthermore, the group had indicated a willingness and a desire to cooperate with the writer in applying the experimental procedures. £. Selection of Control Group Teachers Having found teachers to apply experimental procedures made it desirable to select schools and teachers that were as comparable to the Experimental situations as possible. Furthermore, it was highly desirable to find teachers as representative of the traditional view as possible, and yet comparable in teaching experience and teaching ability to the teachers of Experimental Groups, is a result of a series of interviews with administrators and with teachers, in the three schools from which Control Groups were to be provided, six teachers were selected, is a consequence of this selection, the pupils in the geometry classes of these six teachers constituted what will henceforth be referred to as the Control Groups. Description of School Situations A brief description will now be given of the six school situations so as to point out the background setting for the present study. School Ho. I (Experimental) This is a large urban senior high school with an enrollment of approximately 4000 pupils. Nearly fifty per cent of the pupils are of Jewish ancestry, while the more dominant ancestry of the remaining ones includes Swedish, German, Polish, and Russian. The home situations of this school population represent what is commonly referred to as the average middle class with respect to ideals, attitudes, and economic affluence. Less than five per cent of the enrollment represent families in need of urgent financial aid# One small group comes from one of four "Homes” or HQrphanagesM in the neighborhood, while another group is living with families where they have heen placed by the Jewish Home Finding Society. At least fifty per cent of the pupils say they are going to college; however, fewer than twenty-five per cent will probably go# Many of the pupils hold minor jobs outside of school time (a large portion of this group works for spending money rather than from necessity). The school is over-crowded. It is operated on a two-shift basis, classes being in session from eight in the morning until four in the afternoon. Because of this condition teacher conferences with pupils are difficult to arrange, particularly when the janitors and engineers want the building cleared for locking up ten minutes after the last class period terminates. The principal has the utmost faith in his teachers, which fact creates a mutual and wholesome feeling of cooperation. A fine modern building provides unlimited possibilities for an energetic and enthu siastic student-body. At least ninety per cent of the pupils in geometry classes are in the tenth grade. this subject. A textbook is not necessarily required for the study of Consequently a textbook was not used by the Experimental Group teachers to develop the subject. The class periods are of forty- 42 minute duration. Bach full-time teacher, including the Experimental Group teachers, has six classes daily, one study hall, and a division room ranging in size from forty to fifty pupils, while the regular classes average from thirty to forty pupils. School Ho. II (Control) This is likewise a large urban senior high school with an enroll ment of 4100 pupils. Here too, nearly fifty per cent of the pupils are of Jewish ancestry, in fact the number is between fifty and sixty per cent, which is a bit higher than in the preceding school situation. The remaining dominant nationalities are made up of .American, Swedish, German, and Polish ancestry. The home situations likewise represent the average middle class in respect to ideals, attitudes and economic affluence. Between twenty and thirty per cent of the pupils come from homes of business and professional people. More than fifty per cent come from homes of what is commonly referred to as the “working class". At least seventy per cent of the pupils say they are going to college; however, only thirty-five to forty-five per cent will probably go. Many of the pupils hold minor jobs outside of school time (a large portion of this group works for spending money rather than from necessity). This school is likewise overcrowded. halls or study rooms. There is a lack of study The assembly room must be used as a study hall for six periods of the day and therefore cannot-be used for other worthwhile purposes. There are from 150 to 600 pupils studying there during these six periods. The fact that five principals and a large faculty are working together harmoniously enhances the enthusiasm and desire for cooperation expressed by the student body. Hearly eighty per cent of the pupils in geometry classes are in the 43 tenth grade. A textbook in this subject is required of each pupil and it is used as a basis for the course, minute duration. The class periods are of forty- Each teacher has five classes daily, two study halls or other duties, one adjustment service period, and a HdivisionH of forty pupils, The regular classes average from thirty to forty pupils per classroom. School Ho. Ill (Experimental) This is a large suburban high school with an enrollment of more than 2600 pupils. There are no dominant nationalities other than .American. The ancestry of the pupils is well distributed among the various nationalities with, perhaps, a dominance of English and German. The home situations of the student body are well above average in economic affluence. In other words, the school district is largely made up of residents whose income is well above the average. A small group of pupils represents homes whose parents are care takers or hired help for the more affluent. However, the vast majority come from homes whose parents represent business, executive, professional, or highly skilled people. Over ninety per cent of the pupils say they are going to college and nearly seventy per cent actually go. The major apparent school strength lies in its adviser system. A modern building provides unlimited possibilities for the student body. Every effort is made to provide for individual interests and needs of the pupils. A highly selective staff of teachers and the best available equipment for their disposal is provided. The administration is very cooperative with every group in the system with the result that there is the best of harmony. The morale of the entire system (including the student body, teachers, administration, and the board of education) is 44 very excellent. Because of the select quality of the teachers end their maturity - including the effectiveness of the adviser system - the pupils as a whole seem to he very congenial and regard their teachers with utmost respect. With this kind of a background, the school has very little, if any, weakness. It is generally regarded as one of the outstanding secondary schools in the country. At least ninety per cent of the pupils in geometry classes are in the tenth grade. While the mathematics department recommends a basic text in this course, its use is left entirely up to the teacher. Consequently no textbook was used by the Experimental Group teachers in their develop ment of this subject. Outside readings and references to several different texts were made during the developmental process. The class periods are of one-hour duration. Each full-time teacher has four classes, a study hall, an adviser room, and in most cases one or more extra curricular activities. The size of classes ranges between twenty-five and thirty, the school average being twenty-six pupils per class. The study halls range in size from fifty to one hundred fifty pupils per teacher. The adviser groups range between thirty and forty pupils per teacher. School No. IV (Control) This is a large suburban high school with an enrollment of 3900 pupils. There are no dominant nationalities other than American, which constitutes nearly seventy-five per cent of the enrollment. The remaining twenty-five per cent is distributed among other nationalities. At least twenty per cent of the pupils come from a relatively wealthy class of people. Seventy per cent come from homes that are commonly referred to as 45 the average middle class. Hie remaining ten per cent come from homes that are "below average in economic affluence; however, the group taken as a whole is above the average .American secondary school in this respect. This school is likewise regarded as an outstanding high school. Eifty-four per cent of the graduates go to college. of the pupils hold minor jobs outside of school time. Ten per cent Out of this group nearly half do so in order to get spending money rather than as a necessity. The school is too large to have all-school assemblies. assembly of the entire student body at any one time. There is no On the other hand the recognition ascribed to this school for some of its outstanding merits may undoubtedly be attributed to an enthusiastic student body, expert adminis tration, a harmonious faculty, and deep-rooted tradition. The school has an enviable student record in colleges and in situations of employment. Ninety-five per cent of the pupils in geometry classes are in the tenth grade, and a study of plane geometry is required of all pupils who enter the school. A textbook is used as a basis for developing the course and the department is holding rather rigidly to the traditional course. The teachers, however, are given a free hand to use whatever methods they deem best. The pupils are divided into three ability groups. Class periods are of forty-minute duration, and each full-time teacher has a load of five classes daily* one study hall, and a division room averaging forty pupils per teacher. The size of the regular classes is between twenty-five and thirty pupils. School go. V (Experimental) This is a small high school in another state with an enrollment of approximately 300 pupils. There are no dominant nationalities other than .American, which comprises ninety-seven per cent of the enrollment. The pupils are chiefly children of ranchers and farmers who are economically independent; however, their incomes are not sufficient to allow many luxuries. ■busses. Most of the pupils come in from the rural district on school A negligible per cent of pupils hold outside jobs. itself is an old ranch center. The town Its people are almost guilty of ancestor worship, and the keeping alive of traditions. Fifty per cent of the graduates go away to college. The apparent school weakness is largely the lack of a modern building. The present building was erected in 1921 and lacks modern features as well as equipment. More teachers are needed to decrease the size of classes. Then, too, busses leave ten minutes after school closes* making it impos sible to have conferences at this time. On the other hand, the apparent school strength lies in a harmonious faculty and a splendid morale among the pupils as well as in the system as a whole. A very cordial and friendly relationship exists between pupils and teachers. The community exhibits a most hearty and active desire to cooperate with the school. At least ninety per cent of the pupils in geometry classes are in their tenth year, or rather in their tenth grade level. A textbook is not required and therefore was not used as a basis for developing this subject by the Experimental Groups. There is an excellent reference library available for supplementary reading and study. This library was used extensively for this purpose with frequent special assignments meeting the needs of a pupil. Class periods are of fifty-five minute duration, and each teacher has five classes and a home-room. The classes and home-rooms average between thirty-five and forty pupils per teacher per period. There is considerable race prejudice in the community, because of two civil wars that this state has gone through. This also helps to explain 47 why the community holds so steadfastly to traditions of the past. The town, prior to the school year 1937-1938, has maintained three school systems, one for the White race, one for the Brown race, and one for the Black race. This year, namely 1937-1938, the state forced the school system to admit the Brown race. But even then, they (the Brown race) were admitted only when the st&te threatened to remove the school from the accredited list. School No. VI (Control) This is a small high school with an enrollment of approximately 375 pupils. There are no dominant nationalities other than American. There is, however, a dominance of German ancestry, and English ancestry to a lesser degree. The home situations of these pupils are represented economically hy what is commonly referred to as an average middle class. There is only one H. Y. A. student in the school and very few of the families are on a relief status. from this school go to college. Eeerly fifty per cent of the graduates Between twenty-five and thirty-five per cent of the pupils take up commercial studies. Only a small per cent have odd jobs outside of school hours and very few have regular employment. The apparent school weakness lies in the fact that it is "badly overcrowded. Two shifts are necessary. Because of this overcrowded condition there are few conference opportunities and these are generally in halls, on stairways, or on landings. On the other hand, the morale of the pupils and teachers is the apparent school strength, and is therefore responsible for the progress the school is making in meeting the needs of the community. Geometry is an elective subject in the school, therefore eleventh as well as tenth grade pupils make up the classes. However, most of the 48 pupils enrolled in the geometry classes are in their tenth grade level, A geometry workbook is required of the pupils, and it is used as a basis for developing the course. sixty-minute duration. The class periods are of both forty"- and Each teacher is assigned five classes and extrar curricular responsibilities. The size of classes is between twenty-five and thirty pupils. The afore-mentioned condition of overcrowding is to be eliminated by 1939, because a new modern high school is rapidly nearing completion. Characteristics of Pupils Involved in the^S.tudv Characteristics of a pupil, or of a group of pupils, are an integral part of any experimental situation. In order to obtain as much information as possible regarding a pupil's ability and background prior to his study of demonstrative geometry, several paper and pencil instruments were ad ministered. They are as follows? (l) Otis Self-Administering Test of Mental Ability^, (2) Orleans Geometry Prognosis Test4^, (3) Nature of Proof Test^, and (4) an Inventory Questionnaire designed by the writer for the purpose of getting supplementary information about the pupils involved in this study. The tests of mental ability and geometry prognosis, as well as the inventory questionnaire, were administered to all of the pupils 43 Otis, Arthur S,, "Otis Self-Administering Tests of Mental Ability", New York? World Book Company, 1928. 44 Orleans, Joseph B. and Jacob S., "Orleans Geometry Prognosis Test", New York? World Book Company, 1929. 45 Progressive Education Association, "Nature of Proof Test 5.3", Columbus, Ohio? Ohio State University, 1936. 49 involved in the study at the beginning of the school year. The Eature of Proof test was likewise administered at the beginning of the school year; however, it was given only to the Experimental Groups. The reason why this test was not administered to the Control Groups was largely to avoid its possible influence upon traditional teaching procedures. 1. Intelligence 46 Intelligence quotients were obtained for both Experimental and Control Groups in order to provide an objective measure for determining relative abilities of pupils and of groups of pupils# Pigure 1 reveals not only the combined distribution of the Exper imental and Control Group intelligence quotients, but also a comparison of these two major groups comprised of 659 pupils. The longer shaded bars in the lower half of the distribution and the longer solid bars in the upper half of the distribution clearly indicate the superiority of the Control Groups over the Experimental Groups in terms of the I.££. factor. The significance of this superiority may be noted further by comparing the “Means*1 of the two combined distributions. Table II reveals differences between the Hanges, Standard Deviation, and the Means of the Experimental and Control Groups. TABLE II. Group In this table I.ft# COMPARISONS OP EXPERIMENTS AND CONTROL GROUPS___________ ~ Humber of Pupils Eange Standard Deviation Mean Experimental 333 71 to 137 11.4 105.8 Control 326 74 to 146 11.7 110.2 the difference of the two Means is 4.4 and the standard error of difference 46 A complete tabulation of I.Q. *s for the 659 pupils involved may be found in Appendix B, page 367, from which all of the statistical data in this chapter were derived. 50 EXPERIMENTAL GROUPS Pupils CONTROL GROUPS 90 to 99 Pig, 1,— 100 ' 110 1 120 1 11 130 I to to to to 109 119 129 139 I.Qts or Intelligence Quotients 1 COMBINED DISTRIBUTION OP INTELLIGENCE QUOTIENTS FOR EXPERIMENTAL AND CONTROL GROUPS. 51 "between the two means is found to "be 0.9, Statistically, the difference "between the Means of two distributions of like factors is not significant unless the actual difference of the two Means is at least three times the standard error of difference* .According to Garrett^, and others, 11It is I) usually customary to take a cTdiff. (5 meaning the actual difference of two means, and <%if£ meaning the standard error of difference of the two means) of 3 as indicative of complete reliability, since -3<T includes practically all of the cases in the distribution of differences below the _ D mean. A greater. than_.3_ is. to be taken as indicating .lust so much added reliability.H According to Table II, the standard deviation or reliability of the two groups with respect to 1.(1., namely 11.4 and 11.7, is nearly the same. Interpreted statistically, it means that when the variability is below 20.0 the groups taken as a whole are sufficiently homogeneous in the factor under consideration. The differences of the two I.Q. Means between the Experimental and Control Groups is 4.4 and this difference is 4.9 times the standard error of difference 0.9, which tells us that statistically the Control Groups, taken as a whole, have I.Q.!s that are significantly superior to those of the Experimental Groups. Table III reveals an I.Q. comparison of smaller groups in the study, namely, the School Groups* It was mentioned earlier that these schools were paired on the basis of type, size and location. In this table the differences in the Banges and in the Means between the Exper imental and Control School Groups are more pronounced. several reasons. This is due to Eor example, School No. Ill makes special provisions 47 Garrett, Henry E., Statistics in Psychology and Education. New York: Longmans, Green and Company, 1933, p. 133. 52 TABLE III. I.Q.. RANGES AND MEANS T O SCHOOL GROUP Control Groups Experimental Groups School Group No. V No. I No. Ill Range Mean School Group Range 71 to 127 80 to 137 73 to 128 102.9 103.3 112.1 No. VI No. II No. IV 103 to 130 78 to 146 74 to 142 Mean 114.7 111.2 107.8 for the more precocious or gifted pupils and therefore no members of this small but highly selective group were involved in the present study. This fact likewise accounts for the relatively low upper limit in the 1.^. range for this particular School Group* furthermore, geometry is elective in School No, III, whereas in School No. IV some form of geometry is required of all pupils prior to their graduation. In School No. VI geometry is elective, and only pupils who have shown proficiency in mathematics are permitted to elect this subject, while in School No. V all of the pupils are required to study some form of geometry. Table IV reveals a comparison of the Teacher groups* out earlier that Teacher A School No, VI groups. andG groups were likewise School It was pointed No. V and Teacher B, 0, and D groups made up the School No. I group in Table III, while Teacher H and I groups made up the corresponding School No, II group. Teacher E and E groups comprised the School No. Ill group, while Teacher J, K, and L groups comprised the corresponding School No. IVgroup of Table III. The differences between the Teacher groups are likewise pronounced. This difference is due to factors already e^qplained in connection with the preceding table. However, there is another factor introduced at this point which was not apparent in Table III. This factor is "ability grouping". It must be remembered that each School Group represented a rough crosssection of the entire tenth grade enrollment in that school. The only 53 TABLE IV* I.Q,. RANGES AND MEANS PER TEA/HER GROUP Control Groups Experimental Groups Teacher A 3 C and 3 E F Range 71 80 82 73 99 to to to to to 127 133 137 120 128 Mean Teacher 102.9 103.9 102.7 107.3 117.7 G H I J and K L Range 103 89 78 74 90 to to to to to exceptions were the schools where geometry was an elective. Mean 130 146 126 127 142 114.7 112.3 108.3 103.4 110.5 As a consequence, the factor of ability grouping brings out greater differences in the Teacher Groups than in the School Groups. For example, Teacher F had two groups of accelerated pupils whereas Teacher L had a slow, an average, and a superior group. On the other hand, Teacher E had two slower or retarded groups; Teachers J and K each had a slower and an accelerated group. In conclusion, the factor of ’’ability grouping” is largely responsible for the heterogeneity between the Teacher Groups* In order to reveal the heterogeneity between the Teacher Groups in a more detailed manner, Table V was formulated. This table is the summary of a tabulation that was made from the per cent of pupils, under each teacher, who fell into each of the four quartiles of the combined total distribution of both Experimental and Control Groups. The method for determining the data in Table V was to find percentile ranks of the combined I.Q. distribution of both Experimental and Control Groups and then determine the number of pupils under each teacher that fell into each of the four quartiles. The per cent was then obtained by dividing the number of pupils whose I.$. 's fell into a quartile by the total number of pupils in the particular teacher group. The heterogeneity of the Teacher Groups on the basis of I.Gfc. is apparent in Table V, because the range for the per cent of pupils under TABLE V. Teacher DISTRIBUTION OP I.Cfc. «s BY QUARTILES FOR EACH TEACHER GROUP Number and Per Cent of Pupils Palling Into No. of 2nd Quartile 3d Quartile Classes______1st Quartile 4 3 2 2 2 A B and D E P G H I and K L 2 4 2 2 3 4th Quartile 8 13 5 3 37 Experimental Groups 24 or 24.1$ or 8.1$ or 18.1$ 11 or 15.3$ or 8.6$ 9 or 15.5$ or 5.3$ 25 or 44.7$ 8 or 16.6$ or 7713$ 32 16 18 19 2 or or or or or 32.4$ 22.2$ 31.1$ 33.9$ 4.1$ 35 32 26 9 1 or or or or or 35.4$ 44.4$ 44.8$ 16.1$ 2.1$ 16 35 16 11 33 Control Groups or 41.0$ 17 or or 31.2$ 32 or or 37.3$ 6 or or 22.4$ 7 or or 27.7$ 24 or 6 27 12 11 21 or or or or or 15.5$ 24,1$ 27.9$ 22.4$ 25.3$ 0 18 9 20 15 or or or or or 0.0$ 16.1$ 20.9$ 40.7$ 18.1$ 43.5$ 28.6$ 13.9$ 14.5$ 28.9$ • varies widely by quartiles as follows f 1st 2nd 3d and 4th quartile, quartile, quartile, quartile, from from from from 5.3$ 13.9$ 4.1$ 0.0$ to to to to 77.2$ 44.7$ 33.9$ 44.8$. Any attempt to pair Teacher Groups in terms of the I.Q. factor, based upon the above evidence, could hardly be justified. Por example, if pairings were made, Teacher A group would be paired with Teacher G group, Teacher B group with Teacher H group, and so on down the list in Table V. (It was pointed out earlier that the six Experimental Group teachers were selected first and then the six schools and the six Control Group teachers were determined as a result of this selection). It is because of the heterogeneity between these Teacher Groups and the desire to maintain normal experimental conditions that no attempt was made to equate pupils or Teacher Groups in this study. If one were to assume that intelligence quotients are valid instru ments for predicting success in school subjects, then according to the preceding discussion the achievement of the Experimental Groups, in their study of geometry, should be below that of the Control Groups. However, 55 the I.Q. factor, when taken alone, does not always serve as an accurate prognostic measure. Symonds 48 reports correlation coefficients between Intelligence and Geometry as ranging from .52 to .69, from which he con cludes that the probable correlation coefficient between these two factors is *59. The coefficient of correlation between X.Q. 's and Geometry Prognosis scores for the 659 pupils involved in this study is .614 - .016, which closely approximates the coefficient reported by Symonds. A coef ficient of .614, however, is not very high as it implies an overlapping of approximately 35 per cent of the common elements in the two distri butions. In other words this coefficient of correlation implies a pre diction for success in geometry that is only 35 per cent better than a guess. Because of the uncertainty in using intelligence quotients as a sole means for making comparisons between groups of pupils studying geometry, it was found desirable to administer a geometry prognosis test. 2. Geometry Prognosis Geometry prognosis test scores were obtained for all pupils involved m the study by administering the Orleans beginning of the school year. .90, and according to Orleans 4Q Geometry Prognosis Test at the This test has a reliability coefficient of 50 , its coefficient of correlation with any comprehensive objective geometry test runs as high as .80, which is sig48 Symonds, P.M., “Ability Standards for Standardized Achievement Tests in the High School11, New York? Bureau of Publications, Teachers College, Columbia University, 1927, p. 16. 49 Orleans, Joseph B. and Jacob S., op. cit. 50 Orleans, Joseph B. and Jacob S., “Orleans Geometry Prognosis Test“, Manual of Directions. New Yorks World Book Company, 1927, p.4. 56 nificant for prognostic purposes. The differences between the Experimental and Control Group geometry prognosis scores are indicated in the nBar GraphM of the com bined distribution of the two groups as shown by Figure 2. The longer shaded bars in the lower half of the distribution and the longer solid bars in the upper half of the distribution clearly indicate the geometric superiority of the Control Groups over the Experimental Groups. The significance of this superiority may be noted further by comparing the ‘•Means11 of the two combined distributions# Table VI reveals differences between the Ranges, Standard Deviar tions, and the Means of the Experimental and Control Groups. In this table the difference of the two Means is 11.5 and the standard error of difference between the two Means is found to be 2.4. TABLE VI. GEOMETRY PROGNOSIS SCORES COMPARISONS OF EXPERIMENTAL AND CONTROL GROUPS Group Number of Pupils Range Experimental 333 7 to 168 31.9 84.2 Control 326 24 to 168 29.7 95.7 Standard Deviation Mean The standard deviations of the Experimental and Control Groups, 31.9 and 29.7, are nearly the same. However, since this figure is above .20.0 in each case, the groups taken as a whole are statistically hetero geneous with respect to geometry prognosis scores. In other words, there is apparent a wide range in abilities of the pupils to study and to do the usual work in demonstrative geometry. The difference of the two geometry prognosis Means (11.5) is 4.8 times the standard error of dif ference (2.4), which implies that statistically the Control Groups taken as a whole are significantly superior with respect to the factor under consideration. In other words, the ability of the Control ©roup pupils 57 EXPERIMENTAL GROUPS CONTROL GROUPS Pupils 1 i P *— i Fig. 2.— 40 to 59 60 to 79 Geometry 80 to 99 Prognosis 100 120 to to 119 159 Scores COMBINED DISTRIBUTION OF GEOMETRY PROGNOSIS TEST SCORES FOR EXPERIMENTAL AND CONTROL GROUPS. 160 to 179 58 to do the usual work in geometry is statistically superior to the ability of the Experimental Group pupils to do the same work. Table VII indicates how the geometry prognosis scores may be com pared for the smaller Experimental and Control Groups, namely, the School Groups. In this table the differences in Ranges and in Means between the Experimental and Control School Groups are more poonounced than the dif ferences found in Table VI. As previously mentioned, there are several reasons for the wider variations herein. The most significant reason lies in the fact that geometry is elective in School No. Ill, whereas School No. IV includes some form of geometry as a requirement of all pupils for graduation. Demonstrative geometry is likewise elective in School No. VI and only the pupils who have shown proficiency in mathematics are permitted to elect this subject. School No. V requires all pupils to study some form of geometry. TAB1E VII. RANGES ANDMEANS Off GEOMETRY PROGNOSIS SCORES I'QR SCHOOL GROUP Control Groups Experimental Groups School No. V No. I No. Ill Range 7 to 144 21 to 148 30 to 168 Mean School Range Mean 84.9 76.0 93.8 No. VI No. II No. IV 36 to 147 31 to 159 24 to 168 101.1 92.9 97.5 Table VIII reveals a comparison of the smaller Experimental and Control Teacher Groups. It was pointed out earlier that Teacher A and G Groups were likewise School No. V and School No. VI Groups. Teacher B, C, and D Groups constituted School No. I in Table VII, while Teacher H and I Groups made up the corresponding School No. II Group. Teacher E and P Groups comprised the School No. Ill Group, while Teacher J, K, and L Groups comprised the corresponding School No. IV Group of Table III. The differences between the Teacher Groups of Table VIII are TABLE V I I I . RANGES AND MEANS OF GEOMETRY PROGNOSIS SCORES PER TEANHER GROUP Control Groups Experimental Groups Teacher Range Mean Range Teacher A B C and D E 3T 7 25 21 30 68 apparent. This difference is due to factors already explained in connection to to to to to 144 148 138 127 168 84.9 79.8 71.4 69.9 121.7 with the preceding table. G H I J and K L to to to to to 147 159 138 168 166 101.1 94.1 89.1 92.1 100.7 However, there is another factor introduced at this point which was not apparent in Table VII. “ability grouping11. 36 31 32 24 43 Mean This factor is that of It must be remembered that each School Group represen ted a rough cross-section of the entire tenth grade enrollment in that school. The only exceptions, of course, were the schools where geometry is an elective. As a consequence, the factor of ability grouping brings out greater differences in the Teacher Groups than in the School Groups. For example, Teacher 3P had two groups of accelerated pupils, whereas Teacher L had a slow, an average, and a superior group. Teacher E had two slower or retarded groups, and Teachers 2 and 2 each had a slower and an accelerated group. In conclusion, the factor of “ability grouping” is largely responsible for the heterogeneity between the Teacher Groups. In order to reveal the heterogeneity between the Teacher Groups in a more detailed manner, Table IX was formulated. This table is the summary of a tabulation that was made from the per cent of pupils under each teacher who fell into each of the four quartiles of the combined total distribution of both Experimental and Control Groups. The procedure for determining the data in Table IX was to find percentile ranks of the combined geometry prognosis scores distribution of both Experimental and Control Groups and then determine the number of 60 TABLE IX. DISTRIBUTION OP GEOMETRY PROGNOSIS SCORES BY QUARTILES POE e a c h t e a c h e r tmoup 0£ Teacher Classes Number and Per Cent of Pupils Palling Into 1st Quartile 2nd Quartile 3d Quartile 4th Quartile Experimental Groups 4 3 2 2 2 A B and D E 3? G H I and K L 2 4 2 2 3 pupilsunder 24.1$ 25.0$ 22.4$ 7.1$ 35.4$ 26 18 16 21 2 or or or or or 26.3$ 25.0$ 27.6$ 37.5$ 4.1$ 28 23 26 27 0 or or or or or 28.4$ 31.9$ 44.8$ 48.3$ 0.0$ Control Groups 12 or 3o.7$ 33.3$ 24.1$ 37 or 33.1$ 25.6$ 11 or 25.6$ 32.7$ 9 or 18.3$ 25 or 30.1$ 31.3$ 10 28 10 12 19 or or or or or 25.6$ 25.0$ 23.2$ 24.5$ 22.9$ 4 20 11 12 13 or or or or or 10.4$ 17.8$ 25.6$ 24.5$ 15.7$ 21 or 21.2$ 13 or 18.1$ 3 or 5.2$ or 7.1$ 29 or 60.5$ 13 27 11 16 26 or or or or or 24 18 13 4 17 e©jch teacherthat fell into or or or or or each of the four quartiles. The per cent was then obtained by dividing the number of pupils whosegeometry prognosis scores fell into a quartile by the total number of pupils in the particular Teacher Group. The heterogeneity between the Teacher Groups on the basis of Geometry Prognosis scores is apparent in Table IX. The range by quartiles for the per cent of pupils under each teacher in this table is as follows! | , 1st quartile, 2nd quartile, 3d quartile, 4th quartile, from 5*2$ to 60.5$ from 7.1$ to 35.4$ from 4.1$ to 37.5$ from 0.0$ to 48.3$. ! It is for this reason that the pupils or Teacher Groups in this study are not to be equated. 3. Critical Thinking- Abilities Prior to the Study of Demonstrative Geometry In the preceding sections an effort was made to describe certain pupil characteristics, such as intelligence and ability to do the usual work in demonstrative geometry. This section contains a description of pupil critical thinking ability prior to the study of demonstrative geometry. A great deal of research is necessary in the testing field to develop techniques for evaluating critical thinking behavior as it was described in Chapter I. However, a survey of the testing field revealed instruments that did measure critical thinking behavior with respect to at least five of the elements mentioned earlier* The most appropriate instrument for this purpose was the Nature of Proof^ Test 5.3. This test was selected because it contains at least five elements of critical thinking, as follows! (1) Questioning the meaning of terms and seeking satisfactory definitions or descriptions of them. (2) Detecting and questioning underlying assumptions. (3) Searching for more facts and testing these £acts for pertinency to the situation. (4) Discriminating between facts and assumptions. (5) Testing conclusions for consistency with assumptions. This test revealed some evidence of critical thinking on the part of the pupils; however, the results taken as awhole deficiency in this type of thinking. The element indicated apronounced of criticalthinking that stood out at this point (prior to the study of demonstrative geometry) above the others was the first one, "Questioning the meaning of terms". Sensitivity with respect to the other elements was practically negligible. A statistical summary of the results is as follows! Name of the test - Nature of Proof Test 5.3 Number of pupils involved - 333 Possible range of scores - 0 to 135 Actual range of scores — 1 to 30 Standard Deviation of the scores Mean of the distribution 51 Nature of Proof Test 5.3, opi - - 4.4 12.7. ®iit. 62 There are no available Norms for this test, therefore statistical somparisons in this respect cannot “ be made* However, the small actual range compared to the possible Eange, as well as the low standard deviation and low mean score, definitely indicate and lend support to the remark made earlier with regard to the low level of critical thinking ability at this stage of a pupil*s development. This test was not administered to the Control Groups because it was to be used again as a re-test at the end of the school year for the Experimental Groups, and it was possible that this test would influence teaching procedures in the Control Groups. Furthermore, the combined Experimental Group is sufficiently large to permit statistical treatment for comparative purposes. For example, there is every reason to believe the superiority of the Control Groups with respect to Intelligence and Geometry Prognosis would likewise carry over to the Nature of Proof Test. Supplementarv Information Concerning Experimental__and Control. Group Pupils In order to obtain as much information as possible regarding the pupils involved in this study, an Inventory Questionnaire (see appendix C, page 391 ) was devised. This questionnaire was used primarily for guidance purposes and both Experimental and Control Group teachers were free to make use of the information therein. (This questionnaire did reveal a great deal of information about the pupils in connection with the present study, and some of this information will be included at this point. In order to point out explicitly the more important characteristics of pupils with particular reference to the present study, some of the more pertinent questions were selected and discussed separately, as follows: 63 ft uestion No. 1 "What are your favorite subjects? Why?11 The purpose of this question was not only to determine pupil interests, hut also to find out what percentage of pupils include mathematics among their favorite subjects. The responses were grouped into three types, (l) Mathematics, (2) Others, and (3) Ho response to the question. /, (1) Mathematics Group (2) Others (3) Ho Response Experimental 20.9$ 76.9$ 2.2$ Control 41.4$ 58.6$ 0.0$ The per cent of Control Gkoup pupils who include mathematics among their favorite subjects is nearly twice that of the Experimental Groups. This may he due to the inferiority of the Experimental Groups in intelli gence and in ability to do the usual work in geometry, inother reason would he the tendency to respond in such a way as to make an impression on the teacher; however, this factor should influence both groups propor tionately. Question Ho. 2 "In what way do you think mathematics may help you? Why?" The purpose of this question was to obtain the pupil!s interpretation of how he thinks mathematics may help him. This does not imply that the response is entirely original with the pupil, because external influence or external stimuli are involved in the situation. However, the groups are so large there is every reason to believe this factor operates propor tionately in both groups. The responses to this question were grouped into six types, (l) in vocations, (2) in mental discipline, (3) in developing reasoning ability, (4) in meeting college entrance requirements, (5) in no way whatever, and 64 (6) no response. The following summarized tabulation represents the per cent of pupils from each group whose responses fell into one of the six types* <2> Mental Disc. (3) Season Group (i> Voc. Experimental 41.8$ 20.9$ 9.8$ Control 61.6$ 12.1$ 12.5$ (4) College (5) No Way (6) No Response 2.2$ 8.2$ 17.1$ 4.3$ 3.9$ 5.6$ These responses indicate that more Control Group pupils think of mathematics as a channel leading toward vocations than is true of the Experimental Groups. The latter, however, exceed the Control Groups by eight per cent in looking upon mathematics as a mental discipline. These differences appear to be consistent with the differences in intelligence and geometric ability, discovered earlier, because pupils are interested generally in those subjects in which they meet with most success. Question Ho, 3 HIn frhat way do you think demonstrative geometry may help you? Why?*1 The purpose of this question was to find out what impressions pupils may have of this subject prior to its study and to make comparisons with the responses to the preceding question (Question No. 2). The responses to this question fell into one of the following types, (l) in vocations, (2) in mental discipline, (3) in developing reasoning ability, (4) in meeting college entrance requirements, (5) in no way whatever, and (6) no response. The following summarized tabulation represents the per cent of pupils whose responses fell into one of the six types* Group ( 00. Experimental 12.7$ 8.2$ Control 25.0$ 8.6$ Mental^idsc. * (3) heason Coliege No^fay 10.1$ 3.2$ 10.8$ 55.0$ 19.4$ 5.6$ 6.9$ 34.5$ No Response 65 The most significant fact revealed herein is that "both Experimental and Control Group pupils were not familiar with the meaning of demon strative geometry* The difference in responses of the first type, nin vocations11, is consistent with the corresponding responses to Question No* 2. This is largely due to an association of the term **demonstrative geometry1* with mathematics* Question Ho. 4 **Do you think high school sophomores should have an opportunity to reason about the subject matter of geom etry in their own way, or do you think the subject should be developed and learned the way in which it is presented in a text book or in some work book?11 The purpose of this question was to find out in a subtle way how pupils react to this type of question, or in other words to obtain their preliminary reaction to teaching procedures. The former approach was to be adopted by the Experimental Groups; however, the pupils were not aware of this fact at the time the questionnaire was administered. The responses to this question fell into one of the following four types; (l) independent of textbook, (2) dependent upon textbook, (3) com bination, and (4) no response. The following summarized tabulation represents the per cent of pupils whose responses fell into one of the four types. (1) Independent Group Dependent Combination No Hesponse Experimental 33*2$ 40.5$ 14.6$ 11.7$ Control 15.1$ 62.9$ 15.9$ 0.0# Most of the Control Groups were aware of the fact that a textbook would be required, therefore the 62.9 per cent responding to this question were influenced to a great extent by this fact. On the other hand the Experimental Groups were not informed one way or the other regarding a textbook. Therefore, any influence that was exerted upon the 40.5 per cent responding in favor of a textbook was due largely to a supposition on their part based upon tradition. In the light of this evidence it is safe to conclude that the Experimental Group responses are more typical under the circumstances. Furthermore, the responses indicate a tendency on the part of young people to desire opportunity to reason about subject matter in their own way* Question Ho. 5 HHow would you like to have your progress in this course evaluated? (e.g., by a written examination, judgment of your teacher, self-evaluation, evaluation by parents, etc.). Why?11 The purpose of this question was to find out how the Experimental and Control Groups react to evaluation prior to their study of demonstrar* tive geometry. The responses to this question fell into the following five categories? (l) by examination, (2) teacher judgment, (3) selfevaluation, (4) combination, and (5) no response. The following summarized tabulation represents the per cent of pupils whose responses fell into each of the five types. Group Experimental Control (3) Self-Eval* (4) (5) Combination No Response (1) Examination (2) Teach. Judg. 9.4$ 12.4$ 5.7# 47.2$ 25.3$ 15.1$ 26.7$ 5.6# 44.0$ 8.6$ The most significant fact revealed herein is that nearly 50 per cent of both Experimental and Control Group pupils indicated a preference for evaluation based upon a combination of factors. single factor as inadequate. Both groups deemed any One may conclude from this evidence that young people do not trust their own standards of evaluation; however, they feel they should be given some responsibility or at least a part in the process of evaluating their progress* 67 Question Ho, 6 "The concept of proof has always played a significant role in human experience. When in your judgment is something proved?11 The purpose of this Question was to determine whether or not pupils possessed an accurate “concept of proof" prior to their study of demonstrative geometry. The responses to this question fell into the following four types? (l) an accurate concept - for example, indicating a statement is proved if its underlying assumptions are true and if the conclusions reached are consistent with the underlying assumptions; (2) a partly correct concept, if it contains some of the elements of proof; (3) an inaccurate concept, if no elements of proof are implied; and (4) no response. The following summarized tabulation represents the per cent of pupils whose responses fell into each of the four types? , .Accurate Group (2) Partly .Accurate T Inaccurate Ho Response Experimental 1.6$ 20.6$ 46,8$ 31.0$ Control 1.3$ 22.4$ 54.7$ 21.6$ Less than 25 per cent of the pupils, prior to their study of demon strative geometry, possessed to some degree a concept of proof. It is safe to assert that a large majority of pupils are at least definitely deficient in expressing this concept prior to their study of demonstrative geometry. Question Ho. 7 "The kind of society in which we live calls for a type of citizenry that is capable of thinking critically* When in your judgment is a person thinking critically?" The purpose of this question was to find out what sort of concept of critical thinking pupils have at the stage of maturation prior to their study of demonstrative geometry. The responses were grouped into four types? (l) correct, if four or more elements of critical thinking, estab lished in Chapter I, were implied; (2) partly correct, if the responses implied one to three of the elements; (3) incorrect, if none of the elements wer8 implied; and (4) no response. The following summarized tabulation represents the per cent of pupils whose responses fell into one of the four types* (i) Correct Group (2) Partly Correct (3) Incorrect (4) No Response Experimental 0.0# 21.6# 62.5# 15,9# Control 0.0# 29.7# 54.9# 15.4# The results indicate a definite inability in both groups to express the concept of critical thinking in the way in which this concept was formulated in Chapter I* Chanter Summary A description was given of the Experimental and Control Groups, as well as the method by which teachers and schools were selected. This was followed by a brief description of each school with respect to size or enrollment, location, type of community rserved, status of geometry in the school curriculum, and teaching load. In conclusion, it may be said that the Experimental and Control Groups, taken as a whole, represent a rough cross-section of public school situations on the tenth grade level* Several statistical summaries were compiled to point out the characteristics of the pupils involved in this study. The average chronological age was found to be fifteen years for both Experimental and Control Groups. The average 1.^. for the Experimental Groups is slightly less than 106, and for the Control Groups this average is slightly over 110. The difference "between the two I.%. averages was found to be statistically significant, therefore the Control Group, as a whole, is 69 superior in intelligence to the Experimental Group. It was also determined statistically that the difference in geometric ability "between the two groups was likewise significant. The Mean geometry prognosis score for the Experimental Groups is 84.2 and for the Control Groups it is 95.7. Therefore, the Control Groups indicate superior ability to do the usual work in demonstrative geometry. The Nature of Proof test, which includes several elements of critical thinking, definitely indicates a. deficiency in this type of thinking on the part of pupils prior to their study of demonstrative geometry. The supplementary information obtained from the Inventory Ques tionnaire, regarding the Experimental and Control Group pupils prior to their study of demonstrative geometry, reveals several significant facts, as follows? (1) Over 41 per cent of the Control Group pupils included mathematics among their favorite subjects, while only 21 per cent of the Experimental Group pupils did likewise. (2) Most of the Experimental and Control Group pupils indicated that mathematics would help them in their vocations. (3) Both Experimental and Control Groups failed to indicate how demonstrative geometry could help them, because they knew little, or nothing, about this subject, or even the term 11demonstrative11. (4) Both groups indicated the need for a textbook to develop this subject; however, a substantial per cent of pupils indicated their preference for an opportunity to develop some of the subject matter of geometry in their own way. (5) Both groups indicated a strong desire to share in the evaluation of their progress. (6) Over 75 per cent of the pupils in both groups were found lacking in a clear concept of proof; at least they were unable to express this concept. (7) Nearly all of the pupils in both groups were unable to point out the meaning of critical thinking in the way in which it was described in Chapter I. CHAPTER III DESCRIPTION OP PROCEDURES This chapter presents a description of teaching procedures in the Experimental and Control Groups. Since the procedures of the Control Groups are well known, only a brief description of them will he given. Control Group Procedures The teaching procedures used by the Control Group teachers follow a more or less fixed pattern, that of requiring each pupil to secure a textbook from which assignments are made. The major emphasis is directed toward helping pupils understand as much of the subject-matter of geometry as possible, with little or no opportunity for the pupils to reason about the subject-matter in their own way. Little, if any, effort is made to carry the reasoning in geometry over into non-mathematical situations. Assumptions Underlying the Experimental Teaching Procedures There is so much controversy in the field of method that it becomes extremely difficult to justify any one method or procedure in teaching. It is essential, however, at this point to make some assumptions re garding teaching procedures adopted by the Experimental Group teachers. There are four such basis assumptions, as follows? 1) that high school sophomores are capable of thinking critically. 2) that the pupil is a psychological being and should have an opportunity to reason about instructional material in his own way. 3) that worthwhile transfer can take place if there is a distinct effort made to teach for transfer. 4) that it is possible for teachers to study the behavior of pupils in order to become more sensitive to their needs; stimulate their present interests; and help them develop new interests in line with their abilities. 71 The procedures employed in this study have been derived from the above four basis assumptions* Wheeler 52 lends considerable support to the second one when he writes that ’•Learning is not exclusively an inductive process. First impressions are not chaotic and unorganized*.. There is nothing more highly organized than childrens logic, to which impressions are subordinate. Adults do not discover this logic, that is all*1' If certain values claimed for the study of demonstrative geometry are to become a reality, it becomes extremely important that the logical processes of the pupil must be respected. current teaching practices. This, however, is contrary to At least, the observation reports 53 relative to traditional or Control procedures definitely indicate that young people are given a text fashioned by an adult pattern of logic, and little if any opportunity is offered them to think about geometric content in their own way. The imposition of logical patterns of thought upon young minds who frequently do not accept these patterns as logical tends to impair the very outcomes that are being claimed for the study of demonstrative geometry. statement: It was over a decade ago that Young^ wrote the following ’’The trouble is that the authors of practically all of our current textbooks lay all the emphasis on the formal logical side, to the almost complete exclusion of the psychological, the latter of which is without doubt far more important at the beginning of a first course in algebra or geometry. They fail to recognize the fact that the pupil has 52 Wheeler, Raymond H., “The New Psychology of Learning1’, Tenth Yearbook of the National Council of Teachers of Mathematics, Bureau of Publi cations, Teachers College, Columbia University, New York, 1935,pp.237-238. 53 A trained observer was employed to study each group and reports of his observation appear later in this chapter, page 109 ff. 54 Young, John Wesley, Lectures on Fundamental. Concepts of Algebra and Geometry, New York: The Macmillan Company, 1925, page 5. reasoned, and reasoned accurately, on a variety of subjects before he takes up the subject of mathematics, though this reasoning has not per haps been formal. In order to induce a pupil to think about geometry, it is first necessary to arouse his interest and then let him think about the subject in his own way." This statement, and particularly the last sentence, is in keeping with the four basis assumptions underlying the Experimental procedures in this study, because the discovery of any math ematical principle - even though it be a very simple one - by means of the pupil’s pwn initiative leads to true and wholesome educational experience. The four basic assumptions are likewise in keeping with Kilpatrick’s^* statements 11The most widespread and imperative tendency along methodological lines is the insistent demand that we get our students more fully ’into the game* ... As teachers we are concerned not merely with the objective goals reached by the pupils, but quite as truly with the actual searchings themselves. The good teacher of mathematics knows, perhaps as do few others, that to have searched and found, leaves a pupil a different person from what he would be if he merely understands and accepts the results of others’ search and formulation.” Kilpatrick further states as the last step in method that ”A pupil of himself sees in a situation the mathematical relations dominating it and of himself solves the problem he has thus abstracted from the gross situation.” In order to be consistent with the four basic assumptions there can be no definite sequence of theorems arranged in advance. However, the guidance and counsel of a skillful teacher must not be overlooked at this 55 Kilpatrick, William Heard, ’’The Next Step in Method”, The Mathematics Teacher, Vol. XV, No. 1, 1922, pp. 16-25. 73 point* In fact the teacher ought to “ be well acquainted in the field to which the pupil is about to enter and it becomes the duty of this teacher to assist the pupil in developing whatever sequence will give him the highest sense of accomplishment and success. Since the emphasis is to be placed upon the method or process by which theorems are proved, then what ever theorems are covered is not a matter of great concern. There was one exception in this situation, that in all fairness to the pupils contem plating meeting certain college board requirements, it was mutually agreed by the Experimental Group Teachers to utilize and to emphasize, toward the end of the school year, the ten constructions and twenty theorems essential 56 to a study of geometry as proposed by Christofferson • Just how many theorems are to be covered is largely a matter of teacher judgment with respect to the Hlaw of diminishing returns11. There are those pupils who derive a great deal of genuine satisfaction from proving theorems which they have discovered, for themselves through the process of thinking about the subject matter of geometry in their own way, and generally this group wishes to prolong this type of experience. Con sequently, it is through a careful study of the "law of diminishing returns" that the skillful teacher may cope with this problem more intel ligently even to the extent of suggesting the type of content - whether mathematical or nonBmathematical - that will best serve to bring about the desired outcomes the teacher has set up for himself. One may find a great deal of educational value in group consideration - through critical analyses - of theorems discovered by individual pupils. By means of such discussion results achieved by one pupil frequently influence others toward greater accomplishment. __________________________________________ 56 Christofferson, C.C., "Geometry Professionalized for Teachers"« Menasha, Wisconsin, George Banta Publishing Company, 1933, pages 10-12. 74 Description of Experimental Group Procedures .I*. Procedures Common to the Six Experimental Group Teachers Besides mutual agreement upon "basic assumptions and philosophy underlying Experimental Group teaching procedures, previously mentioned, the Experimental Group teachers agreed to begin with pupil interests and abilities. This procedure was facilitated by means of several paper and pencil instruments as follows! (1) An. Inventory Questionnaire - to determine interests (2) Geometry Prognosis Test - to determine geometric ability (3) Intelligence Test - to determine general ability (4) Mature of Proof Test - to determine critical thinking ability. The purpose of the above instruments was described in Chapter II and the information therein was used to describe experimental situations prior to pupils1 study of demonstrative geometry. It may be of interest to mention at this point that while the inventory questionnaire was used in Chapter II to facilitate a description of the experimental situation in its primary stage, its main purpose - in connection with individual pupil interviews - was to: (1) discover the pupil's attitude toward school in general, and toward mathematics in particular in order to help him plan for an adjustment if any needed to be made. (2) discover the pupil's needs in order to provide opportunities which would facilitate meeting these needs. (3) discover the pupil's present interests in order to help him get started on a project in line with his interests and his ability, which as a consequence may lead him into and develop new interests. Each of the six Experimental Group teachers adopted teaching pro cedures that centered around the ten elements of critical thinking des cribed in Chapter I. A summary of the general teaching procedures adopted 75 by the six Experimental Group teachers is as follows: (1) No textbook was used to develop content; however, references were frequently given for supplementary topics. The references were op tional and were in the form of suggestions for those who could profit from further consideration. Each pupil developed a notebook, or ’’text" as it was called by the pupils, and was given an opportunity to develop it in his own way. (2) Motives were sought by pupil and teacher in every situation of concern to the group. (3) Undefined terms were selected by the pupils with little if any assis tance by the teacher. (4) No attempt was made to limit undefined terms to a minimum. (5) Terms in need of definition were selected by the pupils and definitions became an outgrowth of classroom discussion rather than a basis for it. (6) Words or terms needed to express a concept developed by the pupils were frequently supplied by the teacher. (7) Certain propositions which appeared obvious to the pupils were accepted as assumptions unless sufficient inquiry demanded proof. (8) Most of the assumptions were made by the pupils. teacher led to others. Suggestions by the (9) No attempt was made to limit assumptions to any particular number, or to a minimum. :i0) In the process of developing the course, need for generalizations and establishment of facts was found essential. Since the inductive method was used extensively, the element of fact finding or searching for more facts became inherent in each situation. Ill) The preceding procedures led toward the testing of facts for pertinency, because many facts failed to contribute information that would lead to a generalization. !l2) Classroom discussions and procedures necessitated discrimination between facts and assumptions. This element of critical thinking led to the formulation of a criterion by which the group could judge whether a statement was an assumption or a fact. The criterion agreed upon by the group was as follows: 1. All theorems, corollaries, and propositions or statements in general, which were proved would be considered as facts. 2. AL1 unproved propositions or statements would be considered as assumptions until sufficient evidence justified their being classified as facts. 76 (13) Assumptions "behind any situation of concern to the group were sought# This detection of stated as well as hidden or implied assumptions was recognized and considered as important "by the group# (14) In developing the inductive method of reasoning, which constituted a major portion of classroom discussion, no statement of what was to "be proved was given to the pupils# Certain properties ahout geometric figures were assumed and pupils were encouraged to discover the im plications of these assumed properties. (15) Pupils were encouraged to set up tentative hypotheses ahout a geometric figure and to test these hypotheses for logical consistency with the underlying assumptions. (16) Another step in developing the inductive method of reasoning was to avoid giving pupils generalized statements. The class was given an opportunity to study the implications or consequences of properties assumed ahout a geometric figure and then encouraged to formulate their own generalization. (17) In developing the deductive method the theorems suggested by Christofferson^? as essential to a study of demonstrative geometry were utilized. Some of the theorems, however, were developed in the inductive process. Testing conclusions for consistency in this method of reasoning became very prevalent. (18) Tentativeness of the conclusions reached by the group was definitely recognized, because no conclusion reached proved any more than what was contained in the assumptions. In other words, it became apparent that if the assumptions were true then and only then were the con clusions true. (19) Need for group cooperation in developing a Htheory of space11, as it was called by the pupils, necessitated in some instances that pupils evaluate themselves for bias or prejudice. (20) Need for consideration of consequences was brought to the attention of the group when an assumption was made, because the conclusions reached had inherent in them consequences that in some cases were desirable while in others they were undesirable. (21) Every effort was made on the part of each teacher to provide for the individual abilities of their pupils, because assumptions frequently led to theorems that were unimportant for many of the group. Con sequently, references to such theorems became optional and further consideration was suggested for the pupils who could profit from such consideration. (22) Provision for individual differences was accomplished largely through: a) individual projects, b) voluntary contributions to the group, c) diaries, and d) self-evaluations. These are briefly described as follows: 57 Christofferson, 0. C., loc. cit. 77 a) Individual projects were the outgrowth of a conference between pupil and teacher in connection with the inventory questionnaire administered at the beginning of the school year. Suggestions were made by the teacher regarding a project centered around critical thinking. The pupil was given opportunity to develop this project in connection with geometric or with non-geometric content. In other words the only requirement was that the project be developed in line with the pupil's interests and abilities, and that it be the result of critical thinking on the part of the pupil. b) Voluntary contributions were the outgrowth of classroom discussions. Bach pupil was provided with a folder. This folder was kept in a small filing cabinet, .Any materials related to mathematical or non-mathematical content which dealt with critical thinking were filed in the folders. c) Diaries were kept by the pupils of situations they en countered outside of the classroom in connection with critical thinking* d) Self-evaluations were made periodically with respect to progress in the course and with respect to critical thinking. (23) Uon-mathematical content was introduced freely in classroom discussions by both the pupils and teacher* (24) The geometric content, in general, centered around five geometric concepts and the twenty fundamental theorems and constructions pro posed by Christofferson^® as essential to a study of demonstrative geometry. The five central concepts are as follows? a) Congruence b) Parallel lines crossed by transversals c) Principle of continuity d) loci e) Similarity, or similar figures. (25) Finally* the major emphasis throughout classroom discussions and throughout individual as well as group projects was not upon an accumulation of content or upon subject-matter mastery, but upon the method by which content or subject-matter was developed. This method was directed toward improvement of critical thinking abilities, as previously mentioned. 58 Christofferson, 0. C., loc. cit. 78 For a further treatment of teaching procedures involving numerous exercises, tests, and so forth, in line with a development of critical thinking abilities, the reader is referred to .Appendix C, pages 391 to 426, 2. Examples of Teaching Procedures in Terms of the Ten Elements of Critical Thinking Elements of critical thinking were not developed in terms of iso lated units, hut were inherent in each case in the entire situation under consideration* However, for purposes of clarity and emphasis, each element will he treated separately. Examples of teaching procedures will he given to illustrate how teachers proceeded tC attain their objectives. A great deal of overlapping in the elements of critical thinking may he in evidence for reasons just mentioned; however, each example will he directed toward illustrating some particular element. The examples of procedures to develop critical thinking are as follows* (l) The element of searching for motives. This element of "searching for motives'1 was constantly emphasized throughout the course in connection with situations demanding such ques tions as! "Why are we studying geometry?" throat is my fortune. "Why did Bing Crosby say, 'My That's why I smoke Old Colds'?" bring up the subject of the sinking of the Panay?" "Why did Virginia "Why did your teacher draw a figure of this triangle and ask you to prove the sum of the three interior angles is equal to 180°?" Teacher A approached her class by presenting an excerpt from "Teacher Retirement System of Texas", which read as follows! "A. Membership Sec. 1. Any person who is a teacher (see section 4 for definition of teacher) during the school year 1937-38 automatically becomes a member of the Retirement System, the 79 unless he signs within 90 days after September 1, 1937, a waiver, stating that he does not wish to "become a member* If he signs such a waiver, according to the law, he may become a member at the beginning of any future school year but he is excluded from prior-service benefits. Sec. 2* After September 1, 1938 any person employed as a teacher for the first time in Texas becomes a member as a condition of employment. Sec* 3. Anyone who has taught in the State of Texas in accordance with the terms of this Act, but who is not in service during the year in which the Act becomes effective, shall, if he becomes a teacher within two years of the date on which this Act becomes effective, and if he continues as such for a period of five consecutive years, be entitled to receive credit and resulting benefits for prior-service as provided for in this Act* Sec* 4* teacher1 shall mean a person employed on a full time, regular salary basis by boards of common school dis tricts, boards of independent school districts, county school boards, Retirement Board of Trustees, State Board of Education and State Department of Education, boards of regents of colleges and universities, and any other legally constituted board or agency of an educational institution or organization supported wholly or partly by the State. Thus included are all full-time employees, such as clerks, business managers, librarians, as well as classroom teachers, and school supervisors and administrators. The teacher read from Article A, Section 1 as far as the first comma, omitting the parenthesis, and then stopped with the question as to just who would become members. The discussion began and finally a little girl suggested that almost everyone was a teacher. That remark led to the realization on the part of the pupils of a need for a definition of the term ♦'teacher11. The fourth section of the same article was then read through the first sentence, that is, down to the list of persons included. The pupils supplied most of th& list themselves. critical of the term "regular salary". Some of them became They wondered just what that meant | and questioned whether or not the janitors were included* I There was so much interest aroused over the first illustration, according to the teacher, that the rest of the period was devoted to a discussion of the following situation* "On our homeward journey last summer, my room mate and I stopped in Evansville, Indiana, with relatives for two days. It so happens that Evansville is located on the hanks of the Ohio river. When the states of Kentucky and Indiana were formed, it was agreed that the river was to be the boundary. Years later the river changed its course, cutting off a part of Kentucky on the Indiana side. In the meantime, Evansville has built a large water plant in the 'no-man^ land* area, and at frequent intervals Kentucky attempts to sue the city of Evansville for taxes on the water plant." Having about ten minutes to spare, the teacher closed the recitation by asking the pupils if they had any idea as to what motives were behind the two situations presented. Several were immediately proposed and the group became fully aware of the fact that they were going to have oppor tunities for some live discussions. This teacher writes that? "When I catch three boys who are accomplished loafers re maining in their chairs after class to discuss with each other some points brought up in class discussion and re turning later in the day to carry the discussion further, and when the only and spoiled son of a very rich family becomes so enthusiastic that he sits on the edge of his chair, and when a little girl, who answered in your ques tionnaire that she sees no value in mathematics and does not expect it to be of any use to her in the future, comes to me and tells me that she wishes she could answer the questions again, I say when all those things take place, I cannot feel that I am doing a poorer job of teaching than I did in the past using the old textbook method." As the first step in leading pupils to understand what is meant by "critical thinking" it seemed important for the teacher to discuss the element of "motive". The classroom situation was an ideal one for this purpose,' because every statement made had behind it at least one motive. This discussion seemed to help the teacher and pupils realize more fully an informal and cooperative pupil-teacher relationship. For instance, 81 there was little doubt in the minds of the class as to why their teacher presented the two situations as the first step in their instruction, and particularly the necessity for clear-cut definitions. The full value of the discussion might have fallen hy the way side had not the teacher sen sitized the group to the element of motive, and the fact that this element is constantly in operation wherever human thinking takes place; that is, people do things and say things hecau.se there is some underlying motive for doing so. Teachers E and 3? approached the element of motive hy asking the question! HWhich of the following statements would he the more easily prove d? 1. The improved conditions in this country are due to the New Deal program. 2. The sum of the angles of any triangle is 180°. Why do you think your teacher asked this question? Do you suppose your teacher had other motives for doing so?" Lively or enthusiastic discussion was in evidence between each question. The group became aware of the place of motive in any situation and began to sense the need for defined terms. (2) The element of questioning and seeking definitions or precise meaning upon which there is mutual agreement. The previous situation developed by Teacher A was likewise an example of this particular element. This teacher made every effort to get her pupils to question the meaning of terms in mathematical as well as noit-mathematical situations and to help them recognize the importance of being critical in this respect. An example of numerous exercises used to help sensitize pupils to this element is as follows! 82 The Illinois Law on .Agricultural Seeds "In July, 1931, a bill was passed in Illinois which demanded that the agricultural seed must have a label stating the name of the seeds and the approximate percentage by weight of the purity of the seeds, and the approximate percentage by weight of seeds of noxious weeds. (This, of course, is not a complete statement of the law but it will serve the purpose of the exercise.) nThe law defined ■agricultural seed1 as seeds of red clover, mammoth clover, white clover, sweet clover, alfalfa, lespidegas, timothy, blue grasses, frome grass, sudan grass and other grasses, vetches, millets, rape and seed corn including hybrid corn. ■'Noxious weeds were defined as buckhorn, field sorrel, Canada thistle, quack grass, docks, ox-eye daisy, dodders, wild mustard, Johnson grass, and wild carrot. "The violation of this law carried with it a fine of not more than 100 dollars. In each of the following cases, try to determine whether the man who sells the seeds is liable to be fined. 1. Mr. Harker sold a bushel of oat grass seed. On the label he stated the name of the grain, the percentage of purity and the percentage of wild mustard seed which it contained. 2. Mr. Harker's brother sold oa,t grass seed too. It happened that his field had many bull nettles in it. When he made out the label for the seed it was correct in all details except for the fact that he did not mention the bull nettles. 3. A farmer sold some buckwheat seed and maintained that the only thing he needed to put on the label was the name of the seed. 4. Mr. Green agreed to sell some soy bean seed. delivered the container bore no label. When it was 5. When the oats were threshed on Mr. Bingham's farm it was evident that his fields were over-run with European bindweed. When he sold these oats for seed, he did not mention the bindweed on the label." (Though not a part of the problem, it is interesting to note that in June, 1937, the Illinois seed law had to be amended to redefine "agricultural seed" and "noxious weeds".) Teachers B, Q, and D developed critical thinking abilities with respect to this element in a similar way. are as follows: Some examples of exercises 83 H0n all very high mountains there is a line called 1timber line*, above which trees will not grow. -Are the following statements necessarily true? 1. Since Mt. Blue has no trees on its summit, it does not extend above the timber line. 2. Since Mt. Breen has trees on its summit, it does not extend above the timber line. 3. Since Mt. White extends above the timber line, it has no trees on its summit. 4. Since Mt. Black does not extend above the timber line, it has no trees on its summit. It was impossible for you to tell whether the above were true ? or false unless you k n e w _______________ HWhat is a good definition? ... ire the following good definitions? 1. A beer has thick fur. 2. A triangle is a closed geometric figure formed by three straight lines intersecting in three points. 3. A horse is a hoofed quadruped. 4. A triangle has sides but no diagonals. 5. A circle is a closed curved line, all points equidistant from a point called the center. ofwhichare f,What criteria shall we set up for a satisfactory definition? The following criteria (this really shouldbe listed under results of procedures; however, its pertinency tosubsequent examples warrants an introduction at this point) were developed by the group with some guidance by the teacher in each case: 1. A term must be defined in terms more simple than itself. 2. A term defined must be placed in its nearest class. (A drill, for example, is not a piece of iron but is a pointed cutting tool. A rectangle is not a polygon, but a parallelogram with a right angle.) 3. The difference between the term defined and other similar terms must be pointed out. (A quadrilateral is a polygon which has four sides.) 4. A definition must be reversible. (A triangle is a polygon, but a polygon is not a triangle.) 84 Let us take the definition we agreed upon yesterday* *A quadrilateral is a polygon having four sides1. What is a polygon? ... What is a ‘broken line? ... What is a straight line? ... What is the shortest distance? ... Maybe I had "better ask you, ‘What is a line?1 ... length? What is width? ... You give up on ‘line1? ... But what is I do, too. Define point. ... Yes, I think we had "better not try to define it. It cannot get hy the first criterion we set up for good definitions. But, if we do not define these terms, what are we going to do ahout them? Why not devote a page or two in our "textbook* to undefined terms?.. Gan you think of any other undefined terms we may add to this list?. Teachers E and E developed this element of critical thinking hy procedures similar td the ones just described. Class discussion began with such questions as* nWhat is a strike in baseball? football?** What is a touchdown in or "As I was driving into a town I noticed a sign which read ‘Population 5000** What is meant by 'Population 5000'?" The pupils began to experience great difficulty in agreeing upon \ what the terms actually meant. They were far more difficult to agree upon than the group had at first anticipated. Other words and terms began to be questioned as to meaning* for example, time, space, point, line, out side. inside, fixed, equal, adequate. American. hate, love, superior. intelligent, between, quantity. Liquid, anger, fear, motion, democracy. patriot, traitor, and others. This led to the development of criteria for definitions similar to the ones previously mentioned. Development of geometric content, in connection with this element, was somewhat as follows* "In class, we are going to discover the concepts necessary to study about geometric figures. We may find some terms which 85 we may "be forced to classify as *undefined terms1. We shall make our own definitions for the terms that can "be defined and classify them as *defined terms*. As we makeprogress in our thinking about geometry we will find it necessary to make some assumptions. The undefined terms, the defined terms, and the assumptions should be listed in some organized fashion, because they will serve as a basis for further development of ideas about geometry. In other words we will use undefined terms, de fined terms, and assumptions to develop logically other relationships or propositions. and so on. These propositions will be used in turn to develop others, In this manner we will build up our own generalizations and conclusions and thus write what we may call our own *textbook* in this course.11 Examples of procedures and exercises regarding definitions and j | | undefined terms are as follows: Need for definitions in every day arguments: 1. Exercises requiring the listing and an attempt at definitions of terms used in games, in science, in government, and.?o forth, whose meaning must be agreed upon by everyone concerned. Illustrations: f,What terms in the following statements need to be defined before you could accept or reject the following statements? (Define as many of these terms as possible.) 1) ill birds can fly. 2) You may take this course if you have had the prerequisites. 3) It is impossible for a naturalized citizen to bemorae President of the United States. 4) I j The angle of incidence of a ray oflight the angle of reflection. ^ isequal to 5) The total cost of operating ny automobile last year wasfive hundred dollars. 6) The governor, in his campaign,pledgedhimself adequate support of education. to an 2. Exercises requiring conclusions resulting from definitions. Illustrations? 1) Eood is everything which helps to give energy to the "body or helps to build it up. Tea and coffee act on the nervous system, cause a feeling of mental exhilaration which later wears off, and we are left more tired than before. Is either tea or coffee a food? 2) Aa equilateral triangle is a triangle that has three equal sides. Jin isosceles triangle is a triangle that has two equal sides. Is an equilateral triangle an isosceles triangle? Is an isosceles triangle an equilateral triangle? 3. Exercises dealing with definitions and undefined terms in geometry. Illustrations! 1) Name and draw as many plane geometric figures as you can. 2) In studying these figures what are the simplest terms we would need to use? 3) Which of these terms can you define? 4) Write out your definitions. 5) List the terms you cannot define. 6) Make sketches to illustrate various types of angles, and pairs of angles. We will find their names and then define them. 7) Make sketches to illustrate various types of triangles. Can you name them? Can you define them? (3) The element of detecting stated and unstated or implied assumptions. Examples of the approach made by Teacher A in connection with this element of critical thinking are as follows: Dr. Allan Dafoe in his latest report on the diet of the Dionne Quintuplets said, rtWe entertained about a million visitors at Callander this season. Nearly everyone who comes to Dafoe 87 hospital wants to know how much the Dionne Quintuplets weigh, and whether their gro?/th is normal. Ever since their seventh month I*ve seen to it that they have Quaker Oats. Their gains in weight are more than satisfactory. This statement is really an argument for using Quaker Oats. The argument is “based on certain assumptions. What gre these assumptions? List as many of them as you can* It was pointed out to the pupils, in connection with assumptions, that until 1826 no one challenged Euclidean Geometry. It was at this time Lobachevsky challenged Euclid's parallel postulate and pointed out that it could not he proved. In connection with the study of the theorem, "The sum of the interior angles of a triangle is equal to 180°n, it was pointed out tothe pupils how a change in the parallel postulate affects the con clusions reached, and that we can no longer say, "The sum of the interior angles of a triangle is 180°", unless we specify that we are accepting Euclidean assumptions. In fact there are three geometries that are equally true. Examples from procedures hy Teachers B, C, and D are as follows! Our working materials are definitions, undefined terras, and assumptions, hut still we must observe and draw implications while we use them. We heard last semester that the whole is equal to the sum of its parts. (This statement was copied on the hoard.) Did you make any use of it? ... Do you all believe it? •.. Do you accept this assumption under all circumstances? ... Is a committee equal to the sum of its parts? ... If a tree were to he cut down, all the sawdust saved, the branches and leaves all piled together, is the tree now equal to the sum of its parts? •.. Where did such a statement come from if it is not applicable to all situations? It was pointed out to the classes that assumptions of this type have been accepted by people throughout the ages as convenient and workable aids 88 in developing the concept of proof - ••proof" "being considered in its "broad sense, Do you "believe I can prove all triangles are isosceles? You don*t? I am going to try to prove it to you. You must either agree that all triangles are isosceles or else point out definite statements in the proof which you will not accept. .Another example is as follows: John says he smokes Camels for digestion’s sake. Make a list of assumptions he accepts in concluding that Camels aid his digestion. While the responses to the preceding situation belong in a later chapter, nevertheless it may be of interest to include five of the assump tions listed by the pupils. They are as follows: 1. Smoking aids digestion. 2. Camels aid digestion. (3. Camels are the only cigarettes that aid digestion. 4. Every one who suffers from indigestion should smoke Camels. 5. That the manufacturers of Camels have proved this point. .Another example is as follows: Is this a central angle? (Responses were both Yes and No.) Why is it that we cannot agree on the answer? (Some of the pupils were assuming that 0 was the center of the circle while others were not.) On what authority are you taking 0 as the center?... Now in order to avoid further confusion and mis understanding we had better all agree on a definition for a central angle. (Pupils eventually arrived at the definition - A central angle is an angle formed by any two radii of a circle.) Examples from procedures by Teachers E and E to develop this element of critical thinking are as follows: Various exercises in selecting the necessary assumption definitely implied in various arguments* Illustration: Select, from the assumptions below the argument, the one assumption which you think the argument definitely assumes or takes for granted. 89 Arguments No one needs a college education, because plenty of business men have succeeded without it* Assumptions: (a) Colleges should train their students for the business world* (b) College training is never a help in the business world. (c) College training is of use to only a few business men. (d) Any training which does not prepare for a successful business career is unnecessary. (e) Many business men with college training have' failed. Another example is as follows: (Exercises to bring out the importance of detecting assumptions.) In a magazine, Bob saw the enclosed advertisement for Lucky Strike cigarettes, signed by Cary Cooper. Bob was convinced by this advertisement to the point of trying Lucky Strikes. !Ehe following is a list of assumptions that he might have made in allowing the advertisement to persuade him. In the blank pre ceding e&ch assumption, write the letter: if you consider it a major assumption if you consider it a minor assumption. 1. Luckies are a light smoke. 2, A light smoke is to be preferred. 3. What is good for Gary Cooper is good for me. 4. The throat is the only thing to be considered when buying a cigarette. There are certain throat irritants naturally present in all tobacco. 5. 6. G-ary Cooper has enough intelligence to know what is good for his throat. 7. The process "It’s Toasted11 is something very new in removing throat irritants. 8. Other cigarettes are probably not toasted. 9. Artists, radio and opera stars always use the best quality. 10. % taste is the same as that of Gary Cooper. 11. In order to be prominent, I must smoke. Examples of exercises for finding hidden assumptions. Each of the following contains a statement of given facts followed 90 "by a conclusion or implication drawn from these facts. The conclusion is true only if the assumption not mentioned is true. Read each exercise carefully and write down the assump tion that makes the conclusion true. 1. If A weighs the same as B, and Cweighs the same as B, then A is also equal in weight to 0. 2. If baskets X and Z hold the same number of bushels, and Y and W hold the same, and we also know that Z and W have the same capacity, then X and Y have the same capacity. 3. If we assign numerical values to letters, and make 7 s t, and r s 7, then t s r. 4. If v a t and r ^ w, then v + r > t +■ w. 5. If n + 10 s 50, then n 3 4©. (4) The element of searching for more facts. Examples of exercises used by Teacher A to develop this element of critical thinking are as follows? After each conclusion given below write "Yes" if it is true, "Ho" if it is false, and "Doubtful" if it may be either true or false. Back up each of your statements with evidence. 1. Ever since Roosevelt became president, conditions have improved, therefore? a) Roosevelt is responsible for the improvement. b) Roosevelt has nothing to do with the improvement. c) More information is needed before a decision can be made. 2. When an auto is travelling.20 miles, per hour the distance it will go after the brakes are applied is 28 feet. If its speed is 40 rap.h., the distance it will go after the brakes are applied is 112 feet. If its speed is 80 m.p.h., the distance it will go after the brakes are applied is 448 feet, therefore: a) There is no definite relation between the increase in speed and the increase in the distance the car will go after the brakes are applied. b) There is a definite relation between the increase in speed and the increase in distance the car will go after the brakes are applied. Problem! Devise a method by which you could measure the distance between two points on the opposite sides of a b lake or a large pond, or even a grove of trees " with heavy underbrush. Prove that your method will give the distance between the two points A and B. ^ Some examples of exercises used "by Teachers B, Q, and D to de velop this element of critical thinking are as follows! Suppose I tell you that Mr* Brown, upon the recommendation of his druggist, bought a fine bamboo hair brush, and that during the eight years he had owned it he had not lost a hair from his head. If I have stated the truth, would you be ready to buy a like brush? But, suppose I gave you an additional fact, that Mr. Brown was bald when he bought the brush? Would this additional fact alter your decision? It is necessary to habe sufficient facts before reliable con clusions are drawn. Another example! Smith, Jones, and Brown are candidates for the same office. Smith proves conclusively that Jones cannot perform the duties of the position, and that he can. Therefore Smith should be elected. Would you vote for Smith? Other examples! How far will a car travel going 45 miles per hour? How many books can be bought at one dollar apiece? If one assumes that when wages are raised prices will be raised, does it necessarily follow that! 1. If wages are not raised, prices are not raised? 2* If prices are not raised, wages will not be raised? 3. If wages arerraised, prices will be raised? 4. Ifprices are raised, wages are raised? Another example! Harry Jones was recently arrested twij&e within one week on charges of disorderly conduct. The Justice of the Peace fined him two hundred dollars, which he was unable to pay; whereupon he was sentenced to for ty*-two days in the county jail. Numerous comments on the severity of the penalty are listed below. If you consider only the facts given, indicate with a check mark in Column No. 1 those that you consider just. _ 1. The offense is too slight for such a heavy fine. 2. The offense deserves such a fine. 3. Other facts are needed to justify the fine. 4. It is probable that a smaller fine would have been adequate. 92 It happens, however, that at the time of his first arrest Mr* Johns had "been given a fine of one hundred dollars* He explained to the Justice that he was out of work and could not pay the fine, but that he had obtained a job which was to start the following week. Hence he was granted leniency* Does this change your opinion? Place a check in Column Ho.2 opposite the conclusions you now accept. But there seemed something familiar about Mr. Johns to the Justice of the Peace. He seemed to remember a former charge against the man. So he consulted his files and found another one hundred dollar fine against him on the same charge. Place a check in Column Ho. 3 opposite the comment you believe to be justified. Some examples of procedures used by Teachers 33 and F to develop this element of critical thinking are as follows: Could you develop an absolute method for finding the center of a circular disc and prove this method will always determine the center? Could you determine two points on a material sphere, which would be the extremities of a diameter of this sphere? Could you determine the diameter of such a sphere? Another example: In a certain community there had been numerous burglaries which the police were unable to solve. Early one morning several officers arrested two 18 year old boys who were loitering around a Texaco service station in a residential section of the city. Below are listed some of the conclusions given by a number of the neighbors later in the morning, when they learned what had happened. Place a check mark in Column Ho. 1 opposite each conclusion which you will accept if you consider only the facts given above* 1 1. The boys1 actions certainly were suspicious. 2. The officers had no right to arrest the boys on such evidence. 3. It is certain that the boys were connected with the other robberies. 4. It is probable that the boys were connected with the other robberies. 5* More information is needed.to be sure that the boys intended to break into the service station. 6. The given facts have nothing to do with the boys being involved in the other burglaries. 2 3 93 After further investigation the police were told by a milk man that he had seen some hoys early that morning attempting to open several windows to the station. Using this added information with the facts already given, place a check mark in Column No. 2 opposite the preceding conclusions you would now accept. Fingerprints of the hoys were taken and through these it was found that the hoys had heen arrested in connection with for mer rohheries in another city and had heen committed to a re form school for several years. Does this fact change your judg ment as to which <bf the preceding conclusions you would accept? Check in Column No* 3 the conclusions you now accept. 'Phis element permeates nearly all of the thinking involved in the concept of proof. In the process of developing geometric content it is particularly in evidence in solving "originals". (5) The element of testing facts for pertinency to a situation. This element likewise permeates nearly all of the thinking involved in the concept of proof. Example of a particular situation developed in Teacher A !s classroom is as follows* Boh developed a method for determining the distance between two points on opposite sides of a pond. He illustrated his method hy means of the accompanying dia gram, in which he said! "We must measure BF and make AB equal to BF; also measure BD and make BC equal to BD. Now hy lay ing off angle A equal to angle F the point C is determined and if AC is measured it will equal DF.11 Helen pointed out that laying off angle A equal to angle F was irrelevant to the solution of the problem. Bill pointed out that Helen's statement was true if the triangles were to he proved hy the 11s.a.s." method, however, if the "a. s. a." method was to he used then Bob's assumption regarding BD = BC was irrelevant. An exarrple of exercises employed hy Teachers B, C, and D to develop this element of critical thinking is as follows! Five schoolgirls sat for an examination. Their parents (so they 94 thought) showed an undue degree of interest in the result. They therefore agreed that, in writing home ahout the exam ination, each girl should make one true statement, one untrue statement and one irrelevant statement in the results of their examination. The following are passages from their letters: BETTY: “Kitty was second in the examination. Was terrible. I was only third.H My grade ETHEL: “You'll he glad to hear that I was top. Joan was second. I finished my examination before Joan,“ JOAH: “The questions were awfully hard. and poor old Ethel was bottom.11 I was third KITTY: “Mary was only fourth. I was the first one to finish the examination. I came out second. “ MARYL “I didn't have time to finish answering the last question. I was fourth. Top place was taken by Betty.“ What was the order in which the five girls placed? Which of the facts are irrelevant? Teachers E and F employed exercises similar to the onespreviously described in order to develop this element ofcritical thinking. -An example of a geometric exercise in this connection is as follows: Why are the altitudes erected upon the equal sides of an isosceles triangle equal to each other? A Jane presented the following prosSf: It is assumed that AB = AG, CD J_AB and BE _L,AC. In order to prove that BE s CD, I pre sent the following argument: The Argument Steps 1. AB = AC. 2* Angle CDB s Angle BEC. 3. Angle ABC ss Angle ACB. Reasons 1. Assumed. 2. angles and all right angles are equal. 3. Angles opposite the equal sides of an isosceles triangle are equal. Pe 95 Steps 4. Angle DOB s ingle EOG. 5. ingle 6* DOE ^ AngleBOO. Angle EBC = Angle ECO. 7. Angle EBO =Angle 8. Angle A s Angle A. 9. DOB. triangle ABE iscongruent to triangle ADO. 10. BE s GD. Reasons 4. Opposite angles of two inter secting lines ©re equal. 5. Same reason as No.4. 6, If two angles of a triangle are equal, the third angles are equal. 7. Equals subtracted from equals leave equal remainders. 8. Identity. 9. When two right triangles have the hypotenuse and acutd angle of one equal to the hypotenuse and acute angle, the triangles are congruent. 10. Corresponding parts of congruent triangles are equal. Steve criticized the above proof by pointing out that steps Nos. 3, 4, 5, 6 and 7 were irrelevant to the particular proof selected. Some of the steps, however, could have been relevant if triangle BDC was proved congruent to triangle BEC. (6) The element of discriminating between facts and assumptions. This element necessitated agreement on the part of the pupils upon a criterion which could be used as a basis for judgment. This was found necessaryt because great difficulty was experienced by the pupils in dis criminating between facts and assumptions in border-line cases. The criterion agreed upon by the Experimental Groups for judging contents of situations on this basis was as follows! 1. All theorems, corollaries, and propositions or statements in general that were proved by the group would be con sidered as facts. 2. All unproved propositions or statements would be considered as assumptions until sufficient evidence justified their being classified as facts. Examples of exercises used by Teacher A to develop this element of critical thinking are as follows! Sally and George Robinson were arguing about driving a car. Sally contended that women were better drivers than men. George took the opposite side. Mr. Hobinson remarked that such an argument needed evidence to support it and suggested that instead of spending any more time in "bickering, each secure evidence to support his or her point of view. In a few days Sally "brought in the following table which she had taken from a "booklet put out "by the Travelers Insurance Company of Hartford, Connecticut. TABLE OP DATA WITH KEPEaEHCE TO SEX OE IHIVKtS Drivers in Patal Accidents Male Drivers in Hon-Patal Accidents Per Cent 39,170 93.9$ 1,120,220 92*1$ 2,550 6.1$ 96,090 7.9$ 41,720 100.0$ 1,216,310 100.0$ Pemale Total Per Cent Directions? Part 1* Head each statement "below. Is the statement a fact or is it an assumption? Place a check mark (j/) in the appropriate column before the statement. Part 2. Head over again only those statements which you have marked as assumptions. Place a check mark ( j/) after those three assumptions which are necessary to prove that women are better drivers than men (by this table). Pact List of Statements Assumption •• • • 1. The number of men who are drivers in 1..---fatal accidents was over fifteen times as great as the number of women. » *•• »**• 2. There are as many women drivers as there are men drivers. 2..... • • ♦« • • ** 3. Men are usually more daring than women and will take greater risks. 3 ...... t » •« # * •* 4, There are more men truck drivers than women truck drivers. 4 ...... •« 4« • •• • 5. Women drive as many miles as men when weather conditions are bad or dangerous. 5 . .... 6. The mileage driven by men and women in America is exactly the same each year for each sex. 7* The percentage of non-fetal acci dents for women was higher than the number of fatal accidents in which women were drivers. 6. .... • »•• ♦♦•• * **• 7 ..... 97 .An example of exercises -used "by Teachers B, C, and 33 to develop this element of critical thinking is as follows! As a special advertising feature in connection with sn animal film entitled "Porbidden Adventure11 three monkeys were "being shown in the theatre lobby* One of the monkeys, called Mike, was unusually large and had a reputation of being a "bad actor". Several weeks before at Toledo, Ohio, Mike had bitten a person and later ran amuck in the hotel, leaving a path of ruin and destruction until finally he was captured. He was valuable because of his unusual intelligence, so tt was decided to give him another chance. Therefore, he was exhibited in a cage, but the other two monkeys were confined only by a chain. Since his experience at Toledo the owner kept the monkeys in a storage garage at night. When he was about to feed them one night before leaving them, as a special treat he let Mike out of his cage, taking care, however, to fasten two chains to his collar, one of which he held and the other he fastened to a wall radiator. After Mike stepped out and realized he was tied he became very angry. In a short time he figured out the combin ation to the hook-snaps and thus freed himself. He escaped to a vantage point on a sprinkler pipe near the ceiling, thumped his chest, bellowed defiance at his owner and lunged at him with nasty snarls. All efforts to quiet or catch him failed. The owner finally decided that if he himself could not manage Mike, it was not safe to exhibit him, so he called the police and requested them to kill Mike. Directions! Part 1. Read carefully each statement below. Is the statement a fact or is it an assumption? Place a check mark in the appropriate column before the statement. Pact Assumption List of Statements 1. Mike had a reputation as a "bad actor". •*•• # ••t 2. Mike was too dangerous to use for exhibition. • • •* • *•• 3. He bit a man. • ••* •••• 4. When a monkey bares his teeth he is dangerous « t*» **•• 5 . Mike was a valuable animal because of his unusual intelligence. 6. He was an intelligent animal because he was able to discover the combination of the snap hooks on his collar. «»»• •••• • »•• 7. The more intelligent a monkey is the more dangerous he may be. t*t* * ••• 8. He caused considerable damage in the hotel. •««• «•*• 9. He threatened to attack his owner. 98 Directions: Part 2. Suppose you were describing this incident and gave your reasons for agreeing with Mike*s owner. What are all of the necessary steps, "both facts and assumptions, in the complete proof of your argument? Use as many of the preceding statements as are necessary and place the numbers of these statements in their proper order on the line below. Do not use unnecessary statements. in example of exercises used by Teachers E and F to develop this element of critical thinking is as follows: When I was driving through Tennessee two years ago I visited the Dorris Dam project. It is part of the work of the T.V.A. The United States Government, through this agency, has harnessed the waters of several rivers in the Tennessee Valley and it sells the electric power thus generated to the people of this valley at a rate which is a little more than actual cost. Electricity costs less here than anywhere else in the United States. I visited at the home of Mrs. C. S. Knapp of Marysville, Tenn., a city located in the Tennessee Valley. One evening she made the following remark! "Franklin D. Roosevelt is the first president who has done anything for the South since the Oivil War. (She referred to cheap electric power.) Therefore, I say he is the best president the United States has had since 1865." Directions! Head each statement below. Is the statement a fact or is it an assumption? Place a check ( \/) in the appropriate column before the statement. Assumption ---- ---- ••« • *♦•• List of Statements Fact 2. The residents of the Tennessee Valley get electric 1. ... ipower cheaper than people who live elsewhere in the United States. To help the South makes a man a good president. 2. ... .... 3. We like to get electric power cheap. 3. ... • •♦• 4. Maryville, Tennessee is in the Tennessee Valley. 4. ... • •*• 5. Roosevelt wishes to use electric power. 5. ... •♦•• .... 6. Everyone wishes to use electric power. 6. ... • ••• .... 7. Franklin D. Roosevelt is the best president the United States has had since 1865. 7. ... *«*• .... 8. ... ♦ **• .... **•♦ .... Mrs. Knapp knows what each president has done since 1865. 9. Franklin D. Roosevelt was responsible for the development of' the T.V.A. project. 10. The only thing a man needs to do to be the best president is to help the South. •••♦ • •«•* ••** •■• 1. 8. 9. ... 10___ u cth w University Library Head each assumption and check ( \/) after the statement the three assumptions necessary to reach the sane conclusion Mrs* Knapp reached. (7) The element of self-evaluation for bias or prejudice. This element was prevalent in many situations of mathematical as well as non-mathematical nature. It was particularly apparent in situar- tions where two or more pupils would hold tenaciously to conflicting conclusions. Hefusal to examine one's own major premises or even the other person's major premises, in a situation under consideration, is indicative of bias or prejudice. The teachers of the Experimental Groups utilized every opportunity, whenever such exigencies arose, to acquaint pupils with the necessity of self-evaluation for bias or prejudice. Examples of exercises employed by Teachers B, 0, and D to develop this element of critical thinking are as follows: To learn to think critically and yet reach conclusions upon which all of us can agree, a situation must be void of bias or prejudice. Do you know of any subject wherein such situa tions may be in evidence? Yes, you have been told that Geometry is a subject that helps to develop your ability to think and to reason, v- By the way, are you familiar with most of the Intuitive Geometry? (The responses included so many geometric facts that one teacher remarked! "I may as well tell you that our emphasis is not going to be upon an accumulation of geometric facts, but upon the method by which such facts may be developed.11) I shall try you out to see what kind of thinkers you are. I will write the following statement on the board! John said he would go if Henry went. 1. John went. Did Henry go? (You can't tell.') 3. Henry went. Did John go? (Yes, you did very well.) How many of you have heard of 'Logic1? Is it a new word to most of you? It has been described as the 'art of straight thinking', or the science of reasoning. Logic is much more involved than Geometry; in fact, students are usually sophomores in college before studying it. Do you feel you have any need 'to think' before that time? 100 You have already experienced difficulty in thinking in situations where "bias or prejudice, or may I say, where emotions are con cerned* Try this one! If a 3 b and b s c then ? - ? (Unanimous, a = c.) Perhaps you are not aware of the fact that you know some logic* Hot one of you ask, f,What is a? or b or c?" Yet you all knew the answer* Why? The terms a, b, and c in this example are what we call abstract terms. Geometric terms are to you concrete* By means of Geometry we may hope to learn the art of reasoning, tothink more critically - and perhaps acquire somemore logic* I am going to give you another example similar to one you a day or so ago* Julia said she would go only 1* Julia went* Did Hose 2. Hose went. Did Julia 3* Hose did not go. Did Igave if Hose went. go? go? Julia go? There is a very significant word in the first sentence. is it? Does it change our answers when it is omitted? What Try this one! Negroes would be as intelligent as the whites, if given equal opportunity* 1* Booker T. Washington wan a negro. He was not as intelligent as the whites. 2. Booker T. Washington was very intelligent. He was not a negro. 3. Booker T. Washington had opportunities equal to those of the whites. He was as intelligent as the whites. 4. Booker T. Washington did not have opportunities equal to those of the whites. He was not as intelligent as the whites. Examples of exercises and procedure used by Teachers E and F to develop this element of critical thinking are as follows! Which of the following statements would be more easily proved? 1* The improved conditions in this country are due to the New Deal program. 2* The sum of the interior angles of any triangle is equal to 180°. In general, if I am a Democrat, will I be less or more critical 101 of a Democratic official in office, than I am of a Republican? If an .American and a Japanese were equally qualified for a clerical position, would a Californian hesitate between them? What is my attitude in a question involving a negro and a white man, 1. if I am a white? 2. if I am a negro? A friend of mine and a stranger are suspected of theft. would be my attitude in the situation? What Why is it better to use mathematical relationships to develop critical thinking than to use situations imbued with emotions? This led to a very interesting discussion and to the conclusion that mathematics was less colored by emotions. The discussion likewise led toward an attempt to define bias or prejudice, and to develop a criterion by which one could judge whether or not a person was biased or prejudiced in a given situation# (The group did not get very far with their criterion, but did conclude that all human beings are biased or prejudiced and that all one can doin such a situation is the to strive to reduce this element by doing kind of thinking and reasoning that is done in the study of demon strative geometry.) Situations and problems dealing with Religion, Politics, Law, Capital and Labor, Housing, Unemployment, Racial Relationships, Social Security, Socialized Medicine, Distribution of Wealth, International Relationships, Propaganda, Government Control, War, Crime, and sp forth were frequently referred to in the Experimental procedures because of their highly emotional content. Although disagreement was verjr much in evidence in such areas, the element of self-evaluation for bias or prejudice became more pronounced in pupil behavior as the course was developed# 102 (8) The element of testing conclusions for consistency with the assumptions upon which they depend. in example of exercises used as a means to develop this element of critical thinking, other than the usual geometric exercises, is as follows* Lord Logic was looking for a secretary. The job was a good one and he was able, from a host of applicants, to select three very intelligent candidates. Summoning these three to his office, he assembled them all in one room. "You are all intelligent fellows'1, he said, "but I propose, by means of an exercise of my own, to select the most intelligent of you. What I am going to do is this! "I have here" - he opened his palm - "two small pieces of chalk! green chalk and white chalk. I propose to chalk on the forehead of each of you either a green cross or a white cross. As soon as I have done so I want each of you to look at the other two and, if he can see a green cross, to hold up his hand. As soon as any of you has deduced what colour his own cross is, he must take his hand down. If his explanation as to why he thinks his own cross is green or white is a satisfactory one, he will get the job." As soon as he was certain that these instructions were understood, Lord Logic chalked a green cross on each of the three foreheads. All three hands at once shot up and, almost immediately, one of the three candidates, named Sharp, took his down. "Very good, Sharp", said Lord Logic, "and what colour is yours?" "Green, sir." HOI DID SHARP KNOW? Examples of exercises used by Teachers A, B, and C to develop this element of critical thinking are as follows! Only non-partisan publications ©re fair. Hew newspapers are non-partisan. Pew newspapers are fair. If two angles of a triangle are equal the triangle is isosceles. This triangle has two equal angles. This triangle is isosceles. Other examples! Mr. Williams' dog always barks at strangers. The dog is barking now. Which of the following conclusions do you think are logical? 1. There is a stranger at Mr. Williams' house. 103 2. There is not a strangerat Mr. Williams1house. 3. There may "be a strangerat Mr. Williams'house. 4. No conclusions can he logically drawn. Check the reasons 1. 2. 3* 4. below that you feel support your conclusion. Mr* Williams' dog harks at strangers only. Mr. Williams may he playing with the dog. The dog may see a cat. One needs more information than is given to draw a logical conclusion. Examples of exercises used hy Teachers E and E to develop this element of critical thinking are as follows: The following principles were given as criteria to help pupils deter mine whether each of a series of prepared arguments was correct or incorrect. Each pupil was to indicate which principle he used in making the decision. Principle A; Principle If you accept certain assumptions, then you must accept the conclusions which necessarily follow from them. B: Crucial words or phrases must he defined in exact terms, and a change in a definition will produce a change in the conclusion, denending on that definition. Principle C: The validity of an indirect argument depends on whether all of the possibilities have heen considered. Principle D: A logical argument cannot he disproved hy ridiculing the arguer, or his arguments, or hy attsacking his motives. Principle E: If you cannot accept the assumptions upon which a conclusion is based, you cannot accept that conclusion. Directions: Evaluate each of the following arguments as correct or incorrect, and indicate in the space provided after each argument the principle upon which you base your decision. 1..... Since the beginning of this year, 40 people have heen killed end 1430 have heen injured in Chicago. Therefore we need a drivers' license law. Principle .... 2. .... The future of American government is either a dictator ship in the form of communism, fascism, or our present form of democracy. Since .Americans will never submit to a dictatorship, it will have to he a democracy. Principle .... 104 3. .... In Mr. Brown's automobile policy, the insurance company agrees! To pay on behalf of the insured all sums which the insured shall become obligated to pay by reason of the liability imposed on him by law for damages because of injury to or destruction of property, including the loss of use thereof, caused by acci dent and arising out of the ownership, maintenance or use of the automobile. Mr. Brown*s friend, Mr. Clay, borrowed his car and while driving it had an accident which resulted in a five hundred dollar damage to another car. Mr. Brown thought that his company would pay the damage. Principle .... 4 ..... Alice has agreed with her mother that in the cleaning of her knit dress, for which she paid twenty-five dollars, she must have it cleaned as cheaply an possible, but she must take no risk in damaging the dress. The lowest price she finds is at a cleaning establishment where it can be cleaned for thirty-nine cents, but will be insured for only five dollars. She was assured verbally that the dress would probably come out all right from the cleaning process. However, ALice decided not to leave the dress to be cleaned. Principle .... 5..... Walter is planning to be a teacher, but he reads in a paper! 11In 1918, a farmer could get eight hundred dollars for eight hogs, while for that year, the average high school teacher's yearly salary was hine hundred dollars." Therefore, Walter concludes that he will make more money as a farmer than as a teacher. Principle .... 6. .... Several boys were discussing the reasons for wearing a necktie to school. John said! 11If we assume that the school should train us to be courteous gentlemen, capable of making all the contacts we hope to make in the future, and since no boy can be courteous without a necktie, I think we should wear them.11 Bob said, "John is a sissy. He wants to please the teachers. Therefore his argument is no good." Principle .... ! (9) The element of formulating only tentative conclusions, or in other ! words, suspending judgment. j Experimental procedures were definitely directed toward sensitizing ! pupils in this element of critical thinking. Emphasis was repeatedly ' directed toward the idea that if we accept certain assumptions or premises, we must accept the conclusions which follow from them. However, the ques- i l | tioned character of assumptions readily leads to the tentativeness of 105 conclusions. Therefore, every effort was made to develop an appreciation and an understanding for the relative dependence of all conclusions upon the basic premises or assumptions underlying them. Procedures for developing this element of critical thinking centered around a type of thinking which Keyser names "Fostulational Thinking'1, namely the "If - then kind, distinguished from all others hy its characteristic form* if so - and - so, then so- and - so." Dependence of conclusions upon the "If" clause was constantly referred to, and it was pointed out how the relative validity of the "If" clause necessitated ten tativeness of conclusions or the need for suspending judgment. Numerous exercises dealing with tentativeness of conclusions have i heen devised. Examples of some of the "If - then" types of exercises are as follows? Directions? State the resulting conclusion for each of the fol lowing exercises and then give reasons why you would or why you would not accept your conclusion as tentative* 1. Driving an automobile requires concentration, clearness of judgment and quickness of decision, ALcohol dulls our judgment and slows our reactions. Therefore, ____ (Space provided for reasons) 2. If two lines are parallel, the interior angles on the same side of a transversal are supplementary. /$ In the figure at the right the lines A _____ /________ B AB and CD are parallel. 7x Therefore, _______ ________ . 0 /y D 3. What is good for business is good for everyone. The manufacture of armaments has business world. Therefore, _______________ . helped Directions; Select the necessary conclusion resulting from one of the several given statements. Give reasons as to why you would or would not accept your conclusion as tentative. 1* If autumn brings new life to all things of nature, then 59 Keyser, Cassius J.f "Thinking about Thinking", opus cit. pp. 6-45. the 106 (a) the grass gets green in the spring. (h) trees acquire new leaves in the autumn. (Space provided for reasons) 2. If 11the right of citizens of the United States to vote shall not he denied or abridged hy the United States or hy any state on account of race, color, or previous condition of servitude11, then (a) any colored person is a citieen. o>> any person who has lived in the United States for one year can vote. (c) no citizen shall he prohibited from voting on account of having heen a slave. 3* If the diagonals of a parallelogram bisect each other, and a square is a parallelogram* then (a) the diagonals of a square are equal. (h) some squares are parallelograms. (c) the diagonals of a square bisecteach other. (d) all sides of a square are equal. Directions? Label the following conclusions "Yes1* or "Ho" or “Doubtful**. Give reasons why you would or would not accept the given statements as tentative. 1. If a real estate dealer paid $45,000 for three houses, each house cost him $ 1 5 , 0 0 0 . _____________ (Space provided for reasons) 2. The horse failed to keep a place of importance in the modern world. So will higher mathematics fail to keep its place in our schools. _ 3. A study at Yale University showed that 5 per cent of the honor men, 60 per cent of the average men, and 73 per cent of the failures were smokers. Therefore, (a) the majority of honor men in universities do not smoke. ____________ _ (b) only one out of twenty of the honor men at Yale University smoke. „ (c) the majority of average students smoke. . (d) what is true of the students at Yale is true of the students of all universities. ____________ (e) most students who fail, smoke. ,___________ „ (f) at Yale University, most of the failures smoke. _____ 4. We know now that people are not born equal. Therefor®, we cannot expect to have equality of opportunity for all children in obtaining an education. __,_________ (10) The element of evaluating conclusions for individual and social consequences. It is recognized herein that conclusions are consequences of I 107 assumptions; however, the type of "behavior sought in situations illustrative of this element in critical thinking may be found where a definite course of action is being contemplated following some conclusion or decision, „ uonen 60 relates this same idea to any situation when he ssys, "The true rationality or wisdom of any course of conduct obviously depends upon a true estimate of all its consequences,11 Dewey®* in his treatment of ‘’morals'* implies this element of critical thinking when he points out that "morals based upon concern with facts and deriving guidance from knowledge of them would at least locate the points of effective endeavor and would focus available resources upon them", and thus "enable us to approach the always recurring troubles with a fund of growing knowledge which would add sig nificant values to our conduct even when we overtly failed - as we should continue to do." Experimental procedures were definitely directed toward improvement of critical thinking in terms of this element, is an illustration of individual and social consequences, in connection with geometric content, one of Euclid's postulates was selected for discussion. It was pointed out how the challenge of Euclid's parallel postulate by Lobachewsky not only brought him fame, but also paved the way for Einstein's theory of relativity, which has turned out to be a potent intellectual force and is exerting tremendous impact upon current society. The preceding abstractions, however, were not an outgrowth of class room discussion, but served as a basis for it. In fact, the types of questions raised were as follows* 60 Cohen, Morris R., Reason and Nature? New York! Ear court Brace and Company, 1931, p. 26. 61 Dewey, John, Human Nature and Conduct. Hew Yorks Henry Holt and Company, 1922, pp. 12 - 13. 108 1. What effect will conclusions reached in your geometry course have upon your future plans? 2. Will your esperiences in this course have any effect on your parents, your friends, and your community? Various non-mathematical exercises were devised to sensitize pupils to this element of critical thinking, Examples of some of them are as follows* Directions; -Assuming that you have reached the following conclusions, list or discuss consequences to yourself and to others that may ensue as a result of your action toward fulfillment of any ends you may have in view. 1. Mr. Gray is a wise reader “because he does not depend upon one paper for his information of current affairs. (Space provided for listing and discussion) 2. Hi tier* s conquest of Austria must have "been just, because it was successful. 3. Under capitalism, security varies inversely with the degree of specialization. 4. Our tariffs &ould have “been lowered before 1920, because the world war had made the United States a creditor nation, 5. The time may come when there will be little use for either labor or capital. 6. We cannot have prosperity as long as there is inequality between consumption and production, 7. Democracy will be maintained in this country, because there are more than 100,000 forums and discussion groups in the United States. 8. The right to hear and to be heard will guarantee the perpetuation of democracy in the United States. 9. The ,rTuwn Meeting of the Air11 is a worthwhile radio program. 10. Ho American citizen wants this country to become a dictatorship. 11. I will be a good lawyer, because I am very successful with my study of demonstrative geometry, 12. Football should be abolished, because it is a dangerous activity. 109 Observer’s Reports Eelative to Teaching Procedures In order to indicate contrasts between Experimental and Control teaching procedures, an experienced observer was employed. !37his ob server had no prior knowledge as to which groups were Control and which ones were Experimental. Furthermore, there is no evidence available to indicate preference on the part of the observer for any one procedure. However, the observer was requested to be particularly alert for teacher and pupil behavior with respect to the following outline: 1. Is this teacher giving the pupils an opportunity to reason about instructional material in their own way? 2. Is this teacher making a distinct effort to teach for transfer, that is, trying to point out how thinking in geometry carries over into non-mathematical situations? 3. Is this teacher placing major emphasis on the method of thinking about geometry rather than upon knowledge of the theorems or subject matter as such? 4. Is this teacher studying the pupils and thereby trying to become more sensitive to their needs in terms of their interests and abilities? 5. Are the pupils trying to detect motives behind any situation that is of concern to them? 6. Are the pupils questioning the meaning of terms in different situations and seeking satisfactory definitions or meanings for them? 7. Are the pupils detecting and questioning underlying assumptions? (Stated as well as the unstated ones) 8. Are the pupils searching for more facts pertaining to any situation of concern to them? | 9. Arethe pupils testing facts for pertinency to situations? 10. Arethe pupils endeavoring to discriminate between facts and assumptions? 11. Are the pupils evaluating themselves for bias or prejudice? 12. Arethe pupils evaluating conclusions for consistency with their assumptions? 13. Are the pupils recognizing the importance of formulating only tentative conclusions? 14. Are the pupils evaluating their conclusions in terms of individual as well as social consequences? 110 The following is the observer !s own version of the teaching procedures (the groups are marked Experimental or Control in order to help the reader visualize more readily the fundamental contrasts between them). Observation Ho. 1. (Control) "The text used in this class was _________, and the approach was distinctly a logical one. to the regular polygon. The class was working on some problems relative The proofs were quite formal and the reasons given for various steps in the proofs involved textbook precision. The emphasis was entirely upon mastery of geometric principles, particularly in accordance with the logic found in the text, because on one occasion a boy who had used originality in working out and proving a theorem was corrected by his teacher. It seems his proof did not conform to the standard theorem in the text. "In summary, there was very little evidence, if any, that the class had an opportunity to reason about the subject-matter in their own way. There was no apparent evidence that the class was being taught to transfer the thinking in geometry over into non-mathematical situations. The student-teacher relationship, however, was one of harmony and cooperation. The class appeared to be at ease throughout the recitation. If the students were sensitive to motives, importance of definitions, the place of assump tions, searching for facts, evaluating self for bias, forming tentative conclusions, or considering consequences, they were merely thinking in such terras, because there was no verbal evidence during the period of observation, with one exception, namely, seeking geometric facts relevant to the problem or exercise under consideration. Ill Q^sery-ation Ho. 2. (Experiment^,) "This class was visited during a review period. the solution of original exercises. Various methods were presented and suggested “ by the students rather than by the teacher. solution was in evidence. The lesson included Ho one stereotyped In the main, the class has not used a textbook, The following items were prominent in the course of the class recitation! 1. This teacher was giving the pupils an opportunity to reason about instructional material in their own way. 2. This teacher was making a distinct effort to teach for transfer, that is, trying to point out how thinking in geometry carries over into non-mathematical situations. 3. This teacher was placing major emphasis on the method of thinking about the subject-matter of geometry rather than upon learning the subjectmatter of geometry. 4. This teacher was studying her pupils and trying to become more sensitive to their needs in terms of their interests and abilities. 5. The pupils were questioning the meaning of terms in different situa tions and seeking satisfactory definitions or meanings for them. 5. 7. The pupils were detecting and questioning underlying assumptions, The pupils were searching for more facts pertaining to any situation of concern to them. 8. The pupils were testing facts for pertinency to the situation. 9. The pupils were endeavoring to discriminate between facts and assumptions. 10.The pupils were evaluating conclusions for consistency with their assumptions* 11.The pupils were recognizing and considering the tentativeness of I conclusions. | "The students were alert to proposed solutions and showed evidence of thinking critically and analytically about the propositions and exercises. Lack of response to other factors under observation does not indicate that the objective was not attained in the course." 112 Observation No. g. (Control) “This teacher sat at her desk and asked questions about various theorems in the textbook. The book used by the class w a s ________. The class period was centered around some problems about regular polygons. This class was below the school standard of ability and the students had great difficulty in answering the teachers questions. teacher had several problems placed on the board. Therefore, the These were to be ex plained by the pupils who placed them on the board. had considerable difficulty with their explanations. .Again, the students This led the teacher to request that the class repeat certain basic theorems after her, hoping that they would grasp the essential relationships involved. Since the class still experienced great difficulty, the teacher very patiently explained the less clear points for them. Toward the close of the period the teacher had the class turn to an exercise at the end of the book. “In summary, there was very little evidence, if any, that the class had an opportunity to reason about the subject-matter in their own way. It was quite apparent that the class was not being taught to transfer the thinking they were doing in geometry over into non-mathematical situations. There was a pleasant student-teacher relationship, however, there was little, if any, evidence that the students were sensitive to motives, the need for careful definitions, the place of assumptions, self-evaluation for bias, discriminating between facts and assumptions, seeking additional facts, accepting conclusions as tentative and considering consequences. At least there was no verbal evidence of this behavior.” Observation No, 4 (Experimental) “In this class, which was onetff the lower-ability groups, the teacher opened the recitation by summarizing very briefly the discussion 113 of the previous day., which dealt with definitions, assumptions, and facts about parallel lines. No textbook was in evidence and it seems that nearly all of this material was developed by the students, because this is pre cisely what happened during the class period under observation? The teacher opened the discussion with the statement, 'All of you indicated sometime ago that you knew there were 180° in the sum of the interior angles of a triangle. When asked why this was true, some of you referred to a book and others to your teacher's statement. Not one of you was able to give a reason or reasons that would satisfy all of us, and if you recall, 1 said I would give you another chance to do so when you attained a better understanding of proof and developed sufficient facts in geometry.1 The teacher sketched a triangle on the blackboard and labeled it ABC, with the following remarks? 'Can you prove the three interior angles equal to 180°?i "What followed proved to be a very interesting and exciting recitation. It was apparent from the beginning that the group had no previous experience or knowledge concerning the proof of this theorem. Furthermore, the group was permitted to reason in their own way and to challenge each other's statements. into the discussion. It seemdd as if every student entered In the end, every member of the class appeared satisfied with the proof developed for this theorem. "Words cannot depict the tense moments, the excitement, and the questions and answers brought out during this brief period of fifty-five minutes. The following is a rough description of developmental steps in their proof? "One youngster started the discussion by suggesting that lines AB and BD be drawn perpendicular to iiB at the points A and B. The reasons 114 given "by this student indicated he had in mind the f fact that these two right angles formed "by the two perpendiculars added -up to 180°* She discussion from here on was very lively with only a few minor lulls, following nearly thirty minutes of this procedure the teacher became a bit impatient and remarked as follows? 'Since you cannot establish a relationship between the three interior angles of this triangle and the two right angles formed, perhaps I ought to erase the two lines AB and BD and let you try some other approach.1 The teacher was about to do this when one of the girls remarked? 'Why not construct a perpendicular from 0 to AB? Wouldn't this line be parallel to AB and BD?' This was a master stroke because even the teacher acted surprised. Therefore, Of was drawn perpendicular to AB and in a few minutes the group had pointed out the relationship of alternate interior angles formed by the parallel lines, which the class had agreed upon earlier. "In talking to the teacher after class, I learned that this was the first time he had ever encountered this particular proof. Although I had had first-hand contact with geometry for some time, I too had never encountered this particular proof. The teacher also informed me that the young lady who made the major contribution in this proof experienced great difficulty with mathematics and had on several occasions very definitely indicated her dislike for the subject. "In summary, this teacher was giving his class a definite oppor tunity to reason about the subject-*matter in their own way. The teacher 115 was making a distinct effort to teach for transfer, ■because I noticed the class developing their own "textbook1 as they called it, and in it X observed numerous references to advertisements and other non-mathematical materials* It was obvious that the major emphasis was not upon mastery of subject-matter content but upon the method by which this content was developed. The students throughout the recitation were questioning each other regarding the meaning of the words or terms used, the assumptions that were made or implied, the facets thatwere presented, the conclusions that were reached, and the like.1* Observation No, 5 (Control) "This class was made up of students who were having difficulty with geometry, but are taking it because they need it to enter college. They have for the most part been in other sections and because of a low grade have been placed in this special class. As one boy stated to the observer, ’The class on the average isn't so good*1 This teacher has a definite problem to begin with, and is therefore doing all in her power to get the students to learn as many as possible of the basic concepts of the standard geometry course. She is bending every ounce of her ingenuity to find some methods which will facilitate the learning of the principles of geometry as given in the text by ____________ . •'The same boy just referred to, when asked what had been studied, replied, 'Have studied about geometric figures, areas and proportions.1 The teacher has developed some rather novel ideas and methods for the nresentation of formal subject-matter. The material in its purpose is to get across the subject-matter of the text, at least j got across. Various approaches are used on as much as can be the entire class with the i j I hope that at least one of them will take with each child. 116 "Much emphasis is placed on the matter of grades end the fact that grades are necessary and should he got. The class drill on formulas and problems of areas of geometric figures was 'to get you ready for your home work1. And again, 'I am not going to take grades today. I don't want to embarrass you, - this is to help you get ready for tomorrow and Monday.* "In summary, the major emphasis was to get across to the students, by some hook or crook, as much of the subject matter of geometry as possible, with the result that little or no opportunity was given the child to reason about the subject-matter in his own way and to carry this reasoning over into non-mathematical situations. The students exhibited very little, if any, verbal evidence of questioning or inquiry which is essential toward developing a scientific attitude of mind." Observation Ho. 6 (Experimental) "In this class it was apparent that geometry was being developed as a cooperative enterprise. Ho textbooks were in evidence. day was centered ground the topic of 'polygons'. heard on entering the room was, 'Why? The work of the One of the first things I want the reason here first, don't I?* "Challenging statements were made for the class to criticize and correct. As an example! 'A square and a hexagon are similar' led the dis cussion into developing the fact that the answer to this statement would involve establishing what is meant in this respect by 'similar'. Out of this discussion grew the development of a theorem for similar polygons. "In the process there were many cases of error and mistakes. These were taken up and criticized by enthusiastic activity on the part of the students until a satisfactory solution had been evolved. It wasn't the 117 shortest road, to the answer, hut when it was final 1y reached practically all of the members in the class had some part in its development, "At the close of the period an assignment was made for the next day. It consisted in the teacher's presenting some new definitions for the note hook, wherein they had also placed the proposition mutually developed d m ing that class period, and on the basis of these definitions presenting two problems for them to solve for the next day's wfork. "Apparently the class was developing their own 'textbook1, as they called it. The note hook, or text, had in it many non-mathematical situations which indicated that this teacher was likewise guiding the thinking used in her geometry classes over into noiHmatheraatical situations. Observation Ho. 7 (Control) "On the day visited, the class was studying about polygons. Their activity was at first directed toward definitions as a review measure and then relative to the polygon itself. Formulas concerning the values of angles and the number of sides of a polygon were the order of the class exercise. A great deal of emphasis was being placed on the perfection of learning definitions as tools. The class spent a good deal of time doing seat drill on the formula for the sum of the interior angles of a polygon, under the direction of the teacher. "One boy suggested checking the values of the interior angles by using the supplements of the exterior angles and the astonished class registered a profusion of Ahh's. This would indicate that originality was not a common factor to the procedure. geometric content. The emphasis seemed to be on pure There was no evidence of transferring the thought processes or methods to other areas of knowledge or to other non-matheraatical situations. 118 **A text was “being followed very closely. The class was dealing with Theorem Ho. 416 in the text "by___________ . The teacher seemed | to "be doing a fine job of getting the students to leern the propositions as presented in the text. In fact, the major emphasis was upon the mastery of the classical propositions. 11In summary, there was very little, if any, evidence to indicate that this teacher was giving the class an opportunity to reason about subject-matter in their own way. Ho effort was in evidence to point out how the methods of thinking in geometry may carry over into non-mathematical situations.w Observation Ho. 8 (Experimental) HThe lesson in this class concerned the Pythagorean theorem and its applications. The class had apparently, the day before, developed the theorem or formula, for as they were ready to work a problem, the teacher remarked, *The formula we worked out in class yesterday... 1 The students had a list of exercises they had worked using the formula. Several members of the class had experienced some difficulty with square root in the process j of solving their problems. They were referred to several books in the j library and to any standard text in algebra or arithmetic. ttIn thesolution of the problems, permission was granted to use any " ; approach in keeping with sound thinking. j spirited was the discussion. Class contributions were many and In a problem where a ladder was set against the wall and the height of the wall at the point of ladder contact was the quantity to be found, the teacher asked why it was permissible to use the | Pythagorean theorem or formula. Several students immediately pointed out that they were assuming the wall to be perpendicular to the ground at this I point and that this assumption was an integral part in the solution of this problem. nThere was considerable evidence of group thinking. This class was likewise keeping, or rather writing, a note book, which they referred to as their 1text*. All of the evidence seemed to indicate a genuine effort on the part of the teacher to transfer the reasoning of geometry over into non matheipatical situations•" Observation Ho. 9 (Control) f,The text used in this class was be distinctly textbook and work-book method. . and the course seemed to Ho opportunity was given to see the class actually in recitation, as on the day of visitation they were having written review work, working out problem exercises in a work-book. Discussion with the teacher indicated that major emphasis was on the securing of information on geometric principles as related primarily to mathematics. nConcern was felt over the fact that there was apparently so much irrelevant material in the texts so far as the everyday use of the students was concerned. The teacher would have liked to start with the practical mathematical applications and from those work back to the mathematical solutions as applied to geometry* -An example was given of measuring the distance to another building by the use of angles. nThe major emphasis is by far on geometry as mathematics and on the subject matter concept, with a hope that it can be mace more useful iii the solution of mathematical problems. 11In summary* there was very little evidence, if any, that the students had an opportunity to reason about the subject-matter in their own way, because most of their thinking was guided by a text. was primarily a logical one. The approach While teaching for transfer was based on mathematical principles, there seemed to be no effort exerted to teach for 120 transfer into non-raathematical situations.w Conclusion Relative to the Observer !s Henor-frs. The preceding reports, made "by the observer, indicate very definite contrasts between teaching procedures and teaching objectives. The major emphasis in the Experimental procedures was toward the use of geometric subject-matter as a means of improving or developing critical thinking abilities, and every opportunity was offered to the pupils to reason about this subject-matter in their own way. In the Control Groups the major emphasis appeared to be in getting across tor the pupils as much geometric subject-matter as possible, with very little if any effort made to give the pupils an opportunity to reason about this subject-matter in their own way. Chanter Summary This chapter was limited to a description of teaching procedures. Only the more significant phases ‘ in the Control Group teaching procedures were described. This was followed by a statement of assumptions under lying the Experimental procedures. The major portion of this chapter was confined to a description of Experimental Group teaching procedures. Since a complete description nf the procedures of each of the six teachers could not be presented, a sum mary was compiled of the more salient points common to each of the six Experimental Groups. Moreover, examples were given illustrating the way in which the ten elements in critical thinking were introduced to the pupils by each teacher. In order to indicate contrasts between Experimental and Control teaching procedures an experienced observer was employed. This observer had no prior knowledge as to which groups were Control and which ones were 121 Experimental. The observer was requested, to he particularly alert for teacher and pupil behavior within the frame-work of the four basic assumptions underlying the Experimental procedures and the teneLements in critical thinking. The reports by the observer indicate very definite differences between teaching procedures and teaching objectives for the Experimental and Control Groups. 122 chapter xv GEOMETRIC COHTEHT Introduction In this and the next chapter an attempt will he made to describe geometric as well as non-geometric content developed through procedures described in the preceding chapter. Of course the task of separating learning activities from materials, or procedures from content is not an easy one. However, for purposes of describing and clarifying the nature of the subject-matter involved in this study, some form of separation is essential. Only a description of content developed by the Experimental Groups will be considered in this chapter, because the geometric content in the Control Group situations may be found in any standard plane geometry text^. Therefore, the geometric content in the Experimental Groups will be con sidered from the standpoint of, (l) content developed through the coop erative effort of students within particular groups, and (2) content devel oped through individual effort independent of direct group interaction. It is recognized, of course, that much of the content developed through individual effort resulted from group cooperative effort. Geometric Content Developed Through Cooperative Effort The geometric content developed through cooperative effort will be described under the following categories* (l) undefined terms, (2) defined 62 The textbooks used for developing geometric content in the six Control Groups are as follows* 1. Smith, Reeve and Morss, Text and Tests in Plane Geometry. Hew York, Ginn, 1933. 2. Eorrell and .Arnold, Hew Plane Geometry, Hew York, Charles E. Merrill Company, 1924. 3. Sykes, Comstock and Austin, Plane Geometry. Hew York* Hand McHally, 1932. 123 terms, (3) assumptions, (4) facts or theorems, and (5) fundamental constructions. 1. Undefined Terms AL1 of the following terms were not included as undefined by each Experimental Group, hut in order to he listed each term had to he accepted as undefined hy at least one of the groups. method of Classification.) (See Chapter III regarding The underlined terms were common to each of the six Experimental Groups. I i 1. amount 18. less 2. ASgle 19. line 3* area 20. magnitude 4. average 21. opening 5. between 22. outside 6. hounded 23. plane 7. change 24. noint 8. closed 25. quantity 9. curve 26. ray 10* dimension 27. rotation 11. direction 28. side 12. distance 29. solid 13. equal 30. suace 14. fixed 31. straight 15. greater 32. surface 16. horizontal 33. vertical 17. inside 34. volume .Although the preceding terms were listed as undefined hy at least “ 1 one group, nevertheless there was mutual agreement regarding their meaning. 124 3For example, the term MpointH was conceived and agreed upon as having position, hut neither length, hreadth, nor thickness. The term Hinside11 was agreed upon as belonging to a point or a set of points that did not belong to a set of boundary points or external points. Similar agree ments were made regarding the other terms. 2. Defined Terms The following terms were agreed upon and accepted as defined by at least one of the experimental groups. Those that appear in the un defined list are not repeated here. 1. 2. 3. 4. 5. acute angle adjacent angles alternate-exterior angles alternate-interior angles altitude 26. 27. 28. 29. 30. circumscribed polygon closed curve coincide collinear commensurable 6. 7. 8. 9. 10. antecedent apothem arc assumption axiom 31. 32. 33. 34. 35. common measure common tangent complementary angles concave polygon concentric circles 11. 12. 13. 14. 15. axis base base base base 36. 37. 38. 39. 40. conclusion concurrent cone consequent constant 16. 17. 18. 19. 20. base of a pyramid bisect bisector of an angle center of a circle center of a regular polygon 41. 42. 43. 44. 45. converse theorem convex polygon coplanar corollary corresponding angles 21. 22. 23. 24, 25. central angle chord circle circumference circumscribed circle 46. 47. 48. 49. 50. corresponding sides cosine of an angle cube cylinder decagon of symmetry of isosceles triangle of cone of a cylinder 125 51. 52. 53. 54. 55. degree demonstration diameter of a circle diagonal diameter of a sphere 56. distance between parallel lines 57. distance between parallel planes 58. distance between two points in a plane 59. distance between two points on a sphere d0O. division of a line segment externally 91. 92. 93* 94. 95. mean proportion median minimum minor arc nonagon 96. 97. 98. 99. 100. numerical measure oblique angle obtuse angle octagon opposite angles 61. 62. 63. 64. 65. division of a line segment inter edge nally equiangular equidistant equilateral 101. 102. 103. 104. 105. parallel lines parallel planes parallelogram parallelopiped pentagon 66. 67. 68. 69. 70. equivalent exterior angle of a polygon external tangent extreme and mean ratio geometrical figure 106. 107. 108. 109. 110. perigon perimeter perpendicular bisector perpendicular lines perpendicular planes 71. 72. 73. 74. 75. great circle of a sphere heptagon hexagon hypotenuse hypothesis 111. 112. 113. 114. 115. polygon polyhedron principal parts of a triangle prism projection of a line 76. 77. 78. 79. 80. identity incommensurable initial line inscribed circle inscribed angle 116. 117. 118. 119. 120. proportion pyramid quadrant proposition quadrilateral 81. 82. 83. 84. 85. inscribed polygon internal tangent intersecting planes intersecting lines inversion 121. 122. 123. 124. 125. radius of a circle radius of a sphere ratio rectangle regular polygon 86. 87. 88. 89. 90. isosceles triangle line segment locus maximum major arc 126. 127. 128. 129. 130. reflex angle regular polyhedron rhombus right angle right triangle 126 131• 132. 133. 134. 135. secant sector of a circle segment of a circle semicircle skew lines 146. 147. 148. 149. 150. tangent of an angle tangent lines tangent planes terminal line tesseract 136. 137* 138. 139. 140. similar similar similar sine of sphere 151. 152. 153. 154. 155. theorem third proportional transversal trapezoid triangle 141. 142. 143. 144. 145. square straight angle supplementary angles symmetrical figures tangent circles 156. 157. 15S. 159. 160. 161. 162. unit of area unit of volume variable and limit vertex of an angle vertex of a polygon vertex of a polyhedron vertical angles polygons figures triangles an angle The preceding terms were listed as defined if at least one of the Experimental Groups considered the term as meeting the following criteria for a good definition! (1) Does it contain the simplest language? (2) Can it he classified? (3) Does it describe the term in question sufficiently to distinguish it from similar objects or terms? (4) Is it reversible? Examples illustrating definitions of some of the terms are as follows! A square is a rectangle whose sides are equal. A rectangle is a parallelogram whose angles are right angles. A parallelogram is a Quadrilateral whose opposite sides are parallel. A quadrilater&l is a polygon of four sides. A polygon is a cjosed plane surface bounded by straight lines. The last term, namely ’’polygon’1, of course illustrates the place of undefined terms and it was asserted earlier that undefined terms are needed as a basis upon which definitions can be constructed or formulated. The 127 process of refining a definition to meet the above criteria was fre quently an awkward one. For example, the term "tiiangle" 7/as first stated as "a figure with three lines and three angles". One of the groups refined this definition of "triangle" until it became a "polygon of three sides". 5. Assumptions In the preceding chapter the role of assumptions in any logical conclusion was indicated. In the geometric content developed by the pupils, numerous assumptions were made. Some of these, as previously mentioned, were later proved and thus were established as facts. Some of the assump tions listed below were proved in some of the classes; however, they are included here because they were accepted as assumptions by at least one group in the experiment. Those marked with an asterisk (*) were listed as proved propositions in at least one experimental group. We assume: 1. Our study of geometry is confined to Euclidean space. 2. A straight line is theshortest distance between twopoints. 3. A line can be extendedindefinitely or stoppedat any point. 4. Only one line can connect two points. 5. Only one plane can pass through three points that are not collinear. 6. Three non-collinear points determine a circle. 7. Only one plane can pass through a line and an external point. 8. Two points determine a straight line. 9. tfwo intersecting lines determine a point. 10. Two intersecting lines determine a plane. 128 11. !Ewo parallel lines determine a plane* 12. Any number of lines can be passed through apoint. 13. Any number of planes can be passed through a line. 14. 5?wo intersecting planes determine a line. 15. Given a radius and a definite point as center, a circle may be drawn. 16. Given two points, then a straight line may be drawn connecting them. 17. It talces at least three lines to inclose a surface. 18. AL1 straight angles are equal. 19. ill right angles are equal. 20.* Vertical angles are equal. 21.* Equal angles have equal complements. 22.* Equal angles have equal supplements. 23. Vertical dihedral angles are equal. 24. Only one perpendicular can be erected at a point in a line. 25. Only one perpendicular can be drawn to a line from an external point. 26.* In two supplementary adjacent angles, the sides that are not common form a straight line. 27. CPhe sum of the angles about a point equals 360 degrees. 28. Every circle contains 360 degrees of arc. 29. A geometric figure may be moved about in space without altering the form or size. 30. Only one straight line can be passed through an external point parallel to a given line. 31. Only one plane can be passed through an external point parallel to a given plane. 32. If a line intersects one of two parallel lines, it must intersect the other. 129 33* Quantities may be substituted for equal quantities at any time. 34. If equals are added to equals the suns are equal. 35. If unequals of the same order are added to equals, the sums are unequal in that order. 36. If equals are subtracted from equals the remainders are equal. 37. If unequals of the same order are subtracted from equals the remainders are unequal in the opposite, order. 38. If equals are subtracted from unequals in the same odder, the remainders are unequal in that order* 39. 40. If equals are multiplied by equals the If unequals are multiplied by positive unequal" in the same odder* products areequal. equals, the productsare 41* If equals are divided by equals, the quotients are equal. (Division by zero excluded.) 42. If unequals are divided by positive dquals, the quotients are unequal in the same order. (Division by zero excluded.) 43. A line can bisect another line at only one point. 44. Only one line can bisect an angle. 45. 46. Only one plane can bisect a dihedral angle. A geometric figure is equal to the sum of the parts which it contains. 47. A geometric figure in its totality exceeds any one of its parts. 48. If one of three quantities exceeds a second and the second exceeds the third then the first exceeds the third. 49. In comparing two things of the same kind it is permissible to say that the first is either: (1) greater than the second, (2) equal to the second, or ( 3) less than the second. 50. A good definition must: (1) contain the simplest language, (2) be classified, (3) describe the thing or term sufficiently to distinguish it from similar objects or terms, (4) be reversible. 130 51. Oorresponding parts of congruent figures are equal to each other. 52.*The perpendicular distance between a point and a line is the shortest distance. 53.*The perpendicular distance between two parallel lines is the shortest distance. 54.*2he perpendicular distance between two parallel planes is the shortest distance. 55.*If two triangles have, two sides and included angle respectively equal to each other, then they are said to be congruent. 56.*If two triangles have two angles and an included side respectively equal to each other, then they are congruent. 57.*If two triangles have their three sides respectively equal to each other, then they are congruent. 58. Radii of equal circles are equal. 59. A straight line cannot have more than two points in common with a circle. 60. !3?wo circles, having different centers, cannot have more than two points in cbrnmon. 61.*Central angles of circles contain the same number of degrees as their respective arcs. 62.*®ie area of a rectangle equals the product of its base and altitude. 63.*If two triangles have two angles respectively equal to each other, then they are similar. 64.*If two triangles have their three sides proportional respec tively, then they are similar. 65.*If two triangles have two sides proportional and the included angle equal respectively, then they are similar. S6.*Regular polygons having the same number of sides are similar. 67.*The ratio of the circumference of a circle to its diameter is a constant. 68.* $ x 22 to the nearest hundredth. 7 Or jj - 3.1416 to the nearest ten thousandth. 131 69. Conditions whose relationship is not dependent upon the number of sides of a regular polygon hold true for circles if they are true for the regular polygons. 70. If two variables are equal and each variable approaches a limit, then the limits are equal. The above list of assumptions was derived by methods described in the preceding chapter. Some of the statements in their original form were awkwardly worded, but through critical study were refined into the final form, as indicated above. 4. Pacts or Theorems Many facts or theorems were developed as a result of the procedures described in Chapter III. These may be classified into two categories, namely (a) those common toall of the Experimental Groups, end common to at least one of the Experimental Groups. (b) others (a) Theorems common to all of the Experimental Groups are as follows? 1. If a triangle is isosceles, then the angles opposite the equal sides are equal. 2. If two lines in the same plane cross a third so that the alternate interior angles areequal, then the lines are parallel. 3.* If two parallel lines cross a third, then the alternate interior angles are equal. 4. If two lines cross a third so that the corresponding angles are equal, then the lines are parallel. 5.* If two parallel lines cross a third, then the corresponding angles are equal. 6. The sum of the interior angles of a triangle is 180°. 7.* (The sum of the interior angles of a polygon of n sides is equal to (n-2) 180°. 8.* The sum of the exterior angles of any polygon is equal to 360 degrees. 9. If the hypotenuse and a side of two right triangles are respectively equal, then the triangles are equal. 132 10** If two angles and a side of two triangles are respectively equal, then the triangles are congruent. 11. The opposite sides of a parallelogram are equal. 12. If three or more parallels cut off equal segments on one transversal, then they cut off equal segments on any other transversal. 13.* In any proportion the product of the means equals the product of the extremes. 14.* If the product of two quantities equals the product of two other quantities then either pair may he made the means or extremes of a proportion., 15. If a line is drawn parallel to the hase of a triangle, then it divides the other two sides proportionally. 16. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. 17. Two triangles are similar if two angles of one are equal respectively to two angles of the other. 18.* The areas of two similar triangles are to each other as the squares of any two corresponding segments. 19. In any right triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. Corollary I In any right triangle if a perpendicular he dropped from the vertex of the right angle to the hypotenuse: (a) the two right triangles formed are similar to the given triangle and to each other. (h) either leg of the given right triangle is a mean proportional between the whole hypotenuse and the adjacent segment. (c) The perpendicular is the mean proportional between the segments of the hypotenuse. 20. The area of a triangle is equal to half the product of the base times the altitude. 21. The locus of a point equally distant from two points is the perpendicular bisector of the line segment joining them. 22. The locus of a point equally distant from two intersecting lines is the pair of lines which bisect the angles formed by the lines. 133 23. A dimeter perpendicular to a chord "bisects the chord and the arcs of the chord* 24. Jin angle inscribed in a circle is equal to half the central angle having the same arc. 25. A line perpendicular to a radius at its outer extremity is tangent to the circle at that point. 26. If the number of sides of a regular inscribed polygon is in definitely increased, its perimeter and area will both increase, while the perimeter and area of the circumscribed polygon formed by drawing tangents to the circle, at the vertices 2>f the inscribed polygon, will both decrease. The perimeters and areas of both polygons will each approach a limit. 27. The ratio of any circumference to its radius is constant and is equal to 2 sc. 28. The area of a circle is equal to at times the square of the radius. The eight theorems in the above list marked with an asterisk (*) were not included in the list of twenty essential theorems selected by Chris63 tofferson • The twenty unmarked ones represent his list. One of the Experimental teachers went so far as to let her pupils assume even two of these, namely No. 2 and No. 28. (b) Additional theorems that were common to at least one of the Experimental Srouns are listed in accordance with the following methods of tabulation. The column at the right indicates the number of groups for which this established theorem was in common. By number of groups is meant the number of teacher groups under which the theorem was developed. Since all of the teachers had from two to three classes, the classification of Experimental Groups according to teacher was found to be more satisfactory. To avoid repetition, no proposition is listed herein which appeared under assumptions or in the preceding statements of theorems. 63 Christofferson, C. H.t op. cit.» pp. 11-12. 134 Statement of Theorem or Corollary Common to No. of Groups 1* If a triangle has an exterior angle then this angle is greater than either opposite interior angle. 1. _______ 4_____ 2* If two lines in the same plane are each perpendicular to a third line, then they are parallel to each other. 2. _______ 4____ 3* If two parallel lines are crossed hy a transversal, then the interior angles on the same side of the transversal are supplementary. 3. _______ 4 4. If two lines in the same plane are crossed hy a transversal, making the interior angles on the same side of the transversal supplementary, then the lines are parallel. 4. 5. If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other also. 6. If a triangle has an exterior angle, then this angle equals the sum of the two opposite interior angles. 6 . _______2______ 7. If two angles of a triangle are equal respectively to two angles of another triangle, then their third angles are equal to each other. 7. _______ 4_____ 3______ 5. _____ g_____ 8. If two right triangles have the hypotenuse and an acute angle respectively equal to each other, then they are congruent. 8. _______2_____ 9. If two right triangles have a leg and an acute angle respectively equal to each other, then they are congruent. 9.________2_____ 10. If two angles of a triangle are equal, then the triangle is isosceles. 10. _______3_______ 11. If two sides of a triangle are unequal, then the angle opposite the greater side is the greater. 11. _______2 12. If a quadrilateral is a parallelogram, then its diagonals bisect each other. The converse is also true. 12. _______3______ 13. If two sides of a quadrilateral are equal and parallel to each other, the figure is a parallelogram. 13.________2______ 135 Statement of Theorem or Corollary (continued) Common to No.of Groups 14. If a line segment joins the mid-point of two sides of a triangle, then it is parallel to the third side and equal to half of it* 14. 15* If two angles have their sides respectively parallel to each other and extending in the same direction (or in opposite directions? from the vertex, the angles are equal. 15. 16* If two angles have their sides respectively perpendicular to each other, then they are either equal or supplementary. 16. 17. The diameter of a circle is greater than any other chord. 17. 18. If in the same orequal circles two central angles are equal, then their intercepted arcs are equal. The converse is also true. 18. 19* If in the same orequal circles two central angles are unequal, then the arcs are unequal end the greater angle intercepts the greater arc. The converse is also true. 19. 20. If in the same orequal circles two arcs are equal, then their chords are equal. The converse is also true. 20. 21. If in the same or equal circles two chordsare equal, then they are the same distance from their respective center. The converse is also true. 21. 22. If in the same orequal circles two chords are unequal, then the greater chord is nearer to its respective center. The converse is also true. 22* 23. If a line is tangent to a circle, then it: makes a right angle with the radius at the point of contact. The converse is also true. 23. 24. If two tangents are drawn from a point to a circle, then they are equal to each other. 24. 25. The perpendicular bisectors of the sides of a triangle are concurrent. 25. 26. The angle bisectors of a triangle are concurrent. 26. 21. If two parallel lines are tangent to a circle or cross the circle, then they intercept equal arcs. 27. 28* If two circles intersect, then the line of centers is the perpendicular bisector of their common chord* 28. 29. If two central angles of the same or equal circles intercept arcs, then they are proportional to these arcs. 29. 30. If an angle is inscribed in a semicircle, then it is a right angle. 30. 31* If angles are inscribed in the same circular segment or in equal circular segments, then these angles are equal to each other. 31. 32. If an angle is formed by two intersecting chords, then it is numerically equal to onehalf of its intercepted arcs. 32. 33. If an angle is formed outside of a circle by two tangents or two secants, or a tangent and a secant, then it is numerically equal to one-half the difference of its intercepted arcs. 33. 34. If an angle is formed by a tangent and a chord, then it is numerically equal to one-half of its intercepted arcs. 34. 35. If four quantities are in proportion, then they are in proportion by alternation. 35. 36. If four quantities are in proportion, then they are in proportion by inversion. 36. 3 7. If four quantities are in proportion, then they are in proportion by addition. 37. 38. If four quantities are in proportion, then they are in proportion by subtraction. 38. 39. If three terms of a proportion are equal to three terms of another, then the fourth terms are equal. 39. 40. The mean proportional of two quantities is equal to the square root of their product. 40. 41. Ifseveral ratios axe equal to each other, then the sum of the numerators (antecedents) is to the sum of the denominators (conse quents) as any one numerator is to its denominator. 41. 42. Ifa line is parallel to one of the sides of a triangle, then it divides the other two sides proportionally. 42. 43. Ifa line divides two sides of a triangle proportionally, then it is parallel to the third side. 43. 44. Ifan angle of a triangle is bisected, then the bisector divides the opposite side into segments which are proportional to the other two sides. 44. 45. If an exterior angle of a triangle is bisected, then it divides the opposite side extended into two segments which are proportional to the other two sides. 45. 46. If two triangles have their angles respectively equal, then they are similar to each other. 46. 47. If two right triangles have an aoute angle respectively equal, then they are similar. 47. 48. If two polygons are similar, then they can 48. be divided into triangles which are respectively similar. The converse is also true, 49. If two polygons are similar to each other, then any two corresponding lines are to each other as any other two corresponding lines or perimeters. 49. 50. If two chords intersect, then the products of their segments are equal to each other. 50. 51. If a tangent and a secant are drawn from a point to a circle, then the tangent is a mean proportional between the secant and its outer segment, 51. 52. If a secant is drawn from a point to a circle, then the product of this secant and its outer segment is a constant regardless of how it is drawn. 52. 53. If two rectangles have equal altitudes, then their areas are in the same ratio as their hases. 53. 54. The areas of two rectangles are in the same ratio as the products of their "bases and altitudes. 54. 55. The area of a rhomboid is equal to the product of its base and altitude. 55. 56. If two parallelograms have equal bases and altitudes, then they are equal in area. 56. 57. If two triangles have equal bases and altitudes, then they are equal in area. 57. 58. If two parallelograms have equal altitudes, then their areas are in the same ratio as their bases. 58. 59. If two triangles have ea\ial altitudes, then their areas are in the same ratio as their bases. 59. 60. The areas of two parallelograms are in the same ratio as the products of their respective bases and altitudes. 60. 61. The areas of two triangles are in the same ratio as the products of their respective bases and altitudes. 61. 62. The area of a trapezoid is equal to onehalf the product of its altitude and the sum of its bases. 62. 63. The area of a rhombus is equal to one-half the product of its diagonals. 63. 64. If two triangles are similar to each other, then their areas are in the same ratio as the squares of their corresponding linear parts. 64. 65. If two polygons are similar to each other, then their areas are in the same ratio as the squares of their corresponding linear parts. 65. 66. In a right triangle the square on one of the legs is equal to the difference between the squares on the other two sides. 66. 139 67* Ifan equilateral polygon is inscribed in a circle, tlien it is regular. 67. 68* Ifan equilateral hexagon is inscribed in a circle, then its side iequals the radius of the circle* 68. 69* A circle can be circumscribed about any regular polygon. 69. 70. A circle can be inscribed in anybregular polygon. 70. 71* Ifa polygon is regular, then one of its angles at the center equals 360/n, where n represents the number of sides. 71. 72. If a circle is divided into equal arcs, then the chords joining these points of division form a regular inscribed polygon. 72. 73. If a circle is divided into equal arcs, tangents at these points form a regular circumscribed polygon. 73. the 74* If mid-points of the arcs of a regular inscribed polygon are determined and joined with the vertices, then the new polygon formed is a regular inscribed polygon and has twice the number of sides* 74. 75. If mid-points of the arcs of a regular circumscribed polygon are determined and tangents are drawn at these points, then a new regular circumscribed polygon will be formed with twice the number of sides. 75. 76. If two polygons are regular, then their perimeters have the same ratio as their corresponding linear parts. 76. 77. If two polygons are regular, then their areas are in the same ratio as the squares of their corresponding linear parts. 77, 78. The area of a regular polygon is equal to one-half the product of its apothem and perimeter. 78. 79. The ratio of the circumferences of two circles is the same as the ratio of their diameters* 79. 140 80. Tli©length of an arc is in the same ratio with the circumference of the circle as its central angle is with 360 degrees. 80. 81. The areas of two circles are in the same ratio as the squares of their radii, diameters, or circumferences. 81. 83. 82. The area of a sector is in the same ratio with the area of the circle as its central angle is with 360 degrees. 83. The area of a circular segment is the area of the triangle formed hy the two radii and its chtord. 83. 84. The square on the first side of any triangle is equal to the sum of the squares on the other two sides diminished hy twice the product of the second side and the projection of the., third side on the second side. (Law of cosines) 84. 85. 85. ____1 86. In any triangle, the sum of the squares of two sides is equal to twice the square of one-half the third side increased hy twice the square of the median to the third side. 86. 1 _ If a triangle is ohtuse, then the square of the side opposite the ohtuse angle is equal to the sura of the squares of the other two sides increased hy twice the product of one of those sides and the projection of the other upon it. 1 The maximum number of theorems established hy any one group was eighty-five, while the minimum number was forty-three. These figures include the twenty-eight theorems common to all six of the Experimental Groups, hut do not include the theorems generally ascribed to solid geometry. Since solid geometry is offered as a semester course in each of the Experimental schools, very few theorems were developed. In fact only those concepts and extensions were treated which were of profound concern to the pupils. These were as follows: 141 1. If a series of lines is perpendicular to a given line at a fixed point, then these lines must all lie in a plane which is perpendicular to the given line at this point. 2. If two parallel planes are cut by a third plane, then their lines of intersection are parallel* 3. If two planes intersect, their vertical dihedral angles are equal. 4. 5. The volumeof a cube is equal to the cube of one ofits edges* The volumeof a rectangular block (parallelopiped) is equal to the area of its base times its height. 6. The volume of a right cylinder is equal to the area of its circular base times its height. 7. The volume of a sphere is four-thirds & times the radius cubed. 8. The area of the surface of a sphere is four times the area of one of its great circles. 9. The area of a circular section of a sphere is equal to the product of jc and the difference of the squares of the ds.dius of the sphere and the distance the circular section is from the center of the sphere* A = £ (r2 - d^). (c) Examples illustrating inductive and deductive methods__of reasoning used in developing all of the theorems previously listed are, as follows: (l) An example of deductive reasoning "The sum of the interior angles of a triangle is equivalent to one str aight angle.M Why? D Assume: A ABC to be any scalene triangle. To prove: ^ 1 + Z. 2 A C 3 r one straight angle. Argument Statements 1. Draw DE parallel to AC through the point B. Reasons Supporting the Statements 1. One and only one line may be drawn through a given point parallel to a given line. 142 Statements Argument. continued Reasons Supporting the Statements 2. 1 3 = U>. 2. .Alternate interior angles of parallel lines are equal. 3. £4 = a- 3. Same as 2. 4. £4 + £2 + 5. ZJ- + £5 L? * I? s st. 4* Sum of all angles about a point on one side of a straight line is equivalent to a straight angle. s St. 5. A quantity may be substituted for its equal. Q,. E. D. (2) An example of the inductive reasoning “Is there a mathematical rule or formula for finding the sum of the interior angles of any polygon?11 3 sides 1 triangle 4 sides 4 triangles Assume* Polygons of 3, 4 t 5, and n sides. Implication? ? 5 sides 5 triangles n sides £ triangles Argument Statements Reasons Supporting the Statements 1. Draw lines from 0 to each vertex. 2. The afore-mentioned lines form triangles. 1. Only one line may be drawn connecting two points. 2. Definition of a triangle. 3. Each side of the polygon is a base for each triangle, 3. Def. of the base of a triangle. 4. Each triangle is equivalent to a straight angle. 4. Sum of int. (J s of any triangle is equal to 180°. 5. Sum of /.'s around 0 = 2 5. Definition of a perigon. st. U s. 6. Sum of the angles of n triangles is equivalent to n straight £'s. 6. s !s multiplied by ='s give products. 7. Sum of the angles of a polygon of n sides is equivalent to (n - 2) straight angles. 7. s's subtracted from equals give equal remainders. Generalization? The sum of the interior angles of any polygon is (n - 2)180°. 143 (3) in example of ‘both inductive and deductive methods in reasoning, namely testing a hypothesis As sume: ^ ABC -h DEP having AB s BD BE EP and j\B s JJ>. Implications: AB BE = BC EP - AC DP AABC/^ a ® 11heasons Supporting the Statements 1. Place /\BEP on A ABC so th^t /E coincides with ^B; point D will hecome point D 1, and point P will Become point P 1. 1. Superposition axiom. 2. D'P* 2. Corresponding sides of congruent A are =s, that is, the A BD’P* and DEP have S.A.S. »,S.A.S. s 3* AB s BD* DP. BC. BP* 3. Substitution axiom. 4. D 1?* II 4. A line that divides two sides of a A proportionally is II to the third side. 5. J l * /BD'P*. ZP s Z^'D*. 5. Corresponding angles of parallel lines cut by a transversal are equal. 6. Z.A = £Dt /C = £P. 6. Substitution axiom. 7. Draw P*K II AB. 7. Construction. 8. AD*P,K is a parallelogram. 8. Definition of a parallelogram. 9. AK = D*P*. 9. Opposite sides of a parallelogram are equal to each other. 10. AC AK a BC. BP* 10. A line parallel to one side of a triangle divides the other two sides proportionally. 11. AB » BO S3 AO. BD' BP' AK 11. Ratios =? to the same ratio are ss to each other. (See steps 3 & 8.) 12. Ag s BC = AC. ED EP DP 12. Substitution axiom. 13. 13. (l) Corresponding angles are equal. (2) Corresponding sides are proportional. A ® 1* Generalization: Two triangles are similar when two sides of one are proportional to two sides of the other and the included angles are equal. 144 (4) Miscellaneous examples of inductive and deductive reasoning •All of the Experimental groups used arguments or exercises to develop the elements of critical thinking which deal with searching for facts and testing them for relevancy to a situation. An example of such exercises is as follows! Assume! /BAC a /IX)A and AB = BC. Hfhat are some of the implications? Implications listed! (l) AABC a AABC (2) a a b e g a e d c (3) is isosceles. A Argument Statements Eeasons Supporting the Statements 1. ZJBAC a Z.20A, 1. Assumed. 2. AB s DO. 2. Assumed. 3. AC » AO. 3. Identity. 4. A A B C 5. 6. AB £B 7. /BEA AASC. 4. S.A.S. a S.A.S. (which established the first hypothesis). z DC. 5. Assumed. a 6. Corresponding angles of equal triangles are equal. ID. s Z.®®- 7. Opposite or vertical angles are equal. 8. a a b e s A c^®* 8* S.A.A, - S.A.A. (which established the second hypothesis). 9. £DAC 9. Corresponding angles of congruent triangles are equal. a ZPOJL 10.AAEG is isosceles, 10. If two angles of the triangle are equal, then the sides opposite the angles are equal, and the triangle is isosceles. (This establishes the third hypothesis)* 145 The number of geometric arguments of this type, including con structions and the solution of '•originals**, ranged from a minimum of thirty-five per group to a maximum of approximately seventy-five -per group. It must he kept in mind, however, that all of the content developed so far has been the result of group effort. Since provision was made for individual differences, as described in the chapter on procedures, many pupils had developed geometric content far beyond that of their respective group. In fact some of the pupils in groups where only a minimum number of theorems, constructions and exercises were developed had gone ahead and developed their 11texts11 to include more geometric content (in some cases) than was contained in the texts developed by pupils where a maximum number «tas reported. This, then, leads us to a consideration of some of the content developed as the result of individual effort independent of the group, 5. Fundamental Construction Problems? were divided into two categories, namely, (a) those common to all of the Experimental Groups, and (b) others not common to the Experimental Groups. (a) Fundamental constructions (bommon to all of the Experimental Groups are as follows? 1. Construct a circle with a given radius and a given center. 2. Construct a triangle congruent to a given triangle using only the lengths of the three sides. 3. Construct an angle equal to a given angle. 4. Bisect a given angle. 5. Construct a perpendicular to a line at a point on the line. 6. Construct a triangle congruent to a given triangle using only one side and the two adjacent angles. 146 7. Construct a triangle congruent to a given triangle using only two sides and their included angle, 8. Construct a perpendicular “bisector of a given line segment. 9. Construct a perpendicular to a line from a point not on the line. 10. Construct a line parallel to a given line through a given point. 11* Divide a line segment into any given number of equal parts. 12* Divide a line segment into any given number of proportional parts. 13. Inscribe a circle in a triangle. 14* Circumscribe a circle about a triangle. 15. Inscribe a square in a circle. 16. Inscribe a. hexagon in a circle. 17. Construct a fourth proportional to three given segments. Of the seventeen constructions common to all of the groups, the first ten are listed by Christofferson^ ag essential to the study of geometry. / (b) Additional constructions common to at least one of the Experimental Groups are listed as follows: Construction L Common to No.of Groups Construct a common external tangent to two circles. 1 . ________3 2. Construct a common internal tangent to two circles. 2 . _______ 3 3. Bisect an arc of a circle. 3. 4 4. Divide a given line segment internally. 4. 3 5. Divide a given line segment externally. 6* 7. Construct a mwan proportional to two given segments. Construct a tangent from a point to a circle. 5 . ____ 6. 3, 4 7. _______ 4 64 Christofferson, C. H., 11Geometry Professionalised for Teachers”, loci^cit* 147 Construction 8. Common to No.of Groups Constructa tangent to a circle at a given point on the circle. 9. Constructa triangle ecuivalehtto a polygon* . 10 Constructa square equivalent to a triangle. 11. 12. Constructa regular decagon. Constructa regular polygon of fifteen sides. (Construct an angle of 24 degrees.) 8. 4_______ 9. 1 ------------------ 10. 2 11. i__________ 12. 1__________ Geometric Content Developed Through Individual Effort The geometric content developed through individual effort may he divided into two cl asses, namely (l) projects, and (2) voluntary contri butions. 1. Projects Over three hundred projects were developed by the pupils. A classification of the different types of projects engaged in by individual students is presented in Table X. TABLE X. DISTRIBUTION AND CLASSIFICATION OF INDIVIDUAL PROJECTS , Earic m „» . , Per Cent of Type of Project_________________________________ Pupils Involved 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. Critical Analyses of Advertisements Historical (mathematics) Critical Analyses of Speeches and Articles Philosophical - in relation to mathematics and search for truth Geometric Designs Biographical Sketches (Mathematics and Science) Fourth Dimension Architectural Designs Geometric Applications to Science Development of Measurement Applications of Geometry to Art Applications of Geometry to Sports Mathematical -Recreations Development of Numbers and Number Systems Three Famous Problems of Antiquity Mathematical Instruments Exercises in Paper Folding___________ 24.2$ 15.6$ 12.2$ 7.3$ 6.1$ 4.9$ 4.6$ 4.3$ 4.1$ 3.9$ 3.3$ 2.9$ 2.3$ 1.7p 1.4$ 0.9$ 0.3$____ 148 The preceding table indicates that 64 per cent of the projects were of a mathematical nature end 36 per cent non-mathematical. However, many of the mathematical projects were interspersed with non-mathematical materials, such as analyses of advertisements, editorials, news comments, and speeches. As indicated in Table X, the historical projects ranked first among the mathematical types. These included a wide variety of reports; however, most of them were confined to the historical development of geometry. Algebra was treated historically in a few cases, as was arithmetic and trigonometry. An example of the historical type of project may serve to illustrate the nature of the content involved. The following is an excerpt from a project dealing with ''Highlights in the History of Mathematics up to the Present Day". "When primitive men first began to count on their fingers, mathematics, the first of the sciences to reach a formal develop ment, was invented. Today, applied mathematics is the mainspring of our civilization. Without it, things would be without value. Mathematics is an essential tool in the world of business, economr ics and science, for it deals with the quantitative relationships between material objects. It aids us in our daily lives, in tell ing time, making change, and in all other forms of measuring. It is used to keep score in games and helps us in cooking, sewing, gardening, and other occupations of the home. It is necessary to the farmer, the skilled workman, the merchant and those who are in the professions. In its higher forms, mathematics becomes a form of logic, a method of reasoning in which the scholar follows through to his result any assumptions that he lays down as basic. In this, as in every form of mathematics, it is a test and a training in clear, logical, and accurate thinking. "Most likely the first of mathematics had its origin in the counting, weighing and measuring, so necessary to primitive barter. At first, it was closely related to magic, but gradually people shifted from curiosity and awe, to the serious investigation, and thus, the development of mathematics began. We can trace the practical use of mathematics back to the Babylonians. In their country records were found, inscribed on baked, clay tablets. The mathematics of Babylon was, of course, largely commercial and astrological." 149 The philosophical projects dealt largely with a treatment of the inductive and deductive methods of reasoning. One girl pointed out that: "The deductive method is often called a method of authority since i± requires the pupil to accept the knowledge gained "by another. It is also a method of applying knowledge. It is used hy a uerson who is sufficiently mature of judgment that he can determine what rules, definitions or principles apply to the problem in hand.” Another girl wrote about Logic and as an introductory remark she made the statement that: "Logic is pure reasoning, and to be logical is to argue reasonably upon a basis of fact and in the behalf of truth." Some of the other projects included reports on such topics as "The Relativity of Truth", "The Search for Truth", "The Meaning of Mathematics", "Reasoning", "Logic", "Beginnings of a Hew Philosophy", .and "The Power of Sound Argumentation". The projects dealing with Greometric Resigns were largely in the form of constructions end their applications to architecture, and to designs in the home. Several pupils produced pictures of various designs and pointed out the fundamental geometric constructions involved. The biographical projects are self-explanatory in that they in cluded sketches of mathematicians from antiquity to modern times. Many of these were critical, as for example one boy wrote: "Most of this talk about relativity in my opinion is the bunk. I think that except as a matter of interest it has no earthly use and that Einstein could be helping humanity if he applied his great brain toward social progress. On the other hand, no one can foretell future events with precision and unless I can get all of the facts I may be doing him a great injustice because it is possible that his theory may have a great deal of use in the distant future." Most of the fourth dimension projects grew out of a discussion of the book by Slosson, Easy Lessons in Einstein, and from the book Elatland. 150 "by Abbott. One of the girls in the study makes the following statement in her project regarding her observations! ‘•If a man is traveling due north on land from a given place, to find him one would only need to know one dimension* If he was at sea one would have to know his latitude and longitude to locate him* If he was in the air three dimensions would have to he known to find him, namely latitude, longitude, and altitude. If a prospector died somewhere on a mountain one would need to know his latitude, longitude, altitude, and time of death to describe the incident accurately. . . The term fourth dimension came into use from a spiritual source. There have been numerous hints of its existence in the Bible and if we knew more about this phenomenon we could perhaps explain better some of the verses in the Bible* My own opinion is that we can neither prove nor deny its existence* If a physical fourth dimensional body exists, tijen a three dimen sional body could never perceive it in its totality although it may perceive its presence. My conclusions here are that dimensions are inventions of the human mind. Now if some one wants to accept them as real and existential, then he is obligated to accept the fourth dimension as real and existential. I for one still look upon the whole notion of dimensions as intriguing but I am not ready to accept them as real and existing. Perhaps some day I will, but first I will have to have ikwt© convincing evidence.” .Another example of a project worked out by three pupils and written up by one of them is as follows: ”While talking about a line having one dimension, surface two, etc., we began to discuss the fourth dimension and its probabilities. We heard of it as 'time*, but not actually as being a geometric figure. However, we found that it might be possible to draw such a figure if we would know how many points, line segments, squares, and cubes it would contain. The following chart illustrates how our teacher guided us into a consideration of the number of points, lines, squares, and cubes it contains! Let pu. represent a moving point, ii ii it line, lm " it ii ii surface or square, sm H ii it it volume or a cube, vm H it ii ti tesseract, then Dimensions Geometric Figure to be Moved Oja — — j lp 0 . . . .............. 1 .......................Pm > 2P * 1 2 ........ lm > 4p + 41 + s 3 ............ sm ^ 8p +• 12 1 + 6s + v 4 ....................... vm ^ 16p +32 1 ■+24s + 8v +■ t 151 “Thus, with these figures, we tried to draw the tesseract in the second dimension. A drawing of this in the second dimension may he found in Easy Lessons in Einstein*^: however, this does not seem possible, because we cannot represent a surface by a point, or a cube by a line, therefore it is logical to conclude that we could not represent a tesseract in two dimensions. “Our teacher challenged our thinking somewhat further by sug gesting that we study the relationships that we developed and see if we could write a mathematical expression or generalization for an nth dimensional figure. After working for days and getting some help, three of us came out with the fact that the properties of these figures, namely points, lines, surfaces, etc., actually follow in a numerical way the coefficients in the expansion of the Binomial Theorem of the form (2x *h y)n. With the help of our teacher we found that by using the j?th term of this expansion, namely n(n-l) (n-2) . . . (n-r + 2) (2x)n-r + 1 (y)n“r 1. 2 .3 . . . (nr-tf) we could find the properties of a geometric figure of any number of dimensions.11 The projects dealing with architectural design were twelve in number. One in particular was very interesting in that this boy made critical analyses of “home designs'* and pointed out not only defects in certain types of constructions, but also “dead space" and inadequacies of planning to fit modern modes of life. He made a number of sketches of homes on different economic levels and gave many facts supporting his designs. The inventory questionnaire definitely reveals that this boy*s interests are in architecture. Projects dealing with applications of geometry to science were also quite varied. In fact, while there were only a dozen projects directly related to this field, they varied nevertheless from applications of mathematical formulas to the processes of scientific reasoning. One boy in particular pointed out how Einstein used Hiemann*s Geometry to establish his theory of relativity. Others had indicated how astronomers mad physicists establish their hypotheses by selecting mathematical rules, 65 Slosson, Edwin E., Easy Lessons in Einstein. Hew York: Harcourt Brace and Company* 1920. 152 and geometric figures and their relationships to express and support their theories, Projects dealing with measurement were of two types, namely (l) methods of measuring, and (2) units of measurements, An illustration of the first type is a <£uotation hy one of the hoys from his reading of Hogben's Mathematics for the Million, which is as follows! "Mathematics was not invented hy leisurely people. It was an art developed hy people like tax-gatherers, mariners, craftsmen, slaves, through such means as a set-square, plumb-line, waterlevel, and other instruments* An example of an early method was the primitive man's sowing grain which would hear at certain periods of the year* Thus he was forced to take stock of the seasons through such means as the moon and the sun. And so he "began to group days in moons, later months, and so on." The second type of project dealt with units of measure and standards of length. The reports centered around such units as "cubit", "foot", '’inch11, '•yard11, "digit11, "palm", and "rod". One hoy described the "Metric Gage Blocks" and the accuracy with which the first machine of this type detected differences, namely! "one millionth of an inch". Another hoy described the equipment of the Bureau of Standards in Washington, D. C. One of the best examples of a project where geometry was applied to art was one developed hy a girl. Words cannot adequately portray the project which was in itself truly a great work of art. The project was labeled "Compass Designs", and the title page consisted of a Mariner's Compass. This mariner's compass was accurately constructed and then hand painted, in which the colors of black, white, yellow, blue, red, and green were artistically blended. black letters* Even the points were indicated with small from there on the ruler and compass designs were a myriad of color and blending of a modernistic type that only superior talent could create* 153 The projects dealing with an application of geometry to sports varied to some extent; however most of them centered around the laying out of athletic fields, tennis courts, baseball diamonds, and football fields. One boy extended his constructions to a critical analysis of the direction in which a field should be situated. His conclusion was that? Hpor group games like football* baseball, and soccer, the playing field should be extended north and south because in this way a minimum number of participants are affected by the glare of the sun. This is also true for tennis.” The remaining projects are self-explanatory. The “Three Famous Problems of .Antiquity” were, of course, (l) quadrature of the circle, (2) the trisection of any angle by ruler and compasses, and (3) the duplication of the cube. The projects on mathematical instruments cen tered around (l) calculating instruments, and (2) practical geometric tools. The first dealt with the early compass, Uapie^s rods, sector compass, the slide rule, and adding devices, including modern adding machines. The latter dealt with the early Egyptian level, dioptra, quadrant, groma, and square, 2. Voluntary Mathematical Contributions. The pupils turned in numerous proofs of theorems and exercises to be filed away in their folders. next section. This material will be discussed in the This section, however, contains several proofs which appear to be quite original. Since they are individual contributions, it seemed worthwhile to include some of them here. Perhaps the most out standing example of perseverance and logic is a proof of the Pythagorean theorem submitted by a boy. His proof is as follows! Assume: Hight A ABC with squares on the sides: ABED on side AB; BCHJ on side BO and ACFX on side AO. Implication* ACFX m ABED + BCHJ. Argument Statements Facts 1. Extend DBS and HJ so that they meet at Gr. 1. A line may he extended. 2. Draw LJ a AB and then draw LF. 2. A line may connect two points M o If Sf 3. 3. Sides of a square are equal. 4. BC - CJ. 4. Sides of a square are equal. 5. /.ABC s /XJC. 5. All rt. /.‘s are equal. 6. 6* Hyp. and adj. side. g ACJX. 7. XJLM - ADEB. 7. LJ r AD and XJ = AB. Opposite sides of a square are equal. 155 8* pa = u . 8. Opposite sides of a rect. ere equal. 9* G-H a AD* 9. Opposite sides of a rect. are equal. 10* AD a XJ. 10. Subst. Ax. 11. PSGH r LMXJ. 11. Opposite sides of a square are equal. 12. PS OH = ABSB. 12. Subst. Ax. 13. pa s PS. 13. Opposite sides of a square are equal. 14. P A s PX. 14. Opposite sides of a square are equal. 15. ZJFaH - /PSA. 15. All rt. //s are equal. 16. 16. Hyp. and adj. side. 17. a oj 1s = A 0:FS* 17. Identity. 18. BOXC s BOXC. 18. Identity. 19. BOXC + CXJ + OHX - BHJC. 19. The whole is equal to the sum of its parts. 20. BOXC t ABC + OPS + IPS - APXC. 20. The whole is equal to the sum of its parts. 21. BOXC t ABC + AH’S + POS = BOXC + XJC + PGS + SOP. 21. Subst. Ax. 22. ACPX s ABED + BCHJ. 22. Subst. Ax. Another interesting exercise concerning the Pythagorean Theorem was one turned in by a girl which she calls the •‘Chinese Chou Pei*s“. as follows! It is A Assume! lit. ABEP. Implication! (EP)2 s (BP)2 + (BE)2. H T> 156 Instead of giving a complete and rigorous proof for this theorem, as in the preceding case, this girl merely outlined her facts, giving the key steps to the solution, which is as follows* 1. EEB = i rectangle EAEB - § (BFf) (EBo). 2. Square A3CD - Square EEGH +- 4 times EEB, 3. Square JSBCD 3 (®E ;:)2 + 2 (BP)(SB). 4. Jlso square ABCD - (BP)2 + (E3)2 + 2(BP)(EB). 5. So (BP)2 + 2(BP)(BB) a (BP)2 + (BB)2 + s(BP)(BB). 6. Hence (EP)2 = (BP)2 + (EB)2. The outline is a very accurate one; however, a great many facts would need to be brought out before either of the Experimental Groups would accept it. The fourth step in particular, although true, nevertheless, involves several facts before it would be accepted in a rigorous proof. The following construction problem and proof was turned in by another pupil. It is presented here because it is a good example of how a person striving to be logically consistent has solved a problem even though a large part of the work was unnecessary. The problem was nto inscribe a square in a semi-circle and then prove the construction.H A Construction: Draw circle with center at 0. Draw diameter AD making the semi-circle. Make a square ABCD on diameter AD. Draw lines from B and 0, passing through 0 and cutting the circle at E and E respectively. Erom points E and E drop perpendiculars to AD. Then draw EE, thus forming square EEGH. 157 To Prove. EFGH is a square. Proof 1. ABCD is a square. 1. Construction. 2. £D0C = /FOA. 2. Vertical angles are equal. 3. £FG0 = ^CDO. 3. All right angles are equal. 4. a ^ go ^ a c d o . 4. Two angles of one equal to two angles of another. 5. In the same way prove AEOH/v a a b o . 6. /BOO = £F0E. 6. Vertical angles are equal. 7. CP = QE, 7. Badii of the same circle are =, 8. AB s DC. 8. Sides of a square are equal. 9. AO a QD. 9. Season 7. 10. £BAD = /GDA. 10. All right angles are equal. 11. APiO s a c b o . 11. S.A.S. s S.*A.S. 12. BO s 00. 12. 0. P. 0. T. E. 13. A B O Q and A FOE are isosceles. 13. They are triangles having two equal sides. 14. /GBG + ZPCB - £OEF + /031E. 14. Subt. Ax. 15. Z.OBC - £QCB, 15. Base angles of an isosceles tiiangle are equal. ^OEF - £OFE. 16. ^OBC - £OEF. 16. Js of equals are equal. 17. a b o c ^ a ^ e - 17. 2 angles of one - 2 angles of the other. 18. A gf0 + A F0E + A E0H = OEEGK. 18. The whole is equal to the sum of its parts. 19. A ^ ° + A soc + A 05,c = o a b c d . 19. Beason 18. 20. .*. EFGH ^ ABCD. 20. If two polygons are composed of the same number of A . ^ to each other and similarly placed, the polygons are to . 21. EFGH is a square. 21. All squares are similar. 158 The following approach to the measurements of angles with respect to the intercepted arcs of circles was worked out hy one pupil* This pupil used the following theorem and its proof as a hasis for establishing the relationship between angles and the arcs they intercept. The proof is as follows! 11An angle formed hy a tangent and a chord equals numerically onehalf of its intercepted arc.11 Assume! circle 0 with tangent BC and chord AB. Implication! AB. C 1* Draw radius OB. 1. A line may he drawn connecting two points. 2. OB 2. A radius and tangent drawn to the point of tangency are _1_. BC. 3. Draw OH perpendicular to BA. 3. A line may he drawn _1_ to another. 4 . £L 4. Angles whose sides are I to each other are equal or supplementary* - ^0. 5. £.0 s BH. 5. Central angles are equal numerically to their infcereep-fced arcs. 6. BH -HA r |r BA. 6.Halves of equals are equal. 7. Q. s 7. Substitution. s BH a J BA. With the above as a hasis he followed with the case of the in scribed angle, then the angle formed hy two intersecting chords, and 159 finally the angle formed hy two secants, a tangent and a secant, or two tangents. One very excellent piece of work which merits special attention is the development of the "locus" concept hy a girl. The first part was group activity, from which this pupil diverged and used different colors to represent the loci and to take into account all possibilities. The unit on loci is as follows: "In this unit our class has tried to show all the possibilities in combining two locus problems. The locus problems are: 1. The locus of a point a given distance from a point* 2. ti ii ii ii ii 3. ii n ii ii " 4. ii ii ii ii 5. H it ii ii Wo it ii ii " " straight line* equidistant " two given points. ii ii " " parallel lines. ii ii " " intersecting lines. found that the answers to these were: 1. A circle with the given point as center and the given distance as radius. 2. A pair of parallel lines a given distance away from the’given straight line on either side. 3. The perpendicular bisector of the line segment joining the two given points. 4. A lineparallel and half way between the two parallel lines. 5* A pair of lines which bisect the angles made by the intersecting lines. 160 Hext we combined these different problems, thus* 1 with 1 1 with 2, 2 with 2 1 with 3, 2 with 3, 3 with 3 1 with 4, 2 with 4, 3 with 4, 4 with 4 1 with 5, 2 with 5, 3 with 5, 4 with 5, 5 with 5 making fifteen problems illustrated on the following pages.” This girl did an excellent piece of work in working out these fif teen problems in detail and considering their possibilities. She used red and green to indicate the two different loci being expressed in the problem and black to represent the original line. Furthermore, as an introduction to the fifteen locus problems in question she inserts the following! "You are probably wondering by now what 'locus* means. 'Locus' is merely the Latin word for 'pla.ce1, which like many other Latin words has crept into the mathematical language. "While working with locus we were wondering how locus should be used practically. Our teacher showed us that all the hidden treasure stories are based on locus. The fel lowing is an example of such problems* Tide Rook Martin's Rqpk Giant Oak 161 "The treasure ig hidden on Shark*s Island# It is "buried 50 feet from the giant oak, equidistant from the two cross roads and equidistant from Martin's Eock and Tide Eock." Another pupil with artistic talent illustrated the -procedure for proof regarding parallel lines and interior angles of a triangle "bysketching facial expressions of a pupil as the various ideas dawned upon him. The development is without the faces, "but insertions will "be made in place of the facial expressions to indicate its nature# The develop ment is as follows! How We Discovered the Propositions Which Lead Up to the "Sura of the Angles" "The first time that we heard that there was anything important that had to do with the sum of the three angles in a triangle was on a Thursday when Miss . casually said something like this - 'Of course you all know that the sum of the three angles of a triangle equals a straight angle of 180°.* "This "brief statement was greeted "by a number of "blank looks from the class. (At this point the girl has sketched the face of a boy who looks utterly bewildered#) "The next day, since this seemed to need clearing up, we started to prove it# "First of all to illustrate it we cut out the three angles of a triangle and by placing them together, we saw that their base was a straight line and so the three angles together equaled a straight angle (180°)# (The girl has now pictured the same boy on the second day. He looks as if he has just seen a faint ray of light.) 162 “Many Ideas were brought up concerning this. We drew two parallel lines and on the top one placed a point. Through the point another straight line (not parallel) was drawn. We saw that when the line xy rotated about the point, the corresponding interior angles seem constant, X that is, when one increased the other decreased. (At this point the girl \ again sketches the face of the hoy, hut this time there is a smile on his face signifying that it is all beginning to sink in.) "We worked on this and other suggestions for several days. As we went on, we found that we needed to prove that the alternate interior angles of two parallels crossed hy a transversal were equal. it was necessary to prove four other statements. in the third section.) To do this, (These are written up After proving these we went ahead and proved that the sum of the angles of a triangle equals 180°. (We also found that 5A X$I could he used along with the others to prove the proposition.) (At this point the girl has sketched the hoy with a broad smile on his face. She has also written the word 'Success' above the sketch. last he has seen the dawn,)" At 163 Chapter Summary Only a description of geometric content developed "by the Experimental Groups was considered in this chapter. However, ref erences were given for content developed "by the Control Groups. The geometric content was considered from the standpoint of cooperative group effort and of individual effort. Efforts of the combined Experimental Groups yielded the fol lowing results? 1. Thirty-four undefined terms 2. One hundred sixty^-two defined terms 3. Seventy basic assumptions 4. Twenty-eight theorems common to each group 5. Eighty-six additional theorems not common to each group 6. Seventeen basic construction problems common to each group 7. Twelve additional basic construction problems not common to each group. Examples were given illustrating each of the above types of content. Content resulting from individual effort was classified as ,fIndividual Projects11 and ’’Voluntary Mathematical Contributions". A tabulation of projects revealed 64 per cent were of a mathematical nature and 36 per cent were of a non-mathematical nature. Examples were given for the purpose of illustrating (l) each type of project, and (2) the nature of the more significant voluntary contributions. 164 CHAPTER V HGH-lOTHEMATICiL C O M E T Introduction The preceding chapter was devoted to a description of mathematical content involved in the Experimental and Control Groups, Since no appreciable.effort was made hy the Control Group teachers to include non-mathematical materials in their procedures, the present chapter must he limited to a description of non-mathematical content developed through Experimental Group procedures. One difficulty encountered in presenting a description of nonmathematical materials is the expense involved in making photostatic prints of advertisements, cartoons, and other sketches or pictures necessitating this type of work. Consequently, a great deal of material that merits a place in this chapter must he omitted. However, in situer tions where words may describe the pictures or sketches with some degree of accuracy, such descriptive methods will he employed in the situation. As in the preceding chapter, the non-mathematical content is to he considered from the standpoint of: (l) content developed through the cooperative effort of pupils within particular groups, and (2) content developed through individual effort independent of direct group inter action, Eon-Mathematical Content Developed Through Group Effort Numerous projects were developed hy the Experimental Groups in an attempt to reach conclusions or solve various social problems by methods similar to the ones used in developing geometric content. The content 165 was made -up of commercial advertising, speeches, editorials, political issues, compulsory laws for education, pupil-school relations, liquor legislation, socialized medicine, social security, unemployment, the place of youth in a modern society, and many others. Analyses were made of not only the problem situations, "but also potential instruments of ‘'propaganda1', such as the newspaper, magazine, radio, movie and telephone. The term "propaganda" was used in the. sense in which it is described by the Institute for Propaganda Analysis®8, namely that it "is either a means of social control or a method by which an individual or a group works for his or their own interests." Each project was analyzed in terms of the ten elements of critical thinking developed in Chapter I. As an example of "group thinking" in a social situation where emotions and bias tend to color clear thinking on the part of an individual, the following group project on "Capital Punishment" will be described, 1, Group Project on Capital Punishment Toward the close of the first semester one of the Experimental Groups elected to discuss some problem where thinking may be colored by emotions and bias 6? • The group elected to discuss the following situation "There is a wide disagreement among thoughtful people as to whether or not capital punishment is the most effective way of dealing with certain types of crime. In forty-three states the death penalty is legal, while it has been abolished in Kansas, Maine, Michigan, Hhode Island and Wisconsin. Should it be abolished in all the states?" 66 Institute for Propaganda Analysis, "Propaganda", op, (also see pp. 62-63). cit. p.3 67 This situation and others mentioned herein could be included in Chapter III under procedures; however, the nature of the materials and content in question suggest a more adequate treatment at this point. 166 A preliminary vote, expressing each pupil’s attitude toward capital punishment prior to a discussion of the situation, was as follows! 58$ in favor of capital punishment 32$ opposed to capital punishment 10$ suspended judgment. This was followed "by a period of discussion and a critical analysis of the situation, which in turn was followed hy a written assignment* The written assignment was to he a critical analysis of the situation with an attempt, on the part of each individual, to reach a conclusion. A committee of five was appointed to compile the results and to present a summary to the class. The content was summarized and presented in terms of the ten elements of critical thinking mentioned in Chapter I. I. Motives Listed for and against Abolishing Capital Punishment A. Motives for Maintaining Capital Punishment 1. Do away with the expense of food, clothing, and shelter for criminals. (Wouldn’t take up so much room.) 2. Do away with any chance of their escape to do greater crimes. 3. Be a warning to other criminals and make them think twice before committing a crime. 4. Do away with the chance of hard criminals conversing with the young and imbedding evil ideas v/hile serving long terms. 5. It is better to take one life than to endanger the lives of others. Society is better off without them. 6. Satisfy the sense of justice and provide a sense of protection, if carried out successfully. 7. Most criminals are young men. Capital punishment would prevent their having families and thus eliminate the stock of criminals. 167 B. Motives for Abolishing Capital Punishment 1. Be against many religious beliefs, such asJ it is the right of God, and God alone, to take a person's life. 2. Two wrongs do not make a right. die instead of one? Why should two 3. The government doesn't allow a person under 21 years of age to vote or have any other privileges such as holding a legal office; it shouldn't take away his life. When a criminal has been corrected, the country is ahead one more citizen. 4. Would eliminate taking a man's life and later finding him to be innocent. It is possible for a jury to corne to the wrong verdict. 5. Do away with a person's being falsely accused. 6. The death penalty prevents reform. 7. Yfiaen actually innocent, sometimes mental cases confess being guilty. S. It is pointed out that there ere not any more crimes in states which do not have the death penalty than in those in which it is still found. 9. The harder the punishment the more thrilling the criminals find it. It is the harsh punishments that tend to increase the desire for crime. Porbidden fruit is always the sweetest. 10. We should not kill criminals when it is our fault that they are criminals. 11. Death doesn't prevent crimes. 12. People serving on juries feel guilty of taking a, person's life when they resort to capital punishment. II. Terms Upon Which We Need to Agree 1. Y/ide disagreement - To what extent? How wide? 2. Thoughtful people - When are people not thoughtful? By what criterion can we judge thought fulne ss? 3. Most effective way - When does a way become most effective? How effective? 168 4. Certain types of crime - Does it include murder, robbery, adultery, kidnapping, rape, slander, and a multitude of other mis demeanors? The word crime is quite inclusive, and the terms "certain types11 are quite elusive unless they "become identified with a specific form of crime. III. Underlying Assumptions 1. Thatcapital punishment should "be abolished. 2* That capital punishment should not "be abolished. 3. That capital punishment is something the prospective criminal fears. 4. That if a few states abolish it, all of the states should do so. 5. That the so-called 'thoughtful people1 are authorities. 6. That abolishing capital punishment would increase, reduce, or maintain existing crime records. 7. That there is no solution to the problem because thoughtful people disagree. 8. That capital punishment doesn't prevent crime. 9. That criminals and their purposes gre alike. 10. That courts and juries can discriminate between crimes punishable by death and otherwise. 11. That the five states are right. 12. That the forty-three states are in the right. 13. That the majority of the people in the forty-three states are or are not in favor of capital punishment. 14. That the majority of people in the five states favor abolishing capital punishment. IV. Need for More Pacts in 1. the Situation Is murder the only crime which has thelaw ofdeath over it? Kidnapping is subject to thedeathpenalty insome cases if the jury desires it. 2. Was the crime bad enough to be punished by death? 169 3. Did the person do it under normal conditions? 4* In some states are there more crimes than in others? 5* Different states have different methods of punishment, also different laws concerning it. capital 6. Statistical surveys show that there are more people in the United States who "believe in capital punishment than not. 7. Many minors have not had the right kind of chance inlife. 8. The most notablediscovery in crime is the large number of young criminals who began their anti-social career as the result of some seemingly unimportant grievance. 9. Wardens and prison doctors in states where gas is used feel strongly that is the most merciful way yet. 10. Statistics show that only a small percentage of our murderers are executed. 11. Fighting crime is not a battle against any single individual. By getting even with certain people we do not get even with crime. 12. Today the death sentence is given only for treason and murder. ALso a person is not allowed to receive the death punishment if insane, but only when the said person has become sane. 13. Today, the death instruments used are the electric chair, gallows, gas chamber, sword and axe. 14. Does the small percentage of executions in this country warrant abolishing capital punishment? V. Testing Facts for Pertinency to the Situation 1. There are a great many people who don!t think capital punishment should be abolished by the forty-three states still keeping it. Should this have any bearing on whether or not capital punishment should be abolished? 2. Should capital punishment deal in terms of the kind of crime that has been committed? 3. How many of the people in the forty-three states feel the death penalty should never have been abolished? 170 4. How many of the people in the five states feel that the death penalty should never have "been abolished? 5. Should not the facts he tested for pertinency in this light? 6* Does the method of administering the C. P. penalty have anything to do with this situation? 7. Does the age of the criminal have anything to do with this situation? VI. Discriminating Between Pacts and Assumptions 1. It is a fact in this situation that Kansas, Maine, Michigan, Ehode Island, and Wisconsin abolished 0. P. for certain types of crime, 2. It is a fact that the other forty-three states have not done so. 3. It is an assumption that capital punishment should he abolished in the other forty-three states. 4. It is an assumption that capital punishment should not have been abolished in the five states, VII. Self-Evaluation for Bias or Prejudice 1. I am biased in this case and my mind is set as to what I think is right. It would therefore do no good for me to express my opinion as to which way to deal with killers. You must judge by the facts and assumptions I have pre sented for both sides, as to which is the more just. 2. I am biased in this matter as I do punishment. not believe in capital 3. I am biased in this situation because I believe that society makes the criminals that it has, and with this as my basic assumption I am obligated to oppose capital punishment. 4. I am biased in this situation because when a sane man commits a crime, he does it knowingly and therefore should suffer the consequences. VIII. Evaluation of Conclusions for Consistency with the Assumptions 1. I conclude that this subject is so tied up with emotion that one cannot decide it definitely. 171 2* Society makes its own criminals; therefore, my con clusion is true. (I am for abolishing capital punishment.) 3. (Phis is a question that will never he solved to the satisfaction of all concerned. Both sides have good points hut will probably never come to an agreement among them selves that will give complete satisfaction. 4. Since I feel that everyone has a chance to do what is right and since criminals take the lives of others, then they should be apprehended and their lives taken away from them. IX. Heed for Forming Tentative Conclusions 1. It is possible for a jury to come to the wrong verdict. 2* It is possible that the criminal killed in self-defense and does not have the evidence to support his case. 3. It is possible through birth control and through social re-organization to exterminate crime, or at least to reduce it. (That is why I wouldn't commit myself to any legislation for or against capital punishment at the present time. 4. The reason that I suspend judgment in this matter is due to the word crime itself. I ask myself, what is a crime? I find it covers a multitude of 'wrongs1. Then I confine it to murder and I find myself seeking an answer to what murder is. When does some one commit murder? I read in the paper where a man was given the electric chair for killing another man and that another man or a group of men responsible for the death of fifty thousand young men go free. But you say they didn't have to go and I come right back and say they would have been conscripted as some of them were in the last war. I ©m quite confused because murder to me means taking the life of another, and whether it is done directly or indirectly, it is still murder to me. Hot being clear at this point forces me to suspend judgment, in other words to make my conclusions tentative. X. Evaluation of Individual and Social Consequences That May Arise as a Result of the Conclusion Reached 1. Maintaining capital punishment prevents prison escapes and repeated crimes. 2. Maintaining capital punishment reduces upkeep; the upkeep of feeding, housing, and clothing criminals. 3. Maintenance of capital punishment constantly serves as a reminder for the criminally inclined that their last day is at hand when they get caught after the crime. 172 4. Maintenance of capital punishment constantly reminds all criminally minded persons that 'crime does not pay*. 5. Abolishment of capital punishment could serve as a means of getting a lot of work and service to the States as a result of the criminals' labor in prison. S. Abolishment of capital punishment would more than counteract feeding, housing, and clothing, because of the work or labor performed by the criminal. 7. Abolishment of capital punishment would be a constant reminder for the criminally minded that life imprisonment with labor is not a cheerful outlook. 8. Abolishment of capital punishment would also be a constant reminder that 'crime does not pay'* Conclusions by the Group Kegarding Capital Punishment Before Open Discussion After Open Discussion 58$ in favor 32$ opposed 10$ suspended judgment 11$ in favor 20$ opposed 69$ suspended judgment (however, suggested reorganization, reforms, social planning, birth control, etc.) The preceding is an illustration of how facts when presented from both sides of a situation may change the thinking of young people from one point of view to another. However, there is still a lack of unanimity in this situation, because of disagreement upon basic assumptions, or because of thinking colored by emotions, or in some cases because of deep-rooted prejudice. 2. Group Project on Compulsory Education Another Experimental Group developed an interesting project in connection with the problem of compulsory education, which is as follows* "We have in this country certain laws which compel all young people up to a certain age to attend an organised school. Some 173 people believe that such laws are most desirable while others feel their operation accomplished little if anything in improving the culture of our people.11 A preliminary vote expressing each pupil!s attitude toward compulsory education was taken and the results were as followst 32$ were opposed to compulsory education laws 65$ were in favor of compulsory education laws 3$ were undecided. This was followed by a period of discussion and a critical analysis of the situation, which in turn was followed by a written assignment. The written assignment was to be a critical examination of the situation with an attempt on the part of each pupil to reach a logical conclusion. A com mittee of five was appointed to compile the results for the group and to present a summary to the class* The content of the written assignment was summarized and presented to the class in terms of the elements of critical thinking mentioned in Chapter I, as followsJ I* Motives Listed 3Por and Against Compulsory Education Laws A, Motives 3For Compulsory Education Laws 1. 2* 3. 4. To improve living conditions for everyone To maintain the principles of democracy To raise the cultural level of the nation To prepare everyone for fulfilling his duties in citizenship 5. To give everyone a basic knowledge in reading, writing, and arithmetic 6. To make people more intelligent about their own affairs and those of others 7. To strengthen the bonds of freedom 8. To dispel the evils of ignorance, exploitation, poverty, and disease 9. To give everyone a better background of his chosen vocation 10. To give everyone a chance to think for himself ' 11* To preserve the democratic ideal. B* Motives Against Compulsory Education Laws (Seven motives were listed by the committee) 174 II. Definitions or Agreed-Upon Meanings "by the Group 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Certain laws - meaning laws pertaining to compulsory education. Massachusetts had compulsory education laws before 1865. Compel - meaning to force or require by law. Young people -meaning immature persons in the early stages of growth, not fully grown. Certain age - meaning a person between the ages of six and sixteen. Organized school - meaning a school given permission by the state to carry on activities which will develop each child mentally, physically, and socially. Some people - meaning in this situation the majority, because the majority rules in our country. If "some11 meant the minority, in this case, the laws would soon change. Others - meaning the minority. Desirable - meaning for the good of themajority. Improving the culture - meaning respect for each other, a combination of grace and poise, and a desire to create for the good of all. Our people - meaning citizens of the United States. By education - meaning "by schooling". III. Assumptions Involved 1. [The majority of the people are in favor of compulsory education laws. 2. The people against compulsory education laws are: a. uneducated themselves b. those who want to work and are under 16 c. those who want their children to support them d. those who live in a district without good schools e. those who live in a district too far from school f. those burdened by taxes who feel it wouldbe cheaper to send their children to private schools. 3. The people in favor of compulsory education laws are: a. educated b. those who through lack of their own education have come to realize its importance for their children c. those who realize that the best educated person gets the best job d. those who feel that education raises the standard of living e. those who are altruistic or socially minded f. those who feel that education wipes out ignorance, exploitation, poverty, and disease. 4. In our present industrial civilization, schooling is an essential element in the success of nearly ell of our young people. 5. An educated child will become a better citizen than an uneducated one. 6. The people who do not want compulsory education laws have a better plan for improving the minds of future American citizens. 7. The majority of people against compulsory education lav/s are those who have reasons or facts to back them up. 8. People against compulsory education laws are the oneswho can earn their living only by using their muscles, not their brains. 9. Foreigners who are not used to these laws and therefore cannot adapt themselves to them are against compulsory education. 10. Schooling is all of education. 11. Education helps one in the business world. 12. Education helps one socially. IV. Facts Concerning Compulsory Education 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. There are laws whichcompel young people to attendschool. There are people who disagree with these laws. A majority approve such laws or they would be changed. There are families in dire need of assistance which older children can give. Schooling is a necessary background for nearly all of the professions. Education in public schools is open to all, rich and poor alike. Compulsory educationis a burden to tax payers. School teaches other things besides reading, writing, and arithmetic - things that will help a child to get along fcith people. Poor people derive more benefits from compulsory education than wealthy people. More people attend colleges and universities every year. Schools offer more opportunities now than thirty years ago when our parents attended them. Young people in schools today will become our citizens of tomorrow. Educated people made these laws. Most people want economic security. Educated people receive higher incomes. Abraham Lincoln had very little schooling. It is the constitutional right of every American to learn the three H*s. Many people feel they are burdened with taxes. V. Tsdting Facts for Pertinency The committee agreed that the situation in question is an issue between the desirability and undesirability of compulsory education laws. Some of the facts selected as being pertinent are as follows? 176 1* 2. 3. 4. There are laws which compel young people to attend school. There are people who disagree with these laws. A majority approve such laws or they would he changed. Schooling is a necessary background for nearly all of the professions. 5.‘Young people in schools today will become our citizens of tomorrow. 6. It is the constitutional right of every .American citizen to learn the three E fs. The committee agreed that many of the facts were irrelevant to the situation. Some of these are as follows! 1. Most people want economic security. 2. There have been marked increases in college and university enrollments. 3. Many people feel they are over-burdened with taxes. 4. Schools teach other things besides reading, writing, and arithmetic. 5. All of education is not the result of schooling. 6. Abraham Lincoln had very little schooling. fl. Discriminating Between Pacts and .Assumptions The committee agreed that many assumptions were listed as facts and many facts as assumptions. The following are some assump tions listed as facts! 1. Educated people made these laws. 2. Compulsory education is a burden to the tax payer. 3. Poor people derive more benefits from compulsory education, than the rich. The following are some facts listed as assumptions! 1. Education raises our standard of living. 2. In our present industrial civilization schooling is an essential factor in the success of nearly all of our young people. 3. The majority of people are for compulsory laws in education. VII. Bias or Prejudice The committee found most of the group in favor of compulsory education laws, and that this bias was reflected in the indiv idual analyses. However, the committee indicated that most people opposed to compulsory education laws had special interests which were in conflict with such laws, and therefore their emotions or feelings blinded or colored their thinking toward compulsory education. Por example! UA person of great wealth opposed to democratic processes may send his children to a private school. As a result he may feel eiaotional upset because a portion of his tax money will be used in support of a program of compulsory education." ( 177 VIII* Testing Conclusions for Consistency with. Assumptions The committee agreed that some of the conclusions reached by various members of the group were consistent end others inconsistent with their assumptions. For example, one student remarked! "The best educated person gets the best Job”, and yet at another point he concluded that, "Most jobs require a 'drag1 through some friend, relative, or political affiliation." Another student showed inconsistency when he assumed that: "Uneducated people, and people under sixteen years of age who didn't like school, were opposed to compulsory education." In his conclusion he pointed out that, "The real threat to compulsory education comes from a few wealthy people." However, the committee reported that nearly all of the students agreed that! "Most people are in favor of compulsory education laws," and in most cases the conclusions reached were that "com pulsory education is essential if certain evils such as ignorance, poverty, exploitation, and disease were to be eliminated from our country and the rights of citizenship exerted by each person." IX. Tentativeness of Conclusions The committee found a great deal of evidence of suspended judgment in the different analyses* For example! "The majority of the group emphasized at one point or another that they would be willing to support a better system of education than the present compulsory system, if it could be proved that it was a better one." X. Consequences The committee found numerous statements of consequences on both sides of the issue in question. Some of these are as follows! 1. More people are attending colleges and universities every year as a direct result of our system of compulsory educa tion. Statistics prove this. People are becoming more educationally minded. 2. Schools offer more opportunities for developing our interests today than they did thirty years ago. This is in part due to our compulsory system of education. 3. Conpulsory education is preserving the democratic prin ciples of our country, because it brings:together the poor and the rich on an equal footing, at least in the classroom. 4. Compulsory education has resulted in help being given to families in dire financial circumstances so that their children may attend school rather than seek employment. 5. If compulsory education laws were removed, only children from rich homes would derive benefits from schooling. A comparison of conclusions reached by the group before an open discussion of this situation and after the discussion is as follows! Before an Open Discussion 32$ opposed to compulsory education 65$ compulsory education 3$ undecided After an Open Discussion 2$ opposed to compulsory education 81$ favored, compulsory education 17$ undecided 3. Other Examples of Group Projects In order to study changes in group thinking as the result of group discussion, ta series of non-mathematical exercises purpose. intervals. 68 \7ere designed for this These were selected by the teacher and administered at periodic The'following exercise was administered by the Experimental Group teachers very early in the first semester, discussed by the group the next day, and on the following day administered again! In a certain community there had been numerous burglaries which the police were unable to solve. Early one morning several officers arrested the 18-year-old boys who were loitering around a Texaco service station in a residential section of the city. Below are listed some of the conclusions given by a number ofthe neighbors later in the morning, when they learned what had happened. Place a check mark in column I opposite each conclusion which you will accept if you consider the facts given above. After further investigation the police were told by a milk man that he had seen some boys that morning attempting to open several windows of the station. Using this added information with- the facts already given, place a mark in column ho. 2 opposite the preceding conclusions you would now accept. 68 It isV difficult to describe content of this type without implying procedures on the one hand and evaluation on the other. The nature of the content, however, merits consideration at this point since it was not mentioned in the description of procedures. 179 Per Cent Responding After Group Discussion 1 2 o 1 2 18.4# 60.5# 60.5# 46.1# 96.3# 76.3$ 10.5# 5.3# 58.3# 1.7# c) It is certain that the hoys were connected with the other robberies. 0.0# 7.9# 36.8# 0.0# o o Per Cent Responding Prior to Group Discussion d) It is probable that the hoys were connected with the other rohheries. 7.9# 26.3# 60.5# 1.2# 12.5# 89.4# 86.8# 76.3# 42.1# 98.7# 89.0# 76.2# f) The given facts have nothing 26.3# to do with the hoys heing involved in the other burglaries. 10.5# 0.0# 1.3# e) More information is needed to he sure that the hoys in tended to break into the service station. i '° fc'V .° O -x j h) The officers had no right to arrest the hoys on such evidence. 100# 18.1# I a) The "boys* actions were cer tainly suspicious* 3 0.0# 0.0# fingerprints of the hoys were taken and through these it was found that the hoys had heen arrested in connection with former rohheries in another city and had heen committed to a reform school for several years. Does this fact change your judgment as to 7/hich of the preceding conclusions you would accept? Check in column No. 3 the conclusions you accept now. The ahove was again followed up ahout two months later with a slightly different exercise. in terms of percentages. The number of responses is again indicated The exercise is as follows! Harry Johns was recently arrested twice within one week on charges of disorderly conduct. The Justice of the Peace fined him 200 dollars, which he was unable to pay; whereupon he was sentenced to 42 days in the county jail. Numerous comments on the severity of the penalty are listed below. If you consider only the facts given below, indicate with a check mark ( \J) in column No. 1 those that you consider just. It happens, however, that at the time of his first arrest Mr. Johns had heen given a fine of 100 dollars. He explained to the Justice that he was out of work and could not pay the fine, hut that he had obtained a job which was to start the i 1) The offense is too slight for such a heavy fine. 2 3 1.2$ 2) The offense deserves such a fine. 95.5$ 3) Other facts are needed to justify the fine. 4) It is provable that a smaller fine would have heen adequate* 98.2$ 95.4$ 1.8$ 3.4$ 4.5$ following week. Hence he was granted leniency. Does this change your opinion? Place a check (^/) in column Ho. 2 opposite the conclusions you now accept. But there seemed something familiar about Mr, Johns to the Justice of the Peace. He seemed to remember a former charge against the man. So he consulted his files and found another 100 dollar fine against him on the same charge. Place a check (v/) in column No. 3 opposite the comment you now believe to be justified. Toward the middle of the second semester the preceding exercises were again followed up with an exercise developed by Fawcett follows: 69 , which is as (The number d»f responses in terms of percentage are again recorded.) Mrs. Lewis Seymour was recently struck by an automobile and instantly killed. The driver of the car did not stop and while a man did see the accident he failed to. see the number of the license plates of the car. However, he did notice that the right headlight was broken and that a tire blew out at the time of the accident. He reported these facts to the police and twelve hours later they found a car with a flat tire and a right headlight broken. This car was parked behind the house of Hezekiah Berry and belonged to him. Numerous conclusions considered by the police are stated below. Place a plus sign in column 1 opposite each conclusion which you will accept from a consideration of only the facts given above. 1 a) It is certain that the car which struck Mrs. Seymour belonged to Hekekiah Berry. 3 2 "l.3$ 12.5$ b) The given facts are irrelevant to the problem of discovering who owned the car that struck Mrs. Seymour. c) It is certain that Hezekiah Berry was not driving the car that struck Mrs. Seymour. 69 Fawcett, Harold P., "Thirteenth Yearbook11, cit. pp. 82-33. 4 43.3$ 181 d) Other facts are needed before it can be definitely proved that Hezekiah Berry was the driver of the car that struck Mrs* Seymour* e) It is probable that the car that struck Mrs, Seymour belonged to Hezekiah Berry. 100$ 100$ 100$ 100$ 99.4$ 98.7$ 87.5$ 56.7$ f) It is certain that the car which struck Mrs. Seymour did not belong to Hezekiah Berr yg) It is certain that Hezekiah Berry was the driver of the car that struck Mrs* Seymour. Through further study of the problem the police found that the glass at the scene of the accident was of the same pattern as that in the broken headlight on Mr. Berry’s car. Using this added fact in connection with those already given, indicate in column 2 which of the preceding conclusions you would accept. It was also established that v/hen Mrs. Seymour was struck she was carrying a quart of potato soup. Some of this soup was found at the scene of the accident while traces of the same kind of soup were found on Mr* Berry's car. Does this added fact change your judgment as to which of the preceding conclusions you would accept? Indicate in column 3 those conclusions which you believe are now definitely established by the known facts. The police also found strands of hair on the broken headlight of Mr* Berry’s automobile. How considering all these facts, will you indicate in column 4 which of the conclusions you believe ±o be justified? It is of particular significance to note the differences in the responses to column 3 before discussion and after discussion. inother group attempted to select candidates for the Seventh Senatorial District in Illinois on the basis of the responses the candidates made to the following situation and question? The Situation A study of the amounts paid school districts from State Treasuries for the school year ending June 30, 1936, showed that the payment per pupil in average daily attendance grades 1-12 inclusive, was? average for the United States, $26.12; average for Illinois, $11.19; rank of Illinois in this respedt, 36th. 132 The Question What is your attitude toward providing in "both grade and high schools a distribution of state funds to the schools of Illinois comparable to the school support of at least ah average state? The Replies From Candidates For Senator 1* “I am in favor of more State Aid for education and feel that present State revenues are large enough to justify a large incr ease." 2* f,It is my belief that Illinois, a state that ranks in all other matters, should at least be up to average with respect to education, both grade and high school, and it is my inten tion, if elected to the state senate, to give my support to the furtherance of that cause.11 3# ’’Should I be elected to the General Assembly I am 100 per cent in favor of state aid to both grade schools and high schools in Illinois." 4. "The State’s distribution to the grade schools should be materially increased and help given high schools." The Replies From Candidates For Representatives In General Assembly 1. "I am in favor of distribution of State Funds to the schools of Illinois comparable to the school support of an average state." 2. "Providing both grade and high schools a distribution of state funds to the schools of Illinois comparable to the school support of at least an average state meets with ray whole-hearted approval." 3. "I have consistently worked for and voted for an increase in the State School Distributive Fund. In the 60th Session of the General Assembly after all other amendments for increases had failed, I introduced without success an amendment for an increase of $1,000,000." 4. "Assuming that Illinois1 deficiency in this respect is not overcome by proceeds of local taxation, this question seems almost rhetorical. Illinois should have a school system second to none. We are not only able to do this; we should do all in our power to accomplish this objective; I will be proud to do my part." S."I believe in providing a larger distributive fund and also in providing high school participation. I believe this can be done without increasing the tax burden." 183 6* '‘Definitely in favor of increasing State Subsidies to school districts." Various newspaper articles were brought into the classroom and discussed by the different groups. Examples of these are as follows? 1. The Sinking of the "Panay". 2. Vocational School Theories Hit by Schnackenberg. 3. Dean Doyle Asserts Education hen been in Half-Baked Theorists’ Hands. Then pupils brought in editorials on various topics. Examples of these are as follows? 1. What Japan Wants. 2. The Panay Crisis. 3. The Hew Wire Tapping Decision. 4. Toward an Arbitrary Court. One pupil brought in a newspaper column which is sponsored by a large urban department store. It served an an excellent basis for analysis in terms of the elements of critical thinking. The topic was "Whom Do Cheaters Cheat?" The field of commercial advertising served as an excellent source of non-mathematical materials. Other projectsincluded such topics as? 1. Choosing One's Vocation. 2. Selecting One's College. 3. Liquor Legislation. 4. Awards. 5. Racial Superiority. 6. Court Proceedings (Trials). 7. Student Supervised Study Halls. 184 Many other situations may be brought to be$r out the type of group activity. Many of the group discussions merit mention herein; however, for reasons already indicated it becomes impossible to in clude all of them. Furthermore, in order to describe all that has transpired in each classroom would require a large staff of observers taking notes in shorthand. One of the pupils under teacher E summarized the nature of the procedures, materials, and content of group projects as follows! There are two main purposes of our geometry course, namely to help us think more critically and to learn how to prove pro positions both in geometry and outside of geometry. In order to prove geometric statements we must -agree upon definitions and assumptions, because if this is not done great difficulty will be experienced in working our geometric problems and reaching the same conclusions, ind so in non-mathematical statements, if one wishes to prove them or argue about them, everyone must agree on certain definitions and assumptions. For example, on the next page there is a study about assumptions. On it there are two campaign statements based on the problem of education. In order to decide really who would be the best candidate to vote for, we went through the two speeches and made a list of all of the terms that were used and were not fully explained. In A*s statement he promised to maintain a Reasonable minimum standard of education*. That reasonable standard could mean that the students would have well trained teachers with college degrees, or it could mean that the state would employ teachers with a great deal less education. The very word education could be defined in many different ways. B also made many statements which could be interpreted in any way that he wished after the campaign. This is just one example of how important it is to have a common understanding of the many definitions and assumptions that are used in any argument. .. 4. Analyses of Instruments for Propaganda The Experimental Groups listed the newspaper, periodical, radio, movie, and telephone as the chief instruments for spreading propaganda. These were discussed in terms of the elements of critical thinking previously mentioned. 185 Numerous motives 'behind each of the above instruments were sug gested "by the pupils. 1. 2. 3. 4. 5. 6. Motives Motives Motives Motives Motives Motives Chief among these are: for profit for fostering political ambitions for defending some particular cause for promoting some particular cause for fostering views of an individual or of a group for making profit in another business. "Propaganda", as previously mentioned, was accepted by the group to be "either a means of social control or a method by which an individual or a group works for his or their own interests". Most of the content dealing with advertisements, speeches, and editorials in this chapter was not only analyzed in terms of the elements of critical thinking, but also classified as to the type of propaganda device employed. The list of propaganda devices employed included the listin the November, 1937, issue of "Propaganda Analysis"^, as follows: 1. Name Calling - a device to make us form a judgment without examining the evidence on which it should be based. 2. Glittering Generalities - a device by which the propagandist identifies his program with virtue by use of "virtue words". 3* Transfer - a device by which the propagandist carries over the authority, sanction, and prestige of something we respect and revere to something he would have us accept. 4. Testimonial - a device to make us accept anything from a patent medicine or a cigarette to a program of national policy. 5. Plain Polks - a device used by politicians, labor leaders, business men, and even by ministers and educators to win our confidence by appearing to be people like ourselves - "Just plain folks among the neighbors". 6. Card Stacking - a device in which the propagandist employs all the arts of deception to win our support for himself, his group, nation, race, policy, practice, belief or ideal. 7. The Band Wagon - a device to make us follow the crowd, to accept the propagandist’s program en masse. 70 Institute for Propaganda Analysis, Incorporated, "How to ^Detect Propaganda"! Propaganda .Analysis, Volb. I, No. 2, Nov. 1937, up. 1-3. 136 Examples of the preceding devices are as follows* 1* An educator*s assertion that 11American education has "been too long in the hands of 1half-baked1 theorists. 2. Glittering Generalities - a portion of Hi tier*s speech where he says, “The Aryan Fatherland, which has nursed the souls of heroes, calls upon you for the supreme sacrifice which you, in whom flows heroic hlood, will not fail, and which will echo forever down the corridors of history.” 3. An example of the transfer device is Schnackenberg*s recent speech regarding Vocational School (Theories, in which he defends Americanism and its traditions when he says* "7/e find that even the great American who once walked the streets of this fair city and later successfully carried the responsibility through years of civil war, of maintaining the American republic, is not immune from scurrilous attacks from these sources”. (Meaning certain professors in private colleges and state universities.) 4* Examples of the testimonial device may he found in commercial advertising through the newspaper, periodical, radio and movie. Critical analyses of this technique have been pointed out in several places. It is especially popular with advertisers of cigarettes, toothpastes, cosmetics, soaps, hosiety, and so forth. 5. An example of the Plain Polks device was particularly apparent in the stories and pictures recently published in a large urban community of a candidate running for a political office and backed by this particular paper. (The stories and pictures included descriptions of simple home life, of Sunday school and church attendance, kindness and sympathy for children, and friendly relations with neighbors. 6. An example of the Card S tacking device may be found in a news paper column labeled "Moving Forward”. (This column is paid for by a large urban department store and contains many interesting articles of the type "Whom Do Cheaters Cheat?”, and so forth. 7* Numerous examples may be given of the Band Wagon device. An outstanding example of this technique was the last presidential election and the support given Landon by Eepublican newspapers. In a certain metropolitan area prior to the election, where Eepublican newspapers 'Biere dominant, the papers made it appear that everyone was for Landon. Non-Mathematical Content Developed Through Individual Effort Noir-mathematical content developed through individual effort will be described under the following categories! (l) content dealing with 137 individual projects, and (2) content consisting of voluntary contributions. Difficulties encountered in describing non-mathematical content have been previously indicated, therefore the following situations are not represen tative of all of the materials submitted. 1. Individual Projects It was pointed out in the preceding chapter that 24.2 per cent of the individual projects dealt with advertisements, 12.2 per cent dealt with written articles or editorials and speeches, and approximately one-third of the remaining ones had some space devoted to non-mathematical materials. In other words, approximately 40 per cent of the Experimental Group pupils elected to develop individual projects of a non-mathematical nature. Since some of the pupils made their individual projects a part of their "textbook", one boy writes as follows! It may strike you as strange in a book on geometry to find references to 'Critical Thinking1. But really, it has a definite bearing on the subject. In both Critical Thinking and Geometry, it is necessary to start with two things in mind or else no satis factory conclusion will ever be reached. For example, one day we were talking about Argument in class and someone said that in order to have a satisfactory argument, say on the subject of whether or not a certain individual is a socialist or not, one must first agree on certain definitions, in this case, what a socialist is, and also on certain assumptions. So there we have the two necessities of a good argument. But, aren't they equally necessary in geometry? The class cannot hold a discussion on straight lines if some ‘dodo* doesn't know what a line is. This, of course, is exaggerated, but the fact remains that, in either geometry or Critical Thinking, one must have two things, definitions and assumptions. So now you see how Critical Thinking fits into this book. Advertisements! advertisements. One girl selected as her project an analysis of She clipped different ones out of their context and classified them as poor, average*! and good. She discriminated between the 188 good and "bad on the "basis of whether or not the advertisement tried to deceive the reader. Eor example, she referred to one “&d“ as "being good, "because it had an attractive "bust of a well-groomed man with a variety of ties radially situated about the bust, and a small palm tree in the upper left hand corner. Below were the following words! “We have every pattern and color known to PALM BEACH TIES $1 (Washable and wrinkle resisting) Qo. Ties, Eirst Eloor" Her reasons for selecting this 11ad" as a good one are as follows! The way this advertisement is arranged is very good. Et is well balanced and has everything needed to be said, and it is said in a very fbw words. Of course ‘every pattern and color known1 covers a great deal of territory, but that is a good point, be cause the price is given and it will cause many readers togo down to see the patterns and colors for themselves. In fact, it is a sort of a challenge for the critically minded person. One boy developed a project on ‘‘Using Your Head in Buying Everyday Heeds", or “Are You a Sucker for Adve rtisements?Some of the excerpts from his “magnum opus“, as he calls it, are as follows! One of the biggest jobs for Mr. and Mrs. Average American is to find out what articles like toothpaste, soap, radios, auto mobiles are the best for him to buy. Their problem is more dif ficult nowadays because of the great advertising pressure put on them. This advertising is found in the daily papers, magazines and on the radio. Many of these highly advertised articles have been found to contain harmful ingredients or defective parts. You may ask who has found these things out? The answer is that large non-profit organizations like the Consumers Research and Corasumers Union have been organized to inform the public of the good and bad articles. But perhaps you don't believe that these institutions are 'on the level'. In that case you may ask a doctor whom you believe to tell you and he will usually tell you the same thing that the other organizations have said* 189 The two things that Mr. and Mrs. Average .American are primarily looking for are maximum quality and minimum price in a product. Anyone can find out the price of an article "by asking the clerk in the drugstore, hut most people cannot discover the quality of it. It may also he discovered that the more expensive articles are not necessarily the best in quality. This can he demonstrated in the following exampleft In the Woolworth's stores a soap called 'Gondola Hoating' sells for 16 cents per pound. Also there is the widely advertised 'Camay* soap. This sells for 31 cents per pound. Under laboratory tests the two are rated equal in quality. Here is -part of a Camay *adf. (A very attractive one was pasted below, on which was the statement, 'To look your loveliest you must have lovely skin'.) You may ask why the price of Camay is so high. A very logical answer is that Camay is advertised at tremendous cost in magazines all over the country. The other is not advertised at all. Today everyone must watch out for each phrase used in adver tising and also out-and-out false claims. Look at the example on the bottom of the page. This and other similar ads illustrate the power of advertising. Through these 'ads' America has been made bad-breath conscious. So conscious in fact that many people make their breath noticeable by using strong-smelling toothpastes. In the 'ad* it says it stops stagnate saliva odors. It is a fact that saliva continuously flows into our mouths and that bad breath signifies that something is really wrong with your insides. It is true that the toothpaste stops odors for a few minutes, but when new saliva flows into your mouth the rest is swallowed or evaporated so that the toothpaste is carried away. And so toothpaste should not be bought for the purpose of relieving bad breath, but for cleaning teeth. Here is another interesting 'ad1! (Here is a picture of two partially opened cigarettes and the following words under them! 'Which is which! The tobacco of these two 15# brands looks alike and tastes alike. CU tests showed that most smokers not only cannot distinguish between 15# brands of the seme type, but ere equally incapable of distinguishing between 15# and 10# brands. Moral! If you're a devotee of one of tha 15-centers and want to save money, forget what the ads tell you and try a cheaper brand.') I believe that people ought to try the lower-priced article and actually see if it isn't just as good as the higher priced article they have been using. In a great many cases this can be done be cause the main factor in keeping some things priced high is the large cost of advertising. 190 If this ’ad* below does make lonely girls are there so many other face powders selling? wouldn’t fall in love with a girl just because She has to have something else to go under the fact has been proved since life began with the so beautiful, why Personally, I of her face pov/der. face powder. That caveman. This chewing gum business is being carried too far in my estimation. I know I am not alone in saying that it is pityful or disgusting or any way you want to put it to see a woman chewing gum like a cow chewing its cud. Chewing gum is distracting to the ’chewer* as well as the onlooker. One cannot perform anything well while doing something else at the same time. Of course, this doesn’t bother some people, but such people are the perpetual gum chewers who can’t live without it. It is not healthful to have saliva running into the mouth at full capacity all the time, and it is distracting to have a taste in one’s mouth, be it good or bad, all the time. It is not wise to make chewing gum or anything else, such as smoking or drinking, a regular habit. These things should be preserved as a pleasure. (Here he has an 'ad' which says ’Millions of women chew Double Mint Gum dailyi) In summing this up, we, the average man and woman, should not buy things blindly. We should reason things out, for example: should I buy tomato soup for 30 cents or 20 cents? They both taste the same. We shouldn’t form definite habits on the strength of an advertisement like chewing gum. We should realize that widelyadvertised merchandise must be' priced higher than non-advert ised goods, even though they are the same. Also a doctor is a better person to consult about falling hair and skin disease than the advertising section of your favorite magazine. Another interesting project turned in by a girl involves critical analyses of words used in advertisements. It starts out with the statement of an advertisement as follows* "Do conflicting claims of dandruff remedies bewilder you? Then you will be glad to know there is one logical, scientifically sound treatment, proved again and again in laboratory and clinic. Listerine Antiseptic and massage. "Recently, in the most intensive research of its kind ever undertaken, scientists proved that dandruff is a germ disease. And,in test after test, Listerine Antiseptic, famous for more than 25 years as a germicidal mouth wash and gargle, mastered dandruff by killing the queer, bottle-shaped dandruff germ Fityrosporum ovale. "At one famous skin clinic patients were instructed to use the Listerine Antiseptic treatment once a day. Within two weeks, 191 on the average, a substantial number had obtained marked relief.i At another clinic, patients were told to use this same Listerine Antiseptic treatment twice a day. By the end of a month 76^ showed either complete disappearance of, or marked improvement in, the syraptoms.11 Undefined terms in this advertisement In the first paragraph the advertisement mentions the con flicting claims of dandruff remedies, and then offers a scien tifically sound treatment for this disease. However, the makers of Listerine Antiseptic seem to have overlooked the fact that there is a great deal of difference between a remedy, which guarantees either immediate or gradual cure, and a treatment, which only offers relief for a limited amount of time. In the second paragraph the makers of this treatment tell us that dandruff is a germ disease, and that Listerine has been known for over 25 years as a germicidal mouth wash. Prom this I con cluded that either we must have the same germs in our mouths as are found on our scalps, or that Listerine will kill all germs* The third paragraph offers the fact that patients at one clinic were instructed to use this treatment once daily. The makers do not tell us what the patients at the clinic used the treatment for, but assuming that they used it for dandruff, we still cannot accept Listerine as a proven cure, even after reading that within two weeks a substantial number had obtained marked relief as we don*t know what the maker considers a substantial number of patients, and we also are confused by the term marked relief. The next sentence says that by the end of a month, 76?j of the patients who used the treatment showed either complete disappearance of, or marked improvement., in, the svmutoms. Because the symptoms of the disease disappear or improve does not prove that the disease itself is cured or improved. Lastly, the advertisement tells us nothing of the patients. The climate or other conditions which exist around them may have a great deal to do with the case. Also, we do not know how serious were the cases of dandruff that the patients receded treatment for... Another advertisement is analyzed by the same girl as follows: "A cream so remarkable that it is as world-famous as Mme. Rubinstein1s name. Specially conceived to benefit all skins, its special ingredients and unique consistency make it the supreme one-cream beauty treatment! It cleanses deeply, sweeping away impurities and grey weariness ... animates the complexion to a lasting radiance ... guards against lines, wrinkles and dryness ... softens and helps smooth away fatigue marks. Use it morning and night; use it to whiten and smooth chapped hands; use wherever the skin is dry or roughened, for it leaves the texture exquisitely clear and velvety.11 192 Undefined terms in this advertisement Madame Hubenstein1s cream, the advertisement states, is con ceived to benefit all skins. In what manner this cream actually benefits all skins we are not told, Yfhen speaking of all skins, we assume that this cream will improve in some way even the most unusual skins with the most unusual ailments. We also must assume that Madame Hubenstein is considering such factors as the many and varied types of defects found in normal skin, and the different climates to which different skins ar^ exposed. Madame Hubenstein mentions special ingredients, but does not tell what they are, or in what way they are so special. Her cream is also said to sweep away impurities* We do not know exactly what is meant here by impurities, or for how long a time one must apply the cream before one obtains these results. The cream is also said to sweep away grey weariness. I couldn’t help but wonder if the Header were supposed to conclude that skin turns grey when it is weary or not. At any rate, Madame Hubenstein’s claim that her cream will wipe away weariness leaves us to conclude that this cream, when applied, will immediately give its applier renewed energy. The advertisement goes on to say that the cream animates the complexion into a lasting radiance. Before we can accept this statement, we must know Madame Hubenstein1s definition of the words lasting radiance. On reading further, we find that the cream guards against lines. Prom this statement it seems logical that if one wears this cream constantly, one will have fewer lines in one’s face regardless of wrinkles which develop from habitual actions, such as laughing, frowning, and screwing up the forehead. The cream is also said to smooth away marks of fatigue, but leaves us to imagine what the marks could be. ALso, the reader wonders what the maker would consider fatigue. If the statement that the cream, when removed, leaves the face clear is true, then it follows that the cream must also remove blemishes, blackheads, and other skin defects. • • Another advertisement by the same girl is as follows! "In these, as in every Mayflower dress, you’ll find all occasion smartness... a flare for fit, a talent for tailoring, a clue to spring color ... the ultimate in value and smartness at a budget “price. Depend upon the fabric quality ... checktested and approved for complete satisfaction in use. Left iris blue, desert rust, dusty rose, gray blue, aqua mist, navy. Sizes 14 to 42. Right - solid color skirt, Velveray striped top. In iris blue, aqua mist, desert ruse, gray blue. Sizes 14 to 20. $6.50." 193 Undefined terms in this advertisement When the makers of Mayflower dresses speak of the flare for -fit they must have a very unusual make "because they evidently are taking into consideration all the unusual shapes of figures. Also mentioned is the fact that these dresses are the ultimate in value and smartness at a budget nr ice. We must first compare notes on what is considered a budget price. The advertisement states that these dresses give complete satisfaction, in use. It carefully avoids mentioning the lasting quality of the dresses. The dresses are said to be appropriate for miss and matron. Again the advertisement overlooks the fact that there are many different types of women included under the two mentioned. Another advertisement. "Feet tired at the end of the long day? Fains occasionally shoot up your legs? All you need is a pair of smart-looking Wright Arch Preserver Shoes with their four patented features. Metatarsal and long arches of feet receive scientific support. Plat forepart, crosswise, eliminates pinched toes and squeezed nerves. Arch, as well as foot, receives individual fitting. Visit your nearest dealer. One hundred and twenty-two models say, ’Weill keep your feet from aching] 1 Or write for free foot test." Undefined terms in this advertisement The makers of Martin’s shoes seem to be confident that all foot ailments are caused by shoes that ere badly made. They next assume that their shoes will cure the foot ailments of all people, in spite of unusual sizes, etc. .. Further down in the ad, it says that the arch receives individual fitting. It seems very improbable to me that one model of a shoe could be satisfactory fit for all the different types of arches. The ad then says, 'One hundred and twenty-two models say, "We will keep your feet from aching"'. It doesn't matter how many types of shoes this company produces, as far as I can see. The makers have no right to claim their shoes will stop feet from all types of aching. The same student includes this analysis of an advertisement** " •Whites1 are in season again]" But do you know exactly when and where it's correct to wear white shoes? Or two-tones? Or crepe-soles? Or plain tips? 194 "You can learn, quickly and authentically, with a glance at our Jarman Styjbe Charts. As published every month in Esquire Magazine, the Jarman Style Charts give accurate infor mation on which shoes to wear with what. "These helpful charts are exclusive with Jqrman. Only at our stores can you find them. And only at our stores can you find the extra value, the style leadership that is evident in every shoe in the Jarman line, "Come in today. See our complete line of distinctive Jarmans. And let us give you an authoritative check on your suit-shoe com binations for the coming season." In the ad above, which is for men's shoes, I think that more information is needed for you to determine whether these style charts are authentic and give accurate information. Also if Esquire Magazine is an authority on men's fashions. Another advertisement. "Perfect All Purpose stocking with true crepe twist con struction, which means more elasticity and greater resistance to common snags. Start your budget stretching now with this beautiful long-lived hose. In summer's most popular shades." In the above ad, I question the words "perfect all purpose", especially the word "all". Certainly these stockings cannot be worn for all things and still be a beautiful long-lived hose." Critical Analysis of Piction Another interesting excerpt from a project is one which deals with a critical analysis of a "Photocrime" murder story, which this girl calls "Panic Has Ho Eyes". Briefly, it is stated as follows? Eeba wept wildly as Inspector Hannibal Cobb examined the body of her husband, Paul Horner. He lay half in the hall and half in Heba's bedroom, his feet near an overturned chair. An automatic lay a few feet to the dead man's right on the floor. Two years ago wealthy Paul Horner had wooed chorus girl Eeba Hand. How she was a leading actress. She told how they had settled down to business, how happy they had been, until Paul became infatuated with his secretary, Ann Porter. At first Eeba said she had ignored it, but lately Paul had been living apart from her*. Last 'week she had gone to Ann's apartment, and in a frenzy at losing him, beaten and clawed Ann. Tonight Paul had called. On her knees she had begged him to return. He refused, said his love was dead. 'Crazed at 195 losing him, X said I would kill nayself. X went to my room and got my gun. Paul., followed, grabbed my gun. It went off twice. I felt him slump in ray arms. I tried to hold him tut he fell, where you found him. I telephoned the police, then fainted. I came to as you rang the "bell. Nothing has "been moved.' After Horner's body was removed a chip out of the baseboard caught cobb's eye. Inspector Cobb's Solution Had Heba's story been true there would have been blood on her pajamas (from having held Paul a moment in her arms). There was none. The impression of Paul's bloody hand (Photo 8) pointed in the opposite direction to that in which he was found (proving he had been moved after falling). Keba shot him in a jealous rage in the hall outside her room. Then, thinking it would be more logical if the fictional struggle had taken place in her bedroom where friends knew she kept the gun, she turned Paul on the hall runner (using newspapers which she later burned). A beauty-proof jury found her guilty of second degree murder. Faults I found in this conclusion This murder is placed on Heba without enough facts to support the conclusion reached. Inspector Cobb bases his conclusion on the assumption that had Paul fallen in Heba's arms, her lounging pajamas would have had blood stains on them. He overlooked the fact that heba's pajamas might not have been bloody had Paul fallen backwards. Paul was shot in the left shoulder and below the heart. The impression of Paul's bloody hand pointed to an opposite direction in which he was found, proving that he had been moved. Still, his death might have been suicide. He might have fallen on his face, and his hand might have hit the floor first. Haising himself to his knees, he might have tried to get up, and, too weak, 'fallen backwards. In this way his body would be turned the opposite direction of his hand. Inspector Cobb mentioned no fingerprints, but even if he had, Heba's argument was foolproof, as she admitted tailing the gun in her hands to kill herself. This photocrime is ridiculous because its conclusion is based on shaky assumptions, and it mentions a piece of wood chipped from the baseboard, but the piece of wood, brought in at the last, in an important manner, evidently had nothing to do with the case ... Cobb's telegram to the police commissioner follows! "Police commissioner: 196 Eeba Hand claims husband Paul Horner accidentally shot in left shoulder and below heart during a struggle for auto matic. One button torn from her rumpled but otherwise immaculate and costly blue lounging pajamas; bruise on right index finger and left third finger. Claims Horner grabbed her in bedroom as about shoot self through despair at losing him. Asserts she held Horner in arms moment while he said quote Goodbye, Eeba. I'm glad it was me unquote. Accompanying photographs and this report reveal heba's two mistakes made in unreasoning panic. Booked her for murder." Critical Analysis of Political Speeches Another boy brings in the following introduction to his project which he developed in connection with his "textbook". It is as follows: A Preface to Political Speeches You may be wondering how political speeches got into a geometry class and what bearing it has on geometry. The way they got in is that our teacher brought them in. But they really have quite a bearing on geometry because they show why you need definitions. If the speaker says he111 do something that sounds good, but you don't know the meaning of, he has achieved his point. His idea is to say something so that it will make you vote for him, but maybe isn't so good after all. This is where it enters into geometry. It shows that to understand fully what the politician is saying these phrases must be cleared up so that you can understand them. .. Logical Seasoning Another boy's project, which deals with a variety of things in connection with reasoning, developed into a very interesting report, from which some of the excerpts are as follows? In this geometry class we are taught logical thinking, that is,we must learn to analyze statements, political platforms, pleas to juries and what-not. This teaching prepares us for later life when perhaps we might be obliged to serve on a jury, become a candidate for a political position, or later become a lawyer. We then, after analyzing the given statement or statements, as the case may be, agree on a conclusion. We then formulate reasons why our conclusion is logical. In some statements there need not be just one logical conclusion, but many, which are equally good if you accept the assumptions as true and have logical reasons for your conclusions. 197 The following statements should he accepted as true. These following statements give you an example of logic when the state ments are tnue, "but the conclusions although logical, is wrong and untrue, The mistake in this logic is in the use of words which have two or more meanings, following are a few examples of this. The statements? Feathers are light. Light comes from the sun. Conclusion! Feathers come from the sun. •Although you see that my conclusion is the only possible one from such a statement, it is not true. The reason for this is the word light. Light is used twice hut does not have the same meaning hoth times. This then is the misuse of a word with more than one definition. Below is a statement that a man made. “There is no Truth.11 First we wish to know what he means hy truth because there are a variety of meanings for this word. Then if we accept this as true there would he no truth and because of this the statement would he false. Therefore, there is no truth. Below is a statement that a man made while giving a speech. Statement! How can you doubt the world's greatest physicist believes in God. Conclusion! You must believe in God. What right has he to make this statement in the first place? A physicist is in no better position to say there is a God than anybody else. Statement! Ho cat has nine tails. Any cat has one tail more than no cat. Conclusion! Any cat has ten tails. The conclusion is wrong although if you accept the first two statements as true, then the conclusion is logical. The mistake comes in where 'no cat* is used twice but with different meanings. 198 3. Voluntary Contributions This type of content consisted of critical analyses of such items as advertisements, editorials, speeches, books, and cartoons not included in the projects. Each item submitted by a pupil was strictly voluntary on his part because this procedure was not a compulsory requirement. Out of the 333 Experimental Group pupils nearly 40 per cent contributed items which they felt would be of interest to the group. contributed 30 items or more. Ten per cent Since 2,987 items were contributed, the average number per Experimental Group pupil is 9. Table XI was devised to present a compiled summary of the items and a distribution of the different types of items. TABLE XI. DISTRIBUTION 0E VOLUNTARY CONTRIBUTIONS Rank No. of Items 1. 1989 Critical Analyses of Advertisements 66.5$ 2. 492 Critical Analyses of Current Events 16.6$ 3. 235 Short Essays on Speeches, Books, Articles 7.9$ 4. 147 Critical Analyses of Editorials 4.9$ 5. 124- Critical Analyses of Cartoons 4.1$ Type of Item Per Cent of Total Illustrations of critical analyses of advertisements voluntarily contributed are as follows! Item 1! While riding on the North Shore train I noticed an advertisement of the North Shore and Milwaukee Railroad. It read as follows! "Walking time from anywhere in the Loop to a North Shore train." An analysis of this advertisement is as follows: The obvious motive of the advertisement is to encourage more people to ride the North Shore to and from Chicago. 199 The terms needing definitions are walking time, and anywhere in the Loop. Are they taking into consideration that some people walk slower or faster than average? Also ©re they taking into consideration that anywhere in the Loop might "be on the top story of a 36 story "building, and that it might take 10 minutes to get out of the building. The advertisement is obviously not a fact and could be accepted only by one who does not analyze the words. While riding on the train I called the attention of a friend to the advertisement. I asked him if he saw anything wrong with it. He read it and said, "Ho, What?s wrong with it?" I explained my point and he agreed with me that the term anywhere was used quite loosed and would have to be defined. Item 2! "I Have a Telephone" "True, this executive has a ’telephone1, (l) But to fight today’s business battle effectively, he must be more fully equipped. (2) Complete business control requires dual telephone facilities one system for handling in-and-out city calls: the other, a F-A-X system for quick accurate automatic interior communication. "(3) Such ’double-track' service relieves your switchboard operator of inside traffic - keeps outside communication channels open. (4) Customers never have to be put off or called back. With P-A-X you can secure information instantly, while your party hoibds the wire. "(5) P-A-X is much more than an interior telephone system. Through its 'special services* you can hold conferences — page persons by automatic code call - give emergency alarms - supervise watchmen in short, direct every activity of your company without qnce leaving your desk.* "F-A-X is made by Automatic Electric Company - originators of the automatic telephone and for 25 years specialists in urivate telephone communication for business and industry. Organizations now using P-A-X number well over 2000. For the reasons behind this widespread use, write American Automatic Electric Sales Company, 1033 W. Van Buren Street., Chicago, or consult your local electrical wholesaler." The words underlined are those I question. The motives behind this ad are to get business men to install a F-ArX, par ticularly those men who already have one phone. The first and second sentences are assumptions as well as the fourth. The third sentence sounds logical, while the fifth contradicts the third. I've indicated these with a large circle around each. 200 Item 3: Statements That Make Assumptions "AHGQBA KHIT-TEX $35. Its fabric is "based on the soft, silken hair of the Angora goat which lives in regions where the days are hot and the nights cold. Therefore* the protective hair of this animal which is used in this coat is adaptable to varying degrees of temperature." This article makes the assumption that the hair of the goat will have the same qualities after it is dead as when it is alive. Item 45 "For All Occasions, All Kinds of Weather 10 Months in the Year." What kind of unusual weather do we have the remaining two months? All kinds of weather includes twelve months. Item 5J "MOTHER - Lucky for you. Your child wants Ralston!11 “You see, Ralston does for your child what no ordinary cereal can do! IT'S HOT! More satisfying. More sustaining. A better way to start the day! IT'S TOOLE WHEAT! Provides minerals for strong bones and sound teeth - proteins for healthy flesh - carbohydrates for body heat and energy. IT'S DOUBLE RICH in the important Vitamin B - the vitamin that aids digestion, creates eager appetites, helps prevent nervousness, promotes all-around growth. "Order Ralston. Encourage your child to eat it regularly. Cooks in 5 minutes. Costs less than 1 cent a serving - a small investment for such a big return in health and happiness!" Above, you see an advertisement which with a small thrilling story accompanying it appeared in the comic section of the Chicago Tribune. I presume their intentions of advertising it in the funny paper were to interest small children to read the adver tisement, also the contest which is included, and to go to their mothers to get their opinion. The advertiser knows that by now the child is completely sold on the idea of eating Ralston, not because he likes it or knows how it tastes, but because he would like to enter the contest and receive a prize. How the only thing that remains is to sell the product to the mothers, so they put this article in, as shown above. If she were a wise person, these thoughts would enter her mind while reading it. It tells only that Ralston is good for her child and nothing about adults, so that it would not be worth while to buy it just for her child. It also says that it costs less than 1 cent a serving; but how much is a serving? Looking in the dictionary you will find that a serving equals a pocrtion, 201 and a portion of anything is a parcel. A parcel as defined in the dictionary means a fragment, or a small detached por tion of anything. Summing all this up, it might take four or five of one cent servings to equal the definition of what she pictures as a serving. I question whether this cereal or any other eereal is worth purchasing if contests are necessary to persuade people to "buy the products. Item 6; "fascinating Lady (Until she smiles)" (Ehis conclusion is "based on the assumption that her smile detracts from her fascination. Item 7i "Listerine for the positive relief of Dandruff" Since Listerine may take the dandruff from some hair, they assume that it will take it out of every"bodyfs hair. Item 8: "Waltham, first -American Watchmaker." I would want more than the Waltham word that they were the first watchmakers in this country. (There may have been others. I would like to have them define 'first1. Item 9! "Only $1.00 a week for my Corona ... and it got me a $2.00 raiseJ Everywhere, young women and young men are finding wavs to earn extra income with a Corona, outside or inside their business offices. Are you applying for a job? Your application should be typed, in neat business-like form, ire you on the road? Send in your reports typewritten, and see the difference! faking work home, or studying to improve yourself? Just figure how much a Corona will help! "Corona, the first successful -portable (over two million in use today) is likewise the ONLY -portable with the effortlesa 'floating Shift*. Saves work, saves time, ensures more accurate alignment. "If siyyear-olds can learn quickly to use a Corona., you certainly can!" In connection with e Corona ad it was stated that 'If sixyear-olds can learn quickly to use a Corona, you certainly can!1 I would like to know what they mean by the word use. It seems to me that a six-year-old child and an adult would find entirely different uses for a typewriter. Item 105 "Oldsmobile - The car that has everything" The term "everything" should be defined because it might mean the body or chassis alone. Item 11: On the pictures on the opposite page they are advertising lux Toilet Soap and Lifebuoy. Both of these soaps are manufactured by the same concern, yet both claim that they are good for the skin. This very fact shows that it can't be as good as it says it is. for one thing, movie stars advertise Lux, and since many have beautiful skins anyway, it doesn't prove anything except that they were paid very highly for their advertising. Item 12: In regard to that newspaper clipping about using the funny papers for advertising cigarettes, etc., I think this clipping is both for and against the argument. Parents and children both read the comic weekly. The adver tisements influence many children to get their parents to buy that certain brand. (This is based on the assumption that his article is true). It does not necessarily mean that the children will want to smoke the cigarettes advertised there. But, because the subjects of these advertisements are made up as heroes, it might make the children want to smoke them, reasoning out in their minds that if these people smoked them and became heroes and heroines, why would it not be possible for them to be heroes also, if they smoked them. Of course, this is based on the assumption that the average child takes to heart everything he reads. By the average child, I mean anywhere from 7 to 12. On the other hand, it might make the children want their parents to smoke the certain brand and, as in this case, (in the clipping) parents nine times out of ten will try to please their children. Also, none can deny that advertisements in the form of stories attract the eyes of the elders (say 18, on up) and make them very apt to smoke the certain cigarette or eat the food just for a trial and then stick to it. In the clipping it shows the statistics, which show a definite improvement, and I am inclined to think that this is true because in a foot note they say they will give you the actual name of the advertiser upon request. Of course, it depends on the manufacturer if it was small, they might be paid to say this but if it was large I do not think the company would bother because the price they 203 would probably make would not make it worth it* The very fact that the name is hidden, but can be had on request, practically proves, in my mind, that it is true. Item 13! ••The Tires that ITeyer Uear Smooth11 They are making an assumption that tires never wear smooth. But if you run long enough the tires will eventually wear smooth. Unless, perhaps they are implying that it will blow out before it gets smooth. It® m 14* ‘•Sorry, we want a younger man. ** Assumptions made by the makers of Kreml are that! 1) Men who have bald heads have trouble in securing a job. 2) Kreml removes dandruff. 3) Kreml stops falling hair. 4) Bald-headed men look older* 5) Kreml stops itching. 6) Kreml keeps hair neatly groomed without giving it a slick look. Terms that need defining! l) "Satisfied users". Item 15! "Among busy people, the pause that refreshes with icecold Coca-Cola, is America's favorite moment. It sends you back to work refreshed." This ad assumes that! 1) 2) 3) 4) 5) 6) 7) 8) 9) Item 16! People are busy. People pause once in a while. Coca-Cola is refreshing. American people drink Coca-Cola* Coca-Cola is served ice-cold. People feel refreshed after drinking Coca-Cola. People go back to work after a moment's pause. People enjoy cold Coca-Cola. Ice-cold CocarCola is America's favorite moment. "The Beer That Made Milwaukee Famous" In this statement they are assuming! 1) Milwaukee is famous. 2) A beer made it so. 204 Item 17: This advertisement announces a 'beautiful de luxe holder FHEE in large letters. To one side and in smaller print you find you must "buy two boxes of Kleenex to obtain it. Item 18: ••We had these frames made up to our own careful specifications, and they are t6 be had only here. Beautifully finished - backs and easels completely covered, rings for hanging. Five sizes.11 They are assuming because they had them made to their specifi cations and because they can be bought in their store only that these are better frames than can be purchased elsewhere. Item 19: ••Concentrated Super Suds in the blue box gives me the cleanest whitest wash I*ve ever had.1 And it removes most germs, as well as dirt ... gets my clothes really clean - hospital-clean.1” Assumptions are that: 1. 2. 3. 4. 5. 6. Super Suds comes in other colored boxes. If it works wonders for one, it will help everybody else. There are germs and dirt in clothes. Hospitals are noted for being clean. The person talking has used other soaps. Ho other soap will make clothes so white and clean. Item 20* One can't tell if this is a good ad or not. The first thing you see on the paper is •FHEE*. The things that are free are watches, each of which is supposed to cost sixteen dollars. Implication - You must purchase $39.50 worth of furniture. How could a $16 watch be given free with a purchase of $39.50? It would take a lot of mathematics to figure that out, or are you being overcharged? It says also that you save as much as sixty per cent on new radios. That would probably net them a loss, unless the regular price was extra high. My conclusion is that a furniture house which overcharges you is usually unreliable. 205 Item 21! This ad states specifically that the radio is $54.95 with the automatic tuning. It also says that you are guaranteed to get Europe. Implication - The short wave is five dollars additional. You have noticed the price on top and that Europe was guaranteed, hut did you notice that to get Europe it costs five dollars extra? Item 22* "Avoid Irritating the Smoke Zone. Spuds are soothing,11 What do they mean "by "Smoke Zone"? Assumptions! 1. Spuds do not irritate the throat. 2. Other cigarettes irritate the "Smoke Zone". Item 23! "Eor digestion's sake - smoke Camels". Assumptions! 1. Camels do not impair digestion. 2. Camels help your digestion. Item 24! With reference to an'fe^-Lax* advertisement, onepupil that the'Ex-lax* people points out imply that their product meets thefollowing tests! 1. 2. 3. 4. 5. 6. It is dependable. It is mild and gentle. It is thorough. Its merit has been proven by the test of time. It is not habit-forming. It does not over-act. It does not cause stomach pains. 8. It does not nauseate. 9. It does not cause upset digestion. This pupil concludes that these statements would need to be proved before he could accept them. Item 25! In connection with 'Tomasco' Quality Elying Model kits, there is a label attached to each kit which contains the following statement! 206 "Contents of this kit guaranteed only when this seal is not "broken." This is a clever way of stating their guarantee as it practically eliminates all possibilities for replacement on their part. The buyer has to assume that the contents were thoroughly inspected, and that the article was not "broken in the process of packing or shipping. It would he interesting to open the kit in some way without breaking the seal and then see if the company would replace a defective part. Illustrations of critical analyses of current events (voluntarily contributed) are as follows! Item li "COURT SPOTLIGHT OH JUDGE BLACK IN SESSION TODAY" "JUSTICE BLACK SITS IN OH DENIAL OP WIRE REVIEW" ■Assumption* People watch Black more, especially because of his once belonging to the Ku Klux Klan. Item 2** "LAG IH BUSIMESS ALTERS SPECIAL SESSION COURSE" "Battle Lines Porm against Roosevelt Policies" Writer assumes that! People are just beginning to disagree with Roosevelt. People were with him until now. People believe the lag in business is due to his policies. Item 3! "READY TO SAIL, HE RINDS HIS AIMS MISUNDERSTOOD. U.S. Press Caused Plight of Windsor, British Claim. Denies He backs Any Industrial System or Racial Doctrine." This incident shows how a person, like the Duke of Windsor, may in reality want to do the right thing, but because of his failure to make other people see his views, all his aims are misunderstood. Item 4! "SLUMP IH STOCKS CHECKED" In this statement you assume that the stocks will not sink any lower. Item 5! "JAPAN*S HEED OE MONEY AIDS PEACE EPEQET" 207 Ehis whole sentence or statement is an implication, hut "Japan's Need of Money" is an assumption. You assume that Japan needs money. It is implied that since Japan hasno money it can't continue the war. Thus, we have peace. Item 65 "BRITAIN TO SEND 3 BATTLESHIPS TO CHINA Y/ATikS London Hears U.S. will mass fleet in Pacific." Implies that British and American people are working together and their opinions are the same, also that England wants America to punish Japan for the sinking of our ship. Implies that Japan may try to turn China into a preferential area. If they send their ships it also may mean war, also that British powerin the Mediter ranean and North Sea is greater than other countries and Italy and Germany would like the power. Item 75 "LOUISIANA GIRL IS CONVICTED OP GOSSIP SLAYING" I Believe that the jury on this case, which was composed of members of the county, might he prejudiced toward Jessie Pepper. Gossip spreads quickly in a small town. The members of the jury might have heard something about her previous to the crime and therefore be prejudiced toward her. Their feelings toward her would have a lot to do with their verdict. Item 85 "PONDER NEW CHALLENGE OF BLACK'S RIGHT" I believe that those other eight justices of the Supreme Court might be prejudiced toward Justice Black. Since the news that Black was a member of the EU KLUX KLAN was disclosed, the Supreme Court and President Roosevelt have been ridiculed from all sides. Item 9 5 "A small square bottle, containing a quarter inch of white substance was introduced in evidence today at Anna Marie Kahn's trial for the murder of Jacob Wagner. The bottle was found in Mrs. Hahn's basement by Det. Capt. Patrick Hays after Mrs. Hahn was implicated in what the state calls 'the greatest mass murder in the history of the country*. A big, husky man, well over six feet tall end weighing 200 pounds, Capt. Hayes said under direct examination he found the bottle, which the prosecution charged in its opening statement contained arsenic, partially hidden in the rafters in the stairway leading to the cellar in the house where Mrs. Hahn lived." The bottle is the key to the case. 208 Item 10: 1S 00 WINE OHS WIlT33SQttS? CITY JOTITTEB." Assumed* The Windsors will come to Chicago and will accept invitations. People are excited about it. Illustrations of essays on speeches, books, and articles (voluntarily contributed) are as follows: Item i: A report was made by a boy who experienced a great deal of difficulty in mathematics with reference to "Berkeley's Theory of Space". Space is a mental construction due to the gradual coordination of sensations, especially of sight and motion: this is Berkeley's view and the usual conception of recent psychologists. It is held in two forms, some authorities maintaining that space is merely a product of the coordination of sense, experience, others that it is a quality of the sensations themselves. This same boy follows this with an attempt to give his own version of space, as follows: What I Think of Space In talking of'space' we know what we are talking about, but we are not able to define it. The reason we cannot define it is that we know nothing about it. Nobody knows much about it, except that it is there. Somebody might ask, "Y/here?" and most sane answers would be, "Everywhere". I do not see why space would have to be bounded by lines. The infinity of space may have no boundaries at all. Straight lines and curved lines, etc., may work on the earth, but if there are no boundaries to space how can there be any lines on the infinity of space? Geometry is very largely based on space. Triangles, rectan gles, quadrilaterals, etc., have space between their boundary lines. Space is also on the outside of the boundary lines. Even artists when they paint usually use the three dimensions. There are three kinds of geometry, rectilinear, elliptic and hyperbolic ... The men who worked out these geometry theorems based them on their own theory of space. I have talked to a friend of mine on the way to school about space. He said that "space is just nothingness", meaning that there is not anything there. But there is something there. If you think, you will know that nothing is something. It would be like trying to define suase. 209 Item 2* -Another pupil wrote about "Easy Lessons in Einstein" as follows? Unfortunately, I had only time enough to read one chapter from an intensely interesting hook, namely Easy Lessons in Einstein "by E. E. Slosson. I opened the book and it so happened that my eyes fell on page 15. The heading of this page was ‘Paradoxes of Relativity*. Seeing a few words and sentences that looked familiar from our class discussions, I began to read. After the third sentence, each line became so involved that I took time to stop and think at each period. I have often h&ard about Einstein*s ideas, but never imagined I could come near under standihgthem. There were a few things I was particularly interested in. Einstein says parallel lines may meet] According to Uewton, the action of gravitation is instantaneous through all space. Einstein now says, 'Uo action can exceed the velocity of light.1 The book went on to say, "If the theory of relativity is right, there can be no such thing as absolute time or ways of finding whether clocks in different places are synchronous." All of his thinking is critical, I believe, bechdse he would already have had to think to question Euclid and Hev/ton, but no4r to work out a new way which has proved to be more accurate is indeed thinking.1 In other words, to go beyond what others have done requires critical thinking of the highest form. He became very critical when he stated that "a yard stick may vary according to how we hold it and the weight of a body may depend upon its velocity." Another statement he has made is, "The sliortest distance between two points may not be a straight line." All of these are a part of Einstein's theories of relativity. He did more than just make these discoveries as assumptions? he made them into mathematical laws and they have been proved by tests. From all of these theories, it is pointed out how impossible it is to tell if you are moving or not, when on a train or boat, unless you are sure of some object's being stationary. He also points out from Herbert Spencer's illustration, the sea captain who is walking west on the deck of a ship that is selling east at the same rate. The question is? Is the man moving, or not? If you are on the same boat, you would say yes, but if on the shore when the ship is passing, no; he's "just marking time". I believe 210 that it all depends on theposition you are in; however, "book said, "It all dependson the point of view.11 the Another point that he brought out was how useless it is to make statements about timeand space as being absolutewhen we have seen neither. It made me realize even more what a critical thinker he must be when he asked the question, 11If a yard varies in summer and winter, how do we know it doesn't change in length when we hold it upright or lay it horizontally." To my knowledge we have no way of ever finding out, but who would have been critical enough to recognize it but Einstein? Item 3* A Hews Broadcast A few nights ago I was listening to a radio broadcast on the "Hews of the World". The speaker was talking about the Sino-Jap war. He said words to the effect that "The British are now kicking to the Jap government about bombing noncombatants in war, for they say it is an inhumane thing to do," But he says, "Look back ten to twenty years when they were having trouble in India; they went back into the interior of that country and bombed a whole city because they thought a rebel leader was hiding there. They did the same thing that they are raising such a roar about today." This to my mind is a very weak argument unless you make the assump tion that civilization stands still and does not move forward. The way he is reasoning, just because you could shoot an Indian one hundred years ago and not get punished for it, you can do the same thing today. Item 4: What Are Grades? Why is it that sometimes John gets poor grades? Grades are a poor measurement of one's ability and standard in school work, I think. In class he is timid and is afraid to volunteer or give any of his own ideas. He also is afraid to ask questions for he might be laughed at or scorned by the teacher. When he is asked a question he cannot organize his thoughts well. In doing a test he becomes mervous and can hardly put anything into writing. It is hard for him to get the subject. He doesn't understand the subject or the teacher. There are other people who just naturally do not have to study very hard, for the subject comes easily to them. There are still others who head and lead in sports and offices but have no time for their school work. If there weren't any grades we would also avoid the competition to work just for grades 211 alone and perhaps we would get more out of our education. I don't "believe we will "be able to do without grades unless the colleges would he willing to accept anyone. ■ Perhaps John would also lose some of his nervousness end shyness. Item 5• In my English class a few days ago, a girl, while reciting, remarked that she thought Latin helped her in studying ancient history. Because of the following statements I will show her statements to he false. First of all, the majority of persons taking Latin their first year also include history, which I found out was true in her case. In your first year of Latin you learn more about the fundamentals of the language and do very little important trans lating. In your second year of Latin you begin the translating of Caesar and other great leaders of Rome at that time. It is already too late for your Latin to help you in history because you have completed it, but your knowledge of history will now be able to help you in translating Latin, thus contradicting her statement, and showing that the knowledge of ancient history enables you to translate Latin quicker and easier. If it is not clear to you why history helps in translating Latin, it is because you have a good idea of the story and will know when your sentences make sense or not. Item 7i An Illustration of Clear Thinking A university graduate who prided himself on his wonderful education was talking to a farmer boy who was equally proud of himself because of his clear thinking. The university man was telling him about a great chemist that discovered a chemical which would dissolve anything. The farmer thought about it for quite some time and then asked, "Where does he keep it?" This little story shows that though the university graduate was very well educated, he didn't have the inquiring nature, and took everything that people told him for granted. The farmer, though he didn't have an education, thought clearly and looked into things before believing them. Item 7: Another pupil wrote a brief sketch regarding assumptions, impli cations, and prejudice. 212 Assumption Every statement of any type is “based on one or more assump tions. If you accept a statement as true you must also accept its assumptions as true, For a statement is only as true as its assumptions. Take the following for an example. (From Assump tion Sheet) "This coat is a "bargain because it is so cheap." The assumption behind that statement - if a thing is cheap it is a bargain. If you accept that assumption as true you therefore accept the statement as true. Implication An implication of a sentence is something that is hinted at. Take this statement for example. (From Assumption Sheet) "I am voting for the Socialist Ticket because I want equality of oppor tunity for everyone." The implication would be! by voting for the Socialist Ticket I will help gain equality of opportunity for everyone. And the assumption underlying this statement is! the Socialist Ticket standa for equality of opportunity for everyone. Prejudice Y/hen speaking on almost any subject the problem of prejudice will come up* For behind almost everything there is usually some prejudice. Y/hen talking about the recent Sino-Japanese war, some people might support China because they are prejudiced against Japan, and vice versa. Even when discussing our own 1National Politics*, there may be people who are prejudiced toward President .Roosevelt or certain other men in today’s politics. Geometry is one of the few subjects we can discuss without having the idea, of prejudice behind any statement. Item 8! Modern Publicity. Edited by F. A. Mercer and Gaunt This is a book on Poster, Press Advertising, Direct Mail on packages, and how they should look, be set up, etc. (Poster Number l) is the Normandie. Y/hen you turn the page the boat seems to come out at you. The picture shows the Normandie to be large, clean, and fast. (Advertising a car) The car seems to be rushing along, giving it a streamlined effect and increasing the speed. (A tree with the title ’plan your home') This is very good because it is a sturdy tree and you want a sturdy home. It is a peaceful tree too. A poster for a zoo with a hippopotamus standing on the word "zoo". This is a very catchy sign. Item 9* Advertising to Women, "by Garl A. Neether I enjoyed this book very much and although I didn’t read it all I found what I did read very enlightening. On the first page of the introduction it says that women buy 96 per cent of the dry goods, 67 per cent of the good stuffs, and 76 per cent of theautomobiles. I always thought that the men bought the automobiles, Now I think I see why car manufacturers advertise style and beauty rather than clutches and gears. One chapter in this book that interested me especially was the one with the catch slogans. One is "Yftiy not a ’vacation’ for wives", and it advertises electric vacuum cleaners. When the lady looks at this and sees the word ’vacation1 she immediately, out of self pity, wants one, which is just what the company knows she will. If she already has one, it merely means she probably got caught last month or the month before. Another question for advertising is, "Are your gums suffering from a life-long slumber?" The person reading this begins to wonder if her gums are "suffering from life-long slumber". Perhaps she brushed her gums the night before with some body’s toothpaste and today they ere sore. Do you suppose the average reader will recall this? No, she will rush out end buy another kind that will "wake up her gums". "At last I know that 'a hidden well of poison' is a cavity." A title spread across a page such as this would make anybody stop. An advertisement reads, "Every woman is a rainbow". This is direct flattery and the woman reads on to see why. It caught me, too, and it's because she uses Babani perfume. I found that to advertise almost anything you should use a picture of the person using or wearing the article and seemingvery happy about it. The whole theme through the book was to make the women "get their men" by buying this^i that, and something else. Item 10! The Tyranny of Words % impression of the story is that our teacher was trying to tell how many people use words that they know little or nothing about, or words the meaning of which is not agreed on by the majority of the people. He gave a good example of this by using A speech Hitler made. In place of every semantic blank the word ’blab’ was placed. There were more blabs than words left in the sentence. Our teacher said the more one thinks of words and their meaning the more befuddled one grows. I agree with him on this 214 statement heartily. He says that all languages are taken too much for granted. He also said that v/hen lie read. philoso'Qhy the great words went round and round in his head until he "became dizzy. He couldn't understand whether it was his fault or that of the language itself. He thinks that there are three classes for names of things! labels for common objects, labels for collections of things, such as mankind - labels for essences and qualities - freedom and truth. He feels that a great deal of our misunderstanding in economics or politics is because we are not able to define our terms. He cites some of the following recent examples. The merits of Tugwell vs Landon (1926), Supreme Court controversy (1937), He enumerates certain abstractions, such as the nation, government, constitution, Supreme Court, the law, flag, fascism, and communism. He ends with a happy note by saying people are not dumb because they lack mental equipment; they are dumb because they lack an adequate method for the use of that equipment. I enjoyed this article very much after reading it through several times, because it is so closely related to what we are trying to do in our class. I tem 11? "Hew locations for Youth'1 (A. speech) In conclusion to his analysis of this speech, the boy writes as follows? My chief criticism of this talk is that the speaker failed to emphasize the need for more creative thinking - the kind of critical thinking that will create new jobs for the unemployed. It em 12! List of other miscellaneous topics critically examined and voluntarily contributed is as follows! 1. ''Education for Living". 2. "Has the Day Always Been Divided into 24 Hours?" 3. An essay on errors made in newspaper accounts, as for example excavators of Egyptian pyramids finding mummies, etc., and "coin dated 350 B.C." 4. Can you think of anything lower than using the "funny papers" to try to influence children to smoke cigarettes? Points out the dual implication of the tobacco company and the newspaper. 5. When is a person insane? After citing many definitions of the 215 terra, this youngster arrives at the conclusion that "one-half of the people are sane and the other half are insanewiththe result that your guess is as good as mine with referenceto either half.” 6. It has "been stated that people -under the age of 21 or over 60 should not he permitted to drive. This boy brings in many fa.cts and concludes that a medical and driving test ought to be the basis for such decisions. 7. Did Brutus dothe right thing in murdering Caesar, or should he have gone against the conspirators? 8. Geometry in Engineering Drawing. 9. Need for Thtee Dimensions to Form Conclusions. 10. The Nature of Numbers. 11. Double Features in Movies. 12. The Status of Immigration. 13. Why Riemann Developed His Elliptic Geometry. 14. Why Our Grading System Should Be Abolished. 15. The Role of Definitions in Baseball, and Many Other Sports. Illustrations of critical analyses of editorials (voluntarily con tributed) are as follows! Item 1! BOOKS ARE NOT CULTURE LOYOLA SCHOLAR WARNS '■Culture11, he stated, "is an attitude of mind by which one seeks beauty in all things with which he makes contact. It is a conscious attempt to see the beauty of the world around us." There is a good definition for culture in this item. This man is inferring that not all people who read good books are cultured. Item 2! DEAD WAGE-HOUR BILL SOUGHT FASCIST CONTROL OVER LABOR Implies that people are getting tired of the Presidents methods. The bill about work and wages makes the employer a dictator. Congressmen are afraid to vote against Mr. Wallace when he asks for money from the treasurer. That the labor bill will not go far. The people agree to the farm bill but will not agree on the wa$e hill. 216 Item 3i “DUKE A DUD AT SUNNING HIS OWN LIFE", SAYS PEGLEd I think this is a very good example of prejudice. Fegler has no right to say these things, even if they might he true, because everybody can't be perfect. In this article, Pegler didn't mention anything but his bad points. Item 4! MAKES PUBLIC SENATOR BESSY'S Pi:IDE IN TENNESSEE NEIGHBORS Pegler's articles are usually statements of prejudice. Pegler points out that jihe man is a millionnaire president of a union and therefore a wealthy labor leader. To me this seems prejudiced because he always mentions the word "rich" as if a labor leader was not worthy of a high income. I suppose that Fegler himself is a poor man. Item 5i PALESTINE TO LIFT CURFEW TO EASE TENSION This is prejudice. they are prejudiced. The Arabs are attacking the Jews because Illustrations of critical analyses of cartoons (voluntarily con tributed) are as follows! (It would be desirable to include the cartoon itself in each of the following analyses, but for reasons mentioned earlier it is impossible to do so.) Cartoon No. 1! “The Octopus". In this cartoon the octopus is the political spender and his legs, which are all wrapped up around a man representing the public, represent the different taxes. Since the scene takes place under water, the pupil's comments are as follows! (l) This cartoon assumes that the taxpayer is already pulled under water by the taxes, and (2) it implies that the taxpayer will soon be drowned by the taxes. Cartoon No. 2! "Can't Talk" depicts a Japanese (a Jap soldier) standing before a table, "Fruits of Victory" and saying "m-m-mouth1s fulli m-mr-m" as two hands holding a telephone and receiver, representing the "Brussels Conference", are trying to hand him the telephone which represents "Discussion of Japan's Aggression". The pupil's comments are as follows! (l) This cartoon assumes that the Brussels Conference will 217 have no effect on Japan as long as she is victorious, and (2) it implies that Japan is too busy enjoying the fruits of victory to pajr attention to the Brussels Conference. Cartoon Ho. 3! “Says the General as he Leads Another Attack11, referring to President Roosevelt, the Chinese-Japanese wan, business, and taxation, one pupil remarks as follows: This cartoon implies that F.D.R. has enough to take care of in the U.S.A. without interfering with the trouble in China. Cartoon Ho*4« ’‘Still Depending on the Old Volunteer Fire Fighter” points out, as one pupil states it, "That the League of Rations assumes that the U.S.A. will help them out of their Chinese difficulties." Cartoon Ho, 5* “Beating Them to the Good Deed” pictures a ruffian, hearing the swasticka, leading a lady representing World Reajce across a street representing the Asiatic Crisis. The good hoy scouts in the background are the "Brussels Conference”. This pupil points out that: 1. Germany wishes to lead in world peace. 2. Germany wishes to control world peace. 3. Germany has no consideration for other countries. 4. Germany acts more quickly than other countries. 5. Germany wishes to settle the Asiatic crisis alone. Chapter Summary Since no appreciable effort was made by the Control Group teachers to include non-mathematical materials in their procedures, the content in this chapter was confined to the Experimental Groups. Some difficulty was encountered in describing situations that demanded inclusion of pictures, sketches, or cartoons. Descriptions were given of group projects which included content dealing with such materials as advertisements, court decisions, speeches, editorials, political issues, compulsory laws for education, uupil-school 218 relations, liquor legislation, socialized medicine, capital punishment, social security, unemployment, youth in a modern society, and instruments of "propaganda’1. Descriptions of individual projects revealed materials very similar to the group projects* 24,2 per cent of the individual projects dealt with critical analyses of advertisements, and 12.2 per cent dealt with other items, such as written articles, editorials, court decisions, and speeches. Nearly 40 per cent of the Experimental Group pupils elected non-mathematical projects in preference to mathematical ones. In fact, nearly 70 per cent of the Experimental Group pupils included some non-mathematical content in their projects. Nearly 40 per cent of the 332 Experimental Group pupils submitted voluntary written analyses of situations such as advertisements, editorials, speeches, books, and cartoons not included in the projects. Sen per cent of these pupils submitted 30 or more critical analyses of items that they thought would be of interest to the group. were voluntarily contributed. In fact, 2,987 items These ranged from a small paragraph to a five- or si:x-page typewritten assay. 219 CHAPTER VI EVALUATION Introduction I Up to this point the discussion has "been confined to a descrip tion of (l) the nature of the present study, (2) the Experimental and Control Group situations in their preliminary stage, (3) the teaching procedures employed, and (4) the nature of the content involved. The purpose of this chapter and the two following ones (VII and I VIII) is to present an evaluation of outcomes attained through the Ex perimental procedures and to compare them with outcomes secured through Control procedures* Outcomes to he evaluated include the following: 1. Knowledge of geometric facts; 2* Understanding of a geometric proof* 3* Skill in manipulating geometric content. 4. Pupil reactions to their course in demonstrative geometry. I ! ! 5. Heading interests. i 6. Abilities in critical thinking in hoth geometric and nongeometric situations. Evidence will he presented in terms of the ten elements of critical thinking developed in Chapter I. These elements are: | ! | a. Is this pupil trying to detect motives behind any situation? (By any situation is meant any situation of concern to the pupil.) b. Is this pupil questioning the meaning of terras in the situation and seeking satisfactory definitions or meanings of such terms? c. Is this pupil detecting and questioning underlying assumptions in the situation? (Stated as well as unstated ones*} 220 d. Is this pupil searching for more facts pertaining to the situation? e* Is this pupil testing facts for pertinency to the situation? f* Is this pupil endeavoring to discriminate between facts and assumptions in the situation? g* Is this pupil evaluating himself for bias or prejudice in the situation? h* Is this pupil testing conclusions for consistency with under* lying assumptions in the situation? i* Is this pupil recognizing the importance of formulating ten tative conclusions in the situation? j. Is this pupil evaluating conclusions in the situation in terms of individual as well as social consequences? Although evidence will he presented on each of the above, major attention will he given to those concerned with critical thinking* It is recognized, of course, that these outcomes are not separate or dis tinct, hut are interrelated* She problem of evaluating outcomes in terms of knowledge of geometric facts, or skill in manipulating geometric content, or an understanding of a geometric proof, can accomplished quite easily since there are a number of satisfactory examinations designed for this purpose* However, the problem of evaluating pupil attitudes and pupil behavior in critical thinking is far more difficult* As fyler^ has stated* 11It is harder to evaluate a continuously changing plan of education than one which is static. poses can be rather easily appraised* In the latter case educational pur The better paper-and-pencil exam inations give evidence of the information the pupil has memorized, and 71 Tyler, Ralph W., 11Appraising Progressive Schools11, Educational Method. Vol. XV, Ho* 8, May 1936, pp. 413-413. 221 of his facility with the skills involved in reading* writing* and mathematics* The evaluation of his development in terms of interest* attitudes* good thinking* and so forth, is much more difficult. * Bases Boon Which Outcomes ire to Be Evaluated Because of the Intricate and variable nature of the outcomes sought in connection with the elements of critical thinking* they can not he evaluated by prescribing a plan of uniform examinations or a series of standardized tests* In fact* no one method for evaluating outcomes in critical thinking abilities will suffice* because as Tyler?2 has Indicated* "An adequate evaluation involves the collection of approp riate evidence as to the changes taking place in pupils in the various directions which are important for educational development ••• It may include records of observations of young people* the collection of prodructs of their work ... and evidences regarding purposes"* which an Individual or a group may deem important in their considerations# The following techniques were used in evaluating the outcomes mentioned at the beginning of this chapter* 1 , Written Tests and Inventory Questionnaire* namely: (a) Orleans Geometry Feognosis Test (b) Columbia Research Bureau Plane Geometry .Achievement Test (c) Progressive Education Association Nature of Proof Test, number §3 (d) Test in Critical Thinking e) Periodic exercises and tests f) Inventory Questionnaire i 2* Direct Observations of Critical Thinking Behavior (a) Reports of observations by 165 classroom teachers (b) Reports of observations by experienced observer 72 Tyler, Ralph W.* "Evaluation: A Challenge to Progressive Education"* Educational Research Bulletin* Vol* XIV* No. 1* 1935* pp. 9-16* 222 3. .Anecdotal Records 4. Analyses of Written Work Other Shan Seats (a) English themes (b) Required written reports of reading (c) Voluntary written reports 5* Pupil Diaries 6* Analyses of Reading Interests 7* Individual and Group Projects 8. Self-Evaluations by Pupils IfcUjfffl-ffegtg. and Iw m W & SpptttvwaXr*73 She written tests and inventory questionnaire will be considered from the standpoint of both initial and final performance by the groups* because it is obvious that an evaluation of performance is dependent upon different stages of development* She following were administered for purposes of getting initial data* (1) Otis Self-Administering Test of Mental Ability (2) Orleans Geometry Prognosis Seat (3) Progressive Education Association Mature of Proof Sest number 5*3 (4) Inventory Questionnaire* She results were interpreted in Chapter II in order to describe ptq>il abilities in terms of these factors prior to their study of demonstrar tive geometry* She following were administered for purposes of securing infor mation at the end of the school year* 73 A complete* statistical tabulation of results from these tests and the Inventory questionnaire may be found in Appendix tt3ff* 223 (1) Colombia Research Bureau Plane Geometry Achievement Test (2) Progressive Education .Association Nature of Proof Test number 5*3 (3) Test in Critical Thinking (4) Inventory Questionnaire, Other exercises and tests were given during intermediate stages* 1. The Columbia Res earch Bureau Plane Geometry Achievement Test This test was administered to the Experimental and Control Groups at the end of the school year. Since it was designed to measure know* ledge of geometric facts* understanding of geometric proof* and skill in manipulating geometric content* it serves as a satisfactory means for evaluating these outcomes* Since it was found in Chapter II that the Control (koups were sig nificantly superior in mental ability and in geometric ability prior to their study of demonstrative geometry* there is every reason to believe that this superiority should carry over into geometry achievement at the end of the school year* It so happens that this is true* because the Control Groups were superior in geometric achievement* as indicated in Table XII* TABLE XII. COMPARISON OF EXPERIMENTAL AND CONTROL GROUPS IN TERMS OF I.Q. *S# GEOMETRY PROGNOSIS SCORES. AND GEOMETRY ACHIEVEMENT SCORES Item Intelligence Quotients Experimental Grouns Range Standard Mean _ _ JQeviation Control GrouDs Range Standard Mean Deviation 71-137 11*4 105*8 74-146 11.7 110.2 Geometry Prognosis 7-168 31*9 84.2 24-168 29.7 95.7 Geometry Achievement 0-77 13.7 26*4 7-70 14*4 31*2 234 Table XXX reveals the relative performance and homogeneity of the Experimental and Control Groups in terms of intelligence quotients* geometry prognosis scores* and geometry achievement scores* The Control Groups1 means for each of the items are higher than the corresponding Experimental Group means* Since the Control Groups were found statis tically superior to the Experimental Groups in Intelligence and in geomet ric ability prior to a study of geometry, it remains to be proved that the difference between the geometry achievement means is statistically significant* Specifically the Geometry .Achievement mean is 31*2 for the Control Groups and 26*4 for the Experimental Groups. The actual difference between the two means is 4*8* and this difference was found to be 4,4 times the standard error of difference between the two means, the standard error being 1*10* Statistically, the abilities of the Control (houps to manipulate geometric content turned out to be superior to those of the Experimental Groups. However* a close scrutiny of the amdunt of significance neces sitates some explanation and interpretation, particularly when the super iority in Geometry Achievement of the Gontrol Groups dropped 10*2 per cent in significance* ?or example, the outcomes in pupil ability to manipulate geometric content show a slight gain of the Experimental Groups over the Control Groups. This is made more evident by a comparison of ratios of the actual differences with the standard error of difference in each of the three factors* as follows* Otis X.CL D actual difference standard error of diff. Geometry Prognosis 1 Geometry Achievement D aa s 4*9 *90 s 11*5 s 3*4 s 4*8 s lil 4*8* 4*4* 225 The difference of 0*5 in significance represents a drop of approximately’ 10 per cent* The point at which the Experimental Groups gained over the Control Groups is not apparent in a comparison between the means of the Geometry .Achievement scores* In order to find a reason for the decline in statistical signifi cance of the Control Group superiority over the Experimental Group, it was found convenient to group the scores of the Geometry Prognosis test and the Geometry Achievement test into deciles* The data for this group ing were secured from the tabulation of data in Appendix ,fBn. Table XIII, therefore, represents the decile distribution of the Geometry Prognosis scores and the Geometry Achievement scores* TABLE XIII* Decile 1st End 3d 4th 5th 6th 7th 8th 9th 10th DECILE DISTRIBUTION OP SCORES ON THE GEOMETRY PROGNOSIS AND ACHIEVEMENT TESTS Experimental Groups Geom* Prog* Geom, Achiev* 27 28 28 27 27 24 26 29 38 35 35 25 34 45 41 46 40 36 46 29 Gctttfrvd Groups Geom. Pro£* Geom. ichiev, 39 38 41 36 33 38 31 23 28 19 44 42 39 36 45 22 27 20 28 33 In order to interpret the above table more accurately it seems advisable first to include Table XIV. Orleans74 says, "Table XIV indicates the expectation of success in geometry from pupil's standings on the prognosis test, provided achievement in geometry is measured by a com prehensive, objective achievement test* __________________________ 74 Orleans, J* B* and J* S., ''Orleans Geometry Prognosis Test", Manual of Directions. New York? World Book Company, 1927, page 4* 226 TABLE XIV* SHOWING THE EXPECTATION OP SUCCESS IN GEOMETRY ERQM STANDINGS IN THE GEOMETRY PROGNOSIS TEST Tenths 1st 2nd 3d 4th 5th 6th 7th 8th 9th 10th . 10th 1* 3 5 11 23 56 9th 8_th 7th 6th 1* 3 5 9 14 20 26 23 1* 3 5 9 13 17 20 20 11 3* 5 9 14 16 18 17 14 5 1* 5 9 14 16 17 16 13 9 3 Standimr in Achievement in Geom* 5th 4th 3d 2nd 1st 5# 23* 56* 3* U* 9 14 20 23 26 13 17 20 20 U 18 16 17 14 5 17 3 16 9 13 14 16 9 5 1 14 9 3 5 9 5 3 1 3 5 1 1 The table (meaning Table XIV) ie interpreted as follows* if the number of pupils is divided into tenths* then of those pupils who are In the highest tenth on the prognosis test* 56 per-.aenfcofeill^peobihlyl:be in the highest tenth in geometry achievement, 23 per cent in the second highest tenth* 11 per cent in the third highest tenth* 5 per cent in the fourth* 3 per cent in the fifth* and 1 per cent in the sixth; of those pupils who are in the second highest tenth on the prognosis test* 23 per cent will probably be in the highest tenth in geometry achievement* 26 per cent in the beghnd highest tenth* 20 per cent in the third highest tenth* and so forth* * On the basis of Table XIV* at least 56 per cent of the pupils in the first decile on the prognosis test should be in the first decile on the achievement test* In Table XIII there is a slight increase in the first decile for both groups with perhaps a slightly greater increase for the Control froups; however, this difference is lost in the second decile* which implies that those pupils of simperior ability seemdd to develop normally with respect to their ability in the study of geometry* The significant fact to note here* however* is that the last or tenth decile produces rather significant differences* 227 According to Table XIV, at least 56 per cent of pupils in the tenth decile on the prognosis test ought to be in the tenth decile. This means that out of the forty-six Experimental Group pupils in the tenth decile, 26 of them under normal development would probably remain in this decile on the achievement test, while of the 19 Control Group pupils 11 of them would remain in the same decile. Table XZII shows that the expectation for the Experimental Groups was three more than the prediction, while in the Qontrol Groups there is an increase of 22 pupils in this decile. One, of course, should be extremely cautious in malcing generalizations on the basis of a correlation coefficient of *77 ± ,025, which happens to be the reported one for this prognosis test and any standardized achievement test. However, since a correlation coefficient of *80 is considered statistically significant for prognosis purposes, one may conclude that on the basis of statistical evidence the Experimental Group teachers made better provision for individual differences in the lower end of the ability scale than did the Control Group teachers. Furthermore the two highest scores on the Columbia Research Bureau Geometry Achievement Test were made by Experimental Group pupils and three Experimental Group pupils made scores above 70, whereas only one Control Group pupil achieved this distinction. In addition, at the lower end of ability, only six Experimental Group pupils made scores below five, while sixteen Control Group pupils fell below this mark and out of the sixteen there were four negative scores. There were only two Experimental Group pupils with I*^U *s above 130 and less than 140, while there were ten Control (hroup pupils with I.Q. *s above 130 and three of these were above 140* Only Jfcwo Experimental Group pupils 228 had Geometry Prognosis scores above 160, while five Control pupils had scores above 160* Growp (See tabulation, Appendix ^B11*) Since this study is not concerned primarily with geometric achievement, but mainly with development of critical thinking abilities, it was found necessary to analyze the geometry prognosis and geometry achievement tests for elements in critical thinking* She analysis revealed at least five of the elements developed in Chapter Z, as follows) 1* Seeking definitions and questioning the meanings of geometric terms* 2• Recognizing the place of geometric assumptions* 3* Searching for geometric facts* 4* Testing geometric facts for pertinency* 5* Evaluating geometric conclusions 'for consistency with assumptions* In conclusion it may be said that both Experimental and Control Groups indicate improvement in critical thinking abilities in terms of the above five elements when applied to geometric situations* The reason for this improvement is based upon a comparison of the results on the geometry prognosis and geometry achievement tests and the ^Norms11 sub* mitted for these tests* 2* Other Mathematical Achievement It was found convenient at this point to mention the results of a follow-up study, which was made one year after the experiment was com pleted* This follow-up was made with regard to 166 of the Experimental Group pupils* Forty-nine per cent of this number had elected to go on into higher algebra, solid geometry or trigonometry* The following is 229 a tabulation of the marks they received in the more advanced courses* Subject Algebra III jdeom* III Trig* IT No* A»s No* B*s No* 0*8 No* D ’s No* P*s No* A*s No* B's No* C ’s 12 6 2 26 2 0 34 8 0 21 2 0 7 0 0 4 2 0 10 2 0 8 0 0 No* D ‘s 2 0 0 No, T'i 0 0 0 Out of this group who elected the more advanced mathematics courses 75 per cent were hoys and 25 per cent were girls. These figufes of course mean that the election of these more advanced courses was made in the junior year of high school* Some pupils elect higher algebra and solid geometry or trigonometry in their senior year* The significant point brought out here is that those who elected courses of an advanced nature met with better than average success* and this is particularly true of the pupils in solid geometry and trigonometry* where no failures are recorded* However, the number of cases is so small that further evidence would be required to justify fully this position. It must also be kept in mind that only twenty-eight theorems and seventeen constructions were common to the Experimental Ckoups. This minimum number has apparently given the pupils an adequate control over the subject matter of geometry, at least a sufficient background for courses in solid geometry and trigonometry* 3* Nature of Proof7^ Test Number 5.3 This test was administered at the beginning of the school year and again at the end of the school year* The primary purpose of this test was to measure growth or development In critical thinking abilities from 75 Nature of Proof Test 5*3, Progressive Education .Association, duos cit* 230 one stage of maturation to a later stage# This test did not measure critical thinking abilities in terms of all ten of the elements developed in Chapter I; however, it did measure these abilities in terms of at least five of the elements, as follows* 1* Is this pupil questioning the meaning of termst 2* Is this pupil questioning and detecting underlying assumptions? 3* Is this pupil searching for more facts? 4# Is this pupil discriminating between facts and assumptions? 5* Is this pupil testing conclusions for consistency with assumptions? 6# Is this pupil formulating logical conclusions? The test was not administered to the Control Croups, because the present study is primarily concerned with a description and evaluation of certain experimental procedures indicated in Chapter III# Furthermore, it was felt that Control Group results on this test would add very little significance to this study, because as Fawcett^® has pointed out, the usual course in geometry does not improve abilities of young people in terms of this type of test# The following is a summary of the scores made by the Experimental Groups on this Nature of Proof Test Number 5#3 at the beginning and at the end of the school year# Nature of Proof Test Nqmher 5.3 . Beginning of the School Tear At the end of the School Tear 76 A summary of the results follows! Number of Fu-oils Hanse Standard Deviation _ Mean 333 1-30 4.4 12.7 333 8*65 8.9 30.2 Pawcett, Harold P., •Thirteenth Yearbook^ op* cit# p# 103. 231 The difference 'between the two means (17*5) was found to be highly significant. This difference is approximately 32 times the standard error of difference (*543). Considering the fact that an actual difference of two means is considered statistically significant when it is 3 times the standard error of difference between the two means, an actual difference which is 32 times its standard error of difference becomes very significant. One may conclude that great improvement in critical thinking abilities in terms of at least five of the elements is very much in evidence. One needs to be cautious in making interpretations in terms of statistical evidence, because the apparent growth may be due to a variety of factors, such as maturation and procedures in other class rooms. However, in the light of the evidence presented, it is safe to conclude that the procedures outlined in Chapter III were responsible at least in part for the improvement students made in their response to this test. In order to illustrate the differences between responses on this test at the beginning of the school year and at the end of the school year, a compiled summary of the more pertinent ones was made for each question* is an illustration^* question Ho. 1 of the test is here reproduced, after which examples are given of responses to the question in the preliminary test and to the seme question in the final test. ftfom read that the death rate in the United States Navy during the war with Spain was 9 per 1000. This means that 9 men out of each 1000 died. The death rate in Hew York City for the same period was 16 per 1000. Tom concluded that it is safer to be a sailor in the Navy during a war than to be an ordinary citizen in Hew York City. What things must be taken for granted in order that this conclusion be accepted?... Write them in the space below. 77 See Jppendix "BH for all of the responses compiled for the final Nature of Proof Test Humber 5.3, 232 A* List of responses to the above question compiled in the preliminary test* 1* The populations in the Navy and in New York City were of the same types* 2* Modern wars would not take as many lives* 3* There were the same number of people in New York City as in the Navy. 4* The Navyis a safer place during a war. 5* New York City would not be attacked in modernwarfare. B* List of responses to the above question compiled in the final test* 1* The war with Spain was as bad as any war that might come along* 2* The Navy is the safest place to be in a war. 3. The Navy represented a cross section of New York City's population. 4* He must take for granted that any war would have the same death rate or a lower one than New York City. 5* That there wasn11 an epidemic of any kind going on in New York at the time. 6. New York City is safe from war and would never be in a war sons. ?• Modern war weapons will not increase the number of deaths* 8. Modern medical science will not decrease deaths* 9* Modern traffic problems would not affect this ratio* 10* The men in the Spanish war were all accounted for. 11. The number of people in the Navy and the number of people in New York City were the same. 12. The percentage given in New York did not include women* children and misfits, but only the physically fit young men. 13* The New York deaths were accidents and did not include disease. 14. What Tom read was the truth. 233 15* Sailors are a select group of physically fit manhood. 16. There was active fighting going on during the entire period* 17, He also assumed that the next war would he no worse than the war with Spain* The preliminary responses were less comprehensive, less accurate, and in many instances no response was even attempted. These factors, how ever, were practically eliminated in the final test, because it was pointed out earlier that the differences in the two nmean8H, namely the prelim inary test mean and the final test mean were statistically very significant* 4* Test in Critical Thinking In order to make the evaluation of outcomes in critical thinking abilities as comprehensive as possible, it was found necessary to construct a test^® that would in some way imply all of the elements of critical thinking set forth in Chapter I* That the test was entirely successful in evaluating all of the critical thinking abilities may be seriously questioned* However, it did serve the purpose for which it was designed, namely to get all ten of the elements upon which this study depends into the picture* This test was administered to both the Experimental and Control Groups at the end of the school year* In order to bring out more clearly a comparison of the outcomes of this test between the Experimental and Control Groups, the results are tabulated as follows* Groups Humber of Pupil s Kange of Scores Experimental 333 31-148 Control 326 12-96 Standard Deviation. 20*4 16*4 Mean Score 81*2 54.4 78 A copy of this test may be found in Appendix ttCn, and a tabulation of scores and the pupils1 percentile ranks for these scores may be found in Jppendix HBff* 234 The actual difference between the two means Is 26*8 and the standard error of difference was found to he 1*4* Statistically 1) g 3 la <fd considered a significant difference between the two means* In this case s 26*8 s 19*1 is very significant* 1*4 ed Therefore* one may con elude that the Experimental Group responses to this test are signif icantly superior to the responses made by the Control Groups* In the preceding section it was pointed out that the standard ised tests in geometry involved at least five elements of critical thinking* Therefore* it seemed logical to analyze and to determine* if possible* how the two groups compared in the individual items of the critical thinking test* which implied all ten of the elements* This was done and the results tabulated in Table X7, as follows* TABLE XT* TABULATION OF MEANS, PROBABLE ERROR OF MEANS* AND RANGES OF RESPONSES TO EACH QUESTION ON THE CRITICAL THINKING TEST Question Number 1 2 3a 3b 4 5 6 7 8 9 10 11 Experimental Group Mean 4*3 11*4 2*0 6*5 6*1 6* 3 10.2 9*3 4*8 6*9 6*2 7.6 Control Group Mean 1*3 5*2 1*4 4.2 4*0 4*4 9.7 5*8 3*8 4*2 4.5 3.1 Ave* P.S* Mean Max* F.E. Mean Exp* Group Range Control Group Range 4*4 ±.6 ±.0 4.1 4*1 4*55 4.0 4.7 ±•6 ±•6 4.0 4*1 4.1 4*05 4*0 ±•5 4.0 4*15 4,15 4*5 2-12 1-20 0-4 2-14 2-14 4-10 6-13 4-12 0-9 2-10 2-18 5r35 0-8 0-30 0-4 0-14 0-12 1-9 4-12 0-12 0-9 1-8 0-14 0-15 t.o 4*15 ±*15 t i.o The major differences between the two groups seem to lie in the sensitivity to elements involved in questions No* 1* 2* 3* 7* 9, 10 and 11* The results represented by comparing the means be interpreted to imply that Control Group pupils are not sensitive to elements in 235 critical thinking as set up in this experiment, because a study of the ranges of the responses to the first ten questions reveals that there were maximum scores in Control Groups which approach the corresponding maximum scorOs in the Experimental Groups* The results suggest that the average Experimental Gfcoup pupil is more sensitive to the elements of critical thinking than the average Control Gkoup pupil; furthermoref that this average Experimental Grotp pupil is particularly more sensitive to the elements of critical thinking involved in questions 1, 2, 3, 7, 9* 10, and 11, as follows! 1* 2* 3* 7. 9* 10. detecting motives questioning meanings and seeking satisfactory definitions detecting stated and unstated assteoptions evaluating self for hias or prejudice recognizing need for suspended judgnent evaluating conclusions in terms of individual and social consequences* Question 11, of course, was an essay type of question involving all ten of the elements* Differences in the above five elements would also be present in question 11, and this is mainly the reason why such a significant difference exists between the means or average response to this question* In order to improve the reliability of scoring this test, because of the fact that a part of it was of a subjective nature, two other persons were enlisted to cooperate in framing a method sf scoring, and in actually assigning scores to a random sampling of thirty papers from each group* One of the scorers represented the Control Croups, one the 79 Experimental Groups, and the third had no connection with either group. The criterion for scoring each item in the critical thinking test, 79 See acknowledgments for the names of the scorers* 236 as agreed upon "by the three scorers# is as follows* Question No. Point Value and Description of the Item in Bach Question I* 2 points for each motive that appears relevant to the situation* II* 1 point for each pertinent underline, and 1point each reason that is pertinent to the underline, for III* a* 2 points for dach of the two correct responses* h* 2 points for each assumption that appears relevant to this situation, 17* 2 points V. 1 point for each correct response, 71, 1 point for 711, 2 points for each response if reason given is consistent with the check mark, 7111, 9 points if all three of the responses are correct; no points unless all three are correct. XX, 1 point for X. 2 points for each pertinent consequence, XI, 5 points for each element of critical thinking involved. for each fact that appears relevant to the situation, each correct check mark (\/), each response marked (IT). Since each scorer worked independently# some variations in scoring were found. These were treated statistically and maximum as well as average probable errors of the means for each question were determined, These were recorded in Table XV, from which one may conclude that the fluctuations in scoring were not sufficiently significant to affect the interpretations presented. In conclusion it may he said that the sensitivity of pupils to elements of critical thinking was statistically far superior on the part of the Experimental Groups taken as a whole over that of the Control Gkoups, While the validity of the test may he questioned as to whether 237 or not it tested critical thinking behavior, nevertheless one factor is quite prominent, namely that the test did take into account all ten of the elements set up in Chapter I. This is particularly true of the eleventh question, which involved all ten of the elements; and because of its subjective nature it offered each pupil an opportunity to exercise his critical thinking ability* The differences between the two groups were very significant with respect to this eleventh question, yet regardless of its subjectivity the process of scoring was very much in agreement by the three scorers* 5* Periodic Hxercises and Tests Numerous exercises and tests were given to both Experimental and Control $roups* However, the Experimental Croups were given many non* geometric tests and exercises**® in connection with their geometry course* The geometry achievement test, previously mentioned, measured pupil abilities to manipulate this type of content* Therefore, in order to avoid repetition, this section will be limited to an evaluation of critical thinking abilities through non-mathematical exercises and tests* in effective way of bringing out the changes taking place in pupil responses to such exercises and tests is to describe the various stages of development in terms of each element of critical thinking, as foilows* (l) Detecting Motives* This type of behavior was difficult to test by means of paper and pencil in the earlier stages of development* in effort 80 A number of non-geometric exercises and portions of tests of a nonmathematical nature were illustrated in Chapter III* Several exercises were mentioned in Chapter IV* Some of the better paper and pencil tests of a non-mathematical nature are included in Appendix *&H* 238 was made to present situations in the field of advertising* and so forth, whereby a pupil was asked to make critical analyses. In the earlier stages of the course, this element of motives was overlooked by the pupils. Statements of motives, however, began to appear in such situations after the youngsters were made sensitive to them by means of such questions as3 "Why am X giving you this test?11 •What are the motives behind this advertisement?H •Why was the Dies committee formed?1* It was after a series of direct questions pertaining to motives that evidence of this type of critical thinking behavior began to appear in paper and pencil exercises where no direct response to this element was indicated £ priori* For example, when asked to make a critical analysis of Bing Crosbyls statements "Hjr Throat is % the element of motive earlier stages of the the course nearly all this element in their Fortune That's Why I Smoke Old Golds," was left out by nearly everyone in the course. In similar situations later in of the Experimental Group pupils included analyses, (2) Questioning the Meaning of Terms, In the earlier exercises and teats, only the words which were uncommon to the experiences of young people were underlined or questioned as to meaning. For example, words like inimical, rational, criterion, revenue, philosophical, inductive. deductive, explicit, implicit, and others were questioned in the earlier stages# As a result of the procedures outlined in Chapter III, the later stages of the course found pupils questioning such common terms as all, everyone, adequate, any, &Sm$ no one. Patriotic, proper education. r.eas.pflable state,, ,aA&> end so forth. In like manner the periodic paper and pencil instruments pro duced evidence of development in critical thinking abilities with respect to the remaining eight elements. One significant fact ought to be mentioned at this point, that greater changes were apparent in critical thinking abilities during the earlier stages of the course than in the later ones. This, of course, varied with different pupils and there was evidence of continued development to the very end. The major differences between the earlier and later responses were significant not only in quantity but also in quality. For example, in the situation where a picture of a beautify! girl with an attractive smile was represented as saying3 239 "I'd wished a thousand times for a brighter smile* One tube of Colgate's gave it to me* It was so annoying to see other girls with lovely smiles get all the dates* Then I tried Colgate's. Now my smiles are bright too.0 A pupil in the earlier stages of the course would list assumptions behind this situation as follows: "If you use Colgate's you will get many dates.* "She actually had a brighter smile after using Colgate's." "She had wished a thousand times for a brighter smile." In later stages of development the responses with respect to underlying assumptions would be somewhat as follows: "Colgate's is the only dental cream that can give one a lovely smile*" " H I of the other girls with lovely smiles used Colgate's*" "Only girls with lovely smiles get dates*" "She was actually annoyed to see other gitls get all the dates*" "She actually did use Colgate's." "Other girls did get all the dates." "She actually wished for a brighter smile." "Colgate's is the best dental cream." "If she tried Colgate's and if it brightened her smile, that one tube of Colgate's actually gave her a brighter smile*" "She wished a thousand times for a brighter smile." The elements in which greatest evidence of improvement was exhibited among the Experimental Group pupils in the periodic exercises and tests, are as follows: 1* Ability in detecting motives* 2* Ability in testing conclusions for consistency with assumptions* 3* Ability in detecting stated and unstated assumptions* 4* Ability inevaluating self for bias or prejudice. 5* Ability inrecognizing the tdntativeness of conclusions* 6* Ability in evaluating individual and social consequences* Elements in which the pupils showed less improvement are as follows: 1* Ability in detecting words and phrases that need defining. 240 2* Ability in searching for facts* 3* Ability in testing facts for pertinency* 4* Ability in discriminating between facts and assumptions* 6* Inventory Questionnairef As previously mentioned, the Inventory Questionnaire administered the first day of the school year was primarily Intended for guidance purposes* However, after a follow-up form was administered at the end of the school year the differences in the responses seemed to merit con sideration at this point. In order to bring out these differences more explicitly, it was found convenient to discuss the more pertinent questions separately, as follows* Question Ho* 1* wfhat are your favorite subjects? Why?* The responses to this question were grouped into namely (l) Mathematics, (2) Others, and A sunmary of the responses is as follows? threetypes, (3) no response tothe question* In the preliminary form only 20*9 per cent of the pupils in the Experimental Groups mentioned math ematics, while 41*4 per cent of the pupils in the Gontrol Groups did likewise. In the follow-up form 35*8 per cent of the pupils in the Experimental Groups mentioned mathematics, while only 29*4 per cent of the pupils in the Control Croups did likewise. The Experimental Groups show an increase of 15 per cent, indicating a liking for mathematics, while the Control Cfroups show a drop of 12 per cent* Preliminary Porm Pinal Porm Preliminary Porm T W Tor* (l) Mathematics (2) Others Experimental Groups 20.9$ 76*9$ 35.8$ 63.2$ Control Groups 41*4$ 58*6$ _ 29.4* 70.64, (3) Ho Response 2*2$ 1.0$ 0.0$ 0.0* _ _ 241 Too many variable factors are involved for an accurate interpretation of the differences. However, in the light of the observers reports in Chapter III, there is reason to believe that the difference is due in large measure to the teaching procedures used. Question Ho* 2* “In what way do you think mathematics may help you?n “Why?* The responses to this question, in each instance, fell into one of the following types, namely (l) Vocationally, (2) Mental Discipline, (3) Thinking or Reasoning, (4) College Entrance, (5) In Ho lay, and (6) Ho Response. A summary of these responses is as follows: Preliminary Form Final Form (4) (2) (3) Men.Disc. Thinking College Exuerimental Grou 41.856 20.9$ 2.256 978$ 46.156 2.6)6 0.356 4 2.956 Preliminary Form Final Form 61. 656 46.8j6 (T) Vocat. (5) Ho Way (6) Ho Response 8.256 5.556 17.1)6 3.956 10.956 5.556 6.156 2.656 Control Grouns 12.156 15.356 12.556 16.556 4.356 4.456 The above sunmary is self-explanatory; however, there are three or four differences that merit special consideration. For example, the 15.8 per cent drop in (l) the Control Gkoups with respect to Vocational implications for mathematics and a oouesponding increase of 7 per cent in (5) for the seme groups, suggests at least one possibility, namely that some of the Control Group pupils were losing interest in mathematics, which is a logical inference from the first question. The other significant differences may be found in columns (2) and (3) for the Experimental Groups. The reason for this difference is undoubtedly due to the emphasis placed upon thinking and reasoning in the Experimental Gkoups. Question Ho, 3 * “In what way do you think that demonstrative geometry may help you? Why?® 242 The responses to this question fell into one of the following types, namely (l) Vocational, (2) Mental discipline, (3) Thinking or Reasoning, (4) College Entrance, (5) In Ho Way, and (6) Wo Response* A summary of these responses is as follows* (T5 (55 (3) (4) (5) Ci5 Vocat* Men*Disc* Thinking College Ho Way Ho Response , Experimental Grovroa ^ Preliminary Torn 12.7# 8.2# 10.1# 3.2# 10.8# 56.0# Pinal P o m 11.0# 0.0# 76.8# 1.3# 9.0# 1.9# Control Groups Preliminary P o m 25.0# 8.5# 19.4# 6.6# 6.9# 34.5# Pinal P o m _______ 12.1# 12.9# 30.S# 4.4# 21.8# 18.6# _____ ___ The above summary is again self-explanatory* However, there are four or five significant differences that merit special mention* Bor example, one inference that may he drawn from column (3) is that bdth groups have shown an increase regarding the fact that geometry has helped them to think or reason hotter, and the greatest increase in this respect is in the Experimental Groups, which would imply that there was some factor in operation in the former that is not present in the latter (probably the treatment of non-geometric materials)* In column (5) there was in increase of nearly 15 per cent in the Control Groups from which one may conclude that over one-fifth of the pupils in these groups stated specifically that they felt the study of geometry did not help them in any way* Column (6) implies at least two things, namely (a) lack of knowledge regarding the meaning of ^demonstrative**, and (b) deliberate non-commitrtient of possible consequences* because It is more likely, however, that the first case is the major factor because of the great drop in the Experimental Groups from 55*9 per cent on the preliminary form to 1*9 per cent in the final form* Of course, wDeraonstretionB as a form of argument or 243 proof was the core of the S^erimentaX Group procedures* The latter possibility, that of non-commitment is rather remote, because there was no evidence of any feeling between teachers and pupils that would in any way inhibit the responses to this question* Question No. 4* 11Do you think that high school pupils should have an opportunity to reason about the subject matter of geometry in their own way or do you think that the subject should be learned in the way in which it is presented in a textbook or in some work book? The responses to this question seemed to fall into one of the following four groups, namely (l) Independent of textbook, (2) Dependent upon textbook, (3) Combination, and (4) No Besponse* A summary of these responses is as follows* (1) ______________ Preliminary Borm Pinal form Preliminary form Pinal Porm (2) (3) (4) IndependentDependent Combination No Response Experimental Groups 33.2$ 40.5$ 14.6$ 11*7$ 81.0$ 9*7$ 6.7$ 2.6$ Control Groups 15.1$ 62.9$ 16.9$ 0.0$ 40.3$ 38.3$ 20.2$________ 1.2$ The above summary produces differences that warrant some explanation. | Por example, in column (l), both groups seem to prefer their own treatment i j jj of subject matter. The increase in the Experimental Groups to this response ■was nearly 48 per cent while the Control (koup increase was approximately ii 26 per cent. The fact of actually having experienced the opportunity to work without the text has undoubtedly influenced the Experimental Group response; on the other hand one can infer from the results of the Control Groups that either they had some opportunity to think independently, or ! they definitely felt that they should have this opportunity. Both groups 244 have registered a marked decrease with respect to the dependence upon a textbook from their preliminary responses* Question Ho* 5s HHow would you like to have your progress In this course evaluated? (that is* by a written examinationt judgment of your teacher* self-evaluation* your parents' evaluation* and so forth* or a combination of these?)11 The responses to this question seemed to fall into one of the fol lowing four groups! (l) Examination and Teacher Judgment* (2) Self-Evaluation, (3) Combination, and (4) No Response. A summary of these responses is as follows: Preliminary Form Final Form (1) (2) Exam* & Tcb*J\xd£* Self-Bval. Experimental Qrouos 21.8# 5.756 26.556 7.4$ Control Groups Preliminary Form Final Form | 41.856 49.256 (3) "(4) Combination No Response 5.656 3.256 47.256 64.8*6 25.356 1.356 44.056 40.7;6 8.656 6.956 The above summary definitely Indicates that pupils feel that self- | evaluation is not sufficient, nor is parents* evaluation* Not a single Ipupil indicated a desire for parents* evaluation* and the few that did I comment In this respect pointed out that their parents were not informed I sufficiently to make a valid or unbiased evaluation. j| Some did include parents* evaluation in connection with column (3), namely as a combination* II j The increase in responses in the Experimental Groups for a com- ibination of factors in their evaluation is probably due to the fact that ij three of the Experimental Gkoups actually had the experience of being j evaluated on this basis as described in the chapter on procedures. ! In conclusion, one might say that there is at least a strong trend on the part of the pupils to feel that they should share in the process of their evaluation* 245 Question Mo* 65 "The concept of proof ***** always played a very significant role in human experience* When in your judgment is something proved?11 The responses to this question fell into one of the following groups* namely (l) a fairly accurate concept* (2) partly correct concept* that is* possessing some of the elements of proof* (3) inadequate concept* and (4) no response* A summary of these responses is as follows5 (1) Correct Preliminary Worm Pinal Pornt Preliminary Porm Pinal Porm________ 1*6$ 45.8$ 1*3$ 0.8$ (2) (3) Partly Correct Incorrect Experimental Groups 20*6$ 46*8$ 40*0$ 6.1$ Control arcupfl 22*4$ 54*7$ 27.0$ 57.7$ (4) Mo Response 31*0$ 8.1$ 21*6$ 14.5$ She significant fact here is that there was very little improvement registered on the part of the Control Groups with respect to the concept of proof, after dealing with it for one school year. Groups show a very marked improvement in this respect. She Experimental She concept of proof was considered correct in this case, if the pupil's response contained 81 or in some way implied that it involved, according to Pswcett , the fol lowing: 111. The place and significance of undefined concepts in proving any conclusion. 2# The necessity for clearly defined terms and their effect on the conclusion* 3* The necessity for assumptions or unproved propositions. 4. That no demonstration proves anything that is not implied hy the assumptions." A pupil's response was partly correct if it inplied at least some part of the above criteria, and finally it was incorrect if it failed to imply any part of the above criteria* 81 Mawcett* Harold P., "Thirteenth Yearbook", op. cit., p. 10* 246 Question No* 7* *©16 kind of society in which we live calls for a type of citizenry that is capable of thinking critically* When in your Judgment is a person thinking critically?11 The responses to this question fell into one of the following groups* (l) a fairly accurate concept if it involved four or more of the elements of critical thinking mentioned in Chapter I, (2) partly correct if it implied from one to three of the elements* ^3) incorrect if none of the elements were implied* and (4) no response* A summary of these responses is as follows* fI5 Correct |Preliminary Ibra !final fora 0.0# 30.6# |Preliminary Porm final fora 0*0$ 1.2# (15 (3) Partly Correct Incorrect Experimental terming 21.6# 62.5# 51.3# 9.7# Control Ckouns 29*7$ 54.9$ 33.5# 56.0# (4) No Response 15.9# 8.4# 15.4$ 9.3# The preliminary differences are not significant enough to state definitely that one group had a hatter understanding of critical thinking than the other; however* the final form definitely produces wide differ ences between the two groups. These differences* of course, are due to methods of procedure involved* There is one inference that can he made relative to responses to |this question and also to the preceding one, which lends a great deal of support to the assumption made by the Experimental Croup teachers in ||Ch£pter III, namely ffthat high school sophomores are capable of thinking |critically11. While this was one of the assumptions in this study* one could now admit it as a fact with reference to the criterion set forth, because both groups had some notion of proof and critical thinking prior :i to their taking this course in geometry* The only difference lies in the 347 fact that the Experimental Groups have shown more improvement in the development of these concepts* Question No* 83 ttMake out a list of articles (newspaper or magazine), or a list of hooks that you have read of your own accord in connection with your course in geometry this school year.* Since this question has hearing only upon the activity within the school year, there is no tabulation to the preliminary form of the ques tionnaire. It must he kept in mind that the responses here were limited only to voluntary activity on the part of the pupil* If the pupil1s responses are colored in any way, then the same is probably true for both groups* She responses herein are grouped on a quantitative basis, namely (l) over three articles or books, (2) three articles or books, (3) two articles or books, and (4) one article or book, also (5) no reading what ever in this connection* A summary of these responses is as follows3 (1) 4 or more Readings Experimental Groups 11.9$ Control Ckoups 0*0$ j (8) 3 Readings (3) 2 Readings (4) 1 Reading 18.4# 6.1# 38.1# 14.1# 14.3# 1.6# Here again the differences are marked* (5) None 27.4# 78.2# in explanation for the |j increased voluntary reading on the part of the Experimental Group pupils j lies in the method of procedure and particularly that part of the pro- !cedure which dealt with individual projects and folders. At least one !can imply that this part of the procedure served as a stimulus to the pupil's reading* Conclusions concerning the Inventory Questionnaire are summarized Ias follows3 ■ I 348 1* Experimental groups Indicated a 15 per cent increase in selecting mathematics among subjects best liked* where 13 per cent of the Control Croups showed a 12 per cent drop after studying geometry for nine months* 2* Experimental Groups indicated a 35 per cent increase with regard to mathematics as helping them to think and reason more clearly* while the Control Gkoups showed only a 4 per cent increase with respect to this element* 3* Experimental Croups indicated a 67 per cent increase with regard to demonstrative geometry as helping them to think and reason more clearly* while the Oontrol Croups indicated only a 11 per cent increase in this respect* 4* Experimental Groups indicated a 48 per cent increase (from 33 per cent on the initial form to 81 per cent on the final form) in favor of developing their study of demonstrative geometry independent tof a textbook* having actually experienced such a procedure* Control Croups indicated a 25 per cent increase (from 15 per cent on the initial form to 40 per cent on the final form) in favor of developing their study of demonstrative geometry independent of a textbook* however* not having experienced such a procedure* 5* Two-thirds of the Experimental Group pupils favored a combination of evaluating techniques* while two-fifths of the Control Group pupils did likewise* This was probably due to the fact that over half of the former pupils had first hand experience with such procedures* There was a trend in the responses of both groups indicating a desire on the part of the pupils to share in the process of evaluating their progress* 6* Nearly 64 per cent of the Experimental Group pupils showed improvement in acquiring the concept of proof* whereas only 5 per cent of the Control Croup pupils had shown similar improvement* 7* Nearly 52 per cent of the Experimental Group pupils indicated an understanding of critical thinking in terms of the ten elements developed in this fetudy* while only 13 per cent of the Control Croup pupils had shown a similar understanding* 8* Nearly 67 per cent of the Experimental Group pupils indicated voluntary reading in connection with their geometry course* while only 22 per cent of the Control Croup pupils did likewise* 249 Chapter g^nrnfiny In this chapter an evaluation was made of outcomes hy means of written tests and an inventory questionnaire* The outcomes evaluated were as follows: 1* Knowledge of geometric facts* 3* Understanding of a geometric proof* 3* Skill in manipulating geometric content* 4* Pupil reactions to their course in demonstrative geometry* 5. Reading Interests* 6* Critical thinking ability in terms of the ten elements developed in Chapter I* In order to study the differences in outcomes between the Exper imental and Control Groups, it was found necessary to compare the two major groups in terms of mental ability and in terms of ability to do the usual work in demonstrative geometry prior to a study of this subject* The Control Groups were found to be statistically superior to the Exper imental Groups in mental ability as measured by the Otis Test of Mental Ability, and in geometric ability as measured by the Orleans Geometry Prognosis Test* The geometry achievement of the Control Groups at the end of the 1school year was found to be statistically superior to the Experimental Groups as measured by the Columbia Research Bureau Geometry Test* The !significance of the Control Grotp superiority in geometric ability, however, was reduced by more than ten per cent on the basis of this test* The gain on the part of the Experimental Groups was more pronounced in the lower decile of the distribution of scores* More adequate provision |for individual differences in the Experimental procedures wan given as |one reason for this gain* The fact that the geometry prognosis and 250 geometry achievement tests were analyzed for elements in critical thinking and at least five were found to he present, and the the Experimental procedures were particularly designed to improve critical thinking j abilities suggests another reason for this gain* In connection with outcomes concerning mathematical achievement* |a follow-up study of 166 Ibcperimental Gkoup pupils revealed that 49 per cent of this number elected to study higher algebra and solid geometry |or trigonometry in their junior year. The group as a whole achieved |better than average success in these subjects. Since there were only 28 |theorems and 17 fundamental construction problems common to the Experimental j Groups, this minimum number has apparently given the pupils an adequate control over the subject matter of geometry. j The Hature of Proof Test 5.3 administered at the beginning and at It the end of the school year to the Experimental Groups revealed highly significant differences in the responses. Since this test was analyzed ji Ifor elements in critical thinking and at least five were found, the reason |offered for this great difference in responses was attributed at least i in part to the procedures outlined in Chapter III. It was pointed out, however, that one needs to be extremely cautious in reaching conclusions 1 |based upon statistical evidence alone, because the significant difference jmay likewise be due to other factors, such as maturation and procedures in other classrooms* The test in critical thinking was administered to the Ibcperimental ' T 71T '' ^ 7 |and Control Gro'ups at the end of their study of demonstrative geometry, [since this test was designed to include all of the elements of Critical i |thinking developed in Chapter I, it served as an objective means for 251 comparing the outcomes in this type of thinking of the two major groups. She responses of the Experimental Group were found to he very superior to those of the Control Group. Jbr example, the actual difference b e tween the actual “means** for the two groups was 26.8, and this figure was found to he 19.1 times the standard error of difference (1.4) of the two means. Since a portion of this test was of a subjective nature, two other persons were enlisted for the purpose of formulating a method of scoring and Actually assigning scores to a random sampling of thirty papers from each group, The fluctuations were found to he so statis tically insignificant that the above differences were ascribed to factors other than subjectivity in scoring, namely teaching procedures and materials used. Records of progress, measured by periodic exercises and tests in critical thinking were kept only for the Experimental Groups. Evidence of improvement was exhibited in terms of each of the tit elements. The inventory questionnaire, administered at the beginning and at the end of the school year, revealed several inportant facts, as follows J 1. Experimental Groups indicated a 15 per cent increase in including mathematics among their best liked subjects, while the Control Groins registered a 12 per cent decrease in this respect after studying geometry for nine months. 2. Experimental Groups indicated a 33 per cent increase with regard to mathematics as helping them to think or reason more clearly, while the Control Groups registered only a 4 per cent increase. 3. Experimental Ckoxps indicated a 67 per cent increase with regard to demonstrative geometry as helping them to think and reason more clearly, while the Control (koups registered only an 11 per cent increase. 4. Experimental Groups indicated a 48 per cent increase (from 33 per cent on the initial form to 81 per cent on the final 252 form) in favor of developing their study of demonstrative geometry independent of a textbook, having actually ex perienced such a procedure. Control {Iroups indicated a 25 per cent increase (from 15 per cent on the initial form to 40 per cent on the final form) in favor of developing their study of demonstrative geometry independent of a textbook, not having experienced such a procedure. 5* Two-thirds of the Experimental Gkoup pupils favored a com bination of evaluation* techniques, while two-fifths of the Control Groups did likewise. The responses are indicative of some desire on the part of pupils to share in the eval uation of their progress* 6. Nearly 64 per cent of the Experimental Group pupils in dicated improvement in acquiring the concept of proof, whereas only five per cent of the Control (koup pupils had shown similar improvement. 7. .Approximately 52 per cent of the Experimental Group pupils indicated an understanding of critical thinking in terms of the ten elements developed in this study, while only 13 per cent of the Control (koup pupils had shown similar understanding. 8. Nearly 67 per cent of the Experimental Group pupils in dicated voluntary reading in connection with their geometry course, while only 22 per cent of the Control Group pupils did likewise. 253 CHAPTER VII EVALUATION (Continued) Introduction In the preceding chapter the bases for evaluating outcomes were confined to results secured from written tests and an inventory Ques tionnaire, In this chapter the outcomes will be evaluated in terms of data secured from the following: 1. 2, 3* 4# 5* 6# Direct observations Anecdotal records Analyses of written work other than tests Pupil diaries Analyses of reading interests Individual and group projects* With the exception of the first technique, namely direct obser vations, the remaining ones were confined to the Experimental Ckoups, because facilities for securing other types of information for the Control Groups were inadequate# Direct Observations Evaluation based upon results of direct observation for critical thinking will be considered under three headings, namely: (l) obser vational reports by 165 classroom teachers, (2) reports by a trained observer, and (3) reports by parents and classroom visitors* 1* Reports Prom 165 Classroom Teaflhera All of the teachers with whom the Experimental Group pupils came in contact were interviewed by the six Experimental Group teachers* The purpose of the Interviews was to acquaint these teachers with the present 254 study and to solicit their services as observers of the Experimental Group pupils under their surveillance. More than two hundred teachers were interviewed, and of this number one hundred sixty-five agreed to cooperate in observing the pupils of Experimental Groups for improvement in critical thinking in terms of the ten elements set forth in Chapter X* They were also requested to present a brief report of their findings at the conclusion of the period of observation. The period of observation ranged from a minimum of three months to a maximum of nine months. The reports®^ submitted by the teachers ranged from one to ten in number per pupil; depending of course upon the number of curricular and extrarcurricular activities entered into by each pupil, as well as the number of observers reporting on each pupil. The average number of reports submitted per pupil was between four and five. Examples illus trating the variable nature of these reports are as follows: Pupil No. 315 Eeport by Miss J. W(D.C.) has made very definite progress in? | 1. Trying to detect motives 2* Questioning the meaning of terms and seeking satisfactory definitions 3* Detecting assumptions 4. Searching for more facts 5. Discriminating between facts and assumptions 6. Evaluating herself for bias or prejudice 7. Suspending judgment 5. Evaluating conclusions in terms of individual and social consequences. i There is some doubt about her improvement in the following elements? | 1. Testing facts for relevancy 2. Testing conclusions for consistency with assumptions. | j In general* (D.G.) considers seriously such school problems as jay-walking* behavior in assemblies, and recently the idea ad vanced in student council for use of cafeteria as a recreational 82 A summary of these reports is tabulated for each Experimental Group pupil in ippendix HB11, column 12. room. Although not a member of the student council, she con siders each problem from a broad point of view.** Same Fupil Eeport by Miss E. B(D.C.) has shown a great deal of improvement in the following: 1. Detecting motives in situations 2* Questioning meanings of terms and seeking definitions 3. Detecting and questioning underlying assumptions 4. Searching for more facts 5. Testing facts for ped&nency 6. Discriminating between facts and assumptions 7. Formulating tentative conclusions 8. Testing conclusions for consistency with assumptions 9. Evaluating herself for bias or prejudice 10* Evaluating conclusions in terms of individual and social consequences.N Fupil No. 279 Eeport by Mr. J. H(S.E.) is level-headed enough to think through any situation of concern to him. His improvement is particularly noticeable in the following elements: 1. Detecting motives 2m Questioning meaning of terms 3. Detecting and questioning underlying assumptions 4. Discriminating between facts and assumptions 5. Evaluating himself for bias or prejudice 6. Considering conclusions as tentative 7. Considering Individual as well as social consequences. I have had no occasion to observe (S.E.) in situations involving the following elements: 1. Searching for more facts pertaining to a situation 2. Testing facts for pertinency 3. Evaluating conclusions for consistency with assumptions.H Same Fupil Eeport by Mr. S. is a clear thinker, very dependable, self-directive, a good competitor in athletics, a good sportsman, and a fine allaround boy. He has shown a great deal of improvement in the following elements you mentioned some time ago, namely: 1. 2. 3. 4. 5. 6. Detecting motives Questioning meanings of words or terms Detecting and questioning underlying assumptions Searching for more facts Testing facts for relevancy Constantly evaluating himself for bias or prejudice 256 7* Evaluating conclusions for consistency with, assumptions 8* Considering individual and social consequences* I have failed to observe (S«R*) in situations that demanded his reaction to the followings 1* Discriminating between facts and assumptions 3* Suspending judgment.n Same Pupil Eeport by Mrs, H* *1 feel that (S«R*) has shown considerable improvement in ay Latin class, particularly in terms of the following elements* 1* 2* 3* 4* 5* 6, 7* 8* Detecting motives Questioning meanings of terms and seeking definitions Detecting and questioning underlying assumptions Searching for additional facts Discriminating between facts and assumptions Evaluating self for bias or prejudice Formulating tentative conclusions faking into account individual and social consequences. I have no evidence to judge whether or not (S.E.) has Improved testing facts for pertinency and evaluating conclusions for con sistency with assumptions* Last week (S*R«) volunteered to take part in a debate on whether or not Home was justified in improving her superior culture on others, and he did an admirable piece of work,* Fupil No* 282 Eeport by Hr* f* M(E*M.) has shown very little if any Improvement in critical thinking* He is highly sensitive to motives, and any evidence favorable to big business or to the Republican party is readily digested by him* AL1 other evidence is not worthy of consid eration in his opinion* He has failed to evaluate himself for bias and fights the idea of suspended judgment* Social con sequences have no meaning for him* He is the most unsocial student I've met for some time,* Same Fupil Report by Mr* S* h(R*M*)# in my judgment, has shown remarkable improvement in critical thinking* For example, while officiating a game in which (E«H*) was playing it became necessary to call several technical fouls on him because of his remarks after the fouls were called* Later in this game it became necessary to call similar fouls on his opponents. His reaction became favorable. He has shown much improvement in game situations since this time, which to me indicates that he is evaluating his own conduct as well as others’ and profiting by it. I would say that he has improved in the following* 1* 2# 3* 4* 5* 6* Detecting motives Searching for more facts Discriminating between facts and assumptions Evaluating himself for bias or prejudice Evaluating conclusions for consistency with assumptions Considering individual as well as social consequences. I have no evidence upon which 1 may judge whether or not (E.M.) has shown improvement in the following* 1. 2. 3. 4. Questioning the meanings of terms Detecting and questioning underlying assumptions Testing facts for pertinency Formulating tentative conclusions.tt Same Pupil Report by Mr. P. MIn my own estimation, (R.M.) is not socially adjusted. He seems to feel that the world is against him. His attitude is sullen. He resents being corrected. Apparently he hasn’t studied the social significance of his own behavior to realise how it con ditions the reaction of others to him. I would say that he has definitely failed to think critically in terms of the following* 1. 2. 3. 4. 5. Detecting motives Evaluating himself for bias or prejudice Evaluating conclusions for consistency with assumptions Suspending judgment Evaluating in terms of individual and social consequences. X would say that his improvement is doubtful in the following* 1* 2. 3. 4. 5. Questioning the meaning of terms Detecting and questioning underlying assumptions Searching for more facts Testing facts for pertinency Discriminating between facts and assumptions.w The observational reports by the classroom teachers are, of course, subjective* however, it Was felt that some evidence of improvement in critical thinking abilities might be secured if several different ob servers could judge each pupil within a given frame of reference^ namely the ten elements developed in Chapter I. The following is a summary of reports by classroom teachers tabulated for each of the three pupils mentioned in the above illustration* Reports by Observers Pupil I*<4. No* 315 No* 279 No* 282 117 106 91 Number of Number of Positive Observers Renorting .. _Elements 8 6 7 55 47 19 Number of Doubtful Elements Number of Negative Elements 0 0 15 25 13 36 In order to interpret the above tabulation, Pupil No* 315 is to be considered as follows* (a) Shis pupil's intelligence quotient is 117* (b) Bight different teachers submitted reports with respect to each of the ten elements in critical thinking for this popil* (c) She “number of positive elements" indicates that out of a possible 80 points, this pupil was rated 55* Shis may be interpreted to mean that the eight observers considered this pupil as having shown definite evidence of improvement in nearly three-four the of the elements in critical thinking* (d) She "number of doubtful elements" indicates a rating of 25 for this pupil, which in terms of the preceding explanation means that no evidence was available, in the remaining onefourth of the elements, to serve as a basis for judgment* (e) A i Pupil "number of negative elements" indicates a rating of "sero" for this pupil, which means that none of the observers had rated this pupil as failing to show improvement in any of the elements in critical thinking* similar interpretation may be made for Pupil No* 279 and for No* 282. The former was considered by the six observers as having iexhibited evidence of improvement in at least three-fourths of the elements | Iin improvement in critical thinking. The outcomes of improvement in critical thinking abilities for the latter pupil (No* 282) are doubtful because the seven observers have Indicated nearly as hi^a a rating for failure on the part of this pupil to exhibit improvement in critical thinking as for evidence of definite improvement. In other words, there is little, if any, evidence of improvement in critical thinking ability 259 on the part of Pupil Ho* 282* In Appendix HBM* column 12* there is a similar tabulation for each of the Experimental Group pupils involved in the study* | Out of a total of 333 pupils in the Experimental Groups* 294* or nearly 88 per cent* were reported as showing definite improvement in |critical thinking abilities* It should not he assumed* however* that the remaining 12 per cent have completely failed to reveal some evidence of critical thinking ability* The 88 per cent showing definite improvement in this ability means only that out of the total number of reports sub mitted for each pupil the number of negative ratings for evidence in Icritical thinking exceeded the number of positive Ones* The reports show !that no pupil was rated positive in all ten of the elements by more than 1two observers* nor was any pupil rated negative in all of the ten elements* One case* namely Pupil Ho* 205* was rated positive in all of the elements; however* only pro observers reported on this pupil. The sane is true for Pupil Ho* 183* where only one observer reported* In conclusion it may be said that on the basis of the reports submitted by the 165 observers there was evidence of definite improvement in critical thinking abilities for nearly all of the Experimental Group Ipupils* Each of the six Experimental Group teachers found that improve- i ment in critical thinking abilities was far more pronounced in the i |earlier stages of development of these abilities than in the later stages* Although the exact reasons for this are not clear* it is believed that at least two factors may have contributed. One possibility is that the !Experimental Group pupils had very little* if any* experience in critical I thinking prior to their course in demonstrative geometry. Another 260 possibility is that the methods and materials during the later etages of development were not as challenging, in terms of the ten elements, as in the earlier stages* It is the writer's belief that the first reason cited is the more plausible since there appeared to be no dim inution in students' interests as the work progressed* 2,...Reports by_an_teerlenced Observer Reports by an experienced observer, in connection with Experimental and Control procedures, were presented in Chapter III, pages 109-120* in analysis of these reports reveals the presence of elements in critical thinking* It must be kept in mind, however, that the observer had no prior knowledge or information as to which groups were Experimental and which were Control* This observer indicated in his reports that in the Experimental Groups there was evidence of the followings az (1) Many pupils were trying to detect motives in different situations* (2) Most of the pupils were questioning the meaning of terms, and thereby trying to reach a common understanding of them* (3) Many pupils were detecting and questioning underlying assumptions* (4) Most of the pupils were searching for more facts in support of their conclusions, (5) Many of the pupils were testing facts for pertinency to a situation* (6) Many pupils were attempting to discriminate between facts and assumptions* (?) Many pupils were evaluating themselves for bias or prejudice* (8) Many pupils were trying to test conclusions for consistency with their assumptions. (9) Many pupils were suspending judgment* (10) Many pupils were considering individual as well as social consequences* A consideration of consequences in connection with Euclidean and non-Euclidean geometry in one of the classes was evidence of this element. 83 Most means nearly 100 per cent of the group in question. Many means over 50 per cent of the group* Some means less than 25 per cent of the group* 261 This observer Indicates in bis reports that in the Control Groups* (1) There was some evidence that pupils questioned the meaning of terms* (2) Many, pupils were searching for facts in support of the theorem or problem under consideration* (3) There was moms evidence that pupils were Seating facts for pertinency. (4) There was some evidence of attempts by pupils to discriminate between facts and assumptions. 3. Effports by Parents and Classroom Visitors Because of the subjective nature of reports by parents and class room visitors, the results secured are open to question* Furthermore, this evidence is available only for the writer^ classroom procedures* However, since parents and visitors do make comments regarding their observations of pupils in the home and in connection with school situations, a few examples will be given to illustrate the nature of this information. Example No* 1* The following was presented by the adviser of a boy whose sister was in the writerfs class* 11Mr. . your influence in your geometry class seems to have made quite an impression upon some of the pupils in your class and particularly upon S.M* Mrs* M* says her daughter has become very inquisitive and wants to know the Khv and Wherefore of everything* It seems that several mothers have compared notes and all of them feel that your method of getting pupils to think critically is highly gratifying*■ Example No* 2* During sophomore ?«T.d* night more than 60 parents of the writer*s Experimental Group pupils were interviewed. In the three to five minute interviews with the parents there was one statement that resembled a coined phrase, namely* *'You certainly did something to my child. AL1 we hear at home is Why? Where is the evidence? That*a not a,fact. This i.g.-.fifl assumption. What do you mean by this word? Whai^ia^yqur yaefl motive! Example No* 3* There were approximately a dozen students from Northwestern University who visited the writer fs classes. Comments by these students indicated enthusiasm for the work being done. Some of the comments were as follows* 262 “I never dreamed that a study of geometry could he made so fascinating* It seemed to me as if every student entered into the discussion* I was particularly impressed about the way in which they questioned each other with reference to meaning of terms, relevancy of facts, underlying assumptions, and tested facts for consistency with assumptions*11 "X didn’t believe high school pupils could detect motives behind advertising, or state the hidden assumptions underlying adver tisements* X was particularly impressed by the critical analysis made by the little girl in the fifth row, regarding Angora Knit-Tex* “Nowhere in my experience with mathematics can X recall discussions dealing with miteMt SLQlf*W&Ma&QlU&X-iliaa, consideration of iafliYiduftl ■and. gpciaL c.pjtf.qqa^ac.qg> and particularly the .taafat.taanesfl of conclusions. X was taught that mathematics was an exact science and your emphasis upon conclusions as being tentative really was a surprise.*1 Example No. 4i This illustration is not an example of critical thinking, but of the procedure used to develop this type of thinking. It represents a comment made by a high school teacher of mathematics from Los Angeles, This teacher was making a study of teaching procedures in mathematics and was par ticularly interested in the thirty schools involved in the eight year study of the Progressive Education Association* She was on her way East and after visiting the writerfs class remarked somewhat as follows! ‘•This is the first mathematics class I have visited in which X feel there is a distinct divergence from traditional procedures in the treatment of geometric subjectmatter* Tour emphasis upon certain elements in critical thinking has created a unique learning situation* • Anecdotal Becords Among the reports of pupil progress in critical thinking, submitted by the 165 classroom teachers who acted as observers, nearly 300 anecdotes were presented regarding specific situations wherein this behavior was in evidence* Anecdotal records are one form of what Tyler 84 refers to as “the collection of appropriate evidence ... in the various directions which are important for educational development.11 In order to point out contrasts and variations in the different 84 Tyler, Balph W., oops cit* pp. 9-16* 263 situations involved, eleven anecdotes were selected for this purpose* They are as follows i Example Ho* II This is an illustration of a pupil questioning the meaning of terms and testing facts for pertinency to a situation* *1 gave a test recently to my class in machine shop and one of the questions was! 'What is the mechanical device used in the common automobile engine that admits gasses into the combustion chamber?1 The answer to the question was 'the carburetor1• Charles _ _ _ _ _ (Pupil Ho* 275) , one of my pupils. Insisted that the answer should be the 1intake valve1, and to prove his point he presented the following arguments 1There are three parts of the engine that may qualify for the answer to this question, namely the carburetor, intake manifold, and the intake valve, (a) The intake manifold is eliminated because it is not a mechanical device; (b) the carburetor is eliminated because it is a degice for mixing air with gasoline vapor and it admits the gasses into the intake manifold; (c) the only remaining alternative if the intake valve* iThis fits all of the requirements because it is a mechanical device and it admits gasses into the eoohbustion chamber in all common automobile engines*111 Bbcamole Ho* 2% This is an illustration of a pupil attempting to detect motives and searching for more facts* "Duane •** (Pupil Ho* 144) surprised me by dropping in to see me after school one day* He never impressed me as a boy that could be very serious about anything* It seems that I made a remark in connection with a discussion of tunnels in our geography class* The remark had something to do with the train whistle waking up some of the passengers while going through a tunnel* "Doane had the impression that I said the engineer should not blow the whistle in a tunnel because it may waken some of the passengers* I don't remember which way I really did say it, but what surprised me was the fact that this boy was actually thinking. He informed me that in situations of this sort the natural motive on the part of the engineer would be to blow the whistle in order to frighten any animals that may be seeking shelter, or to warn trespassers that a train was approaching* When I asked him how he knew all of this he said he didn't know but that he intended to find out for sure." Ha- a: This anecdote is an illustration of several elements of 264 critical thinking, namely: (l) questioning the meaning of terms, (2) questioning assumptions, (3) seeking more facts by bringing in more evidence, (4) testing an hypothesis, (5) testing facts, (6) formulating a logical conclusion, and (?) evaluation for blast "In my biology class recent Bill **• (Pupil Bo* 324) did a job of thinking that I didn't believe was possible for him to do* We were studying the carrot and reached the conclusion that carrots store starches in their roots instead of sugars* One of the other boys, John « said he read somewhere that starches were soluble, because they could penetrate a membrane and be tested* Before I had a chance to question John, Bill came right out and said that he was 'under the impression that John had his wires crossed1• Recognizing the possibility for a clean-cut argument and at the same time being surprised at Bill's sudden awakening, I permitted the two boys to settle the argument in their own way* It seems that John was a bit disturbed and indignant toward Bill's questions and soon began to show evidence of not liking to be pushed back in this way* I was about to settle their argument when Bill said: 'I'll tell you what I'll do. If Mr. (meaning the teacher) will let me have that carrot on his desk and also some of the starch solution, I will prove that starch will not pass through its membrane, and that it is insoluble*' (There was only one thing for me to do and that was to let Bill go ahead* He proceeded to take the core out of the carrot, then partially filling a beaker with the starch solution, he inserted the carrot so that its open top protruded about a half inch above the solution* He then put some distilled water into the hollow core of the carrot, saying it would tend to maintain its equilibrium a little better* He then concluded with the statement that if any of the starch penetrated the membrane, the water in the core of the carrot would test positive and if we let this stand over night we would be able to find out for sure whether or not John was right* This turned out to be one of the most interesting classes and Bill actually verified his experiment, besides bringing to class a science book containing written evidence showing that starch was insoluble under these conditions*11 Example Ho* 4: (This anecdote is an illustration of several elements of critical thinking,namely: (l) questioning assumptions, (2) seeking facts pertaining to the situation, (3) formulating logical conclusions, and (4) testing conclusions for consistency with assumptions* "Virginia *«* (Ptqoil Ho* 161) has become very critical in my English class lately. Por example, I remarked that in order 265 to prove that compound -sentences with commas before the conjunction were easier to read, it woiild he necessary to make up two tests of compound sentences* one test with commas hefore the conjunc* tion and the other one without the commas* Using a stop watch* one could determine in this way which reader would finish first* 1 concluded that the one reading with the commas hefore the con junction would most likely finish first which* of course* would prove the proposition* "Virginia was quite troubled with the word prove and definitely insisted that this was merely an assumption on my part because I didn't prove the proposition* She was so hard to convince she even suggested that we try out the experiment for ourselves* It appeared as if she was making amountain out of a mole hill* Her major point of contention was that the difference in the reading rates of two people would probably influence the results to a large extent* "To satisfy the curiosity of the class as a result of this dis cussion* we performed the experiment* Our results were rather irregular* but we did conclude that commas before the conjunction made the reading easier*" Example Ho* 5: This anecdote is an illustration of several elements of critical thinking* namelyi (l) questioning assumptions, (2} searching for more facts* (3) testing facts for pertinency to the situation* (4) eval uating self for bias* and (5) suspending judgment. This situation is particularly illustrative of the element of suspending judgment* "I was explaining to my pupils in Business Correspondence how to fold letters before putting them into envelopes* when Stank (Pupil Ho* 322) said' *1 have five samples and two books at home which are on letter writing and how to fold letters, yet not one of them corresponds to the way in which you have described it." "Having in my possession a very recent book on the subject* I opened it and read the passage to the class regarding letter folding and it agreed with my description* However* I did make the mistake in saying that this was the latest method* because my book was quite a recent edition* Anyway, Trank came in the next day with one of his books and showed me where our methods differed* When I agredd with him that the method was used* but that it was probably an older usage* he asked me what the copyright of my book was* When I looked it up I found it bo be 1935* so Trank proceeded to show me that his was 1937* He seemed to be a tactful and openminded youngster* because he immediately remarked that the author of his book might not be aware of newer methods 266 or that lie might have a personal bias in the matter* He farther concluded that the date of copyrights had very little to do with hest or even conventional ways of doing things* In my judgment this hoy is doing a splendid job of thinking.0 jbosmrple No* 6? This anecdote is an illustration of the following elements of critical thinking, namelyI (l) seeking and questioning the meaning of terms, (2) questioning assumptions, (2) testing conclusions for consis tency, (4) discriminating between facts and assumptions, and (5) searching for facts pertaining to the situation* HX assigned a Latin lesson to my class which included a short story on the City of Borne* Hext day one of my pupils, Bosalind _ _ (Pupil Ho* 259) gave me one of the most critical descriptions of this story ever presented by a pupil. Zn fact it was the first one of Its kind that I had had* It was somewhat as follows ? *Where it mentions Home today as a 0modern city0 I begin to wonder if such a city really can exist, and if so are not all cities of today modern? Then where it says that Borne had become one of the most 0powerful0 nations, I think powerful needs to be defined. Certaini$ this is only an assumption on the part of the author because he fails to support his statement with evidence* The author goes on to say dogmatically that ttthe first settlement was on the Palatine Hill0* To me it seems that he would have been more correct if he had said that Hin the light of the best evidence available, the first settlement was on Palatine Hill0* Unless he qualifies his statement it remains an assumption* It seems to me that he is taking too many things for granted, because there is no evidence to support his statement that there were no other people there before 763 B*C*10 Ifrfiynpie Ho* 7: This is an illustration of the following elements of critical thinking, namely? (l) questioning the meaning of terms, (2) testing conclusions for consistency with assumptions, (3) searching for more facts, (4) evaluating in terms of consequences* 9t thought you would be interested in a brief description of an incident which occurred in my history class last week. We were discussing Aristotle and his influence on inductive and deductive reasoning* I gave some examples of each and stated his so-called three laws of thought} in connection with these three laws 1 made 267 some comment relative to their being the basis for all clear thinking, when lo! and behold! one of my pupils called my hand. (Pupil No* 29l). She remarked somewhat as follows* 'In my own thinking all human beings are subject to error and since men are considered to be human beings, then men are subject to error. Now, laws are made by men, therefore laws are subject to be in error. Don't you think that your illustration of Aristotle's logic proves that logic itself is subject to error?' "What a question for a youngster to ask a teacher! She seemed to be well versed on this particular phase of our discussion, because she mentioned that in her reading a book by Bell on the Search for Truth. reference was made to many valued logics and that it was proved only in 1920 that Aristotle's second law is not necessary for consistent reasoning." Example No. 8* This anecdote is an illustration of two of the elements in critical thinking, (by a pupil of exceptionally low ability), namely* (l) questioning assumptions, and (2) searching for more facts pertaining to a situation. "One of the pupils, Ralph . (Pupil Hoi 24), whom you asked me to observe with regard to critical thinking, made what I con sidered to be a contribution to the class. I say this because he is of very low ability and because he generally site back and sleeps. On this particular occasion we were discussing the topic of 'plants and their value'. Some pupil remarked that plants actually have no value other than furnishing us with food and beautifying our surroundings. Ralph took exception to this statement so suddenly that the entire class was dumbfounded. He said* 'That's all right, but there is more to it, because without plant life there would be no oxygen and we could notlife without oxygen. In fact there would be no living thing on this earth. Where would we be if it wasn't for the simple one-celled plant? How could we breathe without plants?' "This was really remarkable for this serve as a stimulus for his thinking Example No. 9* boy and I believe itwill from here on." This anecdote illustrates several elements in critical thinking, namely* (l) questioning the meaning of terms, (2) questioning assunrptions, (3) searching for facts, (4) searching for motives, and 268 (5) formulating conclusions. Shis teacher remarksi "In my English class something came up about Caesar and Brutus. It was in connection with a Question in the book9 namely: *014 Brutus do the right thing in murdering C&esar, or should he have gone against the conspirators in defense of Caesar.1 One of the girls in the class (Pupil Ho. 140) remarked: 'This is really a silly Question for our day and age, because our conception of right and wrong has certainly changed in all this time. Besides, what may have been right at that time may be wrong today. Hhat we really need to do is agree upon what we mean by right and then find out what the people in Caesar1s time considered to be right, and if our notes were in agreement then we could decide the question. But, of course, we will never know exactly what people thought was right at that time except from what we read in books, and that is merely a lot of information handed down by biased individuals. finally, there is one more point that we have to take into consideration and this is the motive Brutus had for the act. Since he alone knows that motive, it is doubly foolish for us to try answering this question. * NX thought you would be interested in this little note because you asked me to be on the lookout for this kind of behavior.M Example Ho. 10? Shis anecdote illustrates the following elements in critical thinking, namely? (l) questioning assumptions, (2) discriminating between facts and assumptions, (3) searching for facts, (4) formulating conclusions, and (&) considering consequences. This adviser remarks? »In our home room, we were discussing the topic of ,safety,9 and one of the boys had a newspaper clipping stating that ’people under the age of 21 and over the age of 60 should not be permitted to drive*• "One of the best discussions came from a boy (Pupil Ho. 144) whom you asked me to observe for critical thinking behavior. This boy brought out the following points? ’The person who published that statement is assuming that young people, due to their lack of experience and foresight, have caused many automobile accidents. Certainly he has failed to produce facts to this effect. He also assumes that people over 60 have many physical and mental defects, so that when driving a oar they are unable to react to an emergency as quickly, igain there are no facts to support his statement. 269 He also assumes that limiting the ages of people for driving cars will cut down the number of accidents* •Assuming that this man's ideas are good because he doesn't offer any other possibilities and that they would become a law, let's take a look at the consequences* First of all* psychologists tell us that certain skills are learned easily early in life* So if this is true one cannot learn to drive as well after 21 as he could have before 21* This also would mean that anyone between 21 and 6o could drive, which includes physically deformed as well as mentally deficient persons* •My conclusion to this problem would be to enact a law re quiring a rigid and fool-proof driving text following a medical examination* I believe that the solution to this problem must be a scientific one*1" Example Ho. 11? This anecdote is an illustration of what one observer considered as "negative" evidence in critical thinking* Zn other words, the pupil in question has failed to exhibit evidence in this type of thinking* "Sometime ago you asked me to observe several pupils for improve ment in critical thinking abilities* One of these pupils. Bob . (Pupil Ho* 193) is a complete 'wash out' in my opinion* He has average ability and actually did some work during the first two weeks, but ever since then he has come to class for no good reason whatever* He breaks into discussion at any time with some silly or cute remark* After being dismissed from class twice, he still is inconsiderate of other members in the group. He has become very sullen and indifferent. Z checked up and found some of the other teachers are having similar trouble with him* Surely he is not thinking critically in your terms, because if he was one would think he would become more considerate of others and of con sequences." An analysis of the 293 anecdotal records submitted revealed that 270 or approximately 92 per cent were indicative of critical thinking. The remaining eight per cent failed to reveal evidence of this type of behavior. Analyses of Punil Diaries Pupils from four of the six Experimental Groups were asked to keep diaries during the month of January 1938. Although the assignment was not 270 compulsory, nearly all of the pupils submitted diaries at the end of the month# Since this technique was not applied during the first few weeks of the course, it was impossible to note changes that may have taken place in critical thinking abilities. Therefore, this particular tech nique provides data that may be analyzed only for the presence of elements in critical thinking and not for the amount of improvement in this ability# The “diaries11 were not confined to any particular type of situar tion, and each pupil was free to Include any items he felt should be recorded. The following examples were selected from over 200 diaries, and may serve to Illustrate the variety of types of experiences encountered by the pupils# Example No. H Illustrates the elements of searching for motives and making generalizations based on facts# “Last night while reading the paper X noticed a liquor advertisement saying that people should pay their bills and get the necessities that they need before buying liquor# This seemed like critical thinking to me because it didn't encourage people who need their money to waste it, as most advertisements do# When I thought it over again, however, I could see that the motive behind it was to get people over on their side. I think this is the 'soft soap' type of propaganda we discussed in class, the type some politicians use when they start going to church just before an election#" Example No. 2S Illustrates the elements of searching for more facts and I suspending judgment# I "In my English a paper was returned to me and on it a check after each comma, meaning that a space should be left# I looked at my typing book which X used to take typing the year before, and another which my brother had during his typing course# Neither of these books placed a space after the comma# With these two books and other sources of information I went to my English teacher and told her X thought that there should be no space after the comma# After showing her my evidence she looked at her book and found that she had made the mistake# My conclusion to this is that you should investigate something which you think is wrong before you accept someone's word for it#" 271 Bxscap.le Ho* 3t Illustrates the elements of seeking motives* questioning assumptions and searching for facts* *I have seen many signs that have a person on them who says, 'I would Walk a mile for a Camel*' H0n my way to school today X noticed a picture on a sign of a very attractive lady, who in my judgment appeared as if she never had to walk very far* If I were to helieve this advertisement I would like to know if she was paid to make this statement and if she really smokes* If she does smoke I would like to know what hr and; if they were Camels, how far she actually walks for them, and finally, I would like to know if she really would walk a mile for one?1* jxample Mo* 4* Illustrates the elements of questioning meanings, recog nising the place of assumptions and testing conclusions for consistency with assumptions* RIn the report of a certain student on the governments of the world, it was stated that Fascism, Communism and Socialism have proved to he unsuccessful* Before accepting this statement, as I know many who heard it did, I would go into it, thinking very critically* First of all I would want to know what a successful government was* I do not helieve that any form of government can he entirely successful* So how can anyone prove that the above-mentioned governments are unsuccessful? Unsuccessful in what? Supposing everyone did have the same conception of successful* Every government functions upon a set of basic assumptions. One government may he successful in terms of its standards where another may he a failure on this same basis*11 Erapple Bo* Si Illustrates the elements of detecting assumptions, testing conclusions for consistency with assumptions, searching for facts and suspending judgment* 111 happened to he home listening to the radio last night when I heard the announcer talking about Maxwell House Coffee* He said that Maxwell House Coffee is 'refreshing' and that it also 'gives you energy'• I happened to think, if what he says is true, why doesn't everybody drink it if it is so wonderful? I assumed that he was taking a lot for granted by saying it would do those things. Maybe It won't give you energy and maybe it won't be as refreshing as he said*" §72 Example Ho. 6t Illustrates the element of searching for more facts. "Last night I heard an advertisement over the radio for Lux Toilet Soap. The announcer said that all over the United States mothers of twins and triplets are using Lux. Then he read what was supposed to have been a letter written hy a mother of twins. It told of how and why she used Lux for the twins. The announcer did not mention her name nor where she lived. He failed to give evidence that mothers of twins all over the United States were using this soap and gave no proof as to whether or not that was a real letter he read. Why do they use twins for an example? Wouldn't it he just as good to use two children that are not twins hut are in the same family? ■ Example Ho. 78 Illustrates the elements of questioning the meaning of terms, discriminating between facts and assumptions, searching for more facts, and suspending judgment. "On the radio I happened to turn the dial to the 'Lone Hanger* program and I heard the announcer say, 'Get Silvercup, the world's finest bread, at your grocer's tomorrow.1 I think this is an attempt to state a fact that could not he proved. What do they mean hy finest bread and when taking the world into account, how can a test be made of every kind of bread?" E m m i e Ho. 8: Illustrates the elements of searching for facts and for mulating generalizations, based upon many facts. "The adviser room students were discussing should be a forty-five minute period. The cussion was a yes or no, and kept up until all the possible Information and then make basis of facts.'* Example No, 9: whether or not there first part of the dis I said, 'Let's gather our decision on the Illustrates the elements of suspending judgment and searching for more facts. "I tead an *ad* that said, 'Clean your teeth the dentfcst's way with powder. * I asked my dentist yesterday if all dentists used powder to clean teeth and she said that she didn't think there were many dentists who used powder alone. They generally use a powdered substance and add a paste to it- They clean your teeth mostly with a paste. I guess this doesn't prove much except that they don't use powder entirely." 273 Example Ho, 108 Illustrates the elements of searching for more facts, self-evaluation, testing conclusions for consistency vith assumptions, suspending judgnent, and considering consequences, *In my English class nearly everyone fools around and partic ularly me. The teacher is an exchange teacher and he is very liberal, very seldom giving discipline notes, I talk out when I'm not called on and laugh too frequently and make a joke out of a lot of things. He has talked to me several times in the course of an hour, telling me to be quiet* This has been going on since the first day I came into his class, I decided this conduct most cease because I disturb the rest of the class and lead them astray, 1 take advantage of the teacher's better nature; it hinders my work. This teacher has given me several breaks and it is because of these facts that I decided to cooperate •«, Everything goes along better now, at least it has in the last two days, since I started the new deal*11 Example Ho, 111 Illustrates the elements of detecting assumptions, discriminating between fact and assumption:, and suspending judgment, "In the newspaper last night I read a statement that said, 'We will win out in the end because we are in the right,1 To me not all people who are in the right win out. Take China and Japan, or Italy and Ethiopia, Is Japan in the right? Was Italy in the right? It all depends on which side of the fence you are on,H | Eprfimple Ho. lai Illustrates the elements of detecting assumptions, discriminating between fact and assumption, self-evaluation for bias, | searching for more facts, and suspending judgment, | | I | ! I 0During a discussion of government in one of my classes, one person said, 'A democracy is the best form of government,1 I immediately challenged this statement. The person became infuriated because of my challenge; however, she failed to give evidence in support of her statement, and I have never seen or heard of anyone who can prove it. One boy said, 'WellJ Democracy is working in this country, isn't it?' In my way of thinking that's not a proof of the statement*4 democracy is the best form of government,1 I still think this is an assumption, even though I am a firm believer in democracy.11 j ! Byampiftct Nos. 13 arid 14: In order to give the reader a more complete 274 picture of the content and progressive stages of a diary over the indicated period of time, two pupils1 diaries were selected* Since separate analyses were made of each excerpt in the preceding situations, it was felt that such procedure would result not only in repetition, hut also in detraction from actual context of the diaries to he presented* Therefore, it may suffice to.point out that the elements in critical thinking present in each of the two diaries to he presented, namely those of Pupil No* 259 and Pupil No* 279, are as follows: 1# 2* 3* 4* 5* 6* 7* 8* 9* 10* Seeking motives Questioning the meaning of terms and seeking definitions Detecting and questioning underlying assumptions Searching for more facts Testing facts for relevancy Discriminating between facts and assumptions Testing conclusions for consistency with assumptions ©valuation for bias Suspending judgment Considering consequences* Bxamole No* 15: Excerpts from the diary of Pupil No. 259 (girl) are as follows: Wed., Jan. 5, 1938 "Tonight it so happened that we had a discussion of communism at our dinner table. It was all started by my brother, who absentmindedly and for the lack of another word said, 'OhJ that guy is a communist.1 "Although he, my brother, is only thirteen years old, he believes he has enough information and knowledge to cell an adulf a com munist. I began to sit up and take note, because I felt some pretty careless statements were coming that would fall right into my path. X asked him if he knew what a communist was. He said it was a person who believed in communism. When I asked him what communism was, he looked even more queerly and doubtfully. Then X asked my father. He said it was the belief in a common place for grain, or something to that effect. This, of course, didn't get me any place. I did ask my father if he could draw a line and say this m^n is a communist and this man isn't, if it were put on a scale measured by degrees of their beliefs. He admitted that he couldn't do that, although he insisted he knew what a communist was. 275 HHe didn't exactly understand what I meant by the scale and I probably haven't made it very clear here, either. I tried to explain it hy using a scale, and asking him if he believed a person could he shown hy a scale. The ignorant people at the bottom of one end, the genius at the other end or top, with the normal people in the varying degrees in between. I also asked if genius wasn't just a few more degrees of normalness on the scale, and ignorance a few less, and if this was the only difference between the people. He said he thought so, Then I compared the scale to communism and asked him if he could draw the line saying, 'This man is a communist and this man isn't, "* Degrees of Becoming A Communist Non-Believer in Communism Stronger Believer in Communism Jan, 4, 1938 "from the oral reports given on Famous Greeks in our history class, we got on the subject of the differences in the beliefs and preachings of philosophers, Epicurus and the Stoics were the important men we discussed. Both of them figured out what theytthought was the best way to live, the way to get the most out of life* This is what I consider critical thinking to be, Epicurus believed in happiness first and before all other things. The Stoics believed that no matter if it did you pain, you should do it and take things as they come, (my history teacher) illustrated this point by a picture of a mountain, the name of which was 'A Good Life', The question all philosophers try to answer is how to live the best life, "Then we got on the subject of reasoning - deductive and inductive, Aristotle said these were the two ways of thinking or using know ledge. As an illustration of deductive reasoning, Mr. _ _ asked for a true statement. 'The earth revolves around the sun* was given. The class decided this wasn't true because it couldn't be proved, Mr. ______ wrote, 'It is dark in our part of the earth' as a pro position, From this they decided that 'the sun is on the other side of the earth' (assumption). T o r inductive reasoning you start with an observation and work down to the establishment of facts« 276 1* 2* 3* 4. observation assumption testing facts This is what he put on the hoard* He pointed out that induction was the method used hy detectives* Someone else offered a good example of poor reasoning* 'The Joneses live in a red brick house on ninth Street* 1 live in a red brick house on Ninth Streets therefore my name is Jones*' HI suggested that there might have been more than one red house on Ninth Street* Another said his name might also be Jones, but a different family entirely.11 Nri., Jan* 7t 1938 M ’It always has been and always will be difficult for a strong people to rule weaker peoples fairly*1 "This is quoted from my history book, Ancient and Medieval History, by Magoffin and Duncalf* It certainly is a strong assumption on their part, to say that anything will always be so, or can never happen* No one knows what will happen in the future* I believe the motive behind this statement is to bring to our attention the element of selfishness in human beings* Of course I would like to know what they mean by 'difficult','rule', 'always*, 'strong', •weaker1, and 'fairly** nXf anyone were to have something to say about what has £one before or what the future will hold, it ought to be the historian; however, some may be biased and I would want the opinion of several historians.8 Tues*, Jan* 12, 1938 "In my Latin book there is a forward headed 'To the Teacher'. One of the sentences runs as follows* 'One of the most hopeful signs in the educational field today is the attention which is being given to superior pupils* * Words that need defining are* hopeful sijns (hopeful for what?), educational field - (is too all-inclusive), superior -pupils - (superior in what way?)*8 Sri*. Jan. 14, 1938 "Tonight at the dinner table my brother began talking about a little paper he and another boy were getting up* He mentioned that they were going to quote a student from a speech he had made in the student council* They were quite sure they remembered but they wouldn't print it until they had gone to two or three other boys who also had attended the meeting* ttI don't believe this to be very deep critical thinking, but it's along the lines of what we discussed incdlass, about getting the opinion of more than one or two people*8 277 Wed., Jan. 12, 1938 "The other night we had a man at our house for dinner. He was supposed to he a very intelligent person and I was trying to listen for conclusions he would state on different matters and how he felt toward them. "Because our ferally invariably gets started on some worldly problem we started on the subject of popular books. He remarked that How To Win friends and Influence People was the most absurd thing he had ever heard of. Whose were the very words he used. He said anyone that would read or believe that kind of stuff was very ignorant. This is not exactly what he said, but it is the same idea... Anyway, if X could have asked him how he arrived at such a conclusion he couldn't have answered very intelligently (an assumption on my part). I found from just what he said of his own accord that he hadn't read the book... "Ilnally the topic of progressive education! He said he thought it was ridiculous and absurd. About this time X was ready to pull my hair and X guess X must have looked the part. Hot being able to hold it in any longer, I stated how marvelous X thought it was for some children. He named a man who was almost in favor of it at one time and then went around to visit progressive schools and now is strongly opposed to it. He was taking one person's opinion and didn't have facts to support his statements. He said students were permitted to do anything they felt like doing and nothing they didn't want to do, and that that was ridiculous. He said that in a certain town the students took whatever subjects they felt like taking. It seems to me that this man has made a generalization based upon one man's opinion, which to me is a very weak one, and that he is biased Against progressive education. * Example Ho. 14: Exeerpts from the diary of Pupil Ho, 379 (boy) are as follows: Jan. 4, 1938 "Was planning a trip to the north woods to ski, but after weighing all of the possible facts decided that it was out of the question. Discussion in English about ways of communication led to the telephone and to the difference between talking direct and talking over the phone. The difference is that one costs money, the other doesn't; one you merely hear the voice and the other you can see the facial expression. "I was thinking over the rules in basketball and have been trying to figure out all the different kinds of fouls. I enjoyed watch ing the H. U. and Wisconsin game. X tried to follow the referees a lot of the time, and get a real insight of all the decisions instead of just watching the players alone. 278 HI listened to the advertisements over the radio last night and picked out many hidden assumptions ... They were quite obvious, such as 'World's finest bread1. "X corrected Rosalind in geometry regarding her statement that the exterior angles of a triangle equal 180°. The correction was that the sum of the exterior angles of a triangle is 360° and in general the sum of the exterior angles of any convex polygon is 180° and with the proper algebraic signs this theorem holds for all polygons* * Jan. 5, 1938 “Tried to contribute today in geometry in correcting others as well as offering original suggestions* Read the Tyranny of Words again and it takes on more meaning every time I read it* I've reached the same conclusion that Stuart Chase has as to the meaningless labels attached to words like 'socialist*, 'communist^ and others* "Tor the last few days X have been thinking a lot about what X am going to do and what X want to be when X am out of school* I've got a lot of ideas but X can't quite make up my mind as yet** Jan* 6, 1938 "Contributed in geometry today about the statement concerning Mr* Harrison* Also added several facts and assumptions and pro positions to the discussion we were having. X had to criticise orally themes in Rnglish today. X criticized two of them in a critical way, and gave some pertinent suggestions to other members of the class* * Jan* 7, 1938 "Criticized the basketball game for strategy. The strategy near the end was particularly good, X thought, because they held on to the ball until they had a break or a set-up* "X became Critical about my diving and decided that X must atart working hard* Today my dives didn't suit me nor anyone else and X knew it* "After the discussion on fourth dimensions in class, X've been doing a lot of heavy thinking. Xt seems that the more you think about the place of assumptions, the more you appreciate the conclusions reached* * Jan. 10, 1938 "My thinking on Triday as to my diving helped quite a bit. X was more satisfied with my dives today than any other day for a long time* 279 *1 saw an advertisement today for the Quaker Oats Company. It read* Actually Blown from Guns. I was curious so I looked up the Quaker Oats history and found that instead of being actually blown from guns* it was heated and then blown out of a tube with terrific force* just as I had first believed. The 'gun1 part, I finally concluded* was the compressed air being let out sud denly* resulting An a loud BANG.11 Jan. 11* 1938 "Continued discussion on parallelograms in class today. Lots of critical thinking done* I thought. Had a talk with my teacher after school. Had to do some thinking on the questions he asked me. I did some criticizing in English of oral themes. I did a thing yesterday which was an example of not thinking critically. X sent for some Skis and when X thought it over 1 recalled that X had forgotten to put down the length X wanted* But on the other hand* X did think critically when X looked at a lot of skis before sending for them. X consulted several people as to what variety of ski to buy.® Jan. 12* 1938 "Last night while listening to a news broadcast X heard several things that interested me* about a big clipper ship that crashed. The commentator reported that the plane burned up while in the air. He then contradicted his statement by saying the last word that was heard from the ship was that they had had a little motor trouble and were returning to the field. X do not believe that the plane could have burned up in the air* because after reading the papers today X seem to find that too much of the plane was left to have burned up as the announcer put it. If no word had been heard from them after the final message* then X don't believe any one could really tell what happened. X have decided that X will look up and try to find all the facts X can on this question. "I also heard over the air an advertisement which read 'Use Spry, creamier and more wholesome.# X talked to a number of my mother's friends about it and they said either that they had never tried Spry or that it was not any better than other shortenings. X think the words 'creamier' and 'wholesome* need explaining. Creamier than what? More wholesome than what? Xt could be creamier than any low rate fat and the statement could be true. Or more wholesome than bacon grease and it would be true. Actually* I believe their motive is to get the listener to think that their product is better than that of their competitors.® Jan. 13* 1938 ®I came home from school today and found my sister* who is ill* listening to the rddio. I went in to visit with her and listened to a few programs. X decided that a great deal of the broadcasting 280 is very 'bad* They seem to me a waste of time, senseless and very silly* Some of them are improbable* The thing I object to most is that they consume so much time one could he spending in a more profitable manner, and there is very little to be gained hy listening to them* If X was planning a complete program X would have fewer shits and more of the educational things like the 'Man on the Street*, 'Professor Quiz'* More good news broadcasts* the 'Town Meetings'* and all things like that* .Amateur hours and operas (both light and grand). Since so many of the people in the If. S* listen to the radio programs X believe we ought to cut down on the daily skits and have more good programs for the benefit of the people in our country as a whole* Other things on the air X like to hear are some of the good comedies, but not too many, and a good oystery story every now and then* Some of the good dance bands should be put on at regular intervals at night I believe* After thinking the problem oven, X believe our nation would be more highly intelligent if we had more of the better types of programs over tha air*" Jan* 14, 1938 "Tonight X went down to the basketball game and found it very rough but exciting, plus interesting* After X got home X began to think about what had happened* While X was there X thought of the fun X was having pushing and knocking people around in the fight which followed, and thought that our opponents played such a dirty game that they got all they deserved and some besides* "Suddenly it occurred to me that we were the home team and that we had not shown very good hospitality. Just because two boys who were mad had started a little fight was certainly no reason that the spectators should join in* There was also great danger that some one might have been hurt, particularly when the man jumped out of the balcony onto the crowd* "X always have thought that one of the most beneficial things gained in games was learning to be a good sport. This did not seem to be evidence of good sportsmanship, because we were put in a position of being angry because we had lost* Then, too, there is also the grave chance of being dropped from the sub urban conference because of our unsportsmanlike attitude* The consequence would be that the school's reputation would be lowered and we would be called all sorts of names unbecoming for a school that has gained the respect that ours has*" Jan* 17, 1938 "I was thinking about the ski meet I saw Sunday and have decided that it is one of the greatest sports for courage and determination* jp I watched those men come down the slide I couldn't help thinking of the grade, balance and courage of every man who had entered the competition. When I had thought over the past of the men, I came 281 to the conclusion that to he a real skier one must almost devote his life to it* However, there are lots of consequences one mus t take into consideration to he a good skier* First is that you can't he an expert skier all your life, for some day you'll get too old to compete with the younger, more active skiers. Secondf you have to take the chance of a hard fall and possibly a serious injury* (Third, during the summer you have to transport yourself to colder regions in order to keep on prac ticing* "I myself love to ski, hut taking the facts in hand I helieve I would rather he a gentleman skier as it were rather than an expert who has to do it for a living* On the other hand, if 1 lived somewhere where there was a lot of snow all the time 1 can see why people in that position will do it for a living* *1 helieve the large turn-outs at the meets signifies the great popularity of the sport* If ski interest keeps on increasing the way it has for the last five years it will he one of the prominent national sports of America, hut this is only an assumption. * Jan. 18, 1938 "When I got home tonight as I came in the door I was met hy my mother, who was already well into a lecture ahout my not writing a certain Christmas letter* (This little lecture set me to thinking that procrastination is one of the chief faults in Anerica today* If we could get to the place that we didn't have any of it, think how much happier everyone would he* It certainly would he much easier for everyone if we would stop. I've made a critical analysis of myself and have found that I am a great proerastinator* I let my homework go until late at night and as a consequence I have to hurn the midnight oil most of the time to get it done* I could have gotten some sleep as well as some enjoyment out of the rest of the evening* Another thing I put off is calling some one I'm not particularly Interested in and find out latdr that I've missed a nice invitation* Just think what would happen if you procras tinated ahout putting gasoline in your car* Tou know as well as I do what the result would he. After thinking the thing through I find that if you have to do a thing eventually, why not do it in the beginning and not have it hanging over you?*1 "Today when I was coming home from school I saw a crowd ahead of me* When I got up closer I realized that one car had skidded into another, with a dented fender as a result* I inquired from some one in the crowd what had happened. I was told that Mr* A had dented Mr* 3* 's fender a trifle, as well as his own. Mr, 3* was very excited, waving his arms around and demanding payment, while Mr* A was quite calm and dignified* As I watched and listened I felt Mr* A* had complete control of the situation, as well as the sympathy of the crowd, while Mr* B* looked very foolish* 282 "As I thought over this incident tonight X made these conclusions* Mr* A* had used his intellect and Mr* B, had used his emotions* Mr* A* could influence people better than Mr, B. and also X helieve X would entrust myself in the hands of Mr, A. more readily than Mr* B, From all this X began to wonder what kind of a person X was, X feel that X am calm under most circumstances and do not lose my temper easily." Jan, 20* 1938 "Added quite a few suggestions today in geometry class, This morning* while walking over from the corner, X picked up with a hoy whom X had met through & friend of mine. The conversation being at a standstill X asked, merely for the lack of anything else, what he did all his afternoons after shhool was out. He answered and told me that he always went home and either read, studied or sat around, When X asked him if he went out for any sports, he replied that he couldn't find time and didn't like sports anyway. At first X thought this was the most foolish thing X had ever heard anyone say, but as X thought about it during school X came to the conclusion that there really are two distinct types of hoys. As X thought a little more, X formed in my mind the difference between the two. The active boy would make many more contacts* would be much more relaxed in social life, while the grind would not make half as many contacts and be almost hopeless in social life for M s being afraid of everything and everybody. The first type would probably be more physically fit and prepared for a fuller life as an adult, while the second type would probably be Inclined to get sick and would be physically weak* Ho* 2 would in most cases, however* get a better grade than Ho* 1, but X don't believe grades alone warrant going home every night after school, Exercise is good for everyone. Ho* 1 type will have a much better appetite in nine out of ten cases* He is developing his muscles and learning to cooperate with a group of people* He gets more fresh air, while the boy who goes home probably stays in the house. Of the two X try to be in the Ho* 1 class, and X believe X m getting more out of life than the boy X was talking to, but maybe I'm wrong, for he probably gets just as much fun out of what he does as what X do with myself. Every one's entitled to his opinion and X think X will stick to mine. It's good to go out for school sports and takt part in activities, rather than be a grind and stay at home." Analyses of Written Work Analysis of written work was another effective way of evaluating critical thinking abilities, because the pupil was not conscious of the fact that his work was being evaluated in terms of the ten elements. There was a great deal of evidence of critical thinking available for the 283 Experimental Gcowpa in connection with their written work. Some of this evidence has already “been presented in Chapters IV and V* particularly in connection with individual projects and voluntary contributions which were filed in folders* Since the latter are to he discussed later* only the other types of free written work will he examined in this section* These include themes* written reports in English, social studies* science, and other subjects, and reports regarding lectures and reading. In order to illustrate evidence in critical thinking in the types of situations just mentioned* the following examples are submitted* 1* Excerpts from Themes Pupil No* 244 writes* "One evil the Indsutrial Revolution brought with it* which was an unfair burden for the working class* was the 'sweat shop' (meaning employment dealing with drudgery* overwork, and low wages), !Phis was the time before the labor unions put a stop to them. Men* women* and even children were worked for six* teen to seventeen hours a day* and received starvation wages for their efforts,.• "Only a powerful force for justice such as the labor unions could put a stop to this. Of course* this is an assumption on rry part; however* it is a fact that the labor unions did this very thing in the early 1900's*" The above pupil appears to be sensitive to several elements in critical thinking, namely (l) meaning of terms, (2) searching for facts* (2) recognizing assumptions* (4) discriminating between facts and assump tions* and (5) considering consequences* Pupil No* 315 writes* (,Qne thing a person should have in order to get along with other people is respect and consideration for their interests* needs* and beliefs. One should be open-minded or unbiased and avoid condemnation of others, because in a free humanistic society like our own country every man, woman and child is entitled to his or her own opinion. We should have respect for each other's points of view and not try to force our own views upon some one else." The above pupil is sensitive to the element of bias as well as 284 formulating conclusions that are consistent with his assumptions* Pupil No* 125 writes* flIf we permit foreigners to enter our country on a large scale, then our cities will become more crowded* Many of them will not get Jobs; others may take jobs away from those who now have jobs, because they may work for less money. We already have too many people on relief and this would make our problem more complicated* "If we do not let any foreign people come into our country, then this would be unfair, because we were foreigners at one time and not so very long ago at that* Foreigners ought to be per* mitted to come here in order to improve thftir way of living and ours, too, if possible ••• MI believe this is a problem for experts who are not biased and who will study all of the aspects of the problem scientifically* " The above pupil appears to be sensitive to the place of assumptions in a situation and their implications for conclusions* The conclusions reached by this pupil are consistent with his assumptions* Furthermore, this pupil is evaluating himself for bias, because facts are being pre sented on both sides* Finally, this pupil suspends judgnent by pointing out the need for an extended study of the problem. Pupil Ho. 259 wrote a theme entitled "On the War-Path11 from which several excerpts were taken as follows* "This afternoon a college boy came to our house selling magazine subscriptions* He told the girl who is helping mother that he was working his way through college **• "He was very rude and most annoying. Hot only did he force the door open with the salutation that it was very important and he had to see Mrs* (my mother), but he pushed the girl back and entered the house before he was told anything. When told that mother was ill* and that she would tell mother he was here, he followed the girl up to mother*s room, where he asked many inquisitive questions* He could not be gotten rid of unless dealt with rather severely. *• "It seems to me that students are sent to college to receive a 1higher education* and if this is the type they tdrn out after three years (this young man having mentioned being a college senior) the schools are certainly failing somewhere. Of course, I am assuming that this boy actually is a student in the college he mentioned. I intend to look up his name; however, he may be using a fictitious one* "If he is a college student I am beginning to wonder why he 285 continuing in school, for it is evident that he hasn't learned the fundamental steps for living as yet# May"be he feels going to college is a fad, or the thing to do in order to get a job# As far as I can see college has done very little for him, except that it has made him feel very important in the world# His motives are certainly selfish and surely he has failed to consider con sequences of his acts# "I agree with one writer, namely Touts, when he remarked that one of the functions of .American colleges and universities is to teach students how to live#" The above situation brings out a consideration of assumptions, facts, conclusions, motives and consequences# 2. I»sen>t8 from Britten Beading Reports Pupil Ho. 161 reports on the Tyranny of Words hy Stuart Chase, in which he says * "This article, although 1 read it several times, was very difficult to understand# "Words and meanings, very frequently, are Just taken for granted# Two people might argue on one matter and both be right because the thing that they were arguing about was not correctly defined, or else one of the persons thought that it meant some thing entirely different from the other one. "Everyone has his own idea of what something means; but what some people think might be wrong, for they have not found out what the word meant when it was completely defined* HXt seems that Stuart Chase has raised a vital question in the mind of every reader of this article* To express ourselves clearly in terms of what we actually mean is very important# To be fully aware of motives, meanings, and the usage of language by others is also very important, because as Stuart Chase points out, words are most dangerous, particularly when they may be put together by propagandists to say that 'the war is on1 when in reality it isn't# In Germany, Russia, Italy and other places the thinking of the masses is well under control by means of propaganda, that is, they are warping the minds of children as the Chinese have bound the feet of little girls# "I believe that in order to get full benefit out of this article one should study it, and if interested, then one should read some of the other books mentioned by the author The above report implies several elements in critical thinking, searching for motives, questioning meanings, searching for more facts, evaluating arguments, and suspending Judgment* 286 Pupil No. 259 reports on Education for Living “by B. Touts from the November 1937 Forum, page 254, In which she says* "This article was the best I have ever read. I found it by chance, while looking for another article. "It gets down to assumptions that many people really have thought but have either been afraid to say so or thought it must be all right if the situation has gone on for so long already. It not only says what's wrong with our present schools, but offers in its place a good substitute. His ideas may seem a little radical at first to some, but the more you think of it the better you like it. You feel certain that the author didn't just write the article to have something to do, but to try to express his idea on an important problem, for the benefit of everyone. Is I have said, he didn't just knock the present system, but offered another which he was in favor of. "Here are a few statements from his article: 'Colleges and universities are generally run for faculties, not for students.' 'They are designed to provide a pleasant academic retreat for professors who are deeply interested in their chosen subject but who care very little about the job of education.' 'At most universities the student is the forgotten man*' 'The function of Anerican college and university education is to teach the students how to live.1 "These are all very strong statements. I do believe, though, that there is more than a little truth in them. "The article points out, too, that the world is changing one hundred times faster today than it was during the time of our grandparents. "Anything that calls for as complete a change as his plan would take many years to convince the people. Of course, taking a long time to make a complete change is often a very good thing; however, in this case I think it is too bad* (I have made many assumptions. )$ The preceding report implies several elements in critical thinking. The most dominant one is the recognition and place of assumptions in any argument* However, motives and consequences are likewise inplied. an attempt is made based Finally, on the part of thepupil to formulate a conclusion upon factual evidence; however, in this conclusion there is implied the element of suspended judgment. Pupil No. 279 reports on a movie, pointing out the merits of Chevrolet cars. This pupil writes* 287 HThe motives ‘ behind the movie are rather obvious* I think* The major one, I believe, is to advertise and promote sales of their car* Another is to show the public how their car is super ior to other makes of automobiles* Another is for educational purposes and the like* #1 believe before I'd buy the car I would like to ride over logs and see if the car would ride as smoothly as the picture showed, 1 would also want to see the 'no draft ventilation1 work as it did in the movie* I would like to have the friction explained in full to me, and why parts couldn't be made to use water for lubricant instead of oil* "I thought that the Cinderella comedy was to attract the attention of the observer; this was done through color, story, and excitement, so that he would notice all the modern things on this new car* They assume that almost everyone enjoys a comedy and will look to see what's going on and take in all that's being said and done* This is one of their assumptions, I believe* "I would like to secure more facts in the situation of the clutch as given in the movie* I would also like to know what 'best material available* means in their way of thinking* Best for the price of the best that can be bought? T,I m a little prejudiced in this situation as we have never owned a Chevrolet, but am trying to be as fair as possible* I have talked since with a number of people who own this make of car* Some say they aren't so good, while others say theirs are fine* Some say their cars get loose very easily, while others sqy they never have a bit of trouble* I would say that it is the treatment you give a car and not so much its make, but that is only my assumption. * The above report is Illustrative of several elements in critical thinking, namely (l) searching for motives, (2) questioning the meaning of terms used, (3) detecting hidden assumptions, (4) searching for more facts, (5) self-evaluation for prejudice, (6) attempting to discriminate between facts and assumptions, and (7) formulating tentative conclusions or sus pending judgment* Pupil Ho* 259 reports on the book Easy Bessons in Einstein by Slosson, in which she writes: "The following are a few of the best and most important ideas from Slosson's Easy Lessons in Einstein (my opinion): uSpace and time are usually considered undefinables. Slosson describes them as 'merely forms of thought - a frame-work of ideas fixed to suit our needs'. "To my knowledge, this iu the best description of these words I have read* They are quite difficult to define and X am sure it 288 will be sdme time ‘before a better description is found. "Another very good description was, 'A dimension is only measurable direction.1 Slosson points out that we can only actually see two dimensions - no more - no less. We have con cepts of dimensions from one to three. How we have Einstein bring out time as the fourth, and even concepts of five on up to & dimensions. "Another interesting idea was that we can't see a mathematic ian's point, because it has no dimension or size at all. therefore such a position or point cannot exist. "To fix the position of any event requires four dimensions. For instance, a man is shot. Where? At the corner of Seventh Avenue and Forty-Second Street. Hew York. This fixes the place by two coordinates crossing at right angles in a plane. But was it above or below this, on the twentieth floor of the Times Building or in the subway? Knowing this fixes the third dimension, but we still have to fix its position inaa fourth dimension, time. Was it today or last week, and what hour? If. then, we find out all four we can distinguish this shooting from any that may have occurred in other places at the same time or at other times in the same place. Tesseract 289 "One more thing of Importance is Einstein's theory that 'no man, wlhek or any other material thing can travel with the speed of light, for it would require an infinite force to give the smallest particle such a velocity. * *light like everything else1, Slosson says, lfollows the easiest way and this is not always the straight and narrow path. A river takes the easiest, not the shortest way to the sea, and this leads it through many meander ings.1 "The hypercuhe (fourth dimensional figure) named the 'tesseract1 is hounded hy eight cubes, has twenty-four square faces, thirty-two edges and sixteen right angular corners (see illustration). This sketch, however, is fictitious, because one cannot represent a geometric figure of four dimensions in a plane of two dimensions for the same reason that one cannot represent a three dimensional solid hy a line of one dimension. I do helieve that if someone could develop a screen of three dimensions upon which one could project a four dimensional mass, then the diagram could he made to resemble eight perfect cubes. This, of course, is an assumption on my part; however, I'm not so sure that it cannot he done. At least, any person intelligent enough and lucky enough to accomplish this feat would become very famous and perhaps we would gain an understanding about space that would help to explain many nthings to us which at the present time are a total mystery.11 This preceding situation is again illustrative of several elements in critical thinking: (l) the place of undefined terms and definitions in any argument, (2^ the place of assumptions in inductive reasoning, (3) presenting facts, (4) discriminating between facts and assumptions, (5) evaluating conclusions reached in terms of consequences, and (6) suspending judgment. Analyses of Heading Interests This technique is particularly effective for evaluating improvement in critical thinking when an inventory of reading interests is made at various stages in a pupil's development. Unfortunately, this procedure was not used in the earlier stages of the course, except for that portion of the "Inventory Questionnaire" which dealt with reading interests. (See Appendix C )# Since this questionnaire was administered at the beginning and at the end of the school year it represents two stages in a 290 pupil's development, namely (l) the stage prior to his study of demon strative geometry, and (2) the stage nine months laterf that is, after completing the course in demonstrative geometry* Since the responses to the inventory questionnaire were discussed in the preceding chapter for "both Experimental and Control Groups, only the responses to that portion of the questionnaire which deals with reading interests will he considered in this section. In the preliminary form the responses to the question, "List the kind of hooks that you like to read11, fell into five categories, as follows? (l) adventure, (2) mystery stories, (3) science, (4) art, and (5) mis cellaneous* .Adventure and mystery stories were mentioned in approximately 80 per cent of the responses* Only ahout 20 per cent of the pupils listed the other three types of hooks. This distribution was approximately the same for hoth Experimental and Control Gkoups* The responses to the question, "List the kind of reading that is of most interest to you in your newspaper1*, fell into five groups, as follows? (l) comics, (2) sport, (3) war news, (4) local news, and (5) miscellaneous. "Comics** were listed hy nearly all of the pupils. Approx imately 50 per cent of the pupils listed "sports", while nearly 30 per cent listed "war news" and "local news". Less than five per cent listed such items as "foreign news", "society news", "editorials", "lahor problems", and the like* Little difference was found in the responses of Experimental and Control Croups* In the follow-up form of the inventory questionnaire administered at the end of the school year, pupils were asked to make a list of articles they had read from newspapers, periodicals, or hooks* This list was to 291 include only the articles read in connection with their course in geometry during the school year, (the results showed that the pupils of the Control Groups read meagerly whereas the pupils of the Experimental Groups read not only a large volume, hut also a large variety of types of material. Since the Control Groups did not consider non-mathematical materials, the question of course stimulated the two groups differently, therefore, in order to make the results on this question more comparable between the Experimental and Control groups, the non-mathematical readings listed by pupils of the Experimental Groups were omitted in the comparison, The responses were grouped on a quantitative basis under five headings, namely (l) four or more articles or books, (2) three articles or books, (3^ two articles or books, (4) only one article or book, and (5) no reading in this connection. A summary of the results in terms of the percentage of pupils from each group responding is tabulated as follows: Groups Experimental Control (1) Four or More Headings 11.9$ 0.0# (2) Three Readings 14.2# 1.6# (3) (4) (6) One Two No Headings Heading Headings 18.4# 6.1# 28.1# 14.1# 27.4# 78.2# The above tabulation indicates that more than 70 per cent of the pupils of the Experimental Groups engaged in some voluntary reading of a mathematical nature in connection with their course in geometry, while only 22 per cent of the Control Group pupils did likewise. Furthermore, the Experimental Group reading list was far more comprehensive than the above tabulation indicates, since their course in geometry included nonmathematical materials. Even though the list of readings was limited to mathematical materials, the Experimental Gro^s read a much larger amount. 292 She difference in the amount of reading done by the Experimental end Control Groups becomes more significant when consideration is given to the fact that the Experimental Group was statistically inferior in terms of intelligence quotients. Studies®^ show coefficients of correlation between X,Q. and Heading as hi^i as £ s •783 ±.026, It would appear jus tifiable, therefore, to attribute the larger amount of reading of the Experimental Groups to the differences in procedures used# Bach of the books in the following list was mentioned by at least three pupils as voluntary reading in connection with the mathematical part of their course in geometry. She first book.! Mathematics for the Million, was listed by 43 pupils, 41 of whom were members of the Experimental Groups* Ho book was named by as many as three pupils from the Control Groups* She following books are ranked according to popularity. 1. Hogben 2. A Short History of Mathematics Sanford 3* Easy Lessons in Einstein Slosson 4. Platlsnd. A Eomnnce of Manv Dimensions Abbott 5* A History, of Oajori 6. Mathematical wrinklea Jones 7. Search for Sruth Bell 8. Mathematical Recreations Ball 9. Amisements in Mathematics Dudeny 10. History of Mathematics Pink 11. Svranny of Words Chase 12. Poetry of Mathematics Smith 85 St. John, Charles W,, Educational Achievement in Relation to Intelligence. Cambridge: Harvard University Press, 1930, pp. 38-102. 293 13* Thinking About Thinking Keyser 14, Wueen of the Sciences Bell 15. Geometry Exercises in Paper Bolding How 16. Number Storiea of Long Ago Smith 17* The Art of Straight Thinking Clarke 18. Wonderful Wonders of One.Two.Three Smith 19. Men of Mathematics Bell (3 pupils) Individual and Group Projects Examples and illustrations of hoth individual and group projects may he found in Chapters IV and Y. Changes in ability to think critic ally were particularly in evidence in the group projects* because these were developed periodically. Since these centered around the elements in critical thinking* one may expect to find evidence of this type of thinking. In order to avoid repetition, it will suffice to summarize the outcomes developed through this phase of the procedures* as follows* All of the elements in critical thinking were observed in both individual and group projects and in mathematical as well as non-mathematic al situations. The most conspicuous evidence of critical thinking appeared in the projects dealing with analyses of advertisements. In the group projects, changes in critical thinking were far more pronounced in the earlier stages of the course then in the later. Chapter Summary In this chapter an evaluation was made of critical thinking abilities of pupils by means of the following: 294 1. Direct observations of critical thinking behavior a* Eeports of observations by 165 classroom teachers b. Eeports of observations by an experienced observer c. Eeports by parents and classroom visitors 2* Anecdotal records 3* Analyses of written work other than tests, such as! a, English themes b. Kequired written reports of reading c* Voluntary written reports 4* Pupil diaries 5. Analyses of reading interests 6. Individual and group projects. Direct observations of critical thinking behavior reported by the 165 classroom teachers indicate that out of the 333 pupils in the Exper imental Croups, 294 or nearly 86 per cent exhibited definite improvement in critical thinking abilities. It should not be assumed, however, that the remaining 12 per cent have completely failed to reveal some evidence of this ability. Since the 165 classroom teachers observed only the Experimental Groups, an experienced observer was employed to observe each of the classroom situations for both groups. Although this obsesver had no prior knowledge or information as to which groups were Experimental and which groups were Control, his reports reveal significant differences between the two. This observer found far more evidence of critical thinking in the Experimental Group classrooms than in the Control Group classrooms. Because of the subjective nature of reports by parents and class room visitors, the results secured are open to question. Therefore, only a few exanqples of situations in critical thinking were submitted and 295 these were confined to the writer*s classes. The comments submitted were indicative of changes in the direction of critical thinking behavior* ■Anecdotal records were submitted by nearly all of the 165 class room teachers who acted as observers# A total of 293 anecdotes were presented, and an analysis of these revealed that 270 or approximately 92 per cent were indicative of critical thinking behavior as conceived in this study. Analyses of truoil diaries, numbering more than 200, revealed not only a wide variety of different types of experiences encountered by the Experimental Gkoup pupils, but also the presence of elements in critical thinking in nearly all of the situations described* Analyses of written work submitted in situations outside of the geometry classroom were particularly effective, because pupils were not conscious of the fact that their work was being evaluated in terms of the elements of critical thinking* Nearly all of the situations analyzed exhibited some of the terminology and the presence of elements in critical thinking emphasized in the geometry classroom of the Eaqperimental Groups. Analyses of reading interests revealed that the Control Group pupils read meagerly whereas the Experimental Ckoup pupils reed not only a large volume but also a large variety of types of material in connection with their course in geometry. Since individual and group projects were developed around the elements in critical thinking, one may expect considerable evidence of this type of thinking. The group projects, in particular, reveal changes that have taken place, because these were developed periodically. The pupils were far more critical in the later stages than they were at first# 296 C R A P M VIII EVALUATION (Continued) Introduction Chapters VI and VII presented an evaluation of outcomes by means of written tests* observations* anecdotal records* pupils1 diaries* individual and group projects* and heading interests. This chapter describes the evaluations made by pupils themselves* and indicates how thby reacted to their course in demonstrative geometry. Evaluation by Pupils Only pupils of the Experimental Groups attempted to evaluate their progress. Self-evaluations were submitted at the end of each semester by one of the groups, at the end of the first six weeks and at the end of each semester by three of the groups* and at the end of each of the six ^six-weeks*1 periods by the two remaining groups. In evaluating themselves, pupils made an effort to describe their progress not only in the geometry course* but also in ability to think critically. An analysis of the evaluations as a whole revealed that they emphasized five major topics* namely (1) the kind and amount of reading done (2) the effect the geometry course had upon pupil's thinking in connection with their reading* conversation* or any other experiences they cared to mention (3) the achievements in manipulation of geometric content and in situations of a non-raathematical nature (4) the amount of time devoted outside of the classroom to situations directly related to the geometry course (5) specific examples as to how the course had helped pupils with their personal problems. The first attempts at self-evaluation were not only awkward but 297 also very general* They frequently consisted of such unsupported statements asi "This course has helped me in my ability to reason more accurately11. The later attempts, however, were not only replete with specific accounts of experiences in which the geometry course had influenced pupils hut also were more fluent in the expression of these accounts* This may he illustrated hy an excerpt from one of the self- evaluations in which the general statement is justified hy the addition of a definite instance of critical thinking and the fluency involved in the description of this situations •'This course has helped me to think critically and also not to helieve all that I hear or read* It has also helped me develop a questioning attitude of mind ... % father, heing a "buyer of Men's Wear in a department store, comes in contact with sales men* Some of them misrepresent their merchandise and they try to put something over on the buyer. There is one particular instance when I used my geometric training and was conscious of it ... I happened to he present one afternoon when a hosiery salesman came in. He offered my father a hose which he called 'chardinized silk', much cheaper than he could buy pure silk hose. How this hose carried a label which read 'Pure Chardinized Silk*. The salesman claimed that his hose would outwear any other silk hose. I asked him how it was made and if it was better why it sold for less. The salesman tried to explain that the hose went under a very detailed and a more economical pro cess. Asking what the process was he explained it to me. I al ready knew how rayon hose were manufactured, end comparing the processes X found them exactly the same ... I questioned him further and asked if his hose was not a rayon. He admitted it was, therefore I concluded that 'Pure Chardinized Silk* was nothing more than a fancy name for rayon." r Since the total number of evaluations by pupils was more than 1000, some of them being five and six pages in length, it was necessary to choose but a few for purposes of illustration. Examples of each of the three Experimental §roup situations in which two, three, and evaluations were submitted are presented. ,86 86. Other examples may be found in Appendix "E". 298 A fourth self-evaluation was selected because it illustrates the reactions of a pupil who studied geometry in a traditional situation during the first semester and was transferred to one of the Experimental groups at the end of that semester. This type represents only 15 per cent of the evaluations submitted* and the pupils were not included among the 335 Experimental group pupils in this study* 1» An Example Where Two Evaluations Were Submitted Two self-evaluations were submitted by Pupil No. 129, whose intelligence quotient (I. ) is 97* The first was made at the end of the first semester and the other at the end of the second semester. There is little evidence of critical thinking in the first one and the state ments are very general in character, as follows: HI never realized until now, just how little I ever stopped to think about things* But I'm glad I can say that now when I pick up a newspaper or book, or hear someone talk, I turn over in my mind what has been said or written. I feel that this course has accomplished what it has been trying to teach me." At the end of the second semester this pupilL submitted & selfevaluation that was not only indicative of critical thinking, but also more fluent, more descriptive, and more specific. At least three elements in critical thinking are present, namely (l) need for clearly defined terms, (2) the place of assumptions in an argument, and (3) evaluation for bias or prejudice* This pupil wrote, as follows: "This course has affected my reading, thinking, and conversation a great deal* When I read something, I do not just glance at it as I did before. Now I have learned to read it carefully and thoughtfully, and I look for all of the assumptions and impli cations to be found. Aa for thinking, I now look at both sides of the story. I try not to let bias or prejudice interfere, and I do my best to think twice before I speak. I am learning to think more critically and I find myself getting along much better than before* I think that this course should be compul sory for every student, because it not only helps you to think and talk your way out of many difficulties now, but will also help to do this later in life. "I'm going to give the following example of what I mean by being more critical, because it is a statement that I always accepted without questioning until lately. This is the example: A man was speaking of another man. He said, 'He's a good man*. The man speaking meant that the other was good in his work. A person may be a good worker and still not be a good man. The word good needs to be defined in this situation." 2. An Example Where Three Evaluations Were Submitted The three self-evaluations to be considered here were submitted by Pupil No. 199, whose intelligence quotient (I. Q.) is 114. The first evaluation by this pupil was made at the end of the first six-weeks period. The other two were made at the end of the first and second se mesters. In her first evaluation this pupil reveals very little evidence in critical thinking. facts are mentioned. The statements are very general; however, several Por example, this pupil mentions such things as the influence of geometry on her appreciation of bridge designs, her understanding of geometric terms, her interest in the use of geometry in other fields, such as physics. This pupil wrote: "Now, after I have studied geometry I unconsciously look for things such as triangles in the bridges as I go along the street. I notice the many instances every day where I hear geometric terms. The other day I was listening to a conver sation about sailboats and the words perpendicular. adjacent, and parallel came to my ears on several occasions. On the radio in the question bee programs many questions concerning geometry were asked and I believe the greater number of times they were answered incorrectly. It is a subject in which few people are well informed. The other day as I was reading the Life of Madame Curie I saw how much she used geometry and all forms of mathematics for her great discoveries in physics. They were absolutely necessary for her experiments." At the end of the first semester this pupil (1) wrote more fluently, (2) mentioned several books she read, (3) gave specific examples of how the course helped her with some personal problems, and (4) exhibited more evidence in critical thinking, such as searching for more facts and formulating conclusions consistent with assumptions. This pupil wrote as follows: "When I first began taking geometry I had an entirely different conception of it from that I have now. I believed it to be a rather boring subject in which there would be no value to a girl like me who would have no material purpose for it. Now, these old i’dea£ have changed and I look on geometry in a new light. I find it intensely interesting and therefore I try to do my best work in it. I have found that it develops one's perspective and ability for clear thinking and analysis and 1 have come to like geometry very much. "I read on an average of two to three books a month. I can't find time to read more, although I would like to. I enjoy all sorts of books and some I have read lately are "A Princess of Thule" by Black, "The Blue Window" by Temple Bailor, "Hamona" by Jackson and "Showboat" by Edna Perber. believe this course has helped me to think more clearly than before; it has helped me in my by giving specific facts and examples to back up that people find.them to be true instead.of being suppositions. straighter and conversation statements so merely "As for my accomplishments in this course, I have tried to keep a notebook that I can refer to years from now and still find a help. I have gained a great deal of knowledge from the course proper and it has helped me to increase my vocabulary. "I spend about a half hour on geometry a day; sometimes I spend much less time and sometimes more. I simply prepare the lesson for the next day and then I am through* "I believe geometry has helped me with my personal problems* I feel I am fortunate in being able to continue with this new method next semester instead of having to change to a teacher who teaches the old way. I think I will derive more benefit from the course because of this advantage. "Some specific examples of the way this course has helped me are: I was able to analyse my needs for the subjects I chose to take next semester. I talked it over seriously with my parents and I was able to think back of the subjects and X realized the help Ihad obtained from geometry. I was also called upon to make a personal decision which X do not want to mention here and I thought through it clearly and sanely and I believe my decision is working out for the best." 301 At the end of the second semester this pupil appeared to he more specific and more to the point in making generalizations* Several ele ments in critical thinking are in evidence; for example, stating facts in support of her argument, searching for facts, testing conclusions for consistency with assumptions, and possibly suspending judgment. In her evaluation this pupil wrote: uAs I have gone on with the study of geometry, I realize more and more how much it has helped me to think clearly and accur ately. I don*t believe the geometry itself will help me, for I do not intend to go on with mathematics* I would like to coir* tinue the study, but I have other subjects to take in the next two years* I really don*t think geometry will help anyone who has no desire to go into some type of engineering, but the benefits unconsciously derived will help us all* HI think it is fun to take a problem you know nothing about, except what is given in the hypothesis, and work around with it for a solution* It is like working out a puzzle, only here we have to have a reason for every piece we put together* Geometry is a fascinating subject, much more interesting than algebra, because here you can express your own ideas and in algebra you follow a set pattern given to y8u* "Recently I was able to put my benefits from geometry to use* Hy father received a circular advertising some sort of encyclo pedias at an exceptionally low price. He brought it to me, think ing this might be a grand opportunity to buy some new books* We sat down and began to examine the advertisement carefully* It really sounded fine, and we got down to the last page and there seemed to be no catch* ifter we really thought it over we de cided there must be something wrong somewhere. Just for the idea of testing the truth of our assumptions, my father called the company* Then we discovered we could obtain the books at this price with the purchase of an expensive bookcase, and really the cost of the case would more than cover the deduction on the books* I hope no one was roped in on this offer* I'd really like to find out how many people are being taken advantage of by 'chisel ing* merchants because they cannot see through a fraud*11 3* An Example WhereJSix Evaluation* Were,JubmitteA The six evaluations to be considered here were submitted by Pupil Ho* 259, whose intelligence quotient (I* <4*) is 110. These self-evaluar 302 tions are particularly illustrative of (l) changes taking place at more frequent intervals» (2) the tendency to make more specific and more precise statements in subsequent evaluations , and (3) improvement in critical thinking from one period to the next* Unlike the preceding examples, this pupil reveals some evidence in critical thinking hy the end of the first six-weeks period, namely the elements of questioning the meaning of terms and detecting assumptions* Uention is also made of the way in which the course has affected the reading being done* £his pupil, at the end of the first six weeks, wrote as follows* outside reading usually consists of newspapers and magazines* When I read October's Header's Digest I found that unconsciously I was picking out sentences that contained assump tions* Also there were words which I didn't believe were specific* "I enjoy talking about the work we do in this class because it's interesting and so different from other classes* This makes me more conscious of this work when reading or conversing* "I have brought to class some clippings from newspapers and magazines containing statements with assumptions and indefinite words, after which I have written my reaction, from advertisements 1 have seen how many assumptions the stores make to persuade you to buy this or that* "The time 1 spend at home on geometry is not the same every night, I think of geometry as fun instead of work and therefore do it when 1 don't have long nightly assignments that must be in the next day* Naturally, Saturday afternoons and Sundays are when 1 do it or when I'm tdted of working on another subject* UI m sure I have gotten a great deal out of the course already* X don't just 'accept things' as much as t did* Of course I realize I have a long way to go before this even becomes noticeable to outsiders, but I'm sure I have an idea of what you have in mind, but even if I haven't gotten all of it, I like it. I have been working on definitions and constructions, the latter of which I'm now sure of*11 There was more evidence of reading presented by this pupil at the end of the second six-weeks period* Two more elements in critical thinking 303 appear to have "been introduced, namely (l) discriminating between facts and assumptions) and (2) testing conclusions for consistency with assump tions. This pupil wrote: "is I pointed out in my last evaluation, I do a great deal of reading of the daily papers and many magazines, such as Life, fortune. Saturday Evening Post. Reader's Digest and the J o u r n a l among the list. I don't read books for the fun of it unless I hear of a book that is along the lines in which Z am interested, although the reading I do does give me a good opportunity to find assumptions, undefined words and loose statements. "Because I am interested in the field of advertising, I notice advertisements quite often. X believe that the more critical people get the more facts advertisements will have to have, which will tend to improve the quality of the products. Many things that to me were facts at one time are nothing more than assumptions to me at this time. X don't believe things X read as much as X did before. X am more skeptical and X look for and examine the assumption. "X will make the assumption that X am thinking more critically. Often I hear things in class that need defining or assumptions that need to be pointed out. There isn't enough time to stop the teacher or student and ask for these things in class. If there were, they'd probably wonder what your idea was and think you were acting funny or trying to waste time. "Critical thinking has helped me see for myself how important definitions and assumptions are. It has also helped to straighten out some of the differences between facts and assumptions. Xt has shown me not to believe conclusions unless they are consistentwith the assumptions upon which they are based. "It would be even harder for me to judge myself but I do feel I have learned a great deal." this six weeks, There was less evidence of outside reading presented at the end of the third six weeks period by this pupil. However, mention is made con cerning undefined terms and another element in critical thinking is intro duced, namely "the tentativeness of conclusions". This pupil wrote: "I have had more homework this six weeks than ever before, which made my time for pleasure reading almost disappear. I did do some reading in Easy Lessons in Einstein, which I enjoyed enough to make up for all the rest. Everything I found was connected with what we have been discussing in class about non-Euclidean geometry. 304 "The emphasis on recognizing that ail conclusions are tentative and that nothing is absolute was strengthened again* "I am sure that there has been a decided change in my critical thinking since I have started geometry* 1 hear many people making loose statements; the papers are full of undefined terms which I never before thought of and 1 realize so much more now, how important it is to have facts and to prove things by having them tested many times; also to get as many ideas on one subject as possible, instead of one or two* "1 have really enjoyed the small part of this book so much that 1 want to read the whole thing so as to find out more about the fourth dimension and the "Tesseract'• "*Ehe most important thing that I have learned is that con clusions are tentative* I used to think that when a great math ematician established a mathematical law, it was absolutely true and could never be otherwise.11 This pupil reports more reading in the fourth six-weeks period than in the preceding one* Mention is made of the influence the geometry course has had upon this pupil*s thinking in connection with conclusions reached* .Another element in critical thinking becomes apparent, namely "discriminating between facts and assumptions"• achievement in geometry is also given* Some indication of This pupil wrote as follows! "I have read Silas Marner by George Eliot, The Header's Digest* The Doctor by Binehart, and of course the Tribune daily paper, particularly on Sundays* I haven*t had any time to read for the enjoyment of reading due to the unusual amount of homework I have had* "To think & lot more about things is what I have done. This may not be due to this course, but the conclusions I arrive at are, I'm sure* I see more and more how unimportant things are that I once thought tobe most vital* The statement you made quite a while ago about '-All things done by human beings are subject to error* ha« made me do quite a bit of thinking* I have wondered why we do some things that cause so much trouble and time and effort besides being something we dislike when after all they were only sey up by a person or a group of people who are very likely to be wrong. I know this sounds quite radical but 1 don't mean it in a destructive sense* ".Another thing I have been thinking about is, how relative everything is to everything else. Of course this has something 305 to do with this course "because this idea never would have dawned upon me until my old age probably i "I haven't spent anywhere near the time on geometry this six weeks as I did last* While looking through a magazine or paper, I have found a few things which I thought needed defining or that I recognized as an assumption that someone else had thought to be a fact. Of course it has taken some time to copy my notes and diagrams over too* I haven't much idea how much time outside of school I have actually spent on geometry. f,I do believe I am learning to think critically, slow as it may seem* Of course during the first six weeks of geometry I learned more about critical thinking than I ever had before, but this six weeks probably ranks second* Jithough I have been doing quite a bit of the non-geometric work, X haven't got the raathematical part as well as X could have with more work*" less fluency in writing is in evidence in the fifth six weeks period; however, the statements made are more specific than in the pre ceding period* More evidence of outside reading was presented* Greater effort appears to have been made toward a better understanding of geometric content* Besides the elements in critical thinking mentioned earlier, there is another one in evidence, namely "searching for motives". This pupil wrote* "In my mathematics notebook I have entered all my notes on Similar Polygons. Also I've filled all the required assignments. The notes which I entered contained some of the proofs we worked out in class, some definitions and propositions* Things X have done in connection with this course are reading sections, more sections in Easy lessons in Einstein by Slosson. "I have read Shakespeer's Julius Caesar for English, the Reader's Digest and the Story of a Country Bo.v by Dawn Powell, also the magazine life each week* "I have had things happen which had critical thinking involved and have read quite a few things in which I could question almost every term and find many motives, assumptions and underlying terms. Unfortunately I can't remember all of them. "I drew a diagram of a 'Tesseract' and wrote a report on it which you placed in my folder." In the final or sixth self-evaluation, this pupil mentions several 306 personal factors, such as enjoying the year’s work and the like* However, several specific statements were made regarding progress awd achievement in the course* Mention is made of most of the elements of critical thinking and especially two that were not present in ,the earlier reports, namely a consideration of consequences and testing conclusions for con sistency with assumptions* This pupil wrote as follows* "This past year of geometry has opened a new and different field to me from any I have ever experienced "before* This new field is of course 'critical thinking1 or thinking about thinking and reasoning* X have enjoyed this more than the regular formal method of teaching mathematics not only because we didn't have to remember a lot of facts that I'm sure would have been forgotten within a year, but because we did our own thinking and reasoned the problems ourselves* "It is evident that we could get more out of learning to reason and prove propositions by reasoning and doing our own thinking than we could by memorizing proofs which have already been worked and we might not have occasion to see or use again* By learning to reason and think well in one field we are able to think more clearly in others* These are a few of the reasons why I have enjoyed this year of geometry much more than I believe I would have in a formal class* "Of course to me the non-mathematical part was the most in teresting and important part of this year's work* The only way it could be better would be to have more of this and less formal geometry, and to have it in more schools. Certainly it's im portant enough, for it enters into all the other fields and every day of your life* "Not until it was brought out in this class did X recognize that things which are considered established facts are quite tentative* It is quite evident now that all conclusions are ten tative and that not much progress can be made until this is recognized* "This course has affected my reading because X now like to read about a subject which is new to me* X enjoy books that describe how a person can arrive at a conclusion from a system of logic built from fundamental assumptions* "It also affected my conversation because in discussing this course with older people I have had some very interesting conversations and have heard a few very intelligent ideas expressed* 307 I ! | "is you already know, I have read lasv Lessons In Einstein and have written a report# Also 1 have written comments on a few ,ade* from newspapers and magazines# | "This class has helped me a great deal to recognize all that I have said once before — tentative conclusions, thinking about consequences, not jumping at conclusions, and questioning the meaning of words as well as looking for motives behind statements and actions*" | i i*— Selfrlvaluations. by a Pupil transferred to an Experimental Group at the End of the H r at Semester* This pupil, whose intelligence quotient (I.ty#) is 115, was not a member of one of the Experimental Groups during the first semester but was transferred to one at the beginning of the second semester# Since the evaluations submitted by this pupil reveal contrasts between Exper| imental and traditional procedures, and since they represent approximately | 15 per cent of all of the evaluations submitted, it was felt that thqy merit consideration at this point* | At the end of the first six-weeks period of the second semester I this pupil indicates the difficulties she experienced with mathematics and the inferiority complex developed as a result of this experience# This is followed by a brief description of the adjustment made to the Ebcperimental procedures, particularly the ones dealing with non-mathematical situations# Several specific examples are given of situations where thinking or reasoning-was influenced by the geometry course# At least | three elements of critical thinking were present in the examples given, namely (l) questioning assumptions, (s) testing conclusions for con- [ sistency with assumptions, and (3) discriminating between facts and i assumptions# | This pupil wrote as follows* "This summer, before school started in the fall, 1 was carefully thinking over my course that I planned to have this year. 308 I liked 35nglish, especially the literature part of it, so that wouldn't bother me, Ify marks in that subject were usually my best anyway, so I didn't worry about it. I insisted on Applied Arts II, and much as I wanted to get Medieval History in, I would not substitute it for my favorite subject. Then there was French, It would be my first association with any foreign lan* guage (never having had Latin) and the thought of it rather excited than worried me because of the newness of it. Of course I must tfcke dear old geometry. Thinking of my shameful marks in algebra, and how much harder geometry would be (sr so my sophomore friends informed me) it made me feel very downcast, and 1 did everything I could to erase the whole business from my mind and worry about it when the time came. "When I discovered that I was to have the same teacher for geometry that I had for algebra, * really didn't know whether to be glad or sorry, I would have liked a change but still I could see no serious reason why I should bother my adviser with unnec essary burdens, so I left it as it was, and decided to do my best, "I am a person with the best intentions in the world, but that is often as far as I get. It wasn't long, and my marks began to parallel my algebra marks during my freshman year. It disgusted me immensely, because it wasn't as if 1 despised the subject, or hated the very thought of its name. I really liked it in a way, because the thought of reasoning things out and being able to prove things true, step by step, really fascinated me. Still there was something missing, and I determined to ask my teacher every little thing I didn't understand. I would have it on the tip of my tongue, and suddenly decide that I simply couldn't ask him such a simple question. He would think me hopelessly ignorant, so I would let it go. After a long time of this, I developed an infer iority complex on the subject, and decided that I was immune to the comprehension of such a subject. My low marks continued, and I assumed a rather hopeless attitude. I was sure that my teacher, much as I liked him, was plainly disgusted with my marks, and I dreaded geometry. I even decided that I would digest my lunch better if I had any other subject except geometry after it. I suppose it sounds silly, but nevertheless it is true. A geometry test spelled horror to me, and it took a long time for me to overcome it, "When I found out that I was to have a new teacher for the second semester, I was very curious to see whether this change would have any effect on me. Tour new method of teaching interested me at once, for it was very different from anything I had yet heard of. I decided to start all over again and get what I could from your ideas, and apply them. The non-mathematical side of it, and reasoning in outside subjects, interested me particularly. "In my English class we were studying Silas Marner. In the near front of the book, Silas Marner is robbed of his gold, while 309 he was away from the cottage* The village folk decided that a peddler had stolen the gold because 'not long ago, a foreign looking peddler had stolen some money in the village and foreign looking men, especially peddlers, were usually robbing or plundering anyway.1 Therefore, the villagers sat back and accepted their reasoning as a fact. "Immediately I detected the faulty reasoning on their part. If a peddler who was foreign looking had once stolen some money in the village, what sign was that that it had happened again? The villagers could relate only one incident to which the situar tion applied* Even if they had mentioned two, it would not have been enough proof that every time money was stolen in the village, it was 'probably taken by some foreign looking peddler.' "When I awoke to the fact that these people and other characters in the book were making countless unreasonable state ments which had no backing, I enjoyed the whole plot much more* Also, there were many cases which were too much taken for granted, and not looked into* "I was conversing with my aunt the other day, and she made this statement: 'Ho, 1 don't care for Sinclair Lewis' books and most people don't either*1 Immediately I asked her why, and she gave me a very suitable answer. Still not satisfied, 1 said, 'That was rather a radical statement you made about "most people not liking his books". Why did you say that? How do you know that most people don't like them?' She seemed rather surprised at my direct questioning, but she answered with a light tone of indignation, 'It's what I've heard, of course. You know as well as 1 do that few people care for his books.' But I persisted, 'What you*ve heard isn't very substantial evidence, is it?' She looked at me rather queerly - as if I had told her she had five minutes to live. Presently she said, 'Ho, I suppose that isn't much evidence. Say, are you suddenly going to turn to that lofty profession, attorney at law, or something?' I smiled sweetly and said, %}fy geometry is taking effect on me.' She gave ma a very blank look and went out of the room. "It wasn't until I had finished my conversation that I really woke up to the fact that my geometry had a great deal to do with that little conversation. "These are just a few examples of how my outside work is being done unconsciously with this interesting light thrown upon it. 1 have begun to take mental note of things that sound un reasonable to me. I know that this new method of learning geom etry has a great deal to do with it. Geometry holds no such horrors for me as it formerly did, and outside of learning and understanding the subject much more thoroughly, the whole idea is fun* I can see now some of the errors I have made during this first six weeks. Because I am really interested in the non- 310 mathematical side of it, I am ashamed not to have written down my reactions and handed them in. However, now that I am thor oughly acquainted with your methods, I will carry on differently in the time to come." At the end of the second six-weeks period this pupil mentions again her difficulties with mathematics prior to her study of geometry with the Experimental Group, Her self—evaluation at this stage contains several specific instances of how the classroom procedures have influenced her thinking. Several elements of critical thinking were in evidence, such as questioning the meaning of terms, questioning assumptions, discriminating between facts and assumptions, and testing conclusions for consistency with assumptions, Shis pupil seems to feel that she is learning to apply life interests and problems to geometry instead of learning to apply geometry to life situations. Her second self-evaluation is as follows* "Mathematics of any kind had always been the most detested subje&t of my whole school program* She reason I hated the very thought of it was that X didn't understand it. Another reason was that it was always required, X believe that many people, not just pupils, dislike their work because they have to do it. Often, when people do not understand the-thing they are supposed to do, they unconsciously become afraid of it, and it is human nature to hate a thing one is afraid of, This was the case with me. When ^r* _ marked the reasons on my report card for receiving D's there was one which predominated all through my first year and the first semester of my second year in his class, Xt was ilack of fundamentals', X don't know whether to blame the teachers that I had in the lower grades or blame myself. Anyway, that Is beside the point, but it worried me. Mother began to notice that this reason was on all of my report cards too, so we started drilling on the fundamentals of arithmetic a certain portion of each day. X seemed to know my arithmetic rather well, but X was too slow in getting my answers. By just drilling on a few 'do-in-your-head* arithmetic problems each morning before school, X soon found that I reacted much more quickly to the geometry problems in school. This encouraged me a great deal, but my marks stayed in the D's, As X said in my last six-weeks self-analysis, I developed an inferiority complex about geometry, and all mathematics in general. I would do a test, and when I went out of the room, feel quite proud of myself for finishing the test completely. I would expect a 0 at least when my paper was returned, but no, it was always a D or a 0— , When I would ask about it I was usually told, 'Ho, not 311 quite a C* but almost.1 1 was satisfied with the answer, hut when the same thing happened again and again, I became very disgusted with myself for being so ignorant, Also these marks made me fear every hour spent in my geometry class. I was almost afraid to look my adviser straight in the face each morning, for fear she might catch my eye, which would remind her to give me another low notice to take hone. »Qnce or twice X thought of geometry as being useless to me. Here I was* going toggo to a school for girls for two years* where X would concentrate on Jfcench* literature* and art. Then X would spend two years at an art school, and try to become an illustrator and possibly a portrait painter. Where in the world would geometry ever fit into that schedule? It seemed a waste of time. It never entered my head that you could apply geometry to real life problems* until you became my teacher. For the first few weeks* it seemed much more like a psychology class than a geometry class. X found it becoming much more interesting and very new. Unconsciously X began to analyze sentences X read in the newspaper* but mostly things that people said. X began to notice how many people * people who are considered very intelligent - jumped to conclusions. Also* X began to analyze not just words tpt things and people themselves. X met a very interesting person just recently and her profession was identical to the one 1 % most interested in. After she told me a good deal about herself, X mentally began to compare her with myself. What opportunities did she have? Do X have more? Didshe make the best of her opportunities? X didn't think so. She istoo limited in what she can draw. She had the chance to learn more. What schools did she go to? Were her choices wise? Would X be successful if X had her education? I reached the conclusion that she was a very fine young woman* she drew well, but she could draw only certain things. She would have a higher position if she was just as talented in drawing* sketching, and painting many things instead of a few. She regrets her mistake and is very conscious of it. However* X resolved then and there, not to limit myself to a few things* but be able to do many and do them well. Perhaps this seems far away from geometry* but a few days after my meeting with this artist, I realized that in my own way, geometry had a great deal to do with my thoughts. Why don't I reason a little more in geometry? It worked backwards with me.I learned of applying geometry to life interests and problems*tbut^&nstead* I'm applying life interests and problems to geometry, and am very grateful for all I've learned." The final self-evaluation by this pupil is a summary of how this course has affected her thinking. Host of the report centers around her major interest, namely 'art', and she endeavors to point out how critical thinking enters into the work of an artist. 312 "I sincerely believe that 1 received much more from geometry when it is taught in the way of improving critical thinking than in the use of plain, formal mathematics. I know and will frankly admit that there are just a few theorems that X could sit down and prove without a flaw - a very few. However, this doesnft seem so important to me. Xf I go to an art school when X finish high school, X will have very little occasion to use geometry, hut X will have to he ahle to think very critically. So many people believe that to paint a picture, the artist does something that he sees before him and paints it. He does It better than a lawyer, doctor, musician, or anyone else would do it because he possesses the title of 1artist*• How untrue this all is! People do not understand that the work isn't done with the hands, hut the mind. The artist paints exactly what he sees in his mind, and he neverstarts th paint until that picture is completely analyzed, like a geometry theorem or any life problem. The simplest part of an artist*8 job is to paint the picture; the hardest part is to study it, and analyze it while it is still in the painter*s mind. One difficulty is that you do not then have the advantage of taking notes and writing down the steps and reasons. "All of this seems rather off the subject, but it really proves that X am going to have to think very critically if I ever expect to become successful. This is where this critical thinking course comes in. X must admit that my marks have not been what one would chil good. However, X have gained some knowledge in the non-mathematical unit which is very important and helpful to me, even though my test grades are not exceptional. One of these things is that marks do not frighten me as they used to. X have ibearned that my marks are not the important thing but it is what X get out of the schooling that counts. There are many things that have never come up in tests which X'vd retained. X am sorry to say that it is not mathematics that I've gotten so much out of, but the non-mathematical work. Xbr instance, when X have com pleted a mathematics test, X hope and pray that X have passed it and receive a decent mark. Xt is different with the non-math ematical tests. When I have completed one of these, X feel that the period was well spent and am rather pleased with my work. I have no thought of grades, but only wish X could spend another period on the test to see what else X could do with it. HThese are the things X have gained in this course. X do not feel as I did in the first semester of formal geometry, that X couldn't go on with it and feel that my time was being wasted. However, X knew I had to finish geometry to go to college. (Since then I've changed my mind, and decided that time was too precious to spend two years in a girl's school 'gaining culture'. I feel my time would be too precious to lose, so I plan to spend all my time in art school* However, I'm grateful to this course, and sincerely believe I got a lot out of it." 313 The four examples of self-evaluations which have been presented show the influence of the content and procedures described in Chapters III, IV, and V* The progress made in critical thinking varies greatly in each individual case. The greatest progress seems to have been made during the first twelve weeks of the course, although in most cases, there appeared to be evidence of gradual improvement in ability to think critically throughout the year* Probably the major criticisms against these evaluations are that they are too general and that there is a lack of sufficient evidence submitted by pupils to illustrate specifically what they meant by some of the statements presented. The fact, however, that most of the elements of critical thinking were in evidence in later stages and not in the first stage is indicative of improvement in this type of ability. Although the evaluations by the pupils were highly subjective, they reveal two sig nificant facts, namely 1* the influence of Experimental procedures and materials upon the thinking of young people 2# the improvement in critical thinking abilities of young people resulting from these procedures. Finally the values derived fhrough experience in making a written evaluation of progress may be of considerable importance to pupils. ^ Pupil Reactions to Their Course jj^JjgpmeJto Pupil reactions to their course in geometry were implied in several techniques previously mentioned, particularly in the diaries and self-evaluations. However, these techniques were confined only to the iESxperimental Groups. In order to study the reactions of both Experimental end Control Groups to their courses in geometry, twenty of the questions which had heen previously used in the preliminary form of the inventory questionnaire were included in modified form in Part II of the follow-up form. The questions and a summary of the responses to each question for hoth groups may be found in Table XVI* The major differences in the responses of the Experimental and Control Groups were found to be in questions; 1, 2, 3, 6, 7, 18, 19 and 20. Responses to question 1, 11Do you like geometry as presented in this course?*1 were undoubtedly influenced by pupils1 desire to please their I II teachers, in affirmative answer was given by 83.4 per cent of the pupils of the Experimental Groups, and by 73.5 per cent of the Control Groups. The fact, however, that Experimental Groiq>s were encouraged to think critically in any situation suggests that their approval may be due, at least in part, to the differences in the teaching procedures that were used The affirmative responses of the Experimental Groups to question 2, "Has the study of geometry helped you in any way so far?*1 exceeded those of the Control Groups by 35 per cent. The emphasis placed by teachers of the Experimental Groups upon reasoning in non-geometric as well as geometric Jsituations is undoubtedly partially responsible for the differences in (the responses of the groups. The use of non-geometric materials may j^ccount also fhr the fact that a larger percentage of Experimental Group jjpupils indicated that they found the study of geometry interesting jt question 3 - Experimental Group 80.1 per cent, Control Group 67.9 per cent) ji j ! Greater percentages of Experimental Group pupils gave negative Responses to question 6, "Do you like final examinations in mathematics?" land question 7, "Do you think that mathematics examinations are generally 315 TABLE XVI* SUMMARY OF BESFOHSES TO FART II OF THE FOLLOW-UP FORM OF _________ THE IHVEHTQRY Q.UESTIOHNAIEE___________________________ Besnonses Experimental Control Group Ho. Question leg 3 Yes i Ho Ho 1. Do you like geometry as presented In this course? 83.4 16.6 73.5 36.5 2. Has the study of geometry helped you in any way so far? ................................... 86.6 13.4 51.9 48.1 3. Do you find the study of geometry interesting? 80.1 19.9 67.9 22.1 4. Do you find the study of geometry very difficult? 50.5 49.5 55.4 44.6 5. Did you ever worry or feel uncertain about passing this c o u r s e ? ......... .......... . 53.1 46.9 52.0 48.0 6. Do you like final examinations in mathematics? 17.3 82.7 30.1 69.9 7* Do you think that mathematics examinations are generally fair? ............. . . . . . . . 78.2 21.8 86.7 13.3 Do you think the marks your teacher gives you depend on favoritism?.................... 4.9 95.1 4.4 95.6 95.4 4.6 97.6 2*4 Does your teacher talk or lecture too much in the classroom? ............................ 8.8 91.2 6.0 94.0 Does your teacher give you opportunities to express your own Opinions? • 98.0 2.0 98.4 1.6 3.9 96.1 2.0 98.0 96.7 3.3 98.0 2.0 17.3 82.7 24.9 75.1 Is your teacher usually willing to talk with you about your difficulties and give you advice? 96.4 3.6 98.8 1.2 116. ire you given an opportunity to express or tell what you know in this class?............... 96.4 3.6 98.0 2.0 I17* ' ! Does your teacher resent having a pupil express an opinion which is different from his or her own belief or from that of the text book? 5.2 94.8 5.2 94.8 j8. 9* 10. 11. 12. 13. 14. 15. i Is your teacher willing to explain certain topics you do not understand more than once? Does your teacher permit only a few pupils to do all of the reciting? ............... . Does your teacher try to make the class Is your teacher more interested in mathematics 316 18* Do you accept or believe all thatjyour. . . . teacher tells you? 19, 20* 24.1 75,9 55.0 45.0 Do you accept or believe everything in a textbook?................................ 20.8 79.2 58.2 41.8 Do you feel that you should be given an opportunity to think independently even though, your thinking is out of harmony with the opinion of your teacher or a textbook? 91.2 8.8 78.3 21.7 fair?" It is possible that inferiority in intelligence and in geometric ability on the part of pupils of the Experimental Groups had developed a sense of insecurity in taking examinations and distrust in the outcomes measured. To question 18, 11Do you accept or believe all that your teacher tells you?1* the negative responses of the Experimental Groups exceeded those of the Control Groups by more than 30 per cent. This difference is indicative, at least in part, of the influence that emphasis on critical thinking has had upon the Experimental Groups. There were pronounced differences in the responses of Experimental and Control Groups t4 question 19, "Do you accept or believe everything in |your textbook?11 and to question 26, "Do you feel that you should be given |an opportunity to think independently even though your thinking is out of harmony with the opinion of your teacher or a textbook?1'. These dif ferences are probably due to the types of teaching procedures used and Ito the fact that a textbook was not used by the Experimental Groups. I Eesponses of pupils of the Experimental and Control Groups tfc ,! the remaining twelve questions listed in Table XVT did not show sig nificant differences. Since these questions had a direct bearing upon 317 pupil- teacher relationships, the responses serve to support the obser vations of the trained observer who reported superior teaching and wholesome pupil-teacher relationships in both the Experimental and Control Grotqos. In order to secure more information concerning Experimental Croup pupils * reactions to their course in geometry, and particularly to some of the more specific phases of the procedures, two requests were formulated and presented to the groups# The two requests were as follows* 1* List in order of preference the things you liked best about this course* 2. List in order of unpopularity the things you disliked about this course# The pupils were asked to refrain from affixing their names or signatures to the responses® The responses to the first request were grouped and tabulated in the order of their popularity# The percentage of pupils considering each item as first choice, is as follows* Rank 1« 2# |3# 4. 5# I 6# 7. 8* 9# 0. Items Liked Presentation (geometric and non-geometric)....... *.* Evaluation based upon several factors.............. Writing our own textbook......................... Informal classroom discussions .................... Self-evaluations .............. Non-mathematical tests .... ....................... Opportunity to voice our own opinion ........ Only occasional homework (required assignments) No textbook ............................... Diaries ................................ Per Cent 28,9 26*1 14.6 9*8 6.0 4*3 4.2 3.5 1.4 1.2 Ehe above tabulation indicates a wide variety of preferences# The jjtwo most favored were (l) the way in which the geometric and non-geometric developed, and (2) evaluation of progress based upon factors 318 supplementing the usual examination and teacher judgment, The large number of responses to the latter may he due in part to the importance many of the Experimental Group pupils attached to the opportunity to share in the process of evaluating their own progress. The responses to the request* ttList in order of unpopularity things you disliked ahout this course11, were tabulated as follows* Hank 1* 2. 3. 4. 5. 6. Items Disliked Per Cent Geometric content................. .............................. Self-evaluations Writing our own textbook......................... Diaries .................................. . . . Miscellaneous (examinations* homework and the like) Ho response........................... . . . . . . 18.7 5.0 4*1 2,1 1.8 68,3 These responses indicate a variety of items disliked in the course. Unpopularity of geometric content may be attributed to the inability of some pupils to do the usual work in geometry since the inferior mental ability and geometric ability of the Experimental Groups was explained in Chapter IX. In general, one may conclude that the reactions of Experimental Group pupils to their course in geometry were favorable in terms of the objectives set forth in Chapter I, and in terms of the basic assumptions underlying the Experimental procedures. The reactions of the Control Groups in terms of the objectives and assumptions underlying their pro cedures were not as favorable. Table XVI revealed differences between the two major groups which seem to indicate that better provision for i individual differences was made in the Experimental Groups. 319 Chanter Summary This chapter was confined to self—evaluations by pupils and to their, reactions to the respective courses in geometry. The self—evaluations were submitted only by the Experimental Group pupils and indicated the following* 1# the kind and amount of reading done by the pupils 2. how the course in geometry had affected pupils1 thinking in connection with their reading, their conversation, and other experiences that they wished to mention 3* their achievements in manipulation of geometric consent and in situations of a non-mathematical nature 4. the amount of time pupils devoted outside of the classroom to situations directly related to the geometry course 5* how the course helped pupils with their personal problems* Several pupil evaluations were presented to illustrate the general nature of the contents of these reports* Since the self-evaluations were submitted at different stages of pupils' development, certain differences were apparent* 3Por example, the evaluations toward the end of the course were more specific than at the beginning, and critical thinking was more in evidence in the later stages* The reactions of Experimental Group pupils to their course in geometry were found to be favorable in terms of the objectives set forth in Chapter I, and in terms of the basic assumptions underlying the pro cedures in Chapter III. The reactions of the Control Groins in terms of the objectives and assumptions underlying their procedures were not as favorable* Some evidence was presented which indicates that better provisions were made for individual differences by the teachers of the Experimental Groups* 320 chapter ix G-ENERJkL SUMMARY 0? THE STUDY AND CONCLUSIONS General Summary of the Study The problem of this study was to compare Experimental and Control teaching procedures in demonstrative geometry for the purpose of deter mining their relative effectiveness in developing critical thinking abilities of high school pupils* The need for developing more effective critical thinking abilities among high school pupils is generally recognized* This need has been brought to the educational foreground as a result of one of man's great periods of transition* Modern life, during the past several decades, has become extremely complex and greatly enhanced through the medium of in vention and discovery* Increased facilities for transportation make it possible to travel extensively in a relatively short period of time* Highly improved ways and means of communication facilitate the exchange of ideas* Modern production and distribution are fostered by widespread advertising* In a democratic form of society such as that in which we live, it is an important responsibility of the school to prepare boys and girls for intelligent citizenship* The school ought to help young people not only to become aware of the problems of their times, but also to think through these problems, to discover underlying causes, to judge the soundness of opinions expressed over the radio, in the press, and on the lecture plat form* In other words, if young people are to participate intelligently in a society such as our own, it is of utmost importance that their behavior be based upon the best thinking of which they are capable. 321 When the schools attempt to meet this challenge, questions arise concerning the nature of the curriculum and the teaching procedures to he used* Since demonstrative geometry is one of the subjects commonly offered on the secondary school level, the nature of the subject matter and the teaching procedures used should he examined in order to determine whether or not this subject contributes to the development of critical thinking* It was the purpose of this study, therefore, to seek answers to the following questions! 1* Can demonstrative geometry be used by mathematics teachers as a medium or a means to develop more effective critical thinking abilities in high school pupils? 2* Does the usual course in demonstrative geometry accomplish this purpose? 3* How do pupils in geometry classes where critical thinking is stressed compare in this ability with pupils in geometry classes where critical thinking is not emphasized? 4* How do pupils in geometry classes where critical thinking is emphasized compare in knowledge of the usual subject matter of geometry with pupils in geometry classes where this objective is not emphasized? 5* How do pupils react or respond to situations where improvement of critical thinking is the major teaching objective as against situations where this objective is not emphasized? In order to delimit the problem, it was necessary to define critical thinking and to develop a frame of reference by which this type of thinking could be identified and evaluated* Thus, critical thinking was defined as a process of criticising and becoming aware of the thinking that had already taken place, or, in other words, a process of thinking about thinking from the point of view of a critic. Since such a definition was too broad, an analysis of twenty different situations was made to discover common elements of critical thinking. These common elements 322 served as a frame of reference for the study. In terms of pupil behavior, they are as follows: 1. He will try to detect motives behind any situation of concern to Mm* 2* He will question the meaning of terms in the situation and seek satisfactory definitions or descriptions of them. 3. He will detect and question underlying assumptions in the situation, 4. He will search for more facts pertaining to the situation. 5. He will test these facts for pertinency to the situation. 6. He will endeavor to discriminate between fact and assumption. 7. He will evaluate himself and others for bias or prejudice in the situation. 8. He will test conclusions for consistency with underlying assumptions in the situation. 9. He will recognize the importance of formulating only tentative conclusions in the situation. 10. He will consider individual and social consequences in the situation. In order to bring about an effective solution of the problem, experimentation was carried on in six public high schools, two of which were large urban, two suburban, and two small schools. Twelve teachers and more than 700 (this number was reduced to 659) tenth grade pupils, averaging slightly more than fifteen years of age, participated in the i study. The schools were paired according to location and size, and approximately 350 pupils (including six teachers) comprised each of the two major groups of the study, namely the Control Group and the Exper imental Group. Prior to the study of demonstrative geometry, the Control Gkoups were found to be statistically superior to' the Experimental Groups in 333 mental ability and in the ability to do the usual course work in geometry# The degree of superiority was not very great; however, the differences of the means between the two groups were statistically significant. Both groups revealed at this stage of their development a deficiency in their understanding of the concept of proof and of critical thinking# Since the Control Gkoups were taught by traditional methods, only the Experimental teaching procedures wSre described# The assumptions underlying the Experimental procedures, as set forth by the six teachers of these groups, were as follows; 1# That high school sophomores are capable of. thinking critically# 3# That the pupil is a psychological being and should have an opportunity to reason about instructional material in his own way# 3# That worthwhile transfer can take place if there is a distinct effort to teach for transfer. 4# That it is possible for teachers to study the behavior of pupils in order to (a) become more sensitive to their needs, (b) stimulate their present interests, and (c) help them ’ develop new interests in line with their abilities. The following is a summary of the more important characteristics of the Experimental &*oup teaching procedures; 1# No textbook was used to develop content; however, references were frequently given for stqjplementary topics# The references were optional and were in the form of suggestions for those who could profit from further consideration# Each pupil developed a notebook, or f,textw as it was called by the pupils, and was given an opportunity to develop it in his own way# 2# Motives were sought by pupil and teacher in every situation of concern to the group# 3# Undefined terms were selected by the pupils, with little if any assistance by the teacher* 4. Nb attempt was made to limit undefined terms to a minimum. 334 5. Terms in need of definition were selected by the pupils, and definitions became an outgrowth of classroom discussion rather than a basis for it* 6# Words or terms needed to express a concept developed by the pupils were frequently supplied by the teacher. 7. Certain propositions which appeared obvious to the pupils were accepted as assumptions unless sufficient inquiry demanded proof. 8* Most of the assumptions were made by the pupils. by the teacher led to others. Suggestions 9. No attempt was made to limit assumptions to any particular number, or to a minimum. 10. In the process of developing the course, need for generalizations and establishment of facts was found essential. Since the in ductive method was used extensively, the element of fact-finding . or searching for more facts became inherent in each situation. 11. The preceding procedure led toward the testing of facts for per tinency, because many facts failed to contribute information that would lead to a generalization. 12. Classroom discussions and procedures necessitated discrimination between fact and assumption. This element of critical thinking led to the formulation of a criterion by which the group could judge whether a statement was an assumption or a fact. The criterion agreed upon by the group was as follows? 1. All theorems, corollaries, and propositions or statements in general, which were proved, would be considered as facts. 2. All unproved propositions or statements would be considered as assumptions until sufficient evidence justified their being classified as facts. 13. Assumptions behind any situation of concern to the group were sought. This detection of stated as well as hidden or implied assumptions was recognized by the group and considered as important. 14. Developing the inductive method of reasoning constituted a major portion of classroom discussion. No statement of what was to be proved was given to the pupils* Certain properties about geometric figures were assumed, and pupils were encouraged to discover the implications of these assumed properties. 15. Pupils were encouraged to set up tentative hypotheses about geometric figures and to test these hypotheses for logical consistency with the underlying assumptions* 325 16, .Another step in developing the inductive method of reasoning was to avoid giving pupils generalized statements* The class was given an opportunity to study the implications or con clusions of properties assumed about a geometric figure and then encouraged to formulate their own generalizations, 17* In developing the deductive method, the theorems suggested by Christofferson as essential to a study of demonstrative geometry were utilized* Some of the theorems, however, were de veloped by the inductive process. Testing conclusions for consistency became very prevalent* 18* The tentativeness of the conclusions reached by the group was definitely recognized, because no conclusion reached proved any more than what was contained in the assumptions* In other words, it became apparent that if the assumptions were true, then and only then were the conclusions true. 19* Need for group cooperation in developing a Mtheory of spaced as it was called by the pupils, necessitated in some instances that pupils evaluate themselves for bias or prejudice. 20. Need for consideration of consequences was brought to the at tention of the group when an assumption was made, because the conclusions reached had inherent in them consequences that in some cases were desirable, while in others they were undesirable* 21* Every effort was made on the part of each teacher to provide for the individual abilities of his pupils, because assumptions frequently led to theorems that were unimportant for many of the group. Consequently, references to such theorems became optional, and further consideration was suggested for the pupils who could profit from such consideration. 22. Provision for individual differences was accomplished largely through (a) individual projects, (b) voluntary contributions to the group, (c) diaries, and (d) self-evaluations. These are briefly described as follows* a* Individual projects were the outgrowth of a conference between pupil and teacher in connection with the inven tory questionnaire administered at the beginning of the school year. Suggestions were made by the teacher re garding a project centered around critical thinking. The pupil was given opportunity to develop this project in connection with geometric or with non-geometric content. In other words, the only requirement was that the project be developed in line with the pupil*s interests and abilities, and that it be the result of critical thinking on the part of the pupil. b. Voluntary contributions were the outgrowth of classroom 326 discussions* Bach pupil was provided with a folder, Shis folder was kept in a small filing cabinet* Any materials related to mathematical or nonrmathematical content which dealt with critical thinking were filed in the folders* c*Diaries were kept by the pupils of situations they en countered outside of the classroom which involved critical thinking* d.Self-evaluations were made periodically *ith respect to progress in the course and with respect to critical thinking* 23* Non-mathematical content was introduced freely in classroom discussions by both the pupils and teacher* 24* fhe geometric content, in general, centered around five geomr* etric concepts and the twenty fundamental theorems and con structions proposed by Chris tofferson as essential to a stufly of demonstrative geometry, !Ebe five central concepts are as follows* a* b. c* d. e» Congruence Parallel lines crossed by transversals Principle of continuity Loci Similarity, or similar figures* 25* Finally, the major emphasis throughout classroom discussions and throughout individual as well as group projects was not upon an accumulation of content or upon subject matter mastery, but upon the method by which content or subject matter was developed* Shis method was directed toward improvement of critical thinking abilities, as previously mentioned. Illustrative examples of exercises or materials dealing with both mathematical and nonSmathematical content were given in Chapters IV and V. $he geometric content developed through the Experimental procedures may be summarized as follows* 1* A totalof 34 terms was found to be classified as undefined. Out of this list ten were cotaQon. to each of the six Experimental Croups* 2* A total of 162 terms was found to be classified as defined. of this list 93 were common to each of thd iix Experimental Groups* Out 327 3. A total of 70 assumptions was mad© "by the six Experimental Groups as a "basis for their study of geometry* Out of this list 17 were proved by at least one of the Experimental Groups. 4# A total of 114 theorems was developed by the six Experimental Groups. Out of this number only 28 were common to each of the six groups. Nine additional theorems in solid geometry were developed by two of the groups. 5* A total of 29 fundamental construction problems was developed by the six Eaqperimental Groups. Out of this group 17 were common to each of the six groups. Each pupil of the Experimental Groups developed a project sup plementing classroom discussions. to be of a mathematical nature* popularity. Sixty-four per cent of these were found Historical projects ranked first in Other topics chosen included geometric designs, biographical sketches* fourth dimensional space, architectural designs, geometrical applications to science, and development of measurement. Numerous proofs of theorems and original exercises were submitted voluntarily by pupils in connection with their projects and classroom discussions, ©airty-six per cent of the individual projects contained only non-raathematical materials. Commercial advertising was the most popular subject selected for purposes of making critical analyses; cartoons, political speeches, and editorials were next in order. Non-mathematical content developed through group and individual effort included a wide variety of materials and experiences. (Ehis content was made up of such topics as commercial advertising, speeches, editorials, political issues, compulsory laws for education, pupil-school relations, liquor legislation, socialized medicine, social security, unemployment, and the place of youth in modem society. .Analyses were made not only of the problem situations just mentioned, but also of potential instruments of 328 “propaganda11, such as newspapers, periodicals, the radio, movies, and the telephone* In the case of groxp projects, in this connection, considerable disagreement was in evidence, because nearly all of the conclusions were colored by emotions* In evaluating the study, the following outcomes were considered? 1* 2* 3* 4* 5* 6* Knowledge of geometric facts Understanding of a geometric proof Skill in manipulating geometric content Pupil reactions to their course in geometry Heading interests Critical thinking ability in terms of the ten elements* Hesults of the evaluation of outcomes in terms of which Experimental and Control Gkoups were compared may be sunmarised as follows? 1. In Jgpgfl&fttee 0f g.eope.falp fafftfl, in understanding of geometric pr£o£, and in skill in manipulating geometric content, the Control Ckoups surpassed the Experimental Groups as measured by the Columbia Research Bureau Geometry Test* It mast be remembered, however, that the Control Groups were statistically superior in mental ability and in geometric ability prior to the study of this subject* Moreover, the superiority in geometric ability of the Control Gkoups was reduced during the nine-month period of experimentation* 2* The reactions of moils to their course u^e.ometry. as determined by the Inventory Questionnaire, were more favorable among pupils of the Experimental Groups than among pupils of the Control Groups* 3. -Analyses of reading interests revealed that Control Group pupils read meagerly, in connection with their geometry course, whereas the Experimental Group pupils read not only a large amount of material, but also a wide variety of different types of material. 4. In critical thinking abilities* as determined by (l) the test in critical thinking, (2) the reports by the trained observer, and (3) the inventory questionnaire, the pupils of the Experimental Groups definitely surpassed the pupils of the Control Groups* 5* On the basis of (l) reports by 165 classroom teachers, (2)com ments made by parents and classroom visitors, (3) diaries kept over a period of time, (4) written work submitted in classes other than geometry, (5) group projects, (6) self-evaluations by pupils, (7) anecdotal records, and (8) individual projects, nearly all of the pupils of the Experimental Groups gave evidence of ability to think critically and made definite improvement in this ability. 329 Conclusions On the has is of the results of the evaluation we are now in a position to answer the questions raised earlier in this study* The questions and their probable answers are as follows5 l*NCan demonstrative geometry be used by mathematics teachers as a means to develop more effective critical thinking abilities of high school pupils?M In terms of the evidence presented by this study, an affirmative answer appears to be justified. It was definitely pointed ait that the Experimental procedures stimulated and affected the thinking of young people not only in the geometry classroom, but also in situations outside the geometry classroom* 2* HDoes the usual course in demonstrative geometry accomplish this purpose?11 (Cn order to answer this question, it is necessary to consider mathematical and non-mathematicel situations separately. In terms of the ten elements of critical thinking in non-mathematicel situations, there is very little if any evidence that Control procedures accomplished this purpose. In the case of mathematical situations, there was evidence of improvement in critical thinking in terms of five of the elements* 3* wEow do pupils in classes where critical thinking is stressed compare in this ability with pupils in geometry classes where critical thinking is not emphasized?** In terms of the ten elements of critical thinking, pupils in classes where critical thinking was stressed were found to be superior in this ability in the nonrmathematical situations. In mathematical situations, there was very little difference between the Experimental and Control Groups in terms of five of the elements. No mathematical test was available for 330 the purpose of measuring critical thinking abilities in terms of the remaining five elements. 4* °How do pupils trained in critical thinking compare in knowledge and in manipulation of geometric subject matter with pupils where critical thinking was not emphasized?0 The answer to this question is that they compare favorably, The evidence presented indicates that emphasis upon a few carefully selected theorems gave the Experimental Group pupils the essential subject matter knowledge and skills of geometry as effectively as those secured by the Control Group pupils. Finally* it is worthwhile, for those who are con cerned with the geometric achievement of pupils, to note that 49 per cent of the Experimental Group pupils elected more advanced courses in math ematics during their junior year. This group made an average mark of B-, (The scale used for grading or marking pupils was A, B, C, D, and F.) 5. HHow do pupils themselves react in a situation where geometry is used as a means to develop critical thinking as against the procedure where knowledge and manipulation of only geometric content is stressed?H The Experimental groups as a whole reacted very favorably to the procedures used. The vast majority of this group enjoyed the presentation of non-mathematical materials, the opportunity to share in the evaluation of their own progress, and in the chance to develop geometric content in their own way. The reactions of the Control Groups were less favorable toward their'courses in geometry. In conclusion, it may be said that major emphasis upon the devel opment of critical thinking, rather than upon acquisition of knowledge and manipulation of geometric content, has given the students of the Experimental 331 Groups not only a satisfactory understanding of the subject matter of geometry but also a more effective method of thinking through problems encountered in non-mathematical situations. This conclusion is con sistent with the recent trends toward the reorganization of the curriculum and the development of methods of teaching that are more dynamic, functional, and life-like, and that provide better opportunities for the growth and development of young people who are capable of a high degree of intelligent self-direction. 332 SELECTED BIBLIOGRAPHY Abbott, Edwin A., Hatland. A Romance of Many Dimensions. Little. Brown and Company, Boston, 1929. Beatley, Ralph, ®The Second Report of the Committee on Geometry®. Mathematics Teacher, vol. XXVI, No. 6, 1933. 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H., ®The Educational Value of Logical Geometry®, The Mathematics Teacher, vol. XXXII, No. 4, 1939. Breslich, E. S., “Secondary School Mathematics and the Changing Curriculum®. The Mathematics Teacher, vol. XXVI, No. 6, 1933* Ireslieh, E. R,, “The Nature and Plane of Objectives in Teaching Geometry®. The Mathematics Teacher, vol. XXXI, No. 7, 1938. Breslich, E. R., The Technique of Teaching Secondary School Mathematics. The University of Chicago Press, Chicago, 1930. Brink, William G., Directing Study Activities in Secondary Schools. Doubleday, Doran and Company, New York, 1937. Chase, Stuart, The Tyranny of Words. Hare our t, Brace and Company, New York, 1938. Childs, John L., Education and the Philosophy of Sxperimentalism. The Century Company, New York, 1931. 333 Christofferson, H. C., "Geometry A Way of Thinking". Teacher, vol. XXXI, Ho, 4, 1938, The Mathematics Christofferson, H,. C,, Geometry Professionalized for Teachers, Banta Publishing Company, Menasha, Wisconsin, 1933, Clarke, E. L., The Art of Straight Thinking, Hew York, 1934* Coe, Georjje A,, 1934. George .Appleton, Century Company, What Ails Our Youth, Charles Scribner and Sons, Hew York, Cohen, Morris R., Reason and Hature. 1931. Earcourt Brace and Company, Hew York, Cohen, Morris R. and Nagel, Ernest, An Introduction to Logic and Scientific Method. Harcourt, Brace and Company, Hew York, 1934. Columbia Associates in Philosophy, An Introduction to Reflective Thinking. Houghton Mifflin Company, Boston, 1923. Commission on the Secondary School Curriculum of the Progressive Education Association, Mathematics in General Education. Tentative Report of the Mathematics Committee, Columbus, Ohio, June 1938, Cooley, Hollis R. and others, Introduction to Mathematics. and Company, Hew York, 1937. Dewey, John, Characters and Events. Houghton Mifflin Henry Holt and Company, New York, 1929. Dewey, John, Experience and Education. The Dewey, John, How We Think. Dewey, John, Human Hature and Conduct. Henry Holt and Company,Hew York, 1922. D. C. Heath and Macmillan Company,Hew York,1938. Company, New York, 1933. Dewey, John, Logic, the Theory of Inquiry. Henry Holt and Company, Hew York, 1938. Dresden, Arnold, "Mathematics in a Changing World". vol. 38, June, 1934. Scientific Monthly. Durrell, PIetcher, "Value and Logic in Elementary Mathematics". Mathematics Teacher, vol. XXIII, Ho. 1, 1930. The Darrell, Hatcher and Arnold, E. S., Hew Plane Geometry. Charles B. Merrill Company, Hew York, 1934. Everett, Samuel, Democracy Paces the Future. Columbia University Press, Hew York, 1935. Everett, Samuel, and others, A Challenge to Secondary Education. D. AppletonCentury Company, New York, 1935. 334 Fawcett, Harold P., "The Nature of Proof1*, Thirteenth Yearbook of the National Council of Teacher a of Mathematics. Bureau of Pub lications, Teachers College, Columbia University, 1938, Fawcett, Harold P,, "Teaching for Transfer**. vol. XXUI, No. 8, 1935. The Mathematics Teacher, Garrett, Henry E., Statistics in Psychology and Education. Green and Company, New York, 1926. Longmans, Hall, Elizabeth L., “Applying Geometric Methods of Thinking to Life Situations", The Mathematics Teacher, vol. XXXI, No. 8, 1938. Hartung, Maurice L., "Mathematics in Progressive Education". Teacher, vol. XXXII, No. 6, 1939. Hartung, Maurice L.,**Some Problems in Evaluation". vol. XXXI, No. 4, 1938. The Mathematics The Mathematics Teacher, Hassler, J. 0. and Smith, Holland K., The Teaching of Secondary Mathematics. The Macmillan Company, New York, 1930. H&wkes, Herbert E. and Wood, Ben D., Columbia Research Bureau Plane, Geometry Test. World Book Company, New York, 1926. Hawkes, Herbert E. and Wood, Ben D., Manual of Directions. for Columbia Research Bureau Plane _Geometry^Te^t. World Book Company, New York, 1926. Hedrick, E. E,, "The Meaning of Mathematics". October 1935. Scientific Monthly, vol. 41, Hedrick, E. B,, "Teaching for Transfer of Training in Mathematics". Mathematics Teacher, vol. XXX, No. 2, 1937. Hogben, Lancelot, Mathematics for the Million. New York, 1937. The W. W, Norton and Company, Institute for Propaganda Analysis, "Propaganda, How to Recognize It and Deal with It". New York, 1938. Keyxer, Cassius J., The Human Worth of Rigorous Thinking. Columbia University Press, New York, 1925. Keyser, Cassius J., "The Meaning of Mathematics". vol. I, 1932. Scriuta Mathematics, Keyser, Cassius J., Thinking About Thinking. E. P. Dutton and Company, New York, 1926. Kilpatrick, W. H., Remaking the Curriculum. Newson and Company, New York, 1936. 335 Kilpatrick, W, H., 11The Next Step in Method". vol. XV, No. 1, 1923. The Mathematics Teacher, Lazar, Nathan, "The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry", The Mathematics Teacher, vol. XXXI, Nos. 3, 4 and 5, 1938. Leisenring, Kenneth B., "Geometry and Life". vol. XXX, No. 7, 1937. The Mathematics Teacher, Longley, W. R., "What Shall We Teach in Geometry". Pifth Yearbook of the National Council of Teachers, of Mathematics. Bureau of Publications, Teachers College, Columbia University, 1930. Manning, Henry P., Geometry of Four Dimensions. The Macmillan Company, New York, 1928. Manning, Henry P., Non-Kuclidean Geometry. Ginn and Company, Boston, 1901. Mathematical Association, The Teaching of Geometry in Schools. Sons, Ltd., London, 1929. G. Bell and Mathematical Association of .America, and the National Council of Teachers of Mathematics, A Preliminary Report? The Place of Mathematics in Secondary Education. Edwards Brothers, Inc., Ann Arbor, Michigan, 1938. National Committee on Mathematical Requirements, The Reorganization of Mathematics_in Secondary, Education. Houghton Mifflin and Company, Boston, 1923. Ogden, O.K., and Richards, I. A.# The Meaning of Meaning. Harcourt, Brace and Company, New York, 1936. Orata, Pedro T., "Transfer of Training and Educational Pseudo-Science". The Mathematics Teacher, vol. XXVIII, No. 5, 1935. Orleans, Joseph B. and Jacob S., Orleans.geometry Prognosis Test.. Book Company, New York, 1929. World Orleans, Noseph B. and Jacob S., Manual of Direct ions for Orleans Geometry Prognosis Test. World Book Company, New York, 1929. Otis, Arthur S., Otis. .S.elfrAdministering Tests of Mental Ability. Book Company, New York, 1928. World Otis, Arthur S., Manual of Directions for Otis. Self-Administering Tests of Mental Ability. World Book Company, New York, 1928. Parker, Elsie, "Teaching Pupils the Conscious Use of a Technique of Thinking". The Mathematics Teacher, vol. XVII, No. 4, 1924. Perry, Winona, A Study in the Psychology of Learning in Geometry. Bureau of Publications, Teachers College, Columbia University, New York, 1925. 336 Progressive Education Association. Tentative Report of the Mathematics Committee, Mathematics In General Education. Commission on the Secondary School Curriculum, Columbus, Ohio, June 1938, Progressive Education Association, Nature of Proof Test 5.3. Ohio State University, 1936, Rainey, Homer P., How Pare American Youth, New York, 1937. D* Appleton-Centnry Company, Reeve, W. D., "At tacks on Mathematics and How to Meet Them1*. Eleventh Yfiflpbook p£_.t.he National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, New York, 1936, Reeve, W. D,, "Modern Curriculum Problems in the Teaching of Mathematics in Secondary Schools'1• The Mathematics Teacher, vol. XXXII, No, 3, 1939. Reeve, W. D., "The Teaching of Geometry". Fifth Yearbook of the National .Councils of Teachers of Mathematics. Bureau of Pullications, Teachers College, Columbia University, New York, 1930* Robinson, James Harvey, The Mind in the Making. Publishers, New York, 1921. Harper and Brothers Russell, Bertrand, Education and the Good Life. Boni and Liveright, Inc., New York, 1926* Russell, Bertrand, Introduction to Mathematical Philosophy. and Unwin, Ltd., London, 1919* George Allen Russell, Bertrand, The A B 0 of Relativity, Harper and Brothers, New York, 1925* j Sanford, Vera, "Why Teach Geometry". S No. 5, 1935. The Mathematics Teacher, vol. XXVIII, j Sanford, Vera, A Short History of Mathematics. Boston, 1930, Houghton Mifflin fiompany, | Schaaf, William L., Mathematics for Junior High SchoolTeachers. ‘ Publishing Company, New York, 1931. Johnson Schultze, Arthur, The Teaching of Mathematics in Secondary Sehoola. Macmillan Company, New York, 1927. I Shibli, J., Recent Developments in the Teaching of Geometry. State College, State College, 1932. The Pennsylvania Sitomer, J. T., "If-Then in Plane Geometry". The Mathematics Teacher. vol. XXXI, No. 7, 1938. 337 Slosson, Edwin E. # Easy Lessons in Einstein* New York, 1920. Harcourt Brae© and Company# Smith, David Eugene, The Teaching of Geometry. Ginn and Company, Boston, 1911. Smith, David Eugene and Reeve, William D., The Teaching of Junior High School Mathematics. Ginn and Company, Boston, 1927. Smith, Reeve and Moras, Text and Tests in Plane Geometry. Boston, 1933. Ginn and Company, Stamper, Alvin W., The History of the Teaching of Elementary Geometry, Bureau of Publications# Teachers College, Columbia University, Hew York, 1909. Stewart, Maxwell S., "Youth in the World Today". Public Affairs Pamphlets. Public Affairs Committee, Inc*, No. 22# New York, 1938. St. John, Charles W., Educational. Achievement in Relation to Intelligence. Harvard University Press# Cambridge# 1930. Strong# Theodore, "Teaching Geometry without a Text Book". Teacher, vol. XIX, No. 2# 1926. The Mathematics Studebaker, John W.# "New Trends in the Teaching of Mathematics". Mathematics Teacher, vol. XXXII, No. 5, 1939. The Sykes, Comstock, and Austin, M., Plane Geometry. Hand McNally and Company# 1932. Symonds, P. M., Ability. .foanfleEdg.. fP.r,gtt^flarfli.z.ed. jglvteymgRjL.Xes.tB. jfoft High School. Bureau of Publications, Teachers College# Columbia University, New York, 1927. Taba, Hilda# The Dynamics of Education. 1932. Hareourt Brace and Company# New York, Taylor, E. H.# "The Introduction of Demonstrative Geometry", Teacher, vol. XXIII, No. 4, 1930. Tyler, Ralph W., "Appraising Progressive Schools". No. 8# May 1936. The Mathematics Educational Method. vol.XV, Tyler, Ralph W.# "Evaluations A Challenge to Progeessive Education". Educational Research Bulletin. Ohio State University, Columbus, vol. XIV, No. 1# 1935. Tyler, Ralph W,, "Techniques for Evaluating Behavior^". Educational Research Bulletin. Ohio State University, Columbus, vol. XIII, No. 1, 1934. 338 Ulmer, Gilbert, *leaching Geometry for the Purpose of Developing .Ability to Do Logical Thinking”* The Mathematics Teacher, vol* XXX, No* 8, 1937* Upton, Clifford B., HThe Use of Indirect Proof in Geometry and Life”. U l t h Xearbook of the.,National Council of Teachers of Mathematics,. Bureau of Publications, Teachers College, Columbia University, New York, 1930. Webb, L* W*# "The Transfer of Training*'* Chapter XIII in Educational Psychology* edited by Skinner, Charles E., Prentice-Hall, New York, 1936. Witty, Paul A*, **Intelligence5 Its Nature, Development and Measurement". Chapter XVI in Educational Psychology* edited by Skinner, Charles E*, Prentice-Hall, New York, 1936. Wheeler, Eaymond H., "The New Psychology of Learning", Tenth Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, New York, 1935# Young, Jacob W. A., The Teaching of Mathematics. New York, 1934. Longmans, Green and Company, Young, John Wesley, Lectures on .Fundamental Concepts of Algebra^nd Geometry. The Macmillan Company, New York, 1925# .Analyses of Situations for Elements of Critical Thinking 340 Situation No, 1 Recently, M.S. was asked first to analyze and then state the fundamentalassumption underlying the following statement: "Mr. Harrison is a teacher; therefore, he nrust have gone to college." M.S* responded as follows: "To he a teacher one The teacher must attend college." remarked that M.S. *8 assumption could he atated more pre cisely and more to the point. M.S. remarked: "I don't see how it could he stated any clearer." Through the ensuing discussion, M.S. asked the teacher to make a clearer statement of this assumption. The teacher responded as follows: "Teachers must have attended college" and pointed out that this statement and M.S.'s involve a difference in tense. M.S4- then replied: "I still fail to see any difference between your statement of the assumption and my own. I am going to ask some of my other teachers or authorities about it." M.S. consulted several authorities and arrived at a tentative conclusion that there was a difference in tense regarding the two statements and that her teacher's statement was more to the point. However, she still main tained that her statement inferred the same assumption. Of course, the authorities whom she consulted might he questioned, also her own bias and prejudice in the matter might likewise he questioned. The fact, how ever, that M.S. questioned her teacher and then of her own volition sought further information from other sources is in itself indicative of critical thinking elements. 341 Situation No. 2 In discussing the method of induction, the teacher led his pupils from the properties of a point to those of a fourth dimensional figure. It was then pointed out that lay this method one could readily determine the properties of a fifth, sixth, and nth dimensional figure. Most of the class hecaime very interested. Some of the pupils insisted that one eouldn*t describe properties of a fourth dimensional figure if one could not see it, let alone a fifth, sixth, and so on up the scale* R.B. remarked that she understood her hr other, who was studying chemistry at the time, to tell their father that the properties of some of the elements in chemistry were known long before the element was dis covered or isolated. S.B. said that this was also true in astronomy and physics, She teacher finally asked the doubtful pupils if their confidence in the inductive method would he strengthened if they could see a picture of a tesseract, a fourth dimensional figure. Since the eagerness of the reply was in the affirmative, they were then referred to Slosson*s Easy Lessons in Einstein. W.M, came into the teacher*s room after school quite excited. He said, ttI just looked up the tesseract and found that it had the exact number of points, lines, squares, and cubes that we figured out induc tively that it would have. Is it possible to work out a formula for the properties of an nth dimensional figure in the same way?1* lEhe teacher replied that to the extent of his knowledge no one had done this par ticular thing, but that through the inductive method one should be able to make such a generalization. W.M. also remarked that the cubes in the 342 tesseract did not look like cubes* Che teacher asked him if he could eaqplain why, and added that if W.M* could answer these questions and perhaps devise some ingenious construction of a tesseract, such that it would give the appearance of eight perfect cubes to the eye, then his name would undoubtedly go down in history as one of the world's renowned contributors to knowledge* Many elements of critical thinking entered into this situation, when the pupils questioned the terms used, the assumptions behind the inductive method, the desire for more facts, the motive behind the teacher’s presentation, W.M*’s inquisitiveness, the conclusions reached and their tentativeness, their consequences to the individual, and consequences to the group. W.M. was thinking quite critically in this situation and if he continues to do so he may turn out a bit of creative knowledge as a result of this inquisitiveness and questioning. Situation Ho. 3 While reading a newspaper one evening, W.C.’s attention was directed to an overcoat advertisement. It read as follows* HIts fabric (referring to the Angora Knit-tex) is based on the soft, silken hair of the Angora goat which lives in regions where the days are hot and the nights cold. There fore, the protective hair of this animal which is used in this coaA is adaptable to varying degrees of temperature.11 W.C. recognized that this statement was the same type of deductive reasoning that he encountered in proving some of the theorems in his geom etry class* He brought the advertisement to class and pointed out how he doubted that certain living qualities of the Angora pelt, subjected to varying degrees of temperature, were actually retained in the Knit-tex 343 material after the animal was killed* V.M. remarked that W.C. was making an assumption himself, because he did not know whether or not this was true. W.C. replied, however, that facts in support of the conclusion of this advertisement were lacking and furthermore the advertiser was trying to sell his product regardless of the facts that were necessary to support his conclusion* Had W.C* or any other member of the class searched for more per tinent facts in this situation, consulted experts, looked up references relative to the fur industry and made a detailed analysis of Knit-tex, its meaning, the nature of the material, the manufacturing process, and so forth, they would have exhibited a much higher degree of critical thinking. Situation Ho. 4 B.B. reported that in an English class they were discussing in ductive and deductive reasoning* / The following statement was made: Jonses live in a red brick house on Ninth Street. HThe I live in a red brick house on Ninth Streets therefore my name is Jones.11 B.B, was thinking critically when she remarked that there could have beennore than one red brick house on Ninth Street and also more than one Jones living in a red brick house on that street. Situation No* 5 ! B.C. r©narked that last night she heard Ipana Toothpaste adver- !tised over the radio. As a part of the broadcast, the testimonial tech- !nique of propaganda was used when a man said that he heard so much about !Ipana that he asked his dentist about it. He said that his dentist told 344 hip that Ipana was a very good toothpaste, therefore whatever the adver tiser said about the product was really tnue. B.C. was thinking critically when she detected the fact that the dentist's name was not mentioned and that there was a possibility that the man was paid by the advertisers to make this testimonial. B.C. would have thought far more critically, had she of her own accord raised this question with several dentists and then weighed the facts before arriving at a more definite conclusion. Situation Ho. 6 (Che question of the sinking of the Panav was brought up in class. R.M. remarked that "all of the papers in the country had played up the event as a bit of propaganda to stir up the emotions of the people." D.L. replied, "You are making an assumption, because you have not looked into all of the papers, and besides we take the Christian Science Monitor and it certainly failed to play up this event." Critical thinking was exhibited in this situation; however, there Iwas very little evidence presented and no one seemed to be seeking more Ifacts in the situation, or getting at the motives for the newspapers in |playing up this incident, or studying the consequences involved as a result I of such an incident, particularly itseffect on public opinion, j Two days later V.M. informed her teacher that she had gone to a | friend of her father who had spent several years in China. -According to i V.M., he stated that the Panav belonged to the Standard Oil Company and | because of river pirates, the United States government, some time ago, had given the Standard Oil Company protection. He also remarked that since river pirates did not use airplanes, the Standard Oil Company certainly 345 anticipated attacks by Japanese airplanes, because there were anti-aircraft guns aboard and they were used in this instance, H.M. remarked, “that's right because our government wouldn’t own a pile of junk like the Panav.n There was considerable evidence of critical thinking in this situation; however, V,M, failed to think very critically when she con sulted only one source and even here she failed to corroborate the author ity of this source as well as to evaluate herself and her authority for bias and prejudice. The fact, however, that she did seek some further information is in itself an element of critical thinking. While various members of the class contributed many facts in this situation, nevertheless the problem remained unsolved, although most of the class arrived at a tentative conclusion that the Panav was a government boat, V.M. did remark that she would write to the War Department to determine whether or not the Panav was a government boat. Of course, such information may be very confidential, and whatever conclusions ere derived from this situation will, in all probability, remain tentative. Situation No. 7 D.R. 5 ’’The coach of the Prosh-Soph basketball team gave us a good talk on not having enough fight. He didn’t stop to tell us just what he meant by ’fight1. He left it up to us to decide for ourselves. I thought by the way he used the word in reference to basketball, it meant playing with all you’ve got and trying to get possession of the ball and make as many baskets as possible without committing fouls. CUf course if you did without teamwork, you would be referred to as a ’hog’, yet you would be playing for the team and playing to win, which after all is one of the 346 objects of the game. I would like to ask the coach what he means by ‘fight1, but knowing him as I do I think I had better keep it to myself." D«R. made his own analysis of a situation that has in it some elements of critical thinking. Tor example, he tries to get at the meaning that his coach attaches to the term ‘fight1. He is trying to determine the coach* s motive in making this request. seek further information in this situation. He has failed to However, D.B. has considered consequences as far as his knowledge of the coach is concerned, and he also has arrived at some tentative conclusion. Situation Ho. 8 Recently, in a geometry class, it was necessary to make an assump tion before a certain fact involving inequalities could be established. V.M. remarked that if equals multiplied by equals gave equal products, then why couldn't we assume that if we multiply ten greater than five by two equal to two, then twenty will still be greater than ten in the same way. !Ehe class agreed that this was a valid assumption. Following a critical discussion, the group agreed to frame the assumption as follows! "If unequals are multiplied by equals, the products will be unequal in the same way." .After a few minutes S.R* remarked that this statement was not always true. The teacher wondered if someone would discover this subtle difference and in order to make the situation more complex, remarked that [ this statement could be found in asny textbooks almost verbatim to the j j way in which this group had framed it. Now, either the large number of textbooks were wrong or S.P.. was wrong. S.R., however, still insisted S47 that the statement of this assumption was not always true and that he believed he could prove it* When pressed by several members of the class to produce his evidence* S*E* stepped Quietly to the blackboard and used V.M.’s illustration except that hs multiplied ten greater than five by minus two equal to minus two and obtained the products of minus twenty less than minus ten, which naturally gives the opposite order. | !Phis was one of the outstanding examples of critical thinking in class UP to this point. The next day, several pupils came to class, in cluding S.H. and V.M., and each reported that he or she had found this assumption in several textbooks. This includes several elements of cri tical thinking, especially the one in whichthere is a searchfor more facts and not relying upon one authority. Situation ffo. 9 Mary read that singeing the stems of cut flowers before putting them into water would make them keep longer. duct an experiment to prove this statement. She thought she would con She did and concluded that her experiment proved that singeing the stems of cut flowers makes them stay fresh longer. She was so elated over the results of her experiment that she had to tell her science teacher all about it. The teacher, of course, wanted to know how the eaqperiment was performed, so Mary described it as follows J ! ttI*ast Sunday evening I cut some roses and placed them in a vase of water. Monday morning I cut some sweet peas and after singeing the stems I put them into some cold water from the refrigerator. X then set the vase containing the sweet peas in the library beside the vase of roses. Wednesday morning I noticed that the roses were very wilted although the sweet peas still looked fresh. The sweet peas did not begin to show signs of wilting until Thursday evening. Since the sweet peas remained 348 fresh for a day longer than the roses, I proved that singeing the stems of cut flowers "before putting them into water will make them keep longer.11 To her great disappointment the teacher very graciously informed her that the results of her experiment proved nothing. Mary was thinking critically only when she decided to verify the statement Toy means of an experiment of her own. Had she made a critical analysis of her method of procedure in the first place or repeated the Experiment with this in mind, she would be exercising a far higher degree of critical thinking. Situation Ho. 10 The following bulletin was issued3n the advisory room of one of the high schools. The bulletin read as follows: Hln the basketball game with _______ high school an epidemic of whistling broke out which was annoying to many spectators and so seriously interfered with the progress of the game that the official stopped the game while the crowd was asked to stop whistling. The crowd very nicely desisted, but had the game been a close one and had the officials en forced the rules which hold the home team responsible for the conduct of the crowd, our team would have lost because of the crowd. Please see that this does not happen again.11 In a discussion of this situation, most of the members of the class felt that the rule was not fair to the players. R.M., B.M., S.R., and D.R. dido^t believe there was such a rule, although R.M. and B.M. insisted that the referee could call only one technical foul against the captain of the home team, S.R. remarked that the referee could call any number of technical fouls. B.R. insisted that the referee could award the game outright to the offended team. Very little critical thinking was going on in this situation until S.R., R.M., and D.R., of their own accord, actually looked up the rule and 349 discussed its limitations* They also made an analysis of the situation and the motives behind this rule as they applied to,this particular situation* Situation Ho* 11 Recently, a certain class in geometry, having had no previous ex perience ot knowledge of the proof concerning the theorem, ''The sum of the interior angles of a triangle is equal to 180Ofl, proved this theorem by a method unknown to the writer, and to the extent of his knowledge it is not recorded in any geometry textbook* This situation is an excellent example of critical thinking in geometry, because the group was permitted to reason in their own way, to challenge and question each other's terminology and assumptions, to test facts for their pertinency, and thus through mutual agreement they arrived at a tentative conclusion! "If certain definitions and assumptions are accepted, then the sum of the interior angles of a triangle is 180°*11 The following analysis of this situation reveals several elements , \ I of critical thinking! (l) The teacher's motive was to teach the pupils some geometry and at the same time give them an opportunity to think critically* Several pupils recognized this motive* Then, too, there were pupil motives involved, motives appearing in the form of a desire to solve a perplexity and to gain recognition or even a reward for effort. The latter motive, of course, has little educational value if it becomes extrinsic in nature* (2^ The meanings of various terms, such as vertex, transversal, alternate interior angles, supplementary angles, point, line. aagl.e, right, §50 parallel lines, and opposite angles, were questioned by the different members of the class until mutual agreement ensued* (3) The following assumptions were questioned during this disuussionl A quantity may he substituted for its equal in any expression* he extended, or it may he limited at any point* A line may Only one perpendicular can he drawn to a line from an external point, or only one can he erected from a point in the line* geometry. We are assuming the postulates of Euclidean Largely because of the last assumption it was possible to get nearly all of the group to agree upon the others* (4) The group brought out a number of related facts, some of which were as follows* If two or more parallel lines are cut by a third, the corres ponding angles are equal* If two or more parallel lines are cut by a third, the alternate interior angles are equal. If the interior angles on the same side of the transversal are supplementary, the lines are parallel. Lines perpendicular to the same line are parallel to each other* (5) She group tested the above facts for pertinency to the situation and agreed that only the second and fourth facts applied. (6) During the discussion the observer noted that there was a tendency to discriminate between facts and assumptions* For example, “Lines perpen dicular to the same line are parallel11 was regarded as an assumption by some and a fact by others. only after it was proved. 'The group agreed to call a statement a fact Since this statement had been proved earlier, the group agreed to call it a fact. Two other statements were questioned in the same way* “90° in a right angle”, and “If two straight lines intersect, the opposite angles are equal.11 There was a lack of unanimous agreement on the former, but in the end it was agreed that this was a fact* 351 (7) Some pupils “became somewhat stubborn and refused to relinquish their position without offering evidence in support of their belief. Por example, M.S. insisted that ”Iiines perpendicular to the same line are parallel” is an assumption, yet she failed to give adequate reasons that would satisfy the rest of the group. In other words she either needed to evaluate herself for bias or prejudice, or she should have been able to convince at least some of the members in the group that her position was a valid one, (8) There was also the element of formulating a conclusion or generalization, and the testing of this conclusion for consistency with its assumption. Some of the pupils raised questions as follows* Does the conclusion follow logically from the premises or the assumptions that were made? same conclusion be reached using different assumptions? Could the Different facts? (9) Another element seemed to be prevalent in the discussion, the recog nition of the tentativeness of conclusions. This element grew out of the previous one, because the pupils recognized that the conclusions are true only if their underlying assumptions are true. Since all assumptions are relative, then so are the conclusions. (10) The conclusions reached in any area of thought imply individual as well as social consequences. In this situation the following question was raised* What are the limitations of this Euclidean conclusion? Why did Unstein use Eiemann*s geometry in establishing his theory of relativity? What is the essential difference between the Euclidean, elliptic, and hyperbolic geometries with respect to this conclusion? Answers to these questions were somewhat beyond the maturation level of this group* they did grasp the idea that consequences were involved* However, 352 Situation Up, 12 During the early part of the course several pupils asked the customary question: “Why do we need to study mathematics?” Of course, several reasons were given at the time. Recently, the teacher came across the following statement which fitted in very nicely with this situation# “Mathematics has furnished the principal tools for discovering, testing, and stating the laws of nature. Without it, our material civilization would crumble into dust, and man would return to a state of savagery. These undisputed facts alone would warrant a mandatory emphasis on mathematical training in our schools.“ (l) In discussing this situation, R#B# wanted to know who made this state ment and why it was being made. This remark suggests the element for seeking motives# (3) D.W. and several others wanted to know what the different words and phrases meant, for example Hlaws of nature11, ‘‘undisputed facts*1, “mandatory emphasis on mathematical training’*, and so forth. One observes the element of seeking meanings or definitions in this inquiry. (3) R.M* pointed out that the person who made this statement was assuming that mathematics is the essential foundation for our civilization. The element of detecting and questioning assumptions is prevalent# (4) S.R. remarked that he thought the statement was based On many facts that were not stated. He thought that radio, aviation, engineering, science, astronomy, machinery, all depend on mathematics and that many facts could be secured to support this statement. (5) M.H. wanted to know in what way radio, or ainplanes, influence man datory emphasis on mathematical training for all high school pupils. remafck brings out the element for testing facts for pertinency# This 353 (6) M.S. pointed out that we ought to determine whether the statement, "Without mathematics, our material civilization would crumble into dust, and man would return to a state of savagery^1 was a fact or an assumption. Some had remarked earlier in the discussion that this was an assumption# ^erein is the element of distinguishing facts from assumptions. (7) V#M. insisted that "no subject in school warranted mandatory emphasis and that one should have the right to study any subject one wishes to choose." R.M* replied, "That's all well and good, but just because you can't get mathematics is no reason why others should not be required to take it.11 In this situation, the element for self-evaluation with respect to bias or prejudice is quite apparent. (8) S.R. thought that if mathematics was a foundation for our civilization, then it would warrant mandatory training in our schools. He seemed to think that this was the only conclusion that could be consistent with the assump tions. However, if we should find that mathematics is not such a foundation, then we may have a different conclusion. The element of testing conclusions or generalizations for consistency with their assumptions is apparent in this situation. (9) The preceding situation likewise suggests the element of recognizing tentative conclusions, because S.H. intimated that when we change assumptions we may get different conclusions, provided, of course, that we are consis tent. (10) The teacher asked the question? nWhat would be the effect upon indi viduals and society if every person were required to study algebra and geometry?" The responses were numerous and quite varied. The important point here is the revelation of another element, that conclusions when acted upon need to be tested for individual as well as social consequences. 354 Situation No. IS Recently, in a geometry class at School A, M.S. remarked that he heard one of the faculty members make the statement that “School ex hibited very poor sportsmanship after their game with School B last Friday night. “ (This statement refers to the free-for-all fighting that followed the game.) M.S. was immediately challenged by several members in the class relative to this statement. that- made this remark. The group wanted to know who it was Seeing an opportunity for testing the critical thinking abilities of the class, the teacher left the subject open for discussion. The following elements of critical thinking appeared to stand out in this situation* (1) Seeking to determine motive behind this situation or statement. B.M.: “Is this teacher partial toward School B?“ B.C.I “Is this teacher trying to prevent another outbreak similar to this one?“ S.E.! “Did this teacher make this remark simply to humiliate School A.“ (2) Seeking and questioning the meaning of the terms involved. D.L* wanted to know if School A meant the players, the student body, the teachers, spectators, or a combination of all persons connected with the school. V.M, and D.E. and others wanted to know what was really meant by suor tsmanshin. (3) Detecting underlying assumptions. Most of the members of the class thought that this teacher assumed that School A players started the fight. B.C. remarked that the teacher was assuming that that followed the game had a direct bearing upon sportsman ship* V.M., however, remarked that B.C. was making an assumption, be cause sportsmanship, as generally understood by the public applies to all situations and at all times. 355 (4) Seeking more facts* Nearly all of the members of thecUtss thought that more facts were needed in this situation "before any conclusions could be drawn* R.B. thought that the teacher in question should have determined who started the fight and how it had started before making such a positive statement. (5) Testing facts for pertinency. T.Q. thought that some of the suggested facts had nothing to do with the fight. for example, School A*s victory over School B in their swimming meet, that afternoon, had nothing to do with stimulation of the fight. (6) Distinguishing facts from assumptions. for example, D.E. said, H0ne of the School B players kicked one of the School A players and then the latter struck the former with his fist." B.C. asked D.R. if he actually saw the former kick the latter. D.R. said, wNo, but the player told me that this was the case.** B.C. then remarked that D.R. was not presenting a fact but stating an assumption or opin&pn, because no one in the class actually saw the former kick the latter and only a few thought they saw the School ^ player strike the School B player and then they were not absolutely certain that he was actually struck by the School A player. (?) Self-Evaluation for bias or prejudice* The teacher remarked that in a situation of this sort it was very difficult to reach unbiased decisions or conclusions* M.S. r©narked that one ought to be loyal to his school regardless of circumstances, but that in situ ations like this, where the reputation of the entire group was at stake as a result of certain overt acts on the part of one individual or a small minority, one needed to be more open-minded and consider the facts from an 356 unbiased point of view, (8) Evaluating conclusions for consistency. The group agreed that if School £ representatives started the fight then all of the participants in this brawl, including School B, were poor sports- S,R, remarked that this conclusion was true only if this concept was included in the meaning of sportsmanship when we first agreed to our assumptions, (9) Heed for tentative conclusions, S,R, further remarked that while we had many facts we still did not have enough of them and therefore our conclusions could only be tentative, (10) Evaluating conclusions for their individual and group consequences, P,H, thought that a group discussion of this situation helped him develop a different point of view, because he had originally felt that it was all School B*s fault. He also said that if all of the students from both high schools had looked at the situation in this way they would not have fights that reflect upon the entire student body, the faculty, and the community. Situation Ho, 14 When presented with the statement, wMany people believe that the white race is superior to the colored race", a certain group of young people gave numerous and vgried responses. asked were: HWho made the statement? many people? The most prevalent questions What is meant by many people? What was the motive for making this statement? How Do colored people believe this?" (l) Perhaps the primary element in this situation is the search for motives. Several pupils even wanted to know what the teacher *s motive was 357 for presenting this situation to the group. (g) W.M. pointed out an element of critical thinking when she said that 1qanv -people probably meant many white neo-ole, and that the terms white rase and colored race needed to he defined. B.F* thought that superior needed defining, because of the wide differences in physical and mental traits of people. (3) Nearly all of the class agreed that the many people, who believe this way, are making an assumption because there was very little, if any, scientific evidence to support the claim. Consequently, we find here the element for detecting assumptions. (4) The previous item leads us to another element, the need for seeking more facts. Such names as Booker T. Washington, Jesse Owens, and Joe Louis were cited as examples. V.M. remarked that some of the colored pupils in our own school ranked at the very top in athletics as well as in intellectual matters. B.C. also brought out that in this country there were more white people than people of any other race and that he remembered reading somewhere that the white races control the major por tion of natural resources. more opportunities. R-.B. pointed out that the white race had many In this country the white race has far superior educational opportunities, although this ratio has been reduced as a result of the civil war. (5) B.M. in his remark leads to the element of testing facts for pertin ency to the situation when he said that "population has little or no effect upon racial superiority11. (6) The following statements lead to the element of discrimination between facts and assumptions. C.C. remarked, "Colored people are just as smart 358 as the white people*0 VT.M* said* °This is an assumption because the white race is head and shoulders above the colored race. I say this because the white race has had so many more opportunities to develop and to spread out.11 (7) M.L.S* remarked, °Colored races that live in America would be far better off at home where their race belongs*0 D.R* • °How do you know they would? Can you tell us in what way they would be better off?0 M.L.S. again responded, °I just know they would be and so do you.0 this To replied, °I disagree with you for the simple reason that it takes facts and many of them to convince me* It seems that you are prejudiced against the colored race.0 Many fruitless ideas were exchanged in this situation, with very little evidence or basic facts to support arguments either way* The element of bias and prejudice was rather obvious in M*L*S« *s responses. (8) S.E. remarked that if we assume is, having that all men arecreated equally, that equality of opportunity, then by means of facts we should arrive at some conclusion* "If we should happen to arrive at the conclusion that the white race should be superior to the colored race then in all pro bability there are some inconsistencies in our argument. That there are numerous and complex factors cannot be denied; however, one point stands outs that we either need to get more facts or we need to change our fun damental assumption.0 This statement is suggestive of a need in testing conclusions for consistency with their assumption. (9) V*M. J °I don*t think we can arrive at any final conclusion to this problem. It seems to me that new and perhaps additional facts may be uncovered which will change the situation.0 This statement has in it the 359 element of recognizing the tentativeness of conclusions, (lO) throughout the discussion several statements were made that had in them the element of testing conclusions for possible consequences. R.C* brought out the point that colored people would make far greater strides if given the same opportunities (social and economic') as are at the present time accorded to the white race. Situation No. 15 When presented with the situation* "Many people in the United States have recently stated that they would never bear arms in any way. Others feel that it is their duty to support their government at all times whether in peace or in war11, a group of pupils gave a wide variety of responses which had in them many elements of critical thinking. Some of these are as follows: (1) The element of seeking motives became apparent when h*B, asked this questions "Who made this statement? What is he driving at?" (2) The element of seeking the meaning of terms was pointed out by several pupils, for example: "How many people? bear arms in any war? How recently? How many would never Does he mean financial or -physical support? It seems that he ought to explain what he means by duty. When are two nations at War? Are China and Japan at war? Does government mean all of the people in a nation, or only the few in control?" (3) The element of detecting assumptions became apparent when D.W, remarked that the group representing the former statement assumed that all wars were unnecessary while the latter assumed that war might be necessary and that loyalty to onefs country should be the cardinal virtue of every citizen. 360 (4) The element of seeking more facts was brought out when W.C. remarked, "We ought to have more facts relative to both points of view.11 Several facts were presented as follows: V.M. said* "The pastor of our church delivered a sermon recently in which the Nye report of the Munitions investigation was mentioned* The .American Munitions interests were making it possible for the Germans to kill our own soldiers with .American made products and for German soldiers to be killed with German made products.1* R.C. said* '’Certain interests utilized the press to spread propaganda in order to stir the emotions of the people for war. This was achieved in the last World War and it seems that it is being tried today. 3Por some reason or other, however* the people have not fallen for this propaganda as yet.*1 B.C. remarked, "Your government is made up only of people that you put in office, therefore if they decide it is best for the country to declare war you should support the idea or you are a traitor." R.M* said* "I would agree with B*C. if the persons who decide that war is best for their country would themselves be the first to enlist." (5) The element of testing facts for pertinency was brought out by S.E. when he said, "We can get many facts in favor of war and many facts opposed to war. What we are trying to decide here is whether or not it is one's duty to support one's government at all times and to clarify what we mean by support." (6) The element of discriminating between facts and assumptions was brought out when V.M* said that B.C. was merely expressing an opinion or rather making an assumption and not stating a fact* (7) The element of bias or prejudice entered into this situation several times. D.R* said, "Our fore-fathers' fought for democracy, so it is our 361 duty to preserve it tot future generations. Any one that wouldn’t fight for the American flag is unpatriotic.11 V.M. replied, ”What do you mean by patriotic or unpatriotic, or by fight? when you make such statements. You are not thinking critically S.E. has already pointed out how we failed in our fight for democracy in the last war. Besides, you know as well as the rest of us that European dictatorships have imperiled democracy more than ever. I think you are biased because you are considering only the facts In favor of your own point of view.” (8) The element for testing conclusions for consistency is apparent in this situation. In either event the basic assumptions underlying peace or war should determine the nature of the conclusions reached. Eor example, D. Me. said, ttIf no one would bear arms we would not have wars.” His con clusion appears to be consistent with his assumption. On the other hand, S.E. remarked, ”If we assume that some wars are just, then under such circumstances it becomes the duty of every able-bodied citizen to support his government in such a crisis.” This conclusion likewise appears to be consistent with its assumption. (9) The element of recognizing a need for formulating only tentative conr elusions was brought out by E.Mc. when he said, ”It is difficult to con ceive a situation wherein no one would bear arms. On the other hand, it is also probable that a supposedly just war may be found to be unjust. Therefore, any conclusions that we reach must be only tentative because changing conditions may bring out facts that may necessitate changes in our assumptions.” (10) The element of testing conclusions for their individual and social consequences was implied in the previous item. E.E. remarked, ”If it is 362 the duty of every citizen to serve his country to the hest of his know ledge, then going to war is not to the "best knowledge of a great many people. Those that declare war should be the first to enlist. Further more, it most be remembered that war destroys many men that could make great Contributions to the general welfare of society. Finally, the outcomes of war have seriously handicapped human welfare and human pro gress in the past.'1 Situation Ho. 16 This situation and the remaining four have been analyzed and they reveal the same common elements of critical thinking. These will be stated and their analyses omitted at this time. "Some pupils in school feel that there should be a set of fixed criteria for awards and once a pupil has satisfied these criteria he should be granted an award. Others feel that the problem of awards is more complex and that any set of fixed criteria cannot possibly take into account all factors in any given situation.11 Situation 3Sfo» 17 "From early colonial days t^e problem of how best to deal with liquor has been before the people of the United States. Some people feel that there should be no attempt to control the use of liquor while others believe that Congress should prohibit its sale." Situation Uo. 18 "There is a wide disagreement among thoughtful people as to whether or not capital punishment is the most effective way of dealing with certain types of crime. *n forty-three states the death penalty is legal, while it has been abolished in Kansas, Maine, Michigan, Ehode 363 Island, and Wisconsin. Should it he abolished in all the states?1* Situation No. 19 "We have in this country certain laws which compel all young people up to a certain age to attend an organized school. Some people believe that such laws are most desirable, while others feel that their operation accomplishes little if anything in improving the culture of our people.*' Situation Wo. 20 "In a certain community there had been numerous burglaries which the police were unable to solve. Early one morning several officers arrested two eighteen-year-old boys who were loitering around a Texaco service station in a residential section of the city." 364 APPENDIX B I* Complete Tabulation of Tests Results and Reports by Observers for Bach Pupil in Experimental and Control Groups II* Summary of Responses to the Initial Form of the Inventory Questionnaire Administered to Both Groups at the Beginning of the School Year III* Summary of Responses to the Follow-up Form of the Inventory Questionnaire Administered to Both Groups at the End of the School Year 365 APPENDIX B I. COMPLETE TABULATION OP TESTS RESULTS AND REPOETS BY OBSERVERS OP CRITICAL THINKING BEHAVIOR PCE EACH PUPIL. Interpretation Key Column Item Interpretation fl) Pupil Pupils were assigned numbers to take the place of their names♦ (2) Otis I.^U Intelligence quotients were obtained for all the pupils participating in the study* (3) Otis I.Q«, P.R* The 659 pupils were ranked on the basis of their intelligence quotients and then assigned percentile ranks. (4) Geom.Prog. Score These were the scores made by the pupils on the Odeans Prognosis Test of Geometric Ability administered at the beginning of the school year* (5) GeonuFrog* ,P.R. The 659 pupils were ranked on the basis of their geometry prognosis test scores and then assigned percentile ranks. (6) Geora.Ach.,Score These were the scores made by pupils on the Columbia Research Bureau Geometry Achievement Test administered at the end of the school year* (?) Geom.Ach., P.R. The 659 pupils were ranked on the basis of their geometry achievement test scores and then assigned percentile ranks* (8) C.T. Test, Score These were the scores made by pupils on the Critical Thinking Test administered at the end of the school year. (9) C.T. Test, P.R* The 659 pupils were ranked on the,basis of their critical thinking test scores and then assigned percentile ranks* The following information was obtained only for the Experimental Groups. (10) N.P* Test 5*3 Scores made by Experimental Group pupils prior to the study of demonstrative geometry* 366 Column Item Interpretation (XI) N.P. Test 5*3 Scores made "by Experimental Group pupils on the same test nine months later, that is, after studying demonstrative geometry. (is) Reports hy Observers These were summaries of reports submitted by 165 classroom teachers indicating whether or not there was evidence of improvement in critical thinking abilities of the Experi mental Group pupils in terms of the ten elements* No* Obs* This column indicates the number of observers reporting on each pupil* No. «• (Plus) This column indicates the summarized rating ofl definite improvement in some of the elements of critical thinking* No* 0 (Doubtful) This column indicates not only doubt of definite improvement but also Inability to observe critical thinking behavior with regard to some of the elements. No. - (Minus) This column indicates a definite lack of improvement in some of the elements of critical thinking. Pinal Ratfg* This column indicates whether or not the number of plus ratings exceeded the number of minus ratings with respect to improvement in terms of the ten elements. 367 COMPLETE TABULATION OP RESULTS PROM TESTS, PERCENTILE RANKS, AND OBSERVERS * REPORTS, Part I, (Experimental Group Pupils) (1) (2) (3) (4) Otis Geom Otis lift; Prog Pupil I,ft, P«R, Score (5) (6) (7) (8)' (9) (10) Geora Geom'Geom C.T, C,T, N,P, Prog Aoh. Aoh* Test Test TAst P.R. Score P,R, Seore P,R, 5,3 (ll) N,P. R6-TNo, 5,3 Obs, (12) No, No, No, Final + 0 - Rat’g. Teacher A 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 117 107 108 108 98 95 102 89 116 102 87 104 110 120 113 94 105 124 91 85 114 112 104 88 95 114 85 84 114 98 103 107 114 102 102 107 103 114 108 114 8o;s 45;7 48,2 45;7 20,6 / 14i7 29,7 6,5 76i5 29,7 / 4i7 36,4 53.5 88i2 64,3 12i9 39.3 93i5 9;i 3,2 / 68i2 60i7 36;4 5.5 14,7 68;2 3.2 2i7 68.2 20,6 33i4 45;7 68;2 29;7 29,7 / 4517 33i4 68i2 48.2 115 137 124 86 91 44 62 71 80 68,2 68 4 118 88 57 144 58 43 49 65 102 94 65 78 96 114 99 109 74 98 46 140 137 97 123 40 91 96 35 22 100 111 8i;o 48.6 i7;o 96;0 18,4 6;5 nil 24i8 64i7 56,2 s 24i8 36i7 58i3 77i2 61,4 72i0 33i0 60i5 8.5 94,6 92i3 59i4 84.3 5i5 52,2 58i3 3.3 0i6 62i5 74,8 78i0 92;3 85;2 45;5 52,2 7;3 22i3 30i6 38i6 27.3 4 4 34 13 12 38 21 9 25 33 18 18 51 21 24 42 38 5 19 29 18 34 26 28 37 13 44 22 31 11 29 24 2 45 27 14 27 10 24 26 22 26 69;o 16i8 14i9 77i8 33,8 / 10il 44.0 66i5 27i6 27.6 93;0 33i8 4i;3 84;5 77,8 4i5 30;3 56i5 27i6 69,0 ✓ 47i0 53;o 76il is;8 87,0 / 36.3 62i2 13i0 56i5 41,3 lie 88i0 5o;i is;s 50.1 u;s 4i;3 47;o 36i3 47.0 4 4 102 99 97 133 66 71 69 61 95 72 93 56 81 97 65 60 72 124 69 79 81 96 82 63 68 68 50 57 83 78 77 107 66 76 87 - 83 66 94 81 97 92;o 90.3 89;0 99i7 50,3 58.0 54i8 42.4 87i0 59.5 84.9 33i3 7i;e 89;o 48.2 40i5 59i5 99;o 54.8 69,3 7i;e 87i7 73i0 45.5 53.0 53i0 24il 35.5 74;2 67.6 66i2 94.8 so;3 65i3 79.0 74;2 50i3 85i9 71i6 89.0 4 4 4 4 17 15 22 20 11 21 18 15 19 17 17 10 15 17 10 13 19 20 15 17 15 16 18 12 16 21 14 9 15 13 23 15 13 6 18 8 9 20 12 15 45 40 30 44 47 24 22 48 36 32 50 39 33 40 38 28 37 41 24 25 20 30 35 36 30 40 22 26 32 39 35 44 16 15 49 41 32 51 47 28 3 4 4 3 3 3 5 3 4 3 4 3 4 3 4 4 4 3 2 4 4 3 1 2 3 2 5 1 4 2 3 2 3 3 3 3 4 4 4 2 22 7 1 + 23 15 2 + 28 10 2 * 23 5 2 + 13 8 9 + 2 23 5 • 5 34 11 7 19 4 + 10 23 7 + 22 5 3 + 20 10 10 + 13 7 10 + 19 17 4 + 13 14 3 + 15 23 2 * 20 17 3 14 18 8 + 23 7 0 + 1 13 6 31 6 3 + 22 13 5 + 23 7 0 + 5 4 1 + 8 10 2 + 20 9 1 + 6 9 5 + 15 25 10 + 0 3 7 26 9 5 + 12 8 0 + 11 8 11 o 12 7 1 + 19 6 5 + 7 17 6 + 14 16 0 + 12 7 11 + 14 18 8 + 26 9 5 + 22 12 6 + 11 5 4 + 368 Part I* (Experimental Group Pupils) (1) (2) (3) (4) (5) (6) (7) (9). (9). (10) Continued (U) (12) 41 42 43 44 45 Ofcis Geom Geom Geom Geom C.T. C.T. N.P. N.P. gePorts Otis I«Q« Prog Pf*og Ach. A6h'* Test T6si Test R6-T Ko* No* I.Q* P»R* Score P*R* Seore P*R« Score P*R* 5*5 5*5 Obs* + Teacher A (Continued) 82 40*1 109 51*0 3 3 35 10 lli5 72 59iS 14 9 3 101 27i6 38 16 22;6 75 63*5 15 67 26;6 27 109 51.0 121 83;Q 3 10 76 65:3 14 16 22:6 0 35 46:7 17 64 4;7 1 27:6 20 87 0;S 18 32 4 23 96 87.7 18 34 69*0 104 36*4 85 44*0 17 13 13 2 11 10 8 7 8 6 46 47 48 49 50 97 95 101 91 106 Pupil i9;o 87 14;? 81 27;6 79 9'*1 70 42*8 120 47:2 39:4 37:6 29:3 82*4 Observers— No* No* Final 0 - Rat*g* - + + - + *► ♦ 25 44:0 15 20;6 20 32:0 11 13:0 31 62*2 89 48 82 50 112 8i;9 2i:4 73:0 24.1 96*3 13 13 21 11 30 36 40 38 27 43 2 4 1 2 3 5 17 3 6 23 14 13 7 11 7 1 10 0 3 0 22 36:3 7 7:6 24 41:3 46 89:0 19 30*3 85 64 80 88 72 75:7 46;7 70:6 80.4 59*5 14 12 7 9 17 34 43 22 15 37 4 3 1 2 3 20 7 1 8 10 13 17 6 10 12 7 6 3 2 8 14 15 19 13 15 37 24 40 50 45 4 2 3 4 3 12 9 24 34 23 18 8 6 5 6 10 3 0 1 1 + + + 53:0 16 67:6 21 53:0 12 60'.8 19 81.9 15 34 37 21 43 45 5 4 4 3 4 16 13 1 14 27 20 18 16 14 10 14 9 23 2 3 + + + + 9 + + ♦ 4 51 52 53 54 55 56 57 58 59 60 61 62 1 63 | 64 65 85 104 97 95 88 3;2 36:4 19 ;o i4;7 5*5 100 113 97 108 114 25:8 64;3 19:o 48:2 68*2 105 112 88 103 108 / 3p*3 60:7 5:5 33:4 48*2 60 53 55 111 80 82 59 118 124 112 20:0 i3:s is:6 74:8 38*6 / 40:i 19:3 8i:o 85:2 75*6 39:4 12:9 8:5 20:0 62*5 49 10 23 33 18 9i;i 1U5 39:0 66:5 27*6 18 36 16 25 11 27:6 74:2 22:6 44:0 13*0 18 30 24 27 37 27:6 59:8 4i;s 4 45:7 66 67 68 69 70 107 71 115 113 120 71 72 73 74 75 81 96 104 114 92 36;4 76 77 78 79 80 108 104 127 102 107 o:i 72:7 64:3 88*2 98 79 86 116 143 60:5 37:6 45:5 79:i 95*6 68 78 68 73 89 4 76.1 70 65 108 65 115 56:6 48:2 95:i 48:2 97.3 17 7 16 22 9 41 37 40 32 28 3 1 4 3 3 0 24 17 20 13 5 14 12 10 8 5 2 1 0 + • + + + 86 39 93 71 77 77:2 10:8 84:9 58;o 66.2 13 13 21 17 9 55 31 31 30 26 5 20 7 4 4 22 4 1 3 10 22 27 13 6 16 8 6 5 0 4 + + + 80 79 89 99 72 70:6 69:3 13 15 14 17 13 19 34 40 32 27 2 3 3 5 20 11 19 16 0 9 4 3 2 + + + + 50:1 24:4 10*3 3i;6 32:2 64:7 28*2 35 1 28 23 6 72*0 r.l 53;0 39:0 6*3 48:2 36:4 97:0 29:7 45*7 132 79 133 85 90 90:2 37:6 90:9 44;0 51.0 34 38 33 17 16 69:0 77:8 66:5 24:8 22.6 68:2 64 46:7 79 69:3 100 90:9 75 63:5 86 77.2 4 64 72 73 102 69 2:i 16:7 mm + + 4 * 81 52 46 60 100 + + 4 4 81:9 90:3 59.5 3 3 15 1 15 8 12 - 369 Part I# (l) (2) (3) Otis Otis i;Q; Pupil I.Q. P.R. (4) Geom Prog Score (Experimental Group Pupils) Continued (5) (6) (7) <S) (9) Geom Geom'Geom' C.T* C.T. Pfrog Ach. Adh. Test TOst P.R. Score P.R. Score P.R. (10) (ll) (12)7 N.P* N.P. Reports by Observers TAst R4-T No. No. No. No. Final 5.3 5.5 Ob3. * 0 Rat*g. Teacher A (Continued) 81 82 83 84 85 97 111 105 115 109 19i0 105 67i2 56.6 90 5i;o 39;3 108 70i3 72;7 95 57;4 51.0 114 77.2 86 100 25;s 87 73 o;3 88 115 72;7 89 112 eo;7 90 96 16.7 18;6 18 ;6 56.5 27.6 41.3 72 86 97 77 89 59;5 77i2 89;o 66.2 81.9 17 17 15 16 19 36 29 28 34 33 3 4 4 3 3 7 0 94 56;2 20 24 o;s 11 94 56.2 26 31 i;s 0 7 0.1 6 32 ;o 96 59 69 42 58 87i7 13 39;o 9 54.8 18 14;6 18 37.4 8 35 31 40 28 27 3 14 15 3 9 14 3 16 12 1 3 5 4 21 15 1 53 93 51 69 60 29;i 13 84;9 20 26;o 6 54;8 13 40.5 6 35 1 13 37 31 6 3 10 16 2 20 3 7 18 3 2 6 10 50 2 4 U 7 56i5 106 94i0 13 85;7 110 95;8 16 91.1 80 70.6 15 23 26 27 34 3 2 4 1 0 19 11 9 5 6 4 21 15 4 5 1 10 9 12 15 12 22 23 20 19 23 29 29 5 4 5 5 5 5 15 11 1 13 14 19 25 22 32 25 22 20 10 7 17 12 14 11 13 13 14 9 14 22 31 24 29 23 5 4 4 4 4 24 12 0 12 4 26 20 21 25 18 0 8 19 3 18 4 91 92 93 94 95 80 i;7 99 23;i 88 5i5 95 14.7 80 1.7 29 1*3 4 70 29;3 20 57 17i0 32 81 59.4 17 43 6.5 9 96 94 12 ;9 47 9;4 97 116 76;s 126 86;5 98 119 85.8 110 73i 7 99 111 56.6 132 90.2 10 29 43 49 i3;o 47.0 o;8 6.3 4 3.2 32.0 64i 3 24.8 10.1 n; s 33 12 17 17 13 23 12 19 20 13 6 4 3 14 14 29 18 24 0 4 7 2 2 4 4 4 8 5 4 ♦ ♦ + ♦ + + + + + + - + - + + • + + + Teacher B 100 101 102 103 104 105 99 99 93 80 88 120 2 3 ;i 23a 1U 2 i;7 5.5 8 8 .2 51 97 71 54 80 115 12;4 61 .4 30;6 i4 ;s 3 8 .6 7 8 .0 36;3 56;5 3 6 i3 27;6 4 4 .0 5 9 .8 81 73 44 69 77 82 28 25 16 13 28 53 ;o 4 4 ;0 22 ;6 1 6 .8 5 3 .0 90 78 62 102 75 8 2 .5 67;6 44 ;o 92 ;o 63.5 4 4 106 107 108 109 110 95.5 5.5 6 5 i6 1 7 .0 n ;i 23i5 6 5 .6 7 i; 6 60 .8 ie ; 5 5 4 .8 66;2 7 3 .0 22 29 22 18 25 30 + + + 0 + + 124 88 90 104 98 7;6 3 6 .4 2 0 s.6 103 57 49 63 103 111 112 113 114 115 108 110 106 84 113 4 8 .2 5 3 i6 42 i8 2 i7 6 4 .3 92 60 124 92 68 53 ;7 2o ;o 85;2 5 3 ;7 2 7 .3 31 20 32 5 17 62 ;2 32 ;o 64;3 4 ;s 2 4 .8 102 71 51 62 89 92;0 58;0 26;0 4 4 ;0 8 1 .9 17 6 10 11 17 38 24 13 32 39 4 4 4 3 4 24 5 3 8 3 16 31 34 22 29 0 4 3 0 8 + + 116 117 118 119 120 91 113 113 111 86 9 ;i 64; 3 64 ;3 5 6 .6 4 .1 31 112 76 91 25 i;s 75;6 35;2 5 2 .2 1.1 8 43 24 23 8 8 ;8 8 5 ;7 4 i;3 39 ;0 8.8 65 73 106 70 58 48;2 60;8 9 4 .0 56;6 3 7 .4 16 22 1 19 5 26 24 26 25 15 4 4 4 4 4 20 13 8 28 2 10 27 26 12 15 10 0 6 0 23 + 4 4 ♦ + — 0 + — + + + - 370 Part I* (1) (2) (3) (4) Otis Geom Otis I'.Q; Prog Pupil I.Q. P.R. Score (5) Geom Prog P.R. (Experimental Group Pupils) (6) (7) Geom'Geom' Aeh. A6hi Score PJR. Continued (6)' (9) (10) (11) (12) C.T. C.T. N.P. N.P. Reports by Observers Test Test Tdst R6-T No. No. No. No. Pinal Score P.R. 5.35.3 Obs. +0 Rat*g< 121 122 125 124 125 112 122 99 121 119 60;7 91 .4 2 3 il 90;2 85.8 104 125 33 148 116 66;5 85;8 2;9 97;g 79.1 Teacher B 31 62 ;2 34 69 ;o 14 i8 ;6 69;0 34 12 14.9 126 127 128 129 130 103 97 100 97 112 33 ;4 i9 ;o 25.8 i9 ;o 6 0 .7 118 69 50 89 127 8 i; o 28.2 ii;8 so ;o 87 .3 30 12 25 27 41 59;8 u ;9 44;0 5 0 ;i 83.0 77 66;2 70 56.6 66 50i3 65 48 ;2 84 75.0 4 9 14 8 15 26 25 20 26 27 1 5 3 2 5 0 7 27 10 12 10 27 3 10 38 0 16 0 0 0 131 132 133 134 138 100 93 95 82 118 25i8 n ;2 u ;7 2 ;s 83.6 74 68 64 36 92 33;0 27;3 i4 ; s 3 .6 53 .7 23 18 7 3 29 39;0 27;6 7; 6 2 .3 56*5 80 65 68 57 86 70i6 48.2 53;o 35.5 77.2 16 19 9 9 19 20 19 25 16 20 4 5 4 4 5 8 10 19 5 20 26 29 10 33 16 6 11 11 2 14 + 136 137 158 139 140 102 107 113 105 122 29 i7 4 5 .7 64;3 39.3 91.4 47 77 106 92 115 9;4 36;0 68 ;o 53;7 78.0 5 21 38 24 41 4;5 33;8 77;8 4 i;3 83 .0 79 67 98 84 109 69i3 5U7 89;9 75;0 95.3 11 11 10 15 11 22 21 24 22 29 5 4 5 5 5 16 9 25 20 26 27 30 25 16 20 7 1 0 14 4 + + + + + + (Continued) 79 98 91 73 75 69.3 89*9 83;3 60;8 63.5 12 15 9 8 16 24 31 23 27 27 5 5 5 4 5 23 27 10 5 26 17 16 29 34 21 10 7 11 1 3 + + + + 0 + + ♦ ♦ + + + 4 141 142 143 144 145 106 92 95 94 104 42 ;8 106 47 io ;3 66 14; 7 12 ;9 38 36.4 58 68;0 s;4 25;6 30;5 18.4 36 16 20 12 13 74;2 22;6 32 ;o u ;9 16.8 75 73 61 88 102 63.5 60;8 42.4 80U 92.0 16 18 8 14 12 30 24 18 28 33 4 6 2 5 4 10 27 16 11 10 26 12 4 20 28 4 21 0 19 2 + 146 147 148 149 150 112 88 116 96 123 60;7 s;s 76.5 16;7 92.4 91 36 96 38 67 52;2 3 ;s 58.3 4;6 26.6 25 17 37 5 29 44;o 24.8 7 6 ;i 4 ;s 56.6 51 51 78 67 106 26;0 26;0 67.6 5i;7 94.0 13 25 7 16 17 23 15 22 23 36 4 4 4 4 5 4 4 7 7 26 23 15 30 27 24 13 21 3 6 0 a* + + + 151 152 153 154 155 99 86 97 94 97 2 3 ;i 4 ;1 i9 ;o 12.9 19 .0 74 72 62 93 100 33;o 3 i; e 22;3 5 s;o 62.5 21 25 20 26 32 33.8 44;0 32 ;o 47;0 64.3 109 65 64 65 78 95;3 48.2 46;7 48.2 67.6 10 49 7 16 5 16 9 22 6 26 2 4 5 4 4 10 10 13 20 22 2 21 37 20 17 8 9 0 0 1 + + + 156 157 158 159 160 107 123 94 111 92 45;7 92.4 12;9 56.6 10.3 92 77 44 100 71 53i7 36;0 7;3 62.5 30.6 22 23 14 36 24 36;3 39;0 18.6 74;2 41.3 102 122 60 86 80 92 ;o 16 46 4 3 3 5 4 16 12 9 34 15 15 18 12 16 23 9 0 9 0 2 + + O + + 98i5 40;5 77;2 70.6 14 11 13 13 31 26 32 24 m + 371 Part I* (Experimental Group Pupils) Continued "Tlj 161 162 163 164 165 109 96 105 97 106 • 5 i;o 16;7 35 .4 i9 ; o 42 .8 166 167 168 169 170 171 118 108 93 128 135 105 8 3 .6 48 .2 u ;2 97;5 98;7 39.3 (8) (9) C.T. C.T. Test Test Score P.R. (10) N.P. Te^t 5.5 84 62 84 32 112 42 i4 2 2 ;s 4 2 .4 2;4 75.6 Teacher B (Continued) 31 62;2 112 96.3 10 80 70.6 14 27;6 18 56.6 29 82 73i0 9 18.6 6 14 69 54.8 69.0 34 78 67.6 11 137 87 56 29 139 114 92;3 47'*2 16;3 1 .3 94.0 77.2 34 33 23 18 46 23 69;o 66;5 39;0 27'*6 89 ;o 39.0 88 81 68 93 75 94 80;4 7T.6 53i0 84;9 63;5 85.9 (ll) (12) N.P. Reports fry Observers Re-T No. No. No. No. Fina 5.5 Obs. + 0 Rat* rl &0j (2) (3 ) (4 ) (6) (6) (7) Otifc Geom Geom Geom Geom' Otis I.Q. ProgPfrog Ach. Ach. Pupil I.Q. F.R. Score P.R. Score P.R. 28 26 20 17 30 10 7 18 11 2 0 0 3 16 0 + + + f t 30 33 19 3 38 4 4 4 3 - + 19 17 8 11 10 19 20 34 23 43 29 22 4 5 4 4 1 2 15 22 18 24 8 8 18 24 21 16 2 12 7 4 1 0 0 0 + + •f + + + Teachers C and D 172 173 174 175 115 103 93 115 72i7 33.4 11.2 72.7 95 54 45 88 57i4 14:5 7;9 48 .6 28 12 6 27 53;o 14i9 6;3 50.1 60 54 67 50 40i5 30;3 5 i;7 24.1 6 8 9 18 18 22 22 16 3 4 4 4 25 5 9 10 5 14 30 10 0 21 1 10 + + 0 176 177 178 179 180 92 98 86 99 90 10:3 2o ;e 4 .1 23 .1 7 .6 42 96 76 83 83 5 .9 58.3 35 ;2 40.8 40.8 6 29 16 11 37 6;3 56.5 22.6 13iO 76.1 56 61 43 69 52 33.3 42 ;4 15.8 54.8 27.6 6 15 9 8 8 16 22 18 15 20 1 4 3 3 4 1 10 8 12 21 9 24 22 18 19 0 6 0 0 0 + + + + + 181 182 183 184 185 95 109 103 89 107 14i7 5 i; o 33.4 6i5 4 5 .7 66 58 84 43 66 25;6 18 i4 42 ;4 6 ;s 25.6 12 8 28 8 8 14;9 8'.8 53;o 8.8 8.8 106 82 95 40 69 94;o 73;o 87.0 11U 54.8 7 8 9 6 11 30 33 36 27 17 4 4 1 3 4 14 27 10 9 16 26 13 0 21 24 0 0 0 0 0 + + + + + 186 187 188 189 190 91 93 99 103 82 9 ;i n ;2 2 3 il 33;4 2 .3 48 50 65 48 21 io ;3 1U 8 24i8 10:3 0.5 6 4 16 16 13 6;3 3;2 22;6 22;6 16.8 70 50 83 61 38 56i6 2 4 ;i 74;2 42:4 9.2 9 19 12 11 8 18 37 16 18 19 3 3 4 4 4 3 3 15 10 8 17 27 21 26 32 10 0 4 4 0 + + + 107 104 92 106 91 45;7 36:4 io ;3 42 .8 9 .1 56 46 47 109 50 16i3 8'.5 9 .4 72;o 11.8 19 9 6 39 12 30;3 io ;i 6.3 7914 14.9 57 42 39 31 57 35'.5 13 8 14:6 io :s 10 3:7 15 35.5 14 25 33 24 14 29 4 4 4 3 3 39 1 30 10 7 20 15 15 9 21 0 0 13 0 0 58 109 61 87 91 18 ;4 72 ;o 2 i;2 47;2 52.2 16 29 12 29 41 22;6 56:5 14:9 56:5 83.0 76 54 83 64 83 65:3 30:3 74:2 46:7 74.2 10 10 14 21 8 20 19 24 33 33 4 33 4 6 3 24 0 24 4 0 2 20 23 15 4 3 0 16 0 2 191 192 193 194 195 + 4 + + + s 196 197 198 199 200 103 111 103 114 110 33;4 56 .6 33.4 68:2 53 .6 + + - + + 372 Part I* (1) (2) (3) (4) Ofcik Geom Obis I4Q. Prog Pupil I.%. P.R. Soore (5) Geom Prog P.R. (6) (7) Geom'Geom' Ach. Ach'. Score P.R. 4 201 202 203 204 205 102 97 90 105 95 86 4 5 .5 89 5o;o 70 2 9 .3 91 52.2 93 5 5 .0 Teachers C and D ,(Continued) 7742 543 3545 4244 80.4 15 37 7 22 6 41 20 30 13 29 5 21 3 8 4 32 3 15 2 20 29 18 8 14 0 7 18 22 24 25 746 2746 3643 4143 4 4 .0 67 48 45 46 61 5147 2144 1743 1843 42.4 16 2 12 18 17 37 34 52 4 4 4 4 4 5 0 12 16 21 25 26 28 16 15 10 14 0 8 4 + 4+ 36 8 5 19 47 7442 848 445 3043 90.0 85 41 55 74 44 7547 1247 3142 6240 16.5 13 12 12 7 4 4 4 4 3 36 4 2 26 5 35 6 29 16 14 0 12 0 5 + + 4* 10 28 25 15 27 43 6940 9542 6348 1244 21.2 35 47 30 17 18 7240 9040 5948 2448 27.6 98 60 91 86 89 8949 4045 8343 7742 81.9 16 9 15 18 8 34 26 38 26 29 4 4 4 4 4 17 21 36 8 36 23 19 115 7Z'm7 95 5744 66 2546 108 4 8 i 2 108 4 8 ; 2 116 7941 743 95 1447 44 59 19.3 99 2 3 .1 43 13 26 5 14 8547 1648 4740 445 18.6 71 87 95 63 5840 7940 8740 4545 50.3 8 17 16 9 26 31 31 27 21 8 25 3 4 4 4 4 8 27 40 27 24 40 24 28 40 42 38 102 93 91 104 114 2 9 .7 u ;2 9 .1 36;4 68.2 41 78 38 86 130 126 90 88 99 137 9 6 .1 746 5.5 2341 99.2 5;8 36;7 4 .6 45i5 89.3 4 104 38 34 52 70 118 126 119 94 103 8 5 .6 9 6 ;i 85 .8 12:9 3 3 .4 107 138 101 51 61 4 221 222 223 224 225 66.5 4 ;6 3 .2 1249 2 9 .3 4 100 106 115 85 2 s;e 4 2 .8 72i7 3 .2 32 99 81 32 244 6144 3944 2 .4 18 a 4 0 1 0 + + + + + 4 0 + 4 / 66 21 3348 13 15 22 1648 2046 31 4 0 0 0 1 0 + + + + 0 1 0 1 10 ♦ + 8 7 3 28 20 31 23 4 4 4 4 20 20 26 14 28 22 0 0 ♦ 2 1 + 5 3 5 5 5 0 25 25 3 17 10 9 33 8 9 14 27 3 22 25 12 45 3 11 20 4 + ♦ 4 4 4 226 227 228 229 0 86 33 57 61 88 i;e 4 216 217 218 219 220 (8)' (10) (ll) O 2) C.T. H.P. H.P. Reports by-Observers T&st T6st R6-T Ho. Ho. Ho. Ho. Final P.R. 5.35.5 Obs. +0 Rat*g. 33'.8 2046 3643 3 2 .0 4 211 212 213 214 215 (8)' C.T* Test Soore 2 21 15 22 20 2 9 .7 1 9 i0 ?;6 39 .5 1 4 .7 4 206 207 208 209 210 (Experimental Group Pupils) Continued 75 91 79 56 6345 8343 6943 33.3 99 72 91 103 63 96 9043 5945 8343 9248 4545 87.7 36.3 11 16 9 10 17 + 4- Te&oher E 4 4 230 231 232 233 234 235 108 106 105 98 101 113 48; 2 42;8 39 .3 20;6 27;6 6 4 .3 80 83 75 62 30 73 3846 4048 3349 2243 145 32.2 72 72 70 94 117 3146 3146 2943 5642 80.2 19 24 34 28 17 26 115 73 102 105 112 72;7 0 .3 2947 3943 6 0 .7 37 43 41 41 29 7641 8547 8340 8340 56.5 18 15 13 16 8 10 6 + + •f ♦ 4 4 4 236 237 238 239 240 3043 4143 6940 5340 2448 4 7 .0 73 49 63 79 117 6048 2248 4545 6943 97.8 11 10 10 5 21 28 15 30 23 58 6 13 42 5 3 5 3 4 12 20 8 10 8 12 0 8 17 18 22 15 + 4* + + 4- 373 Fart I. (1) (2) (3) Otis Otis I.Qi Pupil I.Q. P.R. (4) Geom Frog Score (Experimental Group Pupils) Continued (5) Geom Frog P«R. (6) (7) Geom'Geom' Ach. A6h. Soore P.R. (6)C.T. Test Soore , Teaoher E 112 60;7 109 5 1 .0 114 68.2 115 72i7 91 9 #1 / 246 113 6 4 .3 247 114 68 '.2 248 108 48 i2 249 105 39.3 250 114 68.2 241 2421 243 244 245 57 57 98 70 37 127 87 61 76 66 / 251 252 253 254 255 117 114 77 105 109 256 257 258 259 260 106 99 109 no 95 80.5 68i2 o;9 3 9 .3 5 1 .0 * 42 ; s 2 3 il 5 i; o 53.6 1 4 .7 i7 ; o 1 7 i0 60^5 2 9 .3 4 4a 27 22 53 26 19 87i3 4 7i2 21'.2 35.2 25*6 4 34 17 22 20 34 SOU 36;3 94 i7 4 7 i0 30.3 (Continued) 97 125 79 100 61 89;o 99.2 6943 9o;§ 42 .4 12 40 11 37 7 28 12 29 5 13 6 7 5 6 5 18 33 15 37 2 36 28 22 20 28 6 9 13 3 25 •*• + ♦ + - 126 91 100 63 81 99.4 8343 90;9 45.5 71.6 12 47 11 22 14 24 13 28 8 22 8 4 5 4 6 60 8 16 15 17 18 26 26 21 33 2 6 8 4 10 + + 22 42 16 42 14 19 20 6 18 16 6 2 7 14 6 + ♦ 4 69i0 24.8 3e;3 32i0 6 9 .0 114 105 90 101 113 68.2 39i3 7i6 27i6 6 4 .3 + + 4 4 2 .4 40 8;5 15 4 4 .0 17 12;9 13 1 0 /.3 22 30i6 29 4 5 .5 49 25«6 22 17i0 17 16*3 24 8 i; o 20;6 24.8 16.8 36.3 125 85 88 119 86 99i2 75;7 80.4 9840 77.2 13 44 10 20 6 21 14 44 22 39 7 6 4 4 4 56.5 9 ia 36.3 24;8 4 1 .3 107 97 103 111 113 94;8 89i0 92.8 96;0 96.8 6 41 14 31 10 39 6 65 15 31 6 34 7 36 5 4 5 40 6 46 24 23 42 3 4 2 11 4 7 0 «*• + 0 + ♦ 27 19 27 14 16 sea 30.3 50a 18 ;6 22.6 4 87 87 101 91 93 79i0 79i0 9 ia 83;3 84.9 12 5 13 16 15 37 27 42 45 37 7 35 6 18 6 42 6 20 6 49 35 30 16 40 9 0 12 2 0 2 + + + 5 12 22 22 54 4;5 i4 ;o 36.3 36;3 9 5 /.4 81 71 96 94 102 11 7 9 10 13 37 36 33 47 56 0 5 10 24 7 25 9 17 9 24 33 55 22 73 45 17 21 23 0 21 • ♦ + + + 23;5 19 17 79a 7;9 12 42;4 25 33.9 50 / s ;9 28 42 49 u ; i 17 5.5 17 40 69 28i2 55 101 63.8 29 30;3 24;8 14i9 4 4 .0 91*9 7 i;6 58 ;o 87;7 85;9 92 /.0 113 96i8 106 94;o 97 89iO 71 58i0 94 85.9 9 20 9 6 15 39 36 40 23 37 6 40 5 22 8 41 4 15 10 38 20 28 13 16 46 0 0 26 9 16 + + 98;2 22.8 87;o 99;9 95.1 7 16 9 10 11 42 27 20 46 38 10 0 22 47 17 48 34 36 13 33 2 16 12 0 0 84 46 85 52 48 71 86 66 57 56 62 66 63 58 63 4 266 267 268 269 270 108 115 109 106 118 4 8 .2 72i7 s i;o 42 ;8 83 .6 63 62 65 31 98 271 272 273 274 275 112 111 106 106 112 60;7 56.6 42 ;8 42i8 6 0 .7 63 116 45 84 75 276 277 278 279 280 107 106 97 106 113 45;7 4 2 .8 i9 ; o 42 ;8 6 4 .3 22;3 2S;6 23.5 18 i4 23.5 / 23i5 2 2i3 24.8 i;e 60.5 + + 4 4 261 262 263 264 265 (9)' (10) (ll) (12) C.T. N.P. N.P. Reportsby' Observers, _ T&st Ttst R6-T No; No. No. No. Pinal' P.R. 5.35.3 Obs. + 0 - Rat*g. •f + + 4 53;0 24;8 2418 96;2 56.5 120 49 95 148 108 6 5 7 6 5 - + + + 374 Part I, (X) (2) (3) (4) Ofcis Geom Pupil Otis I;q; ftrog !»§♦ P^R. Soore 281 282 283 284 285 120 91 111 111 110 88;2 9 ;i 56;6 56;e 53 .6 68 63 50 121 44 4 286 287 288 289 290 125 119 120 123 116 95 ;o 85; 8 86;2 92 i4 76.5 126 99 115 119 118 9 6 ;i 2 3 ;i 72;7 85.8 83.6 4 296 297 298 299 300 301 302 303 304 305 (5) (6) (7) GeomGeom'Geom' Pfcog Ach. Ach; P.R. Soore P.R. 2 7i3 23.5 1U 8 83;o 7 .3 4 116 97 123 129 100 4 291 292 293 294 295 (Experimental Group Pupils) 7 9 ;i 59.4 84;3 88.8 62.5 98.5 48;e 99;0 99.9 97.5 4 68 26 60 52 37 99i2 47 i0 9 7 ;i 93;8 76.1 76;5 92 ;4 83;e 29;7 80.5 126 123 138 118 123 86;s 25 84 i3 37 93;2 53 8 i;o 40 84.3 35 44;0 7 6 ;i 94i7 8 i;o 72.0 120 117 116 119 112 88;s 80.5 76i5 85.8 60 .7 97 92 96 156 109 59;4 53;7 58i3 98;7 72.0 53i0 64.3 69;o 84;s 88.0 28 32 34 42 45 38 36 40 33 38 7 7 6 5 7 23 19 35 22 22 42 36 21 20 28 5 15 4 8 20 18 25 26 42 25 5 8 10 8 7 19 14 51 36 14 27 58 25 37 38 4 8 24 7 18 ♦ 4 114 91 123 111 104 4 116 123 118 102 117 4 (8)' (9)' (10) (ll) (12) C.T. C.T. N.P. N.P. Reportsb y Observers __ Test Tbst Test CeeT No. No* No. No. Final Soore P.R. 5.35.3 Obs. + 0 Rat*g. Teacher E (Continued) 35 72 ;o 72 59;5 13 20 32 ;o 95 87;0 15 5 o ;i 27 76 65;3 11 90 .0 119 98 ;o 47 7 10 11.5 86 77.2 14 4 Teach&r F 54 95i4 85 75;7 14 36 94 85i9 13 74V2 64.3 103 92.8 32 7 87;0 133 99;7 21 44 37 76.1 7 89 81.9 4 154 88 160 168 148 Corrtinued 97i0 83.5 98i8 96i0 93.0 + + + + + + - 11 9 14 13 12 36 27 39 38 29 7 39 7 53 8 73 6 25 6 4 3 28 4 13 0 7 13 22 0 34 + + + 10 11 12 11 14 27 36 30 33 26 9 6 5 6 6 47 22 27 39 46 43 38 23 15 14 0 0 0 6 0 + + + + + + 4 116 116 75 86 85 97;5 97i5 63i5 77;2 75.5 107 97 83 129 89 94i8 89i0 74i2 99'.5 81.9 14 18 10 13 12 35 33 28 35 25 5 7 7 6 5 39 35 44 56 40 11 31 19 4 10 0 4 7 0 0 + 27 29 28 24 32 5 21 7 45 8 56 5 21 2 14 22 25 23 26 5 7 0 1 3 1 + + + + + 4 + + + 4 4 306 307 308 309 310 109 116 119 116 113 5 i; o 76i5 85.8 76i5 64 .5 118 142 94 98 149 8 i;o 24 95 ;4 52 56;2 30 6o;s 40 98.0 44 4 i; 3 93;8 59;8 8 i;o 87.0 83 88 92 87 114 74 ;2 80.4 84;0 79i0 97.0 10 14 13 11 14 311 312 313 314 315 115 72i7 114 68;2 121 90;2 116 76;5 117 80 .5 105 108 148 121 137 67;2 70;3 97;s 83;0 92.3 30 47 48 45 26 59 i8 9o;o 9o;s 88;0 4 7 .0 76 78 104 72 97 65;3 67;6 93;0 59;5 89.0 7 16 11 24 12 25 11 15 19 40 6 16 34 5 27 23 5 23 26 6 20 24 8 55 25 10 0 1 16 0 + ♦ + + + 99 68 89 164 128 6 i;4 27i3 so ;o 99 ;4 88.0 45 24 43 71 50 88;0 41.3 85;7 99;s 91.9 94 81 92 120 76 85;9 7 i;6 84;0 98;2 65.3 7 23 15 17 12 25 16 31 19 34 8 7 4 6 9 35 22 24 35 39 0 11 0 1 1 + + 4 316 317 318 319 320 116 103 111 128 123 76;s 33.4 66;6 97;5 92.4 45 37 16 24 50 + + 375 Part I • (S) Otis otife i ;q ; Pupil I.Q. P.R. (1) (2) (4) Seem r*og Score 4 321 322 323 324 325 117 116 120 119 126 80;5 76;s 88 i2 85i8 96*1 124 121 126 109 117 (5) Geom Ffrofc P.R. 4 130 98 109 128 150 4 326 327 1 328 329 330 (Experimental Group Pupils) 89.3 60i5 72;0 88*0 98*3 (6) (7) Geom" Geom" Ach. A6h; Score P.R. Teaohpr 59 43 35 73 41 93.5 90;g 9S;0 51i0 80.5 115 138 105 109 110 (8)' (9)' (10) C .T. C.I. N.P. Test T6st T6st Soore P.R. 5.3 F (Continued) 97i0 85;7 72;o 99;8 83.0 106 69 105 100 114 94;o 54.8 93;5 90;9 97.0 4 4 78 ;o 93;2 67*2 72;o 73.7 101 86 88 105 94 9i;4 77;2 80;4 93;5 85.9 4 331 332 333 120 116 116 (11) _ (12) N-P- Reports by' Observers Re-T Ho; No. Ho. Ho. Final' 0 5.3 Obs. + - Rat'g. 17 8 13 5 8 33 16 23 16 29 5 26 8 44 5 25 5 44 10 76 23 26 25 6 23 1 10 0 0 1 + + + + 22 13 13 2 16 35 28 24 19 24 8 44 7 62 6 34 6 50 6 14 34 8 25 10 34 2 0 1 0 12 + + + + 13 10 8 32 27 26 6 41 5 30 7 28 19 18 34 0 2 8 + + + 4 85;7 99;9 93;s 93;0 18.6 43 77 52 51 14 Continued •¥ 4 88;2 76;5 76.5 140 141 129 94i6 95i2 88.8 95.4 9s;o 74.2 54 67 36 124 100 112 99;c 90;9 96.3 , Experimental Group Means* Standard Deviations* and. Ranges Geom.Proe. Crit. T. Geom.Achiev. N. P. , N.P. 4 H&an S*D. Range 105.8 11*4 71 - 137 81.2 20.4 31 148 26.4 13. 7 0 - 77 84;2 31.9 7 - 168 - 4 Part II. (1) I.Q. (3) Ofcifc i;q; P*R* (4) Geom Prog Score 126 114 117 103 118 96;i 68 i2 8o;s 33;4 83.6 147 122 129 109 134 (2) obib Pupil .. . . . . . . . . . . . 401 402 403 404 405 (Control Group Pupils) (6) Geom' Ach. Score P*R* Teacher G 97;0 54 33 83i5 50 88;8 72; 0 51 91.1 36 (S) Geom (7) Geom Adh; PJU (8) (9) C.T. C.T. Test «*4E Score P.R. j j ,4 95.4 66;5 9U9 93;0 74.2 69 62 49 69 60 64;3 69;0 62 ;2 33.8 87 74 39 67 47 83;0 84.5 36;3 39 ;o 79.4 47 70 68 55 86 54.8 44;0 22;8 54.8 40.5 4 406 407 408 409 410 116 111 118 117 119 76;5 56i6 83;6 8o;s 85*8 104 144 121 88 128 66.5 96;c 83.0 48 ;6 88.0 32 40 34 31 21 66;5 41 42 8i;o 79;0 62;o io;8 si;7 19.4 4 411 412 413 414 415 115 115 112 111 122 72i7 72i7 60;7 56;6 91.4 104 117 88 95 136 80;2 48.6 57i4 91.6 22 23 39 19;4 56'.6 53i0 31.2 77.2 i2;7 4.4 30 1 - 30;2 8.9 8-65 376 Part II# (1) Pupil (2) Otis I.Q# (3) Otis i ;q ; p .r . (Control Pupils) Continued (4) (5) Geom Prog Score Geom Prog PJt. 113 114 111 111 112 421 422 423 424 426 (7) (8) C.T. 0) C.T. Test Score Test PJ U 23 19 26 33 42 39.0 30i3 47;0 66;5 84.5 72 58 67 94 65 ' 59;5 37.4 5U7 85;9 48.2 33;9 82 ;o 76;4 66.5 90.9 11 47 30 29 49 i3;o 90.0 59i8 56.5 91.1 54 49 51 55 80 6.0 22i8 26.0 31i2 70.6 63 128 91 84 36 23;5 88^0 52.2 42 ;4 3.6 26 21 5 22 ■7 47.0 33;8 4;5 36;3 0.1 46 47 29 55 28 18.3 19;4 2i7 3i;2 2.3 56;6 5i;o 88.2 68.2 76.5 106 75 102 88 84 68.0 33i9 64.7 48 ;6 42.4 10 11 7 6 8 n;& i3;o 7.6 6.3 8.8 35 34 69 46 51 6.7 6.0 54.8 18;3 26.0 45 ;7 83;6 53;6 48.2 78 96 59 63 36;7 58.3 19 ;3 23.5 9 19 4 6 io ;i 30.3 3.2 6.3 37 58 52 31 8*1 37;4 27i6 3.7 24.1 62.0 58 ;o 26.0 12;7 16.5 Teacher G 416 417 418 419 420 (6) Geom Geom Ach. Ach. Score PJt. (Continued) - 64«3 68 ;2 56;6 56;6 60#7 115 84 91 106 84 78;o 42;4 52;2 68i0 42.4 103 119 116 125 130 33;4 85.8 76 95;o 98 #1 75 119 113 104 133 426 427 428 429 430 106 119 112 114 103 42 ;s 85.8 60;7 68;2 33#4 431 432 4333 434 435 111 109 120 114 116 436 437 438 439 107 118 110 108 4 Teacher H 440 441 442 443 444 445 114 120 138 112 100 105 68.2 88.2 99i4 60;7 25;8 39.3 94 97 145 107 55 43 56.2 59.4 96.3 69 ;o 15.6 6.5 34 28 45 38 5 14 69.0 53;o 88.0 77.8 4.5 18.6 50 74 71 51 41 446 447 448 449 450 116 97 87 38 74 58 81 47.2 4.6 33.0 18.4 39.4 12 -1 8 10 29 14.9 0.6 8.8 1U5 41 29 97 108 76;5 i9;o 23;i 19 ;o 48.2 451 452 453 454 455 115 98 98 110 120 72i7 2G;6 20.6 53;6 88.2 123 48 88 100 145 84.3 io;3 48;6 62.5 96.3 53 18 36 6 57 94i7 27.6 74i2 6.3 96.7 99 56.5 44 33 37 41 36 48 44 39 49 12.7 2.7 5.3 8.1 12.7 7;3 21.4 16.5 10.8 22.8 377 Part II. (1) Pupil 456 457 458 459 460 (3 ) Otifc I.Q. (3 ) Otis i .q ; PJR. 112 123 119 115 120 60;7 9 2 .4 8 5 .8 72;? 88 .2 4 461 462 463 464 465 98 127 94 114 114 20;6 97;0 12 ;9 68 .2 68 .2 (Control Group Pupils) (4) Geom P*og Score (5) Geom Ffrog P.R. Teacher H 65 .6 103 141 95 ;2 82 ;o 119 s o ;o 89 6 1 .4 99 60 140 51 108 117 2o ; i 94 i6 12 ;4 70;3 80.2 Continued (6 ) (7 ) Geom Geom Ach. Afch. Soore P.H. (Continued) (8) C.T. Test Score (9 ) C.T. TSst P.H. - 39 48 27 38 30 79;4 90;5 5 o ;i 77;8 5 9 .8 58 62 47 58 66 37;4 4 4 ;0 19 ;4 37;4 5 0 .3 17 52 15 35 72 24;8 93i8 2oi6 72;o 9 9 .7 41 73 32 66 43 i2 ; 7 60;8 4 ;5 50;3 1 5 .8 / 466 467 468 469 470 101 98 104 131 105 2 7 i6 20;6 3 6 .4 98 ;4 3 9 .3 31 71 84 159 40 i; 8 30;6 4 2 .4 98;8 5 .5 9 35 26 53 2 io ;i 72 ;o 4 7 .0 9 4 .7 1 .6 37 42 58 57 33 8 ;i i4;6 37 i4 35;5 5 .3 471 472 473 474 475 116 116 136 113 124 76;5 76i5 99;o 6 4 .3 93.5 107 110 149 76 111 69;o 73i7 98 ;o 35 ;2 74.8 32 31 60 26 39 64;3 62 ;2 9 7 ;i 47;0 79 .4 59 68 91 32 55 39;o 53;0 83;3 4 .5 31.2 476 477 478 479 480 111 116 125 106 111 56;6 76;5 9 s ;o 4 2 .8 5 6 .6 91 61 116 84 89 5212 21i2 79.1 42 '.4 5 0 .0 30 36 39 54 26 59 ;8 74;2 79;4 95 ;4 4 7 .0 45 63 53 59 42 17; 3 4 5 ;s 29a 39;0 1 4 .6 481 482 483 484 485 106 117 98 115 97 42;8 80.5 2o ;e 72 i7 1 9 .0 39 101 102 95 58 s ;2 63;8 64'.7 57 i4 18.4 10 41 33 27 30 u ;s 83;0 66;5 5 o ;i 59 .8 38 45 43 42 35 9;2 i7 ; 3 15;8 i4 ; 6 6 .7 486 487 488 489 490 118 146 112 139 112 83;6 99;9 60 ;7 99'.5 6 0 .7 75 137 78 141 126 33;9 92 ;3 3617 95 ;2 8 6 .5 19 46 30 61 29 30;3 89;0 59;8 97;5 56.5 37 86 47 79 46 8 ;i 77;2 i9 ; 4 6 9 .3 18 .3 76 146 101 86 109 35;2 96;8 6 3 ;s 45 ;5 72 .0 35 42 35 21 54 72 ;o 84;s 72.0 33.8 95 .4 57 68 48 41 51 3 5 ;s 53;0 2 i; 4 12;7 2 6 .0 4 4 491 492 493 494 495 105 131 104 115 111 59.3 98;4 36.4 72;7 5 6 .6 378 Part II# (1) Pupil 496 497 498 499 600 (2) 0bik I.Q* 109 124 118 105 94 (Control Group Pupils) Continued (3) Otis i :q : P*R. (4) Geom Prog Soore 4 Teacher H 5i;o 93.5 83;6 39U 12.9 (5) Geom Pfrog P*R. (6) (7) Geooi Geom Ach. Ach'. Score P.R. (Continued) (8)' C.T. Test Score ($)' C.T. T6st P.R. 4 < mz 90 127 119 93 70 51i0 87;3 82;o 5s;o 29.3 36 40 38 26 34 146 67 76 54 114 96'.8 26:6 45 15 18 14 35 88 :o 8 11 40 30 27 8:8 13.0 8i;o si:o 77:8 47:0 69.0 60 73 57 30 42 40:5 74 32 38 41 61 62.0 4.5 9:2 12:7 42.4 58 21 63 35 56 37:4 0*3 45.5 6:7 33.3 63:5 39.0 19:4 24;i 1.2 45:5 44:0 3:i 45:5 24.1 60:8 35:5 3:1 14.6 4 501 502 503 504 505 136 94 101 103 127 506 1 507 508 509 ' 510 98 97 112 108 103 20;6 i9;o 60*7 48 ;2 33,4 511 512 513 514 515 129 114 115 119 125 97,8 68.2 72;7 85;8 95.0 123 102 79 149 128 84.3 64;7 37:6 98;0 88.0 30 31 28 49 34 59:8 62:2 53.0 69.0 75 59 47 50 26 516 517 518 519 520 96 107 103 112 114 16 ;7 45;7 33i4 60;7 68.2 49 75 53 110 98 li:i 33.9 13:5 73:7 60.5 -5 33 9 43 40 0.4 66:5 10.1 85:7 81.0 63 62 30 63 50 99;0 12:9 27;6 33;4 97.0 35:2 14:5 77.2 20:6 27:6 18:6 66.5 / 55 57 89 87 60 15:6 17:0 50:0 47:2 20.0 59:8 50.1 / 91:1 i 4 16:5 521 522 523 524 525 107 115 94 115 89 45;7 72i7 12.9 72i7 6.5 80 94 67 95 70 38:6 56:2 26:6 57;4 29.3 27 35 33 24 38 SOU 72:0 66:5 4i;3 77.8 44 53 27 41 71 526 527 528 529 530 118 106 110 105 96 83:6 42i8 53i6 39;3 16.7 105 148 107 76 53 67:2 97:5 69 :o 35:2 13.5 41 64 31 20 4 83:0 98:5 62'.2 32:0 3.2 62 63 60 41 49 44:0 45'.5 40:5 12'.7 22.8 531 532 533 534 535 113 124 103 110 122 64^3 93.5 33:4 53;6 91.4 76 98 85 91 106 35;2 60.5 44t0 52:2 68.0 52 27:6 536 539 538 105 111 111 39;3 56i6 56.6 86 95 91 45:5 57:4 52.2 29:1 1.8 12:7 58.0 4 23 !* 29 36 39:0 m 56:5 72.0 1% 48 51 21.4 26.0 75 57 48 63:5 35 .5 21.4 4 27 28 29 so;i 53:0 56.5 n ■ j 379 Part II# (Control Group Pupils) (1) (S) Pupil Otis I.Q. (3) OtiS I.d. P.R. 4 539 540 118 112 83.6 60.7 541 542 543 544 545 101 102 100 121 105 27.6 29i7 25.8 90;2 39.3 (4) Geom Prog Soore Continued (5) Geom Prog P.R. (6) (7) Geom Geom Ach. Achi Score P.R. Teacher'H 97 59i4 118 81.0 (Continued) 29 56i5 22 36.3 (8) C.T. Test Score 0) C.T. Test P.R. * 37 39 8.1 10.8 4 58 73 89 138 84 is;4 32 ;2 5o;o 93;2 42.4 11 34 30 41 26 i3;o 69;o 59.8 83;0 47.0 39 42 67 58 44 10.8 14i6 5i;7 37;4 16.5 114 91 138 88 108 135 7?;2 52;2 93;2 48i6 70.3 91.3 44 27 47 31 33 56 87'.0 50U 9o;o 62 ;2 66;5 96.5 53 32 58 42 70 88 29a 35 23 5 18 72;0 39;0 4.5 27.6 86 53 30 32 77i2 29.1 3;i 4.5 4 546 547 548 549 550 551 116 111 121 105 115 135 76i5 56.6 90i2 39i3 72i7 98.8 4;s 37;4 14.6 56 ;6 80.4 Teaoher I 552 553 554 555 556 557 558 559 560 126 102 90 78 111 99 117 111 103 70;2 42 '.4 13.5 44.0 96;i 29;7 7.6 I#1 132 84 53 85 56i6 23.1 80;5 56;6 33.4 32 93 109 82 118 2;4 55i0 72;o 4o;i 81.0 3 7 47 28 29 2;3 7;e §o;o 53;o 56,5 48 41 56 50 34 2i;4 i2;7 33;3 24'.1 6.0 55 no 38 86 36 i5;e 73'.7 4.6 45.5 3.6 0 55 2 13 -5 o;8 96.2 i;« 16;8 0.4 23 74 26 52 28 o;s 62;0 i;2 27;6 2.3 4 / 561 562 563 564 565 95 112 89 117 79 14;7 60;7 6;5 80;5 1.2 566 567 568 569 570 100 117 125 106 122 25i8 80.5 95i0 42 ;8 91.4 4 4 4 93 125 126 47 113 5s;o 85.8 86.5 9i4 76.4 15 29 62 10 28 20;6 56;s 97i8 n;5 53.0 57 54 51 46 50 35i5 30;3 26;0 i8;3 24.1 58 26 32 42 56 37;4 1.2 4;s 14;6 33.3 37 36 26 27 31 8;i 7;3 i;2 i;8 3.7 4 4 571 572 573 574 575 117 86 96 102 116 8o;s 4;i 16;7 29;7 76.5 138 59 68 61 108 93;2 19*3 27;3 2i;2 70.3 25 2 4 18 28 44.0 i;e 3;2 27;6 53.0 576 577 578 579 580 100 116 102 102 111 25;8 76;s 29;7 29;7 56.6 69 108 61 84 33 28;2 70;3 2i;2 42.4 2.9 27 28 18 23 3 so;i 53;0 27;6 39 ;o 2.3 4 380 Part! U # (1) (2 ) Hipil 0514 I.Q. (3) Otis i .q ; P.R. / 581 582 583 584 585 117 117 100 117 117 8 o ;s 8 0 .5 2 5 .8 80;5 8 0 .5 (Control. Group Pupils)> Continued (4 ) Geom Prog Soore (5 ) Geom Prog P.R. Teacher I 93;2 138 72;o 109 69 28;2 86 45'.5 125 8 5 .8 (6 ) Geom Ach. Score (7) Geom Ach. P.R. (Continued) 4 4 .0 25 37 76U 25 4 4 ;o 20 i6 15 6 2 .2 31 (8 )' C.T. Test Score (9 )' C.T. Test P.R. 4 38 41 24 40 51 9 .2 12;7 o;8 1U4 2 6 .0 4 586 587 588 589 590 591 592 593 594 122 125 99 106 111 91 i4 95;o 2 3 il 42 ;8 56 .6 103 112 126 100 3 3 .4 60;7 96il 2 5 .8 113 126 93 47 82 76 ;4 86i5 55 ;o 9 .4 4 0 .1 30 50 9 12 30 59;8 9 i; 9 io ;i 1 4 .9 5 9 .8 38 59 12 28 34 9;2 3 9 i0 o ;i 2'.3 6 .0 118 110 132 93 8 i;o 73;7 90;2 5 5 .0 31 43 35 15 62i2 85;7 72 ;o 2 0 .6 27 35 53 23 i;8 6;7 2 9 ;i 0 .5 / Teachers J and K 4 4 595 596 597 598 599 600 103 92 86 90 96 80 3 3 .4 io ; 3 4 ;i 7;6 16'.7 1 .7 47 44 86 62 120 24 9;4 7 ;s 4 5 .5 22;3 8 2 .4 0.8 6 12 5 4 31 13 6.3 14;9 4 ;5 3 .2 62 ;2 16 .8 55 41 41 57 43 38 3 i;2 12;7 12; 7 35;5 is ;8 9 .2 36'.3 3 6 .3 5 o ;i 3 2 ;o 2 4 .8 53 58 56 52 47 2 9 ;i 37;4 33;3 27 i6 1 9 .4 4 4 601 602 603 604 605 98 108 102 101 96 20;6 4812 29 ;7 2 7 i6 1 6 .7 76 133 99 71 61 35;2 90;9 6 i; 4 30;6 2 1 .2 22 22 27 20 17 606 607 608 609 610 97 89 104 94 74 19 ;o 6 .5 36.4 12i9 0 .6 78 57 54 103 49 36;7 i7 ; o h ;5 65*6 1 1 .1 20 15 12 2 10 611 612 613 614 615 91 94 97 85 105 9;i 12;9 i9 ; o 3 i2 3 9 .3 4 4 32;0 20;6 i4 ; 9 i;e 11.5 51 39 60 65 31 26;o io ;8 4 0 ;5 4 8 .2 3 .7 4 4 4 83 44 75 48 94 4 o ;e 7 ;s 33 i9 io ; 3 5 6 .2 25 4 19 7 28 44 ;o 3;2 30i3 7;e 5 3 .0 45 49 56 37 61 i7;3 22;8 33.3 8.1 4 2 .4 53 106 114 83 67 is ;5 68;0 77i2 4 0 .8 2 6 .6 7 21 34 22 5 7;6 33.8 69;0 36;3 4 .5 41 47 84 26 48 12;7 i9 ; 4 75;0 1 .2 2 1 .4 4 616 617 618 619 620 80 114 99 74 103 i;7 68;2 2 3 ;i 0 .6 33.4 381 Part II, (1) (2 ) (3V Otis I.Qi P.R. (Control Group Pupils) Continued (4 ) Geom Prog Score (5 ) (6 ) Geom Geom Prog Ach* P.R. Score (7 ) Geom Achi P.R. (8 )' C.T. Test Score (9 )' C.T. Test P.R, Pupil Otis I.Q. 621 622 626 624 625 98 85 116 125 116 ' Teachers J and K 20;6 74 33;0 3 ,2 89 5 0 .0 7 6 ,5 108 7 0 .3 95*0 146 9 6 .8 76*5 88 4 8 .6 (Continued) 27 so ;i 30 5 9 .8 15 2 0 .6 58 9 6 .9 35 7 2 .0 43 48 52 75 86 15;8 2 1 .4 2 7 .6 6 3 .5 77 .2 626 627 628 629 630 108 111 114 117 119 48*2 56*6 68*2 80*5 8 5 ,8 120 126 100 88 129 8 2 .4 8 6 .5 6 2 ,5 4 8 .6 8 8 .8 32 41 26 35 62 6 4 .3 8 3 .0 4 7 .0 7 2 .0 9 7 .8 56 82 61 70 78 3 3 .3 7 3 .0 4 2 .4 5 6 .6 6 7 .6 631 632 633 634 635 101 103 114 120 117 2 7 .6 3 3 ,4 6 8 .2 8 8 .2 8 0 .5 125 54 105 101 116 8 5 ,8 1 4 .5 6 7 .2 6 3 .8 7 9 .1 32 34 26 28 44 6 4 .3 6 9 .0 4 7 .0 5 3 .0 8 7 .0 66 65 61 53 88 5 0 .3 4 8 .2 4 2 .4 2 9 .1 8 0 .4 636 637 638 639 640 113 120 109 113 106 6 4 ,3 8 8 .2 5 1 .0 6 4 .3 4 2 .8 123 131 70 128 112 8 4 .3 8 9 .9 2 9 .3 8 8 .0 7 5 .6 41 62 16 39 32 8 3 .0 9 7 .8 2 2 .6 79.4 6 4 .3 82 78 52 75 58 7 3 .0 6 7 .6 2 7 .6 6 3 .5 3 7 .4 641 642 643 126 127 116 9 6 .1 9 7 .0 76 .5 168 128 140 9 9 .9 8 8 .0 9 4 .6 40 51 52 8 1 .0 9 3 .0 9 3 .8 91 60 75 8 3 .3 4 0 .5 6 3 .5 Teacher L 644 645 110 115 5 3 .6 7 2 .7 103 145 6 5 .6 9 6 .3 48 63 9 0 .5 9 8 .2 83 75 7 4 .2 63 .5 646 647 648 649 650 108 100 100 103 112 4 8 .2 2 5 .8 2 5 .8 3 3 .4 6 0 .7 110 111 55 43 125 7 3 .7 7 4 .8 1 5 .6 6 .5 8 5 .8 32 57 34 32 41 6 4 .3 9 6 .7 6 9 .0 6 4 .3 8 3 .1 66 70 65 56 82 5 0 .3 5 6 .6 4 8 .2 3 3 .3 7 3 .0 651 652 653 654 655 109 128 117 117 120 5 1 .0 9 7 .5 8 0 .5 8 0 .5 8 8 .2 81 136 164 140 135 3 9 .4 9 1 .6 9 9 .4 9 4 .6 9 1 .3 26 52 40 52 62 4 7 .0 93 .8 8 1 .0 9 3 .8 9 7 .8 82 66 96 75 78 7 3 .0 5 0 .3 8 7 .7 6 3 .5 6 7 .6 656 657 658 659 660 140 116 112 105 111 9 9 .7 76 .5 6 0 .7 3 9 ,3 5 6 .6 123 100 70 71 92 8 4 .3 62.5 2 9 .3 3 0 .6 5 3 .7 43 26 46 16 35 8 5 .7 4 7 .0 8 9 .0 2 2 .6 7 2 .0 87 61 52 70 86 7 9 .0 4 2 .4 2 7 .6 5 6 .6 77.2 382 Part IX. (1) Pupil (2) Otis I.Q* (3 ) Otis I*Q* P.H. (Control Group Pupils] (i) Geom Prog Score (« ) Geom Prog PJEU Teacher L (5) Geom Ach* Score Continued (* ) Geom Ach* P *E. #3) C.I. Test Score (9) C.T. Test p*a. (Continued) 661 662 663 664 665 110 124 114 117 120 53*6 93*5 68*2 80*5 8 8 .2 93 138 103 116 112 5 5 .0 9 3 .2 6 5 .6 79 .1 75.6 26 67 32 44 42 4 7 .0 9 9 .0 64 .3 8 7 .0 8 4 .5 93 77 66 88 66 84*9 66*2 5 0 .3 8 0 .4 5 0 .3 666 667 668 669 670 117 131 110 142 103 8 0 .5 98*4 5 3 .6 9 9 .8 3 3 .4 131 166 108 149 63 8 9 .9 9 9 .7 7 0 .3 9 8 .0 2 3 .5 51 64 43 70 11 9 3 .0 98.5 8 5 .7 9 9 .4 1 5 .0 60 93 48 69 47 4 0 .5 84*9 2 1 .4 5 4 .8 1 9 .4 671 672 673 674 675 114 100 111 94 110 6 8 .2 2 5 .8 5 6 .6 1 2 .9 5 3 .6 101 46 97 85 117 6 3 .8 8 .5 5 9 .4 4 4 .0 8 0 .2 28 5 52 20 37 5 3 .0 4 .5 93.8 3 2 .0 7 6 .1 53 30 33 52 79 2 9 .1 3 .1 5 .3 2 7 .6 6 9 .3 676 677 678 679 680 113 100 102 112 101 6 4 .3 2 5 .8 2 9 .7 6 0 .7 2 7 .6 109 79 109 108 91 7 2 .0 3 7 .6 7 2 .0 7 0 .3 5 2 .2 68 38 29 42 39 99.2 77.8 56 .5 8 4.5 7 9 .4 57 66 40 55 65 35 .5 5 0 .3 1 1 .4 31 .2 4 8 .2 681 682 683 684 685 93 115 106 108 98 11*2 7 2 .7 4 2 .8 4 8 .2 2 0 .6 78 100 106 78 89 3 6 .7 6 2 .5 6 8 .0 36*7 5 0 .0 16 24 40 32 39 2 2 .6 4 1 .3 8 1 .0 6 4 .3 79.4 38 69 62 58 35 9 .2 5 4 .8 4 4 .0 3 7 .4 2 9 .1 686 687 688 689 690 112 96 96 98 105 6 0 .7 1 6 .7 1 6 .7 2 0 .6 3 9 .3 100 54 60 50 87 6 2 .5 1 4 .5 2 0 .0 11 .8 4 7 .2 36 12 26 24 29 74.2 14 .9 4 7 .0 4 1 .3 56*3 56 42 55 54 56 33*3 1 4.6 31 .2 3 0 .3 3 3 .3 691 692 693 694 695 102 130 110 90 117 29*7 98*1 5 3 .6 7 .6 8 0 .5 86 163 121 53 115 4 5 .5 9 9 .2 8 3 .0 13.5 7 8 .0 14 64 38 9 31 18 .6 98.5 77.8 1 0.1 62.2 47 70 79 56 59 19 .4 5 6 .6 6 9 .3 3 3 .3 3 9 .0 696 697 698 699 700 103 118 99 106 123 3 3 .4 83*6 2 3 .1 4 2 .8 92*4 88 101 60 110 161 4 8 .6 63 .8 20*0 7 3 .7 9 9 .1 14 25 22 38 39 18*6 4 4 .0 36.3 77.8 79.4 46 72 46 39 65 18*3 5 9 .5 18*3 10.8 4 8 .2 383 Part II. (1) (2) Pupil Otis I»Q. (Control Group Pupils) (3) otis I*Q. P*R. (4) Goes Prog Soore (5) Geom Prog P.R. (6) Geom Rohs Score Continued (7) Geom Rchg P.R. (8) C.T. Test Score (9) C.T. Test P.R. i'Continued) 18 27.6 29 56.5 37 76.1 24 41.3 30 59.8 46 40 76 76 61 18.3 11.4 65.3 65.3 42.4 701 702 703 704 705 90 103 117 120 102 7*6 33*4 80,5 88*2 29 #7 Teacher L 85 44.0 54 14.5 112 75.6 108 70.3 86 45.5 706 707 708 709 710 114 96 109 121 105 68*2 16.7 51.0 90.2 39.3 136 77 83 110 63 91.6 36.0 40.8 73.7 23.5 51 23 33 20 14 93.0 39.0 66.5 32.0 18.6 53 46 55 63 51 29.1 18.3 31.2 45.5 26.0 711 712 713 714 715 90 120 107 113 111 7.6 88.2 46.7 64.3 56.6 101 112 86 107 122 63.8 75.6 45.5 69.0 83.5 35 44 34 35 50 72.0 87.0 69.0 72.0 91.9 36 59 56 68 54 7.3 39.0 33.3 53.0 30.3 716 717 718 719 720 117 121 107 109 99 80.5 90.2 45.7 51.0 23.1 143 108 114 72 58 95.6 70.3 77.2 31.6 18.4 39 40 36 17 28 79.4 81.0 74.2 24.8 53.0 40 87 49 55 62 11.4 79.0 22.8 31.2 44.0 721 722 723 724 ! 725 I 726 122 111 96 113 99 98 91.4 56.6 16.7 64.3 23.1 20.6 117 92 79 131 86 73 80.2 53.7 37.6 1.8 45.5 32.2 38 33 12 50 19 17 77.8 66.5 14.9 91.9 30.3 24.8 58 61 48 59 53 38 37.4 42.4 21.4 39.0 29.1 9.2 Control Group Means* Standard Deviations, and Manges I.Q. Means S.Ds. Manges 110.2 11.7 74 - 146 GeonuProg. 95.7 29.7 24 - 168 Geom. Aehiev. 31.2 14.4 -7 - 70 Crit. Think. 54.4 16.4 12 - 96 APPENDIX B (Continued) II. Summary of Responses to the Initial Form of the Inventory Questionnaire Administered to Both at the Beginning of the School Year Groups 385 The following is a grouping of the responses to the first seven questions in the initial form of the Inventory Questionnaire as per Teacher Group* Part 1 is a tabulation for the Experimental Groups and Part 2 is a tabulation for the Control Groups* Part 1 Question Type of No* Response (Experimental Groups) % Under % Under % Under't Under t Under Tch. F. Per Cent Total 22.6 75;s 1.9 28.8 67.3 3.9 20;9 76.9 2.2 38.3 20;6 9.7 2;7 4.1 24.6 50.9 22.6 7.5 o;o 5.8 13.2 44.3 23;i 13.4 0.0 5.8 13.4 41.8 20.9 9;8 2.2 8.2 17.1 7.5 is ;i i3;2 3.7 U9 58.6 5.5 8i2 13;7 10.9 6i;7 0*0 I7;i s;7 7i5 3.7 13.2 54.8 15.4 9.6 u;5 3.9 5;8 53.8 12:7 8.2 10.1 3.2 10.8 55.0 Exam*-Teach*Judg • 24,7 SeIf-Evaluati on 7.1 60.0 Combination 8.2 No Response 26.4 7.5 33;9 32.2 9.7 6.8 49.3 34.2 26.4 o;o 32.2 41.4 25.0 5:8 52;0 17.2 21.8 5.7 47.2 25.3 5* 4i;2 Independent Dependent on Text 34.2 15.2 Combination 9.4 No Response 28;3 54.8 9.4 7.5 49.3 17.8 17.8 15.1 28.3 43.4 13.2 15.1 7'.7 65;4 15.4 11.5 33.2 40.5 14.6 11.7 6. Accurate Concept i;2 2U2 Partly Accurate Inaccurate Concept 55.2 22*4 No Response 3.7 17.1 60;4 18.8 1.4 15.1 46^6 36.9 0.0 30.3 18.8 50.9 1.9 2i:s 48.0 28.8 1.6 20.6 46.8 31.0 7. o;o Accurate Concept Partly Accurate 34*1 Inaccurate Concept 50*7 15.2 No Response 0.0 24.5 54;8 20.7 0.0 30;2 56.2 13.6 0.0 22;6 54.8 22.6 0.0 34.6 59.6 5.8 0.0 29.7 54;9 15.4 Tch. A Tch. B Mathematics Other Subjects No Response 25 i8 73*0 1.2 I5;i 84;9 0.0 12 i3 83*6 4.1 2* Vocations Mental Discipline Develop Reasoning Enter College In No Way No Response 35 ;3 20:0 3*5 4;7 17;7 18*8 45 ;5 is ;8 18.8 1.9 3.7 11.3 3. 17;7 Vocations Mental Discipline 5*9 Develop Reasoning 5*9 4i7 Enter College \r.i In No Way No Response 48*1 4. 1. 2* - Tch.C,D<, Tch. E 386 Part 2 Question Type of No. Response (Control Groups) % Under % Under % Under Tch. ft Tch* 6 Tch* 1 % Under % Under Tch.J.K Tch. L Per Cent Total 1. Mathematics Other Subjects No Response 38 ;s 61i5 0,0 48*8 5i;2 0.0 45.8 54i2 0*0 34.8 65;2 0.0 34.2 65;8 0.0 41.4 58*6 0.0 2* Vocations Mental Discipline Develop Reasoning Enter College In No Way No Response 77.0 11.5 o;o 7;7 o;o 3.8 53.6 is;i 12.8 2.3 6;9 9.3 54*2 16*6 12;5 8.3 4;2 4.2 78.4 8.7 4;3 4.3 o;o 4.3 63;0 8.3 19.2 4;i 2;7 2.7 61.6 12a 12.5 4i3 3.9 5.6 3. Vocations Mental Discipline Develop Reasoning Enter College In No Way No Response 30;7 23.2 11.5 11.5 7;7 15.4 17;4 10.4 22.2 2*3 5*8 41.9 29;2 12;5 8.3 16;7 8.3 25.0 2i;7 4.3 26.1 4.3 13.2 30.4 31.5 1.4 20.6 4.1 5.5 36.9 25*0 8.6 19.4 5.6 6.9 34.5 4* Exam.-Teach.Judg. Self-Evaluation Combination No Response 38;5 0.0 46*1 15.4 51.2 6.9 37.2 4.7 54.2 8.3 29.2 8.3 47.9 8.7 39.1 4.3 26.1 4.1 57.5 12.3 41.8 5.6 44.0 8.6 5* Independent^)f Text 19*3 Dependent on Text ei;s is ;4 Combination 3*8 No Response 23*3 55.8 12.8 8.1 12 ;s 62;5 16.7 8.3 8.7 78.3 8.7 4.3 6.9 67il 21.9 4.1 15*1 62.9 15.9 6.1 6. o;o Accurate Concept 30*7 Partly Accurate Inaccurate Concept 57*8 11.5 No Response o;o 25;5 52;3 22.2 8.3 16.7 45 i8 29*2 0.0 21.7 47;9 30.4 1.4 17.8 61.6 19.2 1.3 22;4 54.7 21.6 7. o;o Accurate Concept i9;2 Partly Accurate Inaccurate Concept 73*1 7.7 No Response 0.0 22a 64.0 13.9 0.0 8.3 70.9 20.8 O'.O 4.3 60.9 34.8 o;o 3U 6 54.8 13*7 0.0 2i;6 62.5 15.9 APPENDIX III# B (Continued) Summary of Responses to the Followr-up Form of the Inventory Questionnaire Administered to Both Groups at the End of the School Year 388 Tho following is a grouping of the responses to the first eight questions in the final or follow—up form of the Inventory Questionnaire as per Teacher Group* Part 1 is a tabulation for the Experimental Groups and Part 2 is a tabulation for the Control Groups* Part 1 Question No# Type of Response (Experimental Groups) $ Under % Under Tch* A Tch. B % Under % Under % Under Tch.C,D Tch. E Tch. P Per Cent Total 1. Mathematics Other Subjects No Response 4o;o 6o;o 0.0 41i9 58il 0.0 24:6 75;4 0.0 33:9 64:3 1.8 35:6 6o;o 4.4 35:8 63:2 1*0 2* Vocations 42;3 Mental Discipline r.i Develop Reasoning 46*6 Enter College 2.2 In No Way 6.7 No Response 1.1 45i2 o;o 43^6 4;s 55'.2 0:0 30;4 0.0 10; 7 3.7 46;7 0:0 37:8 8.7 0.0 8.8 46:1 0:3 0.0 43:8 0;0 52:6 0:0 1.8 1.8 42.9 2:6 5:5 2.6 Vocations 8;9 Mental Discipline o;o Develop Reasoning 83*3 Enter College 1.1 e;? In No Way 0.0 No Response i4:s 15:7 o;o 67:7 3.2 9.7 4.9 o;o 70.1 1.8 7.1 5*3 3',7 0;0 84)0 0.0 12.3 0*0 13.3 0;0 75:7 0,0 li:0 0.0 11.0 0:0 76.8 1.3 9.0 1*9 Exam.-Teach *Judg » Self-Evaluat ion Combination No Response 10'.7 0:0 87*4 1.9 40:0 0:0 57:8 2.2 26:5 87.4 5.3 OlO 77:9 8:9 8:9 81.0 UP 2.2 1.3 3* 4. 5* 6. 6:4 2i;i 3;4 75 ;5 0.0 37:1 28:i 17.7 4U9 3*3 15:7 88i9 Independent Dependent on Text 4:5 4.5 Combination 2.1 No Response 79:0 66:6 12.9 8.1 0.0 19:4 Accurate Concept 31.1 6i;i Partly Accurate Inaccurate Concept s;5 2.3 No Response 33:8 43:6 11.3 11.3 36:8 42.0 7:i 14.1 83.9 7;i 3.7 5.3 55.5 31.3 2.2 11.0 45*8 40.0 6.1 8.1 56;2 0.0 14.0 0.0 7:4 64*8 1.3 9:7 6:7 4 7. Accurate Concept Partly Accurate Inaccurate Concept No Response 28*9 55;5 5i6 10.0 33:8 58:0 3*3 4.9 21.1 52:5 19.3 7.1 46.4 41.0 8:9 3.7 22.4 44.4 15.5 17.7 30.6 51.3 9.7 8.4 8* More than Three Three Two One No Reading 10:0 15:5 16*6 24.4 33.5 9*7 11.3 12:9 14.5 51.6 14.1 19 :s 26.3 28:1 12.2 19:6 14*4 25*0 32.1 8.9 6.7 8.9 11.1 49.0 24.3 11.9 14.2 18.4 28.1 27.4 389 Part 2 Question No* Type of Response (Control Groups) % Under % Under % Under % Under % Under Tch. G Tch. H Tch. I Tch.J,K Tch. L Per Cent Total 1. Mathematics Other Subjects No Response 30;7 69;3 0 .0 34;o 66 ;o 0 .0 6 ;7 9 3 .3 0 .0 2 6 .6 7 3 .4 0 .0 32.5 67;5 0 .0 2 9 .4 70.6 0 .0 2. Vocations Mental Discipline Develop Reasoning Enter College In No Way No Response 34;6 2 3 i0 7 .7 1 5 .4 11 .6 7 .7 38 ;8 16i5 2 2 .3 3 .9 i3 ;s 4 .9 6o;o 13.4 io ; o 3 .3 3 .3 1 0 .0 5 3 .3 33'.3 0 .0 0 .0 o ;o 1 3 .4 5 5 .4 8 .1 17;6 2 .7 12.2 4 .0 4 6 .8 15.3 16.5 4 .4 10.9 6 .1 3. Vocations Mental Discipline Develop Reasoning Enter College In No Way No Response n ;s 3 2 .0 1 5 .4 11 .6 23;o 1 5 .4 12 ;s 10.8 40 ;7 3 .9 13.6 18 .4 1 0 .0 2 0 .0 13 .4 6 .7 2 6.6 2 3 .3 o ;o 20.2 26 .6 o ;o 26.6 2 6 .6 14 .9 8 .1 28 ;4 2 .7 2 9 .7 16.2 12 .1 12.9 30 .2 4.4 21.8 1 8 .6 4* Exam .-Teach*Judg• Self-Evaluation Combination No Response 53;9 11.5 3 0 .7 3 .9 4 2 i7 6;8 4 8 .6 1 .9 66;6 io ; o 23.4 0 .0 4 6 i6 o ;o 4 6 .6 6 .8 6o ;o 5 .4 3 9.2 5 .4 49;2 3;2 4 0 ;7 6 .9 5* Independent of Text 46i2 2 6 .9 Dependent on Text 26;9 Combination 0 .0 No Response 39 ;8 40;7 16.6 2 .9 43 ;3 33;3 23 .4 0 .0 20;2 73.2 6 .6 0 .0 42 ;o 33i8 24^2 0 .0 4 0 .3 38.3 20.2 1 .2 6* Accurate Concept Partly Accurate Inaccurate Concept No Response o;o 3 8 .4 5 3 .9 7.7 i;o 26.2 6 0 .1 11.8 0 .0 23 .4 5o ;o 26.6 o ;o 2o ;o 6 0 .0 2 0 .0 0 .0 2 7 .0 5 8 .0 1 5 .0 0 .8 27*0 5 7 .7 14.5 7* Accurate Concept Partly Accurate Inaccurate Concept No Response 0 .0 38 .4 5 7 .7 3 .9 1 .0 3 2 .0 6 1 .1 5 .9 o ;o 3 0 .0 4 3 .4 26.6 0 .0 2 6 .6 6 0 .0 1 3 .4 2 .7 36.5 5 2 .7 8.1 1 .2 33.5 5 6 .0 9 .3 8. More than Three Three Two One No Reading 0 .0 0 .0 3 .9 7.7 8 8 .4 0 .0 1 .9 3 .9 15.6 78.6 0 .0 3.3 1 6 .7 26 ;s 53.4 o;o 0 .0 1 3 .4 13 .4 73.2 0 .0 1 .4 4 .1 9 .4 8 5 .1 0 .0 1.6 6.1 14.1 78.2 -APPENDIX C Inventory Questionnaire (Preliminary Form) Otis Self-Administering Tests of Mental Ability Orleans Geometry Prognosis Test Progressive Education Association Nature of Proof Test 5*3 Columbia Research Bureau Plane Geometry Test Critical Thinking Test Inventory Questionnaire (Pinal or Follow-up Form) Examples of Periodical Tests 391 INVENTORY (tfJESTIQNlUIHE !• (Preliminary Porm) Name Age last birthday Address ........................... Street and Number Have you studied geometry beforei *___ City If so, how many months? DirectiongS This is not an examination but merely a questionnaire to find out what your reactions are toward some factors that may reveal your interests and needs so that the course may be modified if necessary to help meet your individual needs. how. Please answer each question the best way you know If you do not understand some of the questions, ask your teacher to explain them more clearly for you. 1. What are your favorite subjects? Why? 2. In what way do you think that a study of mathematics may help you? Why? 3. In what way do you think that a study of demonstrative geometry may help you? Why? 4. How would you like to have your progress evaluated in this course? (that is, by a written examination, self-evaluation of your mental growth, your parents* evaluation of your mental growth, judgment of your teachers, and so forth, or a combination of these?) Why? 5. Doyou think that high school pupils are capable of developing their own concepts of geometry, that is, developing their own theory of space, or do you think that a textbook is absolutely necessary to develop such a theory? 392 6. The concept of proof has always played a very significant role in human experience* When, in your opinion, is a statement proved? 7. The kind of society in which we live calls for a type of citizenry that is capable of thinking critically. When is a person thinking critically? 8. What do you expect to do when you leave school? Why? 9. What led you to make these plans? 10* What is the attitude of your parents or guardian toward these plans?Why? 11* What areyour favorite hobbies? collecting, gardening, (that is, etanp dancing, reading, raising chickens, cooking, and so forth.) 12* What areyour major vocational interests? (that is, engineering, architecture, singing, playing in an orchestra, aviation, law, medicine, journalism, and so forth). 13. How many magazines are regularly taken at your home? ..... list their names, 14. About how many books are in your home? list the kind of books that you like to read (that is, adventure, mystery, science, romance, travel, art, religion, etiquette, and others). 393 15# Do you have a daily paper in your home? ..... Do you read a daily paper? ..... List the kind of reading that is of most interest to you (that is* sports, local news, foreign news, society columns, editorials, comics, war news, lahor problems, strikes, unemployment, and so forth). 16. Do you have a radio in your home? ..... List your favorite programs, (that is, opera organ music, news reviews, speeches, sermons, true stories, drama, sports broadcasts, and so forth). 17. What do you think that the school could do to make your life more pleasant, more meaningful, and more profitable to you? (Peel free to say anything that you might have in mind.) $he following questions are an attempt to give you an opportunity to express your feelings regarding some of the factors concerning your education and perhaps this course. Please answer each question exastly as you yourself feel about it and not in terms of what others might think about your answers. All of your answers are strictly confidential and will be used only as a means of further developing this course. If your answer is TBS, draw a line under the word "YES", and if your answer is NO, draw a line under the word "NO". Samples (a) Do you like to play yourfavorite games? (b) Do you like to be scolded forsomethingyou didn't do? Yes No Yes No 1. Do you like high school?..................................... ^es 2. Doyou expect to be graduated?................................ ^es 394 3. Da you like mathematics?...................................... Yes No 4. Has the study of mathematics helped you in any way? ....... Yes No 5. Do you dislike any of the subjects you are now studying? . . . . . If your answer is Yes, name them. Why do you dislike them? Yes No 6. Are you required to take subjects that you dislike? ............ Yes No 7. Do you think that there are too many required subjects?.........Yes No 8. Would you like more freedom in choosing your own studies? . . . . Yes No 9. Do you think that your high school training will do you much good unless you attend college?............................... Yes No 10. Are you planning to go to college? Yes No 11. Are there subjects you would like to takeif they wereoffered? Yes No 12. Do you ever feel that you would like to quit school and go to work? Yes No 13. Are most of your studies interesting?.................. .. Yes No 14. Are there any subjects in which you don't care whether or not you do any work? Yes No 15. Do you ever worry or feel uncertain about passing in school? 16. Do you like to master a difficult task or a difficult subject? . . YesNo Yes No 17. Are most of your textbooks interesting andeasy to read?. . . . Yes No 18. Do you feel that most of your subjects willbe agreat help you after you finish school? Yes No to 19. Do you expect to quit school as soon as possible? Yes No 20. Do you dread being called upon to recite? Yes No 21. Do you feel that too much emphasis is placed upon good order and discipline in school? Yes No 22. Does this discipline (in question No. 21) or the accompanying rules interfere with your activities? Yes No 23. Do you think there are too many rules to be followed in school? Yes No 24. Do you like final examinations in your subjects?............. Yes N0 25. Doyou think that examinations ar^e generally f a i r ? ............. Yes No 26. Do you like all of your teachers?.......................... , Yes No 27* Do you think that your teachers usually understand your difficulties?.......................... ................... Yes No 28. Are all of your teachers thoughtful and considerate of your work? Yes No 29. Do you "believe that the marks a teacher gives depend upon favoritism?................................................. Yes No 30. Do all of your teachers treat you as a f r i e n d ? Yes No 81. Are all of your teachers willing to explain certain topics, you do not understand, more than once?.............................Yes No 32. Do any of your teachers talk or lecture too much in the classroom?Yes No 33. Do all of your teachers give you opportunities to express your own opinions? Yes No 34. Do any of your teachers permit only a few pupils to do all of the reciting? . . . . . . . . . . . . Yes No 35. Do most of your teachers try to make the class interesting? .. • Yes No 36. Are any of your teachers more interested in their subjects than in their p u p i l s ? .................... * ................. ^es 37. Are your teachers usually willing to talk with you about your difficulties and give you good advice? . . . . . * • • • • * Yes No 38. Do any of your teachers ever embarrass you before the class? . • Yes No 39. Are you given a chance to express or tell what you know in your classes'? Tes Bo 40. Bo you think that any of your teachers are too s t r i c t t ........ Tes Ho 41. Do any of your teachers resent having a pupil express an opinion which differs from their own or from that of the textbook?Yes No 42. Do you accept all that your teachers tell y o u ? ............... Yes No 43. Do you accept all that you read in your textbooks? Yes No 44. Do you think that you should be given an opportunity to think independently even though your thinking is out of harmony with the opinions of your teachers or your textbooks?.............. 45. Do you think that pupils who work the hardest should get the best marks?................................................ 46. Do you like to excel or beat others in their class work? wo 8 . . . . Yes No 396 47# Do you like to volunteer to recite? .......................... Jfes No 48* Do you feel free to express your opinions among your fellow pupils?................................. 49. Do you feel that most of your classmates are superior to you in school w o r k ? Yes • Yes No 50# Do you feel that your ideas and opinions are as good as those of your classmates?....................................... YesNo 51. Do you sometimes feel that the things you do are of minor ............................ importance? YesNo 52* Do you think that your school work is quite monotonous?......... Yes No 53. Do you feel that you are quite a success in the things you do? Yes No 54. Do your parents ever praise you for exceptionally good work? Yes No 55* Do Yes No your teachers ever praise you for exceptionally good work? 56* ire you as successful in your school work as your parents expect you to h e ? ............................ 57. ire your parents satisfied with your school work? . . . . . . . . 58* Do you like to ask your parents for advice or help? ^es Yes No ........... Yes No 59. Do your parents think that most of your teachers are good teachers?..................................................... Tes No 60. Do you like to ask your teachers dor advice or h e l p ? ........... Tee No 61. Do your parents require you to perform many tasks around your h o m e ? ................... .......... . ^ 62. Do your parents want you to do many things that you dislike? . . Yes N0 63. Do your teachers want you to do many things that you dislike? . .Yes No 64. Do you earn money through part time employment during the school year? ............ * ................................ 65. ire your economic circumstances such that you must seek part time employment? ............................... ^ ^ OCXS SELF-AmilNISKtlNG TESTS OF UEfflm ABILITY. OTIS SELF-ADMINISTERING TESTS OF MENTAL ABILITY By A r th u r S. O tis Formerly Development Specialist with Advisory Board, General Staff, United States War Department HIGHER EXAMINATION; F O R M C 20 For High Schools and Colleges Read this page. Score.... Do what it tells you to do. Do not open this paper, or turn it over, until you are told to do so. Fill these blanks, giving your name, age, birthday, etc. Write plainly. Name. Age last.birthday.... years F ir s t nam e, in itia l, a n d la s t nam e Birthday...................... Class............. Date.............10... . M o n th D ay School or College........................... City......................... This is a test to see how well you can think. It contains questions of different kinds. Here is a sample question already answered correctly. Notice how the question is answered : Which one of the five words below tells what an apple is? i flower, 2 tree, 3 vegetable, 4 fruit, 5 animal.................. (^ ) The right answer, of course, is “fruit” ; so the word “fruit” isunderlined. And the word “fruit” isNo. 4; so a figure 4 isplaced in the parentheses at the end of the dotted line. This isthe way you are to answer the questions. Try this sample question yourself. Do not write the answer; just draw a line under it and then put its number in the parentheses: Which one of the five words below means the opposite of north? 1 pole, 2 equator, 3 south, 4 east, 5 west..................... ( ) The answer, of course, is “south” ; so you should have drawn a line under the word “south” and put a figure 3 in the parentheses. Try this one: A foot is to a man and a paw is to a cat the same as a hoof is to a — what ? 1 dog, 2 horse, 3 shoe, 4 blacksmith, 5 saddle ( ) The answer, of course, is “horse”; so you should have drawn a line under the word “horse” and put a figure 2 in the parentheses. Try this one: At four cents each, how many cents will 6 pencils cost? ( ) The answer, of course, is 24, and there isnothing to underline 5 so just put the 24 in the parentheses. If the answer to any question is a number or a letter, put the number or letter in the parentheses without underlining anything. Make allletters like printed capitals. The test contains 75 questions. You are not expected to be able to answer all of them, but do the best you can. You will be allowed half an hour after the examiner tells you to begin. Try to get as many right as possible. Be careful not to go so fast that you make mistakes. Do not spend too much time on any one question. No questions about the test will be answered by the examiner after the test begins. Lay your pencil down. Do not turn this page until you are told to begin. P ublished b y W orld Book C om pany, Y onkers-on-H udson, N ew Y ork, an d Chicago, Illinois C opyright b y W orld Book C om pany, C o p yright in G reat B ritain . A ll rights reserved. 1928 osatua:he:o2I This te st is copyrighted. The reproduction o f any part of H by mimeograph, hectograph, or in any other way, whether the reproductions are sold or are furnished fre e fo r use, is a violation of the copyright law. S. A. Higher E x a m in a t io n b e g in s here. 1. The opposite of defeat is (?) t . ,, . t _ t _ (Do not write on these dotted lines.) i glory, 2 honor, 3 victory, 4 success, 5 nope . ...... ........ 2 . If 3 pencils cost 1 0 cents, how many pencils can be bought for 5 0 cents?. . ..................... ......... 3 . A dog does not always have (?) 1 eyes, 2 bones, 3 a nose, 4 a collar, 5 lungs.................................................................. 4 . The opposite of strange is (?) 1 peculiar, 2 familiar, 3 unusual, 4 quaint, 5 extraordinary. ..................................... 5 . A lion most resembles a (?) 1 dog, 2 goat, 3 cat, 4 cow, 5 horse 6 . Sound is related to quiet in the same way that sunlight is to (?) 1 darkness, 2 evaporation, 3 bright, 4 a cellar, 5 noise. 7 . A party consisted of a man and his wife, his three sons and their wives, and two children in each ............................ son’s family. How many were there in the party?. . 8 . A man always has (?) ................................................... 1 children, 2 nerves, 3 teeth, 4 home, 5 wife 9 . The opposite of stingy is (?) 1 wealthy, 2 extravagant, 3 poor, 4 economical, 5 generous........................................ 1 0 . Lead is cheaper than silver because it is (?) ' 1 duller, 2 more plentiful, 3 softer, 4 uglier, 5 less useful. ................... 1 1 . Which one of the six statements below tells the meaning of the following proverb? “Let sleep ing dogs lie.” . ................... ....................................................................................... 1. 2. 3. 4. 5. 6. 12. 13. Eat heartily at a good feast. Only exceptional misfortunes harm all concerned. Don’t invite trouble by stirring it up. Strong winds blow harder than weak ones. Too much of anything is no better than a sufficiency. Tired dogs need lots of sleep. Which statement above tells the meaning of this proverb? “ Enough is as good as a feast.” Which statement above explains this proverb ? “ It’s an ill wind that blows nobody good.” . 14. A radio is related to a telephone as (?) is to a railroad train. 1 a highway, 2 an airplane, 3 gasoline, 4 speed, 5 noise. . . 15. If a boy can run at the rate of 8 feet in of a second, how far can he run in 1 0 seconds? . . . . 16. A debate always involves (?) 1 an audience, 2 judges, 3 a prize, a controversy, 4 5 an auditorium ... .... ............ 17. Of the five words below, four are alike in a certain way. 1 walk, 2 run, 3 kneel, 4 skip, 5 jump. 18. The opposite of frequently is (?) 1 seldom, 2 occasionally, 3 never, 19. A thermometer is related to temperature as a speedometer is to (?) 1 fast, 2 automobile, 3 velocity, 4 time, 5 heat. 20. Which word makes the truest sentence ? Women are (?) shorter than their husbands. 1 always, 2 usually, 3 much, 4 rarely, 5 never. ................... 21. One number is wrong in the following series. 1 6 2 7 3 8 4 4 sometimes, 9 Which one is not like these four? . . 5 often. .. What should that number be ? 5 10 7 11. ....................... .. . 22. If the first two statements following are true, the third is (?) All children in this class are good students. John is not a good student. John is a member of this class. 1 true, 2 false, 3 not certain. ........... .. 2 3 . A boat race always has (?) 1 oars, 2 spectators, 3 victory, 4 contestants, 5 sails............................. ........................ 24. Which number in this row appears a second time nearest the beginning? 4 2 3 1 5 6 8 7 3 4 6 6 4 3 2 5 1 8 6 7 9 . ...................................................... ................ 25. The sun is related to the earth as the earth is to (?) 1 clouds, 2 rotation, 3 the universe, 4 the moon, 5 circumference. . . . .......... 2 6 . Which word makes the truest sentence ? A youth is (?) wiser than his father. 1 never, 2 rarely, 3 much, 4 usually, 5 always..................................... .. .................... Do not stop. Go right on with the next page. M S. A. Higher 2 7 . T h e o p p o s i t e o f g r a c e f u l is ( ? ) 1 w eak, 2 u g l y , 3 s lo w , 4 a w k w a rd , 5 uncanny. .. ( 28. A g r a n d m o t h e r is a lw a y s ( ? ) t h a n h e r g r a n d d a u g h t e r . 1 s m a rte r, 2 m o r e q u i e t , 3 o ld e r , 4 s m a ll e r , 5 s l o w e r .............................. ( 29. W h i c h o n e o f t h e s ix s t a t e m e n t s b e lo w t e lls t h e m e a n i n g o f t h e f o llo w in g p r o v e r b ? t h a t e n d s w e l l.” . . . . . . .................................................... 1. 2. 3. 4. 5. 6. “ A l l ’s w e ll .. E v e n th e d a r k e s t s itu a tio n s h a v e th e ir b r ig h t a s p e c ts . T h e f in a l r e s u l t is m o r e i m p o r t a n t t h a n t h e i n t e r m e d i a t e s te p s . H a n d s o m e p e r s o n s a lw a y s d o p l e a s i n g t h in g s . A ll c o m e s o u t w e ll i n t h e e n d . P e r s o n s w h o s e a c t i o n s p l e a s e u s s e e m g o o d - lo o k in g . C l o u d s s h i m m e r a s if t h e y w e re m a d e o f s ilv e r. 30. W h ic h s t a t e m e n t a b o v e e x p la i n s t h i s p r o v e r b ? “ E v e r y c lo u d h a s a s ilv e r l i n i n g . ” 31 . W h ic h s t a t e m e n t a b o v e e x p la i n s t h i s p r o v e r b ? “ H a n d s o m e is t h a t h a n d s o m e d o e s .” . . . ( ( 32. I f t h e s e t t l e m e n t o f a d if f e r e n c e b e tw e e n tw o p a r t i e s is m a d e b y a t h i r d p a r t y , i t is c a l l e d ( ? ) 1a c o m p ro m is e , 2 a tru c e , 3 a p r o m is e , 4 a n i n j u n c t i o n , 5 a n a r b itr a tio n . .... 33. ( O il is t o t o i l a s ( ? ) is t o h a t e . 1 lo v e , 2 w o r k , 3 b o il, 4 a te , 5 h a t. . ( ( 3 4 . O f t h e f iv e w o r d s b e lo w , f o u r a r e a li k e i n a c e r t a i n w a y . 1 p ush, 2 h o ld , 3 lift, 4 d ra g , 5 p u l l . ‘ W h ic h o n e is n o t l ik e t h e s e f o u r ? . . . . . . . ( 35 . I f 10 b o x e s f u l l o f a p p l e s w e ig h 3 0 0 p o u n d s a n d e a c h b o x w h e n e m p t y w e ig h s 3 p o u n d s , h o w m a n y p o u n d s d o a l l t h e a p p l e s w e ig h ? ............................ . . . . . . . ( 3 6 . T h e o p p o s i t e o f s o r r o w is ( ? ) 1f u n , 2 s u c c e s s , 3 h o p e ,4 p r o s p e r ity , ( 37. 5 j oy. . . ....................................... . . . . I f a ll t h e o d d - n u m b e r e d l e t t e r s in t h e a l p h a b e t w e re c r o s s e d o u t , w h a t w o u l d b e t h e t w e l f t h l e t t e r n o t c r o s s e d o u t ? P r i n t i t . Do not mark the alphabet. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ..................................... ( 3 8 . W h a t l e t t e r i n t h e w o r d u n f o r t u n a t e l y is t h e s a m e n u m b e r i n t h e w o r d ( c o u n ti n g f r o m t h e b e g in n i n g ) a s i t is i n t h e a l p h a b e t ? P r i n t i t . ( 39. 40. S u c h t r a i t s a s h o n e s t y , s i n c e r i t y , a n d l o y a l t y c o n s t i t u t e o n e ’s ( ? ) 1 p e rs o n a lity , 2 r e p u ta tio n , 3 w is d o m , 4 c h a r a c t e r , 5 su ccess. I f 3 ^ y a r d s o f c l o t h c o s t 25 c e n t s , w h a t w ill 10 y a r d s c o s t ? ( ....................................... ... . .... ( 4 1 . I f t h e w o r d s b e lo w w e r e a r r a n g e d t o m a k e a g o o d s e n t e n c e , w i t h w h a t l e t t e r w o u ld t h e s e c o n d w o r d o f t h e s e n t e n c e b e g in ? M a k e i t lik e a p r i n t e d c a p i t a l . s a m e m e a n s s m a ll l i t t l e th e a s . . . 42. 43. I f t h e f i r s t t w o s t a t e m e n t s fo llo w in g a r e t r u e , t h e t h i r d is ( ? ) G e o r g e is y o u n g e r t h a n F r a n k . J a m e s is y o u n g e r t h a n G e o r g e . 1 tru e , 2 f a ls e , 3 n o t c e r t a i n . . . F r a n k is o l d e r t h a n J a m e s . . . . . . . ( S u p p o s e t h a t t h e f i r s t a n d s e c o n d l e t t e r s i n t h e w o r d a b o l i t i o n i s t w e re i n t e r c h a n g e d , a ls o t h e t h i r d a n d f o u r t h l e t t e r s , t h e f i f t h a n d s i x t h , e tc . P r i n t t h e l e t t e r t h a t w o u l d b e t h e t e n t h l e t t e r c o u n tin g to th e r i g h t. . , •. ( 4 4 . O n e n u m b e r is w r o n g i n t h e fo llo w in g s e rie s . 0 1 3 6 10 15 21 45. W h a t s h o u ld t h a t n u m b e r b e ? 29 36. I f 3 -| y a r d s o f c l o t h c o s t 70 c e n t s , w h a t w ill 4 I y a r d s c o s t ? . . . ( .. . ( . , . . . . , . ( 4 6 . A p e r s o n w h o n e v e r p r e t e n d s t o b e a n y t h i n g o t h e r t h a n w h a t h e is, is s a i d t o b e ( ? ) 1 lo y a l, 2 h y p o c r itic a l, 3 s in c e r e , 4 m e e k , 5 c o u rag e o u s 47. W h ic h o f t h e s e w o r d s is r e l a t e d t o m a n y a s e x c e p ti o n a l is t o o r d i n a r y ? 1 none, 2 each, 3 m o re, 4 m u ch , 5 fe w ..................................................................................................... 4 8 . T h e o p p o s i t e o f c o w a r d l y is ( ? ) 1 b ra v e, 2 s tr o n g , 3 tre a c h e ro u s , 4 lo y a l, 5 frie n d ly . ( I f t h e f i r s t t w o s t a t e m e n t s fo llo w in g a r e t r u e , t h e t h i r d is ( ? ) S o m e o f B r o w n ’s f r i e n d s a r e C a th o l i c s . S o m e o f B r o w n ’s f r i e n d s a r e la w y e r s . Som e of B r o w n ’s f r i e n d s a r e C a th o l i c la w y e r s . 1 tru e , 2 f a ls e , 3 n o t c e rta in . ( 51. H o w m a n y o f t h e f o llo w in g w o r d s c a n b e m a d e f r o m t h e l e t t e r s i n t h e w o r d a n y l e t t e r a n y n u m b e r o f t im e s ? g re a te s t, t a n g l e , g a r a g e , s tr e s s e s , 52. ( ( 4 9 . W h ic h o n e o f t h e fiv e w o r d s b e lo w is m o s t u n l ik e t h e o t h e r f o u r ? 1 fa s t, 2 a g ile , 3 q u i c k , 4 r u n , 5 sp eed y . 50. ( r e la te d ,g re a se , T o i n s i s t t h a t t r e e s c a n t a l k t o o n e a n o t h e r is ( ? ) 1 a b su rd , 2 m is le a d in g , 3 im p r o b a b l e , 4 u n f a i r , [ ,i n e a re s t, re e fin g stra ng le, . . . . 5 w ic k e d .............................................. . Do not stop. u sin g ........ Go on with the next page. ( ( S. A. Higher 53. O f t h e t h in g s f o llo w in g , f o u r a r e a l i k e i n a c e r t a i n w a y . W h i c h o n e is n o t l ik e t h e s e f o u r ? 1 sn o w , 2 s o o t, 3 c o tto n , 4 iv o ry , 5 m ilk . . . . . . . . . . . . . . 54. A s q u a r e is r e l a t e d t o a c ir c le i n t h e s a m e w a y i n w h i c h a p y r a m i d is r e l a t e d t o ( ? ) ............................. 1 a s o lid , 2 E g y p t, 3 h e ig h t, 4 a c o n e , 5 a c ir c u m f e r e n c e 55. ( ( I f t h e f o llo w in g w o r d s w e r e s e e n o n a w a l l b y l o o k in g i n a m i r r o r o n t h e o p p o s i t e w a ll, w h i c h w o r d w o u l d a p p e a r e x a c t l y t h e s a m e a s if s e e n d i r e c t l y ? I MEET, 2 ROTOR, 3 MAMA, 4 DEED, 5 TOOT. . ................................ ( I f a s t r i p o f c l o t h 32 in c h e s l o n g w ill s h r i n k t o 28 i n c h e s w h e n w a s h e d , h o w m a n y in c h e s l o n g w ill a 2 4 -in c h s t r i p o f t h e s a m e c l o t h b e a f t e r s h r i n k i n g ? . . . . . . . . . . . ............................. ( W h i c h o f t h e f o llo w in g is a t r a i t o f c h a r a c t e r ? 1 a b ility , 2 r e p u ta tio n , 3 h a te , 4 s t i n g in e s s , .. .. ( F i n d t h e tw o l e t t e r s i n t h e w o r d c o m in g w h ic h h a v e j u s t a s m a n y l e t t e r s b e t w e e n t h e m i n t h e w o rd a s in th e a lp h a b e t. P r i n t th e o n e o f th e s e l e tte r s t h a t c o m e s f irs t in th e a lp h a b e t ( 59 . M o d e r n is t o a n c i e n t a s ( ? ) is t o y e s t e r d a y . 1 to m o r r o w , 2 tim e , 3 u p - t o - d a t e , 4 h is to ry , ( 60. O n e n u m b e r is w r o n g i n t h e f o llo w in g s e rie s . 1 2 4 8 16 32 64 61 . I f G e o r g e c a n r i d e a b i c y c l e 4 0 f e e t w h ile F r a n k r u n s 3 0 f e e t , h o w f a r c a n G e o r g e r i d e w h ile F r a n k ru n s 45 f e e t? . ............................... ... ................................ . . . ................ ( C o u n t e a c h L i n t h i s s e r ie s t h a t is f o llo w e d b y a U n e x t t o i t if t h e U is n o t f o llo w e d b y a n R n e x t t o i t . T e l l h o w m a n y L ’s y o u c o u n t . L U L R V E L U R E U L U U L V E L L U V L U R U L O E V L U E . ( 56. 57. 58. 62. 5 n e r v o u s n e s s ..................... 5 t o d a y ............................. .... . W h a t s h o u ld t h a t n u m b e r b e ? 96. 6 3 . A m a n w h o is i n f a v o r o f m a r k e d c h a n g e is s a i d t o b e ( ? ) 1 d e m o c ra tic , 2 c o n s e rv a tiv e , 3 r a d ic a l, 4 a n a r c h is tic , 5 r e p u b lic a n . ( . . .. . . . 64. P r i n t t h e l e t t e r w h i c h is t h e f o u r t h l e t t e r t o t h e l e f t o f t h e l e t t e r m i d w a y b e t w e e n N R i n t h e a l p h a b e t ................................ . . . 65. W h a t n u m b e r is i n t h e s p a c e w h ic h is in t h e r e c t a n g l e b u t n o t i n t h e t r i a n g l e o r i n t h e c i r c l e ? 66. What number is in the same geometrical figure or figures (and no other) as the number 3 ?.. . How many spaces are there that are in any one but only one geometrical figure? ... A line is related to a surface as a point is to a (?) 1 circle, 2 line, 3 solid, 4 dot, 5 intersection. If the first two statements are true, the third is (?) One cannot become a good lawyer without diligent study. George studies law diligently. George will become a good lawyer. 1 true, 2 false, 3 not certain. If the words below are arranged to make the best sentence, with what letter will the last word of the sentence end? Print the letter as a capital. honesty traits Generosity character of desirable and are. A man who carefully considers all available information before making a decision is said to be (?) 1 influential, 2 prejudiced, 3 decisive, 4 hypocritical, 5impartial. . . A hotel serves a mixture of 2 parts cream and 3 parts milk. How manypints of milk will it take to make 25 pints of the mixture?. . . . What is related to stars as physiology is to blood? 1 telescope, 2 darkness, 3 astronomy, 4 light waves, 5 chemistry. A statement based upon a supposition is said to be (?) 1 erroneous, 2 ambiguous, 3 distorted, 4 hypothetical, 5 doubtful. If a wire 4 0 inches long is to be cut so that one piece is f as long as the other piece, how many inches long must the shorter piece be ?. I f you finish before the time is up, go back and make sure that every answer is right. 67. 68. 69. 70. 71. 72. 73. 74. 75- [4] ( and ( ( 398 ORLEANS GEOUEJEY PROGNOSIS TEST ORLEANS GEOMETRY PROGNOSIS TEST By Jo sep h B. O r le a n s , A.M. Chairman of the Mathematics Department George Washington High School, New York City and Jacob S. O r le a n s , P h .D. Formerly of the Educational Measurements Bureau New York State Department of Education EXAMINATION: FORM A For High Schools and Colleges Do not open this booklet, or turn it over, until you are told to do so. Fill in these blanks, giving your name, age, etc. Write plainly. N a m e .............. (First name, Age last birthday initial, and last name) .years. Date ...... Grade or class. Teacher. School. City Have you ever studied geometry before ?. If so, how long ?. This is a test to see whether you can learn geometry easily. It contains a number of lessons in geometry, each followed by a test to see what you have learned in the lesson. You will be given a certain time to study each lesson, then a cer tain time to take the test. Study each lesson carefully and try to find out what it teaches, so that you can pass the test on it. Give your complete attention to your work so that you will not waste any time. If you finish any lesson or test before the time for it is up, go back over it until you are sure you have learned the lesson correctly, or have done all the examples in the test correctly. Do not turn to the following lesson or test until you are told to. Ask no questions after the test begins. Do not turn the page yet. 19 T est S cohe 1 2 3 4 5 6 7 8 9 10 T otal Published by World Book Company, Yonkers-on-Hudson, New York, and Chicago, Illinois Copyright 1929 by World Book Company. Copyright in Great Britain A ll righ ts reserved. OGPT: a~ 5 T h is t e s t is c o p y rig h te d . T h e rep ro d u c tio n o f an y p a r i o f it b y m im eo g ra p h , h ectograph , or in a n y o th e r w a y , w h e th e r th e r e p ro d u c tio n s a re s o ld or are fu r n is h e d f r e e f o r u se , is a v iolation o f th e c o p yrig h t law . Geometry Prognosis: A LESSON 1 D ir e c tio n s . Study this lesson very carefully, so that you will know how to do the test on the next page. Read the following statements carefu y an be sure that you understand what they mean. (1) If things that are equal are added to things that are equal, the sums are equal. For example: If two bags of flour have the same weight and 5 pounds of flour are added to each bag, their weights will still be equal. If John is as old as Tom, their ages will again be equal at the end of the next 10 years. Ifa — b, then (adding 5 to both a and b) a + 5 = b + 5. (You know from algebra that letters are used to represent numbers.) (2) Ifthings that are equal are subtracted from things thatare equal, the remainders are equal. For example : If each of two loaves of bread weighs 8 pounds and a 2-pound piece is cut from each loaf, the remaining parts will be equal in weight. (3) Doubles of equals are equal. For example: If the cost of a 5-pound bag of sugar equals 35 cents, the cost of a bag con taining twice as much sugar, or 10 pounds, will equal twice as much, or 70 cents. (4) Halves of equals are equal. For example: The line A B (from the point A to the point B )isas long as the line C D (from the point C to the point D ). Then AR, which is one half of the line AB> is equal to CS, which is one half of the line CD. (See note below.) A ________ R ________ B C________ S ________ D (5) Things equal to the same thing are equal to each other. For example: If Mary isas old as Ann and Sarah isas old as Ann, then Mary and Sarah are equal in age. If the length of one stick is 3 feet and the length of a second stick is also 3 feet, then the two sticks are equal in length. NOTE. In geometry it is customary to name the ends of a line by means of letters. Thus, in Figure 1 line K L means the k l length of the line from the point K to the point L. ' ' In Figure 2 line M N means the part of the line from point M to point N . N ____ £ Line M P means the whole line from point M to point P. Fig'2 Geometry Prognosis: A TEST 1 Read each of the following items. Then, in the parentheses after each item, write the number of the statement in Lesson 1 to which it refers. For example, the first question illustrates Statement 4 on the preceding page. You should, therefore, write 4 in the parentheses for Question 1 . D ir e c tio n s . Ansu' crs 1. If a = b, then \ a = \ b. ( 2. If k — m, then k — 4 = m — 4. )1 )2 3. If r = s and r )3 then s = v. 4. If x = y, then x + 2 = y + 2. 5. If c = d, then 2 c = 2d. — v, )4 )s 6. The two lines R L and F N are equal. If R M istwice as long as RL, and F P istwice as long as FN, then the line R M has the same length as the line FP. R M K N 7. If the line GH has the same length as the line LM , and the line J K has the same length as the line LM , then line GH equals line JK . G 8. L______ M H )6 )7 J _____ K If the line A B is extended f inch to the right to point C and f inch to the left to point D, then the distance from point A to point C is the same as the distance from B to D. D_______ A____ £_______ C 9. The lines AC and ac have the same length. If piece BC is removed from one end of line AC and a piece just as long (be) is removed from the end of line ac, then the remaining parts ( A B and ab) are equal. A B C a b 10. If the two oblongs are equal, then part A equals part B. 11. If part A equals part B, then the first circle equals the second circle. □ □ )9 e )10 )ll 12. If the square marked A equals the square marked C, then A + B equals B + C. 13. If A is 4 times as large as B, and C is 4 times as large as B, then A has the same area as C. )12 ) 13 14. The surface of the whole square equals the surface of the whole circle. If part A of the square equals part B of the circle, then part C equals part D. 15. If part A in the square equals part a in the oblong, and part B equals part b, then the square equals the oblong. ) 14 ) 15 N um ber right (Score, Test 1 ) Geometry Prognosis: A LESSON 2 D ir e c tio n s . Study this lesson very carefully so that you will know how to do the test on the next page. Read the following statements carefully and _____________ be sure you understand what they mean. ( 1 ) If a line is held fast at one end and turned around that end, then the amount the line is turned from its old to its new position is called an angle. Thus, if line A B in Figure 1 (with the end A held fast) is turned to the new position AC, then the amount that the line has been turned from A B to A C is called angle B A C or angle CAB. The same angle would F ig. 1 be formed if the line were at A C and turned to the new position AB. (A curved arrow is used to show the amount the line is turned.) In Figure 2 the angle is R S T or TSR, meaning that the line R S was rotated to the new position TS, or that the line TS was rotated to the new position RS. In Figure 3 the angle is M H P or P H M , meaning that the line M H was rotated to the new position PH, or that the line P H was rotated to the new position M H . F ig . 2 NOTE that in naming the angle, the letter at the point where the lines meet is placed between F ig. 3 the letters that refer to the other ends of the lines. (2 ) Look at Figure 4. You see that the two lines A B and D C cross at E. Four angles are form ed: angle A E D or D E A (with 1 in i t ) ; angle D E B or B E D (with 2 in i t ) ; angle B E C or C E B (with 3 in i t ) ; and angle C E A or A E C (with 4 in it). You may think of angle C E B as being formed by the rotation of line E B to the position E C (or of line E C to the position EB). In the same way you may think of angle C E A as being formed by the rotation of the line from E A to E C (or from E C to EA). Angle A E D would be formed by the rotation of the line from F ig . 4 E A to E D (or from E D to EA), and angle D E B by the rotation of the line from E D to E B (or from E B to ED). You may also think of the angles B E C and A E D being formed at the same time by the rotation of the line A B around to the new position D C about the point E, which is kept fixed. In the same way you may also think of angles A E C and D E B being formed at the same time by the further rotation of the line A B around to the position D C about the point E, which is kept fixed. (3) An angle is formed, therefore, when two lines meet or cross each other, Thus, in Figure 5 you see several lines that meet or cross, and a number of angles are formed at the points marked E, F ,G, K, P, and T. For example: angle E F T or T F E (with 1 in it) angle E P T or T P E (with 2 in it) angle T G K or K G T (with 3 in it) angle K E G or G E K (with 4 in it) angle F E G or G E F (with 5 in it), etc. N O T E that angle F E K is made up of angles F E G and GEK. E F P and P F K together make up angle EFK. \ 4 1 angles Fig. 5 In the same way Geometry Prognosis: A TEST 2 In the figure you see a collec tion of angles. The ends of the lines are marked with capital letters. In some of the angles small numbers have been inserted. D ir e c tio n s . In the parentheses after each question, write the answer to that question. You may refer back to Lesson 2 if you need to. A n sw ers 1. Name the angle which contains the number 1 . 2. Name the angle which contains the number 2 . 3. Name the angle which contains the number 3 . 4. Name the angle which contains the number 5 . 5. Name the angle which contains the number 8 . 6. What number does angle D E A contain ? 7. What number does angle EAC contain? 8. What number does angle ECU contain? 9. Name an angle which does not contain any number. 10. Name another angle which.does not contain any number. 11. Name the angle that is made up of angles 5 and 8 . 12. Angle D A C is made up of what two angles? 13. Angle A E D is formed by the rotation of line A B to the new position CD. What other angle is formed at the same time? 14. What angle is made up of angles A DC and CDB ? 15. Insert, in the figure, the number 15 in angle EC A. N u m ber right. )1 )2 )3 )4 )5 )G ;7 ) 8 ) 9 )io ■ ) ) 13 , 14 . (Score, T est 2) LESSON 3 ir e c t io n s . Study this lesson very carefully so that you will know how to do the test on the next page. D (1 ) Just as lines are measured by means of a unit of length (e.g., an inch), so angles are measured by means of degrees. An angle is said to have a certain number of degrees. You will have occasion to read about angles of 30 degrees, 60 degrees, 4 5 degrees, 90 degrees, 180 degrees, and others. (These may also be written 30°, 60°, 45°, 90°, 180°.) (2 ) If the line A B is held fast at the end A in the figure and is turned through a complete circle, the angle formed has S60J. (3 ) An angle whicn is one fourth of a complete turn (or rota tion) has 90° and is called a right angle (like angle BAC). (4 ) An angle which is one half of a complete turn has 180° and is called a straight angle (like angle BAD). ' (5 ) An angle which is less than a fourth of a complete turn has less than 90° and is called an acute angle (like angle BAE). (6 ) An angle which is more than a fourth but less than half of a complete turn has between 90° and 180° and is called an obtuse angle (like angle BAF). f5 1 11 12 Geometry P rognosis: A TEST 3 Directions. Y o u may look back to Lesson 3 if you need to. In the paren theses after each of the following angles, write what kind of an angle it is. Use the letter r for right angle ac for acute angle s for straight angle ob for obtuse angle 1. Angle of 45c ( )i 2. Angle of 72° ( )2 3. Angle of 180c 4. Angle of 115c ( ( 7. Angle of 123° ( ( ( )12 13. ( ) 13 14. ( ) 14 15. ( )15 16. ( ) 16 17. ( ) 17 18. ( ) 18 ( ) 19 ( )20 )3 )4 5. Angle of 90c 6. 12. ) 6 )7 8. 9. 10. 19. ( )10 ( )H 20. N u m b e r r ig h t ...... (Score, T est 3) Geometry Prognosis: A LESSON 4 Study this lesson very carefully so that you will know how to do the test on the next page. D ir e c tio n s . ( 1 ) If the sum of two angles is a right angle, tary angles. For example: Angles of 70° and 2 0 ° are complementary Angles of 35° and 55° are complementary Angles of 63° and 27° are complementary or 90°, they are called complemen (because 70° + 2 0 ° = 90°). (because 35° + 55° = 90°). (because 63° + 27° = 90°). How would you find the complement of 24° ? Answer. Subtract 24° from 90°. The complement of 24° is 6 6 °. M D Figure 1 is an illustration of two complementary angles, numbered 1 and 2 , forming together the right angle CRM (or MRC). F ig . 1 (2 ) If the sum of two angles is a straight angle, or 180°, they are called supple mentary angles. For example: Angles of 140° and 40° are supplementary (because 140° + 40° = 180°). Angles of 25° and 155° are supplementary (because 25° -f- 155° = 180°). Angles of 62° and 118° are supplementary (because 62° + 118° = 180°). How would you find the supplement of 130° ? Answer. Subtract 130° from 180°. The supplement of 130° is 50°. N E Figure 2 is an illustration of two supplementary angles, numbered 1 and 2 , together forming the straight angle D GED (or DEG). F ig . 2 Figure 3 also shows two supplementary angles, num bered 3 and 4, together forming the straight angle 1 K D S (or SDK). K F ig . 3 B I Figure 4 also shows two supplementary angles, numbered w 5 and 6 , together forming the straight angle E B V (or VBE). F ig . 4 [ 7 ] Geometry Prognosis: A TEST 4 In the parentheses after each question, write the answer to that You may refer back to Lesson 4 if you need to. Answers D ir e c tio n s . question. 1. What is the complement of 60° ? ( ) 2. What is the complement of 23° ? ( ) 2 3. What is the supplement of 120° ? ( ) 3 4. What is the supplement of 147° ? ( ) 4 3. What is the supplement of 39° ? ( )5 6. What is the complement of 78° ? ( ) 6 7. What is the complement of 45°? ( 17 8. ( ) What is the supplement of 2 ° ? 9. ________J ^"3 4 10. Name two more angles n Figure supplementary. Fig‘ 1 X Fig- Name two angles in Figure ( mentary. 2 c | B l l . Name an angle in Figure plement of angle 8 . 2 12. Name an angle in Figure plement of angle 7. 2 13. Name an angle in Figure plement of angle 6 . 2 1 1 1 8 that are supple )9 that are ( ) ( )n ( ) ( ) 13 ( ) 14 ( ) 15 10 that is the sup that is the sup 12 that is the sup- 14. Name another angle in Figure 2 that is the supplement of angle 6. 10 n ^ — ~1---------------- -A Fig. 3 15. Name an angle in Figure plement of angle 4. 16. Name another angle in Figure 17. Name an angle in Figure 18. Name 1 1 1 that is the sup- that is the supplement of angle that is the supplement of angle another angle in Figure 1 1. 4 .( ( 1 .( 2, angle 8 contains 130°. ( ) 19 ( ) 20 How many degrees are there in angle 5 ? 21. How many degrees are there in angle ) 18 If angle 9 equals 30°, how many degrees are there in angle 10 ? 20. In Figure ) 17 that is the supplement of angle 19. Angle A D C in Figure 3 is a right angle. ) 16 7 ? [8 ] N u m b e r r i g h t. ( ) 21 (Score, T e st 4) Geometry Prognosis: A LESSON 5 Study this lesson very carefully, so that you will know how to do the test on the next page. D ir e c tio n s . (1 ) F i g u r e 1 is c a l l e d B F ig . 1 a triangle. It is named triangle ABC. It has three sides, called A B , BC, and CA, which may or may not be equal in length. It also has three angles, numbered 1 , 2 , and 3. They may or may not be equal; that is, they may or may not have the same number of degrees. Each angle is said to be opposite a side, and each side is said to be opposite an angle. For example, side A B is opposite angle 3, and angle 2 is opposite side AC. Each angle is said to be included by two sides, and each side is said to be in cluded by two angles. For example, angle 1 is included by the sides A B and AC, and side BC is included by angles 2 and 3. (2 ) Figure 2 is called a square. It is read R S T V . It has four equal sides, read RS, ST, TV, and VR. It also has four angles which are right angles. Draw a line from S to V. This line is called a diagonal. Draw a line from R to T. This line is also called a diagonal. F ig . 2 (3 ) In Figure 3 the two lines cross, forming four angles marked 1 , 2, 3, 4. The two angles that are opposite each other are called a pair of vertical angles. Thus, angles 1 and 3 are one pair of vertical angles, and angles 2 and 4 are another pair of vertical angles. The vertical angles of a pair may be shown to be equal; that is, angle 1 has the same number of degrees as angle 3, and angle 2 has the same number of degrees as angle 4. F ig . 3 Geometry Prognosis: A TEST 5 This test is on two pages. In the parentheses after each question, write the answer to that question. You may refer to Lesson 5 if you need to. D ir e c tio n s . Answers ( )i ( )2 In Figure 1 , what line is opposite angle 2? 2 . In Figure 1 , what angle is opposite line S T ? 3. In Figure 1 , what angle is included between lines R T and TS? 4. In Figure 1 , line S T is included between what angles ? 1. F ig. 1 ( )3 ( )4 In Figure 2, it is true that if two sides are equal, the angles opposite those sides are equal. M 5. If side M N equals side MO, which are the equal angles ? 6 . If side M N equals side NO, which are the equal angles ? F ig . 2 )5 )« In Figure 3, it is true that if two angles are equal the sides opposite the angles are equal. L 7. If angle 7 equals angle 8 , which are the equal sides ? 8 . If angle 8 equals angle 9, which are the equal sides P H )7 F ig . 3 9. The expression two sides and the included angle is used with refer ence to Figure 3. If L H is one of the sides referred to and angle 8 is the angle, which is the other of the two sides ? )9 10. The expression two angles and the included side is used with refer ence to Figure 3. If angles 9 and 7 are the angles referred to, which is the included side ? 11. If angle 8 is one of the angles and FIK is the included side, which is the other angle ? In Figure 4, K L M N is a square. square are equal. F ig . 4 M )10 ( It is true that the diagonals of a 12. Which lines in the figure are equal for that reason ? ( 13. There are two pairs of equal angles in Figure 4 (other than the angles at the corners of the square). Name a pair of angles that are equal. ( 14. )H Name another pair of angles that are equal. ( ) 12 ) 13 ) 14 Go right on w ith th e n ext p a g e , w here T est 5 is con tin u ed. 1 10 1 Geometry Prognosis : A In Figure 5, the angles of the triangle have the numbers of degrees indicated. It is true that the side opposite the angle of 30° is equal to one half the side included by the angles of 30° and 60°. 15. Write the correct letters in each of the fol lowing parentheses : F ig . 5 Line ( ) equals one half of line ( )15 Figure 6 is a rectangle. It is like a square except that instead of all four sides being equal, only the opposite sides are equal. B c 16. Name a pair of sides that are equal. 17. Name two sides that are not equal. 18. What kind of angles does a rectangle have ? ( ( ( ) ig ) 17 ) 18 F ig . 6 Figure 7 is a rhombus. It is like a square except that the angles are not right angles, but the opposite angles are equal. 19. How many sides of the rhombus are equal in length ? ( 20. Name a pair of equal angles in Figure 7. ( 21. Name another pair of equal angles. ( )19 )20 )21 F ig . 7 N um ber right [ 11 ' (Score, Test 5) Geometry Prognosis: A LESSON 6 Study this lesson very carefully, so that you will know how to do the test on the next page. D ir e c tio n s . (1) To bisect means to divide into two equal parts. Thus, in the line A B (Fig, 1) B point 0 is the midpoint; that is, the distance from A to 0 is the same as from 0 to B or the line AO equals the line OB, Therefore any line Pig. i that would cross the line A B at point 0 would bisect the line A B . (2) If the two lines that cross each other in Figure 2 are each divided into two equal parts at the same time, they are said to bisect ^ each other. Line R S isbisected at point T and at the same time K L is bisected at the point T. Then K L bisects RS, and R S bisects K L \ that is, K L and R S bisect each other. Therefore there are two pairs of equal lines in the figure; namely, R T = TS and K T = TL. ( 3) I n th e a n g le ABC (F ig . 3) t h e l i n e B D m a k e s t h e t w o a n g l e s ( n u m b e r e d 1 a n d 2 ) equal; that is, angle 1 and angle 2 have the same number of degrees. Therefore B D is said to bisect angle A B C . B F ig . 3 [ 12 ] Geometry Prognosis: A TEST 6 In the parentheses after each question, write the answer to th at You may refer back to Lesson 6 if you need to. D ir e c tio n s . question. Answers V P w F ig . 1 In Figure 2, c 1. In Figure 1 , line V W bisects line PT. What two lines are therefore equal? )1 2. In Figure 1 , line P T bisects line VW. What other two lines are therefore equal ? )2 CD bisects angle ACB. CD also bisects the side opposite angle C. 3. Which angles do you know are equal ? )3 4. Which lines do you know are equal ? )4 F ig . 2 In the square in Figure 3, the diagonal K H bisects angle GHJ and also angle GKJ. 5. Name one pair of angles that are there fore equal. 6 . Name another pair of angles that are therefore equal. ) 5 )6 F ig . 3 Figure 4 contains two diagonals. (Diagonal in this figure means the same as diagonal in a square.) It is true that the diagonals bi sect each other. R^ 7. Name one pair oflines that arethere^ fore equal. 8 . Name another pair of lines that are F ig . 4 therefore equal. Figure 5 is a circle. A. Point A in the circle is called the center. line CD is bisected at the center. The 9. Name the lines that are equal. F ig . In Figure 0, the line GK bisects two sides of the triangle. 10. Name one pair of equal lines in the figure. 11. Name another pair of equal lines. 12. HF ig . 6 )io ) ll It is true that GK equals one half of the side that it does not bisect. Which line is twice as long as GK? )12 In Figure 7, lines A B and CD cross at E. Line FE bisects angle A EC. D 13 N ame two angles that are therefore equal. 14. Continue line FE through E to point G and you have bisected angle DEB. Name the two angles that are therefore equal. F ig . 7 13 N um ber right. 13 ( )14 (Score, Test 6) Geometry Prognosis: A LESSON 7 Study this lesson very carefully, so that you will know how to do the test on the next page. D ir e c tio n s . ( 1 ) It is customary to indicate that two lines in a figure are equal by marking them in the same way. The same is done with angles c F that are equal. Thus Figure 1 shows that angle B in triangle A B C is equal to angle E in triangle / DEF. (They are both marked with one cross b ^ -VL line.) C D Fig. 2 In Figure 3 line A B = line D E line B C = line E F line A C = line DF Figure 2 shows that line A B is equal to line D E , since each is marked with one short lin e ; and line AC equals line DF, since each is marked with two short lines. angle B A C = angle E D F angle A C B = angle D F E angle C B A = angle F E D B j £z c F ig. 3 (2 ) Figure 4 shows two triangles in which two sides and the included angle in one C F triangle are equal respectively to two sides and the included angle in the other triangle. That is, sides A B and B C in the first are respectively equal to sides E D and E F in 3 E the second (as shown by the way they are F ig . 4 marked), and angle B equals angle E (as shown by the way they are marked). Figure 5 shows a parallelogram in which the diagonals bisect each other. To show that the diagonals are bisected, the two parts of each diagonal are marked in the same way, showing that they are equal. F ig . 5 NOTE that when two triangles are being compared, they may be separated from one another as in Figure 6 , or they may touch in various ways as in Figures 7 and 8 . Note that when they touch as in Figure 8 , one of the sides ( R T in this figure) belongs to both triangles at the same time. H K J F ig. 6 F ig. 7 [ 14 1 Geometry Prognosis: A TEST 7 Each of the following statements is illustrated by one or more of the diagrams below. (Consider carefully the way in which the lines and angles are marked.) In the parentheses after each diagram write the letter of the statement which it illustrates. For example, Statement a is, Two straight lines crossing each other and making opposite angles that are equal. The first diagram does not illustrate this statement, but the second diagram does. So write the letter a in the parentheses after the second diagram. Look at each of the other diagrams to see if it also illustrates this first statement and put the letter a in the parentheses after each one that does. Then do the same with statement b, and then with each of the other statements. D ir e c tio n s . а. Two straightlinescrossing each other and making opposite angles thatare equal. б. Two straight lines crossing at right angles. c. Two triangles in which two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. d. Two triangles in which two angles and the included side of one triangle are equal to two angles and the included side of the other triangle. e. Two triangles in which the three sides of one triangle are equal to the three sides of the other triangle. /. Two triangles in which two angles and the side opposite one of them of one triangle are equal to two angles and the side opposite one of them of the other triangle. N um ber rig h t ...... (Score, Test 7) [ 15 ] Geometry Prognosis: A LESSON 8 D ir e c tio n s . Study this lesson very carefully, so that you will know how to do the test on the following page. The statement / 4 If iron is heated, it expands,” is made up of two p arts: the conditional part, “ if iron is heated,” and the conclusion, “ it expands.” The same idea is expressed in the words, “ Iron which is heated expands.” Here the condition is implied in the words “ which is heated,” and the conclusion is “ iron. . .expands.” The statement, “ Water that is very cold freezes,” is made up of a conditional part, “ that is very cold,” and a conclusion, “ water. .freezes.” The statement, “ If two lines are perpendicular to each other, the angles they form are equal,” is made up of the conditional part, “ If two lines are perpendicular to each other,” and the conclusion, “ the angles they form are equal.” The same idea is expressed in the statement, “ The angles formed by perpendicu lar lines are equal.” The conditional part is “ formed by perpendicular lines,” and the conclusion is “ the angles. .are equal.” TEST 8 D ir e c tio n s . Draw one line under the conditional part only in each of the following statements. You may look back to Lesson 8 if you need to. 1. If he is well, he will come. 2. He will earn a great deal provided he works hard. 3. If two sides of a triangle are equal, the angles opposite those sides are equal. 4. Vertical angles are equal. 5. The diagonals are equal in a square. 6. Two lines that are parallel are everywhere equidistant. 7. The three angles are equal in a triangle which has three equal sides. 8. The sum of the three angles of a triangle is 180°. 9. From a point without a line there can be only one perpendicular drawn to that line. 10. Two angles whose corresponding sides are parallel and extend in the same direc tion are equal. 11. Two straight lines which are parallel to a third straight line are parallel to each other. 12. The opposite angles are equal in a parallelogram. 13. Parallel lines are equal that are included between parallel lines. 14. A line which joins the midpoints of two sides of a triangle is parallel to the third side and equal to half of it. N u m ber riqht [ 16 ] (Score, T est S) Geometry Prognosis: A LESSON 9 ir e c t io n s . Study this lesson very carefully, so that you will know to do the problems on the following two pages. D how A. First read and learn the following facts : a. If a side of one triangle isthe same line as the side of another triangle, then these two sides are equal. b. If a triangle has two equal sides, itiscalled an isosceles triangle. Therefore in any isosceles triangle two of the sides are equal. c. If a line bisects an angle, itdivides the angle into two equal angles. d. If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are equal. B. Now study the following problem and the way in which it is worked. If you understand it,you will be able to do the problems on the next page. ^ In the accompanying figure, the following facts are true : Triangle M N P is an isosceles triangle. Line M R bisects angle N M P . Two triangles are formed; namely, triangle M R N and triangle M R P . Problem. To show that triangle M R N is equal to triangle MRP. Four statements about the above figure are given in the first column below. In the parentheses after each statement, you are to write the letter that is in front of the reason (at the top part of the page) that tells why the statement is true. For example, the first statement below, “ Line M N equals line M P is true because an isosceles triangle has two equal sides. This is the second reason at the top of the page, with the letter b in front of it. So a letter b is written in the paren theses after the first statement. The second statement, “ Angle 1 equals angle 2,” is true because of the third reason given at the top of the page. So a letter c is written in the parentheses after the second statement below. The third statement, “ Line M R in triangle M R N equals line M R in triangle M R P ,” is true because of the first reason given at the top of the page. So write a letter a in the parentheses after the third statement. The last statement is true because you have shown that two sides and the in cluded angle in triangle M R N are equal to two sides and the included angle in tri angle M R P . This is the last reason given at the top of the page; so write a letter d in the parentheses after the fourth statement. Statem ents 1. Line M N equals line M P. Why ? 2. A n g l e 1 equals angle 2. Why? 3. Line M R in triangle M R N equals lineM R intriangle M R P . 4. Therefore triangle M R N equals triangleM R P . Why? R ea so n s Why ? ( ( ( ( b c )i )2 )3 )4 Study this lesson very carefully. You must understand it and how to write your answers, if you are to be able to do the test on the next two pages. [ 17 ] Geometry Prognosis: A TEST 9 D ir e c tio n s . This test is on two pages. You may look back to Lesson 9 if you need to. P r o b le m I. Read the following problem carefully. In the parentheses after each statement, write the letter of the fact (given below) that tells the reason why the statement is true. In the figure D B and A C are perpendicular to each other. Point B is the midpoint of AC. Problem, To show that triangle A B D is equal to triangle CBD. R S ta te m e n ts 1. D B in triangle A B D equals D B in triangle CBD. 2. Angle 1 and angle 2 are right angles. Why? 3. Angle 1 equals angle 2. Why ? 4. Line A B equals line BC. Why? 5. Triangle A B D equals triangle CBD. Why? Why? ea so n s )l )2 )3 )4 )5 a. If two lines are perpendicular to each other, they form right angles. b. If a sideof one triangle isthe same line as the side of another triangle, then these two sides are equal. c. If a line bisects an angle, it divides the angle into two equal angles. d. The midpoint of a line divides the line into two equal parts. e. All right angles are equal. /• If two of the sides and the angle between them in one triangle are equal to two of the sides and the angle between them in another triangle, then the triangles are equal. Vertical angles are equal. P r o b le m A E II. Read the following problem carefully. In the parentheses after each statement, write the letter of the fact (given at the top of the next page) that tellsthe reason why the state ment is true. In the figure A D bisects B E . B E bisects A D . Problem. To show that triangle A C B equals tri angle ECD. S tatem ents R 6. Angle 3 and angle 4 are vertical angles. Why ? 7. Angle 3 equals angle 4. Why? 8. Line AC equals line CD. Why? 9. Line BC equals line CE. Why? 10. Triangle A C B equals triangle ECD. Why? The fa c ts fo r this p ro b lem are a t th e top o f p a g e 19. [ IB ] ea so n s )6 )7 )8 )9 ) io Geometry Prognosis: A a. b. c. d. Vertical angles are equal. If a line bisects an angle, it divides the angle into two equal angles. When two straight lines intersect, vertical angles are formed. If two of the sides and the angle between them in one triangle are equal to two of the sides and the angle between them in another triangle, then the triangles are equal. e. When one line bisects another, it divides it into two equal parts. / . If two of the angles and the side between them in one triangle are equal to two of the angles and the side between them in another triangle, then the triangles are equal. P r o b le m A D III. Read the following problem carefully. In the parentheses after each statement, write the letter of the fact (given below) B that tells the reason why the statement is true. In the figure ABC D is a rectangle. D B is a diagonal connecting D and B . c Problem. To show that triangle A D B equals tri angle CDB. R S tatem ents ea so n s 11. Line A B equals line DC. Why? )n 12. Line A D equals line BC. Why ? )12 13. Angle A and angle C are right angles. 14. Angle A equals angle C. Why? Why ? )13 )14 15. Triangle A D B equals triangle CDB. Why? ) 15 a. If two of the sides and the angle between them in one triangle are equal to two of the sides and the angle between them in another triangle, then the triangles are equal. b. The opposite sides of a rectangle are equal. c. If two of the angles and the side between them in one triangle are equal to two of the angles and the side between them in another triangle, then the triangles are equal. d. If a line bisects an angle, it divides the angle into two equal angles. e. All right angles are equal. / . The four angles of a rectangle are all right angles. Number right X 2 [ 19 ] (Score, Test 9) 399 4. Test 5*3 Nasi© School Grade________________________________ Date NATURE Qg PROOF Directions; Below are a number of situations in which a conclusion is reached. Accept all of the written statements as true. A person who accepts the conclusion must then take for granted or assume certain statements which are not made. You are to write out in your own words (in the space following each question) the assumptions which you think must he accepted in order that the conclusion he accepted. (If you do not have enough space to write your assumptions, use the hack of the sheet, heing careful to number the answers to match the number of the problem.) Sample! The board of a teeter-totter was perfectly balanced. John and Paul got on the teeter-totter and sometime later noticed that the hoard would remain perfectly balanced when their feet were not touching the ground. John said, HLook, Paul, we must be exactly the same weight." - What must be taken for granted if this conclusion is accepted? Answers? 1. Both boys were the same distance from the place at which the teeter-totter was balanced. 2. There is a relationship between balancing a teeter-totter and the weight of the person on each end. Note? You might not word your answers just like those above, hut that's not so important. Be sure to write all of the things which must he taken for granted if the conclusion is to he correct. Progressive Education Association Columbus, Ohio 1936 400 1« Tom read that the death rate in the United States Navy during the war with Spain was 9 per 1000* This means that 9 men out of each 1000 died* The death rate in New York City for the same period was 16 per 1000. Tom concluded that it is safer to he a sailor in the Navy during a war than to he an ordinary citizen in New York City. What things must he taken for granted in order that this conclusion he acceptedT - Write them in the space helow. 2. One of Jean's teachers was very quick-tempered. This teacher was red headed* Jean also recalled that she knew two or three other people who were quick-tempered and had red hair. She decided that people who have red hair are generally quick-tempered* - What must he taken for granted if this conclusion is to he accepted as correct? 3* Boh Henry received the same score on an examination. Up to that time, Henry's reputation as a scholar had heen better than Boh's. Boh now argued that the examination proved that he was as good a student as Henry. - What must he assumed if this conclusion is to he accepted as correct? 401 4* Norman lived on a farm* The clover in one field did not grow well. Norman learned in agriculture class that clover does not grow well if the soil has an "acid reaction*** He had a sample of the soil tested and the result showed the soil had an **acid reaction**. Norman con cluded that if the soil were chemically treated so that it would no longer have an "ajcid reaction*' the clover would then grow well. What must he taken for granted in order to accept this decision as correct? 5* The following is a common test for the presence of scids. A piece of hlue litmus naner when placed in contact with acid turns pink. Phil accidentally dropped a small piece of hlue paper into a dish con taining acid. The paper turned pink. Phil concluded the piece of paper was litmus paper* - What must he taken for granted in order for this conclusion t6 he accepted as correct? 6. The familiar fact that some substances are heavier than others may he made precise hy comparing their densities. Density is the mass (some times called "weight") divided hy the volume. On a trip in Colorado, Jim found a small piece of heavy "rock". Thinking it might he some valuable mineral, he decided to try to find out what it was. He carefully found the weight of the rock in grams and also found its volume in cubic centimeters. He divided the weight hy the volume and found the density was 8.2. He then looked in a hook which gave the density of many different substances. He found that the density of sylvanite. a mineral which contains gold, was 8.2. He concluded that the "rock" he found was a piece of sylvanite. - What things must he taken for granted in order for this conclusion to he correct? 402 7* When a straight stick is held at an angle and partly under water it appears to he bent at the surface of the water. This may he explained hy supposing that when rays of light pass at an angle from water into air, the rays are hent. Similarly, when rays of light enter many dif ferent substances (grains of sand, for example) at an angle, the rays appear to hend and travel at a new angle. By suitable apparatus these angles can he measured and a number called the index of refraction can he found for each substance. During a robbery in Madison, Wisconsin, a man was shot. The criminal ran down an alley and escaped. Later a man was arrested and brought to trial for the crime. He claimed he was not guilty. The following evidence was presented. Just before the crime, the alley down which the criminal fled had been freshly covered by fine sand of a kind found only in a few places. Only one set of tracks was found in the sand. All of the m a n ^ shoes had been soaked in water, the soles had then been scraped, and some fine particles of sand obtained. When measured (under the micro scope) by a scientific observation the index of refraction for this sand was the same as that of sand found in the alley. Suppose the jury decided the man was certainly guilty. - What things must be taken for granted in order for this conclusion to be accepted? 8. A man decided to replace the dirt floor in his garage with a concrete floor. He decided to do the work himself and purchased sand, gravel, cement, and borrowed a smell mixer. He was not able to do the work until well into the winter. He knew that in freezing weather concrete may freeze before it sets and decided to guard against this. He knew that the addition of salt to water will lower the temperature at which water freezes. When he mixed the concrete he placed a large amount of salt into the mixing water, concluding that this would prevent the freezing of the concrete. - What things must be taken for granted in order for this conclusion to be accepted? 403 9* A magazine advertisement for a chain of newspapers pictured a man who was supposed to "be saying, 111 wholly disagree with what you say, hut I will defend to the death your right to say it." In the advertise ment a famous Frenchman named Voltaire was given credit for first making this statement. A man wrote to the editor of the magazine and claimed that this statement could not he found in any of Voltaire's writings. She men who wrote the advertisement replied that they had found that Voltaire had heen given credit for this statement in many hooks and articles and also in speeches printed in the Congressional Record. What must he taken for granted or assumed in order that the line of argument used hy the men in defending their advertisements will result in a correct proof that Voltaire made the statement? 10. Below is part of an advertisement for a chain of newspapers in the magazine Time. 11An American reporter interviewed a typical family in a dictator ridden European country, from which liberty had heen banished. Hands gnarled from hitter toil, cheeks sallow from privation, clad in tatters, ... this family looked at our reported with genuine pity, 'You are an Amer ican? How sad.1 They said, 'We know that hundreds of your people are being shot monthly hy capitalist controlled police. We know that thou sands of your workers are dying from hunger. We know that no one in your country has the comfort or the security we have. We know that, in all the world, we are the most fortunate people. We know all these things because we read them in our newspapers, which speak the truth because our government publishes them.1,1 "You can't fool us. We're the world's most fortunate people." What assumptions must be made in order that the conclusion of the last sentence be accepted? 404 Part II Before an election each candidate for office tries to explain his pi&ns to the voters. Below is part of a statement made hy a candidate for Governor, Head it and select the important words or phrases which you think should he clearly defined in order to really understand the program* Write them on the lines below. Statement "I propose a new deal for the schools of Ohio, with an adequate state program of state revenues, to he distributed hy the state to make up the deficiency in local revenues* "No patriotic American could take any other position. There is no issue as to the necessity of reasonable state aid for the schools of Ohio. The only question involved is that of a definite program, plus aggressive leadership in the Governor to drive through for a prompt solution. There must he a program developed, and there must he determined leadership to put it through, I propose to give both. "We have an acute problem of finances. The state can not agree to pay all the cost of the schools and should not do so. But it must provide a program of tax revenues to guarantee a reasonable and proper education to the children of Ohio." 405 Fart III The following quotation is an editorial published by a newspaper. Head it and jjbick out the main topic that the author has written about and also the main ideas he has stated relative to that topic. "As was anticipated, the commission appointed to make recommendations as to where authority over permits for gasoline stations should be lodged reports that the authority should be transferred from the Council to an administrative body, the Board of Zoning Appeals. "There has been much criticism of the manner in which permits have been handled. The Board of Zoning Appeals was given control of these permits in the original zoning law, but the Council amended the ordinance to get the power in its own hands. Its action in the matter was severely criti cized at the time, and its subsequent handling of applications for permits has given additional basis for objection to continuance of the present arrangement. It has assumed an authority that, the commission reports, exists in the legislative body of no other city which operated under a zoning law. This is sufficient evidence that the situation here is anomalous and that public dissatisfaction is justified. "The commission suggests a number of regulations to govern issue of permits in the future under the Board of Zoning Appeals. If these regular tions are adopted and published, there should be little likelihood that future handling of permits will be open to criticism. Before these regula tions can be put into practice, however, authority over permits must be got away from the Council. A number of members of the Council have indi cated that they would approve such a step. With public sentiment to back them, they should be able to effect this desirable reform." What is the main-topic that the author has written about? Write it here. What are the main ideas which he has stated relative to this main topic. Write them here. 406 COLUMBIA RESEARCH BUREAU PLAHE GEOKEffitY TEST COLUMBIA RESEARCH BUREAU PLANE GEOMETRY TEST B y H erbert E. H awkes, P h .D. Professor of M athematics and Dean of Columbia College and B en D . W ood, P h .D. Associate Professor of and Director Bureau of Collegiate Educational Research Columbia College, Columbia University TEST: F O R M B For High Schools and Colleges Do not open this paper, or turn it over, until you are told to do so. giving your name, age, etc. Write plainly. Name Fill these blanks, .......................................................................................................................... (F irst nam e, in itia l, and la st nam e) Age last birthday........................ years. C la s s ..................................................... School or college .. ........................ . . ...................................................... .............................................................. C ity ..................................................................................................................................................... Name of teacher.............................................................................................................................. Date of this exam ination. . How many months have you studied plane geometry ? P art ............... ..................... . . ................................... Sc o r e I II Total Rating Classification This examination consists of two parts and requires minutes of working time. The directions for each part are printed at the beginning of the part. Read them carefully and proceed at once to answer the questions. There is a time limit for each part. You are not expected to answer all the questions in either part in the time limit, but if you should finish Part I and are sure that your answers are right, you may go on to Part II- If you have not finished Part I when the time is up, stop work on that part and pro ceed at once to Part II. Ask no questions after the examination has begun. You need two sharpened lead pencils and ample scratch paper. Rules and compasses are not needed. Do not turn the page yet. G e n e r a l D ir e c tio n s . 60 Published by World Book Company, Yonkers-on-Hudson, New York, and 212 6 Prairie Avenue, Chicago Copyright 1926 by World Book Company. Copyright in Great Britain A ll rig h ts reserved. H W :crbpgt :b- i PRINTED IN U.S.A. £gp- T h is te s t is c o p yrig h ted . T h e rep ro d u ctio n o f a n y p a r t o f it b y m im e o g ra p h , h ectograph , o r in a n y o th e r w a y, w h e th e r th e rep ro d u c tio n s a re s o ld or are fu r n is h e d f r e e fo r u se , is a vio la tio n o f th e c o p yrig h t law . 2 Columbia Research Bureau Plane Geometry Test: Form B PART I. TRUE AND FALSE STATEMENTS D irections. If a statement is true, put a plus sign ( + ) in the parentheses after i t ; if it is false, put a zero (0), as shown in the samples. One point is given for each correct marking; one point is subtracted from your score for each in correct marking. Unless a statement is true, wholly and without exception, it must be marked false. For example, the second sample is false, because such a parallelogram might be a rectangle and not a square. You may draw figures anywhere in the margins; if more space is needed, use page 8 . Time lim it: 20 minutes. amples. f The four sides of a square are e q u a l ................................... ( | ^ parallelogram whose angles are right angles is a square , + , (Q ( 1. A diameter of a circle is a chord greater than any chord of the same circle which is not a diameter ..............................................................( 2. All straight angles are e q u a l.................................................................( 3. If two angles of a triangle are equal, the sides opposite those angles are e q u a l ........................................................ ......................................... ( ' 4. Tangents to a circle at the extremities of a diameter are parallel to each o t h e r .................................................................................................( 5. Two perpendicular diameters divide a circle into four equal arcs , ( 6 . An axiom is a statement which is accepted as true without Pro° f ................................................................................................................( 7. Any straight line whose extremities lie on a circle and which bisects the circle is a d i a m e t e r ....................................................................... ( 8 . Any acute angle of a quadrilateral is less than 90 degrees . . , . ( 9. If a line passes through the center of a circle and is perpendicular to a chord, it bisects the chord and the arc subtended by it . . . . ( 10. The sum of the angles of any quadrilateral is equal to four straight a n g l e s ......................................................................................................( 11 . If the centers of any two circles which lie in the same plane be made to coincide, the circles will c o i n c id e .................................................. ( 12. In two unequal circles, equal central angles intercept unequal numbers of degrees of a r c ......................................................................./ 13. The difference between two sides of a triangle is less than the third s i d e .............................................. r 14. If a line bisects one non-parallel side of a trapezoid and is parallel to one base, it bisects the other non-parallel side also . . . f 15. One angle formed by the bisectors of two angles of an equilateral triangle is double the third angle of the equilateral triangle ( 16. Any two similar polygons of more than three sides can be divided into the same number of triangles which are similar each to each ( 17. The bisectors of the three angles of a triangle are concurrent . ( 18. If unequals are subtracted from equals, the results are unequal in the reverse ord er................................................... , 19. If the lines A B and'CII intersect, the sum of AB and CD is less than the sum of A C and DB / ] ] Columbia Research Bureau Plane Geometry Test: Form B 20. Adjacent angles have the same vertex and a common side . . . . ( 21. If two intersecting lines are cut by three or more parallels, the cor responding segments of the two transversals are proportional. . . ( 22. An equilateral polygon circumscribed about a circle is regular . . ( 23. The line of centers of two externally tangent circles is equal to the sum of the two d iam eters...............................................................................( 24. The perpendicular bisectors of the three sides of a triangle meet in a p o in t .................................................................................................................... ( 25. In two similar triangles any two corresponding sides are proportional to two corresponding a lt it u d e s .................................................................... ( 26. If a circle is divided into three equal arcs, any chord joining adjacent points of division is equal to the radius of the c ir c le ................................ ( 27. If four terms are in proportion in order, the first is to the third as the second is to the fo u r th .................................................................................... ( 28. If the diagonals in a quadrilateral meet at right angles, the figure is a r e c t a n g l e ......................................................................................................... ( 29. In any two circles, equal chords subtend equal central angles. . . ( 30. The sum of the lines joining the midpoints of the sides of atriangle is equal to one half the perimeter of the tr ia n g le .............................. ( 31. If the sum of the interior angles of a convex polygon is twice the sum of the exterior angles, the polygon hasfive s i d e s ..........................( 32. Two circles may be placed in such a relative position that two and only two common tangents may be d r a w n ........................... . . . ( 33. If a median of a triangle is perpendicular to one side, the triangle is eq u ilateral......................................................................................................... ( 34. The altitudes of two similar trapezoids are proportional to any two corresponding s id e s ....................................................................................( 35. If the sum of the exterior angles of a polygon equals the sum of the interior angles, the polygon has six sid es...............................................( 36. The line joining the midpoints of the non-paralld sides of a trapezoid bisects the diagonals and is equal to half their s u m ......................... ( 3 7 . In any two circles equal chords subtend equal a r c s ..........................( 38. If central angles in two or more circles intercept arcs of unequal length, the longer arcs subtend the greater a n g l e s ..........................( 3 9 . If a circle is drawn with its center at the midpoint of the hypote nuse of any right triangle and with a radius equal to the shortest side, the circle will pass through the vertices of the triangle . . . ( 40. If the sides of two triangles are respectively proportional, their corresponding angles are e q u a l ...............................................................( 41. If n is the number of sides of a convex polygon and s is the sum of its interior angles, then 5 equals (2 n — 2 ) right a n g le s ..........................( 42. The corresponding sides of two triangles are in proportion if their sides are parallel each to e a c h ...............................................................( 43. The lines joining the midpoints of the adjacent sides of a quadrilateral form a r h o m b u s ......................................................................................... ( 44. On one side of A B as a hypotenuse many right triangles can be c o n s t r u c t e d .............................................................................................. ( 4 5 . All equivalent parallelograms are similar............................................... ( 4 Columbia Research Bureau Plane Geometry Test: Form B 46. The opposite angles of a quadrilateral inscribed in a circle are e q u a l............................................f .................................................................. 47. If a chord of a circle bisects the angle between a tangent and a chord drawn from the point of contact, it bisects the intercepted arc . 48. A square may be constructed equivalent to the difference of two given s q u a r e s .................................................................................................. 49. If the two non-parallel sides of a trapezoid are equal, its opposite angles are e q u a l ............................................................................................. 50. Given a line A B 6 inches long. A point moves so that its distance from the nearest point in the line is always i inch. The locus of the point is a rectangle 2 by 8 in c h e s ................................................. 51. All the diagonals of an equilateral pentagon are e q u a l ...................... 52. In the formula for the area of a triangle in terms of the sides, the symbol 5 stands for the sum of the sides of the triangle...................... 53. The locus of the vertex of the right angle of all right triangles con structed on a given hypotenuse is a line parallel to the hypotenuse . 54. If two angles have their sides parallel each to each, they are equal or c o m p le m e n ta r y ............................................................................................. 55. If three points are in the same plane, a circle can pass through them 56. The area of a sector is to the area of a circle as the length of its arc is to the circumference of the c i r c l e ............................................................ 57. The perimeters of two regular polygons of the same number of sides have the same ratio as their r a d i i ............................................................ 68. Given two parallel lines A B and K R g inches apart; the locus of a point between them and twice as far from A B as from K R is a line parallel to A B and K R , 6 inches from A B and 3 inches from K R . 59. The apothem of a regular polygon is equal to the radius of the cir cumscribed circle ............................................................................................ 60. The area of a triangle is one half the product of its perimeter and the radius of its circumscribed c i r c l e ............................................................ 61. In any triangle the product of two sides equals the square of the altitude on the third side plus the square of the radius of the cir cumscribed circle ............................................................................................. 62. The angle between two tangents to a circle is measured by the supplement of the smaller of the intercepted a rcs................................. 63. The areas of two circles are to each other as their circumferences 64. Two equilateral polygons of the same number of sides may have their sides proportional each to each and not be s i m i l a r ............................ 65. In order to compute what fractional part the area of a sector is of the area of the whole circle, we must know the angle of the sector and the radius of the c ir c le ................................................. Number right _ Number wrong _ Number omitted _ Sum should be 65. Number right Number wrong Score (;Score = rig h ts — w ro n g s) Columbia Research Bureau Plane Geometry Test: Form B PART II. 5 PROBLEMS D ir e c tio n s . Find the answers to these problems as quickly as you can. If necessary, do your figuring in the blank space on pages 7 and 8 , but put the answers in the parentheses on this page at the right of each problem. Do not spend too much time on any one problem. If you find one difficult, skip it and then go back to it if you have time. In this test you must show your geometrical ability by finding and stating exactly certain arithmetical relations. This means that you must check your arithmetical operations carefully before putting down an answer. Wherever possible, save time by indicating operations instead of working them out completely. Thus, if the answer to a problem happens to be one-seventh of the square root of the product of 13 and 9 1 , you should not do any further computing, but write y \ / i 3 - 9 1 . Time lim it: 40 minutes. S a m p le . 2. 5. 6. 7. 8. 9. 10. 11. 12. How many degrees are there in two right angles ? • In this triangle, angle 1 equals 5 0 degrees and angle 2 equals 1 0 0 degrees; how many degrees are there in angle 4 ? .......................................... If three diameters divide a circle into six equal arcs, how many degrees are there in each of the six angles at the center ? . . . . 3. In the figure, the two parallel lines A B and CD are cut by the transversal K R , making angle 2 equal 6 0 degrees. What is the num ber of degrees in angle 4 ? ................................ 4. How many degrees are there in the supplement of angle 8 ? ........................................................... One acute angle of a right triangle is four times the other; what is the number of degrees in the larger acute angle ? ................................ The angle A of a parallelogram ABC D is 6 0 degrees; its including sides are 3 and 4 inches; what is the length of the longer including side of angle C ? .......................................................................................... The sides of an angle whose vertex is at the center of the circle 0 cut the circle at A and B ; the angle is 3 5 degrees; how many degrees are there in the minor arc A B ? ............................................................... How many degrees are there in the angle made by a diameter and a tangent to the circle drawn at one extremity of the diameter ? . . An exterior angle at the base of an isosceles triangle is 1 3 0 degrees; how many degrees are there in the obtuse angle formed by the inter section of the bisectors of the base angles of the triangle ? . . . . Two consecutive angles of an inscribed quadrilateral are 6 0 and 7 0 degrees, respectively; how many degrees are there in the next angle in order ? .......................................................................................... An angle A B C is inscribed in a semicircle; how many degrees are there in the supplement of angle A B C ? ................................................ If a circle whose diameter is 1 4 inches is divided into six equal arcs, how many inches are there in any chord joining adjacent points of d i v i s i o n ? ................................................................................................ (f80 ) 6 Columbia Research Bureau Plane Geometry Test: Form B 13. The sides of a triangle are 6 , 8 , and io inches; lines are drawn con necting the midpoints of the three sides; how many inches are there in the perimeter of the small triangle thus formed ? ..............................( 14. In a right triangle having a 3 0 -degree angle, the shorter side of the right angle is 4 1 inches; how many inches are there in the median from the vertex of the right angle to the h y p o te n u s e ? ...................... 15. Find the mean proportional of 2 and 4 0 . 5 ............................................ 16. What fraction of the area of a circle is the area of a sector in it whose angle is 3 0 d egrees?....................................................................................... 17. If arc A R in the figure at the left is 2 9 degrees and arc B K is 71 degrees, how many degrees are there in angle ROB ? ................................................................. 18. Flow many fines are there in the locus of all points whose distance from one end of a fine segment equals their distance from the other end of the same fine segment ? . . . 19. How many triangles can be drawn upon a fixed base A B , 2 inches long and having an altitude of 1 inch upon A B ? ............................................ 20. How many degrees are there in each exterior angle of a regular octagon ? ....................................................................................................... 21. What is the circumference (to four decimal places) of a circle whose radius is 1 ? .................................................................................................. 2 2 . The triangle A B C and the triangle A B K have a common base, A B , and are on the same side of A B . The fine C K is one-half A B , and is parallel to A B . What is the ratio of the area of triangle A B C to the area of triangle A B K ? ....................................................................... 23. The hypotenuse of a right triangle is 5 8 . Another side is 4 0 . Find the area of the triangle.................................................................................. 24. One interior angle of a regular polygon contains 1 6 8 degrees; how many sides has the polygon ? .................................................................. 25. The area of a circle is 1 0 0 square feet. How many square feet are there in a sector of 3 6 d e g r e es? .................................................................. 26. The sum of the interior angles of a convex polygon is 3 6 right angles; how many sides has the polygon ? ................................................. . 27. If the radius of a circle is equal to what is the length of the side of an inscribed square ? ............................................................................ 28. Find the length of an arc of 2 0 degrees in a circle whose radius is 1 .................................................................................................................. 29. Two sides of a parallelogram are 1 0 and 2 0 inches and one angle is 4 5 degrees. What is the area ? ....................................................................... 30. The sides of a trapezoid are 1 2 , 1 5 , 1 6 , and 3 5 in order; sides 1 5 and 3 5 are parallel; the other two sides are extended to meet. Find the perimeter of the smaller triangle thus form ed................................. 31. In a triangle A B C , angle A is 3 0 degrees, angle B is 4 5 degrees, and A C is 1 0 inches; how many inches are there in CB ? Cjjs. 32. In the accompanying figure, angles A C B and / A K C are right triangles. If A K is 1 2 and / A C is 2 0 , what is A B ? ......................................... ( A K b 33. A fine 8 inches long makes an angle of 3 0 Columbia Research Bureau Plane Geometry Test: Form B degrees with another line. Find the length of the projection of the first line on the secon d .................................................................................. ( 34. The radii of two regular polygons of the same number of sides are 1 0 and 8; what is the quotient of the apothem of the larger polygon divided by the apothem of the smaller? . . . . “ .............................( 35. If a , b, and c are the sides of a triangle and a is opposite an acute angle and (c— x) is the projection of b upon c, what is a2 in terms of b, c, and x ? ........... '.............................. ( Number right (Score) (You may use this space and page 8 for figuring.) 8 Columbia Research Bureau Plane Geometry Test: Form B (You may use this page for figuring.) 407 6« CRITICAL THINKING TEST 1Tame11.......................... School............. Teacher I. Helen saw an advertisement for a cold remedy. It said, "Bu£ a bottle at our risk. The next time you have a cold, take Xenex* If you do not feel better soon, bring the bottle back and we will refund your money." Make a list of as many motives as you can in this situation, giving a briefly stated reason for each one. II. Not many years ago the Supreme Court of the United States rendered a decision concerning the relationship of the state to the schools of the state. In a unanimous decision the court ruled that the state had power over all schools in respect to the following matters and that it was the responsibility of the state* 1) To require "that all children of proper age attend some school." 2) To require "that teachers shall be of good moral character and patriotic disposition." 3) To require "that certain studies plainly essential to good citizenship must be taught." 4) To require "that nothing be taught which is manifestly inimical to public welfare." Read this decision carefully and thoughtfully. What, in your opinion, are the words and phrases which need careful definition in order to make the decision clear? Underline them. In the space below give your reasons for under lining the words and phrases. Be as btfief as possible* 408 III. Direction?: In parts (a) and (b) select from the choices given that one which correctly completes the statement* Place a check mark ( y/) on the dotted line "before the number that correctly completes the statement. (a) This newspaper will never gain a wide circulation for it refuses to print sensational material. This assumes that 1. Any newspaper that prints sensational material will gain a wide circulation. 2. No newspaper that does not print sensational material will gain a wide circulation. 3. Most people like spicy news. 4. Newspapers cater to the lowest type of readers. 5. All newspapers are instruments for propaganda. (b) Socialized medicine would never succeed in America because we want to have some right to choose our wwn doctor. This assumes that ...... 1* Uhder socialized medicine we could have no choiceofdoctors. ...... 2. Americans are an independentpeople. •..... 3. Choosing the doctor we want is an American ......4. right. Socialized medicine would bethe entering wedge forsocialism. ....... 5. The reason socialized medicine would not work here is that we want to choose our own doctors. (c) While looking over a magazine, Jim read the following advertisement: "Spud - Menthol-cooled cigarettes - for folks who don't like their fun confined. Spud took the limits off smoking pleasure when it took the tar-laden heat out of smoke. If you are one of those who likes his smoking fun in big measures, you and Spud ought to get acquainted.11 This argument is based on certain assumptions. State as many of these assumptions as you can in the space provided below. How would you rate the argument in this advertisement? 409 IV* In advertising the breakfast cereal "Wheaties" the radio announcer made the following statements? "Wheaties give yon as much body heat as a hot cereal. Wheaties give yon as many heat producing units for body warmth in cold weather as a hot cereal. Buy Wheaties.11 What facts would need to be proved before yon would accept the argument in this announcement? Also what facts would need to be presented? List as many of them as yon can in the space provided below. V. In a certain classroom the problem of Highway Safety was under discussion. At the end of the period the teacher asked the pupils to find out specifically how far a car will go before coming to a stop - assuming that the car has excellent 4~wheel brakes and that it is traveling on a dry concrete pavement. Place a check mark ( ^) on the dotted line before the statements that yon believe are pertinent to the above question. • 1* Bill reported that Jim's old Model-T Pord had stopped in alittle less than 100 ft. from a speed of 30 miles per hour. . 2. Mary reported that she read an article where people shouldnot be permitted to drive over 50 miles per hour. . 3. Paul brought a National Safety Council bulletin to class and it had a speed and stopping distance chart. This chart revealed the results of experiments that were performed with cars on a dry concrete pavement and with excellent 4-wheel brakes, . 4. Jan© reported that her father tried out the stopping distance for their new Buick on the concrete highway in front of their home and found that it stopped within 68 ft. from a speed of 40 miles per hour. • 5, Jane's chum checked this data with a Speed and Stopping Distance Chart and found that it was approximately the same. However, she did note that this did not include the mental reaction in an unexpected stop. She found in this chart that at 40 miles per hour the car would go 44 ft. before the brakes would be applied. . 6. Tom experimented with the family car (1937 Chrysler) and reported that he brought it to a dead stop only 32 ft. from the point where he applied the brakes while going 32 miles per hour. . 7. John reported that the stopping distance depends largely upon the size and weight of a car. . 8, Betty reported that the physical condition of the driver was a very important factor. . 9, Harry reported that heavy busses and trucks could not stop as quickly as a car. .10. Ed seemed to think that the kind Cf tires and the amount of air in them (air pressure) had something to do with the stopping distance, 410 VI* READ CAREFULLY A 'biology teacher described the following experiment for his pupils! A willow tree was grown for five years in a large pot of soil. The plant was regularly supplied with pure water and ample sunlight. No additional soil or materials other than water were added to the pot. At the beginning of the experiment the willow tree weighed five pounds; at the end of five years it weighed 164 pounds. The loss in weight of the soil was two ounces. The class discussed this experiment and came to the following conclusion! **Growing willow trees derive most of their increase in weight, not from the soil, but from factors other than the soil.11 DIRECTIONS: Head each statement below. Is the statement a FACT, or is it an ASSUMPTION? Place a check mark ( </) in the appropriate column after each statement. List of Statements Fact Assumption 1) At the end of five years the tree weighed 164 lbs, 2) Willow trees will grow in any kind of soil. 3) The willow tree was regularly supplied with pure water and ample sunlight. 4) The soil at the beginning of the experiment con tained the same amount of water as at the end of the experiment. 5) At the beginning of the experiment the willow, tree weighed five pounds. 6) The soil weighed two ounces more at the beginning than at the end of the experiment. 7) The willow tree was grown in a pot of soil five years. 8) Factors other than the soil are responsible for most of the increase in weight of growing willow trees. 9) The willow tree increased in weight by 159 pounds. 10) The growth of this willow tree is typical of growingrwillow trees. 11) The soil is the most important factor in the increase in weight of growing willow trees. 12) Most of the increase in weight of this willow tree was brought about by factors other than the soil. VII. A. Assuming that your school is host to a youth conference and you had the choice of entertaining one of two delegates in your home. Check (^/) in the parenthesis your choice in each of the following three cases: I would select and entertain (1) a) the well dressed delegate(white race) ............... ( ) b) the shabily dressed delegate(white) ................ ( ) (2) a) the Negro delegate.... ............................. ( b) the Uhite delegate ............................. ( (3) a) the Indian delegate b) the White delegate ............................ ( .......................... ••\ ) ) ) ) In the space provided below, discuss briefly whether or not you are biased in each of the three cases above and give your reasons for same. (Use back side of this page if more room is needed.) 411 VII# (Continued) B. If Japan and England were a,t war <and you were an .American munitions manufacturer, indicate your choice by placing a check mark ( >/) in the parenthesis for each case as you did in Part A. (1) If my profits were the same from either country, I would ship munitions to! a) England ... ( ), or b) Japan ... ( ). (2) If my profits were greater from Japan, I would ship munitions to! a) England ... ( ), or b) Japan ... ( ). (3) Regardless of profits, I would not ship munitions to either nation ( ). Discuss bfiefly whether or not you are "biased in each of the three cases and give your reasons in the space provided "below. VIII. Three gentlemen A, B, and C, engage in a conversation; from it we are to decide whether each is a Noble or a Hunter. Wehave two theorems to guide us! Theorem 1. Theorem 2. A Noble always tells the A Hunter always lies. truth. •A* "begins the conversation and says either "I am a Hunter.11 We do not know which of these 111 am a Noble11or he says two statementshe made. •B1 says to*A*: "You said you were aHunter.0 'B* "You are a Noble.0 says to•C ** *0* says to 'A1! "You are a Noble." CONCLUSION 'A1 is a .... ‘B* is a .... *01 is a .... IX. DIRECTIONS! Place a ( T ) in the parenthesis following each statement if it is an absolute truth, and an ( N ) in the parenthesis ifit is only a tentative conclusion. 1. It is self-evident that the whole is equal to the sum of its parts 2. Light travels at a velocity of 186,000 miles per second......... 3. Men are stronger than women................................... 4. The sum of the interior angles of a triangle is 1 8 0 ° ........... 5. The best team always w i n s ................................... . 6# Parallel lines never meet no matter howfar they are extended . . 7. Man descended from monkeys....... .......................... 8. Democracy is the best form of government...................... 9. A statement is absolutely true if it has been proved........... 10. Water boils at 212 degrees Fahrenheit ........................ 412 X. Let us assume that you represent your school or your community in some activity or contest, and that you lose your self-control to the extent that you commit some flagrant offence upon your opponent. !• State the individual consequences that may ensue from such an act. 2m State the social consequences that may ensue from such an act. XI. DirtEOT IONS; Make a critical analysis of the following quotation. Try to make this analysis as complete as you possibly can. Use the hack side of this page if you need more room. “Prom early colonial days the problem of how best to deal with liquor has been before the people of the United States, Some people feel that there should be to attempt to control the use of liquor while others believe that Congress should prohibit its sale.11 413 7. INVENTORY qUESTIONNAlHE Name ....................................... When is your next birthday -Address (Final or Follow-up Form) Age last birthday.......... How long have you studied Geometry...... ........................................................ Street and Humber City __ DI&EQTIOHS; This is not ©n examination but merely a information with regard to certain reactions that toward mathematics* Please answer each question to the best of how you actually feel about it. Your response ever upon your semester grade. (Use back side if your responses.) questionnaire to obtain high school pupils may have of your ability and in terms will have no bearing what you need more room for 1. What are your favorite subjects?r Why? 2. In what way do you think that mathematics may help you? Why? 3. In what way do you think that demonstrative geometry may help you? Why? 4. Do you think that high school pupils should have an opportunity to reason about the subject matter of geometry in their own way, or do you think that the subject should be learned in the way in which it is presented in a textbook or in some work book? I5. How would you like to have your progress in this course evaluated? (that I ] is, by a written examination, self-evaluation, your parents1 evaluation, | judgment of your teacher, and so forth, or a combination of these?) 6. The concept of proof has always played a very significant role in human experience. When is something proved? 7, The kind of society in which we live calls for a type of citizenry that is capable of thinking critically. When is a person thinking critically? 8* Make out a list of articles (newspaper or magazine), or a list of books that you have read in connection with your course in mathematics this school year 414 DEMOTIONS: FJfiS II Fleas© answer the following Questions in terms of your personal reaction to mathematics as it has been presented to you this school year* If your answer is YES, underline the word YES, and if your answer is NO, underline the word NO, 1. Do you like geometry as presented in this course?.............YES NO 2* Has the study of geometry helped you in any way so far? . . ♦ . YES NO If so, how? 3. Do you find the study of geometry interesting?........... YES NO 4* Do you find the study of geometry very difficult? . YES NO 5* Did you ever, worry or feel uncertain about passingthis course? YES NO 6• Do you like final examinations in mathematics?........... . YES NO 7. Do you think that examinations are generally f a i r ? ........ YES NO 8. Do you think that the marks your teacher gives depend on favoritism? YES NO 9. Is your teacher willing to explain certain topics, you do not understand, more than o n c e ? ................................ * YES NO 10* Does your teacher talk or lecture too much in the classroom? YES NO 11* Does your teacher give you opportunities to express your own opinions? YES NO 12. Does your teacher permit only a few pupils to do all of the reciting? ................................... . . . . . . . YES NO 13* Does your teacher try to make the class interesting? YES NO .... 14. Is your teacher more interested in mathematics than in the pupils? . . . . . YES NO 15* Is your teacher usually willing to talk with you about your difficulties and give you good advice? YES NO 16. Are you given a chance to express or tell what you know in this c l a s s ? YES NO 17. Does your teacher resent having a pupil express aa opinion which differs from his own or from that of the textbook? . . . YES NO 18. Do you accept all that your teacher tells y o u ? YES NO 19. Do you accept all that you read in your t e x t b o o k ? YES NO 20. Do you think that you should be given an opportunity to think independently even though your thinking is out of harmony with the opinion of your teacher or your textbook? . . . . . . . . YES NO 415 J. EXAMPLES 021 PERIODICAL TESTS A. Test on Assumptions B. Test on Assumptions C. Test on Assumptions D* Test* Need of Assumptions in Geometry E. Test on Conclusions E. Test on Evaluation of Arguments G. Test on Advertisements 416 A. Test on Assumptions 1» Axioms are truths which do not need to be assumed* —— — (True or False) 2. Can we accept Einstein*s theory of relativity as absolute truth? Explain your answer* 3. State one assumption in any of the following sciences* biology, chemistry, physics, electricity. 4. Why are assumptions necessary in any science? 5. Who made the assumptions we are using in geometry? 6. Why is it possible for two persons, who reason logically, to arrive at conclusions which are contradictory? 7. Definitions are 8. Do you disagree what part? man-made. --(True or False) with the following definition of capitalism? Ifso, Why? "My definition of capitalism iss a human system whereby an employer class plan and supervise the production and distribution df wealth for profit, and who direct an employee class whom they guarantee a stipulated remuneration for their services in the form of wages. 9. Why are definitions necessary in the study of any science? 10. Ham© some terms which are undefinable in geometry. B. Test on .Assumptions I. State the assunption definitely implied in each of the following statements* 1. William Randolph Hearst is a friend of Anerice because he preaches patriotism. 2. Prosperity is returning because prices are going up. 3. Hi tier* s Conquest of Austria must have been just because it was ________ successful.______________ . — __________ — ----— -------------1. Alfred Lawson, Financial ism Against Capitalism, page 2. 417 4. Philip cannot take solid geometry "because he has not completed plane geometry* 5. Mr, Grayson cannot "become President of the United States "because he is not a naturalized citizen of the United States* 6. The "boy who has had training in the R.O.T.C. will know how to command "because he has Dferned to obey. 7* This cylinder oil certainly needs changing "because it is not clear. 8 . According to the sixteenth amendment, you will have to pay a federal income tax, 9, Since the tax rate has been lowered, our taxes will be less this year, 10. The bath-room was filled with steam because the water running from the faucet was boiling. 11. Our newspapers have improved because of the influence of the radio* 12. Mr. Walters can answer any question because he is an authority on science, 13. If you vote for La Follette, you vote for socialism. 14. The New Deal has failed to solve our economic problems because many inportant positions were filled by college processor*. 15. Mr. Gray is a wise reader because he does not depend upon one paper for his information on current affairs# II. State the fundamental assumption or assumptions upon which each of the following statements is based* 1. Under capitalism, security varies inversely with the degree of specialization. 2. Our tariffs should have been lowered before 1980, because the world war has made the United States a creditor nation. 418 8. Stuart Chase says that the time may come when there will he little use for either lahor or capital. 4. In 1929, the ratio of total capacity to production to con sumption was roughly 5?4J3* 5. We cannot have prosperity as long as there is inequality between consumption and production* 0. Test 6n Assumptions Select from the list of suggested assumptions below each of the fol lowing arguments the one which you think the argument definitely assumes or takes for granted. I* Democracy will be maintained in this country because there are more than 100,000 forums and discussion groups in the United States. This argument assumes that 1. It is the right of every citizen of the United States to hear and to be heard. 2. Hard times have made the .American people more intelligent. 3. The right to hear and to be heard will guarantee the perpetuation of democracy in the United States. 4. The ’’Town Meeting of the JUr'* is a worthwhile radio program. II. He is eligible for President of the United States since he is a natural born citizen of the United States, over thirty-five years old, and has lived in this country fourteen years. This argument assumes that 1. No one who is not a natural born citizen of theUnitedStates should be allowed to become President of the UnitedStates. 419 2. Aay citizen who is ovef thirty—five years of age and who has lived in this country fourteen years is eligible for President. 3. There are only three requirements to he eligible for the presidency of the United States. The candidate must he a natural horn citizen of the United States, over thirty-five I years old, and oust have lived in this country fourteen years. 4. The President of the United States should he well qualified for the position. III. If you do not want a dictatorship in this country, write to your congressman requesting him to vote against the Reorganization Bill. This argument assumes that 1. Our Country is headed for a dictatorship. 2. President Roosevelt desires to "become g. dictator. 3. The Reorganization Bill would create a dictatorship in this country. 4. Uo one wants this country to become a dictatorship. IIII. He will he a good lawyer because hw was an excellent geometry student. This argument assumes that 1. The ability to reason is a necessary qualification to he a good lawyer. 2. Good work in geometry insures success in the practice of law. 3. Grades in high school subjects predict success in later life. 4. A good mathematician will he a good lawyer. 5. A good lawyer must understand geometry. 7. He must he a coward to he such an extreme pacifist. This argument assumes that 420 1* Pacifists do not "believe in war. 2. All cowards are pacifists. 3. The extreme pacifist is a coward. 4* His religious "beliefs make him a coward. D. Tesll— Need of Assumptions _in_ geometry What facts must be given about this figure to make the following statements true? 1. AD - EC because halves of equals are equal. 2. AD s EC because if equals are subtracted from equals the remainders are equal. 3. AB s BC because if equals are added to equals the sums are equal. II. If angle D + angle E = angle G, what must be true to prove that angle K + angle E - angle G? III. If Q. + 4* /3 + £4 - 180°, what must be true to prove that ^2 f If r 2 k, what must be true to prove that r + x IV. r z 90°? k + y? What must be given about this drawing in order to make the following statement true? B A ingle 3 a angle 4 because supplements of equal angles are equal. What two things in this figure must be equal to make the following statement true? Angle 1 - angle 2 because base angles of an A M B isosceles triangle are equal. VII. 1. In this figure, if AB s BC, and if .... = .... then triangle ABD is congruent to triangle BDC by "two sides and included angle". 421 2. In the preceding figure, if angle ABD - angle CBD, and AB = BC, then triangles ABD and BDC are congruent hy "two angles and included side". S. Test on Conclusions State the conclusion which necessarily follows from the two given statements in each of the following exercises* I. 1. Capital is wealth. 2. Wealth is anything and everything made valuable by human effort. Therefore* ------------------------- II. 1. A polygon that is both equiangular and equilateral is a regular polygon. 2* A square is equiangular and equilateral. Therefore* ----------------------III. 1. Logic is the study of argument. 2. Argument is discourse containing inference. Therefore* — ---------------------- - IV. 1. .Angle D is either greater than, less than or equal to angle E. 2. It has been proved that angle D can be neither greater than angle E nor less than angle E, Therefore* -----------------------V. 1. If the product of two numbers is equal to theproduct other numbers, then a proportion can be made by of two using one pair of numbers as the means, and the other pair as the extremes. 2, k . 1 s r . b Therefore* ..................... . 422 VI* 1. Any one who cannot do military service should not he allowed to vote. 2* Women are incapable of doing military service# Therefore! -------------------- VII. 1. To serve Hitler is to serve Germany. 2. To serve Germany is to serve God# Therefore* -------------------- IIX# 1. If a line divides two sides of a triangle proportionally then it is parallel to the third side# 2# In triangle ABG, IB intersects the sides AB and BC so that BB is 5 inches, AD is 4 inches, BE is 10 inches, EG is 8 inches# Therefore:--------------------- IX# 1. Those who prefer the best want Springbrook butter, 2* I*m sure that you prefer the best. Therefore: ---------- ---------- X. 1. Newspapers, subject to the influence of advertisers, will print nothing detrimental to the products of their advertisers. 2. The harmful effects of Virginia Cigarettes, advertised in the Morning Press, have been recently discovered by science# Therefore! P. ----------------- Test on Evaluation of Arguments After each of the following arguments, write! valid, if it is a good argument, and invalid, if it is a poor argument. 1. 11America hates war. Anerica hopes for peace. actively engages in the search for peace". Therefore, Anerica P. D. Roosevelt------- 423 2* Since this statement is time, its converse is true. 3* A quadrilateral is a polygon having four sides. has four sides. 4. — *----- A parallelogram Therefore a parallelogram is a quadrilateral.----- All regular polygons are equiangular. A rectangle is equiangular# Therefore a rectangle is a regular polygon. -------5. If one of three possible relationships must be true, and two of them have been proved impossible, then the third must be true.----6. If angle D could be proved equal to angle E in thetriangle ADE, then triangle ABE could be proved isisceles because of the theorem! Base angles of an isisceles triangle are equal.-------7. Only a citizen can vote. Mr. Brown is a citizen. Therefore, Mr. Brown voted in the last election.--------8. Only a citizen can vote. Mrs* Carey voted in the last election. Therefore Mrs. Carey is a citizen. 9. A high school boy reasons thus! --------Although Lindbergh failed in of his college courses, he became a very successful aviator. some Since my chief interest is in aviation, I do not need to worry about my failing grades. --------- 10. More students in Kenosha High School fail in the social studies than in mathematics. Therefore, the social studies must be more difficult than mathematics. ------ 11. IfJohn is the brother of Jessie, Jessie is the brother of John.----12. If the scientists theory is correct, all experiments testing it will confirm the theory. ---------- 13. If Walter's method for trisecting an angle gives accurate results for five different angles, then his method is correct for tri secting any angle. ---------- 424 S 14. Given: {Criangle ABC To prove: /A + /B + £C : 180°. Proof 1. Draw xy through A. 2* Q. - £B because alternate interior angles of parallels crossed by a transversal are equal. 3. £3 s Ip because of reason in No. 2. 4. £2 = /A. 5. £L + £.2 +• Z.3 s 180° because the sum of all the successive adjacent angles around a point on one side of a straight line passing through the point equals one straight angle. 6. (2 4> £A +> /G = 180°, by substitution. ------- 15. If DE is parallel to AB and G3P is parallel to AB, then DE is parallel to G3P, by, substitution. -------- 16. Since you are not a modernist, then you are a fundamentalist.------ 17. "A planless life is like a ship without a compass. because it lacks direction." It gets nowhere The young man who has no plans for his future is apt to be a "drifter" because he has no aims. ------- 18. We will always have the liquor problem because it has always existed.---19. Since government regulation of economic conditions has increased the depression in every country where it has been tried, we cannot hope for recovery until political influence returns to its own sphere. ------20. His conclusion must be correct because every step in his reasoning follows logically from the preceding step.------ - 425 Test on .Advertisements I. The following is an advertisement for cigarettes: "Virginia cigarettes are preferred by the best people, and are made of costly tobacco. Smoke the best* The cheapest cigarette is not the best." A. Do you. conclude that Virginia cigarettes are among the best? 1. Yes -----2. No ------ B. If your conclusion is number 1, place a check mark to the left of the statements below which support that conclusion. If your conclusion is number 2, place a check mark to the right of the statements below which support that conclusion. Do not mark the irrelevant statements. — a. Price determines the quality of an article.---b. Price does not determine the quality of an article.---Costly tohaccoes are the best tobaccoes. ---d. The best people are not necessarily those who can afford the highest priced cigarettes. ----- -— — e. Wealthy people would not choose an inferior cigarette.---f. What is best for one person may not be best for another.---g. Some of our best people do not smoke at all. ---- II. In a radio advertisement, this argument was presented: tells you that you must eat a hearty breakfast. "Your doctor Southern Syrup and pancakes will start the day out right for you and supply you with energy needed for a strenuous forenoon's work." A. Does this reasoning prove that you should buy Southern Syrup? 1. Yes ---2. No ---- 426 B. Why? Check the statements below which support your answer in part A, — — 1* The factory and office worker usually does his hardest work in the forenoon, 2. Some people get along very well without eating any breakfast, 3. A doctor understands the principles of diet much better than the average person, 4 # A. hearty breakfast does not need to be a heavy breakfast. 5. Pancakes and syrup supply one with a large number of calories. — 5. Your doctor does not say that you should have pancakes for breakfast. 7. All high school students should eat a hearty breakfast because the fore-noon session is four hours, while the afternoon session is only two hours, — 8. Pancakes and syrup constitute a hearty breakfast. APPENDIX D Analyses of Literature on the Teaching of Mathematics for Objectives 428 APPENDIX D (1) Books on the teaching of mathematics by David E. Smith, J.W.A, Young, Arthur Schultze, Ernst R. Breslich, J.Q. Hassler and R.R. Smith, William L. Schaaf, David E. Smith and Wm. D. Reeve, Raleigh Schorling, and the Fifth Yearbook of the National Council of Teachers of Mathematics. (2) Reports of various committees on the teaching of mathematics! National Committee of Fifteen on Geometry Syllabus, 1912! National Committee on Mathematical Requirements, 1923; First Committee on Geometry, 1929; Second Committee on Geometry, 1930; Third Committee on Geometry, 1932; Tentative Report of the Mathematics Committee of the Progressive Education Association, 1938; A Preliminary Report by the Joint Commission of the Mathematical Association of America, Inc. and the National Council of Teachers of Mathematics, 1938. (3) Studies by Christofferson and Fawcett. (l) Books on the Teaching of Mathematics Young 2 classifies the aims for teaching mathematics by grouping the objectives under three general headings! (A) practical values, (B) math ematics as a way of thinking, and (C) functional values. In (A) he refers to the intimate connection of the subject with everyday life, its use in various occupations, its informational value, and its values in coping with nature. IN (B) he refers to mathematics as exemplifying certain modes of thought, such as grasping or comprehending a situation and drawing intelligent conclusions. In (C) he refers to functional values as follows! (l) Generalizing conceptions, (2) Information and use of 2. Young, J. W. A., fffle Teaching of Mathematics. New York! Longmans, Green and Company, 1924, pp. 9-52. 429 symbolic language, (3) The finished form in treatment of topics, (4) Early discoveries, (5) Knowledge for its own sake, (6) Cultivation of reverence for truth, (7) Cultivating the habit of self-scrutiny, (8) The aesthetic side of mathematics, (9) Development of imagination, (lO) Culti vating power of attention, and (ll) Fostering habits of neatness. Schultze likewise classifies the aims for teaching mathematics by grouping the objectives under three general headings! (A) the practical value, (B) the disciplinary value, and (C) minor functions. IH (A) he emphasizes the use of mathematics in science, the influence of mathematics upon life, and the value of mathematics to the individual. In (B) he feels that, “Mathematical instruction in a secondary school is - or rather should be - principally a systematic training in reasoning, and not an imparting of information... The reasoning in mathematical work is of a peculiar kind, possessing characteristics that make it especially fitted for training the minds of the students. Some of these characteristics are the following! (l) simplicity, (2) accuracy, (3) certainty of results, (4) originality, (5) similarity to the reasoning of life, and (6) amount of reasoning.H In (0) he emphasizes! (l) development of the power of concentration, (2) development of the constructive imagination or the inventive faculty, (3) growth of mental self-reliance, (4) development of character, (5) increased ability to use English correctly, and (6) increase in general culture. He summarizes the fundamental principle of mathematical teaching by stating that, “Mathematics is primarily taught on account of the mental training it affords, and only secondarily on account of the knowledge of facts it imparts. The true end of math ematical teaching is power and not knowledge." 3. Schultze, Arthur, The Teaching of Mathematics. Hew Yorki The Macmillan Company, 1927, pp. 15-29. 430 Breslich^ gives one hundred and sixty-two specific objectives for teaching of secondary school mathematics. He classifies these under six general headings! I. Power to think and to do, II. appreciations, III. Understandings, 17. Attitudes, V. Habits and ideals, and VI. Skills. Space does not permit a restatement of all of these objectives, but for purposes of this study a few of the more pertinent ones are stated as follows! I. Power to think logically and accurately through a problem, to draw correct inferences, to do original thinking, to analyze complex sit uations, to use correct speech in discussions and proofs of geometric facts and principles, to reason correctly, to establish geometric facts by proof, and to use a variety of methods of proof; II, .Appreciations of the relation of mathematics to the pupils* environment, of the mathematical modes of thinking, of dependence and relationships of facts in everyday life, of the value of a logical reasoning, and of the mathematical proof; and IV. Attitudes - desire to think logically, desire to grow mentally, desire to (Understand, desire to generalize, and desire to assume responsibility for an assigned task. HSssler and Smith5 give four general objectives for the teaching of !demonstrative geometry. They are as follows! (l) "We should teach demon strative geometry in the senior high school mainly (though not exclusively) as a course in reasoning and aim to develop powers and habits of careful, accurate, and independent thinking rather than to present the subject as a finished model of deductive logic.” (2) “Demonstrative geometry may be made the basis of the study of methods of reasoning which will be of use in any field where the necessary facts are at hand." (3) "We should connect 4. Breslich, Ernst H., The Technique of Teaching Secondar?/--School Mathematics. Chicago, The University of Chicago Press, 1930, pp. 203-208. 5. Hassler, Jasper 0, and Smith, Holland R., "The Teaching of Secondary Mathematics1 ,1 Hew York! The Macmillan Company, 1930, pp. 297-301 and p. 100. 431 the geometric theory by means of practical exercises, numerical and oeonstructive, with all the applications of the subject we can discover.M (4) nWe should try to make our pupils appreciate the worth of the sub ject in its vital connection with the development of useful learning in the progress of civilization." The authors emphasize in their conclusions that the most important value in the study of geometry is that "it can show the pupil the nature of logical thinking and the bases for scientific reasoning." Schaaf® in referring to teaching objectives points out that the genuine values derivable from a study of demonstrative geometry may be summed up in five general purposes, as follows* (l) To secure an under standing of the meaning and nature of a demonstration or logical proof; (2) to appreciate the logical interrelationships between truths, that is, the force of a deduction; (3) to realize the importance of assumptions; (4) to realize the significance of provisional conclusions; and (5) to secure the ability to present a simple, straightforward, logical, and coherent argument. S\orthermore, he points out that "it is both feasible and desirable to secure on the part of the pupils attitudes similar to the following* (l) a discrimination between the true and the false; (2) a discrimination between that which has been Assumed' and that which has been 'demonstrated1; and (3) a discrimination between accurate draw ings and rough, freehand sketches." David E. Smith and William D. Reeve7 hold that the "real purpose of demonstrative geometry is suggested by the word 'demonstrative' rather than by 'geometry'. The mere utilities of geometry have already been 6. Schaaf, William L., Mathematics for Junior High School Teachers. Richmond* Johnson Publishing Company, 1931, pp. 77-79. 7. Smith, David E., and Reeve, William D., The Teaching of Junior High School Mathematics. Boston* G-inn and Company, 1927, pp. 229S230. 432 acquired "before the pupil "begins, if he ever does, the work in what is to him an entirely new; field - that of logical proof* Nowhere in his pre vious training, nowhere else in his elementary education, does he come in close contact with a logical proof. The chief purpose of this part of mathematics, then, is to lead a pupil to understand what it is to demon strate something, to prove a statement logically, to 1stand upon the vantage ground of truth*. He sees a sequence of theorems built up into a logical system and he sees how this system is constructed, the result being a basis of proved statements which he can use for establishing further proofs, precisely as a lawyer proceeds to construct his case of a speaker to construct an argument." Q Beeve° points out that, "The purpose of geometry is to make clear to the pupil the meaning of demonstration, the meaning of mathematical pre cision, and the pleasure of discovering absolute truth. If demonstrative geometry is not taught in order to enable the pupil to have the satisfac tion of proving something, to train him in deductive thinking, to give him power to prove his own statements, then it is not worth teaching at all." 9 Longley of Yale also places emphasis on reasoning, pointing out that demonstrative geometry can develop a scientific attitude of mind. However, he also stresses practical values of knowledge in the use of formulas, facts, relations, and methods in geometry. Birkhoff and Beatley^, both Harvard men, point out that, "In demons 8. Beeve, William D., "The Teaching of Geometry", New York: Fifth Yearbook of the National Council of Teachers of Mathematics, Bureau of Publi cations, Teachers College, Columbia University, 1930, pp. 13-14. 9. Longley, W.K., "What Shall We Teach in Geometry", New York: Fifth Year book of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, 1930, p. 29. 10.Birkhoff, G. D. and Beatley, Ralph, "JL New Approach to Elementary Geometry", Now York: Fifth Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, 1930, p. 86. 433 strative geometry the emphasis is on reasoning. This is all the more important because it deepens geometric insight. To the extent that the subject fails to develop the power to reason and to yield an appreciation of scientific method in reasoning, its fundamental value for purposes of instruction is lessened. There are, to be sure, many geometric facts of importance quite apart from its logical structure. The bulk of these belong properly in the intuitive geometry of grades VII and VIII, and are not the chief end of our instruction in demonstrative geometry in the senior high school.” Upton^ firmly believes that, “The reason we teach demonstrative 1geometry in our high schools today is to give pupils certain ideas about the nature of proof. same point of view. The great majority of teachers of geometry hold this Some teachers may at first think our purpose in teaching geometry is to acquaint pupils with a certain body of geometric facts or theorems, or with the applications of these theorems in everyday life, but on second r eflection they will probably agree that our great purpose in teaching geometry is to show pupils how facts are proved... The purpose in teaching geometry is not only to acquaint pupils with methods of proving geometric facts, but also to familiarize them with that rigorous Ikind of thinking which Professor Keyser*^ has so aptly called the *If-Then |kind, a type of thinking whifah is distinguished from all others by its j characteristic form: If this is so, then that is so.1 Our great aim in the |tenth year is to teach the nature of deductive proof and to furnish pupils j with a model of all their life thinking.” 111. Upton, C."b 7 , ”The Use of Indirect Proof in Geometry and Life”. New I York! Fifth Yearbook of the National Council of Teachers of Math| ematics. Bureau of Publications, Teachers College, Columbia University, 1930, pp. 131-132. !12. Keyser, Cassius J., Thinking About Thinking. I and Company, 1926, page 18. New York: E. P. Button 434 Finally, Schlauch^ tells us that, “Geometry seems, of all school subjects, the best adapted to initiate a student into the meaning of mathematics as a science of necessary conclusions*” (2) SgQPrts of Various Committees on the Teaching nf Mathematics The first report having significant hearing upon the teaching of mathematics was the report by the National Committee on Mathematical iequirements • This committee classifies the general aims and values of secondary^-school mathematics into three groups* I. Practical, II* Discip linary, and III* Cultural. The following is a summarized outline of these aims as presented by the committee. I. Practical Aims 1. The immediate and undisputed utility of the fundamental processes of arithmetic. a) A progressive increase of understanding of the nature of the fun damental operations and power to apply them in new situations b) Exercise of common sense and judgment in computing from approxi mate data, familiarity with the effect of small errors in measure ment, the determination of figures to be used in computing and to be retained in the result c) The development of self-reliance in the handling of numerical problems through the consistent use of checks 2* An understanding of the language of algebra 3. A study of the fundamental laws of algebra 4. The ability to understand and7interpret correctly graphic represen tations 5. Familiarity with the geometric forms common in nature, industry, and life; mensuration of these forms; development of space perception; exercise of spatial imagination Cl* Disciplinary aims 1* The acquisition in precise form of the ideas or concepts in terms of which the quantitative thinking of the world is done 2* Development of the ability to think clearly in terms of such ideas and concepts. This involves training in analysis of a complex situa tion, recognition of logical relations, and generalization 3* Acquisition of mental habits and attitudes 4* The idea of relationship and dependence -3* Schlauch, W. S., "The Analytic Method in the Teaching of Geometry”, New York* Fifth Yearbook of the National Council of Teachers of Mathematics* Bureau of Publications, Teachers College, Columbia University, 1930, p. 134. .4* National Committee on Mathematical Requirements, The Reorganization of Mathematics in Secondary Education. Boston* Houghton Mifflin Company, 1923, pp. 6-12. 435 III* Cultural aims 1 * Acquisition of appreciation of beauty in geometrical forms 2 * Ideals of perfection as to a logical structure, precision of statement and thought, logical reasoning, discrimination between true and false 3* Appreciation of the power of mathematics The committee states the point of view governing instruction as follows* "The practical aims enumerated above, in spite of their vital importance, may without danger be given a secondary position in seeking to formulate the general point of view which should govern the teacher, provided only that they receive due recognition in the selection of material and that the necessary minimum of technical drill is insisted upon.11 Therefore, "The primary purposes of the teaching of mathematics should be to develop those powers of understanding and of analyzing relations of quantity and of space which are necessary to an insight into and control over our environment and to an appreciation of the progress of civilization in its various aspects, and to develop those habits of thought and of action whichwill make these powers be effective in the life of the individual." The Third Heport of the Committee on Geometry3,5 presents forty-seven statements in the form of a summary of the major ideas derived from a synopses of all available sources that are of significance for the teaching of geometry. Seven of these statements are selected because of their relationship to this study. They are as follows: 11(l) The important facts of geometry can be learned below the tenth grade, in informal geometry. (2) The main outcomes of demonstrative geometry pertain to logical thinking. (3) We wish pupils to develop the conscious use of a technique of thinking. (4) Demonstrative geometry ought to call attention 15. Beatley, Ralph, "Third Report of the Committee on Geometry". Mathematics Teacher, volume XXVIII (October and November, 1935). pp. 331-333, p. 336, p. 342, p. 343, and p. 449. 436 to logical chains of theorems; to gaps in Euclid's logic; to the nature of a mathematical system, the need of undefined terms, the arbitrariness of assumptions, and the possibility of other arrangements of propositions than that given in any one test. (5) The educational possibilities of demonstrative geometry depend upon the transfer* of the logical thinking of geometry to situations outside geometry. We ought to teach so as to insure this transfer. (6) All book theorems should be treated as originals. (7) The distinction between book theorems and originals on college en trance examinations ought to be abandoned." The committee report goes on to say, "There is almost unanimous agreement that demonstrative geometry can be so taught that it will develop the power to reason logically more readily than other school subjects, and that the degree of transfer of this logical training to situations outside geometry is a fair measure of the efficacy of the instruction. However,great the partisan bias in this expression of opinion, the question 'Do teachers of geometry ordinarily teach in such a way as to secure the transfer of those methods, attitudes, end appre ciations which are commonly said to be most easily transferable?1 elicits an almost unanimous but sorrowful 'NO'." This is followed by another statement, that "the opinion is growing among teachers that many of the students' difficulties spring from their failure to appreciate the logical intent of the subject, and that appreciation of what it is all about tends to remove these difficulties. Furthermore, if we do not make every effort to see that our pupils obtain the logical understanding and appreciation so widely heralded as an important outcome of their study of demonstrative geometry, how can we continue to urge these logical aspects in support of our plea for the retention of this subject in the 427 course of study of the secondary school?" One member of the committee points out* "We should take up appli cations with every group of theorems, hot because these applications are valuable in life, but partly because they motivate the work by seeming valuable to the pupil and partly because they bring the geometry nearer to life and help the transfer to life situations. Then I think an occasional non-geometric application of reasoning to some life situation is valuable. I would like to see a fairly large number of schools try consciously to carry over geometry to life situations by asking questions on non-geometric material and attempting to get the pupils to apply their geometric types of reasoning to these problems. Perhaps a good collection of life situations could be worked out to which geometric reasoning could be applied with a minimum of tacit assumptions. A question of College Entrance or Regents examinations on this sort of thing, at first optional and later required, would stimulate a more active attempt at transfer." Finally, the committee report quotes some of the replies to the questionnaire that was administered to over one hundred supposedly leading teachers of mathematics. The most significant replies to the question, l,How ought they (the teachers of mathematics) to modify their ordinary methods in order to secure this transfer?" are as follows: "Bring logical method to the forefront of consciousness; teach for transfer." uciously teach the things we want." etry to other fields." "Con^ "Actually do transferring from geom "Appiy forms of reasoning to non-mathematicgl i i iituations." "Know both applications and appreciations." "Point out parallels between thinking in geometry and in other fields, and practice :>oth." "Bring in illustrations to show the place of logical thinking in ,ife". "Pay more attention to original^ and to analysis." 438 The Commission on the Secondary School Curriculum of the Progressive 16 Education Association proposes three major ideals as the objectives of general education and then points out the implications for teaching math ematics in the light of these objectives. These objectives of general education are as follows! I. Recognition of the dignity and worth of the individual, involving his a) Creativeness b) Appreciation c) Widening range of interests II. Reciprocal individual and group responsibilityi'for promoting common concerns involving a) Social sensitivity b) Tolerance c) Cooperativeness III. The use of intelligence, involving a) Analyzing problem situations b) Readiness to act on the basis of tentative judgments c) Self-direction. The committee further remarks that, "As generalized concepts of the nature of mathematics become more clearly developed in the minds of teachers and pupils, those broader objectives should become increasingly the determining factors in shaping the work of the schoolroom." The following are the implications for mathematical instruction! "(l) Mathematical instruction should aim to contribute directly to the achievement of the objectives of general education. (2) Instruction in mathematics can make a worthwhile contribution to the achievement of certain objectives of general education through the use of appropriate classroom methods. (3) Instruction in mathematics can make a worthwhile contribution toward the achievement of certain objectives of general education by the appropriate choice of concrete problems which call for mathematical treatment. (4) It is 16 Tentative Report of the Mathematics Committee, "Mathematics in General Education", Commission on the Secondary School Curriculum of the Progressive Education Association, June 1938, pp. 1-6 to 1-23. 439 desirable to organize instruction about a framework which enphasizes the contributions of mathematics to general education. (5) Individual differences between pupils, teachers and schools make it desirable to formulate a program which is flexible enough to allow a wide range of adjustment to given situations. (6) Mathematical education should aim to develop in the student a broad understanding of mathematics as a method of thinking." 17 The Joint Commission of the Mathematical Association of America and the National Council of Teachers of Mathematics states! "Educational objectives,in the last analysis, will center around three permanent factors, namely Nature. Society, and the Child. of reference of the educational process... Thpy are invariant frames Accordingly, objectives may be regarded as having either a factual and impersonal aspect or a personal, psychological bearing. Thus, when we study a given domain in a purely scientific way, irrespective of the learner's personal reactions, we are mainly interested in facts, skills, organized knowledge, accurate concepts, and the like. If, on the other hand, we scrutinize the way in which the pupil behaves in a given situation, or his modes of reaction, we are led to such categories as habits of work or study, attitudes, interests, insight, nodes of thinking, types of appreciation, creativeness, and the like." The Commission presents the following as objectives of the second type. I. Ability to think clearly a) Gathering and organizing data b) Representing data c) Drawing conclusions d) Establishing and judging claims of proof _______ ______________ 17. A Preliminary Report by the Joint Commission of the Mathematical Association of America, Inc., and the National Council of Teachers of Mathematics. The Place of Mathematics in Secondary Mucation. Ann Arbor, Michigan! Edwards Brothers, Inc., 1938. pp. 10-24. 440 Hr t0 Use inforG1ation, concepts, and general princi-oles III. Ability to use fundamental skills IV. Besir able attitudes a) Respest for knowledge b) Respect for good workmanship c) Respect for understanding d) Social-mindedness e) Open-mindedness V. Interests and appreciations VI. Other objectives, such aB health, citizenship, and worthy home membership.H (3) Studies b.v Christofferson and Fawcett Christofferson 1 ft classifies the objectives of demonstrative geometry into two groups: (l) Practical, immediate, or direct aims; and (2) Indirect, transcendent, or cone oraltant aims. He points out that, "Geometry achieves its highest possibilities if, in addition to its direct and practical usefulness, it can establish a patternof reasoning; if it can develop the power to think clearly in geometric situations, and to use the same discrimination in nan—geometric situations; if it can develop the power to generalize with caution from specific cases, and to realize the force and all-inclusiveness of deductive statements; if it can develop an appreciation of the place and function of definitions and postulates in the proof of any conclusion, geometric or non-geometric; if it can develop an attitude of mind which tends always to analyze situations, to under stand their inter-relationships, to question hasty conclusions, to express clearly, precisely, and accurately non-geometric as well as geometric ideas." Pawcett,^ in his study, develops the concept of proof using demon strative geometry as a means. He points out that, "A pupil understands the nature of deductive proof when he understands* (l) The place and 18. Christogferson, Halbert C., Geometry Professionalized for Teachers. Menasha, Wisconsin: George Banta Publishing Company, 1933. pp.27-28. 19. Pawcett, Harold P., "The Nature of Proof". New York: Thirteenth Year book of the National Council of Teachers of Mathematics. Bureau of ^tyblications, Teachers College, Columbia University, 1938, pp. 1012 . 441 significance of undefined concepts in proving any conclusions. (2) The necessity for clearly defined terms and their effect on the conclusion* (3) The necessity for assumptions or unproved propositions. (4) That no demonstration proves anything that is not implied by the assumptions.w He further states that if a pupil clearly understands the aspects of the nature of proof f,his "behavior will he marked hy the following char acteristics? (l) He will select the significant words and phrases in any statement that is important to him and ask that they he carefully defined. (2) He will require evidence in support of any conclusion he is pressed to accept. (3) He will analyze that evidence and distinguish fact from assumption. (4) He will recognize stated and unstated assump tions essential to the conclusion. (5) He will evaluate these assump tions, accepting some and rejecting others. (6) He will evaluate the argument, accepting or rejecting the conclusion. (7) He will constantly re-examine the assumptions which are behind his beliefs and which guide his actions.11 This, then, brings to a close the major sources^ having a direct hearing upon the aims or objectives in teaching mathematics and par ticularly that of demonstrative geometry. 20. For a further treatment of sources that are related to the teaching of geometry the reader is referred to Beatley, Halph, op. cit., .Appendix "A«, pp. 344-379, also pp. 401-440. (This is an annotated bibliography.) appendix e Examples of Self-Evaluations by Pupils of Experimental Groups 443 ■APPENDIX E (l) Evaluations Toy Pupils Pupil_ No. 310 How This Course in Geometry Has Helped Me (First Semester) I think that we have learned more in our class by having to work the proofs of the different theorems and problems by ourselves, I never did like the idea of having the teacher work them out for you as I know is true of some of the other classes which my friends attend. They even have to memorize this stuff. I know that by our method we remember the theorems that we prove much better and that is why I feel I can work problems better than my friehds. This course has helped me to think more clearly and especially when reading newspapers and books. When I read newspapers I now have a more questioning attitude and I think I am able to form better opinions. (Same pupil at end of second semester) I think that this course in jgeometry not only helps us think better, but encourages us to think more. Before I took this course, I didn!t like reading, and used to shirk making booksreports, but after this course I seem to be able to understand the reading material better, and I am now able to pick out the important parts, and express myself in a way that there is no doubt as to what I mean. I also seem to get more out of lectures and educational motion pictures. This course has also helped me convince others of the facts that I know. At a recent meeting of our debate club, I convinced the members, of whom 85$ "believe the communistic teachings and principles, that communism is an idle dream and valueless. Without this course I would not have been able to accomplish this taik. Pupil No. 129 Self-Evaluation (First Semester) I think during this course I have accomplished a great deal toward the way of thinking clearly, speaking my thoughts accurately, and not talking before I think. I feel that this course has accomplished what it has been trying to teach me. (Same pupil end of second semester) This course has effected my reading, thinking, and conversation a great deal- When I read something, I do not just glance at it as I did before. Now I have learned to read it carefully, and thoughtfully, and 444 I look for all of the assumptions and implications to he found. As for thinking, I now look at two sides of a story, I try not to let -prejudice interfere, and I do my best to think twice "before I speak. I am learning to think more critically and I find myself getting along much "better than before. I think that this course should be compulsory for every student, because it does not only help you now, but it also prepares you to think or talk logically your way out of many difficulties, if necessary, in later life. Pupil Ho, 221 Self-Evaluation (First semester) I never realized until now, just how little I ever stopped to think about things. But I'm glad I can say that I'm beginning to wake up. And I also can say that now when X pick up a newspaper, or a book, or hear someone talk, I turn over in my mind what has been said or written. At the beginning of the semester I didn't like this way of teaching geometry, but that was because I didn't know what it was all about, or was prejudiced, but now I think differently about it. So far this course has been a help to me, (Same pupil end of second semester) Although I am not positive, X think this course is the cause of ray being more critical than I used to be. In this notebook I wrote about an expression I had heard, for nonmatheraatical material. I'm going to use this expression as an example of what I mean by my being more critical. I have heard this said time and time again and have always accepted it without questioning it, until lately. This is the example: A man was speaking of another man. He said, "He's a good man." The man speaking meant that the man was good in his work. A person may be a good worker and still not be a good man. The word good needs to be defined in this situation. Pupil No. 24 What I Have Learned In Geometry (First Semester) In some ways I like Geometry. I think it's very interesting to try to solve propositions. But there are some things about Geometry that I don't careifor. I can't seem to understand construction to a point where I can make it without someone else's help. I can't devote more than half an hour at home to my geometry. I have loads of other homework, a dentist's appointment in addition to three or four hours of practicing ray violin. I am learning to question statements and to think before I speak so that I'll not make a statement I have no proof for. This course has taught me to question the statements of others, to beck ray statements with evidence. 445 (Same pupil at end of second semester) I have learned to think more clearly* with a better understanding* I read better books, books of value, books that teach one something* This book is about history, people, and many interesting things* In respect to conversation it has also done a great deal for me. I bring you an example. I belong to an organization which is run through the country. At each meeting half the meeting is devoted to culture. We discuss politics, current events, history of various places, lives of great men and women and a score of others. At first I took very little part in the conversation or discussions. 1 couldn't express myself well enough to bring out my point or what was of value. How, if I must say so myself (though I dohlt care to brag) I am considered a valuable member; I have power and hold an important office. In discussions with ray father on long end serious problems of today on various subjects, he has found that I ask more sensible questions, questions of more intelligence. This my father has noticed since I have been taking geometry. I am more critical than before, also I take more interest in the daily newspaper. My progress in geometry itself you would say isn't very much, but knowing myself I find that from the geometry itself I have learned a great deal and I *111 very glad of that. Pupil No. Ill My Candid Statement (First Semester) This method of geometry has helped me to think out problems more thoroughly. Yfhen I first came into geometry I didn't know how to do a problem unless the figure was drawn, but now I am able to do a problem with just a statement given. (Same pupil at end of secondsemester) The course in geometry has helped me to weigh statements before I accept them, therefore I think more critically. It has taught me to choose the assumptions and implications of certain statements which makes me look at* it from all view points. This method has also taught me to be less prejudiced. I think anyone gets more out of any subject when they have to tmrik it out by themselves as we do. When you see you get something by your self I think you have then accomplished something. | It makes things outside of school also interesting because you always are alert to motives, meanings of words, implications, assumptions, and conclusions. Pupil Ho. 235 Self-Evaluation (First Semester) In my opinion if you do not get the geometry at the beginning of the 446 semester it is harder all the way through to get on to the rest of the work* And now that the semester is drawing to a close I wonder whether or not I got anything Out of geometry. I also wonder whether it has helped me to think more clearly and reason things out as it is supposed to do. At times I think it has helped me and then at times 1 don't think it has. As far as geometry alone is concerned I have learned something, not overly much. I don't especially like geometry and I think that has influenced me a lot as far as learning geometry is con cerned, for one thing I do not like to make up any notebooks, but I don't mind writing the problems as long as I understand them. And as X said before sometimes I think geometry has helped me, (although I don't see where 1*11 need it when I graduate) and sometimes I don't think so. (Same pupil at end of second semester) it. When I first started geometry, X was very much prejudiced against After completing a year of it, I certainly changed ray mind. Instead, 1 believe my course in geometry has been very helpful to me. First, it has taught me that being prejudiced about something I know little about is foolish. Secondly, it has taught me to be more skeptical of things I read, not only articles in the newspaper, but advertisements. It has taught me to be able to judge a good advertise ment when X see one. It has shown me the importance of insignificant little words as* and, no, so, but, etc., words that can change the meaning of a sentence. I believe it has made me a slight degree more intelligent because I judge things impartially. Lastly, my course in geometry has taught me geometry, and it was taught in such a way as to make it fun to take it. Pupil No, 153 Self-Evaluation (First Semester) This course has helped my thinking and reading. It has taught me to look for the assumption and see what the sentence is based on. It has taught me to criticize a sentence, paragraph or a story by seeing if its assumption is correct. This course has taught me how to prove triangles congruently although I can't do them very good* I believe I am getting to think more critically. I can now analyze a statement more clearly than before because this course has taught me where to look for the assumption, or the implication. (Same pupil at end of second semester) This course has helped to develop ray thinking on non-mathematical things and a little on Geometry but I still have raj/ doubts about Geometry. Even though ray opinion matters little I believe that this type of course should be inaugurated with all Geometry teachers as it is very difficult for the student to adjust himself to the different ways of teaching. 447 I think that they should use more time on analyzing statements because to the studenti it is interesting and also enlightening. It teaches him to he careful of what he says, how he says, and to organize his thinking, I think I can analyze statements more clearly than I could first semester, hut as far as Geometry is concerned it is something which I cannot adjust myself to, Pun.il No, 95 .What I Think of Geometry (Pirst Semester) I don’t think geometry has helped me any, I don’t think it has made any''difference in the way I think or the way I thought, (Same pupil end of second semester) This new way of teaching geometry has not helped me in any way I can think of. Maybe later on I will find out this course has helped me in some way* If we had books I think I could have learned geometry better than having the problems on the board or trying to work out ourselves without ever seeing a problem like it. Pupil No. 161 Self-Evaluation (Pirst Marking Period) The course has helped me in many ways, (a) Kot to believe all that I hear but to try to prove the statements made one way or the other. (bj It has helped me develop my mode of thinking, so that when I do homework, I can concentrate on one subject at a time and hence obtain more knowledge in a shorter amount of time, (c) It has helped me organize my work, in other words "I first plan my work and then work my plan.'1 (d) It has helped me express myself clearly, and to bring forth the point which I wish to emphasize, (Same pupil at end of second marking period) This course has helped me to think critically and also to not believe all that I hear or read. It has also helped me develop a questioning attitude in my mind. It has also caused me to think more clearly end more rapidly. I am able to state the following examples to prove my above statements. My father, being buyer of Men's Wear in a department store, comes in contact with salesmen. Some of them misrepresent their merchandise and Ithey try to put something over on the buyer. There is one particular’ instance when I used ray geometric training and was conscious of it. There may have been more of which I was unaware. I happened to be present one afternoon when a hosierysalesman came in. This salesman offered ray father a hosewhich he called"Chardinized" silk, much cheaper than he could buy a pure silk hose. 448 Tills hose carried a label which read "j?ure Chardinized Silk11. The salesman claimed that his hose would outwear any silk hose. I asked him how the hose was made and if it was better why **it sold for less’1. The salesman tried to explain that the hose went under a very detailed process. -Asking what that process was, he explained it to me. I already knew how rayon hose were manufactured and comparing the process I found them exactly the same. I questioned him further and asked him if his hose was not a rayon. He admitted that it was a rayon hose under a fancy name, “Chardinized Silk". Another afternoon a shirt salesman offered my father a men's shirt two dollars per dozen cheaper than other shirts on the market. He claimed his shirt was full cut, standard tailored, pearl buttons, vat dye, and pre shrunk. Hot seeing how it was possible my father and I compared this sample shirt with one of our regular stock. A man's shirt usually has seven or eight buttons, this shirt having only six. We also discovered that instead of the tail being square it was cut round, thus saving material but cheapening the shirt. When the tail is round, the shirt frequently hangs loose instead of fitting snug. This is not only uncomfortable but impractical. This course has taught me the scientific method of attaining knowledge. When one uses this method it is possible to learn more in less time. The method is as follows; We must first have a problem. For example the Law of Gravity. We see an apple fall and form a theory. We must then gather all the data and material possible. For example the size of the apple, the distance it falls, etc. When we are gathering this data we must not have the theory in mind but must take the material just as it is so as not to form false con clusions. We must, however, have the problem in mind at all times. With the data we prove the theory if possible. We then reapply the theory to see if it will work in ary case. If it does work in any case it is a law, if not, we must modify the theory to fit all cases. (S£tae pupil at end of third marking period) This course has taught me to think critically. This does not mwan that I do not believe what might sound logical but I do not accept same as a fact until definite proof is given. This course has also taught me how to prove my point and express myself in an understandable language. I will take the following example to prove my statement. It is a fact that my brother is color-blind. It is a fact that my mother's brother is color-blind. It is also a fact that I am not color-blind. I read in a Zoology book which is considered an accurate source thfe following! "In cases of color-blindness, in which the affected person cannot distinguish red from green, a color blind father mated to a normal mother has no color-blind children, since XY zygotes that develop into males possess one normal X-chroraosome and orie Y-chromosome. Similarly in this 449 case* the X zygotes develop into normal females since only one Xchromosome hears the gene for color-blindness, which is recessive to t e normal condition. In the Eg generation, however, one half of the grandsons and one-half of the granddaughters carry the gene for colorhlindness as a recessive. In other words, when a normal father is mated to a color-blind mother all of the sons hut none of the daughters are color-blind, and one-half of the grandsons and half of the granddaughters are likewise affected, I therefore concluded that my mother she was color-blind all her sons would be is not true; hut that my mother’s mother, blind, because her son is color-blind and grandsons are color-blind. Pupil No. 165 is not color-blind because if in turn color-blind and that or my grandmother, is color also because one-half of her Self-Evaluation (First marking period) Geometry has helped me in many ways. Since I have been taking this subject I find that I am more capable of thinking things out and reasoning with myself about whatever problem that happens to arise. Not Only in Geometry do I not think of things from all angles before making a. state ment, but in many of ray other subjects also. For instance, in English, we are now studying the different kinds of clauses, their modifiers, and how the clause is used in the sentence. This requires a great deal of reasoning and thoughtfulness. Then too, at times you wonder what you are studying a certain subject for, and you immediately look for all the motives which will explain it to you; Just as in Geometry you will look for the i theorem or assumption that will help you to solve a problem. From this course the greatest accomplishments that I have made are: 1. To reason or think more deeply but clearly. 2. To reason more quickly. (Same pupil second marking period) As I have said in my last “Self-Evaluation”, that Geometry has helped me to think more clearly and qjiickLy, I now repeat it. Of course, I cannot say that I now can figure out any problem that should arise, for I cannot. To prove that I have progressed in my thinking ability, however, I will relate an incident that occurred to me in Grammar school. I always enjoyed Arithmetic in the lower grades but when we began to have “thought problems” (the problems which are written out in words instead of in numbers) I Just detested Arithmetic. Naturally 1 would detest it because I was^not able to work that kind of problems. I am now able to attack these problems and I generally obtain the correct answers. (Same pupil third marking period) During this last quarter I don!t believe that I have made such pro gress in my thinking ability although I know that I have progressed a little, for the work becomes more difficult and I am still quite capable of keeping it up to date. 450 I now for me and minutes of great deal at times. as well as spend more time on my Geometry homework as it is more difficult requires more time. I spend, on the average, about forty-five my time working on Geometry, every day. Y/hile preparing a of my homework, it is necessary for me to do some deep thinking, This aids me in thinking more deeply in my other studies in Geometry. I read mostly fiction hooks, outside of school. They are either mystery, adventure, or wartime stories. I read a great deal. Pupil Ho. 155 Self-Evaluation (First Marking Period) To think clearly, brightly and factually is sometimes a hard thing to do. Because of this Geometry is a subject every student should take as it helps you to think more clearly and accurately. One day I went to a court and took special notice of the lawyers. Everything they said or did had a reason so full of meaning that you had to believe every statement they said. X have learned to reason things a little bit better and have taken pride in it. (Same pupil second marking period) It does not take very long for a teacher to find out about her pupils; if they are naturally intelligent, stupid or just not wanting to learn. The pupil has the same instinct as the teacher and judges her; if she teaches so you understand, if she is giving you the mark you think you deserve and other little things she would least suspect. Perhaps in a teacherfs opinion this has nothing to do with self-eval uation but to the student who judges him or herself by certain char acteristics it means a great deal. As soon as you begin the study of geometry it becomes a hase to you, but gradually sifter learning a few facts and getting something straightened out you take an interest ini it. You then feel good when you can answer a question that the teacher asks you. I can say truth fully that I never cared extremely for geometry but this ye?r I have tried to do my tpery best and in my honest opinion I think I can be proud of my accomplishments. The one statement I can write backing up my reason is that I tried to the best of my ability. Perhaps the teacher does not think so and she probably has her reasons but so has the pupil. Thus I have stated my reason for the previous statements. Receiving a mark of good this month was what I deserved, no more and no less. I feel that in the future I will be able to raise my grade to an E or S but this is not important, the main thing being that I understand geometry and what it will mean to me in the future. (Same pupil third marking period) The other day I began thinking over the work X had done in geometry. 451 I found that I had taken a bigger interest in it since I understood geometry so much better. My work seemed to be the same with perhaps a more willing attitude. I found that in my other classes I had tried to improve myself by clear thinking interest and more thought to ray work. Most of the time when I came home from school I did ray homework because I was compelled to but now it seems that instead of thinking it a compliance I have found it exciting to be able to see what you can accomplish in a certain length of time, I have come to realize that jgeometry not only helps you when taking it but lays a foundation for most of your work. Pupil No. 199 i Self-Evaluation (I’irst marking period) The more I go into the study of mathematics, the more I realise the little I know. Geometry is such a vast subject. It embraces and supports all the sciences. Practically everything one can think of has its roots in mathematics - engineering, architecture, the industries using chemicals and many other things. i Now, after I have studied geometry, I unconsciously look for things such as triangles in the bridges as I go along the street. I notice the many instances every day where I hear geometric terms. The other day I |was listening to a conversation about sailboats and the words perpendicular, adjacent, and parallel came to my ears many times, ^n the radio in the question bee program* many questions concerning geometry are asked and I believe the greater number of times they are answered incorrectly. It is a subject few people are educated in. The other day as I was reading the life of Madame Curie I saw how much she used geometry and all forms of mathematics for her great discoveries in physics. They were absolutely necessary for her experiments. j (Same pupil second marking period) When I first began taking geometry I had an entirely different |conception of it, than I have now. I believed it to be a, rather boring j subject in which there would be no value to a girl like me who would ihave no material purpose for it. Now, these old ideas have changed and I look on geometry in a new light. I find it intensely interesting and therefore I try to do my best work in it. I have found that it develop! one*s perspective and ability for clear thinking and analysis !and I have come to like geometry very much. I read on an average of two to three books a month. I can*t find time to read more, although I would like to. I enjoy all sorts of books and some I have read lately are "A Princess of Thule" by Black, "The Blue Window” by Temple Bailey, "Ramona" by Jackson and "Showboat" by Edna Perber. I believe this course has helped me to think straighter and clearer 452 than "before; it has helped me in my conversation by giving specific facts and examples to back up statements so the people find them to be true instead of being merely suppositions. As for my accomplishments from this course, I have tried to keep a notebook that I can refer to years from novr and still find a help. I have gained a great deal of knowledge fxsom the course proper and it has helped me to increase my vocabulary. I spend about a half hour on geometry a day; sometimes I spend much less and sometimes more. I simply prepare the lesson for the next day and then I am thropgh. I believe geometry has helped me with my personal problems. I was fortunate to be able to continue with this new method this semester instead of having to change to a teacher who teaches the old way. I think I will derive more benefit from the course because of this advantage. Some specific examples of the way this course has helped me arei I was able to analyze my needs for the subjects X chose to take next semester. I talked it over seriously with my parents and I was able to think back of the subjects and I realized the help I had obtained from geometry. I was also called upon to make a personal decision which I do not want to mention here and I thought through it clearly and sanely and I believe my decision is working out for the best* (Same pupil third marking period) As J go on with the study of geometry, I realize more and more how much it is helping me to think clearly and accurately. I don't believe the geometry itself will help me for I do not intend to go on with mathematics. I would like to continue the study, but I have other subjects to take in the next two years. I really don't think geometry will help anyone who has no desire to go into some type of engineering, but the benefits unconsciously derived will help us all. I think it is fun to take a problem you know nothing about except what is given in the hypothesis and work around with it for a solution. It is like wwrking out a puzzle, only here we have to have a reason for every piece we put together. Geometry is a fascinating subject, much more interesting than algebra, because here you can express your own ideas and in algebra you followed a set pattenn given to you. Recently I was able to put my benefits from geometry to use. ky !!father received a circular advertising some sort of encyclopedias at an jexceptionally low price. He brought it to me thinking this might be a grand opportunity to buy some new books. We sat down and began to examine the advertisement carefully. It really sounded fine, and we got down to the last page and there seemed to be no catch. After we really thought it over we decided there must be something wrong someplace. Just for the idea of testing the truth of our assumptions, my father called the company. Then we discovered we could obtain the books at this price with the purchase 453 of an expensive "bookcase and really the cost of the case would more than cover the deduction on the hooks. I hope no one was roped in on this dffer and I'd really like to find out how many people are being taken advantage of by chiseling merchants "because they cannot see through a fraud. The following self-evaluations are from a pupil not involved in the experiment hut transferred at the beginning of the second semester from a formal class over into one of the Experimental Groups. Pupil* No. (transferred from a formal class) (First Marking Period) This summer, before school started in the fall, I was carefully thinking over my course that I planned to have this year, I liked English, especially the literature part of it, so that wouldn’t bother me. My marks*in that subject were usually my best anyway, so X didn’t worry about it. I insisted on Applied .Arts II, and much as X wanted to get Medieval History in, I would not substitute it for my most favorite subject. Then there was French, It would be my first association with any foreign language (never having had Latin) and the thought of it rather excited than WDrriid me because of the newness of it. Of course I must take dear old geometry. Thinking of my shameful marks in algebra, and how much harder geometry would be (so my sophomore friends informed me) it made me feel very downcast, and X did everything I could to erase the whole business from my* mind, and worry about it when the time came. When I discovered that I was to have the same teacher for geometry that I had for algebra, I really didn't know whether to be glad or sorry. I would have liked a change but still I could see no serious reason why I should bother my advisor with unnecessary burdens, so I left it as it was, and decided to do my best. I am a person with the best intentions in the world, but that is often as far as I /get. It wasn’t long, and ray marks began to parallel with my algebra marks during my freshman year. It disgusted me immensely, because it wasn't as if I despised the subject, or hated -the very thought of its name. I really liked it in a way, because the thought of reason ing things out and being able to prove things true, step by step, really fascinated me. Still there was something missing, and I determined to ask my teacher every little thing I didn't understand. I would have it on the tip of my tongue, and suddenly decide that I simply couldn't ask him such a simple question. He would think me hopelessly ignorant, so I would let it go. After a long time of this, I developed an inferiority complex on the subject, and decided that I was immune to the comprehension of such a subject. My low marks continued, and I assumed a rather hopeless attitude. I was sure that my teacher much as I liked him, was plainly disgusted with my marks, and I dredded geometry. I even decided that I would digest my lunch much better, if X only had any other subject except 454 geometry after it* I suppose it sounds silly, "but nevertheless it is true. A geometry test spelled horror to me, and it took a long time for me to overcome it. When I found out that I was to have a new teaoher for the second semester, I was very curious to see whether this change would have any effect on me. Your new method of teaching interested me at once, for it is very different from anything I have yet heard. I decided to start j all over again and get what I could from your ideas, and apply them. I The nonrmathematical side of it, and reasoning in outside subjects interested me particularly. In my English class we were studying SilaB Marner. In the near front of the hook, Silas Marner is robhed of his gold, while he was away from the cottage. The village folk decided that a peddler had stolen the gold because "not long ago, a foreign looking peddler had stolen some money in the village and foreign looking men, especially peddlers, were usually robbing or plundering anyway." Therefore, the villagers sat back and accepted thie reasoning as a fact. Immediately I detected the faulty reasoning on their part. If a peddler who was foreign looking had once stolen some money inJthe village, what sign was that that it had happened again? The villagers could only relate one incident to which the situation applied. Even if they had mentioned two, it would not have been enough proof that every time some money was stolen in the village it was "probably taken by some foreign looking peddler." When I awoke to the fact that these people, and other charactersin the book were making countless unreasonable statements which had no backing, I enjoyed the whole plot much more. ALso, there were many cases which were too much taken for granted, and not looked into, I was conversing with ray aunt the other day, and she made this statement. "No, Idon't care for Sinclair Lewis' books and most people don't either." Immediately I asked her why, and she gave me a very suitable answer. Still not satisfied, I said, "That was rather a radical statement you made about 'most people not liking his books'. Why did you say that? How do you know that most people don't like them?" She seemed rather surprised at my direct questioning, but she answered with a light tone of indignation, "It's what I've heard of course. You know as well as I do that few people care for his books." But I persisted. "What you've heard isn't very substantial evidence, is it?" She looked at me rather queerly - as if I had told her she had 5 minutes to live. Presently she said, "No, I suppose that isn't much evidence. Say, are you suddenly going to turn to that lofty profession, attorney at law, or something?" I smiled sweetly, and said, "My geometry is taking effect on me." She gave me a very blank look and went out of the room. It wasn't until I had finished my conversation that I really woke up to the fact that my geometry had a great deal to do with that little wituation. 455 These are just a few examples of how my outside work is being done unconsciously with this interesting light thrown upon it. I have begun to take mental note of things that sound unreasonable to me. I know that this new method of learning geometry has a great deal to do with this* Geometry holds no such horrors for me as it formerly did, and outside of learning and understanding the subject much more thoroughly, the whole idea is fun. I can see now some of the errors I have made during this first six weeks. Because I am really interested in the nonmathematieal side of it, I am ashamed not to have written down my reactions and handed them in. However, now that I am thoroughly ac quainted with your methods, I will carry on differently in the time to dome* (Same pupil second marking period) Mathematics of any kind had always been the most detested subject on my whole school program. The reason I hated the very thought of it was because I didn't understand it. Another reason was, that it was always required. X believe that many people, not just pupils, dislike their work because they have to do it. Often, when people do not under stand the thing they are supposed to do, they unconsciously become afraid of it, and it is human nature to hate a thing one is afraid of. This was the case with me. When Mr. _______ marked the reasons on my report card for receiving D's, there was one which predominated all through my Freshman and the first semester of my second year in his class. It was "lack of fundamentals". X don't know whether to blame the teachers that I had in the lower grades or bleme myself. .Anywgy, that is beside the point, but it worried me. Mother began to notice that this reason was on all of my report cards too, so we started drilling on the fundamentals of arithmetic a certain portion of each day. I seemdd to know my arithmetic rather well, but I was too slow in getting my answers. By just drilling on a few "do in your head" arithmetic problems each morning before school, I soon found that I reacted much quicker to the algebra problems in school. This encouraged me a great deal, but my marks stayed in the D's. As I said in ray last six weeks self analysis, I developed an inferiority complex about geometry, and all math in general. I would do a test, and when I went out of the room, feel quite proud of myself for finishing the test, completely. I would expect a C at least when my paper was returned, but no, it was always a D or a C-. When I would ask about it I was usually told, "No, not quite & C, but almost." I was satisfied with this answer, but when the same thing happened again and again, I became very disgusted with myself for being so ignorant. Also these marks made me fear every hour spent in ray geometry class. I was almost afraid to look at ray adviser straight in the face each morning, for fear she might catch my eye which would remind her to give me another low notice to take home. Once or twice I thought of geometry as being useless to me. Here I was, going to go to a school for girls for two years, where I would conr centrate on French, literature, and art. Then I would spend two years at an art school, and try to become an illustrator and possibly a portrait painter* Where in the world would geometry ever fit into that schedule? 456 It seemed a waste of time. It never entered my head that you could apply geometry to real life problems, until you became my teacher. 3Tor the first few weeks, it seemed much more like a psychology class than a geometry class. I found it becoming much more interesting and very new. Unconsciously I began to analyze sentences I read in the newspaper, but mostly things that people said. I began to notice how many people - people who are considered very intelligent - jumped to conclusions. .Also I began lb analyze, not just words, but things and people themselves. I met a very interesting person just recently and her profession was identical to the one I'm most interested in. After she told me a good deal about herself, I mentally began to compare her with myself. Ifihat opportunities did she have? Do I have more? Did she make the best of her opportunities? I didn't think so. She is too limited in what she can draw. She had the chance to learn more. What schools did she go to? Were her choices wise? Would I be successful if 1 had her education? I reached the conclusion that she was a very nice young woman, she drew well, but she could only draw certain things. She would have a higher position if she was just as talented in drawing, sketching, and painting many things instead of just a few. She regrets her mistake and is very conscious of it. Howevdr, I resolved then and there, not to limit myself to a few things, but be able to do many and do them well. Perhaps this seems far away from geometry, but a few days after ray meeting with this artist, I realized that in ray own way geometry had a great deal to do with my thoughts. Why don't I reason a little more in geometry? It worked backwards with me. I learned of applying geometry to life problems and interests, but instead, I'm applying life interests and problems to geometry, and am very grateful for all I've learned. I | (Same pupil third marking period) ! I I sincerely believe that I received much more from geometry when it is taught in the way of critical thinking than in the use of plain, formal math. I know and will frankly admit that there are just a few theorems that 1 could sit down and prove without a flaw - a very few. However, this doesn't seem so important to me. If I go on to an art school when I finish high school, I will have absolutely no occasion to use geometry, but I will have to be able to think very critically. So many people believe that to paint a picture, the artist does something that he sees before him and paints it. He does it better than a lawyer, doctor, musician, or pnyone else would do it because he possesses the oitle of "artist". How untrue this all isJ People do not understand that the work isn't done with the hands, but the mind. The artist paints exactly what he sees in his mind, and he never starts to paint until that picture is completely analyzed like a geometry theorem or any life problem. The simplest part I of an artist's job is to paint the picture; the hardest part is to study it, and analyze it while it is still in the painter's mind. One difficulty is that you do not then have the advantage of taking notes and writing down the steps and reasons. All of this seems rather off the subject, out it really proves that I am going to have to think very critically if I ever expect to become 457 successful. This is where this critical thinking course comes in. 1 must admit that my marks have not heen what one would call good. However, I have gained some knowledge in the non-malhematical unit which is very important and helpful to me, even though my test grades were not exceptional. One of these things is that marks do not frighten me as they used to. I have learned that my marks are not the important thing but it is what I get out of the schooling that counts. There are many things that have never come up in tests which I've retained* I am sorry to say that it is not mathematics that I've gotten so much out of, but the non-mathematical work. For instance, v/hen I have completed a math test, I hope and pray that 1 have passed it and receive a decent mark. It is different with the non-mathematical tests. When I have completed one of these, I feel that the period was well spent and am rather pleased with my work. I have no thought of grades, but only wish I could spend another periodon the test to see what else I could dowith it* These are the things Ihave gained in this course. I do not feel as I did in the first semester of formal geometry, that I couldn't go on with it and feel that my time was being wasted* However, I knew I had to finish geometry to go to college. (Since then I've changed my mind, end decided that time was too precious to spend two years in a girl's school "gaining culture". I feel my time would be too precious to lose, so I plan to spend all ray time in art school. However, I'm grateful to this course, and sincerely believe 1 got a lot out of it. Funil No* 279 Self Evaluation (First Marking Period) Since entering this course I have been reading and getting a lot more o£t of my Popular Science and other scientific magazines than I ever did before. I seem to catch the meaning or pick out the vague parts much more quickly than I could before taking up this Geometry. I find too that I express myself in conversation much better and don't use so many words I can't define. This year 1 had a hard time with football signals but having advanced in thinking a little at least I find they don't look so complicated. I get a clearer picture in my mind as to what the play is designed for. I jcan think the plays out more clearly in a game, rather than forgetting j everything I ever knew. I I I get a pleasure now out of finding hidden assumptions in the advertisements in the paper and on big signboards, taking headlines apart and finding out what they really mean. I tried to group together J a lot of the geometric symbols; I have also studied the constructions we llhave had closely and tried to keep an accurate list of definables, undefinables, assumptions and propositions. I have tried to enter into the class discussions as much as I can, and contribute anytning I can to the class for its benefit. I believe I could have done setter in the mimeographed material although I have looked most of it over but have neglected to write down some of it. I don't really enjoy doing that type of thing as well as others. To sura the thing up I am getting as much out 458 of this course as I amputting into I have into what Xhave done. it*However I amputting everything (Same pupil second marking period) This period I have really "been thinking a lot more critically than any other six weeks period as ray Daily Diaries have shown. I have been doing a lot of thinking about the problems the world is faced with today. I have tried to contribute any suggestions that 1 thought added to the discussions in other classes. I don't say so many things that afterwards prove to be wrong but think out a question before I offer an answer. I have tried to pay attention in class and understand what's going on all of the time. I think that ray suggestions in the class this six weeks have helped out in the discussion or at least some of them did. I have tried to offer anything of interest I had to the class or any knowledge that 1 mighthave had. My notebook is asneat as Ican make it and I believe it contains most of the facts we have had this six weeks. I have gone through it quite often putting in things where I think they belong. I believe X understand now. I have really gotten it.I am enjoying this way of many advantages in it over ©11 or most of the geometry we had up till as much out of thA course as X have putinto studying geometry very much and can see the old way. I don't believe I am nearly so ready to sw&llow facts without looking at them from all angles, as I was at the end of the last period. The course has helped me think more clearly in respect to the languages I am taking. I don't make half as many small error® as I used to* i I have been reading a lot of old classics lately that I thought were uninteresting before now but find that they are really very interesting. I have tried to be more accurate in what X write and in what X have to Isay. My English oral themes have been offered in a clearer and more effective manner than ever before. All in all I believe that I have taken a big step in critics! thinking, (same pupil third marking period, semester I) During this six weeks period I believe X have advanced in critical jthinking more than in any other period X can remember. I read the article, "The Tyranny of Words", and although I didn't understand all of it X got a vast amount from it which X wouldn't have gotten before, if I had read it at any time in the past. I have worked constantly on my folder and now I believe I have some thing to show for ray efforts. I probably could have done more in the nonmathematical situations but I was too much interested in the other materials 459 I plan to do a lot more of the former group next month. I collected eight newspaper clippings and thought that was enough for this six weeks "because I had so many last period. I also believe that they are not too important because all they are for is to help you learn how to get under the face and see the inferior side. .Also for you to see if I can recognize the hidden assumptions in articles and adver tisements. I have tried to offer whatever and whenever I could to the class-room discussions and tried to contribute anything of interest to the class that I possessed. I try to get as much acut of the class as I can and I feel you ©re really getting somewhere in the way you are teaching geometry to us. If I had geometry taught to me the old way I know I’d be getting on way "fcelow average. You have kept me thoroughly interested so far and I believe you will keep on. I have found that I can talk with more assurance in class than at any time before. My conversation at home and elsewhere has been improved very much, that is to say X can express myself better, think clearer and not use so many dangling words. X can reason out what I want to say and then say it more sensibly than ever before. I ’ve tried to reason out what you’re driving at in this course and I believe I have caught on. I can focus my thought on one item as I never could before. I am beginning to fathom v/here I stand in the world today. I can think ahead more readily and understand better what we are faced with in the United States, and the rest of the world. I have enjoyed this course in every way and I certainly hope you will include my name on your list for second semester. (same pupil fourth marking period) During this six weeks period X thought the work we accomplished was by far the most interesting and the most fun that we have done so far this year. All during the period I have tried to keep up and understand what went on in class. At this point in the work I believe I have understood and grasped everything which went on. I have tried to put everything of importance in my notebook end kept it as neat as possible. During our class discussions X have been able to see how the problem is worked and have it all figured out except for yesterday and then I was really trying to figure the problem out from a different approach than the class was taking. I think that the work in this class has developed my thinking very much this period. Without it I don’t think Ioould have seen the solutions to a lot of the problems we worked out. I have tried to contribute whenever I could in the classroom discussions and offered suggestions that aided in the work we are doing at the time. In my opinion I have kept up the standard that I have set for myself and kept up the same grade of work that I have been doing in the last grade periods with pessibly a few improvements. 450 (Same pupil fifth marking period) During this six weeks period I have enjoyed the work we have done more than any other topic we have studied up to now. Measurement of angles waw very interesting hut I have enjoyed the study of similar triangles much more, as far as we have gone. One reason is that it is more difficult to understand and when you do a certain problem you feel as though you've done something. It certainly brings in lots of new jconceptions and I can see now why you've said that many problems can be solved easier by use of the similar polygon formula. I One day during the last week of the period I examined a surveyor's instrument at school and found it very interesting. I was so interested |that I went to a civil engineer's office whom I know and asked him to (let me look at some of their instruments. It's really very surprising |what the surveyor can do with all his equipment, j j I can't seem to get interested in the non-math side of geometry. |I've tried to but I can't do it the way the rest of the class can. As ja consequence I have not done very much in the non-math field. I I have again tried to offer any suggestions whifch might be of value jto the class. i j This six we9ks in my note book I have: the unit on Areas of Polygons |and similar triangles. In areas of polygons I have a written proof for, j the area of a rectangle A ss BH, a square A = 5^, a rhomboid A s BH, a itriangle A s J BH, a rhombus A ss dd'/'2, end a trapezoid A - |-h(b+b'). JAlso a formal proof for the Pythagorean Theorem c2 s a^ + b2. In the Iunit on similar triangles I have written out 3 ways of finding unknown jheights of trees, etc. I have proofs for six facts which we developed Ion proving /§W. 1.3£-2£/§\ ar 5<xl»2•& £ s a 2 corr* are^. |3. An acute ^ of Rt to corr. acute /_ of Bt are^. 4. 2 sides jproportional to 2 corr. side and included £ are^v/. 5. 3 sides Iproportional to 3 sides are /\>. 5. 2 have sides parallel to each !'other ^ are ro . One new construction developed in class. I believe i I have made my notebook neater than any other period, which is an j!improvement. I believe my notebook contains all but maybe one or two !of the theorems which we have developed this six weeks. I may have 'missed these the day I was absent. It also contains a number of extra i theorems and problems* ! (Same pupil sixth marking period) ii ! During the past school year I have enjoyed the course more than any ' ■other I have taken in New Trier. When I came in to it last fall I thought II was goin£ to dread it but since then I have changed my mind. This course jhas made me a much clearer thinker. I can reason things out much better !than I ever could have without taking this course. I have this six weeks |as,well as the past ones tried to contribute anything that might be interjesting or helpful that I knew. When we made our diaries I tried to reason 461 out critically what happened in my life day by day. I have tried to keep my notebook as complete as I could and I think it includes all of the important things we have done this year* -Any test results have been good during the whole year. I have tried to pay attention in class and to get everything of value out of it. When I came into this class X took in almost anything that was said or done. I believed almost everything I saw in a textbook or heard a teacher say* After completing this course I do more thinking for myself* I don’t so readily swallow everything I read or hear. I have read a few things^ this year on this course. A couple of chapters in Easy Lessons in Einstein and Tyranny of Words which I found very interesting after reading it through four times. I believe I am able to express myself in writing and speaking much better now after this year's work* I can speak with more confidence among older people than before. I don't use as many undefined terms as before. I try to pick out words that mean the same thing to the people with whom I am dealing. To sum up my year in this class I will say that I got out of it what I put in. I'll say agfein I cannot find words to express how much I enjoyed this class. Punil No. 240 Self-Evaluation (First Marking Period) My motives for taking geometry probably differ greatly from the motives of the majority of the class. In the first place I have nfo par ticular interest in the formal aspects of geometry. I have never liked any type of mathematics, furthermore I found math in the grades difficult and I still find it so, which is probably the main reason I don't like it. I am taking it merely because the college I wish to attend requires the erddit* I wish we could devote more class time to non-mathematical work. I think that it is interesting, and gives you a chance to see if geometry on the whole has increased your ability for critical thinking. (Same pupil second marking period) I think that writing a paper like this is quite difficult, but, nevertheless, I will proceed. My geometry work has helped me in many various ways. First: A few weeks ago in my Medieval History class we were discussing the development of men's minds in the Middle Ages. These men, although they knew quite a bit about the arts, etc., didn't know much about logic. This subject was their favorite and by all means should have been their best, but, the "major premise of a problem" and they, so to speak, take it for granted because the old and ancient men said that some of these things were true. They didn't question the old teachings one bit and so they didn't get very far with this study. We spent ^uite a while 462 on this and discussed many problems. Through this studying I have been very careful to look at the so-called “major premise” of a problem. I think that if I hadn't had my geometry this part in Medieval History would have been very vague and unimportant, and I probably could not have seen why these men didn't progress very much. Second, I think that when I enjoy my geometry the most is in class. There honestly isn't anything I like better than trying to figure out a problem. My work in class may not always be right but I like to con tribute anyway. have I really haven't done as much with ray notebook as I would like to but I think that ib has been improved somewhat. Tfifhen I first came into this class I didn't think that I was going to enjoy it at all but now my mind has all bean changed around. Possibly it took me longer to realize what you were going to do and have us do. But now everything is quite clear and I know I will accomplish something now and for myself later on, (Same pupil third marking period) I think that probably the Daily Diaries that we had to write did me more good than anything else in the class. I will try to explain it to you; whenever any problem came up in ray classes or anywhere else I would try and figure it out in a sensible way so that I might put it in my diary. It seems that all the time we had the diaries I was thinking <f what people had said that needed critical thinking. I don't, to be fdank, think that I would have been bothered to figure out what people said, etc., if we hadn't our diaries, and that is why I think it is a good thing. I also believe that it would be a good idea if they were continued. If I had been in a regular geometry class, I don't think that I would have learned half as much. I really didn't like mathematics at all until I started in this class. I enjoy working out problems in class. .Another lthings I like is class discussions, and I enjoy contributing to the class Iwhether it is right or not. i (Same pupil fourth marking period) This semester's work, in my estimation, is much more difficult than last. As you probably know I enjoy the nonrmiGth problems and discussions in class a lot but I just can't seem to \uiderstand the formal part of geometry. The “developing of the mind" as you called it, I think, has really helped me a lot, especially in my history; but as far as formal geometry goes the study is very hard. I have never really been interested in mathematics but I thought that the way the class was run last semester was an excellent way to do it. In the first place I think that the non-math is going to help me a 463 lot more than the other. 1*11 tell you why - This is the last of the math courses that I am going to take and I want to learn something that will do me thd most good# I don't think I'd remember the other for more than two months after I finished the course, anyway. 2. The non-math is a lot more interesting to me. I doh*t think 1*11 ever have the opportunity to use the other. I don*t think girls as a rule, care for geometry or any other math as far as that*s concerned (assumption). I *11 tell you how this has helped $e in my Modern History* We are studying the "French Revolution" and "Napoleorf in History at present. Our class was asked what brought on the Revolution in the way of social problems. There were a lot of hidden reasons, and they would have been hard to detect unless I had some conceptions of what went on in those poor peasants' minds, or at least,what methods we thought they were thinking of. I think that the people of that time did some very critical thinking in order to know that they, by rehellion, could have what really belonged to them (assumption). Another thing that we had to figure out, according to the circum stances was what went on in Napoleon's mind when he became the leader of almost all of Europe, He thought that he was supreme and no one could tell him anything. When he was young he looked ahead with remarkable ability and saw what he should do; he was very lucky (assumption). When he thought that he couldn't be told anything I don't think that he used any critical thinking. (Same pupil fifth marking period) Again comes the time when I must write this paper, and I will try and do it with the best of my ability. I think as you get along in a study of geometry it becomes a lot easier to figure out problems of any sort. When I first was in the class X didn't understand about the motives, etc. and how to detect them. All this has helped me the most in my study of Modern History, I am able to understand why certain people did different things, for instance, why Bismark, a German chancellor, wanted to unite his country, and the motives behind the way he did it. It is also easier to tell whether people are biased by the things that they do. I know I should get better grades in my formal geometrywork but I think the main reason is that it is not compulsory to do the work, Neverthe less I will see how I can improve the next six weeks. j (Same pupil sixth and final narking period) ji | This uaper completes my study of geometry, and, as a matter of fact, all the mathematics that I am going to take* Fowsibly later on I will not work with math or remember the formal geometry but I think that I will remember the other parts of this course that have helped me. X don t see how I can forget them. 464 t took me a while to catfh on to what you called •'critical thinking" hut I believe that I understand it now. I have tried in ray other subjects, besides geometry, to apply this method of thinking. In ray Spanish for instance, you have to figure out sentences. I have learned that it is a very bad policy to make a "stab in the dark" when you are doing this, for mostly always you will be wrong. You have to take each word out and think clearly how it is used in the sentence and everything about it, or else you will probably be wrong. I think this takes a good deal of thinking on the part of any student. I have also tried this policy in ray history. In this way it was easier to discuss how angt why things were done and for what reason they did them. I honestly believe that this course has been a positive benefit to me now and prdbably will con tinue to be so hereafter. Pupil No. 259 Self-Bvaluation (First marking period, semester I) My outside reading usually consists of newspapers and magazines. When I read October's Reader's Digest I found that unconsciously I was picking out sentences that contained assumptions. Also there were words which I didn't believe were specific. I enjoy talking about the work we do in this class because it's interesting and so different from other classes. This makes me more conscious of this work when reading or conversing. i I have brought to class some clippings, for newspapers and magazines containing statements with assumptions and indefinite words after which I have written ray* reaction. From ads I have seen how many assumptionsthe stores make to persuade you to buy this or that. The time I sppnd at home on geometry is not the seme every night. I think of geometry as fun instead of work and therefore do it when I don't have long nightly assignments that must be in the next day. INaturally, Saturday afternoons and Sundays are when I do it or when I'm jtired of working on another subject. I I am sure I have gotten a great deal out of the course already. !'I don't just "accept things" as much as I did. Of course I realize ]I have a long way to go before this even becomes noticeable to outsiders, but I'm sure I have an idea of what you have in mind, buteven if I haven't gotten all of it, I like it, I have been workingon definitions and constructions, the latter of which I'm now sure of. (Same pupil second marking period semester I) i I As I pointed out in ray last evaluation I do a great deal of reading ilas the daily papers and many magazines such as' "Life", "Fortune", I!"Saturday Evening Post", "Reader's Digest" and the "Journal" among the jlist. I don't read books for the fun of it unless I hear of a book that is along the lines I am interested. Although the reading I do, does give me a good opportunity to find assumptions, undefined words and loose statements. 465 Because I am interested in the field of advertising I notice ads quite often as you may have guessed, I believe that the most critical people get, the more facts ads will have to have, which will make it necessary to have the product good. Many more things are assumptions to me that were facts before. I don't believe things I read without having the assumptions pointed out too, as much as X did before, I will make the assumption that I am thinking more critically.Often I hear things in class that need defining of the assumptionsin backof |them. There isn't enough time to stop the teacher or student and ask for jthese things in class* If there were, they'd probably wonder what your |idea was and think you Here acting funny or trying to waste time. Critical thinking has helped me see for myself how important definitions and assumptions are. It has also straightened out the difference between facts and assumptions. It has shown me not to believe conclusions without the assumptions they are based upon. j I do ! i It would be even harder for feel I have ldarned a great me to judge myself this six deal. weeks,but (Same pupil third marking period) I have had more homework this six weeks than ever before, which made my time for pleasure reading almost disappear. I did do some reading in Easy Lessons in Einstein, which I enjoyed enough to make up for all the rest. Everything I found, was connected with what we have been dis cussing in class about non-Euclidean geometry. i The emphasis on recognizing that all conclusions are tentative and that nothing is absolute was strengthened again. I am sure that there has been a decided change in ray critical thinking since I have started geometry. I hear so many people making loose state ments; the papers are full of undefined terms which I never before thought of and I realize so much more now, how important it is to have facts and to prove things by having them tested many times. Also to get as many ideas on one subject as possible, instead of one or two. I have really enjoyed the small pert of this book so much that I want to read the whole thing so as to find out more about the fourth dimension and Einstein's "Tesseracts". The most important thing that I have learned is that conclusions are tentative. I used to think that when a great mathematician set down a ’mathematical law, it was absolutely true and couldn't ever be different. Also I see how totally impossible i t is to prove two clocks in two different cities being together to the very fraction of a second. 466 (Seme pupil fourth marking period) I have read “Silas Msrner" hy George Elliot, "The Header’s Digest", and “The Doctor", "by Hinehart. Of course the “Tribune" daily paper — particularly on Sundays. I haven’t had any time to read for the enjoy ment of reading due to the unusual amount of homework I have had. To think a lot more about things is what I have done. This may not ibe due to this course, but the conclusions I arrive at have, I'm sure. 11 see more and more how unimportant things are that I once thought to be most vital. The statement you made quite atohile ago about “AL1 things done by human beings are subject to error", has made me do quite a bit of thinking. I have wondered why we do something that causes so much trouble and time and effort besides being something we dislike when after all it was only set up by a person or a group of people who are very likely to be wrong. I know this sounds quite radical but I don’t mean it in a destructive sense. .Another thing I have been thinking about is, how relative everything is to everything else. Of course this has something to do with this course because this idea never would have dawned upon me until my old Sage probably.' I haven’t spent anywhere near the time on geometry this six weeks as I did last. Vifhile looking through a magazine or paper, I have found a few things which I thought needed defining or that I recognized as an assumption that someone else had thought to be a fact. Of course it has |taken some time to copy ray notes and diagrams over too. I haven't much idea how much time outside of school I have actually spent on geometry. I do believe I am learning to think critically, slow as it may seem. The reasons being the change in my attitudes and reactions to things. Of course the first six weeks of geometry I learned more about critical thinking than I ever had before, but this six weeks probably ranks second. Although I have been doing quite a bit of the non-geometric work, I haven't got the mathematical part as well as X could with more work. (Same pupil fifth marking period) | In my math notebook I have entered all my notes on Similar Polygons. iALso I ’ve filled all the required assignments. The notes which I entered ’contained some of the proofs we worked out in class, some definitions ijand proportions. Things I have done in connection with this course are |jreading sections, more sections in "Basy Lessons in Einstein" by Slosson. I have read Shakespear's “Julius Caesar" for English, The "Header's Digest" and the “Story of a Country Boy" by Dawn Powell. Also the magazine “Life" each week. I have had things happen which had critical thinking involved and have read quite a few things that I could question almost every term and 467 find many motives, assumptions and underlying terms. I can't remember all of them. Unfortunately I drew a diagram of a "Tesseract" and wrote a report on it which you placed in ray folder. j (Same pupil sixth marking period) I Evaluation of the Year's Y/ork j j This past year of geometry has opened a new and different field to jme than I had ever experienced before. This new field is of course "critical thinking" or thinking about thinking and reasoning. I have enjoyed this more than the regular formal method of teaching math, not only because we didn't have to remember a lot of facts, that I'm sure would have been forgotten within a year, but because we did our own thinking and reasoned the problems ourselves. j It is evident that we could get more out of learning to reason and iprove propositions by reasoning and doing our own thinking than we could jby memorizing proofs which have already been worked and we might not jhave occasion to see or use again. By learning to reason and think well in one field we are able to |think more clearly in others. i| These are a few of the reasons why I have enjoyed this year of |geometry much more than I believe I would have in a formal class* Of course to me the non-math was the most interesting and important jpart of this year's work. The only way it could be better would be to jhave more of this and less formal geometry, and to have it in more schools. Certainly it's important enough, for it enters into all the jother fields and every day of your life. | Uot until it was brought out in this class did I recognize that Ithings which are considered facts, established facts, are quite tentative. It is quite evident now that all conclusions are tentative and that not [much progress can be made until this is recognized. | I This course has affected my reeding because I now like to read about [a subject which is new to me. Books that describe how a person can arrive 'at a conclusion from a system of logic built from fundamental assumptions. ! ! it has also affected ny conversation because in discussing this course with older people I have had some very interesting conversations end have ijheard a few very intelligent ideas expressed. i As you already know, I have read "Easy Lessons in Einstein" and have jwritten a report. Also I have written comments on a few ads from newspapers and magazines. j! This class has helped me a great deal to recognize all that I have said once before - tentative conclusions, thinking about consequences, 458 not jumping at conclusions and questioning the meaning of words as well as looking for all the motives behind statements and actions. j The following is another self-evaluation from a pupil who was not | involved in the Experimental Group, but who was transferred fr©m a formal ! j geometry class at the end of the first semester. This re-oort is included III because of the superior work which the pupil did during the first semester. Self Evaluation When I first entered 2A geometry I entered with an intense dislike |for the subject although I had received very good marks in my fceooftd or 2B j semester. I still had that feeling as if I didn't care whether I learned |geometry or not. Por ray first experience had proven very distasteful, iI loved algebra and all the mathematics I had previously had and naturally expected to feel the same way about geometry but was very disappointed for !instead of thinking out the problem for ourselves we learned straight from Ithe book and all that we had to do was memorize it and recite it for the | j next day in class. i il ji I have been very frank about how I felt at first and am thankful |j that I may honestly say I've changed my mind. When I first realized we liwere going to learn geometry almost entirely without our books I was afraid that I would fall - I was afraid that I couldn't use my awn mind, not for solving geometry theorems at any rate. I | That too has been changed. Perhaps now I am egotistical or vain |but I have acquired a confidence in my ability to think that had not ibeen very evident before. I may have always had that ability and never j ! realized it. 469 APPENDIX I Enrichment Materials Related to Mathematics for High School Pupils 1, References of a Historical and Cultural Nature 2. References Relating to Social and Practical Uses of Mathematics, and to Other School Subjects S. References Relating to the Concept of Number 4, References Relating to the Concept of Measurement 5. References Relating to the Concept of Function 6* References Relating to the Concept of Proof 7. References Relating to Mathematical Instruments 8. References Relating to Recreations and Extra-Curricular Activities 470 1, References of a Historical and Cultural Nature Abbot, Edwin A*, Elatland. A Romance of Many Dimensions. and Company, Boston, 1929. Adler, Claire E., “Calculus Versus Geometry". vol. XXXI, No. 1, January 1938. Little, Brown The Mathematics Teacher, Andrews, Frances E., "The Romance of Logarithms". Mathematics, vol. XXVIII, E ebruary 1928. School Science and Ashford, Sir Cyril, "The Contribution of Mathematics to Education". Eleventh Yearbook. National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, New York, 1936. Ball, W. W. R., Primer of the History of Mathematics. Company, New York, 1922. i The Macmillan Bell, Eric T., Men of Mathematics. Simon and Schuster, New York, 1937. Bell, Eric T., The Queen of the Sciences. Williams and Wilkins Company, Baltimore, 1931. Bell, Eric T., The Search for Truth. Williams and Wilkins Company, j| Baltimore, 1934. i j!Betz, William, "The Origin of Mathematics". The Mathematics Teacher.. ,! vol. XV, No. 5, May 1922. I IBurgess, E. G., "Mathematics". School Science and Mathematics, vol. XXIV, I March 1924. I' i!Cajori, Elorian, History of Mathematics. The Macmillan Company, New York, ij 1914. j! 'Carmichael, R. E., "The Larger Human Worth of Mathematics", Scientific |j Monthly, vol. XIV, May1922. !Chase, Stuart, The Tyranny of Words. York, 1938. Harcourt, Brace end Company, New Clarke, E. L., The Art of Straight Thinking. New York, 1934. Appleton Century Company, ICook, A. J., "An Historical Excursion", The Mathematics Teacher, vol. XXX, | No# 2, February 1937. ICooley, Hollis R. and others, Introduction to Mathematics.. |i Mifflin and Company, New York, 1937. I I Houghton 471 Cor drey, William A., "Ancient Mathematics and the Development of Primitive Culture", The Mathematics Teacher, vol. XXXII, N o / 2, February 1939. Dantzig, Tobiaw/, Number - _The Language of Science. New York, 1930. Dresden, A., "Why Study Mathematics?" October 30, 1920. The Macmillan Company, School and Society, vol. XII, Evans, George W., "Ratio as Multiplier", No. 3, March 1938. The Mathematics Teacher, vol. XXI, Fergusen, Zoe, "A Thread of Mathematical History and Some Lessons". Science and Mathematics, vol. XXIV, January 1924. School Hedrick, Earl R., "The Reality of Mathematical Processes". Third Yearbook. National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, New York, 1928. Hedrick, Earl R., "What Mathematics Means to the World". Teacher. vol. XXV, No. 5, May 1932. The Mathematics Hickey, May, "The Efficiency of Certain Shapes in Nature and Technology", The Mathematics Teacher, vol. XXXII, No. 3, March 1939. Hogben, Lancelot, Mathematics for the Million. W. W. Norton and Company, New York, 1937. Hotfclllhg, Harold, "Some Little Known Applications of Mathematics". The Mathematics Teacher, vol. XXIX, No. 4, April 1936. Kane, Sr. M. Gabriel, "The Cultural Value of Mathematics". Teacher, vol. XV, No. 4, April 1922. The Mathematics Karpinski, L. C., "The Methods and Aims of Mathematical Science". Science and Mathematics, vol. XXII, November 1922. Kempner, Aubrey J., "The Cultural Value of Mathematics". Teacher, vol. XXII, No. 3* March 1929. Keyser, Cassius J., Pastures of Wonder. New York, 1929# School The Mathematics Columbia University Press, Keyser, Cassius J., "The Human Worth of Rigorous Thinking". College Record. May 1917. Teachers Keyser, Cassius J., "The Humanistic Bearing of Mathematics". Sixth Yearbook. National Council of Teachers of Mathematics.. Bureau of Publications, Teachers College, Columbia University, 1931. Keyser, Cassius J., Thinking about Thinking. New York, 1926 E. P. Dutton Company, Kimmel, H., "The Status of Mathematics and Mathematical Instruction during the Colonial Period". School and Society, vol. IX, February 15,1919. 472 Lennes, N. J. “Mathematics for Culture". May 1914. Educational Review. vol. XLVII, Lifcber, Hugh G. and Lillian R., Non-Euclidean Geometry. New York, 1931. .Academy Press, Lovitt, W, V., "Continuity in Mathematics and Everyday Life". Teacher, vol. XVII, No. 1, January 1924. Lovitt, W. V., "Imagination in Mathematics". vol. XVII, No. 5, May 1924. The Mathematics Teacher, Manning, Henry P., Geometry of Four Dimensions. New York, 1928. Manning, Henry P., Non-Euclidean Geometry. The Mathematics The Macmillan Company, Ginn and Company, Boston, 1901. Miller, G* A., "Mathematical Shortcomings of the Greeks". and Mathematics, vol. XXIV, March 1924. School Science Miller, G. A., "Primary Pacts of the History ofMathematics". Teacher, vol. XXXII, No, 5, May 1939. The Mathematics 1 jMinnick, John H., "The Cultural Value of Secondary Mathematics". The ! Mathematics Teacher, vol. XVI, No. 1, January 1923. I |Moulton, E. J., "Mathematics on the Offense". The Mathematics Teacher. i! vol. XXIX, 'No. 6,’ October 1936. i |Parkinson, G. A., "Mathematics and Civilization". Mathematics, vol. XXXIII, October 1933. i j School Science and Paterson, Edith Bruce, "Everyman's Visit to the Land of the Mathematicians", The Mathematics Teacher, vol. XXXI, No* 1, January 1938. Rankin, W. W., "The Cultural Value of Mathematics". The Mathematics Teacher, vol. XXII, No. 4, April 1929. i t IReeve, W. D., "The Universality of Mathematics". The Mathematics Teacher, j! vol. XXIII, No. 2, February 1930. Hiter, H. E., "The Enrichment of the Mathematics Course". | Teacher, vol. XXXI, No. 1, January 1938. The Mathematics !j |iRussell, Bertrand, The A B C of Relativity. i; 1925. Harper and Brothers, New York, !;Sanford, Vera, A Short History of Mathematics,. Houghton Mifflin Company, ij Boston, 1930. ijSchlauch, W. S., "Mathematics as an Interpreter of Life". !i National Council of Teapher^ of Mathematics., Bureau of Publications, I Teachers College, Columbia University, 1928* 473 Slaught, H. E,, "Mathematics and Sunshine". vol. XXI, No. 5f May 1928. Slaught, H. E., "Romance of Mathematics". vol. XX, No. 6, October 1927. The Mathematics Teacher, The Mathematics Teacher, jSlosson, Edwin E., Easy Lessons in Einstein. Harcourt Brane and Company, | New York, 1920. I jSmith, D. E.,"Aesthetics and Mathematics".The MathematicsTeacher. ! vol. XX, No. 8, December 1927. Smith, D. E.,History of Mathematics. Ginnand Company, Boston, 1923, Smith, D. E.,"Mathematics in the Training for Citizenship",Third Yearbook, National Council of Teachers of Mathematics. Bureau of Publication?, Teachers College, Columbia University, 1928. Smith, D. E., "Mathematics and Religion", Sixth Yearbook. National Council j of Teachers of Mathematics. Bureau of Publications, Teachers College, I Columbia University, 1931. ijSmith, D. E,, "The Contribution of Mathematics to Civilization". Eleventh i Yearbook. National Council of Teachers of Mathematics. Bureau of | Publications, Teachers College, ColumbiaUniversity, 1936. t ,!Smith, D. E., "The Poetry of Mathematics and other Essays". Scripts jj Mathamatica. New York, 1934. ! ! Swann, William P. G-., The Architecture of the Universe. The Macmillan j Company, New York, 1934. jjWren, P. L. end Rossmann, R., "Mathematics Used by American Indians North |j of Mexico". School Science and Mathematics, vol.XXXIII, April 1933. 474 2. References Relating to Social Uses, Practical Uses, and to Other School Subjects Maras, A. S., "Civic Values in the Study of Mathematics". Teacher. vol. XXI, Ho. 1, January 1928. Ahern, Lorella, "Art in Geometry". No. 4, April 1939. i The Mathematics The Mathematics Teacher, vol. XXXII, jBarker, E. H., "Applied Mathematics for High Schools". Mathematics, vol. XX, January 1920. Barton, S. G., "The Uses for Mathematics". School Science and Science, vol. XL, November 13,1914. Blair, L., "Mathematical References in General Periodical Literature". Master's thesis, University of Chicago, 1923. | Breckenridge, William E., "Applied Mathematics in High School". The j Mathematics Teacher, vol. XII, No. 6, September 1919. i j Bryant, Carroll W., "Mathematics in Relation to Physics". The Mathematics jj Teacher. vol. XXX, No. 8, December 1937. i jj Camp, C. C., "Contributions of Mathematics to Modern Life". j Teacher, vol. XXI, No. 4, April 1928. j ! The Mathematics Carslaw, Horatio S,, The Elements of Non-Euclidean Plane Geometry and Trigonometry. Longmans, Green and Company,London, 1916. | jj jj Clarke, Edith, "Mathematics in Modern Business". vol. XXI, No. 5, May 1928. The Mathematics Teacher, !* Cohlins, J, V., "Calculations by Geometry of Astronomical Distances", j School Science and Mathematics, vol. XX, May 1920. ] Graver, Mary E., "The Mathematical Foundations of Architecture". j Mathematics Teacher, vol. XXXII, No. 4, April 1939. i General Motors, "The Mathematics of the Automobile". [j vol. XXXI, No. 5, May 1938. The The Mathematics Teacher. j ! Harold, H. R . , "A Study of the Mathematics Involved in the Field of j Auto Mechanics". Master's Thesis,University of Chicago, 1925. i | Heckert, W. W., "Mathematics in Industrial Chemical Research". The jj Mathematics Teacher, vol. XXXII, No. 3, March 1939. ! Hester, F. 0., "Economics in the Gourse in Mathematics from the Standpoint of the High School". School Science and Mathematics, vol. XIII, jj December 1913. Karelitz, George B., "Mathematics in Mechanical Engineering". Teacher, vol. XXXII, No. 2, February 1939. The Mathematics 475 Karpinski, Louis G., “Mathematics and the Progress of Science", Science and Mathematics, vol. XXIX, February 1929. School !Kelly, T. S., “Elementary Statistics in High School Mathematics as a Socializing Agency". School and Society, vol. XI, February 21, 1930. i Kins ell a, John and Bradley, A. Day, “Air Navigation and Secondary School Mathematics". The Mathematics Teacher, vol. XXXII, No. 2, February 1S39. ii jjheonerd, C. J., “Mathematics in Industry". |j vol. XXIX, March 1929. School Sciehce and Mathematics. |jMason, Thomas E., “The Relation of Mathematics to the Natural Sciences". j| Science, vol. XIIV, December 15, 1916. ii ' IMilne, W. P., "Mathematics and the Pivotal Industries". Mathematical j Gazette, vol. IX, March 1919. jj Moore, C. N., "The Contributions of Mathematics j! Educational Review, vol. EVIII, December !jMoore, C. N., “Mathematics and the Future". j vol. XXII, No. 4, April 1929. toWorld Progress", 1919. The Mathematics Teacher, j!Moritz, Robert, “On the Relations of Mathematics to Commerce". |j Science and Mathematics, vol. XIX, April 1919. School ||R eagan, G. W., “Mathematics Involved in Solving High School Physics Problems", jj School Science and Mathematics, vol. XXV, March 1925. ! I H jiRogers, Charles F., “The Mathematics of Elementary Chemical Calculations". | The Mathematics Teen her, vol. XXXII, No.1, January 1937. Rosander, A. C., “Mathematical Analysis in the Social Sciences". Mathematics Teacher, vol. XXIX, No. 6, October 1936. The jl j!Shelly, S. L., “The Slide Rule in Business". The Mathematics Teacher. |i vol. XIV, No. 5, May 1921. I! !lShepherd, C. 0., “Mathematics from the View Point of an Actuary". School jj Science and Mathematics, vol. XXIII, March 1923. jj i*Shirk J. A. G,, "Contributions of Commerce to Mathematics". Teacher. vol. XXXII, No, 5, May 1939. The Mathematics ;Shuster, Carl N. and Bedford, Fred L., Field Work_ in Matj^ematics. jj Anerican Book Company, New York, 1935. j! Smith, D. E., "Mathematics in the Training for Citizenship". jj College Record, vol. XVIII, May 1917, Teachers 476 Tyler, H. H., "Mathematics in Science". vol. XXI, May 1928. The Mathematics Teacher, Wellings, Balph E., "Graphs in Chemistry". vol. XXXIII, May 1933. School Science and Mathematics . Williams, H. B . , "Mathematics for the Physiologist and Physician". The Mathematics Teacher, vol. XIII, No. 3, March 1920. 477 3. References Relating to the Concept of Humber -American Council of Education, "The Story of Numbers". Civilization. No. 2, Washington, B.C. Andrews, 3P. E., New ^urnbers. [Bell, Eric T., Numerology. Harcourt, Brace and Company, New York, 1935. Williams and Wilkins Company, Baltimore, 1933. jConant, Levi L., Number Concent. TheMacmillan Company, New York, 1930. IDantzig, Tobias, Number - The Language of Science. j New York, 1930. i II |jNygaard, || No. Contributions to P. H., ‘‘Repeating Decimals*1. 7, November, 1938. jjsanford, Vera, "Roman Numerals". || No. 1, January 1931. The Macmillan Company, The Mathematics Teacher, vol. XXXI, The Mathematics Teacher, vol. XXIV, i! jiSanford, Vera, Short History of Mathematics. Houghton Mifflin Company, | New York, 1930. j jiSmith, David E., History of Mathematics. Ginn and Company, Boston, 1923. 11 jl j! Smith, David E., Number Stories of Long Ago. Ginn and Company, Boston,1919. jj Smith, David E., The Wonderful Wonders of One. Two. Three. McFarlane, || Warde, McFarlane, New York, 1937. ljSmith and Ginsburg, "Numbers end Numerals". The Mathematics Teacher. l! vol. XXX, No. 2, February 1937. 11 I! j : Young, T. W., Fundamental Concents of Algebra and. Geometry. The Macmillan ji Company, New York, 1925, 45?8 4. References Relating to the Concept of Measurement -American Council of Education, "The Story of Weights and Measures". Contributions to Civilization. No. 3, Washington, D. C. Bakst, A. A., "Approximate Computation". Twelfth Yearbook, National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, New York, 193?. Finley, George, "Measurement and Computation". Third Yearbook. National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University,: New York, 1928. Hogben, Lancelot, Mathematics for the Million. New York, 1337. W. W. Norton Company, National Bureau of Standards, "Units of Heights and Measures". Department of Commerce, National Bureau of Standards U.S. Sanford, Vera, Brief History of Mathematics. Houghton Mifflin Company, New York, 1930. Shuster, Carl N., Field Work in Mathematics. 1935. American Book Company, New Yorki 479 5« References Relating to the Concept of Function jBennett, I i Albert A., 11Algebra as a Language11. vol. XXX, No. 7, November 1937. The Mathematics teacher. I Hogben, Lancelot, Mathematics for the Million. New York, 1937. W. W. Norton Company, j ! IIMiller, G. A., "The Development of the Function Concept". | and Mathematics, vol. XXVIII, May 1928. j School Science Miller, G. A., "The Development of the Graph for Expressing Functionality". School Science and Mathematics, vol. XXVIII, November 1928. | |Nordgaard, M. A., "Introductory Calculus as a High School Subject". Third j Yearbook. National Council of Teachers of Mathematics. Bureau of jj Publications, Teachers College, Columbia University, New York, 1928. :i jSchlauch, W, S., "Mathematics and Measuring World Trends and Forces." i Seventh Yearbook. National Council of Teachers of Mathematics, ji Bureau of Publications, Teachers College, Columbia University, jI New York, 1932. i| |i j Swenson, John, "Selected Topics in Calculus for the High School." Third jj Yearbook.. National Council of Teachers,.of Mathematics. Bureau of jj Publications, Teachers College, Columbia University, New York, 1928. i!Walker, Helen, Mathematics Essential for Elementary Statistics., j! Henry Holt and Company, New York, 1934. jjwellings, Ralph E., "Graphs in Chemistry". j: vol. XXXIII, May 1933. School Science and Mathematics^ 480 6. Heferences Helating to the Concept of Proof Bell, Eric T., The Search for Truth. Baltimore, 1934. Williams and Wilkins Company, Clarke, E# L., The Art of Straight Thinking, New York, 1934. Appleton Centuny Company, Cohen, Morris II. and Nagel, Ernest, An Introduction to Logic and Scientific Method. Harcourt, Brace and Company* New York, 1934. Fawcett, Harold P., '‘Nature of Proof". Thirteenth Yearbook. National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, 1938. Hall, E. L., "Applying Geometric Methods of Thinking to Life Situations". The Mathematics Teacher, vol. XXXI, No. 8, December 1938. Hogben, Lancelot, Mathematics for the Million. Y/. W. Norton and Company, New York, 1937. Institute for Propaganda Analysis, "Propaganda, How to Hecognize It and Deal with It". New York, 1938. Keyser, Cassius J., Thinking about Thinking. New York, 1926. Budman, Barnet, "The Future of Geometry". vol. XXIV, No. 1, January 1931. E. P. Dutton and Company, The Mathematics Teacher, Saxe, Irving, "Proof of the Pythagorean Theorem". Mathematics, vol. XXXIII, November 1933. Slosson, Edwin E., Easy Lessons in Einstein. New York, 1921. Stabler, E. Bussell, "Assumptions and Proofs". vol. XXI, No. 1, January 1928. School Science and Harcourt, Brace and Company, The Mathematics Teacher, Ulmer, Gilbert, "Teaching Geometry for the Purpose of Developing Ability to Do Logical Thinking". The Mathematics Teacher, vol. XXX, No. 8, December 1937. 481 17. i j Heferences Relating to Mathematical Instruments |American Council of Education, "The Story of Time Telling and The | Story of the Calendar". Washington, D. 0* jBreslich, E. R. end Stone, C. A., The Slide Rule. University of Chicago ! Press, Chicago, 1929. jjDawson, Eugene P., "The Slide Rule - An Elementary Instruction and ij Drill Manual". University of Oklahoma. |Pretwell, M* B,, "The Development of the Thermometer". I Teacher, vol. XXX, No. 2, February 1937. ijHogben, Lancelot, Mathematics for the Million. | New York, 1937. The Mathematics W. W. Norton Company, ji |i !jHorton, E. M., "Calculating Machines and the Mathematics Teacher". ii Mathematics Teacher, vol. XXX, No. 6, October 1937* jKeuffel and Esser Company, "Slide Rule Manual". The 127 Fulton Street, New York. [Laboratory Specialties Company, 131 W. Market St., Wabash, Indiana. jiLafayette Instrument Company, 252 Lafayette St., New York. ji ! 'i j iiSkilling, William T., "The New Advance in Astronomy". School Science and || Mathematics, vol. XXXIII, October 1933. [! I! jj Thompson, J. Edgar, Manual of the. Slide Rule. D. Van Nostrand Company, j New York, 1930. 482 8# References Relating to Recreations and Ex trarCurricular .Activities Anning, Norman, HSocrates Teaches Mathematics", School Science and Mathematics, vol. m i l , June 1923. (A play) Ball, W* W. R . , Mathematical Recreations and Essays. Mhe Macmillan Company, ! New York, 1914. I I jBooth, A. L., "A Mathematical Recreation11. The Mathematics Teacher. | vol. XI, June 1919. J Colwell, R. 0., 11A Rule to Square Numbers Mentally"♦ School Science and Mathematics, vol. XIV, November 1921. | Cowley, Elizabeth B., "A Mathematical Fantasy". j vol. XXXIII, May 1933. (A play) School Science and Mathematics, Crawford, Alma E,, "A Little Journey to the Land of Mathematics". Mathematics Teacher, vol. XVII, October 1924. (A play) !Graff, Margaret, "Euclid Dramatized". I vol. XXI, April 1921. (A play) The School Science and Mathematics. J Gulden, M., "Mathematics Club Program". The Mathematics Teacher, vol. XVII, II October 1924. ii I Hansen, Lena B., "Creating Interest in Mathematics through Special Topics". II The Mathematics Teacher, vol.XXIII, January 1930. 11 iHarding, P. H. and others, "A Mathematical Victory". School Science and ! Mathematics, vol. XVII,June1917. (A play) j| jj Harris, Isabel, "A Geometric Recreation". School Science and Mathematics, j vol. XX, November 1920* i ! |!Hartswick, F. G., The Tangram Book. Simon and Shuster, New York, 1925. }i ji Hatton, Mary 0., "A Mathematics Club". The Mathematics Teacher, vol. XX, ii January 1927. IjHeath, Royal V., "The Magic Clock". i| February 1937. The Mathematics Teacher, vol. XXX, i! |!Hilsenrath, Joseph, "Linkages". The Mathematics Teacher, vol. XXX, j October 1937. h IiJones, S. I., Mathematical Wrinkles. Published by S. I. Jones, Nsnhville, I Tennessee, 1923. !j HKrathwohl, William C., "Helping Mathematics with An Exhibit". j| Teacher, vol. XXXI, February 1938. The Mathematics 483 McLaughlin, H. P., 11Algebraic Magic Squares11. The Mathematics Teacher. vol. XXV, February 1921. Miller, Florence B., 11A Near Tragedy11. December 1929. (A play) The Mathematics Teacher, vol. XXII, Miller, Florence B., “Out of the Past11. I December 1937. (A play) The Mathematics Teacher, vol. XXX, Newhall, 0. W., “Recreations in Secondary Mathematics11. Mathematics, vol. XV, -April 1915. 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