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Demonstrative geometry as a means for improving critical thinking

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NORTHWESTERN UNIVERSITY LIBRARY
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This thesis by
.....................
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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Unpublished theses submitted Tor the M a s t e r ’s and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
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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
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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.
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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
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I
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. - 1
■ ■
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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
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The
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Williams and Wilkins Company,
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NfltAflflaJ,.
iff
Mathematics. Bureau of Publications, Teachers College, Columbia
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Hall, Elizabeth L., “Applying Geometric Methods of Thinking to Life
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Hawkes, Herbert E. and Wood, Ben D., Manual of Directions. for Columbia
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Institute for Propaganda Analysis, "Propaganda, How to Recognize It and
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Keyser, Cassius J., "The Meaning of Mathematics".
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Scriuta Mathematics,
Keyser, Cassius J., Thinking About Thinking. E. P. Dutton and Company,
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.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!
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