# An experimental comparison of two methods for alternating mathematics and science

код для вставкиСкачатьAN EXPERIMENTAL COMPARISON OF TWO METHODS FOR ALTERNATING MATHEMATICS AND SCIENCE A Thesis Presented to the Faculty of the School of Education University of Southern California In Partial Fulfillment of the Requirements for the Degree Master of Science in Education by Arthur Chesley Francis January 1940 UMI Number: EP53814 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI EP53814 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 <3t f t This thesis, written under the direction of the Chairman of the candidate’s Guidance C om m it tee and approved by all members of the C om mittee, has been presented to and accepted by the Faculty of the School of Education in partial fulfillment of the requirements for the degree of M aster of Science in Education. D ate... Guidance C om m ittee 0. C. Crawford Chairman 0. H. Hull William G. Campbell TABLE OF CONTENTS CHAPTER PAGE I. 1 THE PROBLEM.................. Statement of the p r o b l e m ....................... 2 Importance of the problem....................... 3 II. PROCEDURE. . . . ................ ............... 4 Procedure of the experiment.................... ,4 Groups of students participating in the e x p e r i m e n t .............. ................. 4 The school subject to be measured............. 5 Mathematics studied by thegroups.............. 6 The test used.................................. 6 Scoring the tests.............................. 7 Procedure of tabulation.......... # ............. 7 Comparability of groups...................... 11 Gains in test scores for eachgroup........... 11 Comparisons of gains for B8 and A8 groups se p a r a t e l y ................................. 12 Comparisons of gains for combined B8 and AQ groups................................... 13 III. RESULTS OF THE B8 GROUPS . .................... 15 Comparability of groups........................ 16 Gains made on tests by each B8group*........... 23 iii CHAPTER PAGE Comparisons of the three pairs of groups in terms of g a i n s ............................. 34 Comparisons of the ten-week groups, as a single group, with the semi-weekly alter nation groups, as a single group, in terms of g a i n s ....................................... 44 Statistical comparisons of B8 groups for prediction purposes.............................47 IV. RESULTS OF THE AB GROUPS ......................... 56 Comparability of groups...................... . . 57 Gains made ontests by each A8 g r o u p ............ 63 Comparisons of the two pairs of groups in terms of g a i n s ................................. 66 Comparisons of the ten-week groups, as a single group, with the semi-weekly alternation groups, as a single group, in terms of gains..................... .................... 76 Statistical comparisons of A8 groups for prediction purposes •. • • V. .................. 81 RESULTS OF B8 AND AQ GROUPS C O M B I N E D .............. 88 Comparisons in terms of gains of the combined ten-week groups as a single group, with the combined semi-weekly alternation groups as a single g r o u p ......................... 89 iv CHAPTER PAGE Statistical comparison of gains of these same two large combined groups for the purpose of showing predictions ............. 94 Statistical comparison of these two large groups showing a second method of making predictions. ..................... VI. . . . . . 96 CONCLUSIONS AND RECOMMENDATIONS.................. 100 Conclusions..................................... 101 Recommendations.......... 101 LIST OF TABLES TABLE PAGE I. Intelligence Quotients of Groups IA and IB, B8 Classes . . . . . . . ........................ 18 II. Intelligence Quotients of Groups IIA and IIB, B8 C l a s s e s ............ 21 III. Intelligence Quotients of Groups IIIA and IIIB, B8 C l a s s e s ..................................... . 2 4 IV. Gains Made by Group IA, Ten-Weeks, B8 Class. . . . 27 V. Gains Made by Group IB, Semi-Weekly Alternation, B8 C l a s s ................ VI. VII. Gains Made by Group IIA, Ten-Weeks, B8 Class . . . 28 29 Gains Made by Group IIB, Semi-Weekly Alternation, B8 C l a s s ......................................... 51 VIII. IX. Gains Made by Group IIIA, Ten-Weeks, B8 Class. . . 32 Gains Made by Group IIIB, Semi-Weekly Alternation, B8 C l a s s ......................................... 33 X. Comparisons of the Two Methods in Terms of Gains for B8 Groups............................. . 3 5 XI. Comparisons of Ten-Week Groups as a Single Group, with Semi-Weekly Alternation Groups as a Single Group, in Terms of Gains for B8 Classes 45 XII. Statistical Comparison of Gains of B8 Groups Show ing Predictions for Separate Class Groups and for the Total Groups as Single Large Distribu tions. ........................................... 51 TABLE PAGE XIII. Statistical Comparison of B8 Groups Showing a Second Method of Making Predictions by Different Computations of the Means . . . . . . 54 XIV. Intelligence Quotients of Groups IVA and IYB, A8 Classes......................... 58 XV. Intelligence Quotients of Groups VA and VB, A8 C l a s s e s . ............................... XVI. Gains XVII. Gains 61 Made by Group IVA, Ten-rWeeks, A8 Class. . . 64 Made by Group IVB, Semi-Weekly Alternation, A8 Class......................................... 65 XVIII. Gains XIX. Gains Made by Group YA, Ten4Weeks* rA8 Class . . .67 Made by Group VB, Semi-Weekly Alternation, A8 Class. ..................................... 68 XX. Comparisons of the Two Methods in Terms of Gains for A8 G r o u p s .................. 69 XXI. Comparisons of Ten-Week Groups as a Single Group, with Semi-Weekly Alternation Groups as a Single Group, in Terms of Gains for A8 C l a s s e s ..................................... . . 77 XXII. Statistical Comparison of Gains of A8 Groups Showing Predictions for Separate Class Groups and for the Total Groups as Single Large Dis tributions................ 83 vii TABLE PAGE XXIII. Statistical Comparison of A8 Groups Showing a Second Method of Making Predictions by Different Computations ofthe M e a n s ........... 86 XXIV. Comparisons of Ten-Week Groups as a Single Group, with Semi-Weekly Alternation Groups as a Single Group, in Terms of Gains for the Combined B8 and A8 Classes..................... 90 XXV. Statistical Comparison of Gains of B8 and A8 Groups Combined Showing Predictions for the Total Ten-Week Groups and for the Total SemiWeekly Alternation Groups as Single Large Distributions.......... .. ................. 95 XXVI. Statistical Comparison of B8 and A8 Groups Showing a Second Method of Making Predic tions by Different Computations of the Means. 98 LIST OF FIGURES FIGURE 1. PAGE Distribution of the Intelligence Quotients of Group IA, Ten-Week B8 Class, and of Group IB, Semi-Weekly Alternation 2# B8 Class...................19 Distribution of the Intelligence Quotients of Group IIA, Ten-Week B8 Class, and of Group IIB, Semi-Weekly Alternation 3. B8 Class.......... 22 Distribution of the Intelligence Quotients of Group IIIA, Ten-Week B8 Class, and of Group IIIB, Semi-Weekly Alternation B8 Class. .............. 25 4. Distribution of Gains Made on Tests by the Ten-Week B8 Group IA, and the Semi-Weekly Alternation B8 Group IB........................................... 38 5. Distribution of Gains Made on Tests by the Ten-Week B8 Group IIA, and the Semi-Weekly Alternation B8 Group IIB............................... 40 6. Distribution of Gains Made, on Tests by the Ten-Week B8 Group IIIA, and the Semi-Weekly Alternation B8 Group IIIB...................................... 43 7. Distribution of Gains Made on Tests by the Com bined Ten-Week B8 Groups, I, II, IIIA, and the Combined Semi-Weekly Alternation B8 Groups, I, II, IIIB........................... 48 FIGURE PAGE 8. Distribution of the Intelligence Quotients of Group IVA, Ten-Week A8 Class, and of Group IVB, Semi-Weekly Alternation A8 Class . . 59 9. distribution of the Intelligence Quotients of Group VA, Ten-Week A8 Class, and of Group VB, Semi-Weekly Alternation A8 Class............ 62 10. Distribution of Gains Made on Tests by the Ten^.. Week A8 Group IVA, and the Semi-Weekly Alterna tion A8 Group IVB................................ 73 11. Distribution of Gains Made on Tests by the TenWeek A8 Group Vt, and the Semi-Weekly Alter nation A8 Group VB. .. ......................... 75 12. Distribution of Gains Made on Tests by the Com bined Ten-¥/eek A8 Groups, IV,VA, and the Com bined Semi-Weekly Alternation A8 Groups, IV,VB. 80 13. Distribution of Gains Made on Tests by the Com bined Ten-Week B8 and A8 Groups, I, II, III, IV, VA, and the Combined Semi-Weekly Alterna tion B8 and A8 Groups, I, II, III, IV, VB • • 93 CHAPTER I THE PROBLEM About five years ago, the subject of general scienee was introduced into the eighth grade course of study in the junior high schools of Los Angeles. Up to that time its study had been confined to grade nine. It was the feeling at that time that much of the content of mathematics for grade eight was not of practical value to the student. There fore, half of the time which had been allotted to mathematics was given to the new subject of general scienee. It was left to each junior high school to revise its mathematics-general science program in accordance with this new time allotment. / Some schools chose to teach each subject ten consecutive weeks during the semester while others chose to teach the two subjects on a semi-weekly alternation basis throughout the semester. ter method. The Edison Junior High School adopted the lat But after using it for five years, the mathematics science teachers of this school felt that a change was advis able. Since one subject was taught two days and the other the remaining three days of each week, the continuity of both was broken and it seemed that the students were not making suf ficient progress in either subject. Reports had come from teachers in other junior high schools that the progress made by their students in mathematics and z science was satisfactory due to the fact that each subject was taught ten consecutive weeks, thus giving continuity in the presentation of subject matter* The Edison teachers felt that these hearsay reports were not sufficient proof of the superiority of the ten-week method td warrant its adoption in their school. Therefore, the mathematics-science teachers of Edison Junior High School decided to compare these two methods of teaching the two subjects by actually using them experiment ally in their own school. It is the purpose of this thesis to show the evidence obtained from this experiment and to draw conclusions as to the relative merits of both these methods. In this chapter we shall present the problem involved in this experimental investigation, and also show the impor tance of its solution in making the mathematics-science pro gram for the eighth grade in the Edison school. Statement of the problem. It was the purpose of 'this study to compare experimentally by a standardized arithmetic test the achievement of several pairs of comparable groups. Each pair had studied mathematics by two different methods, each on a different time-basis, in order to determine which method was the better. The two different time-bases for these methods were (1) one of the groups of each pair studied math ematics for ten consecutive weeks, and science the other ten, 3 and (2) the other group of each pair studied mathematics on a half-week basis, alternating with science the other half* Importance of the problem* This study was of special importance to Edison Junior High School in that the results of the experiment determined the time-basis to be adopted for the teaching of mathematics and science in the eighth grade. It is hoped that the findings will be of value in other Los Angeles schools in selecting a time-basis for the teach ing of these two grade eight subjects. Also the findings of this experiment may shed light on the psychological element of time-space learning where there are specified intermissions of time in the learning process. CHAPTER II PROCEDURE In accordance with the plan of solving the problem stated in the preceding chapter, the mathematics teachers conducted an experiment. The procedure which was followed in conducting this experiment and in tabulating the result ing data is related in this chapter. I. PROCEDURE OF THE EXPERIMENT In describing the experimental procedure, the follow ing points will be discussed: groups of students participa ting in the experiment; the school subject to be measured; the mathematics studied by the groups; the test used; and the scoring of tests. Croups of students participating in the experiment. When students entered Edison Junior High School, they were divided into three main groups, slow, average, and high on the basis of their intelligence quotients. This procedure was followed in grouping students for each of their respective classes. Hence all groups in each of these three classifi cations were assumed to be reasonably comparable. With this student classification already accomplished, the mathematics teachers decided that (1) every teacher who had two average 5 or high groups in grade eight mathematics should participate in the experiment by having one group study on the ten-week basis, and the other on the semi-weekly alternation plan, (2) the slow groups should be omitted since they were doing special remedial work. On this basis, ten groups under five teachers were selected for the experiment, six of which were B8, and four were A8. There was a total of 330 students involved, 198 of whom were B8, and 132 were A8. With this arrangement, it was felt that the chances for success of the two methods were placed on an equal foot ing because each pair of groups was (1) reasonably compar able, (2) instructed by the same teacher, and (3) given the same amount of mathematical content. The school subject to be measured. The course of study in general science for grade eight as set up for the Los Angeles Schools was informal and allowed a choice of subjects best adapted to each particular class. Since all science classes were not studying the same topics no standardized test could be found to measure equally all content studied. The mathematics course of study called for a definite assign ment however, and there were standardized tests available which would measure the content of the prescribed course of study. Hence it was decided that mathematics hlone was to be measured. 6 Mathematics studied by the groups. In further prep aration for the experiment, the mathematics teachers together with the school administrators planned the content of* work for both the B8 and A8 groups. The text1 adopted for all grade eight students in Los Angeles was used. But in order that all B8 groups would receive instruction in the same topics in connection with this text, an outline of work to be covered was given to each B8 teacher. The same procedure was followed for the A8 groups. The test used. Since the superiority, if any, of either method was to be determined by comparing the gains made by the groups using one method with the gains made by the groups of the other, it was necessary to give both a pretest and a final test. Therefore, on the first day of the experiment each teacher gave a pretest to his or her two groups. The stand ardized test given was the New Stanford Arithmetic Test, Form W, for grades £-9, which because of its wide range in grades would include most of the mathematical experiences of grade eight. It has five pages and is divided into two parts, Arithmetic Reasoning, and Arithmetic Computation, the time 1 £.J. Brueckner, C.J. Anderson, and G.O. Banting, Mathematics for the Eighth Grade. (Sacramento: California State Printing Office, 1932)• 380 pp. 7 required for the former being 20 minutes and for the latter 30 minutes. Both parts were given consecutively to each class, thus requiring only one period for the entire test. The teachers carefully adhered to the directions for admin istering the test as given on a separate sheet. The test together with the directions for administering it are given following this page. The students studying mathematics for ten consecutive weeks were given a final test at the end of that time, and those on a semi-weekly alternation basis received their final test during the last week of the semester. In giving Ithe final the same form was used as had been used for the pretest. Scoring the tests. Each teacher scored the "answers" of his or her groups according to the specific directions given by the standardized test. A careful recheck of all tests was made by the school counselor and afterwards by the writer. The actual score made by each student was used in pref erence to either the arithmetic age or school grade because it was more adaptable to statistical purposes. II. PROCEDURE OF TABULATION In describing the procedure of tabulation, the follow ing points will be discussed: comparability of groups; gains New Stanford Arithmetic Test e By Truman L. Kelley, Giles M. Ruch, and Lewis M. Terman TEST: FORM W FOR GRADES 2-9 Grade.B oy or Name.......................................................... • girl...... # A ge...................When is your next birthday ?......................... : ........... . . How old will you be then ?. . .. Name of school.......................................................................................... D ate................................. ................. Score Arith. School1 Score Age Grade 120 119 118 117 116 H5 114 113 112 111 110' 109 108 107 106 105 104 103 102 101 19-2 18-11 18-8 18-5 18-2 17-11 17-8 17-6 17-4 17-2 17-0 16-10 16-8 16-6 16-5 16-3 1 6-2 1 6-0 15-11 10.0 1 5 -9 , 9.8 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 Arith. School1 Score 1Arith. School1 Score Age Grade Age Grade 15-8 15-6 15-4 15-2 15-0 14-10 14—8 T4—6^ 14-4 14-1 13-11 13-9 13-7 13-5 13-3 13-1 12-11 12-10 12-8 12-7 9.7 9.5 9.3 9.2 9.0 8.9 8.7 8.5 8.4 8.2 8.1 7.9 7.8 7.6 7.5 7.4 7.2 7.1 7.0 6.8 80 79 78 77 76 75 74 73. 72 71 70 69 68 67 66 65 64 63 62 61 12-6 12-4 12-3 12-2 12-0 11-11 11-10 11-9 11-8 1 1-7 11-6 1 1-5 1 1-4 11-3 11-2 11-1 11-0 10-11 10-10 10-9 6.7 6.6 6.4 6.3 6.2 6.1 6.0 5.9 5.8 5.7 5.7 5.6 5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 Arith. School1 Score Age Grade 10-8 10-7 10-6 10-6 10-5 10-4 10-3 10-2 10-1 10-0 9-11 9-11 9-1 0 9 -9 9 -8 9 -7 9 -6 9 -5 9 -4 9 -3 4.7 4.6 4.6 4.5 4.4 4.4 4.3 4.3 4.2 4.1 4.1 4.0 4.0 3.9 3.9 3.8 3.7 3.6 3.6 3.5 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 Arith. School1 Age Grade 9 -3 9 -2 9-1 9 -0 8-11 8-10 8 -9 8-8 8 -7 8-6 8 -5 8 -4 8 -3 8-2 8-1 8—0 7-11 7 -10 •7-8 7 -6 7-5 3.4 3.4 3.3 3.3 3.2 3.2 3.1 3.1 3.1 3.0 3.0 2.9 2.9 2.8 2.8 2.8 2.7 2.7 . 2.6 2.6 2.6 1 G rade defined as in th e ta b le in th e D irections for A dm inistering. * A rith m etic ages a b o v e th is p o in t a re ex tra p o la ted values. To th e E x a m in e r . D o not administer this test without first reading carefully the Directions for Administering. T S core est A r it h . A ge S chool Grade Arith. Reas. Arith. Comp. Total (Average) Arith.1 1 T h e T o ta l A rith m etic Score is th e average of th e scores on th e tw o te sts. Published by World Book Company, Yonkers-on-Hudson, New York, and Chicago, Illinois Copyright 1929 by World Book Company. Copyright in Great Britain. All rights reserved, nsat : w-16 PRINTED IN U.S.A. [l] BSP" This te s t is copyrigh ted. The reproduction o f any p a rt o f it b y m im eograph, hectograph, or in any o ther w ay, w h eth er th e reprodu ctions are so ld or are fu rn ish e d fr e e fo r u se, is a violation o f th e copyright law . N ew s ta n f. A rith . w 9 TEST 1. ARITHM ETIC REASONING—Continued 3 21 Frances sold $156 worth of books. She received a com m ission of 40%. H ow much did she earn? Answer............ 32 H ow much more is earned each day by a man w orking 6 days for $45 than by a man w orking 6 days for $32.40? Answer............ 22 In a class of 38 pupils, only 19 had perfect attendance records for a month. -What per cent' of all the class had per fect attendance records? Answer............ 33 H ow m any dollars’ worth of mer chandise must a clerk sell at a com m is sion of 2 / % to earn a salary of $1,000 23 If a man uses 25 gallons of gasoline in driving 375 miles, how far can he drive on 8 gallons, assum ing that he will obtain the same m ileage per gallon? Answer ...... 24 W hen $3 w ill buy 5 yards of g in g ham, how much w ill 7 yards cost? Answer......... 25 John buys papers for $1.80 per hun dred and sells them at 3 cents each. H ow much does he make on 250 papers? Answer............ 26 H ow much cheaper is a bill of $100 worth of goods bought at a 20% dis count than one of the same amount bought at tw o successive discounts of 10% and 10% ? Answer. 27 A man’s autom obile will go 110 miles on 10 gallons of gasoline. If gasoline costs 22 cents per gallon, w hat is the cost of the gasoline per mile? Answer............ 2® Ice is .92 as heavy as water. A cubic foot of water w eighs 62.5 lbs. W hat is the w eight of a cubic foot of ice ? Answer............ 29 If 6 men can build a house in 180 days, how long w ill it take 8 men to build it? Answer............ 30 A broker charges $25 com m ission on every sale, plus 5% on all over $200. W hat would be his com m ission on a 500-dollar sale? Answer. 31 Fred w ill sell his bicycle for $24. That is 34 less than it cost him. W hat did it cost him? A ? Answer.......... Go right on to the next column. D IR EC TIO N S: Get the answers to these examples as quickly as you can without making mistakes. Look carefully at each example to see what you are to do. Begin here. (1) 1 2 a >'ear? 5+3= (2) (3) (4) (5) Add 0 4 Add 4 6 Subtract 5 2 Subtract 7 7 5 --- - ---- Answer............ 34 A train makes a run of 159 miles in 7 hours. One trip it was delayed and made only 63 m iles the first 4 hours. A t w hat average rate per hour must it go the re mainder of the distance in order to arrive on tim e? Answer........ N ew Stanf. A rith . w TEST 2. ARITHM ETIC COMPUTATION 4 3 (6) (8) Subtract 17 3 (7) Subtract 12 4' Add 24 7 (9) Subtract 89 76 (10) (11) (12) (13) 2)6 Subtract 895 437 Add 87 654-2 754290 389364 (16) (17) 7)56 8 -4 = 35 A certain house was assessed at $5,000. T he tax on it was $125. W hat was the tax rate? Answer........ 36 Tulip bulbs should be planted 4 to the square foot. A square plot should be how many feet on a side in order to hold 36 bulbs? Answer............ 4X 8 = 37 W hat actual rate of interest would be obtained if you bought 6% preferred stock at $75 per share? (Par value $ 100-> Answer....:... (14) 38 For $90 each a man bought 5 shares of a preferred stock paying 5% . (Par value $100.) A fter his first dividends he sold his stock for par value. Ignoring brokerage charges, how many dollars did he make on the transaction ? Answer............ 6X 7 = J 15) M ultiply 452 4 (18) (19) (20) 39 If they have the same thickness, a pancake 6 inches in diameter is how m any tim es as large as one 3 inches in diameter?Answer........ 9)77 Subtract Subtract 40 A house and lot were valued at $5,000. T he taxes amounted to $60 a year. It cost $200 annually for depreciation and incidental expenses. For what must it rent per m onth in order that the owner m ay clear 8 per cent on its value? Answer............ End of Test 1. Look over your work. . (21) (22) „ %X H = 3)14.1 1000 % 42 5 ■% (23) (24) (25) Subtract 5 2% 2 7 Ys Add 3 5% 2 4 y3 ___________ 37) 8 5 3 5 9 ' Go right on to the next page. 'N e w Stanf. A rith . W 5 T E ST 2 .‘A R IT H M E T IC CO M PUTATIO N— Continued (26) (27) 6 A X % = A dd % % , (31) Subtract 73% 5% Subtract 31 % 15% T E ST 2 . ARITHM ETICS C O M P U T A T IO N — Cohcluded (29) Add 7 8% 129% (28) Subtract * 31% 19y4 (30) 6 (32) (47) (48) A dd % 7/io . (49) Add 3 T ons 1000 4 T ons 500 300 7 T ons 1750 2 T ons 800 lb. lb. lb. lb. lb. 4 . 6 5 is w hat per cent of 1 5 . 5 0 ?- N ew Stanf. A rith . W (50) ‘ 239)1 4 4 8 3 4 j A nsw er = (33) Add 54% 2 1 %2 (51) (52) 4 5 is 1 5 % of w hat number? ’ (53) A dd 7 - 9 (4 )! = • (54) 2 5216 5 12 8 -8 (35) (34) Subtract 55% 18% Subtract 32% 24% % + 5 Number = (37) (36) -------- % + % + % + % (55) H ow many degrees are there in angle B A C ? A (38-39) H EIG H T IN FEET O F F A M OU S W A T E R F A L L S — to L tn 0> S ® <P o O O § 8 8 8 8 o A ccording to the graph, what is the approximate height in feet of iy w O o o o NIAGARA |_ UPPER Y O S EM ITE | N ia g a r a .................. SHOSHONE LOWER YELLOWSTONE (57) Find the value of F in the following expression if k equals 8 and h equals 5. Multiply 42n —10 n F = kh2 H tarn H i VICTORIA 36* 90° (56) Upper Y osem ite 1 ...... F = Answer = (60) (40) 65.3 2 - 4 7 . 2 = (43) (42) (41) Subtract 5 2% 46% %+%+ 6.2-3.895 (59) (58) . Find the volume of . this figure. Simplify the following expression: %2 V 1 5 1 2 9 8 x + 6 y — \2 x — 3 y ) A nsw er = (44) (45) M ultiply 78.94 3.04 Subtract % % ' . (46) Answer = Sales amount to $5240. Commission rate is 10%. Find amount of commission. Answ er = Turn the page and go right on. End of Test 2. Look over your work. Number right Score 1| 2| 31 4| 5| 6| 7| 8| 91 10| HI 12) 13 14| 15 16) 17) 18 191 20121) 22123| 24| 25 26| 27) 28) 29) 301 | 3 7[11|18 23 28 31 33 36 38| 4 0 1 43) 46) 491 5 2 1 551 57| 59) 601 611 63) 641 651 67| 691 711 72| 74) 75) 771 79| 10| Number right 1 311 321 33| 34| 351 36| 371 38| 39) 40| 411 42| 43| 44| 451 46| 47| 48) 49) 50 . 51 1 52) 53) 541 55| 56| 571 58| 59) 60) Score 181| 84| 86| 88 | 90| 92 94| 96) 98|101|103|106|109|111|112|113| 113)114|114|115|115)116)116|117)117|118|119| 1211123|125| New Stanford Arithmetic Test 10 B y Truman L. Kelley, Giles M. Ruch, and Lewis M. Terman DIRECTIONS FOR ADM INISTERING The following instructions are all that are needed to give the tests. The instructions for scoring the tests are given with the scoring key. More detailed information concerning the construction, validity, and reliability of the tests; the norms; and the interpretation and uses of the test results are given in the complete Guide for Interpreting the New Stanford Achievement Test. The person in charge of .the testing program for a school or school system will need a copy of the complete Guide for Interpreting the New Stanford Achievement Test in order to have the test results interpreted and used properly. (The Guide must be ordered separately.) GENERAL DIRECTIONS N.B. i The teacher should become thoroughly acquainted with all the directions in this booklet before attempting to give the test. C o n d it io n s o f t h e T e s t The New Stanford Arithmetic Test can be given satis factorily by any teacher or principal who is willing to follow the directions in this manual conscientiously and who is reasonably skillful in discipline. The tester should possess a pleasing personality and be able to speak the necessary directions clearly and distinctly enough that every one in the test group may hear. Pupils in Grades 2 and 3 should be tested in ordinary classroom groups. Pupils in Grades 4 to 9 may be tested in ordinary classroom groups or in larger groups of a hundred or more, if proper controls are provided. Pupils in Grades 4 to 9 inclusive may be grouped for testing purposes. Good testing conditions demand that there should be quiet throughout the testing period. Strict obedience and attention on the part of the pupils are absolutely necessary. No questions should be permitted after the testing begins. There should be a spirit of rapport between the tester and the pupils. No visitors should be allowed. Sufficient assistants should be provided, when large groups are being .tested, to see that every one understands what he is to do and that he has the neces sary materials with which to do it, and in order to discourage copying or giving assistance in any way. The person administering the test' must speak dis tinctly and at a moderate speed. Undue stress and levity are to be avoided. An agreeable manner, but one suggestive of authority, is essential. Give all commands in a quick, energetic voice distinct enough for all those for whom it is intended to, hear. Avoid shouting. Give all directions slowly, with careful attention to emphasis where it is needed. Follow the directions exactly. Be watchful, and in so far as possible prevent disturbances within or without the room which might in any way interfere with the work of any pupil. 'Permit no whis pering or copying. Distribute test booklets and admin ister the tests with dispatch. Systematize the work to avoid delay in administration. Adhere carefully to the time limits. A stop watch i s . desirable but riot absolutely necessary. (The time limits are liberal. Probably many pupils will finish before time is called.) Plan your procedure in detail before attempting to ad minister the tests. Their administration is easy, but it is necessary to know exactly what to do at each moment of the testing time. The total working time for the arithmetic test, Grades 2 to 9 inclusive, is 50 minutes. The gross time is slightly longer. The test may be given in a single sitting, or each of the two parts may be given in a separate sitting. DIRECTIONS FOR ADMINISTERING {Identical for all five forms. To be followed verbatim) “ Here is a test to show how much you have learned. I will give each of you a test book. Do not write on it or open it until I tell you to.” (See that this is obeyed.) After all are provided with test books and pencils: “ Now fill the blanks at the top of the first page, here. (Hold up test book and point to the blanks.) Do it as quickly as you can, but write plainly. On the first line where it says N am e , write your name. (Pause.) After the word Grade , write the number that tells what grade you are in. (Name the grade.) After Boy or GirZ, write the word that tells which you are. (Pause.) On the next line, where it says Age, tell how old you are now. (Pause.) Then tell when your next birthday will come and how old you will be then. (Pause.) On the next line write the name of this school. (Give the name.) Published by World Book Company, Yonkers-on-Hudson, New York, and Chicago, Illinois Copyright 1929 by World Book Company. Copyright in Great Britain. All rights reserved PRINTED IN U.S.A. New Stanford Arithm etic Test: Directions for Administering "\ At the end of the same line write the date.” (Name the date. Give to the younger children any necessary assistance.) After the blanks have been filled : “ Now, listen care fully and do just what I tell you to do. Do not begin until I say Go. The very second I say Stop, you must stop and hold your pencils up. After we have begun, you must not ask questions. If you break your pencil, hold up your hand and I will give you another. Do your best and do not pay any attention to what any one else is doing. “ Now turn the page over to Test 1. It says Test 1 at the top of the page.” T e s t 1. A rith m e tic R e a s o n in g “ Read the directions at the top of the page: Find all the answers as quickly as you can. Write the an swers on the dotted lines. Use the margins of the paper to figure on. (Pause slightly.) The test has two pages. As soon as you have finished the first page, go right on to the next. Ready — Go.” , (See that pupils do not Stop at the end of the first page.) Allow 20 m inutes; then say : “ Stop. Turn to Test 2 on the next page. Be sure to turn just one leaf.” (Pause and make sure that all the pupils have found the place.) T e s t 2. A rith m e tic C o m p u ta tio n “ Read the directions at the top of the page: Get answers to these examples as quickly as you can without making mistakes. Look carefully at each example to see what you are to do. (Pause slightly.) You may use the margins of the paper to figure on if you need to. There are three pages of this test. As soon as you have finished the first page, go right on to the next. Ready — Go.” (See that pupils do not stop at the end of the first or second page.) . Allow 30 minutes; then s a y : “ Stop. Close your books.” Collect all books immediately. DIRECTIONS FOR SCORING The Directions for Scoring are given in full with the . scoring keys enclosed in each package of tests. DIRECTIONS FOR INTERPRETING TEST SCORES 1 By means of the parallel rows of figures across the bottom of each test, the pupil’s achievement on any test is automatically given an equated value related to the table of norms. For example, the rows of figures across the bottom of the Arithmetic Reasoning Test in Form V appear as follows : Sum . . 0 1 2 3 S c o re , . . 3 12 21 29 i 22 23 24 25 , 96 98 100 102 The upper row represents the number of blanks to be filled in the test. The lower row represents the equated value, in terms of the norms, for having satisfactorily completed any number of those blanks. If, for instance, a pupil successfully filled 23 blanks in the examination, the scorer checks the number 23 in the upper row of figures and directly under the 23 is 98 , which represents the equated value of the actual achievement. Thus, g8 is the score (not 23 ), and is the only score to be used to represent the pupil’s achievement in any further refer ence made to it. Age and grade norms. On the front page of the Arith metic Test a table of norms is given by means of which pupils’ scores may be changed to arithmetic ages and grades. T able of F r a c t io n a l P a r t s of Date of testing . . . Sept. Oct. Nov. Dec. Jan. is IS is IS 15 Annual promotions Grade, a low section . Grade, a high section .0 .0 ■S .1 .1 .6 .2 .2 •7 •3 •3. .8 G rades C o m pleted Feb. Mar. April May June is IS iS 15 iS ■S •4 .4 Midyear .0 .9 promotions .5 .6 .1 .6 ■7 .2 •7 .8 ■3 .8 •9 •4 •9 There is provided in each package of booklets a Class Record which may be used if it is desired to bring together on one sheet the complete record of scores of the pupils of a grade or class for convenient reference. The names of the pupils may be entered either alphabetically or in order of the total scores, according to preference. A tabular form is provided on the back of the Class Record for tabulating the distributions of scores. 1 T h e p erson in charge of a te stin g p ro g ram for a school or school system will need a copy of th e com plete. Guide for Interpreting the New Stanford Achievement Test in o rd e r to h a v e th e te s t re su lts in te rp re te d a n d used p ro p e rly . 11 in test scores for each group; comparisons of the two methods in terms of gains for B8 and A8 separately; and finally, com parisons of the two methods in terms of gains for these two grades combined. Comparability of groups. As previously stated in this chapter each pair of groups in the experiment was assumed to be reasonably comparable due to the school classification of average or high. However, there was no proof of the extent to which they were comparable. Therefore, it was necessary to compute the mean of the intelligence quotients of every group in order to show the extent of comparability of each pair instructed by the five teachers respectively. The in telligence quotients were obtained from the school counselor’s office. With these data tables were made and the mean intel ligence quotient was computed for each group, thus enabling the extent of comparability of each pair of groups to be shown. This statistical evidence of comparability of each pair of groups was supported by the construction of graphs. Gains in test scores for each group. Having proved the comparability of each pair of groups, the next step was to tabulate the gains made by the students in each group preparatory to comparing the gains made by each pair of groups studying the two different methods. Therefore, a table was made for each of the ten groups showing the pretest 12 score, the final score, and the gain of each student in the group. For example, a pretest score of 112 and a final score of 120 would give a gain of 8; also a pretest score of 110 and a final score of 104 would give a gain of -6. Comparisons of gains for B8 and AS groups separately. In order to arrive at some conclusion regarding the superior ity of one method over the other, it was necessary to compare the gains made by these different groups. were made. To this end, tables In one table the gains of each pair of groups in the B8 level were compared, and the mean, standard deviation, and standard error of the mean of each group were calculated. In another the gains of the B8 ten-week groups were combined and compared with those of the combined B8 semi-weekly alter nation groups, and the mean, standard deviation, and standard error of the mean of each combination were calculated. These resulting data were then c ompiled in still another table in order to serve as a basis for the following calculations: (a) the chances of each group using one method being superior to the group using the other method; and (b) the chances of the combined groups of one method being superior to the com bined groups of the other. In a table following this, the reliability of the results obtained in the previous one was checked. Not only were the gains of B8 groups compared statis tically but also graphically. A graph showing this comparison 13 was made for each pair of groups. Another one was drawn comparing the gains of the combined ten-week groups with those of the combined semi-weekly alternation groups. The same procedure was followed for the A8 groups both in the tabulation of gains and in their graphic repre sentation. Comparisons of gains for combined B8 and J8 groups. Having compared the gains of the B8 and A8 groups separately, it was then necessary to combine these two major groups in order to compare the gains of the entire ten-week groups in the experiment with those of the entire semi-weekly alter nation groups. This was very important as the results of the comparison would show the solution to the problem as to which method was the better of the two* In order to accomplish this, tables were made. First, the gains of the combined B8 and A8 ten-week groups were com pared with those of the combined B8 and A8 semi-weekly alter nation groups, and the mean, standard deviation, and standard error of the mean of each combination were calculated. These resulting data were then compiled in another table in order to serve as a basis for the calculation of what chances the combined groups using one method had of being superior to the combined groups using the other method. This was followed by still another table which checked the reliability of the 14 results obtained in the previous one. Also a graph was drawn showing a comparison of the gains of the combined ten-week groups with those of the com bined semi-weekly alternation groups. All these statistical results and graphic representa tions will be shovm in the next three chapters. CHAPTER III RESULTS OF THE B8 GROUPS Having described the method of procedure of this ex periment and also the method of tabulation, we shall now state and analyze in detail the results obtained in the B8 groups. This is only part of the total results of the experiment. The \two chapters following will deal with the remainder, namely, the results in A8 and the results of the combined B8 and A8 groups, respectively. These results, on which depend the relative merits of the two teaching methods, will be based upon the gains made by the students on two tests given, one a pretest at the begin ning, the other a final test at the end of the experiment. It was stated in the preceding chapter that there were six groups of students in B8. This means that there were three pairs of groups, each pair representing the two methods of Reaching and instructed by the same teacher. In reporting the B8 results, the three pairs of groups will be designated as IA, IB; IIA, IIB; and IIIA, IIIB. The A-groups were those who had studied mathematics on the tenweek basis, and the B-groups those who had studied the same subject on the semi-weekly alternation basis* In this chapter we shall discuss the B8 results in the following order: comparability of groups; gains made on tests 16 by each of the six groups; comparisons of the gains of these six groups arrangedin pairs; comparisons of the ten-week groups, as a single group, with the semi-weekly alternation groups, as a single group, in terms of gains; and finally, comparisons of all groups statistically for prediction pur poses in order to determine the relative merits of both teaching methods for the B8*s. All statistical computations used in this study follow the formulas given in Statistics for Teachers. by Tiegs and Crawford.^Comparability of groups♦ One of the necessary require ments for the successful completion of the experiment was that each pair of groups studying the two methods be compar able in every possible way, leaving only the variable of the two different time-bases. They were comparable in that each pair of groups was taught by the same teacher, was scheduled for an equal length of class period, and had the same number of periods of elass instruction in mathematics during the semester. The final and very necessary comparability would have to be that of ability or intelligence. Although, as stated in the preceding chapter, each pair of groups tapght by the same teacher was assumed to be comparable because they ^I.¥l. Tiegs and C, C. Crawford, Statistics for Teachers. (Boston: Houghton Mifflin Co., 1930). 212 pp. 17 were either average or high according to the school classifi cation, it was necessary to prove this comparability in order to carry out the experiment successfully. To this end the intelligence quotient of each individual in all six B8 groups was obtained from the school counselor’s office and compiled in tables, one for each pair of groups. In these tables, the students were listed alphabetically but for ob vious reasons the names have been omitted in this report. Table I shows the intelligence quotients of groups IA and IB. It is seen that the size of the groups is reason ably comparable due totheir respective 33. membership of 35 and The derived mean intelligence quotients of 114.50 and 112.65, respectively, prove a high degree of comparability in intelligence. Figure I was constructed to show a graphic comparison of the intelligence quotients of these same groups. Although the groups were not equal in number, the slight difference of two students did not call for the construction of the graph on a percentage basis. In this graphic presentation, large class intervals of 20 were used for the reason that a smaller class interval presented the difficulty of too many ’’saw-teeth” which would lessen the visual effect of comparability in the frequency polygon. This figure shows at a glance that both groups have the same range since they meet the base line in the same class intervals. The areas are practically the same. 18 TABU I INTELLIGENCE QUOTIENTS OF GROUPS IA AND IB, B8 CLASSES Pupil no. 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 ]$ean Group IA Group IB 131 115 114 105 148 106 103 102 116 117 130 111 108 133 108 116 99 115 102 135 108 105 111 110 99 118 122 109 118 105 133 121 89 101 131 116 110 108 116 109 123 112 102 112 138 125 99 114 124 117 110 118 131 127 98 97 113 103 123 98 107 101 125 106 107 98 106 122 114,50 112.65 imimi&sMWm: M 61 20 The slight difference which occurs in the height is partly due to the fact that group IA,1represented by the solid line, has two more students than group IB, Hence the figure shows the two groups to be highly comparable in intelligence. Table II shows the intelligence quotients of groups IIA and IIB with 39 and 33 students, respectively. Although there are six more students in the former groups, yet the de rived mean intelligence quotients of 110.32 and 108.41, re spectively are very close and prove the comparability of these groups in intelligence. Figure 2 presents the graphic distribution of the intelligence quotients of these two groups. The difference of six between the numbers of the two groups did not warrant the construction of the graph on a percentage basis. It is seen that the same large class intervals of 20 were used here and for the same reason as in the preceding graph. This figure shows that both groups have the same range since they meet the base line in the same class intervals. The area of the polygon of group IIA, represented by the solid line, is slightly larger than that of group IIB. This is probably due to the fact that group IIA has six more students than group IIB. By studying the figure closely, we see that group IIA exceeds group IIB by two students in the 120-140 interval$ three students in the 100-120 interval and one student in the 80-100 interval. If these particular students were the six 21 TABLE II INTELLIGENCE QUOTIENTS OF GROUPS IIA.AND IIB* B8 CLASSES Pupil no. 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 Mean Group \ IIA Group IIB 118 99 108 100 115 125 104 107 109 102 104 130 107 111 138 106 115 104 120 105 95 108 106 114 104 110 89 125 120 103 109 92 121 123 93 107 118 108 120 118 110 98 87 116 111 107 112 109 108 107 113 102 119 116 106 106 98 97 114 109 118 109 103 104 123 96 101 92 117 106 110 94 110.32 108.41 m 58^ 23 to be taken out to make the groups.equal in number, the poly gons would be identical* Hence it is reasonable to conclude that the two groups are highly comparable. fable III shows the intelligence quotients of group IIIA with 28 pupils and of group IIIB with 30. The computed mean intelligence quotients of 99.29 and 101.00 prove the first phase of their comparability in intelligence. second phase is shown graphically in Figure 3. The As in Figure 1 , the only difference between the two groups is two students, which fact did not require the graph to be constructed on a percentage basis. This fugure shows that although the areas of the polygons are approximately equal, the intelligence quotients of group IIIB, the broken line, are slightly higher than those of group IIIA. However, this difference is not sufficient to prevent them from being considered comparable and being entered in this experiment. The same class inter vals of 20 were used here as in preceding figures. These tables and graphs with their statistical and visual proof of comparability in each pair of groups of the B8 class leaves only, as has been stated previously, the variable of the two time-bases which are the distinguishing elements of the two teaching methods. Gains made on tests by each B8 group. Each group was given a pretest at the beginning of the experiment and a final test at the end. Obviously the gain of each student was his 24 TABLE III INTELLIGENCE QUOTIENTS OF GROUPS IIIA AND IIIB, B8 CLASSES Pupil no. 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 Mean Group IIIA Group IIIB 99 100 98 108 96 107 96 102 92 102 108 84 103 111 109 104 87 92 97 96 80 95 92 118 82 105 106 97 96 112 97 103 97 98 ;9i 98 101 100 101 102 100 98 93 93 94 101 103 114 111 104 94 112 102 99 101 104 102 104 99.29 101.00 26 final score minus that of his pretest. Since the relative - merits of the two methods of teaching were to he determined by the comparison of the gains made by the students on the two tests, it was first necessary to tabulate the gains for each group in order that they might serve as a statistical basis for these later comparisons. In these tables, the scores are placed, not from the highest to the lowest in consecutive order, but according to the names of the students arranged alphabetically. The names are omitted as in the former tables. Table 17 shows the pretest score, the final score, and the gain of each student in group IA using the ten-week method* The range of the pretest score is from 76 to 111, of the final score from 73 to 117, and of the gains from -5 to 23. ^here were five students who made losses which are expressed as negative gains in the table. Table V shows the pretest score, the final score, and the gain of each student in group IB using the semi-weekly alternation method. This group and IA were taught by the same teacher. - The range of the pretest score is from 68 to 111, of the final score from 70 to 112, and of the gains from -3 to 15. In this group there were only two students who made losses. Table 71 shows the pretest score, the final score, and the gain of each student in group IIA using the ten-week 27 TABLE IV GAINS MADE- BY GROUP IA, T1N-WEEKS* B8 CLASS Pupil no. 1 a 3 4 5 6 7 8 9 10 11 ia 13 14 15 16 17 18 19 ao 81 22 83 ft4 85 86 87 88 89 30 31 38 33 34 35 Pre test score Final test score 110 88 93 80 109 94 87 79 90 100 88 76 76 103 94 107 107 96 98 86 99 88 89 93 87 100 97 79 97 91 103 108 88 99 111 113 99 110 94 117 99 97 88 104 105 96 89 73 110 109 110 104 105 105 109 107 100 106 106 88 111 103 89 109 89 103 108 90 96 113 Ga in 3 11 17 14 8 5 10 9 14 5 8 13 - 3 7 15 3 - 3 9 7 83 8 18 17 13 - 5 11 6 10 IS - 8 0 6 8 - 3 8 28 TABLE. 7 GAINS MADE BY GROUP IB* SMI-WEEKLY ALTERNATION, B8 CLASS Pupil no. Pre test score Einal test score Gain 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 94 73 97 96 98 82 102 90 76 102 94 68 100 90 97 79 111 98 104 75 87 89 86 93 77 92 77 93 93 81 92 97 103 85 108 105 109 88 104 97 87 112 100 70 101 100 97 90 112 107 108 82 98 88 88 99 81 95 81 97 108 88 89 105 9 12 11 9 11 6 2 7 11 10 6 2 1 10 0 11 1 9 4 7 11 - 1 2 6 4 3 4 4 15 7 - 3 8 29 TABLE VI GAINS MADE BY GKOUP IIA, TEN-WEEKS, B8 CLASS Pupil no. Pre test score 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 30 318 32 33 34 35 36 37 38 39 96 94 88 83 78 94 83 883 81 85 84 91 92 101 107J 103 91 93 91 91 90 91 100 89 91 83 89 103 104 94 107 68 96 106 75 107 87 96 107 Final test score 108 103 96 86 81 100 98 86 92 81 97 93 98 95 111 107 99 104 96 98 93 97 102 93 99 88 91 • 101 107 103 112 81 94 107 76 107 105 103 10*7 Gain 12 9 8 3 3 6 15 3 11 - 4 13 2 6 - 6 4 4 8 11 5 7 3 6 2 4 8 5 2 - 2 3 9 5 13 - 2 1 1 0 18 7 0 30 method. The range of the pretest score is from 75 to 107, of the final score from 76 to 112, and of the gains from -6 to 18. In this group there were four students who made losses. Table VII shows the pretest score, the final score, and the gain of each student in group IIB using the semi-weekly alternation method. second teacher. This group and IIA were taught by the The range of the pretest score is from 53-to 104, of the final score from 62 to 107, and of the gains from -11 to 20. There were five students who made losses in this group. Table VIII shows the pretest seore, the final score, and the gain of each student in group IIIA using the ten-week method. The range of the pretest score is from 63 to 106, of the final score froiji 64 to 104, and of the gains from -7 to 12. There were seven students who made losses in this group. Table IX shows the pretest score, the final score, and the gain of each student in group IIIB using the semi-weekly alternation method. third teacher. This group and IIIA were taught by the The range of the pretest score is from 71 to 104, of the final score from 73 to 107, and of the gains from -4 to 11. group. There were ten students who made losses in this 31 TABLE 711 GAINS MADE BY GROUP IIB, SEMI-WEEKLY ALTERNATION, B8 CLASS/ ; Pupil no. Pre test score Pinalr test score Gain 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 86 89 80 89 87 93 86 91 87 95 93 81 77 96 76 75 91 80 82 96 89 96 80 86 53 104 70 89 99 102 94 94 91 106 83 81 78 96 103 93 100 99 104 105 86 86 104 89 79 96 81 81 102 95 107 83 96 62 106 71 91 97 105 94 105 87 20 - 6 1 -11 9 10 7 9 12 9 12 5 9 8 13 4 5 1 - 1 6 6 11 3 10 9 2 1 2 - 2 3 0 11 - 4 32 TABLE VIII GAINS MADE BY GROUP IIIA, TIN-WEEKS, B8 CLASS Pupil no. Pre test score Final test score Gain 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 87 93 98 77 78 86 75 93 95 80 97 95 81 72 79 74 89 75 68 63' 86 75 82 106 70 81 72 70 80 104 102 86 75 92 76 98 100 88 100 92 79 84 88 78 94 76 77 71 82 87 85 102 71 83 75 64 - 7 11 4 9 - 3 6 1 5 5 8 3 - 3 - 2 12 9 4 5 1 9 8 - 4 12 3 - 4 1 2 3 - 6 33 TABLE IX GAINS MADE BY GROUP IIIB, SEMI-WEEKLY ALTERNATION, B8 CLASS Pupil no, 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 Pretest; score 100 88 86 99 88 75 71 74 99 86 91 104 78 72 79 87 75 81 80 93 89 84 81 98 81 74 80 . 75 81 101 Final test score Gain 99 85 82 95 89 75 73 80 96 85 93 107 79 76 78 91 81 84 88 90 100 95 77 98 88 84 79 •76 82 104 • r- 1 3 4 4 1 0 2 6 - 3 - 1 2 3 1 4 - 1 4 6 3 8 - 3 11 11 - 4 0 7 10 - 1 1 1 3 34 Comparisons of the three pairs of groups in terms of gains* Having tabulated the gains of each of these six groups, the next step was to make comparisons in terms of gainst of these six groups arranged in pairs, each pair representing both methods of teaching and taught by the same teacher* To this end Tgble X was made* The three pairs of groups are arranged in three para llel columns. The students of each group are arranged in a frequency distribution according to the number making each amount of gain as shown in the corresponding class interval* The gains arranged consecutively in class intervals of two range from (-12)-(-11) to 22-23. The reasons for arranging the groups in pairs were: (a) that the number of students in one group making each amount of gain might be compared with the number of students in the corresponding group making the same amount of gain; (b) that the mean, standard deviation, and standard error of the mean might be calculated from each frequency distribution; (c) that the obtained means of the corresponding groups could be compared; and (d) that the mean, standard deviation, and standard error of the mean might be used as bases in statis tical comparisons to be made later. The means of groups IA and IB being 8*14 and 6.70, respectively, show a difference of 1.44. This is equivalent to each student in group IA making a gain of 1.44 more than 35 TABLE X COMPARISONS OF THE TWO METHODS IN TERMS OF GAINS FOR B8 GROUPS Gains in Number of pupils showing each amount of ga in class Ten weeks Semi weekly Ten weeks Groups intervals IA 22-23 20 - 21 18 - 19 16 - 17 14 - 15 12 - 13 10 - 11 8 - 9 6 - 7 4 - 5 2 - 3 0 - 1 (- 2} (- ,1) (- 4) - (- 3) C- 6) - (- 5) C- 8} - (-7) (-10) - ( -9) (-12) - (-11) 1 Number Mean S.D. S . E Jvl. IB Semi weekly Ten weeks Groups IIA IIB Semi weekly Groups IIIA IIIB 1 i 2 3 3 4 5 4 2 4 1 1 3 1 1 1 1 7 4 7 4 4 3 1 1 1 3 2 5 5 6 8 4 2 1 1 3 4 6 3 3 4 4 2 1 1 2 1 5 1 5 4 3 1 4 1 1 3 1 3 2 5 6 4 6 1 35 8.14 6.66 1.13 39 33 6.70 5.72 4.99 4.18 •73 ' .80 33 5.73 6.17 1.07 28 3.71 5.35 1.01 30 2.40 4.39 .80- S.D. is Standard Deviation, and S.E. M. is the Standard Error of the Mean. IA and IB are the groups taught by one teacher; IIA and IIB are the groups taught by another teacher; and IIIA and IIIB are taught by still another teacher. 36 each student in group IB* This obtained difference is only slightly in favor of group IA representing the ten-week method• Although the standard deviation and the standard error of the mean of each group was calculated for the chief purpose of serving as a basis for statistical comparisons to *. be made later, we shall at this time point out their signifi cance in relation to the means of these two groups. We shall do the same for the other groups when we arrive at a discussion of theif results. Group IA has a standard deviation of 6.66. This sig nifies that approximately two-thirds of all the gains, or S3 gains, would be included between the limits of 8.14 (the mean) plus 6.66, or 14.8G, and 8.14 minus 6.66, or 1.48. This is verified by observing the distribution itself. The standard error of the mean of group IA is 1.13. This signifies that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than 1.13 from the observed mean of 8 ‘ .14. That is, two-thirds of all our future means of this group would lie between the gains of 7.01 and 9.S7. In group IB, with a standard deviation of 4.18, approx imately two-thirds of all the gains, or 22 gains, would lie between the limits 6.70 (the mean) plus 4.18i, or 10.88, and 6.70 minus 4.18, or 2.52. 37 The standard error of the mean of group IB is ,73. This means that if we gave this group a large number of tests -and calculated the gains, two-thirds of the time their means would not deviate more than .73 from the mean of 6.70. That is, two-thirds of all the future means of this group would lie between the gains of 5.97 and 7.43. Figure 4 shows a graphic comparison of the gains made by these same two groups. Here we see that group IA of the ten-week method has a greater range of gains than group IB. If we add the excess high gains and the excess low gains of group IA, the result would offset the excess height of the polygon of group IB. Thus we can reasonably conclude that the areas of the two polygons are approximately equal. The difference of two students between the two groups affects the comparison of the areas but little. The figure also shows that in the two extremes of the gains made by group IA, the excess of high gains was greater than the excess of low gains. This gives the slight advantage to the group of the ten-week method in this case. The small difference of .01 between the means of groups IIA and IIB is to be ignored. Consequently, their means are considered to be equal and neither of the two methods had any advantage. Group IIA has a standard deviation of 4.99. Two-thirds of all the gains, then, would be included between the limits 39 of 5.72 (the mean) plus 4.99, or 10.71, and 5.72 minus 4.99, or .73. Approximately 26 gains would be found in this range. The standard error of the mean of group IIA being .80 means that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .80 from the mean of 5.73. That is, two-thirds of all our future means of this group would lie between the gains of 4.93 and 6.53. Group IIB with a standard deviation of 6.17 signifies that approximately two-thirds of all the gains, or 22 gains, would be included between the limits 5.73 (the mean) plus 6.17, or 11.90, and 5.73 minus 6.17, or -.44. The standard error of the mean of group IIB being 1.07 means that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than 1.07 from the mean of 5.73. That is, two-thirds of all our future means of this group would lie between the gains of 4.66 and 6.80. Figure 5 compares the gains of groups IIA and IIB in a graphic way. Class intervals of 4 were used for the gains as in the preceding figure. A^ a glance we see that the greater range of gains was made by group IIB, a semi-weekly alternation group, the broken line. Group IIA has a larger number of students centering around the mean as shown by the height of its graph, the solid line. This is partly due to BIS jshe fact that group IIA has six more students than the other group* If these six students were excluded from its polygon we could reasonably conclude that the resulting excess height of the polygon of group IIA would be offset by the sum of the excess high gains and the excess low gains of group IIA* the areas of the polygons would be approximately equal* Thus It also shows that in the two extremes of the range of gains made by group IIB, the excess high gains approximately nullify the excess low gains, thus giving neither of the two methods the advantage. Table X shows the means of groups IIIA and IIIB to be 5.71 and 2.40, respectively. The difference of 1.31 is equiv alent to each student in group IIIA making a gain of 1.31 more than each student in group IIIB. This obtained difference slightly favors the ten-week method represented by group IIIA. Group IIIA has a standard deviation of 5.35. This means that approximately two-thirds of all the gains, or 19 gains, would be included between the limits 3.71 (the mean) plus 5.35, or 9.06, and 3.71 minus 5.35, or -1.64. The standard error of the mean of group IIIA is 1.01. This signifies that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than 1.01 from the mean of That is, two-thirds of all our future means of this group would lie between the gains of 2*70 and 4.72. 3.71. Group IIIB has a standard deviation of 4.39, which means that approximately two-thirds of all the gains, or 20 gains, would be included within.the limits 2.40 (the mean) plus 4.39, or 6.79, and 2.40 minus 4.39, or -1.99. The standard error of the mean of group IIIB is .80 signifying that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .80 from the mean of 2.40. That is, two-thirds of all our future means of this group would lie between the gains of 1.60 and 3.20. Figure 6 is a graphic presentation of the gains of groups IIIA and IIIB. Class intervals of 4 were used for the gains as in the two preceding figures. A first glance at this figure might indicate that the areas of the polygons are un equal. By studying it more closely, however, we see that if we add the excess high gains and the excess low gains of group IIIA, the result would about equal the excess height of the polygon of group IIIB. Consequently, the areas of the polygons are approximately equal. The difference of two students between the two groups affects the areas so slight ly that it can be practically ignored. greater range of gains. Group IIIA has the The excess of high gains is greater than the excess of low gains giving this group, representing the ten-week method, the slight advantage. 44 Comparisons of the ten-week groups, as ja single group, with the semi-weekly alternation groups. as £ single group, in terms of gains. Having made comparisons of the six groups arranged in pairs, the next step was to put the same three ten-week groups together as a single group and compare their gains with those of the same three semi-weekly alternation groups placed together as a single group. The reason for do ing this was that, according to statistics, the greater the numbers of cases examined the higher the reliability of the results. To this end, Table XI was made. The main point was to calculate the mean, the standard deviation, and the standard error of the mean of each of the combined groups. The purpose of this was to*see in particular if the means came within close range of each other. If they should, then there would be very little difference between the two methods but if not, then a superiority of one method over the other would be the result. The derived mean, standard deviation, and standard error of the mean will not only be used in the present compar ison but will also serve as some of the statistical bases for further comparisons to be made later. The two combined groups are arranged in this table in two parallel columns. The students of each combined group are arranged in a frequency distribution according to the number making each amount of gain as shown in the corresponding class 45 TABLE XI COMPARISONS OF TEN-WEEK GROUPS AS A SINGLE GROUP, WITH SM I-WEEKLY ALTERNATION GROUPS AS A SINGLE GROUP, IN TERMS OF GAINS FOR B8 CLASSES Gains in Number of pupils showing each amount of gain class Ten* week groups Semi weekly groups intervals I,II, IIIA I,II, IIIB 22 20 18 16 14 12 10 8 6 4 2 0 ( 4 (4 4) C- 6) C- 8) (-10) (-12) 2 - Number Mean S*D. S.E.M. 23 21 19 17 15 13 11 9 7 55 3 1 ) c- i) f- 3) (- 5} (- 7) (- 9) (-11) 1 1 2 2 4 8 7 15 10 13 16 8 4 8 3 1 1 4 14 11 13 9 13 13 7 8 1 1 102 6.00 5.97 .59 96 5.02 5.33 .54 S.D. is the Standard Deviation, and S.E.M. is the Standard Error of the Mean* 46 interval. The gains arranged consecutively in class intervals of two and range from (-12)-(-11) to 22-23. It is seen also that the number of students is practically equal for both methods, that of the ten-week method-being 102 and that of the senond method being 96. The means of the ten-week method and the semi-weekly alternation method are 6.00 and 5.02, respectively. This signifies that they are within close range of each other. The small difference of .98 is slightly in favor of the tenweek method. We notice that the derived standard deviations of the two methods are within close range of each other, as are also the standard errors of the means. The standard deviation of the ten-wefek group being 5.97 signifies that approximately two-thirds of all the gains, or 68 gains, would be included between the limits 6.00 (the mean) plus 5.97, or 11.97, and 6.00 minus 5.97, or .03. The standard error of the mean of the ten-week group being .59 means that if we gave this large group an infinite number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .59 from the mean of 6.00. That is, two-thirds of all our future means of this group would lie between the gains of 5.41 and 6.59. The semi-weekly group having a standard deviation of 47 5.35 signifies that approximately two-thirds of all the gains, or 64 gains, would be included between the limits 5.02 (the mean) plus 5.33, or 10.35, and 5.02 minus 5.33, or -.31. The standard error of the mean of the semi-weekly group being .54 means that if we gave this group alarge number of. tests and calculated the gains, two-thirds of the time our means would not deviate more than .54 from the mean of 5.02. That is, two-thirds of all our future means of this group would liebetween the gains of 4.48 and 5.56. Figure 7 shows a graphic comparison of the gains of these two large groups. At a glance we see that one polygon is practically superimposed upon the other, the distribution of gains of one group following closely that of the other. Hence we can very reasonably say that the areas of the poly gons are equal. Observing the graph more closely, we see that the ten-week group has more high gains and fewer low gains than the semi-weekly group. This indicates a slight advantage in favor of the ten-week method. Statistical comparisons of B8 groups for prediction purposes. In the first place, we shall compare (a) the three pairs of groups, IA and IB, IIA and IIB, and IIIA and IIIB in order to determine for each pair the chances that the group of one teaching method would have of being superior to the group of the other teaching method; and (b) the three tenweek groups I, II, IIIA, as a single group, with the three II 49 semi-weekly groups, I, IIJ,IIIB, also as a single group, in order to determine the chance that one of these combinations would have of being superior to the other. The pairs of groups and the two combinations will be seen in the same table. From the means and standard errors of the means of each pair and also of each combination will be derived other statistical data in order that the chances of superiority may be determined. In the second place, we shall compare the same three ten-week groups, I, II, IIIA, as a single group, with the same three semi-weekly groups, I, II, IIIB, as a single group, not only for the purpose of prediction, but also to serve as a check on the results of the first comparison. The means used here as a basis for comparing the com bined groups are very different from those used in the first comparison. The means used in the first comparison were those obtained as a result of making a frequency distribution of each of the combined groups as a single separate group, while here the original means of each of the groups, I, II, IIIA, the tenweek groups, are simply added and used as a basis of comparison, the same being done with the original means of each of the groups, I, II, IIIB. In addition to these two sums of the means, other data will be calculated in order that the chances of superiority of one combined group over the other may be shown. 50 Having stated the two ways in which these comparisons will be made, we shall now discuss the first with the use of Table XXI. This table shows each of the three pairs of groups and each of the combined groups as previously mentioned. The table indicates that the obtained difference be tween the two means of groups IA and IB is 1.44. The signif icance of this difference is determined by dividing 1.44 by the standard error of the difference, or 1.35, which yields 1.07, or the ratio. By referring to a standard-error table,1 this ratio was found to signify that the chances are only 6 to 1 that group IA, the ten-week method, would be superior to group IB, the semi-weekly method. In order to be practically certain that there is a real difference in merit between these two groups, the difference between the means should be at least three times as great as the standard error of the o difference. In the present case, the difference, 1.44, is just a shade larger than the standard error of the difference, 1.35. On the basis of these data (as revealed by the table referred to) we can say that the chances are very slight that group IA, the ten-week method, would actually make higher ^ . W . Tiegs and C.C. Crawford. Statistics for Teachers, (Boston: Houghton Mifflin Co., 1930). p. 137. 2 H.W. Tiegs, Tests and Measurements for Teachers, (Boston: Hpughton Mifflin Co., 1 9 3 1 ) . p . 234. TABLE XII STATISTICAL COMPARISON OP GAINS OF B8 GROUPS SHOWING PREDICTIONS FOR SEPARATE CLASS GROUPS AND FOR THE TOTAL GROUPS AS SINGLE LARGE DISTRIBUTIONS Groups compared Means (Ten weeks} Means (Semi weekly) S.E.M. (Ten weeks) S.E.M. (Semiweekly) Diff. EP Ratio i 1.07 Chancei i IA vs. IB 8.14 6.70 1.13 .73 1.44 1.35 IIA vs. I IB 5.72 5.73 .80 1.07 .01 1.34 IIIA vs.IIIB 3.71 2.40 1.01 .80 1.31 1.29 1.02 5.5to 1 All students in groups I,11,IIIA 6.00 .98 .80 1.23 8.2 to 1 1 to 1 .59 vs. All students in groups I,IIfIIIB .007 6 to 1 5.02 .54 IA,IIA, and IIIA are the ten-week groups, IB,IIB# and IIIB are the semi weekly alternation groups. S.E.M. is the Standard Error of the Mean, and is the Standard Error of the difference. 5a gains in mathematics than group IB, the semi-weekly method* The difference between the two means of groups IIA and IIB was *01* The significance of this difference was deter mined by dividing *01 by the standard error of the difference, or 1.34, which yields .007, or the ratio. In the standard- error table we find this small ratio to signify that the chances are 1 to 1 that the semi-weekly group IIB (the group whose mean was the larger by .01) is superior to group IIA, the other method. On the basis of these data we can say that the chances are even that one of these two groups would make high er gains in mathematics than the other. The difference between the means of IIIA and IIIB was 1.31. By dividing this difference by the standard error of the difference, or 1.29, we obtain the ratio, or 1.02. The standard-error table reveals this ratio to indicate that the chances are 5.5 to 1 that the ten-week group IIIA is superior to the semi-weekly group IIIB. On the basis of these data we can say that the chances are very slight that group IIIA would actually make higher gains in mathematics than group IIIB. Finally, in this table the three ten-week groups, as a single group, were compared with the three semi-weekly groups, as a single group. The difference between the means of these two large groups is .98. By dividing this difference of .98 by .80, the standard error of the difference, we obtain the ratio of 1.23. The standard-error table shows this ratio to 53 signify that the chances are 8,2 to 1 that the I* II. IIIA combination is superior to the I, II, IIIB combination. On the basis of these data we can say that the chances are very slight that the ten-week combination would make high er gainsin mathematics than the semi-weekly combination. As a result of combining the ten-week groups and also the semi-weekly groups, we see that the chances of superiority of the ten-week groups were slightly increased. The ratio of the combined groups, namely, 8.2 to 1, is slightly larger than each of the ratios obtained in the comparison of the respective pairs. The second method of comparing the three combined tenweek groups with the three combined semi-weekly groups is shown in Table XIII. As was stated previously, the means of IA, IIA, and IIIA, respectively, were added as were those of IB, IIB, and IIIB. vThe obtained sums of the means then be came the basis on which further calculations were made in order to arrive at the chances of superiority of one combined group over the other. This table shows the obtained difference between the sums of the means to be 2.74. Dividing this quantity by the standard error of the difference between the sums of the means, or 2.30, we obtain the ratio of 1.19. This ratio signifies that the chances are 7.6 to 1 that the I, II, IIIA combination is superior to the I, II, IIIB combination. With these data 54 TABLE XIII STATISTICAL COMPARISON OF B8 GROUPS SHOWING A SECOND METHOD OF MAKING PREDICTIONS BY DIFFERENT COMPUTATIONS OF THE MEANS B8 groups Ten-week groups Items Mean % Sum of means groups A and B E of sum of means, groups A and B Semi-weekly groups IA IIA IIIA IB IIB IIIB 3.14 1.13 5.72 .80 3.71 1.01 6.70 .73 5.73 1.07 2.40 .80 17.57 14.83 1.71 1.52 Diff. between sums of means, group A minus groups B 2.74 E of diff. be tween sums of means of groups A and B 2.30 Eatio 1.19 Chances D/ED 7. 6 to 1 E|£ is the Standard Error of the Mean; E is the Standard Error; D is the difference, and ED.the Standard Error of the difference. 55 as a basis, we ean say that the 'chances are again very slight that the ten-week combination would make higher gains in mathematics than the semi-weekly combination. In both these ways of comparing the two combined groups, we found that the results obtained were practically the same, as the ratios were 8.S to 1, and 7.6 to 1, respectively, and both gave the ten-week method the advantage, but of no great significance. CHAPTER IV RESULTS OF TEE A8 GROUPS Having shown the results obtained in the B8 groups, we shall now state the results of the A8 groups both in tab ular and- in graphic form pointing out the significant features of such as we proceed. As in the case of the B 8 ’s, so these A8 results will be based upon the gains made by the students on the two tests given, one a pretest at the beginning, the other a final test at the end of the experiment. These gains will determine the relative merits of the two different methods. It was stated in Chapter II that there were four groups of students in A8, This means that there were two pairs of groups, each pair representing the two different time-bases of teaching and instructed by the same teacher. In relating the A8 results, the two pairs of groups will be designated as IVA, IVB, and VA, VB. The A-groups were those who had studied mathematics on the ten-week basis, and the B-groups those who had studied the same subject on the semi-weekly alternation basis. We shall show the A8 results in this chapter in the following order: comparability of groups; gains made on tests by each of the four groups; comparisons of the gains of these four groups arranged in pairs; comparisons of the ten-week 57 groups, as a single group, with the semi-weekly alternation groups, as a single group, in terms of gains; and finally, comparisons of all groups statistically for prediction pur poses in order to determine the relative merits of both teach ing methods for the A 8 ’s. Comparability of groups. The comparability of the 48 groups was obtained in the same manner as that of the B 8 ’s. The A8 groups were comparable in that each pair of groups was taught by the same teacher, was scheduled for an equal length of class period, and had the same number of periods of class instruction in mathematics during the semester leaving only the comparability of intelligence to be proved. To this end the intelligence quotient of each individual in all four A8 groups was obtained from the school counselor’s office and compiled in tables, one for each pair of groups. In these tables, the students were listed alphabetically but for obvious reasons the names haVe been omitted in this report. Table XIV shows the intelligence quotients of groups IVA and IVB. Although there are five more students in the former group, yet the derived mean intelligence quotients of 99.07 and 94.83, respectively, are reasonably close and prove the comparability of these two groups in intelligence. Figure 8 is a graphic comparison of the intelligence quotients of these same two groups. Large class intervals of 20 were used here for the same reason as stated in the 58 TABLE XIV INTELLIGENCE QUOTIENTS OF GROUPS IVA AND IVB, A8 CLASSES Pupil no. 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 Mean Group IVA Group IVB 96 89 85 112 104 110 111 101 93 100 95 97 82 117 116 92 96 91 113 107 95 90 92 105 77 86 92 80 97 110 99 110 87 118 88 79 80 95 101 95 103 84 87 83 80 98 106 97 89 81 94 107 85 101 81 103 82 91 88' 113 106 90 113 106 105 99.07 94.83 59 :H:E 60 description of the corresponding B8 groups. The difference of five students between the two groups did not warrant the drawing of the graph on a percentage basis. It is seen at a glance that both groups have the same range in intelligence since they meet the base line in the same class intervals. The polygons are very similar. The slight difference between the areas is due to the fact that group TVA has five more stu dents than group IVB. Hence the figure shows the two groups to be highly comparable in intelligence. Table XV shows the intelligence quotients of groups VA and VB. The size of the groups is reasonably comparable due to their respective memberships of 35 and 32. The derived mean intelligence quotients of 1G5.21 and 108.75, respective ly, prove a high degree of comparability in intelligence. Figure 9 shows a graphic distribution of the intelli gence quotients of these same two groups. The slight differ ence of three students did not call for the construction of the graph on a percentage basis. Large class intervals of 20 were used as in the preceding graph. The fact that both polygons meet the base line in the s^ame class intervals shows that both groups have the same range in intelligence. The slight difference in area is principally due to the fact that group VA, represented by the solid line, has three more * students than group VB. Hence it is reasonable to conclude from this graph that both groups are highly comparable in 61 TABLE XV INTELLIGENCE Q U O T I M T S OF GROUPS VA AND VB, A8 CLASSES Pupil no. 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 Mean Group VA Group VB 103 113 108 103 98 113 101 112 96 114 109 112 92 111 101 104 87 106 107 103 116 98 108 87 114 112 98 107 122 106 108 101 112 93 104 103 99 102 101 110 95 84 106 101 109 117 114 118 107 119 99 121 116 109 103 114 112 125 112 117 95 116 116 112 102 103 111 105,21 108.75 63 intelligence. These tables and graphs with their proof of compara bility in each pair of groups of the A8 class leave only the variable of the two time-bases which are the distinguishing elements of the two teaching methods. Gains made on tests by each A8 group. Following the same procedure used with the B8 classes, each A8 group was given a pretest at the beginning of the experiment and a final test at the end. This was done in order made by each student. to show the gain It was then necessary to tabulate the gains made by each group as they were the determinants of the relative merits of the two different methods of teaching. In these tables, the scores are placed according to the names of the students arranged alphabetically and not from the highest to the lowest in consecutive order. . For obvious reasons the names of the students are omitted. Table XVI shows the pretest score, the final score, and the gain of each student in group TVA using the ten-week meth od. The range of the pretest score is from 73 to 107, of the final score from 75 to 113, and of the gains from -7 to 24. There were seven students who made losses which are expressed as negative gains in the table. Table XVII shows the pretest score, the final score, and the gain of each student in group IVB using the semi-weekly 64 TABLE XVI GAINS MADE BY GBOUP IVA, TEN-?/EEKE, A8 CLASS Pupil no* Pretest score Final test score Gain 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 91 96 89 100 93 87 86 89 80 92 101 107 79 90 102 81 90 82 87 99 92 85 98 87 73 75 96 79 84 94 104 83 94 95 98 98 108 94 102 95 90 96 100 89 97 97 112 80 107 100 96 93 83 92 98 88 98 113 100 75 77 97 84 108 97 97 80 105 100 97 7 12 5 2 2 3 10 11 9 5 - 4 5 1 17 - 2 15 .3 1 5 - 1 - 4 13 15 13 2 2 ’ 1 5 24 3 - 7 - 3 11 5 - 1 65 TABLE 2711 GAINS MADE BY GROUP IVB, SEMI-WEEKLY ALTERNATION, A8 CLASS Pupil no. pre test score Final test score Gain 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 73 102 80 96 86 73 74 77 88 74 93 93 91 95 75 97 97 79 •83 88 89 86 75 92 103 90 94 109 83 101 82 100 * 81 196 90 84 83 81 80 73 98 91 102 96 85 93 106 94 92 86 109 96 75 96 103 92 99 109 94 104 9 - 2 1 10 4 11 9 4 - 8 - 1 5 - 2 11 1 10 - 4 9 15 9 - 2 20 10 0 4 0 2 5 0 11 3 66 alternation method. same (teacher. This group and IVA were taught by the The range of the pretest score happens to be the same as that of the final score, namely, from 73 to 109. The gains range from -8 to £0. In this group of 30 students there were three who showed no gain, and sis who made losses. Table XVIII shows the pretest score, the final score, and the gain of each student in group VA using the ten-week method. The range of the pretest score is from 65 to 113, of the final score from 68 to 115, and of the gains from -10 to 10. In this group of 35 students there were three who showed no gain, and ten who made losses. Table XIX shows the pretest score, the final score, and the gain of each student in group VB using the semi-weekly alternation method. teacher. This group and VA were taught by the same The range of the pretest score is from 66 to 110, of the final score from 70 to 110, and of the gains from -8 to 1£. In this group of 3£ students there were two who show ed no gain, and ten who made losses. Comparisons of the two pairs of groups in terms of gains. Having tabulated the gains of each of these four groups, the next step was to arrange these four groups in pairs in order to make comparisons in terms of gains. pair represented both methods of teaching the same teacher. Each and was taught by Table XX shows this arrangement. 67 TABLE XVIII GAINS MADE BY GROUP VA, TEN-WEEKS, A8 CLASS Pupil no. Pre test score Final test score Gain 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 94 91 107 102 68 106 108 96 80 102 65 72 87 97 90 89 85 105 82 96 113 "'96 90 80 97 97 105 104 90 90 90 95 92 90 80 89 93 110 109 68 112 98 94 86 105 70 77 83 97 98 90 94 108 89 91 115 102 84 73 106 103 105 97 100 100 91 89 97 86 88 - 5 2 3 7 0 6 -10 - 2 6 3 5 5 - 4 0 8 1 9 3 7 - 5 2 6 - 6 - 7 9 6 0 - 7 10 10 1 - 6 5 - 4 8 68 TABLE XIX GAINS MADE BY GROUP VB, 8 M M W 5 B K X Y ALTERNATION, A8 CLASS Pupil no. Pre test score Einal test score Gain 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 84 110 102 85 110 87 85 101 103 106 99 84 88 89 66 79 93 98 90 100 92 68 89 109 90 90 94 78 107 95 89 87 86 103 99 87 110 84 77 104 100 110 108 88 94 90 75 82 101 110 93 98 96 70 100 102 95 83 98 73 107 93 90 90 2 - 7 - 3 2 0 - 3 5 8 3 - 3 4 9 4 6 1 9 3 8 12 3 - 2 4 2 11 - 7 5 - 7 4 - 5 0 - 2 1 3 69 TABLE IX COMPARISONS OP THE TWO METHODS IN TERMS OP GAINS FOR A8 GROUPS Gains in Number of pupils showing each amount of gain class Ten Semiweeks weekly Groups IVA IVB intervals 24 22 20 18 16 14 12 10 8 6 4 2 0 (- 2) (- 4) (- 6) (- 8) (-10) Number Mean S.D. SiE.M. - 25 23 21 19 17 15 13 11 9 7 5 .3 1 (((((- Ten weeks Semi weekly Groups VB VA 1 1 1} 3) 5) 7} 9) 1 2 3 2 1 1 6 7 3 3 3 5 2 5 4 1 1 1 35 5.57 6.70 1.13 1 6 4 30 5.33 5.98 1.09 2 4 7 3 5 5 1 2 4 1 1 35 2.83 5.35 .90 1 1 3 1 5 7 4 2 3 1 4 32 2.00 5.29 .94 S.D. is Standard Deviation, end S.E.M. is the Standard Error of the Mean# IVA and IVB are the groups taught by one teacher; VA and VB are the groups taught by another teacher. 70 In this table the two pairs of groups are shown in two parallel columns. The students of each group are arranged in a frequency distribution according to the number making each amount of gain as shown in the corresponding class interval# The gains are arranged consecutively in class intervals of two; the class intervals range from (-10)-(-9) to 24-25. The arrangement of the groups in pairs in this table were for the following purposes: (a) that the number of students in one group making each amount of gain might be com pared with the number of students in the corresponding group making the same amount of gain; (b) that the mean, standard deviation, and standard error of the mean might he calculated from each frequency distribution; (c) that the obtained means of the corresponding groups could be compared; and (d) that the mean, standard deviation, and standard error of the mean might be used as bases in statistical comparisons to be made later• The means of groups IVA snd IVB being 5.57 and 5.33, respectively, show a difference of .24. This is equivalent to each student in group IVA making a gain of .24 more than each student in group IVB. This obtained difference, however, is very small and is slightly in favor of group IVA represent ing the ten-week method. Although the standard deviation and the standard error of the mean of each group was calculated to serve as a basis 71 for statistical comparisons to be made later, we shall at this time point out their significance in relation to the means of these two groups. We shall do the same for the other groups when we arrive at a discussion of their results. Group IVA has a standard deviation of 6,70, This sig nifies that approximately two-thirds of all the gains, or 23 gains, would be included between the limits of 5,57 (the mean) plus 6,70, or 12*27, and 5,57 minus 6,70, or -1,13, By ob serving the distribution closely we verify this. The standard error of the mean of group IVA is 1,13, This signifies that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than 1.13 from the observed mean of 5.57. That is, two-thirds of all the future means of this group would lie between the gains of 4,44 and 6.70# In group IVB, with a standard deviation of 5.98, approx imately two-thirds of all the gains, or 20 gains, would lie between the limits of 5.33 (the mean) plus 5.98, or 11.31, and 5.33 minus 5.98, or -.65. The standard error of the mean of group IVB is 1.09. This means that if we gave this group a large number of tests and calculated the gains, two-thirds of the time their means would not deviate more than 1.09 from the mean of 5.33. That is, two-thirds of all the future means of this group would lie between the gains of 4.24 and 6.42. 72 Figure 10 is a graphic presentation of the gains of groups IVA and IVB# The base line shows that class intervals of 4 were used for the gains. than group IVB# Group IVA has five more students If these five students were excluded from group IVA, represented by the solid line polygon, and then the polygons placed one upon the other in such a way that the right side of one would fall on the left side of the other, we would find that there would be quite a similarity in shape, thus rendering the eye to see more clearly that the areas of both polygons are approximately equal. In connection with the range of gains, we find that the low gains or losses of one group are exactly the same as those of the other as both polygons meet the base line in the same class interval on the left side of the graph; there is a small excess of high gains, however, on the right side of the graph as the solid line repre senting the ten-week method extends slightly farther than the broken line. This graph shows that there is a slight advantage in favor of the group of the ten-week method# This table also shows the means of groups VA and VB to be 2#83 and 2s #00, respectively. The differnce of .83 is equivalent to each student in group VA making a gain of .83 more than each student in group VB# This obtained difference slightly favors the ten-week method represented by group VA. Group VA has a standard deviation of 5.35. This means that approximately two-thirds of all the gains, or 23 gains, m 74 would be included between the limits 2*83 (the mean) plus 5*35, or 8.18, and 2.83 minus 5.35, or -2.52. The standard error of the mean of group VA is .90. This signifies that if we gave this groupa large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .90 from the mean of 2.83. That is to say that two-thirds of all our future means of this group would lie between the gains of 1.93 and 3.73. Group VB has a standard deviation of 5.29. Two-thirds of all the gains, or 21 gains, would be included between the limits of 2.00 (the mean) plus 5.29, or 7.29, and 2.00 minus 5.29, or -3.29. The standard error of the mean of group VB is .94, signifying that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .94 from the mean of 2.00. is, two-thirds of all the future means of this group That would lie between the gains of 1.06 and 2.94. Figure 11 shows a graphic comparison of the gains made by groups VA and VB. Class intervals of 4 were used for the gains as in the preceding figure. students than group VB. Group VA has three more If we exclude these three students from group VA, represented by the solid line polygon, then we can reasonably say that the graph shows the areas of the two polygons to be approximately equal. Group VA has the greater li !«■! 76 range.of gains. The excess of high gains is greater than the excess of low gains giving this group, representing the tenweek method, the slight advantage. Comparisons of the ten-week groups« as a single group, with the semi-weekly alternation groups, as a single group, in terms of gains. Having made comparisons of the four groups arranged in pairs, the next step was to put the same two tenweek groups together as a single group and compare their gains with those of the same two semi-weekly alternation groups placed together as a single group. The purpose for doing this was that, according to statistics, the reliability of the results is higher when the number of eases is increased. To this end, Table XXX was made. In this table the two ten-week groups, TVA and VA, are combined and placed in one column, and parallel to it in another column is a combination of the two semi-weekly alternation groups, IVB and VB. The students of each combined group are arranged in a frequency distribution according to the number making each amount of gain as shown in the corresponding class interval. The gains are arranged consecutively in class inter vals of two; the class intervals range from (-10)-(-9) to 24-35. It is seen that the number of students is practically equal for both methods, that of the ten-week method being 70 and that of the second method being 62. 77 TABLE XXI COMPARISONS OF TEN-WEEK GROUPS AS A SINGLE GROUP, WITH SEMI-WEEKLY ALTERNATION GROUPS AS A SINGLE GROUP, IN TERMS OF GAINS FOR A8 CLASSES Gains in Number of pupils showing each amount of ga in class Tenweek groups Semi weekly groups intervals IV, VA IV,VB 24— -25 22 - 23 20 - 21 18 - 19 16 - 17 14 - 15 12 - 13 10 - 11 8 - 9 6 - 7 4 - 5 2 - 3 0 - 1 U-M>) - ((- 4) - ((- a) - ((- 8) - ((-10) - (Number Mean S.D. S.E.M. 1 1) 3) 5) 7) 9) 1 2 3 5 5 8 9 12 8 4 5 4 2 1 1 1 7 7 1 10 9 9 6 4 1 5 70 4.20 6.22 .74 62 3.61 5.88 .75 S.D, is the Standard Deviation, and S.E,M. is the Standard Error of the Mean. 78 First, the means of both groups were derived. means play a very important part as statistical data. The If they came within close range of each other then there would be very little difference between the two methods but if not, then a superiority of one method over the other would be shown. The table shows the derived means of the ten-week method and the semi-weekly alternation method to be 4.20 and 3.61, respective ly. This signifies that they are within close range of each other. The small difference of .59 is slightly in favor of the ten-week method. Second, the standard deviation and the standard error of the mean of each of the two groups were derived. The table shows the standard deviations of the two groups to be within close range of each other, as are also the standard errors of the means. These calculations, namely, the mean, the standard deviation, and the standard error of the mean, will not only be used in the present comparison but will also serve as some of the statistical bases for further comparisons to be made later. We shall now point out the present significance of the standard deviation and the standard error of the mean of each of the two groups. The standard deviation of the combined ten-week group being 6.22 signifies that approximately two-thirds of all the 79 gains, or 47 gains, would be included between the limits 4.20 (the mean) plus 6.22, or 10.42, and 4.20 minus 6.22, or -2.02. The standard error of the mean of this ten-week group being .74 means that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .74 from the mean of 4.20. That is, two-thirds of all our future means of this group would lie between the gains of 3.46 and 4.94. The combined semi-weekly group having a standard devia tion of 5.88 signifies that approximately two-thirds of all the gains, or 41 gains, would be included between the limits 3.61 (the mean) plus 5.88, or 9.49, and 3.61 minus 5.88, or -2.27. The standard error of the mean of the same semi-weekly group being .75 means that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .75 from the mean of 3.61. That is, two-thirds of all our future means of this group would lie between the gains of 2.86 and 4.36. Figure 12 shows the gains of these two large groups in a graphic way. Class intervals of 4 were used here as in the two preceding graphs in this chapter. The two combined ten- week groups, represented by the solid line, has eight more students than the two semi-weekly combined groups. If these eight students were excluded, we can see at a glance that the 81 polygons would be equal in area as the distribution of the gains of one group followsapproximately that of the other. The losses or low gains of both large groups are about the same. The combined ten-week group, however, has more hi*gh gains than the other group but not tola large extent. gives the ten-week method a slight advantage. This Statistical comparisons of A8 groups for prediction purposes. In the first jblace, we shall compare (a) the two pairs of groups, IVA, IVB, -and VA, VB, in order to determine for each pair the chances that the group of one teaching meth od would have of being superior to the other; and (b) the two ten-week groups, IVA, VA, as a single group, with the two semi-weekly groups, IVB, VB, also as a single group, in order to determine the chances that one of these combinations would have of being superior to the other. The means used in this first comparison will be the original means obtained for each group of the two pairs and for each of the combined groups. We shall elucidate this more fully later. In the second place, we shall eompare the same two tenweek groups, IVA, VA, as a single group, with the same two semi-weekly groups, IVB, VB, as a single group, not only for the purpose of prediction but also to serve as a check on the results of the first comparison. The means used here as a basis for comparing the com bined groups are very different from those used in the first comparison. The means used for the combined groups in the first comparison were obtained by making a frequency distri bution of the gains of each of the combined groups as a single separate group, while here the original means of each of the two groups, IVA and VA, are simply added together to be used as a basis of comparison, the same being done with the original means of each of the two groups, IVB and VB* Having stated the two ways in which these comparisons will be made, we shall now discuss the first with the use of Table XXII* This table shows each of the two pairs of groups and each of the combined groups as previously mentioned. The means used as the basis in this comparison, as we have already pointed out, are the original means obtained by having made a frequency distribution for each group of the two pairs, and for each of the combined groups, as a single group; the means of each of the pairs of groups are taken from Table XX, and those of the combined groups are taken from Table XXI. From the means and standard error of the means of each pair and also of each combination of groups will be derived other statistical data in order that the chances of superiority may be determined. Table XXII shows the obtained difference between the TABLE XXII STATISTICAL COMPARISON OF GAINS OF A8 GROUPS SHOWING PREDICTIONS FOR SEPARATE CLASS GROUPS AND FOR THE TOTAL GROUPS AS SINGLE LARGE DISTRIBUTIONS Biff. Groups compared i/feans (Ten fireeks) .Means (Semiweekly) S.l.M. (Ten weeks) IVA vs, IVB 5.57 '5.33 1.13 1.09 1.24 1.57 .15 1.3 to 1 VA S.83 2.00 .90 .94 .83 1,30 .64 2.8 to 1 .59 1.05 .56 2.4 to 1 vs. VB All students in groups IV,VA 4.20 S.E.M. (Semi weekly) . Ratio Chances .74 vs. All studdnts in groups IV,VB SD 3.61 .75 IVA and VA are the ten-week groups; IVB and VB are the semi-weekly alternation groups, S.E.M. is the Standard Error of the Mean, and Ep is the Standard Error of the difference. 84 two means of groups IVA and IVB to be .24. The signifieance of this difference is determined by dividing .24 by the stand ard error of the difference, or 1.57, which yields .15, or the ratio. By referring to the same standard-error table mentioned in Chapter III, this ratio was found to signify that the chances are only 1.3 to 1 that group IVA, the ten-week method, would be superior to group IVB, the semi-weekly method. Befer- ence was also made in the last chapter to the statement that in order to be practically certain that there is a real differ ence in merit between two groups, the difference between the means should be at least three times as great as the standard error of the difference. In the present case, the difference .24 is much smaller than the standard error of the difference 1.57. On the basis of these data we can say that the chances are indeed very slight that group IVA, the ten-week method, would actually make higher gains in mathemamatics than group IVB, the semi-weekly method. As a matter of fact the chances are about even for both groups since the ratio is 1.3 to 1. The difference between the means of Va and VB was .83. By dividing this difference by the standard error of the dif ference, or 1.30, we obtain the ratio, or .64. The standard- error table reveals this ratio to indicate that the chances are 2.8 to 1 that the ten-week group VA is superior to the semi-weekly group VB. On the basis of these data we can say that the chances are slight that group VA would actually make 85 higher gains in mathematics than group VB. And finally in this table the two ten-week groups, as a single group, were compared with the two semi-weekly groups, as a single group. The difference between the means of these two large groups.is .59. By dividing this difference of .59 by 1.05, the standard error of the difference, we obtain the ratio of .56. The standard-error table shows this ratio to signify that the chances are 2.4 to 1 that the IT, VA combi nation is superior to the IV, VB combination. On the basis of these data we can say that the chances are very slight that the ten-week combination would make higher gains in math ematics than the semi-weekly combination. As a result of combining the ten-week groups and also the semi-weekly groups, we see that the chances of superiority of the combined ten-week groups were slightly greater than those of the pair of groups IVA and IVB and slightly fewer than those of the pair VA and VB. Ihble XKIII shows the second method of comparing the two combined ten-week groups with the two combined semi-weekly groups. As was stated previously, the original means of XVA and VA, respectively, were added together as were those of IVB and VB. The obtained sums of the means then became the basis on which further calculations were made in order to ar rive at the chances of superiority of one combined group over the other. 86 TABLE XXIII STATISTICAL COMPARISON OF A8 GROUPS SHOWING A SECOND METHOD OF MAKING PREDICTIONS BY DIFFERENT COMPUTATIONS OF THE MEANS A8 groups Items Mean % Ten-•week groups Semi-weekly groups TVA ~IVB VA' 5.57 1.13 2.83 .90 VB 5.33 1.09 2.00 .94 Sum of means, groups A and B 8.40 7.33 E of sum of means, groups A and B 1.44 1.44 i Diff. between sums of means, groups A minus groups B 1.07 E of diff. be tween sums of means of groups A and B 2.04 Ratio Chances D/ED .52 2.3 to 1 Ej$ is the Standard Error of the mean; E is the Stand ard Error; D is the difference, and ED the standard Error of the difference. 8? This table shows the obtained difference between the sums of the means to be 1.07. Dividing this quantity by the standard error of the difference between the sums of the means, or 2.04, we obtain the ratio of .52. According to the standard error tabic this ratio signifies that the chances are 2.3 to 1 that the IV, VA combination is superior to the IV, VB combina tion. With these data as a basis, we can say that the chances are again very slight that the ten-week combination would make higher gains in mathematics than the semi-weekly combination. In both these comparisons of the two combined groups, we found that the results obtained were almost identically the same, as the ratios were 2.4 to 1, and 2.3 to 1, respectively, and both gave the ten-week method the advantage, but of very little significance. CHAPTER V RESULTS OF THE B8 M D A8 GROUPS COMBINED In the two preceding chapters we discussed and analyzed the results of the B8 and A8 groups, respectively. One of the important items shown in the discussion of the B8 results was the comparison in terms of gains of the three combined tenweek groups, as a single group, with the three combined semi weekly groups, as a single group. Likewise in the A8 results we showed a comparison of the two combined ten-week groups, as a single group, with the two combined semi-weekly groups, as a single group. Now we shall compare two larger combinations of, groups in terms of gains. combined as All the B8 and A8 ten-week groups will be a large single group and compared with all semi weekly groups also combined as a large single group of the same two grades. Since these two large combinations include all the stu dents who participated in the experiment, the results of the comparison should show more clearly the relative merits of the teaching methods than when the B8 or the A8 combinations of groups were compared separately. in this chapter we shall discuss the results of the combined B8 and A8 groups in the following order: comparisons in terms of gains of the combined ten-week groups as a single 89 group, with the combined semi-weekly alternation groups as a single group; statistical comparison of gains of these same two large combined groups for the purpose of showing predic tions; and finally, statistical comparison of these two large groups showing a second method of making predictions. Comparisons in terms of gains of the combined ten-week groups as a single group, with the combined semi-weekly alter nation groups as a single group. The five ten-week groups to be combined as a single groups are IA, IIA, IIIA, IVA, and YJL, and the five semi-weekly groups to be combined as a single group are IB, IIB, IIIB, XVB, and VB, In each of these two combinations of groups, the first three are B8*s, and the remaining two are A 8 fs. Table XXXV shows the comparison in terms of gains of these two large combinations of groups. In this table the five ten-week groups, as a single group, are placed in one column, and parallel to it in another column are the five semi-weekly alternation groups, as a single group. The stu dents of each combined group are arranged in a frequency dis tribution according to the number making each amount of gain as shown in the corresponding class interval. The gains are arranged consecutively in class intervals of two; the class intervals range from (-IS)-(-11) to 24-25. It is seen that the number of students for the tenweek method is 172, and for the semi-weekly method is 158. 90 TABLE XXIV COMPARISONS OF TEN-WEEK GROUPS AS A SINGLE GROUP, WITH SEMI-WEEKLY ALTERNATION GROUPS AS A SINGLE GROUP, IN TERMS OF GAINS FOR THE COMBINED B8 AND A8 CLASSES Gains in Number of %>upils showing ejach amount of gain class Ten- / week groups intervals 24 - 25 22-23 20 - 21 18 - 19 16 - 17 14 - 15 12 - 13 10 - 11 8 - 9 6 - 7 4 - 5 2 - ..-3 0 - 1 (- 2) - (- 1) t- 4) - (- 3} (- 6) - (- 5) (- 8) - (- 7) (-10) - (- 9) (-12) (-11) Number Mean S.D. S.E.M. I,II,III, IV,VA Semi weekly groups I,II,III, IV,VB 1 1 2 2 3 6 11 12 20 18 22 28 16 8 13 7 3 1 2 5 21 18 14 19 22 22 13 12 2 5 1 172 5.27 6.14 .47 158 4.47 5.59 .45 S.D. is the Standard Deviation, and S.E.M. is the Standard Error of the Mean. 91 First, the means of these two large groups were derived. The table shows the means of the ten-week method and the semiweekly alternation method to be 5.27 and 4.47, respectively, indicating that they are within close range of each other. This signifies that as far as the means are concerned there is very little difference between the two methods. The small difference of .80 is slightly in favor of the ten-week method. Second, the standard deviation and the standard error of the mean of each of the two groups were derived. The table shows the standard deviations of the two large groups to be within close range of each other. The standard errors of the means show a still closer range. These calculations, namely, the mean, the standard deviation, and the standard error of the mean, will not only be used in the present comparison but also will serve as some of the statistical bases for further comparisons to be made later in this chapter. We shall now point out the present significance of the standard deviation and the standard error of the mean of each of the two groups. The standard deviation of the combined ten-week group being 6.14 signifies that approximately two-thirds of all the gains, or 115 gains, would be included between the limits 5.27 (the mean) plus 6.14, or 11.41, and 5.27 minus 6.14, or -.87. 92 The standard error of the mean of this ten-week com bined group being .47 signifies that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .47 from the mean of 5.27. That is, two-thirds of all our future means of this combined group would lie between the gains of 4.80 and 5.74* The combined semi-weekly group having a standard devia tion of 5.59 signifies that approximately tto-thirds of all the gains, or 105 gains, would be included between the limits 4.47 (the mean) plus 5.59, or 10.06, and 4*47 minus 5.59, or -1 .12 . The standard error of the mean of the same semi-weekly group being .45 signifies that if we gave this group a large number of tests and calculated the gains, two-thirds of the time our means would not deviate more than .45 from the mean of 4.47. That is, two-thirds of all our future means of this group would lie between the gains of 4.02 and 4.92. Figure 13 shows a graphic distribution of the gains of these same two large groups. The base line shows that class intervals of 4 were used for the gains. At a glance we see that one polygon is practically superimposed upon the other. Hence we can very reasonably say that the areas of the poly gons are equal. Observing the graph very closely, we can see that both groups have almost identical losses (negative 94 gains) but that the ten-week group, represented by the solid line, has a few more high gains than the semi-weekly group. This indicates a slight advantage in favor of the ten-week method. Statistical comparison of gains of these same two large combined groups for the purpose of showing predictions. We shall now compare the five ten-week groups, IA, IIA, IIIA, IVA, and VA, as a single group, with the five semi-weekly groups IB, IIB, IIIB, IVB, and VB, as a single group, in order to determine the chances that one of these combinations would have of being superior to the other. The means used as a basis in this comparison are the original means obtained by having made a frequency distribu tion for each of the combined groups, as a single group. These means are taken from Table XXIV, as are also the standard errors of the means. From the means and standard error of the means of each combination of groups will be derived other statistical data in order that the chances of superiority may be determined. These data are shown in Table XXV. In this table we see the statistical comparison in terms of gains of the combined ten-week groups as a single group, and of the combined semi-weekly alternation groups as a single group. Examining the data in this table we find that the TABUS XXV STATISTICAL COMPARISON OF GAINS OF B8 AND A8 GROUPS. COMBINED- SHOWING PREDICTIONS FOR THE TOTAL TEN-WEEK GROUPS AND FOR THE TOTAL SEMIWEEKLY ALTERNATION GROUPS AS SINGLE LARGE DISTRIBUTIONS Groups compared Means All students in groups I,II,III, IV,VA 5.27 (Ten weeks)• S .E .M . E jj Ratio Chances .47 • o go - vs. All students in groups I,II,III, IV, VB 4.47 (Semi-weekly) Diff. •65 1.23 8.2 to 1 .45 I,II,III,IV,VA are the ten-week groups, and I,II,III,IT,VB are the semi-weekly alternation groups. S.E.M. is the Standard Error of the Mean, and Ep is the Standard Error of the difference. 96 obtained difference between the means of these two large groups is .80* The significance of this difference is deter mined by dividing .80 by the standard error of the difference, or .65, which yields 1.23, or the ratio. By referring to the same standard-error table mentioned in Chapters III and IT, this ratio was found to signify the chances are only 8.2 to 1 that the combined ten-week group would be superior to the combined semi-weekly group. Reference was also made in the last two chapters to the statement that in order to be prac tically certain that there is a real difference in merit be tween two groups, the difference between the means should be at least three times as great as the standard error of the difference. In the present case, the difference between the means, or .80, is just a shade larger than the standard error of the difference, or .65. On the basis of these data we can say that the chances * are very slight that the combined ten-week group, I, II, III, IV, VA, would actually make higher gains in mathematics than the combined semi-weekly group, I, II, III, IV, VB. Statistical comparison of these two large groups show ing a second method of making predictions. The reason for this second comparison is to check the reliability of the re sults of the first. The means used as a basis in this second comparison of the combined groups are very different from those used in the first. The means used in the first com parison were obtained by making a frequency distribution of 97 the gains of each of the combined groups as one single sep arate group* In this second comparison, however, the orig inal means of each of the five ten-week groups, IA, IIA, IIIA, IVA, and VA, are simply added together, and this sum used as a basis. The same is done with the original means of each of the five semi-weekly groups, IB, IIB, IIIB, IVB, and VB* From the sum of the means and standard error of the sum of the means of each combination of groups will be de rived other statistical data in order that the chances of superiority may be determined. These data are shown in Table XXVI. In this table we see the statistical comparison in terms of gains of the combined B8 and A8 ten-week groups as a single group, and of the combined B8 and A8 semi-weekly alternation groups as a single group. The respective means of the five ten-week groups are shown, as is also the sum of the means. The same items are shown for the semi-weekly groups. Examining the data in this table we find that the ob tained difference between the sums of the means of the two large groups to be 3.81. Dividing this quantity by the stand ard error of the difference between the sums of ^he means, or 3.07, we obtain the ratio of 1.24. According to the standard error table this ratio signifies that the chances are 8.3 to 1 that the ten-week combination is superior to the semi-weekly combination. TABLE XXVI STATISTICAL COMPARISON OF B8 AM) A8 GROUPS SHOWING A SECOND METHOD OF MAKING PREDICTIONS BY DIFFERENT COMPUTATIONS OF THE MEANS Ten-week B8 and A8 groups ! Semi-weekly B8 and A8 groups Items IA IIA IIIA \ IVA VA IB IIB IIIB Mean % 3.14 L. 13 5.72 .80 3.71 1.01 5.57 1.13 2.83 .90 6.70 .73 5.73 1.07 2.40 5.33 .80 ‘ 1.09 Sum of means, groups A and B E of sums of means, groups A and B Diff. of sums o f means, groups A-B E of diff. between sums of means of groups A and B Ratio Chances U/ED 25.97 22.16 2.24 2.10 IVB VB 2.00 .94 3.81 3.07 1.24 8.3 to 1 Ew is the Standard Error of the Mean; E is the Standard Error; D is the differ ence, and ED the Standard Error of the difference. The B8 groups are I,II,IIIB. The A8 groups are IV,VA and IV,VB. to CD 99 With these data as a basis, we can say that the chances are again very slight that the ten-week groups would make high er gains in mathematics than the semi-weekly groups. In both these comparisons of the two large groups, we found that the results obtained were almost identical, as the ratios were 8.2 to 1, and 8.3 to 1, respectively. Both com parisons gave the ten-week method the advantage, but of very little significance. CHAPTER VI CONCLUSIONS AND R1COMMENDATIONS As stated in the problem chapter, it was the purpose of this study to compare experimentally by a standardized arithmetic test the achievement of several pairs of compar able groups in the eighth grade. Each pair had studied math ematics by two different methods, each on a different timebasis, in order to determine which method was the better. The two different time-bases for these methods were (1) one of the groups of each pair studied mathematics for ten con secutive weeks, and science the other ten, and (S) the other group of each pair studied mathematics on a half-week basis, alternating with science the other half. Science was not included in the measurement for reasons stated in the procedure chapter. As previously stated the standardized test was given both at the beginning and at the end of the experiment in order to determine the gains made by the students. Chapters III, IV, and V show the results of this ex periment in terms of gains both in a statistical and graphic manner. It is the purpose of this chapter to state the con clusions and recommendations resulting from this study. 101 Conclusions. The statistical and graphic results of the experiment lead to the following conclusions: fl) That for ©11 practical purposes either of the two methods could be used with the same expectancy of results. The data were practically tied throughout. (2) That the continuity of the ten-week method of studying mathematics in grade eight had no particular advan tage, as shown by the experiment, over the "space-learning” or half-weekly method. Recommendations. In the light of these conclusions, the following recommendations are made: (1) So far as this experiment is able to reveal facts, it is recommended that Edison Junior High School may use either method for the teaching of these two subjects. (2) Since science alternates with mathematics in grade eight in this school, it is recommended that the science teach ers prepare a test based upon the science course of study for this grade and conduct an experiment in this subject parallel to the mathematics experiment in order to determine which would be the better method for the teaching of science. (3) That other junior high schools conduct an experi ment in grade eight mathematics similar to that of this study not only to check the results obtained in this experiment but also to help standardize a method for the teaching of mathe matics and science in this grade throughout the city school system.

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