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Investigation of learning by discovery Bergsma, Marshall 1968

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AN INVESTIGATION OF LEARNING BY DISCOVERY by MARSHALL BERGSMA B.Sc, University of B r i t i s h Columbia,. I96I A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of EDUCATION We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n -t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e fyri/Sj W* i ABSTRACT The problem was to determine whether or not a d i s -covery method of teaching f o r transfer was superior to an expository- method of teaching f o r t r a n s f e r . The experiment was a comparison of the transfer effects of an unverbalized awareness and a verbal reception method of i n s t r u c t i o n . I t was expscted to have implications regarding the theories of Hendrix and Ausubel about transfer of t r a i n i n g . The central problem was to compare the transfer effects of eliminating or delaying the v e r b a l i z a t i o n by students of t h e i r discoveries, with those obtained by didactic presentation using Ausubel's "introductory organizers". The experiment involved ten eighth grade classes taught by six teachers. A l l classes and teachers were from the public school system of B r i t i s h Columbia. Four classes were taught by each of the experimental methods and two classes comprised a:: control group. The teaching involved four days of instruc-t i o n on some of the introductory aspects of the language of sets. A test f o r transfer was administered on the f i f t h day. An attempt was made to ensure that the essential difference between the experimental methods lay i n the unverbalized awareness - verbal reception variable rather than i n some other v a r i a b l e . For example, the amount of teaching time, the sequence of presentation of the topics, the number and kinds of examples used, and the d a i l y assignments were the same for each group. The mental age and the previous mathematics achievement of the groups were covaried. The verbal reception samples i i scored higher on the t rans fer task than the unverbal ized awareness samples, when i n i t i a l di f ferences on these covar-ia t e s were s t a t i s t i c a l l y e l iminated . The di f ferences were not s i g n i f i c a n t at the f i v e percent l e v e l . I t was concluded that , under conditions which existed i n t h i s experiment, there was no bas is for considering e i ther method to be superior i n f a c i l i t a t i n g t r a n s f e r . i i i TABLE OF CONTENTS CHAPTER Page I O u t l i n e o f t h e Problem 1 I I Review of t h e L i t e r a t u r e 5 I I I P r o c e d u r e 21 I V A n a l y s i s o f t h e Data 30 V Summary and C o n c l u s i o n s 40 B i b l i o g r a p h y 43 Appendix I The L e s s o n P l a n s 4$ Appendix I I The T e a c h i n g Notes 62 A. G e n e r a l T e a c h i n g Notes 63 B. S p e c i f i c T e a c h i n g Notes 65 Appendix I I I The S u p e r v i s e d Assignments 68 Appendix I V The A d d i t i o n a l A ssignments 78" Appendix V The T r a n s f e r T e s t 89 Appendix V I T a b l e s 96 LIST OF TABLES iv Table Page I Subject Matter 25 II C l a s s i f i c a t i o n of Test Items by Range of Item V a l i d i t y Indices (Flanagan's r) for the T r i a l Run 31 III Estimates of the R e l i a b i l i t y ox the Transfer Test 30 IV C l a s s i f i c a t i o n of Subjects Involved i n the Experimental Run 32 V C l a s s i f i c a t i o n of Test Items by Range of Item V a l i d i t y Indices (Flanagan's r) for the Experimental Run 33 VI Estimates of the R e l i a b i l i t y of the Transfer Test 34 VII Correlations 34 VIII The Analysis of Covariance - Treatment Group versus Control Group 35 IX The Analysis of Covariance - Part I of the Transfer Test 37 X The Analysis of Covariance - Part II of the Transfer Test 37 XI Adjusted Mean Scores - Treatment Group versus Control Group 3& XII Adjusted Mean Scores on the Transfer Test 39 VI -1 P r e t r i a l Scores Obtained on Part I of the Transfer Test 97 VI-2 P r e t r i a l Scores Obtained on Part II of the Transfer Test 99 VI -3 Item V a l i d i t y Indices (Flanagan's r) for the T r i a l Run 101 VI -4 Experimental Scores on Part I of the Transfer Test 102 LIST OF TABLES (Continued) TABLE Page VI - 5 Experimental Scores on Part II of the Transfer Test 111 VI - 6 Scores f o r Analysis of Covariance 115 VI - 7 Item V a l i d i t y Indices (Flanagan's r) fo r the Experimental Run 121 v i LIST OF FIGURES Figure Page 1 Dimensions of Learning 16 2- Experimental Design 21 ACKNOWLEDGEMENT The writer acknowledges, with thanks, the generous assistance given by the p a r t i c i p a t i n g teachers and by Dr. T.D.M. McKie. P a r t i c u l a r gratitude i s rendered to Dr. E r i c D. MacPherson f o r his patient help and encouragement. CHAPTER I OUTLINE OF THE PROBLEM Statement of the Problem There i s considerable discussion at the present time about the possible advantages of discovery methods of teach-ing versus expository methods of teaching with respect to transfer, retention, and learning how to le a r n . This inves t i g a t i o n sought to determine whether or not a defined discovery teaching method was act u a l l y superior to a defined expository teaching method with respect to transfer. The Experiment The unverbalized awareness procedure, proposed by Hendrix, i s a discovery method of teaching which has attracted wide i n t e r e s t and acceptance. Hendrix claims that i t i s the best of a l l discovery methods f o r the f a c i l i t a t i o n of trans-fer."^" This method was chosen as an exemplar of a discovery method of teaching f o r t r a n s f e r . Ausubel has c r i t i c i z e d many of Hendrix' theories re-garding the unverbalized awareness method of teaching.^ He has proposed an expository method of teaching for transfer and has suggested t h e o r e t i c a l reasons why i t might be almost as ef f e c t i v e as the unverbalized awareness method for f a c i l i t a t i n g Gertrude Hendrix, "Learning by Discovery", Mathematics  Teacher, 54, 1961, pp. 291-299. o David P. Ausubel, "Learning by Discovery: Rationale and Mystique", B u l l e t i n of-the National Association of Second- ary School P r i n c i p a l s , V o l . 45, No. 269, 1961, pp. 18-58. 2 3 tr a n s f e r . Ausubel's teaching method w i l l be c a l l e d the verbal reception procedure. This method was chosen as an exemplar of an expository method of teaching. Thus the experiment sought to determine the r e l a t i v e effects of these two teaching methods on the performance of a transfer task following related i n s t r u c t i o n . D e f i n i t i o n of Terms The unverbalized awareness method of teaching f o r transfer i s , according to Hendrix, a method wherein students are discouraged from v e r b a l i z i n g t h e i r discoveries. I f i t i s necessary to verbalize a discovered generalization, then such verb a l i z a t i o n i s l e f t u n t i l well after^" the o r i g i n a l discovery has occurred. In the verbal reception method of teaching f o r t r a n s f e r the teacher attempts to make a learning task meaningful to learners by didactic means. A s i g n i f i c a n t feature of the method i s the use of what Ausubel c a l l s introductory organiz-ers.^ In t h i s study, transfer of t r a i n i n g w i l l be defined as •^David P. Ausubel, Psychology of Meaningful Verbal  Learning. New York and London, Grune and Stratton, 1963, pp. 15-33. ^"Gertrude Hendrix, "Nonverbal Awareness i n the Learning of Mathematics", Research Problems i n Mathematics Education;  Cooperative Research Monograph No. - 3 , O f f i c e of Education, Washington, I960, pp. 57-61. -'Ausubel, The Psychology of Meaningful Verbal Learning, pp. 81-83. 3 i n the Taxonomy of Educational Objectives 0 under "Application". According to t h i s taxonomy, a test f o r transfer must involve something new to the student or i t must contain new elements as compared to the learning s i t u a t i o n . The authors say, I f the situations presented the student to test " a pplication" are old ones i n which he o r i g i n a l l y learned the meaning of the abstraction, the student does not have to "apply" the abstraction. Rather, he needs merely to r e c a l l the o r i g i n a l situations i n which he learned the abstraction, a The d e f i n i t i o n of transfer given i n the Taxonomy i s not p a r t i c u l a r l y new. In 1937, Orata said, "Thus we may define transfer of t r a i n i n g as that process of using or applying previously acquired information, habit, s k i l l , attitude or d i d e a l i n dealing with a r e l a t i v e l y new or novel s i t u a t i o n . " Bruner defines two types of transfer. One of these, s p e c i f i c transfer, i s defined as "... s p e c i f i c a p p l i c a b i l i t y to tasks that are highly s i m i l a r to those we o r i g i n a l l y learn-Q ed to perform". Bruner suggests that t h i s type of transfer i s u t i l i z e d mainly i n the form of s k i l l s . Benjamin S. Bloom, ed. Taxonomy of Educational  Objectives Handbook I; Cognitive Domain. Toronto, David McKay, 1956, pp. 120-143. 7 I b i d . , p. 125. ^Pedro T. Orata, "Transfer of Training and Reconstruction of Experience", Mathematics Teacher, 30, 1937, p. 99. ^Jerome S. Bruner, The Process of Education. Cambridge, Mass., Harvard University Press, 1961, p. 17. ~ a l e v e l i n comprehension. knowledge or The second type of transfer defined by Bruner i s 4 the transfer of p r i n c i p l e s and a t t i t u d e s . He c a l l s t h i s non-s p e c i f i c transfer and says i t "... consists of learning i n i t i a l l y not a s k i l l but a general idea, which can then be used as a basis f o r recognizing subsequent problems as special cases of the idea o r i g i n a l l y mastered. "~^ I t seems clear that Bruner's " s p e c i f i c transfer" would be l i s t e d i n the Taxonomy of Educational Objectives as know-ledge or a l e v e l i n comprehension. I t may be that there i s no difference between what i s defined as transfer i n the Taxonomy and what Bruner defines as nonspecific t r a n s f e r . The d e f i n i t i o n i n the Taxonomy i s oriented towards tes t i n g , while Bruner's seems more descrip-t i v e . The Hypothesis The hypothesis that was tested i s : a f t e r two groups of pupils have been taught some i n i t i a l concepts of set theory, one group by the unverbalized awareness procedure and one group by a procedure of verbal reception learning, there w i l l be no s i g n i f i c a n t difference between the mean scores of the two groups on a transfer task. The alternative to the above hypothesis was that the experimental methods -would effect unequal amounts of t r a n s f e r . The experimenter's b e l i e f was that the n u l l hypothesis would be accepted. I b i d . CHAPTER II REVIEW OF THE LITERATURE Introduction The major studies which suggest superiority of the un-verbalized awareness method are those by Hendrix^" and by 2 Haslerud and Meyers . These studies are discussed i n t h i s chapter. The major studies which would suggest that unverbal-ized awareness i s not the best method fo r the f a c i l i t a t i o n of 3 L transfer are the studies by Craig and by K i t t e l l . Since K i t t e l l ' s more detailed study was very similar to Craig's e a r l i e r study only K i t t e l l ' s study i s discussed here. Some of Hendrix' opinions regarding the unverbalized awareness method are discussed. Opinions by Bruner, Rosskopf, and Beberman are included i n d i c a t i n g widespread acceptance of Hendrix' p o s i t i o n . Ausubel has suggested what may be several flaws i n the case supporting the unverbalized awareness method and i n the Gertrude Hendrix, "A New Clue to Transfer of Training", Elementary School Journal, . 4 8 , 1947, pp. 197-208. 2G.M. Haslerud and Shirley Meyers, "The Transfer Value of Given and Indi v i d u a l l y Derived P r i n c i p l e s " , Journal of  Educational Psychology, 49, 1958, pp. 293-298. ^R.C. Craig, The Transfer Value of Guided Learning. New York, Bureau of Publications, Teachers College, Columbia University, 1953. ^Jack E. K i t t e l l , "An Experimental Study of the Effects of External Direction During Learning on Transfer and Retention of P r i n c i p l e s " , Journal of Educational Psychology, 48, 1957, pp. 391-405. • 6 supporting studies. Many of his opinion are stated here. Henderson, who sees some merit i n discovery techniques, suggests a possible weakness i n the case f o r discovery methods. The Unverbalized Awareness Procedure The teacher who uses Hendrix' unverbalized awareness method presents, f o r each p r i n c i p l e to be learned, c a r e f u l l y selected sequences of experiences. From these sequences of experiences each learner i s to abstract what Hendrix c a l l s a subverb'al understanding of the p r i n c i p l e . ^ Hendrix and other 6 7 writers ' agree that t h i s i s a discovery method of teaching. Lessons prepared according to t h i s method are highly organized i n that the material to be learned and the learning experiences are preselected by the teacher. In t h i s respect i t d i f f e r s from another disdovery method which Hendrix c a l l s the i n c i d e n t a l or a c t i v i t y method. The unverbalized aware-ness method i s a guided discovery method. Another important c h a r a c t e r i s t i c of the unverbalized awareness method i s that v e r b a l i z a t i o n of p r i n c i p l e s i s avoided as long as possible. I f such verbalization cannot be avoided altogether, then the verb a l i z a t i o n i s l e f t u n t i l well af t e r the o r i g i n a l discovery of the p r i n c i p l e has occurred. There i s no general agreement as to how long Gertrude Hendrix, "Learning by Discovery," Mathematics  Teacher, 54, 1961, pp. 2 9 0 - 2 ^ . -David P. Ausubel, The Psychology of Meaningful Verbal  Learning, New York and London, Grune and Stratton, 1963, • pp. 147-149. 7 Haslerud and Meyers, op.cit., pp. 293-298. "well a f t e r " i s . Hendrix suggests that i t should be l e f t to a d l a t e r l e s s o n . 0 The Verbal Reception Procedure The verbal reception procedure i s an expository method of teaching wherein a special attempt i s made to make the learning meaningful.^ To do t h i s the teacher generally uses what Ausubel c a l l s advance organizers. For Ausubel, these organizers are stable elements i n the existing cognitive structure of the learner,that can be c a l l e d upon to a s s i s t i n the integration of new knowledge. 1^ He says, "These organ-i z e r s are introduced i n advance of the learning material i t s e l f , and are also presented at a higher l e v e l of abstract-11 ion, generality, and ihclusiveness...." Ausubel points out that these organizers are not the same as summaries and overviews. He says, "Summaries and overviews, are o r d i n a r i l y presented at the same l e v e l of abstraction, generality, and inclusiveness as the learning material i t s e l f . " 1 2 Ausubel says, "... a given organizer or series of organizers i s selected on the basis of t h e i r s u i t a b i l i t y f o r Gertrude Hendrix, "Learning by Discovery", Mathematics  Teacher, 54, 1 9 6 1 , p. 2 9 2 . ^David P. Ausubel, The Psychology of Meaningful Verbal  Learning, 1963, p p . 3 4 - 4 7 . 1 0 I b i d . , pp. 8 1 - 8 3 . 1 1 I b i d i , p. 8 1 . 1 2 I b i d . , p. 8 1 . 8 explaining, integrating, and i n t e r r e l a t i n g the material they 13 precede...." ^ Research Evidence Supporting the Unverbalized Awareness Method Hendrix - The research studies most relevant to t h i s thesis are those completed by Hendrix. 1^ In each of these studies the learning task was the generalization that the sum of the f i r s t Tn' odd numbers i s n . The experiment was con-ducted f i r s t with eleven college students and was then r e p l i -cated with twenty-nine eleventh and twelfth grade boys. In each case the subjects were divided into three treatment groups. One group was taught according to an expository method, one group was taught according to the unverbalized awareness method and one group was taught according to an ordinary discovery method. Under the ordinary discovery method pupils were required to state the generalization once they discovered i t . The maximum'teaching time to which any subject was exposed was one class period. The transfer, task i n Hendrix 1 experiment consisted of questions that could be solved either by d i r e c t application of the generalization or by the more laborious procedure of counting or adding. The questions were mixed i n with many unrelated questions and the whole examination was administered about two weeks a f t e r the o r i g i n a l i n s t r u c t i o n was completed. 1 3 I b i d . , p. 81. 1 Z fGertrude Hendrix, "A New Clue to Transfer of Train-ing", Elementary School Journal, 4 8 , 1 9 4 7 , pp. 1 9 7 - 2 0 8 . 9 As the c r i t e r i o n for transfer Hendrix used the r a t i o of the number of questions solved by application of the short cut, to the t o t a l number of correct answers. The scores on t h i s transfer test showed the. unverbal-ized awareness group highest with a r a t i o of and the expository group lowest with a r a t i o of . 6 9 . The r a t i o f o r the ordinary discovery method was .75. Hendrix reports that the difference between the extreme groups, unverbalized awareness and expository,was "... s i g n i f i c a n t at the 12 15 percent l e v e l , The difference between the unverbal-ized awareness group and the other discovery group "... would a r i s e by chance about one time i n three, ,..." l D Hendrix reports a l a t e r unpublished r e p l i c a t i o n of t h i s experiment 17 i n which the same conclusions were reached. As a r e s u l t of her studies Hendrix concluded: "1. For generation of transfer power, the unverbalized awareness method of learn-ing a generalization i s better than a method i n which an authoritative state-ment of the generalization comes f i r s t . 2. Verbalizing a generalization immediately a f t e r discovery does not increase trans-f e r power. 1 5 I b i d . , p. 205. l 6I_bid., p. 205. •^Gertrude Hendrix. Letter to the writer, 30th of November, 1 9 6 4 . 10 3. Verbalizing a generalization immediately a f t e r discovery may. .actually decrease transfer p o w e r . - f \ Hendrix i s of the opinion that language enters the picture only f o r secondary purposes such as communication, c l a s s i f y i n g or recording. She says, " I t i s the subverbal awareness which y i e l d s the power of transfer, whether that power be prediction, problem-solving, or explanation." 1^ Furthermore, "The separation of discovery phenomena from the process of composing sentences which express those ideas i s 20 the big new breakthrough i n pedagogical theory." Ausubel has been c r i t i c a l of Hendrix' experiment and has contrary opinions about discovery learning. He suggests there may have been inadequate experimental controls i n 21 Hendrix' experiment. Hendrix herself says that there was 22 need of more rigorous controls i n the experiment. Ausubel disagrees with Hendrix' theory that transfer power reaches i t s maximum at the moment that the subverbal awareness dawns. He says that the re s u l t s of her experiments "merely show that a r e l a t i v e l y clear subverbal insight, even when only 18 Gertrude Hendrix, "A New Clue to Transfer of Training," Elementary School Journal, 48, 1947, p. 198. 19 Gertrude Hendrix, "Prerequisite to Meaning," Mathem- a t i c s Teacher, 43, 1950, p.-337. 20 Gertrude Hendrix, "Learning by Discovery," Mathematics Teacher, 54, 1961, p. 290. -21 David P. Ausubel, The Psychology of Meaningful Verbal  Learning. New York and London, Grune and Stratton, 1963, p.169. 22 Gertrude Hendrix, "A New Clue to Transfer of Training," Elementary School Journal, 48 , 1947, p. 198. 11 p a r t i a l l y consolidated, i s more functional and transferable than an ambiguous, inept, and marginally competent verbally 23 expressed idea." The learning task i n Hendrix' experiments was a short-cut to a f a i r l y lengthy procedure. The pupils i n the discovery groups were i n possession of a technique f o r obtaining the required answers but i t was a laborious one. I t seems that the discovery of a simple short-cut would be a s t r i k i n g experience f o r these pupils. On the other hand, with the expository group, the short-cut was presented f i r s t , followed by an explanation of the more laborious procedure. I t seems that the short cut would not be nearly as s t r i k i n g f o r t h i s group. Using the d e f i n i t i o n of transfer from the Taxonomy of Educational Objectives, i t seems doubtful that Hendrix tested f o r transfer behavior. The pupils i n her experiment had to r e c a l l the formula that they had learned and apply i t d i r e c t l y to the solution of the problem. No restructuring of the problem was required. This type of behavior i s c l a s s i f i e d i n the Taxonomy as, "knowledge or a l e v e l i n comprehension." 24 Haslerud and Meyers - Haslerud and Meyers ' support 23 ^David P. Ausubel, The Psychology of Meaningful Verbal  Learning. New York and London, Grune and Stratton, 1963, p. 150. 2Zt-G.M. Haslerud and Shirley Meyers, "The Transfer Value of Given and I n d i v i d u a l l y Derived P r i n c i p l e s , " Journal  of Educational Psychology, 49, 1953, pp. 293-298. . 12 Hendrix* findings regarding unverbalized awareness learning. Their experiment was designed to investigate the issue raised by the contradictory conclusions of studies by Hendrix and Katona ? on the one hand and by Craig and K i t t e l l on the other. Katona had concluded that verbal formulation of a general p r i n c i p l e was not indispensable f o r achieving trans-f e r . Hendrix stated that immediate verbalization of a 29 discovery might a c t u a l l y decrease transfer power. On the other hand, Craig concluded that "The more guidance a learner 30 receives, the more learning and transfer w i l l occur." K i t t e l l said, "... furnishing learners with information i n 31 the form of underlying p r i n c i p l e s promotes tran s f e r . . . . " 25 G. Katona, Organizing and Memorizing. Studies i n the Psychology of Learning and Teaching, New York, Columbia Univer-s i t y Press, 1 9 4 0 . 26 R.C. Craig, The Transfer Value of Guided Learning. New York, Bureau of Publications, Teachers College, Columbia University, 1 9 5 3 . 27 Jack E. K i t t e l l , "An Experimental Study of the Effects of External Direction During Learning on Transfer and Retention of P r i n c i p l e s , " Journal of Educational Psychology, 4 8 , 1957 , pp. 3 9 1 - 4 0 5 . -2S Craig, op. c i t . , p. 8 9 . 29 Gertrude Hendrix, "A New Clue to Transfer of Training," Elementary School Journal, 4 8 , 1947 , p. 1 9 8 . 3°Craig, op. c i t . , p. 7 2 . - ^ K i t t e l l , op. c i t . , p. 4 0 4 . 13 Haslerud and Meyers formulated the hypothesis that "... p r i n c i p l e s derived by the learner solely from concrete instances w i l l be more r e a d i l y used i n a new situation than those given to him i n the form of a statement of p r i n c i p l e ,,32 and an instance." The i n s t r u c t i o n was given i n the form of a coding test i n which the subjects were required to break a series of twenty codes and to translate an English sentence into each of the twenty codes. One mark was given f o r each successful t r a n s l a t i o n of the sentence into code. Each subject was re-quired to discover the coding p r i n c i p l e for ten of the problems but was given a written statement of the remaining ten coding p r i n c i p l e s . Thus each subject was involved i n the "given" (expository) and "derived" (discovery) treatment groups. I t was found that the subjects obtained s i g n i f i c a n t l y higher scores on the problems f o r which the coding p r i n c i p l e s were given. The transfer test consisted of twenty di f f e r e n t English sentences — one for each of the twenty codes studied pre-vio u s l y . Following each sentence was one t r a n s l a t i o n into code and three d i s t r a c t o r s . The subjects were required to select the properly coded sentence. There was a one-to-one correspondence between the codes on the transfer test and the Haslerud and Meyers, op. c i t . , p. 2 9 3 . 14 codes on the i n s t r u c t i o n a l t e s t . Thus i t was possible to obtain transfer scores, f o r each subject, relevant to the two modes of i n s t r u c t i o n . There was found to be no s i g n i f i -cant difference between these two sets of transfer scores. Haslerud and Meyers then examined the score gains made from the i n s t r u c t i o n a l test to the transfer t e s t . They found that the subjects made s i g n i f i c a n t l y greater gains on items where they had to derive the coding p r i n c i p l e than on items where the coding p r i n c i p l e was given. On thi s basis the authors concluded that the discovery method was superior f o r the f a c i l i t a t i o n of tra n s f e r . I t may be that the c r i t e r i o n of transfer used by Haslerud and Meyers i s i n v a l i d . The two differences i n scores may be non-comparable. The subjects might have achieved only p a r t i a l understanding of some of the coding p r i n c i p l e s that were to be discovered. P a r t i a l knowledge of a coding p r i n c i p l e would not enable a subject to score on the i n s t r u c t i o n a l t e s t . I t would enable him to score on the transfer t e s t . Thus the subject's "discovery" score might increase considerably from the i n s t r u c t i o n a l t e s t to the transfer t e s t . P a r t i a l knowledge would not be such a s i g n i f -icant factor on the i n s t r u c t i o n a l test f o r the coding p r i n c i p -l e s that were given. Errors here would probably be caused by misconception of the coding p r i n c i p l e or by carelessness. Thus the differences i n scores observed by Haslerud and Meyers might be related to a difference i n the bias of the two tests and not to a difference i n transfer e f f e c t s . Ausubel considers the important result i n t h i s experiment 15 to be that the transfer test scores were the same. He con-siders the r e l a t i v e standing of the two groups on the learning 33 task to be i r r e l e v a n t . I f the d i f f e r e n t scores used by Haslerud and Meyers are non-comparable, as suggested above, then Ausubel Ts point seems v a l i d . The study by Haslerud and Meyers has another possible weakness. Since a l l subjects were given i n s t r u c t i o n accord-ing to both of the experimental methods, there may have been in t e r a c t i o n effects not considered i n the report of t h e i r study. Other Studies;- Studies which are frequently taken to support the discovery position with respect to transfer are those by McConneH,-^ Thiele,35 Swenson,^0 and Katona.-^ Ausubel claims that i n these studies the rote-meaningful 33 David P. Ausubel, The Psychology of Meaningful  Verbal Learning. 1963, p. 170. 34 T.R. McConnell, "Discovery versus Authoritative I d e n t i f i c a t i o n i n the Learning of Children." University  of Iowa Studies i n Education, 9, No. 5, 1934, pp. 11-62. ^C.L. Thiele, The Contribution of Generalization  to the Learning of the Addition Facts. New York, Bureau of Publications, Teachers College, Columbia University, 1938. -^Esther J . Swenson, "Organization and Generalization as Factors i n Learning, Transfer, and Retroactive I n h i b i t i o n , " Learning Theory, i n School Situations, University of Minnesota-Studies i n Education, Minneapolis, University of Minnesota Press, 1949, pp. 3-39. 37 G. Katona, Organizing and Memorizing. Studies i n the Psychology of Learning and Teaching, New York, Columbia University Press, 1940. 16 dimension of learning was not c o n t r o l l e d . ^ 0 Ausubel states that both discovery and expository teach-39 ing can be rote or meaningful as i n Figure 1. He claims that discovery enthusiasts have usually designed experiments or Figure 1: Dimensions of Learning Rote Meaningful Expository A B Discovery A,B studies that contrast the methods i n the c e l l s marked 'A*, whereas the c r u c i a l comparisons should be between the methods i n the c e l l s marked * B' Opinion Supporting the Unverbalized Awareness Method: Bruner says that i t i s important to arrange c u r r i c u l a so that discovery i s possible.^'*'' He says, "Practice i n d i s -covery f o r oneself teaches one to acquire information i n a way that makes the information more readi l y viable i n problem s o l v i n g . " ^ 2 Also, "I am objecting to..., the premature use 3$ David P. Ausubel, The Psychology of Meaningful Verbal  Learning, 1963, pp. 167-163. 39 7 I b i d . , p. 13. ^ I b i d . , p. 166. ^Jerome S. Bruner, The Process of Education. Cambridge, Mass., Harvard University Press, 1961, p. 20. i 2 Jerome S. Bruner, "The Act of Discovery," Harvard  Educational Review, 31, 1961, p. 26. 17 of the language of mathematics, ...." Rosskopf i s another advocate of discovery learning. He says, "Discovery and exploration through many examples that use the same concept should be the means of i n s t r u c t i o n i n a general education mathematics course and the end should be applications of the non-verbalized concepts to new problems. w^ 45 " Rosskopf quotes Hendrix' study i n support of his opinion. Beberman also supports Hendrix* theories. He emphas-izes the matter of i n t e r e s t . He says, "The very f a c t that a student can discover techniques f o r himself i s a l l the evidence he needs to see mathematics as a human endeavor that demands creative energy.... Because of the students* independence of rote rules and routines, i t (discovery) develops v e r s a t i l i t y 47 i n applying mathematics." Hendrix has vigorously promoted the unverbalized awareness method. She says, "..., i n contrast to a tell-them-and-- drill-them approach the inductive lesson seems a revelation i n pedagogy. the authoritarian method at i t s best Jerome S. Bruner, "On Learning Mathematics," Mathem- a t i c s Teacher, 5 3 , I 9 6 0 , p. 6 1 4 . Myron F. Rosskopf, "Transfer of Training," The Learn- ing of Mathematics I t s Theory and Practice. Twenty-First Year book, National Council of Teachers of Mathematics, 1 9 5 3 , p. 2 2 1 , 4 5 I b i d . t pp. 218, 2 2 6 - 2 2 7 . Max Beberman, An Emerging Program of Secondary School  Mathematics. Cambridge, Mass., Harvard University Press, 1 9 5 8 , pp. 5 , 2 4 - 3 9 . 4 7 I b i d . , pp. 3 8 - 3 9 . i a i s a feeble best i n comparison with the inductive method. And the inductive method at i t s best i s a feeble best i n comparison with a genuine discovery-approach which promotes and recognizes a nonverbal awareness stage i n creative thought."2*-8 Research Evidence Opposed to the Unverbalized Awareness Method K i t t e l l ' s s t u d y , ^ which i s an extension of an e a r l i e r 50 study by Craig,^ i s sometimes taken as evidence opposed to the claims f o r the unverbalized awareness method. K i t t e l l used the type of verbal problem i n which the pupil i s to select a word that does not belong with four other words. He says, "Three d i f f e r e n t combinations of clues to the p r i n c i p l e s determining correct responses to the t r a i n i n g items were incorporated into three sets of t r a i n i n g materials, which when administered, each to a different* group of Ss, effected d i f f e r e n t treatments by providing thEee d i f f e r e n t amounts of di r e c t i o n during learning."-' 1 The subjects i n K i t t e l l ' s minimum dire c t i o n group were t o l d only that there was an underlying p r i n c i p l e f o r each group of three items. The subjects i n the intermediate direc-t i o n group were provided with a l l the information supplied i d Gertrude Hendrix, "Learning by Discovery," Mathematics  Teacher, 54, 1961, p. 296. -49 Jack E. K i t t e l l , "An Experimental Study of the Effects of External Direction During Learning on Transfer and Retention of P r i n c i p l e s , " Journal of Educational Psychology, 43, 1957, pp. 391-405. 50 R.C. Craig, The Transfer Value of Guided Learning. New York, Bureau of Publications, Teachers College, Columbia -University, 1953. 51 K i t t e l l , op. c i t . , p. 395. 19 to those i n the minimum di r e c t i o n group plus a verbal state-ment of the p r i n c i p l e involved. The subjects i n the maximum dir e c t i o n group were provided with a l l the information supplied to those i n the intermediate group plus oral statements of the correct responses to a l l items. The transfer test used by K i t t e l l was based on the same f i f t e e n selection p r i n c i p l e s used i n the t r a i n i n g period but the groups of words used were d i f f e r e n t . The intermediate dir e c t i o n group did s i g n i f i c a n t l y better on t h i s test than either of the other groups. K i t t e l l concluded "that furnish-ing learners with information i n the form of underlying p r i n c i p l e s promotes transfer ...."-^ Haslerud and Meyers have suggested that the "low number of successful solutions (means only 4.59 f o r intermediate d i r e c t i o n and 1.93 f o r minimal d i r e c t i o n out of 15 p r i n c i p l e s ) suggests that the problems ... may have been too d i f f i c u l t f o r the Ss. I f that were the case, then following directions i n the stated p r i n c i p l e was about the only way to solve the problem when unprovided with s u f f i c i e n t apperceptive mass and experience Opinion Opposed to the Unverbalized Awareness Method Ausubel claims that discovery proponents have made K i t t e l l , op. c i t . , p. 404. -^ G.M. Haslerud and Shir l e y Meyers, "The Transfer Value of Given and In d i v i d u a l l y Derived P r i n c i p l e s , " Journal  of Educational Psychology, 49, 1958, pp. 293-294. -20 discovery methods a "fad and a r e l i g i o n . " He says that t h e i r "claims f o r i t s uses and effic a c y go f a r beyond the evidence 54 as well as f a r beyond a l l reason." He accuses these people of using the "straw man" technique i n order to disparage 55 " expository teaching. Henderson, who supports many aspects of discovery teaching, agrees with Ausubel*s claim that discovery proponents have used the "straw man" technique.^ He says, "So often the discovery method i s made to g l i t t e r by comparing i t with the drab kind of teaching that sometimes i s done when the t e l l i n g method i s used. ... using unfa i r comparisons, one could just 57 as e a s i l y make the user of the discovery method look bad." Summary of the Research and Opinion I t does not seem that Hendrix* claims f o r superiority of the unverbalized awareness method of teaching f o r transfer are f u l l y supported by published research evidence. Oh the other hand, no good contrary evidence was found either. Since her opinions on t h i s topic have been quite i n f l u e n t i a l i n the f i e l d of education, p a r t i c u l a r l y i n math-ematics education, there i s clear j u s t i f i c a t i o n f o r further research. 54 David P. Ausubel, "Learning by Discovery: Rationale and Mystique," B u l l e t i n of the National Association of Secondary  School P r i n c i p a l s , v o l . 4 5 , no. 2 6 9 , 1 9 6 1 , pp. 1 8 - 5 8 . 55 David P. Ausubel, The Psychology of Meaningful Verbal  Learning, 1 9 6 3 , pp. 1 7 -18 . 56 J Kenneth B. Henderson, "Anent the Discovery Method," Mathematics Teacher, 5 0 , 1 9 5 7 , p. 2 8 9 . 5 7 I b i d . , p. 2 8 9 . CHAPTER III PROCEDURE Introduction The design of the experiment i s summarized i n Figure 2 . Figure 2- : Experimental Design Group I (Unverbalized) -(Awareness) -Pairs of nearly- Screening equivalent classes procedure Four days of - each class i n a to detect i n s t r u c t i o n on given p a i r with pupils who the subject TRANSFER the same teacher had studied matter (sets) sets pre- TASK viously (Verbal Reception) GROUP II GROUP III (No instruction) (Control) ( o n sets ) The subjects involved i n the f i n a l run of the experiment were a l l eighth grade pupils. The learning task was some of the introductory ideas of the language of sets. The actual lesson plans are shown i n Appendix I and a set of teaching notes to accompany them i s shown i n Appendix I I . The trans-f e r task, shown i n Appendix V, was a written examination composed of questions of the type that usually accompany the study of Venn diagrams. P r e t r i a l The p r e t r i a l f o r the experiment was conducted by the experimenter with a s i x t h grade class and a seventh grade class. The p r e t r i a l had four main purposes. The f i r s t was 22 to determine the s u i t a b i l i t y of the lesson materials. The lesson plans were revised on the basis of pupil.lreaction and by pupil achievement on d a i l y assignments. The second purpose of the p r e t r i a l was to discover where and how the teacher might unwittingly mix the methods of i n s t r u c t i o n . At the conclusion of each lesson the experi-menter recorded his observations of places where such mixing might be most l i k e l y to occur. These records were used i n preparing written i n s t r u c t i o n to the teachers. Thirdly, the p r e t r i a l was used to c o l l e c t data neces-sary f o r item analysis of the transfer t e s t . Both parts of the transfer test were administered on the f i f t h day of the p r e t r i a l . The two parts were administered separately but i n the same period. Both classes wrote the transfer test at the same time. The fourth purpose of the p r e t r i a l was to adjust each lesson and the transfer test to the regular f i f t y minute classroom period. Throughout the p r e t r i a l each lesson was timed to the nearest minute. The experimenter also noted the amount of time taken for the f i r s t and for the l a s t pupil i n each class to complete the part of the lesson devoted to a supervised assignment. The time provided f o r the transfer test was found to be adequate. Selection of the Experimental Groups Four pairs of classes were selected. A selection c r i t e r i o n was that both classes i n each pair were to have the same teacher. Other c r i t e r i a used i n attempting to 23 select equivalent classes were scholastic aptitude, previous achievement i n mathematics, d i s t r i b u t i o n of sexes, and class s i z e . Two of the pairs of classes were i n a public school at Terrace, B.C., and the other two pairs were i n a public school at Prince Rupert, B.C. The assignment of pupils to classes and of classes to teachers was not under the control of the experimenter. In Prince Rupert four classes remained unassigned to either experimental method. Two of these classes were schol-a s t i c a l l y comparable to the experimental classes and so were useable as a control group. C o l l e c t i o n of Data The pupils of one school had written the Henmon-Nelson Test of Mental A b i l i t y (grade 6-9, form B) less than two months before the experiment. These scores were made available to the experimenter. This same test was administered to a l l other pupils involved during the week following the experi-ment . The Christmas mathematics examination scores of a l l pupils were collected and recorded shortly before the experi-ment was conducted. Assignment of the Treatments Each teacher was asked to name the class, out of his two, that he believed to be the more capable of learning new material. Then i n each school one teacher had the teaching methods assigned to his classes by the f l i p of a coin. Where 24 t h i s teacher's more able class was assigned to the unverbal-ized awareness method, the other teacher i n that school was asked to assign his less able class to the unverbalized aware-ness method. These differences i n a b i l i t y to learn new material were probably quite small f o r each given p a i r of classes. Screening of the Participants A short quiz .was administered to a l l pupils who took part i n the experiment. Three of the questions were as follows. (a) What i s the largest subset of (1, 2, 3, 4, 10)? (b) I f U = (4, 8, a, b) and A = (4), what i s A? (c) Find (1, 5, 7)f! (5, 7, 9) These questions could be answered correctly only by persons who had previously studied the language of sets. Pupils who answered any of these three questions correctly, or who indicated p a r t i a l knowledge of them, were interviewed a f t e r the experiment to determine whether or not they had studied the language of sets previously. Also, at the con-clusion of the experiment, each teacher asked'if any p u p i l had studied the language of sets before. A l i s t of the names of pupils who were repeating the grade was prepared. These pupils had studied the language of sets previous to the experiment. Thus a l i s t was compiled of the names of a l l pupils who had studied the language of sets p r i o r to the experiment. 2 5 The scores of these pupils were not included i n the s t a t i s -t i c a l treatment of the data. Subject Matter Content The content, arranged i n the sequence i n which i t appeared, i s shown below. Table I: Subject Matter Item Lesson i n which i t F i r s t Appears Concepts and P r i n c i p l e s 1 Lesson 1 Set 2 Lesson 1 Subset 3 Lesson 1 Empty Set 4 Lesson 1 Naming of Sets 5 Lesson 2 Union of Sets 6 Lesson 3 Universal Sets 7 Lesson 3 Complementary Sets 8 Lesson 4 Intersection of Sets Analysis of the Methods The Unverbalized Awareness Method - In the unverbal-ized awareness method the teachers presented a variety of instances of each concept. Sometimes, but not alv/ays, t h i s was followed by a short i n t e r v a l of oral questioning. During the questioning, pupils were asked either to use or to give further examples of the concept. Pupils were not allowed to give anything even remotely resembling an explanation of the concept. Each lesson concluded with a supervised assignment. For examples of t h i s method see the lesson plans i n Appendix I . An accompanying set of teaching notes i s i n Appendix I I . 2 6 During each lesson the teachers were careful to write key examples of each new concept on the chalkboard. These examples were l e f t on the chalkboard at the conclusion of the lesson and pupils who had d i f f i c u l t y with the written assignment were directed to study the examples so as to discover the concept for themselves. A l l i n d i v i d u a l atten-t i o n was given i n the unverbalized awareness format. Each lesson, other than the f i r s t , started with a short review which was also i n the unverbalized awareness format. With the unverbalized awareness group the naming of each concept was delayed for as long as practicable. The intention was to give the pupils an opportunity to under-stand the concept before a name was given to i t . None of the concepts were defined since t h i s would have constituted expository i n s t r u c t i o n . In p a r t i c u l a r , the union operation, the i n t e r s e c t i o n operation, and the operation of finding the complement of a set were not defined at any time during the experiment. A l l questions on the transfer test involved these three concepts. 2 The Verbal Reception Method - In the verbal reception method an introductory organizer preceded the development of each new concept. Immediately following the discussion of For examples of t h i s method see the lesson plans i n Appendix I . An accompanying set of teaching notes i s i n Appendix I I . 27 the introductory organizer the concept was named and then explained. Examples of the use of the concept were then given. A l l examples were accompanied by an explanation of the underlying concept. This procedure was followed i n o r a l response as w e l l . Without these explanations the lessons would have become nearly i d e n t i c a l to the discovery lessons f o r much of the time. At the conclusion of each lesson the chalkboard was erased. No p u p i l was allowed to take notes. These moves prevented the pupils from gaining discovery experience by studying the examples that had been written out. A super-vised assignment was given at the end of each lesson. A l l remedial attention was given i n the form of further explana-t i o n of the various concepts. Each lesson, other than the f i r s t , started with a short review which was given i n the verbal reception format. Instruction to the Teachers The teachers involved were given detailed instructions i n every phase of the experiment. Every teacher was given a set of lesson plans and a set of related teaching notes. These are reproduced i n Appendices I and I I , respectively. The two Terrace teachers and the experimenter held da i l y discussions during which the lesson plan and teaching notes f o r the following lesson were studied point by point. One of the Prince Rupert teachers and the experimenter d i s -cussed the experiment i n d e t a i l . This teacher then accepted the r e s p o n s i b i l i t y of discussing the lesson plans and teaching 28 notes with the other teacher i n his school. The Daily Assignments At the conclusion of each lesson the pupils were given a supervised assignment relevant to the lesson. The same assignments were given to both of the experimental groups^. They were designed so that a l l pupils could complete them during the class period i n which they were given. Since the assignments were collected as soon as they were completed, they provided a check on attendance. The assignments are shown i n Appendix I I I . The Additional Assignments A second set of assignments i s shown i n Appendix IV. these assignments contain work that i s of a review nature and i s not related to the work on sets. Their purpose was to serve as ad d i t i o n a l work f o r the pupils who had completed the assignment on sets. Administration of the Transfer Test The transfer test was administered i n two parts. Both parts are shown i n Appendix V. The time l i m i t s were generous i n order to ensure that the instrument was a power t e s t . Pupils were not allowed to hand in the f i r s t part u n t i l twenty-five minutes had elapsed. They were permitted to work f o r longer than twenty-five minutes i f they wished. Once a pupi l handed i n the f i r s t part of the transfer test he was given the second part. The f i r s t part of the transfer test was administered 29 i n Prince Rupert to a group of eighth grade pupils who had not studied the language of sets. The control group did not write part II of the transfer test because the vocabulary-was such that most scores would have been zero. Arrangements were made to have the test administered to a l l pupils within two consecutive class periods. This was done to prevent any pupil from gaining advance warning or information about the t e s t . Further Controls In an e f f o r t to reduce the Hawthorne effect pupils were not t o l d that an experiment was i n progress. Teachers were informed of the teaching methods well ahead of time. They were asked to use the methods i n t h e i r own lessons so that pupils would be accustomed to them by the time the experiment was conducted. This, i n i t s e l f , i s an experi-mental v a r i a b l e . The i n s t r u c t i o n given to the treatment groups was made equivalent with respect to sequence of presentation, amount of teaching time, and the number of examples used. A one-to-one correspondence was established between the concrete examples used i n the two treatments. The da i l y assignments were i d e n t i c a l f o r the two groups. CHAPTER IV ANALYSIS OF THE DATA P r e t r i a l The p r e t r i a l data, shown i n tables VI-1 and VI-2 of Appendix VI, were used f o r the purpose of item an a l y s i s . Indices of discrimination were obtained using Flanagan's table of estimates of a b i s e r i a l r . This analysis was carried out independently f o r parts I and II of the transfer test and separate analyses were carried out with the data from each treatment group. The indices obtained are shown i n table VI-3 of Appendix VI. A summary of these s t a t i s t i c s i s shown i n table I I , below. Changes were made i n items f o r which the value of Flanagan's r f e l l below .30 for both treatment groups. Item eight was modified to correct a possible misunderstanding. The items which were altered are starred i n table I I . Item 5b of part II of the transfer test was deleted. The r e l i a b i l i t y of each part of the transfer test and of the tra n s f e r test as a whole was calculated using Kuder-Richardson formula twenty. The r e l i a b i l i t i e s obtained are shown i n table I I I , below. Table I I I : Estimates of the R e l i a b i l i t y of the Transfer Test Transfer Test Number of Items R e l i a b i l i t y Part I 22 . 7 9 Part II 8 .55 Combined Scores 30 .82 (Part I and Part II) 31 Table I I : C l a s s i f i c a t i o n of Test Items by Range of Item V a l i d i t y Indices (Flanagan's r) for the T r i a l Run Range of Flanagan's r Part I of the Transfer Test Part II of the Transfer Test Range of Flanagan's r Unverbalized Awareness Group Verbal Reception Group Unverbalized Awareness Group Verbal Reception Group .90-1.00 4b 1, 2 .80-.89 5b, 6b, 16b 6b, 8b*, 16b 5a 1, 2 .70-.79 4b, l i b , 16c 5b, l i b , 16c, I6d 3, 6 5a .60-.69 5a,,t8b*, 11c 14c', loa 13b 4b 3, 4b .50-.59 11a, I6d 7, 8c*, 11a 13a, 14c", 16a 4a 4a, 6 .40-.49 4a, 7, 8a'r, 8c* 13b 11c .30-.39 4a, 5a .00-.29 6a*,^13a, 14a"', 14b 6a*, da*, 14a , 14b v 5b 5b * Items marked with an asterisk were modified 32 Experimental Run The raw data from the experiment are given i n tables VI-4 and VI-5 of Appendix VI. A summary of t h i s information and of other information needed for analysis of covariance i s shown i n table VI-6 of Appendix VI. Table IV, below, i d e n t i -f i e s the subjects according to location, treatment method, and teacher. Table IV: C l a s s i f i c a t i o n of Subjects Involved i n the Experimental Run Student Location Treatment Teacher Numbers Method 1-23 Prince Rupert Unverbalized Awareness I 24-38 Prince Rupert Unverbalized Awareness II 39-53 Terrace Unverbalized Awareness III 59-85 Terrace Unverbalized Awareness IV 86-113 Prince Rupert Verbal Reception I 114-130 Prince Rupert Verbal Reception II 131-154 Terrace Verbal Reception III 155-172 Terrace Verbal Reception IV 173-209 Prince Rupert Control V 210-240 Prince Rupert Control VI To determine the effectiveness of the modifications i n the transfer test, the item analysis was repeated using the data from the experimental run. The indices obtained are shown i n table VI-7 of Appendix VI, and t h i s information i s summarized i n table V. A l l items were found to remain s a t i s -factory using the e a r l i e r c r i t e r i a . An estimate of the r e l i a b i l i t y was found f o r each part of the transfer test and for the transfer t e s t as a whole using Kuder-Richardson formula twenty. The r e l i a b i l i t i e s obtained are shown i n table VI, below. 33 Table V: C l a s s i f i c a t i o n of Test Items by Ranges of Item V a l i d i t y Indices (Flanagan's r) f o r the Experimental Run CO t n a o o3 txO CD ct3 hO c C 03 OS rH Part I of the Transfer Test Part II of the Transfer Test Unverbalized Awareness Group Verbal Reception Group Unverbalized Awareness Group Verbal Reception Group '.80-1.00 l i b , 13b 16b 1 .70-.79 5a, 8a, 8c - 11a, 16c 4a, 5b, 6b, 8a 11a, l i b , 11c, 13b, 16c, I6d 3, 6 4a 4b, 5a ,60-.69 11c 4b, 5a, 8c 2, 5a 3, 6 ' .50-. 59 4a, 4b, 5b, 6a, 6b, 7 16a, 16b, I6d 8b, 14b, 16a 1, 2 ' .40- . 49 13a 6a, 13a '.30-. 39 8b, 14a, 14c 7, 14c 4a - .00- . 29 14b 14a 34 Table VI: Estimates of the R e l i a b i l i t y of the Transfer Test Transfer Test Number of Items R e l i a b i l i t y Part I 22 .33 Part II 7 .64 Combined Scores 29 .35 (Part I and Part II) Each of these estimates of r e l i a b i l i t y was higher than the corresponding estimate obtained using the p r e t r i a l data. None of these differences were s i g n i f i c a n t at the f i v e per-cent l e v e l . Correlations The various correlations involving the covariates and the c r i t e r i o n scores are summarized i n table VII. Table VII: Correlations Scores Considered Correlation Number of Cases Mental Age vs. T i .43 172 Mental Age vs. T 2 .37 172 Previous Math. vs. T i .55 172 Previous Math. vs. T 2 .53 172 Mental Age vs. Previous Math. .52 172 T x vs. T 2 .54 172 Tj_ refers to scores on part I of the transfer t e s t . T 2 refers to scores on part II of the transfer t e s t . Previous Math, refers to scores on the Christmas mathematics t e s t . The multiple correlation c o e f f i c i e n t f o r part I of the transfer test against mental age and the Christmas mathem-a t i c s test score was .58. This was s i g n i f i c a n t l y higher, at the one percent l e v e l , than the corr e l a t i o n between part I of 35 the transfer test and i t s best single predictor. This resulted i n the decision to conduct the analysis of covariance using both predictor variables. The Christmas Mathematics Score as a Covariate Since the Terrace and Prince Rupert groups wrote s l i g h t l y d i f f e r i n g Christmas mathematics tests i t was decided to conduct the analysis separately f o r these groups. The Transfer Test The v a l i d i t y of the f i r s t part of the transfer test was checked by comparing the performance by the treatment group with the performance by the control group. The control group had continued t h e i r normal mathematics lessons under t h e i r usual teachers. They did not study anything dealing with the language of sets p r i o r to the completion of the experiment. Table VIII: The Analysis of Covariance -Treatment Group versus Control Group Source of Degrees of Adjusted Sum Mean F Va r i a t i o n Freedom of Squares Squares Method 1 , 21.510 21.510 1.52 Error 147 2086.7 14.196 C r i t i c a l Value of F = 1.62 (©C= .10, one t a i l e d test) An alpha l e v e l of ten percent was chosen since i t was f e l t that a beta type error could be almost as serious as an alpha type error. The beta type error would cast doubt on 36 what might be in t e r e s t i n g conclusions regarding teaching mathematics f o r transfer. The obtained value of F was less than the c r i t i c a l value at the ten percent l e v e l f o r a one t a i l e d t e s t . Hence there was some doubt as to whether or not part I of the trans f e r test was a v a l i d test f o r tra n s f e r . Part II of the transfer test was assumed to be a v a l i d test f o r tr a n s f e r . The Test of the Hypothesis The n u l l hypothesis was that a f t e r two groups of pupil have been taught some i n i t i a l concepts of set theory; one group by the unverbalized awareness method and one group by a procedure of verbal reception learning, there w i l l be no s i g n i f i c a n t difference between the mean scores of the two groups on a transfer task. The alternative hypothesis was that the transfer effects would be unequal. Analysis of covariance was employed i n order to adjust f o r i n i t i a l differences between the two groups i n mental age and previous achievement i n mathematics. Separate analyses were carried out for the groups of subjects from Prince Rupert and from Terrace. This was done because of concern about the v a l i d i t y of using the Christmas mathematics scores i n any analysis involving the t o t a l group of subjects. This analysis was done separately f o r parts I and II of the c r i t e r i o n t e s t . The res u l t s are shown i n table IX and X. 37 Table IX: The Analysis of Covariance -Part I of the Transfer Test A. Prince Rupert Group Source of Var i a t i o n Degrees of Freedom Adjusted Sum of Squares Mean Squares F Method Error 1 79 3.9966 1096.0 3.9966 13.873 .29 C r i t i c a l Value of F = 3.97 ( 0 d » .05) B. Terrace Group Source of Variat i o n Degrees of Freedom Adjusted Sum of Squares Mean Squares F Method Error 1 85 18.797 979.69 18.797 11.526 1.63 C r i t i c a l Value of F = 3.97 ( * • .05) The adjusted mean score was higher f o r the verbal reception group than for the unverbalized awareness group i n both cases , but the n u l l hypothesis was accepted each time. Table X: The Analysis of Covariance -Part II of the Transfer • Test A. Prince Rupert Group Source of Variat i o n Degrees of Freedom Adjusted Sum of Squares Mean Squares . F . Method Error 1 79 1.0719 183.32 1.0719 2.3205 .46 C r i t i c a l Value of F = 3.97 .05) 38 B. Terrace Group Source of : Degrees o f : : Adjusted Sum ' ' : Mean : F Vari a t i o n Freedom of Squares Squares -Method 1 2 . 1 0 3 5 2 . 1 0 3 5 1 . 6 6 Error 85 1 0 7 . 9 2 1 . 2 6 9 7 C r i t i c a l Value of F • 3 . 9 7 (<*= . 0 5 ) As before, the adjusted mean score was higher i n both cases f o r the verbal reception group than f o r the unverbal-ized awareness group, but the n u l l hypothesis was again accepted. The Adjusted Scores Table XI shows the adjusted mean scores on part I of the transfer test for the treatment and control groups. Table XI: Adjusted MeanrScores -Treatment Group versus Control Group Group Teachers Adjusted Mean Difference Treatment I, II 7 . 0 8 8 6 . 7 9 9 7 Control V, VI 6 . 2 8 8 9 Table separate class XII shows the of students. adjusted mean scores f o r each In a l l but one case the score of the verbal reception group was higher than that of the corresponding unverbalized awareness group. Table XII: Adjusted Mean Scores on the Transfer Test • Part of the Column 1 Column 2 Teacher Transfer Unverbalized Verbal Reception Difference Test Awareness Group Group (Col.2-Col.l) I I 7 . 9 6 2 8 7 . 9 9 7 0 . 0 3 4 2 II I 6 . 9 5 3 6 8 . 0 3 7 5 1 . 0 8 3 9 III I 7 . 0 9 H 7 . 1 3 6 9 . 0 4 5 5 IV I 6 . 0 6 2 9 7 . 8 4 3 8 1 . 7 8 0 9 I II 2 . 6 6 6 7 2 . 6 1 0 0 - . 0 5 6 7 II II 2 . 0 3 3 1 2.7112 . 6 7 8 1 III II 1 . 8 7 3 8 2 . 4 0 9 8 . 5 3 6 0 IV II 2 . 3 0 5 7 2 . 4 6 8 8 . 1 6 3 1 CHAPTER- V SUMMARY AND CONCLUSIONS The Problem Gertrude Hendrix has outlined a guided discovery method which she c a l l s the unverbalized awareness method. She states that t h i s method i s superior to a l l other teaching methods f o r the f a c i l i t a t i o n of tra n s f e r . I t appears that there i s i n -adequate experimental evidence to support t h i s claim. Ausubel has suggested that the most important fa c t o r influencing transfer might be whether or not the method i s meaningful. He suggests that i f a discovery method were com-pared to a meaningful expository method, the discovery method would be but s l i g h t l y better for ef f e c t i n g t r a n s f e r . Ausubel has outlined the verbal reception method as a meaningful expository method. The central problem was to determine which of these two methods of i n s t r u c t i o n , Hendrix* or Ausubel* s, would be super-i o r f o r f a c i l i t a t i n g t ransfer. Since Hendrix" views have been quite i n f l u e n t i a l i n the f i e l d of mathematics education, t h i s question i s of some importance. The Experiment The f i n a l run of the experiment involved ten grade eight classes taught by six teachers, i n two schools. The subject matter was some of the introductory aspects of the language of sets. Four teachers each taught two classes -one class by the unverbalized awareness method and one class 41 by the verbal reception method. The remaining two classes,and the control group, received no i n s t r u c t i o n related to sets. Four regular classroom periods were a l l o t t e d to instruc-t i o n and one period was a l l o t t e d to the administration of a transfer t e s t . The transfer test consisted of two parts which were administered separately. The control group wrote part I only of the transfer t e s t . The Conclusion There was a tendency f o r the verbal reception groups to score higher than the unverbalized awareness groups, (table X I P ) , but the differences were not s i g n i f i c a n t at the f i v e percent l e v e l . I t was concluded that there was no basis f o r deciding that either of the experimental methods was superior for f a c i l i t a t i n g transfer. Discussion of the Conclusion Hendrix reached her conclusions regarding the unverbal-ized awareness method through an experiment involving a short cut. Since the experiment of t h i s thesis did not involve the learning of a short cut, the conclusion reached here may not necessarily be taken as contradicting Hendrix' findings. I t i s suggested that i f r e p l i c a t i o n of Hendrix' study should substantiate her claims then the conclusions should not be generalized beyond the learning of short cuts. This experiment gives some support to Ausubel's theory that whether or not the method of i n s t r u c t i o n i s meaningful 42 i s the most s i g n i f i c a n t factor influencing transfer. Regarding the V a l i d i t y of Part I of the Transfer Test The lack of s i g n i f i c a n t difference between the t r e a t -ment and control groups on part I of the transfer test must be accepted as i t stands; however i t may be noted that the test i n g situations were not i d e n t i c a l . The treatment group knew that they were to write the second part of the transfer test immediately a f t e r completion of the f i r s t part. The control group knew that they were to write a single test -part I of the transfer t e s t . The eff e c t of t h i s would probably be to depress the mean score of the treatment group. Such an effect would cast some doubt on the v a l i d i t y of part I iof the transfer t e s t . Further Questions I t may well be that the optimum method of in s t r u c t i o n , for f a c i l i t a t i n g transfer, depends upon the type of material to be learned. Thus i t i s possible that Ausubel*s method i s best for material of the type i n t h i s thesis while Hendrix* method might be best for the learning of a short cut. Further research should be conducted to c l a r i f y t h i s point. THE BIBLIOGRAPHY BIBLIOGRAPHY Ausubel, D.P. The Psychology of Meaningful Verbal Learning. New York and London, Grime and Stratton, 1963. Ausubel, D.P. "Learning by Discovery: Rationale and Mystique," Bulletin,, of the National Association of Secondary School -P r i n c i p a l s , V o l . 4 5 . No. 269, 1961. PP. 1 8 - 5 8 . Ausubel, D.P. "The Use of Advance Organizers i n the Learning-and Retention of Meaningful Verbal M a t e r i a l , " Journal of  Educational Psychology, 51, I 9 6 0 , pp. 2 6 7 - 2 7 2 . Beberman, M. An Emerging Program of Secondary School Mathematics. Cambridge, Mass., Harvard University Press, 1 9 5 ^ Bloom, B.S., ed. Taxonomy of Educational Objectives Handbook  I: Cognitive~Domain. Toronto, David McKay, 1956. " Brownell, W.A. and Hendrickson, G. "How Children Learn Information, Concepts and Generalizations," Learning  and Instruction. Forty-Ninth Yearbook, National Society f o r Study of Education, Part I, 49 , 1950, pp. 9 2 - 1 2 8 . Bruner, J.S. "On Learning Mathematics," Mathematics Teacher, 53, I 9 6 0 , pp. 6 1 0 - 6 1 9 . Bruner, J.S. "The Act of Discovery," Harvard Educational  Review, 31, 1961, pp. 21-32. Bruner, J.S. The Process of Education. Cambridge, Mass., Harvard University Press, 1 9 6 1 . Butts, D.P. "Does Experience Equal Understanding?," Science Teacher, 30, 1963, pp . 81-82. Corman, B.R. "The Ef f e c t of Varying Amounts and Kinds of Information as Guidance i n Problem Solving," Psychological Monographs, 7 1 , No. 2 (Whole No. 431) , 1957. Craig, R.C. "Directed versus Independent Discovery of Established Relations," Journal of Educational  Psychology, 47, 1956, pp. 223 -234. Craig, R.C. The Transfer Value of Guided Learning. New York, Bureau of Publications, Teachers College, Columbia University, 1953. Crannell, C.W. "Transfer i n Problem Solution as Related to Type of Training," Journal of General Psychology, 54, 1956, pp. 3 - 1 4 . 45 Duncan, C P . "Recent Research of Human Problem Solving," Psychological B u l l e t i n , 56, 1959, pp. 397-429. Forgus, R.H., and Schwartz, R.J. " E f f i c i e n t Retention and Transfer as Affected by Learning Method," Journal of  Psychology, 43, 1957, pp. 135-139. Haslerud, G.M., and Meyers, S h i r l e y . "The Transfer Value of Given and Ind i v i d u a l l y Derived P r i n c i p l e s , " Journal of  Educational Psychology, 49, 1958, pp. 2 9 3 - 2 9 C - " Henderson, K.B. "Anent the Discovery Method," Mathematics  Teacher, 50,-1957, pp. 287-291. Henderson, K.B. "Research on Teaching Secondary School Mathematics," Handbook of Research on Teaching, ed. N.L. Gage, Chicago, Rand McNally; 1963, pp. 1014-1029. Hendrickson, G., and Schroeder, W.H. "Transfer of Training i n Learning to Hit a Submerged Target," Journal of  Educational Psychology, 32, 1941, pp. 205-213. Hendrix, G. "A New Clue to Transfer of Training," Elementary School Journal, 48, 1947, PP. 197-208. Hendrix, G. "Learning by Discovery," Mathematics Teacher, 54, 1961, pp. 290-299. Hendrix, G. "Nonverbal Awareness i n the Learning of Mathematics," Research Problems i n Mathematics Education. U.S. O f f i c e of Education, Cooperative Research Monograph No. 3, I960, pp. 57-61. Hendrix, G. "Prerequisite to Meaning," Mathematics  Teacher, 43, 1950, pp. 334-339. -Hendrix, G. Letter to the writer. 30th November, 1964. Hilgard, E.R., Edgren, R.D., and.Irvine, R.P. "Errors i n Transfer Following Learning with Understanding: Further Studies with Katona's Card Trick Experiments," Journal  of Experimental Psychology, 47, 1954, pp. 457-464. Judd, C.H. "The Relation of Special Training to General In t e l l i g e n c e , " Educational Review, 36, 1908, pp. 28-42. Katona, G. Organizing and Memorizing. Studies i n the Psychology of Learning and Teaching, New York, Columbia University Press, 1940. Kersh, B.Y. "The Adequacy of "Meaning" as an Explanation f o r the Superiority of Learning by Independent Discovery," Journal of Educational Psychology, 49, 1958, pp. 282-292. 46 K i t t e l l , J.E. "An Experimental Study of the E f f e c t s of External Direction During Learning on Transfer and Retention of P r i n c i p l e s , " Journal of Educational  Psychology, 48, 1957, pp. 3 9 1 - 4 0 5 . Lindquist, E.F., ed. Educational Measurement. Menasha, George Banta, 1 9 5 1 . McConnell, T.R. "Discovery versus Authoritative I d e n t i f i -cation i n the Learning of Children," University of Iowa  Studies i n Education,. 9 , No. 5 , 1 9 3 4 , pp. 11-62. Orata, P.T. "Transfer of Training and Reconstruction of Experience," Mathematics Teacher, 30, 1 9 3 7 , pp. 9 9 - 1 0 9 . Rosskopf, M.F. "Transfer of Training," The Learning of  Mathematics I t s Theory and P r a c t i c e . Twenty-First Year-book, National Council of Teachers of Mathematics, 1 9 5 3 , PP. 2 0 5 - 2 2 7 1 Sassenrath, J.M. "Learning Without Awareness and Transfer of Learning Sets," Journal of Educational Psychology, 5 0 , 1 9 5 9 , PP. 2 0 5 - 2 1 2 . Schulz, R.W. "Problem Solving Behavior and Transfer," Harvard Educational Review, 3 0 , I 9 6 0 , pp. 61 - 7 7 . -Stephens,J.M. "Transfer of Learning," Transfer of Learning. eds. R.F. Grose and R.C. Birney,-Princeton, Van Nostrand, 1 9 6 3 , PP. 1 0 3 - 1 2 1 . Sw.enson, E.J. "Organization and Generalization as Factors i n Learning, Transfer, and Retroactive I n h i b i t i o n , " Learning Theory i n School Situations, University of Iowa Studies i n Education, Minneapolis, University of Minnesota Press, 1 9 4 9 , pp. 3 - 3 9 . Taba, H. "Learning by Discovery: Psychological and Educa-t i o n a l Rationale," Elementary School Journal, 6 4 , 1 9 6 3 , pp. 3 0 8 - 3 1 6 . Thiele, C L . The Contribution of Generalization to the Learning of the Addition Facts. New York, Bureau of Publications, Teachers College, Columbia University, 1938. Warren, J.M. "Intertask Transfer i n Code Substitution Learning,'' Journal of Genetic Psychology, 8 9 , 1 9 5 6 , pp. 6 5 - 7 0 . Winer, B.J. S t a t i s t i c a l P r i n c i p l e s i n Experimental Design. Toronto, McGraw-Hill, 1 9 6 2 . 47 Wittroek, M.C., and Twelker, P.$;. "Prompting and Feedback i n the Learning, Retention, and Transfer of Concepts," B r i t i s h Journal of Educational Psychology, 34, 1964, -pp. 1 0 - 1 8 . APPENDIX I The Lesson Plans 49 LESSON I Sets and Subsets Materials: Various classroom materials such as a pen, a pen c i l , a r u l e r etc. Unverbalized Awareness Verbal Reception Introduction (about 5 mins.) 1. Show 3 sets physically by placing the objects to-gether on the desk. Name them as you do so. 2. Show 3 sets of objects by naming them and pointing at them. 3 . Ask f o r several ex-amples from the class, (e.g. Say, "can you do what I am doing.") 4 . Give 2 sets of numbers and 2 sets of l e t t e r s . Write them on the board, and leave them there. Introduction (5 mins. or less) 1. Discuss "what i s the simplest mathematical idea?" Possible answers - addition, subtraction, counting, etc. 2. Say, "another simple mathematical idea i s 'set*." 3. Use the word "set" i n context and t a l k about the common use of the word, e.g. Set of glasses Set of weights Set of colours -( i . e . paint set) . 4 . Say, "hence 'set 1 i s used to mean a c o l l e c t i o n of things or ideas. Of course i n mathematics we are usually interested i n sets of numbers 50 Note: Emphasize the arb i t r a r y nature of the selection of the members of the above sets. Sets (about 10 mins.) 5 . Name 1 set of objects and write the set on the board. 6. Name 2 sets of pupils and write them on the board. 7 . Name the idea as "set" and show how the various sets on the board could be named. Point out the use of the brackets and the commas. S. I l l u s t r a t e what i s meant by "the members of a set." 9. Give further examples of sets i f necessary. but we can and do t a l k about other things, such as sets of objects." Sets (about 10 mins.) 5v Give an example of a set of objects, display them, state what the members of the set are. Explain how the set could be written on the board and named. Explain that the order of naming the objects does not matter. 6. Get pupils to name 3 other sets of objects i n the room. 7. Name and write on the board 2 sets of numbers, 2 sets of l e t t e r s , and 2 sets of pupils. £. Explain that sets can have no members. Obtain the word "empty set" from the cla s s . 51 1 0 , Give examples of empty sets. e.g. Set of a l l pupils over 35 years old. 1 1 . Name these sets as "empty sets." Subsets (10 mins.) 1 2 . Write 6 subsets of the various sets that are on the board. 1 3 . Let the pupils give 4 examples of subsets. 1 4 . Give examples of empty sets for each of the sets that are on the board. Also, i l l u s t r a t e that the " f u l l " set can be eonsider-ed to be a subset. 1 5 . Name the concept as "subset". Subsets (10 mins.) 9 . Explain that once we have sets or groups of objects etc., then i t i s l o g i c a l to start t a l k i n g about parts of the group. Explain "subset". 1 0 . Point out 6 subsets that could be formed from the ex-amples on the board. 1 1 . Explain the inc l u s i o n of the empty set and of the " f u l l set" as subsets. 1 2 . F i l l out the time by having the pupils give ex-amples of subsets. 52 16• Administer Assignment I . Note: Leave a l l your examples on the board. 13. Administer Assignment I. Note: A l l materials on the board should be erased. 53 LESSON II Union of Sets Materials: Same as l a s t day. Unverbalized Awareness Verbal Reception Review (5 mins.) 1. Review the concept of set, the writing of sets, and the naming of sets. 2. Review the concept of subset. Note: Be sure to use the unverbalized awareness procedure. Introduction (5 mins.) 3 . Give several examples of the union operation with sets of pupils. 4. Give many chalkboard i l l u s t r a t i o n s of the union operation. (Do not name i t yet.) Have pupils t r y to copy what you are doing. Review (5 mins.) 1. Review the same points as i n the unverbalized aware-ness lesson. Note: Be sure to use expository techniques. 2. Leave 2 sets of numbers on the board. Introduction (5 mins.) 3 . Develop an awareness of the fact that the operations of arithmetic are binary. Emphasize the fact that addition i s a binary operation. 4. Mention the fact that there are binary operations with sets. 54 5 . Discuss the common use of the word "union". Union of Sets (10 min.) 5 . Write 2 small sets of student names and write t h e i r union. 6 . Write 2 sets of objects and write t h e i r union. 7 . Write 2 sets of numbers and write t h e i r union. £. Write the union of an empty set with a set of l e t t e r s . 9 . Write the union of a set of student names with i t s e l f . (This i s to em-phasize that each member i s named only once.) Union of Sets (10 min.) 6. Show how the union operation works with the sets that you have on the board. 7. Show how the union of the 2 sets i s written, e.g. A U B = (a,b,c) <3. Give several examples of the union operation by using sets of pup i l s . Be certain to explain each step. 9 . Write 2 small sets of student names and write the union of the 2 sets. (Make sure that at lea s t one student i s i n both sets. Explain why we would not write his name twice.) 55 1 0 . Name the operation as the "union" of sets and review the various unions written on the board. Note: Leave a l l of your examples on the board. 1 1 . Administer assignment I I . 1 0 . Write 2 sets of objects and show how the union i s found, 1 1 . Repeat with 2 sets of numbers, 1 2 . Repeat with an empty set and a set of l e t t e r s . 1 3 . Explain how the union of a set with i t s e l f would be found. Note: Erase a l l work from the board. 1 4 , Administer Assignment II 56 LESSON III Universal and Complementary Sets Materials: Same as l a s t day. Unverbalized Awareness Review (5 mins.) 1. L i s t set (1,2) on the board and have pupils give a l l possible subsets of i t . 2. L i s t set (a,b,c) on the board and have pupils give various subsets of i t . 3 . Have pupils give ex-amples of the union of some of the subsets. Verbal Reception Review (5 mins.) 1. Follow the unverbalized awareness guide but add ex-planations as you go. Note: Erase the part on union but leave the two main sets and a l l subsets on the board. Universal Set (3 mins.) 4. Give the name "universal set" to the appropriate sets above. Also, use the form Universal Set (3 mins.) 4. Discuss the meaning of the word "universe". 5 7 U = (a,b,c). T e l l which of the above sets are not universal sets. 5 . Explain why certain sets are c a l l e d universal sets. T e l l which of the above sets are universal sets. Complementary sets ( 1 0 - 1 5 min 5 . Write the complements of each of the above subsets. (Do not give the name "com-plement" or the symbol just yet.) 6 . Write the set ( l , 5 , 7 , a , b ) and give 6 subsets of i t . Have students give the com-plement of each subset. 7 . Repeat with universal set (Tom, B i l l , Joe). 8. Repeat with universal set (a). 9 . State that the subsets that you have been fi n d i n g are c a l l e d complementary sets. Go through each of )Complementary Sets ( 1 0 - 1 5 mins.) 6 . Discuss the idea of a unary operation. Reciprocals are a suitable i l l u s t r a t i o n 7 . T e l l students that there i s a unary operation with sets. Explain the basic idea of the complement of a set and explain how i t can be found. Use the symbol f o r complements. 8. Explain and demonstrate how to f i n d the complement of each of the above subsets. 9 . Write the set ( l , 5 , 7 , a , b ) and give 6 subsets. Have students give the complement of each subset. Explain how the answers were found or 53 the above examples and write the symbol for the comple-mentary sets. Note; Leave a l l examples on the board. 10. Administer Assignment I I I . have the pupils explain how as they are doing the problems. 10. Repeat with the univer-sal set (Tom, B i l l , Joe). 11. Repeat with the univer-sal set (a). Note: Erase everything on the board. 12. Administer Assignment I I I . V 59 LESSON IV Intersection of Sets Materials: Same as l a s t day. Unverbalized Awareness Verbal Reception Review (5 mins.) 1. L i s t the following sets on the board. A = (a,b,c,d) D = (c,d,e,f,g) B = ( ) E = (1,5,a,b) C = (1,3,5) F = (2,c,5) 2. Have the pupils give a l l possible subsets of set C. Use set C as a universal set and have the pupils give the complement of each subset. Note: Leave sets A to F on the board. Review (5 mins.) 1. L i s t sets A to F (see column one) on the board. 2. Review the same points as i n column one. However, be sure to use expository tech-niques. Note: Leave sets A to F on the board. Intersection of Sets (10-15 mins.) 3. L i s t 2 intersections of the above sets. Introduction (2-3 mins.) 3. Review the concept of binary operation as i t applies to arithmetic and to the union of 6 0 4 . Give several physical examples of the in t e r s e c t i o n operation with sets of pu p i l s . . 5. Give some more i l l u s -t r a t i o n s of the i n t e r -section operation with the sets A-F which are on the board. 6 . Name the operation as "in t e r s e c t i o n " of sets and l a b e l each of the i n t e r -sections which are on the board. e.g. A/1 E = (a,b) 7. Record several more examples of the i n t e r -section operation on the board. 8. Leave a l l of your examples on the board. 9. Administer Assignment IV. sets. Point out that sub-tra c t i o n i s also a binary operation. Explain that there i s a second binary operation with sets and that i t i s cal l e d 'intersection'. 4 . Refer to the interse c t i o n of two streets i n order to explain how the i n t e r s e c t i o n of sets w i l l be found. Intersection of Sets (10 mins.) 5. Explain how you w i l l f i n d the i n t e r s e c t i o n of sets A and D. Then record the i n t e r -section on the board. 6 . Write the symbol f o r i n t e r -section and explain that i t i s the opposite of the symbol f o r union. e.g. Afl D = (c,d) 7. Follow the same procedure as i n step 5 with sets A and E. 61 #. Give several physical examples of the intersection operation with sets of pupils. Be sure to explain each step. 9. Record more examples of the i n t e r s e c t i o n operation on the board. I f you have the pupils give some i l l u s t r a t i o n s be sure that they explain how they w i l l get t h e i r answer be-fore they a c t u a l l y state the answer i t s e l f . 10. Erase everything which has been written on the board. 11. Administer Assignment IV. APPENDIX II The Teaching Notes General Teaching Notes S p e c i f i c Teaching Notes 63 A. General Teaching Notes Unverbalized Awareness Verbal Reception 1. Be very careful that pupils do not "give the game away". I f a pupil begins to off e r an explanation of any-thing then stop him immedi-a t e l y . I f necessary, t e l l the pupils that they are not to "give the game away". (This might tend to come up eas i l y during review work.) 2. Plan, before each lesson, what examples you w i l l leave on the board at the end of the lesson. Pupils who ask for help at the conclusion of the lesson should be directed to study the examples on the board. I f you must give i n d i v i d u a l attention 1. Be very careful to avoid using discovery techniques. For instance you must not give i l l u s t r a t i o n s of a con-cept unless those i l l u s t r a -tions are preceded by and accompanied by explanation of the concept. 2. The procedure i n item 1, above, should be followed by the pupils when they are giv-ing examples of various con-cepts. For example, suppose you ask a pu p i l to give the union of two sets. The pupil should explain how he w i l l f i n d the union before he states what the union i s . 3 . Be very careful with re-view work. Be sure to con-tinue using the verbal 64 to any pup i l then you must continue to use the unverbal-ized awareness format. 3. Be very careful, when you have pupils answering questions, that you do not add explanations of what the pupils are doing. Be careful, too, that you do not add explanation i n the way of hints i f a pupi l does not get the correct answer immediately. 4 . Review work must be handled very c a r e f u l l y . Be certain that i t i s done i n the unverbalized aware-ness format. reception format. I f you merely l i s t i l l u s t r a t i o n s of the various concepts without adding explanations then you are using the unverbalized awareness format. 4 . At the conclusion of each lesson erase a l l work which i s on the board. 5 . When you administer the supervised assignment be sure that each pup i l works on his own. I f any p u p i l has d i f f i -culty with the assignment then give him i n d i v i d u a l assistance but be sure that i t i s i n the verbal reception format. 65 B. S p e c i f i c Teaching Notes Unverbalized Awareness Lesson I: 1. You may explain how to write sets or how to name them but you should not explain how the members are selected. For instance you could ask a pupil to give a set of l e t t e r s . When he i s f i n i s h -ed you would say, "Yes, and here are some ways that we could write that set". Set A = (a,b,c) or set A = (c,b,a) etc. Notice that we do not say, "these sets are equal because...." Instead we just say, "these sets are equal." 2. Select sets of various c a r d i n a l i t i e s . Verbal Reception Lesson I: 1. Be very careful of giving reasons f o r the selection of subsets. Pupils tend to think that there must always be a reason f o r the se l e c t i o n . Also, they tend to think that two subsets with the same members are di f f e r e n t i f there are d i f f e r e n t reasons f o r th e i r selections. I t seems that the pupils confuse the reason f o r the selection with the subset i t s e l f . 2. Select sets of various c a r d i n a l i t i e s . 3. Be careful of giving 66 explanations f o r subsets, e.g. "Well, the members of the subset must come from t h i s set here." Lesson I I : 1. When you select sets of pupils i n order to show the union operation be sure to take some examples where one or more pupils are i n both sets. e.g. A=(Bob, B i l l ) B=(Bob, Joe, Dave) Then the absurdity of A B=(Bob, Bob, B i l l , Dave) should be apparent. Lesson I I : 1. When discussing the idea of a "union" as i n a logger's union, emphasize the idea that the men are joining or uniting together. Lesson I I I : Lesson I I I : 1. Name each of the various 1. Be very careful with t h i s subsets that you write on the lesson. I t i s an easy one i n board. which to s l i p up and use e.g. A = (1,3) discovery techniques. Then, when you t e l l the pupils the word "complement", you w i l l be able to run 67 r i g h t through the examples on the board and write the symbol f o r complement next to each one. Lesson IV: 1. Do not warn the pupils of the transfer test which i s to be administered next period. Lesson IV: 1. Do not warn the pupils of the" transfer test which i s to be administered next period. APPENDIX III The Supervised Assignments 6 9 NO.l SETS Name: Divi s i o n : Teacher's Name: Write a l l of your answers on t h i s sheet. 1-. Write 2 d i f f e r e n t sets of objects, each with 5 members, choose a l l members from v i s i b l e things i n t h i s room, at t h i s time. Set A = ( ) Set B = ( ) 2. Write 2 d i f f e r e n t sets, each with 4 members, a l l members to be chosen from things not i n t h i s room, at t h i s time. Set A = ( ) Set B = ( ) 3. How many diff e r e n t sets, with exactly one member each, could you obtain from: a. the complete alphabet? Answer b. the l e t t e r s past "C"? Answer 4 . What i s the very le a s t number of members that any set can Have? Answer 5. How many members are there right now i n the row i n which you s i t ? Answer 70 6. Draw a cross through the pairs of sets that are the same. Set A = (book, pen, pencil) Set M = (dog, cat, mouse) Set B = (pen, pen c i l , book) Set N = (dog, cat, rat) Set 1 = (a,b,c,d) Set S = (A,B,C,D) 7 . Name the set ( 1 , 2 , 3 ) i n 5 di f f e r e n t ways. ( ) ( ) ( ) ( ) ( ) 8 . How many diff e r e n t subsets can be found i n the set ( 1 , 2 ) ? Answer 9. Write 5 d i f f e r e n t subsets of the set of names (Joe, B i l l , Henry, Jim). A = ( ) D = ( ) B = ( ) E = ( ) C - ( ) 1 0 . L i s t a l l possible subsets of (book, pen, p e n c i l ) . 1 1 . What i s the largest subset that can be found i n the set ( 1 , 2 , 3 , 4 , 5 ) ? Answer 1 2 . What i s the smallest subset that can be found i n the set ( 1 , 2 , 3 , 4 , 5 ) ? Answer 71 NO. 2 SETS Name: Divis i o n : Teacher's Name: Write a l l of your answers on t h i s sheet. 1. I f set A = (Jim, B i l l , Joe) and set B = (Dave, Jim). What i s the union of Set A and Set B? 2. What i s the union of (a,b,c,d,e) with (a,b,c,d,e)? 3. What i s (a,b,c) union with the empty set? 4 . Name any set with 4 members i n i t . Name another set with § members i n i t . Now name the union of the two sets. 5. I f set A = (book, pen, pencil) then what i s the union of t h i s set with any subset of i t s e l f ? 6. What i s (Dog, Cat, Cow) union ( B i l l , Joe, John)? 7. I f Set R i s (Tom, Don, Ron) and set T i s (Ron, Don, Tom) then what i s set R union set T? £. Name any set. Then name any other set. Now name th e i r union. 7 2 9. Write three subsets of Set A i f Set A = (book, dog, car). 1 0 . Name the largest subset of (cat, dog, house, car). 1 1 . Name 5- d i f f e r e n t subsets of the set (a,b,c,d). 1 2 . How many members are there i n an empty set? 1 3 . From set (a,b,c,d,e) l i s t a l l the subsets that have exactly four members. What i s the union of any two of them? 1 4 . I f set S union T equals set T, then what do you know about set S? 73 NO.. 3 SETS Name: Divis i o n : Teacher's Name: Write a l l of your answers on t h i s sheet. 1. Universal set "U" = (cup, knife, spoon, plate, t a b l e ) . a. Give the complement of the set (cup, spoon). b. I f set A = (plate, table) what i s set A? c. I f set B has members (cup, knife, spoon, plate) what i s the complement of B? d. The complement of set C i s (kni f e ) . What i s set C i t s e l f ? 2. Given U = (a,b,c,d,e,f) f i n d the complement of each of the sets l i s t e d below. (a,b,c) ( ) (a,b,c,d,e,f) 3. I f a universal set i s given as U = (car, bus, t r a i n ) . What i s U? What i s the complement of the empty set? 4. A universal set - (1,3,5,7,9) and set B = (1,9). What i s set B? 74 5. Name a universal set with 5 members. Name 3 subsets of i t . Give the complement of each subset. U = ( ) A = ( ) A - ( ) B = I ) B = ( ) C = ( ) C = ( ) 6 . Set U = (Ray, Ron, Don, B i l l , Jim). Set A = (Ray, Ron) Set A = (Don, B i l l , Jim) Set B - (Don, B i l l , Jim) Set B = (Ray, Ron) Set C = (Ray, Ron, B i l l , Jim) Set C = (Don) Find: a. the union of set A and set B. b. B union C. c. the union of set A and Set C. d. the union of set A and set C. e. the union of set B and set A. 7. What i s the union of two empty sets? Name 4 d i f f e r e n t subsets of the set (A,B,C) NO. 4 SETS Name: Divis i o n : Teacher 1s Name: Write a l l of your answers on t h i s sheet. 1 . I f set A = (boat, water, f i s h , man) and set B = (water, car) then what i s set A int e r s e c t i o n set B? 2 . What i s ( 1 , 2 , 3 , 4 , 5 ) i n t e r s e c t i o n ( 1 , 3 , 5 , 7 , 9 ) ? 3 . What i s (a,b,c,d,e) i n t e r s e c t i o n (f,g,h,i)? 4 . What i s (book, pen, pencil) i n t e r s e c t i o n (pen, p e n c i l , book)? 5 . What i s (dog, mouse, cat) in t e r s e c t i o n with the empty set? 6 . I f set R = ( 1 , 5 , 7 , 1 1 ) and set B - ( 2 , 4 , 5 , 6 , 7 ) then what i s R union B? What i s R inte r s e c t i o n B? 76 7. A certain universal set i s (l,m,n,o,p). Give the complements of (l,m,n,o), (n), (p). a. b. c. 8. Set A i s (dog, cat). Set A i s (man, boy, g i r l ) . a. What i s the universal set? b. What i s A union A? c. What i s A i n t e r s e c t i o n A? 9. Given set A = (1,5,7,9), set B = (2,4,6), set C = (1,2,3,4,5). Find each of the following. a. A union B d. A int e r s e c t i o n C b. A in t e r s e c t i o n B e. B union C c. A union C f . C interse c t i o n B 10. Name two d i f f e r e n t sets. Give t h e i r union and then give t h e i r i n t e r s e c t i o n . A = ( ) B = ( ) a. A union B = b. A interse c t i o n B = 11. Set A union B equals set B. Set A in t e r s e c t i o n set B equals set A. What do you know about sets A and B? 77 12. Set R = (a,b,c, ... x,y,z). Set S = (a,b,c). Set T = (a,c,e,f). a. S i n t e r s e c t i o n T = b. S inte r s e c t i o n R = ,— c. R inte r s e c t i o n T = APPENDIX IV The Additional Assignments Worksheet 1 MATHEMATICS 8 Name: A . We agree to perform m u l t i p l i c a t i o n f i r s t and then addition - unless we are shown d i f f e r e n t l y by brackets. e.g. 3+2*4 = 11; 3+2x4 f 20 but (3+2)x4 = 20 Perform the following operations! a. 3+4x3 = . g. 7+2x3 = b. 8+4x3+2 = h. (3+4)x3 = c. (3+2)x(4+3) = i . 9x3+3 = d. (3+l)x3x2 = j . £+£xi = e. ( i + 2)xs - k. 3+0x2 = f . 19x14x0x2-1-4 = 1. 2 + i x 2 = B. In each of the following choose the symbol that represents the largest number. a. 1/4, 1/8, 5/16 f . 35/100, 3/10, 2/10 b. 1/2, 2/4, 9/16 g. 3/10, 4/10, .39 c. . 3 , .1 , . 0 9 h. 17/100, .18, 2/10 d. . 2 , . 0 2 , 2.0 i . 2/15, 2 . 0 , .2 e. 1 .5, .51, 5 . 1 . J. 131/132, 124/125, 1 C. Find the missing d i g i t ( s ) . a. 370 + 20_ - 573 c. 223+_0_ = 524 b. 6 + 3_2 = 485 d. 321+ - 462 Find the value of the S22 x2P PPP 2TT 2TTP l e t t e r s : 3A3 2A F3C2 BDB  D232 81 Worksheet 2 MATHEMATICS 8 Name: A. Use the symbols > (greater than) and < (less than) to show the relationship between the following pairs of numbers. a. 3 4 d. 3+6 7+3 g. .6 .06 b. 8 9 e. 8+4 4+7 h. .3+.1 5/10 c. 1/2 1/4 f . 1/2+1/4 1/4+1/4 i . .73 7/10+3/10 B. Complete the following. a. ' 130 + 225 = _00 + 0 +• b. 242 + 306 = _00 + _0 +• c. 371 + 121 ? _00 + _0 +• d. 4_0 + 36_ = 700 + 80 +• 3 e. + 261 = 300 + 60 + 3 f . 204 + = 400 + 0 + 9 g. 121 + 222 + 420 = _00 + _0 +• C. In an arithmetic modulo 4> the number 4 would be c a l l e d a 0 number. What would the following be called? a. 7 b. 9 c. 2 d. 5 In a modulo 3 system, what would the following be called? 82 D. Indicate which symbol should come next i n each of the following. a. 2, 4, 6, 8, 10, t>. 1. 3, 5, 7, 9, 11, c. 1, 5, 9, 13, 17, d. 2, 3, 5, 7, 11, 13, 17, e. 0, 1, 3, 6, 10, 15, 21, 1. 3, 9, 27, 83 Worksheet 3 MATHEMATICS 8 Name: A. In each of the following decide which number comes between the other two i n si z e , (e.g. with 1, 3, % then 2 i s between 1 and 3.) a. 1/2, 1, 1/4 b. .3, .33, .333 . c. .7, .6, .65 d. 1/8, 3/16, 3/32 T T T e. 2, 6", 4 (T fi 0) f . 1/4, 1/8, 3/8 g. .5, .8, . 4 h. .9, 18, .81 i . .3, .36, .37 j . Kx2, Kx6, Kxl (K / 0) B . Write a numeral, i n each of the following bases, that w i l l represent the number of dots i n the box. a. base 10 b. base 9 c. base 8 d. base 7 e. base 6 f . base 5 g. base 4 h. base 3 i . base 2 Present day electronic computors use base 2 to perform calcul a t i o n s . Solve each of the following i n base 2. Addition Subtraction M u l t i p l i c a t i o n a. 1011 c. 11011 e. 10110 1101 10110 10' b. I l l d. 111111 f . 11001 101 11110 11 011 100 8 5 Worksheet 4 MATHEMATICS 8 Name: A. Use only the symbols "<n, ">", or " = " to make true statements with the following. a. 1 / 8 1 / 7 e. 19/2 28 /3 i . T T 3 4 , TXO b. 7 / 1 5 1/2 f . 1 / 5 7 8 1 / 5 7 9 j . 1 / 1 0 0 0/2 c. 0/2 0 / 3 g. 3 / 4 4 5 / 6 0 k. 4 / 1 0 / 1 0 d. 2 / 3 3 / 5 h. 6 / 6 8 / 8 1 . 5 0 / 5 1 5 1 / 5 0 B. In our number system we use place value. In the follow-ing numerals write the value of the underlined d i g i t s . a. 3 1 3 e . 7 6 2 i . 4 2 1 b. 2 3 3 f. 2 2 8 j . 5 6 6 . 2 c. 3 3 - 2 g. 2 4 1 k. 222 d. 5 . 0 5 . h. 4 . 0 6 1 . j>04 C. Find answers to the following. (Remember the order of operations.) a. 1/2 4- 3 / 4 x 4 / 3 « e. (3/2 + 7/2) x 3 = b. 13 + 4 / 8 x 2 = f . 4 + 7 - 2 x 2 = c. 3 + 7 = g. 8 - 2 - 1 - 3 x 4 = 2 d. 5 + 6 x 3 = h. 3 x 2 3 = D. Perform the following operations i n Base 4 . Add 13 Subtract 202 Multiply 12 13 _ ? J _ 1 22 86 Worksheet 5 MATHEMATICS 8 Name: A. The d i s t r i b u t i v e rule i s stated; a x(b+c) = ab + ac. This means that 2(3+4)-2x3+2x4=6+8=14. Use the d i s t r i b u t i v e p r i n c i p l e to expand the following, a. 3(8+3) = x -r x - + s  b. 8(4+5) = x +- x - -f = c. hilH) = x +- x = + = d. _(2+6) = x + x = -4- = e. M(R+V) = x + x = -+- = B. Find answers to the following by using the d i s t r i b u t i v e and associative p r i n c i p l e s . a. = d. 41.7+(48.3+22.9) = b. 9x81+9x19 = e. 13x99+13x1 = c. 3.7+5.2-t-4.3+-5.S = f . 281x99 = (hint 99=100-1 C. Complete the following i l l u s t r a t i o n s of the d i s t r i b u t i v e p r i n c i p l e . a. 5(6+3) = x 6+5 x = 30+ = 45 b. 4 ( 3 + _ J = 4 x + 4 x = + 28 = 40 c 3(5+7) = x +• x - _ + • = d. 4(__+__) = 4 x + 4 x = 20+-24 = e. (3+D = 5 x + x =15+-5 = 87 Worksheet 6 A. MATHEMATICS 8 Name: Decide whether the following numerals represent even (E) or odd (0) numbers. Do not write the sums or products themselves. f . 41 + 22 4- 463 g. 53 +• 46 + 886 h. 570 x 881 x 923 i . 37 x 41 x 13 j . 83 x 40 x 22 a. 330 + 10793 b. 74 x 3028 c. 41 + 3749 d. 999 x 333 e. 886 x 23 B. L i s t a l l the d i v i s o r s of the following, e.g. 56: 1, 2, 4, 7, 8, 14, 28, 56 a. 81: b. 48: c. 29: d. 72: e. 144: c. L i s t only the prime divi s o r s of the following, (1 i s not a prime number.) a. 15 b. 24 c. 48 d. 29 e. 56 f . 36 g. 42 h. 50 i . 75 j . 90 88 D. Write each of the following as a product of t h e i r prime d i v i s o r s . a. 18 = e. 25 = b. 27 = f . 32 = c. 49 = g. 24 = d. 58 = h. 60 = APPENDIX V The Transfer Test MATHEMATICS 8 Name: Part I Div i s i o n Teacher's Name: Write a l l answers on t h i s sheet i n the spaces provided. Rough work may be done on t h i s sheet. 2. Write the set of odd numbers les s than 8. 3. Name any subset of (a,b,c,d,e). 4. Out of 19 people i n a room 11 smoke and 12 wear glasses. I f 4 smokers wear glasses then: a. How many smokers do not wear glasses? b. How many non-smokers wear glasses? 5. Out of 23 men at a party 19 wear black shoes and 11 wear red socks. I f 12 men wear black shoes but do not wear red socks then: a. How many men wear black shoes and red socks? b. How many wear red socks but do not wear 6. Out of a l l the people at a party 8 smoke and 15 chew gum. Exactly 5 people do both. a. How many people smoke or chew gum? b. How many people smoke but do not chew gum? 1. Is set (1,5,7) = (7,1,5)? black shoes? 7. A certain night school class has 18 pu p i l s . In t h i s group of pupils 13 drive a car and 9 pupils smoke. Some do both. How many pupils do not drive a car? 8. In a room f u l l of boys exactly 11 play f o o t b a l l and exactly 9 play basketball. Exactly 6 boys play both sports. Exactly 5 boys do not play either game. a. How many boys are i n the room? b. How many boys play f o o t b a l l but not basketball? c. How many boys play f o o t b a l l or basketball? 9. What i s ( 1 ,3 ,5 ,7 ,9 )0 (2 ,3 ,5 ,7)? 10. Find (2 , 3 , 4 , 5 ) U (1 ,4 ,3,12). 11. Ten men entered a restaurant to have something to eat and drink. Four men had coffee, the rest had milk. Seven men had hamburgers, the rest had chips. Exactly 2 of the men that had coffee had hamburgers also. a. How many men had coffee and chips? b. How many had hamburgers and milk? c. How many men had chips and milk? 12. Find (a,b,c)U empty set. 92 13. A certain number of men work together i n an o f f i c e . Exactly 7 men smoke. Exactly 3 men drive Ford cars. Exactly 4 men play ginlf. Exactly 3 of the 1 smokers drive Fords. Exactly 4 of the 1 smokers play g o l f . Exactly 1 of the men who golfs drives a Ford. a. How many men smoke or drive Fords or play golf? b. How many men smoke and drive a Ford and play golf? 14. B i l l has 14 marbles i n his hand; some have two colours and some have only one colour. There are 3 that have green on them. There are 8 that have blue on them. There are 9 that have red on them. No marble has any other colour. No marble has blue and green together. A l l except 1 of the green marbles has some red on them. There are 3 pure red marbles. a. How many are red and blue? b. How many are pure blue? c. How many are red and blue or are red and green? 15. How many members has the empty set? 93 16. Twenty passengers enter an s i t next to windows. Four of the bus do not s i t next people on the l e f t side of window. a. How many people do not b. How many people on the a window? c. How many people on the next to a window? d. How many people s i t on bus? empty bus and 12 people people on the right side to windows. Seven the bus s i t next to a s i t next to a window? right s i t next to l e f t do not s i t the l e f t of the 17. Find ( ).n (1,3,5) 94 MATHEMATICS 8 Name: Part II Divi s i o n : _, Teacher's Name: ANSWERS 1. Set M has 1 members. Set N has 8 members. The union of the two sets has 13 members. How many members has the interse c t i o n of the two sets? 2. Set P has 9 members and set Q has 6 members. The int e r s e c t i o n of the two sets has 2 members. How many members has the union of the two sets? 3. Set E has 5 members and i t s complement, E, has 9 members. I f set F has 4 members then how many members has set F? 4. Set A has 4 members. Set B has 4 members. Set C has 5 members. Set A interse c t i o n set B has 1 member. Set A int e r s e c t i o n set C has 0 members. Set B int e r s e c t i o n set C has 1 member. a. How many members are i n A int e r s e c t i o n B inte r s e c t i o n C? b. How many members are i n A union B union C? 95 5. Set X has 6 members and i t s complement, X, has 8. Set Y has 7 members and i t s complement, Y, has 7. Set X int e r s e c t i o n Y has 2 members. a. How many members are there i n the universal set? 6. Set A has 5 members and i t s complement, A, has 4 members. Set B (complement of B) has 7 members. What i s the largest number of members of Set A that could also be i n Set B? APPENDIX Tables Table VI-1: P r e t r i a l Scores Obtained on Part I- of the Transfer Test Part A: Unverbalized Awareness Group Student Item Number Number 4a 4b 5a 5b 6a 6b 7 8a 8b 8c 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c I6d 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1 10 1 1 1 1 1 1 1 11 1 1 1 1 1 12 1 1 1 1 1 13 1 1 1 14 1 1 1 1 1 15 1 1 1 1 1 16 1 1 1 1 17 1 1 1 18 1 1 1 19 1 • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 21 1 1 22 1 1 23 1 1 24 1 25 1 Table KI-1 (Continued) Part B: Verbal Reception Group Student Item Number Number 4a 4b 5a 5b 6a 6b 7 8a 8b 8c 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c l 6 d 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 4 1 1 1 1 1 5 1 1 1 1 1 6 1 1 1 1 1 1 7 1 1 1 1 1 1 1 8 1 1 1 1 1 9 1 1 1 1 10 1 1 1 1 11 1 1 1 1 1 12 1 1 1 1 1 13 1 1 1 14 1 1 1 15 1 1 1 1 16 1 1 1 1 17 1 1 1 18 1 1 1 19 1 1 1 20 1 1 1 21 1 1 1 22 1 1 23 1 1 24 1 1 25 1 1 26 1 27 1 28 1 29 1 30 1 1 1 1 • 1 • • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Table VT-2: i P r e t r i a l Scores Obtained on Part II of the Transfer Test Part A: Unverbalized Awareness Group Student Item Number Number 1 2 3 4a 4b 5b 6 1 1 1 1 1 1 1 2 1 1 1 1 1 1 3 1 1 4 1 5 1 1 6 1 1 1 7 1 1 1 8 1 1 1 9 10 1 1 1 11 12 1 1 1 1 13 1 1 14 1 15 1 16 1 1 17 •18 19 1 20 1 21 1 1 22 1 1 1 23 1 24 Table VI-2 (continued) Part B: Verbal Reception Group Student Item Number Number 1 2 3 4a 4b 5a 5b 6 1 1 1 2 1 1 1 1 3 1 1 4 1 1 1 5 1 l 1 6 1 7 8 1 1 9 1 1 10 1 11 1 12 l 1 13 14 15 1 1 16 1 1 17 18 l 19 20 1 1 1 21 22 23 1 1 24 1 1 1 25 l 26 1 27 2 1 28 1 29 1 1 30 1 1 l 101 Table V I - 3 : Item Validity Indices (Flanagan1s r) for the T r i a l Run Section of the Item Unverbalized Verbal Reception Transfer Test Awareness Group Group •I 4a • .40 .31 ! I 4b .73 .94 I 5a .60 .38 I 5b .83 .76 I 6a .14 0 I 6b .81 .85 I 7 .40 .53 : I 8a .40 .18 I 8b .69 .85 I 8c .41 .52 I 11a .53 .53 I l i b .70 .73 I 11c .60 .43 I 13a 0 .52 : i 13b .40 .65 i 14a 0 0 i 14b 0 0 ! I 14c .68 .52 I 16a .61 .53 I 16b .86 .83 I 16c .76 .76 I I6d .52 .76 I I 1 .90 .89 II 2 .94 .85 ' II 3 .75 .65 II 4a .55 .52 II 4b .69 .65 II 5a .85 .73 II 5b 0 0 • II 6 .75 .52 Table VI-4: Experimental scores on Part I of the Transfer Test Part A: Unverbalized Awareness Group. Student Number 4a 4b ? a 6a 6b 7 8a 8b 8c Item Number 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c I6d 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 1 1 <7 1 1 1 1 1 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1 1 1 1 1 10 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 13 1 1 1 1 1 1 1 1 1 14 1 1 1 1 1 1 1 1 15 1 1 1 1 1 1 1 1 16 1 1 1 1 1 1 1 17 1 1 1 1 1 1 1 18 1 1 1 1 1 1 19 1 1 1 1 1 1 20 1 1 1 1 1 21 1 1 1 1 22 1 1 1 1 23 1 1 24 1 1 1 1 1 1 1 1 1 1 1 25 1 1 1 1 1 1 1 1 1 2$ 1 1 1 1 1 1 27 1 1 1 1 1 1 28 1 1 1 1 1 29 1 1 1 1 30 1 1 1 1 Table V I - 4 (Continued) Part A: Unverbalized Awareness Group. Student Number 4a 4b 5a 5b 6a 6b 7 8a 8b 8c 11a Item Number l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c l 6 d 31 l i i X X 32 1 1 1 1 33 1 1 1 34 1 1 35 1 1 36 1 1 37 1 38 39 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 40 1 1 1 1 j . 1 1 1. 1 1 1 1 1 1 1 41 1 1 1 l 1 1 1 1 1 1 1 1 42 1 1 1 1 1 1 1 1 43 1 1 1 1 1 1 1 44 1 1 1 1 1 45 1 1 1 1 1 46 1 1 1 1 1 47 1 1 1 1 1 48 1 1 1 l 49 1 1 1 1 50 1 1 1 1 51 1 1 1 1 52 1 1 1 1 53 1 1 1 1 54 1 1 1 55 1 1 1 56 1 1 57 1 58 1 59 1 1 1 1 1 l 1 1 1 1 1 1 1 1 1 60 1 1 1 1 1 l 1 1 1 1 1 1 1 Table VI -4 (Continued) Part A: Unverbalized Awareness Group. Student Item Number Number 4a Lb 5a 5b 6a 6b 7 8a 8b 8c 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c l 6 d 61 1 1 •1 1 1 1 1 1 1 1 1 - 1 62 1 1 1 1 1 1 63 1 1 1 1 1 1 1 1 1 64 1 1 1 1 1 1 1 1 65 1 1 1 1 1 1 1 1 66 1 1 1 1 1 1 1 1 67 1 1 1 1 1 1 1 1 68 1 1 1 1 1 1 1 69 1 1 1 1 1 1 70 1 1 1 1 1 1 71 1 1 1 1 1 1 72 1 1 1 1 1 1 73 1 1 1 1 1 74 1 1 1 1 1 75 1 1 1 1 1 76 1 1 1 1 1 77 1 1 1 1 78 1 1 1 1 79 1 1 1 1 80 1 1 1 1 81 1 1 1 82 1 1 83 1 1 84 1 1 85 1 1 Table VI - 4 (continued) Part B: Verbal Reception Group. Student Item Number Number 4a 4b 5a 5b 6a 6b 7 8a 8b 8c 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c I6d 86 1 1 1 •1 1 1 1 1 . i . i 1 1 • 1 1 .1 . 1 1 • 1 1 37 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 88 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 89 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 90 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 91 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 92 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 93 1 1 1 1 1 1 1 1 1 1 1 1 1 94 1 1 1 1 1 1 1 1 1 1 1 1 95 1 1 1 1 1 1 1 1 1 1 1 1 96 1 1 1 1 1 1 1 1 1 1 1 97 1 1 1 1 1 1 1 1 1 1 1 98 1 1 1 1 1 1 1 1 1 1 99 1 1 1 1 1 1 1 1 1 1 100 1 1 1 1 1 1 1 1 1 101 1 1 1 1 1 1 1 1 102 1 1 1 1 1 1 1 1 103 1 1 1 1 1 1 1 1 104 1 1 1 1 1 1 1 105 1 1 1 1 1 1 1 10.6:: 10? ' 1 1 1 1 1 1 1 1 1 1 1 1 108 1 1 1 1 109 1 1 1 1 110 1 1 1 111 1 1 112 1 1 113 1 1 114 1 1 1 1 1 1 1 1 1 1 1 1 1 1 115 1 1 1 1 1 1 1 1 1 1 1 1 1 Table VI -4 (Continued) Part B: Verbal Reception Group. Student Item Number Number 4a 4b 5a 5b 6a 6b 7 8a 8b 8c 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c I6d 116: 1 1 1 1 1 1 1 1 1 1 1 1 117 1 1 1 1 1 1 1 1 1 1 1 1 118 1 1 1 1 1 1 1 119 1 1 1 1 1 1 1 120 1 1 1 1 l 1 1 121 1 1 1 1 1 1 122 1 1 1 1 1 123 1 1 1 1 124 1 1 1 1 125 1 1 l 1 126 1 1 1 127 1 1 1 128 1 1 1 129 1 1 1 130 131 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 132 1 1 1 1 1 1 1 1 1 1 1 1 1 1 133 1 1 1 1 1 1 l 1 1 1 1 1 1 134 1 1 1 1 1 l 1 1 1 1 1 1 135 1 1 1 1 1 1 1 1 1 1 1 1 136 1 1 1 1 1 l 1 1 1 1 1 1 137 1 1 1 1 1 1 1 1 1 1 1 138 1 1 l 1 1 1 1 1 1 1 139 1 1 1 1 1 l 1 1 1 1 140 1 1 1 1 1 1 1 1 1 1 141 1 1 1 1 1 1 1 1 1 1 142 1 1 1 1 1 1 1 1 1 143 1 1 1 1 1 1 1 1 144 1 1 1 1 1 1 1 1 145 1 1 1 1 1 l 1 1 Table VI -4 (Continued) Part B: Verbal Reception Group. Student Item Number Number 4a 4b 5a 5b 6a 6b 7 8a 8b 8c 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c l 6 d 146 1 1 T~ ~ 1 1 1 1 1 147 1 1 1 1 1 1 1 ' 148 1 1 1 1 1 1 1 149 1 1 1 1 1 1 150 1 1 1 1 151 1 1 1 1 152 1 1 1 1 153 1 1 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 1 1 171 172 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Table VI - 4 (Continued) Part C: Control Group. Student Number 4a 4b 5a 5b 6a 6b 173 1 174 1 175 176 1 1 l 1 1 177 1 1 178 1 1 179 1 1 l 180 1 1 1 181 1 182 1 1 1 l 183 1 1 184 1 1 185 1 1 186 187 1 1 1 l 188 1 1 1 189 1 1 190 1 191 1 1 1 192 1 1 193 1 1 1 194 1 1 1 1 195 1 1 1 l 1 196 1 1 197 1 198 1 1 1 199 200 1 l 1 201 1 l 202 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l l l 1 1 1 1 1 1 1 l l l l 1 1 1 1 1 1 1 l l l l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Table VI-4 (Continued) Part C: Control Group. Student Number 4a 4b Item Number 8a 8b 8 c 11a l i b 5b 6a 6b 7 11c 13a 13b 14a 14b 14c 16a 16b 16c l 6 d 203 1 1 1 . 1 1 204 1 1 1 1 1 1 20$ 1 1 1 1 1 1 206 1 1 1 207 1 1 1 1 1 1 208 1 1 1 1 1 1 1 1 1 1 1 1 1 209 1 1 210 1 211 1 1 1 1 1 1 1 1 1 212 1 1 1 1 1 1 1 1 1 1 213 1 214 1 1 1 215 1 216 217 1 1 1 1 1 1 1 1 1 1 1 1 218 1 1 219 1 220 1 1 221 1 1 1 222 1 1 1 1 1 1 1 1 1 1 1 1 1 1 223 1 1 224 1 1 1 1 1 225 1 1 226 1 1 1 1 1 1 1 1 1 1 1 227 1 1 1 1 1 1 1 228 1 1 1 1 1 1 1 1 1 1 1 229 1 1 1 230 1 1 231 1 232 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Table VI-4 (Continued) Part C: Control Group. Student . Item Number Number 4a 4b 5a 5b 6a 6b 7 8a 8b 8c 11a l i b 11c 13a 13b 14a 14b 14c 16a 16b 16c I6d 233 1 1 1 1 1 1 1 1 1 1 234 1 1 1 1 1 1 1 235 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 236 1 1 1 1 237 1 1 1 1 1 1 1 1 1 1 1 1 1 238 1 1 1 1 1 1 1 1 1 1 1 1 1 239 240 1 1 1 1 1 I l l Table VI-5: Experimental Scores on Part II of the Transfer Test Part A: Unverbalized Awareness Group. Student Item Number Number 1 2 3 4a 4b 5a 6 .1 . . 1 1 . 1 . i 2 1 1 1 1 1 3 1 4 1 1 1 1 1 1 5 1 1 1 1 6 1 1 7 1 1 1 1 1 1 8 1 1 9 1 1 1 1 1 1 1 1G 1 1 11 1 1 12 1 1 1 13 1 1 14 1 1 1 1 1 1 15 16 1 1 1 17 1 1 18 1 1 1 19 1 1 1 1 20 1 1 1 21 22 23 1 1 24 1 1 25 1 1 26 1 1 27 28 1 1 29 30 1 31 1 1 32 1 1 33 1 1 34 35 1 36 1 37 38 39 1 1 1 1 1 40 1 1 1 41 1 1 1 42 43 1 1 1 44 1 45 112 Table VI -5 (Continued) Part A: Unverbalized Awareness Group. Student Item Number Number 1 2 3 4a 4b 5a 6 / 346 • .1 47 1 1 48 1 1 1 49 50 51 1 1 52 1 53 1 54 1 55 56 57 1 58 1 59 1 1 1 1 1 60 1 1 61 1 1 1 62 1 1 1 1 1 63 1 1 1 64 1 1 65 1 1 66 1 1 67 1 s l 68 1 1 69 1 1 70 1 1 1 71 1 72 1 1 1 73 74 1 1 75 1 1 1 76 1 1 77 1 1 78 1 1 79 1 1 80 1 1 81 1 1 82 1 1 83 1 1 84 1 1 1 85 1 1 113 Table 71-5 (Continued) Part B: Verbal Reception Group. Student Number T f om MnmV.oy» 1 2 3 4a 4b 5a 6 86 1 1 1 1 37 1 1 1 1 1 88 1 1 1 1 89 1 1 1 1 1 90 1 1 1 1 1 91 1 1 1 1 1 92 1 1 93 1 1 94 1 1 1 1 1 1 95 1 1 1 96 1 1 97 1 1 1 1 98 1 1 99 1 1 1 1 1 100 1 1 1 101 1 1 102 1 1 1 103 1 1 1 1 104 1 1 1 1 105 1 1 1 1 106 107 1 1 108 1 109 1 1 1 110 111 1 1 112 1 1 1 1 113 1 1 1 114 1 1 115 1 116 1 1 1 1 1 117 1 1 118 1 1 1 1 1 119 1 1 120 1 1 1 121 122 1 123 1 124 1 1 125 1 1 1 126 1 1 127 128 1 129 1 1 1 1 130 1 1 1 131 1 1 1 1 1 132 1 1 1 133 1 1 1 1 4 Part B: Verbal Recept Student Number 1 2 1 3 4 1 1 1 3 5 1 1 136 1 1 1 3 7 1 138 1 1 3 9 1 1 140 1 1 141 1 1 1 4 2 1 1 4 3 1 1 1 4 4 1 1 1 4 5 1 1 146 1 1 147 1 1 148 1 1 1 4 9 1 1 150 1 151 1 5 2 1 1 1 5 3 1 1 1 5 4 1 1 1 5 5 1 1 156 1 1 1 5 7 1 1 158 1 1 1 5 9 1 160 161 1 1 162 1 1 163 1 1 164 1 1 1 6 5 1 1 166 1 I 6 7 1 1 168 1 169 170 1 1 7 1 1 1 172 1 1 Table VI-5 (Continued) Item Number J 4a 1 1 4b 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 115 Table VT-6: Scores for Analysis of Covariance Part A; Unverbalized Awareness Group. Student Mental Age Raw Score:Christmas Transfer Test Number (years-months) Mathematics Test Part 1 Part II Total 1 15-4 781 15 4 19 2 18-0 82 15 5 20 3 19-0 69 13 1 14 4 18-0 84 13 6 19 5 19-6 81 13 4 17 6 18-0 67 12 2 14 7 15-6 79 12 6 18 8 18-0 75 11 2 13 9 19-6 91 10 7 17 10 15-1 74 10 2 12 11 16-6 49 9 2 11 12 14-6 50 9 3 12 13 17-3 67 9 2 11 14 13-4 81 8 6 14 15 15-3 49 8 0 8 16 17-3 60 7 3 10 17 18-6 62 7 2 9 18 14-4 61 6 3 9 19 16-0 64 6 4 10 20 16-0 60 5 3 8 21 16-6 73 4 0 4 22 18-0 70 4 0 4 23 15-9 48 2 2 4 24 14-6 46 11 2 13 25 14-0 41 9 2 11 26 14-9 44 6 2 8 •27 15-1 52 6 0 6 28 14-0 76 5 2 7 29 14-9 44 4 0 4 30 15-1 59 4 1 5 31 14-6 63 4 2 6 32 12-9 36 4 2 6 33 14-4 46 3 2 5 34 13-3 47 2 0 2 35 14-1 46 2 1 3 36 15-6 37 2 1 3 37 14-6 48 1 0 1 38 14-1 53 0 0 0 39 13-9 83 18 5 23 40 15-8 38 14 3 17 41 14-0 59 12 3 15 42 13-9 45 8 0 8 43 13-0 58 7 3 10 6 44 13-4 59 5 1 45 15-2 48 5 0 5 116 Table VI -6 (Continued) Part A: Unverbalized Awareness Group. Student Mental Age Raw Score:Christmas Transfer Test imber (years-months) Mathematics Test Part I Part II Total 46 14-2 79 • 5 - 1 6 47 15-2 51 5 2 7 48" 14-11 42 4 3 7 49 14-10 45 4 0 4 50 13-6 46 4 0 4 51 13-1 54 4 2 6 52 13-11 49 4 1 5 53 13-2 40 4 1 5 54 13-3 53 3 1 4 55 13-6 41 3 0 3 56 13-1 44 2 0 2 57 14-1 59 1 1 2 58 13-3 50 1 1 2 59 14-0 71 15 5 20 60 13-0 68 13 2 15 61 13-6 65 11 3 14 62 13-2 75 10 5 15 63 13-3 58 9 3 12 64 13-11 63 8 2 10 65 14-1 60 8 2 10 66 14-4 67 8 2 10 67 13-0 69 8 2 10 68 13-8 52 7 2 9 69 16-4 46 6 2 8 70 13-7 70 6 3 9 71 14-0 44 6 1 7 72 14-9 54 6 3 9 73 14-0 56 5 0 5 74 14-4 48 5 2 7 75 13-3 47 5 3 8 76 14-1 54 5 2 7 77 14-9 65 4 2 6 78 13-9 39 4 • 2 6 79 13-5 48 4 2 6 80 14-2 47 4 2 6 31 13-1 65 3 2 5 82 13-6 68 2 2 4 83 13-0 43 2 2 4 34 13-4 36 2 3 5 85 13-2 49 2 2 4 117 Part B: Table VI-6 (Continued) Verbal Reception Group. Student Number 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 Mental Age rears-months! 17-t 17-3 17- 9 18- 6 15- 6 16- 0 14- 6 17- 9 16- 9 18- 0 15- 4 15-0 15-3 17- 6 17- 9 15- 1 16- 3 18- 6 15-9 18- 3 19- 3 15- 9 14- 9 16- 9 15- 0 13- 9 15-4 14- 4 14-9 14-1 17- 9 14- 3 17-6 15- 0 16- 3 14- 6 15- 3 14-6 13- 7 14- 6 14- 4 15- 6 16- 6 16-0 Raw Score:Christmas Mathematics Test 92 75 87 65 87 86 64 87 87 69 81 71 73 75 65 60 84 78 58 91 80 80 76 74 58 75 37 54 67 44 82 63 57 60 62 34 56 45 31 45 48 51 67 64 Transfer Test Part I Part II Total 19 • • 4 23 18 5 23 17 4 21 16 5 21 15 5 20 15 5 20 15 2 17 13 2 15 12 6 18 12 3 15 11 2 13 11 4 15 10 2 12 10 5 15 9 3 12 8 2 10 8 3 11 8 4 12 7 4 11 7 4 11 7 0 7 5 2 7 4 1 5 4 3 7 3 0 3 2 2 4 2 4 6 2 3 5 14 2 16 13 0 13 12 6 18 11 2 13 8 5 13 7 2 9 7 3 10 6 0 6 5 1 6 4 1 5 4 2 6 4 3 7 3 2 5 3 0 3 3 1 4 3 3 6 118 T a b l e V I - 6 (Continued) P a r t B: V e r b a l R e c e p t i o n Group. •Mental Age S t u d e n t Number Raw S c o r e : C h r i s t m a s T r a n s f e r T e s t M a t h e m a t i c s Test P a r t I P a r t I I T o t a l 130 14-6 31 0 .3. . ... . .3 131 13-7 80 18 6 24 132 14-3 69 14 3 17 133 13-3 61 13 2 15 134 13-3 78 12 7 19 135 13-3 63 12 4 16 136 13-8 70 12 4 16 137 12-11 67 11 3 14 138 13-9 77 10 3 13 139 13-3 67 10 2 12 140 13-7 49 10 2 12 141 14-7 67 10 2 12 142 13-11 72 9 3 12 143 13-0 77 8 3 11 144 13-1 77 8 8 2 10 145 13-10 72 4 12 146 13-3 66 8 6 14 147 13-2 54 7 3 10 148 14-6 66 7 3 10 149 13-5 59 6 2 3 150 13-11 60 4 1 5 151 13-9 45 4 0 4 152 13-2 69 4 2 6 153 14-0 62 4 3 7 154 13-6 68 2 2 4 155 13-1 66 15 2 17 156 13-10 57 14 2 16 157 12-11 65 12 3 15 158 13-10 48 11 2 13 159 15-8 56 10 3 13 160 14-7 52 9 2 11 161 15-8 33 8 2 10 162 15-9 63 $ 2 10 163 13-10 43 6 3 9 164 13-3 68 5 4 9 165 13-4 51 6 2 8 166 15-3 51 5 2 7 167 13-3 62 5 2 7 > 168 13-3 52 5 1 6 169 15-5 34 3 0 3 170 14-9 50 3 3 6 171 14-1 51 1 2 3 172 13-8 46 0 2 2 119 Table VI -6 (Continued) Part C: Control Group. Student •Mental Age Raw Score:Christmas Transfer Test Number (years-months) Mathematics Test Part I 173 15-3 - 62 6 174 14-9 75 2 175 11-7 55 1 176 14-6 59 10 177 16-0 62 9 178 15-3 55 3 179 17-6 68 3 180 13-10 61 4 181 14-1 51 1 182 14-9 48 6 183 14-4 61 2 184 17-6 69 10 185 14-4 66 3 186 13-7 40 0 187 13-7 62 6 188 16-0 71 11 189 13-10 40 2 190 15-9 55 3 191 16-0 63 4 $92 14-1 52 2 193 15-1 45 3 194 16-0 67 10 195 17-0 61 10 196 12-10 24 2 197 13-7 45 3 198 17-0 77 10 199 10-11 25 0 200 14-1 61 5 201 13-9 61 5 202 15-0 62 3 203 13-6 46 5 204 17-6 85 6 205 16-6 70 6 206 14-6 46 3 207 15-4 62 6 208 14-1 73 13 209 15-0 57 2 210 14-1 51 1 211 16-0 53 9 212 13-9 57 10 213 11-9 31 1 214 17-0 76 3 215 13-7 36 1 216 15-3 70 0 120 Table VI-6 (Continued) Part C: Control Group. Student Numb er Mental Age (years-months) Raw Score:Christmas Mathematics Test Transfer Part I 217 . . . . . . 15-3 67 12 , 218 13-7 17 2 219 14-9 43 1 220 16-3 39 2 221 15-1 45 3 222 15-6 66 14 223 13-7 43 2 224 15-6 72 5 225 13-7 25 2 226 16-6 43 11 227 14-9 32 7 228 16-6 46 11 229 14-3 36 3 230 15-0 41 2 231 15-3 61 1 232 17-6 88 18 233 16-0 73 10 234 15-1 60 7 235 15-0 54 15 236 15-4 56 4 237 15-9 47 13 238 13-10 56 13 239 15-1 57 0 240 15-9 63 5 121 Table VI-7: Item V a l i d i t y Indices (Flanagan*s r) for the Experimental Run Section of the Item Unverbalized Verbal Reception• f e r Test Awareness Group Group I 4a .58 .73 I 4b .53 .63 I 5a .72 .62 I 5b .57 .76 I • 6a .59 .48 I 6b' .59 .76 I 7 .50 .37 I 8a .73 .76 I 8b .36 .54 I 8c .73 .64 I 11a .74 .79 I l i b .84 .70 I 11c .67 .70 I 13a .48 .48 I 13b .82 .72 I 14a .37 .23 I 14b 0 .53 I 14c .37 .37 I 16a .59 .59 I 16b .59 .88 I 16c .78 .76 I I 6 d .55 .74 II 1 .89 .56 II 2 .69 .51 II . . 3 .74 .69 II. 4a .37 .73 II 4b .83 .73 II 5a .69 .73 II 6 .76 .67 • 

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