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Effects of instruction in groups on individual equation writing Underwood, Barry Richard 1971

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THE EFFECTS OF INSTRUCTION IN GROUPS ON INDIVIDUAL EQUATION WRITING by BARRY RICHARD UNDERWOOD B. Sc. , University of Manitoba, 1965 B. Ed., University of Manitoba, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics Education We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Jul y , 1971 In present ing th i s thes i s in p a r t i a l f u l f i lmen t of the requirements for an advanced degree at the Un iver s i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y ava i l ab le for reference and study. I f u r ther agree that permission for extens ive copying of th i s thes i s fo r s cho l a r l y purposes may be granted by the Head of my Department or by h i s representat ives . It is understood that copying or pub l i ca t i on of t h i s thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of fTlA-TH-g ^ yVT_7CS» t?7- O et-t-T-Toy\J The Un ive r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date 3TVL W ^ I Q 7 / ABSTRACT This study was motivated by the writer's b e l i e f that youngsters do have a tendency to group, and that t h i s propen-s i t y , no matter how emphemeral and v a c i l l a t i n g i t may be at times, should be taken in t o account i n the design of teacher s t r a t e g i e s . Grade four students were assigned to two groups at random, and then, i n one group, subgroups of four students were randomly made up. A l l students were instructed by f i l m loops for three days on wri t i n g an equation f o r a d i v i s i o n problem. On the fourth day of the experiment, the students wrote a c r i t e r i o n t e s t of twenty-five d i v i s i o n problems. The i n v e s t i -gation of student-student i n t e r a c t i o n was done by comparing the e f f e c t s of i n s t r u c t i o n to groups of four students with those of i n s t r u c t i o n to the i n d i v i d u a l l y taught students. A two-tailed t - t e s t was used to t e s t the s i g n i f i c a n c e between the means of the two groups and a F-test was employed to t e s t the dif f e r e n c e i n the variances of the two groups. There was no s i g n i f i c a n t difference between the individual-taught group and the group-taught group i n terms of ei t h e r mean or variance. The conclusion was drawn that the use of small groups to teach students to write equations for d i v i s i o n problems d i d i i not improve the i n s t r u c t i o n . But i t was f e l t that further research using d i f f e r e n t dependent variables i s both warranted and des i r a b l e . i i i TABLE OF CONTENTS CHAPTER Page I. THE PROBLEM 1 Background • • • • • • 1 Statement of the Problem 4 Review of the L i t e r a t u r e 4 Statement of the Hypothesis . 12 I I . THE DESIGN OF THE STUDY 13 Introduction 13 Formation of the Groups 15 The Population 15 The Sample 15 Assignment of Subjects 15 Development of Materials 16 The I n s t r u c t i o n a l Device 16 The Test Instrument 17 Procedure 18 S t a t i s t i c a l Analysis 21 Data ••>••• 21 N u l l Hypotheses . 22 S t a t i s t i c a l Treatment 22 II I . ANALYSIS OF THE RESULTS 24 Testing of Hypotheses 24 Hypothesis One 25 Hypothesis Two 25 i v CHAPTER Page Conclusions 25 Analysis of Ad d i t i o n a l Data 26 Answers on the C r i t e r i o n Test 26 Equations and Answers During Instruction ... 26 IV. IMPLICATIONS OF THE STUDY 28 Introduction 28 Discussion of Conclusions 28 The Content of the Lessons 28 The Method of Instruction 29 Length of Experiment 30 Subgroup Makeup 30 Limitations of the Study 31 The Film Loops 31 A Broader Sample 32 Q u o t i t i v e - P a r t i t i v e Approach .. ... 33 Suggestions for Future Studies 33 Summary 35 FOOTNOTES 36 BIBLIOGRAPHY 39 APPENDIX A . 44 APPENDIX B . 52 APPENDIX C 58 APPENDIX D 63 V LIST OF TABLES TABLE Page I. S t a t i s t i c s for Equation Writing 24 v i LIST OF FIGURES FIGURE Page 1. The E x p e r i m e n t a l P r o c e d u r e 14 v i i ACKNOWLEDGEMENT The author wishes to thank the p r i n c i p a l , teachers, and students who pa r t i c i p a t e d i n the experiment and the members of his thesis committee—Dr. E. D. MacPherson (Chairman), Mr. P. Olley, and Mr. T. B a t e s — f o r t h e i r cooperation and assistance. CHAPTER I THE PROBLEM I. BACKGROUND Teachers have occasionally used concrete objects to introduce or develop c e r t a i n concepts and algorithms. They have probably f e l t that t h i s procedure a s s i s t s the student i n the i n t e r n a l i z a t i o n or abstraction of a concept because he i s able to r e l a t e i t to a r e a l state or action. In deciding on a method of presentation, i t seems reasonable that an i n s t r u c t o r take account of the f a c t that the classroom i s an extremely complex, s h i f t i n g web of interpersonal r e l a t i o n s , and t r y to maximize the components involved which improve learning. The d i s c i p l i n e s of psychology and sociology have provided a wide range of information about the forces that mediate learning i n groups of people. Studies on the e f f e c t s 1 2 of classroom i n t e r a c t i o n , c l i m a t i c influences, discrimina-3 4 . 5 t i o n , motivation, information independence, and communica-t i o n structure** have presented evidence that s i g n i f i c a n t group factors e x i s t . I t seems reasonable that school achievement by i n d i v i d u a l s would be improved i f group influences i n the classroom were better understood and manipulated. 2. I t has often been stated that students learn best i f they p a r t i c i p a t e a c t i v e l y i n the learning process. I t can be argued that group work provides an opportunity for involvement. In a group s i t u a t i o n , the "onus probani" l i e s with the students. Where they cooperatively provide the solution to a problem, i t i s most l i k e l y that opportunities for i n t e r a c t i o n and involve-ment are maximized. In addition, an increase i n group work by students may lead to the achievement of an important goal of education—the development of i n d i v i d u a l s capable of l i v i n g and working with others. Within a group, a student has a chance to experience s o c i a l i n t e r a c t i o n involving a wide range of personality and 7 s k i l l c h a r a c t e r i s t i c s . As Wilhelms stated: The layman and the unsophisticated teacher may and do continue to think of each subgroup as "homogeneous"; the expert knows i t i s rampantly heterogeneous, concealing tremendous ranges on a l l but the one variable chosen as a basis for d i v i s i o n . I t also seems l i k e l y that work i n groups provides an opportunity f o r immediate reinforcement not given i n s t r a i g h t -forward d i d a c t i c presentations v i a lecture or textbook, because the students are able to check t h e i r reasoning step by step with g the other students i n the group. And i f Skinner's dictum "the lapse of only a few seconds between the response and rei n f o r c e -ment destroys most of i t s e f f e c t s " i s even p a r t i a l l y true, then we have a further reason for examining the i n s t r u c t i o n of students i n small groups. 3. 9 A d d i t i o n a l l y , Piaget's model of learning appears to lend i t s e l f to the use of groups. He maintains that to acquire a concept some cognitive reorganization must take place. This readjustment, the e q u i l i b r a t i o n process, centering upon the temporary imbalance between the major functional a c t i v i t i e s of a s s i m i l a t i o n and accommodation, i s usually conceived of as invoking some form of c o n f l i c t . This c o n f l i c t develops between the c h i l d ' s p a r t i a l l y established schemas and new environmental demands which are meaningful to the current l e v e l of psycholog-i c a l functioning. I t seems that peer rel a t i o n s h i p s may be one means of s e t t i n g natural c o n f l i c t forces i n motion to f a c i l i t a t e reorganization of the thinking of the i n d i v i d u a l p a r t i c i p a n t s . Even though group work appears to be valuable, the question remains of whether or not i t lends i t s e l f to work involving material objects. In reply, Piaget"^ seems to argue that they should be combined, for he states: When I say "active," I mean i t i n two senses. One i s acting on material things. But the other means doing things i n s o c i a l c o l l a b o r a t i o n , i n a group e f f o r t . This leads to a c r i t i c a l frame of mind, where ch i l d r e n must communicate with each other. This i s an e s s e n t i a l factor i n i n t e l l e c t u a l development. Cooperation i s indeed co-operation. If he desires that children be engaged i n the learning process, and wishes that "active" have the two senses, then, seemingly, he advocates the concurrent use of material objects and groups. 4. A further j u s t i f i c a t i o n for the concomitant employ-ment of group work and work with material objects i s that material objects serve as a stimulus f o r group i n t e r a c t i o n . They are concrete referents about which discussion can take place. Thus, i t follows that group work with material objects may provide an opportunity for the student to develop unique a b i l i t i e s and s a t i s f y some broader objectives of edu-cati o n . I t appears important that we examine c l o s e l y the factors operating i n group learning, so that they can be optimized. Statement of the Problem Does the teacher strategy i n which physical objects are handled only by the teacher become more e f f e c t i v e with increased student-student interaction? I I . REVIEW OF THE LITERATURE At an e a r l i e r time, many people have been tempted to assert that groups are generally more successful than i n d i v i d u a l s . A number of experimenters, Gournee,"^ Klugman,^" 13 14 Perlmutter and de Montmollin, Hudgins, and Taylor and 15 Faust, have found that groups have produced more correct solutions than comparable subjects working as i n d i v i d u a l s . 5. The g r e a t e r p r o d u c t i v i t y o f g r o u p s , as compared w i t h i n d i v i d u a l s , was a t t i m e s a t t r i b u t e d t o the most c a p a b l e s t u d e n t . That i s , groups were s u p e r i o r , n o t because o f some " s p e c i a l " group e f f e c t s , b u t because group performance a c h i e v e d t h e l e v e l o f i t s most p r o f i c i e n t member. 16 Gournee, who s t u d i e d the e f f e c t i v e n e s s o f c o l l e c -t i v e a c t i o n i n the absence o f a l l v e r b a l i n t e r s t i m u l a t i o n , found t h a t group s c o r e s were s i m i l i a r t o t h e most c a p a b l e s t u d e n t ' s s c o r e . W i t h f i f t y - t h r e e u n d e r g r a d u a t e s i n t h e i n d i v i d u a l s i t u a t i o n and s i x t y - s i x s t u d e n t s i n the group s i t u -a t i o n , h i s c o m p a r i s o n o f i n d i v i d u a l and c o l l e c t i v e judgements r e v e a l e d t h a t group s c o r e s were a p p r o x i m a t e d by the s c o r e s o f the b e s t members. 17 P e r l m u t t e r and de M o n t m o l l i n ' s d a t a a l s o showed t h a t t h e group s c o r e was a d u p l i c a t e o f the s c o r e o f the most p r o f i c i e n t member. W i t h t w e n t y - t h r e e s m a l l groups s t u d y i n g "nonsense" words e i t h e r i n a group f i r s t and t h e n i n d i v i d u a l l y , o r i n d i v i d u a l l y and t h e n i n a group, he found no d i f f e r e n c e between t h e g r o u p - i n d i v i d u a l groups and the b e s t group-i n d i v i d u a l p e r s o n i n each group. But t h e r e were i n s t a n c e s where groups a t t a i n e d b e t t e r s u c c e s s t h a n any component member, a c t i n g a l o n e , was a b l e t o do. 18 For example, F a u s t c o n d u c t e d two e x p e r i m e n t s w i t h s t u d e n t s i n an i n t r o d u c t o r y p s y c h o l o g y c o u r s e . One s t u d y had f i f t e e n groups 6. of four, and forty-one i n d i v i d u a l s , while the other had seven-teen groups of four and sixty-seven i n d i v i d u a l s . In addition, he created nominal groups—groups made up by random assignment from people who had worked i n d i v i d u a l l y , and to which c r e d i t i s given f o r a correct s o l u t i o n i f one or more of the i n d i v i d -uals i n the group solved the problem. On four s p a t i a l problems, the performance of his r e a l four-man groups was well matched by the performance of nominal four-man c o l l e c t i o n s , but on the anagram problems, the r e a l groups solved more problems than did nominal groups. Results having s i m i l i a r s i g n i f i c a n c e are presented 19 by Anderson, based on a task of making as many words as possible from the l e t t e r s " a f l i y b a t " i n a 15 minute time period. Anderson's two and three person groups, composed of junior high students, exceeded the output of the best i n d i v i d u a l s i n com-parable nominal groups. The r e a l groups were equal to the nominal groups when the l a t t e r was credited with a l l the d i f -ferent words produced by the component i n d i v i d u a l s . I t i s noted that t h i s r e s u l t i s e n t i r e l y consistent with a pooling view of the r e a l group's performance, i n that the assumption--i f one or more persons solve a problem then they w i l l be able to convince the others—appears s a t i s f i e d . The author should l i k e to avoid confusion by explaining why the pooling of complementary s k i l l s enables r e a l groups to surpass nominal groups i n Faust's experiment but not i n 7. Anderson's. The reason seems to be rela t e d to the differ e n c e between the tasks of the two experiments. In Anderson's "nominal" groups, c r e d i t was received for every l e g a l word any one of the two or three i n d i v i d u a l s was able to discover; while Faust's task required solving each word i n the l i s t before being allowed to proceed to the next one. A person who might have solved one of the l a t e r words never had a chance to do so i f , working alone, he missed an e a r l i e r one. For t h i s reason, i t appears that Faust's method of giving c r e d i t to a nominal group does not adequately r e f l e c t the po t e n t i a l gains to be derived from pooling of resources i n group a c t i v i t y . 20 In contrast, Davis and Restle found that group per-formance was below the l e v e l of the most p r o f i c i e n t member. This was p a r t i c u l a r l y true for two problems that were rather long and required working through a sequence of ideas i n order to a r r i v e at the correct answer. Similar r e s u l t s emerge from 21 the Lorge and Solomon research on the T a r t a g l i a problem. I t appeared that the group processes handicapped the most p r o f i -c ient member. Using the information previously mentioned, Lindzey 22 and Aronson have hypothesized that the pro f i c i e n c y of groups i n problem solving, as compared to i n d i v i d u a l s , depends on the type of problem undertaken. They maintain that i f the questions are such that the answers prove to be e a s i l y v e r i f i e d , are amenable to wide acceptance, provide a basis f o r more c o n f i -dent advocacy, and tend to be presented by a competent person, 8. t h e n t h e most c a p a b l e p e r s o n w i l l e x e r t g r e a t e r i n f l u e n c e . From t h i s h y p o t h e s i s , i t appears t h a t t h e most c a p a b l e s t u d e n t ' s i n f l u e n c e may be, i n c e r t a i n s i t u a t i o n s , c o n t r o l l e d by c h o o s i n g a p a r t i c u l a r t y p e o f a c t i v i t y . I n o t h e r words, i f the m a t e r i a l was s e l e c t e d so t h a t i t s a t i s f i e d t h e c r i t e r i a o f L i n d z e y and A r o n s o n , t h e n i t can be argued t h a t t h e most c a p a b l e s t u d e n t c o u l d be e x t r e m e l y i n f l u e n t i a l . A n o t h e r approach has been t a k e n t o e x p l a i n why group s c o r e s were h i g h e r t h a n i n d i v i d u a l s c o r e s . I t was m a i n t a i n e d t h a t t h e s t u d e n t s i n a group p e r f o r m b e t t e r because t h e d i s -agreement among group members about a s o l u t i o n i n i t i a t e s a r e v i e w p r o c e s s d u r i n g w h i c h a t t e n t i o n i s d i r e c t e d t o t h e v a r i o u s c r i t i c a l s t e p s i n t h e problem. The group's c o n t r i b u t i o n t o s u c c e s s f u l p r oblem s o l v i n g was t h a t i t " i n s t r u c t e d " the p a r t i c -i p a n t s i n p r o p e r p r o b l e m s o l v i n g p r o c e d u r e . T h a t i s , t h e group's i n f l u e n c e was t h a t i t i n v o k e d a p r o c e s s w h i c h one would n o t o r d i n a r i l y u se. 23 But Hudgms found no e v i d e n c e t o s u p p o r t t h e above mentioned argument. H i s s u b j e c t s were one hundred and twenty-e i g h t f i f t h grade s t u d e n t s i n f o u r S t . L o u i s s c h o o l s . U s i n g t h e C a l i f o r n i a T e s t o f M e n t a l M a t u r i t y t o c o n t r o l m e n t a l a b i l i t y , and the C a l i f o r n i a A r i t h m e t i c T e s t s 4-6 form W t o measure the s t u d e n t ' s p r o b l e m s o l v i n g a b i l i t y , h i s s u b j e c t ' s r e s u l t s showed t h a t the s c o r e s of s t u d e n t s who s p e c i f i e d t h e s t e p s t h e y used i n s o l v i n g a p r o b l e m — t h i s was the a c t i v i t y w h i c h Hudgins c h o s e t o s i m u l a t e t h e r e v i e w p r o c e s s o f t h e 9. g r o u p — w e r e n o t s i g n i f i c a n t l y d i f f e r e n t from t h e s c o r e s o f t h o s e who s o l v e d the problems i n t h e u s u a l manner. When s u b j e c t s worked a l o n e a f t e r a group s e s s i o n , 24 D u n n e t t e , C a m p b e l l and J a a s t a d found t h a t a l a r g e r number o f i d e a s o r s o l u t i o n s were produced. A s t u d y by P e r l m u t t e r and 25 de M o n t m o l l i n i n w h i c h s t u d e n t s memorized two s y l l a b l e nonsense words a l s o showed t h a t p e o p l e who f i r s t worked i n a group and t h e n a l o n e had a b e t t e r r a t e o f r e c a l l t h a n t h o s e who had o n l y worked a l o n e . From t h e l a t t e r r e s u l t s , i t appeared t h a t t h e i n d i v -i d u a l s a c q u i r e d i n f o r m a t i o n w h i l e t h e y were w o r k i n g i n t h e 26 group w h i c h t h e y c o u l d a p p l y l a t e r . B u t Hudgins p r o d u c e d c o n t r a d i c t o r y e v i d e n c e . I n the second p a r t o f the e x p e r i m e n t mentioned b e f o r e , h i s r e s u l t s showed t h a t group work on mathe-m a t i c a l p r oblems had no more e f f e c t on subsequent performance t h a n i n d i v i d u a l work. I t i s n o t e d t h a t t h e r e was an a p p a r e n t heavy use o f nonsense s y l l a b l e s i n e a r l i e r r e s e a r c h , and i t seems q u i t e l i k e l y t h a t s uch m a t e r i a l m i g h t p r e c l u d e group e f f e c t s . Even though groups have performed b e t t e r than i n d i v i d u a l s on such t a s k s , t h i s m i g h t be e x p l a i n e d by the v i e w t h a t a group i s l e s s l i k e l y t o f o r g e t o r t h a t an i m p l i c i t o r g a n i z a t i o n i s e s t a b l i s h e d w i t h i n the group where s t u d e n t s are a l l o t t e d the nonsense s y l l a b l e s . I t was t h e r e f o r e t h o u g h t t h a t i t would be o f i n t e r e s t t o t e s t s u b j e c t s a f t e r a group s e s s i o n i n w h i c h 10. material, believed conducive to the development of group e f f e c t s , was used. 27 Even though Hudgins discovered that no transfer of t r a i n i n g took place a f t e r students had worked together i n groups on mathematical problems, i t seems reasonable to suspect that i f the a c t i v i t y of the group involved the learning of new concepts as well as working with learned ones, then group work might produce differences s i g n i f i c a n t l y greater than those from i n d i v i d u a l work. When the i n s t r u c t i o n of small groups i s considered, the question a r i s e s of whether or not self-pacing i s e s s e n t i a l . 2 8 Sawris examined t h i s problem using programmed i n s t r u c t i o n . He divided a sample of one hundred and twenty-four students i n the t h i r d form at a technical school i n the following way: i n d i v i d u a l s , homogeneous p a i r s , two heterogeneous groups of size eight, and two heterogeneous groups of sixteen. Using ANOVA to analyze the differences between pretest and posttest scores, he found the following scale of scores proceeding from the highest to the lowest: homogeneous pa i r s , heterogeneous p a i r s , heterogeneous eights and sixteens, and i n d i v i d u a l s . He concluded that group learning gives r e s u l t s comparable to indiv-idual self-paced i n s t r u c t i o n , and that i t was possible to present programs to groups of eight pupils and achieve r e s u l t s s i m i l a r to those from a self-paced student, provided the stu-dents are allowed enough time to respond. 11. A d d i t i o n a l information about self-pacing i s supplied 29 by Moore. Seventy students of ages twelve and t h i r t e e n were categorized by IQ and by speed on a previous self-paced program, so that the students using the teaching machines were matched with those working on booklets. The students on the system worked as i n d i v i d u a l s i n groups. The pace of the machine was determined by a percentage of students getting the correct answer. He determined that the groups working on the machines learned f a s t e r during the f i r s t three weeks than the students working on the booklets, and that there was no difference between the self-paced and paced student; with homogeneous subgroups showing more s i g n i f i c a n t r e s u l t s . From the information available on self-pacing, i t seems reasonable that we may discount the e f f e c t s of forced pacing i n studying the e f f e c t s of group learning. Accordingly, i t follows that groups may produce more solutions than i n d i v i d u a l s , and that t h i s could be a t t r i b u t e d to the most capable student—provided the problems are of a c e r t a i n t y p e — o r to the motivational aspects of a group. And although Hudgins^ found that grade f i v e students d i d not acquire any more techniques which could be applied l a t e r from group work than i n d i v i d u a l work, i t i s s t i l l of i n t e r e s t to know whether students taught i n groups would learn a lesson better than t h e i r i n d i v i d u a l counterparts. Statement of Hypothesis I t i s hypothesized that grade four students i n groups of four, taught at a s p e c i f i c pace how to write p a r t i t i v e and qu o t i t i v e d i v i s i o n equations, w i l l produce more cor rec t equations and have a smaller range of scores on a sub-sequent task than comparable grade four students, taught at the same pace, but taught as i n d i v i d u a l s . CHAPTER II THE DESIGN OF THE STUDY I. INTRODUCTION A set of f i l m loops was used to teach two groups of grade four students how to write equations for d i v i s i o n s prob-lems. The treatment method for each group varied only with respect to student-student i n t e r a c t i o n . One group of students was inst r u c t e d i n d i v i d u a l l y , while the subgroups of the other group were taught together. Four days for each group were required to obtain the data for the experiment. On the f i r s t day, the students were shown eight q u o t i t i v e sequences. On the second day, the stu-dents were instructed with eight p a r t i t i v e sequences. The t h i r d day of i n s t r u c t i o n consisted of repeating sequences f i v e to eight of each d i v i s i o n type i n an unordered manner. On the fourth day, a c r i t e r i o n test was administered. The experi-mental procedure i s summarized by Figure 1. The means of the c r i t e r i o n test scores for each group were compared using a two-sample t - t e s t , while the variances of the c r i t e r i o n t e s t scores were examined using a F-test. FIGURE 1 THE EXPERIMENTAL PROCEDURE Day 1 QUOTITIVE Day 2 PARTITIVE 3 CALENDAR DAYS Day 3 QUOTITIVE-PARTITIVE Day 4 CRITERION TEST 15. I I . FORMATION OF THE GROUPS The Population The population consisted of grade four students from elementary schools i n Greater Vancouver. The students were on the regular B r i t i s h Columbia program. Grade four students could be expected to have s u f f i c i e n t background for the w r i t i n g of equations for d i v i s i o n problems, but would have had l i t t l e opportunity for a concentrated study of equation writ i n g . The Sample The sample was made up of a l l the grade four students from an elementary school. The students came from two c l a s s -rooms. From the r e s u l t s of a p i l o t study, a sample size of f o r t y students was determined adequate for s i g n i f i c a n t experi-mental r e s u l t s . Accordingly, the sample siz e was forty-nine. A student was omitted from the study i f he was absent for more than one treatment or i f he was away for the c r i t e r i o n t e s t . A t o t a l of fo r t y students were used i n the analysis of the data. Assignment of Subjects The students were assigned to the two groups at ran-31 dom, using a table of random numbers. Random assignment was also employed i n making up the subgroups within one of the 16. experimental groups. I t was f e l t that close friends might have a tendency to gather together i n the subgroups, and that t h i s rearrangement would destroy some of the randomness. I I I . DEVELOPMENT OF MATERIALS The I n s t r u c t i o n a l Device Since i t i s highly u n l i k e l y that e i t h e r two teachers presenting the same material or one teacher repeating a lesson give the same lesson, i n s t r u c t i o n was given by means of a f i l m loop projector. Seven cartridges of 8 mm. colored f i l m were displayed on a 5'x5' screen by a Technicolor 800 Instant Movie Projector. There were eight sequences of p a r t i t i v e d i v i s i o n problems and eight sequences of q u o t i t i v e d i v i s i o n problems. A d e s c r i p t i o n of the p a r t i t i v e sequences i s given i n Appendix A. The q u o t i t i v e sequences were p a r a l l e l to the p a r t i t i v e sequences. The sequences were designed to i n s t r u c t the students how to write equations for the problems depicted i n the sequences. As the students moved through the sequences, they received pro-gressively less e x p l i c i t information and e x p l i c i t d i r e c t i o n , so that the seventh and eighth sequences of each type gave no e x p l i c i t information or d i r e c t i o n to the students. 17. The Test Instrument The c r i t e r i o n t e s t consisted of twenty-five d i v i s i o n problems presented i n words (see Appendix B). Thirteen of the problems were of the p a r t i t i v e type, and the remaining items were q u o t i t i v e . The two types had been scattered throughout the t e s t to insure that there were not too many successive items of the same type. To assure the content v a l i d i t y of the te s t , the items were ei t h e r chosen d i r e c t l y from the supplementary textbooks for the course or were modifications of such items. The format of the problems was checked for f a m i l i a r i t y and the words were alt e r e d where necessary to keep the problems at an appropriate reading l e v e l . To provide the r e l i a b i l i t y data for the t e s t , a p i l o t experiment was run one month before the experiment. Before they wrote the t e s t , sixty-three students i n grade four at an elementary school i n Vancouver were instructed by the f i l m loops for two periods on separate days. On the t h i r d day, they wrote the test with an adequate amount of time for completion by a l l students. The r e l i a b i l i t y of the tes t i s presented i n terms of a KR20. Because the test was measuring a c e r t a i n state and not a change over time, an i n t e r n a l consistency measure was chosen. In addition, since the t e s t was composed of p a r t i t i v e and 18. q u o t i t i v e items, and the KR20 i s intended to be used on homo-geneous t e s t s , a KR20 was found for the qu o t i t i v e items and a separate KR20 was found for the p a r t i t i v e items. The item analysis, given i n Appendix B, produced the following r e s u l t s : the subtest of the twelve q u o t i t i v e items had a KR20 of .74, and a l l items had a point b i s e r i a l c o r r e l a -tion above .2 and a p v a l u e — t h e per cent of people who got the item correct—between .05 and .95. On the p a r t i t i v e subtest, the KR20 was .86, and as with the q u o t i t i v e items, no items f a i l e d to meet the c r i t e r i a for r and p mentioned above. From the r e s u l t s of the item analysis on the data from the p i l o t experiment, i t was decided that the test used i n t h i s experiment should be exactly the same as the one used i n the p i l o t experiment. IV. PROCEDURE The experimenter met with the p a r t i c i p a t i n g p r i n c i p a l and teachers two weeks before the study began. At that time, the nature of the experiment was discussed and the appropriate times were scheduled for the experimental sessions. In addition, i t was decided that the experiment would take place i n a separate room instead of a regular classroom to allow for the e f f i c i e n t separation of the two groups. On each day of the experiment, the students who were taught i n d i v i d u a l l y were i n the experimental room from 9:00 A.M. 19. to 9:45 A.M., and the students who were taught i n the groups of four were i n i t from 9:45 A.M. to 10:30 A.M. The a c t i v i t i e s of the students i n both groups were supervised by the experi-menter. The teachers of the classes were only observers. On Day 1 (Wednesday), a f t e r introducing himself to each group, the experimenter informed the students that for the next three periods they would be taught by a movie pro-j e c t o r . He also said that the projector was going to teach them how to write an equation for a d i v i s i o n problem. The students were then t o l d that i n a fourth period they would write a t e s t , so that we could see i f the movie projector was a good teacher. They were t o l d that today they were going to see eight examples on w r i t i n g an equation for a d i v i s i o n prob-lem, and although they would be also asked to f i n d the answer to the problem, they should make sure that they could write the equation c o r r e c t l y . They were then t o l d that the experi-menter would work through the f i r s t example with them and that they should use the answer sheet provided and should work according to the experimenter's i n s t r u c t i o n s . They were also t o l d that they would probably f i n d the f i r s t few examples easy because the equation was given to them, but they should work along according to the i n s t r u c t i o n s because i n the l a t e r examples a l l of the equation would not be given to them. As the experimenter went through the f i r s t example with the subjects, he read out i n s t r u c t i o n s . The i n s t r u c t i o n s are given i n Appendix C. The students who always worked i n d i v i d u a l l y received the Individual-taught Instructions, while the students who worked i n the subgroups were given the Group-taught Instructions. The sentences contained i n the brackets under each i n s t r u c t i o n i n Appendix C describe what happened before the students received the next i n s t r u c t i o n . The next day (Day 2, Thursday), both groups were t o l d that they were going to see another eight films on w r i t i n g an equation for a d i v i s i o n problem. They were also t o l d that the examples would be s i m i l a r to the ones they saw yesterday, except that the questions today would have d i f f e r e n t equations, and that they should watch c l o s e l y for the difference. They were then t o l d that the experimenter would work through the f i r s t example with them i n the same way as he did before. The experimenter proceeded i n the same manner as outlined i n Day 1, and showed the eight p a r t i t i v e sequences. Because the p a r t i t i v e and q u o t i t i v e f i l m loops were organized i n the same way, only the problems and the type of equation were d i f f e r e n t from those on Day 1. On Day 3, the following Monday, the students were reminded that they had seen films on two d i f f e r e n t equations for d i v i s i o n problems, and that now they were going to see some of the same films again, but t h i s time the two kinds of problems would be mixed up. They were then t o l d to work through the films in the same way as they did before. 21. The experimenter went through an example i n the same way as was done on Day 1, but the order i n which the loops were presented was d i f f e r e n t . The order of presentation was the following: Quotitive #5, P a r t i t i v e #7, Quotitive #8, P a r t i t i v e #5, P a r t i t i v e #6, Quotitive #6, Quotitive #7, and P a r t i t i v e #8. On the next day (Day 4 , Tuesday), the c r i t e r i o n test was administered. Before the students wrote the t e s t , they were t o l d that they should write the equation and answer for each problem i n the space provided on the te s t paper. They were also t o l d that they had the whole period to write the t e s t , but i f they f i n i s h e d early they could hand i n t h e i r papers. V. STATISTICAL ANALYSIS Data For each student, the correct number of equations on the c r i t e r i o n t e s t was recorded. Although the students were asked to f i n d the answers to the problems, these re s u l t s were not analyzed because the objective of the i n s t r u c t i o n was to teach the students to write equations to d i v i s i o n problems. The students were asked to f i n d answers because i t was f e l t that i t would be d i f f i c u l t to keep the problems i n a form with which the students were f a m i l i a r i f t h i s was not done. Part (a) and Part (b) i n Appendix D l i s t the experimental data. 22. N u l l Hypotheses Hi. There i s no s i g n i f i c a n t difference between the mean of the group of students who were taught i n d i v i d u a l l y and the mean of the group of students who were taught i n groups of four. H2. There i s no s i g n i f i c a n t difference between the v a r i -ance of the group of students who were taught i n d i v -i d u a l l y and the variance of the group of students who were taught i n groups of four. S t a t i s t i c a l Treatment The following s t a t i s t i c s were calculated from the scores on the c r i t e r i o n t e s t : Individual-taught Group-taught Mean Score X T 5L, Standard Deviation s T s_ Number i n Group n T n Hypothesis One. The two-sample t value was deter-mined by: X - Xj. 2 (n - l)sg + (n - 1) s 2 t = where s = ~ ^7T—TT—T p n c + n i " 2 S p d / r i j . + l / n G) and compared with the tabulated value for n^ , + n_ - 2 degrees of freedom. Hypothesis Two. The F r a t i o was computed thus: s 2 Lar er F = ^-x—ar9er where m = n T -1 and n = n_ , .. -1 s 2 Smaller L a r * e r Smaller and compared with the tabulated value for (m,n) degrees of freedom. The means were compared using a two-tailed t e s t , as what was being tested was whether there was a difference between the groups. The variances were also compared using a two-t a i l e d t e s t because there was no evidence to indicate that a one-tailed t e s t could be employed. CHAPTER III ANALYSIS OF THE RESULTS I. TESTING OF HYPOTHESES Hypothesis One Hypothesis One was that there would be no s t a t i s t i -c a l l y s i g n i f i c a n t difference between the mean of the group of students who were taught i n d i v i d u a l l y and the mean of the group of students who were taught i n groups of four. The table below summarizes the r e s u l t s obtained on the twenty-five questions of the c r i t e r i o n t e s t . TABLE I STATISTICS FOR EQUATION WRITING Individual-taught Group-taught Mean Score 13.75 13.80 Standard Deviation 3.54 3.46 Number i n Group 20 20 The c r i t i c a l value of t at the .05 l e v e l of s i g n i f -icance with 38 degrees of freedom and a two-tailed test was 2.02. Since the t value obtained was .044, which did not exceed the c r i t i c a l value, i t was concluded that there was no s t a t i s t i c a l l y s i g n i f i c a n t difference between the groups as regards the number of equations c o r r e c t l y written. Hypothesis Two Hypothesis Two was that there would be no s i g n i f i c a n t d i f f e r e n c e between the variance of the group of students who were taught i n d i v i d u a l l y and the variance of the group of students who were taught i n groups of four. The r e s u l t s given i n Table I were used to obtain a F value of 1.05. This obtained value was compared with the c r i t i c a l value of F with (19,19) degrees of freedom at the .05 l e v e l of si g n i f i c a n c e equal to 2.6. Since the obtained value was less than the c r i t i c a l value, i t was concluded that there was no s i g n i f i c a n t difference i n the variances of the two groups. I I . CONCLUSIONS There was no s t a t i s t i c a l l y s i g n i f i c a n t difference between the group of students who were taught i n d i v i d u a l l y and the group of students who were taught i n groups of four as regards t h e i r performance and v a r i a b i l i t y on the c r i t e r i o n t e s t . That both groups averaged scores around f i f t y percent showed that the a b i l i t y to di s t i n g u i s h between the two types of d i v i s i o n problems had generally not been successfully taught, for the students could have merely guessed at each 26. question or written one type of equation for a l l the questions. In addition, the sameness of the r e s u l t s seem to indicate that neither method taught the students very wel l . Evidently, the teaching of these students i n groups of four was neither a handicap nor a help to them v i s - a - v i s mastery of the subject matter taught. I I I . ANALYSIS OF ADDITIONAL DATA Answers on the C r i t e r i o n Test In addition to writing the equation for each problem on the c r i t e r i o n t e s t , the students found the answer to the problem or, i n other words, the value for which the frame i n the equation stood. The mean and the standard deviation for the individual-taught group were 22.95 and 2.06, respectively, while the corresponding values for the group-taught group were 22.55 and 2.67. The data can be seen i n Appendix D. As the t e s t contained twenty-five items, i t appeared that the students d i d not f i n d the numbers with which they had to work above t h e i r l e v e l of knowledge. Equations and Answers During Instruction In Appendix D the responses made by the students on the problems presented by the f i l m loops are summarized. For each item the f i r s t p o s i t i o n indicates whether the correct equation was written, and the other p o s i t i o n s i g n i f i e s whether the correct answer was given. 27. The group-taught data was arranged i n the same order, as the students were randomly assigned to t h e i r respective subgroups at the beginning of the experiment. But because of absent students, not a l l subgroups were made up of four students. The table below outlines the number and the students i n each subgroup. Subgroup # Student # (inclusive) 1 1 - 3 2 4 - 7 3 8 - 1 1 4 1 2 - 1 3 5 1 4 - 1 6 6 1 7 - 2 0 Upon examining the r e s u l t s of the day to day i n s t r u c -t i o n , i t seems that the students were able to follow the f i l m loops i n most cases. But the data indicates that there were instances where the students i n the subgroups f a i l e d to agree on a group answer before they received the correct answer from the f i l m . S p e c i f i c a l l y , i n some subgroups the students have d i f f e r e n t answers and equations. CHAPTER IV IMPLICATIONS OF THE STUDY I. INTRODUCTION The hypothesis involved i n thi s study was that i n s t r u c t i o n a l use of subgroups of four students would s i g n i f -i c a n t l y improve achievement among the students. A non-signif-icant difference between the performances of the groups was found. I t was necessary, therefore, to consider the possible reasons for the s i m i l a r i t y of the means and variances of the two groups i n the experiment. I I . DISCUSSION OF THE CONCLUSIONS The Content of the Lessons In choosing the material for i n s t r u c t i o n , the experi-menter paid s p e c i a l attention as to whether i t involved a broad range of ideas, so that i t would lend i t s e l f to i n t e r a c t i o n between the students. I t was f e l t that a large range of ideas go into making up an equation, and, accordingly, that t h i s material would be conducive to group discussion. But i t may have been that the equation w r i t i n g for these p a r t i c u l a r problems did not tend to produce i n t e r a c t i o n between the students. 2 9 . The Method of Instruction The effectiveness of the group method may depend greatly on making people adopt a work or performance goal or a change i n t h e i r behavior. Because i t was considered important to give the students immediate reinforcement, the correct equation and answer were provided aft e r each problem. I t may have been that with the continual reliance of the students on the screen for the correct equation and answer, they may not have f e l t the need to work along very c l o s e l y with t h e i r fellow members. The large range of scores i n some of the subgroups indicates that the weaker students may have r e l i e d on the screen for the correct answer and equation. Although the students were instructed to agree on a group equation and answer before they received the correct ones, there were some of the members of the same subgroup who had d i f f e r e n t equations. Here again, i t may have been that the students depended on the screen for t h e i r answers and not t h e i r fellow group members. The l e v e l of the arithmetic facts i n the i n s t r u c t i o n a l material was low, so that there would be l i t t l e chance of the students missing an equation because they were not f a m i l i a r with the numbers i n the problem. But i t may have been that these numbers discouraged i n t e r a c t i o n . In a number of instances, the students informed the experimenter that the questions were too easy. The students maintained that these numbers were for 30. the grade ones and twos. Thus, the l e v e l of the arithmetic facts may have lead the weaker students to think that they understood the material and did not need any help from the other students. Length of the Experiment I t has been argued that s u f f i c i e n t time i s needed for group i n t e r a c t i o n to develop. To account f o r t h i s , the experi-ment was set up with the students working i n t h e i r respective subgroups for three days. A d d i t i o n a l l y , with examples done for the students and the correct equation and answer given immedi-ately a f t e r each problem, i t was expected that the students would have a good idea of what was being taught and, conse-quently, would quickly and confidently partake i n group d i s -cussions. But i t i s possible that even with these adjustments, the groups needed more time together. Subgroup Makeup Although random assignment was necessary i n order to have equivalent subgroups, i t may have been that the lack of "naturalness" i n the subgroups prevented the development of group cohesiveness. S p e c i f i c a l l y , there were a number of subgroups which contained students who did not wish to work with the other students i n t h e i r subgroup. The resentment on the part of some of the students who saw t h e i r friends i n "good" groups may have caused a negative attitude toward group work; leading these students to work as i n d i v i d u a l s . In addition, because of absent students, the number of students i n some of the subgroups varied from day to day. On c e r t a i n days, there were only two students i n some sub-groups. Consequently, some of the students, even by being i n a subgroup, were not experiencing the f u l l e f f e c t s of group work. I I I . LIMITATIONS OF THE STUDY The Film Loops The f i l m loops used to control the teacher variable were s i l e n t movies. To get the students started each day, the experimenter did one example with the students. Then he remained s i l e n t during the showing of the other sequences. This meant that any questions which the students had about the lesson presentation could not be answered. Obviously, there might be d i f f e r e n t r e s u l t s i f sound were to accompany the movies. In the f i r s t two days of the experiment, the students were shown qu o t i t i v e and p a r t i t i v e sequences, respectively. Because only a c e r t a i n number of sequences were avail a b l e to the experimenter, some of the sequences were repeated on the t h i r d day. I t i s f e l t that i t would have been better to have had some d i f f e r e n t examples t o show the s t u d e n t s on t h e t h i r d day, as t h e y were l e a r n i n g a new t a s k — d i s t i n g u i s h i n g between the two t y p e s o f d i v i s i q n p roblems. -A f u r t h e r l i m i t a t i o n o f t h e f i l m l o o p s was t h a t t hey d i d n o t a l l o w d a y l i g h t v i e w i n g . The f i l m f o r the sequences was p l a c e d i n c a r t r i d g e s w h i c h c o u l d f i t i n t o a d a y l i g h t p r o j e c t o r , b u t t h e p i c t u r e e m i t t e d by t h i s p r o j e c t o r was n o t l a r g e enough f o r a group of the s i z e used i n t h i s e x p e r i m e n t . To c i r c u m v e n t t h i s p r o b l e m , a p r o j e c t o r and a s c r e e n were used. A c c o r d i n g l y , t h e room was d a r k e n e d enough so t h a t t h e s t u d e n t s c o u l d see t h e p i c t u r e , and was k e p t l i g h t enough so t h a t t h e s t u d e n t s c o u l d see what they were w r i t i n g . T h i s c o u l d have a f f e c t e d the p e r -formance o f the s t u d e n t s , and hence the r e s u l t s o b t a i n e d . I t appears d e s i r a b l e t o have a d a y l i g h t p r o j e c t o r . A B r o a d e r Sample The s t u d e n t s i n t h i s e x p e r i m e n t a l l came from t h e same s c h o o l . As t h i s s c h o o l g e n e r a l l y has s t u d e n t s o f above average a b i l i t y , the p a r t i c i p a n t s were p r o b a b l y not a r e p r e s e n t a t i v e sample of the grade f o u r p o p u l a t i o n . I t would be i n t e r e s t i n g t o see i f s i m i l a r r e s u l t s would be o b t a i n e d w i t h l e s s a b l e s t u d e n t s . I t i s q u i t e p o s s i b l e t h a t group work o f t h e t y p e c a r r i e d o u t i n t h i s e x p e r i m e n t i s l e s s s u i t a b l e f o r more c a p a b l e s t u d e n t s . Q u o t i t i v e - P a r t i t i v e Approach The a p p r o a c h t a k e n i n t h e i n s t r u c t i o n o f the e x p e r i -ment was t o w r i t e two d i f f e r e n t e q u a t i o n s f o r t h e two d i f f e r e n t t y p e s o f d i v i s i o n problems w h i c h e x i s t i n the r e a l w o r l d . The a d o p t i o n o f t h i s p o i n t o f v i e w i s now l e f t up t o the t e a c h e r s i n t he s c h o o l s , and the s c h o o l i n w h i c h t h e e x p e r i m e n t was r u n had n o t been s t r i c t l y f o l l o w i n g t h i s approach. C o n s e q u e n t l y , the s t u d e n t s c o u l d have had a d i f f i c u l t t i me a d j u s t i n g t o t h e d i f f e r e n t p r e s e n t a t i o n . IV. SUGGESTIONS FOR FUTURE STUDIES I n f o r m i n g the subgroups by random a s s i g n m e n t , t h e p o s s i b i l i t y o f subgroups w i t h i n c o m p a t i b l e members was c r e a t e d . I t seems a d v i s a b l e t h a t t h e r e s h o u l d be a d i a g n o s i s o f the i n d i v i d u a l needs and a b i l i t i e s , and t h e n a s s i g n m e n t o f s t u d e n t s t o groups i n such a way t h a t each group c o n t a i n s t h e n e c e s s a r y r ange o f s k i l l s w i t h a minimum o f d u p l i c a t i o n i n d i f f e r e n t i n d i v i d u a l s . T h i s would mean t h a t s o c i o l o g i c a l f a c t o r s , as w e l l as a c h i e v e m e n t , s h o u l d be used i n p l a c i n g the s t u d e n t s i n t h e i r r e s p e c t i v e subgroups, so t h a t the group s i t u a t i o n i s more l i k e l y t o p r o v i d e p e r s o n a l r e l a t i o n s more i n l i n e w i t h what the members o f the group d e s i r e . The c r i t e r i o n v a r i a b l e i n t h i s e x p e r i m e n t was the a b i l i t y o f the s t u d e n t t o r e c o g n i z e whether a d i v i s i o n problem 34. was p a r t i t i v e or q u o t i t i v e , and to write the corresponding equation. However, i t may be that student-student i n t e r a c t i o n has no s t a t i s t i c a l l y discriminable e f f e c t on the acquiring of the above-mentioned a b i l i t y . Since i t i s safe to say that student-student i n t e r a c t i o n does have a decided e f f e c t on such variables as perception of an objective stimulus, l i k i n g other students, and spontaneity of behavior, i t i s suggested that any future work with t h i s teaching strategy should include the measurement of variables most l i k e l y to involve the above. Improved group performance may e n t a i l the development of working r e l a t i o n s with a c e r t a i n group of people. I t may be that grade four students do not possess the s o c i a l s k i l l s necessary to e s t a b l i s h the s o c i a l organization of the group. In t h i s respect, i t i s recommended that an older group of students be used i n future studies. The answer was given a f t e r each problem because i t was f e l t that this immediate reinforcement would help the students learn. But i t seemed that the students tended to s i t back and wait for the answer to appear on the screen, instead of asking the other students i n t h e i r group. I t would be of i n t e r e s t to know that i f the students worked together for a ce r t a i n length of time before they received any correct solu-tions, whether the weaker students would depend more on the other students from whom valuable problem solving techniques might be obtained. V. SUMMARY This study was designed to investigate the e f f e c t s of i n s t r u c t i o n i n small groups. Findings by psychologists and so c i o l o g i s t s indicated that s i g n i f i c a n t group forces e x i s t which could improve learning. The present study examined the a b i l i t y of students to write equations for d i v i s i o n problems a f t e r they had been taught i n groups of four. The students were instructed for three days. In order to have equivalent i n s t r u c t i o n for the two groups, f i l m loops presented the material of the lessons. In one group, a l l the students were taught i n d i v i d u a l l y , while i n the other group, the students were taught i n groups of four. The students were given a c r i t e r i o n t e s t of twenty-f i v e d i v i s i o n problems to which they wrote the equation and answer. A l l items i n the tes t had s a t i s f i e d the required c r i t e r i a during an item analysis r e s u l t i n g from a p i l o t run i n the previous month. A two-tailed t - t e s t was used to test the sig n i f i c a n c e between the mean of the individual-taught group and the mean of the group-taught group. A two-tailed F-test was used to test the s i g n i f i c a n c e between the variances of the two groups. The r e s u l t s showed that there was no s i g n i f i c a n t d i f -ference between the individual-taught group and the group-taught group with respect to mean and variance. FOOTNOTES Everett W. Bovard, "The Psychology of Classroom Interaction," The Journal of Educational Research, 45:215-224, 1951. 2 H.V. Perkins, "Climatic Influences of Group Learning," The Journal of Education Research, 45:115-119, 1951. 3 B.O. Bergum, and D.J. Fehr, "Eff e c t s of Authori-tarianism on Vigilance Performance," The Journal of Applied  Psychology, 47:75-77, 1963. 4 R.B. Zajonc, and S.M. Sales, "Social F a c i l i t a t i o n of Dominant and Subordinate Responses," The Journal of Experi-mental S o c i a l Psychology, 2:160-168, 1966. ^E.E. Jones, and H.B. Gerard, Foundations of S o c i a l  Psychology (New York: Wiley, 1967), pp.1-154. ^ J . Macy J r . , L.S. C h r i s t i e , and R.D. Luce, "Coding Noise i n a Task Oriented Group," The Journal of Abnormal Social  Psychology, 48:401-409, 1953. 7 Fred T. Wilhelms, and Dorothy Westby-Gibson, "Grouping Research Offers Leads," Educational Leadership, 18:410-413, 1961. 8 B.F. Skinner, The Technology of Teaching (New York: Century Psychological Series, 1968), p.16. 9 Irving E. S i g e l , "The Piagetian System and the World of Education," Studies i n Cognitive Development: Essays i n the  Honor of Jean Piaget, David Elkind and John H. F l a v e l l , Editors (New York: Oxford University Press, 1969), p.473. "^Jean Piaget, "Three Lectures," Piaget Rediscovered, R.E. Ripple and V.N. Rockcastle, editors (Ithaca, New York: Cor n e l l University Press, 1964), p.4. "''"'"Herbert Gournee, "A Comparison of C o l l e c t i v e and Individual Judgements of Fact," The Journal of Experimental  Psychology, 21:106-112, 1937. 12 Samuel F. Klugman, "Cooperative Versus Individual E f f i c i e n c y i n Problem Solving," The Journal of Educational Psychology, 35:91-100, 1944. 37. 13 H.V. Perlmutter, and Germaine de Montmollin, "Group Learning of Nonsense S y l l a b l e s , " The Journal of  Abnormal S o c i a l Psychology, 47:762-769, 1953. 14 " Bryce B. Hudgins, "Eff e c t s of Group Experience on Individual Problem Solving," The Journal of Educational Psychology, 51:37-42, 1960. 15 D.W. Taylor, and W.L. Faust, "Twenty Questions: E f f i c i e n c y i n Problem Solving as a Function of Size of Groups," The Journal of Experimental Psychology, 44:360-368, 1952. 1 6 T V . , Ibid. 1 7 T V., Ibid. 18 W.L. Faust, "Group Versus Individual Problem Solving," The Journal of Abnormal Social Psychology, 59: 68-72, 1959. 19 N.H. Anderson, "Group Performance i n an Anagram Task," The Journal of S o c i a l Psychology, 55:67-75, 1961. 20 James H. Davis and Frank Restle, "The Analysis of Problems and Pr e d i c t i o n of Group Problem Solving," The  Journal of Abnormal and S o c i a l Psychology, 66:103-116, 1963. 21 I. Lorge and H. Solomon, "Individual Performance and Group Performance i n Problem Solving Related to Group Size and Previous Exposure to the Problem," The Journal of Psychology, 48:107-114, 1959. 22 Gardner Lindzey and E l l i o t Aronson, "Group Psychol-ogy and Phenomena of Interaction," The Handbook of S o c i a l  Psychology, 1-694, 1969. 23 Ibid. 24 M.D. Dunnette, J . Campbell, and Kay Jaastad, "The E f f e c t of Group P a r t i c i p a t i o n on Brainstorming Effectiveness for Two I n d u s t r i a l Samples," The Journal of Applied Psychology, 47:30-37, 1963. 25T, . , Ibid. 38. 26 T,., Ibid. 27 ^ ' i b i d . 2 8 M.J. Sawiris, "An Experimental Study of Individual and Group Learning Using a Linear Geometry Program," Programmed  Learning, 3:146-153, 1966. 29 D.L. Moore, "Group Teaching by Programmed Instruc-t i o n , " Programmed Learning and Educational Technology, 4:37-46, 1967. 3 0 I b i d . Roger E. Kirk, Experimental Design: Procedures for  the Behavioral Sciences, (Belmont: Brooks Cole Publishing Company, 1968), pp.520-521. BIBLIOGRAPHY Anderson, N.H. "Group Performance i n an Anagram Task," The Journal of S o c i a l Psychology, 55:67-75, 1961. Balow, Irving H. "The E f f e c t s of Homogeneous Grouping i n Seventh Grade Arithmetic," The Arithmetic Teacher, 11: 186-191, 1964. Banghart, F.W., and H.S. Spraker. "Group Influence on C r e a t i v i t y i n Mathematics," The Journal of Experimental  Education, 31:257-263, 1963. Bergum, B.O., and I.J. Fehr. "Eff e c t s of Authoritarianism on Vigilance Performance," The Journal of Applied Psychology, 47:75-77, 1963. Bovard, Everett W. "The Psychology of Classroom Interaction," The Journal of Educational Research, 45:215-224, 1951. Brewer, Emery. "A Survey of Arithmetic Intra-class Grouping Practices i n the Elementary Schools of Ohio," Dis s e r t a t i o n  Abstracts, 24: 4466-4467, 1964. Dewar, John A. "Grouping for Arithmetic Instruction i n Grade Six," The Elementary School Journal, 63:266-270, 1963. Davis, James H., and Frank Restle. "The Analysis of Problems and Pre d i c t i o n of Group Problem Solving," The Journal of  Abnormal arid S o c i a l Psychology, 66:103-116 , 1963. Dunnette, M.D., J . Campbell, and Kay Jaastad. "The E f f e c t of Group P a r t i c i p a t i o n on Brainstorming Effectiveness for Two I n d u s t r i a l Samples," The Journal of Applied Psychology, 47:30-37, 1963. Davis, O.E. J r . , and N. Tracy. "Arithmetic Achievement and I n s t r u c t i o n a l Grouping," The Arithmetic Teacher, 10: 12-17, 1963. D u r r e l l , Donald D. 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"Individual Performance and Group Performance i n Problem Solving Related to Group Size and Previous Exposure to the Problem," The Journal of  Psychology, 48:107-114, 1959. Marguart, Dorthy Irene. "Group Problem Solving," The Journal  of S o c i a l Psychology, 41:103-113, 1955. Macy, J . , J r . , L.S. C h r i s t i e , and R.D. Luce. "Coding Noise i n a Task Oriented Group," The Journal of Abnormal S o c i a l  Psychology, 48:401-409, 1953. McNemar, Quinn. Psychological S t a t i s t i c s . New York: John Wiley and Sons, Inc., 1962. Moore, D.L. "Group Teaching by Programmed Instruction," Programmed Learning and Educational Technology, 4:37-46, 1967. McHugh, Walter Joseph. "Pupil Team Learning i n S k i l l s Subjects i n Intermediate Grades," Di s s e r t a t i o n Abstracts, 21: 1460-1461, 1960. Piaget, Jean. "Three Lectures," Piaget Rediscovered, R.E. Ripple and V.N. Rockcastle, e d i t o r s . I t h i c a , New York: C o r n e l l University Press, 1964, pp. 1-27. Pask, G., and B.N. Lewis. "An Adaptive Automation for Teaching Small Groups," Perceptual and Motor S k i l l s , 14:183-18 8, 1962. Perkins, H.V. "Climate Influences Group Learning," The Journal  of Educational Research, 45:115-119, 1951. Perlmutter, H.V.,and G. de Montmollin. "Group Learning of Nonsense S y l l a b l e s , " The Journal of Abnormal S o c i a l  Psychology, 47:762-769, 1952. Roseborough, Mary E. "Experimental Studies of Small Groups," The Psychological B u l l e t i n , 50:277-303, 1953. 42. Sawiris, M.Y. "An Experimental Study of Individual and Group Learning Using a Linear Geometry Program," Programmed  Learning, 3:146-153, 1966. Shaw, Marjorie E. "A Comparison o f Individuals and Small Groups in the Rational Solution of Complex Problems," The American  Journal of Psychology, 44:491-504, 1920. S i g e l , Irving E. "The Piagetian System and the World of Education," Studies i n Cognitive Development: Essays i n the Honor of Jean Piaget, David Elkind and John H. F l a v e l l , e d i t o r s . New York: Oxford University Press, 1969, pp.462-487. Skinner, B.F. The Technology of Teaching. New York: The Century Psychological Series, 1968. Smith, B.O., and A.J. Doliv. "Recent Developments i n Grouping— A Minimum Bibliography," Educational Leadership, 4: 403-520, 1947. Smith, W.M. "The E f f e c t s of Intraclass A b i l i t y Grouping," Dissertation Abstracts, 21: 563-564, 1960. Sommers, Mildred Emily. "A Comparative Study of Two Grouping Procedures i n Junior High School," D i s s e r t a t i o n Abstracts, 21:1115-1116, 1960. Stern, Carolyn. " A c q u i s i t i o n of Problem Solving Strategies i n Young Children and Its Relation to Verbalization," The Journal of Educational Psychology, 58:245-252, 1967. Taba, Hilda. With Perspective on Human Relations. Washington: American Council on Education, 1955. Taylor, D.W., and W.L. Faust. "Twenty Questions: E f f i c i e n c y i n Problem Solving as a Function of Size of Groups," The Journal of Experimental Psychology, 44:360-368, 1952. Thelen, Herbert A. "Group Dynamics i n Instruction: P r i n c i p l e of Least Group Size," The School Review, 58:139-148, 1949. Thomas, E.J., and C F . Fink. "The E f f e c t s of Group Size," The Psychological B u l l e t i n , 60:371-384, 1963. Thorndike, Robert L. "The E f f e c t of Discussion Upon the Correctness of Group Decisions, When the Factor of Majority Influence i s Allowed f o r , " The Journal of  Soc i a l Psychology, 9:343-362, 1938. Wallen, Norman E., and Robert 0. Vowles. "The E f f e c t of Intraclass A b i l i t y Grouping on Arithmetic Achievement," The Journal of Educational Psychology, 51:159-163, 1960. Watson, Goodwin B. "Do Groups Think More E f f e c t i v e l y Than Individuals," The Journal of Abnormal and Social Psychol-ogy, 23:328-336, 1929. Wert, James E., Charles 0. Neidt, and J. Stanley Ahmann. S t a t i s t i c a l Methods i n Educational And Psychological  Research. New York: Appleton-Century-Crafts, Inc., 1954. West, J e f f , and C a l l i e Sievers. "Experiments i n Cross Grouping," The Journal of Educational Research, 54: 70-71, 1960. Wilhelms, Fred T., and Dorthy Westby-Gibson. "Grouping: Research Offers Leads," Educational Leadership, 18: 410-413, 1961. Zajonc, R.B., and S.M. Sales. " S o c i a l F a c i l i t a t i o n of Dominant and Subordinate Responses," The Journal of  Experimental S o c i a l Psychology, 2:160-168, 1966. Zimmerman, Donald. "Teaching T h i r t y , Like Teaching One," Education, 85: 364-370, 1965. A P P E N D I C E S APPENDIX A THE INSTRUCTIONAL DEVICE 44. PARTITIVE SEQUENCE 1 P i c t o r i a l Description Word Description E I G H T B L O C K S 6 ooag THE Sft/^E. NUMBER oi BLOCKS ARE: IN E>ICH CUP 6-*- • H o w MKNV EACH CUP ? E Q U A T I O N < 3 t ° = ^  ' N u h B f - R o-f "BLOCKS = ? 1. The words "EIGHT BLOCKS" were pointed at. 2. The "8" was brought down. 3. The blocks were put i n the box. 4. Four blocks were placed i n each cup, although the viewer of the f i l m could not observe how many. 5. The words on the l e f t were changed. 6. The v Q was brought down. 7, 8, The 11 =2" was brought down. The words on the l e f t were changed. The picture on the l e f t below appeared on the screen u n t i l the students had achieved a solution, then the picture on the r i g h t below was shown. E Q U A T / O N N u M 6 £ * o f B LOCKS - 4 " 45. PARTITIVE SEQUENCE 2 P i c t o r i a l Description Word Description TWENTY - ONE. 2 1 .STRAWS F_ACH BuNDLC H-_os THE. S/^fAe: NUMBER STRAWS How MANY STRAWS IN E>CH B-WD-E ? 2 1 3 E Q N ^ B E L R o f S T R A W S - ? 1. The words "TWENTY-ONE STRAWS* were pointed at. 2. The "21" was brought down. 3. The straws were put into three bundles. 4. The words on the l e f t were changed. 5. The T O was brought into place. 6. The "=3" was brought down. 7. The words on the l e f t were changed. 8. The picture below on the l e f t appeared on the screen while the students achieved a solu-t i o n ; then the picture below on the r i g h t was shown. EIQ U A T ! O N ei - • = 5 • N U M B E R « f S T R A W S - 7 46. PARTITIVE SEQUENCE 3 P i c t o r i a l Description Word Description P E N N I E S 2 0 E A C H C U P H O U D S THE. S A . P \ C Ny/-\i3fR o-F P E N N I E S 2 0 ? How AA.MY TCN/MIES IN EACH C U P ? 2 . 0 ? - 4 LTQUAT/ON 2 0 = 4 N •P PEN 1. The words "TWENTY PENNIES" were pointed at. 2. The "20" was brought down. 3. The pennies were placed into a bag, so that they were out of sight. 5, 6, Five pennies were put i n each cup without the students knowing how many. The words on the l e f t were changed. The "?" was brought down. 7 8 The "=4" was brought down. The words on the l e f t were changed. The picture below on the l e f t appeared on the screen while the students achieved a solu-t i o n ; then the picture on the r i g h t was shown. E Q U A T / O / V 20 + • - 4 N u M 6 E l R o ? PENNIES- 5 4 7 . PARTITIVE SEQUENCE 4 P i c t o r i a l Description Word Description O U T E E N INCHES < R IBBON C U T I N T O F O U R P i e c e s ° f E I Q U ^ V L . L e . N G . T H How A"\ A N Y I N C H E S I N lb ? ? ft a . E A C H R E . C E . ? E Q U r \T/ O N lb NunQE.R o T ' I N C H E S 1. The words "SIXTEEN INCHES OF RIBBON" were pointed at. 2. The "16" was brought down. 3. The ribbon was cut into four equal pieces. 4. The words on the l e f t were changed. 5. The "?" was brought down. 6. The second "?" was brought down. 7. The words on the l e f t were changed. 8. The picture below on the l e f t appeared on the screen while the students achieved a solu-t i o n ; then the picture on the r i g h t was shown. E - Q u A T / O / V / b ^ a = 4 NUMBEJR I N C H E S 4 -48. PARTITIVE SEQUENCE 5 P i c t o r i a l Description Word Description T ~ / c K £ T S 27 PUT THE SAME NUMBER »f TICKETS EACH PiuE 2 7 ? 1. The words "TWENTY-SEVEN TICKETS" were pointed at. 2. The "27" was brought down. 3. The t i c k e t s were put into three p i l e s . 4. The words on the l e f t were changed. 5. The "?" was brought down. How fWyT ICKETS Is EACH AUE ? 2 7 ? ? 6. The second "?" was brought down. 7. The words on the l e f t were changed. 8. The picture on the l e f t below appeared on the screen while the students achieved a solu-tion; then the picture below on the r i g h t was shown. 27 U A T / O N N U M B E R o f I I C K E T S = ? U r V T / O N M u r i 8 e : R «f T I C K E T S ^ 9 49. PARTITIVE SEQUENCE 6 P i c t o r i a l Description Word Description EJG,HT££/S/ Ounces of WATER ( 6 0 BCHTEEN OUNCES O? WATER / 6 ? 0 i 3 0 HOW^IANV OUNCES of WATER /N £TACW G-ASS NunBER o f O U N C E S - ? 1. The words "EIGHTEEN OUNCES OF WATER" were pointed at. 2. The "18" was brought down. 3. Three glasses were f i l l e d with water. 4. The "?" was brought down. 5. The second "?" was brought down. 6. The words on the l e f t were changed. 7. The picture on the l e f t below appeared on the screen u n t i l the students achieved a solu-t i o n ; then the sign below was shown. E Q U A T / O N / Q + • = 3 A/QMSER o f OUNCES - 6 50. PARTITIVE SEQUENCE 7 P i c t o r i a l Description Word Description T H I R T Y B O O K S THIRTY BOOKS 8 0 0 K S E A C H PILE ? — , U A T / O N NUMQER of BO O K S 1. The words "THIRTY BOOKS" were pointed at. 2. The books were divided into three equal p i l e s . 3. The words on the l e f t were changed. 4. The picture below on the l e f t appeared on the screen u n t i l the students achieved a solution; then the sign below was shown. U A T 7 0 / V 3 0 - • = 3 NUMBER « f B ° O K S = 1 0 51. PARTITIVE SEQUENCE 8 P i c t o r i a l Description Word Description 3 > of CHEESE r/ow M M y Ou^C£.S IN EACH RECE. ? t QUKT/ ON N U M 8 E R o f OUNCES"? 1. The words "FIFTEEN OUNCES OF CHEESE" were pointed at, 2. The cheese was cut into three equal pieces. 3. The words on the l e f t were changed. 4. The picture below on the l e f t appeared on the screen u n t i l the students achieved a solution; then the picture below was shown. EQU/\T70/\7 15 + a - 3 N U M 6 E : R ° f O U N C E S - 5 APPENDIX B The Test Instrument P a r t i t i v e and Quotitive Items Quotitive Items P a r t i t i v e Items 52. (a) THE TEST INSTRUMENT In the spaces provided, write the equation and the answer for each problem. 1) Mary bought 24 flowers, she put them into bunches of 8. How many bunches of flowers did she make? 2) B i l l sold 20 cents worth of t i c k e t s for the class r a f f l e . The t i c k e t s were 5 cents each. How many did B i l l s e l l ? 3) Mrs. Jones had 24 cabbages i n her patch. There were 6 equal rows i n the patch. How many cabbages were i n each row? 4) Jim caught 15 f i s h . He put the same number of f i s h i n each of 5 bags. How many f i s h i n each bag? 5) Joan has 27 peanuts i n a bag. Joan wants to give the same number of peanuts to each of her friends. How many peanuts does each f r i e n d get i f she has three friends? 6) Mrs. Smith has 24 eggs. Her family eats 6 eggs at each meal. How many meals of eggs can her family have? 7) There are 36 boys playing baseball i n the park with 9 boys on a team. How many teams are playing? 8) A mother t o l d her 3 sons that she would divide 12 apples evenly between them. How many apples did each get? 9) 28 chi l d r e n went to a hockey game i n 4 cars containing the same number of c h i l d r e n . How many ch i l d r e n were i n each car? 10) A restaurant cook has 32 ounces of hamburger meat. He uses 4 ounces of meat for every hamburger that he makes. How many hamburgers can he make with t h i s meat? 11) John went to the store to buy toy cars. He had 32 cents with him. The cars each cost 8 cents. How many cars d i d he buy? 12) A vase has 12 flowers in i t . If the same number of flowers are put into 4 smaller vases, how many flowers are i n each vase? 13) A farmer owns 30 pigs. He decides to put 5 pigs i n each pen. How many pens does he have? 14) The 20 children i n Miss Smith's cla s s formed 4 relay teams of the same s i z e . How many children were on each relay team? 15) In a garden, there are 5 rows. Each row has the same number of corn plants. If there are 15 corn plants i n the garden, how many corn plants are there i n each row? 16) Father w i l l have 21 days for his holidays t h i s year. How many weeks of holidays does he have? 17) Bob uses 3 wheels for each airplane. If he has 24 wheels, how many a i r -planes can he make? 18) A grocer had 30 candies i n a bag. To s e l l the candies, he put them i n smaller bags holding 5 candies each. How many small bags of candy d i d he have to s e l l ? 19) Five g i r l s picked the same number of quarts of strawberries for a farmer. If the farmer had 35 quarts of strawberries, how much did each g i r l pick? 20) A grocer put 26 cans of beans on 2 shelves. If the shelves each held the same number of cans, how many cans were on each shelf? 21) Clara gathered 36 roses into bunches. She put 6 roses i n each bunch. How many bunches did she make? 22) John had 16 pictures to put i n an album. He used 4 pages and placed the same number of pictures on each page. How many pictures did he put on each page? — 23) Some g i r l s shared the cost of a 25 cent bag of potato chips. If each g i r l paid a n i c k e l , how many g i r l s were there? 24) Jean divided 18 ta r t s equally among the 6 g i r l s at her party. How many ta r t s d i d she give to each g i r l ? 25) Jim wanted to put his f i s h into 4 f i s h bowls. If Jim had 28 f i s h , how many did he put i n each f i s h bowl,so that each bowl had the same number of f i s h i n i t ? 55. (b) PARTITIVE AND QUOTITIVE ITEMS Item No. Point B i s e r i a l P Variance 1 0.1550 0.7302 0.1970 2 0.2862 0.6825 0.2167 3 0.4455 0.3402 0.2273 4 0.5341 0.3968 0.2394 5 0.3286 0.4921 0.2499 6 0.2741 0.7619 0.1814 7 0.1815 0.7143 0.2041 8 0.5054 0.3175 0.2167 9 0.3096 0.4127 0.2424 10 0.3046 0.8095 0.1542 11 0.3240 0.7619 0.1814 12 0.4565 0.3016 0.2106 13 0.1842 0.7937 0.1638 14 0.5663 0.3175 0.2167 15 0.5993 0.3968 0.2394 16 0.4033 0.4444 0.2469 17 0.2581 0.6667 0.2222 18 0.1863 0.6508 0.2273 19 0.4163 0.2381 0.1814 20 0.5580 0.3810 0.2358 21 0.4325 0.7143 0.2041 22 0.5568 0.3016 0.2106 23 0.3953 0.5556 0.2469 24 0.5794 0.2698 0.1970 25 0.4277 0.2698 0.1970 ***The Mean i s 12 .7302 ***The KR20 i s 0 . 7661 Results of Task Number 3 Items Omitted If Any. New KR20 1 0.7706 18 0.7790 13 0.7857 7 0.7959 6 0.8020 2 0.8102 11 0.8180 17 0.8288 10 0.8359 21 0.8471 56. (c) QUOTITIVE ITEMS Item No. Point B i s e r i a l P Variance 1 0.6194 0.7302 0.1970 2 0.5396 0.6825 0.2167 6 0.5165 0.7619 0.1814 7 0.5868 0.7143 0.2041 10 0.4745 0.8095 0.1542 11 0.6514 0.7619 0.1814 13 0.5496 0.7937 0.1638 16 0.2891 0.4444 0.2469 17 0.4144 0.6667 0.2222 18 0.5217 0.6508 0.2273 21 0.6886 0.7143 0.2041 23 0.3353 0.5556 0.2469 ***The Mean i s 8. 2857 ***The KR20 i s 0. 7413 Results of Task Number 3 Items Omitted If Any. New KR20 N i l 0.7413 57. (d) PARTITIVE ITEMS Item No. Point B i s e r i a l P Variance 3 0.6532 0.3492 0.2273 4 0.7465 0.3968 0.2394 5 0.3762 0.4921 0.2499 8 0.6853 0.3175 0.2167 9 0.5567 0.4127 0.2424 12 0.5612 0.3016 0.2106 14 0.7319 0.3175 0.2167 15 0.6423 0.3968 0.2394 19 0.3928 0.2381 0.1814 20 0.6860 0.3810 0.2358 22 0.6816 0.3016 0.2106 24 0.7133 0.2698 0.1970 25 0.5217 0.2698 0.1970 ***The Mean i s 4 .4444 ***The KR20 i s 0 .8606 Results of Task Number 3 Items Omitted I f Any. New KR20 N i l 0.8606 APPENDIX C (a) The Daily Work Sheet (b) The Individual-taught Instructions (c) The Group-taught Instructions (a) THE DAILY WORK SHEET Name Room Equation Answer 59. (b) THE INDIVIDUAL-TAUGHT INSTRUCTIONS 1. "Watch c l o s e l y as the problem i s given to you on the screen." (Six eggs, the words "SIX EGGS" and a "6" were shown; then the sequence was paused.) 2. "Write down the '6' on your paper a f t e r the number one." (The eggs were paired, the words "PAIRED THE EGGS" were shown, and a "T 2" was placed i n the equation; then the sequence was paused.) 3. "Put the ' T- 2' i n i t s correct p o s i t i o n . " (The words "HOW MANY PAIRS ARE THERE?" were shown, and the "=CJ" was brought down; then the sequence was paused.) 4. "Place the ' = • * i n your equation, and then write what the frame stands f o r i n the l a s t column of your paper." (The equation "6 -f 2 =•" and the words "NUMBER OF PAIRS=?" appeared on the screen; then the sequence was paused.) 5. "At th i s point i n each f i l m , we w i l l have a short pause. In the films that are coming, you may need extra time to change your equation or answer; i f so, do i t during t h i s pause." (The equation "6 * 2 =•" and the words "NUMBER OF PAIRS=3" were shown; then the sequence was paused.) 60. 6. "Mark whether you were r i g h t or wrong, and get ready for the next sequence. You are going to work through i t by yourself, and you should work i n the same way as we just did." (The f i l m was progressed to the next sequence.) (The equation "6 * 2 = • " and the words "NUMBER OF PAIRS =3" were shown, then the sequence was paused.) "Mark whether your group was r i g h t or wrong, and get ready for the next f i l m . You are going to work through i t by yourselves, and you should work i n the same way as we just d i d . " (The f i l m was progressed to the next sequence.) APPENDIX D (a) Equations for Individual-taught Group (b) Equations for Group-taught Group (c) Answers for Individual-taught Group (d) Answers for Group-taught Group (e) Day One Individual-taught Group (f) Day Two Individual-taught Group (g) Day Three Individual-taught Group (h) Day One Group-taught Group (i) Day Two Group-taught Group (j) Day Three Group-taught Group (a) EQUATIONS FOR INDIVIDUAL-TAUGHT GROUP SCORES STUDENT NUMBER FOR EACH ITEM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 18 20 1. 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 2. 1 1 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 3. 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 4. 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 1 5. 1 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 6. 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 7. 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 0 8. 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 9. 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 10. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 11. 1 1 1 1. 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 12. 1 1 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 13. 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1 1 1 14 . 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 15. 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 16. 0 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 17. 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 18. 0 1 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 0 19. 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 20. 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 21. 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 22. 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 23. 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 24. 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 25. 1 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 15 18 11 17 13 12 16 25 13 11 16 13 12 11 11 11 9 15 13 13 (b) EQUATIONS FOR GROUP-TAUGHT GROUP SCORES STUDENT NUMBER FOR EACH ITEM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1. 1 1 1 0 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 0 1 1 1 1 1 1 1 3. 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 4. 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 5. 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 6. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 7. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8. 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 9. 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 10. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11. 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 12. 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 13. 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 14. 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 15. 0 1 1 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 16. 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 17. 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 18. 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 19. 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 20. 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 21. 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 22. 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 23. 0 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 1 1 0 1 24. 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 25. 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 16 14 14 18 11 11 12 12 10 18 17 12 15 24 13 11 11 14 10 13 (c) ANSWERS FOR INDIVIDUAL-TAUGHT GROUP SCORES STUDENT NUMBER FOR EACH ITEM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1. 1 1 1 1 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 1 1 1 1 1 3. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 4. 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5. 1 1 1 1 1 0 1 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 1 1 1 1 1 1 1 1 7. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 8. 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 9. 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 10. 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 11. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12. 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 13. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14. 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15. 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 16. 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 17. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 18. 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 19. 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1 1 1 20. 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 21. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 22. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 23. 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 24. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 25. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 25 24 22 23 25 21 25 25 21 21 23 25 21 25 23 23 18 25 20 24 (d) ANSWERS FOR GROUP-TAUGHT GROUP SCORES STUDENT NUMBER FOR EACH :tem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 l . 1 1 1 0 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 1 1 1 1 1 3. 1 1 1 1 1 1 1 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 1 1 1 1 1 , . 1 1 5. 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 6. 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 7. 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8. 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 9. 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 10. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 11. 1 1 1 . 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 12. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 13. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 14. 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 15. 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 16. 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 17. 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 18. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 19. 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 20. 1 1 0 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 0 21. 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 22. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 23. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 24. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 25. 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 1 25 24 23 22 23 23 22 25 18 25 23 24 23 25 25 14 21 23 21 22 67. (e) DAY ONE INDIVIDUAL-TAUGHT GROUP STUDENT ITEM NUMBER JMBER 1 2 3 4 5 6 7 8 1. 11 11 11 11 11 01 11 11 2. o 11 11 11 11 11 11 11 00 J . 4. 11 11 11 11 11 11 11 11 5. 11 11 11 11 11 11 11 11 6. 11 11 11 11 11 11 11 00 7. * 11 11 11 11 11 11 11 11 8. 11 11 11 11 11 11 11 11 9. 11 11 11 11 11 11 11 11 10. 11 11 11 11 11 11 01 11 11. 11 11 11 11 11 11 11 11 12. 11 11 11 11 11 11 11 11 13. 11 11 11 11 11 11 01 11 14. 11 11 11 11 11 01 11 11 15. 11 11 11 11 11 01 11 11 16. 11 11 11 11 11 11 11 11 17. 11 11 11 11 11 01 11 11 18. 11 11 11 11 11 11 11 11 19. 11 11 11 11 11 11 00 11 20. 11 11 11 11 11 11 11 11 (f) DAY TWO INDIVIDUAL-TAUGHT GROUP STUDENT ITEM NUMBER NUMBER 1 2 3 4 5 6 7 8 1. 01 11 11 11 01 11 11 11 2. 11 11 11 11 01 11 01 11 3. 11 11 11 11 11 00 11 11 4. 11 11 11 11 00 11 00 11 5. 11 11 11 11 11 11 11 11 6. 11 11 11 11 11 11 11 01 7. 11 11 11 11 11 11 11 11 8. 11 11 11 11 11 11 11 11 9. 11 11 11 11 11 11 11 11 10. 11 11 11 01 11 11 11 11 11. 11 11 11 11 01 11 11 11 12. 11 11 11 11 11 11 11 11 13. 01 01 01 01 01 01 01 01 14. 11 11 11 11 11 11 11 11 15. 11 11 11 11 11 11 11 11 16. 11 11 11 11 11 11 11 11 17. 11 11 11 11 11 11 11 11 18. 11 11 11 11 11 11 11 11 19. 11 11 11 11 11 11 11 11 20. 11 11 11 11 11 11 11 11 (g) DAY THREE INDIVIDUAL-TAUGHT GROUP ITEM NUMBER STUDENT NUMBER 1 2 3 4 5 6 1. 11 11 11 11 11 11 11 11 2. 11 01 00 11 00 11 11 01 3. 11 11 11 11 11 11 11 01 4. 11 11 11 11 01 11 11 11 5. 11 01 11 01 01 11 11 01 6. 11 11 11 11 11 11 11 11 7. 11 01 11 11 11 11 11 11 8. 11 11 11 11 11 11 11 11 9. 11 11 11 11 11 11 11 11 10. 11 01 01 11 11 11 11 01 11. 11 11 11 11 11 11 11 11 12. 11 11 11 11 01 11 11 01 13. 11 11 11 11 11 11 11 11 14. 11 11 11 11 11 11 11 11 15. 11 11 11 11 11 11 11 11 16. 11 11 11 11 11 11 11 11 17. 11 11 11 11 11 11 11 01 18. 11 11 11 11 11 11 11 11 19. 11 11 11 11 11 00 11 00 20. 11 11 11 01 01 01 11 01 (h) DAY ONE GROUP-TAUGHT GROUP STUDENT ITEM NUMBER NUMBER 1 2 3 4 5 6 7 8 1. 11 11 11 11 11 11 11 11 2. 11 11 11 11 11 11 11 11 3. 11 11 11 11 11 11 11 11 4. 11 11 11 11 11 11 11 11 5. 11 11 11 11 11 11 11 11 6. •7 11 11 11 01 11 11 11 11 / • 8. 11 11 11 11 11 11 11 11 9. 11 11 11 11 11 11 11 11 10. 11 11 11 11 11 11 11 11 11. 11 11 11 11 11 11 11 11 12. 11 11 11 11 11 11 11 11 13. 11 11 11 11 11 11 11 11 14. 11 11 11 11 11 11 11 11 15. 11 11 11 11 11 11 11 11 16. 11 11 11 11 11 11 11 11 17. 01 11 11 11 11 11 11 01 18. 19. 11 11 11 11 11 11 11 11 20. 11 11 11 11 11 11 11 11 71. (i) DAY TWO GROUP-TAUGHT GROUP STUDENT ITEM NUMBER NUMBER 1 2 3 4 - 5 6 7 8 1. 11 11 11 11 11 11 11 11 2. 11 11 11 11 11 11 11 11 3. 11 11 11 11 11 11 11 11 4. 11 11 11 11 11 01 11 11 5. 11 11 11 11 11 11 11 11 6. 11 11 11 11 11 11 11 11 7. 11 11 11 11 11 11 11 11 8. 11 11 11 11 11 11 11 11 9. 11 11 11 11 11 11 11 11 10. 11 11 11 11 11 11 11 11 11. 11 11 11 11 11 11 11 11 12. 11 11 01 11 11 01 11 11 13. 11 11 11 11 11 11 11 11 14. 11 11 11 11 11 11 11 11 15. 11 11 11 11 11 11 11 11 16. 11 11 11 11 11 11 11 11 17. 11 11 11 11 11 11 01 11 18. 11 11 11 11 01 11 11 11 19. 20. 11 11 11 11 11 11 11 11 (j) DAY THREE GROUP-TAUGHT GROUP STUDENT ITEM NUMBER NUMBER 1 2 3 4 5 6 7 8 1. 11 11 11 11 11 11 11 11 2. 11 11 11 11 11 11 11 11 3. 11 11 11 11 11 11 11 11 4. 11 11 01 11 11 11 01 00 5. 11 11 11 11 11 00 00 00 6. 11 11 10 11 11 10 10 11 7. 11 11 10 11 11 11 11 01 8. 11 01 01 11 11 11 11 11 9. 11 11 11 00 11 11 01 11 10. 11 01 01 11. 11 01 11 11 11. 11 11 11 11 11 01 11 11 12. 11 01 11 01 01 11 11 01 13. 11 11 11 11 11 11 11 11 14. 11 11 11 11 11 11 11 11 15. 11 11 11 01 11 11 11 01 16. 11 11 11 11 11 11 11 11 17. 11 11 11 11 11 11 11 11 18. 11 11 11 01 11 11 11 11 19. 11 11 11 11 01 11 11 11 20. 11 11 11 11 01 11 11 11 

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