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An investigation of the effects of convergent/divergent teaching methods on the mathematical problem-solving… Koe, Carryl Diane 1979

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AN INVESTIGATION OF THE EFFECTS OF CONVERGENT/DIVERGENT TEACHING METHODS ON THE MATHEMATICAL PROBLEM-SOLVING ABILITIES OF GRADE TEN STUDENTS by CARRYL DIANE KOE B. Sc., University of C a l i f o r n i a , Davis, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Department ofEducation[Mathe_atics]) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, 197 9 (c) Carryl Diane Koe, 1979 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I a g ree t h a t the L ibrary shal l make it f ree ly ava i l ab le for r e f e r e n c e and s t u d y . I further agree that permission for extensive copying o f th i s thesis for scho lar ly purposes may be granted by the Head o f my Department o r by his representat ives. It is understood that copying o r p u b l i c a t i o n o f th is thesis for f inanc ia l gain sha l l not be allowed without my written permission. Department of E d u c a t i o n The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 ABSTRACT Research Supervisor: Dr. James M. S h e r r i l l I t was the purpose of t h i s study to investigate the e f f e c t s of convergent/divergent teaching methods on student performance on two mathematical problem solving tasks (routine/non-routine problems). A concurrent purpose was to investigate the in t e r a c t i o n between the convergent/ divergent teaching methods and the thinking s t y l e (either convergent or divergent) of the learner. Four grade ten classes were randomly selected from the eleven academic mathematics classes i n the secondary school involved i n the study. Due to subject absenteeism a t o t a l of s i x t y - s i x subjects were used for the analyses. Each subject was given the Watson-Glaser Test of C r i t i c a l Thinking (Form YM) and the Torrance test of Thinking Creatively With Words (Booklet A) to determine t h e i r l e v e l on the independent measures of convergent and divergent thinking, respectively. Each subject was taught by one teacher using one method for approximately two hours. The content of these lessons involved the Fibonacci Sequence and Pascal's Triangle. At the end of treatment, each subject received a test on the dependent measures Croutine/non-routine problems). Trained observers were used to ensure consistency of teaching method. Analysis of covariance using the regression model was performed with convergent/divergent thinking styles as the covariates. There was no s i g n i f i c a n t difference between convergent teaching methods and divergent teaching methods (p f__ 0.05) . Convergent thinkers scored s i g n i f i c a n t l y higher than did divergent thinkers on both dependent measures. However, as convergent thinking i s far more highly correlated with i n t e l l i g e n c e than i s divergent thinking, t h i s r e s u l t may have been confounded by i n t e l l i g e n c e . Therefore, i n further studies i n t h i s area, the variance i n problem solving due to i n t e l l i g e n c e should be p a r t i a l l e d out. Only one of eight in t e r a c t i o n e f f e c t s was s i g n i f i c a n t (p _r 0.05). This suggested that non-divergent thinkers did better with convergent (as opposed to divergent) teaching methods and that non-convergent thinkers did better with divergent (as opposed to convergent) teaching methods. The lack of other s i g n i f i c a n t interactions indicated that i n t e l l i g e n c e may have been a confounding e f f e c t i n t h i s study. i v TABLE OF CONTENTS page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENT v i i i Chapter I. THE PROBLEM 1 Background of the Study 3 Statement of the Problem 7 Research Questions 7 II. REVIEW OF THE PERTINENT LITERATURE 11 Convergent/Divergent Thinking 11 Problem Solving i n Mathematics 18 Interaction between Problem Solving and Teaching Methods i n Mathematics 20 Summary 2 5 I I I . DESIGN OF THE STUDY 27 De f i n i t i o n of Terms 27 Sample 35 Teachers 36 Procedures 38 S t a t i s t i c a l Hypotheses 41 V Chapter page IV. ANALYSES AND RESULTS 44 Confounding Ef f e c t s 44 Results of the Study 6 3 V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 82 Summary of the Study 82 Findings 83 Implications of the Study 8 6 Limitations of the Study 87 Recommendations for Further Research 8 7 LITERATURE CITED 90 APPENDICES A. CONVERGENT TEACHING LESSONS 96 B. DIVERGENT TEACHING LESSONS 105 C. ROUTINE PROBLEMS 110 D. NON-ROUTINE PROBLEMS AND KEY FOR DIVERGENT / QUESTION IN NON-ROUTINE PROBLEMS 114 E. EXAMPLE OF STUDENT'S NOTES 119 v i LIST OF TABLES Table page 3.1 Design 38 4.1 Means and Standard Deviations for Teacher E f f e c t s on Dependent Measures 46 4.2 Results of t - tests for Teacher Ef f e c t s on Dependent Measures 47 4.3 Means and Standard Deviations for Day Ef f e c t s on Dependent Measures 49 4.4 Results of t - tests for Day Effe c t s on Dependent Measures 50 4.5 Observer's Checklist 52 4.6 Means and Standard Deviations for Watson-Glaser and Torrance Tests 66 4.7 Group Comparison of Means on Watson-Glaser and Torrance Tests 67 4.8 Means and Standard Deviations of A l l Variables 70 4.9 Correlation Matrix for Dependent and Independent Variables 71 4.10 Results of Regression Analysis with Routine Problems as Dependent Variable (ANOVA) 76 4.11 Results of Regression Analysis with Non-Routine Problems as Dependent Variable (ANOVA) . 77 4.12 Results of Regression Analysis with Total Problem Set as Dependent Variable (ANOVA) 78 4.13 Results of Regression Analysis with Routine Problems as Dependent Variable (ANOCOVA) 79 4.14 Results of Regression Analysis with Non-Routine Problems as Dependent Variable (ANOCOVA) 80 4.15 Results of Regression Analysis with Total Problem Set as Dependent Variable (ANOCOVA) 81 v i i LIST OF FIGURES Figure page 2 . 1 Guilford's Model of the I n t e l l e c t 1 2 ACKNOWLEDGEMENT I wish to thank the members of my thesis committee, Dr. David R o b i t a i l l e and Dr. G a i l S p i t l e r , for t h e i r guidance and support. I p a r t i c u l a r l y wish to thank my chairman, Dr. James S h e r r i l l , for his unending patience and urging which made the completion of t h i s thesis possible. I would also l i k e to thank several people who have contributed i n various ways to t h i s t h e s i s : Dr. Robert Conry, for his help i n the s t a t i s t i c a l analysis; the students and teachers involved i n the study, for t h e i r cooperation; and my husband, Gerald Koe, who while working on his own di s s e r t a t i o n has found the time to give both encouragement and support. 1 CHAPTER I THE PROBLEM Many teachers of mathematics are concerned with finding better teaching methods, in p a r t i c u l a r better methods of teaching problem solving. Suggested methods have ranged from d r i l l i n g students on p a r t i c u l a r classes of problems, to programmed learning, where a student i s lead by small steps to the desired conclusions, and to discovery learning, where examples are presented i n such numbers that the student discovers a correct method of solution. The difference i n these methods i s a major philosophical issue i n education: the difference between i n s t r u c t i o n and teaching. The d i s t i n c t i o n i s e s s e n t i a l l y one of d i r e c t i v e versus discovery teaching. It i s generally assumed that to ensure the best transfer of t r a i n i n g , the method should correspond with the intended product. Kersh (1958) aptly states: If meaningful learning i s the key concept, i t should make no difference whether learning occurs with or without d i r e c t i o n , so long as the learner becomes cognizant of the e s s e n t i a l relationships. However, some learning procedures may be superior to others simply because they are more l i k e l y to cause the learner to become cognizant of the relationships, (p. 282) In the case of problem solving i t i s e s s e n t i a l that the learner develop methods or processes for attacking problems. The m u l t i p l i c i t y of possible problems precludes teaching problem solving from a d i r e c t i v e approach. Gagne 2 (1966) p o i n t s out: Problem s o l v i n g , when c o n s i d e r e d as a form of l e a r n i n g , r e q u i r e s d i s c o v e r y , s i n c e the l e a r n e r i s expected to generate a novel combination of p r e v i o u s l y l e a r n e d p r i n c i p l e s , (p. 150) The emphasis i n on the l e a r n e r combining h i s l e a r n e d s k i l l s to become a problem s o l v e r . The problem f o r the teacher i s how to e f f e c t i v e l y f a c i l i t a t e these combinations of p r e v i o u s i n f o r m a t i o n so t h a t the l e a r n e r w i l l become an e f f i c i e n t problem s o l v e r . "There i s no c o n v i n c i n g evidence t h a t one can l e a r n ^ t o be a d i s c o v e r e r , i n a g e n e r a l sense; but the q u e s t i o n remains an open one." (Gagne, 1966, p. 150). Many s t u d i e s have been done to t r y to determine what f a c t o r s a f f e c t the t e a c h i n g of problem s o l v i n g and, hence, help to c r e a t e b e t t e r problem s o l v e r s . Memory (Gagne, 1960) and IQ ( K l i n e , 1960) of the l e a r n e r , p e r s o n a l i t y of the teacher (Torrance, 1962; McNary, 1967), m o t i v a t i o n a l l e v e l o f the l e a r n e r (Kersh, 1958; Brown, 1975) and c o g n i t i v e s t y l e ( M e r r i f i e l d , e t . a l . , 1960, 1967) seem to be some of the f a c t o r s which i n f l u e n c e the t e a c h i n g of problem s o l v i n g i n mathematics. One f a c t o r t h a t has been i n v e s t i g a t e d i s the r e l a t i o n s h i p of c r e a t i v i t y to problem s o l v i n g . Many s t u d i e s have been done i n v o l v i n g c r e a t i v i t y and problem s o l v i n g ( C l a r k , 1967; Behr, 1970; Maier, 1970; Ruse, e t . a l . , 1976). These s t u d i e s show t h a t c r e a t i v i t y has a p o s i t i v e r e l a t i o n s h i p to problem s o l v i n g . T h i s lends support to Gagne's p r e v i o u s l y s t a t e d p o s i t i o n . 3 However, one c r i t i c a l factor as yet uninvestigated i s the interaction between c r e a t i v i t y and teaching method: s p e c i f i c a l l y , between the thinking style of the learner and the teaching method of the teacher. Perhaps one type of thinker i s a better problem solver, no matter i n what teaching s i t u a t i o n he/she i s involved. Perhaps one type of teaching method tends to create better problem solvers, regardless of i n i t i a l differences i n thinking s t y l e . Perhaps i t i s the inte r a c t i o n between thinking style and teaching method which i s the important factor i n producing better problem solvers. It i s to t h i s l a s t premise that t h i s research i s addressed. Background of the Study In teaching mathematics, one endeavours to transmit both facts and processes. The teacher hopes that the student, through the use of these tools, w i l l be able to recognize and solve problems which occur i n his d a i l y l i f e which are mathematical i n nature. This expectation of mathematics teachers does not simply mean the a b i l i t y of students to achieve correct answers for specialized mathematical textbook problems. It rather purports that " ... the re a l aim of learning mathematics l i e s i n the a b i l i t y to apply i t s methods to new situations." (Avital and Shettleworth, 1968, p. 3). Curriculum guides and various study groups (CEEB, 1959, p. 2; CCSM, 1963, p. 7) l i s t problem solving as one of the major outcomes of the mathematics curriculum. As Lucas (1974) states: 4 The basis of mathematics i s problem solving; therefore, i f the cause of mathematical education i s to be served, e f f e c t i v e means of teaching problem solving must be c l a r i f i e d , (p. 45) Before e f f e c t i v e teaching of problem solving can occur, i t i s necessary to determine what problem solving i n mathematics e n t a i l s . Scandura (1968) states that: ... most meaningful learning including problem solving, may involve the recom-bination of previously learned rules into new patterns ... In e f f e c t , that problem solving be viewed simply as a form of transfer, (p. 9) This transfer may just involve synthesis of relevant data or i t may involve the a b i l i t y to take the pertinent data and test i t against many models (either previously e x i s t i n g or being created to meet the need) which w i l l o f f e r possible solutions for the given problem. If transfer i s a necessary component of problem solving, as Guilford (1965) suggests, then thinking styles which enhance transfer should have a positi v e e f f e c t on problem-solving a b i l i t y . Therefore, students who possess cert a i n c h a r a c t e r i s t i c s - divergent thinking, f l e x i b i l i t y , synthesis or convergent thinking -(Guilford, 1965) should be better problem solvers than those who do not possess these c h a r a c t e r i s t i c s or a combination thereof. M e r r i f i e l d , Christensen, Guilford and Fric k (1960) showed that divergent production of semantic transformation (hereafter termed o r i g i n a l i t y ) had a high c o r r e l a t i o n with (1) the a b i l i t y to think of attributes of a desired goal 5 and (2) the a b i l i t y to deduce l o g i c a l l y s u f f i c i e n t antecedents. Problem solving can be thought of as, f i r s t , an awareness that there i s a problem; secondly, a thinking of many possible solutions; and, t h i r d l y , a decision as to the p l a u s i b i l i t y and effectiveness of these solutions (Feldhusen and Treffinger, 1977). I t would seem that a good problem solver exhibits both a b i l i t i e s as described by M e r r i f i e l d , et. a l . (1960). Therefore o r i g i n a l i t y would appear to be one c h a r a c t i s t i c of the good problem solver. Perhaps by promoting o r i g i n a l i t y i n the classroom, problem solving a b i l i t i e s w i l l improve. If t h i s i s the case, one should t r y to create an atmosphere i n mathematics classrooms whereby more students w i l l be encouraged to think divergently as well as to display divergent thinking processes. Producing better problem solvers i s one of the major goals of school learning (Ausubel, 1963; Dirkes, 1975). However, the methods to best achieve t h i s goal are somewhat elusive. Skemp (1971) suggests that the manner of presentation of mathematics should be f i t t e d to the mode of thinking of the learner. While Skemp was making reference to Piagetian lev e l s of operation, i t i s also possible to apply t h i s notion to other models such as Guilford's (1957,1960) Model of the I n t e l l e c t . This could best be exemplified through the convergent/divergent thinking aspect of the Operations axis in Guilford's (1965) model (See Figure 2.1). Gagne (1960) proposed that one way to test Guilford's (1957) Model of the I n t e l l e c t i s to experimentally show that 6 people who have scores on the Contents axis - f i g u r a l , symbolic and semantic - learn better i f content i s presented i n these modes. However, as problem solving i s a process, i t seems more appropriate to use the Operations axis as the source of experimental manipulation. Taylor (1965) suggests that: ... teacher t r a i n i n g programs could focus in turn on various kinds of thinking i n students as well as on subject matter; and we are advocating that both of these kinds of learning should happen simultane-ously i n the classroom, (p. 261) These t h e o r e t i c a l considerations suggest a need to devise a way to test whether one method of teaching i s more ef f e c t i v e i n creating divergent thinkers. However, there i s l i t t l e guarantee that t r a i n i n g teachers i n divergent thinking processes w i l l produce divergent thinking i n t h e i r students. As Hutchinson (1965) pointedly remarks: ... we are faced with the challenge of how to t r a i n teachers so that t h e i r students w i l l display primarily divergent thinking and remain at t h i s thinking l e v e l without hurrying on through convergent to evaluative thinking, (p. 263) Before looking for the means of t r a i n i n g teachers, one f i r s t needs to ensure that teaching w i l l indeed influence student thinking. McNary (1967) indicates that cert a i n teacher c h a r a c t e r i s t i c s did e f f e c t both convergent and divergent thinking of g i f t e d elementary students. Convergent areas were most effected by teachers who were submissive, dependent, a l e r t , cheerful, and seemed to have a natural warmth and l i k i n g of people; while divergent areas were most effected by teachers who were presistent, energetic, 7 emotionally mature, f r i e n d l y , and without fixed methods for gaining s o c i a l approval. The Sutherlin Program (1964) indicates that c r e a t i v i t y i s nurtured i n a student by matching him with a teacher who i s creative. The two studies indicate that teachers can influence students' patterns of thought. I t i s further necessary to determine the r e l a t i o n -ship between teaching methods and student thinking s t y l e s . Based on the l i t e r a t u r e , the following were assumed to be true for the present study: (1) increasing the mathematical problem-solving a b i l i t y of students i s one of the most important tasks set before mathematics teachers and (2) teaching method and thinking style of the learner are factors that may a f f e c t problem-solving s k i l l s i n the mathematics classroom. Given these two assumptions, the present study was designed to investigate two s p e c i f i c teaching methods and t h e i r two corresponding thinking st y l e s . Statement of the Problem The present study has been designed to investigate the relationship between convergent/divergent teaching methods and student performance on a problem solving task. The relationship between these two teaching methods and the students' thinking style (either convergent or divergent) has also been investigated. Research Questions Some studies (Dahmus, 1970; Campbell, 1964) suggest that a directed approach to learning enhances problem-8 solving a b i l i t i e s . Other studies (Torrance, 1962; Flanders, 1965; Clark, 1967) claim that an open-ended approach to learning has a greater influence on the development of problem-solving a b i l i t i e s . Taylor (1965) and Prouse (1967) suggest that both approaches are necessary. To c l a r i f y these c o n f l i c t i n g research findings, the following research question was asked: 1. For which teaching style (convergent/ divergent) do subjects score s i g n i f i c a n t l y more correct answers on the dependent measures? The teaching styles - convergent and divergent - have been defined to r e f l e c t consonance with convergent production and divergent production as defined by Guilford (1960, 1965). Convergent teaching emphasizes correctness of response, v a l i d i t y of inference from given choices and suggestions made by the teacher and correctness of interpretations from a limited group of possible alternatives. It i s teacher-centered and the teacher asks leading questions which w i l l lead the students to the desired response. Divergent teaching emphasizes the thinking process involved i n the problem-solving process, making hypotheses, u t i l i z i n g problem-solving tools and recombining the various hypotheses to form new hypotheses. It i s a student-centered approach and the teacher acts as a catalyst, providing open-ended questions, l i s t i n g student suggestions and rephrasing the question to stimulate further thought. Maier and Janzen (1969) suggest that creative people 9 are " ... superior problem solvers i n that they are more l i k e l y to f i n d correct answers to d i f f i c u l t problems."(p. 100). This would seem to indicate that divergent thinkers (ones who display a high degree of fluency, o r i g i n a l i t y , and elaboration) would be better problem solvers than convergent thinkers (who have a good f a c i l i t y for transforming and redefining problems and for making v a l i d inferences and choosing the best a l t e r n a t i v e ) . However, as Treffinger, Renzulli and Feldhusen (1971) point out: If c r e a t i v i t y i s viewed as a complex kind of human problem-solving (in which case perhaps the term "creative problem solving" would be preferrable), divergent thinking may be a necessary, although not s u f f i c i e n t , component, (p. 108) This b a s i c a l l y implies that a non-divergent thinker i s not necessarily a convergent thinker and, conversely, that a non-convergent thinker i s not necessarily a divergent thinker. As these thinking types may have some overlap and as both may be necessary components of the good problem solver, the following question i s posed: 2. For which thinking style (convergent/ divergent) do subjects score s i g n i f i c a n t l y more correct answers on the dependent measures? While many studies have dealt singularly with either thinking style or teaching method, few have t r i e d to look at the c o r r e l a t i o n between the two. One such study by Vos (1976) showed that low a b i l i t y students performed better when " ... emphasis on presenting a problem and reviewing 10 past knowledge that may be helpful for the problem s i t u a t i o n followed by s p e c i f i c i n s t r u c t i o n i n the concept ..." (p. 274) was used as the teaching procedure. Vos further showed that high a b i l i t y students performed better when the teaching procedure involved only i n s t r u c t i o n and the problem-solving task, rather than these two procedures combined with review of past knowledge which might be helpful to solving the problem. While the Vos study shows an inte r a c t i o n e f f e c t between a b i l i t y and methodology, both methods incorporated an advanced- and post-organizer which used together c e r t a i n l y confounded the eff e c t s (Ausubel and Fi t z g e r a l d , 1962). Skemp (1971) suggests that teachers should f i t t h e i r teaching to the learner's mathematical schemas and that the manner of presentation should be consonant with the learner's mode of thinking. I t was t h i s suggestion which prompted the following research question: 3. Does a s t a t i s t i c a l l y s i g n i f i c a n t i n t e r a c t i o n e x i s t between teaching method and student thinking style (convergent/divergent) i n terms of students' a b i l i t y to solve problems on the dependent measures? It was hoped that the comparison of divergent production and convergent production i n both the process of teaching and the process of thinking would provide some insight into the p a r t i t i o n i n g of the variance i n problem-solving a b i l i t i e s . I t was further hoped that by allowing these two operations to range over the Contents and Products axes (Guilford, 1960,1965) that s i g n i f i c a n t differences would r e s u l t at the p i s 0.05 l e v e l . CHAPTER I I REVIEW OF THE PERTINENT LITERATURE The review of the l i t e r a t u r e i s d i v i d e d i n t o three major areas of i n v e s t i g a t i o n : (1 ) Convergent/Divergent T h i n k i n g , (2) Problem S o l v i n g i n Mathematics and ( 3 ) I n t e r a c t i o n between Problem S o l v i n g and Teaching Methods i n Mathematics. A summary s e c t i o n i s prov i d e d to focus on the i n t e r r e l a t i o n s h i p s among these areas and to d e p i c t how the r e s e a r c h and l i t e r a t u r e have formed a b a s i s f o r the present study. Convergent/Divergent T h i n k i n g The whole concept of convergent and d i v e r g e n t t h i n k i n g has i t s b a s i s i n G u i l f o r d ' s ( 1 9 5 7 , 1 9 6 0 , 1 9 6 5 ) Model of the I n t e l l e c t . T h i s model c o n t a i n s three major axes: (1 ) Operations, (2) Contents, and (3 ) Products. Each o f these axes c o n t a i n s c a t e g o r i e s as i l l u s t r a t e d i n F i g u r e 2 . 1 . These axes and c a t e g o r i e s are d e f i n e d i n G u i l f o r d and M e r r i f i e l d ( 1 9 6 0 ) . Operations are i n t e l l e c t u a l types of processes o f "thing s t h a t the organixm does with the raw m a t e r i a l s o f i n f o r m a t i o n . " (p. 5 ) . The Operations a x i s has f i v e c a t e g o r i e s : c o g n i t i o n ( r e c o g n i t i o n of i n f o r m a t i o n ) , memory ( r e t e n t i o n o f i n f o r m a t i o n ) , d i v e r g e n t p r o d u c t i o n (generating a v a r i e t y o f output from the same sour c e ) , convergent p r o d u c t i o n (generating unique or best outcomes from the giv e n i n f o r m a t i o n ) , and e v a l u a t i o n . The Contents a x i s i s d e f i n e d to be gene r a l v a r i e t i e s of i n f o r m a t i o n . I t con t a i n s f o u r c a t e g o r i e s : f i g u r a l (images), symbolic (signs 1 2 OPERATIONS CONTENTS FIGURE 2 . 1 GUILFORD'S MODEL OF THE INTELLECT and words), semantic (verbal) and behavioral (non-verbal). The Products axis i s defined as "results from the organism's processing of information."(ibid., p. 5). It contains six categories: units (segrated items), classes (items grouped by a common property), relations (recognized connections between uni t s ) , systems (structured aggregates of information), trans-formation (changes i n production) and implications (predictions, extrapolations of information). It i s the Operations axis wherein the a b i l i t i e s involved in learning w i l l be found. "The concept of thinking, i t s e l f , i s to be applied to the three operation categories of divergent production, convergent production, and evaluation." (Guilford and M e r r i f i e l d , 1960, p. 15). Furthermore, creative thinking " c l e a r l y points toward the category of divergent production" ( i b i d . , p. 11) which includes the factors of fluency, f l e x i b i l i t y o r i g i n a l i t y , and elaboration. While Guilford and M e r r i f i e l d (1960) f i r s t t r y to equate creative thinking and divergent production, they l a t e r l i m i t t h e i r statement by commenting that some aspects of creative thinking may involve some convergent production and even evaluation. However, convergent production has more to do with deduction which "implies the drawing of conclusions or the making of inferences ..." ( i b i d . , p. 10). Guilford's (1957, 1960, 1965) Model of the I n t e l l e c t has been used to relate convergent/divergent thinking to problem solving i n the studies by M e r r i f i e l d , Guilford, Christensen 14 and Frick (1960,1967). Their findings suggested which attributes of thinking were accounted for by p a r t i c u l a r blocks within the model. The a b i l i t y to c l a s s i f y objects of semantic classes and the a b i l i t y to fi n d d i f f e r e n t relationships were accounted for by divergent production of semantic units as well as convergent production of semantic classes and implications. Other a b i l i t i e s involved i n the process of problem solving (thinking rapidly of attributes, thinking of alternative outcomes, thinking of attributes of a desired goal and deducing l o g i c a l l y s u f f i c i e n t antecedents) were accounted for by allowing the Product axis to range over i t s p o s s i b i l i t i e s for divergent production x semantic content (Refer to Figure 2.1). Behr (1970) used Guilford's (1965) Model of the I n t e l l e c t i n presenting verbal and f i g u r a l teaching methods of modular arithmetic. Cognition of semantic relations and both teaching methods had a s i g n i f i c a n t interaction (p 0.05) with tests of knowledge of structure. While these results do not d i r e c t l y relate to the thesis of t h i s study, they do show that some aspects of problem solving i n mathematics are affected by teaching. Behr (1970) further suggests: If the factors that suggest a b i l i t y to succeed at "higher order" learning situations can be i d e n t i f i e d , information from tests which measure these factors might be helpful i n predicting a student's chance of being successful at various educational l e v e l s , (p. 39) 15 As s t a t e d e a r l i e r , i t i s the premise of t h i s study t h a t convergent/divergent t h i n k i n g (and teaching) are f a c t o r s i n the process of improving problem-solving s k i l l s . While M e r r i f i e l d , e t . a l . have used a l l three axes of G u i l f o r d ' s (1965) Model of the I n t e l l e c t f o r t h e i r i n v e s t i g a t i o n s , t h i s study i s c o n f i n e d to one a x i s : the Operations a x i s . T h i s study e x p l o r e s the r e l a t i o n s h i p between convergent/divergent t e a c h i n g methods (Refer to Teaching Methods, t h i s chapter and D e f i n i t i o n of Terms s e c t i o n i n Chapter III) and convergent/divergent thought processes i n students (Refer to D e f i n i t i o n of Terms s e c t i o n i n Chapter I I I ) . I t was f e l t t h a t not r e s t i r c t i n g these two o p e r a t i o n s to any p a r t i c u l a r category of the Contents and Products axes would p r o v i d e a f i r s t step towards de v e l o p i n g c h a r a c t e r i s t i c o p e r a t i o n s i n v o l v e d i n the process of problem s o l v i n g i n mathematics. In a study i n v o l v i n g male undergraduate students, s i x t e e n of whom had high scores on the Remote A s s o c i a t e s T e s t and another s i x t e e n who had low s c o r e s , K l e i n and K e l l n e r (1967) found t h a t " h i g h l y c r e a t i v e " students scored s i g n i f i c a n t l y higher than "low c r e a t i v e " students i n f i n d i n g a p a t t e r n i n a p r o b a b i l i t y c h o i c e problem. The i n v e s t i g a t o r s p o i n t e d out t h a t the "high c r e a t i v e " took more time to s h i f t c h o i c e of s o l u t i o n than d i d the "low c r e a t i v e " . T h i s means t h a t once the optimal s o l u t i o n had been reached, the "high c r e a t i v e " were more l i k e l y to m a i n t a i n t h a t p a t t e r n than were the "low c r e a t i v e " who would o f t e n switch to l e s s 16 optimal choices. Ideational fluency appears to be a c r i t i c a l variable i n d i f f e r e n t i a t i n g between creative and non-creative students. In a study investigating the relationship between verbal response heirarchies and problem solving, Staats (1957) reported a positive c o r r e l a t i o n between fluency of verbal response (fluency of verbal response i s one of the measures of divergent thinking as defined by Guilford (1965)) and problem-solving time. This finding, together with those of M e r r i f i e l d , Guilford, Christensen and Fr i c k (1960, 1967) and Behr (1970), has led to the choice of the Torrance test of Thinking Creatively with Words (Booklet A) (hereafter referred to as the Torrance test) as a measure of divergent thinking for the present study. The Torrance test provides r e l i a b l e measures of fluency, f l e x i b i l i t y , and o r i g i n a l i t y of verbal problem solving. The Torrance tests started at the elementary l e v e l and an entire battery of tests developed through the f i r s t year college l e v e l . This progression of test development seemed to provide a stronger base for testing junior high subjects than would the Guilford battery which started at the adult l e v e l and developed downward into the high school area. Prouse (1967) developed divergent and convergent thinking tests, i n mathematics and found: Correlation c o e f f i c i e n t s between scores on the divergent-thinking items and c r e a t i v i t y test composite scores (one of Guilford's) ranged from 0.10 to 0.64, while c o r r e l a t i o n 17 c o e f f i c i e n t s between scores on the convergent-thinking items and c r e a t i v i t y scores ranged from 0.01 to 0.23. This would seem to corroborate Guilford's assertion that the more prominent creative a b i l i t i e s appear to be concentrated i n the divergent-thinking category, (p. 879) While the content of the tests i n the Prouse study were inappropriate for the age l e v e l involved i n the present study, the correlations show indicate that divergent thinking may be used as a measure of c r e a t i v i t y , and therefore reinforces the choice of the use of the Torrance test. In a comprehensive review of a b i l i t y and c r e a t i v i t y i n mathematics, Aiken (1973) discusses both general and mathematical measures of c r e a t i v i t y . He suggests that as IQ tends to be a c a t c h - a l l for most s i g n i f i c a n t differences attributed to the c r e a t i v i t y factor, choosing measures which have low correlations with IQ w i l l help i n eliminating t h i s problem. This suggests that the differences occurring in studies of convergent/divergent thinking i n mathematical problem solving could be attributed to IQ differences, rather than c r e a t i v i t y factors. Again, t h i s i s a j u s t i f i c a t i o n for using the Torrance test as the most appropriate measure of divergent thinking. Studies involving correlations between the Torrance test and IQ show correlations ranging anywhere from 0.16 to 0.32. Of the available tests of convergent thinking, the Watson-Glaser Test of C r i t i c a l Thinking (Form YM) (hereafter referred to as the Watson-Glaser test) was chosen for two primary reasons: (1) i t s constructs of C r i t i c a l Thinking (inference, recognition of assumptions, deduction, i n t e r -pretation and evaluation of arguments) most cl o s e l y resemble Guilford's (1960) convergent production category, and (2) i t has the lowest c o r r e l a t i o n (0.55 - 0.73) with IQ (other tests of convergent thinking had correlations of 0.80 or bett e r ) . Problem Solving i n Mathematics K i l p a t r i c k (1969) reviewed problem solving and creative behaviours i n mathematics ans summarized the various findings by stating: " I t appears that c r e a t i v i t y , though i t may be related to certain facets of problem solving, bears no simple relationship to problem-solving performance." (p. 168). Torrance (1966) suggests that perhaps i t i s the int e r a c t i o n between c r e a t i v i t y and learning modes which bears a more di r e c t relationship to problem solving than just measures of c r e a t i v i t y alone. What i s meant by a problem when one i s discussing problem solving i s often misunderstood. Duckner (1945) suggests a problem i s a circumstance occurring when someone has a goal but does not know how to obtain t h i s goal. Thus many of the so c a l l e d "problems" i n mathematics are r e a l l y not a problem as the student has read i l y available algorithms which he knows apply to t h i s p a r t i c u l a r "problem" s i t u a t i o n . When agreement has been reached as to what are problems, i t would be nice i f one could predict which students would 19 a good problem solver. K i l p a t r i c k (1969) found that: Subjects who attempted to set up equations (they had not yet had an algebra course) were s i g n i f i c a n t l y superior to the others on measures of quantitative a b i l i t y , mathematics achievement, word fluency, general reasoning, l o g i c a l reasoning, and a r e f l e c t i v e conceptual tempo, (p. 165) These subjects obtained higher scores on the problem-solving measures employed and thus students who were better problem solvers displayed superiority on both divergent (word fluency) and convergent ( l o g i c a l reasoning) measures. The question i s , "Which one of these two factors i s more s i g n i f i c a n t as a predictor of good problem solvers?". Scandura (1968) attempted to define problem solving and i t s c h a r a c t e r i s t i c s when he f i r s t pointed out that problem solving "may involve the recombination of previously learned rules into new patterns ... simply a form of transfer." (p.9). He then continued by saying that problem solving i s "more than selection and integration of previously learned rules." (p. 13). In solving a problem a student f i r s t needs to define the problem and then test t r i a l solutions. The development of t r i a l solutions i s one aspect of divergent thinking, while the selection of the most appropriate approach i s part of the convergent thinking process (Covington, 1968). It would seem that perhaps both factors are needed by the good problem solver. It i s int e r e s t i n g to note than many investigators (Gagne, 1959; M e r r i f i e l d , Guilford, Christensen, and Fr i c k , 1960,1967; Polya, 1965; Taba, 1965; Troutman, et. a l . , 1967) agree on the 20 g e n e r a l phases of the problem-solving p r o c e s s : p r e p a r a t i o n , a n a l y s i s , p r o d u c t i o n , v e r i f i c a t i o n , and r e a p p l i c a t i o n ( M e r r i f i e l d , G u i l f o r d , C h r i s t e n s e n , and F r i c k , 1960, p. 2). With t h i s k i n d of g e n e r a l consensus on the process of problem s o l v i n g , i t seems imperative to begin i n v e s t i g a t i o n of the f a c t o r s which c o n t r i b u t e to c r e a t i n g b e t t e r problem s o l v e r s i n mathematics and to i n c o r p o r a t e these f a c t o r s i n t o the e d u c a t i o n a l p r o c e s s . I n t e r a c t i o n between Problem S o l v i n g and  Teaching Methods i n Mathematics The two t e a c h i n g methods used i n the present study are d e f i n e d i n Chapter I I I . However, i t seems a p p r o p r i a t e to d e s c r i b e these two methods (convergent/divergent) p r i o r to p r e s e n t i n g the r e l a t e d l i t e r a t u r e . The convergent t e a c h i n g method u t i l i z e s teacher-formed a l t e r n a t i v e s , stepwise q u e s t i o n i n g and an emphasis on c o r r e c t s o l u t i o n s i n mathematical p r o b l e m - s o l v i n g s i t u a t i o n s . The teacher p r e s e n t s a l t e r n a t i v e methods f o r a t t a c k i n g the problem and asks the students to choose the one they f e e l i s the best. The emphasis i n on the student making c o n s i s t e n t i n f e r e n c e s and i n t e r p r e t a t i o n s of the a l t e r n a t i v e s the teacher i s p r e s e n t i n g . The answer to the problem, not the method of s o l u t i o n , i s emphasized. The d i v e r g e n t t e a c h i n g method u t i l i z e s open-ended q u e s t i o n i n g on the p a r t of the teacher w i t h an emphasis on "We have an i n t e r e s t i n g problem. What means do we have a v a i l a b l e to p o s s i b l y s o l v e t h i s problem?" a t t i t u d e . The teacher q u e s t i o n s the students so t h a t the students make suggestions how to proceed. These suggestions are l i s t e d and then t r i e d . The emphasis i s on the student u t i l i z i n g p r e v i o u s i n f o r m a t i o n r e g a r d i n g mathematical problems, recombining t h i s i n f o r m a t i o n to form new hypotheses about the s o l u t i o n of the given problem. The method of s o l u t i o n , not the answer to the problem, i s emphasized. Taba (1965) suggested t h a t the a b i l i t y to t h i n k i s not an automatic by-product of studying another's d i s c i p l i n e d thought p r o c e s s e s . She f u r t h e r p o i n t e d out t h a t r e f l e c t i v e t h i n k i n g i s not dependent on the volume of f a c t s presented. Freudenthal (1972) r e i n f o r c e d t h i s v i e p o i n t when he s t r e s s e d the need to a l l o w mathematics to be r e i n v e n t e d . The i m p l i c a t i o n f o r t e a c h i n g mathematics seems c l e a r : i f one wants to c r e a t e good problem s o l v e r s , one must al l o w the students to apply mathematics to the world around them. One of the ways to do t h i s seems to be to s t i m u l a t e d i v e r g e n t t h i n k i n g p a t t e r n s (Hutchinson, 1965). Lucas (1974) attempted to do t h i s through h e u r i s t i c t e a c h i n g of t h i r t y u n i v e r s i t y students e n r o l l e d i n two c a l c u l u s c l a s s e s . In the experimental group, problems were d i s c u s s e d to r e i n f o r c e the l e a r n i n g of p r o b l e m - s o l v i n g s t r a t e g i e s , w h ile i n the c o n t r o l group problems were d i s c u s s e d mainly to r e i n f o r c e mathematical concepts. Lucas found s i g n i f i c a n t d i f f e r e n c e s a t the p 0.05 l e v e l u s i n g a c h i -square a n a l y s i s on p r o b l e m - s o l v i n g performance scores between two groups i n favour of the experimental group. However, one should view the s i g n i f i c a n c e of these f i n d i n g s w i t h c a u t i o n due to the low n i n v o l v e d . L i k e Lucas, Gurova (19 69) found t h a t by making f i f t h - and s i x t h - g r a d e r s aware of the processes they used i n a c t u a l l y s o l v i n g problems, t h e i r p roblem-solving a b i l i t i e s improved. I t i s i n t e r e s t i n g to note t h a t r e c e n t s t u d i e s i n v o l v i n g a p t i t u d e - t r e a t m e n t - i n t e r a c t i o n have shown no s i g n i f i c a n t d i f f e r e n c e s f o r e i t h e r the treatment e f f e c t or the a p t i t u d e -t r e a t m e n t - i n t e r a c t i o n (ATI) e f f e c t . Post and Brennan (1976) used 47 p a i r s of s u b j e c t s i n comparing a General H e u r i s t i c Problem-Solving-Procedure and "normal" i n s t r u c t i o n i n geometry. While they found no s i g n i f i c a n t d i f f e r e n c e s f o r the treatment or the i n t e r a c t i o n between treatment and a b i l i t y l e v e l , they d i d suggest t h a t "... i d e n t i f i c a t i o n of ' t y p i c a l ' problem-s o l v i n g behaviors w i t h i n w e l l - d e f i n e d t o p i c a l domains and subsequent a t t e n t i o n to the development and maintenance of those b e h a v i o r s . " (p. 64) may be one way to approach f u t u r e r e s e a r c h i n the f i e l d o f problem s o l v i n g i n mathematics. The p r e s e n t study has attempted to look a t convergent/divergent t e a c h i n g methods and t h e i r subsequent e f f e c t on the p r o b l e m - s o l v i n g behaviours on r o u t i n e / n o n - r o u t i n e problems r e g a r d i n g sequences. Kantowski (1977) i n a study u s i n g e i g h t high a b i l i t y ninth-grade a l g e b r a students i n v e s t i g a t e d the use of h e u r i s t i c s i n problem s o l v i n g i n mathematics. The use of h e u r i s t i c s i s s i m i l a r to the d i v e r g e n t t e a c h i n g method i n t h a t both emphasize the process i n v o l v e d i n s o l v i n g problems as opposed to the product (or s o l u t i o n ) of the problem. She suggested t h a t "The e f f e c t s o f h e u r i s t i c i n s t r u c t i o n versus e x p o s i t o r y i n s t r u c t i o n should be i n v e s t i g a t e d w i t h the use of h e u r i s t i c s as the dependent v a r i a b l e . " (p. 1 7 5 ) . Eastman and Behr ( 1 9 7 7 ) attempted to do something s i m i l a r to t h i s , although they s t i l l used problem s o l v i n g as the dependent v a r i a b l e . T h e i r ATI study i n v o l v e d 2 0 8 ninth-grade a l g e b r a students u s i n g l o g i c a l i n f e r e n c e . The f i g u r a l - i n d u c t i v e group was taught u s i n g f i g u r a l - i n d u c t i v e programmed m a t e r i a l and the symbolic-deductive group was taught u s i n g symbolic-deductive m a t e r i a l . The s u b j e c t s had n i n e t y minutes to study the programmed m a t e r i a l and were t e s t e d both one day l a t e r and two weeks l a t e r . None of the problems used i n the t e s t s were d e f i n e d to be u n f a m i l i a r ( n o n - r o u t i n e ) . No s i g n i f i c a n t d i f f e r e n c e s were found. However, the authors presented q u e s t i o n s r e g a r d i n g : "What a p t i t u d e s are a p p r o p r i a t e f o r p r e d i c t i n g d i f f e r e n t i a l achievement i n mathematics l e a r n i n g ? " (p. 3 8 1 ) . I t i s the premise of the present study t h a t perhaps convergent/divergent, while c l o s e to d e d u c t i v e / i n d u c t i v e , are more accurate d e s c r i p t o r s o f the a c t u a l d i f f e r e n t i a l thought processes of students and may more a c c u r a t e l y d e s c r i b e the b a s i c d i f f e r e n c e s found i n the t e a c h i n g process of mathematical problem s o l v i n g . I t i s i n t e r e s t i n g t h a t the Eastman and Behr ( 1 9 7 7 ) study used t e a c h i n g m a t e r i a l which corresponded to the t h i n k i n g s t y l e o f the l e a r n e r . T h i s i d e a i s i n c o r p o r a t e d i n Skemp's ( 1 9 7 1 ) suggestion t h a t the task of the mathematics 24 teacher i s t h r e e - f o l d : He must f i t the mathematical m a t e r i a l to the s t a t e of development of the l e a r n e r s 1 mathematical schema; he must a l s o f i t h i s manner of p r e s e n t a t i o n to the modes of t h i n k i n g ( i n t u i t i v e and concrete reasoning or i n t u i t i v e , c oncrete reasoning and a l s o formal t h i n k i n g ) o f which h i s p u p i l s are capable; and f i n a l l y he must be g r a d u a l l y i n c r e a s i n g t h e i r a n a l y t i c a b i l i t i e s to the stage a t which they no longer depend on him to p r e - d i g e s t the m a t e r i a l f o r them. (p. 67) These statements suggest t h a t the mode of t e a c h i n g should correspond to the student's t h i n k i n g s t y l e and i n f e r s (from the f i n a l task) t h a t d i v e r g e n t t e a c h i n g may be one way to help e s t a b l i s h t h i s student independence of t h i n k i n g . McNary (1967) attempted to r e l a t e teacher c h a r a c t e r i s t i c s to the degree of change d i s p l a y e d by g i f t e d elementary students i n both the convergent and d i v e r g e n t areas o f t h i n k i n g . While these c h a r a c t e r i s t i c s are r e l a t e d to the p e r s o n a l i t y of the teacher, perhaps they g i v e a c l u e to methodological c o n s i d e r a t i o n s as w e l l . She found t h a t a teacher who was dependent and stood by s o c i e t y ' s standards e f f e c t e d the convergent s t y l e o f t h i n k i n g . Perhaps such a teacher would teach i n accordance with t h i s p e r s o n a l i t y and tend to f o s t e r dependency and c h o i c e s of the best a c c e p t a b l e outcome when t e a c h i n g problem s o l v i n g . S i m i l a r l y , McNary found t h a t teachers without i n f l e x i b l e p a t t e r n s f o r o b t a i n i n g s o c i a l approval i n f l u e n c e d the d i v e r g e n t s t y l e o f t h i n k i n g . Perhaps t h i s teacher would a l s o teach i n accordance with t h i s p e r s o n a l i t y and be f l e x i b l e i n a c c e p t i n g many d i f f e r e n t s uggestions, s o l u t i o n s and methods from students i n a problem-s o l v i n g s i t u a t i o n . 25 Torrance (1962) a l s o supported the concept t h a t t e a c h i n g method may make a d i f f e r e n c e . In a study on under- and over-achievement of f i f t h - g r a d e r s , he found t h a t w i t h a low c r e a t i v e teacher, h i g h l y c r e a t i v e c h i l d r e n were underachievers, while low c r e a t i v e c h i l d r e n tended to overachieve. He a l s o found t h a t with a h i g h l y c r e a t i v e teacher, both types of c h i l d r e n seemed to overachieve. T h i s seems to suggest t h a t t e a c h i n g method, as i n f e r r e d by c h a r a c t e r i s t i c s of the teacher, may i n f l u e n c e performance i n a classroom. The S u t h e r l i n Program (1964) used an i d e a i n v e n t o r y f o r t e a c h i n g the c r e a t i v e . T h e i r study i n v o l v e d students from grades seven through twelve and t h e i r concern was to f i n d ways to nurture c r e a t i v i t y . They e s t a b l i s h e d f i v e t e a c h i n g p r i n c i p l e s which they found encouraged c r e a t i v e t h i n k i n g : (1) t r e a t i n g the students with r e s p e c t , (2) t r e a t i n g i m a g i n a t i v e ideas w i t h r e s p e c t , (3) p l a c i n g value on p u p i l i d e a s , (4) a l l o w i n g students to e x p l o r e l e a r n i n g s i t u a t i o n s without always being evaluated, and (5) t y i n g e v a l u a t i o n w i t h cause and consequence. P r i n c i p l e s (2), (3), and (4) are c e r t a i n l y p a r t of the b a s i s of the d i v e r g e n t t e a c h i n g method. Summary There i s c o n s i d e r a b l e r e s e a r c h suggesting t h a t c o n s i s t e n c y i n t e a c h i n g methods and student t h i n k i n g s t y l e s i s important i n order to produce b e t t e r problem s o l v e r s i n mathematics ( S u t h e r l i n Program, 1964; T a y l o r , 1965; Behr, 1970; Skemp, 1971). Skemp (1971) suggested t h a t mathematics 26 teachers need to p l a n ways of t e a c h i n g which take i n t o account not o n l y the student's p r e v i o u s mathematical experience but a l s o t h e i r l e v e l of t h i n k i n g (p. 114). T h i s seems to i n d i c a t e t h a t there w i l l be an i n t e r a c t i o n between the way i n which something i s taught and the t h i n k i n g s t y l e of the l e a r n e r . T a y l o r (1965) suggested f o c u s i n g on the student, not o n l y what he l e a r n s , but how he l e a r n s . He f u r t h e r suggested the. use of the f a c t o r s i n the Operations a x i s of G u i l f o r d ' s (1960) Model of the I n t e l l e c t as the means of t h i s i n v e s t i g a t i o n . T h i s leads d i r e c t l y to the area of t h i n k i n g and hence the areas of convergent/divergent p r o d u c t i o n . From Torrance's (1962) f i n d i n g s i t would seem t h a t i t i s p a r t i c u l a r l y important f o r the d i v e r g e n t t h i n k e r to be matched with a d i v e r g e n t t e a c h i n g method. However, McNary (1967) i m p l i e s t h a t g i f t e d c h i l d r e n need both convergent and d i v e r g e n t types of t e a c h e r s . T a y l o r o f f e r s a s o l u t i o n : By u s i n g a p p r o p r i a t e c l a s s i f i c a t i o n systems, one c o u l d s t a r t l o g g i n g the responses of the teacher and students to f i n d out what t h i n k i n g and l e a r n i n g processes i n students are evoked by v a r i o u s behaviors and t e a c h i n g methods of the teacher, (p. 258) I f , as the r e s e a r c h i n d i c a t e s , there i s some k i n d of i n t e r a c t i o n between t e a c h i n g method and t h i n k i n g s t y l e o f the student, then there i s a d e f i n i t e need to a c q u a i n t teachers w i t h d i v e r g e n t methods of t e a c h i n g and to c o n s c i o u s l y promote d i v e r g e n t t h i n k i n g p a t t e r n s and methods of s o l u t i o n t o problems i n the mathematics classroom. CHAPTER I I I DESIGN OF THE STUDY The t h i r d chapter i s p a r t i t i o n e d i n t o the f o l l o w i n g s e c t i o n s : ( 1 ) D e f i n i t i o n o f Terms, i n c l u d i n g a c t u a l examples of convergent/divergent t h i n k i n g and te a c h i n g ; ( 2 ) Sample, who was i n v o l v e d , how and why they were chosen; ( 3 ) Teachers, who they were and how they were t r a i n e d ; ( 4 ) Procedures, what a c t u a l l y happened d u r i n g the study; and ( 5 ) S t a t i s t i c a l Hypotheses, nine o n e - t a i l e d hypotheses, each group of three r e l a t i n g to one of the three r e s e a r c h q u e s t i o n s as o u t l i n e d i n Chapter I. D e f i n i t i o n of Terms Convergent Thinker P r i o r to treatment, the Watson-Glaser C r i t i c a l T h i n k i n g  A p p r a i s a l (Form YM) was admi n i s t e r e d to a l l s u b j e c t s . T h i s t e s t i n v o l v e s f i v e s e c t i o n s - i n f e r e n c e , r e c o g n i t i o n o f assumptions, deduction, i n t e r p r e t a t i o n , and e v a l u a t i o n o f arguments. These v a r i a b l e s are c o n s i s t e n t with E b e r l e ' s ( 1 9 6 5 ) d e s c r i p t i o n of convergent t h i n k i n g a c t i v i t i e s which i n v o l v e r e d e f i n i n g the problem, t r a n s f o r m i n g the problem, and r e c o g n i t i o n o f the best or c o n v e n t i o n a l s o l u t i o n to a given problem. While i n d i v i d u a l s u b t e s t r e l i a b i l i t i e s are low ( 0 . 5 3 - 0 . 7 4 ) on the Watson-Glaser t e s t , r e l i a b i l i t y o f the e n t i r e t e s t at the grade ten l e v e l i s 0 . 8 6 . Consequently, o n l y t o t a l scores were c o n s i d e r e d f o r t h i s study. Convergent t h i n k e r s were d e f i n e d to be those s u b j e c t s 28 who scored one standard d e v i a t i o n or more above the mean on the Watson-Glaser t e s t . The normative data d i d not i n c l u d e the time o f t e s t i n g a t the grade ten l e v e l ; t h e r e f o r e , i t was f e l t t h a t i t was b e t t e r to be c o n s e r v a t i v e and use the grade eleven mean and standard d e v i a t i o n as the s u b j e c t s i n the present study were i n t h e i r l a s t q u a r t e r o f t h e i r grade ten year. The grade eleven mean was 64.4 (compared wi t h 61.7 at the grade ten l e v e l ) and the standard d e v i a t i o n was 11.0 (the same as the grade ten l e v e l ) . Any student who had a raw score o f 76 or b e t t e r on the Watson-Glaser t e s t was d e f i n e d to be a convergent t h i n k e r . A non-convergent t h i n k e r was d e f i n e d to be a s u b j e c t who scored one or more standard d e v i a t i o n s below the mean on the Watson-Glaser t e s t . S u b j e c t s with a raw score o f 53 or l e s s were d e f i n e d as non-convergent t h i n k e r s . A non-convergent t h i n k e r should not be equated with a d i v e r g e n t t h i n k e r . A non-convergent t h i n k e r may be thought o f as a person who does not r e a d i l y r e d e f i n e o r tran s f o r m problems and who i s unable to make the bes t c h o i c e o f s e v e r a l a l t e r n a t i v e s . T h i s does not imply t h a t he/she i s abl e to be o r i g i n a l or generate novel s o l u t i o n s as would a d i v e r g e n t t h i n k e r . The f o l l o w i n g i s a h y p o t h e t i c a l example of how a convergent t h i n k e r might s o l v e a problem. The student i s given the sequence of numbers 1, 2, 6, 15, ... and i s asked to f i n d the next number. A convergent t h i n k e r would probably 29 w r i t e s down the sequence, m e n t a l l y note t h a t adding the numbers d i d not y i e l d a constant sum; however, the d i f f e r e n c e between the numbers gave a sequence of 1, 4, 9, ... which appears to be the s e t of p e r f e c t squares; l e t ' s see, the l a s t one i s t h r e e squared, so the next must be f o u r squared which i s 16, so the next number i s 31 (15 p l u s 16), r i g h t ? Good, now I'm f i n i s h e d . Divergent Thinker To continue the example gi v e n above, the d i v e r g e n t t h i n k e r might s o l v e the problem i n a manner s i m i l a r to the f o l l o w i n g . He might w r i t e down the sequence, but above i t he would probably w r i t e the term numbers because he l i k e s to t h i n k of i n d i v i d u a l t h i n g s as p a r t of a whole: 1 2 3 4 5 1 2 6 15 ? He might then note t h a t the f i r s t i s the same as the f i r s t term, the second i s the same as the second term, you double 3 to get 6, you t r i p l e 4 then add 3 to get the f o u r t h term, maybe you quadruple 5 then add 4 (or maybe 6) to get the next term. What e l s e ? L e t ' s see, square 1 to get 1, square 2 and s u b t r a c t 2 to get 2, square 3 and s u b t r a c t 3 to get 6, square 4 and s u b t r a c t (oops) 1 (oh, t h a t ' s OK) f o r the next j u s t square, then square s u b t r a c t 2, square s u b t r a c t 3, square s u b t r a c t 1. Oh, the d i f f e r e n c e between terms g i v e s 1, 4, 9, ... so t h a t would make the next d i f f e r e n c e 16 and the next number 31. Mmm, wonder what e l s e : take the 30 f i r s t m u l t i p l y by 2 to get the second, m u l t i p l y second by 3 to get t h i r d , m u l t i p l y t h i r d by 3 s u b t r a c t 3 to get f o u r t h , the p a t t e r n c o u l d j u s t keep repeating."'' P r i o r to treatment, a l l s u b j e c t s were gi v e n the Torrance t e s t of T h i n k i n g C r e a t i v e l y with Words (Booklet A ) . T h i s t e s t i n v o l v e s seven a c t i v i t i e s : the f i r s t three d e a l w i t h a s k i n g q u e s t i o n s , guessing causes and guessing consequences r e g a r d i n g an e l f i n type c r e a t u r e l o o k i n g a t h i s r e f l e c t i o n ; the f o u r t h i n v o l v e s product improvement of a s t u f f e d toy elephant, unusual uses of cardboard boxes i s the f i f t h a c t i v i t y , w h i l e unusual q u e s t i o n s r e g a r d i n g cardboard boxes i s the s i x t h ; a j u s t suppose q u e s t i o n i n v o l v i n g s t r i n g s attached to clouds i s the f i n a l a c t i v i t y . Each a c t i v i t y i s scored f o r f l u e n c y (the a c t u a l number of responses on an item), f l e x i b i l i t y (changes i n types of response on an item), and o r i g i n a l i t y (as compared to the responses of o t h e r s i n the norming group). The t o t a l score index of a l l three c a t e g o r i e s p r o v i d e s a more s t a b l e index of the c r e a t i v e energy which a s u b j e c t has a v a i l a b l e and/or i s w i l l i n g to use and the r e l i a b i l i t i e s are h i g h e r f o r the t o t a l score than f o r the separate s c o r e s . T o t a l means and t o t a l standard d e v i a t i o n s were not a v a i l a b l e i n the norming data; consequently, the means f o r the three separate scores ( f l u e n c y = 94 .6 , f l e x i b i l i t y = 4 0 . 2 , and o r i g i n a l i t y = 45.2) were added to g i v e a composite mean of 1 I t should be noted t h a t the author does not f i t the d e f i n i t i o n of a d i v e r g e n t t h i n k e r , but t h i s t h i n k i n g i s designed to simulate d i v e r g e n t t h i n k i n g . 31 180. While standard d e v i a t i o n s are not a d d i t i v e , i f a p o s i t i v e c o r r e l a t i o n e x i s t s between s u b t e s t s (as i s the case i n the Torrance t e s t ) , then adding the i n d i v i d u a l standard d e v i a t i o n s would g i v e a sum which would be g r e a t e r than the a c t u a l t o t a l standard d e v i a t i o n (as the c o v a r i a n c e would be s u b t r a c t e d from t h i s sum). T h e r e f o r e , to g i v e a c o n s e r v a t i v e estimate of the t o t a l t e s t standard d e v i a t i o n , the standard d e v i a t i o n s ( f l u e n c y = 32.5, f l e x i b i l i t y = 9.0, and o r i g i n a l i t y = 23.2) were added to g i v e an s = 64.7 (approximately 65). A d i v e r g e n t t h i n k e r was d e f i n e d to be a s u b j e c t who scored one or more standard d e v i a t i o n s above the mean on the Torrance t e s t . Using the above data, t h i s meant t h a t any student who had a raw score of 245 or more on the Torrance t e s t was d e f i n e d to be a d i v e r g e n t t h i n k e r . A non-divergent t h i n k e r was d e f i n e d as a s u b j e c t who had a t o t a l raw score of 115 or l e s s on the Torrance t e s t . Convergent Teaching Method The convergent t e a c h i n g method was based on E b e r l e 1 s (1965) convergent t h i n k i n g a c t i v i t i e s ( t r a n s f o r m a t i o n , r e d e f i n i t i o n , and the a b i l i t y to p i c k the best c h o i c e o f s e v e r a l a l t e r n a t i v e s ) . As these had no mathematical base, the author and one of the committee members d e v i s e d the f o l l o w i n g l i s t o f expected teacher behaviours: (1) The teacher makes suggestions as to how to s o l v e the problems. (2) Questions are l e a d i n g and aiming towards c o r r e c t c o n c l u s i o n s and answer. 32 (3) Emphasis i s on being c o r r e c t , making v a l i d i n f e r e n c e s from p o s s i b l e c h o i c e s , making c o r r e c t i n t e r p r e t a t i o n s from l i m i t e d p o s s i b l e a l t e r n a t i v e s as presented by the t e a c h e r . (4) I f the c l a s s i s slow, the teacher does not rephrase the q u e s t i o n , but r a t h e r t e l l s , through example or many l i t t l e l e a d i n g q u e s t i o n s , how to get the d e s i r e d answer. (5) The tone of the c l a s s i s teacher d i r e c t e d . The content f o r t h i s method was taken from Jacobs' (1970) Mathematics: A Human Endeavor (See Appendix A f o r the Convergent Teaching L e s s o n s ) . The f o l l o w i n g i s a h y p o t h e t i c a l example of how a convergent teacher might approach t e a c h i n g the 1, 2, 6, 15, ... sequence as presented e a r l i e r . The teacher would w r i t e the sequence on the board and then ask f o r guesses of the next number (accepting o n l y 31 as an answer) then a s k i n g f o r the next number (again a c c e p t i n g o n l y 56). The teacher would ask i f anyone knew how to get these answers ( i f no guesses were forthcoming), "What about the o p e r a t i o n of s u b t r a c t i o n ? What i s the d i f f e r e n c e between the f i r s t and second terms? the second and t h i r d ? the t h i r d and f o u r t h ? Do you see a p a t t e r n i n these numbers? R e c a l l the o p e r a t i o n of squaring numbers. (This should be a b i g enough h i n t , but i f not...) 2 2 2 L e t ' s look a t numbers whose squares we know: 1 , 2 , 3 , . . . Th e r e f o r e , when co n f r o n t e d w i t h sequences of t h i s k i n d the best way to t r y to f i h d the p a t t e r n i s to look f o r d i f f e r e n c e s between terms." Divergent Teaching Method The d i v e r g e n t teacher t e a c h i n g the same sequence might 33 approach doing so i n the f o l l o w i n g manner. "Here i s a sequence of numbers. How do you t h i n k t h i s sequence of numbers was a r r i v e d at? (As students answer, the teacher i s a c c e p t i n g and w r i t e s the suggestions on the board.) ( i f the c l a s s i s slow ...) Does everyone know what a sequence i s ? What does the word i t s e l f mean? Do you know of any other sequences i n math? Could the number l i n e be thought of as a sequence? How d i d we develop that? Can you apply t h a t to our p r e s e n t problem? Again, does anyone have a suggestion as to how I a r r i v e d a t t h i s p a r t i c u l a r sequence of numbers?" The d i v e r g e n t t e a c h i n g method was based on E b e r l e 1 s (1965) d i v e r g e n t t h i n k i n g a c t i v i t i e s ( f l u e n c y , f l e x i b i l i t y , o r i g i n a l i t y , and e l a b o r a t i o n ) . The author and one of the committee members d e v i s e d the f o l l o w i n g l i s t o f expected teacher behaviours: (1) The t e a c h e r ' s q u e s t i o n s are open-ended, aiming toward g e t t i n g more suggestions from the students. (2) Students are 'to make the suggestions as to how to s o l v e the g i v e n problems. (3) Emphasis i s on how the t h i n k i n g i s being done, making hypotheses as to how to s o l v e problems, u t i l i z i n g p r e v i o u s i n f o r m a t i o n and t o o l s f o r s o l v i n g problems, recombining these to get new hypotheses. The answer i s n i c e , but the method i s the most i n t e r e s t i n g f a c e t of the s o l u t i o n . (4) I f the c l a s s i s slow, the teacher rephrases the q u e s t i o n , or asks the c l a s s to t e l l what they t h i n k t h a t the q u e s t i o n means, or the teacher asks a d i f f e r e n t q u e s t i o n and then comes back to the o r i g i n a l q u e s t i o n or d e f i n e s a term which the c l a s s may not know and then asks the o r i g i n a l or a rephrased q u e s t i o n . (5) The tone o f the c l a s s i s s t u d e n t - d i r e c t e d . The teacher i s to a c t as a c a t a l y s t and a resource person, and a l s o as a l i s t e r of student suggestions f o r p o s s i b l e s o l u t i o n s . The content f o r t h i s l e s s o n was taken from Jacobs' (1970) Mathematics: A Human Endeavor (See Appendix B f o r Divergent Teaching L e s s o n s ) . Routine Problems T h i r t y - f o u r q u e s t i o n s were c o n s t r u c t e d by the author to r e f l e c t the content taught i n the convergent and d i v e r g e n t l e s s o n s . The ques t i o n s i n c l u d e ten "true-false-sometimes" q u e s t i o n s based on the F i b o n a c c i Sequence and P a s c a l ' s T r i a n g l e ten m u l t i p l e c h o i c e q u e s t i o n s based on the same two areas; seven q u e s t i o n s i n v o l v i n g d i v i s i o n and squaring o f F i b o n a c c i numbers; and seven q u e s t i o n s i n v o l v i n g P a s c a l ' s T r i a n g l e , powers of eleven, and Chinese symbols. These q u e s t i o n s were r o u t i n e as they d i r e c t l y t e s t e d the content as gi v e n i n the two l e s s o n s . While the r o u t i n e q u e s t i o n s were predominantly convergent type q u e s t i o n s , a few d i v e r g e n t ones were i n c l u d e d (See Appendix C). Non-Routine Problems T h i r t y - f o u r q u e s t i o n s were c o n s t r u c t e d by the author to r e f l e c t content s i m i l a r ( i . e . sequence-based) t o t h a t i n the convergent and d i v e r g e n t l e s s o n s . These q u e s t i o n s , again, were mainly convergent i n nature, but i n c l u d e d a q u e s t i o n on magic squares which was d i v e r g e n t and r e q u i r e d a grading system s i m i l a r to the one used by Prouse (1967) f o r f l u e n c y , f l e x i b i l i t y , and o r i g i n a l i t y (See Key f o r Divergent Question i n Non-Routine Problems, Appendix D). These que s t i o n s were non-routine as they d i d not i n c l u d e any content taught i n the convergent and d i v e r g e n t l e s s o n s . The f i r s t ten qu e s t i o n s were "true-false-sometimes" problems based on fo u r sequences ( p o s i t i v e i n t e g e r s , odd whole numbers, even whole numbers, and square whole numbers). The next ten ques t i o n s were m u l t i p l e choice questions based on the same fo u r sequences. The next f i v e q u e s t i o n s i n v o l v e d a coded sequence. The next two quest i o n s i n v o l v e d a magic square ( t h i s i n c l u d e d the s p e c i a l d i v e r g e n t q u e s t i o n mentioned above). The l a s t seven q u e s t i o n s d e a l t with t r i a n g u l a r , square, and pentagonal numbers (See Appendix D). Sample Subjects f o r t h i s study c o n s i s t e d o f students from f o u r grade ten c l a s s e s randomly s e l e c t e d from the eleven academic mathematics c l a s s e s taught a t one secondary s c h o o l . Vos (1976) i n d i c a t e d t h a t "mathematical m a t u r i t y was a d e f i n i t e f a c t o r i n problem-solving a b i l i t y . " (p. 274). He a l s o i n d i c a t e d t h a t students i n a second year a l g e b r a course were s u p e r i o r on problem-solving tasks as compared to non-academic and f i r s t year a l g e b r a students. As the grade ten students s e l e c t e d f o r t h i s study were i n the f i n a l q u a r t e r of t h e i r second year o f a l g e b r a , i t was hoped t h a t t h e i r mathematical maturation l e v e l would be high enough to cope with the probl e m - s o l v i n g tasks presented to them i n the study. At the grade ten l e v e l , normative data were a v a i l a b l e f o r both s t a n d a r d i z e d t e s t s used to measure convergent or d i v e r g e n t t h i n k i n g . Each c l a s s of s u b j e c t s was randomly assigned to t e a c h i n g method ( e i t h e r convergent or divergent) and to teacher ( e i t h e r A or B) by the t o s s of a c o i n . The f i n a l sample i n c l u d e d a l l students who were i n attendance f o r the p r e - t e s t s (Watson-Glaser and T o r r a n c e ) , the treatment, and the p o s t - t e s t which contained both the r o u t i n e and non-routine problems. Group 1 - T h i s group c o n s i s t e d of 15 s u b j e c t s and was taught c o n v e r g e n t l y by Teacher B. Group 2 - T h i s group c o n s i s t e d o f 18 s u b j e c t s and was taught c o n v e r g e n t l y by Teacher A. Group 3 - T h i s group c o n s i s t e d of 16 s u b j e c t s and was taught d i v e r g e n t l y by Teacher A. Group 4 - T h i s group c o n s i s t e d of 17 s u b j e c t s and was taught d i v e r g e n t l y by Teacher B. Teachers The teachers i n v o l v e d i n the present study were both members of the Mathematics Educ a t i o n Department a t the U n i v e r s i t y o f B r i t i s h Columbia. Both had over f i v e years t e a c h i n g experience a t the secondary school l e v e l . Both of these teachers were known to use both convergent and d i v e r g e n t t e a c h i n g techniques i n some of t h e i r c l a s s e s , and t h i s was the major reason f o r choosing them. Two weeks p r i o r to treatment, both t e a c h e r s were gi v e n the convergent and d i v e r g e n t l e s s o n s (See Appendices A and B). They were a l s o given the l i s t o f expected teacher behaviours 3 7 (Refer to D e f i n i t i o n of Terms, t h i s c h a p t e r ) . They were i n s t r u c t e d to become f a m i l a r with both l e s s o n s and to note t h e i r s i m i l a r i t i e s and d i f f e r e n c e s i n s t y l e . In a study done by T a y l o r (1965), teachers were i n s t r u c t e d f o r an hour each day f o r a week p r i o r to beginning i n s t r u c t i o n of students. T h i s t r a i n i n g was to ensure t h a t the t e a c h e r s would t r e a t the students as t h i n k e r s d u r i n g i n s t r u c t i o n , r a t h e r than t r e a t i n g them as l e a r n e r s as was c o n s i d e r e d the c o n v e n t i o n a l manner. Due to c o n f l i c t i n g schedules, one three-and-one-half hour t r a i n i n g s e s s i o n was h e l d f o u r days p r i o r to treatment i n the present study. During the t r a i n i n g , emphasis was given to the d i s c r e p a n c i e s between the q u e s t i o n i n g techniques o f the two methods. In the convergent method a vague g e n e r a l q u e s t i o n c o u l d be broken down i n t o " l i t t l e l e a d i n g q u e s t i o n s " , g u i d i n g students to the d e s i r e d c o n c l u s i o n s . In the d i v e r g e n t t e a c h i n g method, the teacher was not a t l i b e r t y to g i v e h i n t s , but r a t h e r rephrased the q u e s t i o n . Throughout the convergent l e s s o n , emphasis was p l a c e d on c o r r e c t answers and accurate c o n c l u s i o n s drawn from the teacher's suggestions. The emphasis i n the d i v e r g e n t l e s s o n was on the procedures used to answer ques t i o n s with the s o l u t i o n coming from the students. The d i v e r g e n t l e s s o n a l s o p l a c e d emphasis on p a t t e r n s and r e l a t i o n s h i p s r a t h e r than c o r r e c t c o n c l u s i o n s . Each teacher taught one group on one day f o r a p e r i o d of 100 minutes ( the f i r s t two morning p e r i o d s of 50 minutes each). The next day each teacher taught a d i f f e r e n t group 3 8 f o r a p e r i o d of 1 0 0 minutes. Each teacher taught the convergent method one day and the d i v e r g e n t method on the other day (or v i c e v e r s a ; see Table 3 . 1 ) . Procedures A L a t i n Square Design was s e l e c t e d to e l i m i n a t e d i f f e r e n t i a l e f f e c t s due to time and teacher d i f f e r e n c e s . Each teacher taught one c l a s s per day us i n g e i t h e r the convergent o r d i v e r g e n t method. Each c l a s s was assigned to one and onl y one treatment. The treatment p e r i o d l a s t e d f o r the f i r s t two morning p e r i o d s (of approximately 5 0 minutes d u r a t i o n each). During the t h i r d p e r i o d , the t e s t s on the two dependent measures ( r o u t i n e and non-routine problems) were admin i s t e r e d . TABLE 3 . 1 DESIGN DAY TEACHER A TEACHER B 1 C* D** 2 D C *C means convergent t e a c h i n g method **D means d i v e r g e n t t e a c h i n g method A t o s s of a c o i n determined which teacher was desig n a t e d as Teacher A and which was Teacher B. C l a s s e s were randomly assig n e d by the t o s s of a c o i n to t e a c h i n g method and teacher. One h a l f o f each c l a s s f i r s t r e c e i v e d the r o u t i n e and then the non-routine problems, while the other h a l f o f each c l a s s r e c e i v e d the same problems i n r e v e r s e o r d e r . The experimental groups were t e s t e d once on the dependent measures, t h e r e f o r e a s h o r t term s t a b i l i t y estimate 39 of these measures was r e q u i r e d . Hoyt Estimates of R e l i a b i l i t y were c a l c u l a t e d to p r o v i d e an index of shor t term s t a b i l i t y of the dependent measures administered d u r i n g the experiment. There were t h i r t y - f o u r q uestions administered f o r each dependent measure. I t was hoped t h a t a t l e a s t twenty of these would have s u f f i c i e n t l y high c o r r e l a t i o n w i t h the t o t a l dependent measure as to be r e t a i n e d i n the a n a l y s i s . The o v e r a l l Hoyt Estimate of R e l i a b i l i t y was low (0.58) f o r the r o u t i n e problems (Group 1 = 0.53, Group 2 = 0.82, Group 3 = 0.53, and Group 4 = 0.85). An item a n a l y s i s was performed u s i n g the programme LERTAP (Larry R. Nelson, A p r i l , 1974) a v a i l a b l e a t the Computing Centre a t the U n i v e r s i t y of B r i t i s h Columbia. Those items w i t h a negative or zero c o r r e l a t i o n were omitted (items 1, 3, 8, 10, 16, 19, and 22). Subsequent r e l i a b i l i t i e s were o b t a i n e d as f o l l o w s : Group 1 = 0.60, Group 2 = 0.82, Group 3 = 0.60, and Group 4 = 0.71. T h i s gave an o v e r a l l Hoyt Estimate of R e l i a b i l i t y f o r the r o u t i n e problems of 0.74, thus r a i s i n g the o v e r a l l r e l i a b i l i t y to a c c e p t a b l e standards. F u r t h e r a n a l y s i s of the r o u t i n e problems was done with the above omissions being made. On the non-routine problems, no item omissions were necessary and the Hoyt Estimates of R e l i a b i l i t y were as f o l l o w s : Group 1 = 0.85, Group 2 = 0.85, Group 3 = 0.84, and Group 4 = 0.93. T h i s gave an o v e r a l l Hoyt Estimate of R e l i a b i l i t y f o r the non-routine problems of 0.88. There were two observers i n v o l v e d i n the study. They were the author and a f e l l o w graduate student i n mathematics education at the same u n i v e r s i t y as the teachers who taught the l e s s o n s i n the study. Both observers p a r t i c i p a t e d i n the teacher t r a i n i n g s e s s i o n . T h e i r purpose was to observe each l e s s o n to ensure t h a t both teachers taught the same content, as w e l l as to ensure t h a t the method f o r t h a t p a r t i c u l a r l e s s o n was adhered t o . T h i s was s t r i c t l y an e x t r a precau-t i o n a r y measure (See Table 4 . 5 ) . Due to unavoidable d e l a y on the second day, Group 2 was o n l y observed f o r the l a s t twenty minutes. However, d u r i n g t h i s time Teacher A was t e a c h i n g c o n v e r g e n t l y and completing the content as o u t l i n e d i n the l e s s o n . I t was hoped t h a t by having a s h o r t treatment p e r i o d t h a t d i f f e r e n c e s due to m o d a l i t y (convergent/divergent t e a c h i n g methods) would appear as the major c o n t r i b u t o r to the v a r i a n c e of any d i f f e r e n c e s between groups. Spache ( 1 9 7 6 ) suggests: We can r e a d i l y show t h a t d i f f e r e n c e s i n s h o r t term l e a r n i n g , as a s i n g l e l e s s o n of a c e r t a i n type, seem to be r e l a t e d to m o d a l i t y , (p. 7 0 ) While the treatment was of s h o r t d u r a t i o n (approximately two hours), i t was f e l t t h a t i f s i g n i f i c a n t r e s u l t s c o u l d be obtained, these would probably be more due to treatment than to p r e v i o u s mathematical experience. Gagne, Mayor, Garstens and P a r a d i s e ( 1 9 6 2 ) found t h i s to be the case when students were l e a r n i n g a d d i t i o n of i n t e g e r s f o r the f i r s t time. F a t t u , Mech and Kapos ( 1 9 5 4 ) a l s o found t h a t a two hour treatment d i d s i g n i f i c a n t l y a f f e c t the scores 41 of problem s o l v e r s on a s e t of g e a r - t r a i n problems. I t was f e l t t h a t as the present study i s an e x p l o r a t o r y study, the two hour treatment might be s u f f i c i e n t to show s i g n i f i c a n t r e s u l t s , p a r t i c u l a r l y as the p o s t - t e s t was administered immediately a f t e r treatment. The p o s t - t e s t s (dependent measures were admi n i s t e r e d immediately a f t e r the treatment as there i s ample evidence (Postman, 1963,1964; Petersen and Petersen, 1959) to i n d i c a t e t h a t the manner i n which people s t o r e m a t e r i a l changes wi t h time. That i s to say, the longer the p e r i o d of delay, the more l i k e l y the s u b j e c t i s to r e t u r n to h i s / h e r h a b i t u a l thought p a t t e r n s . I f the dependent measures were allowed to be delayed, one might expect the e s t a b l i s h e d p a t t e r n s of coding to take precedence i n the manner to which the s u b j e c t would respond to the t e s t . Postman (1964) found t h a t t h i s coding process c o u l d be expected to i n t e r f e r e w i t h the experimental method i f the t e s t i n g p e r i o d i s delayed beyond the c r i t i c a l p e r i o d (12 - 14 hours) and might confound the r e s u l t s . S t a t i s t i c a l Hypotheses The f o l l o w i n g s t a t i s t i c a l hypotheses were t e s t e d to answer the r e s e a r c h q u e s t i o n s as posed i n Chapter I of t h i s study: (la) Subjects c o n v e r g e n t l y taught w i l l score s i g n i f i c a n t l y more c o r r e c t answers on r o u t i n e problems than s u b j e c t s d i v e r g e n t l y taught. (lb) S u b j e c t s d i v e r g e n t l y taught w i l l score s i g n i f i c a n t l y more c o r r e c t answers on non-routine problems than s u b j e c t s c o n v e r g e n t l y taught. 42 (lc) Subjects d i v e r g e n t l y taught w i l l score s i g n i f i c a n t l y more c o r r e c t answers on the t o t a l problem s e t than s u b j e c t s c o n v e r g e n t l y taught. The above que s t i o n s look a t the t e a c h i n g method as r e l a t e d to q u e s t i o n type and propose t h a t the t e a c h i n g s t y l e which i s consonant w i t h the qu e s t i o n s w i l l more s i g n i f i c a n t l y e f f e c t s c o r e s . The next s e t of ques t i o n s i n d i c a t e t h a t a student's t h i n k i n g s t y l e w i l l e f f e c t h i s a b i l i t y to s o l v e problems whose type i s s i m i l a r to h i s thought p r o c e s s e s : (2a) Convergent t h i n k e r s w i l l score s i g n i f i c a n t l y more c o r r e c t answers on r o u t i n e problems than w i l l d i v e r g e n t t h i n k e r s . (2b) Divergent t h i n k e r s w i l l score s i g n i f i c a n t l y more c o r r e c t answers on non-routine problems than w i l l convergent t h i n k e r s . (2c) Divergent t h i n k e r s w i l l score s i g n i f i c a n t l y more c o r r e c t answers on the t o t a l problem s e t than w i l l convergent t h i n k e r s . The f i n a l s e t of hypotheses have been posed to analyze the i n t e r a c t i o n between t e a c h i n g method and t h i n k i n g s t y l e and i t s p o s s i b l e e f f e c t on problem s o l v i n g : (3a) Convergent t h i n k e r s taught c o n v e r g e n t l y w i l l score s i g n i f i c a n t l y more c o r r e c t answers on r o u t i n e problems than d i v e r g e n t t h i n k e r s taught d i v e r g e n t l y . (3b) Divergent t h i n k e r s taught d i v e r g e n t l y w i l l score s i g n i f i c a n t l y more c o r r e c t answers on non-routine problems than convergent t h i n k e r s taught c o n v e r g e n t l y . 43 (3c) Divergent t h i n k e r s taught d i v e r g e n t l y w i l l score s i g n i f i c a n t l y more c o r r e c t answers on the t o t a l problem s e t than convergent t h i n k e r s taught c o n v e r g e n t l y . A l i n e a r r e g r e s s i o n model was used f o r the s t a t i s t i c a l a n a l y s i s . An a n a l y s i s of v a r i a n c e was f i r s t used i n the r e g r e s s i o n as s u b j e c t s had been randomly assigned to groups and groups were randomly assi g n e d to treatment. However, when checking f o r the p o s s i b l e confounding e f f e c t s (as mentioned e a r l i e r i n t h i s c h a p t e r ) , the author a l s o i n v e s t i g a t e d whether there had been a d i f f e r e n c e between groups on the mean scores of the Watson-Glaser t e s t (convergent) and/or the Torrance t e s t ( d i v e r g e n t ) . There were s i g n i f i c a n t d i f f e r e n c e s found between Group 3 and Group 1 and between Group 3 and Group 4 on the Torrance t e s t (See Table 4.7). For c o n s i s t e n c y with t h e o r e t i c a l c o n s i d e r a t i o n s ( i . e . t h i n k i n g s t y l e to be c o v a r i e d with the dependent measures) a f u r t h e r a n a l y s i s of c o v a r i a n c e was made us i n g both the Watson-Glaser t e s t and the Torrance t e s t as c o v a r i a t e s . The a n a l y s i s of c o v a r i a n c e c o r r e c t e d the f a c t t h a t there had not been an o p p o r t u n i t y to block the c e l l s , as e n t i r e mathematics c l a s s e s were being used. I t should be f u r t h e r noted t h a t the a n a l y s i s of c o v a r i a n c e i s a more s e n s i t i v e a n a l y s i s than the o r i g i n a l a n a l y s i s of v a r i a n c e . The d i r e c t i o n of any s i g n i f i c a n t r e s u l t s w i l l be determined by u s i n g a t - t e s t . CHAPTER IV ANALYSES AND RESULTS Since , f o r s t u d i e s l i k e the present one, there i s g r e a t concern about the r e s u l t s being e x p l a i n e d , not by the s t a t i s t i c a l hypotheses, but by c h a r a c t e r i s t i c s i n h e r e n t i n the design of the study, the present chapter i s d i v i d e d i n t o two major p a r t s : (1) Confounding E f f e c t s and (2) R e s u l t s of the Study. F i r s t the data w i l l be analyzed to ensure t h a t no s i g n i f i c a n t e f f e c t s were e v i d e n t on v a r i a b l e s which might tend to confound the r e s u l t s of the study. Second, the data w i l l be analyzed with r e s p e c t to the s t a t i s t i c a l hypotheses as presented a t the end of Chapter I I I . Confounding E f f e c t s There were fo u r major areas of concern, r e g a r d i n g the design of the study, which might have confounded the r e s u l t s . The f i r s t was t h a t one teacher might be b e t t e r , r e g a r d l e s s of the method used (Teacher E f f e c t ) . The second was t h a t s u b j e c t s might perform b e t t e r on one day of treatment than on the o t h e r day of treatment, r e g a r d l e s s of the teacher of the method used (Time E f f e c t ) . T h i r d l y , there was concern t h a t the l e s s o n content might not be completed and/or t h a t a teacher might change from a d i v e r g e n t to a convergent t e a c h i n g method (or v i c e versa) (Observer's C h e c k l i s t ) . L a s t l y there was some concern t h a t some s u b j e c t s might not be responding to the l e s s o n a c c o r d i n g to the d i r e c t i o n s of the teacher (Student's Notes). 45 Teacher E f f e c t s One of the concerns a r i s i n g from the design of t h i s study was the use of two d i f f e r e n t t e a c h e r s . While the design was balanced f o r t e a c h e r s , there was s t i l l the concern t h a t a teacher might do b e t t e r on one method i:han on the other method. I t was t h e r e f o r e necessary to determine t h a t there were no s i g n i f i c a n t e f f e c t s due to the use of two d i f f e r e n t people as i n s t r u c t o r s . To ensure t h i s , the means and standard d e v i a t i o n s were c a l c u l a t e d f o r Teacher A's and Teacher B's c l a s s e s (these data are presented i n Table 4.1). An F - r a t i o was performed a t the one percent l e v e l of convidence, and no s i g n i f i c a n t d i f f e r e n c e s were found on e i t h e r the r o u t i n e (F(33,31) = 1.19), the non-routine (F(31,33) = 1.20), or the t o t a l problem s e t (F(31,33) = 1.20), which meant t h a t pooled sums c o u l d be used f o r the t - t e s t to determine whether or not there were s i g n i f i c a n t teacher e f f e c t s . The r e s u l t s o f the t - t e s t s are presented i n Table 4.2. A l l of these v a l u e s showed no s i g n i f i c a n t d i f f e r e n c e a t the p_£__ 0.01 l e v e l . Since using d i f f e r e n t t e a c h e r s seemed to have no e f f e c t on the problem-solving a b i l i t i e s o f the s u b j e c t s , f u r t h e r a n a l y s i s of the data can be made. Time E f f e c t s Another primary concern was t h a t there c o u l d be d i f f e r -e n t i a l r e s u l t s due to treatment o c c u r r i n g on two d i f f e r e n t days f o r s u b j e c t s . While there were no obvious f a c t o r s , TABLE 4.1 MEANS AND STANDARD DEVIATIONS FOR TEACHER EFFECTS ON DEPENDENT MEASURES Teacher A* Teacher B** Routine Non-Routine T o t a l Routine Non-Routine T o t a l Mean 12.29 33.42 45.00 11.91 29.91 41.82 Standard D e v i a t i o n 4.41 7.72 10.90 3.70 9.28 11.63 *N = 34 **N = 32 TABLE 4.2 RESULTS OF t-TESTS FOR TEACHER EFFECTS ON DEPENDENT MEASURES Problem Set t - v a l u e * Routine 0.38 0 Non-Routine 1.563 T o t a l 1.300 * d . f . = 64 C r i t i c a l t - v a l u e (p ^ 0 . 01) = 2.390 48 there may have been some unforseen d i f f e r e n c e i n l e a r n i n g which o c c u r r e d because a s u b j e c t was taught on the f i r s t day of treatment as opposed to the second day (or v i c e v e r s a ) . The means and standard d e v i a t i o n s were c a l c u l a t e d f o r Day 1 and Day 2 of treatment (these data are presented i n Table 4.3). An F - r a t i o was performed a t the p -si 0.01 l e v e l o f c o n f i d e n c e to assure homogeneity of v a r i a n c e . T h i s was necessary to enable the use of pooled sums f o r the t - t e s t of the d i f f e r e n c e between means on Day 1 and Day 2 s u b j e c t s . No s i g n i f i c a n t d i f f e r e n c e s were found on e i t h e r the r o u t i n e (F(35,31) = 1.35), the non-routine(F(35,31) = 1.24) or the t o t a l problem s e t (F(35,31) = 1.47). As no s i g n i f i c a n t d i f f e r e n c e s were found, the a p p r o p r i a t e t - t e s t s were performed to determine whether or not the day d i f f e r e n c e s i g n i f i c a n t l y e f f e c t e d the r e s u l t s . The r e s u l t s o f the t - t e s t s f o r s i g n i f i -cant d i f f e r e n c e s between the means are given i n Table 4.4. A l l o f these v a l u e s showed no s i g n i f i c a n t d i f f e r e n c e a t the p 0.01 l e v e l . The treatment o c c u r r i n g on two c o n s e c u t i v e days had no e f f e c t on the problem-solving achievement of the s u b j e c t s ; these r e s u l t s i n d i c a t e t h a t f u r t h e r a n a l y s i s of the data can be made. Observer's C h e c k l i s t The observer c h e c k l i s t was used to ensure t h a t r e l a t i v e l y the same content was taught i n a l l l e s s o n s and t h a t the c o r r e c t method ( e i t h e r convergent o r divergent) was being used to present t h i s content. The observers were the author and a TABLE 4.3 MEANS AND STANDARD DEVIATIONS FOR DAY EFFECTS ON DEPENDENT MEASURES DAY 1* DAY 2** Routine Non-Routine T o t a l Routine Non-Routine T o t a l Mean 12.23 31.61 43.84 12.00 31.62 43.69 Standard D e v i a t i o n 3.41 7.65 8.98 4.60 9.49 13.18 *N = 31 TABLE 4.4 RESULTS OF t-TESTS FOR DAY EFFECTS ON DEPENDENT MEASURES Problem Set t - v a l u e * Routine 0.010 Non-Routine 0.84 9 T o t a l 0.053 * d.f. = 64 C r i t i c a l t - v a l u e (p £r 0.01) = 2.390 f e l l o w graduate student i n mathematics education, both of whom had taught f o r more than three years i n secondary mathematics classrooms and both o f whom were i n attendance d u r i n g the t r a i n i n g s e s s i o n of the teache r s on the convergent and d i v e r g e n t t e a c h i n g methods. The r e s u l t s of the c h e c k l i s t f o r a l l c l a s s e s are presented i n Table 4.5. The f i r s t column o f the c h e c k l i s t r e p r e s e n t s the content taught i n the two les s o n s as presented to each student. While i t would appear t h a t o n l y one c l a s s of the co n v e r g e n t l y taught s u b j e c t s completed the e n t i r e content, t h i s was r e a l l y not the case. Due to unavoidable dalay, one of the observers c o u l d not atte n d the Group 2 l e s s o n u n t i l the l a s t twenty minutes. However, immediately a f t e r the t e s t i n g , the author and Teacher A c a r e f u l l y went over both l e s s o n s and d i s c u s s e d what had happened i n the Group 2 c l a s s . The bracketed twos (2) i n Table 4.5 i n d i c a t e what happened i n the c l a s s a c c o r d i n g to Teacher A and were subsequently v e r i f i e d from the students' notes d u r i n g the l e s s o n . In both c l a s s e s taught c o n v e r g e n t l y (Groups 1 and 2) a l l o f the content was covered f o r both l e s s o n s . The reviews i n both c l a s s e s were t e a c h e r - d i r e c t e d . During the le s s o n s the most f r e q u e n t l y used technique was t h a t o f " l i t t l e l e a d i n g q u e s t i o n s " (See Convergent Teaching Method, Chapter III) to o b t a i n the d e s i r e d content and alg o r i t h m s from the students. The second most f r e q u e n t l y used t e a c h i n g technique was t h a t o f g i v i n g p o s s i b l e s o l u t i o n s to the students and TABLE 4.5 OBSERVER'S CHECKLIST 52 FIBONACCI CHECKLIST CONTENT Gave Choices for Answers L i t t l e Leading Questions Teacher Directed Open Ended Ques. Student Suggest L i s t How to Solve 1. Number Trick 1 (2) 1 (2) 3 3 4 How 2. Number Trick Works (2) 3 4 1 (2) 4 1 3 4 3 4 1 3 4 Change 3. Machine 1 1 (2) 1 (2) 3 3 4 3 4 Dittoed 4. Sheets 1 1 (2) 3 4 Even/Odd 5. Numbers (2) 1 (2) (2) 3 3 4 3 4 Multiples 6. of Numbers 1 (2) 1 (2) 4 3 4 Sum of 1st 7. N Numbers 1 (2) 3 1 (2) 1 3 4 8. Review 1 (2) 3 4 PASCAL CHECKLIST F i r s t 3 1. Rows 1 (2) 1 (2) 3 4 3 4 3 4 Next 2. Row 1 (2) 3 1 (2) 3 4 3 4 3 4 Row 3. Sums (2) 1 (2) 1 (2) 4 3 4 3 4 "Right" 4. Column Sums 1 1 (2) 4 4 Sums of 5. Shapes 2 1 2 1 2 3 Right 3rd 6. Column Sums . 1 2 1 2 4 4 Fibonacci 7. Pascal Diagonal 1 2 1 2 4 1 2 4 4 1 2 8. Review 1 2 4 1 and 2 Convergently taught 3 and 4 Divergently taught having them decide which c h o i c e would be the best f o r the problems as presented i n the content of the l e s s o n . There were o n l y two content items where the teachers i n both convergent c l a s s e s used techniques from the d i v e r g e n t method p o r t i o n of the c h e c k l i s t . These o c c u r r e d i n the d i s c u s s i o n of t r i a n g u l a r -shaped- and diamond-shaped-sums i n Pascal.'s T r i a n g l e p o r t i o n of the l e s s o n when attempting to determine these sums from some other e n t r y i n P a s c a l ' s T r i a n g l e . The d i v e r g e n t technique used was to d i s c u s s how a s o l u t i o n was a r r i v e d .at emphasizing the s i m i l a r i t y to other p a t t e r n s i n P a s c a l ' s T r i a n g l e . However, the s o l u t i o n was t e a c h e r - d i r e c t e d r a t h e r than student-suggested and " l i t t l e l e a d i n g q u e s t i o n s " were giv e n p r i o r to the d i s c u s s i o n of p a t t e r n s . Consequently, w h i l e p a r t of a d i v e r g e n t t e a c h i n g method was used, i t was employed from a convergent viewpoint. Since t h i s departure i n method c o u l d o n l y e f f e c t the data by making s i g n i f i c a n t r e s u l t s harder to o b t a i n , i t was not co n s i d e r e d a major problem. In both c l a s s e s taught d i v e r g e n t l y (Groups 3 and 4) a l l of the content f o r the F i b o n a c c i l e s s o n was completed. However, i n P a s c a l ' s T r i a n g l e l e s s o n , Group 4 d i d not complete the sums of shapes of t r i a n g l e s nor diamond shapes and Group 3 d i d not complete the l a s t three s e c t i o n s ("Right" v e r s i o n of P a s c a l ' s T r i a n g l e , 3rd column sums and adding the d i a g o n a l upwards on P a s c a l ' s T r i a n g l e to o b t a i n the F i b o n a c c i sequence). Group 3 a l s o d i d not r e c e i v e a review o f P a s c a l ' s T r i a n g l e l e s s o n . F a i l u r e to complete the content was due mainly to the 54 f a c t t h a t i n both d i v e r g e n t c l a s s e s the l e s s o n s s t a r t e d s l o w l y . However, as students became i n v o l v e d i n the problem-solving process, they made many suggestions over and above the proposed content i n the F i b o n a c c i Sequence (See d i s c u s s i o n i n Suggestions f o r Future Research, Chapter V ) . The m a j o r i t y of the d i v e r g e n t t e a c h i n g i n v o l v e d the teacher asking open-ended ques t i o n s ("How might we do t h i s ? " , "Does anyone e l s e have an i d e a ? " , "How do you t h i n k I got t h a t answer?", "Does t h a t r e l a t e to anything we've done e a r l i e r ? " ) and l i s t i n g s o l u t i o n s and suggestions as proposed by the students (See Student's Notes, Groups 3 and 4, t h i s C h a pter). There were three i n s t a n c e s i n each d i v e r g e n t c l a s s where the teacher used a convergent technique. These o c c u r r e d when open-ended q u e s t i o n s had been t r i e d and no r e s u l t s were obtained and there was a long pause (anywhere from 25 to 75 seconds of s i l e n c e ) where the students were o b v i o u s l y p u z z l e d as to which d i r e c t i o n to proceed. In both c l a s s e s the s i t u a t i o n f i r s t o c c u r r e d when the students were asked to decide how the teacher had been a b l e to o b t a i n anyone's number sums from o n l y a s k i n g f o r the seventh number. The h i n t s g i v e n i n both c l a s s e s suggested a c h o i c e of u s i n g a l g e b r a to a s s i s t i n s o l v i n g the problem and then i n both c l a s s e s x was chosen as the f i r s t unknown and then d i s c u s s i o n ensued as to j u s t what the x was to r e p r e s e n t , from t h i s p o i n t both c l a s s e s proceeded c o v e r i n g more content than a n t i c i p a t e d or planned f o r i n the l e s s o n . In Group 3 ( d i v e r g e n t l y taught) use of a convergent technique next o c c u r r e d when the teacher suggested s k i p p i n g numbers i n the sequence to t r y and f i n d the answer f o r the sum of the f i r s t n numbers i n the F i b o n a c c i Sequence and o c c u r r e d again when the students found d i f f i c u l t y i n f i n d i n g the f o u r t h row of P a s c a l ' s T r i a n g l e (they had many suggestions - 1 2 3 2 1 2 3 3 2 ; 1 1 3 1 1 ) and the teacher suggested a c h o i c e of 1 and 4 f o r the f i r s t two numbers a t which p o i n t the students completed the row and then began to d i v e r g e i n t o the elevens times t a b l e s p r i o r to going back and de v e l o p i n g more rows i n P a s c a l ' s T r i a n g l e . In Group 4 ( d i v e r g e n t l y t a u g h t ) , the teacher asked a " l i t t l e l e a d i n g q u e s t i o n " (convergent method) i n a i d i n g the students i n f i n d i n g out how the number t r i c k worked i n the F i b o n a c c i l e s s o n . He asked, "What does the f i n a l sum have to do with the seventh number?" He d i d t h i s o n l y a f t e r a s k i n g s e v e r a l open-ended que s t i o n s such as: "How do we g e n e r a l i z e t h i s ? " , "Can you t h i n k of a way to get the f i n a l sum?". He a l s o used a " l i t t l e l e a d i n g q u e s t i o n " when t r y i n g t o get a t the r e l a t i o n s h i p between P a s c a l ' s T r i a n g l e and the F i b o n a c c i Sequence along the d i a g o n a l s by commenting, "Oh, I see a goody! Look at adding i n the rows of the r i g h t t r i a n g l e v e r s i o n . " T h i s l e d to the students suggesting adding upwards and downwards on the d i a g o n a l s to both the r i g h t and the l e f t as w e l l as columns and L-shaped p a t t e r n s . While i t was unfortunate t h a t some d i s c r e p a n c i e s from the d i v e r g e n t t e a c h i n g o c c u r r e d , they were done i n such a way t h a t they acted as a c a t a l y s t f o r students to b r i n g f o r t h more ideas r e g a r d i n g p o s s i b l e s o l u t i o n s ( i d e a t i o n a l f l u e n c y and f l e x i b i l i t y ) r a t h e r than converging on one s o l u t i o n . I t was f e l t t h a t these d i s c r e p a n c i e s d i d not s i g n i f i c a n t l y e f f e c t the o v e r a l l d i v e r g e n t t e a c h i n g method. Student's Notes At the beginning of each l e s s o n a l l s u b j e c t s were gi v e n f o u r pages of blank computer p r i n t - o u t paper. They were i n s t r u c t e d to p l a c e t h e i r c l a s s block on the f i r s t page (and t h e i r name, i f they wished to do s o ) . The purpose of t h i s was t w o - f o l d : to f a c i l i t a t e p a t t e r n searches and involvement i n the a c t u a l l e s s o n s and to p r o v i d e a check f o r the author to ensure t h a t s u b j e c t s were a c t i v e l y i n v o l v e d i n the l e s s o n and t h a t the content of the l e s s o n was being adequately p e r c e i v e d by the s u b j e c t s . These sheets were c o l l e c t e d by the observers a t the end of P a s c a l ' s T r i a n g l e l e s s o n and p r i o r to t e s t i n g on the dependent measures so t h a t any a l g o r i t h m s d e r i v e d i n the l e s s o n ( e i t h e r by the c l a s s or by the i n d i v i d u a l subject) were un-a v a i l a b l e d u r i n g the t e s t i n g s e s s i o n . In Group 1 (convergently t a u g h t ) , a l l s u b j e c t s handed i n completed notes on both the F i b o n a c c i Sequence and P a s c a l ' s T r i a n g l e . Each s u b j e c t ' s notes c o n s i s t e d of the f o l l o w i n g i n f o r m a t i o n : 57 1) Ten numbers i n a column c r e a t e d by adding the f i r s t two to get the t h i r d , second and t h i r d added to get the f o u r t h and so on u n t i l the t e n t h number p l u s the f i n a l sum. (See Appendix E f o r examples of student's a c t u a l n o t e s ) . 2) Ten a l g e b r a i c e x p r e s s i o n s o b t a i n e d i n a s i m i l a r manner to the above beginning w i t h a and b. 3) A t a b l e of cheque v a l u e s , p o s s i b l e pay o f f s and count the number of ways to pay o f f from the change machine wi t h v a l u e s from $0.00 to $5.00. .4) A l i s t i n g o f the F i b o n a c c i Sequence and t h e i r sums us i n g s u b s c r i p t e d n o t a t i o n . 5) The f i r s t f o u r rows of P a s c a l ' s T r i a n g l e . 6) The sum of these f i r s t f o u r rows together w i t h the powers of two i n s i d e by s i d e columns. 7) A " r i g h t " t r i a n g l e v e r s i o n of P a s c a l ' s T r a i n g l e . T h i s had an arrow on the t h i r d column p l u s upward d i a g o n a l sums showing the r e l a t i o n to the F i b o n a c c i Sequence. 8) A r e g u l a r v e r s i o n of P a s c a l ' s T r i a n g l e w i t h both t r i a n g l e shapes and diamond shapes o u t l i n e d w i t h arrows p o i n t i n g to the sums which were e n t r i e s lower down i n P a s c a l ' s T r i a n g l e . There was very l i t t l e evidence of e x t r a guesses or doodles on these notes and, except f o r page placement of the sketches and a few papers which d i d not have the diamond or e l s e the t r i a n g l e sums marked, there was almost i d e n t i c a l notes taken by each s u b j e c t . T h i s would i n d i c a t e t h a t there was a h i g h l e v e l of involvement w i t h the l e s s o n s and t h a t s u b j e c t s were f o l l o w i n g the d i r e c t i o n s g i v e n by the teacher very e x p l i c i t l y . In Group 2 (convergently t a u g h t ) , t h i r t e e n o f the eighteen s u b j e c t s had a complete s e t of notes on both l e s s o n areas; two s u b j e c t s completed o n l y P a s c a l ' s T r i a n g l e p o r t i o n of the l e s s o n ; one s u b j e c t handed i n a s t i l l blank computer p r i n t - o u t sheet. Each s u b j e c t ' s notes c o n s i s t e d of the same i n f o r m a t i o n as o u t l i n e d f o r Group 1 wit h the f o l l o w i n g a d d i t i o n s or d e l e t i o n s : 1) Same. 2) Same, except i n s t e a d of a and b, x and y were used. 3) Same, except a sketch o f the change machine was i n c l u d e d . 4) No s u b s c r i p t e d n o t a t i o n , i n s t e a d comments about odds and evens and d i v i s i b i l i t y r u l e s f o r the F i b o n a c c i Sequence. 5) Same. 6) Same. 7) Same, p l u s the f a c t t h a t the sum i n the columns can be obt a i n e d by going one over and one down from the l a s t number i n the column which was being added i n the sum. 8) Same. Again, there was no evidence of e x t r a guesses or doodle on these notes. Some students a l s o i n c l u d e d notes r e g a r d i n g the t h i r d column adjacent p a i r sums being r e l a t e d to square numbers (4, 9, 16, 25, . . . ) . The c o n s i s t e n c y among s u b j e c t ' notes was hi g h . A l l s i x t e e n s u b j e c t s i n Group 3 ( d i v e r g e n t l y taught) completed the notes f o r both l e s s o n s on the computer p r i n t -out sheets p r o v i d e d . One student engaged i n some d o o d l i n g making a mushroom and o u t l i n i n g some of h i s c o n j e c t u r e s and 59 c o n c l u s i o n s . The n o t e - t a k i n g of t h i s group v a r i e d f a r more than t h a t of the pre v i o u s two groups. Many students made e x t r a c o n j e c t u r e s which were not l i s t e d on the board by the teacher, however, most of these were f a l l a c i o u s i n nature. In g e n e r a l each s u b j e c t ' s notes c o n s i s t e d of i n f o r m a t i o n s i m i l a r to t h a t of the p r e v i o u s groups; however, there were some notable exceptions due to the d i v e r g e n t t e a c h i n g method as i n d i c a t e d below: 1) Same. 2) A l i s t of ideas of how the sum might be ob t a i n e d as per (1) p l u s many other p a t t e r n s or n o t i o n s : a) take the 7th number, add zero to the end of i t and then add the 7th number to t h a t number. b) m u l t i p l y the 7th number by el e v e n . c) t r y a l l p a i r s d) prime number p a i r s and then g e n e r a l i z e e) use a v a r i a b l e to t e s t guess f) l e t x be 7th (no) answer (no) f i r s t number g) 1st p l u s second equals t h i r d h) 10 - x i s the other number i ) use another v a r i a b l e a At t h i s p o i n t students came up wit h (2) as d i d the two p r e v i o u s groups, however without a s s i s t a n c e , the teacher j u s t a c t ed as a r e c o r d e r . L e t t e r s used were x and a. 3) Most omitted change machine e n t i r e l y from t h e i r notes, those who put i t i n j u s t l i s t e d 1, 2, 3, 5 and noted t h a t i t was the same p a t t e r n as b e f o r e . The teacher was a r e c o r d e r of student guesses on the board and 8 d i d come up i n the c l a s s s e s s i o n as the number of ways to make change f o r $5.00. 4) Here the d i t t o e d sheets were used e x c l u s i v e l y and the f o l l o w i n g c o n j e c t u r e s and r e l a t i o n s were noted by the students i n t h e i r notes: a) to get number you would add the two bef o r e b) to get the 3rd add the 1st and 2nd c) i n number of terms column, adding 2 c o n s e c u t i v e numbers g i v e s you the odd numbers d) every 5th number i s d i v i s i b l e by 5 e) every 4th number i s d i v i s i b l e by 3 f) every 6th number i s d i v i s i b l e by 8 60 g) every 7th number i s d i v i s i b l e by 13 h) i f numbers are a, b, c then c - a = b i) a + b - d = -b j) b - c = -a 5) Same, except additional guesses regarding the 4th row occurred: 131, 232, 12321,233, 2332, 11311, and f i n a l l y 14641. 6) Did not occur. 7) Only the ri g h t t r i a n g l e version of Pascal's Triangle occurred. 8) Using both versions of Pascal's Triangle, conjectures were made about summing: a) top number of tr i a n g l e should be 1. b) every number i n the 7th row i s d i v i s i b l e by 7 except the ones. c) t h i r d row you add one more d) sum of the f i r s t 6 rows equals the 7th row e) add diagonally down go one to the l e f t diagonally and w i l l get the sum f) add pairs adjacent and go down one number to get the sum (in the ri g h t version) g) add the ri g h t angles works (this i s the same as the tr i a n g l e pattern i n the convergent groups) adding squares works Far more ideas (between 10 and 12) were generated during the divergent lessons (Groups 3 and 4) than i n the convergent lesson which used only ideas posed by the teacher and did not have student suggestions occurring. In Group 3 most of the content of both lessons was completed, even i f not being e x p l i c i t l y done by the teacher. The student involvement in the lesson was high and there were many student-student interactions taking place, p a r t i c u l a r l y during the times when sections (2), (4), and (8) of the student's notes (as l i s t e d above) were occurring. Even though not requested to do so, most students did take rather thorough notes of the l i s t s that the teacher was 61 r e c o r d i n g on the board. Four of the seventeen s u b j e c t s i n Group 4 ( d i v e r g e n t l y taught) completed notes on the F i b o n a c c i Sequence o n l y . Of these four s u b j e c t s , there was one who o b v i o u s l y played t i c -t a c - t o e with h i m s e l f d u r i n g the l a t t e r p a r t of the l e s s o n . The other t h i r t e e n s u b j e c t s were f a i r l y c o n s i s t e n t i n t h e i r n o t e - t a k i n g . Group 4 s u b j e c t s d i d not d i s p l a y as many e x t r a c o n j e c t u r e s as d i d the s u b j e c t s i n Group 3, but they d i d tend to r e c o r d c o n j e c t u r e s t h a t arose i n c l a s s and were recorded by the teacher. In g e n e r a l , each s u b j e c t ' s notes c o n s i s t e d of the f o l l o w i n g i n f o r m a t i o n : 1) The same. 2) Conjectures were f i r s t made as to how the 7th number r e l a t e d to the t o t a l sum: a) f i r s t and l a s t numbers same then the middle number i s the f i r s t two added together. b) number i s the f i r s t two added together c) 7th number times 3rd equals the sum (vetoed by a vote of the c l a s s ) d) 7th times ten then add i t s e l f e) eleven times the 7th number And then a f t e r d e c i d i n g to use an "open symbol" the a l g e b r a i c sums were d e r i v e d as i n Group 3. 3) The c o i n machine problem was done u s i n g amount, ways and number of ways, although the order of these was d i f f e r e n t f o r d i f f e r e n t s tudents. 4) The d i t t o e d sheet was used, no s u b s c r i p t e d n o t a t i o n was recorded by students; the f o l l o w i n g f a c t s were noted: a) odd-odd-even b) every 1st term i s a m u l t i p l e of one c) every 3rd term i s even d) every 3rd term i s d i v i s i b l e by 2 e) every 4th term i s d i v i s i b l e by 3 f) every 5th term i s d i v i s i b l e by 5 g) every 6 t h term i s d i v i s i b l e by 4 h) every 6 t h term i s d i v i s i b l e by 8 i) every 7 t h term i s d i v i s i b l e by 13 j) every 8 t h by 21 62 5) Same. 6) Same, except student-suggested and not t e a c h e r -d i r e c t e d . 7) Same, i n a d d i t i o n the students a l s o d i s c o v e r e d sums i n downward d i a g o n a l s i f you work from the edge of the r i g h t t r i a n g l e v e r s i o n of P a s c a l ' s T r i a n g l e then the sum i s the number j u s t below the l a s t number you add along the d i a g o n a l . 8) Same, students found the t r i a n g l e s and diamonds, where the diamonds i n the r i g h t v e r s i o n gave the odd numbers as sums i f you s t a r t e d from the top; the t r i a n g l e shape was d i s g u i s e d as the L-shape (as i n Group 3) . Students t r i e d to put columns of zeroes and n e g a t i v e s on the l e f t s i d e of the r i g h t t r i a n g l e v e r s i o n , but t h i n g s began to get too f a r a f i e l d f o r the m a j o r i t y of the c l a s s and time ran out. In examining each group of student's notes (as presented by 1 - 8 f o r Groups 1 - 4 above) i t i s apparent t h a t a l l groups completed approximately the same m a t e r i a l s . I t would a l s o seem t h a t the convergent c l a s s e s (Groups 1 and 2) were taught i n the same manner as t h e i r notes were c o n s i s t e n t . I t appears t h a t the d i v e r g e n t c l a s s e s (Groups 3 and 4) were taught i n a s i m i l a r manner, because t h e i r notes were r e l a t i v e l y c o n s i s t e n t , but not to the high degree of the two convergent c l a s s e s . There were, however, some notable d i f f e r e n c e s between the notes of the two convergent c l a s s e s and the. two d i v e r g e n t c l a s s e s . These d i f f e r e n c e s are p a r t i c u l a r l y e v i d e n t i n items (2), (4), and (8) of the s u b j e c t ' s notes (as l i s t e d above), where the d i v e r g e n t l y taught s u b j e c t s l i s t e d c h o i c e s and the c o n v e r g e n t l y taught s u b j e c t s l i s t e d o n l y the c h o i c e s g i v e n to them by the teacher i n v o l v e d . Based on the o b s e r v a t i o n s and the student's notes, 63 i t appears t h a t the s u b j e c t s were taught (and p e r c e i v e d the teaching) i n the manner to which they were assigned. In t h i s study there was a p o s s i b i l i t y t h a t a teacher c o u l d have changed methods d u r i n g the l e s s o n . T h i s was c o n t r o l l e d f o r by having an observer with a c h e c k l i s t watching the method employed f o r each content item. The a n a l y s i s of the check-l i s t s r e v e a l e d t h a t both the convergent and d i v e r g e n t l e s s o n s were a p p r o p r i a t e l y taught. The content of the students' notes i n d i c a t e d t h a t l e s s o n s had e s s e n t i a l l y covered the same m a t e r i a l . The r e s p e c t i v e s t y l e of students' notes was a l s o c o n s i s t e n t w i t h the t e a c h i n g method (convergent or d i v e r g e n t ) . The nature of these notes r e v e a l e d t h a t the d i v e r g e n t l e s s o n s were taught very d i f f e r e n t l y from the convergent l e s s o n s (as they were supposed to be). There was a l s o the p o s s i b i l i t y t h a t one teacher would do b e t t e r than the o t h e r . Analyses of the dependent measures showed no s i g n i f i c a n t d i f f e r e n c e s (p =^ 0.01) between t e a c h e r s . There was the p o s s i b i l i t y t h a t on one day s u b j e c t s would do b e t t e r than on the other day. Analyses of the dependent measures showed no s i g n i f i c a n t d i f f e r e n c e s (p _^ 0.01) between days. On the b a s i s of these f i n d i n g s i t would appear t h a t the t e a c h e r s adhered to the method and content of the l e s s o n s on both days, and t h a t the t e a c h i n g method (convergent or divergent) was c l e a r l y d i s t i n g u i s h a b l e . R e s u l t s of the Study The r e g r e s s i o n a n a l y s i s as used i n t h i s study i s based on techniques d e s c r i b e d by K e r l i n g e r and Pedhazur (1973) . 64 Regression a n a l y s i s was chosen as i t enabled the use of both continuous and dichotomous v a r i a b l e s as independent measures. T h i s was p a r t i c u l a r l y important i n t h i s study as the Watson-G l a s e r and the Torrance t e s t s r e presented continuous v a r i a b l e s , w h i le the t e a c h i n g method (convergent/divergent) and the i n s t r u c t o r (Teacher A/Teacher B) represented dichotomous v a r i a b l e s . A stepwise r e g r e s s i o n was chosen so t h a t v a r i a b l e s which may have been good p r e d i c t o r s i n the e a r l y stages of the r e g r e s s i o n c o u l d be removed i f they were no longer u s e f u l i n the r e g r e s s i o n equation. The stepwise r e g r e s s i o n a n a l y s i s was performed u s i n g Stepwise Regression (BMD 02R) (Jason Halm, 1974) as adapted from BMD (UCLA) (Dixon, 1970) which i s a v a i l a b l e a t the Computing Centre of the U n i v e r s i t y of B r i t i s h Columbia. Both the Watson-Glaser and Torrance t e s t s were used as c o v a r i a t e s i n the r e g r e s s i o n (See d i s c u s s i o n a t the end of Chapter I I I ) . Since the c o r r e l a t i o n between the two t e s t s was shown to be 0.04, the two t e s t s were t r e a t e d as two v a r i a b l e s independent of each other. A n a l y s i s of c o v a r i a n c e was used s i n c e i t was not p o s s i b l e to block the convergent/ d i v e r g e n t s u b j e c t s i n t o c l a s s e s on an equal b a s i s . Using the scores on the Watson-Glaser t e s t (convergent and the Torrance t e s t (divergent) as c o v a r i a t e s allowed a l l the groups to be equated r e g a r d i n g these two v a r i a b l e s . A n a l y s i s of c o v a r i a n c e was a l s o used to improve the s e n s i t i v i t y of the a n a l y s i s ( K e r l i n g e r and Pedhazur, 1973, p. 266). The assumptions u n d e r l y i n g the use of a n a l y s i s of c o v a r i a n c e were met as academic mathematics students had been randomly assigned to t h e i r p a r t i c u l a r c l a s s e s w i t h i n the school ( w i t h i n the c o n s t r a i n t s of the t i m e t a b l e by the s c h o o l admin-i s t r a t i o n ) and the c l a s s e s used were randomly s e l e c t e d from the academic mathematics c l a s s e s and were randomly assigned to treatment. I t was f u r t h e r assumed t h a t w i t h i n each treatment group the r e s i d u a l s were independently and normally d i s t r i b u t e d with a mean of zero and homogeneous v a r i a n c e . A l i n e a r r e g r e s s i o n of the dependent measures on the independent measures was assumed with homogeneous r e g r e s s i o n c o e f f i c i e n t s . The treatment d i d not have an e f f e c t on the c o v a r i a t e s (Watson-Glaser t e s t and Torrance t e s t ) , as these data were c o l l e c t e d p r i o r to treatment. I f the r e g r e s s i o n c o e f f i c i e n t s are heterogeneous, the F - t e s t performed i n the r e g r e s s i o n a n a l y s i s would be c o n s e r v a t i v e ( f o r f u r t h e r d i s c u s s i o n , see Meyers, 1973, p. 327). The means and standard d e v i a t i o n s f o r each group on both the Watson-Glaser and Torrance t e s t s i s g i v e n i n Table 4.6. A t - t e s t u s i n g pooled sums was employed to t e s t f o r s i g n i f i c a n t d i f f e r e n c e s between the means of each group. While there were no s i g n i f i c a n t d i f f e r e n c e s on the Watson-Glaser t e s t between any of the groups, there were s i g n i f i c a n t d i f f e r e n c e s between Group 3 ( d i v e r g e n t l y taught) and Group 1 (convergently taught) and between Group 3 and Group 4 ( d i v e r g e n t l y taught) on the Torrance t e s t w i t h the mean of Group 3 being s i g n i f i c a n t l y 66 TABLE 4.6 MEANS AND STANDARD DEVIATIONS FOR WATSON-GLASER AND TORRANCE TESTS Group 1 Group 2 Group 3 Group 4 Watson-Glaser N 15 18 16 17 Mean 63.87 63.17 59.06 61.59 Standard D e v i a t i o n 11.31 9.45 9.31 8.66 Torrance N 15 18 16 17 Mean 171.13 202.11 231.25 174.71 Standard D e v i a t i o n 52. 37 74.47 56.31 64.28 TABLE 4.7 GROUP COMPARISON OF MEANS'ON WATSON-GLASER AND TORRANCE TESTS t- v a l u e Watson-Glaser Group 1 vs Group 2 0.189 Group 1 vs Group 3 1.252 Group 1 vs Group 4 0.624 Group 2 vs Group 3 1.2 35 Group 2 vs Group 4 0.499 Group 3 vs Group 4 0.783 Torrance Group 1 vs Group 2 1.314 Group 1 vs Group 3 2.970** Group 1 vs Group 4 0.166 Group 2 vs Group 3 1.23 6 Group 2 vs Group 4 1.129 Group 3 vs Group 4 2.599** ** S i g n i f i c a n t at the p ~ 0.01 l e v e l . 68 higher than e i t h e r the mean of Group 1 or t h a t o f Group 4 (See Table 4.7). T h i s f i n d i n g suggests t h a t there were d i f f e r e n c e s between groups on the measure of d i v e r g e n t t h i n k i n g . As was mentioned e a r l i e r (See d i s c u s s i o n a t the end of Chapter III) i t was o r i g i n a l l y planned to use an a n a l y s i s of v a r i a n c e i n the r e g r e s s i o n a n a l y s i s . However, the d i f f e r e n c e s between groups on the Torrance t e s t made i t necessary t o use the Torrance t e s t r e s u l t s as a c o v a r i a t e . While s i g n i f i c a n t d i f f e r e n c e s were not noted between groups on the Watson-Gl a s e r t e s t , i t was f e l t , f o r the reasons s t a t e d i n the e a r l i e r d i s c u s s i o n (See end of Chapter III) t h a t the Watson-Gl a s e r t e s t would a l s o be used as a c o v a r i a t e . As t h i s study i s l o o k i n g f o r r e l a t i o n s h i p s among te a c h i n g method, t h i n k i n g s t y l e and problem s o l v i n g , another advantage of u s i n g the r e g r e s s i o n a n a l y s i s i s the m u l t i p l e R ob t a i n e d . 2 "R i n d i c a t e s the p o r t i o n o f the t o t a l v a r i a n c e of the dependent v a r i a b l e t h a t the independent v a r i a b l e accounts f o r . . . " ( K e r l i n g e r and Pedhazur, 1973, p. 98). Thus, w i t h i n the a n a l y s i s , i n f o r m a t i o n should be gained r e g a r d i n g the c o n t r i b u t i o n s o f both t e a c h i n g method and t h i n k i n g s t y l e to the t o t a l v a r i a n c e on both r o u t i n e and non-routine problems'. The model f o r the r e g r e s s i o n a n a l y s i s used assumed t h a t each of the dependent measures ( r o u t i n e , non-routine and t o t a l problem set) were a l i n e a r combination of the f o l l o w i n g : the c o v a r i a t e s (Watson-Glaser/Torrance), the dichotomous v a r i a b l e s (method - convergent/divergent and i n s t r u c t o r - Teacher A/ 69 Teacher B), the i n t e r a c t i o n s (Method x I n s t r u c t o r , Watson x Method, Watson x I n s t r u c t o r , Torrance x Method, Torrance x I n s t r u c t o r , Watson x Method x I n s t r u c t o r , and Torrance x Method x I n s t r u c t o r ) , and the e r r o r terms. Means and standard d e v i a t i o n s are p r o v i d e d f o r both the independent and dependent v a r i a b l e s i n Table 4.8. Table 4.9 c o n t a i n s the c o r r e l a t i o n matrix f o r the independent and dependent v a r i a b l e s . Since the s u b j e c t s i n the p r e s e n t study were i n the l a s t q u a r t e r of grade 10 and there was no i n d i c a t i o n when i n the grade 10 year the normative data were c o l l e c t e d f o r the Watson-Glaser t e s t , the grade 11 normative data were used. Though there i s not a s i g n i f i c a n t d i f f e r e n c e between the grade 10 and grade 11 normative data f o r the Watson-Glaser t e s t (Grade 11: X = 64.4, s = 11.0; Grade 10: X = 61.9, s = 11.0) i t was concluded t h a t the more c o n s e r v a t i v e approach was appro-p r i a t e . Since there i s no s i g n i f i c a n t d i f f e r e n c e between the two means, the i n t e r p r e t a t i o n of the r e s u l t s of the present study i s not a f f e c t e d . The means and standard d e v i a t i o n s on the Torrance t e s t were l i k e w i s e c o n s e r v a t i v e l y chosen (Fluency: X = 94.6, s = 32.5; F l e x i b i l i t y : X = 40.2, s =9.0; O r i g i n a l i t y : X = 45.2, s = 23.2) g i v i n g a t o t a l mean of 180.0 and a t o t a l standard d e v i a t i o n * of approximately 65. The mean and standard d e v i a t i o n found * See d i s c u s s i o n , Divergent Thinker, Chapter I I I . 70 TABLE 4.8 MEANS AND OF STANDARD DEVIATIONS ALL VARIABLES VARIABLE MEAN STANDARD DEVIATION Dependent Measures (3) Routine Problems 12.11 4.12 (4) Non-Routine Problems 31. 65 8.78 (5) T o t a l Problem Set 43.76 11.49 Independent Measures C o v a r i a t e s (1) Watson-Glaser 61.89 9.92 (2) Torrance 195.12 67.73 . TABLE 4.9 CORRELATION MATRIX FOR DEPENDENT AND INDEPENDENT VARIABLES* 2 3 4 5 6 7 (1) Watson-Glaser (convergent) .04 .44 .41 .47 -.16 -.08 (2) Torrance (divergent) -.14 .06 -.01 .10 . 32 (3) Routine Problems .53 .76 -.25 .05 (4) Non-Routine Problems .95 -.09 .19 (5) T o t a l Problem Set -.16 .16 (6) Method -.06 (7) I n s t r u c t o r i n t h i s study (X = 195.12 and s = 67.74) were not s i g n i f i c a n t l y d i f f e r e n t (p 0.01) from t h i s c o n s e r v a t i v e estimate from the Torrance normative data, and should t h e r e f o r e not a f f e c t the r e s u l t s of the study. While an a n a l y s i s of c o v a r i a n c e became necessary, the a n a l y s i s of v a r i a n c e had a l r e a d y been performed. T h e r e f o r e , both analyses are presented i n the study f o r completeness. The r e s u l t s of the r e g r e s s i o n analyses of v a r i a n c e are presented i n Tables 4.10-4.12. The r e s u l t s of the r e g r e s s i o n analyses of c o v a r i a n c e are presented i n Tables 4.13-4.15; These t a b l e s w i l l be d i s c u s s e d i n the subsequent a n a l y s i s of the s t a t i s t i c a l hypotheses. S t a t i s t i c a l Hypothesis l a Subjects c o n v e r g e n t l y taught w i l l score s i g n i f i c a n t l y more c o r r e c t answers on r o u t i n e problems than s u b j e c t s d i v e r g e n t l y taught. When us i n g the a n a l y s i s df v a r i a n c e r e g r e s s i o n , the c o n t r i b u t i o n made to the v a r i a n c e of the r o u t i n e problems by the convergent t e a c h i n g method was s i g n i f i c a n t a t the p jj» 0.05 l e v e l of s i g n i f i c a n c e . While the a n a l y s i s i n d i c a t e d t h a t there were s i g n i f i c a n t d i f f e r e n c e s , a f u r t h e r post hoc a n a l y s i s was done to determine the d i r e c t i o n a l i t y . The post hoc a n a l y s i s r e v e a l e d t h a t convergent t e a c h i n g was s u p e r i o r to d i v e r g e n t t e a c h i n g on r o u t i n e problems, which supported the h y p o t h e s i s . However, a f t e r a d j u s t i n g f o r d i f f e r e n c e s due to convergent/divergent t h i n k i n g w i t h i n c l a s s e s , no s i g n i f i c a n t e f f e c t f o r method was found (p 0.05) . In l i g h t of t h i s 73 f u r t h e r a n a l y s i s , the hupothesis was not supported. (Refer to Tables 4.10 and 4.13). S t a t i s t i c a l Hypothesis l b Subjects d i v e r g e n t l y taught w i l l score s i g n i f i c a n t l y more c o r r e c t answers on non-routine problems than s u b j e c t s c o n v e r g e n t l y taught. The hypothesis was not supported (p 0.05) by e i t h e r the a n a l y s i s of v a r i a n c e or c o v a r i a n c e w i t h i n the r e g r e s s i o n a n a l y s i s (Refer t o Tables 4.11 and 4.14). S t a t i s t i c a l Hypothesis l c Subjects d i v e r g e n t l y taught w i l l score s i g n i f i c a n t l y more c o r r e c t answers on the t o t a l problem s e t than s u b j e c t s c o n v e r g e n t l y taught. T h i s hypothesis was a l s o unsupported by e i t h e r the a n a l y s i s o f v a r i a n c e or c o v a r i a n c e w i t h i n the r e g r e s s i o n a n a l y s i s (p!£o.05) (Refer to Tables 4.12 and 4.15). S t a t i s t i c a l Hypothesis 2a Convergent t h i n k e r s w i l l score s i g n i f i c a n t l y more c o r r e c t answers on r o u t i n e problems than w i l l d i v e r g e n t t h i n k e r s . Using the r e g r e s s i o n a n a l y s i s o f c o v a r i a n c e t h i s h y p o thesis was supported a t the p ^ O . O l l e v e l . The Watson-Glaser tes t " accounted f o r approximately 19 percent o f the t o t a l v a r i a n c e of the r o u t i n e problems. T h i s was the h i g h e s t s i n g l e v a r i a b l e c o n t r i b u t o r to the v a r i a n c e o f the dependent measure. The post hoc a n a l y s i s v e r i f i e d t h i s r e s u l t . (Refer to Table 4.13). S t a t i s t i c a l Hypothesis 2b Divergent t h i n k e r s w i l l score s i g n i f i c a n t l y more c o r r e c t answers on non-routine problems than w i l l convergent t h i n k e r s . 74 T h i s hypothesis was not supported i n the a n a l y s i s of c o v a r i a n c e . However, again, the Watson-Glaser was the s i n g l e h i g h e s t c o n t r i b u t o r to the v a r i a n c e on the non-routine problems as w e l l as on the r o u t i n e ; accounting f o r approximately 17 percent of the t o t a l v a r i a n c e a t t r i b u t e d t o the non-routine problems. T h i s i s a s t a t i s t i c a l l y s i g n i f i c a n t r e s u l t . A f u r t h e r post hoc a n a l y s i s confirmed t h a t convergent t h i n k e r s d i d b e t t e r than d i d d i v e r g e n t t h i n k e r s on the non-routine problems (p-TO . 0 5 ) (Refer to Table 4 . 1 4 ) . S t a t i s t i c a l Hypothesis 2c Divergent t h i n k e r s w i l l score s i g n i f i c a n t l y more c o r r e c t answers on the t o t a l problem s e t than w i l l 'convergent t h i n k e r s . The a n a l y s i s of co v a r i a n c e i n d i c a t e d t h a t t h i s was not the case. P r e c i s e l y the o p p o s i t e o c c u r r e d w i t h the Watson-G l a s e r c o n t r i b u t i n g approximately 22 percent of the t o t a l v a r i a n c e a t t r i b u t e d t o the t o t a l problem s e t . T h i s f i n d i n g was again v e r i f i e d by post hoc a n a l y s i s and found to be s i g n i f i c a n t a t the p 0.01 l e v e l i n favour of convergent t h i n k e r s over d i v e r g e n t t h i n k e r s (Refer to Table 4 . 1 5 ) . S t a t i s t i c a l Hypotheses 3 a) Convergent t h i n k e r s taught c o n v e r g e n t l y w i l l score s i g n i f i c a n t l y more c o r r e c t answers on r o u t i n e problems than d i v e r g e n t t h i n k e r s taught d i v e r g e n t l y . b) Divergent t h i n k e r s taught d i v e r g e n t l y w i l l score s i g n i f i c a n t l y more c o r r e c t answers on non-routine problems than convergent t h i n k e r s taught c o n v e r g e n t l y . 75 c) Divergent t h i n k e r s taught d i v e r g e n t l y w i l l score s i g n i f i c a n t l y more c o r r e c t answers on the t o t a l problem s e t than convergent t h i n k e r s taught c o n v e r g e n t l y . As the c o n t r i b u t i o n s made to the v a r i a n c e by the i n t e r a c t i o n v a r i a b l e s (Watson-Glaser x Method and Torrance x Method) ranged from a high of one percent to a low of three hundredths of a percent, i t was f e l t t h a t separate analyses were not i n d i c a t e d . Hypothesis 3 was not supported by the a n a l y s i s of v a r i a n c e or c o v a r i a n c e (Refer to the i n t e r a c t i o n p o r t i o n of Tables 4.10-4.15). There were other i n t e r e s t i n g r e l a t i o n s h i p s with the c o v a r i a n c e r e g r e s s i o n a n a l y s i s . However, as these d i d not bear on the s t a t i s t i c a l hypotheses of t h i s study, these w i l l be d i s c u s s e d i n Chapter V (See F i n d i n g s - 3). TABLE 4.10 RESULTS OF THE REGRESSION ANALYSIS WITH ROUTINE PROBLEMS AS THE DEPENDENT VARIABLE (N = 66) DEPENDENT VARIABLE SOURCE OF VARIATION F-VALUE TO F-OBS.2 ENTER/REMOVE R WITH DEPENDENT VARIABLE ARSQ 3 Routine Instructor 0.1443 0.1574 0.4740 0. 0022 Problems Method 4.0672 4.3281 0.2505 0. 0605* Method x Ins. 0.0812 0.0858 0.2529 0. 0012 Torrance x Ins. 7.7365 7.5401 0.4115 0. 1054** Watson x Ins. 0.3240 0.3219- 0.4169 0. 0045 Torrance x Met. 1.0855 1.0659 0.4344 0. 0149 Watson x Met. 0.9998 0.9801 0.4500 0. 0137 Wat x Met x Ins 0.8932 0.8799 0.4634 0. 0123 Tor x Met x Ins 0.1731 0.1769 0.4660 0. 0024 ^ This represents a step-wise regression using analysis of variance. 2 Thxs i s the F-value used to calculate s t a t i s t i c a l significance. The change i n the square of the multiple correlation represents the proportion of the variance i n the dependent variable accounted for by the independent variable. T A B L E 4 . 1 1 R E S U L T S O F T H E R E G R E S S I O N A N A L Y S I S W I T H N O N - R O U T I N E P R O B L E M S A S T H E D E P E N D E N T V A R I A B L E ( N = 6 6 ) D E P E N D E N T S O U R C E O F F - V A L U E T O F - O B S . 2 R WITH A R S Q 3 V A R I A B L E V A R I A T I O N E N T E R / R E M O V E D E P E N D E N T V A R I A B L E Non- I n s t r u c t o r 2 . 3 2 7 3 2 . 4 0 3 5 0 . 1 8 7 3 0 . 0 3 5 1 Routine Problems Method 0 . 3 9 4 1 0 . 4 1 0 9 0 . 2 0 2 7 0 . 0 0 6 0 Method x Ins. 0 . 0 0 0 3 0 . 0 0 0 0 0 . 2 0 2 7 0 . 0 0 0 0 Torrance x Ins. 5 . 9 2 04 5 . 8 0 6 8 0 . 3 5 4 9 0 . 0 8 4 8 Watson x Ins. 1 . 6 0 3 6 1 . 5 6 1 3 0 . 3 8 5 6 0 . 0 2 2 8 . Torrance x Met. 0 . 0 5 6 2 0 . 0 5 4 8 0 . 3 9 0 5 0 . 0 0 0 8 Watson x Met. 0 . 2 0 9 1 0 . 2 0 5 4 0 . 3 8 9 5 0 . 0 0 3 0 Tor x Met x Ins 1 . 4 1 9 0 1 . 4 1 0 6 0 . 4 1 6 0 0 . 0 2 0 6 Wat x Met x Ins 0 . 6 2 2 3 0 . 6 2 3 1 0 . 4 2 6 8 0 . 0 0 9 1 T h i s r e p r e s e n t s a step-wise r e g r e s s i o n u s i n g a n a l y s i s of v a r i a n c e . T h i s i s the F-value used to c a l c u l a t e s t a t i s t i c a l s i g n i f i c a n c e . The change i n the square o f the m u l t i p l e c o r r e l a t i o n r epresents the p r o p o r t i o n of the v a r i a n c e i n the dependent v a r i a b l e accounted f o r by the independent v a r i a b l e . TABLE 4.12 RESULTS OF THE REGRESSION1 ANALYSIS WITH TOTAL PROBLEM SET AS THE DEPENDENT VARIABLE (N = 66) DEPENDENT VARIABLE SOURCE OF VARIATION F-VALUE TO ENTER/REMOVE F -OBS.2 R WITH DEPENDENT VARIABLE ARSQ 3 Total Instructor 1.6846 1 .8112 0.1601 0.0256 Problem Set Method 1.4355 1 .5353 0.2176 0.0217 Method x Ins. 0.0077 0 .0071 0.2179 0.0001 Torrance x Ins. 8.3630 8 .1223 0.4029 0.1148** Watson x Ins. 1.4208 1 . 3725 0.4263 0.0194 Torrance x Met. 0.0424 0 .0425 0.4346 0.0006 Watson x Met. 0.4791 0 .4670 0.4339 0.0066 Tor x Met x Ins 1.3067 1 .2877 0.4550 0.0182 Wat x Met x Ins 0.0989 0 .0991 0.4566 0.0014 This represents a step-wise regression using analysis of variance. This i s the F-value used to calculate s t a t i s t i c a l significance. The change i n the square of the multiple correlation represents the proportion of the variance i n the dependent variable accounted for by the independent variable. TABLE 4.13 RESULTS OF THE REGRESSION ANALYSIS WITH ROUTINE PROBLEMS AS THE DEPENDENT VARIABLE (N = 66) DEPENDENT VARIABLE SOURCE OF VARIATION F-VALUE TO ENTER/REMOVE F-OBS. 2 R WITH DEPENDENT VARIABLE -2. RSQ 3 Routine Problems Watson Torrance 15.0702 1.9898 15.6111 2.0312 0.4366 0.4641 0.1906** 0.0248 I n s t r u c t o r 1.5516 1.5726 0.4843 0.0192 Method 1.8782 1.8756 0.5073 0.0229 Met X Ins 0.4156 0.4177 0.5124 0.0051 Tor X Ins 2.7132 2.6357 0.5431 0.0324 Wat X Ins 1.1501 1.1221 0.5556 0.0137 Tor X Met 0.9554 0.9337 0.5657 0.0114 Wat X Met 0.0321 0.0328 0.5661 0.0004 Tor X Met x Ins 1.6896 1.6627 0.5837 0.0203 Wat X Met x Ins 0.0003 0.0000 0.5837 0.0000 T h i s r e p r e s e n t s a step-wise r e g r e s s i o n using a n a l y s i s of covariance w i t h the scores of the Watson and Torrance as the c o v a r i a t e s . T h i s i s the F-value used to c a l c u l a t e s t a t i s t i c a l s i g n i f i c a n c e . The change i n the square o f the m u l t i p l e c o r r e l a t i o n r epresents the p r o p o r t i o n of the v a r i a n c e i n the dependent v a r i a b l e accounted f o r by the independent v a r i a b l e . TABLE 4.14 RESULTS OF THE REGRESSION 1 ANALYSIS WITH NON-ROUTINE PROBLEMS AS THE DEPENDENT VARIABLE (N = 66) DEPENDENT VARIABLE SOURCE OF VARIATION F-VALUE TO ENTER/REMOVE F-OBS. 2 R WITH DEPENDENT VARIABLE A R S Q 3 Non-Routine Problems Watson Torrance 13.0781 0.1318 14.5827 0.1461 0.4119 0.4140 0.1697** 0.0017 I n s t r u c t o r 3.7359 4.0474 0.4674 0.0471* Method 0.0017 0.0000 0.4675 0.0000 Met x Ins 0.0296 0.0344 0.4679 0.0004 Tor x Ins 1.6175 1.7444 0.5219 0.0203 Wat x Ins 2.6161 2.8530 0.5021 0.0332 Tor x Met 0.1154 0.1289 0.5233 0.0015 Wat x Met 0.0665 0.0773 0.5241 0.0009 Tor x Met x Ins 5.1597 5.3450 0.5804 0.0622* Wat x Met x Ins 2.9805 2.9836 0.6096 0.0347 1 T h i s r e p r e s e n t s a step-wise r e g r e s s i o n u sing a n a l y s i s of covariance i w i t h the scores of the Watson and Torrance as the c o v a r i a t e s . T h i s i s the F-value used to c a l c u l a t e s t a t i s t i c a l s i g n i f i c a n c e . The change i n the square o f the m u l t i p l e c o r r e l a t i o n r e p r e s e n t s the p r o p o r t i o n o f the v a r i a n c e i n the dependent v a r i a b l e accounted f o r by the independent v a r i a b l e . TABLE 4 . 1 5 RESULTS OF THE REGRESSION ANALYSIS WITH TOTAL PROBLEM SET AS THE DEPENDENT VARIABLE (N = 66 ) DEPENDENT VARIABLE SOURCE OF VARIATION F-VALUE TO ENTER/REMOVE F-OBS. 2 R WITH DEPENDENT VARIABLE 4& RSQ 3 T o t a l Problem Set Watson Torrance 1 8 . 2 6 7 4 0 . 0 4 9 0 . 2 0 . 5 6 6 1 0 . 0 5 5 6 0 . 4 7 1 2 0 . 4 7 1 9 0 . 2 2 2 0 * * 0 . 0 0 0 6 I n s t r u c t o r 3 . 9 3 6 5 4 . 2 9 8 5 0 . 5 1 8 7 0 . 0 4 6 4 * Method 0 . 2 7 9 3 0 . 3 0 5 7 0 . 5 2 1 9 0 . 0 0 3 3 Met X Ins 0 . 1 3 6 3 0 . 1 4 8 2 0 . 5 2 3 5 0 . 0 0 1 6 Tor X Ins 3 . 2 2 0 8 3 . 4 8 3 2 0 . 5 5 8 2 0 . 0 3 7 6 Wat X Ins 2 . 3 0 8 8 2 . 4 4 5 7 0 . 5 8 1 4 0 . 0 2 6 4 Tor X Met 0 . 0 0 5 0 0 . 0 0 9 3 0 . 5 8 1 6 0 . 0 0 0 1 Wat X Met 0 . 0 2 2 2 0 . 0 2 7 8 0 . 5 8 1 6 0 . 0 0 0 3 Tor X Met x Ins 5 . 3 2 1 0 5 . 4 1 0 2 0 . 6 2 9 8 0 . 0 5 8 4 * Wat X Met x Ins 1 . 8 9 3 0 1 . 8 9 0 0 0 . 6 4 5 8 0 . 0 2 0 4 1 T h i s r e p r e s e n t s the step-wise r e g r e s s i o n using a n a l y s i s o f covar i a n c e with the scores o f the Watson and Torrance as the c o v a r i a t e s . T h i s i s the F-value used t o c a l c u l a t e s t a t i s t i c a l s i g n i f i c a n c e . The change i n the square o f the m u l t i p l e c o r r e l a t i o n r e p r e s e n t s the p r o p o r t i o n o f the v a r i a n c e i n the dependent v a r i a b l e accounted f o r by the independent v a r i a b l e . 82 CHAPTER V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS The purpose of t h i s study was to i n v e s t i g a t e the e f f e c t of two te a c h i n g methods (convergent/divergent on student performance on two problem-solving tasks (routine/non-r o u t i n e problems). Would one method be b e t t e r f o r both kinds of problems? Would one method be b e t t e r on one type and the other method be b e t t e r on the other type? The study a l s o i n v e s t i g a t e d the r e l a t i o n s h i p s between the convergent/ d i v e r g e n t t e a c h i n g methods and the student's t h i n k i n g s t y l e ( c o n v e r g e n t / d i v e r g e n t ) . Would students taught by a method s i m i l a r to t h e i r t h i n k i n g s t y l e be b e t t e r than those who were taught by a method d i s s i m i l a r t o t h e i r t h i n k i n g s t y l e ? Would there be an i n t e r a c t i o n among t h i n k i n g s t y l e , t e a c h i n g method, and problem type? Summary of the Study Four grade ten c l a s s e s were randomly s e l e c t e d from the eleven academic mathematics c l a s s e s i n a secondary school i n v o l v e d i n the study. Due to s u b j e c t absenteeism f o r e i t h e r the p r e - t e s t s and/or the treatment, a t o t a l o f s i x t y - s i x s u b j e c t s were i n v o l v e d i n the study. Each s u b j e c t was given the Watson-Glaser T e s t o f C r i t i c a l T h i n k i n g (Form YM) and the Torrance t e s t of T h i n k i n g C r e a t i v e l y w i t h Words (Booklet A) to determine t h e i r l e v e l on the two independent measures of convergent and d i v e r g e n t t h i n k i n g . A L a t i n -design was used to a s s i g n c l a s s e s to treatment (convergent/ 83 divergent teaching) and instructor (Teacher A/Teacher B). Each subject was taught by one teacher using one method for approximately two hours. The content for the lessons involved the Fibonacci Sequence and Pascal 1s Triangle and was taken from Jacobs' (197 0) Mathematics: A Human Endeavor. At the end of the treatment each subject received a test on the dependent measures (routine/non-routine problems). Trained observers were used to ensure consistency of teaching method. Analysis of covariance using the regression model was performed with, convergent/divergent thinking styles as the covariates. Findings 1. Convergent teaching was superior to divergent teaching on routine problems only. When the variance due to thinking style was removed, t h i s finding was no longer s i g n i f i c a n t . The superiority of the convergent teaching method was i n fact due to differences between subjects on the convergent and divergent thinking measures. This finding questions studies that suggest that directed teaching i s superior to non-directed teaching for short term e f f e c t s (Dahmus, 1970; Campbell, 1964). This r e s u l t mayv however, be a socio- c u l t u r a l phenomena i n that current teaching practices tend to be largely convergent and di r e c t i v e i n nature. 2. Convergent thinkers were found to score s i g n i f i c a n t l y more correct answers on a l l the dependent measures than did divergent thinkers. While there i s some evidence that convergent t h i n k i n g i s h i g h l y c o r r e l a t e d w i t h i n t e l l i g e n c e ( F u r s t , 1950), the Watson-Glaser t e s t was o n l y moderately c o r r e l a t e d with i n t e l l i g e n c e (0.55-0.73) (See convergent/ d i v e r g e n t t h i n k i n g , Chapter I I , f o r f u r t h e r d i s c u s s i o n ) . The c o r r e l a t i o n of the dependent measures used i n t h i s study w i t h i n t e l l i g e n c e i s unknown. Th e r e f o r e i t i s unknown to what extent i n t e l l i g e n c e may have i n f l u e n c e d scores on the dependent measures. 3. Only one out of the e i g h t i n t e r a c t i o n e f f e c t s t e s t e d was s t a t i s t i c a l l y s i g n i f i c a n t . T h i s i n t e r a c t i o n was o n l y found f o r r o u t i n e problems and suggested t h a t non-divergent t h i n k e r s d i d b e t t e r with convergent t e a c h i n g as opposed to d i v e r g e n t t e a c h i n g , while non-convergent t h i n k e r s d i d b e t t e r w i t h d i v e r g e n t t e a c h i n g as opposed to convergent t e a c h i n g . U n f o r t u n a t e l y , the n was so l i m i t e d as to make a g e n e r a l -i z a t i o n of t h i s f i n d i n g q u e s t i o n a b l e . I t i s i n t e r e s t i n g to note t h a t a s i m i l a r f i n d i n g was not found when the d e f i n i t i o n of groups was d i v e r g e n t as opposed to non-convergent, which supports the d i v i s i o n of these two groups (See d e f i n i t i o n of non-convergent i n Chapter I I I ) . Only t h i s one f i n d i n g supports the t h e o r e t i c a l h y p othesis of t h i s study t h a t an i n t e r a c t i o n e f f e c t would be found between t e a c h i n g method and t h i n k i n g s t y l e . The f a c t t h a t none of the other f i n d i n g s support the t h e o r e t i c a l h y p othesis g i v e s f u r t h e r i n d i c a t i o n t h a t i n t e l l i g e n c e may have been a confounding f a c t o r , i . e . those who scored low on the Watson-85 G l a s e r and/or Torrance t e s t s were of lower i n t e l l i g e n c e compared to those who had high scores on these measures. 4. When the s i g n i f i c a n t d i f f e r e n c e s f o r method disappeared on the r o u t i n e problems due to changing from an a n a l y s i s o f v a r i a n c e to an a n a l y s i s of co v a r i a n c e w i t h i n the r e g r e s s i o n model, a s i g n i f i c a n t d i f f e r e n c e f o r the Watson-G l a s e r (convergent t h i n k i n g ) appeared. T h i s would seem to i n d i c a t e t h a t i t i s the t h i n k i n g s t y l e r a t h e r than the convergent t e a c h i n g method which has the g r e a t e r e f f e c t on problem s o l v i n g . I t should be noted t h a t t h i s s i g n i f i c a n t e f f e c t f o r convergent t h i n k i n g s t y l e was found f o r a l l the dependent measures. 5. The a n a l y s i s of v a r i a n c e showed a s i g n i f i c a n t e f f e c t f o r the Torrance x I n s t r u c t o r i n t e r a c t i o n . Upon f u r t h e r a n a l y s i s , u s i n g c o v a r i a n c e , t h i s e f f e c t disappeared e n t i r e l y on the r o u t i n e problems and two other s i g n i f i c a n t e f f e c t s were found ( I n s t r u c t o r and Torrance x Method x I n s t r u c t o r ) i n both the non-routine problems and the t o t a l problem s e t . The teacher e f f e c t must be q u a l i f i e d by the f a c t t h a t the Torrance x Method x I n s t r u c t o r e f f e c t a l s o o c c u r r e d . When the post hoc analyses o f the comparison between group means were done (as presented i n Table 4.7) one of the groups taught d i v e r g e n t l y by Teacher A (Group 3) scored s i g n i f i c a n t l y (p £ 0.01) d i f f e r e n t l y from the two groups taught by Teacher B (Groups 1 and 4). T h i s may have i n f l u e n c e d the r e s u l t s and t h i s f a c t has a l i m i t i n g e f f e c t on the g e n e r a l i z a b i l i t y of 86 t h i s study. 6. In a n a l y z i n g the c o r r e l a t i o n matrix, there was a p o s i t i v e c o r r e l a t i o n between the Watson-Glaser t e s t and the r o u t i n e problems (0.4366) and a p o s i t i v e c o r r e l a t i o n between the Watson-Glaser t e s t and the non-routine problems (0.4119). T h i s would i n d i c a t e the the convergent p o r t i o n s of both dependent measures was adequate. However, there was a s l i g h t n egative c o r r e l a t i o n between the Torrance t e s t and the r o u t i n e problems (-0.1404) and a s l i g h t p o s i t i v e c o r r e l a t i o n between the Torrance t e s t and the non-routine problems (0.0576). T h i s seems to i n d i c a t e t h a t some r e v i s i o n i s needed i n the d i v e r g e n t p o r t i o n s on both dependent measures. The c o r r e l a t i o n of the Watson-Glaser t e s t w i t h both problem s e t s may have i n f l u e n c e d the r e s u l t s of the study i n favour of the convergent t h i n k e r s , which may have accounted f o r the l a c k of s i g n i f i c a n t r e s u l t s f o r the d i v e r g e n t t h i n k e r s . I m p l i c a t i o n s of the Study The r e s u l t s of t h i s study suggest t h a t i n a s h o r t term s i t u a t i o n , academic grade ten students w i l l l i k e l y b e n e f i t most from being taught i n a convergent manner. Students who are convergent t h i n k e r s w i l l b e n e f i t most from t h i s type of i n s t r u c t i o n . The r e s u l t s of t h i s study a l s o i n d i c a t e t h a t convergent t h i n k e r s , i n a s h o r t term s i t u a t i o n , do b e t t e r than d i v e r g e n t t h i n k e r s on a mathematical problem-solving t a s k . However, t h i s r e s u l t may be confounded by i n t e l l i g e n c e . 87 L i m i t a t i o n s of the Study 1. The study i n v o l v e d students from academic grade ten mathematics c l a s s e s and t h e r e f o r e g e n e r a l i z a t i o n s should be l i m i t e d to groups from a s i m i l a r p o p u l a t i o n . 2. G e n e r a l i z a b i l i t y of the f i n d i n g s i s l i m i t e d by the f a c t t h a t one teacher seemed to get b e t t e r r e s u l t s than the other teacher i n v o l v e d and t h a t teacher a l s o i n t e r a c t e d w i t h the Torrance t e s t and d i v e r g e n t t e a c h i n g method. 3. The r e s u l t s of the study should o n l y be g e n e r a l i z e d f o r s h o r t term e f f e c t s . 4. Work i s needed on the dependent measures to i n c r e a s e t h e i r c o r r e l a t i o n with the Torrance t e s t and thus b r i n g t h e i r measures of d i v e r g e n t p r o b l e m - s o l v i n g a b i l i t i e s i n l i n e w i t h t h e i r measures o f convergent p r o b l e m - s o l v i n g a b i l i t i e s . 5 . I t should be noted t h a t due to the time c o n s t r a i n t s of t h i s study, convergent t h i n k i n g and convergent t e a c h i n g may have been g i v e n u n f a i r advantage. Convergent l e s s o n s completed more of the s p e c i f i e d m a t e r i a l and had the advantage of r e viewing w i t h the students the important p o i n t s of the l e s s o n . Recommendations f o r F u r t h e r Research I f the q u a l i t y of mathematics t e a c h i n g i s to improve, i t i s e s s e n t i a l t h a t r e s e a r c h i n v o l v i n g problem s o l v i n g be implemented i n such a way as to m a i n t a i n the complexity of the p r o b l e m - s o l v i n g p r o c e s s . T h i s w i l l u l t i m a t e l y i n v o l v e f u r t h e r i n v e s t i g a t i o n and understanding of the t r a i t - t r e a t m e n t 88 i n t e r a c t i o n o f t h i n k i n g s t y l e and t e a c h i n g method as a minimal s t a r t i n g p o i n t . F u r t h e r r e s e a r c h i s needed to s p e c i f y and c l a r i f y the c r i t i c a l v a r i a b l e s i n v o l v e d i n both t h i n k i n g s t y l e and t e a c h i n g method. While the r e s u l t s o f the c u r r e n t study suggest t h a t a convergent t h i n k i n g s t y l e and a convergent t e a c h i n g method f a c i l i t a t e problem s o l v i n g , i t i s important to i n v e s t i g a t e p o s s i b l e confounding v a r i a b l e s . I t i s recommended t h a t any f u r t h e r r e s e a r c h i n the area of convergent/divergent t h i n k i n g s t y l e as r e l a t e d to mathematical problem s o l v i n g be designed to p a r t i a l out the v a r i a n c e a t t r i b u t a b l e to i n t e l l i g e n c e . T h i s i s p a r t i c u l a r l y important i n the l i g h t of the s i g n i f i c a n t i n t e r a c t i o n between non-divergent and non-convergent t h i n k e r s and the convergent/ d i v e r g e n t t e a c h i n g methods. I t i s recommended t h a t any f u r t h e r r e s e a r c h u t i l i z i n g l e s s o n s s i m i l a r to those used i n the study i n c o r p o r a t e some of the content from the F i b o n a c c i Sequence as found by the d i v e r g e n t l y taught s u b j e c t s (See Observer's C h e c k l i s t , Groups 3 and 4, Chapter I V ) . An i n c r e a s e i n the time f a c t o r would probably improve the e f f e c t i v e n e s s of the d i v e r g e n t t e a c h i n g method. I t might a l s o show t h a t there are d i f f e r e n t i a l e f f e c t s over time between t e a c h i n g methods and t h i n k i n g s t y l e s . As T a y l o r (1965) suggests, perhaps both convergent and d i v e r g e n t t e a c h i n g methods need to be used i n c o n j u n c t i o n 89 with one another to t r u l y improve the q u a l i t y o f problem s o l v i n g i n students. One f u r t h e r v a r i a t i o n o f t h i s study might be to use three t e a c h i n g methods: convergent, d i v e r g e n t , and a combination method of the two over a long p e r i o d o f time and study the e f f e c t s t h a t each of these methods have on the problem-solving a b i l i t i e s o f students i n mathematics. 90 LITERATURE CITED Aiken, Lewis R. J r . A b i l i t y and c r e a t i v i t y i n mathematics. Review of Educational Research, 1973, 4_3(4) , 405-431 Ausubel, D. P. The Psychology of Meaningful Verbal Learning. New York, Grune and Stratton, 196 3. Ausubel, D. P. and F i t z g e r a l d , D. Organizer, general back-ground, and antecedent learning variables i n sequential verbal learning. Journal of Educational Psychology, 1962, 53, 243-249. A v i t a l , Shmuel M. and Shettleworth, Sara J . Objectives for mathematics learning: some ideas for the teacher. B u l l e t i n  of the Ontario Institute for Studies i n Education, 1968, No. 3. Behr, Merlyn J . 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J o u r n a l f o r Research i n Mathematics Education, 1976, 7(5), 264-275. 96 APPENDIX A CONVERGENT TEACHING LESSON FIBONACCI NUMBERS I. Introduction Observer introduce the teacher by name, as well as s e l f . Observer take seat at back. Observer i s there to note content covered and consistency with convergent teaching. II . Lesson 1. Pass out a piece of paper to each student. 2. Have each student write the numbers 1-10 i n a column along the l e f t hand margin. 3. Have each student choose two numbers between 1 and 10. 4. Have them write t h e i r f i r s t choice by the (1) i n t h e i r column and the second choice by the (2). 5. The t h i r d number i s the sum of the f i r s t and second. The fourth i s the sum of the second and t h i r d . 6. Have students f i l l i n a l l ten numbers i n a similar way. 7. Have students f i n d the sum of the ten numbers. 8. Steps 3-7 should be done by teacher at the board along with the students to ensure that they understand the procedure. Teacher should determine the sum of his own ten numbers by multiplying the seventh number by 11. Then should check with a few students for th e i r 7th number. 9. Teacher t e l l s the class the sum of the numbers for the students selected. Then asks another student for his seventh number and t e l l s the class the sum. 10. "Notice, I am asking everyone for t h e i r seventh number before t e l l i n g them the sum. You know that numbers may be related to each other through the arithmetic operations (Check to be sure that students know these are addition, subtraction, m u l t i p l i c a t i o n and d i v i s i o n ) . Which of these operations relates the t o t a l sum to the seventh number?" 11. "We have found that t h i s works for many cases. However, in mathematics we often want to fin d out i f something works for any numbers chosen. We can use algebra to represent what we have done for the general case of any two numbers and then deduce whether t h i s number t r i c k does indeed work no matter what numbers we choose." 12. Let a represent the f i r s t number and b represent the second, then our ten numbers look l i k e : (1) a (a) looking at the sum, i s there (2) b any common factor between the (3) a + b two terms? (4) a + 2b (b) r e c a l l i n g the d i s t r i b u t i v e (5) 2a + 3b property ab + ac = a(b+c) (6) 3a + 5b 11(5a) + 11(8b) = 55a + 88b (7) 5a + 8b (c) W i l l our t r i c k work for any (8) 8a + 13b two numbers? (9) 13a + 21b (10) 21a + 34b Sum 55a + 88b 97 13. Consider the c o e f f i c i e n t s (the numbers i n f r o n t ) o f the a's (Agreement: when no number appears i n frong o f a or b we agree the number i s one.) Can you best d e s c r i b e the p a t t e r n of c o e f f i c i e n t s as a product? a sum? or a d i f f e r e n c e ? What reasoning l e d you to t h i s c hoice? How do the d o e f f i c i e n t s o f the a's bets r e l a t e to the c o e f f i c i e n t s fo the corresponding b ' s ? ( S t a r t a t the t h i r d or f o u r t h term). Would you say t h a t they are one ahead? One behind? the same? What l e d you to b e l i e v e your c o n c l u s i o n ? 14. "A c e r t a i n machine makes change f o r any whole number of d o l l a r s . However, i t w i l l o n l y pay out i n one d o l l a r or two d o l l a r b i l l s . The change i n b i l l s may come out i n any order , though. For i n s t a n c e , i f change i s wanted f o r three d o l l a r s , i t may come out i n any one of the f o l l o w i n g 3 ways: $1 f o l l o w e d by another $1 f o l l o w e d by another $1; OR $1 f o l l o w e d by a $2; OR a $2 f o l l o w e d by a $1. I f we agree to denote a $1 b i l l by A and $2 b i l l by B, we get AAA, AB, BA as the three p o s s i b l e ways of making change f o r three d o l l a r s . 15. On the back of your p i e c e o f paper, make the f o l l o w i n g headings: Amount of P o s s i b l e ways of Numbers of ways _ Change to paying out of paying out be p a i d  16. Next t r y a $5 b i l l to change, what are the p o s s i b l e ways of paying out? What are the number of ways? 17. Suppose we had no money. The p o s s i b l e ways to pay out are? How many ways i s t h i s ? 18. F i l l i n the t a b l e up to and i n c l u d i n g $5 to be changed. 19. Look a t the numbers i n the t h i r d column. Do they look f a m i l a r ? R e c a l l the numbers which were i n f r o n t o f the a's and b's i n our t r i c k problem. How d i d we get those numbers? Do you t h i n k t h a t we get these numbers i n the same way? What would you p r e d i c t the number of d i f f e r e n t ways the machine c o u l d make change f o r $10 would be? 20. Pass out d i t t o e d sheets on F i b o n a c c i Sequence. There are many kinds o f t h i n g s we might be ab l e to do wit h t h i s sequence. You may note how i t i s r e l a t e d to our number t r i c k . 98 21. What about even and odd numbers i n the sequence? Could you describe a pattern to t e l l someone else which terms are even and which terms are odd? 22. What about multiples of certain numbers? 2(evens)? 3? (Work out together at the board) Notice that multiples of 4 do not occur every 5th term. Do multiples of 5 occur every 6th term. Notice that multiples of the Fibonacci numbers do occur i n a p a r t i c u l a r pattern. 23. Could we also sum terms (demonstrate what i s meant by this) What i f we wanted to add the f i r s t terms? Should we try to figure out the general case f i r s t ? Or should we take s p e c i f i c cases and look for a pattern? How should we choose these s p e c i f i c cases? Choose anywhere? Start at the beginning? Start at the end? What gave you a hint as to shich cases to choose? (Recall the change machine). 24. Review what has been learned about Fibonacci Sequences. Emphasize correct choices and conclusions. (The following i s an outline of notes as given to the teachers to use during the lesson on the Fibonacci Sequence.) 1.* 1) 5 2) 4 3) 9 4) 13 5) 22 6) 35 7) 57 8) 92 9) 149 10) 241 Sum 627 Trick (11 x 57 = 627 = sum) 11 x 7th # = sum - do some how i s the sum related to the 7th number? numbers can be related by by operations add, sub., mult., div. Now does i t work for a l l numbers? Can we use algebra to represent the numbers? How? These numbers ( 1 - 8 ) refer to the numbers on the Observer's Checklist and Student's Notes, see Chapter IV, Table 4.5. 99 1) a + Common F a c t o r 2) b D i s t r i b u t i v e 3) a + b P a t t e r n of c o e f f i c i e n t s 4) a + 2b need a, b 10? 5) 6) 7) 2a 3a 5a + + + 3b 5b 8b Are c o e f f i c i e n t s sums, products, d i f f e r e n c e s 8) 9) 8a 13a + + 13b 21b What p a t t e r n r e l a t e s a's and b's 10) 21a + 34b Sum 55a + 88b Change machine f o r cheques — ^ change CHANGE cheque V Amount of cheque to be p a i d Do t h i r d 0 1 2 do f i r s t 3 4 do second 5 A = $1 b i l l B = $2 b i l l o r der counts as a separate way of doing change. P o s s i b l e ways of Paying out AA B AAA BA AB AAAA AAB BAA ABA BB AAAAA AAAB AABA ABAA BAAA BBA BAB ABB Number of Ways of Paying out 1 2 3 5 Fou r t h - r e l a t e to c o e f f i c i e n t s i n two (2.) F i f t h - P r e d i c t f o r $10 100 4. Pass out d i t t o e d sheets of F i b o n a c c i Sequence. E x p l a i n . F a m i l a r ? 5. Even and odd numbers - p a t t e r n (even i s every 3rd) 6. M u l t i p l e s o f numbers 2 3 4 5 8 every 3 r d 4 t h g t h 5 f c h g t h Then look a t ONLY m u l t i p l e s of the F i b o n a c c i numbers. Sum of F i b o n a c c i Numbers F 1 1 Sum of f i r s t n n F 2 1 1 1 F 3 2 2 2 F 4 3 4 3 F c 5 7 4 D F c 8 12 5 6 F ? 13 20 6 F g 21 Sum of f i r s t n = F , 0 - 1 n+2 101 THE FIBONACCI SEQUENCE NAME OF TERM NUMBER OF TERM TERM F1 1 1 F 2 2 1 F 3 3 2 F 4 4 3 F 5 5 5 F 6 6 8 F ? 7 13 F g 8 21 F g 9 34 F 1 Q 10 55 F X 1 11 89 F 12 12 1 4 4 F 1 3 13 2 3 3 F 14 14 3 7 7 F 1 5 15 610 F l g 16 987 F 1 7 17 1,597 F 1 8 18 2,584 F 1 Q 19 4,181 F 2 Q 20 6,765 102 PASCAL'S TRIANGLE 1. Watch th i s sequence c l o s e l y . I t builds i n a d i f f e r e n t way than the Fibonacci Sequence did. 1. 1 1 2. 1 2 1 3. 1 3 3 1 2. What things do you notice about the rows so far? How do the numbers between the rows relate to the row number? 3. Do you think that the next row w i l l be 14641? 14541? or 14441? 4. The next row i s 1 4 6 4 1 . Did you notice that the ones remain the same? Notice that the numbers i n the row above are not d i r e c t l y above the numbers i n the row below. Do you see a re l a t i o n s h i p between any 2 numbers i n the previous row and a number in the row below? 5. On the Fibonacci Sequence we found some very inte r e s t i n g patterns, p a r t i c u l a r l y when we looked at sums. Let us consider a row as a minature sequence (subsequence) and add across the rows and look for a pattern. Sum of numbers i n the nth row i s 2 n. What does the shape of our sequence suggest? (triangle) Is there a portion missing? How might we use what we have just found about the sums of numbers i n rows to help us fi n d the answer? Get the top 1 i n the t r i a n g l e , goes with 0 row and 2^ = 1. 6. What i f we changed the shape of the "tr i a n g l e " s l i g h t l y to form a r i g h t triangle? Have we changed the rows? 7. Now we can look at what instead of rows? To fi n d the sum of the f i r s t _ numbers i n any column, go over 1 and down 1 from where you ended i n that column to f i n d the answer. 8. Consider the tr i a n g l e shape i t s e l f (use outer row) Go back to the o r i g i n a l form of the t r i a n g l e . Could we look at the sums i n any other shape? Now consider a diamond shape (use outer edge). 9. Go to the ri g h t t r i a n g l e version. Consider the t h i r d column. What i f instead of adding up a l l the numbers in the column to a certa i n point, what i f you added up any 2 adjoining numbers and looked at the sequence formed? 1 103 10. Which of the following does the question mean? You add up 1 and 3, then add up 6 and 10, then 15 and 21? or You add up 1 and 3, then 3 and 6, then 6 and 10? 11. The sequence formed i s 4, 9, 16, ... What do you notice about these p a r t i c u l a r numbers? Would 22, 3 , 4 , ... be an equivalent sequence? Would 2, 3, 4, ... be an equivalent sequence? 12. Do you see any way i n which t h i s sequence and the Fibonacci sequence are related? 13. We've looked at rows and columns, what i s l e f t to look at for a pattern? Should we look at the upward or downward diagonals? 14. Review the relationships found i n Pascal's Triangle. Again emphasis i s on answers and conclusions. (The following i s an outline of notes as given to the teachers to use during the lesson on Pascal's Triangle.) 1.* 1 1 Next row: 14641? 1 1 3 2 3 1 1 14541? 14441? Relationship between row above and one below. Let's see i f we can f i n d any rela t i o n s h i p s . 2. 1) Get row sums 2) 1 Work to 0 row 3)1 3 * These numbers (1-8) refer to the numbers on the Observer's Checklist and Student's Notes, see Chapter IV, Table 4.5. 104 0) 1 1) 1 1 2) 1 2 1 3) 1 3 3 1 4) 1 4 6 4 1 5) 1 5 10 10 5 1 6) 1 6 15 20 15 6 1 7) 1 7 21 35 35 21 7 Look at the column sums of f i r s t so many one over and one down from l a s t number added. 4. Sums of l i t t l e t riangles Sums of diamonds 1 1 1 1 1 1 1 1 3 1 6 4 5 10 10 6 15 20 yYA^V 4 1 ,1 '/5/10 10 5 Sums \ 1 4 9 16 25 Sums of 3rd column adjacent pairs Interpretation of question 1+3, 6+10,15+21 OR 1+3, 3+6, 6+10 (yes) Sum sequence 4/9,16... i s 2 2, 3 2, 4 2 or 2,3,4 Sums of diagonals upwards (Get Fibonacci Sequence) 7. Review: Answers found i n lesson. 105 APPENDIX B DIVERGENT TEACHING LESSON FIBONACCI NUMBERS 1 - 9 Identical with convergent lesson. 10. How did I know the sum? Yes, I found out about the seventh number, but how does that relate to the sum? How do numbers rel a t e to one another any way? ( l i s t suggestions on the board) Do you think that t h i s t r i c k w i l l work for any two numbers? What suggestions do you have for finding out? (Make l i s t of suggestions on the board) (Have students come up with some form of representation which i s general for any 2 numbers) What do you notice about the numbers that you see i n the general representation? Do they appear to be special i n any way? How are they related to each other? (Li s t suggestions). 13. This content should come out of the above discussion. 14. A cert a i n machine makes change for any whole number of d o l l a r s . However, i t w i l l pay out i n $1 or $2 b i l l s only. The change i n b i l l s may come out i n any order and we want to consider the order that the b i l l s come out i n to be a d i f f e r e n t way of making change even though you may end up with the exact same number of b i l l s - e.g. getting $2 f i r s t and the $1 i s d i f f e r e n t than getting $1 and then a $2. If we want to f i n d out how many d i f f e r e n t ways of cashing a cheque of any p a r t i c u l a r amount, how can we do i t ? ( l i s t suggestions on board) Which should we try f i r s t ? Why are you making that choice? 15-18 The content of the convergent lesson should be brought out, but through p a r t i c i p a t i o n and suggestions<and should not p a r t i c u l a r l y be teacher led. 20. What properties do you know about numbers? (Lis t suggestions f r o j students such as odd, even, multiples, more than, less than,... must be student suggested). Pass out dittoed sheets of Fibonacci Sequence. What kinds of things can we do with 106 21. 22. 23. 24. these numbers? What ways of combining numbers do you know of? What kinds of patterns might you expect? Let's make a l i s t of suggestions which we might make to combine these numbers and which might give a pattern? Are there any patterns which are obvious i n the sequence? Hope that odd/even w i l l come out of t h i s discussion as well as multiples which are Fibonacci numbers. (As per 21) Are there any things that we have done previously that might suggest something to do with the Fibonacci Sequence? (Hope that : F, , F~ + ... + F = F , 1 . c L ' \ 1 + 2 n n+2 comes out of t h i s ) . Review what has been going on i n class regarding Fibonacci Sequence. Emphasis on the ways we found for solving problems, relationships, combining one or more previously learned facts to make a new conjecture, using old tools and applying to new situations, etc. (The following i s an outline of notes as given to the teachers to use during the lesson on the Fibonacci Sequence). 1) 5 2) 4 3) 9 4) 13 5) 22 6) 35 7) 57 8) 92 9) 149 10)241 Ways to fin d out i f t r i c k works for any two ( l i s t suggestions) Trick (11 x 7th = sum) DO NOT TELL Relationships between numbers ( l i s t suggestions) Sum627 2. $2 then $1 i s d i f f e r e n t way to make change than $1 then $2. Find out how many ways to make change for any p a r t i c u l a r cheque (Suggestions? l i s t these.) * These numbers (1-8) refer to the numbers on the Observer's Checklist and Student's Notes, see Chapter IV, Table 4.5. Pass out d i t t o e d sheets of F i b o n a c c i Sequence: a) What p r o p e r t i e s do you know about numbers? ( L i s t suggestions l i k e even/odd, g r e a t e r than, l e s s than, equal t o , m u l t i p l e s , d i v i s i b l e by, constant sum, etc.) b) How do these r e l a t e to the sequence? Look f o r p a t t e r n s . Get content of m u l t i p l e s of F i b o n a c c i F F F F F F *1 2 3 4 5 *6 1 1 2 3 5 8 D i v i s i b l e by 2, r e p e a t i n g every 3rd D i v i s i b l e by 3 p r e s e n t s a p a t t e r n of every 4th How much i s the sum of the 1st n F i b o n a c c i Numbers How can we f i n d out? ( l i s t suggestions) F, + F„ + ... F = F . „ - 1 1 2 n n+2 Review - How have we found r e l a t i o n s h i p s ? How have we s o l v e d problems? What techniques d i d we use? 108 PASCAL'S TRIANGLE 1. Identical to convergent lesson. 2. How do you think t h i s sequence i s formed? L i s t suggestions on board. Just how can we test these hypotheses? What w i l l the next row be given your hypothesis or rule? ( l i s t these beside the suggestions) 3. Suppose I choose the next row to be 1 4 6 4 1? Which of these rules could I use? 4. Did a l l of the hypotheses work? Did more than one of the hypotheses work? What would we do i f none of our hypotheses worked and somebody said that 1 4 6 4 1 was the correct next row of the sequence that they were developing. Using one or more hypotheses, now predict the next row (again l i s t beside suggestions) (If time here to expand - maybe look at some of the other hypotheses and wee what things come out of them.) 5. On the Fibonacci sequence we found some very inte r e s t i n g patterns i n the sequence. What might we look for i n t h i s sequence? (Lis t many suggestions, explore each b r i e f l y , u t i l i z e the mos productive i f time i s short). 6. Identical with convergent lesson. 7. What might we now be interested i n looking at? (Lis t these new suggestions on board, explore). 8. Go back to the o r i g i n a l form of the t r i a n g l e . Could we look at the sums of any other shape? What other shapes do you know? Where should we start? (Lists and conjectures and conclusions on the board). 9. If we want to look at pairs of adjacent numbers, how would we add them (Discussion) How could we go about solving this? Make a guess (hypothesis)(list on board) Let's see i f they work. Try some of these guesses and test workability. 10. Discuss sequence formed. 1D9 10-12 Omit, should get t h i s content with discussion i f not, that's acceptable. 13. Which tria n g l e should we try i n looking for patterns? What made you choose that one? Are there any patterns we might have missed i n our exploration so far? ( l i s t on board) 14, 1.* 1. 2. 3. Review our approach to solving the problems presented so far. Emphasis on procedures, l o o l s used, how we made hypotheses, etc. (The following i s an outline of notes as given to the teachers to use during the lesson on Pascal's Triangle.) 1 1 1 2 1 1 3 3 1 Next Row/Rule Guesses Right tr i a n g l e as l i s t e d i n convergent lesson Suggestions for next row? L i s t What rules do you need to get next row? ( l i s t ) If 1 4 6 4 1 i s next row, which row works? Row sum (may ask question) Add zero row (use t h i s content only i f student suggested) L i s t suggestions a) l i s t suggestions of what might now look at. b) Pairs of adjacent numbers (vertical) How to add? L i s t Suggestions. c) Using 3rd column, what pattern? Relationship between Pascal's Triangle and Fibonacci Sequence? How to f i n d - l i s t suggestions - t r y . Review - Relationships, conjectures, These numbers are assoicated with the Observer's Checklist (See Chapter IV, Table 4.5) where 1 goes with 1 and 2; 2 goes with 4,5 and 6; 3 goes with 7; and 4 goes with 8. 110 APPENDIX C ROUTINE PROBLEMS NAME: BLOCK: REFERENCE FIBONACCI SEQUENCE 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1,598 PASCAL'S TRIANGLE " E q u i l a t e r a l " "Right" 1 1 1 1 1 1 1 2 1 1 2 1 1 3 3 1 1 3 3 1 1 4 6 4 1 1 4 6 4 1 1 5 10 10 5 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 6 15 20 15 6 1 ****************************************** I. For each of the f o l l o w i n g q u e s t i o n s , c i r c l e the ONE answer which BEST f i t s : T i f you t h i n k the statement i s ALWAYS t r u e . ST i f you t h i n k the statement i s SOMETIMES t r u e . F i f you t h i n k the statement i s ALWAYS f a l s e . 1. The f i f t h term of the F i b o n a c c i sequence i s 5. T ST. F 2. The sum of two F i b o n a c c i numbers g i v e s another F i b o n a c c i number. T ST F 3. The sum of the numbers i n any row of P a s c a l ' s T r i a n g l e a f t e r row 1 i s j u s t the row number m u l t i p l i e d by two. T ST F 4. Numbers i n the f i f t h column of the " r i g h t " v e r s i o n o f P a s c a l ' s T r i a n g l e are m u l t i p l e s of f i v e . T ST F 5. One way to get the s i x t h F i b o n a c c i number i s to add one to the sum of the f i r s t f o u r terms. T ST F 6. The sum of two numbers i n the same row of P a s c a l ' s T r i a n g l e w i l l g i v e a number i n the next row of the t r i a n g l e . T ST F 7. I f you know t h a t a c e r t a i n number i s a F i b o n a c c i number, then you know t h a t i t i s a l s o a number i n P a s c a l ' s T r i a n g l e . T ST F 8. In the F i b o n a c c i Sequence every t h i r d term i s a m u l t i p l e of 2. In g e n e r a l , every (n+l)th term i s a m u l t i p l e o f n: e.g. i f n = 3, then every f o u r t h term i s a m u l t i p l e of 3. T ST F I l l 9. I f you add a number i n the f i r s t column of the " r i g h t " v e r s i o n of P a s c a l ' s T r i a n g l e to a number i n the second column the r e s u l t w i l l be a number i n the second column. T ST F 10. I f you know t h a t a p a r t i c u l a r row of P a s c a l ' s T r i a n g l e c o n t a i n s ten numbers, then you know t h a t you are i n the e l e v e n t h row. T ST F **************************************** 11. For each of the f o l l o w i n g problems, choose the BEST answer and p l a c e the l e t t e r o f your c h o i c e i n the column a t the r i g h t . 1. The number of terms i n the F i b o n a c c i Sequence i s i n f i n i t e , y e t o n l y two F i b o n a c c i numbers are squares. They are (a) 1 and 8 (b) 8 and 144 (c) 1 and 144 1. Both 4 and 8 d i v i d e every term of the F i b o n a c c i Sequence. (a) s i x t h (b) f i f t e e n t h (c) t w e n t y - f i r s t 2. I f you take the d i f f e r e n c e between adjacent F i b o n a c c i numbers the answer i s (a) a F i b o n a c c i number (b) the F i b o n a c c i number j u s t b e f o r e the adjacent p a i r (c) the F i b o n a c c i number j u s t a f t e r the adjacent p a i r . 3. When you add the f i r s t nine F i b o n a c c i numbers the sum i s (a) an even number (b) one l e s s than the e l e v e n t h F i b o n a c c i number (c) one more than the e l e v e n t h F i b o n a c c i number. 4. I f you know t h a t a c e r t a i n even number was l e s s than 10 and was a l s o a F i b o n a c c i number and a P a s c a l ' s T r i a n g l e number, then the number would be (a) 2 (b) 8 (c) 2 and 8 5. To get the row above i n the " e q u i l a t e r a l " v e r s i o n of P a s c a l ' s T r i a n g l e you should (a) Take the sum of two adjacent numbers and w r i t e i t in-between and above these numbers. (b) Take the product of two adjacent numbers and w r i t e i t in-between and above these numbers. (c) Take the d i f f e r e n c e of two adjacent numbers and w r i t e i t in-between and above these numbers. 6. In the " r i g h t " v e r s i o n of P a s c a l ' s T r i a n g l e the sum of the f i r s t 5 numbers i n any column may be obtained by (a) going over one and down one from the 5th number. (b) going over one and up one from the 5th number. (c) going over one and down one from the 1st number.7 112 8. In row 3 of P a s c a l ' s T r i a n g l e , the t h i r d number i s (a) 3 (b) 1 (c) 6 8. 9. S t a r t i n g a t the second row, the numbers i n P a s c a l ' s T r i a n g l e from l e f t to r i g h t . (a) i n c r e a s e (b) i n c r e a s e then decrease (c) decrease 9. 10. We have found a p a t t e r n r e l a t i n g to the " r i g h t " v e r s i o n of P a s c a l ' s T r i a n g l e and the F i b o n a c c i numbers by summing along (a) columns (b) d i a g o n a l s (c) rows 10. ********************************************* I I I . When you d i v i d e the numbers i n the F i b o n a c c i Sequence by 4 some i n t e r e s t i n g r e s u l t s occur when you look a t the remainders a f t e r d i v i s i o n , e.g. 1 4 ) 5 4 remainder a f t e r d i v i s i o n The f i r s t f o u r remainders are j u s t the terms themselves, s 1. Complete the f a l l o w i n g t a b l e of remainders by d i v i d i n g 4 i n t o each number of the F i b o n a c c i sequence and w r i t i n g the remainders. \ ^ Number 1 1 2 3 5 8 13 21 34 55  Remainder 1 1 2 3 XI) 2. L i s t the complete p a t t e r n t h a t r e p e a t s . I f you were to " r e - i n v e n t " the F i b o n a c c i Sequence, what two numbers would you s t a r t with? ************************************************************** IV. A, The squares of the. f i r s t e i g h t F i b o n a c c i numbers are: ( F 1 ) . 2 ( F 2 ) 2 ( F 3 ) 2 ( F 4 ) 2 ( F 5 ) 2 ( F 6 ) 2 ( F y ) 2 ( F g ) 2 1 1 4 9 25 64 169 441 By t a k i n g adjacent p a i r s of t h i s square sequence e.g, 1 1, 1 4, 4 9, ... the f o l l o w i n g sequences are formed: ^ Sequence A 2 5 13 34 89 233 610 Sequence B O 3 5 16 39 105 272 1,. How i s Sequence A formed from the square sequence? 2, How i s Sequence B formed from the square sequence? 113 3. Choose e i t h e r Sequence A or Sequence B and p r e d i c t the e i g h t h term 4. E x p l a i n why you made t h a t p r e d i c t i o n , be as e x p l i c i t as p o s s i b l e . •  B. 1. Complete the f o l l o w i n g t a b l e of powers of 11. 2. How do these powers of 11 r e l a t e to P a s c a l ' s T r i a n g l e ? 3. Does your statement i n #2 h o l d t r u e f o r 11 and 11 ? ******************************************** V. T h i s diagram was taken from a Chinese Mathematics book w r i t t e n i n 1303. v 1. What does t h i s diagram'represent? 2. What i s the Chinese symbol f o r 10? 3. What i s the Chinese symbol f o r 15? 4. Use t h i s diagram to make a Chinese symbol f o r 25. 5. Use t h i s diagram to make a Chinese symbol f o r 30. 114 APPENDIX D NON-ROUTINE PROBLEMS NAME: BLOCK: REFERENCES Number of Term (n) 1 2 3 4 5 6 ... SEQUENCE 1 1, 2, 3, 4, 5, 6, ... SEQUENCE 2 1, 3, 5, 7, 9,11, ... SEQUENCE 3 2, 4, 6, 8, 10,12, . . . SEQUENCE 4 1, 4, 9, 16, 25,36, .. . ****************************************** I. Each of the f o l l o w i n g q u e s t i o n s r e f e r t o the sequences above. For each of the f o l l o w i n g q u e s t i o n s c i r c l e the ONE answer which BEST f i t s : T i f you t h i n k the statement i s ALWAYS t r u e . ST i f you t h i n k the statement i s SOMETIMES t r u e . F i f you t h i n k the statement i s ALWAYS f a l s e . 1. The numbers i n SEQUENCE 2 are prime numbers. T ST F 2. I f you have SEQUENCE 2 and want to get SEQUENCE 3 then you j u s t add one to each term i n SEQUENCE 2 T ST F 3. I f you add adjacent p a i r s from SEQUENCE 1 then you get a l l of SEQUENCE 2. T ST F 4. I f you have SEQUENCE 3 and want to get SEQUENCE 4 then you j u s t square each term i n SEQUENCE 3 and s u b t r a c t one. T ST F 5. SEQUENCE 4 i s a sequence o f p e r f e c t squares. T ST F 6. No number i n SEQUENCE 2 i s d i v i s i b l e by 2. T ST F 7. The terms i n SEQUENCE 4 are double the corresponding terms i n SEQUENCE 1. T ST F 8. I f you add adjacent p a i r s o f numbers i n SEQUENCE 3 you get the numbers i n SEQUENCE 2. T ST F 9. SEQUENCE 3 c o n s i s t s e n t i r e l y o f even numbers. T ST F 10. I f you have SEQUENCE 1 and want to get SEQUENCE 2 then j u s t double each term i n SEQUENCE 1, add two and the r e s u l t i s SEQUENCE 2. T ST F *************************************************************** 115 I I . Each of the f o l l w o i n g q u e s t i o n s r e f e r s to Sequence 1 - 4 above. For each of the f o l l o w i n g problems, choose the BEST c h o i c e and p l a c e the l e t t e r o f your c h o i c e i n the column a t the r i g h t . 1. The gen e r a l term (nth term) of SEQUENCE 1 i s (a) 2 n - l (b) n (c) 2n 1. The g e n e r a l term (nth term) of SEQUENCE 2 i s (a) 2 n - l (b) 2n (c) n 2 2. The g e n e r a l term (nth term) of SEOUENCE 3 i s (a) 2 n - l (b) n 2 (c) 2n 3. The ge n e r a l term (nth term) of SEOUENCE 4 i s • (a) 2n (b) nZ (c) 2 n - l 4. To get SEQUENCE 4 from SEQUENCE 3 (a) s u b t r a c t 1 and then square each term i n Seq. 3 (b) d i v i d e by 2 and then square each term i n Seq. 3 (c) square and then s u b t r a c t 1 from each term i n Seq. 3 5. I f you add the f i r s t 3 terms of SEQUENCE 2, you get as a sum (a) the t h i r d term of SEQUENCE 4 (b) the f i f t h term of SEQUENCE 2 (c) both (a) and (b) above. 6. I f you add the f i r s t 5 terms of SEOUENCE 2, you get as a sum (a) the f i f t h term of SEQUENCE 4 (b) the seventh term of SEQUENCE 2 (c) both (a) and (b) above. 7. Sequences which have the same f i r s t term a r e : (a) 1 and 2 (b) 1, 2, 3 (c) 1, 2, 4 8. To get SEQUENCE 1 from both SEOUENCE 2 and SEOUENCE 3 you (a) add the terms of SEQUENCES 2 and 3 (b) choose terms a l t e r n a t e l y from SEQUENCES 2 and 3, beginning w i t h SEQUENCE 2 (c) choose terms a l t e r n a t e l y from SEQUENCES 2 and 3, beginning w i t h SEOUENCE 3 9. 10. To get SEQUENCE 1 from SEOUENCE 4 you (a) take the square r o o t o f each term i n Seq. 4 (b) take the p o s i t i v e square r o o t o f each term i n Seq. 4 (c) halve each term i n Seq. 4 10. *************************************************** 116 I I I . The f o l l o w i n g code was found i n the bottom of a mathematician's trunk: (D © © ® ® © <f> There was a note which s a i d = e l e v e n . 1. What does t h i s sequence represent? 2. What i s the code symbol f o r 9?. 3. What i s the code symbol f o r 5? 4. Use t h i s code to make a symbol f o r 24. 5. Use t h i s code to make a symbol f o r 30. ************************************* IV. The f o l l o w i n g i s a 4 by 4 magic square. I f you take p a r t i c u l a r s e t s o f 4 numbers you w i l l f i n d t h a t they a l l . g i v e the same magic sum. 16 3 2 13 9 6 7 12 5 10 11 8 4 15 14 1 What i s the magic sum? Look f o r p a t t e r n s t h a t g i v e you the magic sum. L i s t as many DIFFERENT p a t t e r n s as you can i n words (not u s i n g numbers) d e s c r i b i n g where the p a t t e r n s t a r t s and ends and how you f o l l o w i t . * T h i s i s the d i v e r g e n t problem whose key f o l l o w s the non-routine problems 117 V. A TRIANGULAR number i s a number which can be represented with dots i n the shape of a t r i a n g l e . 1 3 6 10 » * a » A SQUARE number i s a number which can be r e p r e s e n t e d w i t h dots i n the shape of a square. 1 4 9 16 » • i « • • • • A PENTAGONAL number i s a number which can be r e p r e s e n t e d w i t h dots i n the shape of a pentagon. 1 5 12 22,,% Consider the f o l l o w i n g c h a r t : Number of term (n) 1 2 . 3 4 5 6 ... T r i a n g u l a r number 1 3 6 10 Square number 1 4 9 16 Pentagonal number 1 5 12 22 1. Describe how you would get the next two t r i a n g u l a r numbers u s i n g the p a t t e r n from the f i r s t f o u r : 2. Describe how you would get the next two square numbers usi n g the p a t t e r n from the f i r s t f o u r : 3. D e s c r i b e how you would get the next two pentagonal numbers u s i n g the p a t t e r n from the f i r s t f o u r : 4. What r e l a t i o n s h i p do you see between the t r i a n g u l a r numbers and the square numbers? (a) D e s c r i b e i n words: 5. What r e l a t i o n s h i p do you see between the pentagonal numbers and the other two ( i . e . t r i a n g u l a r and square) (a) Describe i n words: 6. In the space below u s i n g d o t s : (a) Answer 4a (b) Answer 5a 118 KEY FOR DIVERGENT QUESTION IN NON-ROUTINE PROBLEMS FLEXIBILITY FLUENCY ORIGINALITY Row 1 1 1 0 Row 2 0 1 0 Row 3 0 1 0 Row 4 0 1 0 Column 1 0 1 0 Column 2 0 1 0 Column 3 0 1 0 Column 4 0 1 0 Diagonal Top to Bottom 0 1 0 Diagonal Bottom to Top 0 1 0 Bottom r i g h t 4 1 1 1 Bottom l e f t 4 0 1 1 Top r i g h t 4 0 1 1 Top l e f t 4 0 1 1 Centre 4 0 1 1 Column 2 - 1st 2 and Column 3 - 2nd 2 (and the reverse) 1 1 1 Column 1 - 1st 2 and Column 4 - 2nd 2 (and the reverse) 0 1 1 Z-shape (3, 7, 10, 14) 1 1 1 Every 2nd number 1 1 1 4 cor n e r s 1 1 1 Row 1 - 1st 2 and Row 4 - l a s t 2 1 1 1 TOTAL POSSIBLE 7 21 11 * See Magic Square p. 114 APPENDIX E EXAMPLE OF STUDENT'S NOTES R e l a t i o n s 1. 4 1. Add the two before to get the t h i r d 2. 8 2. Every 3rd no. i s t h i r d 3. 12 4. adding 2 c o n s e c u t i v e nos. get c o n s e c u t i v e odd nos. 2 c o n s e c u t i v e odd get even 4. 20 5. 32 5. every 5th number i s d i v i s i b l e by 5 6. 52 6. term column: every 3rd i s d i v i s i b l e by 2 7. 84 7. (number of term) every 4th no. by 3 8. 136 8. every 6th i s d i v i s i b l e by 8 9. 220 9. every 7th no. (term) i s d i v i s i b l e by 13 0. 356 924 1. one more than each other i n sequence 1. 2. x a 3. a + x 4. 2a + x 5. 3a + 2x 6. 5a + 3x 7. 8a + 5x 8. 13a + 8x 9. 21a +13x 0. 34a +21x 11 (8a+5x) 1$ - 1 2$ 1 - 2 2 - 1 3$ 3 - 1 2 - 1 1 - 1 1 2 1 2 0 e v e r y 3 r d d i v b y 2 4 t h 5 t h 6 7 3 5 8 1 3 e a c h t e r m i s o n e m o r e t h a n t h e o n e b e f o r e a d d t w o c o n s e c u t i v e g i v e s o d d ( n o . o f t e r m ) s k i p o n e s u b t r a c t t e r m c o l u m n 3 n o s . i n a r o w c - a = b c - b = a d o i n t h r e e s 3 ' s 4 ' s 1 1 2 3 5 8 1 + 1 + 2 = 4 1 6 - 4 = 1 2 = 1 3 - 1 3 + 5 + 8 = 1 6 = 2 m o r e t h a n 8 = 3 m o r e t h a n 1 3 = 5 m o r e t h a n 2 1 = 8 m o r e t h a n 3 4 1 . 2 . 3 . 4 . 5 , 6 . 2 3 2 ( a d d o n e t o e a c h ) - u p a n d d o w n 2 3 3 - x 2 3 3 2 1 1 3 1 1 p u t n o s . d o w n 1 4 6 4 1 - 1 2 1 x 1 2 1 1 1 Al 1$ *S X\ t t 

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