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The relationship between field-independence and instructional strategy on performance on elementary mathematics… O'Brien, Margaret Anne 1972

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THE RELATIONSHIP BETWEEN FIELD-INDEPENDENCE AND INSTRUCTIONAL STRATEGY ON PERFORMANCE ON ELEMENTARY MATHEMATICS ALGORITHMS by MARGARET ANNE O'BRIEN B.Sc, Saint Francis Xavier University, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics Education We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1972 In present ing th is thes is in p a r t i a l f u l f i l m e n t o f the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y sha l l make it f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying o f th is t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t i ves . It i s understood that copying or p u b l i c a t i o n o f th is t h e s i s fo r f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion . Department of M a t h e m a t i c s E d u c a t i o n The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada Date A u g u s t , 1972 ABSTRACT THE RELATIONSHIP BETWEEN FIELD-INDEPENDENCE AND INSTRUCTIONAL STRATEGY ON PERFORMANCE ON ELEMENTARY MATHEMATICS ALGORITHMS A s tudy was conducted t o determine the i n t e r a c t i o n e f f e c t , i f any, between the f i e l d - i n d e p e n d e n c e c o n s t r u c t and two i n s t r u c t i o n a l s t r a t e g i e s , a p a t t e r n s t r a t e g y which used diagrams e x t e n s i v e l y and an a l g e b r a i c s t r a -tegy which used a l g e b r a i c f i e l d p r o p e r t i e s f a m i l i a r to the c h i l d and was devoid of d iagrams. Two a l g o r i t h m s c l a s s i f i e d as s i m p l e and two a l g o r -ithms c l a s s i f i e d as complex formed the content of the i n s t r u c t i o n a l ma-t e r i a l s . One h a l f the c h i l d r e n i n each of twelve grade f i v e c l a s s e s , which were p a r t i c i p a t i n g i n a s tudy conducted by a d o c t o r a l s t u d e n t , were r a n -domly s e l e c t e d to form the sample of the s t u d y . The C h i l d r e n ' s Embedded F i g u r e s Test was i n d i v i d u a l l y a d m i n i s t e r e d to the sample . Three n u l l hypotheses were t e s t e d each at *< = . 0 5 . These were : (1) 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 i n mean p o s t - t e s t scores between s tudents taught by a p a t t e r n i n s t r u c t i o n a l s t r a t e g y and s tudents taught by an a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y ; (2) There i s no s i g n i f i c a n t d i f -ference i n mean p o s t - t e s t scores between groups of s tudents d i f f e r i n g i n degree of f i e l d independence; (3) There i s no s i g n i f i c a n t i n t e r a c t i o n between s t u d e n t s ' degree of f i e l d independence and i n s t r u c t i o n a l s t r a t e g y . M u l t i p l e l i n e a r r e g r e s s i o n t e c h n i q u e s were used t o a n a l y s e t h e d a t a . The r e s u l t s o f t h e s t u d y i n d i c a t e d t h a t extreme f i e l d i n d e p e n d e n t c h i l d r e n d i d respond d i f f e r e n t l y t o t h e two i n s t r u c t i o n a l s t r a t e g i e s , a l -though f o r t h e sample as a whole t h e two s t r a t e g i e s d i d n o t p r o d u c e s i g n i -f i c a n t l y d i f f e r e n t r e s u l t s . F o r extreme f i e l d i n d e p e n d e n t s t u d e n t s , t h e a l g e b r a i c s t r a t e g y was s u p e r i o r t o t h e p a t t e r n s t r a t e g y . ACKNOWLEDGMENT I would l i k e t o than k my c h a i r m a n , D r. G a i l S p i t l e r , and my com-m i t t e e members, Dr. D o u g l a s Owens and D r . D a v i d R o b i t a i l l e , f o r t h e i r encouragement and e s p e c i a l l y f o r t h e i r humane p a r t i c i p a t i o n i n t h e h e c t i c c o m p l e t i o n o f t h i s t h e s i s . I woul d a l s o l i k e t o than k M a r i a n W e i n s t e i n who d e v e l o p e d t h e i n s t r u c t i o n a l m a t e r i a l s used i n t h i s s t u d y . A s p e c i a l t h a n k s , t o Dr. R o b e r t Conry f o r h i s h e l p b o t h w i t h t h e s t a t i s t i c a l a n a l y -s i s o f t h e d a t a and i n t h e s e t - u p o f t h e computer p r o g r a m s . TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES C h a p t e r 1. THE PROBLEM I n t r o d u c t i o n B ackground o f t h e P r o b l e m The P r o b l e m W i t k i n ' s C o n s t r u c t o f C o g n i t i v e S t y l e H y p o t h e s e s D e f i n i t i o n o f Terms 2. REVIEV7 OF THE LITERATURE I n t r o d u c t i o n I n d i v i d u a l D i f f e r e n c e s i n L e a r n i n g I n d i v i d u a l i z e d I n s t r u c t i o n A p t i t u d e - I n s t r u c t i o n I n t e r a c t i o n i n M a t h e m a t i c s F i e l d - D e p e n d e n c e - I n d e p e n d e n c e S t u d i e s D i s c u s s i o n o f t h e L i t e r a t u r e 3. DESIGN AND PROCEDURE INTRODUCTION P o p u l a t i o n Sample Page INSTRUCTIONAL MATERIALS 27 MEASURING INSTRUMENTS 31 PROCEDURE 39 CONTROLS AO STATISTICAL PROCEDURES 41 4. ANALYSIS OF THE DATA 42 G r a p h i n g o f S i g n i f i c a n t R e s u l t s 51 D i s c u s s i o n o f t h e F i g u r e s 52 D i s c u s s i o n o f t h e R e s u l t s 57 5. CONCLUSIONS AND IMPLICATIONS 60 SUMMARY 60 LIMITATIONS 62 DISCUSSION OF THE RESULTS 63 CONCLUSIONS 65 IMPLICATIONS OF THE STUDY 66 BIBLIOGRAPHY 68 APPENDIXES 72 A. I n s t r u c t i o n a l M a t e r i a l s 73 B. The M e a s u r i n g I n s t r u m e n t s 123 C. E x p e r i m e n t a l D a t a 157 i v L IST OF TABLES T a b l e Page 1. D e s c r i p t i o n o f t h e I t e n s and KR-20 R e l i a b i l i t y 32 C o e f f i c i e n t s f o r t h e F o u r P r e t e s t s 2. Types o f Items o f S I , P r o d u c t o f a F r a c t i o n and a 33 Mix e d Number, C o m p u t a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 3. Types o f Items o f S2, Comparison o f F r a c t i o n s , 33 C o m p u t a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 4. Types o f Items o f C l , C h a n g i n g a F r a c t i o n t o a D e c i m a l , 34 C o m p u t a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 5. Types o f Items o f C2, F i n d i n g t h e Square Root o f a 34 F r a c t i o n , C o m p u t a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 6. Types o f Items o f S I , P r o d u c t o f a F r a c t i o n and a 35 M i x e d Number, G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 7. Types o f Items o f S2, Comparison o f F r a c t i o n s , 36 G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 8. Types o f Items o f C l , C h a n g i n g a F r a c t i o n t o a D e c i m a l , 36 G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 9. Types o f Items o f C2, F i n d i n g t h e Square Root o f a 37 F r a c t i o n , G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t 10. CEFT R e l i a b i l i t y E s t i m a t e s and V a l i d i t y C o e f f i c i e n t s 38 11. A n a l y s i s o f S I C o m p u t a t i o n S c o r e s 45 12. A n a l y s i s o f S2 C o m p u t a t i o n S c o r e s 13. A n a l y s i s o f S I G e n e r a l i z a t i o n S c o r e s 46 47 V T a b l e Page 14. A n a l y s i s o f S2 G e n e r a l i z a t i o n S c o r e s 47 15. A n a l y s i s o f C l C o m p u t a t i o n S c o r e s 48 16. A n a l y s i s o f C2 C o m p u t a t i o n S c o r e s 48 17. A n a l y s i s o f C l G e n e r a l i z a t i o n S c o r e s 49 18. A n a l y s i s o f C2 G e n e r a l i z a t i o n S c o r e s 50 v i LIST OF FIGURES Figure Rage 1. Flow Chart of the Procedure 39 2. Mean Residual Scores on C2 Computation Scores 53 3. Mean Residual Scores on Cl Generalization Scores 54 4. Mean Residual Scores on C2 Generalization Scores 56 CHAPTER I THE PROBLEM Introduction Today, in North American education, there is a revival of interest i n individualized instruction. In mathematics programs such as IPI(Individually Prescribed Instruction) and SAMI(Systematic Approach to Mathematical Instruction) stress an individualized, sequential approach through the extensive use of diagnostic testing, work-week units, multi-media learning centres, teacher-student contracts and self pacing. Yet, as Gage and Unruh have noted, "... the fact i s , that despite several decades of concern with individualization, few, i f any, striking results have been r e p o r t e d . ' Coop and Sigel have noted that "... few, i f any, of these individualized programs have examined carefully the inter-individual v a r i a b i l i t y of the learners, who w i l l be exposed to their educational 2 y stimuli." Yet, Bloom, Cronbach, Gagne, Glaser and Jensen, have sug-gested that there i s no one instructional method which provided 3 optimal learning for a l l students. Cronbach and Snow have also stated ^N. L. Gage and W. R. Unruh, "Theoretical Formulations for Research on Teaching", Review of Educational Research, XXXVTII(1967), 368. 2 Richard H. Coop and Irving E. Sigel, "Cognitive Style: Implica-tions for Learning and Instruction", Psychology in the Schools, VIII, 1971, 152. 3 Glenn H. Bracht, "Experimental Factors Related To Aptitude-Treatment Interactions", Review of Educational Research, XXXX(1971), 627. 2 that "... the search f o r generally superior methods must be supple-4 mented by a search for ways of adapting I n s t r u c t i o n to the i n d i v i d u a l . " Bracht"' and Mitchell*' have strongly advocated i n v e s t i g a t i o n s to discover s i g n i f i c a n t i n t e r a c t i o n s between personological v a r i a b l e s of the learner and a l t e r n a t i v e i n s t r u c t i o n a l routes to a desired educational outcome.(They l a b e l t h i s Aptitude-Treatment I n t e r a c t i o n research.) But which personological v a r i a b l e s are relevant and as Becker'' notes which methods combined with these personological v a r i a b l e s are relevant to which desired educational outcomes? As yet, these questions remain f o r the most part, unanswered i n t h i s area of research. Cronbach has sug-gested that "... we w i l l f i n d these aptitude v a r i a b l e s to be quite unlike g our present aptitude measures ... ". The challenge to researchers, then, i s to search f o r relevant personological v a r i a b l e s . The challenge to curriculum developers i s to design v i a b l e a l t e r n a t i v e i n s t r u c t i o n a l s t r a t e g i e s . -Lee J . Cronbach and Richard E. Snow, " I n d i v i d u a l Differences i n Learning A b i l i t y as a Function of I n s t r u c t i o n a l V a r i a b l e s " F i n a l Report Stanford U n i v e r s i t y , C a l i f o r n i a School of Education, ED 029 001 ^Glenn H. Bracht, "Experimental Factors Related To Aptitude-Treatment In t e r a c t i o n s " , Review of Educational Research, XXXX(1971), 627-41. James V. M i t c h e l l , "Education's Challenge To Psychology: The P r e d i c t i o n Of Behavior From Person-Environment I n t e r a c t i o n s " , Review of  Educational Research, XXXIX(1969), 695 - 721. ^Jerry P. Becker, "Research In Mathematics Education: The Role Of Theory And Of Aptitude-Treatment I n t e r a c t i o n " , Journal for Research In  Mathematics Education, 1(1970), 19 - 27. Lee J . Cronbach, "The Two D i s c i p l i n e s of S c i e n t i f i c Psychology", The American Psychologist, XII(1957), 681. Background o f t h e P r o b l e m E d u c a t o r s and p s y c h o l o g i s t s have r e c e n t l y s u g g e s t e d t h a t t h e p s y c h o l o g i c a l c o n s t r u c t o f c o g n i t i v e s t y l e may be r e l e v a n t t o t h e prob l e m s o f e d u c a t i o n . I t has been s u g g e s t e d t h a t r e s e a r c h i n t o t h e i n t e r a c t i o n between c o g n i t i v e s t y l e and i n s t r u c t i o n a l p r o c e s s e s may p r o -v i d e a t h e o r e t i c a l and e m p i r i c a l b a s i s f o r t h e o p t i m a l a s s i g n m e n t o f 9 10 l e a r n e r s t o a l t e r n a t i v e i n s t r u c t i o n a l p r o c e s s e s . S p i t l e r has f u r t h e r s u g g e s t e d t h a t W i t k i n ' s c o n s t r u c t o f c o g n i t i v e s t y l e ( f i e l d - d e p e n d e n c e -i n d e p e n d e n c e ) may have i m p o r t a n t i m p l i c a t i o n s f o r m a t h e m a t i c s e d u c a t i o n i n terms o f t h e t y p e s o f c u r r i c u l a r m a t e r i a l s u s e d . A c o n s i d e r a b l e p o r t i o n o f e l e m e n t a r y s c h o o l m a t h e m a t i c s i s c o n -c e r n e d w i t h a l g o r i t h m i n s t r u c t i o n . W e i n s t e i n r e v i e w e d r e l e v a n t l i t e r a -t u r e and c o n c l u d e d t h a t t h e m e a n i n g f u l n e s s o f t h e a l g o r i t h m f o r t h e c h i l d d e t e r m i n e s t h e c h i l d ' s a b i l i t y t o r e t a i n and a p p r o p r i a t e l y a p p l y t h e a l g o r i t h m . She a l s o c o n c l u d e d t h a t t h e p r o c e d u r e u s e d t o j u s t i f y an a l g o r i t h m i s one o f t h e m a j o r f a c t o r s i n f l u e n c i n g t h e m e a n i n g f u l n e s s o f t h e a l g o r i t h m f o r t h e c h i l d . ^ -D. J . S a t t e r l y and M. A. B r i m e r , " C o g n i t i v e S t y l e s and S c h o o l L e a r n i n g " , The B r i t i s h J o u r n a l Of E d u c a t i o n a l P s y c h o l o g y , X X X X I ( 1 9 7 1 ) , 294 - 303; Herman A. W i t k i n , "Some I m p l i c a t i o n s o f R e s e a r c h on C o g n i t i v e S t y l e f o r P r o b l e m s o f E d u c a t i o n " , ( r e p r i n t e d f r o m P r o f e s s i o n a l S c h o o l P s y c h o l o g y , V o l . I l l , C o p y r i g h t Grune and S t r a t t o n I n c . , 1969) (mimeo-g r a p h e d ) ; R i c h a r d H. Coop and I r v i n g E. S i g e l , " C o g n i t i v e S t y l e : I m p l i -c a t i o n s f o r L e a r n i n g and I n s t r u c t i o n " , P s y c h o l o g y i n t h e S c h o o l s , V I I I , 1971, 152 - 159. ^ G a i l J . S p i t l e r , " I m p l i c a t i o n s o f R e s e a r c h on C o g n i t i v e S t y l e f o r M a t h e m a t i c s E d u c a t i o n " ( U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , Wayne S t a t e U n i v e r s i t y , 1 9 7 0 ) . ^ M a r i a n S. W e i n s t e i n , "A S t u d y o f Types o f A l g o r i t h m J u s t i f i c a -t i o n i n E l e m e n t a r y S c h o o l M a t h e m a t i c s " ( U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 9 7 2 ) . 4 Weinstein designed a study to investigate the relative effective-ness of two instructional strategies, which she labels pattern>arid alge-braic, in the teaching of simple and complex algorithms. She classified algorithms commonly taught in the elementary grades as simple or com-plex on the basis of the number of prerequisites required for their acqui-si t i o n . She then designed two alternative instructional strategies for each of two examples of simple algorithms and two examples of complex algorithms. Diagrams are used to j u s t i f y the algorithm in the pattern instructional strategy, while appeal to definitions and algebraic f i e l d postulates are used in the algebraic instructional strategy. For each algorithm, a computation post-test and a generalization post-test were developed to measure achievement on the algorithm. The Problem The problem which this researcher sought to investigate was: Do children differing in their degree of f i e l d independence respond differently to the two instructional strategies developed by Weinstein? In other words, i s there an interaction effect between f i e l d independence and instructional strategy? Two questions relevant to the problem were also investigated: Does one of the instructional strategies, on the average result in superior pupil performance? Is there d i f f e r e n t i a l achievement on the algorithms among students differing in their degree of f i e l d inde-pendence? In order to answer these questions, the investigator randomly selected a sample of the students participating in the Weinstein study and measured these students on the f i e l d independence construct. These students participated f u l l y In the Weinstein study, following either a 5 p a t t e r n o r a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y on a s i m p l e a l g o r i t h m and a complex a l g o r i t h m and t a k i n g t h e c o m p u t a t i o n and g e n e r a l i z a t i o n p o s t - t e s t s on t h e s e a l g o r i t h m s . W i t k i n ' s C o n s t r u c t o f C o g n i t i v e S t y l e : F i e l d - D e p e n d e n c e - I n d e p e n d e n c e The term c o g n i t i v e s t y l e i s used i n p s y c h o l o g i c a l l i t e r a t u r e t o de n o t e i n d i v i d u a l c o n s i s t e n c i e s i n modes o f f u n c t i o n i n g o v e r a w i d e r a n g e o f b e h a v i o r a l s i t u a t i o n s . However, s i n c e t h e t e r m i s a p s y c h o l o g i c a l c o n s t r u c t , t h e r e i s some d i s a g r e e m e n t among p s y c h o l o g i s t s as t o w h i c h s p e c i f i c o b s e r v a b l e b e h a v i o r s a r e r e p r e s e n t a t i v e o f t h e t e r m . T h e r t e r m , i s t h e r e f o r e , n e c e s s a r i l y i n v e s t i g a t o r s p e c i f i c . W i t k i n and h i s a s s o c i a t e s have d e v e l o p e d a t h e o r y o f i n d i v i d u a l s e l f - c o n s i s t e n c y a l o n g d i v e r g e n t p s y c h o l o g i c a l g r o w t h p a t t e r n s . "These p a t t e r n s s u g g e s t c o n s i s t e n c y i n p s y c h o l o g i c a l f u n c t i o n i n g w h i c h p e r v a d e s t h e i n d i v i d u a l ' s p e r c e p t u a l , i n t e l l e c t u a l , e m o t i o n a l , m o t i v a t i o n a l , de-12 f e n s i v e and s o c i a l o p e r a t i o n s . " The p e r c e p t u a l and i n t e l l e c t u a l components o f W i t k i n ' s t h e o r y o f " d i f f e r e n t i a t i o n " a r e combined t o f o r m t h e c o g n i t i v e d i m e n s i o n , t h e e x -tremes o f w h i c h a r e r e p r e s e n t e d by an " a n a l y t i c a l f i e l d a p p r o a c h " and a " g l o b a l f i e l d a p p r o a c h " o r " a r t i c u l a t e d " v e r s u s " g l o b a l " . F i e l d - d e p e n -d e n c e - i n d e p e n d e n c e i s an i n d e x o f t h e p e r c e p t u a l component. F i e l d i n d e p e n d e n c e r e p r e s e n t s t h e a b i l i t y t o overcome an embedding c o n t e x t and p e r c e i v e an i t e m as d i s t i n c t f r o m i t s b a c k g r o u n d . The f i e l d - d e p e n -Herman A. W i t k i n e t a l . , P s y c h o l o g i c a l D i f f e r e n t i a t i o n ( N e w Y o r k : W i l e y I n c . , 1 9 6 2 ) , p.A. dence-independence d i m e n s i o n r e f l e c t s t h e i n d i v i d u a l ' s a b i l i t y t o e x p e r i -ence s t i m u l i a n a l y t i c a l l y . W i t k i n and h i s a s s o c i a t e s have d e m o n s t r a t e d t h r o u g h r e s e a r c h t h a t a t e n d e n c y t o e x p e r i e n c e a n a l y t i c a l l y , t h a t i s a t e n d e n c y toward f i e l d i n d e p e n d e n c e , i s s t r o n g l y a s s o c i a t e d w i t h a t e n d e n c y 13 t o s t r u c t u r e e x p e r i e n c e . The a b i l i t y t o b o t h a n a l y s e and s t r u c t u r e e x -p e r i e n c e i s r e f e r r e d t o as an " a r t i c u l a t e d " way o f e x p e r i e n c i n g as opposed t o a " g l o b a l " way o f e x p e r i e n c i n g . A t t h e g l o b a l e x t r e m e " . . . when t h e f i e l d i s s t r u c t u r e d , t h e r e i s a t e n d e n c y f o r i t s o r g a n i z a t i o n , as g i v e n , t o d i c t a t e t h e manner i n w h i c h b o t h t h e f i e l d as a w h o l e and i t s p a r t s a r e e x p e r i e n c e d ; when t h e f i e l d l a c k s s t r u c t u r e , e x p e r i e n c e t e n d s t o be g l o b a l and d i f f u s e . " ^ A t t h e " a r t i c u l a t e d " extreme "... t h e r e i s a t e n d -ency f o r e x p e r i e n c e t o be d e l i n e a t e d and s t r u c t u r e d , even when t h e mate-r i a l l a c k s i n h e r e n t o r g a n i z a t i o n ; p a r t s o f a f i e l d a r e e x p e r i e n c e d as d i s -c r e t e and t h e f i e l d as a w h o l e as o r g a n i s e d . W i t k i n ' s l o n g i t u d i n a l s t u d i e s i n d i c a t e t h a t i n c h i l d r e n f r o m t h e ages o f 5 - 17 t h e r e i s a p r o g r e s s i o n toward g r e a t e r f i e l d i n d e p e n d e n c e , b u t t h a t t h e r e i s a l s o a h i g h d e g r e e o f r e l a t i v e s t a b i l i t y . T h a t i s , c h i l d r e n t e n d t o m a i n t a i n t h e i r p o s i t i o n a l o n g t h e f i e l d - d e p e n d e n c e - I n -dependence d i m e n s i o n i n r e l a t i o n t o t h e i r p e e r s . T h e r e i s a l e v e l l i n g — o f f about t h e age o f 17 and t h e n a g r a d u a l " d e d i f f e r e n t i a t i o n " t o w a r d [ :  1 3 I b i d . , pp. 81 - 114. 14 Herman A. W i t k i n , P h i l i p K. Oltm a n , E v e l y n R a s k i n and St e p h e n A. K a r p , A Manual F o r The Embedded F i g u r e s T e s t s ( P a l o A l t o , C a l i f o r n i a : C o n s u l t i n g P s y c h o l o g i s t s P r e s s , 1 9 7 1 ) , p.7. 1 5 I b i d . f i e l d dependence a f t e r t h e mid t h i r t i e s . ^ R e s e a r c h a l s o i n d i c a t e s t h a t t h e c o n s t r u c t o f f i e l d - d e p e n d e n c e -i n d e p e n d e n c e i s i n d e p e n d e n t o f I . Q. l e v e l . W i t k i n e t a l . f o u n d s i g n i f i -c a n t c o r r e l a t i o n between t h e c o n s t r u c t and a group o f s u b t e s t s o f WISC, B l o c k d e s i g n , O b j e c t a s s e m b l y and P i c t u r e c o m p l e t i o n ! however, t h e r e was n o n - s i g n i f i c a n t c o r r e l a t i o n between t h e c o n s t r u c t and v e r b a l c o m p r e h e n s i o n and a r i t h m e t i c s u b t e s t s c o r e s o f WISC. Thus "... i n t e l l i g e n c e t e s t s c o r e s c a n n o t be i n t e r p r e t e d t o mean t h a t f i e l d - i n d e p e n d e n t c h i l d r e n a r e o f 18 g e n e r a l l y s u p e r i o r i n t e l l i g e n c e . " H y p o t h e s e s C o n j e c t u r e d Outcomes. The i n v e s t i g a t o r c o n j e c t u r e d t h a t s t u -d e n t s w i l l p r e f e r t h e i n s t r u c t i o n a l s t r a t e g y w h i c h i s most c l o s e l y matched t o t h e i r c o g n i t i v e s t y l e , and t h a t , c o n s e q u e n t l y , t h i s p r e f e r e n c e w i l l be d e m o n s t r a t e d by s i g n i f i c a n t d i f f e r e n c e s between mean s c o r e s on t h e c o m p u t a t i o n and g e n e r a l i z a t i o n p o s t - t e s t s between s t u d e n t s w i t h s i m i -l a r c o g n i t i v e s t y l e s t a u g h t by d i f f e r e n t i n s t r u c t i o n a l s t r a t e g i e s . F i r s t , s i n c e t h e p a t t e r n i n s t r u c t i o n a l s t r a t e g y u t i l i z e s an o v e r -r i d i n g p h y s i c a l a n a l o g y f o r j u s t i f i c a t i o n o f t h e a l g o r i t h m and s i n c e f o r t h e f i e l d dependent c h i l d "... t h e o r g a n i z a t i o n o f a f i e l d as a whole d i c -19 t a t e s t h e way i n w h i c h i t s p a r t s a r e e x p e r i e n c e d " , i t i s e x p e c t e d t h a t ^ H e r m a n A. W i t k i n , D o n a l d R. Goodenough and Ste p h e n A. K a r p , " S t a b i l i t y o f C o g n i t i v e S t y l e From C h i l d h o o d To Young A d u l t h o o d " , J o u r n a l o f P e r s o n a l i t y and S o c i a l P s y c h o l o g y , V I I ( 1 9 6 7 ) , 291 - 300. ^H e r m a n A. W i t k i n e t a l . , P s y c h o l o g i c a l D i f f e r e n t i a t i o n (New Y o r k : W i l e y I n c . , 1 9 6 2 ) , 223. 1 8 I b i d . , p.70. Herman A. W i t k i n e t a l . , A Manual F o r The Embedded F i g u r e s T e s t s ( P a l o A l t o , C a l i f o r n i a : C o n s u l t i n g P s y c h o l o g i s t s P r e s s , 1 9 7 1 ) , 7. 8 f i e l d dependent c h i l d r e n w i l l a c h i e v e h i g h e r group p o s t - t e s t s c o r e s on the p a t t e r n a p p r o a c h t h a n on t h e a l g e b r a i c a p p r o a c h . S e c o n d l y , s i n c e t h e a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y i s b a s i c a l l y an a n a l y t i c p r o c e d u r e i n w h i c h i n d i v i d u a l s t e p s must be drawn t o g e t h e r t o f o r m t h e whole and s i n c e f o r t h e f i e l d i n d e p e n d e n t c h i l d "... p a r t s o f a f i e l d a r e e x p e r i e n c e d as d i s c r e t e and t h e f i e l d as a w h o l e as 20 s t r u c t u r e d , " i t i s e x p e c t e d t h a t f i e l d i n d e p e n d e n t c h i l d r e n w i l l a c h i e v e h i g h e r group p o s t - t e s t s c o r e s on t h e a l g e b r a i c a p p r o a c h t h a n on t h e p a t -t e r n a p p r o a c h . N u l l H y p o t h e s e s . The r e s u l t s o f t h e c o m p u t a t i o n and g e n e r a l i z a -t i o n t e s t s were a n a l y z e d s e p a r a t e l y f o r each o f t h e two s i m p l e and two complex a l g o r i t h m s . The f o l l o w i n g n u l l h y p o t h e s e s were t e s t e d a t oC =.05 i n each o f t h e a n a l y s e s . H^: T h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n mean p o s t - t e s t s c o r e s between s t u d e n t s t a u g h t by a p a t t e r n i n s t r u c t i o n a l s t r a t e g y and s t u d e n t s t a u g h t by an a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y . H^: T h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n mean p o s t - t e s t s c o r e s between grou p s o f s t u d e n t s d i f f e r i n g i n degr e e o f f i e l d i n d e p e n d e n c e . H^: T h e r e i s no s i g n i f i c a n t i n t e r a c t i o n between s t u d e n t s ' d e g r e e o f f i e l d i n d e p e n d e n c e and i n s t r u c t i o n a l s t r a t e g y . 2 0 I b i d . D e f i n i t i o n o f Terms A l g e b r a i c I n s t r u c t i o n a l S t r a t e g y — " . . . t h o s e e x p l a n a t i o n s w h i c h c o n s i s t p u r e l y o f a p p e a l s t o d e f i n i t i o n s , t o r u l e s o f l o g i c and t o t h e a l g e b r a i c f i e l d p o s t u l a t e s o r t o c o m b i n a t i o n s t h e r e o f . " 2 1 P a t t e r n I n s t r u c t i o n a l S t r a t e g y — " . . . t h o s e e x p l a n a t i o n s w h i c h use p h y s i c a l a n a l o g s f o r m a t h e m a t i c a l o p e r a t i o n s . " 2 2 S i m p l e A l g o r i t h m s — " . . . t h o s e a l g o r i t h m s w i t h a r e l a t i v e l y s m a l l number o f p r e r e q u i s i t e s f o r t h e i r a c q u i s i t i o n . . . " 2 ^ Complex A l g o r i t h m s — " . . . t h o s e a l g o r i t h m s w h i c h r e q u i r e a s u b s t a n t i a l l y g r e a t e r number of p r e r e q u i s i t e s f o r t h e i r a c q u i s i t i o n . F i e l d Independent C h i l d — a c h i l d whose s c o r e i s above t h e sample mean on t h e Children's Embedded F i g u r e s T e s t F i e l d Dependent C h i l d — a c h i l d whose s c o r e i s below t h e sample mean on t h e C h i l d r e n ' s Embedded F i g u r e s T e s t Degree o f F i e l d I n d e p e n d e n c e — r e l a t i v e p o s i t i o n o f t h e s t u d e n t ' s s c o r e i n t h e d i s t r i b u t i o n o f t h e sample s c o r e s on t h e C h i l d r e n ' s Embedded F i g u r e s T e s t . A h i g h s c o r e r e p r e s e n t s a h i g h d e g r e e o f f i e l d i n d e p e n d e n c e . S t a r i a n S. W e i n s t e i n , "A S t u d y o f Types o f A l g o r i t h m J u s t i f i c a -t i o n i n E l e m e n t a r y S c h o o l M a t h e m a t i c s " ( U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 9 7 2 ) , 4. 22 23 24 I b i d . I b i d . , p.5. I b i d . CHAPTER I I REVIEW OF THE LITERATURE I n t r o d u c t i o n T h i s s e c t i o n i s d i v i d e d i n t o f o u r p a r t s . The f i r s t two, I n d i -v i d u a l D i f f e r e n c e s i n L e a r n i n g and I n d i v i d u a l i z e d I n s t r u c t i o n , a r e a r e -v i e w o f l e a r n e d o p i n i o n s r e g a r d i n g i n d i v i d u a l d i f f e r e n c e s and p r e s e n t a t t e m p t s t o adapt i n s t r u c t i o n t o i n d i v i d u a l d i f f e r e n c e s . The l a s t two s e c t i o n s c o n s i s t o f a r e v i e w o f s t u d i e s i n d i r e c t l y r e l a t e d t o t h i s s t u d y . The t h i r d s e c t i o n , A p t i t u d e - I n s t r u c t i o n I n t e r a c t i o n i n M a t h e m a t i c s , r e -v i e w s s t u d i e s i n w h i c h t h e t r e a t m e n t v a r i a b l e i s i n s t r u c t i o n i n m a t h e m a t i c s . The f o u r t h s e c t i o n , F i e l d - D e p e n d e n c e - I n d e p e n d e n c e S t u d i e s , r e v i e w s s t u d i e s i n w h i c h t h e f i e l d i n d e p e n d e n c e c o n s t r u c t i s used as an a p t i t u d e v a r i a b l e i n an e d u c a t i o n a l s e t t i n g . A t h o r o u g h r e v i e w o f t h e l i t e r a t u r e f a i l e d t o p r o d u c e any s t u d i e s w h i c h i n v e s t i g a t e d t h e r e l a t i o n s h i p between t h e f i e l d i n d e p e n d e n c e c o n -s t r u c t and i n s t r u c t i o n a l s t r a t e g y i n m a t h e m a t i c s . I n d i v i d u a l D i f f e r e n c e s I n L e a r n i n g Gagne'has n o t e d t h a t "... t h e q u e s t i o n o f how p e o p l e d i f f e r i n t h e r a t e , e x t e n t , s t y l e and q u a l i t y o f t h e i r l e a r n i n g i s one w h i c h has c o n -c e r n e d p s y c h o l o g i s t s f o r a g r e a t many y e a r s . Y e t ... we do n o t know much more about i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g t h a n we d i d t h i r t y y e a r s ago." •'•Robert M. Gagne, " L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s i n t r o d u c t i o n t o t h e C o n f e r e n c e " , e d . R. M. Gagae, L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s (Columbus, O h i o : C h a r l e s E. M e r r i l l B o o k s ) , 1967, X I . 11 Cronbach a l s o n o t e s t h e l a c k o f a c o h e r e n t t h e o r y o f i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g . He c a l l s f o r an e x p e r i m e n t a l s t r a t e g y o f a t t e m p t -i n g t o i s o l a t e a p t i t u d e v a r i a b l e s r e l e v a n t t o i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g and d e s i g n i n g a l t e r n a t i v e t r e a t m e n t s t o i n t e r a c t w i t h t h o s e v a r i -a b l e s . He s t a t e s : " I presume t h a t an i n d i v i d u a l has g r e a t e r a p t i t u d e f o r l e a r n i n g , s a y , t o m u l t i p l y from one method o f t e a c h i n g t h a n from a n o t h e r method t h a t i s e q u a l l y good on t h e a v e r a g e . " He b r o a d l y d e f i n e s a p t i t u d e as "... w h a t e v e r promotes t h e p u p i l ' s s u r v i v a l i n a p a r t i c u l a r e d u c a t i o n a l e n v i r o n m e n t , and i t may have as much t o do w i t h s t y l e s o f t h o u g h t and p e r -2 s o n a l i t y v a r i a b l e s as w i t h t h e a b i l i t i e s c o v e r e d i n g e n e r a l t e s t s . " He f u r t h e r a r g u e s t h a t i n s t r u c t i o n a l s t r a t e g y , as opposed t o l e a r n i n g r a t e , i s t h e key t o s u c c e s s f u l i n d i v i d u a l i z a t i o n b e c a u s e a p e r s o n ' s l e a r n i n g r a t e i s dependent upon t h e n a t u r e o f i n s t r u c t i o n . J e n s e n has a l s o n o t e d t h e l a c k o f knowledge c o n c e r n i n g i n d i v i d u a l d i f f e r e n c e s I n l e a r n i n g . I n an a r t i c l e i n w h i c h he a t t e m p t e d t o d e l i n e a t e i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g he c o n c l u d e d : "... i n t h i s a t t e m p t to o f f e r some d e s c r i p t i o n o f t h e domain o f ID's i n l e a r n i n g a t o u r p r e s e n t s t a t e o f knowledge, o r a t l e a s t my ovm s t a t e o f k n o w l e d g e , I f e e l v e r y much l i k e one o f t h e l e g e n d a r y b l i n d men who t r i e d t o d e s c r i b e an e l e p h a n t . At 3 t h i s s t a g e more t h a n one a p p r o a c h i s o b v i o u s l y w a r r a n t e d . " 2 Lee J . C r o n b a c h , "How Can I n s t r u c t i o n Be Ad a p t e d To I n d i v i d u a l D i f f e r e n c e s " , ed. R. M. Gagne, L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s (Columbus, O h i o : C h a r l e s E. M e r r i l l B o o k s ) , 1967, 23 - 24. 3 A r t h u r R. J e n s e n , " V a r i e t i e s Of I n d i v i d u a l D i f f e r e n c e s I n L e a r n i n g " , ed. R. M. Gagne, L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s ( C o l u m b u s , O h i o : C h a r l e s E. M e r r i l l B o o k s ) , 1967, 134. 12 I n d i v i d u a l i z e d I n s t r u c t i o n T y l e r , a t t h e A b i n g t o n C o n f e r e n c e i n 1967, r e v i e w e d t h e c u r r e n t s i t u a t i o n i n i n d i v i d u a l i z e d i n s t r u c t i o n and i d e n t i f i e d s i x m a j o r c o n c e p t s p r e v a l e n t i n i n d i v i d u a l i z e d programs a t t h a t t i m e . He i d e n t i f i e d "... t h e c o n c e p t t h a t t h e i n d i v i d u a l s h o u l d be a b l e t o work a t h i s own r a t e " as "... t h e c o n c e p t most commonly f o l l o w e d i n c u r r e n t e f f o r t s t o i n d i v i d u a l i z e 4 i n s t r u c t i o n . " The o t h e r p o i n t s he m e n t i o n s a r e : (2) P u p i l s h o u l d be a b l e t o work a t t i m e s convenient, t o him. (3) Slow l e a r n e r s h o u l d n o t be embar-r a s s e d by f e e l i n g t h a t he i s much s l o w e r t h a n o t h e r s . (4) L e a r n i n g p r o g r e s s i s l i n e a r . (5) A few f a c t o r s , e a s i l y i d e n t i f i a b l e , i n t e r f e r e w i t h p r o g r e s s . (6) W i t h t h e w e a l t h o f media o f c o m m u n i c a t i o n , a u d i o , v i s u a l , v a r i o u s k i n d s o f f o r m s , s t a t i c and m o ving, t h e l e a r n e r can s e l e c t t h e one o r more w h i c h i s o r a r e e f f e c t i v e f o r him."' T y l e r r a i s e s many o f t h e n a g g i n g q u e s t i o n s c o n c e r n i n g i n d i v i d u a l i z a -t i o n , and f u r t h e r s u g g e s t s t h a t we may be p l a c i n g t h e p r o v e r b i a l c a r t b e f o r e t h e h o r s e . He a s k s : "Have we r e a l l y d e v i s e d t h e s t r a t e g y f o r l e a r n i n g b e f o r e we've t r i e d t o d e v e l o p t h e i n d i v i d u a l i z a t i o n o f i t ? I f we don't do t h a t , t h e n maybe we a r e j u s t assuming t h a t e v e r y b o d y l e a r n s f o l l o w i n g t h e same s t r a t e g y , and we m e r e l y p r o v i d e more o f t h e m a t e r i a l s t h a t we now have r a t h e r t h a n ma-t e r i a l s r e q u i r e d f o r an e f f e c t i v e s t r a t e g y . " * * _ R a l p h W. T y l e r , "New D i r e c t i o n s i n I n d i v i d u a l i z i n g I n s t r u c t i o n " , P r o c e e d i n g s o f t h e A b i n g t o n C o n f e r e n c e '67 ( A b i n g t o n , P e n n s y l v a n i a : The A b i n g t o n C o n f e r e n c e ) , 1968, 4. " ' i b i d . pp. 4 -5 . 6 I b i d . p.7. 13 I P I ( I n d i v i d u a l l y P r e s c r i b e d I n s t r u c t i o n ) i s p e r h a p s t h e most w i d e l y known o f p r e s e n t a t t e m p t s t o i n d i v i d u a l i z e i n s t r u c t i o n a l s y s t e m s . I P I p r o -v i d e s a sequenced i n d i v i d u a l i z e d program f o r g r a d e s K-6 i n f o u r s u b j e c t a r e a s ; m a t h e m a t i c s , r e a d i n g , s c i e n c e and s p e l l i n g . The t h r e e m a j o r compo-n e n t s i n t h e I P I program a r e d e t a i l e d s e q u e n t i a l " b e h a v i o r a l l y - s t a t e d " i n -s t r u c t i o n a l o b j e c t i v e s , d i a g n o s t i c t e s t i n g and t h e i n d i v i d u a l p r e s c r i p t i o n s . The i n d i v i d u a l p r e s c r i p t i o n i s t h e most i m p o r t a n t o f t h e t h r e e . The p r e -s c r i p t i o n i s p r e p a r e d by t h e t e a c h e r based on t h e d i a g n o s t i c t e s t s and cumu-l a t i v e p e r s o n a l r e c o r d o f t h e c h i l d . Once t h e o b j e c t i v e i s d e c i d e d , t h e t e a c h e r chooses from among a number o f i n s t r u c t i o n a l s e t t i n g s , one a p p r o -p r i a t e f o r t h e c h i l d . These i n s t r u c t i o n a l o p t i o n s i n c l u d e t u t o r i n g by t h e t e a c h e r , s m a l l - g r o u p i n s t r u c t i o n , work s h e e t s , l i s t e n i n g t o t a p e s o r d i s c s , and v i e w i n g f i l m s t r i p s . G l a s e r has n o t e d : "At t h e p r e s e n t s t a t e o f o u r knowledge, t h e d e c i s i o n r u l e s f o r g o i n g f r o m measures o f s t u d e n t p e r f o r m a n c e t o i n s t r u c t i o n a l p r e s c r i p t i o n s may n o t be v e r y complex, b u t l i t t l e i s known about t h e amount o f c o m p l e x i t y r e q u i r e d . ... S u s t a i n e d a n a l y s i s o f s u c h i n f o r m a t i o n about i n d i v i d u a l d i f f e r e n c e - l e a r n i n g e n v i r o n m e n t r e l a t i o n s h i p s s h o u l d r e s u l t i n t h e a b i l i t y t o s u p p l y t h e t e a c h e r w i t h t h e k i n d o f d a t a r e -d u c t i o n and i n f o r m a t i o n t h a t w i l l e n a b l e him t o manage t h e t a s k o f a d a p t i n g 9 t o i n d i v i d u a l d i f f e r e n c e s . " P i e r o n e k r e c e n t l y c o n d u c t e d a s u r v e y o f i n d i v i d u a l i z e d r e a d i n g and m a t h e m a t i c s programs. She p e r s o n a l l y v i s i t e d s c h o o l s a c r o s s N o r t h A m e r i c a ^ I n d i v i d u a l l y P r e s c r i b e d I n s t r u c t i o n ( W a s h i n g t o n , D. C. : E d u c a t i o n U. S. A.) 1968, 6. ^ I b i d . p.12. 9 I b i d . pp.12-13. 14 i n w h i c h S A M I ( S y s t e m a t i c A p p r o a c h t o M a t h e m a t i c a l I n s t r u c t i o n ) and I P I were b e i n g u s e d . She i d e n t i f i e s s e l f - p a c i n g as t h e p r e d o m i n a n t f e a t u r e of I P I and n o t e s t h a t I P I has an e x c e l l e n t d e t a i l e d h i e r a r c h i c a l s t r u c t u r e i n mathe-m a t i c s . She f ound SAMI v e r y s i m i l a r t o t h e m a t h e m a t i c s program of I P I . How-e v e r , SAMI a l s o f e a t u r e s s e l f - s e l e c t i o n o f l e a r n i n g m a t e r i a l s . F o r any g i v e n u n i t , t h e s t u d e n t i s f r e e t o choose h i s i n s t r u c t i o n a l m a t e r i a l s from p r e -packaged a l t e r n a t i v e s . ^ The l i t e r a t u r e r e v i e w e d i n t h e above two s e c t i o n s i n d i c a t e s t h a t v e r y l i t t l e p r o g r e s s has been made i n t h e p a s t t h i r t y y e a r s toward a t h e o r y o f i n d i -v i d u a l d i f f e r e n c e s i n l e a r n i n g and t h a t t h e l a c k o f s u c h a t h e o r y i s s e v e r e l y h ampering t h e s u c c e s s o f c u r r e n t a t t e m p t s t o i n d i v i d u a l i z e i n s t r u c t i o n . I t has been s u g g e s t e d by many e d u c a t o r s , i n c l u d i n g C r o n b a c h , Gagne and T y l e r , t h a t a two-pronged e x p e r i m e n t a l s t r a t e g y o f i s o l a t i n g a p t i t u d e v a r i a b l e s r e l e v a n t t o i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g and d e s i g n i n g a l t e r n a t i v e i n s t r u c t i o n a l s t r a t e g i e s t o be t e s t e d f o r i n t e r a c t i o n w i t h t h e s e v a r i a b l e s i s needed f o r t h e development o f an e f f e c t i v e s y s t e m a t i c a p p r o a c h t o t h e a d a p t a t i o n o f e d u c a -t i o n a l s t i m u l i t o l e a r n e r s . A p t i t u d e - I n s t r u c t i o n I n t e r a c t i o n i n M a t h e m a t i c s B e c k e r has commented t h a t "... few s t u d i e s have been d e s i g n e d t o i n v e s t i g a t e t h e i n t e r a c t i o n between a p t i t u d e and I n s t r u c t i o n , " and i n p a r t i -c u l a r , t h a t "... o n l y a s m a l l number o f s u c h s t u d i e s d e a l d i r e c t l y w i t h mathe-m a t i c s l e a r n i n g . " ' ' " ^ He f u r t h e r s t a t e s t h a t t h e r e i s as y e t no e v i d e n c e t o s u p -^ F l o r e n c e T. P i e r o n e k , "A S u r v e y o f I n d i v i d u a l i z e d R e a d i n g and Mathe-m a t i c s P rograms", C a l g a r y C a t h o l i c S c h o o l B o a r d , C a l g a r y , A l b e r t a , 1971, ED 04789^ • ^ J e r r y . P. B e c k e r , " R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n : The R o l e o f T h e o r y and of 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 " , J o u r n a l f o r R e s e a r c h i n Mathe- m a t i c s E d u c a t i o n , I , No. 1, 1970, 24. 15 p o r t t h e g e n e r a l i z a b i l i t y o f r e s e a r c h i n o t h e r a r e a s t o m a t h e m a t i c s . 12 A i k e n a l s o n o t e s t h e p a u c i t y o f s t u d i e s i n m a t h e m a t i c s w h i c h r e -s u l t e d i n s i g n i f i c a n t i n t e r a c t i o n s between a p t i t u d e and i n s t r u c t i o n a l s t r a t e g y . He d o e s , however, r e p o r t t h r e e u n p u b l i s h e d d o c t o r a l s t u d i e s , two o f w h i c h w i l l 13 be d e s c r i b e d below. These s t u d i e s a r e a l s o d i s c u s s e d i n Cronbach's r e p o r t . Cronbach's d i s c u s s i o n o f one o f them i s q u o t e d b e l o w . 14 B e c k e r used two programmed i n s t r u c t i o n a l s t r a t e g i e s t o t e a c h h i g h s c h o o l a l g e b r a s t u d e n t s t o sum s e r i e s . One a p p r o a c h was e x p o s i t o r y , t h e s t u -d e n t s were g i v e n the f o r m u l a s and e x p l a n a t i o n s r e l a t i n g t h e s e t o t h e s e r i e s i n b o t h v e r b a l and s y m b o l i c f o r m . The o t h e r was d i s c o v e r y o r i e n t e d , t h e s t u -d e n t was g i v e n examples o f t h e r e l a t i o n s h i p s s o u g h t and a s k e d t o d i s c o v e r t h e f o r m u l a . Two a p t i t u d e measures were u s e d ( v e r b a l and n u m e r i c a l ) . However, no s i g n i f i c a n t i n t e r a c t i o n s between a p t i t u d e s and i n s t r u c t i o n a l methods were f o u n d . C a r r y ( 1 9 6 7 ) c o n d u c t e d a d i s s e r t a t i o n c o m p a r i n g g e o m e t r i c - g r a p h i c a l v s . a l g e b r a i c - a n a l y t i c a l p r e s e n t a t i o n s u s i n g programmed i n s t r u c t i o n a l m a t e r i a l s i n t h e m a t h e m a t i c s o f q u a d r a t i c i n e q u a l i t i e s . C r i t e r i o n mea-s u r e s r e p r e s e n t i n g b o t h i m m e d i a t e r e c a l l and t r a n s f e r t o new p r o b l e m s were o b t a i n e d f o r 181 h i g h - s c h o o l geometry s t u d e n t s . C a r r y h y p o t h e s i z e d t h a t s p a t i a l v i s u a l i z a t i o n w o u l d be c a l l e d f o r i n t h e g r a p h i c a l t r e a t -ment and so w o u l d p r e d i c t s u c c e s s i n i t , more t h a n i n t h e a l g e b r a i c t r e a t -ment. He h y p o t h e s i z e d a l s o t h a t g e n e r a l r e a s o n i n g w o u l d r e l a t e more h i g h l y t o l e a r n i n g f r o m a l g e b r a i c t h a n f r o m g r a p h i c a l i n s t r u c t i o n . The 12 L e w i s R. A i k e n , J r . , " I n t e l l e c t i v e V a r i a b l e s and M a t h e m a t i c s A c h i e v e m e n t " , J o u r n a l o f S c h o o l P s y c h o l o g y , I X , 1971, 203-212. 13 Lee J . Cronbach and R i c h a r d E. Snow, " I n d i v i d u a l D i f f e r e n c e s i n L e a r n i n g A b i l i t y as a F u n c t i o n of I n s t r u c t i o n a l V a r i a b l e s " F i n a l R e p o r t . S t a n f o r d U n i v e r s i t y , C a l i f o r n i a S c h o o l o f E d u c a t i o n ED 029 0 0 1 . 14 L e w i s R. A i k e n , J r . , " I n t e l l e c t i v e V a r i a b l e s and M a t h e m a t i c s A c h i e v e m e n t " , J o u r n a l o f S c h o o l P s y c h o l o g y , I X , 1971, 203-212 r e p o r t e d J . P. B e c k e r , "An A ttempt t o D e s i g n I n s t r u c t i o n a l T e c h n i q u e s i n M a t h e m a t i c s t o Accommodate D i f f e r e n t P a t t e r n s o f M e n t a l A b i l i t y " ( D o c t o r a l d i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y , Ann A r b o r , M i c h . : U n i v e r s i t y M i c r o f i l m s , 1 9 6 7 ) . 16 d a t a d i d n o t c o n f i r m t h e s e h y p o t h e s e s . No i n t e r a c t i o n s were o b t a i n e d w i t h t h e r e c a l l c r i t e r i o n f o r e i t h e r a p t i t u d e v a r i a b l e . S i g n i f i c a n t i n t e r a c t i o n was d e t e c t e d f o r t h e t r a n s f e r measure, but t h e low i n t e r n a l c o n s i s t e n c y o f t h i s measure made o v e r a l l f i n d i n g s s u s p e c t . A n a l y s e s a t t h e i t e m l e v e l showed two o f t h e e i g h t t r a n s f e r i t e m s i n v o l v e d i n i n t e r a c t i o n s w i t h a p t i t u d e . F o r b o t h i t e m s , t h e r e a s o n i n g measure was f ound p r e d i c t i v e o f r e s p o n s e s i n t h e g r a p h i c a l t r e a t m e n t b u t n o t i n t h e a n a l y t i c t r e a t m e n t . F o r one i t e m , s p a t i a l a p t i t u d e a l s o p r e d i c t e d g r a p h i c b u t n o t a n a l y t i c a c h i e v e m e n t . W i t h o u t c o n f i r m a t i o n , r e s u l t s s u c h as t h e s e a r e u n i n t e r p r e t a b l e . 1 5 K i n g , R o b e r t s and K r o p p ^ ^ a d m i n i s t e r e d a b a t t e r y o f a p t i t u d e t e s t s t o 426 f i f t h and s i x t h g r a d e s t u d e n t s ( f o u r f i f t h g r a d e c l a s s e s and f o u r s i x t h g r a de c l a s s e s ) . The s t u d e n t s were t h e n c l a s s i f i e d as v e r b a l o r f i g u r a l and w i t h i n t h e s e groups were c l a s s i f i e d as d e d u c t i v e o r i n d u c t i v e . F o u r i n -s t r u c t i o n a l s t r a t e g i e s , v e r b a l - d e d u c t i v e , v e r b a l - i n d u c t i v e , f i g u r a l - d e d u c t i v e and f i g u r a l - i n d u c t i v e , were d e v e l o p e d f o r a programmed two day u n i t on e l e -m e n t a r y s e t c o n c e p t s . C l a s s e s were a s s i g n e d i n t a c t t o one o f t h e f o u r s t r a -t e g i e s . The i n v e s t i g a t o r s h y p o t h e s i z e d t h a t v e r b a l a b i l i t y measures w o u l d c o r r e l a t e s i g n i f i c a n t l y h i g h e r t h a n f i g u r a l a b i l i t y measures w i t h a c h i e v e -ment on m a t e r i a l s p r e s e n t e d v e r b a l l y . S i m i l a r e f f e c t s were h y p o t h e s i z e d b e -tween f i g u r a l a b i l i t y measures and m a t e r i a l s p r e s e n t e d f i g u r a l l y , between i n d u c t i v e measures and m a t e r i a l s p r e s e n t e d i n d u c t i v e l y , and between d e d u c t i v e measures and m a t e r i a l s p r e s e n t e d d e d u c t i v e l y . The dependent measure was a c r i t e r i o n t e s t o f 24 i t e m s , 12 p r e s e n t e d Lee J . Cronbach and R i c h a r d E. Snow, " I n d i v i d u a l D i f f e r e n c e s i n L e a r n i n g A b i l i t y as a F u n c t i o n o f I n s t r u c t i o n a l V a r i a b l e s " F i n a l R e p o r t S t a n f o r d U n i v e r s i t y , C a l i f o r n i a S c h o o l o f E d u c a t i o n , ED 029 001, 125. ^ F . J . K i n g , D e n n i s R o b e r t s and R u s s e l l P. K r o p p , " R e l a t i o n s h i p Between A b i l i t y Measures and A c hievement under F o u r Methods o f T e a c h i n g E l e m e n t a r y S e t C o n c e p t s " , J o u r n a l of E d u c a t i o n a l P s y c h o l o g y , LX, 1969, 244-47. 17 v e r b a l l y and 12 p r e s e n t e d f i g u r a l l y . The d a t a was a n a l y s e d u s i n g r e g r e s s i o n a n a l y s i s and t - t e s t s were used t o d e t e c t s i g n i f i c a n t d i f f e r e n c e s between p a i r s o f r e g r e s s i o n c o e f f i c i e n t s . A summary o f t h e r e s u l t s f o l l o w s . None of t h e t r a t i o s f o r t h e v e r b a l - f i g u r a l c o m p a r i s o n s was s i g n i f i -c a n t , so t h e r e was no s u p p o r t f o r ATI e f f e c t s i n the v e r b a l o r f i g u r a l g r o u p s . However, t h e d e d u c t i v e - i n d u c t i v e c o n t r a s t s s u p p o r t t h e hypo-t h e s i s b ecause two o f t h e t r a t i o s were s i g n i f i c a n t a t t h e .05 l e v e l and t h e d i f f e r e n c e s i n b o t h c a s e s were i n t h e h y p o t h e s i z e d d i r e c t i o n . Thus t h e I n f e r e n c e T e s t ( d e d u c t i o n ) was a b e t t e r p r e d i c t o r f o r t h e de-d u c t i v e m a t e r i a l s t h a n f o r t h e i n d u c t i v e m a t e r i a l s . F o r t h e Word G r o u p i n g T e s t ( i n d u c t i o n ) t h e c o n v e r s e was t r u e . " - ^ A r e v i e w of t h e l i t e r a t u r e c o n f i r m e d B e c k e r and A i k e n ' s comments r e -g a r d i n g t h e l a c k o f 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 r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n . O n l y t h r e e s t u d i e s were f o u n d and t h e d i v e r s i t y o f t h e s t u d i e s made i t i m p o s s i b l e t o draw any c o n c l u s i o n s w i t h r e g a r d t o 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 s i n m a t h e m a t i c s e d u c a t i o n . F i e l d - D e p e n d e n c e - I n d e p e n d e n c e S t u d i e s 18 D a v i s and K l a u s m e i e r c o n d u c t e d two s e p a r a t e s t u d i e s i n v o l v i n g c o g -n i t i v e s t y l e and a c o n c e p t i d e n t i f i c a t i o n t a s k . The f i r s t v a r i e d c o g n i t i v e s t y l e and l e v e l o f c o m p l e x i t y o f t h e t a s k . The second v a r i e d c o g n i t i v e s t y l e and t h e t r a i n i n g p r o c e d u r e . I n b o t h s t u d i e s , t h e H i d d e n F i g u r e s T e s t ( w h i c h c o r r e l a t e s r = .62 w i t h W i t k i n ' s Embedded F i g u r e s T e s t ) was u s e d t o c a t e g o r i z e s t u d e n t s on t h e c o g n i t i v e s t y l e f a c t o r . S t u d e n t s were i d e n t i -f i e d as h i g h , m i d d l e o r low a n a l y t i c d e p e n d i n g upon t h e i r p o s i t i o n i n t h e d i s t r i b u t i o n o f s c o r e s on t h e t e s t . H i g h a n a l y t i c r e p r e s e n t e d t h e a b i l i t y 1 7 I b i d . pp. 246-247. 18 J . Kent D a v i s and H e r b e r t J . K l a u s m e i e r , " C o g n i t i v e S t y l e and Concept I d e n t i f i c a t i o n As A F u n c t i o n Of C o m p l e x i t y and T r a i n i n g P r o c e d u r e s " , J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , L X I , 1970, 423-430. 18 t o i d e n t i f y t h e h i d d e n f i g u r e s . I n e x p e r i m e n t one, t h e t h r e e l e v e l s o f c o g n i t i v e s t y l e were used and c o m p l e x i t y was d e f i n e d as t h e n u m b e r ( l , 3 o r 5) o f b i t s o f i r r e l e v a n t I n f o r m a t i o n i n t h e p r o b l e m . N i n e t y s e n i o r h i g h s c h o o l males formed t h e s t u d y ' s sample. The d a t a i n d i c a t e d a main e f f e c t due t o c o g n i t i v e s t y l e . The h i g h a n a l y t i c s i d e n t i f i e d t h e c o n c e p t s w i t h g r e a t e r ease t h a n t h e low a n a l y t i c s . However, t h e r e was no i n t e r a c t i o n between c o g n i t i v e s t y l e and c o m p l e x i t y . I n t h e second e x p e r i m e n t , t h e two extreme l e v e l s o f t h e c o g n i t i v e s t y l e f a c t o r , h i g h a n a l y t i c and low a n a l y t i c , and f o u r t r a i n i n g c o n d i t i o n s were u s e d . The t r a i n i n g c o n d i t i o n s were prompt, v e r b a l o n l y , v e r b a l - p r o m p t , and c o n t r o l , t h e s t a n d a r d p r o c e d u r e f o r t h e t a s k . E i g h t y s e n i o r h i g h s c h o o l males(AO h i g h a n a l y t i c and AO low a n a l y t i c ) formed t h e s a m p l e . As i n t h e f i r s t e x p e r i m e n t , a s i g n i f i c a n t main e f f e c t due t o c o g n i t i v e s t y l e was f o u n d . However, no s i g n i f i c a n t i n t e r a c t i o n e f f e c t was found between c o g n i t i v e s t y l e and t r a i n i n g c o n d i t i o n . D a v i s and K l a u s m e i e r c o n c l u d e d t h a t "... t h e s e t r a i n i n g p r o c e d u r e s do n o t d i f f e r e n t i a l l y i n f l u e n c e c o n c e p t "19 i d e n t i f i c a t i o n f o r i n d i v i d u a l s m a n i f e s t i n g d i f f e r e n t c o g n i t i v e s t y l e s . 20 D a v i s i n v e s t i g a t e d t h e r e l a t i o n s h i p between c o g n i t i v e s t y l e and two d i f f e r e n t c o n c e p t i d e n t i f i c a t i o n t a s k s . The H i d d e n F i g u r e s T e s t was a d m i n i s t e r e d t o 600 s e n i o r h i g h s c h o o l f e m a l e s . T h i r t y - s i x s t u d e n t s who were c l a s s i f i e d as a n a l y t i c ( + 1 s t a n d a r d d e v i a t i o n above t h e mean) and 1 9 I b i d . p. A29. 20 J . Kent D a v i s , " C o g n i t i v e S t y l e and C o n d i t i o n a l Concept L e a r n i n g " (paper r e a d a t t h e a n n u a l m e e t i n g of t h e A m e r i c a n E d u c a t i o n a l R e s e a r c h A s s o c i a t i o n , C h i c a g o , 1 9 7 2 ) . 19 t h i r t y - s i x who were c l a s s i f i e d as g l o b a l ( - l s t a n d a r d d e v i a t i o n b e l o w t h e mean) were chosen t o p a r t i c i p a t e i n t h e s t u d y . The two c o n c e p t i d e n t i -f i c a t i o n t a s k s were l a b e l l e d n o n s i g n - d i f f e r e n t i a t e d ( N S D ) and s i g n - d i f f e r -e n t i a t e d ( S D ) . A p r o b l e m whose s o l u t i o n "... i s n o t based upon a p a r t i -c u l a r s i g n o r cue b u t r a t h e r upon t h e c o n d i t i o n a l r e l a t i o n s h i p between 21 f i g u r e s " was l a b e l l e d as NSD. An SD.problem was one i n w h i c h t h e c o r -r e s p o n d i n g NSD p r o b l e m was changed t o a l l o w f o r i t s s o l u t i o n by i s o -l a t i n g a r e l e v a n t cue. That i s , t h e SD p r o b l e m c o u l d be s o l v e d u s i n g t h e c o n d i t i o n a l r u l e s ( a s was r e q u i r e d f o r t h e c o r r e s p o n d i n g NSD) o r t h e r e l e v a n t cue. D a v i s h y p o t h e s i z e d t h a t g l o b a l s t u d e n t s w o u l d s o l v e t h e NSD p r o b l e m s o o n e r t h a n a n a l y t i c s t u d e n t s b e c a u s e t h e s o l u t i o n r e q u i r e s a g l o b a l s t r a t e g y t o d i s c o v e r t h e r e l a t i o n s h i p between t h e two f i g u r e s . He f u r t h e r h y p o t h e s i z e d t h a t , s i n c e t h e SD p r o b l e m c o u l d be s o l v e d by an a n a l y t i c s t r a t e g y , a n a l y t i c s t u d e n t s w o u l d s o l v e t h e SD p r o b l e m s o o n e r t h a n g l o b a l s t u d e n t s . Two dependent measures were u s e d t o a n a l y s e t h e r e s u l t s : (1) t h e number o f t r i a l s t o c r i t e r i o n and (2) t h e number o f e r r o r s t o c r i t e r i o n . From t h e a n a l y s i s o f v a r i a n c e p e r f o r m e d on t h e d a t a , t h e f o l l o w i n g c o n -c l u s i o n s were drawn: The r e s u l t s o f t h i s s t u d y o n l y p a r t i a l l y s u p p o r t t h e h y p o t h e s e s . The o v e r a l l p e r f o r m a n c e o f t h e a n a l y t i c Ss was s u p e r i o r t o t h e o v e r -a l l p e r f o r m a n c e o f t h e g l o b a l S s . The s i g n i f i c a n t i n t e r a c t i o n o f c o g n i t i v e s t y l e and p r o b l e m t y p e , however, d e m o n s t r a t e d t h a t t h i s s u p e r i o r i t y was r e s t r i c t e d o n l y t o t h e SD p r o b l e m . T h e r e was no d i f -f e r e n c e between c o g n i t i v e s t y l e l e v e l s on t h e NSD p r o b l e m . I t was h y p o t h e s i z e d t h a t t h e a n a l y t i c Ss would p e r f o r m b e s t on t h e SD p r o b l e m and t h a t t h e g l o b a l Ss would p e r f o r m b e s t on t h e NSD p r o b l e m . I t was found t h a t t h e a n a l y t i c Ss p e r f o r m e d s i g n i f i c a n t l y b e t t e r on t h e SD p r o b l e m t h a n t h e y d i d on t h e NSD p r o b l e m , but t h a t I b i d . p.3. 20 t h e g l o b a l Ss d i d n o t p e r f o r m b e t t e r on t h e NSD p r o b l e m . I n f a c t , t h e i r p e r f o r m a n c e on b o t h p r o b l e m s was v i r t u a l l y t h e same and i n g e n e r a l was q u i t e p o o r . " 2 2 23 A s t u d y c o n d u c t e d by D a v i s and G r i e v e e x p l o r e d t h e r e l a t i o n s h i p between c o g n i t i v e s t y l e and i n s t r u c t i o n a l s t r a t e g y i n t h e t e a c h i n g of geo-g r a p h y . Two l e v e l s o f i n s t r u c t i o n a l s t r a t e g y , d i s c o v e r y and e x p o s i t o r y , and two l e v e l s of c o g n i t i v e s t y l e , a n a l y t i c and g l o b a l , were f a c t o r i a l l y combined t o g i v e a 2 x 2 d e s i g n . The H i d d e n F i g u r e s T e s t was a d m i n i s t e r e d t o 117 g r a d e n i n e s t u d e n t s and a median s p l i t was used t o c l a s s i f y s t u d e n t s as a n a l y t i c o r g l o b a l . The two i n s t r u c t i o n a l s t r a t e g i e s each r e q u i r e d e l e v e n h o u r s t o c o m p l e t e and d i f f e r e d o n l y i n the p l a c e m e n t o f t h e v e r b a l i -z a t i o n o f t h e g e n e r a l i z a t i o n t o be drawn from t h e i n s t r u c t i o n a l u n i t . I n t h e d i s c o v e r y method t h e v e r b a l i z a t i o n was d e l a y e d u n t i l t h e end. Whereas, i n t h e e x p o s i t o r y method, t h e v e r b a l i z a t i o n was t h e i n i t i a l s t e p i n t h e s e q u ence. Two dependent measures were u s e d : (1) a m u l t i p l e - c h o i c e t e s t m e a s u r i n g a knowledge o f t h e geography s t u d i e d and (2) a m u l t i p l e - c h o i c e t e s t m e a s u r i n g t h e a b i l i t y t o use g e o g r a p h i c m a t e r i a l s i n new s i t u a t i o n s . The d a t a was a n a l y s e d t w i c e . The f i r s t t i m e , a l l t h e s u b j e c t s were u s e d . The second a n a l y s i s u t i l i z e d o n l y t h o s e s u b j e c t s who had been i d e n -t i f i e d as extreme a n a l y t i c o r extreme g l o b a l . ( 7 4 of t h e 117 s t u d e n t s were c l a s s i f i e d as extreme) The m a j o r c o n c l u s i o n s o f t h e s t u d y w e re: 2 2 I b i d . pp. 10-11. 23 J . Kent D a v i s and T a r r a n c e Don G r i e v e , "The R e l a t i o n s h i p o f C o g n i t i v e S t y l e and Method o f I n s t r u c t i o n t o P e r f o r m a n c e i n N i n t h Grade Geography", The J o u r n a l of E d u c a t i o n a l R e s e a r c h , X L V T I , 1971, 137-141. 21 1. N e i t h e r c o g n i t i v e s t y l e n o r method o f i n s t r u c t i o n had an o v e r -a l l e f f e c t on t h e a c q u i s i t i o n o f knowledge. W i t h r e s p e c t t o t h e e x -treme a n a l y t i c and extreme g l o b a l S s , however, i t was found t h a t e x -treme g l o b a l males r e c e i v i n g t h e e x p o s i t o r y i n s t r u c t i o n e x p e r i e n c e d s i g n i f i c a n t d i f f i c u l t y i n a c q u i r i n g knowledge o f J a p a n ' s g e o g r a p h y . T h i s f i n d i n g s u g g e s t s t h a t t h e e x p o s i t o r y method of i n s t r u c t i o n s h o u l d be a v o i d e d when t e a c h i n g e x t r e m e l y g l o b a l males u n l e s s s u f f i -c i e n t t i m e i s d e v o t e d t o e s t a b l i s h i n g t h o s e d i s c r i m i n a t i o n s w h i c h a r e b a s i c t o t h e g e n e r a l i z a t i o n s t h a t a r e t o be l e a r n e d . 2. An i n d i v i d u a l ' s c o g n i t i v e s t y l e was f o u n d t o d i f f e r e n t i a l l y i n f l u e n c e h i s h i g h e r l e a r n i n g s c o r e s . A n a l y t i c Ss were b e t t e r a b l e t o a p p l y knowledge o f geography t o new s i t u a t i o n s t h a n were g l o b a l S s . N e i t h e r o f t h e methods of i n s t r u c t i o n were f o u n d t o have an o v e r a l l e f f e c t on h i g h e r l e a r n i n g p e r f o r m a n c e , b u t t h e a n a l y s i s o f t h e extreme Ss i n d i c a t e d t h a t g l o b a l rcales r e c e i v i n g t h e e x p o s i t o r y i n s t r u c t i o n e x p e r i e n c e d s i g n i f i c a n t d i f f i c u l t y i n a p p l y i n g knowledge t o new s i t u a t i o n s . 2 4 25 H e s t e r and T a e a t z s t u d y i n v e s t i g a t e d the i n t e r a c t i o n e f f e c t o f c o g n i t i v e s t y l e and i n s t r u c t i o n a l s t r a t e g y on c o n c e p t a t t a i n m e n t . Ten s i m i l a r c o n c e p t i d e n t i f i c a t i o n t a s k s were p r e s e n t e d t o 72 g r a d u a t e e d u c a -t i o n s t u d e n t s by one o f two d i f f e r e n t i n s t r u c t i o n a l s t r a t e g i e s , common-a l i t y o r c o n s e r v a t i v e . These s t u d e n t s had been i d e n t i f i e d as a n a l y t i c o r g l o b a l a c c o r d i n g t o t h e i r r e l a t i v e p o s i t i o n t o t h e median s c o r e o f t h e group on t h e T I P T ( T a g a t z I n f o r m a t i o n P r o c e s s i n g T e s t ) . TIPT has been found t o c o r r e l a t e s i g n i f i c a n t l y w i t h t h e H i d d e n F i g u r e s T e s t a t t h e .01 l e v e l . "Those r e c e i v i n g t h e c o m m o n a l i t y i n s t r u c t i o n were d i r e c t e d t o l o o k a t t h e " f o c u s " i n s t a n c e and t h e e x e m p l a r s , and t o d e t e r m i n e t h e a t -t r i b u t e s common t o t h e s e c a r d s . The c o n s e r v a t i v e s t r a t e g y d i r e c t e d Ss t o compare t h e " y e s " and "no" c a r d s w i t h t h e " f o c u s " i n s t a n c e . A " y e s " c a r d d i f f e r e d by one i r r e l e v a n t a t t r i b u t e and a "no" c a r d d i f f e r e d by I b i d . p.141. 25 F l o r e n c e M. H e s t e r and G l e n n E. T a g a t z , "The E f f e c t s o f C o g n i -t i v e S t y l e and I n s t r u c t i o n a l S t r a t e g y on Concept A t t a i n m e n t " , The J o u r n a l o f G e n e r a l P s y c h o l o g y . L X X X V , 1971, 229-237. 22 26 one r e l e v a n t a t t r i b u t e . " The dependent v a r i a b l e was t h e s u b j e c t s t i m e -t o - c r i t e r i o n s c o r e on each t a s k . The d a t a s u p p o r t e d t h e f o l l o w i n g c o n -c l u s i o n s : (a) Ss d i s p l a y i n g t h e a n a l y t i c c o g n i t i v e s t y l e can e f f i c i e n t l y u t i l i z e e i t h e r t h e c o n s e r v a t i v e o r c o m m o n a l i t y i n s t r u c t i o n a l s t r a t e g y . The a n a l y t i c c o g n i t i v e s t y l e seems t o be an i n h e r e n t o r g a n i s m i c c h a r a c t e r i s t i c t h a t e n a b l e s Ss t o a c h i e v e t h e d i f f e r e n t i a t i o n r e -q u i r e d by t h e more r i g o r o u s c o n s e r v a t i v e i n s t r u c t i o n a l s t r a t e g y . (b) Ss d i s p l a y i n g t h e g l o b a l c o g n i t i v e s t y l e a r e a b l e t o u t i l i z e e f f i c i e n t l y t h e c o m m o n a l i t y i n s t r u c t i o n a l s t r a t e g y , w h i c h does n o t r e q u i r e f i n e d i s c r i m i n a t i o n s w i t h i n t h e s t i m u l u s f i e l d and i s t h e r e -f o r e r e l a t e d t o t h e i r c o g n i t i v e s t y l e . Ss d i s p l a y i n g t h e g l o b a l c o g n i t i v e s t y l e a r e u n a b l e t o u t i l i z e and a r e i n h i b i t e d by t h e more r i g o r o u s c o n s e r v a t i v e i n s t r u c t i o n a l s t r a t e g y . 2 7 28 S a a r n i i n v e s t i g a t e d d i f f e r e n c e s i n p r o b l e m s o l v i n g as a f u n c t i o n o f t h e c o g n i t i v e d e v e l o p m e n t a l l e v e l o f t h e s u b j e c t and t h e s u b j e c t ' s c o g -n i t i v e s t y l e . I t was h y p o t h e s i z e d t h a t P i a g e t ' s t h e o r y o f t h e development o f l o g i c a l t h i n k i n g w o u l d p r o v i d e an o v e r - a l l framework f o r u n d e r s t a n d i n g complex p r o b l e m - s o l v i n g p e r f o r m a n c e s and t h a t W i t k i n ' s c o n s t r u c t o f f i e l d i n d e p e n d e n c e would p r o v e f r u i t f u l i n u n d e r s t a n d i n g i n d i v i d u a l d i f f e r e n c e s w i t h i n each P i a g e t i a n d e v e l o p m e n t a l l e v e l . S i x t y - f o u r s t u d e n t s ( e i g h t male and e i g h t f e m a l e s t u d e n t s r andomly s e l e c t e d from each o f g r a d e s s i x , s e v e n , e i g h t and n i n e ) p a r t i c i p a t e d i n t h e s t u d y . Two P i a g e t i a n t a s k s were u s e d t o i d e n t i f y t h e s t u d e n t s ' p r e -s e n t c o g n i t i v e d e v e l o p m e n t a l l e v e l ( f o r m a l o p e r a t i o n a l , t r a n s i t i o n a l o r c o n c r e t e o p e r a t i o n a l ) . The p o r t a b l e r o d and frame t e s t was used t o i d e n -2 6 I b i d . p.232. 2 7 I b i d . p.236. 28 C a r o l y n I n g r i d S a a r n i , " P i a g e t i a n O p e r a t i o n s and F i e l d I n d e -pendence As F a c t o r s i n C h i l d r e n ' s P r o b l e m S o l v i n g P e r f o r m a n c e " ( p a p e r r e a d a t t h e a n n u a l m e e t i n g o f t h e A m e r i c a n E d u c a t i o n a l R e s e a r c h A s s o c i a t i o n , C h i c a g o , 1 9 7 2 ) . 23 t i f y t h e s t u d e n t ' s l e v e l o f f i e l d i n d e p e n d e n c e . T h r e e l e v e l s o f f i e l d i n d e p endence were u s e d ; low, medium and h i g h . The range o f t h e s e l e v e l s was d e t e r m i n e d by r a n k i n g t h e r o d and frame t e s t s c o r e s and d i v i d i n g t h e r e s u l t i n g d i s t r i b u t i o n i n t o t h i r d s . Two d e t e c t i v e s t o r i e s c o n s t i t u t e d t h e p r o b l e m - s o l v i n g t a s k s , and f o u r dependent measures were u s e d t o measure p e r f o r m a n c e on each o f t h e s t o r i e s . The d a t a was a n a l y s e d u s i n g m u l t i v a r i a t e a n a l y s i s o f v a r i a n c e . Among t h e c o n c l u s i o n s o f t h e s t u d y was t h e f o l l o w i n g : The c o n s t r u c t f i e l d i n d e p e n d e n c e a p p e a r s t o have d o u b t f u l i m p l i -c a t i o n s f o r complex p r o b l e m s o l v i n g p e r f o r m a n c e . The a n a l y s e s i n d i -c a t e t h a t f i e l d i n d e p e n d e n c e w i t h i n e ach P i a g e t i a n l e v e l does n o t a f f e c t complex, m u l t i - s t e p p r o b l e m s o l v i n g p e r f o r m a n c e as m a n i f e s t e d i n t h e P r o d u c t i v e T h i n k i n g p r o b l e m s . T h i s does n o t i n v a l i d a t e t h e r o l e f i e l d i n d e pendence m i g h t have i n d e t e r m i n i n g p e r f o r m a n c e on p r o b l e m s w h i c h a r e more p e r c e p t u a l l y bound and/or r e l a t i v e l y non-v e r b a l . The r e s u l t s o b t a i n e d h e r e , however, c a s t doubt on t h e gen-e r a l i t y o f t h e f i e l d i n d e p e n d e n c e c o n s t r u c t as a " c o g n i t i v e s t y l e " o r as a c o n s i s t e n t c h a r a c t e r i s t i c o f t h e i n d i v i d u a l i n h i s i n -t e l l e c t u a l f u n c t i o n i n g . 2 9 Of t h e f i v e s t u d i e s r e p o r t e d above w h i c h used f i e l d - d e p e n d e n c e as an a p t i t u d e v a r i a b l e , o n l y one, t h e s t u d y by D a v i s and G r i e v e , u t i l i z e d an i n s t r u c t i o n a l s e t t i n g as t h e t r e a t m e n t v a r i a b l e . I n t h i s s t u d y , i t was h y p o t h e s i z e d t h a t s t u d e n t s w o u l d p e r f o r m b e s t when t a u g h t by a method c o n -s i s t e n t w i t h t h e i r c o g n i t i v e s t y l e . The r e s u l t s , however, c o n f i r m e d t h i s h y p o t h e s i s o n l y i n t h e c a s e of extreme g l o b a l s t u d e n t s . Three of t h e s t u d i e s used a c o n c e p t i d e n t i f i c a t i o n t a s k as t h e t r e a t m e n t v a r i a b l e and t h e f o u r t h , t h e s t u d y by S a a r n i , used a p r o b l e m -s o l v i n g t a s k as t h e t r e a t m e n t v a r i a b l e . Of t h e t h r e e s t u d i e s w h i c h used / y l b i d . pp.19-20 24 a c o n c e p t i d e n t i f i c a t i o n t a s k , t h e s t u d i e s by D a v i s and H e s t e r and T a g a t z r e p o r t e d p a r t i a l i n t e r a c t i o n s between f i e l d i n d e p e n d e n c e and t r e a t m e n t . The D a v i s s t u d y h y p o t h e s i z e d t h a t s t u d e n t s would p e r f o r m b e s t on t h e c o n -c e p t i d e n t i f i c a t i o n t a s k f o r w h i c h t h e s o l u t i o n r e q u i r e d a s t r a t e g y most c l o s e l y matched t o t h e s t u d e n t s c o g n i t i v e s t y l e . The H e s t e r and T a g a t z s t u d y h y p o t h e s i z e d t h a t s t u d e n t s w o u l d p e r f o r m b e s t on t h e c o n c e p t i d e n t i -f i c a t i o n t a s k when t h e t r a i n i n g p r o c e d u r e matched t h e s t u d e n t ' s c o g n i t i v e s t y l e . I n t h e D a v i s s t u d y t h e h y p o t h e s i z e d outcome was t r u e o n l y f o r t h e a n a l y t i c s t u d e n t and i n t h e H e s t e r and T a g a t z s t u d y t h e r e v e r s e outcome was o b s e r v e d . D i s c u s s i o n o f t h e L i t e r a t u r e The s t u d e n t i n an i n d i v i d u a l i z e d program i n m a t h e m a t i c s p r o c e e d s a t h i s own p a c e , u s i n g a v a r i e t y of a u d i o - v i s u a l d e v i c e s and p r e - p a c k a g e d i n s t r u c t i o n a l m a t e r i a l s , t h r o u g h a d e t a i l e d h i e r a r c h i c a l model o f mathe-m a t i c s c o n t e n t . I n s t r u c t i o n a l s t r a t e g i e s a r e n o t a d a p t e d t o i n d i v i d u a l l e a r n i n g d i f f e r e n c e s o f t h e c h i l d , b e c a u s e as Gagne' Cronbach and J e n s e n have p o i n t e d o u t , v e r y l i t t l e i s known a t p r e s e n t about i n d i v i d u a l d i f -f e r e n c e s i n l e a r n i n g r e l e v a n t t o an e d u c a t i o n a l s e t t i n g . T y l e r has s u g -g e s t e d , t h a t u n t i l a c o h e r e n t t h e o r y o f i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g i s d e v e l o p e d , s u c c e s s e s i n t h e a r e a o f i n d i v i d u a l i z e d i n s t r u c t i o n w i l l be l e s s s t r i k i n g t h a n a n t i c i p a t e d . O n l y a few i s o l a t e d s t u d i e s o f i n t e r a c t i o n between a p t i t u d e and i n s t r u c t i o n a l s t r a t e g y i n m a t h e m a t i c s were f o u n d . Two o f t h e s e r e p o r t e d s i g n i f i c a n t i n t e r a c t i o n s . I n b o t h o f t h e s e s t u d i e s , the a p t i t u d e v a r i a b l e was r e a s o n i n g measure and t h e i n s t r u c t i o n a l s t r a t e g i e s were p r e s e n t e d i n programmed b o o k l e t s . However, t h e d e a r t h o f s t u d i e s i n t h i s a r e a , makes i t i m p o s s i b l e t o draw any c o n c l u s i o n s c o n c e r n i n g r e l e v a n t a p t i t u d e v a r i a b l e s t o m a t h e m a t i c s i n s t r u c t i o n . F i v e s t u d i e s u s i n g f i e l d i n d e p e n d e n c e as an a p t i t u d e v a r i a b l e were p r e s e n t e d . T h r e e o f t h e s e s t u d i e s , D a v i s , H e s t e r and T a g a t z , and D a v i s and G r i e v e , p r o v i d e p a r t i a l s u p p o r t f o r t h e h y p o t h e s i z e d outcomes o f t h i s s t u d y : s p e c i f i c a l l y t h a t a s t u d e n t ' s p e r f o r m a n c e w i l l be b e s t on t h e i n s t r u c t i o n a l s t r a t e g y w h i c h i s most c l o s e l y matched t o h i s c o g n i t i v e s t y l e . I n c o n c l u s i o n , t h e l i t e r a t u r e i n d i c a t e s t h e need f o r e x t e n s i v e r e s e a r c h b o t h i n t h e a r e a o f i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g and i n t h e a r e a o f a d a p t a t i o n o f i n s t r u c t i o n a l s t r a t e g i e s t o t h e s e i n d i v i d u a l d i f f e r e n c e s . The r e v i e w o f t h e l i t e r a t u r e i n m a t h e m a t i c s e d u c a t i o n p r o -v i d e d no g u i d a n c e as t o an a p p r o p r i a t e a p t i t u d e v a r i a b l e t o be c o n s i d e r e d i n i n s t r u c t i o n i n m a t h e m a t i c s . The l i t e r a t u r e d i d i n d i c a t e t h a t W i t k i n ' s f i e l d - i n d e p e n d e n c e c o n s t r u c t i s a p o t e n t i a l r e l e v a n t a p t i t u d e v a r i a b l e i n an e d u c a t i o n a l s e t t i n g . However, o f t h e f i v e s t u d i e s w h i c h u t i l i z e d f i e l d i n d e p e n d e n c e as an a p t i t u d e v a r i a b l e i n an e d u c a t i o n a l s e t t i n g , t h r e e o f t h e s t u d i e s used a c o n c e p t i d e n t i f i c a t i o n t a s k as t h e t r e a t m e n t v a r i a b l e and a f o u r t h used a p r o b l e m - s o l v i n g t a s k as t h e t r e a t m e n t v a r i a b l e . O n l y one, t h e s t u d y by D a v i s and G r i e v e , used an i n s t r u c t i o n a l s e t t i n g as t h e t r e a t m e n t v a r i a b l e . Thus, w h i l e t h e need f o r r e s e a r c h i n t o t h e i n t e r a c t i o n between a p t i t u d e v a r i a b l e s and i n s t r u c t i o n a l s t r a t e g i e s has been r e c o g n i z e d f o r t h e p a s t d e cade, v e r y few s t u d i e s o f t h i s n a t u r e have been c o n d u c t e d i n a c l a s s r o o m i n s t r u c t i o n a l s e t t i n g . CHAPTER I I I DESIGN AND PROCEDURE INTRODUCTION The s t u d y was c a r r i e d o ut c o n c u r r e n t l y w i t h one c o n d u c t e d by M a r i a n W e i n s t e i n , a d o c t o r a l c a n d i d a t e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . The W e i n s t e i n s t u d y i n v e s t i g a t e d t h e e f f e c t s o f f o u r i n s t r u c t i o n a l s t r a t e g i e s i n t h e t e a c h i n g o f s i m p l e and complex a l g o r i t h m s a t t h e g r a d e f i v e l e v e l . The i n s t r u c t i o n a l s t r a t e g i e s were: (1) an a l g e b r a i c j u s t i f i c a t i o n a p p r o a c h , (2) a p a t t e r n j u s t i f i c a t i o n a p p r o a c h , (3) a mixed a p p r o a c h , p a t t e r n f o l l o w e d by a l g e b r a i c , and (4) a mix e d a p p r o a c h , a l g e b r a i c f o l l o w e d by p a t t e r n . The p r e s e n t s t u d y i n v e s t i g a t e d t h e i n t e r a c t i o n e f f e c t between two o f t h e s e a p -p r o a c h e s , p a t t e r n and a l g e b r a i c , and t h e c h i l d ' s d e g r e e o f f i e l d i n d e p e n -dence. A l l i n s t r u c t i o n a l m a t e r i a l s , t e s t s and c o v a r i a t e s a r e t h o s e o f t h e W e i n s t e i n s t u d y . P o p u l a t i o n The p o p u l a t i o n c o n s i s t e d o f t w e l v e g r a d e f i v e c l a s s e s i n s i x s c h o o l s i n a l o w e r m a i n l a n d s c h o o l d i s t r i c t i n B r i t i s h C o l u m b i a w h i c h were a l s o p a r t i c i p a t i n g i n t h e W e i n s t e i n s t u d y . These c l a s s e s had been a s s i g n e d a t random t o one o f f o u r p o s s i b l e t r e a t m e n t s ; (1) a p a t t e r n a p p r o a c h on s i m p l e and complex a l g o r i t h m s 1, (2) an a l g e b r a i c a p p r o a c h on s i m p l e and complex a l g o r i t h m s 1, (3) a p a t t e r n a p p r o a c h on s i m p l e and complex a l g o r i t h m s 2, and (4) an a l g e b r a i c a p p r o a c h on s i m p l e and complex a l g o r i t h m s 2. T h e r e were t h r e e c l a s s e s i n each o f t h e f o u r t r e a t m e n t s . 27 The grade f i v e l e v e l was chosen because t h e s t u d e n t s a r e u n f a m i l i a r w i t h t h e a l g o r i t h m s u s e d , y e t p o s s e s s t h e n e c e s s a r y p r e r e q u i s i t e s f o r t h e l e a r n i n g o f t h e a l g o r i t h m s . Sample U s i n g a t a b l e o f random numbers, a random sample o f o n e - h a l f o f t h e s t u d e n t s who c o m p l e t e d t h e W e i n s t e i n s t u d y i n each o f t h e t w e l v e c l a s s e s was s e l e c t e d t o p a r t i c i p a t e i n t h e s t u d y . These s t u d e n t s were t h e n t e s t e d on t h e c o g n i t i v e s t y l e f a c t o r . INSTRUCTIONAL MATERIALS F o u r a l g o r i t h m s , two c l a s s i f i e d as s i m p l e and two c l a s s i f i e d as complex, formed t h e b a s i s o f t h e i n s t r u c t i o n a l u n i t s . F o r each a l g o r i t h m , two d i f f e r e n t i n s t r u c t i o n a l s t r a t e g i e s were u s e d ; one c a l l e d a p a t t e r n i n s t r u c t i o n a l s t r a t e g y and t h e o t h e r an a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y . The p a t t e r n i n s t r u c t i o n a l s t r a t e g y used d i a g r a m s e x t e n s i v e l y , whereas t h e a l g e b r a i c a p p r o a c h r e l i e d h e a v i l y on renaming and t h e a s s o -c i a t i v e , d i s t r i b u t i v e and commutative f i e l d p r o p e r t i e s . Diagrams were n e v e r used i n t h e a l g e b r a i c a p p r o a c h and c o n v e r s e l y , renaming o r t h e f i e l d p r o p e r t i e s were n e v e r u s e d i n t h e p a t t e r n a p p r o a c h . The i n s t r u c t i o n a l u n i t s were d i v i d e d i n t o s t a g e s w i t h p r a c t i s e f o r t h e s t u d e n t a t each s t a g e . The o b j e c t i v e s f o r each s t a g e were s t a t e d f o r t h e t e a c h e r as w e l l as t h e s u g g e s t e d c o m p l e t i o n t i m e . The s i m p l e a l g o r i t h m s r e q u i r e d f i v e i n s t r u c t i o n a l p e r i o d s , w h i l e t h e complex a l g o -r i t h m s r e q u i r e d n i n e i n s t r u c t i o n a l p e r i o d s . W o r k s h e e t s were a l s o p r o -v i d e d w i t h s p e c i f i c i n s t r u c t i o n s as t o when t h e y were t o be u s e d . A 28 d e s c r i p t i o n o f each o f t h e u n i t s f o r t h e f o u r a l g o r i t h m s f o l l o w s . S I : P r o d u c t o f a M i x e d Number and a F r a c t i o n S t a g e 1: (a) whole number and a u n i t f r a c t i o n ( l / n ) (b) whole number and a p r o p e r f r a c t i o n ( c ) w h o le number and a mixed number S t a g e 2: (a) u n i t f r a c t i o n and a u n i t f r a c t i o n (b) p r o p e r f r a c t i o n and a p r o p e r f r a c t i o n (c) p r o p e r f r a c t i o n and a mixed number P a t t e r n A p p r o a c h . A t a l l s t a g e s , f i n d i n g t h e a r e a s o f r e c t a n g l e s i s t h e p h y s i c a l a n a l o g y used t o j u s t i f y t h e a l g o r i t h m . S t a g e 1 ( c ) used t h e p r i n c i p l e o f c o n s e r v a t i o n o f a r e a by p a r t i t i o n i n g r e c t a n g l e s . See A p p e n d i x A f o r t h e development o f s t a g e s 2(a) and 2 ( b ) . S t a g e 2 ( c ) i s d e v e l o p e d by p a r t i t i o n i n g r e c t a n g l e s . A l g e b r a i c A p p r o a c h . S t a g e s 1(a) and 1(b) made use o f t h e r e p e a t e d a d d i t i o n model f o r m u l t i p l i c a t i o n ^ x 1/2 = 1/2 + ... + 1/2). S t a g e 1 ( c ) u t i l i z e d r enaming a mixed number as a wh o l e number p l u s a f r a c t i o n and t h e n used t h e d i s t r i b u t i v e p r i n c i p l e . See A p p e n d i x A f o r t h e development of s t a g e s 2 ( a ) and 2 ( b ) . S t a g e 2 ( c ) was d e v e l o p e d t h r o u g h renaming and t h e d i s t r i b u t i v e p r i n c i p l e . S2: Comparison o f F r a c t i o n s U s i n g t h e C r o s s - P r o d u c t S t a g e 1: (a) comparing a f r a c t i o n w i t h 1 (b) comparing a f r a c t i o n w i t h w h ole numbers St a g e 2: (a) g e n e r a t i n g e q u i v a l e n t f r a c t i o n s w i t h t h e common d e n o m i n a t o r t h e p r o d u c t o f t h e two d e n o m i n a t o r s (b) c r o s s - p r o d u c t r u l e : a/b > c/d i f a x d > b x c 29 P a t t e r n A p p roach. S t a g e s 1(a) and 1(b) were d e v e l o p e d by renaming the w h o le numbers v i a d i a g r a m s . S t a g e 2 ( a ) was d e v e l o p e d t h r o u g h d i a g r a m s by c u t t i n g a l l p i e c e s i n t h e o r i g i n a l d i a g r a m i n t h e same manner. The s t u d e n t was t a u g h t t h a t i f he wanted t o compare f o r example t h i r d s and f o u r t h s , he would c u t each o f t h e t h i r d s i n t o f o u r t h s and each of t h e f o u r t h s i n t h i r d s g i v i n g t w e l f t h s i n each d i a g r a m . See A p p e n d i x A f o r t h e development o f s t a g e 2(b) . A l g e b r a i c A p p r o a c h . S t a g e 1(a) was d e v e l o p e d by c h o o s i n g an a p p r o -p r i a t e e q u i v a l e n t name f o r 1 and t h e n r e l y i n g on a m u l t i p l i c a t i o n a r g u -ment. S t a g e 1(b) i n v o l v e d r e naming t h e whole number as t h e whole number x 1 and t h e n c h o o s i n g an a p p r o p r i a t e name f o r 1. S t a g e 2 ( a ) was d e v e l o p e d by w r i t i n g t h e two f r a c t i o n s t o be compared as m u l t i p l i c a t i o n s t a t e m e n t s and t h e n m u l t i p l y i n g by an a p p r o p r i a t e e q u i v a l e n t f o r m o f 1. See A p p e n d i x A f o r t h e development o f s t a g e 2 ( b ) . C l : C hanging a F r a c t i o n t o a D e c i m a l S t a g e 1: P r e r e q u i s i t e s (a) d e c i m a l s y s t e m (b) d i v i s i o n o f d e c i m a l s by whole numbers (c) i n t e r p r e t a t i o n o f a f r a c t i o n as d i v i s i o n ( a / b = a - f - b ) St a g e 2: The A l g o r i t h m (a) t e r m i n a t i n g d e c i m a l s (b) n o n - t e r m i n a t i n g d e c i m a l s The two approaches d i f f e r e d i n t h e i r development o f s t a g e 1 y e t were i d e n t i c a l i n t h e development o f s t a g e 2. 30 Pattern Approach. Stage 1(a) is developed through the use of rectangles divided into tenths and hundredths. See Appendix A for the development of stages 1(b) and 1(c). Algebraic Approach. Stage 1(a) was developed by the renaming of ones, tenths, and hundredths(1 = 10 x 1/10; 1/10 = 10 x 1/100). See Appendix A for the development of stages 1(b) and 1(c). C2: Finding the Square Root of a Fraction Stage 1: Prerequisites (a) multiplication of fractions (b) concept of the square root of a whole number (c) square root of a fraction as /numerator/^/denominator Stage 2: The Algorithm (a) division technique for the square root of wholes (b) division technique for the square root of fractions (c) approximating square roots For each of the approaches, stage 1(a) was developed in the same manner as stages 2(a) and 2(b) of the corresponding approach in algo-rithm SI. See Appendix A. Stage 2(b) is a drawing together of stages 1(c) and 2(a). Pattern Approach. Stage 1(b) was presented as the factors of a number which w i l l produce a square with area that number. Stage 1(c) was developed by dividing a 1 x 1 square into an equal number of parts, each part being 1/denominator of the area. The number of these parts s needed t o make up t h e num e r a t o r were s e l e c t e d and a s q u a r e was made. St a g e 2(a) was j u s t i f i e d by u s i n g an a r e a o f r e c t a n g l e s argument. A p r o -c e s s o f s q u e e z i n g between w i d t h and h e i g h t o f t h e r e c t a n g l e s was demon-s t r a t e d . See A p p e n d i x A f o r t h e development o f s t a g e 2 ( c ) . A l g e b r a i c A p p r o a c h . I n s t a g e 1(b) t h e s q u a r e r o o t was i m m e d i a t e l y d e f i n e d as t h a t number w h i c h when m u l t i p l i e d by i t s e l f g i v e s t h e w h o l e number i n q u e s t i o n . The s q u a r e r o o t i s f o u n d by l i s t i n g a l l t h e f a c t s a s s o c i a t e d w i t h t h a t w h ole number. S t a g e 1 ( c ) was d e v e l o p e d t h r o u g h a renaming o f b o t h w h o l e s and f r a c t i o n s . S t a g e 2 ( a ) used a c l o s i n g - i n a r -gument. The s t u d e n t was t a u g h t t h a t as one f a c t o r g e t s l a r g e r , i t s c o r -r e s p o n d i n g f a c t o r g e t s s m a l l e r . See A p p e n d i x A f o r t h e development o f s t a g e 2 ( c ) . MEASURING INSTRUMENTS C r i t e r i o n P r e t e s t s "Each o f t h e c r i t e r i o n p r e t e s t s f o r t h e f o u r a l g o r i t h m s c o n s i s t e d of f r e e r e s p o n s e i t e m s d e s i g n e d t o t e s t knowledge o f t h a t a l g o r i t h m ' s p r e -r e q u i s i t e s as d e t e r m i n e d by a p a n e l o f m a t h e m a t i c s e d u c a t i o n j u d g e s . The p r e t e s t s were g i v e n i n o r d e r t o a d j u s t c r i t e r i o n s c o r e s f o r d i f f e r e n c e s among c l a s s e s i n ' r e a d i n e s s ' f o r t h e i n s t r u c t i o n a l m a t e r i a l . " ^ A b r e a k -down o f i t e m s o f t h e f o u r p r e t e s t s as w e l l as t h e KR-20 r e l i a b i l i t y c o e f -f i c i e n t s i s g i v e n i n T a b l e 1 b e l o w . H f a r i a n S. W e i n s t e i n , "A Study o f t h e Types o f A l g o r i t h m J u s t i f i c a -t i o n i n E l e m e n t a r y S c h o o l M a t h e m a t i c s " ( U n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1972) 23. T a b l e 1 D e s c r i p t i o n o f t h e I t e m s a n d K R - 2 0 R e l i a b i l i t y C o e f f i c i e n t s f o r t h e F o u r P r e t e s t s P r e t e s t s T y p e o f I t e m S I S 2 C l C 2 M u l t i p l i c a t i o n f a c t s 1 0 1 0 1 0 D i a g r a m m a t i c r e p r e s e n t a t i o n o f f r a c t i o n s 5 5 5 5 R e c i p r o c a l s 3 3 5 2 F r a c t i o n s a s m u l t i p l i c a t i o n s t a t e m e n t s 3 3 3 1 a s t h e m u l t i p l i c a t i v e i d e n t i t y 4 2 3 C o m m u t a t i v i t y a n d a s s o c i a t i v i t y 3 3 D i s t r i b u t i v e l a w 3 2 A r e a f o r m u l a 5 5 L o n g d i v i s i o n a l g o r i t h m 5 5 R e w r i t i n g a m i x e d n u m b e r a s a s u m 4 C o n s e r v a t i o n o f a r e a 2 C o m p a r i n g f r a c t i o n s u s i n g d i a g r a m s 5 F r a c t i o n a l n a m e s f o r 1 3 D i v i s i o n a s s h a r i n g 5 R e v e r s i b i l i t y o f m u l t i p l i c a t i o n a n d d i v i s i o n 5 P r e s e r v a t i o n o f i n e q u a l i t i e s w h e n m u l t i p l i e d b y a p o s i t i v e 3 T o t a l 4 2 3 2 3 0 3 5 K R - 2 0 R e l i a b i l i t y C o e f f i c i e n t s . 9 0 . 8 4 . 9 0 . 9 0 33 C r i t e r i o n C o m p u t a t i o n T e s t s "Each o f t h e c o m p u t a t i o n t e s t s f o r t h e f o u r a l g o r i t h m s c o n s i s t e d o f f r e e r e s p o n s e i t e m s d e s i g n e d t o t e s t t h e s t u d e n t " s a b i l i t y t o p e r f o r m t h a t 2 a l g o r i t h m . " D e s c r i p t i o n s o f t h e i t e m s o f t h e f o u r t e s t s as w e l l as t h e KR-20 r e l i a b i l i t y c o e f f i c i e n t s a r e g i v e n b e l o w i n T a b l e s 2, 3, 4 and 5. T a b l e 2 Types o f Items o f S I , P r o d u c t o f a F r a c t i o n and a M i x e d Number, C o m p u t a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t I t e m Type Number o f Items F r a c t i o n x whole number 6 M i x e d x w h o l e number 6 F r a c t i o n x f r a c t i o n 6 Mi x e d x f r a c t i o n 6 T o t a l 24 KR-20 .94 T a b l e 3 Types o f Items o f S2, Comparison o f F r a c t i o n s , C o m p u t a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t I t e m Type Number o f Items F r a c t i o n and a whole number 9 2 p r o p e r f r a c t i o n s 9 2 i m p r o p e r f r a c t i o n s 9 T o t a l 27 KR-20 .95 2 I b i d p.24 34 Table 4 Types of Items of Cl, Changing a Fraction to a Decimal, Computation Test and KR-20 Reliability Coefficient Item Type Number of Items Converting fractions to terminating decimals 8 Approximating the decimal equivalent of fractions 10 Total 18 KR-20 .90 Table 5 Types of Items of C2, Finding the Square Root of a Fraction, Computation Test and KR-20 Reliability Coefficient Item Type Number of Items Square root of a perfect square fraction 8 Square root of a non-perfect square fraction 7 Total 15 KR-20 .90 35 C r i t e r i o n G e n e r a l i z a t i o n T e s t s "Each o f t h e g e n e r a l i z a t i o n t e s t s f o r t h e f o u r a l g o r i t h m s c o n s i s t e d o f 30 f r e e r e s p o n s e i t e m s . The t e s t s were bas e d on f i v e t y p e s o f i t e m s — t h o s e d e s i g n e d t o measure: t h e a b i l i t y t o s h o r t c u t t h e a l g o r i t h m b e c a u s e o f t h e numbers i n v o l v e d ( T y p e A ) , t h e a b i l i t y t o d e f i n e t h e en-t i r e p r o b l e m when g i v e n t h e s o l u t i o n and a p a r t o f t h e p r o b l e m ( T y p e B ) , t h e a b i l i t y t o e x t e n d t h e a l g o r i t h m t o more t h a n two o p e r a n d s ( T y p e C ) , t h e a b i l i t y t o use t h e a l g o r i t h m w i t h numbers o t h e r t h a n t h e t y p e s t u d i e d ( T y p e D ) , and t h e a b i l i t y t o e x p l a i n an a l t e r n a t e a p p r o a c h t o th e a l g o r i t h m ( T y p e E ) . " 3 The t y p e s o f i t e m s o f t h e f o u r t e s t s as w e l l as t h e KR-20 r e l i a b i l i t y c o e f f i c i e n t s a r e r e p o r t e d i n T a b l e s 6, 7 , 8 and 9 b e l o w . T a b l e 6 Types o f Items o f S I , P r o d u c t o f a F r a c t i o n and a M i x e d Number, G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t I t e m Type Number o f Items A 8 B 6 C 5 D 4 B - C 3 E 4 T o t a l 30 KR-20 .83 3 I b i d . p.25. 36 T a b l e 7 Types o f Items o f S2, Comparison o f F r a c t i o n s , G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t I t e m Type Number o f Items A 8 B 6 C 8 D 4 E 4 T o t a l 30 KR-20 .82 T a b l e 8 Types o f Items o f C l , Changing a F r a c t i o n t o a D e c i m a l , G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t I t e m Type Number o f Items A 8 B 9 D 9 E 4 T o t a l 30 KR-20 .80 37 T a b l e 9 Types o f Items o f C2, F i n d i n g t h e Square Root o f a F r a c t i o n , G e n e r a l i z a t i o n T e s t and KR-20 R e l i a b i l i t y C o e f f i c i e n t I t e m Type Number o f Items A 8 B 9 D 9 E 4 T o t a l 30 KR-20 .88 T e s t i n g on C o g n i t i v e S t y l e F a c t o r The C h i l d r e n ' s Embedded F i g u r e s T e s t i s a m o d i f i e d v e r s i o n o f W i t -k i n ' s Embedded F i g u r e s T e s t . The t w e n t y - f i v e i t e m t e s t c o n s i s t s o f e l e v e n p i c t u r e s o f complex f i g u r e s i n w h i c h a t r i a n g u l a r shape i s embedded and f o u r t e e n p i c t u r e s i n w h i c h a house-shaped form i s embedded. The t e s t i s a m o d i f i c a t i o n by K a r p and K o n s t a d t o f t h e C h i l d r e n ' s Embedded F i g u r e s T e s t d e v e l o p e d by Goodenough and E a g l e i n 1963. Good-enough and E a g l e ' s t e s t , a l t h o u g h i t has h i g h r e l i a b i l i t y was c o n s i d e r e d t o o 4 b u l k y and c o s t l y f o r use i n t h i s s t u d y . The Embedded F i g u r e s T e s t was found t o be t o o d i f f i c u l t and f r u s -t r a t i n g f o r c h i l d r e n aged t e n and und e r and t o r e q u i r e m o d i f i c a t i o n s i n _ -Herman A. W i t k i n e t a l . , A Manual f o r t h e Embedded F i g u r e s T e s t s ( P a l o A l t o , C a l i f o r n i a : C o n s u l t i n g P s y c h o l o g i s t s P r e s s , 1971) pp.22 - 29. administration for c h i l d r e n i n the ten to eleven age bracket. The C h i l -dren's Embedded Figures Test reduced the d i f f i c u l t y and f r u s t r a t i o n of the Embedded Figures Test through the use of simple and complex forms more f a m i l i a r to the c h i l d and through the e l i m i n a t i o n of the pressure of a time l i m i t . The c h i l d i s also given more than one opportunity to locate the f i g u r e , although only f i r s t t r i e s are a c t u a l l y used f o r scoring pur-poses . The test was standardized using one hundred and s i x t y c h i l d r e n , ranging i n age from f i v e to twelve years, randomly selected from elemen-tary schools i n Brooklyn. "Because of the small N's involved, these nor-mative data can be considered only t e n t a t i v e . " ^ R e l i a b i l i t y estimates f o r c h i l d r e n i n the age groups 9-10 and 11-12 are recorded i n Table 10 below. The c h i l d r e n i n the present study were i n the 10-11 age bracket. Table 10 CEFT R e l i a b i l i t y Estimates and V a l i d i t y C o e f f i c i e n t s Age N . I n t e r n a l Consistency r r CEFT, EFT 9-10 11-12 40 40 .88 .87 .71 .85 Sources: Internal Consistency r : A Manual For The Embedded Figures Tests, 1971, Table 4, p.25. r CEFT, EFT: A Manual For The Embedded Figures Tests,1971, Table 5, p.25. ^ I b i d . p.17. 6 I b i d . p.24. 39 PROCEDURE P r i o r t o t h e a c t u a l b e g i n n i n g o f t h e s t u d y i n t h e c l a s s r o o m s , a m e e t i n g was h e l d between t h e r e s e a r c h e r s ( W e i n s t e i n and O ' B r i e n ) and t h e p a r t i c i p a t i n g t e a c h e r s . A t t h i s t i m e , t h e two b a s i c a p p r o a c h e s were o u t -l i n e d and t h e n e c e s s i t y o f f o l l o w i n g t h e m a t e r i a l s c a r e f u l l y was empha-s i z e d . A G e n e r a l I n f o r m a t i o n Sheet was g i v e n t o t h e t e a c h e r s . T h i s s h e e t i n c l u d e d r e m i n d e r s on do's and d o n ' t ' s f o r u s i n g t h e m a t e r i a l s ( a l r e a d y d i s c u s s e d a t t h e m e e t i n g ) and t h e home phone numbers o f t h e r e s e a r c h e r s . An i n d i v i d u a l c l a s s f o l l o w e d e i t h e r an a l g e b r a i c o r p a t t e r n ap-p r o a c h t h r o u g h o u t S I f o l l o w e d by C l o r S2 f o l l o w e d by C2. T h e r e were no mixed sequences o f a l g o r i t h m s . A f l o w c h a r t o f t h e g e n e r a l p r o c e d u r e i s c o n t a i n e d i n F i g u r e 1. F i g u r e 1 F l o w C h a r t o f t h e P r o c e d u r e P r e t e s t on s i m p l e a l g o r i t h m ^ i n s t r u c t i o n on s i m p l e a l g o r i t h m ^ c o m p u t a t i o n and g e n e r a l i z a t i o n t e s t s on s i m p l e a l g o r i t h m -> p r e t e s t on complex a l g o r i t h m ^ i n s t r u c t i o n on complex a l g o r i t h m ^ c o m p u t a t i o n and g e n e r a l i z a t i o n t e s t s on complex a l g o r i t h m ^ a d m i n i s t r a t i o n o f t h e C h i l d r e n ' s Embedded F i g u r e s T e s t The c o r r e c t i n g o f a l l t e s t s , p r e t e s t s , c o m p u t a t i o n and g e n e r a l i -z a t i o n t e s t s , was done by t h e two r e s e a r c h e r s . F o l l o w i n g t h e c o m p l e t i o n o f t h e i n s t r u c t i o n a l and t e s t i n g s e -quence on t h e a l g o r i t h m s , t e s t i n g on t h e c o g n i t i v e s t y l e f a c t o r was c o n -AO d u c t e d by t h i s i n v e s t i g a t o r . The t e s t was a d m i n i s t e r e d a c c o r d i n g t o t h e p r o c e d u r e o u t l i n e d i n t h e t e s t m a n u a l . 7 CONTROLS There were t h r e e c o n t r o l s on t e a c h e r v a r i a t i o n . These were: (1) d t a i l e d d a i l y i n s t r u c t i o n a l g u i d e s and w o r k s h e e t s were p r o v i d e d ; (2) t h e r e s e a r c h e r s v i s i t e d each o f t h e t w e l v e t e a c h e r s e v e r y s e c o n d day t o d i s c u s s t h e p r o g r e s s o f t h e s t u d y and (3) a l l t e s t s were c o r r e c t e d by t h e two r e -s e a r c h e r s . Hawthorne e f f e c t was c o n t r o l l e d by: (1) a l l t e a c h i n g and t e s t i n g on t h e i n s t r u c t i o n a l m a t e r i a l s was c o n d u c t e d by t h e c l a s s r o o m t e a c h e r and (2) t h e i n d i v i d u a l t e s t i n g on t h e c o g n i t i v e s t y l e f a c t o r was c a r r i e d out o n l y a f t e r t h e c o m p l e t i o n o f i n s t r u c t i o n I n and t e s t i n g on t h e two a l g o -r i t h m s . A p o s s i b l e " d i f f e r e n c e s i n r a t e r " p r o b l e m i n t e s t i n g on t h e c o g -n i t i v e s t y l e f a c t o r was overcome by t h e r e s e a r c h e r s o l e l y d o i n g t h e t e s t -i n g . Two s t a t i s t i c a l c o n t r o l s were a l s o u s e d . These were: (1) p r e t e s t s c o r e s were used as c o v a r i a t e s t o a d j u s t f o r i n i t i a l d i f f e r e n c e s among s t u d e n t s and (2) t h r e e c l a s s e s were a s s i g n e d t o each o f t h e f o u r p o s s i b l e t r e a t m e n t s t o h e l p i n t h e c o n t r o l o f t e a c h e r v a r i a t i o n . I b i d . pp.26-28. 41 STATISTICAL PROCEDURES Each o f t h e d a t a s e t s o f t h e e i g h t p o s t - t e s t s was a n a l y z e d s e p a -r a t e l y u s i n g m u l t i p l e l i n e a r r e g r e s s i o n t e c h n i q u e s . The computing f a c i l i -t i e s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a were used and a m u l t i p l e l i n e a r r e g r e s s i o n program c o n t a i n e d i n t h e p e r s o n a l f i l e o f Dr. Seong Soo L e e , F a c u l t y o f E d u c a t i o n o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , was u s e d . CHAPTER I V ANALYSIS OF THE DATA The r e a d e r w i l l r e c a l l t h a t two p o s t - t e s t s , a c o m p u t a t i o n t e s t w h i c h measured a b i l i t y t o p e r f o r m t h e a l g o r i t h m and a g e n e r a l i z a t i o n t e s t w h i c h measured a b i l i t y t o s h o r t c u t , e x t e n d and e x p l a i n t h e a l g o r i t h m , were a d m i n i s t e r e d upon c o m p l e t i o n o f i n s t r u c t i o n on each o f t h e f o u r a l g o r i t h m s . The two s i m p l e a l g o r i t h m s were S I , t h e p r o d u c t o f a f r a c t i o n and a mix e d number, and S2, the c o m p a r i s o n o f f r a c t i o n s . C l , c h a n g i n g a f r a c t i o n t o a d e c i m a l , and C2, f i n d i n g t h e s q u a r e r o o t o f a f r a c t i o n were c l a s s i f i e d as complex a l g o r i t h m s . F o r t h e p u r p o s e s o f a n a l y s i s , each o f t h e s e e i g h t p o s t -t e s t s was t r e a t e d as an i n d e p e n d e n t u n i t o f d a t a . M u l t i p l e l i n e a r r e g r e s -s i o n t e c h n i q u e s were used t o t e s t t h e t h r e e n u l l h y p o t h e s e s f o r each d a t a s e t . Where s i g n i f i c a n t i n t e r a c t i o n s were f o u n d , t h e i n t e r a c t i o n s were graphed. The f o l l o w i n g d i s c u s s i o n o f t h e s t a t i s t i c a l t e c h n i q u e a p p l i e s t o each o f t h e e i g h t s e p a r a t e a n a l y s e s o f t h e d a t a . A l i n e a r r e l a t i o n s h i p between t h e dependent v a r i a b l e , s c o r e on t h e p o s t - t e s t , and i n d e p e n d e n t v a r i a b l e s was assumed. The i n d e p e n d e n t v a r i a b l e s were method o f i n s t r u c t i o n , f i e l d i n d e pendence and i n t e r a c t i o n between meth-od o f i n s t r u c t i o n and f i e l d i n d e p e n d e n c e . T h i s r e l a t i o n s h i p may be e x p r e s s e d a s : ( D Y ^ & + B,vlt + 4 v 2 i + s3v3i + ^ v 4 i + where Y = o b s e r v a t i o n o f i t h s u b j e c t on t h e p o s t - t e s t A3 V,. = o b s e r v a t i o n o f i t h s u b j e c t on t h e c o v a r i a t e l i V„. = o b s e r v a t i o n o f i t h s u b j e c t on t h e method o f i n s t r u c t i o n 2 i = o b s e r v a t i o n o f i t h s u b j e c t on t h e f i e l d i n d e p e n d e n c e measure «• o b s e r v a t i o n o f i t h s u b j e c t on t h e i n t e r a c t i o n between method o f i n s t r u c t i o n and f i e l d i n d e p e n d e n c e £ > ••• » 61 a r e t h e p o p u l a t i o n r e g r e s s i o n c o e f f i c i e n t s lH. i s t h e d e p a r t u r e o f from t h e l i n e a r model S m i l l i e s t a t e s : We may use t h e a n a l y s i s o f v a r i a n c e t o t e s t t h e h y p o t h e s i s t h a t t h e l a s t p-k i n d e p e n d e n t v a r i a b l e s , Xk+1, Xk+2, ... , Xp, f o r some k p do n o t make a s i g n i f i c a n t c o n t r i b u t i o n t o t h e r e g r e s s i o n sum o f s q u a r e s , SSR computed f o r a l l p i n d e p e n d e n t v a r i a b l e s . Suppose t h a t t h e r e g r e s -s i o n sum o f s q u a r e s computed f o r t h e r e g r e s s i o n model w i t h o n l y t h e f i r s t k i n d e p e n d e n t v a r i a b l e s i n c l u d e d i s SSR. Then i t may be shown t h a t t h e d i f f e r e n c e SSR - S S R / i s d i s t r i b u t e d as Tt1 w i t h p-k d e g r e e s o f freedom. Thus t h e r a t i o F = (SSR - SSR 7) / (p-k) SSE / (n-p-1) has an F - d i s t r i b u t i o n w i t h p-k and n-p-1 d e g r e e s o f f r e e d o m , and may be used t o t e s t t h e h y p o t h e s i s t h a t 6^ = hH = Skn ... = Sf = 0. 1 R e s t r i c t e d r e g r e s s i o n models were t h u s d e f i n e d i n o r d e r t o t e s t t h e f o l l o w i n g n u l l h y p o t h e s e s . H^: T h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n mean p o s t - t e s t s c o r e s between s t u -d e n t s t a u g h t by a p a t t e r n i n s t r u c t i o n a l s t r a t e g y and s t u d e n t s t a u g h t by an a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y . was t e s t e d by computing t h e sum o f s q u a r e s o f t h e r e g r e s s i o n c o -e f f i c i e n t s f o r the r e g r e s s i o n model: W. S m i l l i e , An I n t r o d u c t i o n t o R e g r e s s i o n and C o r r e l a t i o n ( T o r o n t o : The R y e r s o n P r e s s , 1 9 6 6 ) , pp. A9-50. 44 (2) Y ± = 04 + 0 ( V U + / ? 3 V 3 i + ^ V A i + 4; ( o m i t t i n g . m e t h o d , V^^) and a p p l y i n g f o r m u l a I . H^: 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 i n mean p o s t - t e s t s c o r e s between groups o f s t u d e n t s d i f f e r i n g i n d e g r e e o f f i e l d i n d e p e n d e n c e . was t e s t e d by computing t h e sum o f s q u a r e s o f t h e r e g r e s s i o n c o e f f i c i e n t s f o r t h e r e g r e s s i o n m o d e l : (3) Y ± = 4 + ^ V 1 ± + 6tV2t + ^ V A i + Hi ( o m i t t i n g f i e l d i n d e p e n d e n c e , v 3 ^ ) and a p p l y i n g f o r m u l a I . H^s There i s no s i g n i f i c a n t i n t e r a c t i o n between s t u d e n t s ' d e g r e e o f f i e l d i n d e p e n d e n c e and i n s t r u c t i o n a l s t r a t e g y . was t e s t e d by computing t h e sum o f s q u a r e s o f t h e r e g r e s s i o n c o e f f i c i e n t s f o r t h e r e g r e s s i o n m o d el: (4) Y ± = 30 + /> (V 1 ± + BzVZi + ^ 3 V 3 i + ( t i ( o m i t t i n g i n t e r -a c t i o n , V ^ ) and a p p l y i n g f o r m u l a I . I n a d d i t i o n , f o r each d a t a s e t , t h e s i g n i f i c a n c e o f t h e c o n t r i b u -t i o n o f t h e c o v a r i a t e t o t h e r e g r e s s i o n sum o f s q u a r e s o f t h e l i n e a r mod-e l chosen was t e s t e d by comp u t i n g t h e sum o f s q u a r e s o f t h e r e g r e s s i o n c o e f f i c i e n t s f o r t h e r e g r e s s i o n m o d el: (5) Y ± - B, + d 2 V 2 i + /5 3V 3 i + ^ V A 1 + U.I ( o m i t t i n g c o -v a r i a t e , V ^ ) and a p p l y i n g f o r m u l a I . The e i g h t s e p a r a t e a n a l y s e s o f t h e d a t a a r e r e p o r t e d i n T a b l e s 11-18 b e l o w . The d i s t r i b u t i o n o f s c o r e s on t h e f i e l d i n d e p e n d e n c e mea-45 s u r e a r e c o n t a i n e d i n A p p e n d i x C. The raw d a t a a r e a l s o c o n t a i n e d i n A p p e n d i x C. I n each o f t h e t a b l e s b e l o w , SSR and SSR r e f e r t o t h e sum o f s q u a r e s o f t h e r e g r e s s i o n c o e f f i c i e n t s o f t h e f u l l m o d e l , ( 1 ) , and t h e sum o f s q u a r e s of t h e r e g r e s s i o n c o e f f i c i e n t s o f t h e r e s t r i c t e d m o d e l s , ( 2 ) , ( 3 ) , ( 4 ) , o r ( 5 ) , r e s p e c t i v e l y . The numbers i n each o f t h e s e c o l -umns c o r r e s p o n d t o t h e r e g r e s s i o n models d e f i n e d above. The computer program u s e s n-p as t h e d e g r e e s o f freedom f o r t h e d e n o m i n a t o r as op-2 posed t o n-p-1. S i x t y - f o u r s u b j e c t s t o o k t h e S I and C l p o s t - t e s t s . Thus, i n t h e a n a l y s e s o f S I and C l c o m p u t a t i o n and g e n e r a l i z a t i o n s c o r e s , t h e computed F - v a l u e s have one d e g r e e o f freedom f o r t h e n u m e r a t o r and s i x t y d e g r e e s o f freedom f o r t h e d e n o m i n a t o r . S e v e n t y - s e v e n s u b j e c t s t o o k t h e S2 and C2 p o s t - t e s t s . Thus, i n t h e a n a l y s e s o f S2 and C2 compu-t a t i o n and g e n e r a l i z a t i o n s c o r e s , t h e computed F - v a l u e s have one d e g r e e o f freedom f o r t h e n u m e r a t o r and s e v e n t y - t h r e e d e g r e e s o f freedom f o r t h e d e n o m i n a t o r . T a b l e 11 A n a l y s i s o f S I C o m p u t a t i o n S c o r e s SSR SSR 7 F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e (1) .08228 (5) .05974 1.4737 .22949 o f c o v a r i a t e S i g n i f i c a n c e (1) .08228 (2) .08217 .0071 .93306 o f method S i g n i f i c a n c e (1) .08228 (3) .06194 1.3295 .25346 o f f i e l d i n d e p . S i g n i f i c a n c e (1) .08228 (4) .08228 0.0 1.00000 of i n t e r a c t i o n w i t h l a r g e N's, such as i n t h e sample o f t h i s s t u d y , t h e r e 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 u s i n g n-p and n-p-1. 46 I n t h e a n a l y s i s o f S I C o m p u t a t i o n S c o r e s p r e s e n t e d i n T a b l e 11, the t h r e e n u l l h y p o t h e s e s o f no s i g n i f i c a n t main e f f e c t due t o i n s t r u c -t i o n a l s t r a t e g y , no s i g n i f i c a n t main e f f e c t due t o degree o f f i e l d i n d e -pendence and no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and d e g r e e o f f i e l d i n dependence were n o t r e j e c t e d a t << = .05. The F - v a l u e f o r t h e c o v a r i a t e was s i g n i f i c a n t a t p = .22949. T a b l e 12 A n a l y s i s o f S2 C o m p u t a t i o n S c o r e s SSR SSR' F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e o f c o v a r i a t e (1) .19443 (5) .16895 2.3092 .13292 S i g n i f i c a n c e o f method (1) .19443 (2) .17842 1.4510 .23224 S i g n i f i c a n c e o f f i e l d i n d e p . (1) .19443 (3) .19426 .0151 .90253 S i g n i f i c a n c e o f i n t e r a c t i o n (1) .19443 (A) .19111 .3008 .58509 From T a b l e 12 i t can be seen t h a t i n t h e a n a l y s i s o f S2 Computa-t i o n S c o r e s , t h e t h r e e n u l l h y p o t h e s e s o f no s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n a l s t r a t e g y , no s i g n i f i c a n t main e f f e c t due t o d e g r e e o f f i e l d i n d e p e n d e n c e and no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and d e g r e e o f f i e l d i n dependence were n o t r e j e c t e d a t <>C = .05. The F-v a l u e f o r t h e c o v a r i a t e was s i g n i f i c a n t a t p = .13292. I n t h e a n a l y s i s o f S I G e n e r a l i z a t i o n S c o r e s p r e s e n t e d i n T a b l e 13, t h e t h r e e n u l l h y p o t h e s e s o f no s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n -47 a l s t r a t e g y , no s i g n i f i c a n t main e f f e c t due t o degr e e o f f i e l d i n d e p e n d e n c e and no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and degr e e o f f i e l d i n d e p e n d e n c e were n o t r e j e c t e d a t o<. = .05. The F - v a l u e f o r t h e c o v a r i a t e was s i g n i f i c a n t a t p = .00808. T a b l e 13 A n a l y s i s o f S I G e n e r a l i z a t i o n S c o r e s SSR S S R 7 F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e o f c o v a r i a t e (1) .33100 (5) .24728 7.5087 .00808 S i g n i f i c a n c e o f method (1) .33100 (2) .33053 .0426 .83723 S i g n i f i c a n c e of f i e l d i n d e p . (1) .33100 (3) .31496 1.4389 .23501 S i g n i f i c a n c e (1) .33100 (*) .32617 .4332 .51294 o f i n t e r a c t i o n T a b l e 14 A n a l y s i s o f S2 G e n e r a l i z a t i o n S c o r e s SSR SSR ' F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e of c o v a r i a t e (1) .18560 (5) .09223 8.3700 .00502 S i g n i f i c a n c e o f method (1) .18560 (2) .16575 1.7793 .18639 S i g n i f i c a n c e o f f i e l d i n d e p . (1) .18560 (3) .16591 1.7654 .18809 S i g n i f i c a n c e (1) .18560 (4) .16221 2.0969 .15187 o f i n t e r a c t i o n 48 I n t h e a n a l y s i s o f S2 G e n e r a l i z a t i o n S c o r e s p r e s e n t e d i n T a b l e 14, t h e t h r e e n u l l h y p o t h e s e s o f no s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n -a l s t r a t e g y , no s i g n i f i c a n t main e f f e c t due t o degr e e o f f i e l d i n d e p e n d e n c e and no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and d e g r e e o f f i e l d i n d ependence were n o t r e j e c t e d a t = .05. The F - v a l u e f o r t h e c o v a r i a t e was s i g n i f i c a n t a t p = .00502. T a b l e 15 A n a l y s i s o f C l C o m p u t a t i o n S c o r e s SSR / SSR F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e o f c o v a r i a t e (1) .34542 (5) .15500 17.4544 .00010 S i g n i f i c a n c e of method (1) .34542 (2) .32787 1.6087 .20956 S i g n i f i c a n c e o f f i e l d i n d e p . (1) .34542 (3) .32587 1.7918 .18576 S i g n i f i c a n c e o f i n t e r a c t i o n (1) .34542 (A) .32518 1.8551 .17826 T a b l e 16 A n a l y s i s o f C2 C o m p u t a t i o n S c o r e s SSR SSR 7 F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e o f c o v a r i a t e (1) .29443 (5) .14279 15.681 .00017 S i g n i f i c a n c e o f method (1) .29443 (2) .28271 1.2128 .27440 S i g n i f i c a n c e of f i e l d i n d e p . (1) .29443 (3) .25451 4.1302 .04576 S i g n i f i c a n c e (1) .29443 (4) .28788 .6773 .41321 o f i n t e r a c t i o n A9 From T a b l e 15 i t can be s e e n t h a t i n the a n a l y s i s o f C l C o m p u t a t i o n S c o r e s , the t h r e e n u l l h y p o t h e s e s of no s i g n i f i c a n t main e f f e c t due t o i n -s t r u c t i o n a l s t r a t e g y , no s i g n i f i c a n t main e f f e c t due t o degree o f f i e l d i n -dependence and no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and d e g r e e of f i e l d i n d e p endence were n o t r e j e c t e d a t = .05. The F - v a l u e f o r t h e c o v a r i a t e was s i g n i f i c a n t a t p = .00010. I n t h e a n a l y s i s o f C2 C o m p u t a t i o n S c o r e s p r e s e n t e d i n T a b l e 16, t h e two n u l l h y p o t h e s e s o f no s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n -a l s t r a t e g y and no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and de-g r e e of f i e l d i n d e p e n d e n c e were n o t r e j e c t e d a t «<= .05. The n u l l h y p o t h e -s i s o f no s i g n i f i c a n t main e f f e c t due t o d e g r e e o f f i e l d i n d e p e n d e n c e was r e j e c t e d a t «C = .05. The F - v a l u e f o r t h e c o v a r i a t e was s i g n i f i c a n t a t p = .00017. T a b l e 17 A n a l y s i s o f C l G e n e r a l i z a t i o n S c o r e s SSR SSR / F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e o f c o v a r i a t e (1) .51876 (5) .3AA71 21.6996 .00002 S i g n i f i c a n c e of method (1) .51876 (2) •A8083 A.7291 .03361 S i g n i f i c a n c e o f f i e l d i n d e p . (1) .51876 (3) .A812A A.6782 .03A5A S i g n i f i c a n c e (1) .51876 (A) .A6123 7.1725 .0095A of i n t e r a c t i o n 50 I n t h e a n a l y s i s o f C l G e n e r a l i z a t i o n S c o r e s p r e s e n t e d i n T a b l e 17, the t h r e e n u l l h y p o t h e s e s o f no s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n -a l s t r a t e g y , no s i g n i f i c a n t main e f f e c t due t o d e g r e e o f f i e l d i n d e p endence and no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and d e g r e e o f f i e l d i n d ependence were r e j e c t e d a t << = .05. The F - v a l u e f o r the c o v a r i a t e was s i g n i f i c a n t a t p = .00002. T a b l e 18 A n a l y s i s o f C2 G e n e r a l i z a t i o n S c o r e s SSR SSR' F - V a l u e P r o b a b i l i t y S i g n i f i c a n c e o f c o v a r i a t e (1) .63736 (5) .21713 84. 5927 .00000 S i g n i f i c a n c e of method (1) .63736 (2) .62134 3. 2247 .07668 S i g n i f i c a n c e of f i e l d i n d e p . (1) .63736 (3) .62623 2. 2401 .13880 S i g n i f i c a n c e o f i n t e r a c t i o n (1) .63736 (A) .60991 5. 5250 .02145 I n t h e a n a l y s i s o f C2 G e n e r a l i z a t i o n S c o r e s p r e s e n t e d i n T a b l e 18, the n u l l h y p o t h e s e s o f no s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n a l s t r a t e g y and no s i g n i f i c a n t main e f f e c t due t o d e g r e e o f f i e l d i n d e p e n d e n c e were n o t r e j e c t e d a t oL = .05. The n u l l h y p o t h e s i s of no s i g n i f i c a n t i n t e r -a c t i o n between method and d e g r e e o f f i e l d i n d e p endence was r e j e c t e d a t <£= .05. The F - v a l u e f o r t h e c o v a r i a t e was s i g n i f i c a n t a t p = .00000. 51 G r a p h i n g of S i g n i f i c a n t R e s u l t s S i n c e t h e n u l l h y p o t h e s i s o f no s i g n i f i c a n t main e f f e c t due t o de-gre e o f f i e l d i n d e pendence was r e j e c t e d i n t h e a n a l y s i s o f C2 c o m p u t a t i o n s c o r e s and s i n c e t he n u l l h y p o t h e s i s o f no s i g n i f i c a n t i n t e r a c t i o n e f f e c t between method and f i e l d i n d e pendence was r e j e c t e d i n t h e a n a l y s i s o f C l and C2 g e n e r a l i z a t i o n s c o r e s , t h e r e s u l t s o f t h e C2 c o m p u t a t i o n t e s t and th e C l and C2 g e n e r a l i z a t i o n t e s t s were graphed t o a i d i n t h e i n t e r p r e t a -t i o n o f t h e r e s u l t s . 3 The UBC computer program CGROUP was used t o i d e n t i f y n a t u r a l r a n g e g r o u p i n g s on t h e f i e l d i n d e p e n d e n c e measure. CGROUP o p t i m a l l y c l u s t e r e d s u b j e c t s by m a t c h i n g s u b j e c t s on f i e l d i n d e p e n d e n c e s c o r e s and dependent v a r i a b l e s c o r e s . i n s u c h a way as t o m i n i m i z e v a r i a t i o n w i t h i n t h e c r e a t e d g r o u p s . The s c o r e s on t h e f i e l d i n d e p e n d e n c e measure r a n g e d f r o m 9-25. F i v e o p t i m a l g r o u p i n g s were i d e n t i f i e d by CGROUP. These were: (1) 9-15, (2) 16-18, (3) 19-20, (A) 21-22 and (5) 23-25. The C2 c o m p u t a t i o n s c o r e s were a d j u s t e d u s i n g t h e C2 p r e t e s t as a c o v a r i a t e , and t h e C l and C2 g e n e r a l i z a t i o n s c o r e s were a d j u s t e d u s i n g t h e C l and C2 p r e t e s t s as c o v a r i a t e s . The means o f t h e r e s u l t i n g r e s i d u a l s c o r e s were c a l c u l a t e d f o r each o f t h e f i v e r ange groups on b o t h t h e p a t -t e r n and a l g e b r a i c a p p r o a c h e s . The graphs o f t h e s e means f o l l o w . The means and t h e number o f o b s e r v a t i o n s f o r each group a r e c o n t a i n e d i n Ap-p e n d i x C. T h i s program i s on f i l e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a Computing C e n t r e . 52 D i s c u s s i o n o f t h e F i g u r e s The r e a d e r s h o u l d b e a r i n mind t h a t t h e g r a p h s o f t h e r e s u l t s do not p r o v i d e s t a t i s t i c a l s u p p o r t on w h i c h t o base c o n c l u s i o n s , b u t r a t h e r s e r v e o n l y as an a i d i n i n t e r p r e t i n g s i g n i f i c a n t r e s u l t s i n d i c a t e d by t h e s t a t i s t i c a l a n a l y s i s o f t h e d a t a . The r e a d e r s h o u l d f u r t h e r keep i n mind t h a t t h e group s i z e s were u n e q u a l . The a n a l y s i s o f C2 c o m p u t a t i o n s c o r e s i n d i c a t e d a s i g n i f i c a n t main e f f e c t due t o d e g r e e o f f i e l d i n d e p e n d e n c e , b u t no s i g n i f i c a n t main e f f e c t s due t o method o r i n t e r a c t i o n between method and d e g r e e o f f i e l d i n d e p e n d e n c e . F i g u r e 2 i n d i c a t e s t h a t on b o t h t h e p a t t e r n and a l g e b r a i c a p p r o a c h e s , e x -c e p t f o r t h e extreme f i e l d i n d e p e n d e n t group on t h e p a t t e r n a p p r o a c h , w i t h an i n c r e a s e i n d e g r e e o f f i e l d i n d e p e n d e n c e t h e r e i s a c o r r e s p o n d i n g i n -c r e a s e i n l e v e l o f a c h i e v e m e n t . The a n a l y s i s o f C l g e n e r a l i z a t i o n s c o r e s i n d i c a t e d a s i g n i f i c a n t main e f f e c t due t o method, a s i g n i f i c a n t main e f f e c t due t o d e g r e e o f f i e l d i n d e p e n d e n c e and a s i g n i f i c a n t i n t e r a c t i o n e f f e c t . F i g u r e 3 i n d i c a t e s t h i s i n t e r a c t i o n e f f e c t i s due t o t h e p e r f o r m a n c e s o f t h e extreme r a n g e s . F o r extreme f i e l d dependent s t u d e n t s , 9-15 r a n g e , t h e p a t t e r n i n s t r u c t i o n a l s t r a t e g y r e s u l t e d i n a h i g h e r l e v e l o f a c h i e v e m e n t t h a n t h e a l g e b r a i c i n -s t r u c t i o n a l s t r a t e g y ; w h i l e f o r extreme f i e l d i n d e p e n d e n t s t u d e n t s , 23-25 r a n g e , t h e a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y r e s u l t e d i n a h i g h e r l e v e l o f achievement t h a n t h e p a t t e r n i n s t r u c t i o n a l s t r a t e g y . The f i g u r e a l s o i n d i -c a t e s t h a t t h e s i g n i f i c a n t main e f f e c t o f method i s due t o t h e h i g h l e v e l o f a c hievement of t h e f i e l d i n d e p e n d e n t group on t h e a l g e b r a i c i n s t r u c t i o n -a l s t r a t e g y . T h i s h i g h l e v e l o f a c h i e v e m e n t o f t h e f i e l d i n d e p e n d e n t group F i g u r e 2 Mean R e s i d u a l S c o r e s on C2 Computation S c o r e s F i g u r e 3 Mean R e s i d u a l S c o r e s on C l G e n e r a l i z a t i o n S c o r e s P a t t e r n A l g e b r a i c Mean R e s i d u a l S c o r e s -3 -4 9 15 16 18 19 20 21 22 23 ' 25~~ F i e l d Independence Ranges on t h e a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y a l s o a c c o u n t s f o r t h e s i g n i f i c a n t main e f f e c t due t o degree o f f i e l d i n d e p e n d e n c e . Due t o t h i s h i g h l e v e l of a chievement of t h e extreme f i e l d i n d e p e n d e n t group t a u g h t by t h e a l -g e b r a i c a p p r o a c h , t h e o v e r a l l mean o f t h e extreme f i e l d i n d e p e n d e n t group i s g r e a t e r t h a n t h e mean o f a l l o t h e r g r o u p s . The a n a l y s i s o f C2 g e n e r a l i z a t i o n s c o r e s i n d i c a t e d a s i g n i f i c a n t i n t e r a c t i o n e f f e c t . F i g u r e 4 i n d i c a t e s t h a t s t u d e n t s i n t h e extreme f i e l d i n d e p e n d e n t g r o u p , 23-25 r a n g e , a c h i e v e d h i g h e r on t h e a v e r a g e t h a n a l l o t h e r s t u d e n t s when t a u g h t by t h e a l g e b r a i c a p p r o a c h and a c h i e v e d l o w e r on the a v e r a g e t h a n a l l o t h e r s t u d e n t s when t a u g h t by t h e p a t t e r n a p p r o a c h . T h e r e i s a s i m i l a r , b u t l e s s e x t r e m e , e f f e c t i n t h e 21-22 range g r o u p . The a n a l y s i s o f t h e d a t a a l s o i n d i c a t e d a t r e n d t o w a r d a main e f f e c t due t o method, but t h e n u l l h y p o t h e s i s was n o t r e j e c t e d . F i g u r e 4 i n d i c a t e s t h a t t h e t r e n d i s t o w a r d t h e a l g e b r a i c a p p r o a c h , but a g a i n t h e p e r f o r m a n c e of t h e extreme f i e l d i n d e p e n d e n t group i n f l a t e s t h e o v e r a l l e f f e c t o f t h e a l g e b r a i c a p p r o a c h and d e f l a t e s t h e o v e r a l l e f f e c t o f t h e p a t t e r n a p p r o a c h . F i g u r e s 3 and 4 i n d i c a t e t h a t , e x c e p t f o r t h e two extreme f i e l d i n d e p e n d e n t r a n g e s , 21-22 and 23-25, t h e i n t e r a c t i o n r e s u l t s a r e i n c o n c l u -s i v e . I n t h e 16-18 r a n g e group and t h e 19-20 range group t h e r e i s no i n t e r a c t i o n w i t h method. F o r t h e extreme f i e l d dependent g r o u p , 9-15 r a n g e , t h e r e i s some e v i d e n c e o f i n t e r a c t i o n s . However, t h e i n t e r a c t i o n s on C l and C2 f o r t h i s group a r e i n t h e o p p o s i t e d i r e c t i o n s and hence no c o n c l u -s i o n s can be drawn f o r t h i s g r oup. The i n c o n s i s t e n t r e s u l t s f o r t h i s group may be due t o t h e s m a l l number of s u b j e c t s i n t h i s r a n g e , s i x on C l and e i g h t on C2. T h i s range c o n t a i n e d t h e s m a l l e s t number o f s u b j e c t s o f a l l t h e r a n g e s of t h e f i e l d i n d e p e n d e n c e d i m e n s i o n . F i g u r e A Mean R e s i d u a l S c o r e s on C2 G e n e r a l i z a t i o n S c o r e s P a t t e r n A l g e b r a i c 57 However, th e r e s u l t s a r e c o n s i s t e n t w i t h r e s p e c t t o t h e two e x -treme f i e l d i n d e p e n d e n t r a n g e s , 21-22 and 23-25. F o r t h e 21-22 r a n g e , w i t h r e g a r d t o p e r f o r m a n c e on t h e g e n e r a l i z a t i o n t e s t s on t h e complex a l -g o r i t h m s , t h e r e i s a t r e n d toward s u p e r i o r i t y o f t h e a l g e b r a i c i n s t r u c t i o n -a l s t r a t e g y . F o r t h e 23-25 r a n g e , w i t h r e g a r d t o p e r f o r m a n c e on t h e g e n e r -a l i z a t i o n t e s t s on t h e complex a l g o r i t h m s , t h e a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y was s t r o n g l y s u p e r i o r t o t h e p a t t e r n i n s t r u c t i o n a l s t r a t e g y . D i s c u s s i o n o f t h e R e s u l t s The h i g h t o t a l sums o f s q u a r e s o f t h e f u l l m o d e l s , SSR i n each o f t h e s e p a r a t e a n a l y s e s i n d i c a t e t h a t t h e model chosen was a p p r o p r i a t e t o t h e d a t a . The s t a t i s t i c a l a n a l y s i s o f t h e d a t a a l s o i n d i c a t e s t h a t t h e xzo-v a r i a t e s were w e l l c h o s e n . I n each o f t h e d a t a a n a l y s e s , t h e c o v a r i a t e a c c o u n t e d f o r t h e l a r g e s t p o r t i o n o f t h e sum o f s q u a r e s o f t h e r e g r e s s i o n c o e f f i c i e n t s and i n each o f t h e d a t a a n a l y s e s , e x c e p t f o r S I and S2 compu-t a t i o n s c o r e s , t h e c o n t r i b u t i o n o f t h e c o v a r i a t e t o t h e t o t a l sum o f s q u a r e s o f t h e r e g r e s s i o n c o e f f i c i e n t s was s i g n i f i c a n t a t = .01. F o r S I t h e c o n -t r i b u t i o n o f t h e c o v a r i a t e was s i g n i f i c a n t a t = .23 and f o r S2 a t ^ = .14. The r e s u l t s o f t h e a n a l y s i s of t h e d a t a i n d i c a t e t h a t f o r t h e two examples o f s i m p l e a l g o r i t h m s c h o s e n , on b o t h t h e c o m p u t a t i o n and g e n e r -a l i z a t i o n p o s t - t e s t s , t h e r e was no s t a t i s t i c a l l y s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n a l s t r a t e g y , no s t a t i s t i c a l l y s i g n i f i c a n t main e f f e c t due t o d e g r e e o f f i e l d i n d e pendence and no s t a t i s t i c a l l y s i g n i f i c a n t i n -t e r a c t i o n e f f e c t between i n s t r u c t i o n a l s t r a t e g y and f i e l d i n d e p e n d e n c e . 58 Thus, f o r the s i m p l e a l g o r i t h m s , t h e two i n s t r u c t i o n a l s t r a t e g i e s d i d n o t produce s i g n i f i c a n t l y d i f f e r e n t r e s u l t s and t h e r e was no d i f f e r e n t i a l a-chievement e f f e c t due t o degr e e o f f i e l d i n d e p e n d e n c e . T h e r e was a l s o no e v i d e n c e o f d i f f e r e n t i a l a chievement on the two I n s t r u c t i o n a l s t r a t e g i e s w i t h i n g r o u p s o f s t u d e n t s s i m i l a r i n t h e i r degree o f f i e l d i n d e p e n d e n c e . The s t a t i s t i c a l a n a l y s i s o f t h e d a t a i n d i c a t e s t h a t f o r p e r f o r -mance on t h e c o m p u t a t i o n t e s t s on t h e complex a l g o r i t h m s , t h e r e i s no s i g -n i f i c a n t main e f f e c t due t o i n s t r u c t i o n a l s t r a t e g y and no s i g n i f i c a n t i n -t e r a c t i o n e f f e c t between i n s t r u c t i o n a l s t r a t e g y and d e g r e e o f f i e l d i n d e -pendence. Thus, w i t h r e g a r d t o p r o d u c i n g c o m p u t a t i o n a l a b i l i t y w i t h t h e complex a l g o r i t h m s , t h e two i n s t r u c t i o n a l s t r a t e g i e s a r e n o t s i g n i f i c a n t l y d i f f e r e n t o v e r a l l , and a l s o do n o t produce s i g n i f i c a n t l y d i f f e r e n t r e s u l t s w i t h i n groups s i m i l a r i n t h e i r d e g r e e o f f i e l d i n d e p e n d e n c e . However, t h e a n a l y s i s o f t h e d a t a , does i n d i c a t e t h a t t h e r e i s d i f f e r e n t i a l a c h i e v e m e n t on t h e C2 c o m p u t a t i o n t e s t among s t u d e n t s d i f f e r i n g i n d e g r e e o f f i e l d i n d e p e n d e n c e . The g r a p h i n g of t h e r e s u l t s o f t h e C2 c o m p u t a t i o n t e s t i n d i -c a t e t h a t w i t h i n c r e a s i n g d e g r e e o f f i e l d i n d e p e n d e n c e , t h e mean l e v e l o f achievement i n c r e a s e s w i t h a s l i g h t t a p e r i n g o f f a t t h e extreme f i e l d i n d e p e n d e n t l e v e l . The s t a t i s t i c a l a n a l y s i s o f C l g e n e r a l i z a t i o n s c o r e s i n d i c a t e s a s i g n i f i c a n t main e f f e c t due t o degr e e of f i e l d i n d e p e n d e n c e . However, t h e a n a l y s i s o f t h e C2 g e n e r a l i z a t i o n s c o r e s d i d n o t s u p p o r t t h i s f i n d i n g . The s t a t i s t i c a l a n a l y s i s o f C l g e n e r a l i z a t i o n s c o r e s i n d i c a t e s a s i g n i f i c a n t main e f f e c t due t o i n s t r u c t i o n a l s t r a t e g y and t h e a n a l y s i s o f C2 g e n e r a l i z a t i o n s c o r e s i n d i c a t e s a s t r o n g t r e n d , F - v a l u e has p = .077, 59 toward a main e f f e c t due t o i n s t r u c t i o n a l s t r a t e g y . The g r a p h s of t h e r e -s u l t s of t h e C l and C2 g e n e r a l i z a t i o n t e s t s i n d i c a t e t h a t t h i s s i g n i f i c a n t main e f f e c t i s due t o t h e h i g h l e v e l o f p e r f o r m a n c e of s t u d e n t s a t t h e two h i g h e s t l e v e l s of t h e f i e l d i n d e p e n d e n c e d i m e n s i o n on t h e a l g e b r a i c i n -s t r u c t i o n a l s t r a t e g y . The s t a t i s t i c a l a n a l y s e s o f t h e C l and C2 g e n e r a l i z a t i o n s c o r e s i n d i c a t e a s i g n i f i c a n t i n t e r a c t i o n e f f e c t between i n s t r u c t i o n a l s t r a t e g y and d e g r e e o f f i e l d i n d e p e n d e n c e . The g r a p h s of t h e s e r e s u l t s , however, p r o v i d e c o n c l u s i v e i n f o r m a t i o n o n l y f o r t h e two extreme l e v e l s o f f i e l d i n d e p e n d e n t s t u d e n t s , 21-22 and 23-25 r a n g e s . F o r t h e s e s t u d e n t s t h e a l -g e b r a i c a p p r o a c h i s s u p e r i o r t o t h e p a t t e r n a p p r o a c h and i s d r a m a t i c a l l y s u p e r i o r f o r t h e s t u d e n t s i n t h e 23-25 r a n g e . CHAPTER V CONCLUSIONS AND IMPLICATIONS SUMMARY A s t u d y was c o n d u c t e d t o d e t e r m i n e t h e i n t e r a c t i o n e f f e c t , i f any, between t h e f i e l d i n d e p e n d e n c e c o n s t r u c t and two i n s t r u c t i o n a l s t r a t e g i e s , a p a t t e r n s t r a t e g y w h i c h used d i a g r a m s e x t e n s i v e l y and an a l g e b r a i c s t r a t e g y w h i c h used a l g e b r a i c f i e l d p r o p e r t i e s f a m i l i a r t o t h e c h i l d and was d e v o i d o f d i a g r a m s . A r e v i e w o f t h e r e l e v a n t l i t e r a t u r e i n d i c a t e d t h a t c o g n i t i v e s t y l e may h o l d t h e key t o u n d e r s t a n d i n g i n d i v i d u a l d i f f e r e n c e s I n l e a r n i n g and a t t h e same t i m e t h a t v e r y few s t u d i e s have i n v e s t i g a t e d t h e i n t e r a c t i o n e f f e c t between c o g n i t i v e s t y l e and i n s t r u c t i o n a l s t r a t e g y . No s t u d i e s w ere found w h i c h i n v e s t i g a t e d t h e i n t e r a c t i o n e f f e c t between f i e l d i n d e p e n d e n c a and i n s t r u c t i o n a l s t r a t e g y i n m a t h e m a t i c s . The s t u d y was d e s i g n e d t o answer t h r e e m a j o r q u e s t i o n s : (1) Do c h i l d r e n d i f f e r i n g i n t h e i r d e g r e e o f f i e l d i n d e p e n d e n c e r e s p o n d d i f f e r -e n t l y t o t h e two i n s t r u c t i o n a l s t r a t e g i e s ? ; (2) I s one o f t h e i n s t r u c t i o n -a l s t r a t e g i e s , on t h e a v e r a g e , s u p e r i o r t o t h e o t h e r ? ; (3) I s t h e r e d i f -f e r e n t i a l a c hievement among s t u d e n t s d i f f e r i n g i n t h e i r d e g r e e o f f i e l d i n d e pendence? Twelve grade f i v e c l a s s e s formed t h e p o p u l a t i o n o f t h e s t u d y . ihc - s e . • c l a s s e s were p a r t o f t h e sample o f a s t u d y b e i n g c o n d u c t e d by M a r i a n Wein-s t e i n , a d o c t o r a l c a n d i d a t e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . Ms. W e i n s t e i n d e v e l o p e d t h e two i n s t r u c t i o n a l s t r a t e g i e s and t h e p r e t e s t s and p o s t - t e s t s used i n t h e s t u d y . Ms. W e i n s t e i n c l a s s i f i e d a l g o r i t h m s as s i m -61 p i e o r complex on t h e b a s i s o f t h e number o f p r e r e q u i s i t e s r e q u i r e d f o r t h e i r a c q u i s i t i o n and two examples o f each t y p e were s e l e c t e d f o r use i n th e s t u d y . The two s i m p l e a l g o r i t h m s chosen were t h e p r o d u c t o f a f r a c -t i o n and a mixed number, S I and t h e c o m p a r i s o n o f f r a c t i o n s , S2. The two complex a l g o r i t h m s chosen were c h a n g i n g a f r a c t i o n t o a d e c i m a l , C l and f i n d i n g t h e s q u a r e r o o t o f a f r a c t i o n , C2. Each c l a s s went t h r o u g h t h e f o l l o w i n g i n s t r u c t i o n a l and t e s t i n g s e q u e n c e , g o i n g t h r o u g h e i t h e r S I and C l o r S2 and C2: P r e t e s t on s i m p l e a l g o r i t h m - > i n s t r u c t i o n on s i m p l e a l g o r i t h m s ( 5 s e s s i o n s ) c o m p u t a t i o n and g e n e r a l i z a t i o n t e s t s on t h e s i m p l e a l g o r i t h m => p r e t e s t on complex a l g o r i t h m => i n s t r u c t i o n on complex a l g o r i t h m ( 9 s e s s i o n s ) > c o m p u t a t i o n and g e n e r a l i z a t i o n t e s t s on complex a l g o r i t h m . I n each o f t h e t w e l v e c l a s s e s , one h a l f o f t h e s t u d e n t s who com-p l e t e d t h e W e i n s t e i n s t u d y were randomly s e l e c t e d t o f o r m t h e sample o f t h i s s t u d y . The C h i l d r e n ' s Embedded F i g u r e s T e s t was t h e n i n d i v i d u a l l y a d m i n i s t e r e d t o t h e s e s t u d e n t s . ^ M u l t i p l e l i n e a r r e g r e s s i o n t e c h n i q u e s were u s e d t o a n a l y s e t h e d a t a . E i g h t s e p a r a t e a n a l y s e s o f t h e d a t a were p e r f o r m e d : t h e r e s u l t s o f t h e c o m p u t a t i o n and g e n e r a l i z a t i o n t e s t s were a n a l y s e d s e p a r a t e l y f o r each o f t h e two s i m p l e and two complex a l g o r i t h m s . T h r e e n u l l h y p o t h e s e s were each t e s t e d a t «x* = .05: (1) T h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n mean p o s t - t e s t s c o r e s between s t u d e n t s t a u g h t by a p a t t e r n i n s t r u c t i o n a l s t r a -t e g y and s t u d e n t s t a u g h t by an a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y ; (2) T h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n mean p o s t - t e s t s c o r e s between gro u p s o f s t u d e n t s d i f f e r i n g i n d e g r e e o f f i e l d i n d e p e n d e n c e ; (3) T h e r e i s no s i g -n i f i c a n t i n t e r a c t i o n between s t u d e n t s ' degree o f f i e l d i n d e p e n d e n c e and i n s t r u c t i o n a l s t r a t e g y . The a n a l y s i s o f t h e d a t a r e s u l t e d i n a c c e p t a n c e o f t h e t h r e e n u l l h y p o t h e s e s f o r t h e s i m p l e a l g o r i t h m s and f o r t h e C l c o m p u t a t i o n t e s t . On the C2 c o m p u t a t i o n t e s t , t h e d a t a r e s u l t e d i n r e j e c t i o n o f t h e n u l l hypo-t h e s i s o f no s i g n i f i c a n t main e f f e c t due t o degr e e o f f i e l d i n d e p e n d e n c e . The a n a l y s i s o f t h e complex a l g o r i t h m g e n e r a l i z a t i o n t e s t s r e s u l t e d i n r e -j e c t i o n o f t h e n u l l h y p o t h e s i s o f no s i g n i f i c a n t i n t e r a c t i o n between i n -s t r u c t i o n a l s t r a t e g y and degr e e o f f i e l d i n d e p e n d e n c e . LIMITATIONS The r e s e a r c h agreement w i t h t h e s c h o o l d i s t r i c t s t i p u l a t e d t h e use of e x i s t i n g i n t a c t g r a d e f i v e c l a s s e s . Thus, o n l y c l a s s e s and n o t su b -j e c t s were randomly a s s i g n e d t o t r e a t m e n t s . T e a c h e r p r e f e r e n c e s f o r mode o f m a t h e m a t i c a l i n s t r u c t i o n was an u n c o n t r o l l e d f a c t o r i n t h e s t u d y . No assessment was made o f t h e t e a c h e r ' s i n s t r u c t i o n a l s t r a t e g y p r e f e r e n c e s n o r o f t h e t e a c h e r s a t t i t u d e toward t h e i n s t r u c t i o n a l s t r a t e g y r a ndomly a s s i g n e d t o them. However, s i n c e t h e t e a -c h e r s p a r t i c i p a t e d on a v o l u n t a r y b a s i s , i t i s assumed t h a t t h e i r a t t i t u d e t owards t h e e x p e r i m e n t a l m a t e r i a l s was p o s i t i v e . The s u b j e c t s ' p r e v i o u s i n s t r u c t i o n a l e x p e r i e n c e s i n m a t h e m a t i c s i s an u n c o n t r o l l e d v a r i a b l e i n t h e s t u d y . No assessment was made o f t h e t y p e s o f i n s t r u c t i o n a l s t r a t e g i e s t o w h i c h t h e s u b j e c t had been exposed i n h i s s c h o o l i n g p r i o r t o t h e s t u d y , n o r o f t h e i n s t r u c t i o n a l s t r a t e g y p r e d o m i -n a n t l y used i n h i s m a t h e m a t i c s e x p e r i e n c e s . 63 The use of t h e f a m i l i a r c l a s s r o o m s e t t i n g and t i m e - s c h e d u l e m i n i -m i z e d t h e Hawthorne E f f e c t . However, t h e amount and l e n g t h o f t h e t e s t i n g done i n t h e s t u d y was a d e p a r t u r e from t h e norm f o r m a t h e m a t i c s and can be c o n s i d e r e d t o have c o n t r i b u t e d t o a Hawthorne E f f e c t . However, t h e t e s t i n g s c h e d u l e was u n i f o r m f o r a l l s u b j e c t s . T h e r e i s an u n c o n t r o l l e d c u m u l a t i v e l e a r n i n g e f f e c t . I t i s impos-s i b l e t o s e p a r a t e t h e e f f e c t s o f i n s t r u c t i o n on t h e s i m p l e a l g o r i t h m from t h e e f f e c t s o f i n s t r u c t i o n " on t h e complex a l g o r i t h m , because i n s t r u c t i o n on t h e s i m p l e a l g o r i t h m p r e c e d e d i n s t r u c t i o n on t h e complex a l g o r i t h m . DISCUSSION OF THE RESULTS The a n a l y s e s o f t h e c o m p u t a t i o n and g e n e r a l i z a t i o n s c o r e s f o r t h e two s i m p l e a l g o r i t h m s , p r o d u c t o f a f r a c t i o n and a mixed number and com-p a r i s o n o f f r a c t i o n s , r e s u l t e d i n n o t r e j e c t i n g t h e n u l l h y p o t h e s e s o f no s i g n i f i c a n t d i f f e r e n c e between t h e two i n s t r u c t i o n a l s t r a t e g i e s and o f no s i g n i f i c a n t i n t e r a c t i o n between d e g r e e o f f i e l d i n d e p e n d e n c e and i n s t r u c -t i o n a l s t r a t e g y . T h e r e a p p e a r s t o be two e q u a l l y p l a u s i b l e e x p l a n a t i o n s f o r t h e s e f i n d i n g s . The f i r s t i s t h a t f o r s i m p l e a l g o r i t h m s , t h e a l g e b r a i c and p a t t e r n i n s t r u c t i o n a l s t r a t e g i e s a r e e q u a l l y e f f e c t i v e b o t h f o r t h e sample as a whole and f o r g r o u p s o f t h e sample d i f f e r i n g i n t h e i r d e g r e e o f f i e l d i n d e p e n d e n c e . The s econd i s t h a t t h e p e r i o d o f e x p o s u r e t o t h e two a p p r o a c h e s , f i v e s e s s i o n s , was i n s u f f i c i e n t t o a l l o w f o r d i f f e r e n t i a l e f f e c t s t o emerge. The a n a l y s i s of t h e C2 c o m p u t a t i o n t e s t s c o r e s i n d i c a t e d t h a t , w i t h t h e e x c e p t i o n o f t h e extreme f i e l d i n d e p e n d e n t s t u d e n t s , as t h e d e g r e e o f f i e l d independence i n c r e a s e d , t h e l e v e l of a c h i e v e m e n t i n c r e a s e d . The d a t a 64 f o r t h e C l c o m p u t a t i o n t e s t , however, d i d n o t s u p p o r t t h i s f i n d i n g . C l o s e r e x a m i n a t i o n o f the two complex a l g o r i t h m s may p r o v i d e an i n s i g h t i n t o t h e s e f i n d i n g s . A l t h o u g h t h e two complex a l g o r i t h m s , c h a n g i n g a f r a c t i o n t o a d e c i m a l and f i n d i n g t h e s q u a r e r o o t o f a f r a c t i o n , a r e s i m i l a r i n t h e i r d egree o f c o m p l e x i t y , d e f i n e d by t h e number o f p r e r e q u i s i t e s r e q u i r e d f o r t h e i r a c q u i s i t i o n , t h e i n v e s t i g a t o r c l a i m s t h a t t h e a l g o r i t h m s d i f f e r i n t h e i r d e g r e e o f a b s t r a c t n e s s . Changing a f r a c t i o n t o a d e c i m a l i n v o l v e s an e x t e n s i o n o f t h e p l a c e v a l u e s y s t e m t o t h e r i g h t and t h e l e a r n i n g o f d i v i s i o n i n t h i s e x t e n d e d s y s t e m . F i n d i n g t h e s q u a r e r o o t o f a f r a c t i o n c annot r e l y on s u c h an e x t e n s i o n of p r e v i o u s l y l e a r n e d m a t h e m a t i c a l c o n -c e p t s , n o r does i t n a t u r a l l y a r i s e o u t o f t h e r e a l w o r l d e x p e r i e n c e s o f t h e c h i l d , t h u s making i t more a b s t r a c t f o r t h e c h i l d t h a n c h a n g i n g a f r a c t i o n t o a d e c i m a l . The i n v e s t i g a t o r t e n t a t i v e l y s u g g e s t s t h a t a t t h e grade f i v e l e v e l , t h e r e may be a p o s i t i v e c o r r e l a t i o n between t h e c h i l d ' s d e g r e e o f f i e l d i n d e pendence and h i s a b i l i t y t o cope w i t h a b s t r a c t c o n -c e p t s . The s i g n i f i c a n t i n t e r a c t i o n e f f e c t s i n d i c a t e d by t h e a n a l y s e s o f t h e complex a l g o r i t h m g e n e r a l i z a t i o n t e s t s p a r t i a l l y s u p p o r t e d t h e i n v e s t -i g a t o r ' s h y p o t h e s i z e d outcomes. The i n v e s t i g a t o r had h y p o t h e s i z e d t h a t c h i l d r e n d i f f e r i n g i n t h e i r d e g r e e o f f i e l d i n d e p e n d e n c e w o u l d r e s p o n d d i f -f e r e n t l y t o t h e two i n s t r u c t i o n a l s t r a t e g i e s . That i s , c h i l d r e n t e n d i n g toward f i e l d dependency would, p e r f o r m b e t t e r on t h e p a t t e r n a p p r o a c h t h a n on t h e a l g e b r a i c a p p r o a c h , and c h i l d r e n t e n d i n g t o w a r d f i e l d i n d e p e n d e n c y would p e r f o r m b e t t e r on t h e a l g e b r a i c a p p r o a c h t h a n on t h e p a t t e r n a p p r o a c h . No c o n c l u s i v e r e s u l t s c o u l d be drawn f o r f i e l d dependent c h i l d r e n . However, 65 f o r c h i l d r e n t e n d i n g toward f i e l d i n d e p e n d e n c y , w i t h r e g a r d t o p e r f o r m a n c e on t h e complex a l g o r i t h m g e n e r a l i z a t i o n t e s t s , t h e a l g e b r a i c a p p r o a c h was s u p e r i o r t o t h e p a t t e r n a p p r o a c h , as had been h y p o t h e s i z e d . I n f a c t , t h e p e r f o r m a n c e o f t h e extreme f i e l d i n d e p e n d e n t s t u d e n t s on t h e complex a l -g o r i t h m g e n e r a l i z a t i o n t e s t s , when t a u g h t by t h e a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y , was s u p e r i o r t o t h e p e r f o r m a n c e o f a l l o t h e r s t u d e n t s . However, t h e p e r f o r m a n c e o f t h e s e s t u d e n t s on t h e complex a l g o r i t h m s g e n e r a l i z a t i o n t e s t s , when t a u g h t by t h e p a t t e r n i n s t r u c t i o n a l s t r a t e g y , was d r a m a t i c a l l y i n f e r i o r t o t h e i r p e r f o r m a n c e when t a u g h t by t h e a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y and on t h e C2 g e n e r a l i z a t i o n t e s t , was d r a m a t i c a l l y i n f e r i o r t o t h e p e r f o r m a n c e o f a l l o t h e r s t u d e n t s . T hus, w h i l e on t h e a v e r a g e t h e p a t t e r n i n s t r u c t i o n a l s t r a t e g y and t h e a l g e b r a i c i n s t r u c t i o n a l s t r a t e g y p r o d u c e e q u i v a l e n t r e s u l t s , t h e y do n o t p r o d u c e e q u i v a l e n t r e s u l t s w i t h r e s p e c t t o f i e l d i n d e p e n d e n t c h i l d r e n . F o r t h e s e c h i l d r e n , t h e a l g e -b r a i c i n s t r u c t i o n a l s t r a t e g y a p p e a r s t o be s u p e r i o r t o t h e p a t t e r n i n -s t r u c t i o n a l s t r a t e g y . CONCLUSIONS The f o l l o w i n g c o n c l u s i o n s were drawn f r o m t h e r e s u l t s o f t h e s t u d y : 1. The f i e l d i n d e p e n d e n c e c o n s t r u c t i s a p r o f i t a b l e a p t i t u d e v a r i a b l e t o be c o n s i d e r e d i n t h e a d a p t a t i o n o f m a t h e m a t i c a l i n s t r u c t i o n a l s t r a t e g i e s t o i n d i v i d u a l l e a r n e r s . 2. F o r s t u d e n t s a t t h e g r a d e f i v e l e v e l , w i t h t h e e x c e p t i o n o f extreme f i e l d i n d e p e n d e n t s t u d e n t s , an i n s t r u c t i o n a l s t r a t e g y w h i c h uses d i a -grams e x t e n s i v e l y i s e q u a l l y e f f e c t i v e as an i n s t r u c t i o n a l s t r a t e g y 66 w h i c h i s d e v o i d o f d i a g r a m s and w h i c h uses a l g e b r a i c p r o p e r t i e s . 3. S t u d e n t s a t t h e g r a d e f i v e l e v e l , w i t h p e r h a p s t h e e x c e p t i o n o f e x -treme f i e l d dependent s t u d e n t s , can cope a d e q u a t e l y w i t h i n s t r u c t i o n a l e x p l a n a t i o n s w h i c h a r e based s o l e l y on a l g e b r a i c p r o p e r t i e s f a m i l i a r t o t h e s t u d e n t s and w h i c h a r e d e v o i d of c o n c r e t e o r p i c t o r i a l a i d s . 4. F o r extreme f i e l d i n d e p e n d e n t s t u d e n t s i n t h i s s t u d y , w i t h r e g a r d t o i n s t r u c t i o n i n complex a l g o r i t h m s , as measured by p u p i l p e r f o r m a n c e , an i n s t r u c t i o n a l s t r a t e g y i n m a t h e m a t i c s w h i c h r e l i e s e x t e n s i v e l y on t h e f i e l d p r o p e r t i e s o f t h e number s y s t e m and w h i c h i s d e v o i d o f d i a -grams i s s u p e r i o r t o an i n s t r u c t i o n a l s t r a t e g y w h i c h r e l i e s e x t e n s i v e -l y on d i a g r a m s . IMPLICATIONS OF THE STUDY The r e v i e w o f t h e l i t e r a t u r e i n m a t h e m a t i c s e d u c a t i o n r e v e a l e d t h e l a c k o f a s y s t e m a t i c a p p r o a c h t o a d a p t i n g m a t h e m a t i c s i n s t r u c t i o n a l s t r a -t e g i e s t o i n d i v i d u a l l e a r n e r s . R e s e a r c h s t u d i e s i n m a t h e m a t i c s have been p r i m a r i l y c o n c e r n e d w i t h t h e s e a r c h f o r t h e s u p e r i o r i n s t r u c t i o n a l s t r a -t e g y f o r m a t h e m a t i c s a t s p e c i f i c g r a de l e v e l s . Y e t t h i s s t u d y s u g g e s t s t h a t a l t h o u g h two i n s t r u c t i o n a l s t r a t e g i e s may p r o d u c e e q u i v a l e n t r e s u l t s on t h e a v e r a g e , t h e y do n o t n e c e s s a r i l y p r o d u c e e q u i v a l e n t r e s u l t s f o r i n d i v i d u a l groups o f s t u d e n t s . T h e r e i s a need f o r a c o n c e n t r a t e d , c o o r d i n a t e d r e s e a r c h e f f o r t t o e s t a b l i s h a t h e o r e t i c a l b a s i s f o r the a s s i g n m e n t o f l e a r n e r s t o i n -s t r u c t i o n a l s t r a t e g i e s i n m a t h e m a t i c s . T h i s w i l l r e q u i r e d r a w i n g on t h e r e s o u r c e s o f b o t h l e a r n i n g t h e o r i s t s and d e v e l o p e r s o f m a t h e m a t i c s c u r r i c u -lum. The l e a r n i n g t h e o r i s t s can a i d i n t h e i s o l a t i o n and c h a r a c t e r i z a t i o n o f r e l e v a n t i n d i v i d u a l d i f f e r e n c e s i n l e a r n e r s . The c u r r i c u l u m d e v e l o p e r s can a i d i n t h e d e s i g n i n g o f a l t e r n a t i v e i n s t r u c t i o n a l s t r a t e g i e s . T o g e t h e r , t h r o u g h r e s e a r c h s t u d i e s s i m i l a r t o t h e one r e p o r t e d i n t h i s p a p e r , b o t h t h e l e a r n i n g t h e o r i s t and t h e c u r r i c u l u m d e v e l o p e r c a n r e f i n e t h e i r c o n -t r i b u t i o n s t o t h i s a r e a o f r e s e a r c h . Out o f t h i s r e f i n e d body o f know-l e d g e new r e s e a r c h s t u d i e s s h o u l d a r i s e and t h e r e f i n i n g p r o c e s s r e p e a t e d . T h i s s t u d y s u g g e s t s t h a t t h e a r e a o f c o g n i t i v e s t y l e , and I n p a r t i c u l a r t h e c o n s t r u c t o f f i e l d i n d e p e n d e n c e , may p r o v e t o be a f r u i t f u l s t a r t i n g p o i n t f o r s u c h a r e s e a r c h e f f o r t . The s t u d y a l s o has i m p l i c a t i o n s f o r m a t h e m a t i c s e d u c a t i o n w i t h r e -s p e c t t o i n s t r u c t i o n o f an a l g e b r a i c n a t u r e , i n p a r t i c u l a r w i t h r e s p e c t t o a l g e b r a " r e a d i n e s s " and t o t h e i n t r o d u c t i o n o f a l g e b r a i c i n s t r u c t i o n a l s t r a t e g i e s i n g r a d e s s i x t o n i n e . T h i s s t u d y s u g g e s t s t h a t f i e l d I n d e p e n -dent c h i l d r e n may be b e t t e r a b l e t o cope w i t h i n s t r u c t i o n a l m a t e r i a l o f an a l g e b r a i c n a t u r e t h a n t h e i r f i e l d dependent c o u n t e r p a r t s , and t h a t know-l e d g e o f t h e c h i l d ' s degree o f f i e l d i n d e p e n d e n c e may p r o v i d e p r o f i t a b l e i n s i g h t i n t o t h e c h i l d ' s r e a d i n e s s f o r a l g e b r a i c i n s t r u c t i o n a l m a t e r i a l s . BIBLIOGRAPHY 69 A i k e n , L e w i s R., J r . " I n t e l l e c t i v e V a r i a b l e s and M a t h e m a t i c s A c h i e v e m e n t , " J o u r n a l o f S c h o o l P s y c h o l o g y , I X ( 1 9 7 1 ) , 202-212. B e c k e r , J e r r y P. " R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n : T he R o l e o f T h e o r y and o f 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 , " J o u r n a l f o r R e s e a r c h i n  M a t h e m a t i c s E d u c a t i o n , 1 ( 1 9 7 0 ) . B r a c h t , G l e n n H. " E x p e r i m e n t a l F a c t o r s R e l a t e d To 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 s " R e v l e w _ o f _ J i d j : i c ^ XXXX(1971) , 627-645. Coop, R i c h a r d H. and S i g e l , I r v i n g E. " C o g n i t i v e S t y l e : I m p l i c a t i o n s f o r L e a r n i n g and I n s t r u c t i o n , " P s y c h o l o g y i n t h e S c h o o l s , V I I I ( 1 9 7 1 ) , 152-161. C r o n b a c h , L e e . J . "The Two D i s c i p l i n e s o f S c i e n t i f i c P s y c h o l o g y , " The  A m e r i c a n P s y c h o l o g i s t , X I I ( 1 9 5 7 ) . Cronbach L e e . J . "How Can I n s t r u c t i o n Be Adapted To I n d i v i d u a l D i f f e r e n c e s , " ed. R. M. Gagne. L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s . Columbus O h i o : C h a r l e s E. M e r r i l l B ooks, 1967. C r o n bach Lee J . and Snow, R i c h a r d E. " I n d i v i d u a l D i f f e r e n c e s i n L e a r n i n g A b i l i t y as a F u n c t i o n o f I n s t r u c t i o n a l V a r i a b l e s , " F i n a l R e p o r t . S t a n f o r d U n i v e r s i t y , C a l i f o r n i a S c h o o l of E d u c a t i o n , ED 029 0 0 1 . D a v i s , J . Kent " C o g n i t i v e S t y l e and C o n d i t i o n a l Concept L e a r n i n g , " P a p e r r e a d a t t h e a n n u a l m e e t i n g of t h e A m e r i c a n E d u c a t i o n a l R e s e a r c h A s s o c -i a t i o n , C h i c a g o , 1972. D a v i s , J . Kent and G r i e v e , T a r r a n c e .Don "The R e l a t i o n s h i p o f C o g n i t i v e S t / ' e and Method of I n s t r u c t i o n t o P e r f o r m a n c e i n N i n t h Grade Geography," The J o u r n a l of E d u c a t i o n a l R e s e a r c h , L X V ( 1 9 7 1 ) , 137-141. D a v i s , J . Kent and K l a u s m e i e r , H e r b e r t J . " C o g n i t i v e S t y l e and Concept I d e n t i f i c a t i o n As A F u n c t i o n o f C o m p l e x i t y and T r a i n i n g P r o c e d u r e s , " J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , L X I ( 1 9 7 0 ) , 423-430. Gage, N. L. and Unruh, W. R. " T h e o r e t i c a l F o r m u l a t i o n s f o r R e s e a r c h on T e a c h i n g " Review of E d u c a t i o n a l R e s e a r c h , X X X V I I I ( 1 9 6 7 ) . Gagne, R o b e r t M. . " L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s : I n t r o d u c t i o n t o t"-.z C o n f e r e n c e , " ed. R. M. Gagne. L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s . Columbus, Ohio: C h a r l e s E. M e r r i l l B ooks, 1967. Gagne, R o b e r t M. ed. L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s . Columbus, O h i o : C h a r l e s E. M e r r i l l B ooks, 1967. 70 J e n s e n , A r t h u r R. " V a r i e t i e s o f I n d i v i d u a l D i f f e r e n c e s i n L e a r n i n g , " ed. R. M. Gagne. L e a r n i n g and I n d i v i d u a l D i f f e r e n c e s . Columbus, O h i o : C h a r l e s E. M e r r i l l Books, 1967. H e s t e r F l o r e n c e M. and T a g a t z , G l e n n E. "The E f f e c t s o f C o g n i t i v e S t y l e and I n s t r u c t i o n a l S t r a t e g y on Concept A t t a i n m e n t , " The J o u r n a l o f G e n e r a l  P s y c h o l o g y . LXXXV(1971), 229-237. K i n g , F. J . and R o b e r t s , D e n n i s , and K r o p p , R u s s e l l P. " R e l a t i o n s h i p Between A b i l i t y Measures and Achievement under Four Methods o f T e a c h i n g Elemen-t a r y S e t C o n c e p t s " J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , L X ( 1 9 6 9 ) , 244-247. M i t c h e l l , James V. " E d u c a t i o n ' s C h a l l e n g e t o P s y c h o l o g y : The P r e d i c t i o n o f B e h a v i o r From P e r s o n - E n v i r o n m e n t I n t e r a c t i o n s , " Review o f E d u c a t i o n a l  R e s e a r c h , X X X I X ( 1 9 6 9 ) , 695-721. P i e r o n e k , F l o r e n c e T. "A S u r v e y o f I n d i v i d u a l i z e d R e a d i n g and M a t h e m a t i c s Programs," C a l g a r y C a t h o l i c S c h o o l B o a r d , C a l g a r y , A l b e r t a , ED 047 894. S a a r n i , C a r o l y n I n g r i d " P i a g e t i a n O p e r a t i o n s and F i e l d Independence As F a c t o r s i n C h i l d r e n ' s P r o b l e m S o l v i n g P e r f o r m a n c e , " P a p e r r e a d a t t h e a n n u a l m e e t i n g o f t h e A m e r i c a n E d u c a t i o n a l R e s e a r c h A s s o c i a t i o n , C h i c a g o , 1972. S a t t e r l y , D. J . and B r i m e r , M. A. " C o g n i t i v e S t y l e s and S c h o o l L e a r n i n g , " The B r i t i s h J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , X X X X I ( 1 9 7 1 ) , 294-303. S p i t l e r , G a i l J . " I m p l i c a t i o n s o f R e s e a r c h on C o g n i t i v e S t y l e f o r Mathe-m a t i c s E d u c a t i o n , " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , Wayne S t a t e U n i v e r s i t y , 1970. S m i l l i e , K. W. An I n t r o d u c t i o n t o R e g r e s s i o n and C o r r e l a t i o n . T o r o n t o : The R y e r s o n P r e s s , 1966. T y l e r , R a l p h W. "New D i r e c t i o n s i n I n d i v i d u a l i z i n g I n s t r u c t i o n , " P r o c e e d i n g s of t h e A b i n g t o n C o n f e r e n c e . A b i n g t o n , P e n n s y l v a n i a : The A b i n g t o n C o n f e r e n c e , 1968. and o t h e r s . P r o c e e d i n g s o f t h e A b i n g t o n C o n f e r e n c e ' 6 7 . A b i n g t o n , P e n n s y l v a n i a : The A b i n g t o n C o n f e r e n c e , 1968. W e i n s t e i n , M a r i a n S. "A Study o f Types o f A l g o r i t h m J u s t i f i c a t i o n I n E l e m e n t a r y S c h o o l M a t h e m a t i c s , " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1972. and o t h e r s . I n d i v i d u a l l y P r e s c r i b e d I n s t r u c t i o n . W a s h i n g t o n , D. C. : E d u c a t i o n U.S.A., 1968. 71 W i t k i n , H. A., Dyk, R. B., F a t e r s o n , H. J , Goodenough, D. R. and K a r p , S. A. P s y c h o l o g i c a l D i f f e r e n t i a t i o n . New Y o r k : W i l e y and Sons, I n c . 1962. W i t k i n , Herman A., Goodenough, D o n a l d R. and K a r p Stephen A. " S t a b i l i t y o f C o g n i t i v e S t y l e From C h i l d h o o d To Young A d u l t h o o d , " J o u r n a l o f  P e r s o n a l i t y and S o c i a l P s y c h o l o g y , V I I ( 1 9 6 7 ) , 291-300. W i t k i n , Herman A. "Some I m p l i c a t i o n s o f R e s e a r c h on C o g n i t i v e S t y l e f o r P r o b l e m s o f E d u c a t i o n , " R e p r i n t e d from P r o f e s s i o n a l S c h o o l P s y c h o l o g y , V o l . I l l , C o p y r i g h t Grune and S t r a t t o n , I n c . , 1969(mimeographed). W i t k i n , Herman A., O l t m a n , P h i l i p K., R a s k i n , E v e l y n , and K a r p , S t e p h a n A. A Manual F o r The Embedded F i g u r e s T e s t s . P a l o A l t o , C a l i f o r n i a : C o n s u l t i n g P s y c h o l o g i s t s P r e s s , 1971. APPENDIX A INSTRUCTIONAL MATERIALS •cr.e oc-arc. — 1 1 « 3 ~ v t o i : tne r.ii, iractions 5-Ler.s suc.i , 2 V 3 5* T the s i d e s as C? THE RECTANGLES I HAV3 SHADED? NOTICE THAT TK2 ORIGINAL SQUARE HAD A R I A 1. 3* A / / ! .18 ans".'"6i" i DV )H CF T H i S E RECTANGLES? WHAT HAPPENS I? I ALSO C I T THE SQUARE THIS WAY? ' :? T •..I-:AT ARE THE D L Z N S I C N S C? HE NEW S E L L E R RECTANGLES FORMED? Ex DOC t the answer 5Y A M U L T I F L I C A T I C N STATEMENT, NAMELY 3;* 3-'.THAT I S THE AREA C? EACH C? THESE SMALLER RECTANGLES? NOTICE THAT I KNCW THE AREA CF THE ENTIRE SQUARE WAS 1 SQUARE UNIT , 30 THE AREA CE i A C r . OF T.iESE RECTANGLES HuST HE A FRACTION CF THAT AND THE i S DETERMINED BY THE NUMBER CF RECTANGLES THERE ARE I N THE .-.2 answer c . — "L A.N3LZ5 ARE THERE CF DIMENSIONS 5Y 3 4. WAY A3 ^ V 3 .2 * d 4. v ; W C U L D ? Z : D UIA.",; A ? . 3 : r - A : : : -1 •1 5--- ± A i < P C U T I T : -f f M M ' J i i i i i MU. V.L-.Vf . V A T . - i t ? . 7 . . - . : : 0? 2AGH? :pect the answers 23 and respectively J 3 - i : JL 5 1 T8- YflLL 33? zjsoect the answer Lo'T'S CHiSK V/iTK A DIACr.iA.1. I CA!w DAA'.7: THE DI.-Z::SIC:;3 C? EACH i av £ 3* 3fe Uw .iv'l.v/j JL.IriA .1.. .iA - . . O — < t ... —J | .... X ..^ivii.WA a A X •J'-Ji .-v 1 r 5 H e r e t h e s t u d f i n d i n g t h e • x 1 o r 1— • X 1 x A 5 7 ** 1 SUPPCSS v/s Y<A:;T TO :-:ULTI?L: 3Y • r? :-~ir".l q i - . ' i ? -.'7 P * ? T ' ' T 1 •TH3 A33A C? A naCTAXSIS WITH TK2S3 DES.'SICSS TO 50LV3 CUR HIC3L31-:. 30 L3T"S START C U T V/ITH A 3QUAR3 A GAEL i? v/3 V:A:;T ess C? THS DI:3:;SIC:;3 TO BS 4 . ••/s CUT IT SC 1 3 XT 9 E x p e c t t h o a n s v o r 6- . Vw.. ..--LEO Z:~-*Jz,'J LA • ± 3 s 3^ . la . S::J\j'£D ?iiC33 AND F I G U R E . L E T ' S F I N D I DRAW: DE:CO:-LTNATC?V T E L L S :-LE T H E TOTAL NAFFER OF i ; 3 5 A r : 3 r , 3. :FE Nul-33?. OF SHADED SQUARES I S 3 x 2 . EACH CF THESE SQUARES •^ fq 3IICC3 THERE ARE 5 x i ' J J. . j i / j A i U i . i i . .-i-v.Li.-i V,-;? T H E R E F O R E , 5 3 is .THAT WOULD x ^ HE? L ... u 5 2 —-U . „ o O OiiO C.ii J n o ^. • 3o WE DC HAVE —•*"- FJ PVT. L'J «i.. i | F> W' .-A 1 . «-\ rtF*» I—T, 5o 2 c (8 = 4 x 2 ) JUST A3 THE DENOMINATOR CF MY PRODUCT WAS THE PRODUCT CF MY DENOMINATORS (33 = 6 x 5). ^2 5 ±_ 3 ar.sv; ere the s t u d e n t - . r i l l i e a r n t f i n d i r . n the areas of a o D r o s r i a t e r e c t a n g l e s v r i t h d imensions l i k e 4? ^ 3 s 5 • 3. 3 3 Fl ^.3 ii c?. DER TO F IND PRC T3 LIME D THE 30 "3 L I : 3 THE •J. FOLLOW y. 1 1 ' /' , 1 1 / ; y ' /, 1 i i 1 1 ~4 x n 1 7- 3 3. RECALLING THE REVIEW WE DID EARLIER ABOUT SPLITTING UP RECTANGLES WITHOUT CHAINING AREA, I REALIZE THAT I CAN SPLIT THIS RECTANGLE INTO TWO PARTS SO 3 AS TO MAKE THE COMPUTATION SIMPLER, ONS RECTANGLE WITH DIMENSIONS AND ANOTHER WITH DIMENSIONS 7^  BY ~^ SO LONG AS I ADD THESE TWO NUMBERS. *f~s 7— BY Z AND THE AREA REMAINS THE SAME 1 / y V//Y7 ~ 1 ^ x 2 1 x 3 E x p l a i n t h a t t h e y m a y a l w a y s g o back t o a r e n a m i n g s t a t e m e n t t o v e r i f y t h e i r answers, ** H e r e t h e s t u d e n t s w i l l l e a r n t o f i n d t h e p r o d u c t s o f u n i t f r a c t i o n s s u c h a s S U P P O S E I W A N T T O F I N D T H E A N S W E R T O A M U L T I P L I C A T I O N Q U E S T I O N W H E R E B O T H O F T H E N U M B E R S T O B E KULTIPL3D A R E F R A C T I O N S . F O R E X A M P L E , W E M I G H T T R Y A x \ W H A T D O Y O U S U G G E S T T H E A N S W E R M I G H T B E ? W a i t f o r a r e s p o n s e o f b e f o r e p r o c e e d i n g . L E T ' S S E E I F T H I S I S R E A S O N A B L E . W H A T D O W E K N O W A B O U T ^ ? O N E T H I N G W E K N O W I S T H A T H *W = \ = 6 x 7 ] = 1. S O , W E K N O W T H A T 6 x =1. T H E R E F O R E , ^ I S A N U M B E R I C A N M U L T I P L Y B Y 6 T O G E T O N E . W I L L T H E R E 3E A N T O T H E R N U M B E R S I C A N M U L T I P L Y B Y 6 T O G E T O N E ? i . I F T H E R E W E R E , T H E Y W O U L D H A V E T O E Q U A L % S I N C E T H A T I S E X A C T L Y W H A T W E M E A N B Y J - _ ( g x -jr = 1. T H E N , I F 6 x ( x ) = 1, T H A T W O U L D K H A N T H A T ~iK i = i S I N C E "t W A S T H E O N L Y N U M B E R I C O U L D M U L T I P L Y B Y 6 T O G E T O N E . H ^ T , 6 X- d% = 3 x 2 < i x = (3 x 4 ) x (2 x 4=. ) = 1 S I N C E I C A N M O V E T H E N U M B E R S A R O U N D W I T H O U T C H A N G I N G T H S A N S W E R N O T I C E T H A T A L L I D I D W A S T O D E C I D E W H E T H E R W A S T O S E E I ? 6 x ( i - ^ ^ = 3 - 3 J o X ^ ) = 1 . JS HOW WOULD WE FIND x WE CHECK TO SEE LF J± x 3UT, 23 x ( V ^ ) = 7 x ^ x ( ^ 4 j ) = (7 x ^ ) x (4 x ^ ) = 1 x 1 1 SO, 2$ x ( \ * ^ ) = 1, SO, 2 # WHAT DO XOU THINK 4 *=- WILL EE? 2> V Fi* SEEING IF 2S x ( i X ~ -7 ) = — _ J+-L -t-L +_L +-L^ L 7 X = T * - 7 ^ ^ T ™ T - 3 = 1 it x 1 = -L +-J. A—L +-L = l Expect the answer ^ 3 LET US CHECK THIS: IF THIS IS SO, THEN 18 x ( ^= * 3 Co BUT, = 6 x 3 x ( ^ * = (6 x % ) x (3 x i ) ) = 1 . 6 x 4 3 x 1 j . , i V L +i v i +-4-(pT |p \0 © to SO, 18 x ( A. ^ ) = 1, so 3 v» DO YOU NOTICE THAT IN EACH OF THESE PROBLEMS, IF I MULTIPLY " T B l T I GST AS AN ANSWER k FRACTION H * ~ S • T H I S SEEMS REASONABLE SINCE Hi EACH CASE IF I MULTIPLIED BY " 5 * ^ , I GOT ( Q.< ^ ) X ( < -= = 1 x 1 = 1. L* Here t h e s t u d e n t v n . l l l e a r n how t o f i n d p r o d u c t s o f any two p r o p e r f r a c t i o n s Ft 5 2, 3 S" 1 5 - P O S E -.- ,T-~. 1 .... 1 • •'•E HP 0,7 THAI S'—Z RE. i : FIND THIS ANSWER. 3 — BE RF:;A:ZD AS 2 x ^ 3 A::D a 3 x if = ( 2 y. 4 ) M 3 x i ) 3 4. O _.. ^  i . ^ .-i ., MULTIPLY MM; AMY ORDER THIS IS TFIZ 3A. 1Z A3 V. s~ J ) x ( A % ) VS ALREADY ....... T -j_ „ A. 1 1 • 7 THAT ( 2 x 3 ) x = THERE? ORE, a. 3 _ t T A " M i LET'S ?I:;D ? x USING THIS APP 5 3 PI • J~ . I.'wli Fj • o CL = ( 3 :•: 4- ) x ( 2 x 4r ) = ( 3 >: 2 ) x ( ^ x % ) = ( 3 x 2 ) JL5 4^  1* 45 1. -- A 5 -15 ::e c .-5 o v.. c:- : Js^ 3o it ( ^ >: J, ( 2 ^ ) ^ ) 3 t f 3 H A. 3 5 o.ie Doar-a x.2 sn.o-.-r ens rer.a: is ZrJiz t n e n u r i e r a ^ c r o i r a c t i o n s they are n u l t i n l v i i i : : s i r n ui .s r a c.c r s t n e i r reria::in£ tas:-cs Q i c rns c f the f r a c t i o n s •—- and as o.iey a r e c o i j . e c " ccries i r o n ce they are c c l l e c t i . i . i n - - denor . ina tc rs s tudent - . T i l l l e a r n to f i n d c r cduc t s of f r a c t i o n s and. n i x e d nun; sucn as * - 2 FIND PRODUCTS LINE A x 2 .3 ^JrrOoR 1 WAN FIND J3 2. x 2 A.i D x 4 _ ^ x 3 AP SPLITTING- THE 1PPXPD PUI-3PR PFERD JUST AS I DID './HEN ' Z iST PROD X T IS V2 'v. J1 X <i "3 A:;D 3 V A : , D 3 A AND Co 1 5~ Have th ree s tudents show t h e i r work on the b o a r d . I f any s tudents use denominators e ther than the products of the g i v e n ones, e x p l a i n t h a t they a rs c o r r e c t ( i f they a r e ) , but ask the.u to use the product as a guar antes t ha t the procedure w i l l work . i . 3 .-or example, f o r — and ,we r i g h t use the denominator ~ , bu t ve c o u l d c e r t a i n l y not do t h i s f o r -A and 3. ; p o i n t out t h a t we can always use the : : r cduc t . 3s sure t h a t the s tudents r e a l i z e t ha t the product c f the two denominators can always be used s i n c e we are i n s u r i n g t h a t each of the s m a l l p ieces i n t o wh ich each of the diagrams i s d i v i d e d can be w r i t t e n as an i n t e g r a l number of the new s u b d i v i s i o n s , and the re fo re the t o t a l f r a c t i o n s we s t a r t w i t h can a l s o be w r i t t e n as i n t e g r a l m u l t i p l e s . 4 *Here the s tudent w i l l l e a r n t h a t > ^ o n l y when y b<c , f o r example, -Q on ly because 3 x 5 ^ 2 x 6 . * * :cw iw§ WE A L R E A D Y R E A L I Z E THAT AU E A S I E R WAY TC CC: : ?ARE TWO FRACTIONS TO DECIDE WHICH I S LARGER I S TO REWRITE THEM A 3 F R A C T I C H S WITH A CCMMC:: D E ; ' . C H I N A , C R . . W E HAVE PRACTICED R E W R I T I N G E Q U I V A L E N T F R A C T I O N S . ,'-.-..-.-.Ru. J_ ,^ I ^ m , i SUPPOSE I WA:;T TC COMPARE n WITH T, TO F I N D WHICH I S G R E A T E R , I C A : ; RE -..7.113 THESE AS E Q U I V A L E N T FRACTIONS WITH THE SAME D E : : : . - Z : ; A T : R . WHAT FRACTIONS WCVLD I USE ? / O -:->-o^  tn-3 answers: — and 1 . X ± < £ _ • v*.'.- rt. v-L- > -» do k oto •EATER? mswe: S <o _TP A : : D 53 ' 1 A - — u.-. x x ^ A : . . r * r t i / 1 - C - L i v C - j J fVj i " « T\ o " r~ "* T - c T ' ^ * * -*rtiJ ^ x JL—»s* — J - / i _L i uA iv 50 I C 1-3_JD IS LARGER. SINCE 1 x 5 < 3 x 2 , THEN uF 1 X /AS 3U3DIVIDSD INTO 5, AND THE x c oujjDi ' / iD^D I. ' i iC 2 . I '.iE 3'w^D_v'j.3IC:.S ARE NCW ALL COUNT THE NUMBERS C? EACH TC TELL WHICH FRACTION ± 3 3_ 4-5 ;AN ,<E Expec t the answer: change both i n t o t w e n t i e t h s and then see i f the numerator of the f i r s t i s g r ea t e r than the one of the second . Or, e x p e c t : m u l t i p l y .3 x 6 and compare i t t o 4 x 5» AND cc: L I S 30 5 S 5 0 COMPARING 3 x 6 "WITH 4 x 5,. S I N C E 3 x 6 SAYS I CHIGIN'ALLY HAD J3 P I S C E S , EACH OF WHICH WAS CUT INTO S I X T H S AND SO NOW I RAVE 3 x 6 , i AND 4 x 5 , THAT I O R I G I N A L L Y HAD 4 P I E C E S , EACH 0? WHICH WAS CUT INTO ' F I F T H S , AND SO I NCW HAVE 4 x 5 P I E C E S , A L L CF THESE CF THE SAME S I Z E . SAME A3 COMPARING THE NUMERATORS IN THE ABOVE METHOD. CAN YOU SEE WHY THIS IS CALLED THE CROSS-PRODUCT RULE 7OR COMPARING FRACTIONS? I TAKE THE PRODUCT ACROSS THE FRACTIONS AND COMPARE THESE: FOR EXAMPLE, IN \ .AND 1 I COMPARED 3 x 6 WITH h x 5 WHICH COKES FROM Ask the s tudents t o f i n d the l a r g e r i n each c i these p a i r s by us i n 3 the c ro s s product r u l e . Have a s tudent show each 01" these cn the beard d i a r r a . ' r - a t i c a l l y e x p l a i n i n g vhy the rule w o r k s . 3 5 CR t 3 OR 4-<? 5 5 lo J_ r \-> 3 3 5 :o sure the students realize that vre exanir.e th nar.e i c r i t o ce aseo. i n renarrJLn;: i.xile, i 7 A " u 7 5 c n i y cecause cniv vrhen a * ^  ? -?*C 3 x 5 > 2 x 6 . ** i o r .>« W i.iA'i v/3 Aiov^rtDi' R:^Ai_.Z_ Tr_A'i' A.I ^.-.JJ L£_! .'.AY T- Cw.rArv— IVi't TO D3CID3 V/HICH 13 IAHGZH. 15 TO ?. 3HAI--Z r*.* r-*- • ' • o * ' ' * T •*~5 ~" " " ^ • l i t . ; . .xi . i A \ > v . _ - . u - i r.-vAT/ . 0 ,-iAv-i rvwio. — _ — — ; w CA.-i GJ r _,~. p r r - n i O T " - ' m " r O » i vO - j _ i i U w o . ; ~ ~ A r » i i - - ^ iL T., „_.._ .,.__ iS i_<A . 1 « A . i r . E . , . - k . ^ id_o_. AO .•.<jj-.i_ri.i--. H i V . • 3X,?.-v.335ICi-,'S V.rITH A i - p . v . • 4 V i • .v.iAT wii__ TH23Z ZAPRZOSICNS 33? the answers : __* ^ * (.%: * ^ and * ( 4 - i). T cv: C'-FC'C THIS : J- • = 1 x 3-- J - -- V x = 1 x ^ - x(6 X \ ^ • = ^ x \ x(2 x = 1 x 6 x ( \ x ^ 4 = 0 x 2 x ( x i _ ) = 6 "x ( i . x i - ) -2- W ) 1:10.., i i-.0'i_t->_i '7.-.A? i A.. CC.rARxi. C - I IH 9 nsv'er : 2. KCTICS THAT TH3 FRACTION WITH 2 AS DENOMINATOR WAS MULTIPLIED B I 6 x -Qy , SI2ICE THE OTHER DENOMINATOR WAS 6 . WHT WAS THE FRACTION WITH 6 AS DENOMINATOR MULTIPLIED 31 2? - 6 C a *: " 9 o trier ceric-inater was 5a. 1 x 6 AND ^ x 2 . It AND I SAY 1 x ^ x SUPPOSE I WANT FIND 31 o 3 UT7 A L , WHAT EXPRESSIONS WOULD I USE? Expec t the a n s w e r s : £*S * ( ^  *-|)ar.d 3 * ^ * ( \ 5 * r J . I CAN CHECK: 3. = 1 x % — 1 — - L x A a. = 1 x 5 x PJ i r^ -rt 4 - ( 5 X 1. <=• ~ 3 x 3 > . V i i : = 3 x 2 x ( 4- x 2. 3 -.1 ID - . z r ^ _3 expect the answer: m u l t i p l y the f i r s t by 6 x J. and the second by or e l s e compare 3 x 6 w i t h 4 x 5« ^ x ^ , % - 3 * i iv^: = 3 x x (6 x ^ ) = 5 x = 5 x -V x (4 x 4 ) = 3 x 6 x ( x = 5 x 4 x ( -L x J L ) OR ELSE WE MIGHT SIMPLY NOTICE THAT THIS IS EXACTLY THE SAME AS COMPARING 3 x 6 WITH 4 x 5 , SINCE 3 x 6 TELLS HE I ORIGINALLY HAD 3 x A FRAC TION AND THEN MULTIPLIED IT BY 6 x 1 AND THE 5 x 4 THAT I ORIGINALLY HAD 5 x ANOTHER FRACTION AND THEN MULTIPLIED IT BY 4 x JL 5 ) . CAN YOU SEE WHY THIS IS CALLED THE CROSS-PRODUCT RULE FOR COMPARING FRACTIONS. I TAKE THE PRODUCT ACROSS THE FRACTIONS AND COMPARE THESE. FOR EXAMPLE, IN H AND I COMPARED 3 x 6 WITH 5 x 4 WHICH COMES FROM zr.Q s t ucen t s t.o u n a tr.e . larger m eacn ox i u c t r u l e . Have a s tudent show each of thes £ z.n" n " w*1** the r u l e works . e on tne ocar-u • 5 S ~ -A . . ? Too ^ T o o Have the s tudents who read cut t h e i r answers e x p l a i n i n terms c f e i t h e r a diagram or an e x p l a n a t i o n of t h e i r rnamir.g hew they got the c o r r e c t answer. Expec t the answers: ones and 5 t en ths and 2 hundredths and 9 thousandths , or 34 ones and 529 thousandths , or 34,529 t housand ths . I f not a l l of these answers come up, mention the ones t h a t d i d n o t . P o i n t cu t the analog;.' t o the hundredths s i t u a t i o n . Hi PARTICULAR, I CAN REWRITE A WHOLE NUMBER, SAY 4, A3 TENTHS CR HUNDRl CR THOUSANDTHS. THAT I S , 4 = 4 .0 = 4.00 = 4.000 , 30 4 CAN 3E READ AS 4, 40 TENTHS, 400 HUNDREDTHS, OR 4000 THOUSANDTHS. Hand s tudents worksheet 1 to comple te . ** Here the s tudents w i l l be i n t r o d u c e d to the i d e a of d i v i d i n g dec imals bp wholes .** SUPPOSE 'WE WOULD LINE TO EIND THE ANSWER TO 1 . 5 - r 3 ? 'WE HN0W THAT 15 - f 3 MEANS TO SEPARATE 15 INTO 3 C-RCUP3 0? T: SIZE AND THEN FIND THE SIZE 0E EACH 07 THESE 3RCUP3. THIS DIAORAm WOULD SHOW WHAT WE MEAN: 000 000 ooc 00 00 00 I S lo ..o i'lUi i.i-.i l'.iio A A 3 'i'.-jj irt.--j HO . " u i U . ' i J —5 iv'ii w rt.ii 3»: 3 OR- Ur5 AI.D i'HEN .•vol-.^ MH^ Jiil. 3 THE TYPE T."_A'i' x5, T.-E A.o.rEH WAS ( i 5 T 3 ^  TENTHS, J r J . . . D l . c T * . . IT - A T D J i o u o.\>i^CI T."^ H.iS.io?. To no? Expect the answer.: 4 tenths or . 4 . LET'S ERAi A PICTURE. 1 .6 = = 16 T E N T H S , THE: ; , TC PU] L6 TENTHS INTO 4 CROUPS, I DRAW: r.ave tr.e- s tucen t s ; a i c t the answers t o : 4 .5 • 9 ! /-.s.-c s tucen t 5 *** ° eao. ou* L^)ifi'_ of these , draw a d i a : r as con : ,s 1 .Oc .•.ar.a stuuer:* i e a v s c e c i - m a i s i o : iver tr.at anot..£ .on i s a i v i s i c n . - s , 3 i _ = i - 2 , a n d so t o s e t t h e 2. QZViO.; 2 ini I n A / i T.-ii E;\ACTiC.; ^ . IST 052 322 WHAT WH0L3 NUHEnR THIS i.3 ANCTHErC i\AHH rCR.. DO YCJ .-'CHOW ALREADY? E x p e c t t h e a n s w e r : 3« I n a n y c a s e , go on w i t h t h e f o l l o w i n g d i s c u s s i o n . n - i n T ** • 'r -* 7—* T ' r"-* — V o ~* s * i • - ' . r> -f-7 — d- . .—. ..... i n i J. rji'o i.iOi . . o r i i v j j OiW.i o i io ~rr • w.n iVr . i —•—-5 p-7 - p*.- T — q n - t-p.--; —1 * E? ""T" — "—J-*. • t J i-d-Et A TNU-*.WIJ J-i-V - J _L.- ~7^ J .-.Y _RA.v_hG J_.DICAT.J2 T.iAT 3 J-3 THE CORRECT A-SWER. NOTICE THAT EACH TIME I WAS FINDING HOW MANY GROUPS CF THE DENOIFINATCR IIHERB '.FFRD IN THS NUMERATOR (GROUPS OF 4 IN 12, GROUPS OF 7 IN 42, AND GROUPS OF 6 IN 12) . VFS ALREADY HNCW THAT FINDING THE NUMBER CF GROUPS OF ONE NUMBER IN ANOTHER IS THE SAME AS DIVIDING THAT SECOND NUMBER 3Y THE FIRST. BUT FINDING THS NUMBER O? GROUPS ON ONE NUMBER THERE ARE IN ANOTHER (LIX3 THE NE__3R CI FOURS IN TWELVE) IS THE SAME AS FIIIDEJG THE SIZE C? EACH OF THAT NUMBER OF GROUPS IN THE LARGER NUMBBR(LIK3 THE SIZE OF EACH OF FOUR GROUPS OF TWELVE IN TOTAL). • 0© ©0 ©0 ©0 ©0- ©0 2 V 3 — J • t , o •. ) ;?3 OF 4 L; 12, J A i o _ rU..D -cp TC SHARE ALL 12 / «. -2> \ ^ J". L r w . i . 'w O _J-L .Li ? J \ 0 .L. ".•32?. OF 3272.15 IN 42 J.S TI-7 .2 3AHS A3 FiNDINC- IH 42 3Y 7 TC FIND T; i 3IZ2 CF EACH OF 7 . i OA.. D.\A.'i( i '0 SHOW HLW .CANY 7 ' s I N 42) OO OO (X) OO QO o b cooo'oo o o o o o o o o o o CAN 21.A -.7 THE DIAGRAM HERB TO SHOW HOW MANY OBJECTS IN EACH OF THE ? ROUPS FOR SHARING T H E 42 OBJECTS, AND I GET THE SAME ANSWER. = 4, ~ 1 7. c_5 O M • = 3 = 15— 6 SINCE I WAS FIMDIMC Z AS FIND G THo SIZE CF EACH OF 6 GRO; wTiONS, SITE A DIViSiON SEN 3 j . .^ ZR ;.A..o F'wH TrH. FRACTI.W.: . /5 s t u : :o rsac. out t r . s t r answers ar.u ;.-.a: ar .c tr . rr : a i i , y e x p l a i n tr.e:. •v.".or. can ar . ;avs oe u.'Vwj.\5 • VA.. D __ _, o_? 2 ^ — L^-iFv • i • , '»_•«_-_.. . CT_JJ »' r v : * i — i J v -» _D w . \ _L r*___. _____ UD __i v _ _ i . i i _ j . u.-«A.__i_ ..i___j i r - j v __-'-•-* ARE USED U?. Xw » _ r _ i » , i _ _ i -O r ^ ' i w _\o i _ --.ii _»rt_-. * _____ • 0 0 0 0 0 1.0 j i z ZJ F_> _ .-__r. i _ i 3 A . » D 2. -_f ' • I —— i t A -_ _. F_» • > 3 -4-4 ^ J 3.0 -3> ^ J 30 T E N T H S 77 0 TSHTH; 0 T E N T H S 7 T_.NT)-V> File/. 3 T O 3.0 ^ 3 »00 IF .3TF_.AD , T F I X O SFJZ. -3 K^A3L.i.\zLcli J. Cr_A_-.3F._D J_T A V - X J PR-_}__.-•__ o< r . _ . . - _ r i x . . J , _ . i , _v_ _. .•___.-i .... }0 O N E S T E P F U R T H E R . ^ J3Tc 4 ) 300 HXiDREDTHS 2oQ H U N D R E D T H S 20 H U N D R E D T H ; 20 H X : D R E D T H S 0 H'-.._L.EDT;i5 ^ U __ w'_ ( J . 0 1 rio 5 H U N D R E D T H S 75 HUF>DREDrH3 = .75 • 75 . T ••n-7> t T- r A L . ' A Y S G E T R I D C F T H E ttVAZ'.DZ?. E V E N T H E H U N D R E D T H S O R T H O U S A N D T H S , B U T VFE C A N K E E P T R Y I N G P L A C E S E I T H E R S T O P O R A R E C C N V I N C E D V G C A N N O T S T . 11 Fl F"„ \ __ D_. FIFJ ,1 \J jli i. F___.Fw • I'V F_ v _-. „ * i i . FFF-j a - ^ s = 25 as 25 J 2.0 —> 2 5 ) 2 . 0 0 25/ _ u J .-J.>i..'._-_/..-.0 } C\J<J . . . £— -5 0 = .0: ^ t o c c r . s l o t e , o x c e c :he answer: I J . z l . .-lave the students who read out their answers explain in terms of renaming how got the correct answer. 3IMIIA.-tLi, j. MIGHT RzAD AN3 '.vRxTS DECIMALS INTO THE THOUSANDTHS. HO,/ V/CULD I HSAD 34.529 ? Expect the answers: 3^ ones and 5 tenths and 2 hundredths and 9 thousandths, or 34 ones and 529 thousandths, cr 34,529 thousandths. If not a l l of these answers come up, nention the ones that did not. Point out the analogy to IH PARTICULAR, I CAN REWRITE A WHOLE NUMBER, SAY 4, A3 TENTHS OR HUNDREDTHS OR THOUSANDTHS. THAT IS, 4 = 4.0 = 4.00 = 4.000, SO 4 CAN BE READ AS 4, 40 TENTHS, 400 HUNDREDTHS, or 4000 THOUSANDTHS. Hand stude; ;s worksheet 1 to ccmolete. 'Here the students fwij .'io tivJlfD U^kL i ^ i . i D i'."w-* H.»5./-1.5-^ -3 = U 1.54- 3 3 = 1.5 . - _ 4__ 1 5 = 0-L. 3 5 -lo \0 \o \a io 10 i° *° \ ° 1 0 ' ° 1 0 l o l t r | ! - 1 0 U x 3 = 15 :• F_."V__ . ._rv f — i _, s-/ . w.j ?c?. EXAMPLE, TO 5CLYE 3 X Q _ = 3 x 5 x 4 , I ?3?LkZZ X O . x 3 = 15 x 4 , , ',•13 SEE _HAT WE CAN REPLACE Q BY 5 x ____ AND THEN O x 3 =<0 x ik\x 3 = 5 x 3 x -_ -0 0 x x 3 A, = 1.5 -lo NOTICE THAT THIS WAS THE 3A.PE A3 F I N D I M - '.THAT I HAD TO MULTIPLY 3 SET 15 AND THEN RE IFE". PEER INS TO MULTIPLY THAT ANSWER 5Y _rv . SO TH A--5J..R WAS Cl5 "f 3 ) TENTHS . --1-G -ET«3 FIND 1.6 -1 -4 . WHAT DO YOU EXPECT THE ANSWER TO BE] he answer: 4 tenths or .4 Q x 4 v x v 16, 4 TENTHS x = 16 TNETH3, 30 XX = Vs.vy e~-._ "reo_.c - -:.e answers zo: rtSK s - u a e n t s t c r e a u cut, ^ n e i r a n s w e r s , tr.ere i s anv :se, rewrite as mu i m u t.ne s o i u t i c r.iYii.-f :C7:;;D 1.25 -4- 57 FIND 'WHAT NU.-^om .Ui i ..Ort..O Q x 5 125 HU:;IH3DT:-13 125 BY 5 AND THE:: 1 A:: DZALINJ WITH HUNDREDTH Ask the students to predict the answers to: . 2 5 - 5 1 . 2 5-r 2 5 2 . 0 0 -f 4 Have three students explain their answers. If there is any difficulty, rename as mutliolication sentences as illustrated above. \~j-,.jLi L < A . ( A i T m A j . ' , V B U d j A L L Y u3S T t .VRIi'E rC3 EXAMPLE, T C - F I N D 4.2 - f 6, I W R I T S : 6 ) 4.2 ^ 6 )42 T1N7TH3 AND TC * » 07 IH3) i'o.-. i .-.o 1 = .? . . I W w \t J -L. • IA — A . j ^-THIS HI CI; LI. I: 4 .5 • 9 P. rr--.v - r - - ~ - ; r ; : 0 H U..N-moD-.i5 .06 nc. st.uas.its sheet 2 to complete , Here the s tudent w i l l now l eave dec ima l s f o r a w h i l e to d i s c o v e r tha ' another i n t e r p r e t a t i o n f o r a f r a c t i o n i s d i v i s i o n . That i s , 3 wi ld , t i e i n t o dec imal s when he d i s c o v e r s t h a t 2- = i ~ 2 , and so to get o.eetmaj. z or , he d i v i d e s 2 i n t o 1. * ' C7 A3 V/H S A I D 3 1 ? .vr.C-jUj IDiiA C? jN.TRCDU HATH A O L LAVE THE FRACTION ~ ^ SUPPOSE 13 ANOTHER NAH3 FOR. DC YCU iC.Ta-7 ALREADY? JJJT 33 SEo .'(HAT .THCiH) ilURHoR TmiS' Expec t the answer:3 . . i n any case , go cn w i t h the f o l l o w i n g d i s u c s s i c n . 12 12 x \ =(3 x \ = 3 x ( 4 x ^ = 3 x 1 = 3 AND SC I SEE Ti i . - . ' i i X .1-lV j i i i X . X vrt , i .-Co.,A.-. y H ^ A -++ *v-i (HAT WHOLE JO i o < , < i i , T SHOULD I HO- ? = £ x 1 -7 = 6 .r.ey GO not surgest t h i s , you renane f c r the.-. l.'C'TIC-lo Th_. Viu.C'L__ NUh 4J_ Expect the answer: 3. 12 x <3 x 6: 5)x = 3 x (6 x __- \ = 3x1 = 3 I c Z ' T * • r i T * * - T « . v * o ~-7 x D = 42 , A ; ; D 6 x U = 1 5 ) . T.__ u.-__.rtATCR 3 E : ; C : - : I : ; A _ C R ( 4 X Q ^ 1 . . . i i,*— Q X 71 Cr» 42 Hr2_ .?. I : . A . . ? I _ . , i ..'A3 ? L ; D I : - ? •= a ci* 42 - f -n =7 3-iom ..A.-_o ( A:CCT;{OA NA.CE) r-?. T H E FRA.C ns.-c J s tuaen ts to r eaa out •vers and tnen e x o o a m tnen ay rer.amin." i n , il / I a w l f J ^ 4 / > a ^ . ^fjs^j5- " i U i o t p I i c a t i o n s e n t e n c e . ** Here the s tudent w i l l l e a r n t h a t another name f o r a f r a c t i o n can always be g the numerator by the denominator , even when the d i v i s i o n i s found by c i v i c not even . ** .»^ .« ,vo . - L r i v o ooo;( jL.-Jii ? Lrt Su.-o r . - t A ^ i i U O , rt.wTooR .-'.-il C.-1 r NA.02 F u t THE FRACTION IS TC DIVIDE THE) DENOIONATCR INTO TH: NO OLD 1ZHE TO FIND CUT I F THIS IS ALWAYS THUS. NUMERATOR 5 I F CUR RUI 3H<VUJUD 3H LHAT NU.-^ioR LA S H O U L D BE IF I FIND THE SOLUTION TC Q x 5 = U , ±4. | LET ' 5 D C THIS. i SUPPOSE \B WRITE: U x 5 = 11. I ? LI = ^ , THEN IT ^ x 5 5 4^ : X 5 • x % > 5 ( 5 x = ) = 11 X 1 = 11 5 x -L-i. | rv. 11/ * JIl!x.'-_LT ^  .TL , 1 1 J- 4 i 5 „ O OJ .-i . _ j — i>_L .-v . i . 2. _J3-n x o r. - _ __/ 3 <3 3 (3 :•: x 3 U x 3 = 3 AND 30 34-3 = \ 5 , 3 * = 5 ? so:".scns to suggest: -5" x k = (5x = e; v x -1 ) = 5 x 1 since [£. no one does this, you writs out this rena.T_.ng scheme. DLL WHY 3 3 - ^  = n n, a x 4 , = ( 3x J ^ f c ^ • L - i 1 ^ . - 3 4? D x 4 = 3 ( ^ x i ) = 3 x 1 = 7 land stuaents worksheet 3 to co.-plete. 'OL. Lie sz--i~a , .'^  ^ » « _oci.-:-.xs anc ii^ac * _..u ecu._.vaxe: f r a c t i o n , 4--TV „*\ — i n • 10 =4 x 4 V 10 = 4 x (10 x ^ J = 4 x 1 = 4 H x 10 D O E S E Q U A L 4 , 3 0 - _ — = d o 10. ! cv: .~-"\.._J. . I - J r._.oo i_.iv^w . ' — i i . v j . 1 i ^ . i A O -n UZ^j .4 . V 7 - L L , 4 — 10 CAI - I B E W R I T T E N A 3 10 J 4 . B U T I ? I T R Y T O D O T H I S D P / I S I O N , I O N L Y S E T A H A N S V / E R O F 0 W I T H R E . 1 A I N D B R 4, W H I C H D O B S --L-? . —__I . _LC_i o R E W R I T E 4 = 4.0 = ~r. [ S I N C E 4 .0 = 4 O N E S = 4 x 1 = 4 x (10 x A ) = (4 x 10 ) x _ _ . = 4 0 T3 . .THS . ) 4 . 10 CAN _E .'iRx'i'T3:i: io p r —> 10 )4 . o 40 TENTHS ll P'-'q 0 TENTHS I 4 3_ d o 3"T ~J3 _L I3 D-i V13 -i V. • F*-E-iO v - j « FJ . . IHZ3 Tr_.vl'J3;I Di ' / I5IC.i; 10 J 10)6.0 -5> 10 ) 6 ;r ' '. 1 60 i _ -.5 j TENTHS 3-2 [ T — 1 2 , 5 WE — ^ > 2 ) l.O - > 2 ) :o T E : ; I H S I O T E N - ."io " ?.-.."T'-3 o T E : ; I E 3 3 -0 . . i.-.0 : I : . - \ L F C R 1-A = .5 E .5 = no ± .o . . i . - .o = o N O T I C E T H A T I C O . , V A R I E D T E E I I N T O I .O G E T A R Y . / H S R S 3 Y D I V I D I N G I 3 Y 2 , •g NOV/ L E T 0 3 T R Y A N O T H E R F R A C T I O N . 3 0 ? ? 0 3 E V /E S T A R T V / I T H "If A N D V /E W A N T T H E D E C I M A L F O R T H I S . R l V i oO i o o i 30 ve WRITE: 3.0 4 )30 TENTHS 2o i i . - a . i o J7 rp—• *. ti—i- - -»i j. IJ;« ± Il D 0 1 WHICH I AH . i ^ i o w..o o,t TO r-ANDLS. C o rOS3l3li j i-TY I S T O C H A N G E 3 I N T O 3.00 I N S T E A D . T H I S 3 3 3 . 0 3 R E A 5 O N A 3 LO . I C H A N G E D I T F R O : : 3 T C 3.0 O N L Y T O A V O I D T H E P R C S L S I O S O F A R E M A I N D E R , S O I M I G H T R T H S R . 4 HTCO —=> 4 J 3 0 0 HUNDREDTHS 2 5 0 •.J/.O^i.lO 2 0 . - . J . i D R E D i . " I 5 7 0 H U N D R E D T H S 20 H/JNDR3DTH3 \ oRoi>...-i3 I 7.? r.-..JRo^'iHS — .75 ou H I D TI" = .75 . , - r . H . i ^a r .d s tudents v o r k s h s e t 3 t o comple t e . f^l0 r i ^ V J ) **Hera the s tudents w i l l l e a r n about app rox ima t ing square r o o t s .of w h o l e s . ** SUPPC53 WE NOW WANT TO FIND A T T . W2 KNOW THAT 2 IS TOO SMALL, SLNCS 2 X 2 = 4 AND 3 IS TOO LARGE SINCE 3 x 3 = 9. SO WE HAVE A PR03LSM W3 HAD NOT PREVIOUS LT FACED. CN2 TECHNIQUE WE MIGHT USE IS TO CONVERT 5 TO A FRACTION AND TRY TO FIND A FRACTION NEAR I T . FOR EXA MPLS, WE KNOW THAT 5 = "^5 , -1023 111 ±d. AND . IS CLOSE TO , WHICH HAS AN "EVEN" SQUARE RCCT NAMELY - 5 -BUT THIS MIGHT PROVE VERY LONG AND TEDIOUS I F WE DON'T FIND THE RIGHT FRACTION RIGHT AWAY. LET'S TRY TO DEVELOP A METHOD. SUPPOSE WE TRY TO DRAW A SQUARE WITH AREA CF 5. WE MIGHT DRAW SOMETHING LIKE:-WHERE SBC TICKS 1, 2 AND 3 HAVE A TOTAL AREA CF 1 UNIT. HOWEVER, WE DO NOT KNOW THE SIZE CF THE H WHICH EXTENDS FROM 2. WHAT IS THE AREA OF SECTION 1? E x p e c t t h e a n s w e r ^ . 2 x O. . WHAT IS THE AREA OF SECTION 2? E x p e c t t h e a n s w e r : 2 x p i WHAT IS THE AREA CF SECTION 3? E x p e c t t h e a n s w e r : Q . x Q . • r i NOW IN ORDER CF S I Z E , SECTIONS 1 AND 2 COME EZFC33 3, SO LET US PRETEND I TEMPORARILY THAT THEY MAKE UP MOST CF THE 1 UNIT THAT SECTION'S 1,2, AND 3 \ M A K E U P A L T O G E T H E R . S O , T H E T O T A L A R E A C ? ' S E C T I O N S 1 A N D 2 I S (2 x Q ) + (2 x Q ) I S A 3 0 U T N O W I F k x Q I S A B O U T 1, W H A T L S D. 1 — E x p e c t t h e a n s w e r : a b o u t H" . r ~ ± r N O W S U P P O S E W E P R E T E N D T H A T Q I S . W H A T I S T H E A R E A O F S E C T I O N 3, j L N O T H E R W O R D S , W H A T I S U L x V_L\ ? S I N C E T H I S N U M B E R I S L E S S T H A N 2^  , W E C A N S A Y T H A T W E H A V E U S E D U P M O S T O F T H E E X T R A 1 U N I T O F A R E A T H A T I S N O T I N T H E 2 B Y 2 S E C T I O N C F T H I S S Q U A R E : , r - i B U T , £2 x Q ^ +(2 x iH") I S A N O T H E R W A Y O F W R I T I N G 4 x Q S I N C E I C A N D R A W : # E x p e c t t h e a n s w e r : - t t - a / - t •- ± A N D S O W S C A N S A Y , T H A T ^ 5 I S A B O U T -2 . W E W R I T S T H I S 4 1 T ~ z.% • W E M I G H T H A V E U S E D T H E D I V I S I O N P R O C E D U R E A G A I N A S W3LL. S O O U R F I R S T G U E S S M I G H T H A V E B E E N 2. T H E N W E W O U L D D I V I D E 5 B Y 2, A N D G E T 2 TJl U N F O R T U N A T E L Y , T H I S G E T S U S I N T O D I V I S I O N l* j 2 1 1 2 . W I T H F R A C T I O N S S I N C E C U R N E X T C H O I C E W O U L D B E A N U M B E R B E T W E E N 2 a n d 0- ~2. A N D S O , W E W I L L A V O I D T H E D I V I S I O N T E C H N I Q U E F O R T H E T I M E B E I N G . S U P P O S E W E W A N T T O F I N D A . 13 WS KNOW THAT THE ANSWER MUST EE BETWEEN 3 AND 4, SINCE 3 x 3 = 9 AND 4 x4 = 16, AND 13 IS BETWEEN 9 AND 16. SO, WS AGAIN DRAW A'SQUARE LIKE SO: ve RAVE USED UP 9 UNITS 0? THE 13 IN THE 3 BY 3 SQUARE, LEAVING US WITH -+ MORS UNITS DISTRIBUTED IN SECTIONS 1,2, AND 3. NOW WHAT IS THE AREA 0? SECTION L ? Expect the answer: 3 x 0 - . WHAT IS THE AREA OF SECTION 2? Expect the answer: 3 x O. • WHAT IS THE AREA OF SECTION 3? Expect the answer: CLx L-~L . AGAIN,WS WILL GO FROM LARGEST TO SMALLEST. IF WS TEMPORARILY IGNORE SECTION 3, VE HAVE USED UP MOST OF OUR 4 UNITS IN SECTIONS 1 AND 2 WITH COMBINED AREA: 3 x Q ' + 3 x Q = 6 x "Q . D A 3 IF 6 x p = k, WHAT IS Q ? •( WE KNOW THAT 6 x = 1 BY DRAWING LT SC .THAT 6 x 4 . = 4 f  ( I ! 1 - i . -! ij | 1 cn I -A J V-AND CERTAINLY W2 NEED TO MULTIPLY BY SOMETHING 4 TIMES A3 BIG A§5jA TO GET j »L • I THEN, IF WS GUESS L L = , WS FIND THAT THE' NEGLECTED AREA IS \ x 7^  = D x Q 1. ±9 WHICH 13 LESS THAN ( WHICH IS ~ ^ ) AND SO WE SAY THAT ^ 1 3 # 3 ^ • | SUPPOSE WS WANT TO FIND -AJ3?. IF WE DID NOT KNOW IN ADVANCE TO TRY A NUMBER i I NEAR 6, WS MIGHT USE CUR DIVISION PROCEDURE TO CLOSE IN ON A NUMBER BETWEEN i i 6 AND 7. I ! FOR EXAMPLE, WE MIGHT DO: | GUESS THE LENGTH OF THE SQURS WITH AREA 37 TO BE 4. THEN THE i | WIDTH =? 37 "f 4 * TPT\ 1 SO WE TRY A NUMBER BETWEEN 4 AND 9 FOR THE LENGTH, SAY 7. SO THE WIDTH » 37 — 7 7 ) 3 7 21 2 SO WE STILL DID NOT GST A SQURE. WS NOT TRY A NUMBER BETWEEN 5 AND 7, 6, IF THE LENGTH = 6, THE 'WIDTH = 37^6, 6 737] 6 AND WE ARE 3AKC TO THE POINT WHERE OUR DIVISION PROCESS CANNOT HELP US. HOWEVER, WE DO KNOW FOR SURE NOW THAT -$37 IS ZZ.T,iE3\i 6 AND 7. 50 LET US DRAW: f l \ WHAT ARE THE AREAS 0? EACH 0? THE SECTIONS 1, 2 AND 3 ? Expect the answers: 6 x Q , 6 x Q , and H x Q SINCE SECTION 3 IS S M A I I 2 S T WE MIGHT IGNORE IT TEMPORARILY AND SEE THAT SECTIONS 1 AND 2 HAVE A TOTAL AREA OF (6 x +(§ x Q ) = 12 x Q 0 0 SINCE THERE IS ONLY ONE UNIT CF T?FE 3? NOT EI THE 6 BY 6 SQUARE, WE HAVE TO ASSUME THAT 12 x Q IS ABOUT 1,' WHAT VALUE IS Q ? Expect tb.9 answer: about THEN WHAT IS THE AREA OF SECTION 3? Expect the answer: D. x G . = - U 4 "37 THIS IS CERTAINLY SMALLERTHAN , SO WE CAN ASSUME THAT A) 3 7 £, (_> oU • NOTICE THAT IF I HAD BEEN TRYING TO FIND -A) 33, I WOULD HAVE DONE THE SAME THING UNTIL I 'SAID THAT 12 x Q IS ABOUT 1. THEN I 'WOULD HAVE SAID THAT 12 x Q. IS ABOUT 2, AND SC D. IS ABOUT 2 x t w 1 -la. ^.3. I STILL WOULD HAVE CESCESD' CL X U. TO SEE THAT IT WAS SELLER THAN , • 2 - 2 . ± AND SINCE ^-gjL 1 2 s y i A L L E : R T H A N 7^ . 1 WOULD HAVE ACCEPTED THIS Ask the s tudents to work out the approximate square r o o t s f o r each o f the f o l l o w i n g and then ask s tudents t o core to the board to show t h e i r work . 8 13 50 67 ** Here the s tudents w i l l l e a r n how t o approximate the square r o o t s o f f r a c t i o n s whose denominators a re p e r f e c t squares , but whose numerators are n o t . ** NOW SUPPOSE I 'WANT TO FIND THE APPROXIMATE SQUARE ROOT FOR V T£ . I HAVE ALREADY LEARNED THAT I CAN SEPARATELY ATTACK THE NUMERATORS AND DENOMINATORS AND FUT TOGETHER THESE SQUARE ROOTS TO FORM THE SQUARE ROOT FRACTION. SUPPOSE VS DO THAT HERE. r — pi WE FOUND THAT Jd«5 ~ X "4- FROM OUR PREVIOUS WORK AND VE KNOW THAT AJ4 = 2 . SO THE APPROXIMATE SQUARE ROOT OF IS - ~ . SUPPOSE WE ARE TRYING TO FIND Al WE KNOW THAT %> 3 b AND 4 ^ = 2. . S O L E T ' S DO THE SAME THING FCR -ij |I WE CAN WRITE THAT THE APPROXIMATE SQUARE ROOT CF Have the s tudents f i n d approximate square r o o t s f o r each c f the f o l l o w i n g ; have a few s tudents show t h e i r work on the b o a r d . -V 32. AND 27 x 27 = 729, so A! 12S\ • &1 \^QcjJ^ia>n&. students worksheet 3 to complete, JXL'J ./j 5"** H ere the students w i l l learn about approximating square roots of wholes. ** SUPPCS3 VS NOW WANT TO FIND Aj 5. VS KNOW THAT 2 IS TOO SMALL, SINCE 2 x 2 = 4 AND 3 LS TOO LARGE SINCE 3 x 3 = 9. SO WE HAVE A PR03LEK WE HAD NOT PREVIOUSLY FACED. CIS TECHNIQUE 'WE MIGHT USE IS TO CONVERT 5 TO A FRACTION AND TRY TO FIND A FRACTION NEAR IT WITH AH "EVEN" SQUARE ROOT. FOR EXAMPLE, WE KNOW THAT 5 = 5 x 1 = 5 x 25 x 1 = 125 x 4K IS CLOSE TO 121 x A WHICH HAS A SQUARE RCOT OF 11 x 4 = ^ • BUT THIS £5 5 3 MIGHT PROVE VERY LONG AND TEDIOUS I F WE DON'T FIND THE RIGHT FRACTION RIGHT AWAY. L E T ' S TRY TO DEVELOP A METHOD. SUPPOSE WE TRY TO FIND WE KNOW THAT THE SOLUTION IS SOMETHING BETWEEN 2 AND J. LET US CALL IT 2 + Q , WHERE U IS SrlALLER THAN 1. THEN, SINCE (2 + XX ) IS THE SQUARE RCCT OF 5, WE HAVE (2+ a) x (2 + a ) = 5. HOW CAN WE MULTIPLY THESE NUMBERS CUT . WELL, JUST AS 4 x (5 + 6) = 4 x 5 + ^  X 6, WE CAN CONSIDER: ( 2 + 0 ) x ( 2 + O ) = (2+CL)x2 + ( 2 + U)x _Q • THEN WE CAN REWRITE EJ HE EXPRESSIONS CN THE RIGHT: (2+\__ )x 2 = 2 x 2 + Q x 2 AND (2+ T J )x n = 2 x E l + Notice that the use of the d is tr ibut ive law here i s very involved and i t may be necessa to go back to consideration of simpler exaaples, l ike 2 x ( 4 + 5 ) = 2 x 4 + 2 x 5 . or (2+3) x (4 + 5 ) = (2+3) x 4 + (2+3) x 5 and then distr ibute a g a i n . T r y t o use u n d e r l i n i n g t o i n d i c a t e t h e common t e r m and make s u r e t h a t t h e e n t i r e f i r s t e x p r e s s i o n i s v i e w e d as t h e common t e r m t h e f i r s t t i m e , b u t as t h e u s u a l addends t h e s e c o n d t i m e t h r o u g h . T h a t i s , t h e f i r s t t i m e (2+ L i ) on t h e l e f t i s t r e a t e d as a u n i t e d q u a n t i t y w h i c h t r a v e l s t o g e t h e r i n o r d e r t o b r e a k up t h e r i g h t hand e x p r e s s i o n , b u t a f t e r wards i t t o o i s b r o k e n up. THEN, WS HAVE THAT ( 2 x 2 ) + (2 x Q ) + ( Q x 2 ) + ( ELx U ) = 5. ! V V _ „ ! SINCE 2 x 2 = 4 , THAT MEANS THAT ( 2 x U ) + ( U x 2 ) + ( "Q x D ) = £ r ^ s J . SUPPCSE WE PRETEND TEMPORARILY THAT WS CAN IGNORE THE Ll^n PART. j THEN WE HAVE (j. x LA*} + ( U x 2 ^ ) = 1. \ BUT 2 x Q = Q x 2 , SO WS HAVE (z x D ^ + (2 x O ^ = 1. I BUT (^ 2 x Q.^ ) + (j2 x U *= 4 x Q USING THE DISTRI3UTIVS PRINCIPLE. SO, Jt x Q = 1 SO H IS • IF WS ASSUME U IS ^ , THEN WE NOTICE THAT Q*Q IS ONLY" WHAT NUMBER? Expect the answer: --y^ SIKCE THIS NUMBER IS LESS THAN % ^ AL-MOST OF THE 1 IN THE OTHER EXPRESSIONS BESIDES THE HI * Q. fj^tl + ^ X6) AND SO WS SAI THAT ^ ^ IS ABOUT 1% .WE WRITS THIS , WE CAN SAY THAT WS HAVE USED UP WE MIGHT HAVE USED THE DIVISION PROCEDURE AGAIN, AS WELL. J SO OUR FIRST GUESS WOULD HAVE BEEN 2. THEN WE WOULD DIVIDE 5 BY 2, AND GST 2) 5 _4_ UNFORTUNATELY, THIS GETS US INTO DIVISION WITH FRACTIONS SINCE CUR NEXT CHOICE WOULD BE A NUMBER BETWEEN 2 AND cD. eP, AND SO WS WILL AVOID THE DIVISION TECHNIQUE FOR THE TIMS BEING. SUPPOSE WE WANT TO FIND *$13 VS KNOW THAT THE ANSWER MUST EE BETWEEN 3 XND 4, SINCE 3 x 3 = 9 AND 4 x 4 = 16, AND 13 IS BETWEEN 9 AND 16. SO AGAIN, 'WE ASSUME THAT THE SQUARE ROOT IS A LITTLE OVER 3, SAY, 3 + Q THEN, BECAUSE IT IS A SQUARE ROOT, (3+ I I ) x (3 + Q ) = 13. Yi__* WAIT. r. I s.-U'~YLt; (3-MI) x ( 3 ^+ Q ) = x 3 + (3+ n ^ x a 9 + ( 3 x Q ) + ( 3 x C T ) + ( Q * Q ^ . THEN, IF 9 + O x Q) +(^x t f ) + (a*Q*) = 13, ( 3 x a) + ( 3 x a ) + (a*n)= 4 z ^ - 0 -AGAIN, LET US IGNORE THE Q x Q TEMPORARILY. IF WE DO, WE GET 0 x a ) x DJ = 4. BUI 6 x Q ) + ^ x U - ) = 6 x Q AND IF 6 x "Q = 4, THEN Q = 4 x 4 = I T SINCE 6 X (\ x = 4 x ( 6 x ^ ) = 4 x l = 4 SO BY IGNORING O * t l , WE GET Q = ^ THEN, IF WE INCLUDE Q K U , WE GET Q*JL_ = * ^ = Tf^ BUT SINCE THIS IS LESS THAN , WE DO NOT WORRY ABOUT IT, AND SAY THAT WE ARE CLOSE ENOUGH TO ES SATISFIED. VS WRITE: T ±_ SUPPOSE WE WANT TO FIND ^jj. IF WE DID NOT KNOW IN ADVANCE TO TRY A NUMBER NEAR 6, WE MIGHT USE OUR DIVISION PROCEDURE TO CLOSE IN ON A NUMBER BETWEEN 6 O ?. : r v.v o.Viv.iri> i , no iU.urix i > v . 4x n = 37 TO S: IF n = 4 . WS DIVIDE: 4 117" 36 AND WS FIND THAT CL L3 ABOUT 9 . SO NON WS TRY A NUMBER FOR Tr SQUARE ROOT BETWEEN 4 AND 9, LIKE 7. SO VE SOLVE: 7 x H = 37 TO SEE IF HI = 7. i 37 5 2 5 AND VE FIND THAT Q IS ABOUT 5 . SO NOW WE TRY A NUMBER BETWEEN ? AND 5. SAY, 6. 65171 # AND WS ARE BACK TO THE POINT WHERE OUR DIVISION PROCESS CANNOT HELP US. HOWEVER, WE DO KNOW FOR SURE NOW THAT Q SITS BETWEEN 6 AND 7. SO WS GO BACK TO OUR PROCEDURE OF BEFORE. WE KNOW THAT SO (6+ Q )x (6 + a ) = 37 BUT, (6+D.) x (6 + H ) = (6+ Q )x 6 = (jsx6^,+ ( a x 6/) +l6x\X) = 36 + 0 $ x C O +(6xO^ + CU^D^) THE, 36 +J6^x +fe x U) = 37, SO (6 x + ( 6 x aw cn^ ) = i ^ 3 7 - ^ ) TEMPORARILY IGNORING THE OxQ, WE FIND THAT 4 X +C6 x EH = 12 x Q IS ABOUT 1, SO J Q IS ABOUT . I F WS TEEN CALCULATE UxjC\ ,WS GET ^ v f H I C H I S M X H SMALLER THAN SO 'WE ARS NOT TOO FAR 0??, AND WE CAN SAY THAT ^JlVl 0^ C - i ^ -NOTICS THAT I? I HAD BEEN TRYING TO FIND A) 33, I WOULD HAVE DONS THE SAME THING 'UNTIL I SAID THAT 12 x Q IS ABOUT 1. THEN I WOULD HAVE SAID THAT 12 x Q IS ABOUT 2, AND SO Q IS ABOUT 2 x i t - = -§^; I STILL WOULD HAVE CHECKED Q * Q TO SES THAT IT WAS SMALLER THAN i , AND SINCE * ~±Q_ IS SMALLER THAN , I WOULD HAVE ACCEPTED THIS APPROXIMATION. ~ * J (fi ^ • ^ f Ask the students to work out the approximate square roots for each of the following and then ask students to cose to the board to show their work. 8 18 50 67 **Here the students w i l l learn how to approximate the square roots of fractions whose denominators are perfect squares, but whose numerators are not. ** OF AI-TT: HCV SUPPOSE I WANT TO FIND THE APPROXIMATE SQUARE ROOT I HAVE AIRSADY LEARNED THAT I CAN SEPARATELY ATTACK THE NUMERATORS AND DENOMINATORS AND PUT TOGETHER THESE SQUARE ROOTS TO FORM THE SQUARE ROOT FRACTION. i SUPPOSE WE DO THAT HERE. 1, WS FOUND THAT A) 5 0^ 1 -q. FROM CUR PREVIOUS WORK AND WE KNOW THAT 4 4 = 2. j SO THE APPROXIMATE SQU JARS ROOT OF ^ \ SUPPOSE I AM TRYING TO FIND c<_ I.IKT "V-K5 AND ^ f ' ^ , SO L E T ' S DO TFiE SAKS FOR v ^ * WE CAN WRITS THAT THE APPROXIMATE 1/ I s Q> 12, 4 7T • Have the students f ind approximate square roots foreach of the following; have a few students show their work on the board. APPENDIX B THE MEASURING INSTRUMENTS Items o f P r o d u c t o f a Mi x e d Number and a F r a c t i o n P r e t e s t 1. 6 x 3 - a 2. 2 x 9 = a 3. 8 x 4 = n 4. 6 x 8 = D 5. 7 x 5 = CI 11. 3 1 / 2 = 3 + a/A 12. 6 1/4 = •+ 1/4 13. 29 5/9 = D + 5/9 14. Cl 6/8 = 5 + 6/8 15. G i v e a f r a c t i o n name 16. Shade i n 1/3 o f t h e d i a g r a m on t h e r 17. G i v e a f r a c t i o n name t o t h e shaded p o r t i o n b e l o w : 18. Draw a d i a g r a m w i t h 2/3 shaded i n on t h e l i n e a t t h e r i g h t . 19. Draw a d i a g r a m w i t h 3/4 shaded i n on t h e l i n e a t t h e r i g h t . 20. 0 x 1 / 5 = 1 21. 0 x 1 / 8 = 1 22. 6 x ate = 1 23. 3/2 = 3 x 6. 9 x 7 - D 7. 6 x 7 - D 8. 8 x 9 = D 9. 6 x 6 = 0 10. 8 x 8 = 0 t o t h e shaded p o r t i o n b e l o w : 125 24. 8/9 = • x 1/9 25. 16/5 = £ 3 x 1 / 5 26. What i s the l e n g t h of a r e c t a n g l e w i t h a w i d t h of 2 f ee t and an a rea of 6 x 2 s q . f t . ? 27. What i s the w i d t h of a r e c t a n g l e w i t h a l e n g t h o f 8 f e e t and an a rea of 8 x 3 s q . ft .? 28. What i s the area of a r e c t a n g l e w i t h w i d t h 3 f t . and l e n g t h 5 f t . ? 29. What o p e r a t i o n would you use to f i n d the area of a r e c t a n g l e w i t h l e n g t h 89 f ee t and w i d t h 38 f ee t? (Would you a d d , s u b t r a c t , m u l t i p l y or d i v i d e ? ) 30. Us ing the i d e a that to f i n d a r e a , you f i n d the number of o n e - u n i t squares i n a f i g u r e , show how you would f i n d the area of the f i g u r e be low: a. j 31. 1 x 6 = D 32. £1 x 8 = 8 33. x 239 = 239 34. 1 x 1 / 4 = C J / A 35. 33 x 24 x 51 = 24 x D x 33 36. 56 x 19 x D = 19 x A x 48 37. 27 x 56 x 65 x 41 = 27 x £3 x 56 x 41 38. 55 x (£3 + 38) = (55 x 42) + (55 x 38) 39. a x (23 + 872) = 8950 40. 6 x (84 + O) - (6 x 84) + (6 x 156) 41. What i s the area of the f i g u r e below marked w i t h the q u e s t i o n mark? 126 42 . What i s the area of the f i g u r e below marked w i t h the q u e s t i o n mark? f ( -4 Items o f P r o d u c t of a M i x e d Number and a F r a c t i o n C o m p u t a t i o n T e s t 1. 2 x 6/7 2. 4 x 3/8 3. 6 x 8/9 4. 5 x 5/7 5. 9 x 8/11 6. 7 x 8/9 7. 2 x 2 3/8 8. 4 x 2 2/13 9. 8 x 4 3/25 10. 6 x 5 4/29 11. 9 x 6 5/70 12.7 x 7 6/50 13. 2/7 x 1/2 14. 2/3 x 8/9 15. 4/6 x 5/6 16. 4/5 x 7/8 17.8/9 x 8/9 18. 5/6 x 9/10 19. 1/2 x 8 6/9 20. 2/3 x 7 4/5 21. 4/5 x 7 4/6 22. 3/4 x 8 7/9 23. 8/9 x 8 6/7 24. 7/9 x 7 8/9 128 Items o f P r o d u c t o f a M i x e d Number and a F r a c t i o n G e n e r a l i z a t i o n T e s t 1. I f you know t h a t 1/81 x 1/63 = 1/5103, you c o u l d a l s o t e l l t h a t 2/81 x 1/63 - n/A-2. I f you know t h a t 53/69 x 48/73 = 2544/5037, f i n d v a l u e s f o r O and A so t h a t 48/69 x 53 / A = D /5037 3. I f you know t h a t 13/64 x 12/15 = 156/960, you c o u l d a l s o t e l l t h a t 13/64 x 12/10 x 15 = O/A. 4. I f you know t h a t 13/5 x 4/27 = 52/135, you c o u l d a l s o t e l l t h a t 13 x 10/5 x 4/27 = A / £7 < 5. W r i t e t h e f o l l o w i n g as a m u l t i p l i c a t i o n s t a t e m e n t . 2 3 / 5 + 2 3/5 + 2 3 / 5 + 2 3/5 6. W r i t e t h e f o l l o w i n g as s t r i c t l y a m u l t i p l i c a t i o n e x p r e s s i o n ( n o t i n -v o l v i n g a d d i t i o n ) . Do n o t compute t h e answer. 5/6 x 2 4/9 + 5/6 x 5/9 7. W r i t e t h e f o l l o w i n g as s t r i c t l y a m u l t i p l i c a t i o n e x p r e s s i o n ( n o t i n -v o l v i n g a d d i t i o n ) . Do n o t compute t h e answer. 3/4 x 2 1/4 + 3/4 x 2 3/4 8. F i n d t h e answers t o a, b, and c. W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . The f i r s t t h r e e answers a r e o n l y meant t o h e l p you w i t h t h e answer t o ( d ) . ( 0 i s a whole number) (a) 3/4 x 4/3 = O (b) 8/9 x 9/8 = • ( c ) 5/6 x 6/5 = O (d) 879/432 x 432/879 = A 129 9. F i n d t h e answers t o a, b, and c. W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . ( flis a w h ole number) (d) A x 2 1/A = 17 10. I f 3 x 4 0/9 = 12 6/9, what i s D 11. I f 5 x 0 A/& = 20 5/8, what a r e fl and A ? 1 2 . I f D x 3 5/16 = 9 15/16, what i s £7? 13. I f 7/2 x D 1/4 = 35/2 + 7/8, what i s O 14. I f 2/3 x 5 a/A = 10/3 + 8/15, what i s fl /A 15. How w o u l d you use t h e r u l e f o r m u l t i p l y i n g a f r a c t i o n by a m i x e d number, f o r example, t o m u l t i p l y : 2 1/4 x 3 1/5 . Show a l l o f y o u r s t e p s i n t h e s p a c e below and w r i t e t h e answer on t h e l i n e t o t h e r i g h t . 16. F i n d 1/2 / 2 x 24 2/5 17. F i n d 2/3 / 4 x 16 9/11 18. F i n d t h e answers t o a , b, and c. W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . The f i r s t t h r e e answers a r e o n l y t o h e l p you w i t h ( d ) . (a) 3 x 2 1/3 = D (b) 4 x 2 1 / 4 = O (c) 9 x 2 1/9 = D (a) 2 1/3 (b) 2 1/3 x 3 1/4 x 4 1/5 6 = 3 x 2 8 = 4 x 2 1 = 3 x 1 / 3 4/3 = 4 x 1/3 2/4 - 2 x 1/4 2/5 = 2 x 1/5 + 1/12 = 1/3 x 1/4 + 1/15 = 1/3 x 1/5 130 ( c ) 3 1/5 (d) 4 1/3 x 1 4/6 x 11 1/5 3 = 1 x 3 A a/6 1/5 - 1 x 1/5 12/6 = 3 x 4 / 6 + 4./Q = 1/5 x 4/6 D A/a 19. What i s (6 x 2/9) x 5 1/3 20. What i s (1/3 x l / 2 ) x 2 5/8 21. What i s 3/4 x (2/3 x 3) 22. What i s (4 x 1/3 x 1/5) x 2 1/6 23. I f (3 x 1/2) x 3 1/4 « D/^ x 9 3/4, what i s A 24. I f (6 x a/a) x 5 1/3 = 30/3 + 6/9, what i s P/A 25. I f (4 x 2/3) x O No = 16/3 + 8/15, what i s H A/<J 26. F i n d v a l u e s f o r d and A so that 4 x £ 3 / 3 x 2 5/A = 40/3 + 25/27 2 7 . S u s i e has s u g g e s t e d a n o t h e r r u l e f o r m u l t i p l y i n g f r a c t i o n s . To m u l t i p l e , f o r example, 2/3 x 4/5, she d o u b l e s t h e f i r s t f r a c t i o n ' s n u m e r a t o r and d o u b l e s t h e se c o n d f r a c t i o n ' s d e n o m i n a t o r and t h e n m u l t i p l i e s . 4/3 x 4/10 = 16/30, so 2/3 x 4/5 = 16/30. She c l a i m s t h a t h e r answer i s e q u i v a l e n t t o t h e one w h i c h t h e t e a c h e r g e t s by th e method t a u g h t i n c l a s s . (2 x 4/3 x 5 = 8/15) W i l l she a l w a y s g e t e q u i v a l e n t answers t o y o u r s ? Why do you t h i n k she s h o u l d ? Use 3/4 x 2/9 as an example. 131 28. Johnny has s u g g e s t e d a n o t h e r r u l e f o r m u l t i p l y i n g f r a c t i o n s . To m u l t i p l y , f o r example, 2/3 x A/5, he f i n d s t h e answer t o 4/3 x 2/5 He s w i t c h e s the n u m e r a t o r s o f t h e f r a c t i o n s and t h e n m u l t i p l i e s . He c l a i m s t h a t h i s answer i s e q u i v a l e n t t o t h e one w h i c h t h e t e a c h e r g e t s by t h e method t a u g h t i n c l a s s . W i l l he a l w a y s g e t e q u i v a l e n t answer t o y o u r s ? Why do you t h i n k he s h o u l d ? Use 3/4 x 2/9 as an example. 29. Sam has s u g g e s t e d a r u l e f o r m u l t i p l y i n g two f r a c t i o n s , t o o . To f i n d t h e answer t o , f o r example, 2/3 x 4/5 Sam f i n d s t h e answer t o 3/2 x 5/4 and t h e n t u r n s t h e f r a c t i o n answer u p s i d e down. I n o t h e r w o r d s , he t u r n s b o t h f r a c t i o n s u p s i d e down, m u l t i p l i e s , and t h e n t u r n s h i s answer u p s i d e down. He c l a i m s t h a t h i s answer i s t h e same as t h e one t h e t e a c h e r g e t s by t h e method t a u g h t i n c l a s s . W i l l he alw a y s g e t t h e same answer as y o u r s ? Why do you t h i n k he s h o u l d ? Use 3/4 x 2/9 as an example. 30. J u d y has s u g g e s t e d a way t o check m u l t i p l i c a t i o n o f f r a c t i o n a n s w e r s . She s a y s t h a t one can be s u r e t h a t a/b x c/d = P / _ ) i f £J — c = a and ^ d = b. F o r example, i t i s t r u e t h a t 3/5 x 2/9 = 6/45 s i n c e 6 r 2 = 3 and 45 T 9 - 5. Can you be s u r e i f y o u r answer i s r i g h t by c h e c k i n g w i t h J u d y ' s method? Why do you t h i n k so? 132 Items o f Comparing F r a c t i o n s P r e t e s t 1. 6 x 3 = 6. 9 x 7 = 2. 2 x 9 = 7. 6 x 7 = 3. 8 x 4 = 8. 8 x 9 = 4 . 6 x 8 = 9 . 6 x 6 = 5. 7 x 5 = 10. 8 x 8 = 11. G i v e a f r a c t i o n name t o t h e shaded p o r t i o n b e l o w : 12. Shade i n 1/3 o f t h e d i a g r a m on t h e r i g h t . 13. G i v e a f r a c t i o n name t o t h e shaded p o r t i o n b e l o w : 14. Draw a d i a g r a m w i t h 2/3 shaded i n on t h e l i n e a t t h e r i g h t . 15. Draw a d i a g r a m w i t h 3/4 shaded i n on t h e l i n e a t t h e r i g h t . 16. I s t h e f r a c t i o n r e p r e s e n t i n g a r e a ( a ) more t h a n , l e s s t h a n , o r t h e same as t h a t r e p r e s e n t i n g a r e a (b) 133 17. I s t h e f r a c t i o n r e p r e s e n t i n g a r e a (a) more t h a n , l e s s t h a n , o r t h e same as t h a t r e p r e s e n t i n g a r e a ( b ) ? (h) I WM/f//J/W 18. I s t h e f r a c t i o n r e p r e s e n t i n g a r e a (a) more t h a n , l e s s t h a n , o r t h e same as t h a t r e p r e s e n t i n g a r e a ( b ) ? 19. I s t h e f r a c t i o n r e p r e s e n t i n g a r e a (a) more t h a n , l e s s t h a n , o r t h e same as t h a t r e p r e s e n t i n g a r e a ( b ) ? 20. I s t h e f r a c t i o n r e p r e s e n t i n g a r e a (a) more t h a n , l e s s t h a n , o r t h e same as t h a t r e p r e s e n t i n g a r e a ( b ) ? A | UI A (V) _ 21. What s t a t e m e n t does t h e d i a g r a m b e l ow s u g g e s t ? F o r example: s u g g e s t s 1/2 = 2/4 22. What s t a t e m e n t does t h e d i a g r a m below s u g g e s t : 134 23. What s t a t e m e n t does t h e d i a g r a m b e l o w s u g g e s t ? 24. 33 x 24 x 51 = 24 x 0 x 33 25. 56 x 19 x 48 = 19 x £J x 48 26. 27 x 56 x 65 x 41 = 27 x p x 56 x 41 27. 0 x 1 / 5 = 1 28. 0 x 1 / 8 - 1 29. 6 x fJ/A = 1 30. 3/2 = 3 x D/A 31. 8/9 = O x 1/9 32. 16/5 - D x 1/5 33. Which i s l a r g e r , 44 x 13 o r 52 x 13 34. Which i s l a r g e r , 69 x 158 o r 32 x 158 35. Which i s g r e a t e r , 15 x 1/4 o r 16 x 1/4 135 Items o f Comparing F r a c t i o n s C o m p u t a t i o n T e s t 1. 18/5 o r 3 15. 5/9 o r 4/6 2. 25/4 Or 2 16. 7/8 o r 8/9 3. 29/10 o r 2 17. 9/10 o r 7/9 4. 33/7 o r 5 18. 6/7 o r 6/8 5. 28/8 o r 4 19. 7/4 o r 4/2 6. 27/5 o r 5 20. 6/2 o r 9/2 7. 45/7 o r 7 21. 3/2 o r 4/3 8. 68/9 o r 8 22. 10/3 o r 7/4 9. 52/5 o r 10 23. 5/3 o r 8/7 10. 3/4 o r 5/6 24. 4/3 o r 10/9 11. 1/2 o r 5/9 25. 7/5 o r 9/8 12. 4/8 o r 2/3 26. 8/6 o r 10/7 13. 6/9 o r 3/6 27. 8/6 o r 9/5 14. 4/6 o r 6/7 136 Items o f Compar i s o n o f F r a c t i o n s G e n e r a l i z a t i o n T e s t 1. Which i s g r e a t e r : D /37 o r O / 3 8 2. Which i s g r e a t e r : 87/0 o r 8 8 / 0 3. F o r e a c h , f i n d w h i c h i s g r e a t e r . W r i t e t h e answer t o (d) on t h e l i n e t o t he r i g h t . The f i r s t t h r e e answers a r e o n l y meant t o h e l p you w i t h the answer t o ( d ) . (a) 2/3 o r 1/2 (b) 3/4 o r 4/5 ( c ) 7/8 o r 6/7 (d) • / • +1 o r • - 1 / O 4. F o r e a c h , f i n d w h i c h i s g r e a t e r . W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . The f i r s t t h r e e answers a r e o n l y meant t o h e l p y ou w i t h t h e answer t o ( d ) . (a) 2/3 o r 2/5 (b) 3/17 o r 3/28 ( c ) 13/53 o r 13/78 (d) D /834 o r • /729 5. F o r e a c h , f i n d w h i c h i s g r e a t e r . W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . The f i r s t t h r e e answers a r e o n l y meant t o h e l p you w i t h t h e answer t o ( d ) . (a) 1/14 o r 2/14 (b) 21/4 o r 33/4 ( c ) 40/58 o r 33/58 (d) 6/Q o r 3 / a 137 6. F o r e a c h , f i n d w h i c h i s g r e a t e r . W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . (a) 33-1/33 o r 33/33+1 (b) 33/33+9 o r 33-9/ 33 (c) 33-18/33 o r 33/33+18 (d) 33/33+ D o r 33- D/33 7. I f 384/529 ^ 384/CJ , what can you say about fl 8. I f 483 x 25 ^ 365 x 18, w h i c h i s g r e a t e r : 2 x 483/365 o r 2 x 18 /25 9. I f 6 x 18 "713 x 8, w h i c h i s g r e a t e r : 6/13 o r 8/18 10. I f 17 x 11 14 x 15, w h i c h i s g r e a t e r : 14/17 o r 11/15 11. I f 82 x 15 ~7 63 x • , w h i c h i s g r e a t e r : 82/a o r 63/15 12. I f 8 x (5 + Q ) 7 6 x 3, w h i c h i s g r e a t e r : 5 + £3 /6 o r 3/8 13. I f 425 x 211 ^.310 x 316, w h i c h i s g r e a t e r : 425-310/310 o r 316-211/211 14. I f 6 x ( 3 - a) P" 2 x 5, w h i c h i s g r e a t e r : 6/5 o r 2 / 3 - 0 15. Use t h e r u l e f o r comparing f r a c t i o n s t o compare two w h o l e numbers, 5 and 7. Show a l l o f y o u r s t e p s i n t h e spa c e below and w r i t e t h e g r e a t e r o f t h e two numbers on t h e l i n e t o t h e r i g h t . ( H i n t : W r i t e them f i r s t as f r a c t i o n s . ) 138 16. Use t h e r u l e f o r comparing f r a c t i o n s t o compare t h e two mixed num-b e r s , 2 1/3 and 2 2/5. Show a l l o f y o u r work i n t h e s p a c e b e l o w and w r i t e t h e g r e a t e r o f t h e two numbers on th e l i n e t o t h e r i g h t . 17. U s e t h e r u l e f o r co m p a r i n g f r a c t i o n s t o compare 2 1/5 and 20/9. W r i t e t h e l a r g e r on t h e l i n e t o t h e r i g h t . 18. Use t h e r u l e f o r comparing f r a c t i o n s t o compare 3 2/3 and 31/9. W r i t e t h e l a r g e r on th e l i n e t o t h e r i g h t . 19. I f 22 x U = 83 x A , i s 22/83 more t h a n , l e s s t h a n , o r e q u a l t o • / A -20. You know t h a t 22 /0 = 8 3 / A i f 22 x A = -21. You know t h a t O IA = 6"/Q I f = . 22. A r r a n g e t h e s e t h r e e f r a c t i o n s f r o m l a r g e s t t o s m a l l e s t : 3/5 19/30 7/12 23. A r r a n g e t h e s e t h r e e f r a c t i o n s from l a r g e s t t o s m a l l e s t : 4/9 2/3 6/11 24. A r r a n g e t h e s e t h r e e f r a c t i o n s from l a r g e s t t o s m a l l e s t : 8/3 25/10 25/9 25. F i n d t h r e e f r a c t i o n s l e s s t h a n 2/5. 26. A r r a n g e t h e s e f r a c t i o n s from l a r g e s t t o s m a l l e s t ; 3/13 6/25 14/52 27. S u s i e has s u g g e s t e d a n o t h e r r u l e f o r comparing f r a c t i o n s . To d e c i d e , f o r example, i f 2/3 ~7 4/9, she compares 2/4 t o 3/9. I f 2/4 >"3/9, t h e n 2/3 C>4/9. She s w i t c h e s t h e f i r s t d e n o m i n a t o r w i t h t h e se c o n d n u m e r a t o r and t h e n compares t h e s e t o d e c i d e i f t h e o r i g i n a l f i r s t f r a c t i o n i s l a r g e r t h a n t h e o r i g i n a l second f r a c t i o n . She c l a i m s t h a t h e r answer i s a l w a y s t h e same as t h e one w h i c h t h e t e a c h e r g e t s by h e r method. W i l l S u s i e a l w a y s g e t t h e r i g h t answer? Why do you t h i n k she s h o u l d ? Show how S u s i e would compare 7/11 and 5/8; e x p l a i n h e r method and why i t w o r k s . 28. Johnny has s u g g e s t e d a method f o r comparing f r a c t i o n s when t h e n u m e r a t o r o f t h e second goes i n t o t h e n u m e r a t o r of t h e f i r s t e x a c t l y and t h e de-n o m i n a t o r o f t h e second goes i n t o t h e d e n o m i n a t o r of t h e f i r s t e x a c t l y . F o r example, h i s method works f o r comparing 15/16 and 3/4 s i n c e 3 goes i n t o 15 e x a c t l y and s o does 4 i n t o 16. Johnny d e c i d e d i f t h e f i r s t n u m e r a t o r d i v i d e d by t h e second i s g r e a t e r t h a n t h e f i r s t d e n o m i n a t o r d i v i d e d by t h e s e c o n d , t h e n t h e f i r s t f r a c t i o n i s l a r g e r t h a n t h e s e c o n d . F o r example, 15 — 3 = 5 and 16 - j - 4 = 4 and 5 ^ 4 , so 15/16 7 3/4. Does John n y ' s method a l w a y s work f o r t h e s e k i n d s o f p r o b l e m s ? E x p l a i n why, u s i n g t h e p r o b l e m o f c o m p a r i n g 24/25 and 4/5 as an example. 29. Sam s a y s t h a t a/b 7 c/d whenever 1/a x d -<d 1/b x c and o n l y t h e n , so t o compare, f o r example, 2/3 and 4/9, he d i s c o v e r s t h a t 1/2 x 9 ^~ 1/3 x 4, so he c o n c l u d e d t h a t t h e f i r s t f r a c t i o n i s g r e a t e r t h a n t h e s e c o n d . Does Sam's method a l w a y s work f o r c o m p a r i n g f r a c t i o n s ? Ex-p l a i n what Sam would do and why he i s c o r r e c t by comparing 7/11 and 5/8 by h i s method. 30. J u d y s a y s t h a t she can t e l l i f a/b 7 c / d r i g h t away, I f a 7 ( b x x ) ~ d , t h e n t h e f i r s t f r a c t i o n i s l a r g e r . F o r example, t o compare 2/3 and 4/9 she s a y s 2 ^ > ( 4 x 3 ) - r - 9 = l 1/3. T h e r e f o r e , t h e f i r s t f r a c t i o n i s g r e a t e r . Does J u d y ' s method a l w a y s work f o r comparing f r a c t i o n s ? E x p l a i n what Jud y w o u l d do and why she i s c o r r e c t by c o m p a r i n g 7/11 and 5/8 by h e r method. 140 Items o f Changing F r a c t i o n s t o D e c i m a l s P r e t e s t 1. G i v e a f r a c t i o n name t o t h e shaded p o r t i o n b e l o w : 2. G i v e a f r a c t i o n name t o t h e shaded p o r t i o n b e l o w : 3. Shade i n 1/5 of t h e d i a g r a m on t h e r i g h t . 4. Draw a d i a g r a m w i t h 3/4 shaded i n on t h e l i n e t o t h e r i g h t . 5. Draw a d i a g r a m w i t h 2/6 shaded i n on t h e l i n e t o t h e r i g h t . 6. What e q u a t i o n w o u l d y o u w r i t e t o d e s c r i b e t h e a c t i o n i n s h a r i n g 27 m a r b l e s among 3 p e o p l e ? 7. What e q u a t i o n would you w r i t e t o d e s c r i b e t h e a c t i o n i n s h a r i n g 18 cupcakes among 6 c h i l d r e n ? 8. What p r o b l e m would you w r i t e t o d e s c r i b e t h e a c t i o n i n s h a r i n g 24 p e n c i l s among 8 s t u d e n t s ? 9. Draw a d i a g r a m below showing why 8 *r 4 = 2. 10. Draw a d i a g r a m showing why 12 -r U = 4. Use t h e s p a c e below f o r t h e d r a w i n g and f i n d t h e v a l u e o f D . 11. D i v i d e 1 1 ) 425 1 4 1 1 2 . D i v i d e 2 0 ) 5 8 3 4 1 3 . D i v i d e 7 J 2 8 5 9 1 4 . D i v i d e 2 5 J 4 6 3 2 1 5 . D i v i d e 6 / 384 1 6 . W r i t e t h e f o l l o w i n g m u l t i p l i c a t i o n s t a t e m e n t as a d i v i s i o n s t a t e m e n t : 5 3 2 x 1 8 = 9 5 7 6 1 7 . W r i t e t h e f o l l o w i n g d i v i s i o n s t a t e m e n t as a m u l t i p l i c a t i o n s t a t e m e n t : 4 9 2 - ^ - 1 2 3 = 4 1 8 . W r i t e t h e f o l l o w i n g d i v i s i o n s t a t e m e n t as a m u l t i p l i c a t i o n s t a t e m e n t : D T 6 = A 1 9 . W r i t e t h e f o l l o w i n g m u l t i p l i c a t i o n s t a t e m e n t as a d i v i s i o n s t a t e m e n t : a x 6 = A 2 0 . W r i t e t h e f o l l o w i n g m u l t i p l i c a t i o n s t a t e m e n t as a d i v i s i o n s t a t e m e n t : O x A = O 2 1 . £ 3 x 1 / 9 = 1 2 2 . 8 x D / A = 1 2 3 . 1 4 / 9 = 1 4 x D / A 2 4 . 3 / 7 = D x 1 / 7 2 5 . 3 8 / 6 = O x 1 / 6 2 6 . 7 4 x 8 3 x 7 7 = D x 7 4 x 7 7 2 7 . 3 1 2 x 2 5 x O x 87 = 43 x £ x 25 x 31 2 28. 45 x 203 x 87 = O x 203 x 87 29. 1 x 576 = O 30. a x (374 + 596) = 374 + 596 143 Items o f Changing F r a c t i o n s t o D e c i m a l s C o m p u t a t i o n T e s t 1. 2/5 = 10. 5/6 ^ 2. 3/4 = 11. 5/7 ^ 3. 1/2 = 12. 3/9 ^ 4. 3/8 = 13. 3/7 ^ 5. 7/10 = 14. 8/9 ^ 6. 13/25 = 15. 5/11 7. 17/20 = 16. 6/13 ~ 8. 37/50 = 17. 9/14 :=S 9. 2/3 -=s 18. 15/17 ^ f 144 Items o f Changing F r a c t i o n s t o D e c i m a l s G e n e r a l i z a t i o n T e s t I f you know t h a t t h e d e c i m a l f o r 4/3 C 1.333, you can say t h a t t h e d e c i m a l f o r 2 x 4/3 *v F o r each p a r t , f i n d t h e d e c i m a l e q u i v a l e n t r e q u i r e d . W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . The f i r s t t h r e e answers a r e o n l y meant t o h e l p you w i t h t h e answer t o ( d ) . (a) I f you know t h a t t h e d e c i m a l f o r 4/6 .666, you can a l s o t e l l t h a t t h e d e c i m a l f o r 4/10 x 6 m (b) I f you know t h a t t h e d e c i m a l f o r 3/5 = .600, you can a l s o t e l l t h a t t h e d e c i m a l f o r 3/10 x 5 = ( c ) I f you know t h a t t h e d e c i m a l f o r 6/7 ~ .857, you can a l s o t e l l t h a t t h e d e c i m a l f o r 6/10 x 7 ~ (d) I f you know t h a t t h e d e c i m a l f o r 6/ CL .295, you can a l s o t e l l t h a t t h e d e c i m a l f o r 6/10 x • ^ F o r each p a r t , f i n d t h e d e c i m a l e q u i v a l e n t r e q u i r e d . W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . Do a l l o t h e r work below. (a) I f you know t h a t t h e d e c i m a l f o r 4/6 qy.666, you can a l s o t e l l t h a t t h e d e c i m a l f o r 10 x 4/6 oz, (b) I f you know t h a t t h e d e c i m a l f o r 3/5 = .600, you can a l s o t e l l t h a t t h e d e c i m a l f o r 10 x 3/5 = ( c ) I f you know t h a t t h e d e c i m a l f o r 6/7 ^ .857, you can a l s o t e l l t h a t t h e d e c i m a l f o r 10 x 6/7 (d) I f you know t h a t t h e d e c i m a l f o r 6/ £7 ~ .295, you can a l s o t e l l t h a t t h e d e c i m a l f o r 10 x 6/ Q 145 4. I f you know t h a t t h e d e c i m a l f o r 7/25 = .28, you can a l s o t e l l t h a t t h e d e c i m a l f o r 7/25 x 2 = 7/50 = ' 5. I f you know t h a t t h e d e c i m a l f o r 3/50 = .06, and t h e d e c i m a l f o r 9/50 = .18, t h e n you can a l s o t e l l t h a t t h e d e c i m a l f o r 3 + 9/50 « 12/50 = 6. I f you know t h a t t h e d e c i m a l f o r 9/15 = .600 and t h e d e c i m a l f o r 7/15 ^ . 4 6 7 , t h e n you can a l s o t e l l t h a t t h e d e c i m a l f o r 9-7/15 » 2/15 ~ 7. I f you know t h a t t h e d e c i m a l f o r 17/16 = 1.0625, t h e n you can a l s o t e l l t h a t t h e d e c i m a l f o r 1/16 = ? 8. I f you know t h a t t h e d e c i m a l f o r 8/14 ~ .571 and t h a t t h e d e c i m a l f o r 7/14 = .5, t h e n you can a l s o t e l l t h a t t h e d e c i m a l f o r 8-7/14 = 1 / 1 4 ^ 9. F o r each p a r t , f i n d t h e d e c i m a l e q u i v a l e n t r e q u i r e d . W r i t e t h e answer t o (d) on t h e l i n e t o t h e r i g h t . Show a l l o t h e r work b e l o w . (a) I f you know t h a t t h e d e c i m a l s f o r 4/9 ^ .444 and 2/9 ^ .222, t h e n you can a l s o t e l l t h a t .666 i s t h e a p p r o x i m a t e d e c i m a l f o r what f r a c t i o n O / A (b) I f you know t h a t t h e d e c i m a l f o r 3/16 ^ .187 and 8/16 = .500, t h e n you can a l s o t e l l t h a t .687 i s t h e a p p r o x i m a t e d e c i m a l f o r what f r a c t i o n O / A ( c ) I f you know t h a t t h e d e c i m a l s f o r 15/20 =. .750 and 4/20 = .200,then you can a l s o t e l l t h a t .950 i s t h e a p p r o x i m a t e d e c i m a l f o r what f r a c t i o n QlA (d) I f you know t h a t t h e d e c i m a l s f o r 3 7 / 0 = .074 and 1 2 / f l = .024, t h e n .098 i s t h e a p p r o x i m a t e d e c i m a l f o r what f r a c t i o n A I Q Do n o t f i n d t h e v a l u e o f Q 146 10. I f you know t h a t t h e d e c i m a l f o r 2/3 s^.666, t h e n you c o u l d a l s o t h a t 3 x O 2: 2, where 0 i s a d e c i m a l 11. I f you know t h a t t h e d e c i m a l f o r 2/5 = .400, t h e n you c o u l d a l s o t e l l t h a t 5 x Q = 2 , where J] i s a d e c i m a l . 12. I f t h e d e c i m a l f o r 0 / l l ~ .45, what whole number i s O 13. I f t h e d e c i m a l f o r 9/ 13 <Z .81, what whole number i s 07 14. I f t h e d e c i m a l f o r 8/ Q 2? .88 and t h e d e c i m a l f o r 5 / 0 ~ .55, t h e n .33 i s t h e a p p r o x i m a t e d e c i m a l f o r what f r a c t i o n A fU Do n o t f i n d t h e v a l u e o f D 15. I f t h e d e c i m a l f o r 2/ 0 = .04, what w h o l e number i s a 16. Suppose t h e d e c i m a l s f o r 0/50 = .56 and f o r <_/50 = .48. I f t h e d e c i m a l (.56 - .48) = .08 i s t h e d e c i m a l e q u i v a l e n t f o r t h e f r a c t i o n tf"/50, what can y o u s a y about t h e r e l a t i o n s h i p between 0 , A , and <T . 17. I f you know t h a t t h e d e c i m a l f o r 0 I Z_ = .89, t h e n what d e c i m a l can 1 m u l t i p l y A by t o g e t a 18. How w o u l d you use t h e r u l e f o r c h a n g i n g a f r a c t i o n t o a d e c i m a l t o f i n d t h e d e c i m a l f o r a mixed number, l i k e 2 1/2 i n two d i f f e r e n t ways. Show a l l o f y o u r work i n t h e s p a c e b e l ow and w r i t e t h e a c t u a l d e c i m a l e q u i -v a l e n t on t h e l i n e t o t h e r i g h t . 19. F i n d t h e d e c i m a l e q u i v a l e n t f o r 2/.3 . N o t i c e t h a t t h e d e n o m i n a t o r i s 0.3 ( o r 3 t e n t h s ) 20. F i n d t h e d e c i m a l e q u i v a l e n t f o r 8/0.2 21. F i n d the d e c i m a l e q u i v a l e n t f o r 0.7/2 22. F i n d t h e d e c i m a l e q u i v a l e n t f o r 1/2 /4 23. F i n d t h e d e c i m a l e q u i v a l e n t f o r 1/3 /2 147 24. F i n d t h e d e c i m a l e q u i v a l e n t f o r 4/9 /2 25. F i n d t h e d e c i m a l e q u i v a l e n t f o r 3/5 / 8/10 26. F i n d t h e d e c i m a l e q u i v a l e n t f o r 4/6 / 5/3 27. S u s i e has s u g g e s t e d a n o t h e r r u l e f o r f i n d i n g the d e c i m a l e q u i v a l e n t o f a f r a c t i o n a/b. She m u l t i p l i e s each o f t h e n u m e r a t o r and d e n o m i n a t o r by t e n , and t h e n she d i v i d e d t h e d e n o m i n a t o r i n t o t h e n u m e r a t o r . F o r example, t o f i n d t h e d e c i m a l f o r 3/5, she d i v i d e d 50 Jlo = 50 J 307b - 50 J300 t e n t h s 300 t e n t h s 6 t e n t h s 0 t e n t h s 6 t e n t h s = .6 She c l a i m s t h a t h e r answer i s a l w a y s c o r r e c t , and t h e same as t h e one t h e t e a c h e r g e t s by u s i n g t h e method t a u g h t i n c l a s s . W i l l S u s i e a l w a y s g e t t h e r i g h t answer? Why do you t h i n k so? Show how S u s i e w o u l d f i n d t h e d e c i m a l f o r 5/8 and e x p l a i n why h e r method seems t o work. 28. Johnny has s u g g e s t e d a n o t h e r method f o r f i n d i n g t h e d e c i m a l e q u i v a l e n t f o r a f r a c t i o n when t h e n u m e r a t o r i s l a r g e r t h a n t h e d e n o m i n a t o r . He s u b t r a c t s t h e d e n o m i n a t o r f r o m t h e n u m e r a t o r and t h e n d i v i d e d t h i s number by t h e d e n o m i n a t o r . F o r example, t o f i n d t h e d e c i m a l f o r 6/4 he w r i t e s : 6 - 4 = 2 , so I d i v i d e 2 by 4 4 J~2 = 4 r O = 4 J20 t e n t h s 20 t e n t h s 5 t e n t h s 0 t e n t h s 5 t e n t h s = .5 He t h e n adds one t o t h e answer, t o g e t .5 + 1 = 1.5. He c l a i m s t h a t h i s method a l w a y s works f o r p r o b l e m s where t h e n u m e r a t o r i s l a r g e r t h a n t h e d e n o m i n a t o r . I s he c o r r e c t : Show how Johhny would f i n d t h e d e c i m a l f o r 8/5 and e x p l a i n why h i s method seems t o work. 148 29. Sam's r u l e f o r f i n d i n g t h e d e c i m a l e q u i v a l e n t t o a g i v e n f r a c t i o n i s v e r y much l i k e S u s i e ' s , but i n s t e a d o f m u l t i p l y i n g n u m e r a t o r and d e n o m i n a t o r by 10, he m u l t i p l i e s e ach by 100 and t h e n d i v i d e s . F o r example, t o f i n d t h e d e c i m a l f o r 3/5, he d i v i d e s 6 t e n t h s 500 J 300 = 500 J 300.0 = 500 ) 3000 t e n t h s 3000 t e n t h s 0 t e n t h s 6 t e n t h s = .6 He c l a i m s t h a t h i s answer i s a l w a y s c o r r e c t . W i l l Sam a l w a y s g e t t h e c o r r e c t answer: Why do you t h i n k so? Show how Sam w o u l d f i n d t h e d e c i m a l f o r 5/8 and e x p l a i n why h i s method seems t o work. 30. J u d y ' s method f o r f i n d i n g t h e d e c i m a l f o r a f r a c t i o n i s v e r y much l i k e J o h n n y ' s , o n l y h e r s works f o r any f r a c t i o n . She adds t h e de-n o m i n a t o r t o t h e n u m e r a t o r and t h e n d i v i d e s t h i s number by t h e numer-a t o r . F o r example, t o f i n d t h e d e c i m a l f o r 6/4, she w r i t e s : 6 + 4 = 10, so I d i v i d e 10 by 4 4 / T O - 4 JlO.O = 4 J100 t e n t h s 100 t e n t h s 25 t e n t h s o t e n t h s 25 t e n t h s Then she s u b t r a c t s one f r o m t h i s answer, t o g e t 2.5 - 1 = 1.5. She c l a i m s t h a t h e r method a l w a y s works f o r problems o f t h i s s o r t . Show how Judy w o u l d f i n d t h e d e c i m a l f o r 5/8 and e x p l a i n why h e r method seems t o work. 149 Items of F i n d i n g t h e Square Root o f a F r a c t i o n P r e t e s t 1. 8 x 7 = a 6. 6 x 8 = a 2. 4 x 9 - O 7. 4 x 5 = Q 3. 6 x 5 = a 8. 9 x 8 = 1 7 4. 9 x 2 = 0 9. 6 x 7 = H 5. 7 x 5 = 0 10. 7 x 7 = D 11. G i v e a f r a c t i o n name t o t h e shaded p o r t i o n b e l o w : 14. Draw a dia g r a m w i t h 3/4 shaded i n on t h e l i n e t o t h e r i g h t . 15. Draw a dia g r a m w i t h 2/6 shaded i n on t h e l i n e t o t h e r i g h t . 16. W r i t e t h e f o l l o w i n g m u l t i p l i c a t i o n s t a t e m e n t as a d i v i s i o n s t a t e m e n t : 532 x 18 = 9576 17. W r i t e t h e f o l l o w i n g d i v i s i o n s t a t e m e n t as a m u l i t p l i c a t i o n s t a t e m e n t : 492 -T-123 = 4 150 18. W r i t e the f o l l o w i n g d i v i s i o n s t atement as a m u l t i p l i c a t i o n s t a t e m e n t : O r 6 = A 19. W r i t e the f o l l o w i n g m u l t i p l i c a t i o n statement as a d i v i s i o n s t a t e m e n t : Q X 6 = A 20. W r i t e the f o l l o w i n g m u l t i p l i c a t i o n statement as a d i v i s i o n s t a t e m e n t : a * A = <r 21. £ 7 x 1 / 9 = 1 22. 8 x D / A =1 23. 14/9 = 14 x • IA 24. 3/7 - O x 1/7 25. 38/6 = £ 7 x 1 / 6 26. 74 x 83 x 77 = Q x 74 x 77 27. 312 x 25 x • x 87 = 43 x A x 25 x 312 28. 45 x 203 x 87 = flx 203 x 87 29. 1 x 576 = a 30. • x (374 + 596) = 374 + 596 31. What i s the l e n g t h o f a r e c t a n g l e w i t h a w i d t h o f 2 f e e t and an a r e a of 6 x 2 s q . f t . 32. What i s the w i d t h o f a r e c t a n g l e w i t h a l e n g t h o f 8 f e e t and an a r e a o f 8 x 3 s q . f t . 33. What i s the a r e a o f a r e c t a n g l e w i t h w i d t h 3 f e e t and l e n g t h 5 f e e t ? 34. What o p e r a t i o n would you use to f i n d the a r e a o f a r e c t a n g l e w i t h l e n g t h 89 f e e t and w i d t h 38 f e e t ? (Would you add, s u b t r a c t , m u l t i p l y o r d i v i d e ? ) 151 35. U s i n g t h e i d e a t h a t t o f i n d a r e a , you f i n d t h e number o f o n e - u n i t s q u a r e s i n a f i g u r e , show how you would f i n d t h e a r e a o f t h e f i g u r e b e low: 36. D i v i d e 37. D i v i d e 38. D i v i d e 39. D i v i d e 40. D i v i d e 11 J 425 20 ) 5834 7 J 2859 25 ) 4632 152 Items o f F i n d i n g t h e Square Root o f a F r a c t i o n C o m p u t a t i o n T e s t 1. J 1089/4 9. J 1369/9 2. ^2764/9 10. / 7 o / 4 ~ 3. J 961/16 11. v/38/9 4. /1849/25 12. v/37/16 5. J 3025/16 13. t/ 22/25 6. ^1521/9 14. JT6/16 7. J~676/16 15. v/~33/25 8. v/~361/25 153 Items of Finding the Square Root of a Fraction Generalization Test 1. If you know that = 13/15, then you can also t e l l that fl x 169/225 = J2/A. 2. If you know that J256/25 = 16/5, then you can also t e l l that J~256/25 x 9 = O/A. 3. I f you know that J9 /16 = 3/4, and l/25/49 = 5/7, you can also t e l l that J~9 x 25/16 x 49 = a/A-4. For each part, find the square root required. Write the answer to (d) on the l ine to the right. The f i r s t three questions are only meant to help you with the answer to (d). (a) I f you know that J 9 / I 6 = 3/4, you can also t e l l that J16/9 = O / A (b) I f you know that Jvb/25 = 4/5, you can also t e l l that/25/16 = O/A (c) If you know that J~81/100 = 9/10, you can also t e l l that /1OO/8I =EJ/A (d) If you know that / x / y = 8/5, you can also t e l l that 5. If you know that J8/9 ^ 2 3/4 /3 , you can also t e l l that J T x 8/9 ~ a / A . 6. If you know that /15/25 ~ 3 7/8 15 , you can also t e l l that Jl5/ 25 x 100 =- CI /A 7. If you know that J~8/9 2 3/4. /3 , you can also t e l l that / T x 8/9 ~ C f / A . 8. If you know that 7l6/25 = 4/5 andv/49/64 = 7/8, you can also t e l l that 16 x 64/25 x 49 = O / A 154 9. F o r each p a r t , f i n d t h e s q u a r e r o o t r e q u i r e d . W r i t e t h e answer t o (d) o n l y on t h e l i n e t o t h e r i g h t , (a) Jl/9 = 1/Q x J T (b) JT/25 = 1/0 x Jls ( c ) J l / 1 0 0 = 1/n x v/lOO (<0 Ji/a = i/iooo x v/: 1000 10. Jo/81 ~ 5/9 11. J49/D = 7/8 12. JflM x 2/9 = 2/6 13. How do a and b compare i f Ja/b i s l a r g e r t h a n 1? 14. I f you know t h a t J 8 / 9 = 2 3/4 /3 and J l l / 2 5 ^ Q /Q/5 t h e n you can a l s o t e l l t h a t (2 3/4 /5 x fJ /Q /5) x (2 3/4 /5 x £J / 0 /5)- - CJ/<n 15. I f you know t h a t JjOx 6/9 - 24/3, what i s • 16. F i n d v a l u e s f o r D and A so t h a t J8 x 0 / A = 12/5 17. I f you know t h a t v/fl~749 = 161/7 and J i U G = 9 / 8 7 , t h e n y ou can a l s o t e l l t h a t 161/7. x 9/87 x 161/7 x 9/87 = Ol <T . Do n o t f i n d v a l u e s f o r CI and Zl . E x p r e s s Q / 0^  i n terms o f O and h . 18. F i n d a f r a c t i o n "C J~U/5 19. F i n d a f r a c t i o n ^ J5/S 20. U s i n g 18/1 as an example, show how you use t h e r u l e f o r f i n d i n g t h e s q u a r e r o o t s o f f r a c t i o n s t o f i n d s q u a r e r o o t s o f w h o l e s ? 21 How would you use t h e r u l e f o r f i n d i n g t h e s q u a r e r o o t s o f f r a c t i o n s t o f i n d t h e s q u a r e r o o t o f a mixed number, l i k e 1 9/16 . Show a l l o f y o u r s t e p s and w r i t e t h e a c t u a l s q u a r e r o o t on t h e l i n e t o t h e r i g h t . 155 22. F i n d / l 6 / 2 5 /A9 23. F i n d J A/25 /6A 2A. F i n d j 8 / 1 6 /2 25. F i n d J 81/25 /A9 26. F i n d J 1 6 / 1 0 0 /25 27. S u s i e has s u g g e s t e d a n o t h e r method f o r f i n d i n g s q u a r e r o o t s o f f r a c t i o n s , She s a y s t h a t i f she wants t o f i n d Ja/b , she f i n d s Jk x a/b and t h e n m u l t i p l i e s t h e answer by 1/2. F o r example, t o f i n d y i 6 / 2 5 , she s a y s t h a t A x 16 = 6A; t h e n , J~6h/25 = 8/5 and 1/2 x 8/5 = 8/10. She c l a i m s t h a t h e r answer i s a l w a y s c o r r e c t , and e q u i v a l e n t t o t h e one t h e t e a -c h e r g e t s by u s i n g t h e method t a u g h t i n c l a s s . W i l l S u s i e a l w a y s g e t an answer e q u i v a l e n t t o t h e t e a c h e r ' s i f she f o l l o w s h e r d i r e c t i o n s c o r r e c t l y ? Show how S u s i e w o u l d f i n d l/36/ 100 u s i n g h e r method and e x p l a i n why i t seems t o work. 28. Johnny s a y s t h a t a n o t h e r way t o f i n d Ja/b i s t o f i n d ^ - a x b and t h e n f i n d a/Jax b. F o r example, t o f i n d / l 6 / 2 5 , he s a y s 16 x 25 = A00, J^OO = 20. T h e r e f o r e , 16/20 = / l 6 / 2 5 . He c l a i m s t h a t he a l w a y s g e t s an answer e q u i v a l e n t t o t h e t e a c h e r ' s u s i n g t h e method t a u g h t i n c l a s s . I s he c o r r e c t : Show how Johnny would f i n d J3f>/100 and e x p l a i n why h i s method seems t o work. 156 29 . Sam s a y s t h a t he has s t i l l a n o t h e r way t o f i n d j / a / b . He s a y s J~ajb = J a x b/b. F o r example, t o f i n d Jib 12 5, he s a y s 16 x 25 = 400 , and t h e n Jlb/25 = / 4 0 0 / 2 5 = 2 0 / 2 5 . He c l a i m s t h a t h i s method i s c o r r e c t . Show how Sam wou I d f i n d l / " 3 6 / 100 and e x p l a i n why h i s method seems t o work. 30. To f i n d Ja/b, J u d y f i n d s (/a/4 x b and t h e n m u l t i p l i e s h e r answer by 2 . F o r example, t o f i n d fie/25, she sa y s 25 x 4 = 100 and t h e n / l 6 / 1 0 0 = 4 /10 and 2 x 4 /10 = 8 / 1 0 . So J l 6 / 2 5 = 8 / 1 0 . She c l a i m s t h a t h e r method i s c o r r e c t . Show how J u d y would f i n d J 36/100 and e x p l a i n why h e r method seems t o work. A P P E N D I X C E X P E R I M E N T A L D A T A f 158 T a b l e 19 S I and C l D a t a S u b j e c t P/A F . I . S I P r e S I Conp. S I Gen. C l P r e C l Comp. C l Gen. 1 A 23 41 21 10 25 17 5 2 A 23 40 22 12 29 16 8 3 A 23 25 17 2 11 3 2 4 A 25 34 24 9 25 18 10 5 A 24 40 22 12 28 14 13 6 A 23 38 23 10 30 15 10 7 A 23 37 23 18 24 18 14 8 P 24 35 12 4 29 17 2 9 P 23 29 18 5 16 3 2 10 P 24 36 24 10 21 14 6 11 P 24 30 16 6 19 3 2 12 P 24 28 5 4 14 12 1 13 P 25 26 14 5 18 11 1 14 P 24 40 19 6 21 4 3 15 P 24 29 10 3 12 13 1 16 P 23 25 9 2 9 9 0 17 P 21 29 13 5 16 7 0 18 P 22 24 23 3 18 4 2 19 P 18 36 10 6 27 9 0 20 P 18 27 15 2 15 4 1 21 P 21 33 17 12 28 5 5 22 P 18 27 24 10 25 17 4 23 P 21 30 23 7 23 16 2 24 P 19 30 16 3 19 6 0 25 P 18 26 23 1 19 11 3 26 P 18 23 12 0 15 6 0 27 P 20 25 19 2 10 4 4 S u b j e c t P/A F . I . S I P r e S I Oomp. S I Gen. C l P r e C l Comp. C l Gen. 28 P 21 30 18 2 16 11 3 29 P 18 20 19 8 9 7 0 30 P 21 21 23 2 19 3 1 31 P 22 24 16 3 19 4 0 32 P 20 35 11 3 23 12 4 33 P 21 30 17 4 15 3 0 34 P 18 25 7 2 12 3 0 35 A 22 25 12 3 14 7 0 36 A 21 33 0 1 15 9 4 37 A 18 33 19 5 22 11 2 38 A 22 40 12 8 27 12 4 39 A 21 39 11 4 25 1 3 40 A 21 35 8 4 11 7 0 41 A 21 40 12 5 23 11 3 42 A 20 30 18 1 11 2 0 43 A 22 30 0 14 29 10 6 44 A 19 28 17 11 22 14 5 45 A 18 24 11 6 20 5 1 46 A 19 24 4 1 17 5 3 47 A 22 23 16 6 24 8 1 48 A 19 29 19 11 25 10 7 49 A 20 28 6 9 23 18 4 50 P 17 28 22 3 13 1 1 51 P 10 27 17 4 14 14 2 52 P 15 20 18 3 11 7 0 53 P 17 23 8 4 21 8 3 54 P 13 25 6 2 16 3 1 55 P 13 22 0 2 8 4 1 56 P 17 33 6 3 9 5 0 57 P 17 15 2 0 5 1 0 58 A 10 15 15 2 10 2 0 59 A 17 33 12 6 23 1 2 60 A 17 30 13 4 15 7 0 160 S u b j e c t P/A F . I . S I P r e S I Comp. S I Gen. C l P r e C l Cornp. C l Gen. 61 A 16 36 18 62 A 17 33 12 63 A 16 33 18 64 A 15 28 5 5 21 12 1 9 25 14 7 9 21 9 7 6 27 10 3 161 T a b l e 19 S2 and C2 D a t a S u b j e c t P/A F . I . S2 P r e S2 Comp. S2 Gen. C2 P r e C2 Comp. C2 Gen. 65 P 23 35 27 22 37 14 9 66 P 23 25 25 12 28 13 6 67 P 24 35 22 9 39 15 13 68 p 23 28 7 4 30 14 7 69 P 23 35 27 13 38 8 3 70 P 23 24 27 2 24 0 2 71 P 24 33 27 18 38 13 12 72 P 24 23 19 11 29 5 1 73 P 24 33 26 18 36 9 11 74 A 23 26 19 6 24 2 7 75 A 24 24 1 10 22 0 1 76 A 24 35 27 24 38 14 22 77 A 25 35 27 24 40 15 22 78 A 24 35 27 11 39 11 19 79 A 23 34 26 21 39 15 17 80 A 23 24 25 13 31 13 16 81 A 23 30 10 8 29 14 11 82 A 24 26 26 15 29 15 11 83 P 22 15 27 13 15 12 3 84 P 18 27 20 14 36 13 14 85 P 21 24 18 1 23 11 8 86 P 19 31 22 7 24 9 8 87 P 20 18 26 10 23 7 2 88 P 20 28 27 5 22 9 0 89 P 19 28 27 7 20 2 3 90 P 18 16 26 4 18 9 2 91 P 22 33 27 4 35 9 8 92 P 21 22 19 11 25 10 9 93 P 20 33 19 10 29 9 5 S u b j e c t P/A F . I . S2 P r e S2 Comp. S2 Gen. C2 P r e C2 Comp. C2 Gen. 94 P 18 33 27 11 33 7 8 95 P 22 ' 31 25 10 30 15 13 96 P 22 31 27 13 40 14 12 97 P 20 26 25 14 25 15 11 98 P 20 35 27 8 39 14 12 99 A 21 26 5 4 24 1 4 100 A 18 25 4 7 16 0 4 101 A 22 30 0 15 26 9 7 102 A 19 34 11 10 26 5 5 103 A 18 29 15 13 28 6 5 104 A 22 27 13 1 27 3 9 105 A 19 33 17 4 35 l i 3 106 A 18 27 14 6 25 2 7 107 A 21 26 2 8 20 0 2 108 A 20 25 17 7 21 7 4 109 A 20 35 18 10 35 9 9 110 A 21 31 27 20 38 13 11 111 A 21 32 26 14 39 14 21 112 A 21 35 27 14 40 15 21 113 A 18 18 26 2 17 2 1 114 A 20 27 13 9 24 12 7 115 A 21 22 9 7 22 11 9' 116 A 20 24 25 6 15 5 1 117 A 20 22 5 7 17 5 2 118 A 21 23 25 14 27 8 7 119 A 21 32 22 15 24 15 9 120 A 21 24 23 10 27 12 11 121 A 19 22 25 9 27 3 8 122 P 14 26 23 8 24 3 3 123 P 16 14 19 12 20 13 8 124 P 9 19 21 13 21 1 2 125 P 17 27 20 4 21 9 8 S u b j e c t P/A F . I . S2 P r e S2 Comp. S2 Gen. C2 P r e C2 Comp. C2 Gen. 126 P 13 24 25 13 22 8 2 127 P 17 24 26 4 24 13 4 128 P 17 26 9 9 22 3 4 129 P 16 25 26 13 25 9 4 130 P 17 28 26 9 29 9 3 131 P 16 27 27 9 27 6 2 132 P 17 28 24 5 17 0 0 133 P 16 22 27 4 19 9 3 134 P 15 24 27 12 33 9 8 135 P 17 23 23 4 22 9 6 136 P 15 21 19 5 20 9 3 137 A 11 26 16 7 27 2 6 138 A 15 31 10 6 37 1 11 139 A 16 35 23 15 34 13 13 140 A 12 22 27 13 25 9 6 141 A 16 22 9 9 25 8 9 164 T a b l e 20 D i s t r i b u t i o n o f S c o r e s on t h e C h i l d r e n ' s Embedded F i g u r e s T e s t S c o r e F r e q u e n c y f 25 3 75 24 15 360 23 16 368 22 12 264 21 21 441 20 14 280 19 9 171 18 16 288 17 13 221 16 8 128 15 5 75 14 1 14 13 3 39 12 1 12 11 1 11 10 2 20 09 1 9 Mean = £.Fk_Xk N = 2776 141 = 19.7 

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