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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 t h i s thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1972  In  presenting  this  an advanced degree the I  Library  further  for  shall  agree  thesis  in p a r t i a l  fulfilment  of  at  University  of  Columbia,  the  make  it  freely  by  his  of  this  representatives. thesis  available  that permission for  s c h o l a r l y purposes may  for  be  It  gain  shall  Education  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  August,  1972  by  the  not  requirements  reference copying o f  Head  i s u n d e r s t o o d that  financial  Mathematics  for  extensive  granted  written permission.  Department o f  British  the  of  I  agree  and this  be a l l o w e d  that  study. thesis  my Department  copying or  for  or  publication  w i t h o u t my  ABSTRACT THE RELATIONSHIP BETWEEN FIELD-INDEPENDENCE AND INSTRUCTIONAL STRATEGY ON PERFORMANCE ON ELEMENTARY MATHEMATICS ALGORITHMS A s t u d y was conducted t o d e t e r m i n e the i n t e r a c t i o n e f f e c t , i f 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  a p a t t e r n s t r a t e g y w h i c h used diagrams e x t e n s i v e l y tegy w h i c h used a l g e b r a i c d e v o i d of d i a g r a m s .  f i e l d properties  any,  strategies,  and an a l g e b r a i c  stra-  f a m i l i a r t o t h e c h i l d and was  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 c o n t e n t  of the i n s t r u c t i o n a l ma-  terials . One h a l f the c h i l d r e n i n each of t w e l v e grade f i v e c l a s s e s , w h i c h were p a r t i c i p a t i n g i n a s t u d y conducted by a d o c t o r a l s t u d e n t , were r a n domly  selected  t o form the sample of the s t u d y .  F i g u r e s T e s t 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 t o the  The C h i l d r e n ' s Embedded sample.  Three n u l l h y p o t h e s e s were t e s t e d each a t *< = . 0 5 . (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  students  i n mean p o s t - t e s t s c o r e s  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  by an a l g e b r a i c  instructional strategy;  (2)  and s t u d e n t s  between s t u d e n t s '  (3)  between taught  There i s no s i g n i f i c a n t  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 of s t u d e n t s degree of f i e l d i n d e p e n d e n c e ;  These w e r e :  There i s no s i g n i f i c a n t  dif-  differing in interaction  degree of f i e l d independence and i n s t r u c t i o n a l  strategy.  Multiple  linear regression The  children  results  t e c h n i q u e s were used t o a n a l y s e the  of the study i n d i c a t e d  d i 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  though f o r the sample ficantly  different  algebraic  strategy  as a w h o l e  results. was  extreme  field  independent  instructional strategies, a l -  t h e two s t r a t e g i e s  For extreme  superior  that  data.  field  to the pattern  d i d not produce  independent students, strategy.  signithe  ACKNOWLEDGMENT  I would  like  t o t h a n k my c h a i r m a n , D r . G a i l  m i t t e e members, D r . D o u g l a s encouragement and e s p e c i a l l y completion o f this  thesis.  Owens a n d 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 f o r t h e i r humane p a r t i c i p a t i o n I would  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 thanks, t o Dr. Robert Conry sis  S p i t l e r , a n d my com-  also  like  m a t e r i a l s used  i n the hectic  t o thank Marian W e i n s t e i n i n this study.  f o r h i s help both with  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  A  special  the s t a t i s t i c a l programs.  analy-  TABLE OF CONTENTS  L I S T OF T A B L E S L I S T OF FIGURES  Chapter 1.  THE PROBLEM Introduction Background o f t h e Problem The P r o b l e m Witkin's  Construct  of Cognitive  Style  Hypotheses D e f i n i t i o n o f Terms 2.  REVIEV7 OF THE L I T E R A T U R E Introduction Individual Differences Individualized  i n Learning  Instruction  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 Mathematics Field-Dependence-Independence Discussion 3.  of the Literature  DESIGN AND PROCEDURE INTRODUCTION Population Sample  Studies  Page INSTRUCTIONAL  4.  5.  MATERIALS  27  MEASURING INSTRUMENTS  31  PROCEDURE  39  CONTROLS  AO  S T A T I S T I C A L PROCEDURES  41  A N A L Y S I S OF THE DATA  42  Graphing of S i g n i f i c a n t Results  51  Discussion  of the Figures  52  Discussion  of the Results  57  CONCLUSIONS AND I M P L I C A T I O N S  60  SUMMARY  60  LIMITATIONS  62  DISCUSSION  OF THE RESULTS  63  CONCLUSIONS  65  I M P L I C A T I O N S OF THE STUDY  66  BIBLIOGRAPHY  68  APPENDIXES  72  A.  Instructional Materials  B.  The M e a s u r i n g I n s t r u m e n t s  123  C.  Experimental  157  Data  73  iv  L I S T OF TABLES Table  Page  1.  D e s c r i p t i o n o f t h e I t e n s a n d 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 f o r the Four P r e t e s t s  32  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 Coefficient  33  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 a n d KR-20 R e l i a b i l i t y C o e f f i c i e n t  33  4.  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 , 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  34  5.  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 , C o m p u t a t i o n T e s t a n d KR-20 R e l i a b i l i t y Coefficient  34  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 a n d KR-20 Reliability Coefficient  35  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 a n d KR-20 R e l i a b i l i t y C o e f f i c i e n t  36  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  36  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 a n d KR-20 R e l i a b i l i t y Coefficient  37  10. CEFT R e l i a b i l i t y 11.  E s t i m a t e s and V a l i d i t y  Coefficients  38  Analysis  of S I Computation Scores  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  46  13. A n a l y s i s  of SI Generalization  47  Scores  V  Table  Page  14.  Analysis  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.  Analysis  of C l Computation  Scores  48  16.  Analysis  o f C2  Scores  48  17.  Analysis of C l G e n e r a l i z a t i o n Scores  49  18.  Analysis  50  Computation  o f C2 G e n e r a l i z a t i o n S c o r e s  vi  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, i n North American education, there i s a r e v i v a l of interest i n individualized instruction.  In mathematics programs such  as I P I ( I n d i v i d u a l l y Prescribed Instruction) and SAMI(Systematic Approach to Mathematical Instruction) stress an i n d i v i d u a l i z e d , sequential approach through the extensive use of diagnostic t e s t i n g , work-week u n i t s , multi-media learning centres, teacher-student  contracts and s e l f pacing.  Yet, as Gage and Unruh have noted, "... the fact i s , that despite several decades of concern with i n d i v i d u a l i z a t i o n , few, i f any, s t r i k i n g r e s u l t s have been  r  e  p  o  r  t  e  d  .  '  Coop and S i g e l have noted that "... few, i f any, of these i n d i v i d u a l i z e d programs have examined c a r e f u l l y the i n t e r - i n d i v i d u a l v a r i a b i l i t y of the learners, who  w i l l be exposed to t h e i r educational  2 stimuli."  y  Yet, Bloom, Cronbach, Gagne, Glaser and Jensen, have sug-  gested that there i s no one i n s t r u c t i o n a l method which provided  3 optimal learning f o r a l l students.  Cronbach and Snow have also stated  ^N. L. Gage and W. R. Unruh, "Theoretical Formulations f o r Research on Teaching", Review of Educational Research, XXXVTII(1967),  368.  2 Richard H. Coop and Irving E. S i g e l , "Cognitive S t y l e : Implications for Learning and I n s t r u c t i o n " , Psychology i n the Schools, VIII, 1971, 152.  3  Glenn H. Bracht, "Experimental Factors Related To AptitudeTreatment Interactions", Review of Educational Research, XXXX(1971), 627.  2 t h a t "...  the s e a r c h  f o r g e n e r a l l y s u p e r i o r methods must be  supple4  mented by  a search  f o r ways of a d a p t i n g  Bracht"' and discover  M i t c h e l l * ' have s t r o n g l y advocated i n v e s t i g a t i o n s to  significant  l e a r n e r and  I n s t r u c t i o n to the i n d i v i d u a l . "  i n t e r a c t i o n s between p e r s o n o l o g i c a l v a r i a b l e s of  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 d e s i r e d  educational  outcome.(They l a b e l t h i s 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 . ) which p e r s o n o l o g i c a l v a r i a b l e s are r e l e v a n t  and  outcomes?  As y e t , t h e s e q u e s t i o n s  the most p a r t , unanswered i n t h i s a r e a g e s t e d t h a t "...  we  w i l l find  But  as Becker'' n o t e s which  methods combined w i t h these p e r s o n o l o g i c a l v a r i a b l e s are r e l e v a n t which d e s i r e d e d u c a t i o n a l  the  of r e s e a r c h .  to  remain f o r  Cronbach has  t h e s e a p t i t u d e v a r i a b l e s t o be q u i t e  sugunlike  g our  present The  a p t i t u d e measures ... challenge  ".  to r e s e a r c h e r s ,  personological variables.  The  t h e n , i s to s e a r c h  challenge  to c u r r i c u l u m  for relevant  developers i s to  design  viable alternative instructional strategies. 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 Report 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 029 001 ^Glenn H. B r a c h t , " 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 Treatment I n t e r a c t i o n s " , Review of E d u c a t i o n a l R e s e a r c h , XXXX(1971), 627-41. James V. M i t c h e l l , " E d u c a t i o n ' s C h a l l e n g e To P s y c h o l o g y : The P r e d i c t i o n Of B e h a v i o r From Person-Environment 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, XXXIX(1969), 695 - 721. ^ J e r r y P. B e c k e r , "Research In Mathematics E d u c a t i o n : The R o l e Of Theory 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 Research In Mathematics E d u c a t i o n , 1(1970), 19 - 27.  The  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 American P s y c h o l o g i s t , X I I ( 1 9 5 7 ) , 681.  Psychology",  Background o f the Problem  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 suggested  psychological construct  o f c o g n i t i v e s t y l e may b e r e l e v a n t  problems of education.  I t has been s u g g e s t e d  that the t othe  that research  into the  i n t e r a c t i o n b e t w e e n c o g n i t i v e s t y l e a n d i n s t r u c t i o n a l p r o c e s s e s may vide  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 rthe optimal  assignment o f  9 learners 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 processes. suggested  that Witkin's  construct  i n d e p e n d e n c e ) may h a v e i m p o r t a n t  of cognitive  A considerable cerned w i t h a l g o r i t h m  instruction.  the  has f u r t h e r  style(field-dependenceeducation  used.  Weinstein  reviewed relevant  the meaningfulness of the algorithm  determines the c h i l d ' s a b i l i t y  algorithm  Spitler  p o r t i o n of elementary school mathematics i s con-  t u r e and c o n c l u d e d t h a t  algorithm.  10  i m p l i c a t i o n s f o r mathematics  i n terms of t h e types o f c u r r i c u l a r m a t e r i a l s  pro-  litera-  f o rthe child  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  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  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  algorithm  f o rthe c h i l d . ^  D. J . S a t t e r l y a n d M. A. B r i m e r , " C o g n i t i v e S t y l e s a n d 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 - 3 0 3 ; H e r m a n 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 o n C o g n i t i v e S t y l e f o r Problems of 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 School 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 G r u n e and S t r a t t o n I n c . , 1969) (mimeog r a p h e d ) ; R i c h a r d H. Coop a n d 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 , 1 9 7 1 , 152 - 1 5 9 . ^ 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 for Mathematics Education"(Unpublished D o c t o r a l d i s s e r t a t i o n , Wayne State U n i v e r s i t y , 1970). ^ M a r i a n S. W e i n s t e i n , "A S t u d y o f T y p e s 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 Elementary School Mathematics"(Unpublished 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 of B r i t i s h Columbia, 1972).  4 Weinstein designed a study to investigate the r e l a t i v e e f f e c t i v e ness of two i n s t r u c t i o n a l s t r a t e g i e s , which she labels pattern>arid algeb r a i c , i n the teaching of simple and complex algorithms.  She  classified  algorithms commonly taught i n the elementary grades as simple or complex on the basis of the number of prerequisites required f o r t h e i r acquisition.  She then designed two a l t e r n a t i v e i n s t r u c t i o n a l strategies f o r  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 i n the pattern  i n s t r u c t i o n a l strategy, while appeal to d e f i n i t i o n s and algebraic f i e l d postulates are used i n the algebraic i n s t r u c t i o n a l strategy. algorithm, a computation  For each  post-test and a generalization post-test were  developed to measure achievement on the algorithm.  The Problem The problem which t h i s researcher sought to investigate  was:  Do children d i f f e r i n g i n t h e i r degree of f i e l d independence respond d i f f e r e n t l y to the two i n s t r u c t i o n a l strategies developed by Weinstein? In other words, i s there an i n t e r a c t i o n e f f e c t between f i e l d independence and i n s t r u c t i o n a l strategy?  Two questions relevant to the problem were  also investigated: Does one of the i n s t r u c t i o n a l s t r a t e g i e s , on the average r e s u l t i n superior p u p i l performance?  Is there d i f f e r e n t i a l achievement  on the algorithms among students d i f f e r i n g i n t h e i r degree of f i e l d independence? In order to answer these questions, the investigator randomly selected a sample of the students p a r t i c i p a t i n g i n 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 pattern  or algebraic instructional strategy  complex a l g o r i t h m on t h e s e  Witkin's  Construct  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  of Cognitive  Style:  term c o g n i t i v e s t y l e  individual consistencies  of b e h a v i o r a l construct,  situations.  there  i s used  Witkin  i n psychological  However, s i n c e  i s some d i s a g r e e m e n t  and h i s a s s o c i a t e s  self-consistency  the  post-tests  along  divergent  suggest consistency  individual's perceptual,  literature to  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 t h e term i s a  range  psychological  among p s y c h o l o g i s t s  therefore, necessarily investigator  patterns  and a  Field-Dependence-Independence  s p e c i f i c observable behaviors are representative is  algorithm  algorithms.  The denote  and t a k i n g  on a s i m p l e  as t o which  of the term.  Therterm,  specific.  have developed a theory psychological  i n psychological intellectual,  of individual  growth p a t t e r n s .  "These  functioning which  pervades  emotional,  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 The  perceptual  "differentiation" tremes  operations." and i n t e l l e c t u a l  by an " a n a l y t i c a l  field  of  a p p r o a c h " and a  approach" or " a r t i c u l a t e d " versus " g l o b a l " .  dence-independence  theory  a r e combined t o form 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 -  of which are represented  "global field  components o f W i t k i n ' s  i s an i n d e x o f t h e p e r c e p t u a l  independence  represents  and p e r c e i v e  an i t e m  the a b i l i t y  as d i s t i n c t  Herman A. W i t k i n W i l e y I n c . , 1 9 6 2 ) , p.A.  Field-depen-  component.  Field  t o overcome an embedding  from i t s background.  The  context  field-depen-  et a l . , Psychological Differentiation(New  York:  dence-independence dimension r e f l e c t s ence s t i m u l i a n a l y t i c a l l y . through  research  the i n d i v i d u a l ' s  W i t k i n and h i s a s s o c i a t e s have  that a tendency t o experience  tendency toward f i e l d  ability  to experi-  demonstrated  analytically,  that i s a  independence, 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 tendency 13  to  structure experience.  perience to  analyse  and s t r u c t u r e e x -  A t t h e g l o b a l e x t r e m e " . . . when t h e  i s s t r u c t u r e d , t h e r e i s a tendency f o r i t s o r g a n i z a t i o n , as g i v e n ,  dictate  t h e manner i n w h i c h b o t h  are experienced;  when t h e f i e l d  g l o b a l and d i f f u s e . " ^ ency f o r experience rial  t o both  i s r e f e r r e d t o a s 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 a s o p p o s e d  a " g l o b a l " way o f e x p e r i e n c i n g .  field to  The a b i l i t y  the f i e l d  as a w h o l e and i t s p a r t s  lacks s t r u c t u r e , experience  At the " a r t i c u l a t e d "  e x t r e m e "... t h e r e i s a  c r e t e and t h e f i e l d  as a whole as  are experienced  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 from the independence,  that there i s a l s o a high degree of r e l a t i v e s t a b i l i t y .  children  tend  to maintain  their position  along  the  dependence dimension i n r e l a t i o n  to their peers.  off  a gradual  a b o u t t h e a g e o f 17 a n d t h e n [ 1 3  Ibid.,  as d i s -  organised.  a g e s o f 5 - 17 t h e r e i s a p r o g r e s s i o n t o w a r d g r e a t e r f i e l d but  tend-  t o b e d e l i n e a t e d a n d s t r u c t u r e d , e v e n when t h e m a t e -  lacks inherent organization; parts of a f i e l d  Witkin's  tends t o be  That i s ,  field-dependence-In-  There i s a  levelling—  "dedifferentiation"  toward  :  p p . 81 - 1 1 4 .  14 Herman A. W i t k i n , P h i l i p K. O l t m a n , E v e l y n R a s k i n A. K a r p , A M a n u a l 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 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), p.7. 1 5  Ibid.  and S t e p h e n California:  field  dependence a f t e r t h e mid  thirties.^  Research a l s o i n d i c a t e s that  the construct  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 . cant c o r r e l a t i o n between t h e c o n s t r u c t Block  design,  Witkin  and a group o f s u b t e s t s  c o r r e l a t i o n between t h e c o n s t r u c t  arithmetic subtest  scores  o f WISC.  c a n n o t b e i n t e r p r e t e d t o mean t h a t  field-dependence-  e t a l . found  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 !  non-significant and  of  o f WISC,  however, t h e r e  and v e r b a l  was  comprehension  T h u s "... i n t e l l i g e n c e  field-independent  signifi-  test  scores  children are of  18 generally superior  intelligence."  Hypotheses Conjectured dents w i l l  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  that  stu-  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  cognitive style,  and t h a t , c o n s e q u e n t l y ,  w i l l be demonstrated by s i g n i f i c a n t  this  preference  d i f f e r e n c e s b e t w e e n mean s c o r e s  the  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  with  lar  c o g n i t i v e s t y l e s taught 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 , since the pattern i n s t r u c t i o n a l strategy u t i l i z e s  r i d i n g p h y s i c a l analogy f o rj u s t i f i c a t i o n the  field  tates  of the algorithm  an  on simi-  over-  and s i n c e f o r  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 a s a w h o l e d i c 19 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 ti s expected that  ^ H e r m a n A. W i t k i n , D o n a l d R. G o o d e n o u g h a n d S t e 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 F r o m C h i l d h o o d To Y o u n g 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 - 3 0 0 .  Wiley  ^ H e r m a n A. W i t k i n I n c . , 1962), 223.  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 :  18 I b i d . , p.70. Herman A. W i t k i n e t a l . , A M a n u a l F o r The Embedded F i g u r e 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.  Tests  8 field  dependent c h i l d r e n w i l l  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  Secondly, since an  achieve higher  group p o s t - t e s t  the algebraic i n s t r u c t i o n a l strategy  to form t h e whole and s i n c e are experienced  f o rthe f i e l d  on  approach.  a n a l y t i c procedure i n which i n d i v i d u a l steps  of a f i e l d  scores  i s basically  must be drawn  independent c h i l d  as d i s c r e t e and t h e f i e l d  together  "... p a r t s  as a whole as  20 structured," higher tern  i ti s expected  group p o s t - t e s t  scores  field  independent c h i l d r e n w i l l  achieve  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 o n t h e p a t -  approach. N u l l Hypotheses.  t i o n t e s t s were a n a l y z e d complex a l g o r i t h m s . in  that  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 separately  f o r e a c h o f t h e two s i m p l e  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  and two a t oC =.05  each o f t h e a n a l y s e s .  H^: T h e r e i s n o 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 t u d e n t s taught by a p a t t e r n  instructional strategy  scores  between  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 n o 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 groups of students d i f f e r i n g H^: T h e r e i s n o s i g n i f i c a n t  i n degree o f f i e l d  2 0  Ibid.  between  independence.  i n t e r a c t i o n between s t u d e n t s '  independence and i n s t r u c t i o n a l s t r a t e g y .  scores  degree of f i e l 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 — " . . . those explanations which c o n s i s t purely of appeals 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 or to combinations thereof." 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 — " . . . those e x p l a n a t i o n s which use p h y s i c a l analogs f o rmathematical operations." 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 prerequisites f o rtheir acquisition..." ^ 2  Complex A l g o r i t h m s — " . . . those 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 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 . Field  I n d e p e n d e n t C h i l d — a c h i l d w h o s e s c o r e i s a b o v e t h e s a m p l e mean o n t h e C h i l d r e n ' s Embedded F i g u r e s T e s t  F i e l d D e p e n d e n t C h i l d — a c h i l d w h o s e s c o r e i s b e l o w t h e s a m p l e mean o n 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  Independence—relative p o s i t i o n of the student's score i n the 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 high score represents a high degree o f f i e l d independence.  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 T y p e s 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 Elementary School Mathematics"(Unpublished Doctoral dissertation, 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  Ibid.  23  I b i d . , p.5.  24  Ibid.  CHAPTER I I  REVIEW OF THE L I T E R A T U R E  Introduction This  section i s divided  vidual Differences view of learned  opinions  attempts t o adapt sections The  third  in  regarding  instruction to individual differences. indirectly  i n which  related to this  two study.  the treatment v a r i a b l e i s i n s t r u c t i o n i n mathematics.  the f i e l d  independence  construct  Studies,  and i n s t r u c t i o n a l s t r a t e g y  Individual Differences  In  variable  failed  t o produce  independence  con-  i n mathematics.  "... t h e q u e s t i o n  s t y l e and q u a l i t y o f t h e i r  cerned psychologists  any s t u d i e s  Learning  Gagne'has n o t e d t h a t  more a b o u t  studies  setting.  which 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  extent,  reviews  i s used as an a p t i t u d e  A thorough review of t h e l i t e r a t u r e  rate,  The l a s t  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 Mathematics, r e -  an e d u c a t i o n a l  struct  two, I n d i -  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  f o u r t h s e c t i o n , Field-Dependence-Independence  i n which  The f i r s t  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 -  consist of a review of studies  views studies The  i n Learning  i n t o four parts.  f o r a great  individual differences  o f how p e o p l e d i f f e r  i n the  l e a r n i n g i s one w h i c h h a s c o n -  many y e a r s .  Y e t ... we d o n o t know much  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 a g o . "  •'•Robert M. G a g n e , " L e a r n i n g a n d 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. G a g a e , L e a r n i n g a n d 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 ) , 1 9 6 7 , X I .  11 Cronbach a l s o notes t h e l a c k o f a coherent theory o f i n d i v i d u a l differences ing  i n learning.  to i s o l a t e aptitude  learning ables.  and d e s i g n i n g  He c a l l s variables  learning, say, to multiply  as  i s equally  relevant  of attempt-  to individual differences i n  a l t e r n a t i v e treatments to i n t e r a c t with  He s t a t e s : " I p r e s u m e t h a t  method t h a t  f o r an e x p e r i m e n t a l s t r a t e g y  an i n d i v i d u a l h a s g r e a t e r  from one method o f t e a c h i n g  good on t h e a v e r a g e . "  vari-  aptitude f o r  than from  He b r o a d l y  defines  "... w h a t e v e r p r o m o t e s 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 n v i r o n m e n t , a n d i t may h a v e a s much t o do w i t h  those  another aptitude educational  s t y l e s o f thought and p e r 2  sonality variables f u r t h e r argues that the  as w i t h  the a b i l i t i e s covered i n general  instructional strategy,  key to s u c c e s s f u l  tests."  He  as opposed t o l e a r n i n g r a t e , i s  i n d i v i d u a l i z a t i o n because a person'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 . Jensen has a l s o noted t h e l a c k o f knowledge concerning i n d i v i d u a l differences  In learning.  individual differences  I n an a r t i c l e  i n w h i c h he attempted t o d e l i n e a t e  i n l e a r n i n g h e 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 d o m a i n o f I D ' 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 k n o w l e d g e , 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 o n e 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  to describe  an e l e p h a n t .  At  3 this  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  warranted."  2 L e e J . C r o n b a c h , "How C a n I n s t r u c t i o n Be A d 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 " , e d . R. M. G a g n e , L e a r n i n g a n d 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 ) , 1 9 6 7 , 23 - 2 4 .  3  A r t h u r R. J e n s e n , " 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 " , e d . R. M. G a g n e , L e a r n i n g a n d 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 ) , 1 9 6 7 , 1 3 4 .  12  Individualized  Instruction  T y l e r , at the Abington situation  in individualized  prevalent i n individualized concept "...  C o n f e r e n c e i n 1967,  instruction  identified  programs at t h a t time.  t h a t t h e i n d i v i d u a l s h o u l d be  the concept  and  reviewed  He  the  current  s i x major  concepts  identified  "...  a b l e t o w o r k a t h i s own  m o s t 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  to  rate"  the  as  individualize  4 instruction."  The  t o work at times r a s s e d by is  (6) W i t h  convenient,  feeling (5)  linear.  o t h e r p o i n t s he m e n t i o n s a r e :  t h a t he  A few  the wealth  of forms,  static  t o him.  (2) P u p i l s h o u l d be  (3) S l o w l e a r n e r s h o u l d n o t be  i s much s l o w e r  factors, easily  than  others.  identifiable,  (4)  Learning  interfere with  embarprogress  progress.  of media of communication, audio, v i s u a l , v a r i o u s  and  moving, t h e l e a r n e r can  select  t h e one  able  kinds  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 t i o n , and  f u r t h e r suggests  the horse.  He  we've t r i e d  to develop  maybe we and  we  terials  asks:  are j u s t  merely  t h a t we  "Have we  may  really  be  questions  concerning  p l a c i n g the p r o v e r b i a l c a r t  devised  assuming t h a t everybody l e a r n s f o l l o w i n g  p r o v i d e m o r e o f t h e m a t e r i a l s t h a t we  _  before  the 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  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  r e q u i r e d f o r an  individualiza-  now  d o n ' t do  that,  t h e same s t r a t e g y ,  have r a t h e r than  e f f e c t i v e strategy."**  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. 6  Ibid.  p.7.  4  -5.  then  ma-  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 known o f p r e s e n t  attempts  to individualize  v i d e s a sequenced i n d i v i d u a l i z e d  p r o g r a m f o r g r a d e s K-6  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 nents  i n the I P I program are d e t a i l e d  spelling.  i s prepared  by  the  t e a c h e r b a s e d on  l a t i v e p e r s o n a l record of the c h i l d . teacher  teacher, small-group viewing  the i n d i v i d u a l  G l a s e r has  a b o u t t h e amount o f c o m p l e x i t y  pre-  t h e d i a g n o s t i c t e s t s and  noted:  "At  one  i n the a b i l i t y  cumuthe  appro-  r e q u i r e d . ...  complex, but  information that w i l l  or  s t a t e of  from measures of student  n o t be v e r y  to supply  to tapes  the present  environment  the  discs, our  performance  little  Sustained a n a l y s i s of  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  d u c t i o n and  The  Once t h e o b j e c t i v e i s d e c i d e d ,  f o r going  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  result  compo-  prescriptions.  of the three.  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  knowledge, the d e c i s i o n r u l e s  should  three major  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  filmstrips.  pro-  i n four subject  c h o o s e s f r o m 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 ,  p r i a t e f o r the c h i l d .  and  and  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  scription  The  IPI  sequential "behaviorally-stated" i n -  structional objectives, diagnostic testing The  i n s t r u c t i o n a l systems.  widely  i s known such  relationships  the teacher w i t h the k i n d of data  enable  him  t o manage t h e t a s k o f  re-  adapting  9 to i n d i v i d u a l d i f f e r e n c e s . " Pieronek  r e c e n t l y conducted  mathematics programs.  She  ^ D.  C.  9  Ibid.  personally visited  of i n d i v i d u a l i z e d  U.  S.  A.)  p.12. pp.12-13.  1968,  6.  reading  schools across North  Individually Prescribed Instruction  : Education ^Ibid.  a survey  and  America  (Washington,  14 i n which SAMI(Systematic Approach to Mathematical I n s t r u c t i o n ) being used. and  She  notes that  matics.  She  I P I has  the  an  s e l f - p a c i n g as  features  the  IPI  predominant feature  the mathematics program of  s e l f - s e l e c t i o n of  learning materials.  IPI  i n matheIPI.  How-  any  given  For  student i s f r e e to 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  were  of  excellent detailed hierarchical structure  f o u n d SAMI v e r y s i m i l a r t o  e v e r , SAMI a l s o unit,  identifies  and  from  pre-  packaged a l t e r n a t i v e s . ^ The little  literature  p r o g r e s s has  has  i n the  b e e n made i n t h e  vidual differences hampering the  reviewed  in learning  success of  b e e n s u g g e s t e d by  and  current  development of  tested  an  B e c k e r has  cular,  that  "...  such a theory i s  of  i n learning  isolating and  aptitude  designing  for interaction with  indi-  severely It  Tyler,  variables  that  relevant  alternative instructional  these v a r i a b l e s  approach to the  i s needed f o r  adaptation  of  the  educa-  learners.  Aptitude-Instruction  i n v e s t i g a t e the  l a c k of  very  toward a t h e o r y of  attempts to i n d i v i d u a l i z e i n s t r u c t i o n .  e f f e c t i v e systematic  t i o n a l s t i m u l i to  the  indicates that  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 , G a g n e and  to i n d i v i d u a l d i f f e r e n c e s t o be  sections  past t h i r t y years  that  a two-pronged e x p e r i m e n t a l s t r a t e g y  strategies  a b o v e two  I n t e r a c t i o n i n Mathematics  commented t h a t  "...  few  studies  i n t e r a c t i o n between a p t i t u d e only  m a t i c s learning."''"^  a s m a l l number o f He  further states  and  have been d e s i g n e d  I n s t r u c t i o n , " and  such s t u d i e s that  there  deal  to  in  directly with  i s as y e t  no  partimathe-  evidence to  sup-  ^ 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 M a t h e m a t i c s P r o g r a m s " , 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 , 1 9 7 1 , 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 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 Research i n Mathem a t i c s E d u c a t i o n , I , No. 1, 1 9 7 0 , 24.  15 port the g e n e r a l i z a b i l i t y  of r e s e a r c h  i n other  areas  to  mathematics.  12 Aiken sulted  a l s o notes  in significant  the p a u c i t y of s t u d i e s i n mathematics which  i n t e r a c t i o n s b e t w e e n a p t i t u d e and  He  does, however, r e p o r t three unpublished  be  d e s c r i b e d below.  d o c t o r a l s t u d i e s , two  These s t u d i e s are a l s o d i s c u s s e d  C r o n b a c h ' s d i s c u s s i o n o f one  instructional  o f them i s q u o t e d  restrategy.  of which  will 13 i n Cronbach's r e p o r t .  below.  14 Becker  u s e d two  school algebra students dents were g i v e n i n both d e n t was formula.  the  v e r b a l and  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  t o sum  formulas  symbolic  series. and  form.  One  a p p r o a c h was  explanations The  relating  o t h e r was  significant  a p t i t u d e measures were u s e d ( v e r b a l  i n t e r a c t i o n s b e t w e e n a p t i t u d e s and  expository, the these  and  stu-  to the s e r i e s  d i s c o v e r y o r i e n t e d , the  g i v e n e x a m p l e s 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 Two  high  stu-  asked to d i s c o v e r  numerical).  However,  the no  i n s t r u c t i o n a l methods were  found. Carry(1967) conducted a d i s s e r t a t i o n comparing 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 the mathematics of 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 meas 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 w e r e o b t a i n e d f o r 181 h i g h - s c h o o l g e o m e t r y s t u d e n t s . Carry hypothesized 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 m e n t and s o w o u l d p r e d i c t s u c c e s s i n i t , m o r e t h a n i n t h e a l g e b r a i c t r e a t m e n t . 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 m o r e h i g h l y to l e a r n i n g from a l g e b r a i c than from 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 Achievement", J o u r n a l of S c h o o l P s y c h o l o g y , I X , 1971, 203-212. 13 L e e J . C r o n b a c h 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. 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 , 1 9 7 1 , 2 0 3 - 2 1 2 r e p o r t e d J . P. B e c k e r , "An A t t e m p t 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  data d i d not confirm these hypotheses. No i n t e r a c t i o n s w e r e o b t a i n e d w i t h the 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 . Significant 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 m e a s u r e , b u t t h e l o w 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 m e a s u r e made o v e r a l l f i n d i n g s s u s p e c t . Analyses a t t h e i t e m l e v e l s h o w e d 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 interactions with aptitude. F o r b o t h i t e m s , t h e r e a s o n i n g m e a s u r e was found p r e d i c t i v e of responses i n the g r a p h i c a l treatment but not i n the a n a l y t i c treatment. 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 but not a n a l y t i c achievement. Without 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 to  426  sixth and  fifth  and  s i x t h grade s t u d e n t s ( f o u r  grade c l a s s e s ) .  w i t h i n these  Kropp^^ administered  The  students  a b a t t e r y of a p t i t u d e  fifth  g r a d e c l a s s e s and  were then c l a s s i f i e d  groups were c l a s s i f i e d  as d e d u c t i v e  tests  four  as v e r b a l o r  or i n d u c t i v e .  figural  Four i n -  structional strategies, verbal-deductive, verbal-inductive, figural-deductive and  f i g u r a l - i n d u c t i v e , were developed  mentary s e t concepts. tegies.  The  C l a s s e s were assigned  i n v e s t i g a t o r s hypothesized  correlate significantly  higher  ment on m a t e r i a l s p r e s e n t e d  measures and The  Learning Stanford  intact  t o one  day of  u n i t on the  verbally.  four  elestra-  t h a t v e r b a l a b i l i t y measures would  than f i g u r a l a b i l i t y measures w i t h  t w e e n f i g u r a l a b i l i t y m e a s u r e s and i n d u c t i v e m e a s u r e s and  f o r a p r o g r a m m e d two  achieve-  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  m a t e r i a l s presented  materials presented  figurally,  i n d u c t i v e l y , and  materials presented  deductively.  d e p e n d e n t m e a s u r e was  a criterion  t e s t o f 24  between  between  items,  12  be-  deductive  presented  L e e J . C r o n b a c h 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 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 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 , 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 B e t w e e n A b i l i t y M e a s u r e s and A c h i e v e m e n t u n d e r F o u r M e t h o d s 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 a n d 12 p r e s e n t e d analysis  figurally.  and t - t e s t s w e r e u s e d  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  to detect significant  p a i r s of regression c o e f f i c i e n t s .  differences  A summary o f t h e r e s u l t s  between  follows.  None o f 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 , s o t h e r e was no s u p p o r t f o r A T I e f f e c t s i n t h e v e r b a l o r f i g u r a l groups. 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 hypot h e s i s b e c a u s e two o f t h e t r a t i o s w e r e 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 w e r e 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 . T h u s 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 d e d u c t i v e m a t e r i a l s than f o r the 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 review of the l i t e r a t u r e confirmed Becker garding  and A i k e n ' s  comments r e -  the lack of aptitude-treatment i n t e r a c t i o n r e s e a r c h i n mathematics  education.  Only  t h r e e s t u d i e s were found  and t h e d i v e r s i t y  of the studies  made i t i m p o s s i b l e t o d r a w a n y 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 interactions  i n mathematics e d u c a t i o n .  Field-Dependence-Independence  Studies 18  D a v i s and K l a u s m e i e r n i t i v e s t y l e and a c o n c e p t  conducted  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  identification  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 . style  and t h e t r a i n i n g  (which c o r r e l a t e s  r =  procedure.  In both  as h i g h , m i d d l e  distribution 1 7  The s e c o n d studies,  factor.  o r low a n a l y t i c depending  o f s c o r e s on t h e t e s t .  Ibid.  The f i r s t  varied  varied  cognitive  cognitive  the Hidden Figures Test  .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  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 fied  task.  High  cog-  Students were  to  identi-  upon t h e i r p o s i t i o n i n t h e  analytic represented the a b i l i t y  pp. 246-247.  18 J . K e n t D a v i s a n d 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 C o n c e p t I d e n t i f i c a t i o n A s 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  to i d e n t i f y  the hidden  figures.  I n e x p e r i m e n t one, and  complexity  Information study's The  was  d e f i n e d as  i n the problem.  sample.  The  high analytics  analytics.  the  data  t h r e e l e v e l s of c o g n i t i v e s t y l e were the number(l,  Ninety  3 or  5)  the  H o w e v e r , t h e r e was  no  of  irrelevant  s e n i o r h i g h s c h o o l males formed  i n d i c a t e d a main e f f e c t  identified  of b i t s  concepts  due  used  the  to c o g n i t i v e s t y l e .  w i t h g r e a t e r ease than  the  low  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  complexity. In the second experiment, style  f a c t o r , h i g h a n a l y t i c and  were used. and  The  training  the low  c o n t r o l , the standard  was  the f i r s t found.  experiment, H o w e v e r , no  c o g n i t i v e s t y l e and t h a t "...  these  analytic,  procedure f o r the  and  four training  AO  low  task.  significant  verbal-prompt, high  a n a l y t i c ) formed the sample.  interaction  condition.  conditions  Eighty senior  a s i g n i f i c a n t main e f f e c t  training  training  extreme l e v e l s of the c o g n i t i v e  c o n d i t i o n s were prompt, v e r b a l o n l y ,  s c h o o l m a l e s ( A O h i g h a n a l y t i c and in  two  Davis  p r o c e d u r e s do n o t  due  to cognitive s t y l e  e f f e c t was and  As  found  Klausmeier  differentially  between  concluded  i n f l u e n c e concept "19  identification  for individuals manifesting different  cognitive styles.  20 Davis two  different  administered  i n v e s t i g a t e d the r e l a t i o n s h i p between c o g n i t i v e s t y l e concept i d e n t i f i c a t i o n t o 600  tasks.  The  senior high school females.  and  Hidden Figures Test  was  T h i r t y - s i x students  who  w e r e 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 a b o v e t h e mean) and I b i d . p. A29. 20 J . K e n t 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 C o n c e p t L e a r n i n g " (paper read at the annual meeting of the American E d u c a t i o n a l Research A s s o c i a t i o n , Chicago, 1972). 1 9  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  mean) w e r e c h o s e n t o p a r t i c i p a t e i n t h e s t u d y . fication  The  two  concept  t a s k s w e r e 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  entiated(SD).  A p r o b l e m whose s o l u t i o n  "...  the  identi-  sign-differ-  i s n o t based upon a  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  parti-  between  21 figures"  was  labelled  r e s p o n d i n g NSD lating  a s NSD.  p r o b l e m was  a relevant  cue.  changed  That  t h e c o n d i t i o n a l r u l e s ( a s was relevant NSD  cue.  was  i s , t h e SD required  problem could  be s o l v e d  using  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 would  or the  solve  the s o l u t i o n  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  cor-  iso-  f o r t h e c o r r e s p o n d i n g NSD)  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  Two  i n which the  t o a l l o w f o r i t s s o l u t i o n by  problem sooner than a n a l y t i c students because  than g l o b a l  one  Davis h y p o t h e s i z e d that g l o b a l s t u d e n t s would  a global strategy He  An S D . p r o b l e m  the  requires  t h e two  figures.  c o u l d be s o l v e d b y  s o l v e t h e SD p r o b l e m  an  sooner  students. dependent  number o f t r i a l s  measures were used t o a n a l y s e the r e s u l t s :  to c r i t e r i o n  and  (2) t h e number o f e r r o r s  to  (1) t h e  criterion.  F r o m 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 l u s i o n s were  con-  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 performance of the g l o b a l Ss. 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 , h o w e v e r , 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 b e t w e e n c o g n i t i v e s t y l e l e v e l s o n 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 S s w o u l d p e r f o r m b e s t o n t h e SD p r o b l e m a n d t h a t t h e g l o b a l Ss w o u l d p e r f o r m b e s t on t h e NSD problem. I t was f o u n d t h a t t h e a n a l y t i c S s 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 , b u t t h a t  Ibid.  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 . In fact, 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 quite poor."  their was  2 2  23 A study  c o n d u c t e d by D a v i s  b e t w e e n c o g n i t i v e s t y l e and graphy. and  two  Two  levels  117  Grieve  l e v e l s of c o g n i t i v e s t y l e ,  grade nine students  and  The  a n a l y t i c and The  differed  only  drawn from the  used.  The  tified  as  geographic  twice.  second a n a l y s i s u t i l i z e d  The  until  as  extreme)  Ibid.  pp.  The  the end.  the i n i t i a l step  verbaliIn  Whereas, in  the  (1) a m u l t i p l e - c h o i c e t e s t (2)  a multiple-choice  m a t e r i a l s i n new  first  only those  situations.  time, a l l the s u b j e c t s were s u b j e c t s who  extreme a n a l y t i c or extreme g l o b a l . ( 7 4 of the  classified 2 2  t o use  analysed  students  instructional unit.  delayed  dependent measures were used:  d a t a was  administered  used to c l a s s i f y  m e a s u r i n g a k n o w l e d g e o f t h e g e o g r a p h y s t u d i e d and  The  geo-  factorially  i n the placement of the  e x p o s i t o r y m e t h o d , t h e v e r b a l i z a t i o n was  t e s t measuring the a b i l i t y  of  expository,  H i d d e n F i g u r e s T e s t was  t h e d i s c o v e r y m e t h o d t h e v e r b a l i z a t i o n was  Two  teaching  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  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  sequence.  relationship  g l o b a l , were  a m e d i a n s p l i t was  two  e l e v e n h o u r s t o c o m p l e t e and  the  the  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  as a n a l y t i c o r g l o b a l .  in  explored  i n s t r u c t i o n a l s t r a t e g y i n the  combined t o g i v e a 2 x 2 d e s i g n . to  and  major c o n c l u s i o n s of  the  117  study  had  been  students  idenwere  were:  10-11.  23 J . K e n t 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 M e t h o d 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 G r a d e G e o g r a p h y " , The J o u r n a l o f E d u c a t i o n a l R e s e a r c h , X L V T I , 1 9 7 1 , 1 3 7 - 1 4 1 .  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 m e t h o d o f i n s t r u c t i o n h a d a n 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 t r e m e a n a l y t i c a n d e x t r e m e g l o b a l S s , h o w e v e r , i t was f o u n d 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 Japan's geography. 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 o f i n s t r u c t i o n s h o u l d b e 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 m a l e s u n l e s s s u f f i c i e n t time i s devoted t o e s t a b l i s h i n g those d i s c r i m i n a t i o n s which are b a s i c t o the 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. A n 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 w a s f o u n d t o d i f f e r e n t i a l l y influence h i shigher learning scores. 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 k n o w l e d g e o f g e o g r a p h y t o new s i t u a t i o n s t h a n w e r e g l o b a l Ss. N e i t h e r o f t h e methods o f i n s t r u c t i o n were found 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 e x t r e m e S s 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 experienced 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 cognitive similar  study investigated  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  or  or conservative.  global  according  t o 72 g r a d u a t e  instructional strategies,  level.  Ten educa-  common-  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  to their relative position  t o t h e median score 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 ) . found t o c o r r e l a t e  effect of  on concept a t t a i n m e n t .  concept 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 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 ality  the i n t e r a c t i o n  TIPT has been  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  "Those r e c e i v i n g  t h e commonality i n s t r u c t i o n were d i r e c t e d  to  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 tributes to  common t o t h e s e c a r d s .  compare t h e " y e s "  and "no" c a r d s w i t h  c a r d d i f f e r e d by one i r r e l e v a n t Ibid.  The c o n s e r v a t i v e s t r a t e g y  attribute  directed  the "focus" instance.  Ss  A"yes"  and a "no" c a r d d i f f e r e d by  p.141.  25 F l o r e n c e M. H e s t e r a n d 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 C o n c e p t 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 Psychology.LXXXV, 1971, 229-237.  22 one  26 attribute."  relevant  to-criterion  s c o r e on  The  d e p e n d e n t v a r i a b l e was  each t a s k .  The  data supported  the the  subjects  time-  following  con-  clusions: ( 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 c a n e f f i c i e n t l y u t i l i z e e i t h e r the c o n s e r v a t i v e or commonality 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 m o r e 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 the commonality 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 not 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 fore r e l a t e d to 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 m o r e rigorous conservative i n s t r u c t i o n a l strategy.27 28 Saarni of  the  c o g n i t i v e developmental l e v e l of  nitive style. of  investigated differences  logical  I t was  hypothesized  t h i n k i n g would provide  w i t h i n each P i a g e t i a n  an  selected the  from each of  study.  Two  subject  tasks  that Witkin's  2 6  Ibid.  p.232.  2 7  Ibid.  p.236.  The  function  subject's the  cog-  development  understanding  construct  of  field  level.  m a l e and  eight  female students  and  rod  and  randomly  nine) participated i n  were used t o i d e n t i f y the  portable  a  individual differences  sent cognitive developmental l e v e l ( f o r m a l o p e r a t i o n a l , concrete operational).  the  o v e r - a l l framework f o r  grades s i x , seven, eight  Piagetian  and  t h e o r y of  i n understanding  developmental  Sixty-four students(eight  the  that Piaget's  c o m p l e x 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 independence would prove f r u i t f u l  i n p r o b l e m s o l v i n g as  students'  pre-  t r a n s i t i o n a l or  f r a m e t e s t was  used to  iden-  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 p e n d e n c e 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 read at the annual meeting of the American E d u c a t i o n a l Research A s s o c i a t i o n , Chicago, 1972).  23 tify  the student's l e v e l of f i e l d  independence.  i n d e p e n d e n c e w e r e u s e d ; l o w , medium and h i g h . was  d e t e r m i n e d by r a n k i n g  resulting  distribution  Two  into  The  frame  The  test  levels  of  field  range of these  levels  s c o r e s and d i v i d i n g  the  the problem-solving tasks,  and  thirds.  detective stories  f o u r dependent stories.  t h e r o d and  Three  constituted  m e a s u r e s w e r e u s e d t o m e a s u r e p e r f o r m a n c e on e a c h o f d a t a was  the  analysed using m u l t i v a r i a t e a n a l y s i s of 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  the  following:  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 h a v e d o u b t f u l i m p l i c a t i o n s f o r complex problem s o l v i n g performance. 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 each 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 the P r o d u c t i v e T h i n k i n g problems. 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 p e n d e n c e 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 a n d / o r r e l a t i v e l y n o n verbal. The r e s u l t s o b t a i n e d h e r e , h o w e v e r , c a s t d o u b t on t h e g e n e r a l i t y o f the f i e l d independence 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 " or 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 of t h e i n d i v i d u a l i n h i s i n tellectual functioning.29 Of  t h e f i v e s t u d i e s r e p o r t e d above which used  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 b y D a v i s and G r i e v e , u t i l i z e d  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 . hypothesized  that s t u d e n t s would  sistent with  their cognitive style.  Three of the s t u d i e s used  solving  The  the f o u r t h ,  lbid.  pp.19-20  r e s u l t s , however,  global  con-  confirmed this  t a s k as  the  t h e s t u d y by S a a r n i , u s e d a p r o b l e m Of  an  s t u d y , i t was  students.  a concept 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 .  / y  In this  p e r f o r m b e s t when t a u g h t b y a m e t h o d  h y p o t h e s i s o n l y i n the case of extreme  t r e a t m e n t v a r i a b l e and  f i e l d - d e p e n d e n c e as  the three studies which  used  24 a concept  identification  reported p a r t i a l The  Davis  cept  t a s k , t h e s t u d i e s by D a v i s  i n t e r a c t i o n s between f i e l d  study hypothesized  identification  fication style.  task f o r which  cognitive style.  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 In the Davis  study  Tagatz  treatment.  b e s t on  and  The  Hester  the  b e s t on  and  o u t c o m e was  Tagatz  study  identi-  cognitive  true only f o r the  the r e v e r s e outcome  observed.  D i s c u s s i o n of the The a t h i s own  Literature  student  i n an  i n d i v i d u a l i z e d program i n mathematics  pace, using a v a r i e t y  i n s t r u c t i o n a l m a t e r i a l s , through matics  content.  o f a u d i o - v i s u a l d e v i c e s and a detailed  proceeds  pre-packaged  h i e r a r c h i c a l model of  I n s t r u c t i o n a l s t r a t e g i e s are not  adapted  to  mathe-  individual  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 a s G a g n e ' C r o n b a c h and have p o i n t e d out, v e r y  little  gested, that u n t i l developed,  less striking Only  a coherent  successes than a few  i n the area of i n d i v i d u a l i z e d  was  dif-  T y l e r has  theory of i n d i v i d u a l d i f f e r e n c e s i n  sug-  learning  instruction w i l l  be  anticipated. isolated  s t u d i e s of i n t e r a c t i o n between a p t i t u d e  i n s t r u c t i o n a l s t r a t e g y i n mathematics were found. significant  Jensen  i s known a t p r e s e n t a b o u t i n d i v i d u a l  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 .  is  con-  Tagatz  the concept  matched the s t u d e n t ' s  the hypothesized  i n the Hester  and  the 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  t h a t students would perform  a n a l y t i c s t u d e n t and was  i n d e p e n d e n c e and  t h a t students would perform  c l o s e l y matched to the s t u d e n t s study hypothesized  and H e s t e r  interactions.  In both  r e a s o n i n g m e a s u r e and  i n programmed b o o k l e t s .  Two  of these s t u d i e s ,  of these  and  reported  the a p t i t u d e v a r i a b l e  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  presented  However, the d e a r t h of s t u d i e s i n t h i s  area,  makes i t i m p o s s i b l e variables  t o d r a w any  to mathematics  Davis  and  of t h i s the  Three of  field these  Grieve, provide  study:  concerning  relevant  aptitude  instruction.  Five studies using were presented.  conclusions  i n d e p e n d e n c e as  aptitude variable  studies, Davis, Hester  partial  specifically  an  support  and  Tagatz,  f o r the hypothesized  that a student's  outcomes  performance w i l l  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  and  be  best  on  to h i s c o g n i t i v e  style. I n c o n c l u s i o n , the research  both i n the  the  of  area  adaptation  differences. v i d e d no in  The  instruction  t o an  literature  appropriate  i n mathematics. construct  an  The  However, of the  a f o u r t h used a p r o b l e m - s o l v i n g  treatment  variable.  and  the past  Thus, w h i l e  decade, very  in  a classroom  few  instructional  educational t a s k as  t a s k as  G r i e v e , u s e d an  b e t w e e n a p t i t u d e v a r i a b l e s and for  considered Witkin's  f i v e studies which u t i l i z e d  a p t i t u d e v a r i a b l e i n an  by D a v i s  pro-  i s a potential relevant aptitude variable i n  and  study  i n mathematics education  l i t e r a t u r e did i n d i c a t e that  s t u d i e s used a concept i d e n t i f i c a t i o n  the  in  individual  a p t i t u d e v a r i a b l e t o be  the  one,  extensive  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  of the  educational setting.  i n d e p e n d e n c e as  i n d i c a t e s the need f o r  of i n s t r u c t i o n a l s t r a t e g i e s to these  review  g u i d a n c e as  field-independence an  area  literature  the need  the  the  setting, treatment  treatment  instructional setting  for research  into  setting.  t h i s nature  the  as  Only the  interaction  been  have been  of  variable  variable.  i n s t r u c t i o n a l s t r a t e g i e s has s t u d i e s of  three  field  recognized  conducted  CHAPTER I I I  DESIGN AND PROCEDURE  INTRODUCTION  The  s t u d y was c a r r i e d  o u t 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  Weinstein, a d o c t o r a l candidate at the University of B r i t i s h Weinstein study in The  investigated  Columbia.  of four i n s t r u c t i o n 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 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  approach,  a l g e b r a i c , and ( 4 ) a m i x e d a p p r o a c h ,  present  study  proaches, dence.  investigated  (3) a mixed approach,  The  strategies  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  (2) a p a t t e r n j u s t i f i c a t i o n by  the effects  by Marian  level. approach,  pattern followed  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  t h e i n t e r a c t i o n e f f e c t b e t w e e n two o f t h e s e a p -  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  indepen-  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  Weinstein  study.  Population The  p o p u l a t i o n c o n s i s t e d of twelve grade f i v e  schools i n a lower mainland also participating  school d i s t r i c t  i n the Weinstein study.  i n British  classes i n s i x Columbia which  These c l a s s e s had been  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 ;  c o m p l e x 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 2, a n d ( 4 ) a n a l g e b r a i c a p p r o a c h  assigned  (1) a p a t t e r n approach  s i m p l e a n d c o m p l e x a l g o r i t h m s 1, ( 2 ) a n a l g e b r a i c a p p r o a c h  on  on s i m p l e and  on s i m p l e and c o m p l e x a l g o r i t h m s  o n s i m p l e a n d c o m p l e x a l g o r i t h m s 2.  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  were  treatments.  There  27 The with  grade f i v e  the algorithms  learning of the  l e v e l was c h o s e n b e c a u s e t h e s t u d e n t s a r e u n f a m i l i a r  used, y e t possess the necessary p r e r e q u i s i t e s  f o rthe  algorithms.  Sample U s i n g a t a b l e o f random numbers, a random s a m p l e o f o n e - h a l f o f the  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 e a c h 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 tested  to p a r t i c i p a t e i n the study.  These s t u d e n t s were  then  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  Four algorithms, complex, formed the b a s i s two  two c l a s s i f i e d  a s s i m p l e a n d two c l a s s i f i e d  of the i n s t r u c t i o n a l u n i t s .  F o r each  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 w e r e u s e d ; one c a l l e d a  instructional strategy The  pattern  i n s t r u c t i o n a l strategy  whereas t h e a l g e b r a i c ciative,  and t h e o t h e r an a l g e b r a i c  approach r e l i e d  d i s t r i b u t i v e and c o m m u t a t i v e f i e l d  never used i n t h e a l g e b r a i c properties The  properties.  i n t o stages with  t h e s t u d e n t a t each s t a g e .  for  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 .  rithms required vided  with  five  Diagrams were  approach.  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  required  extensively,  renaming o r t h e f i e l d  for  algorithms  pattern  on r e n a m i n g and t h e a s s o -  a p p r o a c h and c o n v e r s e l y ,  were never used i n t h e p a t t e r n  algorithm,  instructional strategy.  used diagrams  heavily  as  The o b j e c t i v e s  instructional periods,  nine i n s t r u c t i o n a l periods.  practise  f o r each s t a g e were  while  The  stated  simple  t h e complex  Worksheets were a l s o  s p e c i f i c i n s t r u c t i o n s a s t o when t h e y w e r e t o b e u s e d .  algoproA  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  SI:  f o r the four algorithms  P r o d u c t o f a M i x e d Number a n d a F r a c t i o n S t a g e 1:  (a) whole  number a n d a u n i t  fraction(l/n)  ( b ) w h o l e number a n d a p r o p e r (c) S t a g e 2:  w h o l e number and a m i x e d  (a) u n i t  fraction  number fraction  and a p r o p e r  (c)  and a mixed  proper f r a c t i o n  At a l l stages, finding  t h e p h y s i c a l analogy used  principle  and a u n i t  fraction  (b) p r o p e r f r a c t i o n  P a t t e r n Approach. is  follows.  to justify  by p a r t i t i o n i n g  number  the areas of rectangles  the algorithm.  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  A f o r t h e development  fraction  o f s t a g e s 2 ( a ) and 2 ( b ) .  Stage 1(c) used t h e  rectangles.  See Appendix  Stage 2(c) i s developed  rectangles.  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 ) a n d 1 ( b ) made u s e 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 ^ utilized  x 1/2 = 1/2 + ... + 1 / 2 ) .  r e n a m i n g a m i x e d number a s a w h o l e number p l u s a f r a c t i o n and  then 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 . of  Stage 1(c)  s t a g e s 2 ( a ) and 2 ( b ) .  distributive  See Appendix A f o r t h e development  S t a g e 2 ( c ) was d e v e l o p e d t h r o u g h r e n a m i n g  and t h e  principle.  S2: Comparison  of Fractions Using theCross-Product  S t a g e 1:  (a) comparing  a fraction with  1  (b) c o m p a r i n g a f r a c t i o n w i t h w h o l e S t a g e 2:  numbers  ( 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 r o a c h . S t a g e s 1 ( a ) a n d 1 ( b ) w e r e d e v e l o p e d by the  w h o l e numbers v i a d i a g r a m s .  by c u t t i n g s t u d e n t was fourths,  S t a g e 2 ( a ) was  developed through diagrams  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 m a n n e r . t a u g h t t h a t i f he w a n t e d  t o compare f o r example  he w o u l d c u t e a c h o f t h e t h i r d s  fourths i n thirds giving development  renaming  twelfths  into  The  thirds  and  f o u r t h s and e a c h o f t h e  i n each diagram.  See A p p e n d i x A f o r t h e  of 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 a n  appro-  priate  e q u i v a l e n t name f o r 1 a n d 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.  Stage 1(b) i n v o l v e d  r e n a m i n g t h e w h o l e number a s t h e w h o l e  x 1 a n d t h e n c h o o s i n g a n a p p r o p r i a t e name f o r 1. by w r i t i n g  t h e two f r a c t i o n s  and t h e n m u l t i p l y i n g A f o r the development  Cl:  S t a g e 2:  The  developed  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  b y a n 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  Appendix  of stage 2(b).  Changing a F r a c t i o n S t a g e 1:  S t a g e 2 ( a ) was  number  to a Decimal  Prerequisites (a)  decimal system  (b)  d i v i s i o n o f d e c i m a l s by whole  (c)  interpretation  The  Algorithm  (a)  terminating decimals  (b)  non-terminatingdecimals  two a p p r o a c h e s d i f f e r e d  were i d e n t i c a l i n the development  numbers  o f a f r a c t i o n as d i v i s i o n ( a / b  i n t h e i r development  o f s t a g e 2.  =  a-f-b)  of stage 1 yet  30 Pattern Approach. Stage 1(a) i s developed through the use of rectangles divided into tenths and hundredths.  See Appendix A f o r 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 f o r the development of stages 1(b) and 1 ( c ) . C2: Finding the Square Root of a Fraction Stage 1:  Prerequisites (a) m u l t i p l i c a t i o n of fractions (b) concept of the square root of a whole number (c) square root of a f r a c t i o n as /numerator/^/denominator  Stage 2:  The Algorithm (a) d i v i s i o n technique f o r the square root of wholes (b) d i v i s i o n technique f o r the square root of f r a c t i o n s (c) approximating square roots  For each of the approaches, stage 1(a) was developed i n the same manner as stages 2(a) and 2(b) of the corresponding approach i n algorithm SI. See Appendix A.  Stage 2(b) i s 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  n e e d e d t o make up t h e n u m e r a t o r w e r e s e l e c t e d Stage  2 ( a ) was j u s t i f i e d  a n d a s q u a r e was made.  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 pro-  c e s s o f s q u e e z i n g b e t w e e n w i d t h a n d h e i g h t o f t h e r e c t a n g l e s was demonstrated.  See A p p e n d i x  A f o r the development of stage 2 ( c ) .  A l g e b r a i c Approach.  I n stage 1(b) the square  d e f i n e d a s t h a t number w h i c h number i n q u e s t i o n .  when m u l t i p l i e d  The s q u a r e  Stage  renaming o f b o t h w h o l e s and f r a c t i o n s . T h e s t u d e n t was t a u g h t  responding  f a c t o r gets s m a l l e r .  by i t s e l f  a l l the facts  1 ( c ) was d e v e l o p e d  Stage  2 ( a ) used  immediately  gives the whole  root i s found by l i s t i n g  a s s o c i a t e d w i t h t h a t w h o l e number.  gument.  r o o t was  through  a closing-in ar-  t h a t as one f a c t o r g e t s l a r g e r , See Appendix  a  i t s cor-  A f o r t h e development o f  stage 2 ( c ) .  MEASURING  INSTRUMENTS  Criterion Pretests "Each of 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  free response  requisites  items designed  as determined  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 -  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 .  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  scores f o r differences  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 . " ^ 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 a s w e l l ficients  i s given i n Table  The  A  as t h e KR-20 r e l i a b i l i t y  breakcoef-  1 below.  H f a r i a n S. W e i n s t e i n , "A S t u d y o f t h e T y p e s 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 Elementary School Mathematics"(Unpublished 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) 2 3 .  Table 1 Description o f t h e Items and KR-20 Reliability Coefficients f o r t h e Four Pretests Type of Item  SI  Multiplication facts Diagrammatic representation of fractions Reciprocals Fractions as multiplication statements 1 as t h e multiplicative identity Commutativity and associativity Distributive law Area formula Long division algorithm Rewriting a mixed n u m b e r a s a sum Conservation of area Comparing fractions using diagrams Fractional names f o r 1 Division as sharing Reversibility of multiplication and division Preservation o f inequalities when multiplied by a positive Total  10 5 3 3 4 3 3 5  Pretests S2 Cl 10 5 3 3  5 5 3 2 3 2 5  C2 10 5 2 3 5 5  4 2 5 3  5 5  3 42 32 30 35 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 .90  .84  .90  .90  33 Criterion  Computation  "Each free  Tests  of the computation tests  response items designed t o t e s t  f o r the four  algorithms  the student"s a b i l i t y  consisted  to perform  that  2 algorithm."  Descriptions  KR-20 r e l i a b i l i t y  of the items of the four  c o e f f i c i e n t s a r e given below  tests  as w e l l  as t h e  i n T a b l e s 2, 3, 4 a n d 5.  Table 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 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  Item Type  and  Number o f I t e m s  Fraction  x whole  Mixed x whole Fraction  number  6  number  6  x fraction  6  Mixed x f r a c t i o n  6 Total  24  KR-20  .94  Table 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 Reliability Coefficient  Item Type  Fraction  Number o f I t e m s  and a w h o l e  number  9  2 proper fractions  9  2 improper f r a c t i o n s  9  2  Ibid  p.24  Total  27  KR-20  .95  of  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 Criterion Generalization  Tests  "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 . T h e t e s t s w e r e b a s e d o n f i v e t y p e s o f i t e m s — t h o s e designed t o measure: the 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 e n 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 a n d 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 m o r e 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 u s e 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 the algorithm(Type 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 coefficients  are reported i n Tables  6, 7 , 8  Table  a s w e l l a s t h e KR-20 r e l i a b i l i t y and 9 below.  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 Coefficient  Item Type  Number o f I t e m s  A  8  B  6  C  5  D  4  B - C  3  E  4  3  I b i d . p.25.  Total  30  KR-20  .83  36  Table 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 a n d KR-20 Reliability Coefficient  I t e m Type  Number o f I t e m s  A  8  B  6  C  8  D  4  E  4  Table  Total  30  KR-20  .82  8  Types o f Items o f C l , Changing a F r a c t i o n 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  to a  I t e m Type  Number o f I t e m s  A  8  B  9  D  9  E  4 Total  30  KR-20  .80  37  Table 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  Item Type  Number o f I t e m s  A  8  B  9  D  9  E  4  Testing  on C o g n i t i v e  Style  The C h i l d r e n ' s k i n ' s Embedded F i g u r e s  30  KR-20  .88  Factor  Embedded F i g u r e s Test.  Total  Test i s a modified  The t w e n t y - f i v e  item  version of Wit-  test consists of  eleven  p i c t u r e s o f c o m p l e x 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 s h a p e i s embedded a n d fourteen  p i c t u r e s i n which a house-shaped  f o r m 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.  enough and E a g l e ' s t e s t , a l t h o u g h bulky  and c o s t l y f o r u s e i n t h i s T h e Embedded F i g u r e s  trating  i t has h i g h  Good-  r e l i a b i l i t y was c o n s i d e r e d  too  study.  T e s t was f o u n d t o b e t o o d i f f i c u l t  and  frus-  f o r c h i l d r e n aged t e n and u n d 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 M a n u a l 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.  -  4  a d m i n i s t r a t i o n f o r c h i l d r e n i n the t e n to e l e v e n age b r a c k e t . dren's Embedded F i g u r e s T e s t reduced  the d i f f i c u l t y  Embedded F i g u r e s T e s t through the use f a m i l i a r to the c h i l d time l i m i t .  The  and  of simple  and  only f i r s t  Chil-  f r u s t r a t i o n of  the  complex forms more  through the e l i m i n a t i o n of the p r e s s u r e  c h i l d i s a l s o g i v e n more than one  the f i g u r e , although  and  The  of a  o p p o r t u n i t y to l o c a t e  t r i e s a r e a c t u a l l y used f o r s c o r i n g p u r -  poses . The  t e s t was  r a n g i n g i n age  s t a n d a r d i z e d u s i n g one  sixty  children,  from f i v e to twelve y e a r s , randomly s e l e c t e d from elemen-  tary schools i n Brooklyn. mative d a t a can be  "Because of the s m a l l N's  considered  c h i l d r e n i n the age The  hundred and  only t e n t a t i v e . " ^  groups 9-10  c h i l d r e n i n the p r e s e n t  and  11-12  Reliability  are r e c o r d e d  study were i n the 10-11  Table CEFT R e l i a b i l i t y E s t i m a t e s  i n v o l v e d , these  age  nor-  estimates  i n Table  for  10 below.  bracket.  10 and V a l i d i t y  Coefficients  Age  N .  I n t e r n a l Consistency  9-10  40  .88  .71  11-12  40  .87  .85  r  r CEFT,  EFT  Sources: 1971,  Internal Consistency T a b l e 4, p.25. r CEFT, EFT:  5,  p.25. ^Ibid.  p.17.  Ibid.  p.24.  6  r : A Manual F o r The  A Manual For The  Embedded F i g u r e s T e s t s ,  Embedded F i g u r e s T e s t s , 1 9 7 1 , T a b l e  39  PROCEDURE  Prior  to the a c t u a l beginning  of the study  m e e t i n g was h e l d b e t w e e n t h e r e s e a r c h e r s ( W e i n s t e i n participating lined sized.  teachers.  At t h i s  time,  i n the classrooms,  and O ' B r i e n ) and t h e  t h e two b a s i c a p p r o a c h e s were  A General  discussed  Information  S h e e t was g i v e n  to the teachers.  sheet  the materials(already  individual  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 b y C l o r S2 f o l l o w e d b y C 2 . mixed sequences of a l g o r i t h m s . contained  This  a t t h e m e e t i n g ) a n d t h e home p h o n e n u m b e r s o f t h e r e s e a r c h e r s .  An  i n Figure  algorithm  1  of the Procedure  ^ i n s t r u c t i o n on s i m p l 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 on s i m p l e complex a l g o r i t h m  ^instruction  C h i l d r e n ' s Embedded F i g u r e s  algorithm  algorithm  on 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 on complex a l g o r i t h m  The  procedure i s  1.  Flow Chart  P r e t e s t on simple  There were no  A flow chart of the general  Figure  zation  out-  a n d 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-  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  and  a  ^  -> p r e t e s t o n ^computation  ^ administration of the  Test  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 , computation and g e n e r a l i -  t e s t s , was d o n e b y t h e two r e s e a r c h e r s . Following  the completion  quence on t h e a l g o r i t h m s ,  testing  of the i n s t r u c t i o n a l  and t e s t i n g s e -  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 ducted  by t h i s  procedure  investigator.  outlined  The  i n the test  t e s t was  manual.  administered  according  to the  7  CONTROLS  T h e r e w e r e 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 . tailed  daily  instructional  searchers v i s i t e d the progress  guides  and w o r k s h e e t s w e r e p r o v i d e d ;  each of the twelve  of the study  These were:  teachers  every  (1) d  (2) t h e r e  s e c o n d day t o d i s c u s s  and ( 3 ) a l l t e s t s w e r e c o r r e c t e d b y t h e two r e -  searchers . H a w t h o r n e e f f e c t was  c o n t r o l l e d by:  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 (2) t h e i n d i v i d u a l only after  testing  the completion  conducted  ( 1 ) a l l t e a c h i n g and by t h e c l a s s r o o m  on t h e c o g n i t i v e s t y l e of i n s t r u c t i o n  f a c t o r was  I n and t e s t i n g  testing  teacher carried  o n t h e two  and out  algo-  rithms . 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 " problem i n t e s t i n g nitive  style  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  on t h e  doing  the  cogtest-  ing. Two  statistical  c o n t r o l s were a l s o used.  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 students  and (2) t h r e e c l a s s e s were a s s i g n e d  treatments  to help  Ibid.  i n the c o n t r o l of teacher  pp.26-28.  d i f f e r e n c e s among  t o each of the f o u r p o s s i b l e variation.  41  S T A T I S T I C A L PROCEDURES  Each  of the data  s e t s o f t h e e i g h t p o s t - t e s t s was  rately using multiple linear ties  regression  at the U n i v e r s i t y of B r i t i s h  linear  regression  The  sepa-  computing  facili-  C o l u m b i a w e r e u s e d and a m u l t i p l e  program contained  Lee, F a c u l t y of Education  techniques.  analyzed  i n the personal  file  of the U n i v e r s i t y of B r i t i s h  o f Dr. Seong  C o l u m b i a , was  Soo used.  CHAPTER I V  A N A L Y S I S OF THE DATA  The r e a d e r w i l l  recall  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  test  which measured a b i l i t y  to perform the algorithm  which 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 , w e r e  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  and a g e n e r a l i z a t i o n  on each o f t h e f o u r  The two s i m p l e a l g o r i t h m s w e r e S I , t h e p r o d u c t o f a f r a c t i o n number, and S 2 , t h e c o m p a r i s o n o f f r a c t i o n s . d e c i m a l , and C2, f i n d i n g complex  algorithms.  s i o n t e c h n i q u e s were used t o t e s t  graphed.  and a m i x e d to a  t h e square root o f a f r a c t i o n were c l a s s i f i e d  F o r the purposes of a n a l y s i s ,  Where s i g n i f i c a n t  algorithms.  C l ,changing a f r a c t i o n  t e s t s was t r e a t e d a s a n i n d e p e n d e n t u n i t o f d a t a .  set.  test  each o f t h e s e e i g h t Multiple  linear  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  post-  regres-  t h e t h r e e n u l l hypotheses f o r each  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  as  data  were  technique applies to  each o f the e i g h t s e p a r a t e a n a l y s e s o f the d a t a . A linear  r e l a t i o n s h i p between t h e dependent  p o s t - t e s t , a n d i n d e p e n d e n t v a r i a b l e s was a s s u m e d . were method o f i n s t r u c t i o n ,  field  od o f i n s t r u c t i o n  independence.  and f i e l d  independence  v a r i a b l e , s c o r e on t h e The i n d e p e n d e n t  and i n t e r a c t i o n between meth-  T h i s r e l a t i o n s h i p may b e e x p r e s s e d  as:  (DY^ where Y  & + B,v  lt  +  4v  = observation of i t hsubject  2 i  variables  +  sv 3  3i  +  on t h e p o s t - t e s t  ^v  4 i  +  A3 V,. li  = observation  of i t h s u b j e c t  on  the  covariate  V„. 2i  = observation  of  i t h subject  on  the method o f  = observation  of  i t h subject  on  the  field  instruction  independence measure  «• o b s e r v a t i o n o f i t h s u b j e c t o n t h e i n t e r a c t i o n b e t w e e n m e t h o d 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  are  the p o p u l a t i o n  lH. i s t h e d e p a r t u r e  of  regression  from the  linear  coefficients model  Smillie states: We may u s e 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, X k + 2 , ... , X p , 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 c o m p u t e d 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 . S u p p o s e 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 c o m p u t e d f o r t h e r e g r e s s i o n m o d e l 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. T h e n 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 Tt 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 /  (SSR F  =  - SSR ) / 7  SSE  h a s an F - d i s t r i b u t i o n w i t h p-k be u s e d t o t e s t t h e h y p o t h e s i s Restricted following null H^:  (Toronto:  a n d n-p-1 d e g r e e s o f t h a t 6^ = h = S ... H  f r e e d o m , and = Sf = 0.  may  1  kn  r e g r e s s i o n models were thus d e f i n e d  by  i n order  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  algebraic instructional  efficients  (n-p-1)  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  dents taught  was  /  (p-k)  to test  the  hypotheses.  T h e r e i s no  an  1  t e s t e d by  f o r the  between s t u -  students  taught  by  strategy.  c o m p u t i n g t h e sum  of squares of  the  regression  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 The R y e r s o n P r e s s , 1 9 6 6 ) , p p . A 9 - 5 0 .  and  Correlation  co-  44 (2) Y  = 0  ±  V^^) a n d a p p l y i n g  +  4  0 V (  groups o f students  coefficients  independence,  3  differing  = 4  v 3  ^)  +  ^V  1  ±  and a p p l y i n g  H^s T h e r e i s n o s i g n i f i c a n t  action, V^)  ±  +  6V t  + 4;  (omitting.method,  between  independence.  2t  +  ^ V  A  i  + Hi  (omitting  field  formula I .  i n t e r a c t i o n between  students'  degree of f i e l d  strategy.  t e s t e d b y c o m p 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  =  3  0  +  /> V (  and a p p l y i n g  1±  model: +  BV z  Zi  +  ^ V 3  3 i  + (ti (omitting  inter-  formula I .  I n a d d i t i o n , f o r each data  s e t , the s i g n i f i c a n c e of the contribu-  tion of the covariate to the regression el  A i  model:  f o rthe regression  (4) Y  ^ V  i n 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  coefficients  +  t e s t e d b y c o m p 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  ±  was  3 i  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  f o rthe regression  (3) Y  /? V  formula I .  H^: T h e r e i s n o s i g n i f i c a n t  was  +  U  sum o f s q u a r e s o f t h e l i n e a r  mod-  c h o s e n was t e s t e d b y c o m p 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  coefficients  f o rthe regression  (5) Y variate, V^) The 11-18  below.  ±  -  B, +  d V  and a p p l y i n g  eight separate  2  2 i  model: +  /5 V 3  3i  +  ^V  A  1  + U.I ( o m i t t i n g c o -  formula I . analyses  The d i s t r i b u t i o n  of the data  of scores  are reported  on t h e f i e l d  i n Tables  i n d e p e n d e n c e mea-  45 sure are contained Appendix  i n A p p e n d i x C.  squares of the regression  b e l o w , SSR  coefficients  of squares of the r e g r e s s i o n  (2),  raw d a t a a r e a l s o  contained i n  C. In each of the t a b l e s  sum  The  SSR  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  r e f e r t o t h e sum  of the f u l l model,  coefficients  ( 3 ) , ( 4 ) , or ( 5 ) , r e s p e c t i v e l y .  p r o g r a m u s e s n-p  and  of  ( 1 ) , and  of the r e s t r i c t e d  models,  The n u m b e r s i n e a c h o f t h e s e  models  defined  as t h e d e g r e e s o f f r e e d o m  above.  The  the  col-  computer  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. Thus, the  subjects  post-tests. scores,  F - v a l u e s h a v e one d e g r e e o f f r e e d o m f o r t h e n u m e r a t o r  degrees of freedom f o r the denominator.  t o o k t h e S2 a n d C2 p o s t - t e s t s . t a t i o n and g e n e r a l i z a t i o n of  t o o k t h e S I and C l  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  computed  sixty  Sixty-four  freedom f o r the numerator  Seventy-seven subjects  T h u s , i n t h e a n a l y s e s o f S2 a n d C2  s c o r e s , t h e computed  and  F - v a l u e s have one  compu-  degree  and s e v e n t y - t h r e e d e g r e e s o f f r e e d o m f o r t h e  denominator.  Table Analysis  of SI  11  Computation Scores  SSR  SSR  7  F-Value  Probability  Significance of c o v a r i a t e  (1)  .08228  (5)  .05974  1.4737  .22949  Significance of method  (1)  .08228  (2)  .08217  .0071  .93306  Significance of f i e l d indep.  (1)  .08228  (3)  .06194  1.3295  .25346  Significance of i n t e r a c t i o n  (1)  .08228  (4)  .08228  0.0  1.00000  w i t h l a r g e N's, s u c h 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 n o s i g n i f i c a n t d i f f e r e n c e b e t w e e n u s i n g n-p a n d n - p - 1 .  46 In the  the analysis  of SI Computation Scores presented i n Table 11,  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 m a i n e f f e c t  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 m a i n e f f e c t p e n d e n c e a n d no s i g n i f i c a n t of  field  interaction  c o v a r i a t e was s i g n i f i c a n t  at p =  instruc-  due t o d e g r e e o f f i e l d  effect  independence were not r e j e c t e d  due t o  a t << =  inde-  between method and d e g r e e .05.  The F - v a l u e f o r t h e  .22949.  Table  12  A n a l y s i s o f S2 C o m p u t a t i o n  SSR  Scores  F-Value  SSR'  Probability  Significance of c o v a r i a t e  (1)  .19443  (5)  .16895  2.3092  .13292  Significance of method  (1)  .19443  (2)  .17842  1.4510  .23224  Significance of f i e l d indep.  (1)  .19443  (3)  .19426  .0151  .90253  Significance of interaction  (1)  .19443  (A)  .19111  .3008  .58509  F r o m T a b l e 12 i t c a n b e s e e n 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 m a i n e f f e c t to  instructional  field and  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  i n d e p e n d e n c e a n d no s i g n i f i c a n t  degree of f i e l d  independence were n o t r e j e c t e d  v a l u e f o r t h e c o v a r i a t e was In the  interaction  the analysis  significant  at p =  due  due t o d e g r e e o f  effect  between method  a t <>C =  .05.  The  F-  .13292.  of SI G e n e r a l i z a t i o n Scores presented  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 m a i n e f f e c t  due t o  i n Table  13,  instruction-  47 al  strategy,  no s i g n i f i c a n t m a i n e f f e c t due t o d e g r e e  of f i e l d  and n o 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 b e t w e e n m e t h o d and d e g r e e independence  were n o t r e j e c t e d  was s i g n i f i c a n t a t p =  a t o<. = . 0 5 .  The F - v a l u e  f o r the  independence of  field  covariate  .00808.  Table Analysis  13  of SI Generalization  SSR  SSR  7  Scores  F-Value  Probability  Significance of c o v a r i a t e  (1)  .33100  (5)  .24728  7.5087  .00808  Significance of method  (1)  .33100  (2)  .33053  .0426  .83723  Significance of f i e l d i n d e p .  (1)  .33100  (3)  .31496  1.4389  .23501  Significance of i n t e r a c t i o n  (1)  .33100  (*)  .32617  .4332  .51294  Table Analysis  o f S2  14  Generalization Scores  SSR  SSR '  F-Value  Probability  Significance of c o v a r i a t e  (1)  .18560  (5)  .09223  8.3700  .00502  Significance of method  (1)  .18560  (2)  .16575  1.7793  .18639  Significance of f i e l d i n d e p .  (1)  .18560  (3)  .16591  1.7654  .18809  Significance of i n t e r a c t i o n  (1)  .18560  (4)  .16221  2.0969  .15187  48 In the al  the analysis  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 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 m a i n e f f e c t d u e t o i n s t r u c t i o n strategy,  n o s i g n i f i c a n t m a i n e f f e c t due t o d e g r e e o f f i e l d  independence  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 b e t w e e n method and d e g r e e o f independence were n o t r e j e c t e d was s i g n i f i c a n t a t p =  at  = .05.  field  The F - v a l u e f o r t h e c o v a r i a t e  .00502.  Table Analysis  15  of C l Computation /  SSR  SSR  Scores F-Value  Probability  Significance of c o v a r i a t e  (1)  .34542  (5)  .15500  17.4544  .00010  Significance of method  (1)  .34542  (2)  .32787  1.6087  .20956  Significance of f i e l d indep.  (1)  .34542  (3)  .32587  1.7918  .18576  Significance of i n t e r a c t i o n  (1)  .34542  (A) .32518  1.8551  .17826  Table Analysis  16  o f C2 C o m p u t a t i o n  SSR  SSR  7  Scores F-Value 15.681  Probability  Significance of c o v a r i a t e  (1)  .29443  (5)  .14279  .00017  Significance of method  (1)  .29443  (2)  .28271  1.2128  .27440  Significance of f i e l d i n d e p .  (1)  .29443  (3)  .25451  4.1302  .04576  Significance of i n t e r a c t i o n  (1)  .29443  (4)  .28788  .6773  .41321  A9 F r o m T a b l e 15 i t c a n b e s e e n t h a t  i n the a n a l y s i s  of C l Computation  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 m a i n e f f e c t structional  s t r a t e g y , no s i g n i f i c a n t m a i n e f f e c t  d e p e n d e n c e a n d no s i g n i f i c a n t of  field  independence were not r e j e c t e d  c o v a r i a t e was s i g n i f i c a n t In the al  the analysis  effect  at  =  b e t w e e n m e t h o d and d e g r e e .05.  The F - v a l u e f o r t h e  .00010.  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  s t r a t e g y a n d no s i g n i f i c a n t  interaction  effect  independence were not r e j e c t e d  o f no s i g n i f i c a n t m a i n e f f e c t  rejected p =  at p =  due t o d e g r e e o f f i e l d i n -  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 m a i n e f f e c t  gree of f i e l d sis  interaction  a t «C =  .05.  due t o i n -  i n Table  due t o  16,  instruction-  b e t w e e n method and d e -  a t «<=  .05.  due t o d e g r e e o f f i e l d  T h e F - v a l u e f o r t h e c o v a r i a t e was  The n u l l  hypothe-  i n d e p e n d e n c e was significant  at  .00017.  Table  17  Analysis of C l G e n e r a l i z a t i o n  SSR  SSR  /  Scores  F-Value  Probability  Significance of c o v a r i a t e  (1)  .51876  (5)  .3AA71  21.6996  .00002  Significance of method  (1)  .51876  (2)  •A8083  A.7291  .03361  Significance of f i e l d indep.  (1)  .51876  (3)  .A812A  A.6782  .03A5A  Significance of interaction  (1)  .51876  (A)  .A6123  7.1725  .0095A  50 In the al and  the analysis  of C l Generalization  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 s t r a t e g y , no s i g n i f i c a n t m a i n e f f e c t no s i g n i f i c a n t  interaction  independence were r e j e c t e d significant  at p =  effect  a t << =  Scores presented main e f f e c t  due t o  i n Table  instruction-  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 e t w e e n method and d e g r e e o f  .05.  field  The F - v a l u e f o r t h e c o v a r i a t 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  Scores  SSR'  SSR  F-Value  Probability  Significance of c o v a r i a t e  (1)  .63736  (5)  .21713  84. 5927  .00000  Significance of method  (1)  .63736  (2)  .62134  3. 2247  .07668  Significance of f i e l d i n d e p .  (1)  .63736  (3)  .62623  2. 2 4 0 1  .13880  Significance of interaction  (1)  .63736  (A)  .60991  5. 5 2 5 0  .02145  the  the analysis  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  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 m a i n e f f e c t  s t r a t e g y a n d no s i g n i f i c a n t m a i n e f f e c t were not r e j e c t e d action <£=  a t oL =  .05.  due t o  18,  instructional  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  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  b e t w e e n method and d e g r e e o f f i e l d  .05.  was  .00002.  Table  In  17,  i n d e p e n d e n c e was r e j e c t e d  The F - v a l u e f o r t h e c o v a r i a t e was  significant  at p =  interat  .00000.  51 Graphing of S i g n i f i c a n t Results 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 gree 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  s c o r e s and s i n c e between  i n the analysis  e f f e c t due t o d e o f C2 c o m p u t a t i o n  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 e f f e c t  method and 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  i n the analysis  ofC 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 a n d the  C l a n d C2 g e n e r a l i z a t i o n t e s t s w e r e g r a p h e d  to a i d i n the interpreta-  tion of theresults.  3 The UBC c o m p u t e r p r o g r a m g r o u p i n g s on t h e f i e l d  CGROUP  was u s e d t o i d e n t i f y n a t u r a l  independence measure.  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  CGROUP o p t i m a l l y  Five (2)  The s c o r e s on t h e f i e l d  The covariate, Cl  ( 3 ) 19-20,  (A)  the created  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.  o p t i m a l groupings were i d e n t i f i e d 16-18,  clustered  independence 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 a s 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 groups.  b y CGROUP.  These were:  ( 1 ) 9-15,  21-22 and ( 5 ) 23-25.  C2 c o m p u t a t i o n s c o r e s w e r e a d j u s t e d u s i n g  t h e C2 p r e t e s t  a n d t h e C l a n d C2 g e n e r a l i z a t i o n s c o r e s w e r e a d j u s t e d  a n d C2 p r e t e s t s  as c o v a r i a t e s .  s c o r e s were c a l c u l a t e d  range  as a  using the  T h e means o f t h e r e s u l t i n g r e s i d u a l  f o r each o f t h e f i v e range 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 approaches.  T h e g r a p h s o f t h e s e means f o l l o w .  means a n d t h e number o f o b s e r v a t i o n s f o r e a c h g r o u p a r e c o n t a i n e d  The i n Ap-  p e n d i x C.  T h i s program Computing C e n t r e .  i s on f i l e  at theUniversity of B r i t i s h  Columbia  52  D i s c u s s i o n of the Figures The not  r e a d e r s h o u l d b e a r i n mind  provide s t a t i s t i c a l  a n a l y s i s of the data.  t h a t the group s i z e s The effect due  due  analysis  significant  The  results  rather  indicated  r e a d e r s h o u l d f u r t h e r keep  do  by  the  i n mind  were u n e q u a l . o f C2  computation scores indicated  to degree of f i e l d  t o method o r i n t e r a c t i o n  Figure 2 indicates  the graphs of the r e s u l t s  s u p p o r t on w h i c h t o b a s e c o n c l u s i o n s , b u t  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 statistical  that  a significant  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 m a i n between  m e t h o d and d e g r e e o f f i e l d  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  cept f o r the extreme f i e l d  main effects  independence.  approaches, ex-  i n d e p e n d e n t g r o u p 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  independence  there i s a corresponding i n -  crease i n l e v e l of achievement. The  analysis of C l g e n e r a l i z a t i o n scores indicated  main e f f e c t  due  t o method, a s i g n i f i c a n t  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 interaction extreme  effect  field  dependent  strategy resulted structional  i s due  interaction  effect.  s t u d e n t s , 9-15  t o degree of  Figure  range, the pattern  field  3 indicates  this For  instructional  i n a h i g h e r l e v e l of achievement than the a l g e b r a i c i n -  s t r a t e g y ; w h i l e f o r extreme f i e l d  achievement  due  significant  to the performances of the extreme ranges.  range, the a l g e b r a i c i n s t r u c t i o n a l  cates that  main e f f e c t  a  strategy  than the p a t t e r n i n s t r u c t i o n a l the s i g n i f i c a n t  of  achievement of the f i e l d  al  strategy.  main  effect  independent students,  resulted strategy.  23-25  i n a higher l e v e l of The  o f m e t h o d i s due  figure  also  to the high  i n d e p e n d e n t g r o u p on t h e a l g e b r a i c  This high l e v e l of achievement of the f i e l d  indi-  level  instruction-  independent  group  Figure  2  Mean R e s i d u a l S c o r e s on C2 C o m p u t a t i o n  Scores  Figure 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 Pattern Algebraic  Mean Residual Scores  -3 -4 9  15  16  18  19  20  21  Field  22  23  Independence  '  25~~  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 main e f f e c t of  due  to degree of f i e l d  achievement of the extreme f i e l d  also accounts f o r the  independence.  Due  independent group  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 e x t r e m e is  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 The  to this high  level  t a u g h t by t h e a l -  field  independent  group  groups.  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  interaction effect.  significant  Figure 4 indicates  a  significant  that s t u d e n t s i n the extreme  field  i n d e p e n d e n t g r o u p , 2 3 - 2 5 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 b y t h e a l g e b r a i c a p p r o a c h a n d 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 b y  There i s a s i m i l a r , but l e s s The to  analysis  extreme, e f f e c t  of the d a t a a l s o  indicated  m e t h o d , b u t t h e n u l l h y p o t h e s i s was  the pattern  i n t h e 21-22  a trend  a c h i e v e d l o w e r on approach.  range  group.  toward a main e f f e c t  not r e j e c t e d .  Figure 4  due  indicates  that the trend  i s toward the a l g e b r a i c approach, but again the performance  of  field  the extreme  independent group i n f l a t e s  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 Figures  3 and 4 i n d i c a t e  i n d e p e n d e n t r a n g e s , 21-22 sive.  I n t h e 16-18  a c t i o n w i t h method.  the o v e r a l l  sions  effect  of the p a t t e r n  and 2 3 - 2 5 , t h e i n t e r a c t i o n  r a n g e g r o u p and  t h e 19-20  For the extreme f i e l d  range group  dependent  g r o u p may  be due  group.  The  inconsistent  t o t h e s m a l l number o f s u b j e c t s  and  e i g h t o n C2.  T h i s range contained  all  the ranges of the f i e l d  are  field inconclu-  t h e r e i s no  g r o u p , 9-15  and h e n c e no results  in this  on C l conclu-  for this  r a n g e , s i x on C l  t h e s m a l l e s t number o f s u b j e c t s  independence dimension.  inter  range,  However, the i n t e r a c t i o n s  f o r t h i s group are i n the o p p o s i t e d i r e c t i o n s can be drawn f o r t h i s  results  of the approach.  t h a t , e x c e p t f o r t h e two e x t r e m 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 . and C2  the o v e r a l l e f f e c t  of  Figure  A  Mean R e s i d u a l  S c o r e s on C2  Generalization  Scores Pattern Algebraic  57  However, the  r e s u l t s are  consistent with  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  with  t o p e r f o r m a n c e on  regard  gorithms, there al  strategy.  alization  i s a trend  For  the  t e s t s on  s t r a t e g y was  Discussion  the  high  For  to the  the  21-22  g e n e r a l i z a t i o n t e s t s on  23-25 r a n g e , w i t h  regard  complex a l g o r i t h m s , to the  the  t o t a l sums o f  separate analyses  the  data.  complex a l -  algebraic instructionthe  instructional strategy.  squares of the  i n d i c a t e that  f u l l m o d e l s , SSR  t h e m o d e l c h o s e n was  i n each  appropriate  a n a l y s i s of  the  data also i n d i c a t e s that  v a r i a t e s were w e l l chosen.  In each of  the  data analyses,  f o r the  coefficients  and  tation scores, of  the  ^=  l a r g e s t p o r t i o n of i n each of the  the  regression  t r i b u t i o n of  the  gener-  algebraic instructional  statistical  accounted  ex-  range,  t o p e r f o r m a n c e on  the  pattern  the  two  Results  the  The  23-25.  toward s u p e r i o r i t y of  strongly superior  of  The  the  the  and  respect  sum  of  data analyses,  c o n t r i b u t i o n of c o e f f i c i e n t s was  c o v a r i a t e was  the  the  squares of  significant  to the  the  at  =  =  .23  at  total  .01. and  to xzo-  the  covariate regression  e x c e p t f o r S I and  covariate  significant  the  of  S2  sum  For  of  SI  f o r S2  compusquares  the  con-  at  .14. The  r e s u l t s of  examples of simple  the  a n a l y s i s of  algorithms  a l i z a t i o n p o s t - t e s t s , t h e r e was  c h o s e n , on no  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  due  to degree of  field  the  data i n d i c a t e that  both the  statistically statistically  i n d e p e n d e n c e and  no  c o m p u t a t i o n and  two  gener-  s i g n i f i c a n t main e f f e c t s i g n i f i c a n t main e f f e c t  statistically  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  f o r the  and  field  significant i n independence.  58 Thus,  f o r the simple algorithms,  produce s i g n i f i c a n t l y  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 f f e r e n t r e s u l t s a n d t h e r e was no d i f f e r e n t i a l  c h i e v e m e n t e f f e c t due t o d e g r e e o f f i e l d evidence of d i f f e r e n t i a l within  achievement  independence.  statistical  analysis  o n t h e two I n s t r u c t i o n a l  of the data i n d i c a t e s  mance on t h e c o m p u t a t i o n t e s t s o n t h e c o m p l e x nificant  main  t e r a c t i o n e f f e c t between pendence. complex  algorithms,  different within  Thus, w i t h  instructional strategy  regard  o f t h e d a t a , does  f o r perfor-  t h e r e i s no  and d e g r e e o f f i e l d  to producing computational a b i l i t y  indicate that  inde-  with the  different results  independence.  However, t h e  there i s d i f f e r e n t i a l  achievement  o n 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  field  independence.  The g r a p h i n g o f 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  cate that with  increasing  achievement  increases  independent  level.  The significant analysis  with  statistical main  degree of f i e l d  The significant  e f f e c t due t o d e g r e e o f f i e l d  statistical main  C2 g e n e r a l i z a t i o n  o f f a t t h e extreme  of C l generalization  o f t h e C2 g e n e r a l i z a t i o n analysis  indi-  i n d e p e n d e n c e , t h e mean l e v e l o f  a s l i g h t tapering  analysis  sig-  are not s i g n i f i c a n t l y  do n o t p r o d u c e s i g n i f i c a n t l y  groups s i m i l a r i n t h e i r degree of f i e l d  analysis  strategies  and no s i g n i f i c a n t i n -  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  o v e r a l l , and a l s o  no  independence.  that  algorithms,  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  a-  T h e r e was a l s o  groups of students s i m i l a r i n t h e i r degree of f i e l d The  d i dnot  field  scores indicates  independence.  a  However, t h e  scores d i d not support t h i s f i n d i n g . of C l g e n e r a l i z a t i o n  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 scores indicates  a strong  trend,  scores indicates  a  and t h e a n a l y s i s o f  F - v a l u e has p = .077,  59  toward a main e f f e c t sults  of  the  main e f f e c t highest  The  and  and  i s due  levels  structional  indicate  Cl  of  due C2  to the  to  instructional  generalization the  tests  h i g h l e v e l of  field  statistical  a significant  indicate  graphs of  that  p e r f o r m a n c e of  i n d e p e n d e n c e d i m e n s i o n on  field  a n a l y s e s of interaction  independence.  the  independent students,  21-22  and  gebraic approach i s s u p e r i o r to the  s t u d e n t s i n the  C l and  effect The  provide conclusive information only for  for  The  this  the  significant  s t u d e n t s at the  re-  the  algebraic  two  in-  strategy.  degree of  superior  strategy.  generalization  between i n s t r u c t i o n a l  graphs of the  two  23-25 r a n g e s . the  C2  these r e s u l t s , extreme l e v e l s  For  strategy however, of  these students  p a t t e r n a p p r o a c h and  23-25 r a n g e .  scores  is  field the  al-  dramatically  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 a n y , between t h e f i e l d a pattern  independence construct  s t r a t e g y which used diagrams e x t e n s i v e l y  which used a l g e b r a i c of diagrams.  field  properties  and an a l g e b r a i c  f a m i l i a r to the c h i l d  A review of the relevant  s t y l e may h o l d and  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 ,  literature  a n d was  indicated that  the key t o understanding i n d i v i d u a l d i f f e r e n c e s  a t t h e same t i m e t h a t v e r y  The  were  independenca  i n mathematics.  i n t h e i r degree o f f i e l d  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  (1)  independence respond  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) al  In learning  s t u d y was d e s i g n e d t o a n s w e r t h r e e m a j o r q u e s t i o n s :  children differing  cognitive  No s t u d i e s  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 instructional strategy  devoid  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 .  and  strategy  differ-  I s one o f t h e i n s t r u c t i o n -  to the other?;  f e r e n t i a l a c h i e v e m e n t among s t u d e n t s d i f f e r i n g  Do  (3)  I s there  dif-  i n t h e i r degree of f i e l d  independence? 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  of the study.  ihc-se. •  c l a s s e s were p a r t o f t h e sample o f a study b e i n g conducted by M a r i a n Weinstein,  a doctoral  candidate at the University of B r i t i s h  Columbia.  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 study.  Ms. W e i n s t e i n c l a s s i f i e d  algorithms  as s i m -  61 p i e o r c o m p l e x on t h e b a s i s o f t h e n u m b e r o f p r e r e q u i s i t e s their  required for  a c q u i s i t i o n a n d two e x a m p l e s o f e a c h t y p e w e r e s e l e c t e d f o r u s e i n  the study.  The two s i m p l e a l g o r i t h m s c h o s e n w e r e t h e p r o d u c t  t i o n and a m i x e d n u m b e r , S I a n d t h e c o m p a r i s o n complex a l g o r i t h m s chosen were changing finding  the square  following  i n s t r u c t i o n a l and t e s t i n g  sequence, going  simple algorithm  computation  The  through  the  e i t h e r S I and  > i n s t r u c t i o n on  and g e n e r a l i z a t i o n  > computation  two  t o a d e c i m a l , C l and  => p r e t e s t o n c o m p l e x a l g o r i t h m  complex a l g o r i t h m ( 9 s e s s i o n s )  frac-  E a c h c l a s s went t h r o u g h  C l o r S2 a n d C2: P r e t e s t o n s i m p l e a l g o r i t h m algorithms(5 sessions)  o f f r a c t i o n s , S2.  a fraction  r o o t o f a f r a c t i o n , C2.  of a  simple  t e s t s on t h e  => i n s t r u c t i o n  and g e n e r a l i z a t i o n  on  tests  on complex a l g o r i t h m . I n each of t h e twelve pleted  c l a s s e s , o n e h a l f o f t h e s t u d e n t s who  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 form  t h i s study.  The C h i l d r e n ' s Embedded F i g u r e s T e s t was  administered  t o these students.  then  com-  the sample of individually  ^  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 data.  E i g h t separate analyses of the data were performed:  the computation  the r e s u l t s of  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 a n d two c o m p l e x a l g o r i t h m s .  Three n u l l hypotheses  e a c h t e s t e d a t «x* = . 0 5 : ( 1 ) T h e r e i s no s i g n i f i c a n t p o s t - t e s t s c o r e s between s t u d e n t s  were  d i f f e r e n c e i n mean  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  t e g y and s t u d e n t s  taught  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 b e t w e e n g r o u p s o f  students d i f f e r i n g  by an a l g e b r a i c i n s t r u c t i o n a l  stra-  i n degree of f i e l d  independence;  s t r a t e g y ; (2) There  (3) T h e r e i s no s i g -  nificant  interaction  instructional The  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  strategy.  analysis  of the data resulted  i n acceptance of the three  null  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 . the  C2 c o m p u t a t i o n t e s t ,  thesis The  the data resulted  o f no s i g n i f i c a n t main e f f e c t  analysis  jection  o f t h e complex  i n rejection  due t o d e g r e e o f f i e l d  algorithm  generalization  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  structional  of the n u l l  s t r a t e g y and d e g r e e o f f i e l d  tests  On  hypo-  independence. resulted  interaction  i n re-  between i n -  independence.  LIMITATIONS  The of  existing  jects  r e s e a r c h agreement intact  were randomly  factor  the school d i s t r i c t  grade f i v e c l a s s e s .  stipulated  Thus, o n l y c l a s s e s and n o t sub-  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  i n the study.  strategy preferences nor of the teachers attitude  instructional  s t r a t e g y randomly assigned on a v o l u n t a r y b a s i s ,  t o w a r d s 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 The  of  t o them.  instructional  schooling prior  i n the study.  strategies  their  attitude  positive. experiences i n mathematics i s  No a s s e s s m e n t was made o f t h e t y p e s  t o w h i c h t h e s u b j e c t had been  t o the study, nor of the i n s t r u c t i o n a l  n a n t l y used i n h i s mathematics  toward t h e  However, s i n c e t h e t e a -  i t i s assumed t h a t  subjects' previous instructional  an u n c o n t r o l l e d v a r i a b l e  was a n  No a s s e s s m e n t was made o f t h e t e a c h e r ' s  instructional  chers p a r t i c i p a t e d  the use  assigned t o treatments.  Teacher preferences uncontrolled  with  experiences.  exposed  strategy  in his  predomi-  63 The  use  of the  familiar  mized the Hawthorne E f f e c t . done i n the considered  s t u d y was  uniform  a departure  the on  to separate  simple  the  e f f e c t s of  algorithm  two  simple  analyses  on  cumulative  the  i n s t r u c t i o n on  THE  c o m p u t a t i o n and  two  T h e r e a p p e a r s t o be first  two  these f i n d i n g s .  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 are  i s that  f o r groups of  independence.  The  from  because i n s t r u c t i o n  complex  algorithm.  for  a m i x e d n u m b e r and  the  com-  the  i n d e p e n d e n c e and  f o r simple  of  instruc-  the  algebraic  equally e f f e c t i v e both f o r  the p e r i o d  insufficient  of  to allow  in their  the  degree  exposure to for  no  explanations  algorithms,  sample d i f f e r i n g  no  the  differential  t o emerge. a n a l y s i s of  exception  field  algorithm  equally plausible  second i s t h a t  a p p r o a c h e s , f i v e s e s s i o n s , was  The the  The  simple  i n s t r u c t i o n a l s t r a t e g i e s and  for  effects  testing  r e j e c t i n g the n u l l hypotheses of  i n t e r a c t i o n between degree of f i e l d  a w h o l e and  the  p r o d u c t o f a f r a c t i o n and  significant  two  be  I t i s impos-  g e n e r a l i z a t i o n scores  d i f f e r e n c e between the  of f i e l d  can  RESULTS  significant  s a m p l e as  the  complex a l g o r i t h m ,  f r a c t i o n s , r e s u l t e d i n not  tional strategy.  the t e s t i n g  However, the  learning effect.  i n s t r u c t i o n on  the  preceded  of  algorithms,  p a r i s o n of  of  f r o m t h e n o r m f o r m a t h e m a t i c s and  D I S C U S S I O N OF  The  length  mini-  for a l l subjects.  e f f e c t s of i n s t r u c t i o n " the  amount and  time-schedule  t o a Hawthorne E f f e c t .  T h e r e i s an u n c o n t r o l l e d sible  s e t t i n g and  However, the  t o have c o n t r i b u t e d  s c h e d u l e was  classroom  of the  the  C2  computation t e s t scores  extreme f i e l d  independence i n c r e a s e d ,  the  indicated that,  independent students, level  as  the  degree  of achievement i n c r e a s e d .  The  with of data  64 for  the C l computation t e s t , however, d i d not support t h i s  e x a m i n a t i o n o f t h e two findings.  complex  A l t h o u g h t h e two  d e c i m a l and  finding  a l g o r i t h m s may  complex  their  acquisition,  p r o v i d e an i n s i g h t  are s i m i l a r  i n t h i s extended  cannot r e l y  Changing  system.  a fraction  the  child,  fraction  thus making  level,  degree of f i e l d  The  t h e r e may  independence  to a  i n their  in  to a decimal involves and  the l e a r n i n g  the square root of a  of  fraction  learned mathematical  con-  a r i s e out of the r e a l w o r l d experiences of  i t more a b s t r a c t  to a decimal.  grade f i v e  Finding  on s u c h an e x t e n s i o n o f p r e v i o u s l y  c e p t s , nor does i t n a t u r a l l y  these  the a l g o r i t h m s d i f f e r  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 division  into  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  the i n v e s t i g a t o r claims that  t h e i r degree of a b s t r a c t n e s s .  Closer  algorithms, changing a f r a c t i o n  the square root of a f r a c t i o n ,  degree o f c o m p l e x i t y , d e f i n e d by  finding.  f o r the c h i l d  investigator tentatively be  than changing  suggests that at  a p o s i t i v e c o r r e l a t i o n between the  and h i s a b i l i t y  a  t o cope w i t h a b s t r a c t  the  child's con-  cepts. The the  complex  significant  interaction  algorithm generalization  i g a t o r ' s h y p o t h e s i z e d outcomes. children differing ferently toward  t o t h e two  field  instructional  No  i n d i c a t e d by  tests partially  That  invest-  hypothesized that  independence  strategies.  the analyses of  supported the  i n v e s t i g a t o r had  i n t h e i r degree of f i e l d  would  respond  c h i l d r e n t e n d i n g toward  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  conclusive results  c o u l d be drawn f o r f i e l d  dif-  i s , children tending  d e p e n d e n c y w o u l d , 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  on t h e a l g e b r a i c a p p r o a c h , a n d would  The  effects  field  independency  t h a n on t h e p a t t e r n dependent  than  children.  approach. However,  65 for  c h i l d r e n tending  toward f i e l d  on t h e c o m p l e x a l g o r i t h m superior  to the pattern  independency, with  generalization tests,  regard  t o performance  t h e a l g e b r a i c a p p r o a c h was  a p p r o a c h , as had been h y p o t h e s i z e d .  performance o f t h e extreme  field  In fact, the  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 b y 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 the performance of a l l other  instructional  students.  the 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 t e s t s , when t a u g h t b y t h e p a t t e r n  instructional  generalization  s t r a t e g y , was  inferior  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 b y t h e a l g e b r a i c  strategy  a n d o n 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  the  performance of a l l other  pattern  instructional  produce equivalent respect  to field  strategy  results,  Thus, w h i l e  instructional inferior to  on t h e a v e r a g e t h e  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  strategy  dramatically  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  independent c h i l d r e n .  braic instructional structional  students.  strategy  results  For these children,  appears t o be s u p e r i o r  However,  with  the alge-  t o the pattern i n -  strategy.  CONCLUSIONS  The  following conclusions  were drawn from t h e r e s u l t s  of the  study: 1.  The f i e l d  independence  t o be c o n s i d e r e d strategies 2.  construct  i n the adaptation  to individual  independent students,  grams e x t e n s i v e l y  aptitude  of mathematical  variable  instructional  learners.  For students at the grade f i v e field  i s a profitable  i s equally  level, with  the exception  an i n s t r u c t i o n a l effective  of extreme  s t r a t e g y which uses  as an i n s t r u c t i o n a l  dia-  strategy  66 which i s devoid 3.  o f d i a g r a m s and w h i c h u s e s a l g e b r a i c p r o p e r t i e s .  Students at the grade f i v e l e v e l , w i t h perhaps the exception of extreme 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  e x p l a n a t i o n s which are based s o l e l y to 4.  the students  instruction  an i n s t r u c t i o n a l the  field  of concrete  independent students  i n complex  i n this  study, with regard  to  performance,  p r o p e r t i e s o f t h e n u m b e r s y s t e m and w h i c h i s d e v o i d  of  dia-  strategy which r e l i e s extensive-  on d i a g r a m s .  The  review  STUDY  of the l i t e r a t u r e i n mathematics  education revealed  l a c k of a systematic approach to adapting mathematics tegies to individual primarily  learners.  f o r mathematics at s p e c i f i c two i n s t r u c t i o n a l  instructional  f o r the s u p e r i o r i n s t r u c t i o n a l  grade l e v e l s . s t r a t e g i e s may  Yet t h i s  produce equivalent  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  individual  students.  T h e r e i s a need establish  structional  for a concentrated,  a theoretical strategies  stra-  coordinated  stra-  study suggests  on t h e a v e r a g e , t h e y groups of  the  R e s e a r c h s t u d i e s i n mathematics have been  concerned w i t h the search  that although  to  aids.  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  I M P L I C A T I O N S OF THE  tegy  familiar  or p i c t o r i a l  a l g o r i t h m s , a s m e a s u r e d by p u p i l  g r a m s 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 ly  instructional  on a l g e b r a i c p r o p e r t i e s  and w h i c h a r e d e v o i d  For extreme f i e l d  with  research  results for  effort  b a s i s f o r the assignment of l e a r n e r s t o i n -  i n mathematics.  This w i l l  r e q u i r e d r a w i n g on  the  resources lum.  of both  learning theorists  The 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  can a i d i n the i s o l a t i o n  of 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 . can  a i d i n the designing of alternative  through the  research studies similar  learning theorist  tributions  to this  construct of f i e l d for  such  The spect  study  strategies.  can r e f i n e  Out o f t h i s r e f i n e d  Together,  their  both con-  b o d y o f know-  a r i s e and t h e r e f i n i n g p r o c e s s  that the area of c o g n i t i v e s t y l e ,  repeated.  and I n p a r t i c u l a r t h e  i n d e p e n d e n c e , may p r o v e t o b e a f r u i t f u l  a research  developers  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 ,  area of research.  suggests  instructional  curricu-  and c h a r a c t e r i z a t i o n  The c u r r i c u l u m  and t h e c u r r i c u l u m d e v e l o p e r  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 This study  of mathematics  starting  point  effort.  a l s o has i m p l i c a t i o n s f o r mathematics e d u c a t i o n w i t h r e -  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 grades s i x t o n i n e .  This study  suggests  that f i e l d  Indepen-  d e n t c h i l d r e n may b e b e t t e r a b l e t o c o p e 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 a n algebraic nature  than  ledge of the c h i l d ' s insight  into  their  field  d e p e n d e n t c o u n t e r p a r t s , and t h a t know-  degree of f i e l d  the child's  i n d e p e n d e n c e may p r o v i d e  profitable  readiness f o r algebraic i n s t r u c t i o n a l materials.  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 J o u r n a l of School P s y c h o l o g y , I X ( 1 9 7 1 ) , 202-212.  Achievement,"  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 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 Mathematics Education, 1(1970). B r a c h t , G l e n n H. " E x p e r i m e n t a l actions" Revlew_of_Jidj ic^ :  C o o p , R i c h a r d H. L e a r n i n g and  Theory in  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 XXXX(1971) , 627-645.  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 I n s t r u c t i o n , " P s y c h o l o g y i n the 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 , " American P s y c h o l o g i s t , XII(1957).  The  C r o n b a c h L e e . J . "How Can I n s t r u c t i o n Be A d 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 , " e d . 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. C r o n b a c h L e e 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 o f E d u c a t i o n , ED 029 001. D a v i s , J . K e n t " 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 C o n c e p t L e a r n i n g , " P a p e r read a t the a n n u a l meeting of the American 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 . K e n t 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 M e t h o d 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 G r a d e G e o g r a p h y , " The J o u r n a l o f 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 ) , 1 3 7 - 1 4 1 . D a v i s , J . K e n t 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 C o n c e p t 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 Procedures," 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 , L X I ( 1 9 7 0 ) , 423-430. G a g e , N. L . and U n r u h , 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 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 , XXXVIII(1967).  on  G a g n e , 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 , " e d . R. M. G a g n e . 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 , 1 9 6 7 . G a g n e , R o b e r t M. e d . 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 , 1 9 6 7 .  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 , " e d . R. M. G a g n e . 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 , 1 9 6 7 . H e s t e r F l o r e n c e M. a n d 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 a n d 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 , " 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 . a n d R o b e r t s , D e n n i s , a n d 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 B e t w e e n A b i l i t y Measures and A c h i e v e m e n t under Four Methods o f T e a c h i n g E l e m e n t a r y S e t Concepts" 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 , J a m e s 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 : T h e 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 of E d u c a t i o n a l Research, XXXIX(1969), 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 a n d M a t h e m a t i c s P r o g r a m s , " 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 8 9 4 . 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 Factors i n C h i l d r e n ' s Problem S o l v i n g Performance," Paper read a t t h e annual meeting of t h e American E d u c a t i o n a l Research 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 . a n d B r i m e r , M. A. " C o g n i t i v e S t y l e s a n d 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 ) , 2 9 4 - 3 0 3 . 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 Research on C o g n i t i v e S t y l e f o r Mathem 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 a n d C o r r e l a t i o n . The R y e r s o n P r e s s , 1 9 6 6 .  Toronto:  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 , " Proceedings of the Abington Conference. Abington, Pennsylvania: The A b i n g t o n C o n f e r e n c e , 1 9 6 8 . and o t h e r s . P r o c e e d i n g s 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  of the Abington Conference'67. Conference, 1968.  W e i n s t e i n , M a r i a n S. "A S t u d y o f T y p e s 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 Elementary School Mathematics," Unpublished 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 Columbia, 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., 1 9 6 8 .  71 W i t k i n , H. A., Dyk, R. B., F a t e r s o n , H. J , G o o d e n o u g h , D. R. and K a r p , S. 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 S o n s , I n c . 1962. W i t k i n , Herman A., G o o d e n o u g h , D o n a l d R. and K a r p S t e p h e n A. " S t a b i l i t y C o g n i t i v e S t y l e F r o m C h i l d h o o d To Y o u n g 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 ) , 2 9 1 - 3 0 0 .  A.  of  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 Problems of 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 School 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 G r u n e and S t r a t t o n , I n c . , 1 9 6 9 ( m i m e o g r a p h e d ) . W i t k i n , H e r m a n 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 M a n u a l 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 : Consulting Psychologists Press, 1971.  A.  APPENDIX A INSTRUCTIONAL MATERIALS  •cr.e  1 «3~v  —1  oc-arc. to i :  tne  r.ii, i r a c t i o n s 5-Ler.s s u c . i  ,2  V  3  5*  T  the s i d e s as  C? THE RECTANGLES I HAV3 SHADED? NOTICE THAT TK2 O R I G I N A L SQUARE HAD A R I A  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 T H I S WAY?  '  •..I-:AT ARE THE D L Z N S I C N S C? :? T THE 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? KNCW THE AREA CF THE E N T I R E SQUARE WAS  CE  i A C r . OF iS  T.iESE  RECTANGLES  HuST  1  SQUARE U N I T ,  HE A FRACTION  —  c.  4.  CF THAT  3 0 THE AREA AND THE  DETERMINED BY THE NUMBER CF RECTANGLES THERE A R E I N THE  A.N3LZ5 .-.2 a n s w e r  NOTICE THAT  ARE THERE CF DIMENSIONS  "L  5Y  3  I  1.  WAY A 3  ^ 3 V  4.  .2 * d  v ; WCULD  ?Z:D  UIA.",; A  ?.3:r-A:::-  1  •1  5---  ±  -f f M M '  Ai<P C U T IT :  J iii i  i V.L-.Vf . V A T .  -it?  0? 2AGH?  :pect the answers  MU.  23 and  respectively J3  JL  1  -i:  T-  YflLL 33?  Lo'T'S CHiSK V/iTK A DIACr.iA.1.  I CA!w DAA'.7:  5  8  zjsoect the answer  .7..-.::  3*  i  THE DI.-Z::SIC:;3 C? EACH  av  £  3fe  .iv'l.v/j JL.IriA .1.. .iA  Uw X  - . . O — < t ... —J | .... X  ..^ivii.WA  a  A  •J'-Ji .-v  X  1  Here the stud finding the  r  5•  1  x  1— 1  or  1  x  5  A  **  7  •  SUPPCSS v/s Y<A:;T TO :-:ULTI?L:  3Y  • r? :-~ir".l q i - . ' i ?  -.'7 P  *  ?  T  •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  1  i ? v/3 V:A:;T ess C? THS DI:3:;SIC:;3 TO BS  4 3  9 XT  E x p e c t t h o a n s v o r 6- .  . ••/s CUT IT SC  '  '  T  1  Vw.. ..--LEO Z:~-*Jz,'J LA •  3^  .la. ± 3 s  S: J\j'£D :  DE:CO:-LTNATC?V T E L L S :-LE T H E T O T A L N A F F E R  ? i i C 3 3 AND  OF  i;  FIGURE.  LET'S I  A  3 5  FIND  r :  3  r  ,  DRAW:  3.  :FE  Nul-33?. OF SHADED SQUARES •^fq  i ' J J. . j i / j A  3IICC3 THERE A R E  5 .THAT WOULD  ...  3 x 2  .  EACH CF T H E S E  SQUARES  x  i U i . i i . .-i-v.Li.-i V,-;?  THEREFORE,  L  5  IS  2  u  x  ^  5  3  is  HE?  —-U . „ o O OiiO C.ii J n o ^. •3o  WE  DC HAVE  —•*"- FJ PVT. L'J «i.. i | F> ' .-A 1 . «-\rtF*»I—T, W  5o  2c  JUST A3 THE DENOMINATOR CF MY PRODUCT WAS THE PRODUCT CF  (8 = 4 x 2 )  MY DENOMINATORS (33 = 6 x 5).  ^2  ±_  3  5  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 a r e a s 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 i m e n s i o n s l i k e 5  s  4?  ^  3  •  3. 3. 3  3 Fl  ^.3  ii  c?.DER TO F IND PRC  T3 LIME  "3  D THE 3 0  x n 1  7-  3  L I : 3 THE FOLLOW  •J. y. '  /'  1  1  ,  ' /,  / ;y  i ~4  1 1 1 1  i  1  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 BY Z AND ANOTHER WITH DIMENSIONS  7^ BY ~^  SO LONG AS I ADD THESE TWO NUMBERS. *f~s  1  /  7— y  V//Y7  ~  1  AND THE AREA REMAINS THE SAME  ^  x2  1 x 3  E x p l a i n t h a tt h e y may a l w a y s go  back  to  a  renaming statement to verify their  ** 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  SUPPOSE I WANT TO F I N D THE ANSWER TO A M U L T I P L I C A T I O N Q U E S T I O N WHERE 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 , WE MIGHT T R Y A x \ WHAT DO YOU SUGGEST T H E ANSWER MIGHT BE? Wait f o r a response o f  before  proceeding.  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 . WHAT DO WE KNOW ABOUT ^ ? ONE T H I N G WE KNOW I S T H A T H *W = =6x7] = 1. S O ,  \  WE KNOW T H A T 6 x TO GET ONE.  =1. T H E R E F O R E ,  ^ I S A NUMBER I CAN M U L T I P L Y BY 6  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.  IF THERE MEAN BY THEN, I F "t W A S  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 WE J - _ ( g x -jr = 1. 6x ( x ) = 1, T H A T W O U L D K H A N T H A T ~i i = i SINCE 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 K  %  = 3 x 2 < i = (3 =  x  x  4 ) x (2 x 4=. )  SINCE I CAN MOVE THE NUMBERS WITHOUT CHANGING THS ANSWER  1  N O T I C E T H A T A L L I D I D WAS T O D E C I D E W H E T H E R WAS T O S E E I ?  6 x  (  i  ^ ) = 1 . - ^ ^ =  X  JS  3 - 3  Jo  AROUND  an  HOW WOULD WE FIND  x  WE CHECK TO SEE LF  x  J±  Fi* SEEING IF  2S x ( i  X ~  ) =  -7  3UT, 23 x  (V  ^ )  = 7 x ^ x ( ^  = (7 x =  4 j)  ) x (4 x  ^  1  —  ^  x  )  _ J+-L -t-L +_L  7 X  it  1  =T*-7^^  1  x  =  -L  +-J.  +-L^L  T ™  A—L  -3=1  T  +-L = l  1 SO,  2$ x (  \  * ^  ) = 1,  SO,  2#  WHAT DO XOU THINK  4 *=V  2>  Expect the answer  WILL EE?  LET US CHECK THIS: IF THIS IS SO, THEN  18 x ( ^=  *  3  ) =1 .  Co  BUT,  = 6x 3 x ( ^ = (6 x  %  *  ) x (3 x  i  6 x 4  )  3 x  SO,  ( A.  18 x  3  ^  )  =  1,  j.,i L +i V  (pT  1  |p  \0  I GST AS AN ANSWER k FRACTION  ^  3 L  to  so  H*~S  Hi EACH CASE IF I MULTIPLIED  =  v i +-4-  v»  DO YOU NOTICE THAT IN EACH OF THESE PROBLEMS, IF I MULTIPLY " T  = 1x1  ©  •  T H I S  Bl  SEEMS REASONABLE SINCE  BY " 5 * ^ ,  I GOT ( Q.< ^  ) X (  1.  * Here t h e s t u d e n t v n . l l l e a r n  how  to find  T  p r o d u c t s o f any two p r o p e r  fractions  < -=  2,  Ft  3  5  S" -.-  5-POSE S'—Z  RE.  i:  1  , -~.  1  T  FIND  THIS  •  •'•E HP 0,7  THAI  ANSWER.  — B E RF:;A:ZD AS  3  1  ....  2 x  ^  A::D  3 a 3  x if 4 ) M 3 x i  = ( 2 y. O _.. ^  i . ^ .-i ., M U L T I P L Y MM; V.  VS  3  s~ J ) x (  ALREADY  AMY  A  7 THAT  ( 2 x  a. T ?  A.  3  11  3  _  " M i  x  USING  THIS  3  CL  = ( 3 :•:  4-  )x (2 x )x ( ^ x %  = ( 3 >: 2 =  ( 3 x 2 )  4r ) )  45  JL5 1. -- A -15  4^ 1* ::e c  5  .-5 o v.. c:- : Js^ 3o  it ( ^ >: J,  •  t  J~ . I.'wli Fj • o  TFIZ 3A. 1Z A3  =  ) x  A  5  THIS IS  )  -j_ „  T  L E T ' S ?I:;D  ORDER  %  .......  T H E R E ? ORE,  )  4.  ( 2  ^  ) ^  )  APP  PI •  3tf  A.  3 H  3 5 o.ie Doar-a x.2 sn.o-.-r e n s rer.a:  is ZrJiz  tne nuriera^cr o i  ccries  iron  r a c t i o n s they a r e n u l t i n l v i i i : : s i r ce t h e y a r e n ui .s ra c.c r s  tneir  reria::in£ tas:-cs  i c r n s c f the f r a c t i o n s  •—Q  as  o.iey a r e c o i j . e c " i n -  -  ccllecti.i.  denor.inatcrs  and  student -.Till l e a r n to f i n d  c r c d u c t s of f r a c t i o n s and. n i x e d nun;  s u c n as  * -2 FIND PRODUCTS LINE 1  ^JrrOoR  A x  A.i D  x  2.3  FIND  WAN  J3 2.  x 2  _^x AP  ' Z i S T PROD X T  SPLITTING- THE 1PPXPD PUI-3PR PFERD J U S T AS  IS  V  2  X <i 'v. J  1  3  "3  I DID './HEN  4  3  A:;D V  A  A  :  ,  3  D  1 5~  AND  Co  Have t h r e e s t u d e n t s ether  show t h e i r work on the b o a r d .  t h a n the p r o d u c t s of the g i v e n ones,  I f any s t u d e n t s  use d e n o m i n a t o r s  e x p l a i n t h a t they a r s c o r r e c t ( i f t h e y  b u t ask the.u t o use the p r o d u c t as a guar a n t e s t h a t the p r o c e d u r e w i l l w o r k .  i. .-or  example, f o r  3  —  and  ,we r i g h t use the d e n o m i n a t o r ~ , b u t ve c o u l d  c e r t a i n l y n o t do t h i s f o r  and  -A  3.  ; p o i n t out t h a t we can a l w a y s use  the  ::rcduct. 3s s u r e t h a t the s t u d e n t s  r e a l i z e t h a t the p r o d u c t c f the two d e n o m i n a t o r s 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 i e c e s i n t o w h i c h each o f the diagrams i s d i v i d e d c a n be w r i t t e n as an i n t e g r a l number o f new s u b d i v i s i o n s , and t h e r e f o r e  the  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 c a n 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  * H e r e the s t u d e n t w i l l l e a r n t h a t  -Q  o n l y because  :cw  iw§  > ^  o n l y when  y b<c , f o r e x a m p l e ,  3 5^2x6.** x  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 C C : : ? A R E TWO F R A C T I O N S  TO D E C I D E 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 P R A C T I C E D R E W R I T I N G E Q U I V A L E N T  ,'-.-..-.-.Ru. SUPPOSE I WA:;T TC COMPARE  J_ ^, n WITH T,  ITO ^ mF I,Ni 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 F R A C T I O N S WITH T H E SAME -  WHAT F R A C T I O N S WCVLD I  -:->-o^ tn-3 answers:  / — 1 . X  and  USE ?  O ±<£_  FRACTIONS.  DE:::.-Z:;AT:R.  are),  k  • v*.'.-rt.v-L- > -» do  oto  •EATER?  mswe: S _TP u.-.  x x ^ A:..  r*rti/  1  « T\ -*rtiJ  -C-LivC-j J  o ^  "  —J-/ i  _L i uA iv 50 I C  53  '  1  A  -  —  1 X /AS 3U3DIVIDSD INTO 5, AND THE  fVji  " r~ "* T - c x JL—»s*  <o  A : : D  T  ' ^  **  oujjDi'/iD^D I.'iiC 2 .  x c  I'.iE 3'w^D_v'j.3IC:.S ARE NCW A L L  1-3_JD COUNT THE NUMBERS C? EACH TC TELL WHICH FRACTION  IS LARGER. SINCE  1 x 5 < 3 x 2,  THEN  ±  3  3_  uF  5  ;AN ,<E  4-  E x p e c t the answer: change b o t h i n t o t w e n t i e t h s and t h e n see i f of  the f i r s t  i s g r e a t e r t h a n the one o f the s e c o n d .  .3 x 6 and compare i t t o  AND  LIS  cc:  Or, e x p e c t :  J3 P I S C E S , i AND  ' FIFTHS,  "WITH  4 x 5,. S I N C E  3 x 6  SAYS I  THAT I O R I G I N A L L Y HAD 4 P I E C E S ,  AND SO I NCW HAVE  4 x 5  PIECES,  5  0  CHIGIN'ALLY HAD  EACH OF WHICH WAS C U T INTO S I X T H S AND SO NOW I  4 x 5 ,  multiply  30 S  3 x 6  numerator  4 x 5»  5  COMPARING  the  RAVE  3 x 6 ,  EACH 0? WHICH WAS C U T INTO  A L L C F T H E S E CF T H E SAME  SIZE.  SAME A3 COMPARING THE NUMERATORS IN THE ABOVE METHOD.  CAN YOU SEE WHY THIS I S CALLED THE CROSS-PRODUCT RULE 7OR COMPARING FRACTIONS? I TAKE THE PRODUCT ACROSS THE FRACTIONS AND COMPARE THESE: FOR EXAMPLE, I N I COMPARED  \  .AND  3 x 6 WITH  1  h x 5 WHICH COKES FROM  Ask the s t u d e n t s t o f i n d the l a r g e r i n each c i t h e s e p a i r s by us i n 3 the c r o s s p r o d u c t r u l e .  Have a s t u d e n t  show each 01" t h e s e cn the  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 v h y the rule w o r k s . 3 5  CR  3  OR  <?  5 J_  3  r \->  t 45 lo  3 5  beard  :o sure the students realize that vre exanir.e th nar.e i c r i t o ce aseo. i n renarrJLn;:  cniv vrhen  i.xile,  i  7 "5 A  u  cniy  7  3x5  cecause  i.iA'i v/3 Aiov^rtDi' R:^Ai_.Z_ Tr_A'i'  .>« W  >  ^.-.JJL£_!  A.I  2 x 6 . .'.AY T-  r _,~. p r r - n i O T " - '  O  1  » i  v  O - j  «A.i  _  i  r . E . , . - k . ^  m  i  .  "  Uw o .  id_o_.  ,-iAv-i  0  rvwio.  — _ —  —;  **  lit.;  5  ' ' * T •*~ ~"  ""^ •  CA.-i G J  w  r  ; ~ ~ A r » i i  iL  --^  .•.<jj-.i_ri.i--  AO  ., „_.._ .,.__  T  iS  3X,?.-v.335ICi-,'S V.ITH A  i_<A .  the answers : __* ^ * (.%: * ^ and  H  i  V . •  .v . •  i).  * (4-  T c v : C'-FC'C THIS : - J -  1  x 3-  - V x  -  = 1x  ^-  = 1x6 x ( =  6  "x (  x(6 X \ \  x i . -2-  1:10..,  ^ ^  •=^ x \ 4  x i W  i -p  r  .  .v.iAT wii__ TH23Z ZAPRZOSICNS 33?  J- • =  i or  Cw.rArv— IVi't  TO D3CID3 V/HICH 13 IAHGZH. 15 TO ?.3HAI--Z r*.* r-*- • ' • o * ..xi.i A \ > v . _ - . u - i r.-vAT/  * ^ ? -?*C  a  x(2 x  = 0 x 2 x (  x  ) )  )  i i-.0'i_t->_i '7.-.A? i A.. CC.rARxi.  i_  -IIH  C  nsv'er:  2.  9  4  V  i  •  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 *:  o trier ceric-inater was5  "9  a.  It 1 x 6  AND  AND ^ x 2 .  I SAY  1 x  ^ x  SUPPOSE I WANT  FIND  31 o 3 UT7 A L , WHAT EXPRESSIONS WOULD I USE? E x p e c t the  £ * S * ( ^ *-|)ar.d  answers:  3 * ^ * ( \ 5 * r J .  I CAN CHECK: 3.  = 1 x —1 —  -L  %  A 4a.  x  = 1 x 5 x PJ -.1  ID - . z r ^  i r^-rt  <=• ~ 3 x -(5 X  1.  = 3x 2x (  3 > . V i i :  4- x 2.  3  _3 6 x  e x p e c t the a n s w e r : m u l t i p l y the f i r s t by 3x6  or e l s e compare  J.  and the second by  ^ x ^  ,  x (4 x  4  4 x 5«  with  iv^:  %  - 3  *  i  = 3x  = 5 x x (6 x  = 3 x 6 x (  ^  )  = 5 x  x  = 5x4  -V x (  -L  x  JL  OR ELSE WE MIGHT SIMPLY NOTICE THAT THIS IS EXACTLY THE SAME AS COMPARING 3 x  3 x 6 WITH  4x5,  SINCE  3x6  TELLS HE I ORIGINALLY HAD  A FRAC TION AND THEN MULTIPLIED I T 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 I COMPARED  3x6  zr.Q s t u c e n t s t.o u n a  H  AND  WITH  5x4  tr.e . l a r g e r m  WHICH COMES FROM  eacn ox  Have a s t u d e n t show each o f thes e on tne ocar-u  iuct r u l e .  £ z.n" n " w*** the r u l e works . 1  •5 S  )  )  ~ -A . . ? Too  ^ Have the  Too  s t u d e n t s who read c u t t h e i r answers  e x p l a i n i n terms c f  either  a d i a g r a m 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  ones and 5 t e n t h s and 2 hundredths  E x p e c t the a n s w e r s :  or 34 ones and 529 t h o u s a n d t h s ,  or  34,529 t h o u s a n d t h s .  answers come up, mention the ones t h a t d i d n o t . the hundredths  and 9  answer.  thousandths,  I f n o t a l l of t h e s e  P o i n t c u t the analog;.' t o  situation.  H i 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 t u d e n t s w o r k s h e e t 1 t o c o m p l e t e . ** Here the s t u d e n t s w i l l be i n t r o d u c e d t o the i d e a of d i v i d i n g d e c i m a l s bp wholes .**  SUPPOSE 'WE WOULD LINE TO EIND THE ANSWER TO 'WE HN0W THAT  15 - f 3  1.5-r  MEANS TO SEPARATE 15 INTO  SIZE AND THEN FIND THE SIZE 0E EACH 07 THESE  3 ?  3  C-RCUP3 0? T:  3RCUP3.  THIS DIAORAm WOULD SHOW WHAT WE MEAN:  000 00  000 00  ooc 00  I S  lo  ..o i'lUi i.i-.i l'.iio A A 3 'i'.-jj irt.--j HO  ."uiU.'iJ  —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-AT  DJ  .4  .  E x p e c t the answer.: 4 t e n t h s or LET'S ERAi A PICTURE.  4.5 •  /-.s.-c s t u c e n t 5 L^)ifi'_  of t h e s e ,  I  DRAW:  ; a i c t the answers  9 !  eao. ou*  *** °  draw a d i a : r  = 16 TENTHS,  1.6 =  L6 TENTHS I N T O 4 CROUPS,  r.ave tr.e- s t u c e n t s  i o u o.\>i^CI T."^ H.iS.io?. To no?  as con  to:  THE:;,  T C PU]  :,s  1  .Oc  .•.ar.a s t u u e r : *  ieavs  ceci-mais  io:  iver tr.at anot..£  .on i s a i v i s i c n .  i_  I n A / i T.-ii E;\ACTiC.;  i.3 ANCTHErC i\AHH rCR.. Expect  the  answer:  3«  In  n - i  i  ^  .  - 2,  i  and  so  to  set  the  IST  052 322 WHAT WH0L3 NUHEnR THIS  DO YCJ .-'CHOW ALREADY?  any  case,  n  T  ** • 'r *  J.  rji'o i . i O i . . o  ni  =  2 ini  QZViO.;  2.  3  -s,  —•—-5 p-7 "—J-*. •  go  7—* ' r"*  -  - p*.tJ  T—  on w i t h  T  q  i-d-Et  -  the  — V  r i i v j j  o  following  ~* s  *i  • -  OiW.i  n - t-p.--; — 1 * E? ""T" A TNU-*.WIJ J - i - V - J _L.-  —  ~7^J  '  .  r>  -f-7  discussion.  —  oiio  d-  ~rr  . .—. .....  •  w.n  iVr.i  .-.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 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). THS  •  0© ©0  ©0 ©0 ©0- ©0  2 V 3  — J• ;?3 OF 4  ,o •. )  t  L; 12,  J A i o _ rU..D -cp TC SHARE A L L 12  / «.  -2>  \ ^ J". L r w  .  i . 'w  O  _J-L  .Li  ? J  \  0 .L.  42  7  3Y 7 TC FIND T;  ".•32?. OF 3272.15 IN 42 J.S TI- .2 3AHS A3 FiNDINC- IH i 3IZ2 CF EACH OF 7 .  o  OO OO (X) o o o o OO QO  42)  IN  i OA.. D.\A.'i( i'0 SHOW HLW .CANY 7 ' s  b cooo'oo o  o o oo  CAN 21.A -.7 THE DIAGRAM HERB TO SHOW HOW MANY OBJECTS IN EACH OF THE ? ROUPS FOR SHARING T H E 42  1  =  4,  c_5 O M  ~ 7. = 3  •  OBJECTS, AND I GET THE SAME ANSWER.  =  15— 6  Z AS FIND  SINCE I WAS FIMDIMC  G THo SIZE CF EACH OF 6 GRO;  wTiONS, SITE A DIViSiON SEN j . .^ZR ;.A..o F'wH  TrH. FRACTI.W.:  3 .  /5  :o rsac. out t r . s t r answers ar.u stu:  ;.-.a: a r . c t r . r r  : a i i , y e x p l a i n tr.e:.  •v.".or.  c a n ar.;avs  oe  u.'Vwj.\5 •  i  •  ,  ARE  USED  Xw  »_r_i»  VA..  '»_•«_-_..  .  CT_JJ  D __ _, o_? »' r v * i :  — i  2  J  v  ^ — L^-iFv •  -»  _D  w.\  _____ UD  _L r*___.  __i  v _ _ i . i i_ j .  U?.  , i__ i  0  -O  r^'i  0  w _\o i _  0  --.ii _»rt_-. * _____ •  0  0  u.-«A.__i_  ..i___j  ir-j  v __-'-•-*  1.0  3 j i z  2. -_f  >  ^  3  -4-4  J 3.0  '  • I ——  ^  -3>  it A  J 30  F_> _ .-__r. i  ZJ  -_ _. F_» •  TENTHS  7  7 TENTHS  0  0  TSHTH; T_.NT)-V>  7  ^  File/. 3 T O  3.0  }0 O N E S T E P  IF.3TF_.AD ,  3 »00  A V - X J  PR-_}__.-•__  HUNDREDTHS  20 0  ••n-7> At LTrYS .'A  GET R I  T H E HUNDREDTHS  STOP  ^U  HUNDREDTH;  20  EITHER  SFJZ.-3  K^A3L.i.\zLcli  J. Cr_A_-.3F._D J_T  r._..-_rix..J,_.i,  _v_ _. .•___.-i ....  o<  4 ) 300 HXiDREDTHS 2oQ  EVEN  TFIXO  FURTHER.  ^ J3Tc  T  A.»D  _ i  5  HX:DREDTHS H'-.._L.EDT;i5  75  __ w'_ ( J . 0 1  rio  HUNDREDTHS  HUF>DREDrH3 = .75  • 75 . D C F T H E ttVAZ'.DZ?.  OR T H O U S A N D T H S ,  OR A R E C C N V I N C E D  11  B U T VFE C A N K E E P  TRYING  PLACES  V G CANNOT S T .  Fl F"„ \  __ D_. FIFJ  ,1 \J jli i. F___.Fw •  I'V F_  v _-. „ *  i  i . FFF-j  as  a-^s =  25  25  J  2.0  —>  25)2.00  25/ _ u J  0  =  .-J.>i..'._-_/..-.0 }  C\J<J . . .  .0:  ^  t o ccr.slote,  £—  -5  oxcec  :he answer:  IJ.zl  .  .-lave the students who read out t h e i r answers e x p l a i n i n terms of renaming how got the c o r r e c t answer.  3IMIIA.-tLi, j. MIGHT RzAD AN3 '.vRxTS DECIMALS INTO THE THOUSANDTHS. HO,/ V/CULD I HSAD  Expect the answers:  34.529 ?  3^ ones and 5 tenths and 2 hundredths and 9 thousandths,  or 34 ones and 529 thousandths, cr 34,529 thousandths. answers come up, nention the ones that d i d not.  I f not a l l of these  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  f w i j .'io tivJlfD U^kL i  ^ i . i D i'."w-* H.»5./-  1.5-^-3 =  U  1.54- 3 3 = 1.5 .  35  4__  - _ 1  5  =  -lo  \0  0-L.  Ux3=  15 :•  O. x 3 = AND THEN  4,,  15 x  \a i o  F_."V__ . ._rv f — i  3 X Q_  ?c?. EXAMPLE, TO 5CLYE  \o  =  10 i° *° \ °  _, s-/  3 x 5 x 4 ,  '°  1 0  1 0  l o  l t r  I ?3?LkZZ  X  ',•13 SEE _HAT WE CAN REPLACE  Q  BY 5 x ____  -lo  A , = 1.5  -_-0  x  x  3  NOTICE THAT THIS WAS THE 3A.PE A3 F I N D I M - '.THAT I HAD TO MULTIPLY 3 SET 15 A--5J..R  AND THEN RE IFE". PEER I N S TO MULTIPLY THAT ANSWER 5Y  1.6-1-4.  he answer:  Qx  e~-._  WHAT DO YOU EXPECT THE ANSWER TO BE]  4 tenths or  .4  4  v x v Vs.vy  _rv . --1-G  WAS Cl5 "f 3 ) TENTHS .  -ET«3 F I N D  16,  "reo_.c - - :.e  4 TENTHS x  answers zo:  -  . w.j  O x 3 =<0 x ik\x 3 = 5 x 3 x 0  |!  = 16 TNETH3,  30  XX =  SO TH  1 0  rtSK s - u a e n t s t c r e a u  :se,  cut, ^ n e i r  answers,  tr.ere  rewrite a s mu  imu  r.iYii.-f  FIND  i s anv  :C7:;;D  t.ne s o i u t i c  1.25  -4-  .Ui i  57  ..Ort..O  'WHAT NU.-^om  Q  x  5  125 HU:;IH3DT:-13  125 1 A:: D Z A L I N J  BY  5  AND  THE::  WITH HUNDREDTH  Ask the students to predict the answers to: . 2 5 - 5  1.25-r 2 5 2.00  -f  4  Have three students explain their answers.  If there i s any d i f f i c u l t y , rename  as mutliolication sentences as illustrated above.  \~j-,.jLi  r  C3  EXAMPLE,  TC- FIND  6 ) 4.2  ^  4.2  L< A .(Ai TmAj.' ,VB UdjALLY  -f  6,  I WRITS:  6 ) 4 2 T1N7TH3  AND  TC  IH3)  07  1 . .I W w  \t  J  -L.  i'o.-. i .-.o  • IA — A .j ^-T H I S  = .?  HI CI; LI. I:  4.5  •  9  *  »  u3S  Tt  .VRIi'E  rr--.v-r--~-;r : ;  P.  0  sheet 2 t o  nc. st.uas.its  Here the s t u d e n t  complete,  w i l l now l e a v e d e c i m a l s f o r a w h i l e t o d i s c o v e r  another i n t e r p r e t a t i o n wild, t i e  .06  H U..N-moD-.i5  for a fraction i s d i v i s i o n .  i n t o d e c i m a l s when he d i s c o v e r s  That  that  is,  tha'  3  = i ~ 2 , and so to  2-  get  , he d i v i d e s 2 i n t o 1. * '  o.eetmaj. z or  .vr.C-jUj IDiiA C? jN.TRCDU  C7 A 3 V/H S A I D 3 1 ?  HATH  AOL  SUPPOSE  LAVE THE FRACTION  13 ANOTHER NAH3 FOR.  E x p e c t the a n s w e r : 3 . .  12 x  i.-.'ii  33 SEo .'(HAT .THCiH) ilURHoR TmiS'  JJJT  DC YCU iC. a-7 ALREADY? T  i n any c a s e ,  12  ~^  X  .1-lV  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 .  iii X  j  .  X  vrt,i .-Co.,A.-.  \  =(3 x  \  = 3 x (4 x  ^  H  y  ^  A  -+ *v+  =3 x 1 =3 AND SC I S E E Ti  i  , T SHOULD I (HAT WHOLE  JO i  o<  ,<i  i  HO-  ?  -7 = £ x 1 = 6 .r.ey GO not surgest t h i s , you renane f c r the.-. l.'C'TIC-  4J_  Th_. Viu.C'L__ NUh  lo  Expect the answer: 3. 12  x  <3 x 6: 5)x = 3 x (6 x __- \ = 3x1 = 3  I  c  T  Z ' T *•r i T * *  - «.v  * o ~-  T.__  u.-__.rtATCR  3E:;C:-:I:;A_CR(  7 x  = 42 ,  D  A;;D  6 x U =  1 5  i  Q  i,*—  X 71  Hr2_  - ? •= a  ci*  42 - f - n  =7  X  ) .  ^1 ...  4  Cr»  42  .?.  I:.A..?I_.,  i  ..'A3  ? L ; D I :  Q  iom  ..A.-_o ( A:CCT;{OA NA.CE)  r-?.  T H E FRA.C  3-  ns.-c J s t u a e n t s t o r e a a ,  il / I ^fjs^j 5  a  "w i lU fi Jo t p^I 4i /c >a at i^ o. n  ** Here the  out  •vers and t n e n e x o o a m t n e n ay rer.amin." i n  sentence.  student w i l l l e a r n t h a t another  found by c i v i c  name f o r a f r a c t i o n c a n a l w a y s be  g the numerator by the d e n o m i n a t o r ,  even when the d i v i s i o n  is  not even. **  .»^.« ,vo . - L r i v o  ooo;( jL.-Jii ? Lrt Su.-o r . - t A ^ i i U O ,  NA.02 F u t THE FRACTION NO OLD  1ZHE TO F I N D CUT  I F T H I S I S ALWAYS  IF IF  NU.-^ioR  S H O U L D BE  LA  .-'.-il C.- r 1  I S TC D I V I D E THE) DENOIONATCR  5 LHAT  rt.wTooR  CUR  INTO TH: NUMERATOR  THUS.  RUI  3H<VUJUD 3H  I F I N D THE SOLUTION  TC  Q  x  5 =  ±4.  | LET'5 DC THIS. i  SUPPOSE \B  WRITE:  x  U  ^  4^: •  x  = 11 X  X  5  5  % >  5  (5x  =  5 = 11.  I?  LI =  ^  , THEN  x 5  )  1  = 11 5  x  -L-i. | rv. 11/ * JIl!x.'-_LT ^ .TL ,  „ O OJ .-i . _ j  — i>_L .-v . i .  2.  _J3-  1 1 J-  4i 5  IT  U,  n  x  o r. - _ __/  3  3  <3  x 3 (3 :•:  Ux  3 = 3  34-3  AND 30  =  3  \  5,  * = 5  ?  so:".scns to suggest: -5"  since  x k  = (5x  = e; v =  5  x  x -1  )  1  [£. no one does this, you writs out this rena.T_.ng scheme. DLL  WHY  3  ax  n,  3 4  ^ • L - i  -^ =n  ,  =( x J ^ f c 3  (  =  ^ x i  1^.-3  )  3 x 1  = 7  land stuaents worksheet 3 to co.-plete.  4?  D x  4 = 3  'OL.  Lie  _oci.-:-.xs anc  , .'^ ^ » «  sz--i~a  fraction,  ecu._.vaxe:  * _..u  4—  H ! cv:  •  10  =4 DOES  THIS  4  —  --L-?  . —__I  EQUAL  4.0  =  I  . _LC_i  ONLY  4  io  ^  3 0  =  10.  . '  — i i . v j . 1  -_—  i_.iv^w  A3  10  S E T A H ANSV/ER  J  .  J 4  OF 0  = 4 x 1  WITH  i^.i  = 4  AO  BUT I ?  I  RE.1AINDBR  4,  (4  x  -n UZ^j  WHICH  .  )  DOBS  =  4  x  1  4  =  4.0  =  4  x  =  (10  ~r.  x  A  )  =  x  10  )  __.  =40  T3..THS  10 ) 4 . o  40 TENTHS  ll  P'-'q  0 TENTHS I 4  3_  3"T ~J3 _L I3 D-i V13 -  do  i V. • F*-E-iO v - j «  IHZ3 Tr_.vl'J3;I  10 J  .  TRY TO DO  . 10 CAN _E .'iRx'i'T3:i:  p r —>  .4  o  ONES  4  10 = 4 x (10 x  BE WRITTEN  REWRITE  [SINCE  4 ,  . I - J r._.oo  CAI-I  10  D P / I S I O N ,  4 V  x  do  .~-"\.._J.  V7-LL,  -TV „*\  in  x 10  ii^ac  10)6.0  Di'/I5IC.i;  -5> 10 ) 6' '. 1 60 i _  ;r -.5  j TENTHS  FJ . .  —  32[T  1  2 ,  — ^ >  5  WE  )l.O  2  - >  ):o  2  TE:;I H S TEN-  IO  o  ."io 3  "  T E : ;I E  ?.-.."T'-3  3  -0..i.-.0  .o..i.-.o =  o  1:I:.-\L  A = .5  FCR  .5 =  E  NOTICE  GET  THAT  I  ARY./HSRS  CO.,VARIED  3Y  TEE  D I V I D I N G  I  ±  no I  INTO  3Y  I.O  2 ,  •g NOV/  LET  WANT  03  THE  TRY  ANOTHER  DECIMAL  FOR  FRACTION.  30??03E  V/E  START  V/ITH  AND  "If  V/E  T H I S .  30 ve WRITE:  RlVi  4  3.0  ) 3 0 TENTHS 2o  J7 0  ii.-a.io  rp—• *. ti—i- - -»i j. IJ;« ± Il D  1 oO  i ooi  I S  TO  FRO::  WHICH I AH  CHANGE  3  TC  3  3.0  INTO  ONLY  3.00  TO  .i^i  INSTEAD.  AVOID  o w..o THIS  THE  o,t TO r-ANDLS. 333.03  PRCSLSIOS  OF  Co  rOS3l3liji-TY  R E A 5 O N A 3 LO .  I  A  SO  REMAINDER,  RTHSR.  4 HTCO  —=>  4 J 3 0 0 HUNDREDTHS 250  •.J/.O^i.lO  2 0  .-.J.iDREDi."I5  20 H/JNDR3DTH3 \ oRoi>...-i3 I  70  HUNDREDTHS  7.? r.-..JRo^'iHS — .75  TI" = .75 .  ou H I D  ,-r.H.  CHANGED  I  MIGHT  I T  i  ^ar.d students vorkshset 3 to  complete.  r i ^ V J ) **Hera the s t u d e n t s w i l l l e a r n a b o u t a p p r o x i m a t i n g square r o o t s .of w h o l e s . ** f^ 0 l  SUPPC53 WE NOW WANT TO FIND A T T . W2 KNOW THAT 2 IS TOO SMALL, SLNCS 2 X 2 = 4 AND 3 I S TOO LARGE SINCE 3 x 3 = 9. SO WE HAVE A PR03LSM W3 HAD NOT PREVIOUS LT FACED. CN2 TECHNIQUE WE MIGHT USE I S 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 I S THE AREA OF SECTION 1? Expect the answer^. WHAT  2 x O. .  IS THE AREA OF SECTION 2?  Expect the answer: 2x pi WHAT IS THE AREA CF SECTION 3?  Expectthe answer: 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 \  MAKE UP ALTOGETHER. 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 B U T , £2 x Q ^ C A N DRAW:  ) + (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 #  N O W I F k x Q I S A B O U T 1, W H A T L S D.  4 x Q  ) IS A30UT SINCE I  1  —  E x p e c t t h e a n s w e r : a b o u t H"  .  r~  ±  NOW S U P P O S E WE P R E T E N D T H A T L N O T H E R W O R D S , W H A T I S ULx  Q IS V_L\ ?  r  . W H A T I S T H E A R E A O F S E C T I O N 3, j  Expect t h e answer: 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 MOST OF THE EXTRA 1 UNIT OF AREA THAT I S NOT I N THE 2 BY 2 SECTION C F THIS SQUARE: , r-i -tt-a/ -t  •-  ^5  AND S O WS C A N S A Y , THAT 41T  ~  z.%  I S A B O U T -2  ±  .WE WRITS THIS  •  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, AND G E T 2 TJl UNFORTUNATELY, T H I S GETS US I N T O D I V I S I O N l*  j 2  112  . WITH FRACTIONS SINCE CUR NEXT CHOICE WOULD BE  A N U M B E R B E T W E E N 2 a n d 0- ~2. WE W I L L AVOID T H E D I V I S I O N TECHNIQUE FOR T H E TIME BEING. SUPPOSE WE WANT T O F I N D  A . 13  AND SO,  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: D  3 x Q  + 3 x Q  '  =  A3  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 ij | cn I (  - i. -  1  -A  J  6 x "Q  .  V-  AND CERTAINLY  W2 NEED TO MULTIPLY BY SOMETHING 4 TIMES A3 BIG A§5jA TO GET  j I THEN, IF WS GUESS  »L • ,  LL=  WS FIND THAT  THE' NEGLECTED AREA IS \ 1. ±9 WHICH 13 LESS THAN ( WHICH IS ~ ^ ^13  #  3 ^  x )  7^ =  D x Q  AND SO WE SAY THAT  •  | 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. i  THEN THE  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)37  21  2  SO WE STILL DID NOT GST A SQURE. IF THE LENGTH = 6, THE 'WIDTH = 6 737]  WS NOT TRY A NUMBER BETWEEN 5 AND 7, 6, 37^6,  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:  fl\  WHAT ARE THE AREAS 0? EACH 0? THE SECTIONS 1, 2 AND 3 ? Expect the answers:  6 x Q  SINCE SECTION 3 IS THAT  , 6 x  Q  , and  H x  Q  WE MIGHT IGNORE IT TEMPORARILY AND SEE  S M A I I 2 S T  SECTIONS 1 AND 2 HAVE A TOTAL AREA OF (6 x  +(§ x  Q ) = 12 x  00  SINCE THERE IS ONLY ONE UNIT CF T?FE 3? NOT EI THE 6 BY 6 SQUARE, WE HAVE TO ASSUME THAT Expect tb.9 answer:  12 x Q  IS ABOUT  WHAT VALUE IS  Q  ?  about  THEN WHAT IS THE AREA OF SECTION D. x G .  Expect the answer:  =  3 7  £,  (_> o U  3?  -U4  THIS IS CERTAINLY SMALLERTHAN A)  1,'  "37  , SO WE CAN ASSUME THAT  •  NOTICE THAT IF I HAD BEEN TRYING TO FIND -A) 33, I WOULD HAVE DONE THE SAME THING UNTIL I'SAID THAT SAID THAT  12 x Q .  12 x Q  IS ABOUT 1.  IS ABOUT 2, AND SC  I STILL WOULD HAVE CESCESD' •2-2. AND SINCE ^-gjL 1 2  CL X U. s y i A L L E : R  T H  D.  THEN I 'WOULD HAVE  IS ABOUT  2 x  t w 1 - l a . ^.3.  TO SEE THAT IT WAS SELLER THAN , ± AN 7^ . 1 WOULD HAVE ACCEPTED THIS  Q  Ask t h e s t u d e n t s t o work out the a p p r o x i m a t e square r o o t s f o r each o f the  following  and t h e n a s k s t u d e n t s t o c o r e t o the b o a r d t o show t h e i r w o r k .  8 13 50 67 ** Here the s t u d e n t s w i l l l e a r n how t o a p p r o x i m a t e the square r o o t s o f whose d e n o m i n a t o r s a r e p e r f e c t s q u a r e s ,  fractions  b u t whose n u m e r a t o r s a r e 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 d«5 ~ X "4-  FROM OUR PREVIOUS WORK AND VE KNOW THAT  J  AJ4  =2  .  SO THE APPROXIMATE SQUARE ROOT OF SUPPOSE WE ARE TRYING TO FIND WE KNOW THAT  %> 3 b  - ~  .  Al  AND  L E T ' S DO THE SAME THING FCR  IS  4 ^ = 2.  .SO  -ij |I  WE CAN WRITE THAT THE APPROXIMATE SQUARE ROOT CF  Have the s t u d e n t s f i n d a p p r o x i m a t e s q u a r e r o o t s f o r e a c h c f the f o l l o w i n g ; few  s t u d e n t s show t h e i r work on the  -V  32.  board.  have a  AND \^QcjJ^ia>n&. JXL'J./j  27  729, so  27 =  x  students worksheet 3 to  A!  12S\  •  &1  complete,  5"** ere H the students w i l l learn about approximating square roots of wholes. ** SUPPCS3 VS NOW WANT TO FIND 2 x 2 = 4  Aj 5.  VS KNOW THAT 2 I S TOO SMALL,  AND 3 LS TOO LARGE SINCE  3 x 3 = 9.  SINCE  SO WE HAVE A PR03LEK  WE HAD NOT PREVIOUSLY FACED. CIS TECHNIQUE 'WE MIGHT USE I S TO CONVERT 5 TO A FRACTION AND TRY TO FIND A FRACTION NEAR IT WITH AH "EVEN" SQUARE ROOT. WE KNOW THAT 5 = 5 x 1 = 5 x 25 x 121 x A £5  1  = 125 x  WHICH HAS A SQUARE RCOT OF 11 x 4 5  4K =  FOR EXAMPLE, I S CLOSE TO  ^  •  3  BUT THIS  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 I S SOMETHING BETWEEN 2 AND J. 2 +Q  , WHERE  a)  I S SrlALLER THAN 1.  (2 + XX ) I S THE SQUARE RCCT OF 5, WE HAVE  THEN, SINCE (2+  U  LET US CALL I T  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  (2+0)x (2 + O ) THEN WE CAN REWRITE EJ (2+\__ )x 2 =  2x2  +Qx  CONSIDER:  = (2+CL)x2 + (2+U)x HE EXPRESSIONS CN THE RIGHT: 2  AND  (2+ T J )x  n  _Q •  =2  x El  +  Notice that the use of the d i s t r i b u t i v e law here i s very involved and i t may be necessa to go back to or  (2+3) x  consideration of simpler exaaples, l i k e (4  +  5 ) =  (2+3) x 4  +  (2+3) x 5  2x  (4+5  and then  )  = 2 x 4 + 2 x 5 .  distribute  again. the as  Try t o use underlining  entire  first  to indicate  t h e common t e r m a n d make s u r e  e x p r e s s i o n i s v i e w e d a s t h e common t e r m t h e f i r s t  t h e u s u a l addends t h e second time t h r o u g h .  on t h e l e f t  That i s ,  thef i r s 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 which t r a v e l s  that  time, b u t time  (2+ L i )  together i norder t o  break 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 broken up.  THEN, WS HAVE THAT ( 2 x 2 ) + SINCE  2x2=4,  (2  x Q ) + ( Q x 2 ) + ( ELx U ) = 5.  THAT MEANS THAT  V  ( 2 x U ) + (  V  U x 2 ) + ( "Q x D ) =  SUPPCSE WE PRETEND TEMPORARILY THAT WS CAN IGNORE THE THEN WE HAVE  (j. x  BUT 2 x Q  =  (z x D  SO,  Jt x  Q  IF WS ASSUME Expect the answer:  U  IS  PART.  j  = 1.  \  I  ^ = 1.  + (j2 x U = 1 SO  s  x 2 , SO WS HAVE  Q  ^ + (2 x O  BUT (^2 x Q.^)  (Ux2^)  LA*} +  Ll^n  H  ^  *= 4 x Q IS  USING THE DISTRI3UTIVS PRINCIPLE.  •  , THEN WE NOTICE THAT Q * Q  IS ONLY" WHAT NUMBER?  --y^  SIKCE THIS NUMBER IS LESS THAN % , WE CAN SAY THAT WS HAVE USED UP ^ ALMOST OF THE 1 IN THE OTHER EXPRESSIONS BESIDES THE HI * Q. fj^tl AND SO WS SAI THAT ^ ^  IS ABOUT  1%  +  ^  X  6)  .WE WRITS THIS  WE MIGHT HAVE USED THE DIVISION PROCEDURE AGAIN, AS WELL. J SO OUR FIRST GUESS WOULD HAVE BEEN 2. THEN WE WOULD DIVIDE AND GST  5 BY 2,  UNFORTUNATELY, THIS GETS US INTO DIVISION  2) 5 _4_  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  !  _ „ £ r ^ J . !  *$13  VS KNOW THAT THE ANSWER MUST EE BETWEEN 3 XND 4, SINCE  3 x 3 = 9 AND  4 x 4 = 16, AND 13 IS BETWEEN 9 AND 16. SO AGAIN, W ' E ASSUME THAT THE SQUARE ROOT IS A LITTLE OVER 3, SAY, 3 + Q THEN, BECAUSE IT IS A SQUARE ROOT, (3+ I I ) x (3 Yi__*  WAIT. r.  +  Q  ) = 13.  Q  ) =  I s.-U'~YLt;  (3-MI) x ( 3  ^+  9  THEN, IF  9  (3 x  +Ox  a)  +  0 x  BUI  6x  AND IF  Q)  6 x "Q  x  QxQ  -  6 X (\ x  =  6 x  Q  Q  =  O * t l , WE GET  THEN, IF WE INCLUDE  (Q*Q^  Q  Q K U , WE GET  BUT SINCE THIS IS LESS THAN  IF WE DO, WE GET  4 x 4I =T  =  SINCE =4  ^  Q*JL_  =  * ^  =  Tf^  , WE DO NOT WORRY ABOUT IT, AND SAY THAT  WE ARE CLOSE ENOUGH TO ES SATISFIED. VS WRITE:  ±_  T  SUPPOSE WE WANT TO FIND  ^jj.  IF WE DID NOT KNOW IN ADVANCE TO TRY  A NUMBER NEAR 6, WE MIGHT USE OUR DIVISION PROCEDURE TO CLOSE IN ON A NUMBER BETWEEN 6 O  ?.  .  Q  = 4 x ( 6 x ^ ) = 4 x l  SO BY IGNORING  a  (a**) = 13,  TEMPORARILY.  THEN  4,  +  ^x  = 4.  DJ  + ^xU )  =  (3+ n  + (a*n)= 4 z ^ - 0 -  +(3xa)  a)  +  +  + ( 3 x C T )  (3xQ)  Q ) +(^x t f )  AGAIN, LET US IGNORE THE  3  x  : r v.v o.Viv.iri> i ,  no  iU.urix  i>v.  n=  4x WS DIVIDE:  n  IF  37 TO S:  = 4.  4 117" 36 CL L3 ABOUT 9 .  AND WS FIND THAT  SO NON WS TRY A NUMBER FOR Tr  SQUARE ROOT BETWEEN 4 AND 9, LIKE SO VE SOLVE:  7 x  i 37 2  H  TO SEE I F  = 37  HI = 7.  5 5 5.  AND VE FIND THAT Q I S ABOUT SAY,  7.  SO NOW WE TRY A NUMBER BETWEEN ? AND 5 .  6.  #  65171  AND WS ARE BACK TO THE POINT WHERE OUR DIVISION PROCESS CANNOT HELP U S . HOWEVER, WE DO KNOW FOR SURE NOW THAT Q  SITS BETWEEN 6 AND 7.  SO WS GO BACK TO OUR PROCEDURE OF BEFORE. WE KNOW THAT  (6+ Q )x (6  SO  BUT,  ) = 37  +  H  )  )x 6  = (jsx6^,+ ( a x 6/)  +l6x\X)  +  +(6xO^  =  (6 x  a  (6+D.) x (6  = (6+ Q  THE,  +  36  0$xCO  36 +J6^x +  cn^)  TEMPORARILY IGNORING  THE O x Q ,  IS ABOUT 1, SO  I S ABOUT  JQ  I F WS TEEN CALCULATE  UxjC\  CU^D^)  = 37, SO  +fe x U)  (6x aw  +  = i ^ 3 7 - ^ ) WE FIND THAT  4  +C6 x  X  EH  = 12 x  .  ,WS GET ^  v f H I C H  I  S  M  X  H  SMALLER THAN  Q  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 SAID THAT  12 x Q  * ~±Q_  IS ABOUT 1.  IS ABOUT 2, AND SO  I STILL WOULD HAVE CHECKED AND SINCE  12 x Q  Q*Q  Q  THEN I WOULD HAVE  IS ABOUT  2 x  it-  =  TO SES THAT IT WAS SMALLER THAN  IS SMALLER THAN  -§^  i  ;  ,  , I WOULD HAVE ACCEPTED  THIS APPROXIMATION. ~*  J  ^•^f  (fi Ask the students to work out the approximate  square roots f o r each of the following  and then ask students to cose to the board to show t h e i r work.  8 18  50 67 **Here the students w i l l l e a r n how to approximate  the square roots of  fractions  whose denominators are perfect squares, but whose numerators are n o t . **  HCV SUPPOSE I WANT TO FIND THE APPROXIMATE SQUARE ROOT  OF  AI-TT:  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. WS FOUND THAT A) 5 0^  1,  1 -q.  FROM CUR PREVIOUS WORK AND WE KNOW THAT  4 4 = 2. j SO THE APPROXIMATE SQUJARS ROOT OF  ^ \  4  SUPPOSE I AM TRYING TO FIND c<_  I.IKT  ^ f ' ^  AND  "V-K5  , SO  7T •  L E T ' S DO TFiE SAKS FOR v  ^  *  WE CAN WRITS THAT THE APPROXIMATE  1/  I s  Q> 12,  Have the students f i n d approximate square roots foreach of the following; have a few students show t h e i r work on the board.  APPENDIX B THE MEASURING INSTRUMENTS  I t e m s o f P r o d u c t o f a M i x e d Number  1. 6 x 3 -  a  2. 2 x 9 =  a  3. 8 x 4 =  n  4. 6 x 8 =  D  5. 7 x 5 11.  6. 9 x 7 -  D  -  D  7. 6 x 7  8. 8 x 9 =  D  9. 6 x 6 =  0  10.  = CI  3 1/2=3+  1 2 . 6 1/4 = • +  Pretest  8 x 8 = 0  a/A 1/4  D + 5/9  1 3 . 29 5/9 = 14.  and a F r a c t i o n  Cl 6/8 = 5 + 6/8  15. G i v e a f r a c t i o n  name t o t h e s h a d e d p o r t i o n  below:  1 6 . S h a d e i n 1/3 o f t h e d i a g r a m o n t h e r  17. G i v e a f r a c t i o n  name t o t h e s h a d e d p o r t i o n  below:  1 8 . Draw a d i a g r a m w i t h  2/3 s h a d e d  i n on t h e l i n e a t t h e r i g h t .  1 9 . Draw a d i a g r a m w i t h  3/4 s h a d e d  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  2 3 . 3/2 = 3 x  125 24. 8/9 = • x 1/9 25. 16/5  =£3x1/5  26. What i s t h e 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 . ? 27. What i s t h e 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  of 8 x 3 s q . f t . ? 28. What i s t h e 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 t . and l e n g t h 5 f t . ? 29. What o p e r a t i o n would y o u u s e t o f i n d t h e a r e a o f a r e c t a n g l e l e n g t h 89 f e e t and w i d t h 38 f e e t ?  (Would you a d d , s u b t r a c t ,  with multiply  or d i v i d e ? ) 30. U s i n g t h e i d e a t h a t squares  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  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  below:  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 t h e a r e a o f t h e f i g u r e below marked w i t h t h e q u e s t i o n mark?  126  4 2 . What i s the a r e a o f 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  I t e m s o f P r o d u c t o f a M i x e d Number a n d  a F r a c t i o n Computation  1. 2 x  6/7  1 3 . 2/7  x  1/2  2. 4 x  3/8  14. 2/3  x  8/9  3. 6 x  8/9  1 5 . 4/6  x  5/6  4.  5/7  1 6 . 4/5  x  7/8  5. 9 x  8/11  17.8/9 x  6.  8/9  1 8 . 5/6  x  5 x  7 x  8/9 9/10  7. 2 x 2  3/8  1 9 . 1/2  x 8  6/9  8. 4 x 2  2/13  2 0 . 2/3  x 7  4/5  9.  3/25  2 1 . 4/5  x 7  4/6  8 x 4  10. 6 x 5  4/29  2 2 . 3/4  x 8  7/9  11. 9 x 6  5/70  2 3 . 8/9  x 8  6/7  24.  x 7  8/9  12.7  x 7  6/50  7/9  Test  128 I t e m s 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  1. I f y o u know t h a t  also t e l l  that  n/A-  2/81 x 1/63 2. I f y o u know t h a t so  1/81 x 1/63 = 1 / 5 1 0 3 , y o u c o u l d  Test  53/69 x 4 8 / 7 3 = 2 5 4 4 / 5 0 3 7 , f i n d  t h a t 48/69 x 5 3 / A  3. I f y o u know t h a t  =  D  f o r O and  A  /5037  13/64 x 1 2 / 1 5 = 1 5 6 / 9 6 0 , y o u c o u l d  also t e l l  that  O/A.  13/64 x 12/10 x 15 = 4. I f y o u know t h a t  values  13/5 x 4/27 = 5 2 / 1 3 5 , y o u c o u l d  also  tell  that  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 a s 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 a s s t r i c t l y  a m u l t i p l i c a t i o n expression(not i n -  v o l v i n g a d d i t i o n ) . Do n o t c o m p u t e t h e a n s w e r . 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 a s s t r i c t l y  a m u l t i p l i c a t i o n expression(not i n -  v o l v i n g a d d i t i o n ) . Do n o t c o m p u t e t h e a n s w e r . 3/4 x 2 1/4 + 3/4 x 2 8. F i n d  t h e answers t o a, b , and c.  to the r i g h t . the  x  4/3 =  three  •  =  O  879/432 x 432/879 =  W r i t e t h e answer t o (d) on t h e l i n e  a n s w e r s a r e o n l y meant t o h e l p  i s a w h o l e number)  O  8/9 x 9/8 =  ( c ) 5/6 x 6/5 (d)  The f i r s t  answer t o ( d ) . ( 0  ( a ) 3/4 (b)  3/4  A  you w i t h  129 9. F i n d  t h e answers t o a , b , and c.  to the r i g h t .  (  flis  (a)  =  D  3 x 2  1/3  (b) 4 x 2 1 / 4 =  O  ( c ) 9 x 2 1/9  D  A x 2 1/A  (d)  0/9  10. I f 3 x 4 11.  0  If 5 x  12.If 13.  =  D  /&  x  D  line  D ?  = 9 1 5 / 1 6 , w h a t i s £7?  1/4  O  = 35/2 + 7/8, w h a t i s  = 10/3 + 8/15, w h a t i s fl /A  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  n u m b e r , f o r e x a m p l e , t o m u l t i p l y : 2 1/4 steps  / 2 x 24  2/5  1 7 . F i n d 2/3  / 4 x 16  9/11  18. F i n d  x 3 1/5  . Show a l l o f y o u r  i n t h e s p a c e b e l o w and w r i t e t h e answer on t h e l i n e  1 6 . F i n d 1/2  t h e answers t o a , b, and c.  to the r i g h t . (a) 2  1/3  x 3  1/4  6 =  The  first  2/4 1/12  3 x 2  = 1/3  1/4 x  1/4  right.  line  t h r e e answers are o n l y t o h e l p you w i t h ( d ) . (b)  - 2 x  to the  W r i t e t h e answer t o (d) on t h e  1 = 3 x 1 / 3  +  t o (d) on t h e  17  = 20 5/8, w h a t a r e fl and A  x 5 a/A  1 4 . I f 2/3  t h e answer  a w h o l e number)  = 12 6/9, w h a t i s  x 3 5/16  I f 7/2  1 5 . How  A  =  Write  +  2  1/3  x 4  1/5  8 =  4 x 2  4/3  = 4 x  1/3  2/5 = 2 x  1/5  1/15  = 1/3  x  1/5  130 ( c ) 3 1/5  ( d ) 4 1/3  x 1 4/6  x 1 1 1/5  A  3 = 1 x 3  a  /6  1/5 - 1 x 1/5  +  12/6  = 3 x 4 / 6  4./Q  = 1/5 x 4/6  A/a  D  1 9 . What i s ( 6 x 2/9) x 5 1/3 2 0 . What i s ( 1 / 3 x l / 2 ) x 2 5/8 2 1 . What i s 3/4 x ( 2 / 3 x 3) 2 2 . What i s ( 4 x 1/3 x 1/5) x 2 1/6 2 3 . I f ( 3 x 1/2) x 3 1/4 « D/^ 24.  I f ( 6 x a/a)  No  26. F i n d v a l u e s f o r d £3/3x2  27.Susie has suggested  A/<J  and A s o that 5/A  = 4 0 / 3 + 25/27  another r u l e  and d o u b l e s  multiplies.  P/A  = 16/3 + 8/15, what i s H  f o r multiplying  m u l t i p l e , f o r e x a m p l e , 2/3 x 4 / 5 , s h e d o u b l e s numerator  A  9 3/4, w h a t i s  x 5 1/3 = 30/3 + 6/9, w h a t i s  2 5 . I f (4 x 2/3) x O  4 x  x  t h e second  fractions.  the f i r s t  f r a c t i o n ' s denominator  4/3 x 4/10 = 1 6 / 3 0 , s o 2/3 x 4/5 = 1 6 / 3 0 .  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 method t a u g h t i n c l a s s .  3/4 x 2/9 a s a n e x a m p l e .  fraction's and t h e n She c l a i m s  t h e t e a c h e r g e t s by  ( 2 x 4/3 x 5 = 8/15)  get e q u i v a l e n t answers t o yours?  To  Will  she always  Why d o y o u t h i n k s h e s h o u l d ?  Use  131  28. J o h n n y h a s s u g g e s t e d multiply,  another  rule f o rmultiplying  f o r e x a m p l e , 2/3 x A / 5 , h e f i n d s  He s w i t c h e s t h e n u m e r a t o r s  fractions.  t h e a n s w e r t o 4/3 x 2/5  of the fractions  and t h e n m u l t i p l i e s .  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 g e t s by t h e method t a u g h t i n c l a s s . answer t o yours?  Why  To  W i l l he a l w a y s  do y o u t h i n k he s h o u l d ?  He  the teacher  get equivalent  U s e 3/4 x 2/9 a s a n  example. 29. Sam h a s s u g g e s t e d find  a rule f o rmultiplying  two f r a c t i o n s ,  t h e a n s w e r t o , f o r e x a m p l e , 2/3 x 4/5 Sam f i n d s  3/2 x 5/4 a n d t h e n t u r n s t h e f r a c t i o n  a n s w e r u p s i d e down.  In other  and t h e n  He c l a i m s t h a t h i s a n s w e r 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 b y t h e method t a u g h t i n c l a s s . always Use  ^  g e t t h e same a n s w e r a s y o u r s ?  has suggested  do y o u t h i n k he  he  should?  a way t o c h e c k m u l t i p l i c a t i o n o f f r a c t i o n  s a y s t h a t o n e c a n b e s u r e t h a t a/b x c / d = P / _ ) i f £J — d = b.  F o r e x a m p l e , i t i s t r u e t h a t 3/5 x 2/9 = 6/45  6 r 2 = 3 a n d 45 T 9 - 5. by  Why  Will  3/4 x 2/9 a s a n e x a m p l e .  30. J u d y She  To  t h e answer t o  w o r d s , h e 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 , t u r n s h i s a n s w e r u p s i d e down.  too.  answers.  c = a and since  Can y o u be s u r e i f y o u r answer i s r i g h t  c h e c k i n g w i t h J u d y ' s method?  Why  do y o u t h i n k s o ?  132  Items o f Comparing  Fractions  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 = 11.  Pretest  10. 8 x 8 =  G i v e a f r a c t i o n name t o t h e s h a d e d p o r t i o n  below:  12. S h a d e i n 1/3 o f t h e d i a g r a m o n t h e r i g h t .  1 3 . G i v e a f r a c t i o n name t o t h e s h a d e d p o r t i o n  below:  14. Draw a d i a g r a m w i t h  2/3 s h a d e d  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 s h a d e d  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 as t h a t  representing  a r e a ( a ) more t h a n , l e s s  a r e a (b)  t h a n , o r t h e same  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  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  WM/f//J/W  (a) more t h a n , l e s s  t h a n , o r t h e same  ( 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 as t h a t r e p r e s e n t i n g a r e a ( b ) ?  A  20. I s t h e f r a c t i o n  | U  IA  representing area  ( 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 ) ?  (V)  _  2 1 . What s t a t e m e n t d o e s t h e d i a g r a m b e l o w For  suggest?  example: s u g g e s t s 1/2 =  2 2 . What s t a t e m e n t d o e s  the diagram below  suggest:  2/4  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  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 27.  0x1/5=1  28.  0x1/8-1  x 56 x 41  29. 6 x fJ/A = 1 30. 3/2 = 3 x  D/A  31. 8/9 =  1/9  32. 16/5 33.  Ox D x  1/5  Which i s l a r g e r ,  34. W h i c h i s l a r g e r ,  44  x 13 o r 52 x  69 x 158  35. W h i c h i s g r e a t e r , 15 x 1/4  13  o r 32 x o r 16 x  158 1/4  suggest?  135  Items of Comparing F r a c t i o n s Computation Test  1. 18/5  or 3  1 5 . 5/9  or  4/6  2. 25/4  Or 2  1 6 . 7/8  or  8/9  3. 29/10  or 2  1 7 . 9/10  or  7/9  4. 33/7  or 5  1 8 . 6/7  or  6/8  5. 28/8  or 4  1 9 . 7/4  or  4/2  6.  27/5  or 5  2 0 . 6/2  or  9/2  7. 45/7  or 7  2 1 . 3/2  or  4/3  8. 68/9  or 8  2 2 . 10/3  9.  o r 10  2 3 . 5/3  or  8/7  10. 3/4  o r 5/6  2 4 . 4/3  or  10/9  1 1 . 1/2  o r 5/9  2 5 . 7/5  or  9/8  1 2 . 4/8  o r 2/3  2 6 . 8/6  or  10/7  1 3 . 6/9  o r 3/6  2 7 . 8/6  or  9/5  14. 4/6  o r 6/7  52/5  or  7/4  136  Items o f 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 Test  1. W h i c h i s g r e a t e r : D / 3 7  or  O/38  2. W h i c h i s g r e a t e r : 8 7 / 0  o r 88/0  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 . to  the r i g h t .  The f i r s t  Write  t h e answer t o (d) on t h e l i n e  t h r e e a n s w e r s a r e o n l y meant t o h e l p y o u 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) 4.  • / • +1 o r  • -1/O  For each, f i n d which i s greater. to  the r i g h t .  The f i r s t  Write  t h e answer t o (d) on t h e l i n e  t h r e e a n s w e r s a r e o n l y meant t o h e l p y o u w i t h  the 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 . to  the r i g h t .  The f i r s t  the 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  Write  t h e answer t o (d) on t h e l i n e  t h r e e a n s w e r s a r e o n l y meant t o h e l p y o u w i t h  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 .  Write  t h e answer t o (d) on t h e  line  to the r i g h t . (a)  3 3 - 1 / 3 3 o r 33/33+1  ( b ) 33/33+9 o r 3 3 - 9 / 33 ( c ) 3 3 - 1 8 / 3 3 o r 33/33+18 ( d ) 33/33+ D  D/33  o r 33-  7. I f 3 8 4 / 5 2 9 ^ 3 8 4 / C J  , what can you say about  fl  8. I f 483 x 25 ^ 365 x 1 8 , w h i c h i s g r e a t e r : 2 x 4 8 3 / 3 6 5 o r 2 x 18  /25  9. I f 6 x 18 " 7 1 3 x 8, w h i c h i s g r e a t e r : 6/13 o r 1 0 . I f 17 x 11  8/18  14 x 1 5 , w h i c h i s g r e a t e r :  14/17 o r 1 1 / 1 5 1 1 . I f 82 x 15 ~7 63 x  82/a  £3  , which i s greater:  o r 63/15  12. I f 8 x ( 5 + Q ) 7 6 5 +  •  /6 o r  x 3, w h i c h i s g r e a t e r : 3/8  1 3 . I f 425 x 2 1 1 ^ . 3 1 0 x 3 1 6 , 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 ) 6/5 o r  P" 2 x 5, w h i c h i s g r e a t e r : 2/3-0  15. Use t h e r u l e f o r c o m p a r i n g 5 a n d 7. greater  f r a c t i o n s t o c o m p a r e two w h o l e  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 b e l o w o f t h e two n u m b e r s  them f i r s t  as f r a c t i o n s . )  on t h e l i n e  to the r i g h t .  numbers,  and w r i t e t h e (Hint: Write  138 16.  Use t h e r u l e  f o r comparing  b e r s , 2 1/3 a n d 2 2/5. write 17.  Usethe r u l e  f o r comparing  fractions  on t h e l i n e  Use t h e r u l e f o r comparing  I f 22 x U  •  = 83 x A  to the right. than, or equal to  2 2 / 0 = 8 3 / A i f 22 x A = -  Y o u know t h a t  22.  Arrange these three f r a c t i o n s  O IA  = 6"/Q I f  3/5  4/9  8/3 Find  26.  Arrange these f r a c t i o n s  three fractions  3/13  from l a r g e s t  from l a r g e s t  to smallest:  25/9  l e s s t h a n 2/5. from l a r g e s t  6/25  to smallest;  14/52 f o r comparing  fractions.  i f 2/3 ~7 4/9, s h e c o m p a r e s 2/4 t o 3/9.  t h e n 2/3 C>4/9.  fraction  to smallest:  6/11  S u s i e has suggested another r u l e  numerator  to smallest:  7/12  25/10  25.  .  from l a r g e s t  2/3  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  example,  =  19/30  Arrange these three f r a c t i o n s  her  t o c o m p a r e 3 2/3 a n d 31/9.  /A-  21.  for  t o c o m p a r e 2 1/5 a n d 20/9.  , i s 22/83 m o r e t h a n , l e s s  Y o u know t h a t  27.  to the right.  to the r i g h t .  fractions  20.  23.  num-  Show a l l o f y o u r w o r k i n t h e s p a c e b e l o w a n d  Write the larger onthe l i n e 19.  t o c o m p a r e t h e two m i x e d  t h e g r e a t e r o f t h e two n u m b e r s o n t h e l i n e  Write the larger 18.  fractions  She s w i t c h e s t h e f i r s t  To d e c i d e ,  I f 2/4 >"3/9,  denominator w i t h  t h e second  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 i s larger  than the o r i g i n a l  second  fraction.  first  She c l a i m s  that  a n s w e r i s a l w a y s t h e same a s t h e o n e w h i c h t h e t e a c h e r g e t s b y h e r  method. W i l l Show how  S u s i e would  28. J o h n n y of  Susie always get the r i g h t c o m p a r e 7/11  has s u g g e s t e d a method  t h e second goes  into  example,  into  divided  15/16 7  F o r example, 3/4.  problems?  2 9 . Sam to  15 —  E x p l a i n why,  using  3/4  16 - j - 4 = 4 and  goes  first  denominator than the  5^4,  the problem of comparing  de-  exactly.  since 3  decided i f the  fraction i s larger  c / d w h e n e v e r 1/a x d -<d 1/b 2/3  and  x 4, s o he c o n c l u d e d t h a t  second.  so of  24/25 and  4/5  4/9,  x c and o n l y t h e n , so  he d i s c o v e r s  the f i r s t  that  would  do a n d why  1/2  fraction i s greater  D o e s Sam's m e t h o d a l w a y s w o r k f o r c o m p a r i n g  p l a i n w h a t Sam  he i s c o r r e c t b y  x 9  ^~  than the  fractions?  comparing  7/11  Exand  by h i s method.  30. J u d y s a y s t h a t she can t e l l then the f i r s t she says 2 ^ greater.  >  fraction  ( 4 x 3 )  -  and  i f a/b  i s larger.  r - 9 = l 1/3.  7c/d  right  away, I f a 7 ( b x x )  For example,  t o c o m p a r e 2/3  T h e r e f o r e , the f i r s t  Does J u d y ' s method a l w a y s w o r k f o r c o m p a r i n g  E x p l a i n what Judy would 7/11  and  i s greater than the f i r s t  3 = 5 and  the  of the f i r s t  15/16  Johnny  and  example.  compare, f o r example,  5/8  4 i n t o 16.  exactly  i t works.  numerator  Does J o h n n y ' s method a l w a y s w o r k f o r t h e s e k i n d s  s a y s t h a t a/b 7  1/3  t h i n k she s h o u l d ?  f r a c t i o n s when t h e  of the f i r s t  f o r comparing  the second  do y o u  the denominator  by t h e s e c o n d , t h e n t h e f i r s t  second.  as an  d i v i d e d by  Why  e x p l a i n h e r m e t h o d and why  f o r comparing  into  h i s method works  15 e x a c t l y and s o d o e s  numerator  5/8;  the numerator  n o m i n a t o r o f t h e second goes For  and  answer?  5/8  do a n d why  by h e r m e t h o d .  she i s c o r r e c t by  ~ d ,  and  fraction i s fractions?  comparing  4/9  140  Items o f Changing F r a c t i o n s  to Decimals  1. G i v e a f r a c t i o n  name t o t h e s h a d e d p o r t i o n  below:  2. G i v e a f r a c t i o n  name t o t h e s h a d e d p o r t i o n  below:  Pretest  3. S h a d e i n 1/5 o f t h e d i a g r a m o n t h e r i g h t .  4. D r a w a d i a g r a m w i t h  3/4 s h a d e d  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 s h a d e d  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  to describe  the action i n sharing  27  to describe  the action i n sharing  18  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 w o u l d y o u w r i t e c u p c a k e s among 6 c h i l d r e n ? 8. What p r o b l e m w o u l d y o u w r i t e pencils  to describe  the action i n sharing  24  among 8 s t u d e n t s ?  9. D r a w a d i a g r a m b e l o w s h o w i n g why 8 * r 4 = 2. 1 0 . D r a w a d i a g r a m s h o w i n g why 12 -r U the d r a w i n g and f i n d 11.  Divide 1 1 ) 425  the value of D  = 4. .  Use t h e s p a c e b e l o w f o r  141 12.  Divide  20 ) 5834 13.  Divide  7 J2859 14.  Divide  25 J4632 15.  Divide  6 / 384 16. Write 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  statement:  as a m u l t i p l i c a t i o n  statement:  as a m u l t i p l i c a t i o n  statement:  532 x 18 = 9576 17. Write the following  d i v i s i o n statement  492 -^-123 = 4 18. Write the f o l l o w i n g  d i v i s i o n statement  D T 6  =  A  19. Write the f o l l o w i n g m u l t i p l i c a t i o n  a  x  as a d i v i s i o n  statement:  statement  as a d i v i s i o n  statement:  x6= A  20. Write the f o l l o w i n g m u l t i p l i c a t i o n  O  statement  A =  O  21. £3x1/9=1  22. 23. 24. 25.  8 x D/A = 1 14/9 = 14 x D/A 3/7 = D x 1/7 38/6 = O x 1/6  26. 74 x 83 x 77 = D x 74 x 77 27. 312 x 25 x O x 87 = 43 x £ x 25 x 312  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 of Changing F r a c t i o n s  to Decimals Computation Test  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.  18.  15/17  2/3 -=s  f  ~ :  ^  144  Items o f Changing F r a c t i o n s  If  y o u know t h a t  decimal  the decimal  f o r 4/3 C  Test  1.333, y o u c a n s a y t h a t t h e  f o r 2 x 4/3 * v  For each p a r t , f i n d to  t o Decimals G e n e r a l i z a t i o n  (d) on t h e l i n e  the decimal to the right.  meant t o h e l p y o u w i t h ( a ) I f y o u know t h a t the decimal  equivalent The f i r s t  W r i t e t h e answer  t h r e e answers a r e o n l y  t h e answer t o ( d ) .  the decimal  tell  that  f o r 4/6  .666, you can a l s o  t h e d e c i m a l f o r 3/5 = . 6 0 0 , y o u c a n a l s o t e l l  that  m  f o r 4/10 x 6  ( b ) I f y o u know t h a t  required.  t h e d e c i m a l f o r 3/10 x 5 = ( c ) I f y o u know t h a t  the decimal  t h e d e c i m a l f o r 6/10 x 7 ( d ) I f y o u know t h a t  the decimal  the decimal  tell  that  the decimal equivalent  required.  Do a l l o t h e r w o r k  the decimal  W r i t e t h e answer t o  below.  f o r 4/6 q y . 6 6 6 , y o u c a n a l s o t e l l  that  t h e d e c i m a l f o r 3/5 = . 6 0 0 , y o u c a n a l s o  tell  that  tell  that  f o r 10 x 3/5 =  ( c ) I f y o u know t h a t  the decimal  t h e d e c i m a l f o r 10 x ( d ) I f y o u know t h a t the decimal  .295, you can a l s o  f o r 10 x 4/6 oz,  ( b ) I f y o u know t h a t the decimal  f o r 6/ CL  that  ^  (d) on t h e l i n e t o t h e r i g h t . ( a ) I f y o u know t h a t  .857, y o u c a n a l s o t e l l  ~  t h e d e c i m a l f o r 6/10 x • For each p a r t , f i n d  f o r 6/7 ~  f o r 6/7 ^  .857, y o u c a n a l s o  6/7  the decimal  f o r 10 x 6/  Q  f o r 6/ £7 ~  .295, you c a n a l s o  tell  that  145  4. I f y o u know t h a t  t h e d e c i m a l f o r 7/25 = . 2 8 , y o u c a n a l s o  t h e d e c i m a l f o r 7/25 x 2 = 7/50 = 5. I f y o u know t h a t =  tell  that  '  t h e d e c i m a l f o r 3/50 = . 0 6 , a n d t h e d e c i m a l f o r 9/50  .18, t h e n you c a n a l s o t e l l  that  the decimal f o r  3 + 9/50 « 1 2 / 5 0 = 6. I f y o u know t h a t  t h e d e c i m a l f o r 9/15 = .600 a n d t h e d e c i m a l f o r  7/15 ^ . 4 6 7 , t h e n y o u c a n a l s o 9-7/15 » 2/15 7. I f y o u know t h a t tell  that  tell  ~  t h e d e c i m a l f o r 1/16 = ? t h e d e c i m a l f o r 8/14 ~  7/14 = . 5 , t h e n y o u c a n a l s o t e l l 9. F o r e a c h p a r t , f i n d  ( a ) I f y o u know t h a t can a l s o t e l l  fraction  can also t e l l  fraction  O /  can a l s o t e l l  fraction  t h e d e c i m a l f o r 8-7/14 =  that  1/14^  W r i t e t h e answer  Show a l l o t h e r w o r k b e l o w . f o r 4/9 ^  .444 a n d 2/9 ^  .222, t h e n  .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 w h a t  t h e d e c i m a l f o r 3/16 ^ that  .187 a n d 8/16 = . 5 0 0 , t h e n  .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 w h a t  A  ( c ) I f y o u know t h a t you  the decimals  the decimal f o r  O / A  ( b ) I f y o u know t h a t you  that  .571 a n d t h a t  the decimal equivalent required.  (d) on t h e l i n e t o t h e r i g h t .  you  the decimal f o r  t h e d e c i m a l f o r 17/16 = 1.0625, t h e n y o u c a n a l s o  8. I f y o u know t h a t  to  that  the decimals that  f o r 1 5 / 2 0 =. .750 and 4/20 =  .200,then  .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 w h a t  QlA  (d) I f y o u know t h a t  the decimals  f o r 3 7 / 0 = .074 a n d 1 2 / f l = . 0 2 4 , 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 w h a t f r a c t i o n the value o f  Q  A I Q  Do n o t f i n d  146  10.  I f you O  3 x 11.  I f you that  know t h a t  the decimal 0  2: 2, w h e r e know t h a t  5 x  Q  is a  f o r 2/3  where J]  also  that  f o r 2/5  =  could  also  tell  .400,  I f the decimal  for  13.  I f the decimal  f o r 9/ 13  <Z . 8 1 , w h a t whole  14.  I f the decimal  f o r 8/ Q  2? .88  f o r 2/ 0  I f the decimal  16.  Suppose the d e c i m a l s the decimal  fraction  17.  , A  I f you  18. How  20.  the decimal f o r 5 / 0  =  Find  A  Find  .55, not  then  find  f o r 0/50 -  .48)  =  .56  and  for  <_/50  a  =  .48.  i s the decimal e q u i v a l e n t  what can you  say  about the r e l a t i o n s h i p  t h e d e c i m a l f o r 0 I Z_ = to get  use  f o r the  between  then what d e c i m a l  f o r changing a f r a c t i o n  f o r a mixed number, l i k e  2 1/2  to the  can  i n two  to a decimal  to  d i f f e r e n t ways.  w r i t e the a c t u a l decimal  equivalent  f o r 2/.3  . N o t i c e that the  equivalent  for  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  for  0.7/2  22.  Find  the decimal  equivalent  f o r 1/2  /4  23.  Find  the decimal  equivalent  f o r 1/3  /2  find Show  equi-  right.  tenths)  the decimal  .89,  a  the r u l e  the l i n e  (or 3  Do  .08  by  the decimal  0.3  ~  A fU  .04, w h a t w h o l e number i s  o f y o u r w o r k i n t h e s p a c e b e l o w and  v a l e n t on  is  and  07  number i s  =  know t h a t  the decimal  O  .45, w h a t w h o l e n u m b e r i s  <T .  , and  would you  all  (.56  tf"/50,  1 multiply  19.  ~  D  15.  0  / l l  i s the 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  the value of  If  then you  i s a decimal.  12.  .33  0  could  decimal  the decimal  =2,  s^.666, then you  denominator  147 24.  Find the decimal  e q u i v a l e n t f o r 4/9  25.  Find the decimal  e q u i v a l e n t f o r 3/5  /  26.  Find the decimal  e q u i v a l e n t f o r 4/6  /  27.  S u s i e has  suggested  a fraction t e n , and  a/b.  She  then she  example, to f i n d  Jlo  50  another  rule  multiplies  /2  5/3  for finding  d i v i d e d the denominator i n t o f o r 3/5,  the decimal  J 307b  = 50  - 50 J300  tenths  6  tenths  the  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  get  the r i g h t  f o r 5/8  Johnny has  suggested  do y o u  For  .6  t h e same as t h e  in class.  t h i n k so?  e x p l a i n why  another  by  tenths  a n s w e r i s a l w a y s c o r r e c t , and  and  denominator  the numerator.  6 tenths =  claims that her  Why  of  she d i v i d e d  She  answer?  equivalent  tenths 0  the decimal  the d e c i m a l  e a c h o f t h e n u m e r a t o r and  300  28.  8/10  Show how  Will  Susie  Susie would  one always find  h e r m e t h o d seems t o w o r k .  method f o r f i n d i n g  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  the decimal  than  equivalent  the denominator.  subtracts  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  then  number by  the denominator.  the decimal  For example, to f i n d  divided  He  this 6/4  for  he w r i t e s :  6-4=2, 4 J~2 =  He  then  so I d i v i d e  4 rO  a d d s one  = 4 J20  2  by  4  tenths  20  tenths  0  tenths  5  tenths  5 tenths =  to the answer, to get  .5  +  1 =  .5 1.5.  He  claims  that  h i s method always works f o r p r o b l e m s where the n u m e r a t o r i s l a r g e r the denominator. decimal  f o r 8/5  I s he and  correct:  e x p l a i n why  Show how  Johhny would  find  h i s m e t h o d seems t o w o r k .  the  than  148  29.  Sam's r u l e  for finding  much l i k e S u s i e ' s , b u t by  10, he m u l t i p l i e s  find  the decimal 500 J 300  He  =  the decimal  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  e a c h by  f o r 3/5, 500  equivalent to a given f r a c t i o n  100  and  then  answer:  decimal  f o r 5/8  Why and  J 300.0 =  do y o u  500 ) 3 0 0 0  tenths  3000  tenths  0  tenths  t h i n k so?  e x p l a i n why  30. J u d y ' s m e t h o d f o r f i n d i n g  For  6 + 4 /TO  example, to f i n d  4 = 10, -  6 tenths = Will  Sam  Sam  .6  always get  would f i n d  for a fraction  the decimal  s o I d i v i d e 10 b y  fraction.  i s very  She  tenths  100  tenths  from t h i s  f o r 6/4,  Judy would f i n d  m e t h o d seems t o w o r k .  much  adds t h e the  denumer-  she w r i t e s :  25  tenths  25  tenths  answer, to get  2.5  - 1 =  1.5.  c l a i m s t h a t h e r method a l w a y s w o r k s f o r p r o b l e m s o f t h i s  Show how  the  the  4  4 JlO.O = 4 J100  s u b t r a c t s one  to  tenths  t h e n d i v i d e s t h i s number by  o tenths Then she  example,  h i s m e t h o d seems t o w o r k .  the decimal  n o m i n a t o r t o t h e n u m e r a t o r and  6  Show how  l i k e J o h n n y ' s , o n l y h e r s w o r k s f o r any  ator.  For  denominator  he d i v i d e s  c l a i m s t h a t h i s answer i s always c o r r e c t .  correct  She  divides.  i s very  the decimal  f o r 5/8  and  e x p l a i n why  sort. her  149  Items of F i n d i n g  1. 8 x 7 =  a  2. 4 x 9 -  O  t h e Square Root o f a F r a c t i o n  6. 6 x 8 = 7. 4 x 5 =  a 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 s h a d e d p o r t i o n  Pretest  below:  14. Draw a d i a g r a m w i t h  3/4 s h a d e d  i n on t h e l i n e t o t h e r i g h t .  1 5 . Draw a d i a g r a m w i t h  2/6 s h a d e d  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 = 9 5 7 6 17. W r i t e t h e f o l l o w i n g  division  492 -T-123 = 4  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  statement:  150 18.  Write  the following d i v i s i o n  O 19. W r i t e  Write  the following m u l t i p l i c a t i o n X  6=  £7x1/9  statement:  statement  as a d i v i s i o n  statement:  statement  as a d i v i s i o n  statement:  A  the following m u l t i p l i c a t i o n  a 21.  as a m u l t i p l i c a t i o n  r 6= A  Q 20.  statement  * A = <r  = 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 t h e l e n g t h of  32.  of a rectangle  with  a length  of a rectangle with width  What o p e r a t i o n  would you use t o f i n d  length  and w i d t h  or  a width  o f 2 f e e t and a n  area  of 8 feet  area  and an  8 x 3 sq. f t .  33. What i s t h e a r e a 34.  with  6 x 2 sq. f t .  What i s t h e w i d t h of  of a rectangle  89 f e e t  divide?)  38 f e e t ?  3 feet  the area  and l e n g t h  of a rectangle  5 feet? with  (Would y o u a d d , s u b t r a c t , m u l t i p l y  151 35.  Using  the idea that  squares i n a f i g u r e , below:  36.  Divide  11 J 425 37.  Divide  20 ) 5834 38.  Divide  7 J 2859 39.  Divide  25 ) 4632 40.  Divide  to find  area, you find  show how y o u w o u l d  t h e number o f o n e - u n i t  find  t h e area  of the figure  152  Items o f F i n d i n g  the Square Root o f a F r a c t i o n Computation  1.  J 1089/4  9.  2.  ^2764/9  10.  /7o/4~  3.  J  961/16  11.  v/38/9  4. / 1 8 4 9 / 2 5  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  J 1369/9  Test  153  Items of Finding the Square Root of a Fraction Generalization Test 1.  I f you know that x 169/225 =  fl 2.  = 13/15,  then you can also t e l l that  J2/A.  I f you know that J256/25 = 16/5, then you can also t e l l that O/A.  J~256/25 x 9 =  3. I f you know that J 9 / 1 6 = 3/4, and /25/49 = 5/7, you can also t e l l that l  x 25/16 x 49 =  J~9  a/A-  4. For each part, find the square root required. on the line to the r i g h t .  Write the answer to (d)  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  J9/I6  = 3/4,  (b) I f you know that Jvb/25  =  you can also t e l l that J16/9  =  O / A  4/5, you can also t e l l that/25/16 = O/A  (c) I f you know that J~81/100 = 9/10, you can also t e l l that  /1OO/8I  =EJ/A  (d) I f you know that / x / y = 8/5, you can also t e l l that 5. I f you know that J8/9 ^ 2 3/4 /3 , you can also t e l l that JTx  8/9  ~  a/A.  6. I f you know that /15/25 ~ 3 7/8 15 , you can also t e l l that Jl5/  25  x 100  =- C I / A  7. I f you know that J~8/9 / T x  8/9  2 3/4. /3 , you can also t e l l that  ~ Cf/A.  8. I f 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 e a c h p a r t , f i n d only (a)  on t h e l i n e  Jl/9  x  W r i t e t h e answer t o (d)  J T  (b) JT/25 = 1/0 x  Jls  (c)  v/lOO  Jl/100  required.  to the right,  = 1/Q  (<0 Ji/a 10.  the square root  = 1/n  x  1000 = i/iooo x v/:  Jo/81 ~  11. J 4 9 / D  5/9  = 7/8  12. J f l M x 2/9 = 2/6 1 3 . How do a a n d b c o m p a r e i f Ja/b 14. I f y o u know t h a t J 8 / 9 = tell  JjOx  1 5 . I f y o u know t h a t  2 3/4 /3 a n d J l l / 2 5 ^  f J /Q  t h a t (2 3/4 /5 x  i s l a r g e r t h a n 1?  6/9 -  /5) x (2 3/4 /5 x 24/3, what i s  16. F i n d v a l u e s f o r D a n d A s o t h a t J8 1 7 . I f y o u know t h a t tell  that  v/fl~749  Express Q  18. F i n d  a f r a c t i o n "C  19. F i n d  a fraction  20. U s i n g  find  = 161/7 and  0  / A  JiU  ^  / 0^ i n t e r m s  G  £J /  then you can a l s o /5)- -  0  CJ/<n  • = 12/5 = 9 / 8 7 , then you can a l s o <T .  o f O and h  Do n o t f i n d  values  .  J~U/5 J5/S  18/1 as an example,  square roots 21 How w o u l d  x  161/7. x 9/87 x 1 6 1 / 7 x 9/87 = Ol  f o r CI a n d Zl .  Q /Q/5  s h o w how y o u u s e t h e r u l e f o r f i n d i n g t h e  of f r a c t i o n s to find  square roots  of wholes?  you use the r u l e f o r f i n d i n g the square roots o f f r a c t i o n s t o  the square root  o f a mixed  number, l i k e  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  1 9/16  on t h e l i n e  .  Show a l l o f y o u r  to the r i g h t .  155 22.  Find  /l6/25  23.  Find  J A/25 /6A  2A. F i n d j 8 / 1 6 25.  Find J  26.  Find  27.  Susie  /A9  /2  8 1 / 2 5 /A9  J 1 6 / 1 0 0 /25 h a s 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 square roots  She s a y s t h a t i f s h e w a n t s t o f i n d Ja/b multiplies  t h e a n s w e r b y 1/2.  t h a t A x 16 = 6A; t h e n , J~6h/25  , s h e f i n d s Jk  by u s i n g  t h e method  an answer e q u i v a l e n t correctly?  x a/b a n d t h e n  F o r example, t o f i n d y i 6 / 2 5 ,  she says  = 8/5 a n d 1/2 x 8/5 = 8/10.  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 cher gets  of f r a c t i o n s ,  taught  i n class.  t o the teacher's  She  claims  t o t h e one t h e t e a -  Will  Susie  always get  i f she f o l l o w s h e r d i r e c t i o n s  Show how S u s i e w o u l d f i n d / 3 6 / 100 u s i n g h e r m e t h o d a n d l  e x p l a i n why i t seems t o w o r k . 28.  J o h n n y s a y s t h a t a n o t h e r way t o f i n d Ja/b find  a/Jax  J^OO  = 20.  gets  b.  F o r e x a m p l e , t o f i n d / l 6 / 2 5 , h e s a y s 16 x 25 = A 0 0 ,  Therefore,  16/20 = / l 6 / 2 5 .  an answer e q u i v a l e n t  class.  I s he c o r r e c t :  why h i s m e t h o d  i s t o f i n d ^ - a x b and t h e n  He c l a i m s  t o the teacher's  using  t h a t he a l w a y s  t h e method  Show how J o h n n y w o u l d f i n d J3f>/100  seems t o w o r k .  taught i n  and e x p l a i n  156  29.  Sam s a y s t h a t h e h a s s t i l l J~ajb and  = then  correct.  J a x b/b. =  Jlb/25  a n o t h e r way t o f i n d j / a / b .  F o r example,  t o f i n d Jib 12 5, h e s a y s 16 x 25 = 4 0 0 ,  / 4 0 0 / 2 5 = 20/25.  Show how Sam wou I d f i n d  He s a y s  l  He c l a i m s  t h a t h i s method i s  / " 3 6 / 100 a n d e x p l a i n why h i s m e t h o d  seems t o w o r k . 30.  To f i n d Ja/b,  Judy  f i n d s (/a/4 x b a n d t h e n m u l t i p l i e s h e r a n s w e r b y 2 .  t o f i n d fie/25,  For  example,  and  2 x 4/10 = 8/10.  correct.  So J l 6 / 2 5 = 8 / 1 0 .  Show how J u d y w o u l d  seems t o w o r k .  s h e s a y s 25 x 4 = 100 a n d t h e n / l 6 / 1 0 0 = 4 / 1 0  find J  She c l a i m s  t h a t h e r method i s  3 6 / 1 0 0 a n d e x p l a i n why h e r m e t h o d  APPENDIX EXPERIMENTAL  C DATA  f  158  Table  19  S I and C l  Data  Subject  P/A  F.I.  SI Pre  S I Conp.  S I Gen.  Cl Pre  C l Comp.  Cl  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  Gen.  Subject  P/A  F.I.  SI Pre  S I Oomp.  S I Gen.  C l Pre  C l Comp.  Cl  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  Gen.  160  Subject  P/A  F . I . S I Pre  S I Comp.  S I Gen.  C l Pre  C l Cornp.  C l Gen.  61  A  16  36  18  5  21  12  1  62  A  17  33  12  9  25  14  7  63  A  16  33  18  9  21  9  7  64  A  15  28  5  6  27  10  3  161  T a b l e 19 S2 a n d C2 D a t a  Subject  P/A  F . I . S2 P r e  S2 Comp.  S2 G e n .  C2 P r e  C2 Comp.  C2 G e n .  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  Subject  P/A  F . I . S2 P r e  S2 Comp.  S2 G e n .  C2 P r e  C2 Comp.  C2 G e n .  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  li  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  Subject  P/A  F.I.  S2 P r e  S2 Comp.  S2 G e n .  C2 P r e  C2 Comp.  C2 G e n .  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  11  26  16  7  27  2  6  15  31  10  6  37  1  11  16  35  23  15  34  13  13  12  22  27  13  25  9  6  16  22  9  9  25  8  9  137 138 139 140 141  A A A A A  164  Table Distribution  20  o f S c o r e s o n t h e C h i l d r e n ' s Embedded  Score  Frequency  Figures  Test  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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