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The effect of problem context upon the problem solving processes used by field dependent and independent… Blake, Rick N. 1976

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THE E F F E C T OF PROBLEM CONTEXT UPON THE PROBLEM S O L V I N G PROCESSES USED BY F I E L D DEPENDENT AND INDEPENDENT STUDENTS: A C L I N I C A L STUDY by R i c k N. B l a k e B.A., C h i c o S t a t e C o l l e g e , 1964 M.S., U n i v e r s i t y o f N o t r e Dame, 1969 M.A., W a s h i n g t o n S t a t e U n i v e r s i t y , 1972  A T H E S I S SUBMITTED I N P A R T I A L F U L F I L L M E N T OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF EDUCATION  in the Faculty of Education  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g required standard  t o the  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA February,  1976  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  by h i s of  at make  that  written  thesis  it  freely  may  is  financial  University  of  British  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  March 2. 1976  of  Columbia,  British  by  for  gain  Columbia  shall  the  that  not  requirements I  agree  r e f e r e n c e and copying  t h e Head o f  understood  Eduction  of  of  for extensive  be g r a n t e d  It  fulfilment  available  permission.  Department  Date  partial  permission  purposes  for  in  the U n i v e r s i t y  representatives.  this  The  thesis  of  be a l l o w e d  or  that  study.  this  thesis  my D e p a r t m e n t  copying  for  or  publication  without  my  ii  Abstract  Research Supervisor:  Dr. G a i l J . S p i t l e r  I t was t h e purpose o f t h i s s t u d y t o a n a l y z e t h e p r o c e s s e s s t u d e n t s used i n s o l v i n g m a t h e m a t i c a l word problems of problem c o n t e x t on t h e s e p r o c e s s e s . determine whether s t u d e n t s who  and t o determine t h e e f f e c t  A concomitant purpose was  d i f f e r i n t h e i r degree o f f i e l d  to  independence,  d i f f e r i n t h e p r o c e s s e s t h e y use i n s o l v i n g m a t h e m a t i c a l problems. F o r t y s u b j e c t s o f b o t h sexes, who  were c o m p l e t i n g a grade e l e v e n  academic mathematics program were randomly s e l e c t e d from 14 A l g e b r a I I classes. program  The s u b j e c t s were o f average a b i l i t y f o r s t u d e n t s on t h i s (IQ range 115-125).  The s u b j e c t s were t e s t e d  u s i n g W i t k i n ' s Embedded F i g u r e s T e s t .  individually,  They were matched on t h i s  and randomly a s s i g n e d t o one o f two groups.  One  group was  u s i n g a r e a l w o r l d s e t t i n g , w h i l e t h e o t h e r group was problems u s i n g a m a t h e m a t i c a l s e t t i n g .  given  variable problems  g i v e n t h e same  The s u b j e c t s were i n d i v i d u a l l y  i n t e r v i e w e d and asked t o t h i n k a l o u d as t h e y s o l v e d f i v e m a t h e m a t i c a l word problems. To a n a l y z e t h e problem s o l v i n g p r o c e d u r e s t h e s u b j e c t s ' t a p e r e c o r d e d p r o t o c o l s o f the i n t e r v i e w s were coded by means o f a system based on a model o f m a t h e m a t i c a l problem s o l v i n g by MacPherson. system had two p a r t s :  The c o d i n g  a c o d i n g m a t r i x used t o s e q u e n t i a l l y code problem  s o l v i n g b e h a v i o r , and a summary sheet f o r c o m p i l i n g i n f o r m a t i o n o b t a i n e d from t h e c o d i n g m a t r i x as w e l l as o t h e r d a t a r e l a t e d t o t h e problem s o l v i n g b e h a v i o r . . The c o d i n g system had i n t e r c o d e r of  .80 and i n t r a c o d e r r e l i a b i l i t y o f  .86.  subjects' reliability  iii Problem c o n t e x t p r o v e d t o be u n r e l a t e d t o t h e h e u r i s t i c s  used.  Both t h e t o t a l number o f h e u r i s t i c s used and t h e number o f d i f f e r e n t h e u r i s t i c s used were n o t i n f l u e n c e d by problem s e t t i n g .  S u b j e c t s working  problems i n t h e math w o r l d s e t t i n g had a s l i g h t l y more d i f f i c u l t  time  u n d e r s t a n d i n g t h e problems, but performed as w e l l as t h e o t h e r group. W i t h i n t h e IQ range, 115 t o 125,  f i e l d independence had a marked  e f f e c t on t h e use o f h e u r i s t i c s and on t h e number o f c o r r e c t  solutions  obtained.  The f i e l d independent s u b j e c t s used a g r e a t e r v a r i e t y o f  heuristics  ( r = .33) i n a t t a c k i n g and s o l v i n g problems.  w i l l i n g t o change t h e i r mode o f a t t a c k ( r = .27) g r e a t e r number o f c o r r e c t s o l u t i o n s  They were more  and t h e y o b t a i n e d a  ( r = .30) t h a n t h e i r f i e l d  dependent  counterparts. Both t o t a l number o f h e u r i s t i c s as w e l l as number o f d i f f e r e n t heuristics,  accounted f o r a s i g n i f i c a n t  i n number o f c o r r e c t for  an a d d i t i o n a l 2lf  solutions. 0  (P  <^.0l) amount o f t h e v a r i a n c e  I n p a r t i c u l a r , h e u r i s t i c s accounted  o f v a r i a n c e n o t accounted f o r by c o r e p r o c e d u r e s  ( a l g o r i t h m s , diagramming,  e q u a t i o n s , and g u e s s i n g ) .  The h e u r i s t i c s u s e d  by t h e s u b j e c t s i n t h i s s t u d y added t o t h e i r a b i l i t y t o s o l v e  problems  beyond t h e i r m a t h e m a t i c a l c o r e knowledge. The number o f t i m e s a s u b j e c t  attempted t o s o l v e a problem was  found t o be u n r e l a t e d t o o b t a i n i n g a c o r r e c t  s o l u t i o n , w h i l e changing  one's mode o f a t t a c k i n s o l v i n g a problem was s i g n i f i c a n t l y related t o obtaining a correct  (P  <^.0l)  solution.  When t h e s u b j e c t s were grouped by problem c o n t e x t , b o t h groups e x h i b i t e d t h e same g e n e r a l p a t t e r n o f problem s o l v i n g b e h a v i o r .  The  r e a l w o r l d s t u d e n t s e x p r e s s e d concern f o r s o l u t i o n s o b t a i n e d u s i n g h e u r i s t i c s w h i l e t h o s e s t u d e n t s i n t h e math w o r l d s e t t i n g e x p r e s s e d none.  However, e x p r e s s i o n o f concern f o r s o l u t i o n was u n r e l a t e d t o  iv the  correctness of the solution. When grouped by f i e l d  the  independence  a d i f f e r e n c e was o b s e r v e d i n  o v e r a l l p a t t e r n o f s e q u e n t i a l moves i n t h e problem  solving process.  These d i f f e r e n c e s among t h e groups were n o t t e s t e d f o r s t a t i s t i c a l significance.  The f i e l d  a l l p r o c e d u r e s coded.  independent student moved more f r e e l y  across  He was more concerned w i t h h i s work, and c o n t i n u a l l y  checked b o t h t h e p r o c e d u r e s he was u s i n g as w e l l as h i s s o l u t i o n . The f i e l d  independent s t u d e n t was more w i l l i n g t o check h i s work,  u s u a l l y by r e t r a c i n g h i s s t e p s , whereas t h e f i e l d u s u a l l y r e r e a d t h e problem.  dependent  student  •v:  T A B L E OF  CONTENTS  Page L I S T OF T A B L E S  vi'i  L I S T OF F I G U R E S  viii  ACKNOWIiEDGEMENT  x-  Chapter I.  1  THE PROBLEM  2  Purpose o f t h e Study Analyzing  t h e Problem S o l v i n g  The Need f o r a M o d e l o f P r o b l e m S o l v i n g  4  MacPherson's  5  Model f o r Mathematical Problem S o l v i n g  D e f i n i t i o n o f Terms  20  The P r o b l e m  22  G e n e r a l Hypotheses  23  Significance  24  o f t h e Study  Assumptions and L i m i t a t i o n s II.  III.  2  Process  REVIEW OF RELATED L I T E R A T U R E  25 27  Models o f Problem S o l v i n g  27  F i e l d Independence  42 51  PROCEDURES Subjects  51  Pilot  Study  52  The C o d i n g System  55  The I n t e r v i e w  61  Procedure  The C o d i n g P r o c e d u r e  67  vi Chapter IV.  V.  Page  A N A L Y S I S AND R E S U L T S  73  Research Hypotheses  73  Method o f A n a l y s i s  74  Results  o f A n a l y s i s - Problem Context  80  Results  o f A n a l y s i s - F i e l d Independence  93  Results  o f A n a l y s i s - Core  and H e u r i s t i c P r o c e d u r e s  ....  114  CONCLUSIONS Summary o f t h e E x p e r i m e n t a l S t u d y  «  f o r Education  114  »  Summary a n d C o n c l u s i o n s f o r t h e M o d e l a n d C o d i n g S y s t e m Implications  100  ..  119 122  L i m i t a t i o n s o f t h e Study  125  Implications f o r Research  127  LITERATURE CITED  130  Appendix A.  PROBLEMS USED I N P I L O T STUDY  134  B.  PROBLEMS USED I N THE STUDY  137  C.  THE CODING SYSTEM  141  D.  HISTOGRAMS OF SELECTED V A R I A B L E S  152  vii  L I S T OF T A B L E S  Table 1.  2.  3.  4.  Page Types and Numbers o f D i s a g r e e m e n t s Between D i f f e r e n t Coders  Two 70  Types and Numbers o f D i s a g r e e m e n t Between Code and R e c e d e  71  Means a n d S t a n d a r d D e v i a t i o n s Hypothesized Variables  80  Results  of Regression  f o r Measures o f  A n a l y s i s w i t h Problem  Context  as I n d e p e n d e n t V a r i a b l e 5«  6.  7.  8.  .  Results of Regression Analysis with F i e l d as I n d e p e n d e n t V a r i a b l e I n t e r c o r r e l a t i o n s Among M e a s u r e s o f Variables  Independence .  f o r Procedures  105 Solution 107 Derived  from Coding System 9.  10.  11.  94  Hypothesized  Results of Regression Analysis with Correct as D e p e n d e n t V a r i a b l e Means a n d S t a n d a r d D e v i a t i o n s  82  108  I n t e r c o r r e l a t i o n s Between P a i r s o f V a r i a b l e s Coding System  from  R e g r e s s i o n A n a l y s i s on C o r r e c t S o l u t i o n w i t h Entered Before H e u r i s t i c s  Core  Regression A n a l y s i s on C o r r e c t S o l u t i o n w i t h H e u r i s t i c s Entered Before Core  109  I l l  112  viii  LIST OF FIGURES  Figure  Page  1.  Three F a c e t s o f t h e D i s c i p l i n e o f Mathematics  6  2.  The H e u r i s t i c s  8  3.  A Model o f t h e E v e n t s I n v o l v e d i n M a t h e m a t i c a l Problem  from MacPherson's  Model  19  Solving 4.  The C o d i n g Form  56  5.  Example o f C o d i n g System:  Student #36,  Problem #3  62  6.  Example o f C o d i n g System:  Student #24,  Problem #4  63  7.  Examples  8.  The P r o c e s s M a t r i x w i t h I d e n t i f i e d Areas  77  9.  Process Matrix Categories  78  69  o f Coder E r r o r  10.  Process Matrix:  Problem S o l v i n g P r o c e d u r e s f o r 100 Math World Problems Process Matrix: Frequency D i s t r i b u t i o n o f Change o f  ....  83  11.  Problem S o l v i n g P r o c e d u r e s f o r 100 R e a l World Problems  ....  84  12.  P e r c e n t o f P r o c e d u r e s Used i n Each C a t e g o r y - Math World ....  85  13.  P e r c e n t o f P r o c e d u r e s Used i n Each C a t e g o r y - R e a l World ....  86  14.  Process Matrix:  15.  S o l v i n g P r o c e d u r e s , Counted Once p e r Problem, f o r 100 Math World Problems Process Matrix: Frequency D i s t r i b u t i o n o f Change o f Problem S o l v i n g P r o c e d u r e s , Counted Once p e r Problem, f o r 100 R e a l World Problems  89  Row P e r c e n t a g e o f Moves: Problems  P r o c e s s M a t r i x f o r Math World 91  Row P e r c e n t a g e o f Moves: Problems  P r o c e s s M a t r i x f o r R e a l World  16.  17.  18.  Frequency D i s t r i b u t i o n o f Change o f  Frequency D i s t r i b u t i o n o f Change o f Problem  P r o c e s s M a t r i x : Frequency D i s t r i b u t i o n o f Change o f Problem S o l v i n g P r o c e d u r e s f o r F i e l d Independent Students  88  92  96  ix Figure 19-  20.  21.  22.  23.  24.  25.  26.  Page P r o c e s s M a t r i x : Frequency D i s t r i b u t i o n o f Change o f Problem S o l v i n g P r o c e d u r e s f o r F i e l d Dependent S t u d e n t s P e r c e n t o f P r o c e d u r e s Used i n Each C a t e g o r y Independence  Field  P e r c e n t o f P r o c e d u r e s Used i n Each C a t e g o r y Dependence  Field  ...  97  98  99  P r o c e s s M a t r i x : Frequency D i s t r i b u t i o n o f Changes o f Problem S o l v i n g P r o c e d u r e s , Counted Once p e r Problem, f o r F i e l d Independent S t u d e n t s  101  P r o c e s s M a t r i x : Frequency D i s t r i b u t i o n o f Change o f Problem S o l v i n g P r o c e d u r e s , Counted Once p e r Problem, f o r F i e l d Dependent S t u d e n t s  102  Row Percentage- o f Moves: Independent S t u d e n t s  Process M a t r i x f o r F i e l d 103  Row P e r c e n t a g e o f Moves: Dependent S t u d e n t s  Process Matrix f o r F i e l d  R e l a t i v e C o n t r i b u t i o n s o f Core and H e u r i s t i c s t o C o r r e c t Solution  104  113  X  ACKNOWLEDGEMENT  I w i s h t o thank t h e members o f my d i s s e r t a t i o n  committee,  Dr. Douglas Owens, Dr. James S h e r r i l l , D r . P e t e r O l l e y , and Dr. Robert Conry f o r t h e i r  assistance.  I wish p a r t i c u l a r l y t o thank my chairman, D r . G a i l S p i t l e r , f o r her  p a t i e n c e and guidance i n t h e c o m p l e t i o n o f my  dissertation.  I wish t o thank, t o o , D r . E r i c MacPherson f o r h i s i n s p i r a t i o n and  counsel.  Chapter I THE PROBLEM The I n t e r n a t i o n a l Commission the  summer o f I966, chose t h r e e t o p i c s o f p a r t i c u l a r importance f o r  discussion. for  o f Mathematics, which met i n Moscow i n  These t o p i c s were:  t h e u n i v e r s i t y programs  i n mathematics  p h y s i c i s t s , t h e use o f t h e a x i o m a t i c method i n t h e t e a c h i n g o f mathe-  m a t i c s i n t h e secondary s c h o o l , and t h e r o l e o f problems i n d e v e l o p i n g students' mathematical a c t i v i t y .  The r e p o r t t o t h e commission by t h e  Conference Board o f M a t h e m a t i c a l S c i e n c e s (Begle, I966) i n d i c a t e d t h a t t h e c r e a t i o n o f problem s e t s s h o u l d be a body o f problem m a t e r i a l which p r o v i d e t h e environment f o r i m a g i n a t i v e and c r e a t i v e t h i n k i n g . Cambridge 1963)  Conference on S c h o o l Mathematics  will  The  (Goals f o r S c h o o l Mathematics,  has n o t e d t h a t "the c o m p o s i t i o n o f problem sequence i s one o f t h e  l a r g e s t and one o f t h e most u r g e n t t a s k s i n c u r r i c u l a r development  [p. 2 8 ] . "  Even w i t h the growing emphasis on s o l v i n g more complex and c h a l l e n g i n g problems, l i t t l e  has been done i n d e v e l o p i n g methods t h a t t h e t e a c h e r  use t o improve t h e problem s o l v i n g s k i l l s o f h i s s t u d e n t s .  may  In fact  D e s s a r t and Frandsen ( l ? 7 2 ) have suggested t h a t t h e "major q u e s t i o n o f problems are s o l v e d . . . [ i s o n e ] . . .of t h e t o p i c s f o r which a more i n t e n s i v e s e a r c h f o r fundamental p r i n c i p l e s might b e g i n i n hopes o f b u i l d i n g f o u n d a t i o n s t o support v a l i d r e s e a r c h [ p . 1191]." the  l i m i t e d knowledge  o f how  I n view o f  s t u d e n t s s o l v e problems, p r e c u r s o r s t o any  how  2  l a r g e s c a l e s t u d y i n t h e t e a c h i n g o f problem s o l v i n g s h o u l d be studies of i n d i v i d u a l subjects  (Kilpatrick,  1969,p.  clinical  179)•  Purpose o f t h e Study The q u e s t i o n o f how  m a t h e m a t i c a l problem s o l v i n g may  a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n among mathematics  be improved  educators.  has  I n most  problem s o l v i n g t a s k s , t h e r e i s no simple r e l a t i o n s h i p between t h e s o l u t i o n o b t a i n e d by an i n d i v i d u a l and t h e p r o c e s s used t o a c h i e v e i t .  Measures  such as t h e c o r r e c t n e s s o f a s o l u t i o n o r t h e time needed t o a c h i e v e a c o r r e c t s o l u t i o n do not i n d i c a t e t h e p a r t i c u l a r p r o c e d u r e s t h e i n d i v i d u a l used i n r e a c h i n g a s o l u t i o n .  I n o r d e r t o s t u d y t h e problem  solving  p r o c e s s t e c h n i q u e s must be used which a l l o w t h e s u b j e c t t o g e n e r a t e an o b s e r v a b l e sequence o f b e h a v i o r ( K i l p a t r i c k , It  1967,  p«  4)«  i s a purpose o f t h i s s t u d y t o a n a l y z e an i n d i v i d u a l ' s b e h a v i o r  d u r i n g t h e s o l u t i o n o f a m a t h e m a t i c a l problem and t o determine t h e of v a r y i n g t h e problem c o n t e x t on t h i s b e h a v i o r .  effect  The e x p e r i m e n t e r w i l l  a l s o t r y t o determine i f s t u d e n t s d i f f e r i n g i n t h e i r degree o f f i e l d , independence, d i f f e r i n t h e i r b e h a v i o r w h i l e s o l v i n g m a t h e m a t i c a l problems. F i e l d independence i s a c o n s t r u c t which p r o v i d e s i n f o r m a t i o n about individual differences.  Spitler  (1970, pp. 53-57) contends t h a t  field  independent s t u d e n t s a r e more a b l e t o d e a l w i t h complex g e o m e t r i c problems and w i t h problems which d e a l w i t h p a t t e r n e d s t i m u l i t h a n f i e l d students.  dependent  T h i s would seem t o i m p l y t h a t t h e f i e l d - i n d e p e n d e n t - d e p e n d e n t  c o n s t r u c t i s an i m p o r t a n t v a r i a b l e i n mathematical problem  solving.  A n a l y z i n g the Problem S o l v i n g P r o c e s s A s e a r c h o f t h e l i t e r a t u r e r e v e a l s t h a t many t e c h n i q u e s have been used t o s t u d y t h e problem s o l v i n g p r o c e s s . which  One  of the p r i n c i p a l  must be sought by t h e i n v e s t i g a t o r i s t h a t o f o b t a i n i n g  objectives  and  3 maintaining observable  behavior.  I t i s not enough t o t r y t o d e t e c t which  problem s o l v i n g methods a r e used by s t u d y i n g t h e r e s u l t s o f t h e s t u d e n t s ' work on t h e i r f i n a l p a p e r s . observable  The i n v e s t i g a t o r must t r y t o e l i c i t an  response from t h e student  be d i f f i c u l t  even though t h i s t y p e o f b e h a v i o r  f o r t h e s t u d e n t w o r k i n g a n o v e l problem.  P s y c h o l o g i s t s have used numerous d e v i c e s and t e c h n i q u e s to  may  encourage t h e i r s u b j e c t s t o e x t e r n a l i z e t h e i r thought  i n attempts  processes.  can be found i n Lucas (1972) and K i l p a t r i c k  Comprehensive reviews  (1967).  I n view o f t h e g e n e r a l purpose o f t h i s study a method w e l l s u i t e d f o r e x t e r n a l i z i n g t h e problem s o l v i n g p r o c e d u r e was suggested (1967), who  by K i l p a t r i c k  observed:  T h e r e i s one method o f g e t t i n g a s u b j e c t t o produce s e q u e n t i a l l y - l i n k e d , observable behavior that requires n e i t h e r s k i l l i n s e l f - o b s e r v a t i o n nor the manipulation p f m e c h a n i c a l d e v i c e s : have t h e s u b j e c t t h i n k a l o u d as he works [ p . 6 ] . T h i n k i n g a l o u d r e q u i r e s o n l y t h a t t h e s u b j e c t g i v e an account  of h i s  mental a c t i v i t y as b e s t he c a n . However, t h i s method may have limitations.  certain  Thoughts may come and go t o o q u i c k l y t o be v e r b a l i z e d .  S u b j e c t s may a l s o t e n d t o remain s i l e n t d u r i n g moments o f deepest thought.  More s e r i o u s , however, i s t h e p o s s i b i l i t y t h a t a s u b j e c t may  s o l v e a problem i n a d i f f e r e n t manner when asked t o v e r b a l i z e h i s t h o u g h t s t h a n when he i s l e f t  alone t o s o l v e i t s i l e n t l y  (Kilpatrick,  1967, p» 7)»  Two r e s e a r c h s t u d i e s have examined t h e e f f e c t o f r e q u i r i n g s u b j e c t s to  t h i n k a l o u d when s o l v i n g problems.  F l a h e r t y (1973) found no s i g n i f i c a n t  d i f f e r e n c e w i t h r e s p e c t t o problem s o l v i n g t e s t who were r e q u i r e d t o t h i n k a l o u d and t h o s e to  verbalize overtly.  s u b j e c t s who were n o t r e q u i r e d  However, t h e r e was a s i g n i f i c a n t d i f f e r e n c e  t h e two groups i n t h e a r e a o f c o m p u t a t i o n a l that:  s c o r e s between s u b j e c t s  errors.  between  Flaherty indicated  4 Perhaps t h e tendency o f o v e r t v e r b a l i z a t i o n s u b j e c t s t o make more c o m p u t a t i o n a l e r r o r s t h a n t h e n o n - v e r b a l i z a t i o n s u b j e c t s may be a t t r i b u t e d t o t h e i r b e i n g somewhat d i s t r a c t e d by t h e requirement t o t h i n k a l o u d [ p. 1767 ]  (1966) found t h e r e was no s i g n i f i c a n t d i f f e r e n c e i n t h e number  Roth  o f c o r r e c t s o l u t i o n s o r t h e time r e q u i r e d t o f i n d a s o l u t i o n between s u b j e c t s who and  were r e q u i r e d t o t h i n k a l o u d w h i l e  s u b j e c t s not  required to think  Information-processors procedure.  Paige  and Simon  s o l v i n g r e a s o n i n g problems  aloud.  have made f r e q u e n t use  (1966) used t h e p r o t o c o l s o f s u b j e c t s asked  t o t h i n k a l o u d t o i d e n t i f y some o f t h e important must use The  of the t h i n k i n g aloud  processes  that a  person  i n o r d e r t o do a l g e b r a word problems s u c c e s s f u l l y . method o f t h i n k i n g a l o u d i s b o t h p r o d u c t i v e and  easy t o  use,  o n l y r e q u i r i n g t h e s u b j e c t t o work on t h e s o l u t i o n t o a m a t h e m a t i c a l problem and t o t e l l about i t as he goes a l o n g . i n t e l l i g e n t l y and  c o n s c i e n t i o u s l y , keeping  I f t h e method i s u s e d  i n mind i t s l i m i t a t i o n s , i t  can p r o v i d e i n f o r m a t i o n about t h e d e t a i l e d p r o c e s s i s not  The  problem  so much t o c o l l e c t t h e d a t a as i t i s t o know what t o do w i t h them .  ( M i l l e r , Galanter, The  of thought.  and Pribram,  i960, p.  193).  Need f o r a Model o f Problem S o l v i n g I t i s apparent t h a t a framework f o r c l a s s i f y i n g and  i s necessary.  In f a c t , before undertaking  i n v e s t i g a t o r must d e c i d e which p r o c e d u r e s  analyzing the  a study i n problem s o l v i n g , t h e t o examine and which t o i g n o r e .  T h i s c h o i c e o f f a c t o r s w i l l not o n l y determine t h e u s e f u l n e s s o f t h e  study,  but a l s o i t s l i m i t a t i o n s . I n o r d e r t o m i n i m i z e t h e chances o f o v e r l o o k i n g o r not b e i n g  able  t o account f o r a p r o c e d u r e used by a s u b j e c t , t h e i n v e s t i g a t o r must aware o f t h e model he i s u s i n g .  Begle  data  and W i l s o n  (1970) d e s c r i b e d a  model as " p r o v i d i n g an o r g a n i z a t i o n a l framework; i t r e p r e s e n t s  a  be  5 c a t e g o r i z a t i o n system w i t h some s t a t e d r u l e s and r e l a t i o n s h i p s f o r u s i n g the to  system [ p . 372]."  The f u n c t i o n s o f a model s h o u l d be t o make c l e a r  o t h e r s what one has i n mind and so a i d communication, t o d i s t i n g u i s h  between d e f i n i t i o n and e m p i r i c a l p r o p o s i t i o n s , and t o p r o v i d e a means of  d a t a o r g a n i z a t i o n and i n t e r p r e t a t i o n  (Kaplan, 1964,  ch. 7).  With more emphasis b e i n g p l a c e d on t h e r o l e o f problems i n b o t h t h e c u r r i c u l u m and i n r e s e a r c h , i t i s n e c e s s a r y t o know more about t h e problem s o l v i n g p r o c e s s t h a n t h e u s u a l t h r e e o r f o u r " s t e p s " suggested by models in  t h i s area.  The f o l l o w i n g statement made by Johnson (1944) which  summarized t h e knowledge o f problem s o l v i n g i n t h e f o r t i e s i s s t i l l relevent today: Problem s o l v i n g b e g i n s w i t h t h e i n i t i a l o r i e n t a t i o n and ends w i t h t h e c l o s i n g judgment, but between t h e s e bounds almost a n y t h i n g can happen, i n any sequence [ p .  203]. The w r i t i n g s o f George P o l y a (1957,  1962,  I965) have added a g r e a t  d e a l t o t h e knowledge o f problem s o l v i n g i n mathematics. m a t h e m a t i c a l problem s o l v i n g i n terms o f "methods and  i n v e n t i o n " which he c a l l s h e u r i s t i c s  Polya describes  and r u l e s o f d i s c o v e r y  (1957, P» 112).  However, i t i s  s t i l l "between t h e s e bounds" where a more c a r e f u l l y c o n s t r u c t e d model is  needed.  The p r o c e d u r e s g i v e n i n t h e model must be b r o a d enough t o  account f o r a l l  t h e observed p r o c e s s e s and y e t not so b r o a d t h a t no  d i s t i n c t i o n can be made. such a model.  real  The model used i n t h i s study appears t o be  The model i s p r i m a r i l y t h e work o f E r i c  MacPherson,  p r e s e n t l y Dean o f t h e S c h o o l o f E d u c a t i o n , U n i v e r s i t y o f M a n i t o b a . M a c P h e r s o n s Model f o r M a t h e m a t i c a l Problem S o l v i n g 1  For it  t h e purpose o f d i s c u s s i o n from a problem s o l v i n g p o i n t o f view  i s c o n v e n i e n t t o view t h e d i s c i p l i n e o f mathematics as h a v i n g t h r e e  facets:  A p p l i c a t i o n , Core and D i s c o v e r y  (MacPherson, 1970,  1973).  6 CORE facts APPLICATION lore  categories  DISCOVERY  algorithms  heuristics  notation FIGURE 1 THREE FACETS OF THE DISCIPLINE OF MATHEMATICS  Core.  The c o r e i s t h a t p a r t o f t h e d i s c i p l i n e which i s n o t under a c t i v e  question  at t h e moment.  algorithms.  I t c o n s i s t s o f c a t e g o r i e s , n o t a t i o n , f a c t s and  The c a t e g o r i e s a r e t h e c o n v e n t i o n a l l y agreed upon g r o u p i n g s  o f f a c t s and a l g o r i t h m s . termed a l g e b r a ,  I n mathematics such g r o u p i n g s a r e b r o a d l y  geometry, a n a l y s i s , t o p o l o g y ,  number systems, i n t e g e r s and t h e l i k e .  and more p a r t i c u l a r l y ,  Associated with these  categories  i s a s e t o f f a c t s which a r e p r o p o s i t i o n s about t h e d a t a n o t under a c t i v e question;  and a l g o r i t h m s  a r e t h e p r o c e d u r e s f o r answering  questions,  which, when a p p l i e d c o r r e c t l y , guarantee a s o l u t i o n i n a f i n i t e number of steps. notation Lore.  Associated with (MacPherson,  each o f t h e s e i s u s u a l l y f o u n d  1973).  Lore i s t h e s e t o f a c t s o f f a i t h by which m a t h e m a t i c a l systems a r e  t i e d t o the r e a l world.  The e f f i c a c y o f l o r e i s n o t d e t e r m i n e d by t h e  l o g i c a l c o n s i s t e n c y o f any a l g o r i t h m s acts of f a i t h Discovery. its  core.  1973).  well-defined  (MacPherson, 1970,  used, but by t h e j u s t i f i c a t i o n o f t h e  1973).  A d i s c i p l i n e i n v a r i a b l y has a s e t o f p r o c e d u r e s f o r a d d i n g t o Such p r o c e d u r e s a r e termed h e u r i s t i c s (MacPherson, 1970,  That i s :  a h e u r i s t i c i s a g e n e r a l , n o n - c o r e s t r a t e g y which i s  used f o r t h e purpose o f d i s c o v e r i n g some o r d e r o f m a t h e m a t i c a l g e n e r a l i z a t i o n i n a novel  situation.  Operationally,  h e u r i s t i c s d i f f e r from a l g o r i t h m s  are c h a r a c t e r i z e d by an u n c e r t a i n t y  as t o t h e c h o i c e  i n that h e u r i s t i c s of the i n i t i a l  d a t a and t h e e f f i c a c y o f t h e p r o c e d u r e , whereas a l g o r i t h m s  have a  guarantee o f p r o d u c i n g a s o l u t i o n when a p p l i e d c o r r e c t l y . • I t seems u s e f u l t o c o n s i d e r  four h i e r a r c h i c a l categories  Low, Cases, M i d d l e , and G e n e r a l . t o t h e use o f a l g o r i t h m s  of heuristi  Lower h e u r i s t i c s l e a d more d i r e c t l y  and i n g e n e r a l do n o t c r e a t e new problems,  whereas t h e h i g h e r h e u r i s t i c s u s u a l l y c r e a t e new problems and t h e n l e a d t o use o f t h e l o w e r h e u r i s t i c s o r core p r o c e d u r e s The  h e u r i s t i c s from MacPherson*s model a r e l i s t e d i n F i g u r e 2 and  defined  as f o l l o w s :  Smoothing^" is and  (MacPherson, 1973)<-  The h e u r i s t i c o f smoothing i s used when t h e problem  a l t e r e d i n o r d e r t o o b t a i n some isomorphism between t h e new problem a m a t h e m a t i c a l system.  F o r example, i n t h e problems  What i s t h e l o n g e s t p i e c e o f m e t a l r o d which can be p l a c e d i n a box o f d i m e n s i o n 3 i n c h e s b y 4 i n c h e s by 12 i n c h e s ? One might " i d e a l i z e " o r smooth t h e box t o a r e c t a n g u l a r p a r a l l e l e p i p e d and t h e m e t a l r o d t o a l i n e  segment.  A n o t h e r example, i n s o l v i n g t h e problem: The  p o i n t A i s 50 u n i t s from a s t r a i g h t l i n e CD,  and B i s a p o i n t 80 u n i t s from CD.  Find the  p o i n t X on CD so t h a t t h e d i s t a n c e from A t o X t o B i s as s m a l l as p o s s i b l e .  The d e f i n i t i o n s f o r t h e t w e l v e h e u r i s t i c s i n t h e model were d e t e r m i n e d i n c o n s u l t a t i o n w i t h MacPherson.  Low  1.  Smoothing  2.  Analysis  ,all Cases  1.  Cases  ^^^random someN^^  sequential systematic critical  2.  Templation  .•direct Middle  1.  Deduction ^"^hypothetical  2.  Inverse  deduction fixation  3.  Invariation exclusion  General  4.  Analogy  5.  Symmetry  6.  Preservation  1.  Variation  2.  Extension  FIGURE 2 THE H E U R I S T I C S FROM MACPHERSON'S MODEL  9  A  SO  130  One  c o u l d d i v i d e each o f t h e measurements by 10,  f i g u r e w i t h dimensions o f 5,  13,  involved i n obtaining a solution. problem  and 8.  T h i s may  creating a similar s i m p l i f y the  Once a s o l u t i o n f o r t h e  i s found, t h e s o l u t i o n t o t h e o r i g i n a l problem  arithmetic  simplified  can be found by  m u l t i p l y i n g t h e s i m p l e r s o l u t i o n by t e n . A n a l y s i s - The h e u r i s t i c o f a n a l y s i s i s used when t h e problem i s s e p a r a t e d o r b r o k e n up i n t o subproblems. t h e subproblems s o l u t i o n f o r the  The  s o l u t i o n s o b t a i n e d from  are t h e n used i n s o l v i n g t h e o r i g i n a l problem.  One  problem:  What i s t h e maximum p e r i m e t e r you can o b t a i n by a r r a n g i n g one hundred  one i n c h squares by f o l l o w i n g t h e r u l e t h a t  each time a new  square i s added at l e a s t one o f i t s  s i d e s must be p l a c e d a g a i n s t one s i d e o f a p r e v i o u s l y arranged  square.  i s t o arrange t h e squares i n a s t r a i g h t row. one might  When d e t e r m i n i n g the p e r i m e t e r ,  c o n s i d e r t h e end squares s e p a r a t e from t h e o t h e r s , t h u s  c r e a t i n g two  subproblems.  One  f i n d i n g the p e r i m e t e r f o r t h e two  squares i n v o l v i n g t h r e e s i d e s and a subproblem o f t h e r e m a i n i n g squares I n v o l v i n g two  sides.  end  f o r f i n d i n g the p e r i m e t e r  10 Cases - A case i s renaming the v a r i a b l e s i n a problem as U s i n g t h e h e u r i s t i c s o f cases i s t o c o n s i d e r two of  constants.  o r more c a s e s i n  one  t h e f o l l o w i n g manners: (a)  To c o n s i d e r a l l p o s s i b l e cases, u s u a l l y i f t h e r e i s o n l y a s m a l l number.  (b)  To c o n s i d e r cases at random.  (c)  To c o n s i d e r cases i n some s y s t e m a t i c f a s h i o n , such s e q u e n t i a l l y o r t o determine the c r i t i c a l  as  cases.  and then t o r e c o g n i z e a p a t t e r n ; t h a t i s , t o r e c o g n i z e t h e  characteristics  shared by t h e d a t a c o l l e c t e d from t h e cases c o n s i d e r e d and t h e p r o p e r t i e s and procedures  from c o r e .  C o n s i d e r t h i s example o f t h e use o f  systematic  cases: F i n d the sum The  o f t h e f i r s t one hundred odd n a t u r a l numbers.  f o l l o w i n g cases  ( i n v o l v i n g t h e number o f n a t u r a l numbers i n t h e  sum)  c o u l d be c o n s i d e r e d i n sequence:  1 + 3 = 4 1 + 3 + 5 = 9  1 + 3 + 5 + 7 = 16 There i s a r e l a t i o n s h i p between t h e number o f terms i n t h e sum square to  r o o t o f t h e term on t h e r i g h t o f the e q u a t i o n .  t h e problem i s 1 0 0 One  can a l s o use  are r e c o g n i z e d .  and  Hence, t h e  the solution  squared. s e q u e n t i a l cases when a l l o f t h e p o s s i b l e s o l u t i o n s  F o r example:  A l a d y gave t h e p o s t a g e stamp c l e r k a one d o l l a r b i l l s a i d , "Give me  some two-oent stamps, t e n t i m e s  one-cent stamps, and t h e b a l a n c e clerk f u l f i l l  in fives."  and  as many  How  can  the  t h i s p u z z l i n g request?  In answering t h i s q u e s t i o n , one  c o u l d w r i t e the e q u a t i o n 12X  + 5Y = 1 0 0 ,  f o r w h i c h t h e r e a r e e i g h t p o s s i b l e v a l u e s f o r X. c o n s i d e r e d , o r s y s t e m a t i c c a s e s f o r X = 1,  A l l cases could  be  2, 3,*>*>8 c o u l d b e u s e d  u n t i l a solution i s obtained. S e q u e n t i a l c a s e s c a n b e u s e d i n f i n d i n g a*** w h e r e a / 0 i f one is  f a m i l i a r w i t h t h e use of p o s i t i v e F i n d t h e v a l u e o f a ^ where a /  exponents.  That i s :  0.  The f o l l o w i n g c a s e s c o u l d b e c o n s i d e r e d i n s e q u e n c e of  2: 2  As each exponent is  u s i n g a base  4  =  16  d e c r e a s e s b y 1,  one-half the preceding value.  2 ^ = 1.  t h e v a l u e on t h e r i g h t  of the equation  I f the pattern i s t o hold,  then  O t h e r b a s e s c a n be c o n s i d e r e d , e i t h e r a t random o r s y s t e m a t i c a l l y  obtaining a similar result.  Hence a ^ = 1  T e m p i a t i o n - The h e u r i s t i c  for  a / 0.  of templation i s recalling  of content which i s r e l a t e d t o t h e problem b e i n g solved. the r e c a l l theorems,  This  includes  o f such t h i n g s as a l g o r i t h m s , problem t y p e s , p r o c e d u r e s , and p r o p e r t i e s r e l a t e d t o t h e problem o r problem  The p u r p o s e o f t e m p l a t i n g i s t o r e c a l l r e l a t e d be u s e d i n s o l v i n g t h e p r o b l e m .  sea w a l l ,  CD.  away f r o m a  straight  The c a p t a i n o f t h e y a c h t w i s h e s t o r o w  to the sea wall t o c o l l e c t  a p a s s e n g e r and t h e n t o a  s p e e d - b o a t m o o r e d a t B, 8 0 m e t r e s  from t h e w a l l .  s h o u l d t h e p a s s e n g e r meet t h e c a p t a i n t o  make t h e r o u t e a s s h o r t a s p o s s i b l e ?  area.  core m a t e r i a l which can  F o r example:  A y a c h t i s m o o r e d a t A, 50 m e t r e s  Where  a category  12  One might p i c k a p o i n t X between C and D„then l o o k i n g a t t h e two t r i a n g l e s , c o n s i d e r t h e theorems  r e l a t e d t o r i g h t t r i a n g l e s t o see i f any i n f o r m a t i o n  about t h e p o s i t i o n o f t h e p o i n t X can be  obtained.  D e d u c t i o n - The h e u r i s t i c o f d e d u c t i o n can be s u b - c l a s s i f i e d i n t o two h e u r i s t i c s , d i r e c t and h y p o t h e t i c a l . t o ask what consequences premises.  Hypothetical  To use d i r e c t d e d u c t i o n i s  can be i m p l i e d from a g i v e n premise o r s e t o f d e d u c t i o n i s a s k i n g what consequences  can be  i m p l i e d from a premise o r s e t o f p r e m i s e s assumed by t h e problem In e i t h e r instance,  solver.  an attempt must be made t o answer t h e q u e s t i o n s  F o r an example o f d i r e c t d e d u c t i o n ,  assume t h e f o l l o w i n g d a t a , o r  s e t o f premises a r e g i v e n : G i v e n t r i a n g l e XYZ w i t h A t h e m i d p o i n t o f XY and B t h e m i d p o i n t o f YZ, l e t C and D be p o i n t s on XZ so t h a t q u a d r i l a t e r a l ABCD can be formed by f o l d i n g a l o n g AD so t h a t X i s a p o i n t on CD, f o l d on BC so t h a t Z i s a p o i n t on CD, and f o l d on AB.  posed.  What c o n s e q u e n c e s c a n b e i m p l i e d b y t h e s e t o f p r e m i s e s ? rectangle. of  T h e sum o f a n g l e s  ABCD i s a  X, Y, a n d Z i s 1 8 0 d e g r e e s .  The a r e a  q u a d r i l a t e r a l ABCD i s e q u a l t o o n e - h a l f t h e a r e a o f t r i a n g l e X Y Z .  T h e s e a r e e x a m p l e s o f some o f t h e i m p l i c a t i o n s w h i c h H y p o t h e t i c a l d e d u c t i o n may b e u s e d problem.  i n s o l v i n g t h e previous yacht  I f X i s t h e p o i n t on t h e s e a w a l l where t h e c a p t a i n p i c k s up  the passenger, problem  c o u l d b e made.  t h e n two r i g h t t r i a n g l e s  are formed.  A solution tothe  c a n b e o b t a i n e d b y m i n i m i z i n g t h e sum o f t h e l e n g t h s o f t h e i r  hypotenuse.  I f one assumes t h e h y p o t e n u s e  of a right triangle i s twice  A 8 /  /  /  \  SO  p  c (  ( ^  the longest side If  CX i s l e s s t h a n 50m,  moved c l o s e r t o C.  160m  a n d AX w i l l  XB w i l l  IP  ( w h i c h i t i s n o t ) , what c a n be c o n c l u d e d i n t h i s  X B i s 2 ( 1 3 0 - CX)m. is  i_r  t h e n t h e l e n g t h o f hypotenuse  AX i s 100m a n d  T h e sum o f A X a n d BX i s i n c r e a s e d a s t h e p o i n t X I f XD i s l e s s t h a n 80m, t h e l e n g t h o f X B w i l l  h a v e l e n g t h 2 ( 1 3 0 - XD)m.  be  H e n c e t h e sum o f AX a n d  i n c r e a s e a s t h e p o i n t X i s moved t o w a r d s  t h e p o i n t D.  c o u l d c o n c l u d e t h e m i n i m u m o f t h e sum, A X + X B w i l l 50m f r o m  problem?  So one  o c c u r when X i s  C.  I n v e r s e D e d u c t i o n - The h e u r i s t i c  o f i n v e r s e d e d u c t i o n i s used  one assumes ( o r one i s g i v e n ) a c o n c l u s i o n , antecedents imply i t . backwards'..  and a s k s what p r e m i s e s o r  I n v e r s e d e d u c t i o n means ' w o r k i n g t h e p r o b l e m  F o r example:  when  14 What i s t h e l o n g e s t p i e c e o f m e t a l rod which can be p l a c e d i n a box o f d i m e n s i o n 3 i n c h e s by 4 i n c h e s by 12 i n c h e s ? The  longest piece o f metal  rod  which w i l l f i t i n t o t h e  box  has t h e l e n g t h AD.  Now  AD i s t h e hypotenuse o f r i g h t g  £  t r i a n g l e CDA, and i f t h e  l e n g t h o f DC and AC were known, t h e n t h e Pythagorean Theorem c o u l d be applied  t o f i n d AD.  of AC must be f o u n d . and  CD i s known t o be 4 i n c h e s l o n g , so t h e l e n g t h But AC i s t h e hypotenuse o f r i g h t t r i a n g l e BCA  t h e l e n g t h s o f AB and BC a r e known t o be 3 said 12 i n c h e s  respectively.  So by a p p l y i n g t h e P y t h a g o r e a n Theorem t o t r i a n g l e BCA, AC can be f o u n d . Invariation  - The h e u r i s t i c o f i n v a r i a t i o n  i n t o two h e u r i s t i c s , f i x a t i o n and e x c l u s i o n . a variable  i s renamed as a c o n s t a n t  i n t h e problem ( e x c l u s i o n ) . problem and i t s s o l u t i o n  can be  subclassified  In using invariation  (fixation), or a variable  either  i s excluded  Then an attempt i s made t o s o l v e t h e new  s t u d i e d i n o r d e r t o g a i n some i n s i g h t  into the  g i v e n problem. The the  h e u r i s t i c o f i n v a r i a t i o n - f i x a t i o n can be u s e d i n s o l v i n g  following  problem:  Derive a formula t o f i n d the roots of the general  3 aX For  cubic  2 + bX  + cX + d, where a, b, c, and d a r e r e a l numbers.  t h i s problem, one o r more o f t h e c o n s t a n t s i s f i x e d a t zero and t h e  r o o t s o f t h e new f u n c t i o n  are found.  Then t h e s o l u t i o n  problem i s s t u d i e d i n hopes o f g a i n i n g some i n s i g h t o r i g i n a l problem.  into  o f t h i s new solving the  The h e u r i s t i c o f i n v a r i a t i o n - e x c l u s i o n can be used i n s o l v i n g t h e f o l l o w i n g geometry problem: Construct  a t r i a n g l e ABC g i v e n t h e l e n g t h o f t h e s i d e  BC, t h e measure o f angle A, and t h e l e n g t h o f t h e a l t i t u d e h from A. I n t h i s problem,  one might e x c l u d e t h e measure o f angle A and f o r m u l a t e  the new problem o f c o n s t r u c t i n g a t r i a n g l e ABC g i v e n BC and t h e a l t i t u d e h from A.  A g a i n t h e new problem may be s o l v e d by some method and i t s  s o l u t i o n s t u d i e d t o g a i n some i n s i g h t i n t o t h e o r i g i n a l  problem.  fK  It.  B  c  Analogy - I f t h e r e i s a " q u a s i " isomorphism between t h e problem and a m a t h e m a t i c a l system p r e v i o u s l y s t u d i e d t h e n t h e h e u r i s t i c o f a n a l o g y can be used.  The h e u r i s t i c o f a n a l o g y i n v o l v e s a s k i n g q u e s t i o n s and  c o n s i d e r i n g p r o p e r t i e s based on t h e r e c o g n i t i o n o f t h e isomorphism. A n a l o g y can be used t o s o l v e t h e f o l l o w i n g problem by r e c a l l i n g an analogous s i t u a t i o n from t w o - d i m e n s i o n a l geometry f o r which t h e s o l u t i o n i s known: What i s t h e g r e a t e s t d i s t a n c e between two p o i n t s o f a r e c t a n g u l a r s o l i d o f dimensions 3 by 4 by 12? I f one c o n s i d e r s t h e s i m i l a r i t y between t h i s problem  and t h a t o f f i n d i n g  the l e n g t h o f a d i a g o n a l o f a r e c t a n g l e o f d i m e n s i o n a by b, t h e n , a  + b , the length of the diagonal  of t h i s rectangular or  3/  c o u l d be d  solid  3  3  3  + 4  3  / 2  , y ;3 0  +12  ,2 2 + 4 + 1 2 , 1 0  \J J> + 4 + 12 . 3  3  Symmetry - To use t h e h e u r i s t i c o f symmetry i s t o make use o f t h e inherent  o r constructed  symmetry i n a problem.  F o r example:  A y a c h t i s moored at A , 50 meters away from a s t r a i g h t sea w a l l , CD.  The c a p t a i n o f t h e y a c h t wishes t o row  to the sea wall t o c o l l e c t  a p a s s e n g e r and t h e n row  t o a speed-boat moored a t B, 80 meters from t h e w a l l . Where s h o u l d t h e p a s s e n g e r meet t h e c a p t a i n t o make t h e route  as s h o r t  as p o s s i b l e ?  , 6 IN  N  \  c  p  130  \ N N  Nj  One  3'  might t a k e advantage o f t h e f a c t t h a t once t h e c a p t a i n has moved  from p o i n t A t o t h e s e a w a l l CD at X, t h e d i s t a n c e same as t h e d i s t a n c e  from X t o B i s t h e  from X t o a p o i n t B' (where B' i s t h e r e f l e c t i o n  o f B t h r o u g h t h e l i n e CD.)  S i n c e t h e s h o r t e s t d i s t a n c e between two  p o i n t s i s a s t r a i g h t p a t h , t h e p o i n t X i s e a s i l y found. I n t h e f o l l o w i n g example, t h e problem i s s t a t e d s y m m e t r i c a l l y i n terms o f t h e s e t s A and B: L e t A and B be s e t s such t h a t f o r any s e t , C, A A C = B/\ C and A U C =. BU C. The  Prove A = B.  r o l e o f A and B can be i n t e r c h a n g e d  w i t h o u t changing t h e problem.  17 So  i f one  subset  shows A i s a subset  o f A.  o f each o t h e r ,  S i n c e two A =  o f B, t h e n by the  s e t s are e q u a l i f and  symmetry, B w i l l  o n l y i f t h e y are  subsets  h e u r i s t i c o f p r e s e r v a t i o n i s used when a  e f f o r t i s made t o p r e s e r v e  conscious  the p r o p e r t i e s o f a m a t h e m a t i c a l system when  i t s domain i s extended ( i s o m o r p h i c a l l y ) t o form a new extension  a  B.  P r e s e r v a t i o n - The  i n the  be  o f t h e r e a l numbers t o the  system.  complex, one  t h e f i e l d p r o p e r t i e s o f the  r e a l number system.  The  p r e s e r v a t i o n i s used i f t h e  commutative, a s s o c i a t i v e and  For  example  wishes t o  heuristic  preserve  of  distributive  p r o p e r t i e s are t a k e n i n t o account when d e f i n i n g a d d i t i o n and m u l t i p l i c a t i o n for  the  complex numbers so as t o p r e s e r v e  The  the f i e l d  properties.  h e u r i s t i c o f p r e s e r v a t i o n can be used i n s o l v i n g t h e  following  problem: F i n d a d e f i n i t i o n o f a^ where a ^  0. X  One  Y  o f the p r o p e r t i e s o f exponents i s b /b  conscious  e f f o r t i s made t o p r e s e r v e  t h e h e u r i s t i c o f p r e s e r v a t i o n has  X—Y = b  , f o r X / Y.  t h i s p r o p e r t y when d e f i n i n g a^,  the  constraints  o f the v a r i a b l e s o f the problem are r e l a x e d i n o r d e r t o  e f f e c t on t h e o t h e r v a r i a b l e s .  and Y be t h e m i d p o i n t s o f AC  study  As an example from geometry, l e t X  and AB  r e s p e c t i v e l y i n t r i a n g l e ABC. one  X. >v  can  c o n c l u d e t h a t XY  p a r a l l e l t o BC  and t h e  o f XY  length of  The  i s -g- t h e  Then  is  length BC.  h e u r i s t i c of v a r i a t i o n i s  R  C  J  t h a t X and Y are m i d p o i n t s and between XY  then  been u s e d .  V a r i a t i o n - I n u s i n g the h e u r i s t i c of v a r i a t i o n the on one  If a  and  BC.  used i f one  relaxes the  condition  s t u d i e s t h e e f f e c t on t h e r e l a t i o n s h i p s  18  Extension an e x t e n s i o n  - The  heuristic  of e x t e n s i o n i s p o s i n g  a problem t h a t i s  o r a g e n e r a l i z a t i o n o f t h e g i v e n problem.  " F i n d a n a t u r a l number which i s b o t h a square and the h e u r i s t i c o f e x t e n s i o n  a cube.", one  when p o s i n g t h e q u e s t i o n ,  numbers which a r e b o t h a square and  a cube be  F o r the purposes o f t h i s study,  Given the  'can  problem,  uses  a l l natural  characterized?'.  the sequence o f events u s e d  s o l v i n g a m a t h e m a t i c a l problem i s c h a r a c t e r i z e d by F i g u r e  3.  A  while brief  d e s c r i p t i o n o f t h e model f o l l o w s : Willingness: once he has has  The  s u b j e c t ' s w i l l i n g n e s s t o accept  t h e problem  and  s t a r t e d to solve i t , h i s w i l l i n g n e s s t o continue u n t i l  he  a solution. Sieve:  The  m a t h e m a t i c a l content  and p r o c e s s e s  ( c o r e ) which  are  f a m i l i a r t o the problem s o l v e r . Upon p r e s e n t a t i o n o f t h e problem, t h e problem i s e i t h e r a c c e p t e d rejected  (willingness box).  I f t h e problem i s accepted,  immediately t r i e s t o r e c a l l whatever content  then the  one  t o o b t a i n a s o l u t i o n . . I f the  i n t h e p a t t e r n he has  p a t t e r n i n mind, he fit  confidence  core and  3)  He  obtained:  problem.  i n t h e p a t t e r n he  and d e c i d e t o e i t h e r c o n t i n u e a solution.  confidence ( p r o b a b i l i t y ) l)  Keeping  the  c o u l d r e t u r n t o the s i e v e and t r y t o o b t a i n a b e t t e r  of the p a t t e r n t o the  little  will  e x i t i s from t h e s i e v e t o p a t t e r n ,  o f t h r e e t h i n g s can happen depending upon t h e  the i n d i v i d u a l has  individual  he knows t h a t i s r e l a t e d  t o t h e problem ( s i e v e ) i n hopes o f r e c o g n i z i n g a p a t t e r n which enable him  or  I f the i n d i v i d u a l  could r e t u r n to the w i l l i n g n e s s  working on the problem o r t o q u i t  c o u l d e x i t t o t h e use  l e v e l s or p r o b a b i l i t i e s  2)  o f core a l g o r i t h m s .  sequence o f p r o c e d u r e s u s e d f o r e v e r y problem, one the p a t t e r n and t h e o t h e r c o n c e r n i n g  the a c c u r a c y  concerning of the  box  without Two  ( i n d i c a t e d by d o t t e d l i n e ) a r e a s s i g n e d  has  to  confidence the  the f i t o f  algorithm.  These  FIGURE A M O D E L OF T H E MATHEMATICAL  3  EVENTS INVOLVED I N PROBLEM SOLVING  20 p r o b a b i l i t i e s determine t h e p o s s i b l e r o u t e s from the concern S i m i l a r l y with the  a p p l i c a t i o n o f an a l g o r i t h m ,  attached to the s o l u t i o n .  a reasonableness  is  I f there i s a high p r o b a b i l i t y attached  t h e p a t t e r n and y e t , t h e r e i s some concern f o r t h e use the. problem s o l v e r may  box.  o f an  algorithm,  r e t u r n d i r e c t l y t o t h e a l g o r i t h m box.  t h e r e i s some concern f o r a p a r t i c u l a r p a t t e r n t h e n he may s i e v e ( v i a t h e w i l l i n g n e s s box)  I f however,  return to the  and t r y t o o b t a i n a b e t t e r f i t .  outcome o f t h e a p p l i c a t i o n o f the a l g o r i t h m i s r e a s o n a b l e , i n d i v i d u a l w i l l exit with a s o l u t i o n .  However, i f no  to  I f the  then the  e x i t i s made from  s i e v e t o p a t t e r n , t h e n he must e i t h e r r e t u r n t o the w i l l i n g n e s s box to  t h e use o f h e u r i s t i c s .  a n a l y s i s appear t o be  The  lower h e u r i s t i c s such as smoothing  c l o s e l y r e l a t e d t o the s i e v e so t h e r e may  c o n s i d e r a b l e movement between the s i e v e and h e u r i s t i c s . lower h e u r i s t i c s may  a l s o l e a d more d i r e c t l y t o t h e  The  or  and  be  a  use o f t h e  s i e v e and  whereas the use o f h i g h e r h e u r i s t i c s u s u a l l y c r e a t e s new  the  pattern  problems  and  t h e n l e a d s t o the use o f the l o w e r h e u r i s t i c s .  D e f i n i t i o n o f Terms  C e r t a i n terms o c c u r throughout t h e  study  and  are d e f i n e d  here.  The term degree o f f i e l d independence r e f e r s t o the r e l a t i v e 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 of the experimental  s c o r e s on t h e Embedded F i g u r e s T e s t  ( W i t k i n , 1969)•  See C h a p t e r I I f o r  f u r t h e r d i s c u s s i o n o f t h e c o n s t r u c t o f f i e l d independence. student  Conventionally  a  i s f i e l d independent i f h i s s c o r e i s above t h e sample median on  the Embedded F i g u r e s T e s t median.  sample  To t e s t one  (EFT) and  f i e l d dependent i f h i s s c o r e i s below the  o f t h e hypotheses i n t h i s study,  independent i f h i s s c o r e on the EFT  a subject i s f i e l d  i s i n t h e t o p t h i r d o f the s c o r e s f o r t h e  sample and f i e l d dependent  i f h i s s c o r e i s i n t h e bottom  one-third.  The t e r m problem r e f e r s t o " a s i t u a t i o n i n which one must g i v e a response ( t h a t i s , when he seeks s a t i s f a c t i o n ) and has no h a b i t u a l response which w i l l g i v e t h i s  satisfaction  [ Cronbach,  1948,  p. 3 2 ] . "  d e f i n i t i o n i m p l i e s t h a t what i s a problem f o r one s t u d e n t may a problem f o r a n o t h e r . it  This not be  I t i s not t h e p o s i n g o f a q u e s t i o n t h a t makes  a problem, but t h e w i l l i n g n e s s o f t h e i n d i v i d u a l t o a c c e p t i t as  something he must t r y t o s o l v e .  Furthermore, a q u e s t i o n such a s :  What, i s t h e g r e a t e s t d i s t a n c e between two p o i n t s i n a r e c t a n g u l a r s o l i d o f d i m e n s i o n 3 i n c h e s by 4 i n c h e s by 12 i n c h e s ? i s not a problem t o someone who geometry  i s f a m i l i a r with three-dimensional  and r e c a l l s a f o r m u l a f o r f i n d i n g t h e l e n g t h o f t h e d i a g o n a l  of a rectangular parallelepiped. student who  has s t u d i e d o n l y p l a n e  However, i t may  be a problem f o r t h e  geometry.  The term problem c o n t e x t r e f e r s t o t h e s e t t i n g i n which t h e problem is  s t a t e d , m a t h e m a t i c a l v s . r e a l o r p h y s i c a l w o r l d . . The v o c a b u l a r y u s e d  i n t h e statement o f t h e problem s h o u l d be f a m i l i a r t o a s t u d e n t who i s t a k i n g a grade e l e v e n mathematics.  The terms  " m a t h e m a t i c a l " and  r e f e r t o t h e s i t u a t i o n and t h e v o c a b u l a r y u s e d i n t h e problem. s t a t e d i n each c o n t e x t i s g i v e n below: M a t h e m a t i c a l World o r Math World:  What i s t h e g r e a t e s t  d i s t a n c e between two p o i n t s i n a r e c t a n g u l a r s o l i d o f d i m e n s i o n 3 u n i t s by 4 u n i t s by 12 R e a l World:  units?  What i s t h e l o n g e s t p i e c e o f m e t a l r o d  which can be p l a c e d i n a box o f d i m e n s i o n 3 i n c h e s by 4 i n c h e s by 12 i n c h e s ?  "real" A problem  22 The term  a l g o r i t h m r e f e r s t o a s y s t e m a t i c p r o c e d u r e which, i f c a r r i e d  out c o r r e c t l y , guarantees  a s o l u t i o n i n a f i n i t e number o f s t e p s .  A l g o r i t h m s v a r y i n c o m p l e x i t y from v e r y simple such as t h e a d d i t i o n o f two d i g i t whole numbers w i t h o u t r e g r o u p i n g , t o complex and i n v o l v e d p r o c e d u r e s , l i k e u s i n g t h e e l i m i n a t i o n method t o f i n d t h e s o l u t i o n o f a system  of l i n e a r equations.  An example o f an a l g o r i t h m from t h e  grade  e l e v e n c u r r i c u l u m i s t h e p r o c e d u r e used t o f i n d t h e r o o t s o f t h e q u a d r a t i c 2 4x  - l6x  - 9 by c o m p l e t i n g t h e  square.  The term h e u r i s t i c r e f e r s t o t h e t w e l v e p r o c e d u r e s model.  A h e u r i s t i c d i f f e r s from  d e f i n e d i n the  an a l g o r i t h m i n t h a t h e u r i s t i c s have  a low p r o b a b i l i t y o f g u a r a n t e e i n g any s u c c e s s i n s o l v i n g t h e problem. F o r example, t h e renaming o f a v a r i a b l e as a c o n s t a n t i n o r d e r t o i t s e f f e c t s on t h e o t h e r v a r i a b l e s i n t h e problem one w i l l f i n d a s o l u t i o n , b u t i t may  The  does not guarantee  Problem  t o determine,  by use o f a  procedure, whether t h e r e were any s i g n i f i c a n t d i f f e r e n c e s i n t h e s o l v i n g b e h a v i o r among s u b j e c t s working problems i n e i t h e r a An  attempt  r e l a t i o n s h i p between t h e s e problem and n o n h e u r i s t i c p r o c e d u r e s ,  that  g i v e some i n s i g h t i n t o t h e problem.  I n t h i s study, t h e w r i t e r attempted  or r e a l world s e t t i n g .  study  was  clinical  problem  mathematical  a l s o made t o i n v e s t i g a t e t h e  s o l v i n g b e h a v i o r s , i n terms o f h e u r i s t i c  and how  these behaviors r e l a t e t o  field  independence. The p r e s e n t s t u d y has i n mathematical  problem  as i t s c e n t r a l theme, the use o f h e u r i s t i c s  solving.  a t t a c k and s o l v e m a t h e m a t i c a l play i n t h i s procedure.  Little  i s known about  problems and what p a r t ,  I n o r d e r t o g a i n some i n s i g h t  a study which i s e x p l o r a t o r y i n n a t u r e was  designed.  how  students  i f any,  heuristics  into this The  area,  g e n e r a l aims  o f t h e s t u d y were: 1.  To d e v e l o p a workable system, based on t h e model o f for to  2.  MacPherson,  c o d i n g t h e audio t a p e r e c o r d e d p r o t o c o l s o f s u b j e c t s asked t h i n k a l o u d as t h e y s o l v e m a t h e m a t i c a l w o r l d problems.  To i n v e s t i g a t e t h e r e l a t i o n s h i p o f t h e problem s o l v i n g b e h a v i o r s d e r i v e d from t h e c o d i n g system, t o one a n o t h e r .  3.  To i n v e s t i g a t e t h e e f f e c t o f problem c o n t e x t and f i e l d  independence  on problem s o l v i n g b e h a v i o r s . In 1.  light  o f t h e s e aims, t h e f o l l o w i n g q u e s t i o n s were asked:  What p r o c e d u r e s from c o r e and what h e u r i s t i c s are u s e d by s t u d e n t s i n a t t a c k i n g and s o l v i n g m a t h e m a t i c a l word problems?  2.  Does t h e c o n t e x t o f t h e problem i n f l u e n c e h e u r i s t i c  usage?  3.  What e f f e c t does f i e l d - d e p e n d e n c e - i n d e p e n d e n c e have on t h e problem s o l v i n g p r o c e s s ?  4.  Do s e l e c t e d groups o f i n d i v i d u a l s e x h i b i t p a t t e r n s o f h e u r i s t i c usage when s o l v i n g m a t h e m a t i c a l word problems?  G e n e r a l Hypotheses •  Some o f t h e hypotheses t h a t were t e s t e d i n o r d e r t o h e l p t h e i n v e s t i g a t o r g a i n some i n s i g h t t o t h e answers f o r t h e above q u e s t i o n s a r e as f o l l o w s : HI:  Problem c o n t e x t w i l l not a f f e c t t h e t o t a l number o f h e u r i s t i c s used by a s t u d e n t .  H2:  Problem c o n t e x t w i l l not a f f e c t t h e number o f d i f f e r e n t  heuristics  used by a s t u d e n t . H3*  Problem c o n t e x t w i l l not a f f e c t t h e number o f c o r r e c t  solutions  o b t a i n e d by a s t u d e n t . H4:  F i e l d independence w i l l not a f f e c t t h e number o f t i m e s h e u r i s t i c s  2k  are H"5:  used by a s t u d e n t .  F i e l d independence w i l l n o t a f f e c t t h e number o f d i f f e r e n t  heuristics  used by a s t u d e n t . H6:  F i e l d independence w i l l n o t a f f e c t t h e number o f c o r r e c t  solutions  o b t a i n e d by a s t u d e n t . H7:  The number o f t i m e s h e u r i s t i c s a r e used by a s t u d e n t i s u n r e l a t e d to  H8:  The number o f d i f f e r e n t h e u r i s t i c s used by a s t u d e n t i s u n r e l a t e d t o the  H9:  t h e number o f c o r r e c t s o l u t i o n s he o b t a i n s .  number o f c o r r e c t s o l u t i o n s he o b t a i n s .  The sequence  o f h e u r i s t i c s used b y t h e s u b j e c t s w i l l n o t be a f f e c t e d  by problem c o n t e x t . H10:  The sequence o f h e u r i s t i c s u s e d by t h e s u b j e c t s w i l l n o t be a f f e c t e d by f i e l d  independence.  S i g n i f i c a n c e o f t h e Study Kilpatrick  (1969) s t a t e d i n a r e c e n t r e v i e w o f m a t h e m a t i c a l problem,  solving, ...the researcher...who chooses t o i n v e s t i g a t e problem s o l v i n g i n mathematics i s p r o b a b l y b e s t a d v i s e d t o undertake c l i n i c a l s t u d i e s o f i n d i v i d u a l s u b j e c t s . . . because o u r i g n o r a n c e i n t h i s a r e a demands c l i n i c a l s t u d i e s as p r e c u r s o r s t o l a r g e r e f f o r t s [ p . 179]• The s i g n i f i c a n c e o f t h i s s t u d y i s apparent when one c o n s i d e r s t h a t so  few attempts i n r e s e a r c h have been made t o a n a l y z e i n d i v i d u a l b e h a v i o r  d u r i n g t h e problem s o l v i n g p r o c e s s .  T h i s b e h a v i o r w i l l be a n a l y z e d i n  terms o f a problem s o l v i n g model d e v e l o p e d by MacPherson. i s known about t h e way t h e s e h e u r i s t i c s a r e employed  little  by way o f i n s i g h t i n t o an adequate t e a c h i n g methodology problem s o l v i n g  U n t i l more can be g a i n e d  t o enhance  skills.  T h i s s t u d y i s a l s o s i g n i f i c a n t i n t h a t i t makes use o f complex  25 problems, u s u a l l y not found i n t o d a y ' s t e x t b o o k s .  With a g r e a t e r emphasis  b e i n g p l a c e d on c h a l l e n g i n g problems i n t h e c u r r i c u l u m , t h e r e i s a need for  r e s e a r c h which makes use o f such problems.  (1967):  I n K i l p a t r i c k ' s words  .  The mathematics e d u c a t o r , i n p a r t i c u l a r , sometimes complains that, t h e kind, o f complex, c h a l l e n g i n g problems t h a t are t h e most d i f f i c u l t t o l e a r n how t o s o l v e r a r e l y appear i n t h e research l i t e r a t u r e . He q u e s t i o n s how much one can e x t r a p o l a t e from f i n d i n g s based on l e v e r p u l l i n g and c a r d s o r t i n g t o t h e p r o c e s s e s t h a t u n d e r l i e t h e s e a r c h f o r an e l e g a n t geometric p r o o f or t h e p r o d u c t i o n o f an e q u a t i o n describing a physical situation. The a n a l o g i e s may be d i r e c t o r t h e y may be e x c e e d i n g l y s u b t l e and c o m p l i c a t e d . We have no way o f knowing because complex problems have so seldom been used i n r e s e a r c h [ p . l j . F i n a l l y , t h e most important of  c o n t r i b u t i o n o f t h i s s t u d y may  adding t o t h e c u r r e n t s e l e c t i o n ,  models o f mathematics problem  as s m a l l as i t may  solving.  of theoretical  T h i s model can be used as a  framework f o r e x t e n d i n g r e s e a r c h i n m a t h e m a t i c a l perhaps  be,  problem  solving  F i v e b a s i c assumptions  Limitations  were made i n b e g i n n i n g t h i s s t u d y .  assumed t h a t t h e h e u r i s t i c s from MacPherson's model c o u l d be and used t o c l a s s i f y problem  s o l v i n g procedures.  I t was  It  was  observed  assumed t h e  problems used i n t h i s s t u d y would e l i c i t t h e use o f h e u r i s t i c s . was  and  as a mode f o r t e a c h i n g mathematics.  Assumptions and  it  be t h a t  Also,  assumed t h a t t h e s t u d e n t s would be w i l l i n g t o t r y t o s o l v e t h e  f i v e problems a s s i g n e d t o them, and t h a t t h e y would v e r b a l i z e thoughts.  I t was  a l s o assumed t h a t s t u d e n t s i n grade  their  e l e v e n were mature  enough, b o t h e m o t i o n a l l y and m a t h e m a t i c a l l y , t o p a r t i c i p a t e i n t h i s clinical  study.  D u r i n g the p i l o t t o be t e n a b l e .  The  s t u d y t h e s e assumptions  were examined and  found  s t u d e n t s i n t h e p i l o t s t u d y were w i l l i n g t o work on  all  a s s i g n e d problems  c o d i n g system i t was  and t o v e r b a l i z e t h e i r t h o u g h t s .  In modifying the  found t h a t t h e h e u r i s t i c s c o u l d be o b s e r v e d and  used t o c l a s s i f y problem s o l v i n g p r o c e d u r e s . The l i m i t a t i o n s o f t h i s s t u d y a r e a t t r i b u t a b l e t o : (a)  The problems used.  The problems were not s e l e c t e d from a s p e c i f i c  c a t e g o r y , such as a l g e b r a o r geometry, i n the p i l o t  b u t were s e l e c t e d from t h o s e used  study u s i n g t h e f o l l o w i n g f o u r  criteria:  1.  They were problems not u s u a l l y found i n t h e s c h o o l  2.  They c o u l d be s o l v e d by about h a l f t h e s t u d e n t s  3.  They c o u l d be s o l v e d i n a number o f d i f f e r e n t ways  4.  They e l i c i t e d t h e use o f h e u r i s t i c s  curriculum  A s e t o f d i f f e r e n t problems c o u l d have produced d i f f e r e n t d a t a . (b)  The sample  o f s u b j e c t s used.  The s u b j e c t s were s e l e c t e d from t h e  middle IQ range o f academic grade e l e v e n mathematics  students.  are based on a r e l a t i v e l y homogeneous group o f s t u d e n t s .  The d a t a  I t i s quite  p o s s i b l e t h a t , i f t h e s t u d y had i n c l u d e d a g r e a t e r v a r i e t y o f i n d i v i d u a l s , a g r e a t e r v a r i e t y o f h e u r i s t i c s might have been o b s e r v e d . (c)  The method o f d a t a c o l l e c t i o n .  Having t h e s u b j e c t s t h i n k  as t h e y s o l v e m a t h e m a t i c a l problems may  cause them t o commit  t h e y n o r m a l l y would n o t ; i n f a c t t h e y may  aloud  errors  s o l v e t h e problem i n a d i f f e r e n t  manner when asked t o v e r b a l i z e t h a n when l e f t  alone t o s o l v e  it.  Chapter I I  REVIEW OF RELATED LITERATURE  T h i s c h a p t e r w i l l examine some o f t h e l i t e r a t u r e r e l e v a n t t o t h e study.  R a t h e r t h a n attempt a comprehensive  s o l v i n g r e s e a r c h (see K l e i n m u n t z ,  s u r v e y o f r e c e n t problem  1966; K i l p a t r i c k , 1969; Jerman,  t h e d i s c u s s i o n w i l l be c o n f i n e d t o two r e l e v a n t t o p i c s . concerns t h e problem s o l v i n g p r o c e s s .  The  197l),  first  A s e r i e s o f models o f t h e problem  s o l v i n g p r o c e s s w i l l be d i s c u s s e d w i t h t h e r e l a t e d r e s e a r c h .  The  second  t o p i c d e a l s w i t h t h e c o n s i s t e n t o r i n c o n s i s t e n t problem s o l v i n g b e h a v i o r s e x h i b i t e d by s u b j e c t s .  Recent r e s e a r c h i n which t h e f i e l d  independence  c o n s t r u c t i s used as an a p t i t u d e v a r i a b l e i n a problem s o l v i n g s e t t i n g be  will  cited.  Models o f Problem  Solving  Over t h e y e a r s , s c h o l a r s have sought t o shed l i g h t upon t h e problem s o l v i n g p r o c e s s by s p e c i f y i n g t h e sequence o f b e h a v i o r s t h r o u g h which might p r o c e e d i n s o l v i n g a problem ( D a v i s , 1973, i s g i v e n by Johnson  p. 15)«  One  example  (1944) i n which he i d e n t i f i e s t h r e e p r o c e s s e s o r  groups o f p r o c e s s e s which, he says r e g u l a r l y o c c u r d u r i n g problem (l)  o r i e n t a t i o n t o t h e problem,  (3)  judging. Dewey  (2)  producing relevant material,  (1933) g i v e s a s l i g h t l y d i f f e r e n t a n a l y s i s .  r e f l e c t i v e thinking, there i s l i t t l e solving.  one  solving: and  As he d e f i n e s  d i f f e r e n c e between i t and problem  Hence, h i s a n a l y s i s o f r e f l e c t i v e t h i n k i n g can be t a k e n as an  a n a l y s i s o f t h e p r o c e s s o f problem s o l v i n g (Henderson and P i n g r y ,  1953)*  28  (1933, pp. 107-116) o u t l i n e s f i v e phases o f r e f l e c t i v e t h i n k i n g :  Dewey  1.  "A f e l t  difficulty",  a q u e s t i o n f o r which t h e answer must be  sought 2.  L o c a t i o n and d e f i n i t i o n o f t h e d i f f i c u l t y  3.  "The i d e n t i f i c a t i o n o f v a r i o u s h y p o t h e s e s . . . t o  i n i t i a t e and  guide o b s e r v a t i o n and o t h e r o p e r a t i o n s i n c o l l e c t i o n o f f a c t u a l material [p. 4.  107]."  E l a b o r a t i o n o f each h y p o t h e s i s by r e a s o n i n g and t h e t e s t i n g of the hypothesis  5«  A c t i n g on t h e b a s i s o f t h e p a r t i c u l a r h y p o t h e s i s s e l e c t e d i n s t e p f o u r t o see whether t h e r e s u l t s t h e o r e t i c a l l y actually  occur.  Dewey n o t e s t h e s e f i v e phases o r s t a g e s o f thought one  another  indicated  i n a set order.  do n o t f o l l o w  Each s t e p i n t h i n k i n g does something t o t h e  f o r m a t i o n o f a s u g g e s t i o n t o change i t i n t o an i d e a o r h y p o t h e s i s . Each s t e p a l s o does something t o promote t h e l o c a t i o n and d e f i n i t i o n o f t h e problem.  Each improvement i n t h e i d e a l e a d s t o new o b s e r v a t i o n s  t h a t y i e l d new f a c t s t o h e l p judge t h e r e l e v a n c e o f f a c t s a l r e a d y a t hand.  The e l a b o r a t i o n o f an h y p o t h e s i s does n o t w a i t u n t i l t h e problem  has been d e f i n e d , b u t i t may come a t any i n t e r m e d i a t e t i m e .  As w e l l ,  any e v a l u a t i o n need n o t be t h e f i n a l s t e p i n t h e p r o c e s s . I t may be i n t r o d u c t o r y t o new o b s e r v a t i o n s o r s u g g e s t i o n s , what happens as a consequence o f i t (Dewey, 1933, Other examples o f t h e suggested Burt  (1928)  according t o  p. H 5 ) .  " s t e p s " i n problem s o l v i n g f o l l o w :  Gray (1935)  1.  Occurrence  2.  C l a r i f i c a t i o n of the perplexity  of a perplexity  1.  S e n s i t i v i t y t o t h e problem  2.  Knowledge o f t h e problem conditions, recognition of s i g n i f i c a n t information  3.  Appearance o f suggested solutions  3.  Suggested s o l u t i o n o r hypothesis  4«  Deducing i m p l i c a t i o n s o f suggested s o l u t i o n s  4.  S u b j e c t i v e e v a l u a t i o n , does the proposed s o l u t i o n work?  5.  V e r i f y i n g a c t i o n o r observation.  5.  Objective  6.  Conclusion  Humphrey (1948)  Burock  Directed thinking  involves:  test or generalization.  (1950)  1.  Clear formulation lem  2.  Preliminary survey of a l l concepts o f t h e m a t e r i a l  of the prob-  1.  A problem s i t u a t i o n  2.  Motivating  3.  T r i a l and e r r o r  3.  A n a l y s i s i n t o major v a r i a b l e s  4.  Use o f a s s o c i a t i o n and images  4.  L o c a t i n g a c r u c i a l aspect o f t h e problem  5.  A f l a s h o f i n s i g h t (the p l a c e o f 3, 4» snd 5 v a r i e s w i t h t h e problem)  5.  A p p l i c a t i o n of past  6.  Varied  7.  Control  6.  factors  Some a p p l i c a t i o n s i n action .  Bloom and B r a d e r  (1950)  "problem s o l v i n g characteristics" are: 1.  Understanding o f the n a t u r e o f t h e problem  2.  Understanding o f the ideas contained i n t h e problem  experience  trials  8.  E l i m i n a t i o n o f sources o f error  9.  Visualization.  Kingsley  and G a r r y (1957)  1.  A difficulty i s felt  2.  The problem i s c l a r i f i e d and d e f i n e d  3.  A s e a r c h f o r c l u e s i s made  4.  V a r i o u s s u g g e s t i o n s appear and a r e t r i e d o u t  3.  G e n e r a l approach t o t h e s o l u t i o n o f t h e problem  5.  A suggested s o l u t i o n i s accepted  4.  A t t i t u d e towards t h e s o l u t i o n o f t h e problem.  6.  The s o l u t i o n i s t e s t e d .  Both B r o w n e l l (1942, p. 432) and K i l p a t r i c k (1967, against  p . 20)  caution  t h e tendency t o misuse c o n c e p t u a l frameworks such as t h o s e above  30 by assuming t h a t i n r e a l i t y , sequential stages. of formal  I n the  problem s o l v i n g o c c u r s i n w e l l  1930's s t u d e n t s were t a u g h t t o use  a n a l y s i s t o s o l v e a r i t h m e t i c problems.  t o ask themselves a sequence o f q u e s t i o n s , (2)  given?  questions  (3)  What i s t o be found?  i s a close estimate of the were t o be  any  B r o w n e l l (1942) i n d i c a t e s t h a t f o r m a l p a t t e r n o f t h i n k i n g which may  o r may  the  What i s t h e (4)  What i s t o be done? p. 47]."  What  These  c o m p u t a t i o n a l work was analysis, "represents not  technique  S t u d e n t s were encouraged  such as " ( l )  answer? [Burch, 1953,  asked b e f o r e  on the p a r t o f a d u l t s , but  defined  done. a logical  characterize expert  thinking  c e r t a i n l y has not y e t been shown t o  characterize  good t h i n k i n g on t h e p a r t o f c h i l d r e n [ p . 432]." Formal a n a l y s i s has f o r the  student.  To  been shown t o be i n e f f e c t i v e and  t r a i n e d t h r o u g h t h e use He  a n a l y s i s , B u r c h (1953),  study the e f f e c t of formal  conducted a s t u d y u s i n g 305  elementary s c h o o l c h i l d r e n who  of formal  require formal  a n a l y s i s t h a n on one  t o determine whether s t u d e n t s used f o r m a l t o do  so, Burch s e l e c t e d 51  t h i n k aloud  as t h e y s o l v e d  scores  which d i d .  I n an  sample and  n e v e r used f o r m a l  instances.  required  Out  of  interviews,  F u r t h e r m o r e , when s t u d e n t s  each o f them i n d i c a t e d t h a t  a n a l y s i s except when r e q u i r e d t o do  so.  t h a t when t h e y t r i e d t o use  formal  noted t h a t "one  o f the inadequacy o f f o r m a l  explanation  attempt  asked them t o  a s e r i e s of a r i t h m e t i c problems.  were q u e s t i o n e d f o l l o w i n g the i n t e r v i e w ,  problems.  a n a l y s i s even i f n o t  s t u d e n t s from t h e  s t e p s were used i n o n l y two  been  on t h e t e s t w h i c h  a p p r o x i m a t e l y f i v e hundred problems attempted d u r i n g t h e formal  had  analysis to solve arithmetic  found t h a t s t u d e n t s g e n e r a l l y a t t a i n e d h i g h e r  d i d not  troublesome  Many  reported  a n a l y s i s , t h e y became c o n f u s e d .  t h a t i t fragments t h e problem i n t o i s o l a t e d p a r t s  [p.  a n a l y s i s may 47]."  he  Burch be  One  o f t h e b e s t known models o f problem s o l v i n g was  W a l l a s (1926).  W a l l a s ' s t a g e s o f problem s o l v i n g a r e :  c l a r i f y i n g and d e f i n i n g t h e problem, activity,  (3)  inspiration,  checking the s o l u t i o n . definition, (1945,  1962,  (2)  (l)  Preparation,  i n c u b a t i o n , unconscious mental  s o l u t i o n appears suddenly, and  (4)  verification,  The s t a g e s o f i n c u b a t i o n and i n s p i r a t i o n ,  are u n o b s e r v a b l e m e n t a l p r o c e s s e s . 1965)  g i v e n by-  by  However, b o t h P o l y a  and Hadamard (1945) r e a d i l y acknowledge  the unconscious,  p r e c o n s c i o u s o r sometimes " f r i n g e - c o n s c i o u s " a c t i v i t y l e a d i n g t o t h e s o l u t i o n o f a d i f f i c u l t problem. model may  Green  (1966) observed t h a t t h e W a l l a s  c h a r a c t e r i z e t h e r e s e a r c h e f f o r t s o f s c i e n t i s t s working on  d i f f i c u l t problems, but i t f a i l s t o d e s c r i b e t h e h i g h s c h o o l s t u d e n t t r y i n g t o s o l v e an a l g e b r a problem. W i c k e l g r e n ' s (1974) model o f m a t h e m a t i c a l problem s o l v i n g has i n f l u e n c e d by c u r r e n t work i n a r t i f i c i a l i n t e l l i g e n c e and  been  computer  s i m u l a t i o n o f human problem s o l v i n g done by N e w e l l and Simon.  The  p r i n c i p a l aim o f t h e W i c k e l g r e n model i s t o p r e s e n t t h e e l e m e n t a r y p r i n c i p l e s n e c e s s a r y t o s o l v e m a t h e m a t i c a l problems o f e i t h e r t h e " t o f i n d " o r t h e " t o p r o v e " c h a r a c t e r , but not problems o f d e f i n i n g " m a t h e m a t i c a l l y i n t e r e s t i n g " axiom systems  ( W i c k e l g r e n , 1974,  P»  2).  The p r o c e d u r e s i n W i c k e l g r e n ' s model i n c l u d e i n f e r e n c e , t r i a l  and  e r r o r , s t a t e e v a l u a t i o n , s u b g o a l , c o n t r a d i c t i o n , working backwards, r e c a l l o f r e l a t e d problems.  and  These p r o c e d u r e s are t o be used by t h e  problem s o l v e r o n l y i f he can't s o l v e t h e problem by t h e d i r e c t use o f algorithms.  A d e s c r i p t i o n o f W i c k e l g r e n ' s model f o l l o w s :  Inference: To draw i n f e r e n c e s from e x p l i c i t l y and i m p l i c i t l y p r e s e n t e d i n f o r m a t i o n t h a t s a t i s f y one o r b o t h o f the f o l l o w i n g two c r i t e r i a : (a) t h e i n f e r e n c e s have f r e q u e n t l y been made i n t h e p a s t from t h e same t y p e o f i n f o r m a t i o n ; (b) t h e i n f e r e n c e s are concerned w i t h p r o p e r t i e s ( v a r i a b l e s , terms, e x p r e s s i o n s , and so on) t h a t appear i n the g o a l , t h e g i v e n ,  32 o r i n f e r e n c e s from t h e g o a l and t h e g i v e n s Drawing i n f e r e n c e s i s t h e f i r s t in  attempting  t o s o l v e a problem.  [p. 23].  problem s o l v i n g p r o c e d u r e employed  The g o a l o r t h e g i v e n s  are e s s e n t i a l l y  expanded by b r i n g i n g t o b e a r a l l o f t h e knowledge t h e problem s o l v e r has  concerning  t h e problem.  Random T r i a l and E r r o r : t o a p p l y t h e a l l o w a b l e t h e g i v e n i n a random f a s h i o n [ p . 46]•  operations t o  S y s t e m a t i c T r i a l and E r r o r : t o produce a m u t u a l l y e x c l u s i v e and e x h a u s t i v e l i s t i n g o f a l l sequences o f a c t i o n s up t o some maximum l e n g t h [ p . 47 ! • C l a s s i f i c a t o r y T r i a l and E r r o r : t o o r g a n i z e sequences o f actions i n t o classes that are equivalent with respect t o t h e s o l u t i o n o f t h e problem [ p . 473» I n t h i s case i f one sequence o f a c t i o n s w i t h i n a c l a s s w i l l (not s o l v e ) t h e problem, t h e n a l l t h e o t h e r t h e same c l a s s w i l l p r o b a b l y  also solve  solve  sequences o f a c t i o n s w i t h i n  (not s o l v e ) t h e problem.  S t a t e E v a l u a t i o n and H i l l C l i m b i n g : T h i s method has two parts: (a) d e f i n i n g an ' e v a l u a t i o n f u n c t i o n ' o v e r a l l states (the set o f a l l the expressions that e x i s t i n the w o r l d o f t h e problem a t a g i v e n t i m e ) i n c l u d i n g t h e g o a l s t a t e and (b) c h o o s i n g a c t i o n s a t any g i v e n s t a t e t o achieve a next s t a t e w i t h an e v a l u a t i o n c l o s e r t o t h a t o f t h e g o a l . P i c k i n g an a c t i o n on t h e b a s i s o f such a l o c a l e v a l u a t i o n o f i t s consequences i s known as ' h i l l climbing' [p. 67]. An example o f s t a t e e v a l u a t i o n and h i l l  climbing i s given i n the  f o l l o w i n g example: You have a p i l e o f 24 c o i n s . Twenty-three o f t h e s e c o i n s have t h e same weight, and one i s h e a v i e r . Your t a s k i s t o determine which c o i n i s h e a v i e r and t o do so i n t h e minimum number o f w e i g h i n g s . You a r e g i v e n a beam b a l a n c e ( s c a l e ) , which w i l l compare t h e weight o f any two s e t s o f c o i n s out o f t h e t o t a l s e t o f 24 c o i n s . A s u i t a b l e e v a l u a t i o n f u n c t i o n f o r s o l v i n g t h i s problem would be t h e number o f c o i n s whose c l a s s i f i c a t i o n as heavy o r l i g h t i s known. A t t h e b e g i n n i n g o f t h e p r o b lem, t h e v a l u e o f t h e f u n c t i o n i s zero, s i n c e none o f t h e 24 c o i n s i s known t o be e i t h e r heavy o r l i g h t . I n the g o a l s t a t e , t h e h e a v y - l i g h t c l a s s i f i c a t i o n o f a l l 24 c o i n s i s known, so t h e v a l u e o f t h e f u n c t i o n i s 24.  Thus, a h i l l c l i m b i n g approach would choose an a c t i o n a t each node t h a t m a x i m i z e d t h e number o f c o i n s whose h e a v y - l i g h t c l a s s i f i c a t i o n i s k n o w n [ p . 7l]« Subgoal: t o analyze a problem i n t o subproblems o r t o break i t i n t o p a r t s [ p . 9l]« The  purpose o f t h i s procedure i s t o r e p l a c e  p r o b l e m w i t h two o r more s i m p l e r Contradiction; from t h e givens The  a single  difficult  problems.  P r o v i n g t h e g o a l c o u l d n o t p o s s i b l y be [ p . 109].  obtained  method o f c o n t r a d i c t i o n c a n be a p p l i e d i n t h e f o l l o w i n g f o u r  ways: Indirect Proof: t o assume t h e c o n t r a r y i s t r u e a n d s h o w t h a t t h e contrary statement i n combination w i t h t h e givens, results i n a contradiction [p.Ill], M u l t i p l e Choice - S m a l l Search Space: i n problems i n v o l ving a small set of a l t e r n a t i v e goals, t o systematically a p p l y t h e method o f c o n t r a d i c t i o n t o e v e r y a l t e r n a t i v e g o a l [ p . 115]. • C l a s s i f i c a t o r y C o n t r a d i c t i o n - Large Search Space: t o d e v i s e an e f f e c t i v e s e a r c h p r o c e d u r e t h a t c o n t r a d i c t s large classes of a l t e r n a t i v e goals simultaneously [p. 126]. C l a s s i f i c a t o r y C o n t r a d i c t i o n - I n f i n i t e Search Space: t o d e v i s e an e f f e c t i v e s e a r c h p r o c e d u r e t h a t c o n t r a d i c t s i n f i n i t e l y l a r g e c l a s s e s o n t h e b a s i s o f some common p r o p e r t y [ p . 133]. Working Backwards: t o guess a p r e c e d i n g statement o r statements t h a t , taken together, would imply t h e goal statement [ p . 138]. Wickelgren  describes  four fundamental types  ofrelationships  between problems t h a t c a n be used b y t h e problem s o l v e r . Equivalent Problems: t o recognize t h a t problems d i f f e r only w i t h r e s p e c t t o t h e names a t t a c h e d t o d i f f e r e n t e l e m e n t s , b u t w h o s e r e l a t i o n s a n d o p e r a t i o n s a r e i d e n t i c a l [ p . 156]. S i m i l a r Problems: t o recognize that two problems share common e l e m e n t s , a n d t h e n t o r e c a l l t h e m e t h o d s u s e d t o s o l v e t h e s i m i l a r p r o b l e m [ p . 153]. Simpler Problems: t o pose and s o l v e o r r e c a l l a problem w h i c h i s s i m p l e r o r a s p e c i a l c a s e o f t h e more c o m p l e x p r o b l e m [ p . 157],  34 More Complex Problems t P o s i n g a problem t h a t i s more complex t h a n t h e g i v e n problem and i n which t h e g i v e n problem i s embedded. Then s o l v e t h e more complex problem [ p . 166]. In  MacPherson's  (1973) terms, W i c k e l g r e n ' s model c o n s i s t s o f  p r o c e d u r e s from b o t h c o r e and h e u r i s t i c s . of  random t r i a l  problems  and e r r o r , s u b g o a l s , working backwards,  and examples  and complex  c o r r e s p o n d t o t h e h e u r i s t i c s o f random c a s e s , a n a l y s i s ,  d e d u c t i o n , and e x t e n s i o n , r e s p e c t i v e l y . problems, trial  His definitions  s t a t e e v a l u a t i o n and h i l l  inverse  Rather than procedures f o r s o l v i n g  climbing,  s y s t e m a t i c and c l a s s i f i c a t o r y  and e r r o r , and t h e methods o f c o n t r a d i c t i o n appear t o be o v e r a l l  methods o r p l a n s o f a t t a c k i n g p a r t i c u l a r t y p e s o f problems. T h i s model may have some i m p l i c a t i o n s f o r t e a c h i n g problem but  as a framework f o r c l a s s i f y i n g d a t a i t i s q u i t e r e s t r i c t e d .  of  t h e p r o c e d u r e s , such as s u b g o a l s and w o r k i n g backwards,  in  s o l v i n g many k i n d s o f problems  scheme. to  solving Some  are u s e f u l  and s h o u l d be i n c l u d e d i n a c l a s s i f i c a t o r y  However, many o f W i c k e l g r e n ' s p r o c e d u r e s appear t o be t o o s p e c i f i c  be used t o c h a r a c t e r i z e problem s o l v i n g i n any g e n e r a l s e n s e .  framework f o r c l a s s i f y i n g d a t a s h o u l d be f l e x i b l e  A  enough t o accommodate  b r o a d s t r a t e g i e s o f problem s o l v i n g as w e l l as n a r r o w e r t a c t i c s u s e d t o r e a l i z e these strategies Schwieger  (Kilpatrick,  1967).  (1974) d e s c r i b e s a model o f m a t h e m a t i c a l problem s o l v i n g  b a s e d on t h e i d e n t i f i c a t i o n and d e s c r i p t i o n o f e i g h t b a s i c g e n e r a t e d from t h e l i t e r a t u r e .  components  These b a s i c components a r e :  Classify: t o r e c o g n i z e p e r t i n e n t c h a r a c t e r i s t i c s and a t t r i b u t e s o f m a t h e m a t i c a l problems o r e x p r e s s i o n s and t o s p e c i f y t h e c l a s s o r c l a s s e s t o which t h e y b e l o n g [ p . 3 8 ] • Deduce: t o r e l a t e a s e t o f statements so t h a t acceptance of t h e statements and t h e i r i n t e r r e l a t i o n s h i p s d i c t a t e s ' a p a r t i c u l a r c o n c l u s i o n [ p . 4l]« E s t i m a t e : t o use a v a i l a b l e m a t h e m a t i c a l i n f o r m a t i o n t o make a judgement o f measurement o r o f a r e s u l t o f c a l c u l a t i o n [p. 44]•  35 Generate P a t t e r n : t o put known o r a v a i l a b l e mathematical d a t a i n t o a s y s t e m a t i c arrangement [p. 47]• Hypothesize: to recognize or t o generate c o n d i t i o n a l r e l a t i o n s h i p s between m a t h e m a t i c a l statements [ p . 50]• Translate: t o s u b s t i t u t e f o r one m a t h e m a t i c a l form, an equivalent r e p r e s e n t a t i o n [p. 53]. T r i a l and E r r o r : t o a p p l y knowledge t o a mathematical problem i n an u n o r g a n i z e d manner [ p . 5 6 ] . Verify: t o apply data t o a hypothesis i n t e s t i n g i t s v a l i d i t y [ p . 59], Schwieger a l s o i d e n t i f i e d t h e f o l l o w i n g h i e r a r c h y among t h e b a s i c components ( p . 80).. A s o l i d arrow i n d i c a t e s t h a t t h e a b i l i t y a t t h e t a i l o f t h e arrow i s a p r e r e q u i s i t e t o t h e a b i l i t y at t h e head o f t h e arrow.  The dashed arrows i n d i c a t e some o f t h e more common t a s k - s p e c i f i c _^-_—^»' Classify  T r i a l of E r r o r  Estimate  \  Translate  Generate P a t t e r n s  x * .  Hypothesize  A.  X N  X  Deduce Verify*" " prerequisites, of  patterns.").  (i.e.  " a b i l i t y t o h y p o t h e s i z e may  A problem s o l v e r may  depend on p r i o r  generation  r e t u r n t o a co-nponent h i g h e r  on t h e diagram a t any t i m e . Schwieger to  (pp. 36-39) c l a i m s "These e i g h t components are c o n s i d e r e d  be b a s i c t o any t h i n k i n g i n m a t h e m a t i c a l problem s o l v i n g ,  and any  m a t h e m a t i c a l problem s o l v i n g can be e x p l a i n e d i n terms o f them.", and t h a t , i n f a c t , t h e s e b a s i c components are independent o f each o t h e r . However, t h e r e i s v e r y l i t t l e  e v i d e n c e t o s u p p o r t t h i s c l a i m o f independence.  Schwieger r e p o r t e d t h e a n a l y s i s o f o n l y two problems coded u s i n g t h i s model.  36 If  t h e model i s t o be used t o a n a l y z e  problem s o l v i n g , some o f t h e  components appear t o be n o t d e f i n e d c l e a r l y enough f o r r e l i a b l e  coding.  T h i s i s e s p e c i a l l y t r u e f o r t h e components o f deduce, h y p o t h e s i z e ,  and  estimate. The  c u r r e n t i n t e r e s t i n t h e use o f " h e u r i s t i c s " i n m a t h e m a t i c a l  problem s o l v i n g i s due p r i n c i p a l l y t o P o l y a  (l957t 1962, 1965).  Polya's  model i n c l u d e s a v a r i e t y o f p r o c e d u r e s , b o t h g e n e r a l  and s p e c i f i c , f o r  s o l v i n g m a t h e m a t i c a l problems i n a number o f c o n t e n t  areas.  of Polya's himself  model i s p r e s e n t e d  i n terms o f a l i s t  as he t r i e s t o s o l v e a problem.  t o mental a c t i o n .  Polya's  list  Polya's  one asks  He p o s t u l a t e s t h a t t h e s e  includes different  geared t o d e f i n i n g and a p p r o a c h i n g d i f f i c u l t tasks.  o f questions  A portion  correspond  forms o f q u e s t i o n i n g  and u n f a m i l i a r m a t h e m a t i c a l  (1957) m a t h e m a t i c a l c h e c k l i s t i n c l u d e s : U n d e r s t a n d i n g t h e Problem  What i s t h e unknown? What a r e t h e d a t a ? What i s t h e c o n d i t i o n ? Is i t possible t o s a t i s f y the condition? I s the condition s u f f i c i e n t t o determine t h e unknown? Or i s i t i n s u f f i c i e n t ? Or redundant? Or c o n t r a d i c t o r y ? Draw a f i g u r e . I n t r o d u c e s u i t a b l e n o t a t i o n . Separate the v a r i o u s p a r t s o f t h e c o n d i t i o n . Can you w r i t e them down? Devising a Plan Have y o u seen i t b e f o r e ? Or have you seen t h e same problem i n a s l i g h t l y d i f f e r e n t form? Do y o u know a r e l a t e d problem? Do you know a theorem t h a t c o u l d be u s e f u l ? Look at t h e unknown! And t r y t o t h i n k o f a f a m i l i a r problem h a v i n g t h e same o r a s i m i l a r unknown. Here i s a problem r e l a t e d t o y o u r s and s o l v e d b e f o r e . C o u l d you use i t ? C o u l d you use i t s r e s u l t ? C o u l d y o u u s e i t s method? S h o u l d you i n t r o d u c e some a u x i l i a r y element i n o r d e r t o make i t s use p o s s i b l e ? C o u l d you r e s t a t e t h e problem? C o u l d you r e s t a t e i t s t i l l differently? Go back t o d e f i n i t i o n s . I f you cannot s o l v e t h e proposed problem t r y t o s o l v e f i r s t some r e l a t e d problem. C o u l d you imagine a more a c c e s s i b l e r e l a t e d problem? A more g e n e r a l problem? A more s p e c i a l problem? An analogous problem? C o u l d you s o l v e a p a r t o f t h e problem? Keep o n l y a p a r t o f t h e c o n d i t i o n , drop t h e o t h e r p a r t ; how f a r i s t h e unknown t h e n determined, how can i t v a r y ? C o u l d you  37 d e r i v e something u s e f u l from t h e d a t a ? C o u l d you t h i n k o f o t h e r d a t a a p p r o p r i a t e t o determine t h e unknown? C o u l d you change t h e unknown o r t h e d a t a , o r b o t h i f n e c e s s a r y , so t h a t t h e new unknown and t h e new d a t a are n e a r e r t o each o t h e r ? D i d you use a l l the data? D i d you use t h e whole c o n d i t i o n ? Have you t a k e n i n t o account a l l e s s e n t i a l n o t i o n s i n v o l v e d i n t h e problem? C a r r y i n g Out t h e P l a n C a r r y i n g out your p l a n o f t h e s o l u t i o n , check each s t e p . Can you see c l e a r l y t h a t t h e s t e p i s c o r r e c t ? Can you prove t h a t i t i s correct? L o o k i n g Back Can you check t h e r e s u l t ? Can you check t h e argument? Can you d e r i v e the r e s u l t d i f f e r e n t l y ? Can you see i t a t a g l a n c e ? Can you use t h e r e s u l t , o r t h e method, f o r some o t h e r problem [ p . xvi, x v i i ] . The model c o n s i s t s o f f o u r phases,  but i t i s not t o be i m p l i e d t h a t  i n r e a l i t y t h e s e phases always o c c u r i n sequence, o r t h a t a g i v e n  problem  s o l v e r w i l l e x h i b i t b e h a v i o r c h a r a c t e r i z e d by e v e r y phase d u r i n g t h e s o l u t i o n o f a problem. T h i s model i s expanded, ( P o l y a , 1957) H e u r i s t i c s " and i n two  i n a "Short D i c t i o n a r y of  o f P o l y a * s o t h e r books ( P o l y a , 1962,  The model i n c l u d e s p r o c e d u r e s from drawing Most o f t h e s e are e i t h e r expansions  1965).  figures to generalization.  o f t h e q u e s t i o n s from t h e l i s t  are e x p l a i n e d i n terms o f combinations  o f q u e s t i o n s from the  list.  These p r o c e d u r e s i n c l u d e g e n e r a l i z a t i o n , v a r i a t i o n o f t h e problem, working  or  and  backwards.  Difficulties  i n a t t e m p t i n g t o use P o l y a ' s c h e c k l i s t t o a n a l y z e t h e  p r o t o c o l s o f f i f t y - s i x e i g h t h grade were i n d i c a t e d by K i l p a t r i c k  s t u d e n t s o f above average  (1967) who  ability  stated:  Attempts t o a p p l y t h e c h e c k l i s t t o s e v e r a l p r o t o c o l s from t h e p i l o t s t u d y demonstrated c l e a r l y t h a t whatever m e r i t s P o l y a ' s l i s t has f o r t e a c h i n g problem s o l v i n g , i t i s o f l i m i t e d u s e f u l n e s s , as i t s t a n d s , f o r c h a r a c t e r i z i n g t h e b e h a v i o r o f t h e s e s u b j e c t s . Many o f t h e c a t e g o r i e s were u n o c c u p i e d : s u b j e c t s seemi n g l y d i d not e x h i b i t b e h a v i o r even r e m o t e l y r e s e m b l i n g a c t i o n s suggested by the h e u r i s t i c q u e s t i o n s . F o r example, no s u b j e c t  38 asked themselves a l o u d whether t h e y were u s i n g a l l o f t h e e s s e n t i a l n o t i o n s o f t h e problem. Furthermore, the c a t e g o r i e s were, not d e f i n e d c l e a r l y enough f o r r e l i a b l e c o d i n g [ p . 44] • Kilpatrick system based in  (I967) used a m o d i f i e d c h e c k l i s t and  "Process-sequence"  on P o l y a ' s model t o a n a l y z e t h e p r o t o c o l s o f t h e s u b j e c t s  h i s study.  N o n - s e q u e n t i a l b e h a v i o r , such as "draw f i g u r e " ,  same o r r e l a t e d problem", "uses on t h e c h e c k l i s t t h e f i r s t were not r e c o r d e d .  The  s u c c e s s i v e approximation",  time i t was  observed,  s e q u e n t i a l b e h a v i o r was  Reading and t r y i n g t o u n d e r s t a n d  D  D e d u c t i o n from c o n d i t i o n  E  S e t t i n g up  T  T r i a l and  C  Checking  e t c . , was  and r e p e a t e d  checked  occurrences  translated into  symbols and r e c o r d e d i n t h e i r o r d e r o f o c c u r r e n c e . R  "recalls  coding  These p r o c e d u r e s  were:  t h e problem  equation error  solution  The t h r e e symbols D, E, and T were f o l l o w e d b y a number from 1 t o 5 t h a t r e p r e s e n t e d t h e outcome o f t h e p r o c e s s . for  an i n c o m p l e t e d e d u c t i o n .  can be found i n K i l p a t r i c k ' s  F o r example, DI  stands  A f u l l e x p l a n a t i o n o f t h i s c o d i n g system (1967, Appendix F)  dissertation.  When K i l p a t r i c k a p p l i e d t h i s c o d i n g system t o t h e s t u d e n t s ' t a p e r e c o r d e d p r o t o c o l s , o n l y a few were o b s e r v e d .  of t h e procedures  F o r i n s t a n c e , few  problem o r attempted  from P o l y a ' s model  students v a r i e d the c o n d i t i o n s o f the  t o d e r i v e a s o l u t i o n by another method.  t h e s u b j e c t s drew f i g u r e s w h i l e s o l v i n g t h e problems, but t h e w i t h which f i g u r e s were drawn was to  problem s o l v i n g . related  (P <  .05)  Checking,  (P <  frequency  u n r e l a t e d t o the o t h e r p r o c e d u r e s  s u c c e s s i n s o l v i n g t h e problems.  and s u c c e s s i v e a p p r o x i m a t i o n  Most o f  Both t r i a l  .01)  and e r r o r (P < . 0 5  or  )  were c o r r e l a t e d w i t h s u c c e s s f u l  as i d e n t i f i e d by the c o d i n g system, was  t o t h e number o f c o r r e c t  solutions.  also  K i l p a t r i c k a l s o compared t h e r e s u l t s o f t h e c o d i n g system f o r each s u b j e c t w i t h h i s performance on a b a t t e r y o f t e s t s from t h e N a t i o n a l L o n g i t u d i n a l Study o f M a t h e m a t i c a l t h a t s u b j e c t s who attempted  Abilities  (NLSMA) f i l e s .  He found  t o s e t up and use e q u a t i o n s were s i g n i f i c a n t l y  s u p e r i o r t o t h o s e s t u d e n t s who d i d n o t use e q u a t i o n s on measures o f mathematics achievement, g e n e r a l r e a s o n i n g , word f l u e n c y , q u a n t i t a t i v e ability,  and r e f l e c t i v e - i m p u l s i v e s t y l e .  use e q u a t i o n s , t h o s e t h a t used t r i a l s t u d e n t s who d i d n o t use t r i a l ability.  Of t h o s e s u b j e c t s who d i d n o t  and e r r o r were h i g h e r t h a n  those  and e r r o r i n achievement and q u a n t i t a t i v e  The s u b j e c t s who used t h e l e a s t t r i a l  and e r r o r and d i d n o t  use e q u a t i o n s had t h e most t r o u b l e w i t h word problems, spent t h e l e a s t amount o f time on them, and got t h e fewest number o f c o r r e c t  solutions.  U s i n g a m o d i f i e d v e r s i o n o f K i l p a t r i c k ' s c o d i n g system based on P o l y a ' s model, Webb (1975) a n a l y z e d t h e problem s o l v i n g a b i l i t y o f f o r t y second y e a r h i g h s c h o o l a l g e b r a s t u d e n t s .  The s t u d e n t s were  i n t e r v i e w e d i n d i v i d u a l l y and asked t o t h i n k a l o u d as t h e y s o l v e d e i g h t mathematical  problems.  l a t e r date.  Each student was g i v e n a t o t a l s c o r e from  each o f t h e problems. approach,  The p r o t o c o l s were r e c o r d e d and coded a t a  The s c o r e was based  p l a n , and r e s u l t .  t h e mathematical  zero t o f i v e f o r  on t h e sum o f s u b s c o r e s f o r  U s i n g r e g r e s s i o n a n a l y s i s , Webb found t h a t  achievement component accounted  f o r 50/£ o f t h e v a r i a n c e  i n t h e t o t a l s c o r e and t h e h e u r i s t i c s t r a t e g y components accounted  for  an a d d i t i o n a l 13% o f t h e v a r i a n c e . Kantowski (1975) a n a l y z e d t h e problem s o l v i n g a b i l i t y o f e i g h t above average  a b i l i t y n i n t h grade a l g e b r a s t u d e n t s as t h e y l e a r n e d t o  s o l v e problems i n geometry.  Her study was comprised  of four  phases:  a p r e t e s t , a r e a d i n e s s i n s t r u c t i o n phase i n t h e use o f s e l e c t e d of Polya's,  heuristics  an i n s t r u c t i o n i n geometry phase u s i n g t e a c h i n g s t r a t e g i e s  40 based on t h e same h e u r i s t i c s ,  and a p o s t t e s t .  D u r i n g each o f t h e  phases,  t h e s u b j e c t s were asked t o t h i n k a l o u d as t h e y s o l v e d problems and  their  p r o t o c o l s were r e c o r d e d and t h e n a n a l y z e d u s i n g a m o d i f i e d v e r s i o n o f K i l p a t r i c k * s c o d i n g scheme.  A s c o r e was  a s s i g n e d t o each problem  on t h e p r o c e d u r e s used by the s u b j e c t as w e l l as t h e Kantowski's  s t u d y was  based  solution.  a c l i n i c a l e x p l o r a t o r y study to  determine  t h e p r o c e d u r e s u s e d by s t u d e n t s as t h e y l e a r n e d t o s o l v e problems i n geometry.  The  o b j e c t i v e o f t h e s t u d y was  generate hypotheses conclusions.  t o seek r e g u l a r i t i e s t h a t would  f o r f u r t h e r e x p e r i m e n t a l s t u d y and not t o s t a t e  Kantowski found t h a t t h e i n c r e a s e d use o f h e u r i s t i c s  and t h e development o f problem  s o l v i n g a b i l i t y were p o s i t i v e l y  correlated.  The use o f " l o o k i n g back" s t r a t e g i e s d i d not i n c r e a s e as problem  solving  a b i l i t y d e v e l o p e d n o r d i d i t appear t o be r e l a t e d t o s u c c e s s i n problem solving.  Kantowski n o t e d t h e l e v e l o f r i g o r r e q u i r e d i n t h e use o f t h e s e  s t r a t e g i e s may  be beyond s t u d e n t s who  are j u s t b e g i n n i n g t o s t u d y i n  a content area. P o l y a ' s model has been m o d i f i e d and used i n r e s e a r c h t o a n a l y z e problem  solving.  MacPherson's model o f h e u r i s t i c s may  as a r e f i n e m e n t o f P o l y a s model. r  by a combination o f p r o c e d u r e s of  can be accounted f o r  from P o l y a ' s work.  F o r example, t h e  heuristic  i n v a r i a t i o n - f i x a t i o n can be d e s c r i b e d i n terms o f P o l y a ' s l i s t .  heuristic  o f f i x a t i o n i s t h e renaming o f a v a r i a b l e as a c o n s t a n t  t h e n a t t e m p t i n g t o s o l v e t h e new to  Each h e u r i s t i c  be c o n s i d e r e d  problem  I n terms o f P o l y a ' s model,  t h e f o l l o w i n g q u e s t i o n s would have t o be asked: problem  and  and study i t ' s s o l u t i o n i n o r d e r  g a i n some i n s i g h t i n t o t h e g i v e n problem.  t h e proposed  The  I f you  t r y t o s o l v e some r e l a t e d problem  cannot first.  solve Could  y o u change t h e unknown o r t h e d a t a (rename a v a r i a b l e as a c o n s t a n t ) ? C a r r y out your p l a n (on the new  problem) o f t h e s o l u t i o n .  Can you  use  the r e s u l t ,  o r t h e method, f o r some o t h e r problem  o r i g i n a l one)?  ( i n t h i s case t h e  One o f t h e major advantages o f u s i n g MacPherson's  model  t o a n a l y z e problem s o l v i n g may be t h e r e l a t i v e l y s m a l l number o f h e u r i s t i c s ( t w e l v e ) , as compared t o P o l y a ' s c h e c k l i s t  of t h i r t y - s i x  questions.  Models o f problem s o l v i n g v a r y i n c o m p l e x i t y from t h e 3 o r 4 s t e p s o f " f o r m a l a n a l y s i s " t o t h o s e o f W i c k e l g r e n , Schwieger, and P o l y a who attempt d e t a i l e d d e s c r i p t i o n s o f p r o c e d u r e s used i n m a t h e m a t i c a l problem solving.  Of t h e s e models, P o l y a ' s has been used i n r e c e n t r e s e a r c h t o  a n a l y z e problem s o l v i n g .  The model was m o d i f i e d by K i l p a t r i c k , w i t h  o t h e r m o d i f i c a t i o n s by Webb and Kantowski, and used a t d i f f e r e n t  grade  levels. W i c k e l g r e n ' s model seems i n a p p r o p r i a t e t o use f o r a n a l y z i n g problem solving.  H i s model i n c l u d e s some p r o c e d u r e s s i m i l a r t o P o l y a ' s but many  o f them a r e o v e r a l l p l a n s f o r a t t a c k i n g problems r a t h e r t h a n problem s o l v i n g procedures.  Many o f t h e p r o c e d u r e s i n Schwieger's model appear  t o o v e r l a p and a r e n o t c l e a r l y d e f i n e d enough t o u s e i n a n a l y z i n g problem solving. T h i s review o f t h e l i t e r a t u r e has r a i s e d s e v e r a l i m p o r t a n t q u e s t i o n s i n terms o f t h e u s a b i l i t y o f models o f m a t h e m a t i c a l problem s o l v i n g f o r a n a l y z i n g problem s o l v i n g .  These q u e s t i o n s a r e :  1.  Are t h e p r o c e d u r e s from t h e model u s e d b y t h e s u b j e c t s ?  2.  Do t h e p r o c e d u r e s from t h e model d e s c r i b e t h e problem  solving  p r o c e s s , i . e . , a r e t h e p r o c e d u r e s i n t h e model broad enough t o account f o r a l l o f t h e p r o c e s s e s used by t h e s u b j e c t and y e t n o t so b r o a d t h a t no r e a l d i s t i n c t i o n can be made? 3.  A r e t h e p r o c e d u r e s from t h e model d e f i n e d c l e a r l y enough t o be coded  reliably?  42 I n terms o f t h e u s a b i l i t y o f MacPherson's model t o analyze problem s o l v i n g , an attempt was  made t o answer each o f t h e s e  Field  mathematical  questions.  Independence  I n t h e l a s t decade, p s y c h o l o g i s t s have attempted t o r e v i v e extend One  t h e study o f dominant p a t t e r n s o r modes o f c o g n i t i v e b e h a v i o r .  speaks o f t h e s e dominant p a t t e r n s o r modes as c o g n i t i v e s t y l e  1970,  p. 1)  style. how  and  The  and  (Spitler,  c l a s s i f i e s t o g e t h e r i n d i v i d u a l s w h o . t y p i c a l l y use t h e same  study o f c o g n i t i v e s t y l e s , which began i n o b s e r v a t i o n s  i n d i v i d u a l s p e r c e i v e and  c a t e g o r i z e i n f o r m a t i o n , has  broadened t o i n c l u d e t h e o p e r a t i o n o f t h e s e t a s k s such as p r o b l e m - s o l v i n g Modern e x p e r i m e n t a l his  colleagues  who  found t h a t people  (Kilpatrick,  styles i n intellectual 1967,  pp.  20-21). and  F a t e r s o n , Goodenough, and Karp, 1962),,  d i f f e r i n t h e way  When a s u b j e c t i s s e a t e d i n a t i l t e d independently  gradually  work on c o g n i t i v e s t y l e began w i t h W i t k i n  ( W i t k i n , Dyk,  of  :  t h e y o r i e n t themselves i n space.  c h a i r and t h e room i s t i l t e d  o f t h e c h a i r , t h e c o n f l i c t i n g v i s u a l cues and b o d i l y  s e n s a t i o n s o f t e n make i t d i f f i c u l t t h e room i n t o a v e r t i c a l p o s i t i o n .  f o r him t o b r i n g e i t h e r h i s body o r Subjects  seem t o make c o n s i s t e n t  e r r o r s on t h i s t a s k and t h e i r s c o r e i s c o r r e l a t e d w i t h t h e i r performance on t h e Embedded F i g u r e s T e s t  (EFT), i n which t h e y are asked t o f i n d  p a r t i c u l a r simple f i g u r e w i t h i n a l a r g e r complex d e s i g n  ( W i t k i n , Ch.  a 4)«  A s u b j e c t whose s c o r e i s above t h e median f o r a g i v e n sample i s s a i d t o be f i e l d  independent and t h o s e whose s c o r e i s below t h e median  are s a i d t o be f i e l d dependent.  F i e l d independence-dependence i s an  i n d e x o f p e r c e p t u a l components.  F i e l d independence r e p r e s e n t s  ability  to. overcome an embedding c o n t e x t  from i t s background.  the  and p e r c e i v e an i t e m as  distinct  44  Research i n d i c a t e s (Witkin, of  1962,  1974)  Walsh,  that the construct  f i e l d independence i s not s i g n i f i c a n t l y c o r r e l a t e d w i t h IQ  level.  W i t k i n f o u n d s i g n i f i c a n t c o r r e l a t i o n between t h e c o n s t r u c t o f f i e l d independence and a group o f s u b t e s t s o f WISC ( W e c h s l e r I n t e l l i g e n c e S c a l e f o r C h i l d r e n ) ; B l o c k D e s i g n , O b j e c t Assembly  and P i c t u r e C o m p l e t i o n ;  however, t h e r e was n o n - s i g n i f i c a n t c o r r e l a t i o n between t h e c o n s t r u c t and v e r b a l comprehension  and a r i t h m e t i c s u b t e s t s c o r e s o f WISC.  Hence  " . . . i n t e l l i g e n c e t e s t s c o r e s cannot be i n t e r p r e t e d t o mean f i e l d - i n d e p e n d e n t c h i l d r e n are o f g e n e r a l l y superior i n t e l l i g e n c e  [Witkin,  1962,  70]."  p.  R e s e a r c h has shown t h a t f i e l d independence i s r e l a t e d t o m a t h e m a t i c a l achievement. the  I n a s t u d y o f 100 grade n i n e boys, R o s e n f e l d (1958)  examined  r e l a t i o n s h i p o f m a t h e m a t i c a l a b i l i t y as measured on s c o r e s o f t h e  P r o g r e s s i v e Achievement Correlations of -.56,  T e s t and performance on t h e EFT.  -.32,  -.64,  Significant  (P < . 0 5 ) , were f o u n d r e l a t i n g  independent s c o r e s on t h e EFT  (used t i m e as s u b j e c t ' s s c o r e :  independent below t h e median,  f i e l d dependent  t o t a l mathematics  field  field  above t h e median) and t h e  s c o r e , m a t h e m a t i c a l r e a s o n i n g s c o r e , and t h e m a t h e m a t i c a l  fundamentals s c o r e , r e s p e c t i v e l y .  The p o o r mathematics  s t u d e n t s were:  more f i e l d dependent t h a n t h e good mathematics s t u d e n t s . Few  s t u d i e s have 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 f i e l d i n d e p e n -  dence and problem s o l v i n g i n mathematics.  However, t h e f i e l d  c o n s t r u c t has been used i n s t u d y i n g problem s o l v i n g i n  independent  other areas.  S a a r n i (1972) i n v e s t i g a t e d d i f f e r e n c e s i n problem s o l v i n g as a f u n c t i o n of  cognitive style.  She proposed t h a t P i a g e t ' s development  of l o g i c a l  t h i n k i n g would p r o v i d e an o v e r a l l framework f o r u n d e r s t a n d i n g problem s o l v i n g performance and t h a t W i t k i n ' s f i e l d independence would p r o v i d e 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 s i n problem s o l v i n g b e h a v i o r w i t h i n each P i a g e t i a n d e v e l o p m e n t a l l e v e l .  S i x t y - f o u r students, eight  male and e i g h t f e m a l e , p e r grade were randomly s e l e c t e d f r o m g r a d e s s i x through nine.  Two P r o d u c t i v e T h i n k i n g p r o b l e m s , "The M i s s i n g J e w e l "  a n d "The O l d B l a c k H o u s e " , c o n s t i t u t e d t h e p r o b l e m s o l v i n g t a s k s . s u b j e c t s ' p e r f o r m a n c e o n e a c h p r o b l e m was s c o r e d i n f o u r (a)  number o f r e l e v a n t  (b)  number o f c o r r e c t  clues cited  adequacy  categories:  ( p u z z l i n g f a c t s i n second problem),  a n a l y t i c c h o i c e s made,  ideas generated f o r solution,  The  (c)  number o f p l a u s i b l e  and ( d ) s c o r e o f 1 t o 5 f o r speed and  of attainment of the correct  solution.  The n e s t i n g o f f i e l d  independence w i t h i n P i a g e t i a n l e v e l s d i d n o t y i e l d any s i g n i f i c a n t (P  <.05)  d i f f e rences i n problem s o l v i n g performance.  Saarni concluded  that: 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 i s i n d i c a t e t h a t f i e l d independence 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 problem s o l v i n g p e r formance as m a n i f e s t e d i n t h e P r o d u c t i v e T h i n k i n g p r o b l e m s . This 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 independence might 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 o n p r o b l e m s w h i c h a r e more p e r c e p t u a l l y bound and/or r e l a t i v e l y n o n - v e r b a l [p.22], Farr  (1968) conducted a s t u d y t o determine whether  and p r o b l e m d i f f i c u l t y  s t u d e n t s b e t w e e n t h e a g e s o f 18 a n d 24,  g r a d u a t e e d u c a t i o n c o u r s e s were u s e d i n t h e s t u d y . asked t o s o l v e anagram problems problems  independence  were r e l a t e d t o problem s o l v i n g performance  o r g a n i z e d and d i s o r g a n i z e d r e o r g a n i z a t i o n - t y p e problems. ninety eight  field  representing verbal,  Two  hundred  e n r o l l e d i n underThe s t u d e n t s w e r e and match  stick  representing non-verbal, reorganization-type problems.  Each  t y p e o f p r o b l e m was p r e s e n t e d i n o r g a n i z e d a n d d i s o r g a n i z e d f o r m s , e a s y and d i f f i c u l t  problems i n each form.  a p t i t u d e was h e l d c o n s t a n t , f i e l d  She f o u n d  t h a t when  with  mathematics  independent students g e n e r a l l y  s i g n i f i c a n t l y h i g h e r s c o r e s t h a n f i e l d dependent  on  received  s t u d e n t s on n o n - v e r b a l  b u t n o t on v e r b a l p r o b l e m s , r e g a r d l e s s o f p r o b l e m o r g a n i z a t i o n o r  difficulty.  However, b o t h Walsh (1974) and Cooperman (1974) f o u n d f i e l d independent s t u d e n t s performed s i g n i f i c a n t l y b e t t e r t h a n f i e l d dependent in  s o l v i n g anagram problems o f moderate d i f f i c u l t y .  students  The s t u d e n t s i n  Walsh's s t u d y were 12 and 13 y e a r s o l d w h i l e t h o s e i n Cooperman's s t u d y were 10 y e a r s o l d . The r e s u l t s o f r e s e a r c h i n v o l v i n g f i e l d independence and problem s o l v i n g i n a non-mathematical s e t t i n g are mixed.  Both S a a r n i and F a r r  found n o n - s i g n i f i c a n t d i f f e r e n c e s between f i e l d independence and v e r b a l problem s o l v i n g .  However, s t u d i e s by Walsh and Cooperman f o u n d f i e l d  independent s t u d e n t s performed s i g n i f i c a n t l y b e t t e r t h a n f i e l d s u b j e c t s on v e r b a l problems.  dependent  The r e s u l t s o f F a r r ' s s t u d y a l s o i n d i c a t e  t h a t w i t h m a t h e m a t i c a l a b i l i t y h e l d c o n s t a n t , f i e l d independence i s s i g n i f i c a n t l y r e l a t e d t o problem s o l v i n g i n a n o n - v e r b a l s e t t i n g . Two  r e s e a r c h s t u d i e s have examined t h e r e l a t i o n s h i p o f f i e l d i n d e p e n -  dence and a s u b j e c t ' s a b i l i t y t o change h i s mode o f a t t a c k i n s o l v i n g non-mathematical problems. The E i n s t e l l u n g problems o f L u c h i n s (1942) have been u s e d t o s t u d y the  e f f e c t s o f t h e " s e t " o f a s u b j e c t upon h i s problem s o l v i n g b e h a v i o r .  A s e t i s d e f i n e d as " t h e t e n d e n c y o f an i n d i v i d u a l t o p e r s e v e r a t e i n a g i v e n mode o f a t t a c k [Buetzkow,  1951,  P« 219]."  The problems  require  an i n d i v i d u a l t o o b t a i n a g i v e n q u a n t i t y o f l i q u i d by u s i n g v a r i o u s combinations o f t h r e e j a r s . A, B, and C, w i t h g i v e n volumes. f i v e problems which a l l have t h e same s o l u t i o n , A + B - 2C, a "set".  The  initial  establish  A measure o f t h e s t r e n g t h o f t h e " s e t " i s c o n t a i n e d i n t h e  s u c c e e d i n g two problems which can be s o l v e d by a simple d i r e c t p r o c e d u r e , A - C, o r by t h e complex manner u s e d t o s o l v e t h e f i r s t f i v e problems. F i n a l l y , t h e t e r m i n a l problem which can o n l y be s o l v e d by t h e s i m p l e r  method, i s used as a measure o f a b i l i t y t o overcome t h e " s e t " .  Guetzkow  (1951) c a r r i e d out a study u s i n g L u c h i n s ' E i n s t e l l u n g problems. d i v i d e d h i s s u b j e c t s i n t o two groups a c c o r d i n g t o t h e i r ability.  The s e t b r e a k e r s  were t h o s e who  He  set-breaking  adopted t h e s e t by u s i n g t h e  (A + B - 2 C ) method o f s o l u t i o n , r a t h e r t h a n t h e (A - C) method on t h e c r i t i c a l problems, but who were a b l e t o break t h e s e t on t h e t e r m i n a l problem.  The non-set b r e a k e r s  were t h o s e  break t h e s e t on t h e t e r m i n a l problem.  s u b j e c t s who were u n a b l e t o  He found t h e s e t - b r e a k e r s d i d  s i g n i f i c a n t l y b e t t e r t h a n t h e n o n - s e t - b r e a k e r s on T h u r s t o n e ' s G o t t s c h a l d t  2 Figures Test.  Guetzkow a l s o f o u n d s i g n i f i c a n t r e l a t i o n s h i p between  Thurstone's G o t t s c h a l d t  F i g u r e s T e s t and t h e time r e q u i r e d t o s o l v e t h e  t e r m i n a l problem. Goodman ( i 9 6 0 ) , These s t u d e n t s  conducted a s i m i l a r s t u d y u s i n g c o l l e g e  students.  r e c e i v e d t h e EFT, T h u r s t o n e ' s G o t t s c h a l d t F i g u r e s  and t h e E i n s t e l l u n g t e s t .  Test,  No s i g n i f i c a n t d i f f e r e n c e i n performance o f  e i t h e r o f t h e p e r c e p t u a l t e s t s were found between s t u d e n t s who  solved  t h e c r i t i c a l E i n s t e l l u n g problems by t h e s h o r t method and t h o s e s o l v e d them by t h e l o n g method.  However, s i g n i f i c a n t  who  c o r r e l a t i o n s were  found between b o t h t h e EFT and T h u r s t o n e ' s G o t t s c h a l d t F i g u r e s T e s t and t h e t i m e r e q u i r e d t o s o l v e t h e t e r m i n a l E i n s t e l l u n g problem. These r e s u l t s ,  c o n f i r m i n g and e x t e n d i n g  t h e f i n d i n g s o f Guetzkow,  i n d i c a t e that set-breaking a b i l i t y i n the E i n s t e l l u n g s i t u a t i o n i s related to f i e l d  independence.  A m u l t i v a r i a t e d e s i g n was used by Dodson  (1972) t o t e s t t h e r e l e v a n c e  W i t k i n ' s Embedded F i g u r e s T e s t was adapted from T h u r s t o n e ' s Gottschaldt Figures Test. C o l o r was added t o G o t t s c h a l d t ' s b l a c k and white o u t l i n e complex f i g u r e s t o make them more d i f f i c u l t . Studies c a r r i e d on i n t h e 1950's have a l l shown s i g n i f i c a n t c o r r e l a t i o n s between t h e s e two t e s t s (See W i t k i n 1962, p. 4 0 ) .  o f 77 concomitant v a r i a b l e s f o r t h e a b i l i t y t o s o l v e i n s i g h t f u l mathematics problems and t o determine which o f t h e s e v a r i a b l e s d i s c r i m i n a t e among a b i l i t y groups.  best  I n s i g h t f u l problems a r e problems which cannot be  s o l v e d by simple r e c a l l from memory o r s t a n d a r d  computational  algorithms,  n o r does t h e s o l u t i o n depend on a s p e c i a l t r i c k  (Dodson, 1973.  p» 3 ) .  A random sample o f 1123  grade e l e v e n  s t u d e n t s was s e l e c t e d from  t h o s e s t u d e n t s p a r t i c i p a t i n g i n NLSMA who were c u r r e n t l y e n r o l l e d i n mathematics. The  A l l o f t h e d a t a f o r each s u b j e c t were o b t a i n e d  from NLSMA.  s t u d e n t s were p l a c e d i n one o f s i x a b i l i t y groups depending upon  t h e i r s c o r e on a t e s t o f i n s i g h t f u l mathematics problem s o l v i n g . 77 d e s c r i p t o r v a r i a b l e s were c l a s s i f i e d i n t o f i v e major  The (l)  mathematics a p t i t u d e  variables  and achievement v a r i a b l e s , (2)  ( e . g. a t t i t u d e s , a n x i e t y ,  background v a r i a b l e s , (4) mathematics c u r r i c u l u m  school  categories:  psychological  and c o g n i t i v e f a c t o r s ) , (3)  teacher  and community v a r i a b l e s and (5)  variables.  Dodson found t h e m a t h e m a t i c a l achievement v a r i a b l e s t o be t h e strongest  d i s c r i m i n a t o r s among a b i l i t y groups and t h a t t h e c o g n i t i v e  v a r i a b l e s were second s t r o n g e s t .  Of t h e c o g n i t i v e v a r i a b l e s , Dodson  found t h e b e s t  i n s i g h t f u l mathematics problem s o l v e r s tended t o have t h e  highest  on t h e r e a s o n i n g  scores  t e s t s - v e r b a l and l o g i c a l  as w e l l as t h e n u m e r i c a l r e a s o n i n g  test.  F i e l d independence was found t o d i s c r i m i n a t e as w e l l among t h e a b i l i t y groups as d i d t h e p o o r e s t was i n c l u d e d i n t h e composite l i s t  reasoning  reasoning  of the strongest  (P < . 0 0 1 )  v a r i a b l e s and  characteristics of  a s u c c e s s f u l i n s i g h t f u l mathematics problem s o l v e r (Dodson, 1973. Analyzing school  algebra  t h e problem s o l v i n g p r o c e d u r e s o f 40 second y e a r students,  Webb  P»  122).  high  (1974) s u p p o r t e d t h e f i n d i n g o f Dodson t h a t  m a t h e m a t i c a l achievement i s t h e s t r o n g e s t  component i n a c c o u n t i n g f o r  problem s o l v i n g a b i l i t y .  U s i n g r e g r e s s i o n a n a l y s i s , Webb found t h a t a c c o u n t s f o r 50% o f t h e v a r i a n c e i n t h e t o t a l  m a t h e m a t i c a l achievement  s c o r e s from h i s problem s o l v i n g i n v e n t o r y .  With mathematical  achievement,  v e r b a l r e a s o n i n g and n e g a t i v e a n x i e t y e n t e r e d i n t h e r e g r e s s i o n e q u a t i o n first,  field  independence d i d n o t add s i g n i f i c a n t l y t o t h e amount o f  v a r i a n c e on t h e t o t a l s c o r e a l r e a d y accounted f o r . Research f i n d i n g s i n v o l v i n g t h e f i e l d  independent c o n s t r u c t and  problem s o l v i n g i n non-mathematical s e t t i n g s a r e mixed, favor the f i e l d  independent s u b j e c t .  but i n g e n e r a l  F i e l d independent s t u d e n t s a r e  more a b l e t o overcome a problem s o l v i n g s e t t h a n t h e i r f i e l d counterparts.  dependent  T h i s would seem t o be an i m p o r t a n t c h a r a c t e r i s t i c o f a  good problem s o l v e r i n mathematics, t h a t i s , when s o l v i n g a problem i f a mode o f a t t a c k i s n o t l e a d i n g t o a s o l u t i o n , r a t h e r t h a n c o n t i n u e t o use t h i s mode, change t h e p r o c e d u r e s b e i n g used and t r y a d i f f e r e n t method o f a t t a c k i n g t h e problem. F i e l d independence has been shown t o r e l a t e s i g n i f i c a n t l y t o m a t h e m a t i c a l r e a s o n i n g and i n s i g h t f u l problem s o l v i n g . associated with a f i e l d resist distraction, a problem,  (3)  independent s t u d e n t a r e :  (2)  a b i l i t y t o s e p a r a t e p a r t s from t h e whole and recombine  elements, and (5) Romberg and W i l s o n ,  the  ( l ) ability to  a b i l i t y t o i d e n t i f y t h e c r i t i c a l elements o f  them t o form a new whole,  In  The c h a r a c t e r i s t i c s  (4)  a b i l i t y t o remain independent o f i r r e l e v a n t  a b i l i t y t o overcome a problem s e t ( W i t k i n ,  1962;  1969)»  c o n c l u s i o n , t h e l i t e r a t u r e from two major areas has i n f l u e n c e d  design o f the present study.  The l i t e r a t u r e on models o f problem  s o l v i n g has r a i s e d t h r e e i m p o r t a n t q u e s t i o n s i n terms o f t h e u s a b i l i t y of  models o f m a t h e m a t i c a l problem s o l v i n g f o r a n a l y z i n g problem  behavior.  solving  50 1.  Are t h e p r o c e d u r e s from t h e model used by s u b j e c t s i n s o l v i n g m a t h e m a t i c a l problems?  2.  Can t h e p r o c e d u r e s from t h e model be coded  reliably?  3.  Do t h e p r o c e d u r e s from t h e model d e s c r i b e t h e problem  solving  process? In  terms o f t h e u s a b i l i t y o f MacPherson's  s o l v i n g an attempt was  model f o r a n a l y z i n g problem  made t o answer each o f t h e s e q u e s t i o n s .  The c h a r a c t e r i s t i c s a s s o c i a t e d w i t h a f i e l d  independent s t u d e n t  i n d i c a t e t h a t t h e s e s t u d e n t s are b e t t e r problem s o l v e r s t h a n t h e dependent  students.  I n terms o f t h e outcome o f t h i s  study, t h e  field field  independent s t u d e n t i s expected t o use a g r e a t e r v a r i e t y o f h e u r i s t i c s from MacPherson's  model, t o be more w i l l i n g t o change p r o c e d u r e s i f he  i s not being, s u c c e s s f u l i n s o l v i n g a problem, solutions than the f i e l d  dependent  student.  and t o o b t a i n more c o r r e c t  Chapter I I I  PROCEDURES  Two major c o n s i d e r a t i o n s i n a n a l y z i n g problem s o l v i n g b e h a v i o r a r e t h e c h o i c e o f s u b j e c t s and t h e c h o i c e o f problems.  I f the subjects are  t o o w i d e l y d i v e r s e i n t h e i r background and a b i l i t i e s , t h e r e i s l i t t l e hope o f o b s e r v i n g any common p a t t e r n s o f t h i n k i n g .  The c h o i c e o f t h e  problem m a t e r i a l used must be made w i t h t h e s u b j e c t s i n mind. problems a r e t o o hard t h e r e w i l l be l i t t l e problem  I f the  solving  behavior observed.  On t h e o t h e r hand, i f t h e problems a r e t o o easy,  t h e y may n o t e l i c i t  any problem s o l v i n g b e h a v i o r at a l l ( K i l p a t r i c k ,  1967,  p.  32).  Subjects  The  s u b j e c t s s e l e c t e d f o r t h i s s t u d y were a l l c o m p l e t i n g t h e e l e v e n t h  grade academic mathematics program. girls,  o f average  ability  The sample c o n s i s t e d o f boys and  f o r t h o s e s t u d e n t s e n r o l l e d i n t h i s program,  who were p a r t i c i p a t i n g i n a one semester a l g e b r a c o u r s e .  A popular  modern a l g e b r a and t r i g o n o m e t r y t e x t was used f o r t h i s c o u r s e .  At t h e  time o f t h e problem s o l v i n g i n t e r v i e w s t h e s t u d e n t s were s t u d y i n g t h e chapter r e l a t e d t o t h e quadratic f u n c t i o n . was  One o f t h e aims o f t h e s t u d y  t o a s s e s s t h e h e u r i s t i c s used by s t u d e n t s i n s o l v i n g word problems.  S i n c e a l a r g e p o r t i o n o f t h e content o f second y e a r a l g e b r a d e a l s w i t h word problems, s t u d e n t s who have completed  t h e course s h o u l d have a  s u f f i c i e n t background f o r d e a l i n g w i t h word problems. An  average  IQ range was s e l e c t e d i n o r d e r t o c o n t r o l t h e IQ v a r i a b l e  and h o p e f u l l y s t i l l study had  o b t a i n a f a i r l y wide range o f EFT  scores.  shown t h a t s t u d e n t s i n t h i s a b i l i t y range were not  A  pilot  unduly  t h r e a t e n e d by t h e t a s k o f s o l v i n g d i f f i c u l t m a t h e m a t i c a l problems i n an interview  situation.  To determine t h o s e  s t u d e n t s o f average a b i l i t y i n t h e  m a t h e m a t i c a l program, a sample o f 150  s t u d e n t s was  academic  randomly drawn from  the Algebra I I c l a s s e s of the schools i n v o l v e d i n the study. IQ s c o r e on. t h e C a l i f o r n i a T e s t o f M e n t a l M a t u r i t y f o r t h i s 119.8 of  w i t h a s t a n d a r d d e v i a t i o n o f 10.4.  average a b i l i t y i f h i s IQ was  A student was  The mean sample  was  defined to  be  w i t h i n one-half standard d e v i a t i o n  o f t h e mean. A total  o f 40  s e l e c t e d from 14 (grades  11  and  s t u d e n t s , i n t h e IQ range 115  12)  classes.  no t e a c h e r The  i n t h e g r e a t e r Vancouver a r e a .  The  c l a s s e s and two  randomly schools  Three s t u d e n t s were  s t u d e n t s from each o f t h e  c l a s s e s were t a u g h t by t e n d i f f e r e n t t e a c h e r s ,  h a v i n g more t h a n two c l a s s e s .  mean IQ f o r t h e s t u d e n t s i n t h e s t u d y was  d e v i a t i o n o f 10.2. ranged from 16 age o f 16  was  A l g e b r a I I c l a s s e s i n t h r e e s e n i o r secondary  s e l e c t e d from each o f twelve o t h e r two  t o 125,  Ages o f t h e s t u d e n t s  y e a r s , 4 months, t o 17  y e a r s , 11  months.  The  s t u d y was  conducted  with a  standard  at t h e time o f t h e i n t e r v i e w  y e a r s , 4 months, w i t h a median  sample c o n t a i n e d 22  Pilot  The p i l o t  119*5  boys and  18 g i r l s .  Study  from December, 1973  t o March,  1974,  i n a s e n i o r h i g h s c h o o l i n t h e same g e n e r a l a r e a as t h e s c h o o l s used i n t h e main s t u d y .  The  i n i t i a l phase o f t h e p i l o t  study i n v o l v e d the  use o f w r i t t e n problem s e t s t o a i d i n t h e s e l e c t i o n o f the problems t o be used i n t h e study.  T h i s phase was  f o l l o w e d by a s e r i e s o f  pilot  53 interviews. 1.  The major aims o f t h e p i l o t  Determine i f e l e v e n t h grade  s t u d y were:  s t u d e n t s o f average  a b i l i t y could  respond w e l l t o t h e i n t e r v i e w p r o c e d u r e 2.  S e l e c t t h e problems t o be used i n t h e s t u d y  3.  Modify K i l p a t r i c k ' s  c o d i n g system o r d e v e l o p a new  system u s i n g MacPherson*s model f o r problem  coding  solving  4»  Develop  an i n t e r v i e w format  5.  P r a c t i c e t h e i n t e r v i e w i n g and c o d i n g t e c h n i q u e s  Twelve mathematical  problems s e l e c t e d from t h e areas o f number  t h e o r y , geometry, and a l g e b r a were used i n t h e i n i t i a l t e s t i n g o f t h e pilot  study.  The problems were s t a t e d i n b o t h t h e r e a l w o r l d and  math w o r l d s e t t i n g .  These problems,  potential to e l i c i t experimenter  s e l e c t e d on t h e b a s i s o f t h e i r  t h e use o f c e r t a i n h e u r i s t i c s , were t h o s e t h a t t h e  f e l t c o u l d be s o l v e d by a p p r o x i m a t e l y h a l f o f t h e s t u d e n t s  p a r t i c i p a t i n g i n the study.  The problems were t h e n randomly d i v i d e d  i n t o s e t s o f t h r e e i n t h e same s e t t i n g and a<±irinistered as w r i t t e n examinations  t h r o u g h December, 1973,  who would complete period,  and January,  the A l g e b r a I I course i n February.  11 new problems were added t o t h e l i s t  work on a problem  as l o n g as t h e y wished  t o 90  students  Throughout t h e .  and many o t h e r s were  d e l e t e d because t h e y proved t o be t o o d i f f i c u l t . to  1974,  The s t u d e n t s were  asked  and n o t t o e r a s e a n y t h i n g .  Each student was asked t o r e c o r d t h e amount o f t i m e he spent on each problem.  Appendix A c o n t a i n s t h e problems used i n t h e p i l o t  F i f t e e n problems were s e l e c t e d from t h i s l i s t f o r t h e main study.  d e s i g n e d t o generate as much o b s e r v a b l e problem possible.  as p o s s i b l e  Over t h e next t h r e e months, t w e l v e  i n t e r v i e w e d u s i n g from 3 t o 8 o f t h e 15 problems.  study. candidates  s t u d e n t s were  Each i n t e r v i e w was  s o l v i n g b e h a v i o r as  A room was p r o v i d e d a t t h e h i g h s c h o o l f o r t h e i n t e r v i e w  sessions.  S u b j e c t s were t o l d t h a t t h e s e s s i o n would be t a p e  and t h e y s h o u l d t h i n k a l o u d w h i l e working on t h e problems.  recorded The s t u d e n t s  were a l s o i n s t r u c t e d n o t t o e r a s e a n y t h i n g and i f a diagram was t o be m o d i f i e d , t o draw a new one.  A l l instructions,  except t h e problem  ments, were communicated v e r b a l l y b y t h e i n t e r v i e w e r .  state-  The problems were  t y p e w r i t t e n and p r e s e n t e d t o t h e s t u d e n t s one a t a t i m e . It  was determined  t h a t grade e l e v e n s t u d e n t s o f average  not unduly t h r e a t e n e d by t h e t a s k o f s o l v i n g d i f f i c u l t problems i n t h e i n t e r v i e w s i t u a t i o n . t h e i r thoughts,  mathematical  A l s o , t h e y are a b l e t o v e r b a l i z e  e s p e c i a l l y i f t h e y have some s u c c e s s and u n d e r s t a n d  what t h e i n t e r v i e w e r e x p e c t s . this into  a b i l i t y are  The i n t e r v i e w format was r e v i s e d t o t a k e  account.  The need f o r s e v e r a l p r a c t i c e problems t o a c c l i m a t e t h e s u b j e c t s to  s o l v i n g problems a l o u d became e v i d e n t i n t h e p i l o t  s t u d y when t h e  s t u d e n t s i n v o l v e d became accustomed t o t h i n k i n g a l o u d o n l y a f t e r  attempting  one o r two problems. The  list  o f problems was reduced t o 5 and two p r a c t i c e problems  were added (see Appendix B ) . to  T h i s allowed t h e subject s u f f i c i e n t  work on t h e problems and s t i l l  time  complete t h e i n t e r v i e w i n one s e s s i o n .  Most s t u d e n t s were w i l l i n g t o work f o r up t o 2-|- h o u r s i f g i v e n a s h o r t r e s t p e r i o d between problems. Four o f t h e problems were s e l e c t e d from t h e p i l o t did  s t u d y because t h e y  e l i c i t t h e use o f d i f f e r e n t h e u r i s t i c s and t h e f i f t h problem was  added i n an e f f o r t t o o b t a i n some use o f t h e h e u r i s t i c o f symmetry (problem #1, Appendix B ) . in  a number o f d i f f e r e n t  However, a l l f i v e problems c o u l d be s o l v e d  ways.  A f i n a l f o u r s t u d e n t s were i n t e r v i e w e d u s i n g t h e 7 problems, two s t u d e n t s d o i n g problems i n each o f t h e s e t t i n g s ,  as a check on t h e problems  and t h e i n t e r v i e w method, modified coding  The p r o t o c o l s were a l s o a n a l y z e d u s i n g t h e  system.  The C o d i n g  One  o f t h e main reasons f o r s p a c i n g t h e p i l o t  o v e r t h e t h r e e month p e r i o d was modify  System  t h e c o d i n g system.  to K i l p a t r i c k ' s  interview sessions  t o a l l o w ample time t o e v a l u a t e  The i n i t i a l  (1967, pp. 50-53).  and  c o d i n g forms were q u i t e s i m i l a r  Both t h e c h e c k l i s t  and t h e  sequence had been m o d i f i e d t o t a k e i n t o account t h e p r o c e d u r e s  process from  MacPherson's. model. A f t e r each i n t e r v i e w p e r i o d d u r i n g t h e p i l o t  study, t h e  investigator  used b o t h t h e t a p e r e c o r d e d p r o t o c o l s as w e l l as a l l w r i t t e n work t o h e l p a n a l y z e t h e s u b j e c t s ' problem s o l v i n g b e h a v i o r . b e h a v i o r s were a n a l y z e d i n terms o f process-sequence  The problem  solving  c o d i n g symbols  and  o t h e r events were r e c o r d e d by c h e c k i n g o f f a p p r o p r i a t e c a t e g o r i e s on t h e checklist. D u r i n g t h e attempts the p i l o t  t o a p p l y t h e c o d i n g system t o p r o t o c o l s from  study, d i f f i c u l t i e s and  shortcomings  became apparent.  o f t h e c a t e g o r i e s on t h e c h e c k l i s t were not used.  I n some i n s t a n c e s ,  the p r o c e s s sequence became v e r y l o n g and complex w i t h some nested i n others. of the procedure  position  i n t h e sequence o f t h e problem s o l v i n g p r o c e s s was  and  device f o r r e c o r d i n g the procedures The  combined  c o d i n g sequence i n t o a complete s e q u e n t i a l c o d i n g  The use o f a m a t r i x p r o v i d e d t h e r e s e a r c h e r w i t h a  protocols.  more  occurance.  f i n a l c o d i n g system, used i n t h e a n a l y s i s o f t h i s study,  b o t h t h e check l i s t system.  procedures  More i m p o r t a n t l y , i t became e v i d e n t t h a t t h e  important than i t s frequency o f The  Many  coding matrix,  o b t a i n e d from t h e tape  convenient recorded  (see F i g u r e 4 ) , accounts f o r d i f f e r e n t  Reading  problem  Request d e f n . o f ter-13 Recall Recall  same problem  related  problem  R e c a l l problem  type  Recall related  fact  Draw diagram Modify Identify Setting  up  diagram variable  equations  Algorithms-algebraic Algorithms-arithmetic Guessing Smoothing Analysis Templation Cases-all Cases-random Cases-systematic Cases-critical Cases-sequential Deduction Inverse  deduction  Invariation Analogy Symmetry Obtain  solution  Checking Checking by  part  solution  subst. i n equation by  retracing  steps  by r e a s n b l e / r e a l i s t i c uncodable Exp. Exp.  concern-method  concren-algorithm  Exp.  concern-equation  Exp.  concern-solution  Work s t o p p e d - s o l n . Work s t o p p e d - n o  soln.  FIGURE -* THE CODING FORM (Reduced by 33%)  p r o c e d u r e s used s i m u l t a n e o u s l y ; p r o c e d u r e s which a r e n e s t e d i n o t h e r s , such as t h e use o f d i f f e r e n t a l g o r i t h m s and diagrams,  which might  take  p l a c e w h i l e a s u b j e c t i s u s i n g t h e h e u r i s t i c o f random c a s e s ; and t h e sequential order o f a l l procedures.  At a g l a n c e one can t e l l t h e methods  used by a s u b j e c t , t h e o r d e r i n w h i c h he used them, and how  successful  he was. To t a k e i n t o account t h e sequence o f e v e n t s used i n s o l v i n g a problem,  see F i g u r e 3,  P' 19.  t h e p r o c e d u r e s were grouped  m a t r i x as f o l l o w s : Preparation Reading problem Request d e f i n i t i o n o f terms Sieve R e c a l l same problem R e c a l l r e l a t e d problem R e c a l l problem t y p e Recall related fact Draw diagram M o d i f y diagram Identify variable S e t t i n g up e q u a t i o n s Algorithms-algebraic Algorithms-arithmetic Guessing Heuristics Smoothing Analysis Templation Cases-all Cases-random Cases-systematic Cases-critical Cases-sequential Deduction Inverse deduction Invariation Analogy Symmetry  i n the coding  Solution Obtain  solution  Reasonableness Checking part Checking s o l u t i o n by s u b s t i t u t i n g i n e q u a t i o n s by r e t r a c i n g s t e p s by r e a s o n a b l e / r e a l i s t i c uncodable Concern Express Express Express Express  concern concern concern concern  about method about a l g o r i t h m about e q u a t i o n about s o l u t i o n  Work Stopped Work s t o p p e d - s o l u t i o n Work stopped-no s o l u t i o n As t h e coder i d e n t i f i e s t h e s e p r o c e s s e s from a s u b j e c t ' s p r o t o c o l , he e n t e r s a check  ) i n t h e a p p r o p r i a t e row and f i r s t  empty  column.  I f t h e p r o c e s s i s from t h e h e u r i s t i c c a t e g o r y o r t h e s i e v e ( e x c e p t r e c a l l o f same problem r e l a t e d problem, problem t y p e ) he e n t e r s  either  a 1, 2, o r 3 , depending upon t h e outcome: 1  Incomplete  2  Incorrect  3  Correct  F o r example,  a 2 i n t h e a l g o r i t h m - a l g e b r a i c row i n d i c a t e s an e r r o r  i n . t h e use o f t h e a l g o r i t h m , whereas, a n a l y s i s , f o r example,  a 2 f o r one o f t h e h e u r i s t i c s ,  i n d i c a t e s t h e outcome o f t h e problem a f t e r t h e  use o f a n a l y s i s i s i n c o r r e c t , n o t t h a t t h e s u b j e c t committed i n the h e u r i s t i c  an e r r o r  itself.  I f two o r more p r o c e d u r e s o c c u r s i m u l t a n e o u s l y t h e n a check o r number i s e n t e r e d i n t h e same column f o r each one.  I f a procedure i s  59 c a r r i e d on l o n g e r than entered i n t h e l a s t  a column, i t i s e n c l o s e d i n a box w i t h t h e outcome  column e n c l o s e d .  F o r example, suppose a s u b j e c t i s  u s i n g random cases and c o n s i d e r s t h r e e cases a l o n g w i t h u s i n g a l g o r i t h m , c h e c k i n g p a r t o f h i s work and r e c a l l i n g r e l a t e d f a c t .  In this instance  t h e " b l o c k " f o r random c a s e s would cover t e n columns.  See t h e example  i n F i g u r e 5« Most o f t h e time t h e s u b j e c t , i n c h e c k i n g h i s work o r s o l u t i o n , d i d so by s u b s t i t u t i n g i n an e q u a t i o n , see i f h i s r e s u l t was r e a s o n a b l e , checking h i s s o l u t i o n ,  r e t r a c i n g steps o r checking t o  or realistic.  I f a s u b j e c t was  a check would go i n t h e a p p r o p r i a t e column i n t h e  c h e c k i n g s o l u t i o n row and a check o r numbers i n t h e row which i n d i c a t e d t h e procedure  he used.  A c c o r d i n g t o MacPherson's model, a p r o b a b i l i t y is  (amount o f c o n c e r n )  a t t a c h e d t o t h e outcome from t h e p a t t e r n box, a p p l i c a t i o n o f an  a l g o r i t h m , and t h e s o l u t i o n o b t a i n e d by t h e problem s o l v e r . some f e e l i n g f o r t h e s e p r o b a b i l i t i e s t h e concern was  expressed  To d i s c e r n by a s u b j e c t  coded i n one o f t h e f o u r c a t e g o r i e s . As an i l l u s t r a t i o n o f t h e c o d i n g system, two sample p r o t o c o l s  f o l l o w , w i t h t h e coded forms g i v e n i n F i g u r e s 5 and 6 . Problem #3. A r e c t a n g u l a r lawn i s t o be formed so t h a t one s i d e o f a b a r n s e r v e s as one s i d e o f t h e r e c t a n g l e . You have 300 f e e t o f w i r e t o e n c l o s e t h e o t h e r t h r e e s i d e s . What a r e t h e d i m e n s i o n s o f t h e r e c t a n g l e i f t h e a r e a i s t o be a maximum s i z e ? Why does y o u r s o l u t i o n g i v e t h e maximum area? Protocol (Student #36, problem #3) A r e c t a n g u l a r lawn i s t o be formed so t h a t one s i d e o f a b a r n serves as one s i d e o f t h e r e c t a n g l e . You have 300 f e e t o f w i r e t o e n c l o s e t h e o t h e r t h r e e s i d e s . What a r e t h e dimensions o f t h e r e c t a n g l e i f t h e area i s t o be o f maximum s i z e ? Why does y o u r s o l u t i o n g i v e t h e maximum area? I don't know. O.K., I've got a r e c t a n g u l a r l a w n . . . i s t o be formed so t h a t one s i d e of t h e barn s e r v e s as one s i d e o f t h e lawn. O.K.,  Column #  1  60 2  l e t ' s draw a r e c t a n g l e . . . O . K . b a r n i s h e r e . Covers t h a t whole s i d e . You have 300 f e e t o f w i r e t o e n c l o s e t h e o t h e r t h r e e s i d e s . O.K. L e t ' s see - a r e c t a n g u l a r lawn i s t o be formed so t h a t . . . I don't know how b i g t h i s r e c t a n g l e has t o be, but I have t o g e t t h e most s i z e out o f i t . T h a t ' s 300 f e e t . O.K. L e t ' s see, no e q u a t i o n f o r t h i s one. I ' l l s c r a p t h e equation. O.K., Now, i f i t ' s g o i n g t o need more a r e a f o r my barn, I t h i n k I ' d use somewhat o f a square, o r as c l o s e t o a square as p o s s i b l e . O.K.,. How b i g i s my b a r n ? L e t ' s imagine a 50 f o o t l e n g t h a c r o s s t h e r e , 50 down here - r i g h t , O.K., 50 and 50 from t h e r e i s 250 so 125 f e e t l o n g . Now am I u s i n g as much a r e a as I c o u l d ? Now, l e t ' s see, a r e a i s a 125 t i m e s 50. O.K., so 0 i s 5 i s 5 i s 2, 2 o v e r t h e r e i s 12, i s 1 o v e r t h e 6, 625O square f e e t . O.K., i f I had a 100 t h e r e , so 100, 100, 100, O.K., so 100 t i m e s 100 i s 0, 0, 0 i s 10,000 square f e e t . How am I g o i n g t o get a r e c t a n g u l a r lawn? How r e c t a n g u l a r i s rectangular? Now, t h a t ' s t h e most I can have f o r a square. Now, 0 , 0 , 1 t h a t ' s r i g h t . That would be t h e maximum f o r a square. I want a r e c t a n g l e . O.K., I ' l l g i v e my b a r n 90 f e e t , 0,K., 90 and 90 i s 180, O.K., t h a t ' s 120, t h a t ' s 60 t i m e s 60, 60 t i m e s 90 i s 0, nope, wrong. A...I don't know how I'm g o i n g t o do t h i s . . .  3  4 5 6 7 8 9 10 11  12 13 14 15 16  Problem #4: What i s t h e l o n g e s t p i e c e o f m e t a l r o d which can be p l a c e d i n a box o f dimensions 3 i n c h e s b y 4 i n c h e s by 12 i n c h e s ? Protocol (Student  #24,  problem #4)  Column # What i s t h e l o n g e s t p i e c e o f  m e t a l r o d which can be p l a c e d i n a box o f dimensions 3 i n c h e s by 4 i n c h e s b y 12 i n c h e s ? I t ' s n o t h a r d t o see t h e l o n g e s t p i e c e o f m e t a l r o d w i l l be from o p p o s i t e c o r n e r s , from t h e bottom o p p o s i t e c o r n e r o f one t o t h e top opposite corner o f the other. Now t h e q u e s t i o n i s how l o n g t h a t i s ? I know i t s l o n g e r t h a n 12. L e t ' s see i f I can work out some k i n d o f t r i a n g l e here. The two ends o f t h e boxes l i k e t h e ends o f t h e box a r e 3 by 4, now t h a t would mean t h a t , A, I'm t r y i n g t o form 2 t r i angles here t o f i n d out, so I can f i n d out, a d j a c e n t , o r complementary, o r c o r r e s p o n d i n g p a r t s so I can get t h a t l o n g r o d as a c o r r e s p o n d i n g p a r t i n t h i s box. Now, i f I had a 3 by 4 i n c h box, I'm assuming t h i s box i s r e c t a n g u l a r i n shape. That, I'm j u s t p u t t i n g i n t h e a n g l e s here. I f t h e r e i s r i g h t angles i n t h e box, s i d e - a n g l e s i d e , I f o r g o t how t o do t h a t , oh yeah, I want t o f i n d out i n t h a t box, I want t o f i n d out r i g h t on t h e two ends, a l i n e g o i n g r i g h t t h r o u g h t h e middle t o t h e oppos i t e s i d e s so t h a t one s i d e would connect w i t h t h e l i n e I have put as t h e i m a g i n a r y r o d and t h e r e f o r e form a t r i a n g l e , w e l l , I s a i d t h e a l t i t u d e o f t h e box i s 4 i n c h e s , and t h e w i d t h i s 3 , t h e r e f o r e , I formed a  1  2 3  4 5 6 7  61 t r i a n g l e w i t h two l e g s o f 3 i n c h e s and 4 i n c h e s and a hypotenuse t h a t I do n o t know, and s i n c e i t i s a r i g h t a n g l e t r i a n g l e , s i n c e I'm assuming i t s a r e c t a n g l e , a, 4 squared i s e q u a l t o t h e hypotenuse squared, I ' l l s a y H squared, but s h o u l d be C squared t h a t t h e o l d e q u a t i o n . So t h a t ' s 9 p l u s 16 i s e q u a l t o 25 and t h a t i s 5 squared. T h e r e f o r e , 5 i s e q u a l t o t h e l e n g t h o f t h a t l i n e , 5 i n c h e s . Now, I've found out t h a t one s i d e o f my i m a g i n a r y t r i a n g l e i s 5 i n c h e s . Now I know t h e o t h e r s i d e i s 12 i n c h e s because o f t h e t h i n g . Again, I am assuming I have a r i g h t a n g l e t r i a n g l e , because o f t h e a n g l e s o f t h e box. T h e r e f o r e , a l l I would do i s 5 squared p l u s 12 squared i s e q u a l t o C squared. 5 squared i s 25 and 12 squared i s 144 i s e q u a l t o 169. Now, I know I69 i s 13 squared. T h e r e f o r e , 13 i n c h e s must be t h e l e n g t h o f t h a t r o d . L e t me t h i n k . 13 i n c h e s I'm s a y i n g . That means I've formed a t r i a n g l e from one p a r t o f t h a t box. I ' l l say i t s 13 i n c h e s . I t must be because, l i k e I t r i e d t o form a whole t r i a n g l e and t r y i n g t o f i n d t h e s i d e s o f t h e t r i a n g l e so t h a t I c o u l d assume what t h e l a s t s i d e i s by mathematical e q u a l i t i e s . . . S o 25, 5 squared, t h a t . o n e s i d e p l u s 12 squared which i s 144 i s e q u a l t o 169. And 169 i s 13 squared. T h e r e f o r e t h e answer i s 13, 13 i n c h e s . A c o d i n g form summary sheet was  8  9  10  11 12 13  14,  a l s o used t o summarize t h e major  p a r t s from t h e c o d i n g form and t o count e r r o r s and p r o c e d u r e s The number o f cycles- end changes was  15  also noted here.  each time t h e s u b j e c t attempts t o s o l v e t h e problem.  used.  A c y c l e occurs These  are d e t e r m i n e d  from t h e s u b j e c t ' s t a p e r e c o r d e d p r o t o c o l f o r t h e problem e i t h e r by t h e s u b j e c t h i m s e l f , i n d i c a t i n g t h a t he i s g o i n g t o s t a r t t h e problem o r from t h e sequence  o f p r o c e d u r e s used.  t h e problem each time he c y c l e s .  over  U s u a l l y the subject w i l l read  I f a s u b j e c t a l t e r s h i s method o f  a t t a c k , e i t h e r by c h a n g i n g t h e c o r e p r o c e d u r e s o r h e u r i s t i c s he i s u s i n g o r by changing h i s o v e r a l l p l a n , t h e n a change o c c u r s . o f t h e c o d i n g system i s g i v e n i n Appendix  A f u l l explanation  C.  The I n t e r v i e w Procedure  The Embedded F i g u r e s T e s t ( E F T ) , was h a l f of A p r i l ,  1974» and t h e problem  administered during the l a s t  s o l v i n g i n t e r v i e w s were h e l d  62  Reading p r o b l e m Request d e f n . o f terr.s  V  J  1  R e c a l l same p r o b l e m Recall related  problem  R e c a l l problem  type  Recall related  fact  %  Draw d i a g r a m Modify Identify S e t t i n g up  3  diagram  i  variable  i i  equations  Algorithms-algebraic Algorithms-arithmetic  3  Guessing Smoothing Analysis Templation Cases-all Cases-random Cases-systematic  j j 3 1 i 1  3  1  I  3  y  y  y i  Cases-critical Cases-sequential  i  Deduction Inverse  I  deduction  1 1  Invariation Analogy Symmetry Obtain  solution  Checking  part  V  Checking s o l u t i o n ] by s u b s t . i n e q u a t i o n by r e t r a c i n g  yi  steps  by r e a s n b l e / r e a l i s t i c  1  uncodable Exp. Exp.  concern-method  concren-aleorithm  Exp.  concern-equation  Exp. c o n c e r n - s o l u t i o n Work s t o p p e d - s o l n . Work stopped-no  soln.  FIGURE  5  j E X A M P L E OF C O D I N G S Y S T E M S T U D E N T #36, PROBLEM # 3 ( R e d u c e d b y 33?6)  63 i  Reading p r o b l e m Request d e f n . o f ter~r.3 R e c a l l same p r o b l e m Recall related  problem  R e c a l l problem  type  Recall related  fact  Draw d i a g r a m Modify Identify  J  diagram variable  S e t t i n g up e q u a t i o n s Algorithms-algebraic  3  Algorithms-arithmetic Guessing  3  Analysis Templation  t  !  Smoothing  |  i  3  3  Cases-all Cases-random Cases-systematic Cases-critical  1  Cases-sequential  i i  Deduction Inverse  deduction  3-  Invariation  !  Analogy Symmetry Obtain  solution  Checking  part  V  Checking s o l u t i o n ] by s u b s t . i n e q u a t i o n by r e t r a c i n g  steps  1  i  uncodable Exp. Exp.  |  concern-method  concren-algorithm  Exp.  \/  |  by r e a s n b l e / r e a l i s t l c  concern-equation  —  Exp. c o n c e r n - s o l u t i o n Work  stopped-soln  Work stopped-no  soln  FIGURE  6  EXAMPLE OF CODING SYSTEM STUDENT # 2k, PROBLEM # k (Reduced by 33>)  d u r i n g May and June, 1974, i n t h e s e n i o r secondary s c h o o l s t h e s u b j e c t s attended.  Each s c h o o l made a room a v a i l a b l e f o r b o t h t e s t i n g p e r i o d s .  A l l EFT's and i n t e r v i e w s were conducted by t h e e x p e r i m e n t e r . The  s u b j e c t s were t o l d t h a t at a l a t e r date t h e y would spend about  2|- hours s o l v i n g m a t h e m a t i c a l problems and t h i n k i n g a l o u d w h i l e worked on them.  they  I t was emphasized t h a t t h e purpose was t o l e a r n more  about how grade e l e v e n s t u d e n t s  s o l v e d problems and was  i n d i v i d u a l diagnosis or evaluation.  n o t t o make an  I t was a l s o emphasized t o each  i n d i v i d u a l t h a t t h e outcome o f t h e i n t e r v i e w had n o t h i n g t o do w i t h h i s grade i n A l g e b r a I I n o r would t h e i n f o r m a t i o n be made a v a i l a b l e t o h i s mathematics t e a c h e r . audio t a p e r e c o r d e d  The s u b j e c t s were t o l d t h a t t h e i n t e r v i e w s would be t o a s s i s t the interviewer i n determining  what p r o c e d u r e s  were u s e d . The EFT was a d m i n i s t e r e d period.  i n d i v i d u a l l y t o t h e s u b j e c t s i n a 50 minute  The t e s t m a t e r i a l c o n s i s t s o f a s e t o f 12 c a r d s w i t h  f i g u r e s and a s e t o f 8 c a r d s w i t h a given is  simple  simple  figures.  complex  The t a s k i s t o f i n d  f i g u r e which i s embedded i n a complex p a t t e r n .  asked t o d e s c r i b e t h e complex f i g u r e i n any way he w i s h e s .  simple  The s u b j e c t Then t h e  f i g u r e i s shown t o t h e s u b j e c t and he i s asked t o f i n d i t i n t h e  complex d e s i g n .  When he i n d i c a t e s he has found i t ,  the interviewer  s t o p s t i m i n g and t h e s u b j e c t o u t l i n e s t h e f i g u r e w i t h a s t y l u s .  I f he  is  If  c o r r e c t , t h e time i s n o t e d and t h e next problem i s p r e s e n t e d .  he i s i n c o r r e c t t h e s u b j e c t i s t o l d he i s wrong and may c o n t i n u e t o l o o k f o r t h e sample f i g u r e .  A maximum o f 180 seconds i s a l l o w e d  each o f t h e 12 problems (See W i t k i n , Oltman, R a s k i n , pp.  and Karp,  for  1971,  16, 17). A f t e r a l l 40 s u b j e c t s had t a k e n t h e EFT, t h e s c o r e s were rank  o r d e r e d and p a i r e d .  Each s u b j e c t  t o one o f two groups:  from a p a i r was randomly  t h o s e working problems i n t h e r e a l w o r l d  and t h o s e working problems i n t h e math w o r l d The  assigned setting  setting.  problem s e t s a l o n g w i t h t h e i n s t r u c t i o n s t h a t were g i v e n t o t h e  s t u d e n t s can be f o u n d i n Appendix B.  The problems were o r d e r e d so t h a t  each o f them appeared i n p o s i t i o n s 1 t o 5 e x a c t l y 4 t i m e s and each problem was p r e c e d e d by t h e same problem 4 t i m e s and f o l l o w e d t h e same problem 4 times.  T h i s gave a t o t a l o f twenty d i f f e r e n t o r d e r i n g s  problems.  The o r d e r i n g s  subjects.  Each student was g i v e n t h e same two warm up problems  for  s e t t i n g ) i n t h e same The  1974*  (except  order.  The s u b j e c t s were d i s m i s s e d A l l subjects  viewed i n c o n s e c u t i v e subjects. The  t o t h e twenty p a i r s o f  problem s o l v i n g i n t e r v i e w s were h e l d from May 6, t o June 7,  or afternoon.  the  were randomly a s s i g n e d  of the f i v e  from c l a s s e s f o r e i t h e r t h e morning  from t h e same mathematics c l a s s were i n t e r -  s e s s i o n s i n o r d e r t o minimize d i s c u s s i o n between  No i n t e r v i e w s  were h e l d on F r i d a y  i n t e r v i e w s were t a p e r e c o r d e d  of the subject. everything  afternoons.  w i t h the; microphone i n f u l l  The s u b j e c t was asked t o do h i s t h i n k i n g aloud,  t h a t came t o mind.  w r i t i n g and diagrams.  view  t o say  He was a l s o i n s t r u c t e d t o v e r b a l i z e a l l  Whenever t h e s u b j e c t  f e l l silent  f o r t e n seconds  o r more, t h e i n t e r v i e w e r would ask, "What are you t h i n k i n g now?" o r "Can  you t e l l me what you a r e t h i n k i n g about?" The  them.  o r some s i m i l a r p r o b e .  s u b j e c t s were g i v e n t h e sheet o f d i r e c t i o n s and asked t o read  The importance o f t h i n k i n g a l o u d was s t r e s s e d .  The  subject  was t o l d t h a t he was f r e e t o work t h e problems any way he saw f i t .  It  was i n d i c a t e d t h a t t h e o n l y k i n d o f q u e s t i o n t h e i n t e r v i e w e r would answer would be t o d e f i n e  any terms used i n t h e statement o f t h e problem t h a t  were u n f a m i l i a r t o t h e s u b j e c t *  I t was a l s o s t a t e d t h a t  any d i s c u s s i o n  o f t h e problem w i t h t h e i n t e r v i e w e r , i n c l u d i n g t h e c o r r e c t n e s s o f any s o l u t i o n s o b t a i n e d by t h e s u b j e c t s , would t a k e p l a c e a f t e r t h e i n t e r v i e w was o v e r .  Once a s u b j e c t s t a r t e d work on a problem he was asked t o  c o n t i n u e t o work on t h e problem and n o t t o r e t u r n t o i t a t a l a t e r t i m e . A f t e r t h e i n s t r u c t i o n s were c a r e f u l l y p r e s e n t e d was g i v e n t h e f i r s t  p r a c t i c e problem t o s o l v e .  expressing himself or f e l l  t o t h e s u b j e c t , he  I f t h e s u b j e c t had t r o u b l e  s i l e n t t o o o f t e n d u r i n g t h e time he was w o r k i n g  on t h i s problem, t h e i n t e r v i e w e r would remind t h e s u b j e c t o f t h e purpose o f t h e i n t e r v i e w and g i v e some i n s t r u c t i o n s and examples.  F o r example,  t h e i n t e r v i e w e r might work p a r t o f t h e problem t h i n k i n g a l o u d , o r i f t h e s u b j e c t had been w r i t i n g e q u a t i o n s ,  t h e i n t e r v i e w e r would go back  o v e r t h e work and v e r b a l i z e what t h e s u b j e c t might have b e e n t h i n k i n g . By t h e end o f t h e second problem, a l l t h e s u b j e c t s were t a l k i n g most o f t h e t i m e and i n d i c a t i n g what t h e y were d o i n g . As  soon as t h e s u b j e c t had f i n i s h e d p r a c t i c i n g on t h e two sample  problems and had d i s c u s s e d any r e m a i n i n g given the f i r s t  o f t h e f i v e problems.  questions  on format, he was  Each problem was t y p e w r i t t e n a t  t h e t o p o f t h e page w i t h space u n d e r n e a t h f o r t h e s u b j e c t ' s work. E x t r a paper was a l s o a v a i l a b l e . While t h e s u b j e c t was engaged i n w o r k i n g t h e g i v e n problem, t h e i n t e r v i e w e r o b s e r v e d t h e student  and i f t h e s u b j e c t f e l l  s i l e n t , he was  encouraged t o v o c a l i z e as much o f h i s t h i n k i n g as p o s s i b l e .  When each  problem was completed, t h e s u b j e c t had t h e o p t i o n o f e i t h e r s t o p p i n g temporarily t o rest o r proceding  t o t h e n e x t problem.  Most  subjects  p r e f e r r e d t o work c o n t i n u o u s l y on t h e problem s e t . Once t h e i n t e r v i e w s e s s i o n was completed and t h e s u b j e c t had l e f t , t h e i n t e r v i e w e r p l a y e d t h e t a p e r e c o r d i n g back, matching t h e p r o t o c o l  67 w i t h t h e w r i t t e n work. the  T h i s was done so t h e two c o u l d be combined when  coding took p l a c e .  The C o d i n g P r o c e d u r e  As a f i n a l the  check on t h e c o d i n g system and f o r p r a c t i c e c o d i n g ,  w r i t e r coded 40 problems  chosen from t h e main s t u d y .  These  included  a l l f i v e problems t a k e n from b o t h t h e r e a l w o r l d and math w o r l d At the  settings.  t h e t i m e i t was f e l t t h a t t h e c o d i n g system was i n d e e d a c c o u n t i n g f o r p r o c e d u r e s u s e d by t h e s u b j e c t s i n t h e v a r i o u s problems. The f i v e problems were randomly o r d e r e d as f o l l o w s :  3,  1,  4,  Then t h e p r o t o c o l s o f a l l 40 s u b j e c t s f o r problem number 3 were assigned.  5,  2.  randomly  These p r o t o c o l s were t h e n coded by t h e w r i t e r u s i n g t h e f i n a l  c o d i n g form.  The p r o c e d u r e was r e p e a t e d f o r t h e o t h e r 4 problems,  a t o t a l o f 200 p r o t o c o l s .  giving  E a c h problem was coded on a s e p a r a t e c o d i n g  form. The c o d i n g was done u s i n g t h e t a p e r e c o r d e d p r o t o c o l s and t h e w r i t t e n work o f each s u b j e c t . To a s s e s s t h e r e l i a b i l i t y and i n t r a c o d e r r e l i a b i l i t y  o f the coding, both i n t e r c o d e r  o f t h e c o d i n g were u s e d .  been t r a i n e d d u r i n g t h e p i l o t  study.  A second coder had  She and t h e w r i t e r spent n i n e  hours w o r k i n g t o g e t h e r c o d i n g s i x t e e n problems. coded another 14 problems a l o n e .  reliability  The second coder t h e n  Like the writer,  she was v e r y f a m i l i a r  w i t h t h e h e u r i s t i c s used i n t h e s t u d y and had had s e v e r a l y e a r s e x p e r i e n c e t e a c h i n g secondary For  mathematics.  each o f t h e 5 problems, 4 p r o t o c o l s were s e l e c t e d a t random.  These were t h e n coded by b o t h t h e w r i t e r and t h e second coder.  The  r e s u l t s o f t h e s e c o d i n g s were compared t o t h e w r i t e r ' s o r i g i n a l c o d i n g s of  t h e same p r o t o c o l s .  Four t y p e s o f c o d i n g e r r o r s were i d e n t i f i e d .  These are i l l u s t r a t e d i n F i g u r e 7 1)  and d e s c r i b e d h e r e :  A Blank o c c u r s when a p r o c e d u r e i s coded by one coder and not  the other.  The  assumption  i s that t h i s i s the o n l y e r r o r .  That i s ,  i n t h e example coder A missed t h e p r o c e d u r e o f i d e n t i f y i n g a v a r i a b l e which was 2) is  C a l l i n g a p r o c e d u r e by a d i f f e r e n t name o c c u r s when a p r o c e d u r e  coded d i f f e r e n t l y by t h e two 3)  two  coded by coder B.  coders.  Coding p r o c e d u r e s i n d i f f e r e n t o r d e r o c c u r s when t h e o r d e r o f  o r more s u c c e s s i v e p r o c e d u r e s i s r e v e r s e d by one o f t h e c o d e r s .  I t i s assumed t h a t t h e o n l y e r r o r made i s t h e r e v e r s a l o f t h e p r o c e d u r e s . In  t h e example, r e c a l l o f r e l a t e d f a c t 4)  and t e m p l a t i o n were r e v e r s e d .  The l a s t e r r o r , c a s e s , o c c u r s when one c o d e r i n d i c a t e s t h e  use o f t h e h e u r i s t i c s o f cases, s t o p s , and t h e n codes i t s t a r t i n g a g a i n and t h e o t h e r coder codes i t as a c o n t i n u o u s p r o c e s s . A summary o f t h e p r o c e d u r e s coded d i f f e r e n t l y , under each o f t h e c a t e g o r i e s , by t h e two  c o d e r s , i s found i n T a b l e 1 and t h e  by t h e w r i t e r i s found i n T a b l e  2.  As a measure o f r e l i a b i l i t y , was  code-recode  t h e p e r c e n t a g e o f i t e m s coded  identically  c a l c u l a t e d u s i n g a f o r m u l a d e r i v e d from McGrew (1971» p. 24)..,  The f o r m u l a used i n t h i s s t u d y i s a more c o n s e r v a t i v e measure than t h a t of McGrew because  i t t a k e s i n t o account t h e number o f i t e m s  coded  i d e n t i c a l l y by b o t h coders o n l y once. Let  the number o f items coded by A which agree w i t h t h o s e coded  B e q u a l a, and t h e remainder  e q u a l a'.  Then a + a' = t o t a l coded by  A.  by  69 CODER "A" Recall  related  fact  3  3  /  Draw diagram Modify  diagram  Identify Setting  variable  3  up e q u a t i o n s  T  Guessing  \  Smoothing  3  3  Algorithms-arithmetic  3  1  i  z  Algorithms-algebraic  —  3-  3 3  3  /  V  3 3  Templation Cases-all  /  Cases-random  3  Cases-systematic  Checking  part  .C  o o c. uu  OH  a  •3 0>  w  fact  Draw diagram  Identify  <H  *o p O X5  to H  variable  S e t t i n g up e q u a t i o n s Algorithms-algebraic Algorithms-arithmetic  O  |  -p c/2  I  1  ti  * H | (DO-  W  CD  fl> t>  ft  !  <H -P iH  U j o 3 i  !  |  O  •+-> .  U •H  i  !  8  a  in  uU >  i I  .3. T diagram i  Recall related  Modify  o o  0) -p  >>  M  •  !  c• <D — tJ O n  CODER B "  V  i.  n  w  V  •/  1  !  V 3  %  i  3  1 3 3  3  3  3  3  Guessing Smoothing Analysis Templatiori  -V  3  2  3  3  C a s e s - a l !L  V V  V i  Case3-randor n Cases-systemati(  /  Checking par t  FIGURE 7 EXAMPLES OF CODER ERROR (Reduced by 33$)  3  70  TABLE 1 TYPES AND NUMBERS OF DISAGREEMENTS BETWEEN TWO DIFFERENT CODERS  FIRST CODER  SECOND CODER  BLANKS Reading Problem -2 Modify Diagram -3 I d e n t i f y V a r i a b l e -2 S e t t i n g up Equations -4 Guessing -3 Templation -2 A n a l y s i s -1 Concern -2  Reading Problem -3 R e c a l l R e l a t e d F a c t -2 Modify Diagram -4 I d e n t i f y V a r i a b l e -2 S e t t i n g up E q u a t i o n s -1 Guessing Smoothing -2 Templation -1 Checking P a r t -2 Concern -3  CODING PROCESS DIFFERENTLY Templation S e t t i n g up E q u a t i o n s Diagram Checking P a r t Checking P a r t Templation A r i t h m e t i c Algorithms  R e c a l l R e l a t e d F a c t -2 A l g e b r a i c A l g o r i t h m s -1 Modify Diagram -1 Concern -2 C a s e s ( A l l ) -1 Cases(Random) -1 A l g e b r a i c A l g o r i t h m s -2  ORDER REVERSED Templation - R e c a l l R e l a t e d F a c t -2 Modify Diagram - Smoothing - C a s e s ( S t a r t ) -1  CASES Continuous  - S t a r t - S t o p -3  71  TABLE 2 TYPES AND NUMBERS OF DISAGREEMENTS BETWEEN CODE AND RECODE  RECODE  FIRST CODE  BLANKS Reading Problem -1 R e c a l l Problem Type -1 R e c a l l R e l a t e d Fact -4 I d e n t i f y V a r i a b l e -2 S e t t i n g up E q u a t i o n s -1 A r i t h m e t i c A l g o r i t h m s -3 Guessing -2 T e m p l a t i o n -1 Checking P a r t -2 Concern -3  Reading Problem -3 Modify Diagram -2 I d e n t i f y V a r i a b l e -1 S e t t i n g up E q u a t i o n s -2 A r i t h m e t i c Algorithms -1 Guessing -1 Checking P a r t -2 Concern -1  CODING PROCESS DIFFERENTLY Templation Obtain S o l u t i o n Checking P a r t R e c a l l Related Fact Checking P a r t  R e c a l l R e l a t e d F a c t -2 Checking P a r t -1 A r i t h m e t i c A l g o r i t h m s -1 Templation. -1 Concern -1  ORDER REVERSED Smoothing - Diagram -1 R e c a l l R e l a t e d Fact - T e m p l a t i o n -1  CASES Continuous - S t a r t - S t o p  -1  72 Similarly,  a + b* = t o t a l coded by  Then a + a' + b  B.  = t o t a l number o f d i f f e r e n t  1  and B, c a l l t h i s sum  i t e m s coded by b o t h A  T.  Then a/T = y, t h e percentage o f i t e m s coded i d e n t i c a l l y by b o t h coders i s t h e measure o f For  intercoder  a = 294,  reliability.  reliability:  a' = 40, b' = 35,  T = 369  The p e r c e n t o f i t e m s coded i d e n t i c a l l y by two d i f f e r e n t is For  coders  80. intracoder  a = 307,  reliability:  a' = 27,  T = 355  b' = 2 1 ,  The p e r c e n t o f i t e m s coded i d e n t i c a l l y on t h e code/recode  by t h e  same coder i s 86. These r e l i a b i l i t i e s  were f e l t t o be s u f f i c i e n t l y h i g h .  s e v e r a l i m p o r t a n t f a c t o r s which may being high.  have c o n t r i b u t e d t o t h e  There  are  reliabilities  Both coders were v e r y f a m i l i a r w i t h t h e second y e a r a l g e b r a  course, b o t h h a v i n g s e v e r a l y e a r s t e a c h i n g e x p e r i e n c e at t h i s l e v e l . Both coders were v e r y f a m i l i a r w i t h t h e c o d i n g system heuristics.  I n a d d i t i o n a g r e a t d e a l o f t i m e was  and w i t h t h e  spent on t h e c o d i n g  itself. MacPherson's model appears t o be a r e l i a b l e model t o use f o r a n a l y z i n g m a t h e m a t i c a l problem who:  (l)  solving.  I t can be used e f f e c t i v e l y by a p e r s o n  knows and understands t h e h e u r i s t i c s  c o r e and i s f a m i l i a r w i t h t h e background (3)  i s f a m i l i a r with t h i s coding  system.  well,  (2)  of the subjects,  knows t h e and  Chapter  A N A L Y S I S AND  In The  this  first  Chapter tested  I.  s t u d y two  hypotheses  statistically.  t e s t hypotheses  1-6.  are r e s t a t e d below.  H y p o t h e s e s 7 and  statistically.  A Flanders  i n t o the sequence o f procedures second  the c o d i n g system.  Problem  their intercorrelations  H y p o t h e s e s 9 and  (See Amidon and  were  designed  8 were t e s t e d u s i n g  to  Pearson 10  were  H o u g h , 1967)  not  type  u s e d t o g a i n some i n s i g h t  used by t h e  phase i n v o l v e d p a s t hoc  was  data.  stated i n  H y p o t h e s e s 1-8  A regression analysis procedure  i n t e r a c t i o n m a t r i x ( a p r o c e s s m a t r i x ) was  The  analysis of the  a n a l y s i s of the hypotheses  product-moment c o r r e l a t i o n c o e f f i c i e n t s . tested  RESULTS  phases were i n v o l v e d i n t h e  phase c o n s i s t e d o f t h e The  IV  students i n t h i s  study.  analysis of the data obtained  from  s o l v i n g b e h a v i o r s were examined i n terms  as w e l l as w i t h t h e u s e  Research  of  o f r e g r e s s i o n analysis.,  Hypotheses  HI:  Problem context w i l l not are used by a s t u d e n t .  c o n t r i b u t e t o t h e number o f t i m e s  heuristics  H2:  Problem c o n t e x t w i l l n o t c o n t r i b u t e t o t h e number o f h e u r i s t i c s used by a student.  H3:  Problem context w i l l not obtained by a student.  H4:  F i e l d i n d e p e n d e n c e w i l l n o t c o n t r i b u t e t o t h e number o f h e u r i s t i c s are used by a s t u d e n t .  times  H5:  F i e l d i n d e p e n d e n c e w i l l not c o n t r i b u t e t o t h e number o f h e u r i s t i c s used by a student.  different  H6:  F i e l d i n d e p e n d e n c e w i l l n o t c o n t r i b u t e t o t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d by a s t u d e n t .  different  c o n t r i b u t e t o t h e number o f c o r r e c t s o l u t i o n s  74 H7s  T h e r e i s n o c o r r e l a t i o n b e t w e e n t h e number o f t i m e s h e u r i s t i c s are u s e d b y a s t u d e n t a n d t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d .  HS:  T h e r e i s n o c o r r e l a t i o n between t h e number o f d i f f e r e n t h e u r i s t i c s u s e d b y a s t u d e n t a n d t h e number o f c o r r e c t s o l u t i o n s he o b t a i n e d .  H9:  Problem context w i l l not observably used b y a subject.  H10:  a f f e c t t h e sequence o f procedures  F i e l d independence w i l l not observably procedures used by a subject.  a f f e c t t h e sequence o f  Method o f A n a l y s i s  The  r e g r e s s i o n a n a l y s i s approach employed f o r t h i s study  has been  d e s c r i b e d b y many w r i t e r s ( e . g . K e r l i n g e r a n d P e d h a z u r , 1973)* (1971) d e s c r i b e s conventional (2)  three  advantages t h a t r e g r e s s i o n  analysis of variance:  l e s s data processing  time,  Walberg  analysis has over the  ( l )the use o f continuous v a r i a b l e s ,  a n d (3)  d i r e c t comprehensive  estimates  of t h e magnitude and s i g n i f i c a n c e o f t h e independent v a r i a b l e s e f f e c t s on t h e d e p e n d e n t v a r i a b l e . important  t othis  s m a l l sample s i z e levels  The f i r s t  andt h i r d o f these are e s p e c i a l l y  study.  With t h e u s e o f continuous v a r i a b l e s and a  (N=40),  p r e c i s i o n i n grouping scores  could be l o s t  i n t o two o r three  and t h e sample s i z e i s t o o s m a l l t o m a i n t a i n  cell  s i z e s i f more l e v e l s w e r e added. I n an e x p l o r a t o r y  study  such a s t h i s one, t h e t h i r d  advantage  t o have d i r e c t a p p l i c a t i o n , t h a t i s , t h e u s e o f t h e m u l t i p l e coefficient:  appears  regression  when s q u a r e d i t r e v e a l s d i r e c t l y how much v a r i a n c e  i nthe  dependent v a r i a b l e i s a s s o c i a t e d w i t h o r accounted f o r b y t h e independent variables. A separate  regression equation  o f t h e h y p o t h e s e s 1 t o 6. The  Triangular Regression  was d e f i n e d  f o r t h e a n a l y s i s o f each  The r e g r e s s i o n a n a l y s e s Package  were performed  using  (TRIP) ( B j e r r i n g and Seagraves,  1974)  a v a i l a b l e a t t h e Computing Center o f t h e U n i v e r s i t y o f B r i t i s h  Columbia.  75 A l t h o u g h i n some c a s e s t h e  r e l a t i o n s h i p between the  logarithmic  range of  i n nature, the  small that in  very l i t t l e  f i r s t two  model.  the  the  d e t e r m i n e i f any  significance  an  exploratory  l e v e l which i s not  important implications chosen t o t e s t the a Type  1 e r r o r by  statistical  may  be  lost.  A  hypotheses of t h i s r e j e c t i n g the  significance study.  hypothesized variables,  w r i t e r were o b t a i n e d from the  the  number o f t i m e s a s u b j e c t  and  the  n u m b e r o f t i m e s he protocols  o f t i m e s and  that  two  protocols  the  otherwise  .10  has  p r o b a b i l i t y of  variables of the  attempted to  been  making  given for  each  of i n t e r e s t  to  subjects.  These  begin a problem again  changed h i s method of  do  the  same t h i n g  attack.  It  Many o f t h e s e changes i n v o l v e d  an  attempt t o  g a i n some i n s i g h t  the  subject,  a p r o c e s s m a t r i x was  appeared  the  use  of  They are:  (l)  Core,  tried heuristics  core.  i n t o the  sequence of  developed from the  procedures coding  This process m a t r i x contains s i x major categories f o r  procedures.  are  (cycles)  each time w h i l e o t h e r s  w h i l e o t h e r s were more d i r e c t l y r e l a t e d t o  systems.  of  a  some s u b j e c t s w o u l d a t t e m p t a p r o b l e m a n u m b e r  continue to  a v a r i e t y of methods.  used by  level  mathemat-  exist,  chosen  n u l l h y p o t h e s i s w i l l be  the  In  The  solving  do  s h o u l d be  exists  test.  Besides the  from the  stringent  exploratory  and  study, i f differences too  linear  variables.  difference  attacking  agreed  meaningful  T h i s i s an  between groups o f i n d i v i d u a l s i n methods o f As  level for  so  and  assumed b e t w e e n t h e  considered carefully.  study u s i n g MacPherson's model t o  i c a l problems.  (F- p r o b a b i l i t i e s  appropriate significance  i n t e r p r e t a t i o n m u s t be  was  n o t e d between a l o g a r i t h m i c  H e n c e a l i n e a r r e l a t i o n s h i p was  Deterinining  was  independent v a r i a b l e s  s t a t i s t i c a l difference  d e c i m a l p l a c e s ) was  variables  (2)  Heuristics,  (3)  analyzing Solution,  (4)  Checking, The  matrix if  are taken  Work s t o p p e d  from t h e coding  sheets  (See  Figures)-  and e n t e r e d  i n t h e process  i n t h e a p p r o p r i a t e row and column.  F o r example:  reads t h e p r o b l e m and t h e n draws a diagram, one i s added t o  entry i n the f i r s t t h e student  row,  row, f o u r t h column.  A f t e r drawing t h e diagram,  u s e s t e m p l a t i o n , one i s added t o t h e e n t r y i n t h e f o u r t h  t h i r t e e n t h column.  which are nested the  C o n c e r n , a n d (6)  ( F i g u r e s g a n d 9)  a student  the if  data  (5)  This  system does n o t account f o r t h e p r o c e d u r e s  i n o t h e r s , such as t h e u s e o f a l g o r i t h m s  o r diagrams  s u b j e c t may b e u s i n g w h i l e h e i s u s i n g r a n d o m o r s y s t e m a t i c  A l s o , when a s u b j e c t i s c h e c k i n g  cases.  e i t h e r p a r t o f h i swork o r h i s s o l u t i o n ,  t h e method he i s u s i n g i s n o t n o t e d  i n the matrix.  D i f f e r e n t p a r t s o f t h e m a t r i x i n d i c a t e d i f f e r e n t k i n d s o f problemsolving procedures.  These areas  a r e i d e n t i f i e d i n F i g u r e 8 and d e s c r i b e d  as f o l l o w s : AREA A.  This area i n d i c a t e s t h e problem s o l v i n g procedures r e l a t e d t o the core. A l l procedures are from core t o core.  AREA B.  The c e l l s i n t h i s a r e a i n c l u d e a l l o f t h e c o r e p r o c e d u r e s are followed by t h e use o f a h e u r i s t i c .  AREA C.  T h i s group o f c e l l s i n c l u d e s a l l o f t h e core of a checking procedure.  AREA D.  This area i n d i c a t e s t h e concern a core procedure.  AREA E .  T h i s area c o n s i s t s o f t h e h e u r i s t i c s which a r e f o l l o w e d b y t h e use o f a core procedure.  AREA F.  This area i n d i c a t e s t h e use o f a h e u r i s t i c  AREA G.  These c e l l s i n c l u d e t h e c h e c k i n g procedures w h i c h f o l l o w t h e u s e of a h e u r i s t i c .  AREA H.  This group o f c e l l s of a h e u r i s t i c .  AREA I .  This area represents t h e core procedures used a f t e r a was o b t a i n e d .  AREA J .  T h i s shows a l l o f t h e h e u r i s t i c s u s e d a f t e r a s o l u t i o n  which  followed by the use  w h i c h i s shown a f t e r t h e u s e o f  i n d i c a t e s t h e concern  followed by a h e u r i s t i c .  shown a f t e r t h e u s e  solution  was  77 1  2 3  4  5 6 ? 8  9  10  11 12 13  14  15 16  17  18  19  20  22 23  24  25 26 27  1  2 3 4  5 6  A  B  C  D  F  G  H  I  J  K  L  M  N  0  P  Q  R  S  T  7 8 9  10 11 12 13 14  15 16 17 18 19  20 21 22 23 24  25 26 27  FIGURE 8 THE PROCESS MATRIX WITH IDENIFIED AREAS  78 FIGURE 9 PROCESS MATRIX CATEGORIES CATEGORIES Core  1. Reading the problem 2. R e c a l l problem ( i n c l u d e s : R e c a l l same problem Recall related problem R e c a l l problem type) 3. R e c a l l r e l a t e d f a c t 4. Draws diagram 5. Modify diagram 6. I d e n t i f y v a r i a b l e 7. S e t t i n g up equations 8. A l g o r i t h m s - a l g e b r a i c 9. A l g o r i t h m s - a r i t h m e t i c 10. Guessing  Heuristics  11. 12. 13.  Smoothing Analysis Templation 14. A l l cases 15. Random cases 16. S y s t e m a t i c cases ( i n c l u d e s : S y s t e m a t i c cases Sequential cases) 17. C r i t i c a l cases 18. Deduction 19. I n v e r s e d e d u c t i o n 20. V a r i a t i o n  Solution  21.  Obtain  Checking  22. 23.  Checking Checking  Concern  24. 25. 26. 27.  Express Express Express Express  Work Stopped  28. Work s t o p p e d - s o l u t i o n 29. Work stopped-no s o l u t i o n  solution part solution concern concern concern concern  about method about a l g o r i t h m about e q u a t i o n about s o l u t i o n  79 obtained. AREA K.  The c e l l s i n t h i s a r e a i n d i c a t e t h e o b t a i n i n g o f a followed by a checking procedure.  AREA L .  T h i s a r e a i n d i c a t e s t h e c o n c e r n shown f o l l o w i n g a  AREA M.  These c e l l s r e p r e s e n t t h e use o f a c h e c k i n g procedure by t h e use o f core.  AREA N ,  This i n d i c a t e s t h e use o f h e u r i s t i c s f o l l o w i n g a checking  AREA 0.  The c e l l s i n t h i s a r e a r e p r e s e n t t h e u s e o f two c o n s e c u t i v e checking procedures.  AREA P.  This area i n d i c a t e s t h e concern procedure.  AREA Q.  T h e s e c e l l s show t h e u s e o f a c o r e p r o c e d u r e expression o f concern.  AREA R.  The c e l l s i n t h i s type o f concern.  AREA S.  These c e l l s i n d i c a t e  AREA T.  These c e l l s i n c l u d e t h e c o n s e c u t i v e e x p r e s s i o n o f concern.  The  solution  solution. followed  procedure.  shown f o l l o w i n g a c h e c k i n g  f o l l o w i n g an  a r e a show t h e h e u r i s t i c s u s e d  following  concern f o l l o w e d by a checking  some  procedure.  m a t r i x i n d i c a t e s t h e amount a n d t o some e x t e n t a p a t t e r n o f t h e  procedures  used t o solve mathematical  i n t h e process matrix.  problems according t o t h e c a t e g o r i e s  The m a t r i x a l s o i n d i c a t e s t h e number o f t i m e s  t h a t s u b j e c t s change c o r e and h e u r i s t i c s used w i t h i n each  a n d t h e amount o f p r o c e d u r e s  category.  Students working problems i n t h e mathematical  w o r l d s e t t i n g were  coded 1 and t h o s e w o r k i n g p r o b l e m s i n t h e r e a l w o r l d s e t t i n g were coded 2. " T o t a l H e u r i s t i c s " i s t h e t o t a l number o f a l l h e u r i s t i c s coded f o r a l l f i v e problems.  " D i f f e r e n t H e u r i s t i c s " i s t h e number o f d i f f e r e n t  heuristics  coded f o r a l l f i v e p r o b l e m s and t h e c o r r e c t s o l u t i o n i s t h e t o t a l number of correct  s o l u t i o n s f o r t h e f i v e problems.  T h e means a n d s t a n d a r d  d e v i a t i o n s of; t h e h y p o t h e s i z e d v a r i a b l e s . a r e g i v e n i n T a b l e 3.  The r e a d e r  s h o u l d b e a w a r e o f t h e d i s t r i b u t i o n o f some v a r i a b l e s i n i n t e r p r e t i n g t h e r e s u l t s from t h e r e g r e s s i o n a n a l y s i s . for several selected variables.  Appendix  D contains t h e histograms  E x t r e m e s c o r e s may h a v e i n f l u e n c e d t h e  80 TABLE 3 MEANS AND STANDARD D E V I A T I O N S FOR MEASURES OF HYPOTHESIZED V A R I A B L E S (N =  Field  Independence  Total  Heuristics  Different  8.60  3.00 10.37 1.85 1.15  Heuristics  Changes Solution  s t a t i s t i c a l r e s u l t s , b u t because  o f t h e s m a l l sample s i z e  scores were n o t d e l e t e d from t h e s t a t i s t i c a l i n t e r c o r r e l a t i o n s between t h e s e  Deviation  20.03 7.90 1.70 4.02 2.30 .89  143.53  Cycles  Correct  Standard  Mean  Variable  40)  analysis.  (N = 40)  these  T a b l e 6 shows t h e  variables.  I t was n o t e d i n C h a p t e r I I I t h a t o n e o f t h e r e a s o n s f o r c h o o s i n g these p a r t i c u l a r problems the  subjects.  T h e mean f o r t h e c o r r e c t n u m b e r o f s o l u t i o n s i s 1,15  i m p l i e s t h e problems indicated. the  was t h a t t h e y c o u l d b e s o l v e d b y about h a l f o f  may h a v e b e e n m o r e d i f f i c u l t t h a n t h e p i l o t  Only problems  2 and 5 were answered  study  c o r r e c t l y b y about h a l f o f  subjects. The  areas.  discussion of the results of the analysis i sdivided into three The i n f l u e n c e o f problem  context i s examined f i r s t  r e l a t e d hypotheses.  The second  r e l a t e d hypotheses.  In thethird  from t h e coding system of  which  area d e a l s w i t h f i e l d independence  The s e c t i o n d e a l s w i t h t h e i n f l u e n c e  b o t h c o r e and h e u r i s t i c p r o c e d u r e s on problem  Problem  and i t s  area, t h e r e s u l t s o f behaviors derived  arediscussed.  Results o f A n a l y s i s - Problem  along with the  solving.  Context  c o n t e x t appears t o have had l i t t l e  effect  s o l v i n g procedures used b y t h e subjects i n t h i s  study.  on t h e problem The r e s u l t s o f t h e  81 r e g r e s s i o n a n a l y s i s performed variable  a r e g i v e n i n T a b l e 4.  w i t h problem  context  as t h e  independent  The r e s u l t s o f h y p o t h e s e s  1,  2,  3,  and 9  are g i v e n as f o l l o w s :  1:  Hypothesis  No s i g n i f i c a n t d i f f e r e n c e  (P =  .5195)  was f o u n d  t o t a l number o f d i f f e r e n t h e u r i s t i c s u s e d b y s t u d e n t s in  solving  a r e a l w o r l d s e t t i n g and s t u d e n t s s o l v i n g p r o b l e m s i n a  i nthe  problems  mathematical  setting. Hypothesis  2:  No s i g n i f i c a n t d i f f e r e n c e  (P =  .1556)  was f o u n d  i n the  number o f c o r r e c t s o l u t i o n s o b t a i n e d b y s t u d e n t s s o l v i n g p r o b l e m s i n a r e a l w o r l d s e t t i n g and s t u d e n t s s o l v i n g p r o b l e m s i n a math w o r l d setting.  A l t h o u g h t h e r e was n o s i g n i f i c a n t d i f f e r e n c e f o r t h e h y p o t h e s i s  t e s t e d , t h e .1556 trend. 1.35  p r o b a b i l i t y l e v e l may b e s u f f i c i e n t t o i n d i c a t e  The mean f o r c o r r e c t s o l u t i o n s f o r t h e r e a l w o r l d  c o m p a r e d t o .95  a  subjects i s  f o r t h e math w o r l d s u b j e c t s .  The r e s u l t s o f p o s t h o c a n a l y s e s u s i n g l i n e a r r e g r e s s i o n i n d i c a t e d t h a t problem  c o n t e x t d i d n o t a f f e c t t h e number o f c y c l e s o r changes  made b y a s t u d e n t .  No s i g n i f i c a n t d i f f e r e n c e  between t h e number o f t i m e s ( c y c l e s ) and t h e problem (P = of  .4225)  .6232)  w  as  a student i s w i l l i n g t o a t t a c k a  context.  N o r was t h e r e a s i g n i f i c a n t  between t h e number o f t i m e s  a t t a c k i n g a problem  (P =  and t h e problem  a student  found  problem difference  changes h i s method  context.  T h e p r o c e s s m a t r i c e s f o r h y p o t h e s i s 9 a r e g i v e n i n F i g u r e s 10 with the percent o f procedures 12  a n d 13.  Hypothesis  In  e v a l u a t i n g h y p o t h e s i s 9,  and  used i n each c a t e g o r y g i v e n i n F i g u r e s  9 was n o t t e s t e d f o r s t a t i s t i c a l t h e f o l l o w i n g major areas  significance.  from t h e process  m a t r i c e s were c o n s i d e r e d : 1.  Moves from  core t o core o r t o h e u r i s t i c s  (Areas A and B)  2.  Moves from  h e u r i s t i c s t o core o r t o h e u r i s t i c s  ( A r e a s E a n d F)  11  TABLE 4 RESULTS OF REGRESSION ANALYSIS WITH PROBLEM CONTEXT AS INDEPENDENT VARIABLE (N=40)  F - VALUE TO ENTER/REMOVE  PROBLEM CONTEXT  ,4478  .5145  .1077  .0116  .9762  .3312  .1581  .0250  CYCLES  .2533  .6232  .0802  .0064  CHANGES  .6722  .4225  .1319  .0174  .1556  .2267  .0514  TOTAL HEURISTICS DIFFERENT HEURISTICS  CORRECT SOLUTION  2.06l  2  F - PROB.  R WITH DEPENDENT VARIABLE  RSQ  SOURCE OF VARIATION  DEPENDENT VARIABLE  3  Each of These i s a Separate Simple Regression. P r o b a b i l i t y of Making a Type 1 E r r o r by Rejecting N u l l Hypothesis, that i s , Claiming S t a t i s t i c a l Significance. -^The Proportion of Variance i n the Dependent Variable Accounted f o r by Problem Context.  83  27  3 18  1  1  2  5  6  7  8  71 12  27  87  7  20  2 5  4  4  10  5  3  4  7  4  U  8  7 1 9  5  4  6  2  4  9 10  11 12  26  13 12  15 16  14  17 18 19 20  9 2  21  22  7  14  23-  24  25  26  27  29  29  1  7  6  7  1  5  l  2  8  2 3  8  3  9  4  39  13  19  5  4  1  4  6  10  4  7  44  5 2  8  19  9  10  10  9  11  2  12  1  13  13  3  15  3  1  16  1  3  3 2 26  l  1  3  3  1  2  2 2  2  4  4  1 1  4 1  l  4  61 3  1  7  8  6  12 62 5 5  1  12  1  8  24  6  3  2  31  4  8  2  2  3 43  7  4  4  3  2 21  2  3  2  8  i  5  4  3 48  i  2  1  2  1  1  6  2  4  3  ,3  2  l  1  2 3  3  2  3  1  4  32  2  1 1  1 8  2  6  6  1  1  1 4  8  X  4  1  52  2 1  3  2 3  l  14 1  1  1  1 1  1  1  2  22  1  6  17  1  i 2 1 1  5  3  1  1  18  1  19  i  1  20  1  21  21  3 5 2  22  20  3 5  23  14  2 2  24  3  25  l  26  7  27  9  5 8  3  1  7  1 2  1  4  6 1  3  12  4  1  3 4  1  1  4  4  1  1 1  4  1  3  1  18 2  4  1  3 2  l  8  5 2 1? * x  1 1  1 • 1  J  22  3 9  1  1 1  l  2  1 2  1  2 27  1  2  FIGURE 10 PROCESS MATRIX»FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR 100 MATH WORLD PROBLEMS (N=20)  84  1 2 3 4 5 6 1  21  3 12 64  2 3 4 5  2  6  8  7  19  8  8  2  2  1  6  9  9 17  4  4  l  2  1  1  2 1  5  l  9  12  10  6  -11  5  12  1  13  10  1  5  15  8  1  2  16  1  1  1  2  2  4  2  3  2  3  4  8  36  15  1  1 3  6  11 12  5  1 56  5  8 64  6  67  1  2  1  5  3  1  2  1  3  22  23  24  2  6  15  2  1  1 3  2  l  3  1  1  l  1  17  1  16  14  5  42  11  35  9  5  1  4  6  3  6  7 1  1  1  3  1  9  4  4  3  1  4  1  3  2  1  2  4  25 26 27  28 2 9  4  10  3  1  1  1  1  2 2  3  21  1  2  1  17 18 19 20  2  1  3 24  1  15 16  8  1 2  14  15  2 10  13 7  9  35  13 15  2  4  7 1  1 1 1 3  1  l  2  2  2  5  5  1  3  1  7  1 1  9 10  2  1 1 41  3 1  31 31  1 2 24" 1  6  7  1  3  11  2  6  2  1  1  1  2  3  1  1  1  '  14 2  1  2  1  17 18  2  2  1  18 5  1  1  1  1  1  19  1  20 21  19  22  18  23  15  24  4  25  2  2  2 4 14  3 1  4  2  1 1  1  1  26  2  27  9  3 2  1  9  3  3  4  2  2  1  1  2  5  3 1  2 1 1 1  4  1  2  5  3  1  1  1 1 1 1  3  8  3  4  1  1  4  1  2  49  2  12  1 1  3  2  22 4  1 9  22  1  3  2  3 1  3  2  1  1 5  FIGURE 11 PROCESS MATRIX*FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR 100 REAL WORLD PROBLEMS (N=20)  11  2  85 1 2 3 4 5 6 7 8 9  10  11 12 131** 1516 1718 19 20  22 23  2U  25 26 27  1 2 3 if  5  62  7  5  2  2  1  . 1  2  1  2  1  5  1  3  1  7 8 9  10 11 12 13 14 15  16 17 18 19  20 21 22 23  1  24  25 26 27  FIGURE 12 PERCENT OF PROCEDURES USED IN EACH CATEGORY - MATH WORLD (N=20)  1  86  1  2 3  4  5  6  54  7  8  9 10  11 12  13  14  15 16  17 18 19 20  22  23  24  25  7  3  7,  3  1  •  2  1  4  1  6:  2  1  1  3  1  1  FIGURE 13 PERCENT OF PROCEDURES USED IN EACH CATEGORY - REAL WORLD (N=20)  26  3  27  87 3.  M o v e s c o n c e r n e d w i t h c h e c k i n g ( A r e a s C, G, M, a n d N )  4.  Moves c o n c e r n e d w i t h e x p r e s s i o n o f c o n c e r n  I n comparing  ( A r e a s D, H, Q, a n d R ) .  t h e s e a r e a s , f o r t h e two g r o u p s (math w o r l d and r e a l w o r l d )  the researcher considered the percentage g r o u p as w e l l as comparing  o f moves i n e a c h  which  s t u d e n t s have a tendency  H y p o t h e s i s 9: groups,  area f o r each  t h e n u m b e r a n d t y p e s o f moves f o r e a c h  That i s , f o r a g i v e n procedure, i s t h e r e a procedure  f  procedure.  ( o r procedures)  t o use next?  The p a t t e r n o f p r o c e d u r e s u s e d b y s u b j e c t s i n t h e t w o  r e a l w o r l d a n d m a t h w o r l d , was n o t o b s e r v a b l y d i f f e r e n t .  A l t h o u g h t h e s u b j e c t s w o r k i n g problems i n t h e math w o r l d s e t t i n g a higher percent o f core r e l a t e d procedures i n the real world setting  as compared t o t h e s u b j e c t s  (See A r e a A and p e r c e n t a g e s  a r e a s i n F i g u r e s 12 a n d 13)>  used  o f core  these differences are not great.  related Two  e x c e p t i o n s i n v o l v e t h e n u m b e r o f t i m e s s u b j e c t s move f r o m r e a d i n g ( # l ) t o s e t t i n g u p e q u a t i o n s (#7) the problem.  and from s e t t i n g up e q u a t i o n s t o r e a d i n g  I n b o t h o f t h e s e i n s t a n c e s t h e m a t h w o r l d s u b j e c t s made  m o r e moves t h a n t h e r e a l w o r l d s u b j e c t s , . . . 4 4 t o 19 a n d 87 t o 37 •These d i f f e r e n c e s come m a i n l y f r o m p r o b l e m : n u m b e r 2,  respectively.  where t h e number o f  moves f r o m e q u a t i o n s t o r e a d i n g a n d r e a d i n g t o e q u a t i o n s f o r t h e m a t h w o r l d a n d r e a l w o r l d i s 31 t o 13 a n d 66 t o 23  respectively.  W i t h o v e r h a l f o f t h e moves i n A r e a A, t h e d i v i s i o n s i n t h e p r o c e s s m a t r i x may n o t h a v e b e e n f i n e t h e two g r o u p s .  each  between  A l s o a f e w s t u d e n t s may h a v e c o n t r i b u t e d t h e m a j o r i t y  o f moves i n some a r e a s . f o r each  enough t o d e t e c t any d i f f e r e n c e s  R a t h e r t h a n c o u n t i n g t h e t o t a l n u m b e r o f moves  s u b j e c t , a s e c o n d p r o c e s s m a t r i x was o b t a i n e d w h e r e a move f o r  s u b j e c t was o n l y c o u n t e d o n c e f o r e a c h p r o b l e m  ( s e e F i g u r e s 14 a n d  15).  The e n t r i e s i n t h e s e m a t r i c e s i n d i c a t e t h e number o f p r o b l e m s i n  which  a p a r t i c u l a r move was made.  F i v e areas from t h e core were i d e n t i f i e d :  88  2  to  o  w  to to w CD  S  M  >-)  w  w  Q <  EC  < o  < K CD  K  2  O M E-i  <  to tO  tD  W  W  CD  RECALL  16 11 55 34 32 2 3 14 2  DIAGRAM  24  EQUATION  21 4 1 51 4  READING  8 18  GUESS  6  SMOOTHING  4 2 6  TEMPLATION  6 20  5 1 3  10 1 6 4 4  RANDOM CASES  7 1 2  SYSTEMATIC CASES  1 1 1 2  o  8  3  O O  S  CO  a  CO  < o p  H 2 W <<  8  CHECKING PART"  13  CHECKING SOLUTION  12 1 5 3 4  3 13 3  CONCERN PROCESS  6  2 3 1  CONCERN SOLUTION  9  3  1  o to  M F-i  £ W  S3 to  6 4  CD  C3  I—I  M  o  o  o  o  w w  5  14  to to  w  o o  oc re  2  O M  o w (X  W o  w o  O  o o  o  2 4  1  1  4 5  1 1  2 2 4 2 15  6  1 1  1 3  1  9  9  4  1  2 1 3 1  1  2  2 1 2 4  2  2  1  1  2 4 1  3 38  2 11  1 2 5 3  1 1  3 1  3 6  1 4  2 6  4  19  6-<  o  5 6 13 2 6 2  1  1 1  OBTAIN SOLUTION  o  M F-t  1  2 3 2  1  7 5  FIGURE 14 PROCESS MATRIXiFREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM, FOR 100 MATH WORLD PROBLEMS (N=20)  89 2  CO W CO  S <  2  M P  < W K  READING RECALL  «  W K  M Q  2  2  M  O  M EH  175  <  w  C? W  co  •3 O  O O  2 CO  21 16 65 54 25 8  3 11  4  5  11  6  7  12  1. 7 10  CHECKING SOLUTION  12  2  CONCERN PROCESS  11  1  9  o o a: p<  2 O  M  EH  o  w 2 (X  a: W o  w  2  2  O o  o o  o  13 8  5  3 6 1  1  7  1  3  2 8  4  5  1  2  ^  l  1  1  2  3  1  2  i  2  1  2 25  1 14  3  3 3  1 1 2  4  1 1  2  l  4  4  6  3  4  w  1  2  6  to CO  o  3  TEMPLATION  3 4  >-r-  w  5  1  14  o  &  7 12  4  CHECKING PART"  W EH  w  O  9 12  2  3 6  ss CO  is."*  M  2  SMOOTHING  19  6  W  e>  2  3 2  OBTAIN SOLUTION  E-*  2  1  7  1  M  4  9  1  JE  2  GUESS  SYSTEMATIC CASES  EH  5  8  33 9 3 47  1  H  o D  o CO  o  <  3  2  EQUATION  3 1  PH  o M  <  EH  8  35 14 22  RANDOM CASES  EH  o  EH  11  DIAGRAM  CONCERN SOLUTION  « EH  o M  CO W CO «< o  o M  4  3 1 1  4  1 5  8 2  2  l 1  FIGURE 15 PROCESS MATRIXiFREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM* FOR 100 REAL WORLD PROBLEMS (N=20)  2  90 reading, r e c a l l  r e l a t e d f a c t , working w i t h a diagram  and m o d i f y d i a g r a m ) , w r i t i n g e q u a t i o n s s e t t i n g up e q u a t i o n s ) , and g u e s s i n g . templation,  (includes  (includes i d e n t i f y variable Four o f t h e h e u r i s t i c s :  random c a s e s , and s y s t e m a t i c c a s e s ) were u s e d  p r o b l e m s and a r e i d e n t i f i e d i n t h e m a t r i x . obtaining a solution,  diagram  checking part,  and  smoothing,  across the f i v e  The o t h e r a r e a s i n c l u d e d w e r e  checking solution,  concern  f o r process  ( i n c l u d e s c o n c e r n f o r method, a l g o r i t h m , and e q u a t i o n ) , and c o n c e r n f o r solution.  The p e r c e n t a g e  o f moves was o b t a i n e d f o r e a c h  a c r o s s e a c h r o w ( s e e F i g u r e s 16 a n d 17)• of  t h e problems,  of these  F o r e x a m p l e i n F i g u r e 16,  working w i t h a diagram  followed reading the  across almost  a l l of the procedures  t h e s u b j e c t s i n t h e math w o r l d s e t t i n g used diagramming. to  i n the process matrix while  i t mainly i n connection with  almost  every process.  i n understanding t h e problem.  i n both problem  s e t t i n g s had d i f f i c u l t y  (1967. P«  Kilpatrick  t h e number o f t i m e s a s u b j e c t r e a d s t h e p r o b l e m difficulty  of  The m a t h w o r l d s u b j e c t s made a g r e a t e r p e r c e n t a g e  r e a d i n g from  i n 28%  problem.  The s u b j e c t s i n t h e r e a l w o r l d s e t t i n g u s e d t h e h e u r i s t i c smoothing  procedures  64)  o f moves indicated  i s a measure o f h i s  I t appears  as though s u b j e c t s  i n understanding the  w i t h t h o s e i n t h e math w o r l d s e t t i n g b e i n g t h e h a r d e s t t o  problems,  understand.  A l s o , t h e r e a l w o r l d s u b j e c t s seemed t o b e more c o n f i d e n t i n t h e s o l u t i o n s they obtained using the heuristics, systematic cases  s o l u t i o n s u s i n g random  c o m p a r e d t o 25 f o r t h e m a t h w o r l d s u b j e c t s .  subjects expressed procedures  o b t a i n i n g 21  The math  and world  concern f o r solutions obtained u s i n g both of these  while the r e a l world subjects expressed  none.  B o t h g r o u p s o f s u b j e c t s moved t o a n d f r o m t h e u s e o f t e m p l a t i o n a n d random c a s e s , u s i n g a l l t h e p r o c e d u r e s  i n the process matrix.  Both  of  t h e s e g r o u p s s t o p p e d w o r k w i t h a s o l u t i o n u s i n g a p p r o x i m a t e l y t h e same procedures.  I n g e n e r a l , t h e o v e r a l l p a t t e r n o f moves ( e x c e p t f o r  s m o o t h i n g ) f o r b o t h g r o u p s o f s u b j e c t s i s t h e same.  91 2  o M  to w  to  CO  CO  CO  o  CO  o  w  2  C5  S  P  cc o  < CC  ce  O M  <  GO  to w  c?  Q  W  11  24  17  11  11  17  7  9  7  2  38  6  36  28  12  8  SMOOTHING  22  44  11  TEMPLATION  23  2  13  19  RANDOM  18  6  6  12  9  7  28  RECALL  17  7  DIAGRAM  27  EQUATION  26  GUESS  CASES  SYSTEMATIC  OBTAIN  CASES  SOLUTION  14  CHECKING  PART"  32  7  9  CHECKING  SOLUTION  35  6  9  4  42  CONCERN  SOLUTION  28  12  PERCENTAGE MATH  00  cc  E-«  OF  WORLD  M  M  w  2  O  o  3  9  2  4  4  4  2  3  4  9  12  4  4  4  3  1  1  6  1  6  12  3  5  3  18  6  4  32  MATRIX  (N=20)  1  18  7  7  6  FOR  24  8  19 3  12  29 14  12  3  16  PROBLEMS  2  7  3  MOVESiPROCESS  2  14  3 12  3  2  O O  12  14  12  2  cc w o  -  8  5  8  I  4  5  6  4  2  o  CO  O o  4  15  14  CC w o  w  6 CO  2  M  1  4  9  e>  1  3  9  CO  o  O w  5  5  CH  2  w o o CC P-«  i-3 o  22  FIGURE ROW  o p  5  8  o  g a s  14  4  PROCESS  M  14  24  CONCERN  o  2  o  O  23  READING  o  CC <  2; O M  19  3  6  92 CO  o  CO  EH  w  o 2 M Q  <  o  K  Q  2 O M En  W  n CO CO  EH  w  s  O O  o  READING  8  6 28 17 16  RECALL  6  9 44  2 O M En  Cm  S  w  EH  CO  o  CO  u M  w  < o  EH  <  o  W  <  5H  Q  CO  n  EH  t-H  <;  O CO  PH  C5  o  M  M  O w  O w  o  o  *~  to  CO w  o o  PL,  K  W o  2 O  o  5 3 7 1  \A O W 2  «  W o 2 O o  6  1 2  6  3  3  6 4  4  19  EQUATION  19  GUESS  22  SMOOTHING  13 7 20  TEMPLATION  29 3 17 11 11  6 3 9 3  3  6  RANDOM CASES  21 3 6 24 9  6 3 6 11  6  6  SYSTEMATIC CASES  12 12 12  OBTAIN SOLUTION  22 2 5  CHECKING PART"  25  CHECKING SOLUTION  26 2 11 6  9  CONCERN PROCESS  26  4  CONCERN SOLUTION  39  8  5 49 4 4  10  1 1 5  4  4 11  4  •  12 12  33 3  12  2 5 1  4  5  2 2 4 2 14  3 30 13  6 25 6  13  t—i  £D  3 7  23 8 17 6 19  9 13  O  EH  oo  DIAGRAM .  19  2  12  12  3 44  2 4 9• 6 2 2 6 13 4  9  13 9  2 9 30  2 13 ! 6 2 4 13  6 22  FIGURE 17 ROW PERCENTAGE OF MOVES «PROCESS MATRIX FOR REAL WORLD PROBLEMS (N=20)  Results of Analysis - Field  Independence 115  W i t h I Q r e s t r i c t e d t o one s t a n d a r d d e v i a t i o n , field of  independent-dependent  the variables  v a r i a b l e had a s i g n i f i c a n t  considered i nt h i s  a n a l y s i s performed w i t h f i e l d are  g i v e n i n T a b l e 5.  given  independence  effect  o n many  The r e s u l t s o f t h e r e g r e s s i o n as t h eindependent  T h e r e s u l t s o f h y p o t h e s e s 4,  5,  6,  variable  a n d 10 a r e  below:  H y p o t h e s i s l+:  A significant difference  ^•number o f h e u r i s t i c s u s e d b y f i e l d dependent for  study.  t o 125., t h e  students.  (P=.0110)  was f o u n d i n t h e t o t a l  independent students over  By i t s e l f t h e f i e l d  independence  field  variable  accounts  16% o f t h e v a r i a n c e o f t h e t o t a l n u m b e r o f h e u r i s t i c s .  H y p o t h e s i s 5:  A significant difference  number o f d i f f e r e n t field  dependent  (P=.0385) w a s f o u n d i n t h e  h e u r i s t i c s used by f i e l d  students.  By i t s e l f  independent students over  thefield  independence  variable  a c c o u n t s f o r 11% o f t h e v a r i a n c e o f t h e n u m b e r o f d i f f e r e n t  heuristics  used.  H y p o t h e s i s 6:  .A s i g n i f i c a n t d i f f e r e n c e  (P=.0575) was f o u n d i n t h e  number o f c o r r e c t s o l u t i o n s o b t a i n e d b y f i e l d over f i e l d  dependent  students.  By i t s e l f  independent students  thefield  independent  a c c o u n t e d f o r 9% o f t h e v a r i a n c e o f t h e n u m b e r o f c o r r e c t  variable  solutions.  P o s t h o c a n a l y s e s r e v e a l e d t h a t t h e r e was n o s i g n i f i c a n t  difference  (P=.3263) b e t w e e n t h e n u m b e r o f t i m e s a s u b j e c t i s w i l l i n g t o a t t a c k a problem did  ( c y c l e s ) and f i e l d  have a s i g n i f i c a n t  effect  makes i n a t t a c k i n g a p r o b l e m . the  independence.  However, f i e l d  independence  (P=.0892) o n t h e n u m b e r o f c h a n g e s h e F i e l d independence  v a r i a n c e i n t h e number o f changes.  a c c o u n t s f o r 7% o f  TABLE 5 RESULTS OF -""REGRESSION ANALYSIS WITH FIELD INDEPENDENCE AS INDEPENDENT VARIABLE (N=40)  DEPENDENT VARIABLE  TOTAL HEURISTICS DIFFERENT HEURISTICS CYCLES  SOURCE OF VARIATION  FIELD INDEPENDENCE  F - VALUE TO ENTER/REMOVE  2 F - PROB.  R WITH DEPENDENT VARIABLE  ^RSQ  7.084  .0110  .3964  .1571  4.495  .0385  .3252  .1058  .3263  .1597  .0255  .9957  CHANGES  2.970  .0892  .2693  .0725  CORRECT SOLUTION  3.745  .0575  .2995  .0879  Each of These i s a Separate Simple Regression. P r o b a b i l i t y of Making a Type 1 E r r o r by Rejecting N u l l Hypothesis, that i s , Claiming S t a t i s t i c a l Signifance. 'The Proportion of Variance i n the Dependent Variable Accounted f o r by F i e l d Independence.  95 I n t e s t i n g h y p o t h e s i s 10,  t h e s u b j e c t s were d i v i d e d i n t o  a c c o r d i n g t o t h e i r s c o r e on t h e EFT. the  field  independent group  the  f i e l d dependent group.  were e x c l u d e d from t h i s  (14  The t o p t h i r d  and t h e bottom t h i r d  (14  independence may  be  the  entire  analysis.  T h i s was  between  significance.  e a s i e r t o o b s e r v e u s i n g t h e s e two  a  subjects  as f i e l d  dependent.  I f field  does e f f e c t t h e p a t t e r n o f procedures used, t h e  difference  groups o f s u b j e c t s r a t h e r  than  sample. a p a i r o f p r o c e s s m a t r i c e s were formed  r e s u l t s f r o m t h e c o d i n g f o r m s f o r t h e t o p 14  f o r one m a t r i x ( F i g u r e 1 8 ) a n d t h e b o t t o m f o r t h e o t h e r m a t r i x ( F i g u r e 19)• is  formed  done i n o r d e r t o o b t a i n  and t h o s e c l a s s i f i e d  not t e s t e d f o r s t a t i s t i c a l  To t e s t h y p o t h e s i s 10 the  formed  Those s u b j e c t s i n t h e m i d d l e range o f s c o r e s  as f i e l d i n d e p e n d e n t  H y p o t h e s i s 10 was  subjects)  subjects)  g r e a t e r d i f f e r e n c e i n t h e degree o f f i e l d independence classified  thirds  g i v e n i n F i g u r e s 20 a n d 21.  14  using  f i e l d independent  f i e l d dependent  subjects  subjects  The p e r c e n t a g e o f moves i n e a c h I n e v a l u a t i n g h y p o t h e s i s 10,  the  area following  major a r e a s from t h e p r o c e s s m a t r i x were c o n s i d e r e d *  In  1.  Moves f r o m c o r e t o c o r e o r t o h e u r i s t i c s  2.  Moves f r o m h e u r i s t i c s t o c o r e o r t o h e u r i s t i c s  3.  M o v e s c o n c e r n e d w i t h c h e c k i n g ( A r e a s C, G,  4.  M o v e s c o n c e r n e d w i t h e x p r e s s i o n o f c o n c e r n ( A r e a s D,  comparing t h e s e a r e a s f o r t h e two  groups  (field  ( A r e a s A and  B)  ( A r e a s E and  M and  F)  N) H,  independent  Q and R ) *  and  f i e l d d e p e n d e n t ) t h e r e s e a r c h e r c o n s i d e r e d t h e p e r c e n t a g e o f moves in  each a r e a f o r each group  o f moves f o r e a c h p r o c e d u r e . procedure  as w e l l  as c o m p a r i n g t h e number and t y p e s  That i s , f o r a g i v e n procedure i s t h e r e a  ( o r p r o c e d u r e s ) which s t u d e n t s have a tendency t o use  H y p o t h e s i s 10: independent  The p a t t e r n o f p r o c e d u r e s u s e d b y t h e t o p 14  subjects i s observably d i f f e r e n t than that of t h e  next?  field bottom  96  1  1  2  3  9  3 11 hz  4  2  5 6 7 5 16 1  2  41  8  9 10  11 12 13  14  15 16 17 18  16  8  10  1  2  9  4  2  6  2  8  2  1  2  2  1  5  4  3  5  4  1  5  1  5 6  4  17  12  10  4  2  2  13  7  5  1  3 2  1  l  1  5  2  6  1  3 1  1  27  7  25  3  2  30 39  2  6  8  10  7  33  1  3  1  1  65 8  3  1  2  18  1  1  5  1  1  4  4  10  •7.  3  3  6 1  2  2.  2  2  1  2  1  1  6  7  1  10  5  3  1  1  -11  2  2  5  1  8  3  3  2  3  4  7  l  3  l  l l  3  5  25 26 27  28 2 9  5  4  1  2  1  1  l  l  3  1  24  1  7  1  7  8  12  3  2  29  7  2U  4  1  4  6  1  1  3  1  3  2  3  1  1  3  12  5  4  3  15  6  1  2  16  1  13  22 23  9  3  9  21  3  3  2  19 20  2  2  1  2  3  1  1  1  1  4  1  2 1  2  4  3  2  3  l l  14  1  1  2  1  2  2  17  8  2  1  1  1  1  l  17 18  1  1  1  1  1  1  19 20  1  21  13  1  7  3  7  2  6  3  3  4  3  4  3  1  1  3  22  13  1  7  3  7  2  6 3  3  It  3  4  3  1  1  3  3  23  5  1  4  l  1  1  2  6  l  4  14  24  4  1  3  1  2  1  1  26  4  27  7  3 4  3  2  l  1  2  3  2 1  4  4  1  1 1  1  1  1  4  2  1  25  1  1 1  3 -  1  5  1  1  1  8  1  FIGURE 18 PROCESS MATRIX«FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR FIELD INDEPENDENT STUDENTS (N=l4)  4  97 1 1  25  2  3  5 6 7 8  4  9 45 4 23 52 3 15 32  2  3 4  9 10  11 12 13  1  14  15 16  6  11  4 4  1  3  2 5  1 1 2  1  3 4  1  17  18  1  19  20  21  22 23  24  2  6 10  2  2  1  25  26  27  23  29  1  6  4  8  1  4  1 1  1 4 3 1  25 1 6 4 1 3 3  5  2  6  11  1  7  24  3 1  3  50  2 4  8  13  1  1 7 57  6 6  9  12  2 1  10  6  •11  2  12  1  13  9  1  1  l  2  l  1  1  11  4  3 1  22  5 1  1  29  2  3  3 16  1  2 1  1  2  2  1  2 3 1  1  1 2 22 3  3 2  3  1 1  22 1 28  1  i  1 7  1  2 1  16  2  2 1  1 1  3 1  1  2 2  1  1 2  1  l  1  2 2  A  t  14 15  3  16  l  3  l  1  1  1  1  1 1  1  2  17  12  1  1  18* 19  20 21  16  1 2 3  22  17  1  6  23  16  1 1  2  24  2  25  1  26  1  27  6  2 2  4  2 1  2 1 2 1  1  2  2 23  3 1  1  3  1  1  2  10 __16_  1  t  4  10  1  1 1  A  1  3 l  1  1  3  2  FIGURE 19 PROCESS MATRIX 1FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR FIELD DEPENDENT STUDENTS (N=l4)  11  2  98.  1 2 3 4 5 6 7 8 9  10  11  12  13  14  15 16 17  18  19.20  22 23  24  25  26  1 2 3 4 5 6  49  8  5  3  9  3  1  1  2  1  3  1  5  2  1  1  3  1  2  7 8 9  10 11 12 13 14 15  16 17 18 19  20 21 22 23 24  25 26 27  FIGURE 20 P E R C E N T OF CATEGORY -  PROCEDURES USED I N FIELD INDEPENDENCE  EACH (N=l4)  27  99 1  2 3 *  5  6  7  8  9 10  11 12 13 14 15 lfS 17 18 19 20  22 23  24 25 26 27  5  2  1 2 3 4  5 6  64  5  4  2  7  8 9  10 11 12 13 14  15 16  1  17 18 19 20 21 22  23  3  3 1  6  24  25 26  2  27  FIGURE 21 PERCENT OF PROCEDURES USED IN EACH CATEGORY - FIELD DEPENDENCE (N=l4)  1 1  100 14 f i e l d  dependent s u b j e c t s .  more  e oriented with  field  independent  The f i e l d  a higher percent  subjects  concerned w i t h  (#13)  their  and random c a s e s  (#15)  (#16)  cases  evidence there i s of the higher h e u r i s t i c s .  appear t o use both t e m p l a t i o n  The  I n t h e use o f h e u r i s t i c s , t h e  i n d e p e n d e n t s u b j e c t s make g r e a t e r u s e o f s y s t e m a t i c  g i v e what l i t t l e  a p p e a r t o be  o f p r o c e d u r e s i n A r e a A.  a r e more c o n s i s t e n t l y  w o r k ( s e e A r e a s D, H, Q, R, S, T ) . field  dependent s u b j e c t s  Both  and  groups  across a l l  procedures. Again, the  a second process  f o r each group where  e n t r i e s i n t h e m a t r i x i n d i c a t e t h e number o f p r o b l e m s i n w h i c h  p a r t i c u l a r move i s made. for  m a t r i x was o b t a i n e d  these The  matrices field  (See F i g u r e s  22 a n d 23).  a r e g i v e n i n F i g u r e s 24 a n d 25.  t h e p r o b l e m s w i t h 31 t o 57 p e r c e n t  reading from a m a j o r i t y o f t h e procedures. are t o diagrams, equations concern The  The row p e r c e n t a g e s  dependent s u b j e c t s appear t o have had g r e a t  i n understanding  a  or templation.  difficulty  o f t h e i r moves t o  T h e m a j o r i t y o f t h e i r moves Half o f t h e i r responses t o  i s t o r e r e a d t h e p r o b l e m h o p i n g t o g a i n some i n s i g h t . field  i n d e p e n d e n t s u b j e c t moves more f r e e l y t o a g r e a t e r  o f p r o c e d u r e s and t o t h e f o u r h e u r i s t i c s . h i s work, c o n t i n u a l l y c h e c k i n g about t h e p r o c e s s  He i s m o r e c o n c e r n e d  b o t h h i s work and s o l u t i o n .  variety  with  When c o n c e r n e d  h e u s e s o r h i s s o l u t i o n , he i s more w i l l i n g t o c h e c k  his procedures o r solutions. In general, the f i e l d  i n d e p e n d e n t s u b j e c t moves b e t w e e n a l l t h e  procedures i n t h e matrix while t h e f i e l d great d e a l of time i n the core  correlations  Procedures  f o r h y p o t h e s e s 7 a n d 8 a r e g i v e n i n T a b l e 6.  was u s e d t o d e t e r m i n e i f t h e s e 0.  spends a  areas.  R e s u l t s o f A n a l y s i s - Gore and H e u r i s t i c The  dependent s u b j e c t  An F - T e s t  c o r r e l a t i o n s were s i g n i f i c a n t l y d i f f e r e n t  from  101 2 O M EH  CO  w  CO  CO  W  2  o 2 H Q  < W CC  •-H  < w  <  CC C5  <*:  Q i—i  2 O M  E-t  < c w  M EH  2 M  00 CO  CO  o  w  EH O O  o  0"}  s  «<  -< o  o  t—«  2  W  EH  o  •< CC  EH CC  o  <  O  <  CJ 2 M  e>  O  o w  M EH  s w o  •3 PH  •<  EH {0 >H CO  w  o  CO  2  •T* f—  o  CO CO W o o CC PH 2 CC  o  2 O O  3 7 9 1  RECALL  3 1 6 9 2  6 2 1  2 1  1  DIAGRAM  13 12 14 6 9  7 8 4 3  4 5  1  EQUATION  13 6 1 39 5  6 3 2  6  7 1  5  SMOOTHING  3 2 1  1  4  6 1 1  1 4 4  1 1  TEMPLATION  5 2  3 7  2 3 5 1  RANDOM CASES  5 1 2 5 2  2 1 2 4  SYSTEMATIC CASES  1  OBTAIN SOLUTION  1  1  11 2 5  1 2 4 1  CHECKING PART"  9 1 6 8  3  3 4 3  CHECKING SOLUTION  5 1 6 2 3  2 5 1  CONCERN PROCESS  8  1 3 2  1 2 2  CONCERN SOLUTION  7  4  1 4  3  4  4 1  3  1  1  1  3 1  2  3 1 ;  3 23  3 9  3  2 3 : 4  2 4 2  10 1  2 CC W  o  2 O  6 9 36 28 14  GUESS  t=> •H O CO  w o  READING  8  2 O M EH  7  1  J FIGURE 22 PROCESS MATRIX tFREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM, FOR FIELD INDEPENDENT STUDENTS (N=l4)  ...  .  to w to •<  2  t-H  p  <  M  M EH <<  •H  K  o 2  2 o  P  W  r-H.  to to w O  X EH O O  sto  2 O  r-H  to o w to o <; M O  EH  PL.  w  EH  P  EH  •<  w  EH  S  <  2  o M  to to w o  FH EH  K  PH  2  o  O  to e» 2  2  2  l-l  M  O  O  re I X w w o o o o  iii  w  &  w  to  o  o  5  9  2 O  2 5. 9 1  RECALL  4 1 2 7 4  1 3  6 6 6 8  3 2 1  1 3  1 3  EQUATION  26 4  l  32 4  1 3 2  7 2  GUESS  6  3  1  1 1  SMOOTHING  3  2  1  2  TEMPLATION .  9  4 3 2  RANDOM CASES  3  4 1  SYSTEMATIC CASES  1 2 1  1  1 1  1  16 1 4  CHECKING PART-  11 1  5  CHECKING SOLUTION  11 1 1  1 1  1 3  1  1  CONCERN PROCESS  4 1  CONCERN SOLUTION  6  2 2  1 1  3 6  1  1 •  OBTAIN SOLUTION  1  E> hH  re o to  20 9 40 34 30 24  M  £<  CM  READING  DIAGRAM  ON  102  1 3  1  1  2 2  2 19 1 1  8  1  2  3  1 2  2  FIGURE 23 PROCESS MATRIXiFREQUENCY DISTRIBUTION OF CHANGE OF PR03LEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM, FOR FIELD DEPENDENT STUDENTS (N=l4)  103 o  CO  w  UT  M  CO  O  o 2 M  n  w CC  r-H HH  < o  w CC  2 M  •X M CC  o  < M P  EH  •—\ <  w  O  o  s 00  w  READING  5 7 28 22 11  RECALL  9 3 18 26  M EH  •4  CO CO  O  2  6  d  CO  o  W  2 2 O  CO  PH  t:" r—  W  EH  < o  o  o  11  11  OBTAIN SOLUTION  17  3 8  CHECKING PART"  20  2 13 17  CHECKING SOLUTION  14  3 17  CONCERN PROCESS  26  CONCERN SOLUTION  25  o o  O  6 3  CC 2  O  3  4  6  1  7 3 2  7  6 22 22  SYSTEMATIC CASES  w o  3  6  o  to  w o  9 4  6  &  r-H  2  CC  3  5  ff—<  o M  2  3  5  17 3 7 17 7  CC CM  6  15 10  RANDOM CASES  M o o  3  25  6  >-r"  CO  6  GUESS  9 20  l-H  to  6  6 11  e » 2  n  < CC  7 1 44  14  CO  o w  15  TEMPLATION  a  o  o w  EQUATION  33  < P-,  >-}  EH  14 13 16  SMOOTHING  w  5 7 1 15  8  <  CC  o p  DIAGRAM  7 10  M  EH  to  i  8  20  5 15  6  9 14 3  3 3  7 3 7 14  3  3 10 3 22  11  33 11  3 6 2  5 36  5 14 ;  7  7 9 7  7  4  6  9  6 14  3 10  6  3 6  11  4 14  14  2  3  6 11  11  32  6 4  6 2 5  FIGURE 24 ROW PERCENTAGE OF MOVES »PROCESS MATRIX FOR FIELD INDEPENDENT STUDENTS (N.= l4)  7  1  4  104 CO  M0  CO  FH  M  w  s  2 M  Q  < W  re  < o w K  re  < i— 1  Q  2 O  M  e-t •<  n? c  w  CO CO w  o  o 2  2 O M  H  EH  EH O O 2  CO  CO  o  << o  EH  w CO  << W  *H  E-i  re  RECALL  18  5 9 32 18  5 14  DIAGRAM  39 10 10 10 13  EQUATION  31 5 1 38 5  GUESS  40  20  7  SMOOTHING  38  25  13  TEMPLATION  39  17 13 9  RANDOM CASES  23  31  OBTAIN SOLUTION  30 2 7  CHECKING PART  46  CONCERN PROCESS  57 14  CONCERN; SOLUTION  46  2  2  r-H  f-0  o w  O IV!  CO  o  o  o  re W o  w o  re  o o  o  2 O  5 3 2  2 5  2  1 4 2  8  4 1  7 7  13  7  25  •  4 9  4 4 4 12  14 8  to  E>  HH  2 3  4 4 4 4  CHECKING SOLUTION  e » 2  PH  O  3 5  8  20 4  CO  >-J  ce  < PH  o  o  M EH  w o o a:  M  1 3 5 1  25 25  , 44 4  <  w E--  12 5 23 20 17  25  M  s: • •3 o PH o  READING  SYSTEMATIC CASES  o  to CO  ID  EH  8  4  4  8 15 15 25 4 35 4 4  4 13  15 4  8  14 8 15  FIGURE 25 . ROW PERCENTAGE OF MOVESiPROCESS MATRIX FOR FIELD DEPENDENT STUDENTS (N=l4)  15  TABLE  6  I N T E R C O R R E L A T I O N S AMOUNG M E A S U R E S H Y P O T H E S I Z E D V A R I A B L E S (N=40)  1.  PROBLEM  CONTEXT  2..  FIELD  INDEPENDENCE  3.  TOTAL  HEURISTICS  4.  DIFFERENT  5.  CYCLES  6.  CHANGES  7.  CORRECT  SOLUTION  8..  OFFERED  SOLUTION  3 ,  OF  2  3  4  5  6  7  ,01  .11  .16  -.08  .13  .23  -.02  .40*  .33*  .16  .27  .30  .23  .84  .48  .80  .49  .08  .43  .75  .47  .19  .52  .14  .12  .51**  .20  HEURISTICS  8  .18  i s Equivalent  to the P o i n t - B l s e r i a l Correlation Coefficent. # ## otomous Variable. i t 3 l 2 1 @ P=.05 crit ^° * R  = #  R  = ,  2 6 @  p = =  0 1  c r  o  Hypothesis  7:  A significant positive  (P=.0013)  correlation  was  found  between t h e t o t a l number o f h e u r i s t i c s u s e d b y a s t u d e n t a n d t h e number of  correct  solutions  Hypothesis  8:  he  obtained.  A significant positive  b e t w e e n t h e number o f d i f f e r e n t number o f c o r r e c t  solutions  Post hoc a n a l y s i s  he  (P=.0022)  correlation  obtained.  was p e r f o r m e d , u s i n g l i n e a r r e g r e s s i o n , t o  which c o u l d be accounted f o rby t h e h e u r i s t i c s , a r e g i v e n i n T a b l e 7»  (the  cycles  solutions  and changes.  The t o t a l n u m b e r o f h e u r i s t i c s  accounts f o r 22$ o f t h e v a r i a n c e i n c o r r e c t of d i f f e r e n t  found  h e u r i s t i c s used b y a s t u d e n t and t h e  d e t e r m i n e t h e amount o f v a r i a n c e i n t h e n u m b e r o f c o r r e c t  These r e s u l t s  was  solutions  h e u r i s t i c s used accounted f o r 22%.  w h i l e t h e number  The number o f c y c l e s ,  number o f t i m e s a s u b j e c t a t t e m p t s t o s o l v e a p r o b l e m ) a c c o u n t s  f o r l e s s t h a n 2% o f t h e v a r i a n c e , w h i l e t h e change o f p r o c e d u r e s for 26% of the variance of correct Table 8 l i s t s the  between p a i r s  coding system f o r t h e t o t a l noted that  a correlation  provides additional  solutions.  t h e means a n d s t a n d a r d d e v i a t i o n s  intercorrelations  of variables  and T a b l e 9  sample o f f o r t y s u b j e c t s .  into  lists  d e r i v e d from t h e I t s h o u l d be  c o e f f i c i e n t , which i s s t a t i s t i c a l l y  insight  accounts  significant,  t h e data, but does not allow  causal  i n f e r e n c e t o b e made. Most o f t h e s i g n i f i c a n t The l a r g e  correlations  have o b v i o u s  number o f s i g n i f i c a n t c o r r e l a t i o n s  reflect the fact  that  with cycles  interpretationso and changes  s u b j e c t s who a r e a t t e m p t i n g t o b e g i n p r o b l e m s  a g a i n o r change t h e i r p r o c e d u r e s a r e p e r f o r m i n g more p r o c e s s e s . be  expected, t h e core areas are s i g n i f i c a n t l y c o r r e l a t e d .  which are algebraic  such as, i d e n t i f y i n g v a r i a b l e s ,  As would  Those procedures  s e t t i n g up  equations  TABLE 7 RESULTS OF REGRESSION ANALYSIS WITH CORRECT SOLUTION AS DEPENDENT VARIABLE (N=40)  DEPENDENT VARIABLE  CORRECT SOLUTION  SOURCE OF VARIATION  F - VALUE TO ENTER/REMOVE  *F - PR OB'.  R WITH DEPENDENT VARIABLE  TOTAL HEURISTICS  11.84  .0015  .4873  .2375  DIFFERENT HEURISTICS  10.53  .0025  .4658  .2170  CHANGES  13.33  .0009  .5096  .2597  .3986  .1386  .0192  CYCLES  .7424  -*RSQ  ^•Each of These i s a Separate Simple Regression. 2 P r o b a b i l i t y of Making a Type 1 E r r o r by Rejecting N u l l Hypothesis, that i s Claiming S t a t i s t i c a l Significance. 3  ^The Proportion) of Variance i n the Dependent Variable Accounted Variable.,  f o r by Independent  108  TABLE 8 MEANS AND STANDARD DEVIATIONS FOR PROCEDURES DERIVED FROM CODING SYSTEM (N=40) Variable  Mean  Reading  17.85  Standard  Deviation  6.93  R e c a l l Fact  3.25  2.68  Draw Diagram:  8.12  6.80  Modify Diagram.  2.32  2.29  Identify Variable  2-80  2.33  Set up Equations  9.08  6.85  Algebraic Algorithms  8.28  Arithmetic- Algorithms  31.10  10.53 35.68  Guessing  4.90  2.63  Smoothing  1.08  1.99  Templation.  3.75  3.54  Random Cases  2.35  2.51  Systematic Cases  .95  1.95  Obtain S o l u t i o n  6.15  2.69  Checking Part  5.98  4.44  Checking Solution;  4.42  3.42  Concern Process  2.32  3.12  Concern Solution  2.52  3.13  10.38  Cycles  4.12  1.85  2.30  Stops without S o l u t i o n  1.12  1.22  Correct Solution.  1.15  .89  Changes  ;  Note- Appendix D Contains Histograms  f o r Some Selected Variables.  109  1. READING PR03LEM  .20  2 . RECALL RELATED FACT 3 . DRAW DIAGRAM  .26  .25 «»  .51  *  .28  -.01  .39  .18  -.02  .48  .50  .5**  .44*  .13  .24  .09 .56*  4. KCDIFY DIAGRAM 5.IDENTIFY VARIABLE 6.SETTING UP EQUATIONS  -.09  - . 1 9  -.19  «  .22  .23  .18  .13  .46*  .18  • 50  .04  .05  .44*  .18  .3^*  .59**  .30  .30  .47** .25  .32  .80  .01  .17  - 0 *  .11  -.12  .00  -.11  7.ALG0RITKKS-ALGE3RAIC  -.02 .10  -.08 .14  8 .ALGORXTHIS-ARITHKETIC  -.06 -.13 .62*  .19  9.GUESSING  .38  »  .39  •  •  .79 .03  .21  «W  .14  **  .56 .08  3  .18  .21  .02  .24  4  .45  .60  .45**  .20  ,14  5  .10  .60  ,15  .09  -.13  6  .20  .43*  .13  -.16  7  .12  .70*  ,84** - .27  .47  8  .23  • 32  .18  -.18  .63  .49  .66  -.18  .44  10  .36*  .26  .16  11  .58  .62  -.21  ,58  12  ,68*  .81**-30  .30  13  ,22  14  .45"  .20  .36  .13  .29  .51  .22  .12  .3*  .05  .38*  •• .50 .15  .18  .62  •  -.06  .22  .21  .39 «  .14  .15  .31  -.02  .59  .4l*< .36*  .67  .22  .20 . **  .40*  .42  *  •  .21  . *» -.03 ,** .56 .29 «* .21 .52 , .61 .60  .42  -.05  .27  .48** .52  .40*  «  .16  .04 .18 .3*  .30  .56  .75  .33  -.02  1 5 . CHECKING PART  .41  .02  .18  l4.OBTAIN SOLUTION  .62** - . 0 8  .60  .38*  .26  .58*  13 .CASES-SYSTEMATIC  2  .26  .57  12.CASES-RANDOM  .06  .26  .62*  11 .TEMPLATION  -.03  .05  .28  #  1  .17  .37  .89  -.13  .30 «  .43  »  .19 .17  2 0  .21  .07  .36  10.SMOOTHING  .59*  .21  16. CHECKING SOLUTION  -.08  1  * .32 #  1 9 . CYCLES  f  9  .45" -53'  .74  „5?" - 0 2  .22  15  .25 . * .60  .01  -.28  .21  16  ,49** .16 ,14 - . 1 2  .15  17  -.0?  18  ,52  13.CONCERN-SOLUTION  -.03  .62*  .49*  17. CONCERN-PROCESS  ®°  -.12 -.20  .14  19  .51  20  -.18  21 22  2 0 . CKANGES 21. WORK STOP-NO SOLN. 2 2 . CORRECT SOLUTION  R  crit  .49  • .08  .07  .02  .25  ..08  .14  .,.= . 4 0 2 6  @ P=.01  ^crit TABLE  9  INTERCORRELATION B E T W E E N P A I R S OF V A R I A B L E S FROM C O D I N G S Y S T E M (N=40)  110 and  algebraic  algorithms  are a l l correlated.  As w e l l , t h o s e f r o m t h e  g e o m e t r i c a r e a such as d r a w i n g d i a g r a m s , m o d i f y i n g d i a g r a m s and algorithms part  (mainly  from the use o f formulas) are c o r r e l a t e d .  Checking  o f work i s p o s i t i v e l y c o r r e l a t e d w i t h most o f t h e p r o c e s s  whereas, checking s o l u t i o n s i s s i g n i f i c a n t l y c o r r e l a t e d w i t h  solutions.  obtaining  However, e x p r e s s i n g  a correct  variables,  those  p r o c e d u r e s p e r t a i n i n g t o s o l u t i o n s o r w h i c h were used t o o b t a i n the  arithmetic  most o f  concern f o r s o l u t i o n i s unrelated  to  solution.  Four procedures from t h e coding form accounted f o r 86% o f the solutions obtained:  algebraic  (31%), g u e s s i n g ( 2 8 % ) , 22%).  are  and random c a s e s  Of t h e s e , g u e s s i n g and a l g e b r a i c  correct  (10%), a r i t h m e t i c  algorithm  (17%), ( w i t h s y s t e m a t i c algorithms  s o l u t i o n , whereas b o t h a r i t h m e t i c  significantly correlated with  algorithm  correct  are unrelated and random  cases  solution.  A detailed  analysis  (26)  were o b t a i n e d b y random c a s e s f o l l o w e d  (14)  with  algorithms  f o r 1.  considerably core  Both arithmetic  to  algorithms  o f t h e c o d i n g f o r m s shows t h a t m o r e t h a n h a l f o f t h e c o r r e c t  guessing accounting f o r only  cases,  by arithmetic  6 correct  algorithms  w i t h b o t h random and s y s t e m a t i c  solutions  solutions  algorithms and  algebraic  and diagrams were u s e d cases as w e l l as w i t h  other  procedures. To g a i n  variables,  some f u r t h e r i n s i g h t i n t o t h e r e l a t i o n s h i p b e t w e e n t h e s e  a regression  a n a l y s i s was p e r f o r m e d u s i n g  as t h e d e p e n d e n t v a r i a b l e . into the regression.equation  correct  The v a r i a b l e s f r o m t h e c o r e a r e a w e r e f i r s t , . f o l l o w e d b y the.  correct  the  regression  T h e c o r e p r o c e d u r e s a c c o u n t f o r 40% o f t h e v a r i a n c e  s o l u t i o n , i n a d d i t i o n h e u r i s t i c s a c c o u n t e d f o r a n o t h e r 21%  systematic  entered  heuristic strategies  ( v a r i a b l e s w i t h i n core.and h e u r i s t i c s were a l l o w e d t o e n t e r equation f r e e ) .  solution  c a s e s a c c o u n t i n g f o r 13% o f t h i s  ( s e e T a b l e 10).  i n with  When t h e  Ill  TABLE 10 REGRESSION ANALYSIS ON CORRECT SOLUTION WITH CORE ENTERED BEFORE HEURISTICS (N=40)  SOURCE OF VARIANCE  MULTIPLE R RSQ  ?  ARSQ  F - VALUE TO F-PROB ENTER/REMOVE  1  CORE Arith.Algorithms  .4742  .2249  .2249  11.0246  .0021  Draw Diagram  .5242  .2748  .0499  2.5477  .1151  Reading  .5597  .3133  .0384  2.0153  .1608  R e c a l l Fact  .5829  .3398  .0266  1.4085  .2418  Guessing  .5981  .3577  .0178  .9442  .3402  Modify Diagram.  .6322  .3997  .0420  2.3098  .1343  Alg.  .6340  .4019  .0022  .1194  .7292  Set up Equations  .6343  .4024  .0005  .0248  .8496  Iden. Variables  .6348  .4026  .0002  .0088  .8877  Algorithms  HEURISTICS Systematic. Cases  .7552 .5307  .1281  11.3247  .0023  Random Cases  .7712  .5948  .0641  1.6888  .2018  Templation  .7778  .6058  .0110  .6986  .4154  Smoothing  .7800  .6084  .0026  .2226  .6453  P r o b a b i l i t y of Making a Type 1 E r r o r by Rejecting N u l l Hypothesis,i.e. Claming S t a t i s t i c a l S i g n i f i c a n c e .  2 The Amount of Variance i n Correct Solution Accounted f o r by the Independent V a r i a b l e .  112  TABLE 11 REGRESSION ANALYSIS ON CORRECT SOLUTION WITH HEURISTICS ENTERED BEFORE CORE (N=40)  SOURCE OF VARIANCE  MULTIPLE R RSQ  ?  ARSQ  F - VALUE TO F-PROB 1 ENTER/REMOVE  HEURISTICS I9.5656  Random Cases  .5830  .3399  .3399  Smoothing  .6133  .3760  .0361  2.1427  . 1484  Systematic Cases  .6365  .4051  .0291  1.7580  .1903  Templation  .6366  .4052  .0001  .OO96  .8852  .0001  -  CORE Draw Diagram  .6596  .4351  .0298  1.7946  .1862  Reading  .6952  .4833  .0482  3.0808  .0849  A r i t h . Algorithms . 7 0 9 3  .5031  .0198  1.2731  .2670  Modify Diagram-  .7305  .5336  .0306  2.0315  .1606  Guessing  .7481  .5596  .0260  1.7700  .1905  R e c a l l Fact  .7658  .5865  .0269  1.8846  .1772  Set  up Equations  .7783  .6058  .0193  1.3706  .2504  Alg.  Algorithms  .7793  .6074  .0016  .1089  .7398  .7800  .6084  .0010  .0666  .7867  Iden. Variable  •""Probability of Making a Type 1 E r r o r by Rejecting N u l l Hypothesis,i.e. Claming S t a t i s t i c a l S i g n i f i c a n c e . 2  The Amount of Variance i n Correct Solution Accounted f o r by. the Independent Variable.  The a r e a w i t h i n t h e r e c t a n g l e r e p r e s e n t s t h e t o t a l variance of correct solution. Circles C and H r e p r e s e n t t h e p e r c e n t a g e s o f v a r i a n c e o f correct s o l u t i o n which are c o n t r i b u t e d by core and h e u r i s t i c s , r e s p e c t i v e l y .  FIGURE 26 R E L A T I V E CONTRIBUTIONS OF CORE AND H E U R I S T I C S TO CORRECT SOLUTION  o r d e r was r e v e r s e d i n t h e r e g r e s s i o n e q u a t i o n ( s e e T a b l e heuristics  accounted  f o r 41$  a d d e d a n a d d i t i o n a l 20$.  of the variance.  procedures  I t appears as though f o rt h e s e s u b j e c t s , t h e  i n f l u e n c e o f c o r e and h e u r i s t i c s p r o c e d u r e s 26)  The c o r e  11), t h e  i s equivalent (see Figure  CHAPTER V  CONCLUSIONS  T h i s s t u d y was u n d e r t a k e n t o g a i n some i n s i g h t i n t o problem  solving.  mathematical  C o n f i n i n g a t t e n t i o n t o mathematical word problems,  an  a t t e m p t was made t o u n d e r s t a n d s o m e t h i n g o f t h e s o l u t i o n t e c h n i q u e s u s e d by a group  of senior high school students.  p r o c e d u r e s and h e u r i s t i c s do t h e y u s e ? have on t h e s e p r o c e d u r e s ? usage? exhibit? in  Specifically,  to  core  What e f f e c t d o e s p r o b l e m  Does f i e l d independence  influence  What p a t t e r n s o f h e u r i s t i c u s a g e d o s p e c i f i c  groups  context  heuristic of individuals  To h e l p a n s w e r t h e s e q u e s t i o n s , a c o d i n g s y s t e m was d e v e l o p e d  a f o r m w h i c h a l l owed c o m p a r i s o n  across individuals not only of the  p r o c e d u r e s t h e y used, b u t a l s o o f t h e sequence The  what  c o d i n g system  i n w h i c h t h e y were  appears t o be b o t h u s e f u l and u s a b l e .  the data i n t h i s  applied.  I t ' s application  s t u d y h a s y i e l d e d many i n t e r e s t i n g r e s u l t s  and i m p l i -  cations f o rf u r t h e r researcho  Summary o f t h e E x p e r i m e n t a l S t u d y  F o r t y s t u d e n t s who w e r e j u s t program i n mathematics  c o m p l e t i n g t h e grade  were randomly  The  (IQ range:  academic  s e l e c t e d from f o u r t e e n Algebra I I  classes i n t h r e e s e n i o r secondary schools. average mental a b i l i t y  eleven  115  The s u b j e c t s were o f  t o 125)  f o rt h i s  population.  s u b j e c t s w e r e g i v e n t h e Embedded F i g u r e s T e s t i n d i v i d u a l l y and t h e  s c o r e s r a n k - o r d e r e d and p a i r e d .  The s u b j e c t s f r o m e a c h p a i r w e r e  a s s i g n e d t o one o f two g r o u p s , one g r o u p w o r k i n g p r o b l e m s setting, the other i n a real world setting.  randomly  i n a mathematical  They were t h e n  interviewed  115 i n d i v i d u a l l y and asked t o t h i n k a l o u d as t h e y s o l v e d a s e l e c t i o n o f mathematical  word  problems.  A c o d i n g system  was d e v e l o p e d  b a s e d on t h e m o d e l o f h e u r i s t i c s o f  M a c P h e r s o n a n d was u s e d t o a n a l y z e t h e t a p e subjects. the  The c o d i n g s y s t e m  not only i n d i c a t e d t h e procedures  used by  s u b j e c t s , b u t t h e sequence i n w h i c h t h e y were a p p l i e d . When t h e c o d i n g s y s t e m  protocols, observed of  recorded protocols of the  was  applied t o the subjects' tape  recorded  o n l y a few o f t h e h e u r i s t i c s d e s c r i b e d b y MacPherson were  sufficiently  o f t e n t o be u s e d i n t h e s e q u e n t i a l a n a l y s i s .  Most  t h e s u b j e c t s w e r e v e r y c o r e o r i e n t e d a n d made l i m i t e d u s e o f t h e ...  heuristics. The d a t a c o l l e c t e d f r o m t h e c o d i n g f o r m s a l o n g w i t h t h e u s e o f r e g r e s s i o n a n a l y s i s were used i n answering posed i n Chapter  I.  t h r e e questions-  The i n f o r m a t i o n g a i n e d f r o m t h e p r o c e s s  was u s e d i n a n s w e r i n g in  the f i r s t  the last question.  matrices  Consider t h e four questions  order: 1.  What p r o c e d u r e s  from  c o r e and what h e u r i s t i c s  s t u d e n t s i n a t t a c k i n g and s o l v i n g m a t h e m a t i c a l All  o f t h e core procedures  used by the f o r t y s u b j e c t s .  are used by word problems?  i d e n t i f i e d i n t h e c o d i n g system  As w o u l d be e x p e c t e d ,  most  were  subjects  attempted  t o use a l g e b r a i c o p e r a t i o n s t o s o l v e t h e problems by  variables  a n d a t t e m p t i n g t o s e t u p some e q u a t i o n s .  identifying  The h e u r i s t i c s o f t e m p l a t i o n a n d r a n d o m c a s e s w e r e u s e d b y t h r e e f o u r t h s o f t h e s u b j e c t s o n a t l e a s t one p r o b l e m .  Over half of the  s t u d e n t s u s e d e a c h o f t h e s e h e u r i s t i c s more t h a n o n c e . and  analysis,  s e q u e n t i a l cases were used by o n e - f o u r t h o f t h e s t u d e n t s , w i t h  a n a l y s i s used o n l y on problem of  Smoothing,  t h e use o f c r i t i c a l  n u m b e r 5«  Some e v i d e n c e was  obtained  cases, a l l cases, h y p o t h e t i c a l deduction, i n v e r s e  116 deduction  ( u s e d b y f o u r s t u d e n t s o n p r o b l e m n u m b e r 4),  and  variation.  The o t h e r h e u r i s t i c s were n o t o b s e r v e d . Of t h e f o r t y - s e v e n c o r r e c t  s o l u t i o n s o b t a i n e d , 26  were t h r o u g h t h e  u s e o f random o r s y s t e m a t i c c a s e s , a r i t h m e t i c a l g o r i t h m s m a i n l y f o r 14,  g u e s s i n g f o r 6,  diagrams  and a l g e b r a i c  a l g o r i t h m s f o r 1.  was r e l a t e d t o t h e n u m b e r o f c o r r e c t s o l u t i o n s  of the h e u r i s t i c s .  Four o f t h e problems  most o f t h e s m o o t h i n g  accounted  The u s e o f as w e l l  a s many  were g e o m e t r i c i n n a t u r e , and  a n d t e m p l a t i o n was r e l a t e d t o g e o m e t r i c concepts«  Many o f t h e s t u d e n t s u s e d d i a g r a m s cases and drew a s e p a r a t e diagram  w i t h b o t h random and s y s t e m a t i c f o r each  case.  B o t h t h e t o t a l number o f h e u r i s t i c s u s e d b y a s u b j e c t as w e l l as t h e number o f d i f f e r e n t h e u r i s t i c s u s e d ,  account f o r a s i g n i f i c a n t  o f t h e v a r i a n c e i n t h e number o f c o r r e c t heuristics  solutions.  a c c o u n t e d f o r a n a d d i t i o n a l 21%  I n p a r t i c u l a r , ..."  of the variance not  f o r by core procedures w i t h systematic cases accounting f o r a significant  amount  subjects i n t h i s  of this  (13$).  amount  accounted statistically  Hence, t h e h e u r i s t i c s u s e d b y t h e  s t u d y added t o t h e i r a b i l i t y t o s o l v e problems  beyond  t h e i r m a t h e m a t i c a l c o r e knowledge» S t u d e n t s who u s e d b e t t e r problem  a wide  solvers.  range o f h e u r i s t i c s were on t h e average,  T h e s u b j e c t s who u s e d  u s e d t h e m more e f f e c t i v e l y .  a w i d e r range  of heuristics  T h e s e s t u d e n t s w e r e more w i l l i n g t o c h a n g e  t h e i r mode o f a t t a c k o r t h e p r o c e d u r e s t h e y w e r e u s i n g i f t h e y w e r e n o t having success i n obtaining a s o l u t i o n . s o l v i n g a p r o b l e m was  Changing  o n e ' s mode o f a t t a c k i n  significantly related to obtaining a  correct  solution. 2.  Does t h e c o n t e x t o f t h e p r o b l e m  Problem  i n f l u e n c e h e u r i s t i c usage?  c o n t e x t , as i d e n t i f i e d i n t h i s  t o t h e h e u r i s t i c s used.  s t u d y , p r o v e d t o be  unrelated  B o t h t h e t o t a l number o f h e u r i s t i c s u s e d  and  t h e number o f d i f f e r e n t problem  setting.  h e u r i s t i c s u s e d were n o t i n f l u e n c e d b y t h e  A l t h o u g h i n e a c h c a s e , t h e means o f t h e v a r i a b l e s  the real' world s e t t i n g , t h e d i f f e r e n c e s are not great. problems i n t h e mathematical understanding t h e problems, 3.  The  solving  working  world s e t t i n g had a s l i g h t l y harder b u t performed  time  as w e l l as t h e o t h e r group.  What e f f e c t d o e s f i e l d - d e p e n d e n c e - i n d e p e n d e n c e problem  Subjects  favor  have on t h e  process?  s u b j e c t s used i n t h i s  s t u d y were chosen from t h e m i d d l e I Q  range o f students t a k i n g A l g e b r a I I . independence had a marked e f f e c t number o f c o r r e c t s o l u t i o n s  W i t h i n t h i s IQ range,  on t h e u s e o f h e u r i s t i c s  field  and on t h e  obtained.  F i e l d dependent s t u d e n t s appear t o be v e r y o r i e n t e d t o t h e u s e o f core procedures..  T h e y e x h i b i t some u s e o f t h e l o w e r h e u r i s t i c s , b u t - i n  g e n e r a l t h e y a t t a c k e d most p r o b l e m s f r o m Most o f them drew a diagram to  h e l p s e t up e q u a t i o n s .  an a l g e b r a i c p o i n t o f v i e w .  f o ra problem,  b u t t h i s was u s e d i n o r d e r  Once t h e y h a d a t t e m p t e d  t o solve a  problem  w i t h o u t s u c c e s s , m o s t c o n t i n u e d i n t h e same v e i n u s i n g e x a c t l y t h e same procedures of  as b e f o r e .  t h e stamp problem  Evidence  (problem #2) i n which  unknowns c a n be w r i t t e n , second  equation.  be  field  independent  a single  equation w i t h two  and hence, t h e r e appears t o be a need f o r a  The f i e l d  t h e y o b t a i n e d one w h i c h The  o f this i s exemplified i nthe solving  dependent s t u d e n t s c o n t i n u e d t o do so u n t i l  was i n c o r r e c t o r d e p e n d e n t o n t h e f i r s t  students soon r e a l i z e d  a second  equation.  equation could not  o b t a i n e d a n d c h a n g e d t h e i r mode o f a t t a c k u s u a l l y t o c a s e s . What l i t t l e  e v i d e n c e t h e r e was o f t h e u s e o f t h e h i g h e r  came f r o m t h e f i e l d  independent  greater variety o f both s o l v i n g problems.  students.  heuristics  These s t u d e n t s used a  core and h e u r i s t i c s i n a t t a c k i n g and  T h e y w e r e much m o r e w i l l i n g t o c h a n g e t h e i r mode o f  lid attack. the  They o b t a i n e d a g r e a t e r number o f c o r r e c t s o l u t i o n s .  From  view p o i n t o f procedures used, as w e l l as c o r r e c t s o l u t i o n s , t h e  field  independent student i s a b e t t e r problem 4.  solver.  Do s e l e c t e d g r o u p s o f i n d i v i d u a l s e x h i b i t p a t t e r n s o f h e u r i s t i c u s a g e when s o l v i n g m a t h e m a t i c a l w o r d  Two d i f f e r e n t answer t h i s  criteria  problems?  were u s e d t o group s u b j e c t s i n an attempt t o  q u e s t i o n , b y problem c o n t e x t and b y degree o f f i e l d  S t u d e n t s g r o u p e d b y p r o b l e m c o n t e x t e x h i b i t e d t h e same g e n e r a l of  p r o b l e m s o l v i n g p r o c e d u r e s , b u t when g r o u p e d b y f i e l d  showed an o b s e r v a b l e d i f f e r e n c e . statistical  independence. patterns  independence  These d i f f e r e n c e s were n o t t e s t e d f o r  significance.  When g r o u p e d b y p r o b l e m c o n t e x t , b o t h g r o u p s e x h i b i t e d t h e same g e n e r a l p a t t e r n o f p r o b l e m s o l v i n g moves i n v o l v i n g c o r e p r o c e d u r e s a n d h e u r i s t i c s except f o r smoothing.  Subjects i nthe real world  setting  moved t o s m o o t h i n g f r o m a v a r i e t y o f p r o c e d u r e s w h e r e a s t h o s e i n t h e math w o r l d s e t t i n g u s e d smoothing m a i n l y t h r o u g h diagramming.  Both  g r o u p s o f s t u d e n t s moved t o a n d f r o m t h e u s e o f t e m p l a t i o n a n d random cases t o a l l o f t h e core procedures. B o t h t h e r e a l w o r l d and math w o r l d groups o b t a i n e d s o l u t i o n s the  using  same p r o c e d u r e s a n d s t o p p e d w o r k o n t h e p r o b l e m s f o l l o w i n g t h e same  moves.  T h e s t u d e n t s who e n c o u n t e r e d r e a l w o r l d p r o b l e m s e x p r e s s e d  concern f o r s o l u t i o n s obtained u s i n g h e u r i s t i c s while those students i n t h e math w o r l d s e t t i n g e x p r e s s e d none. for  However, e x p r e s s i o n o f c o n c e r n  s o l u t i o n was f o u n d t o be u n r e l a t e d t o c o r r e c t When g r o u p e d b y f i e l d  independence  solution.  a d i f f e r e n c e was f o u n d i n t h e  o v e r a l l p a t t e r n o f s e q u e n t i a l moves i n t h e p r o b l e m s o l v i n g p r o c e s s . F i e l d dependent s t u d e n t s had g r e a t e r d i f f i c u l t y problems.  i nunderstanding the  I n g e n e r a l t h e y were v e r y c o r e o r i e n t e d , making t h e m a j o r i t y  of  t h e i r moves f r o m c o r e p r o c e d u r e s t o c o r e p r o c e d u r e s .  The c o r e  procedures  u s e d m o s t o f t e n w e r e w r i t i n g a n e q u a t i o n a n d some u s e o f d i a g r a m s . F i e l d d e p e n d e n t s t u d e n t s e x h i b i t e d some u s e o f r a n d o m c a s e s , b u t m a i n l y used t h e h e u r i s t i c  of templation.  templating than the f i e l d The  field  independent  and h e u r i s t i c p r o c e d u r e s . templation, the  I n g e n e r a l , t h e y were p o o r e r a t  independent  student.  s t u d e n t moved m o r e f r e e l y b e t w e e n t h e c o r e He u s e d f o u r o f t h e h e u r i s t i c s :  smoothing,  random and s y s t e m a t i c c a s e s , a c r o s s a l l t h e p r o c e d u r e s i n ,  process matrix.  He was more c o n c e r n e d w i t h h i s w o r k b e c a u s e  c o n t i n u a l l y c h e c k i n g b o t h t h e p r o c e d u r e s h e was u s i n g a s w e l l solution.  When c o n c e r n e d w i t h h i s s o l u t i o n t h e f i e l d  as h i s  independent  was more w i l l i n g t o c h e c k i t , u s u a l l y b y r e t r a c i n g h i s s t e p s , the  field  dependent student r e r e a d t h e problem.  h e was  student  whereas  Perhaps t h e f i e l d  .  d e p e n d e n t s t u d e n t was n o t s u r e o f w h a t t h e p r o b l e m was a s k i n g s In solver.  general, the f i e l d He u s e d  independent  s t u d e n t was a b e t t e r  a greater variety of heuristics  a c r o s s a l l t h e c o r e and h e u r i s t i c p r o c e d u r e s .  problem  a n d moved m o r e  The f i e l d  freely  dependent  s t u d e n t was v e r y c o r e o r i e n t e d i n t h e p r o c e d u r e s h e u s e d .  Summary a n d C o n c l u s i o n s f o r T h e M o d e l a n d C o d i n g  The  System  h e u r i s t i c s d e f i n e d i n t h i s model account f o r t h e non-core  procedures used by t h e f o r t y subjects i n t h e study.  They a r e b r o a d  enough t o account f o r a l l o f t h e o b s e r v e d p r o c e s s e s and y e t , n o t so b r o a d t h a t n o r e a l d i s t i n c t i o n c a n b e made. to and  The h e u r i s t i c s  appear  be t h e n a t u r a l k i n d s o f p r o c e d u r e s t h a t would be used i n a t t a c k i n g solving The  problems.  u s e o f cases p r o v e d t o be an i m p o r t a n t p r o c e d u r e i n o b t a i n i n g  solutions.  One o f t h e s t r o n g a d v a n t a g e s  o f t h i s model over t h o s e  discussed i n Chapter of " t r i a l  II', i s t h e d i s t i n c t i o n between t h e d i f f e r e n t  and e r r o r " o r cases.  involved i n this  kinds  Over t h r e e - f o u r t h s o f t h e students  study used cases  o f some k i n d i n a n a t t e m p t  t o obtain  a s o l u t i o n t o a problem.  I t i s important t o d i s t i n g u i s h t h e type o f  sequence t h e y a r e u s i n g .  I s t h e student j u s t t r y i n g  some n u m b e r s i n  a c o m p l e t e l y random f a s h i o n i n hopes o f h i t t i n g o n t h e s o l u t i o n o r i s he  attempting t o f i n d  a s o l u t i o n i n some k i n d o f s y s t e m a t i c f a s h i o n ?  I s he p e r h a p s u s i n g t h e i n f o r m a t i o n o b t a i n e d f r o m the next The  o r a p p l y i n g t h e d a t a from t h e problem  student might determine  of possible solutions possible solving.  cases.  one c a s e t o  "improve**  i n a sequential fashion?  t h a t t h e r e c a n be o n l y a s m a l l f i n i t e number  (as i n problem  # 2) and hence, c o n s i d e r a l l  A l l o f t h e s e t y p e s o f cases have t h e i r p l a c e i n problem  They a r e d i f f e r e n t procedures  As w o u l d b e e x p e c t e d , t o u s i n g core procedures.  and s h o u l d be c o n s i d e r e d as such.  the subjects i n this  s t u d y were v e r y o r i e n t e d  Although they d i d e x h i b i t use o f the lower  h e u r i s t i c s , t h e y spent t h e m a j o r i t y o f t h e i r time t o s o l v e t h e problem  u s i n g only core m a t e r i a l .  s o l u t i o n s were o b t a i n e d b y t h e s e t h e s t u d e n t s were s t i l l  motivated  and e f f o r t  i n attempting  Even though fewer c o r r e c t  students than those i n the p i l o t  studies,  a n d a p p e a r e d t o do t h e i r b e s t i n w o r k i n g  a l l t h e problems. The  i n e f f e c t i v e use o f t h e h e u r i s t i c  most u s e d h e u r i s t i c ,  o f templation, which  r e v e a l e d t h a t s t u d e n t s d o n o t know w h a t c o r e m a t e r i a l  they can b r i n g t o bear  on a p a r t i c u l a r problem.  c a t e g o r i e s o f i n f o r m a t i o n ; sometimes c o r r e c t l y , recall  specific details.  the. Pythagorean  was t h e  Theorem.  In particular,  They can r e c a l l  general  sometimes n o t ; b u t cannot  only four students could  recall  M a n y o f t h e o t h e r s u b j e c t s knew t h e r e was some  r e l a t i o n s h i p among t h e s i d e s o f a r i g h t t r i a n g l e , w h i l e o t h e r s d i d n o t e v e n comment o n t h e r e l a t i o n s h i p .  T h e way t e m p l a t i o n i s d e f i n e d , i t  121 g i v e s no in  evidence of whether the  t h e p r o b l e m and  bridge the  gap  t h e n t r y i n g t o r o a m o v e r some c o n t e n t  t o the  conclusion or working the  c o n c l u s i o n o f t h e p r o b l e m and h e l p him  achieve  characteristic a forward little As  i t .  T h i s d i s t i n c t i o n may  manner o n l y , o r f r o m t h e  core  obtained  o r i e n t e d as  heuristic  and  information  area t o  o t h e r way,  help  l o o k i n g at  r e v e a l some  Does he  s h o u l d be  will  a t t a c k the problem i n  c o n c l u s i o n , w o r k i n g backwards  are. templation r e d e f i n e d as  heuristic  of  (very both?  a p p e a r s t o be  an  follows:  of d i r e c t templation The  h e u r i s t i c involves r e c a l l i n g a category  which i s related  This includes the  of content  such  r e c a l l of  problem types,  procedures,  premises.  purpose of d i r e c t t e m p l a t i o n i s t o  The  t h e o r e m s , and  use  i s u s e d when  a p r e m i s e o r s e t o f p r e m i s e s i s g i v e n o r assumed.  the premises given.  of  Inverse  recall  core  heuristic  when a c o n c l u s i o n i s g i v e n o r a s s u m e d . r e c a l l i n g a category the problem.  This includes the  problem types, conclusion.  of content  procedures,  The  the  c o n c l u s i o n and Very l i t t l e  heuristics  material gap  use  of the  r e c a l l of such t h i n g s  as  e v i d e n c e was  ( s e e F i g u r e 2,  p.  obtained t o 8).  Few  help  support  of  algorithms,  properties related to  that w i l l  involves  conclusion  recall  the  core  c o n c l u s i o n i n o r d e r t o expand knowledge  determine content  the  i s used  heuristic  which i s related t o the  t h e o r e m s , and  to  premises.  purpose of i n v e r s e templation i s t o  which i s r e l a t e d t o the  the  of i n v e r s e templation The  to  algorithms,  properties related to  expanding knowledge about t h e . T e m p l a t i o n - The  the  t h i n g s as  which i s r e l a t e d t o the premises i n order t o help bridge the c o n c l u s i o n by  the  important  of t h i s ) or i s i t a combination  students  D i r e c t T e m p l a t i o n - The  given  t r y i n g t o determine which content  of the problem s o l v e r .  e v i d e n c e was  important  subject i s l o o k i n g at the  material about  achieve i t .  any  ordering of  the  of the h e u r i s t i c s i n the middle  and  122 g e n e r a l c a t e g o r i e s were observed.  There appeared t o be c o n s i d e r a b l e  movement b e t w e e n t h e s i e v e ( c o r e ) a n d t h e l o w e r h e u r i s t i c s . higher h e u r i s t i c s which a lower h e u r i s t i c  were observed,  o r back t o core.  but t h i s for  S u b j e c t s d i d move f r o m b o t h t h e  The  c o d i n g system  c o r e and h e u r i s t i c  The system  are used w i l l  Concern  procedures.  and c a p a b l e  b u t a l s o i n d i c a t e s t h e sequence i n which  procedures  A  of identifying  not only i d e n t i f i e s t h e procedures  o r d e r t o be a b l e t o i d e n t i f y p r o b l e m which  Q  r e l a t e d t o e x p r e s s i o n o f concern,  p r o v e d t o be r e l i a b l e  individual differences. used,  which  ( F i g u r e 3 , p . 1 )<>  e x p r e s s i o n was u n r e l a t e d t o o b t a i n i n g a s o l u t i o n .  p r o c e s s was r e l a t e d t o b o t h  those  s t u d e n t s u s i n g t h e m d i d move t o  s i e v e and h e u r i s t i c s t o o b t a i n i n g a s o l u t i o n Some e v i d e n c e was o b t a i n e d  From  they  are applied.  In  s o l v i n g s t y l e s , t h e sequence i n  c e r t a i n l y be a p r i n c i p a l s o u r c e o f  information.  Implications f o r Education  The  findings of this  s t u d y i n d i c a t e t h a t s t u d e n t s who a r e c o m p l e t i n g  a n A l g e b r a I I c o u r s e h a v e some f a c i l i t y w i t h t h e h e u r i s t i c s d e f i n e d i n MacPherson's model. suggest  In particular,  t h a t s t u d e n t s who u s e t h e h e u r i s t i c s o f r a n d o m a n d s y s t e m a t i c  c a s e s , when f a c e d w i t h a d i f f i c u l t s u c c e s s f u l i n s o l v i n g t h e problem. use  some e v i d e n c e w a s o b t a i n e d t o  mathematical  problem,  t e n d t o be  Many o f t h e s t u d e n t s who f a i l e d t o  c a s e s may h a v e k n o w n how t o u s e t h i s t e c h n i q u e , b u t a v o i d e d i t b e c a u s e  they thought  i t was i n a p p r o p r i a t e .  u s e d cases d i d s o as a l a s t  I n f a c t , most o f t h e s t u d e n t s  r e s o r t w i t h some i m p l y i n g t h a t ,  t h i s t e c h n i q u e was n o t a c c e p t a b l e t o u s e a s a p r o c e d u r e  although  i n the classroom,  t h e y w o u l d a t l e a s t " t r y some n u m b e r s " t o s e e w h a t w o u l d h a p p e n . m a t h e m a t i c s t e a c h e r s who a r e e a g e r  who  Many  t o have t h e i r s t u d e n t s l e a r n t h e  123 " r i g h t " way o f s o l v i n g p r o b l e m s , use  o f cases. Polya  his he  (1962, p . 26) s u g g e s t s  students from u s i n g t r i a l  that t h e teacher should not discourage  and e r r o r  (cases) - but t o t h e contrary,  should encourage t h e i n t e l l i g e n t use o f t h i s  Many o f t h e g o o d p r o b l e m systematic cases how  p e n a l i z e t h e s t u d e n t s who r e s o r t t o t h e  effectively.  t o u s e t h e method.  very l i t t l e  solvers i nthis  s t u d y d i d u s e b o t h random and  When t r y i n g t o u s e c a s e s , t h e s e  i n f o r m a t i o n about t h e problem.  g i v e n t o t h e use o f cases heuristic  procedure.  Some o f t h e s t u d e n t s d i d n o t u n d e r s t a n d  b o t h mathematics t e a c h e r s and t h e i r  The  fundamental  students  I t would be advantageous t o  s t u d e n t s i f more c o n s i d e r a t i o n were  as an a c c e p t a b l e problem  solving  procedure.  o f t e m p l a t i o n was u s e d b y t h r e e - f o u r t h s o f t h e  s t u d e n t s and y e t h a d no e f f e c t  on t h e s o l u t i o n s o b t a i n e d .  t h i s may b e t h a t s t u d e n t s d o n ' t h a v e t o t e m p l a t e a r e a s k e d t o do i n t h e c l a s s r o o m . r e q u i r e d f o r a problem  A reason  f o r t h e problems  for  they  I n most t e x t b o o k s t h e m a t e r i a l  i s g i v e n i n t h e two o r t h r e e pages  preceeding the exercise.  immediately  I n g e n e r a l , s t u d e n t s don't have t o r e c a l l _  m a t e r i a l from t h e p r e v i o u s y e a r o r even t h e p r e v i o u s c h a p t e r . exercises involve the direct from t h e textbook.  gained  _..  Most  a p p l i c a t i o n o f t h e contents o f a few pages  The s t u d e n t s i n t h i s . s t u d y h a d v e r y p o o r r e c a l l o f t h e  core m a t e r i a l t h e y had s t u d i e d and very l i t t l e t h e y could use i n a problem.  i d e a o f what c o r e m a t e r i a l  T h i s i s n o t s u r p r i s i n g s i n c e most  a r e t o l d what t o do b y b o t h t h e t e x t  and t h e t e a c h e r .  I fthe creation  new p r o b l e m  s e t s i s as important  of  t h i s paper,  t h e n n o t o n l y a r e c u r r i c u l u m d e v e l o p e r s g o i n g t o have t o  but  as suggested  students  of  devote  i n t h e opening  more t i m e t o t h e c r e a t i o n o f i m a g i n a t i v e a n d c r e a t i v e  be p l a c e d i n t h e e x e r c i s e s i m m e d i a t e l y  remarks  problems,  a l s o t o t h e placement o f these problems i n t h e c u r r i c u l u m .  s h o u l d n o t always  v  Problems  following the  .: -  124 p r e s e n t a t i o n o f t h e m a t e r i a l needed t o s o l v e them, b u t t h e y included i n exercises at a l a t e r Another  s h o u l d be  date.  of the findings of this  study i sthat those  s t u d e n t s who  c h a n g e t h e i r mode o f a t t a c k a r e m o r e s u c c e s s f u l i n o b t a i n i n g a s o l u t i o n . Most t e a c h e r s do t e l l t h e i r having d i f f i c u l t y  s t u d e n t s t o t r y a n o t h e r method i f t h e y a r e  i n s o l v i n g a problem,  b u t i f students are going t o be  a b l e t o c h a n g e t h e i r mode o f a t t a c k , t h e y h a v e t o b e a b l e t o r e c a l l t h e core m a t e r i a l t h e y have s t u d i e d (template) o r be able t o a t t a c k t h e problem  by using various  heuristics.  (l953  Henderson and P i n g r y problems and a conscious improve problem problem  p.  238) s t a t e  s o l v i n g which  that practice i n solving  awareness o f t h e problem  s o l v i n g performance.  s o l v i n g p r o b l e m s md 1962,  f  solving process  P o l y a proposes  c o n s i s t s o f two aspects:  a program f o r t e a c h i n g  abundant e x p e r i e n c e i n  serious study o f t h e solution process.  p. v ) expresses t h e need f o r t h e f i r s t o f t h e s e  " S o l v i n g problems i s a p r a c t i c a l a r t , l i k e  will  He ( P o l y a ,  as f o l l o w s :  swimming o r s k i i n g o r p l a y i n g t h e  p i a n o ; y o u can l e a r n i t o n l y b y i m i t a t i o n and p r a c t i c e . " P o l y a a l s o warns t h a t i n problem are n o t s u f f i c i e n t .  s o l v i n g , i m i t a t i o n and p r a c t i c e  N o t o n l y must p r o b l e m s be s o l v e d , b u t t h e l e a r n e r s '  a t t e n t i o n must b e d i r e c t e d t o t h e methods u s e d .  T h e s e must b e g e n e r a l  e n o u g h s o t h a t t h e y become a v a i l a b l e f o r u s e i n s o l v i n g s i m i l a r  problems  in the future. The  h e u r i s t i c s i n MacPherson's model a r e o f a g e n e r a l n a t u r e .  appear t o be t h e n a t u r a l k i n d s o f procedures a t t a c k i n g and s o l v i n g mathematical P i n g r y and P o l y a do i n d e e d  problems.  They  t h a t would be used i n I f t h e programs o f Henderson,  have m e r i t f o r t e a c h i n g problem  solving,  then  perhaps t h e h e u r i s t i c s from t h i s model s h o u l d be i n c l u d e d . Problem ability,  solving,  as c o n s i d e r e d i n t h i s  i s r e l a t e d t o f i e l d independence.  study w i t h students o f average N o t o n l y does t h e f i e l d  125 dependent student  o b t a i n fewer c o r r e c t s o l u t i o n s , he i s a l s o l e s s  w i l l i n g t o change h i s method o f a t t a c k once he s t a r t s t o s o l v e Spitler  (1970) d i s c u s s e d  to help  students  1.  a problem.  f i v e p r o c e d u r e s w h i c h may b e u s e d b y t h e t e a c h e r  l e a r n t o s o l v e p r o b l e m s u s i n g d i f f e r e n t modes o f a t t a c k .  Near t h e end o f each s e t o f problems w h i c h a r e d e s i g n e d t o provide  practice i n a given  which require other 2.  procedure, i n c l u d e s e v e r a l problems  procedures.  P r o b l e m s w h i c h c a n b e o r must b e done u s i n g d i f f e r e n t p r o c e d u r e s should  3.  be i n t e r s p e r s e d t h r o u g h o u t a s e t o f p r o b l e m s .  The t e a c h e r  should  responses w i t h h i s 4.  Superfluous the  5.  regularly discuss t h e influences o f patterned students.  information  should  be i n c l u d e d i n problems t h r o u g h o u t  exercises.  When p r a c t i c i n g c o m p l e x p r o b l e m s o l v i n g , p r o b l e m s w h i c h t h e student  recognizes  should  be i n c l u d e d .  t h a t he c o u l d s o l v e b y a l e s s complex p r o c e d u r e ,  S p i t l e r a d d i t i o n a l l y recommends t h a t f i e l d  dependent students  be  g i v e n p r o b l e m s e t s w h i c h r e q u i r e many d i f f e r e n t m e t h o d s o f s o l u t i o n . The teacher  recommendations i n t h i s  s e c t i o n r e q u i r e t h a t t h e mathematics  c o n t i n u a l l y a s s i g n problems which r e q u i r e d i f f e r e n t procedures  t o o b t a i n a s o l u t i o n and t h a t t h e s e p r o c e d u r e s be d i s c u s s e d students. in the  .  Also the f i e l d  grouping students  with the  i n d e p e n d e n t - d e p e n d e n t c o n s t r u c t may b e u s e f u l  f o r t e a c h i n g mathematical problem s o l v i n g because  problem s o l v i n g behavior  o f students  i n these two groups i n d i c a t e  t h e y may h a v e a n e e d f o r d i f f e r e n t t e a c h i n g s t r a t e g i e s .  Limitations o f t h e Study  There a r e s e v e r a l l i m i t a t i o n s o f t h e present three  study.  .The f i r s t  l i m i t a t i o n s were n o t e d i n C h a p t e r I as i n c o m i n g l i m i t a t i o n s  ofthe  126 study.  These l i m i t a t i o n s a r e r e s t a t e d here a l o n g w i t h o t h e r l i m i t a t i o n s  w h i c h w e r e d e t e r m i n e d a s t h e s t u d y was c a r r i e d o u t . All academic  o f t h e s u b j e c t s were o f average  a b i l i t y f o r s u b j e c t s on an  mathematics  r e g u l a r i t i e s observed give  evidence that  program,  above a v e r a g e o r weak s t u d e n t s w o u l d  problem s o l v i n g behavior. academic  therefore,  e x h i b i t t h e same  The s u b j e c t s w e r e r a n d o m l y  s e l e c t e d from t h e  grade e l e v e n program ( A l g e b r a I I ) and so any  made b a s e d u p o n t h e s e r e s u l t s  are l i m i t e d t o t h i s  no  generalizations  or a similar  population  of students. An o b v i o u s l i m i t a t i o n i s t h e s e l e c t i o n o f problems study.  used i n t h e  The d a t a f o r e a c h s t u d e n t were c o l l e c t e d f r o m one s e s s i o n i n  w h i c h t h e s t u d e n t was a s k e d t o s o l v e f i v e p r o b l e m s .  An  individual  may h a v e e x h i b i t e d d i f f e r e n t p r o b l e m s o l v i n g c h a r a c t e r i s t i c s i f a s k e d t o attempt a d i f f e r e n t  s e t o f problems.  The p r o b l e m s  used i n t h i s  study  were n o t s e l e c t e d f r o m any p a r t i c u l a r c o r e a r e a n o r were t h e y o f any p a r t i c u l a r t y p e , a l t h o u g h f o u r o f t h e m i n v o l v e a maximum o r minimum solution. There were a l s o l i m i t a t i o n s  r e s u l t i n g from t h e use o f the " t h i n k  a l o u d " method i n d e t e r n i i n i n g t h e p r o c e d u r e s u s e d b y t h e s t u d e n t s i n s o l v i n g problems. might  T h e m e t h o d i t s e l f may b e u n r e l i a b l e  r e m a i n s i l e n t d u r i n g moments o f d e e p e s t t h o u g h t .  verbalized  s o l u t i o n c o u l d be e s s e n t i a l l y d i f f e r e n t  silently.  The p r e s e n c e o f a n o b s e r v e r m i g h t i n h i b i t  s i n c e an Moreover,  f r o m one  both K i l p a t r i c k  (1967) a n d  Kantowski  a  affected  a problem s o l v e r i n  s u c h a way t h a t h e m i g h t n o t a t t e m p t s o l u t i o n s w h i c h he f e l t c o n s i d e r e d f o o l i s h b y someone e l s e .  individual  would  be  These l i m i t a t i o n s were n o t e d b y  (1975).  There were a l s o l i m i t a t i o n s i n t h e c o d i n g system used.  The  subjects  i n t h e s t u d y were v e r y o r i e n t e d t o u s i n g c o r e p r o c e d u r e s and s p e n t a  127 great d e a l o f time core procedures  attempting t o understand  t h e problems.  Perhaps i f t h e  used i n t h e coding system had i n c l u d e d a f i n e r  i n t h e r e a d i n g a n d p r e p a r a t i o n c a t e g o r i e s , some i m p o r t a n t c h a r a c t e r i s t i c s may h a v e b e e n  division  problem  solving  identified.  F i n a l l y , t h e a n a l y s i s o f t h e s e q u e n t i a l d a t a was o f l i m i t e d u s e . S e q u e n c e s w e r e o n l y a n a l y z e d i n p a i r s o f c o n s e c u t i v e moves a n d t h i s analysis only involved a non-statistical  comparison o f  percentages.  I f p r o b l e m s o l v i n g s t y l e s a r e t o be i d e n t i f i e d , t h e sequence o f w i l l m o s t c e r t a i n l y o f f e r many i m p o r t a n t  procedures  clues.  There i s a l s o a l i m i t a t i o n i n terms o f t h e i n t e r p r e t a t i o n o f t h e r e s u l t s from t h e r e g r e s s i o n a n a l y s i s .  E x t r e m e s c o r e s o n some o f t h e  v a r i a b l e s may h a v e i n f l u e n c e d t h e s t a t i s t i c a l o f t h e s m a l l sample s i z e statistical of these  analysis.  (N = 40),  these  Appendix D contains t h e histograms  solving process  i n undertaking  Research  studies i n which t h e problem  i s investigated i s that of determining the  procedures  The u s e o f t h e " t h i n k a l o u d " method a s a d a t a g a t h e r i n g  gaining i n popularity.  Roth  f o rseveral  variables.  One p r a c t i c a l d i f f i c u l t y  is  However, because  s c o r e s were n o t d e l e t e d from t h e  Implications for  used.  results.  (1966)  and F l a h e r t y  However, w i t h t h e e x c e p t i o n o f t h e s t u d i e s b y  (1973),  t h e r e l i a b i l i t y o f t h i s method.  little  r e s e a r c h h a s been done t o  Do s t u d e n t s  when a s k e d t o v e r b a l i z e t h e i r t h o u g h t s problem?  technique  T h i s and o t h e r q u e s t i o n s  a t t a c k problems  t h a n when l e f t  determine  differently  alone t o solve t h e  r e l a t e d t o t h e " t h i n k a l o u d " method  need f u r t h e r i n v e s t i g a t i o n . Krutetskii  (1969)  b e l i e v e s t h a t t h e problem s o l v i n g processes  s t u d e n t s w i t h above average o r l o w m a t h e m a t i c a l  aptitude d i f f e r  of  from  128 those  students w i t h average mathematical  aptitudes.  Therefore,  s t u d y s h o u l d be r e p l i c a t e d u s i n g s t u d e n t s o f h i g h e r and l o w e r It  i s p o s s i b l e t h a t t h e problems s e l e c t e d f o rt h e lower  w o u l d have t o be d i f f e r e n t . in  different  content  areas  determine  content such  Since t h e procedures  a r e a s may v a r y , t h i s  as w e l l as t o suggest  used i n s o l v i n g  solving,  In particular,  this  One i n t e r e s t i n g  model i s t h e o r d e r i n g o f t h e h e u r i s t i c s  as t h i s  mathematical  study gives very l i t t l e  information replicated  Also t h e study  r e p l i c a t e d u s i n g open ended problems t o determine  the use of higher h e u r i s t i c s .  as s t u d e n t s  e x p l o r a t o r y s t u d i e s such  on t h e u s e o f t h e h i g h e r h e u r i s t i c s , t h e r e f o r e , i t s h o u l d be  be  of content  area.  s o l v i n g may d i f f e r  u s i n g s t u d e n t s who a r e v e r y m a t h e m a t i c a l l y m a t u r e .  using  and c a l c u l u s t o  s h o u l d be done u s i n g s t u d e n t s a t v a r i o u s s t a g e s o f  sophistication.  problems  s t u d y s h o u l d be r e p e a t e d  s p e c i f i c t o a content  used i n problem  g a i n more e x p e r i e n c e i n p r o b l e m one  students  e x i s t i n h e u r i s t i c usage a c r o s s areas  hypotheses  Since t h e procedures  ability.  ability  as a l g e b r a , geometry, number t h e o r y ,  i f regularities  this  i f they w i l l  may elicit  aspect o f MacPherson's  (seeFigure 2  t  p. 8) i n t o  four  categories. In  order t o determine  i f t h e o r d e r i n g o f t h e h e u r i s t i c s i n MacPherson's  model does i n d e e d h o l d , a s t u d y w i l l have t o be c a r r i e d o u t u s i n g s u b j e c t s who m i g h t d e m o n s t r a t e t h e u s e o f t h e s e h e u r i s t i c s . of  Also, a modification  t h e m o d e l , a d d i t i o n o r d e l e t i o n o f some h e u r i s t i c s ,  out i f t h e d a t a o b t a i n e d from The  coding system  proved  such  c o u l d be  carried  a s t u d y so i n d i c a t e ,  t o b e v e r y r e l i a b l e when u s e d b y  coders  who w e r e b o t h v e r y f a m i l i a r w i t h t h e h e u r i s t i c s a n d t h e c o d i n g The in The  system  was d e s i g n e d p r i m a r i l y w i t h t h e p r o b l e m s u s e d i n t h i s  m i n d a n d h e n c e n e e d s much r e f i n e m e n t , c o d i n g system  e s p e c i a l l y i n t h e core  system. study area.  s h o u l d b e a p p l i e d t o new s t u d e n t s a n d new s e t s o f p r o b l e m s  to  see i f t h e d i f f e r e n c e s observed i n t h e present study are maintained.  I f t h e s y s t e m i s u s e d w i t h s t u d e n t s who a r e m o r e m a t h e m a t i c a l l y s o p h i s t i c a t e d t h a n t h e ones i n t h i s  s t u d y , some o f t h e c a t e g o r i e s m i g h t  show g r e a t e r  frequency o f usage. The  c o d i n g system  coded i n sequence.  a l l o w s f o r a l l o f t h e p r o c e d u r e s u s e d t o be  Some a t t e m p t was made t o a n a l y z e t h e s e q u e n t i a l  d a t a , b u t t h e i n f o r m a t i o n o b t a i n e d was l i m i t e d . s t y l e s a r e g o i n g t o be i d e n t i f i e d , u s e d w i l l be a m a j o r in  t h e sequence  source o f informationo  I f any problem  solving  i n which procedures are  More r e s e a r c h i s needed  d e v e l o p i n g and r e f i n i n g methods o f a n a l y z i n g s e q u e n t i a l  information.  T h i s s t u d y d i d n o t i n c l u d e an i n s t r u c t i o n a l phase because  i t was  d e s i r e d t o o b s e r v e what h e u r i s t i c s were b e i n g u s e d b y s t u d e n t s w i t h o u t any s p e c i a l i n s t r u c t i o n .  Research i s needed which i n c l u d e s  instruction  on t h e u s e o f h e u r i s t i c s t o d e t e r m i n e i f such i n s t r u c t i o n w i l l the  e f f e c t i v e n e s s o f t h e h e u r i s t i c s beyond t h a t  core procedures.  accounted f o r by t h e  Research i s a l s o needed t o determine t h e e f f e c t o f  i n s t r u c t i o n on t h e problem  s o l v i n g processes used by f i e l d  and f i e l d d e p e n d e n t s t u d e n t s . heuristics,  increase  independent  The s t u d e n t s ' knowledge o f b o t h c o r e and  n o m a t t e r how l i m i t e d ,  should be t a k e n i n t o  account i n  d e s i g n i n g e x p e r i m e n t a l s t u d i e s which i n c l u d e an i n s t r u c t i o n a l E x p l o r a t o r y s t u d i e s such as Lucas a n a l y s i s o f t h e problem  (1972)  and Kantowski  phase.  (1975) w h e r e  s o l v i n g p r o c e s s was u n d e r t a k e n d u r i n g t h e s t u d y  i n d i c a t e t h a t i n s t r u c t i o n i n h e u r i s t i c s n o t o n l y a f f e c t s problem performance, students.  but also has a p o s i t i v e  effect  solving  on t h e procedures used b y t h e  A p l a n n e d program o f p r o c e s s r e s e a r c h , where t h e r o l e o f  h e u r i s t i c s i n problem  s o l v i n g i s s y s t e m a t i c a l l y i n v e s t i g a t e d i s needed.  LITERATURE CITED  Amidon, E . J . & Hough, J . B. ( E d s . ) I n t e r a c t i o n a n a l y s i s ; Theory, r e s e a r c h and a p p l i c a t i o n . Reading, Mass. : A d d i s o n - W e s l e y , 1967• B e g l e , E . G., ( E d . ) T h e r o l e o f a x i o m a t i c s a n d p r o b l e m s o l v i n g i n mathematics. Report, Conference Board o f t h e Mathematical Sciences, Boston: G i n n , 1966. B e g l e , E . G. & W i l s o n , J . W. 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J . & F r a n d s e n , H. R e s e a r c h o n t e a c h i n g s e c o n d a r y - s c h o o l mathematics. I n R. M. W. T r a v e r s ( E d . ) , S e c o n d h a n d b o o k o f r e s e a r c h on t e a c h i n g . C h i c a g o : R a n d M c N a l l y , 1973. Dewey, J . How we t h i n k .  (2nd. e d . ) B o s t o n :  Heath,  1933.  131 D o d s o n , J . W. C h a r a c t e r i s t i c s of s u c c e s s f u l i n s i g h t f u l problem s o l v e r s , NLSMA R e p o r t s N o . 31. J . W. W i l s o n & E . G. B e g l e ( E d s . ) , S t a n f o r d : School Mathematics Study Group, 1972. F a r r , R. S. P e r s o n a l i t y v a r i a b l e s and 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 An i n v e s t i g a t i o n o f t h e r e l a t i o n s h i p s b e t w e e n f i e l d - d e p e n d e n c e i n d e p e n d e n c e , s e x - r o l e i d e n t i f i c a t i o n , p r o b l e m d i f f i c u l t y and p r o b l e m s o l v i n g performance. D i s s e r t a t i o n A b s t r a c t s . 1968, 2£, 256I-A. (Abstract) F l a h e r t y , E . G. C o g n i t i v e processes used i n s o l v i n g mathematical problems. D i s s e r t a t i o n A b s t r a c t s , 1973, 34. 1767-A. (Abstract) Goals f o r school mathematics. Report o f t h e Cambridge Conference.on School Mathematics. B o s t o n : H o u g h t o n M i f f l i n , 1963* Goodman, study D. R. Wiley  B. F i e l d d e p e n d e n c e a n d c l o s u r e f a c t o r s , i960. U n p u b l i s h e d q u o t e d i n W i t k i n , H. A., D y k , R. B., F a t e r s o n , H. J . , G o o d e n o u g h , & K a r p , S. A. Psychological differentiation. New Y o r k : and Sons, 1962.  G r a y , J . S. What s o r t o f e d u c a t i o n i s r e q u i r e d f o r a d e m o c r a t i c c i t i z e n ship?. S c h o o l a n d S o c i e t y . 1935 . 4j~, 353-359. G r e e n , B. F . Current t r e n d s i n problem s o l v i n g . I n B. J . K l e i n m u n t - z (Ed.), Problem s o l v i n g : R e s e a r c h , method, and t h e o r y . New Y o r k : W i l e y , 1966. G u s t z k o w , H. behavior.  An a n a l y s i s o f t h e o p e r a t i o n o f s e t i n p r o b l e m - s o l v i n g 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 , 1951, 4_5_, 219-244.  H a d a m a r d , J . The p s y c h o l o g y o f i n v e n t i o n i n t h e m a t h e m a t i c a l f i e l d . Princeton: P r i n c e t o n U n i v e r s i t y P r e s s , 1945«  -  H e n d e r s o n , K. B., & P i n g r y , R. E . Problem s o l v i n g i n mathematics. In H. F. F e h r ( E d . ) , The l e a r n i n g o f m a t h e m a t i c s : I t s t h e o r y and p r a c t i c e . Yearbook, N a t i o n a l C o u n c i l o f Teachers o f Mathematics,  1953,  21, 228-270.  H u m p h r e y , G.  Directed thinking.  New  York:  Dodd-Mead,  194&.  J e r m a n , M. I n s t r u c t i o n i n p r o b l e m s o l v i n g and an a n a l y s i s o f s t r u c t u r a l variables that contribute t o problem-solvingd i f f i c u l t y . Technical R e p o r t N o . 180, N o v e m b e r 12, 1971. Stanford University, Institute for Mathematical Studies i n the Social Sciences. J o h n s o n , D. M. A modern a c c o u n t o f p r o b l e m s o l v i n g . B u l l e t i n . 1944, U±, 201-229.  Psychological  K a n t o w s k i , M. G. Processes i n v o l v e d i n mathematical problem s o l v i n g . Paper presented at the annual meeting o f t h e N a t i o n a l C o u n c i l o f Teachers o f Mathematics, Denver, A p r i l , 1975. K a p l a n , A. The c o n d u c t o f i n q u i r y . P u b l i s h i n g Co., 1964.  Scranton, Pennsylvania:  Chandler  132 K e r l i n g e r , F . N. & P e d h a z u r , E . J . M u l t i p l e r e g r e s s i o n i n b e h a v i o r a l research. New Y o r k : H o l t R i n e h a r t and W i n s t o n , 1973. K i l p a t r i c k , J . A n a l y z i n g t h e s o l u t i o n o f word problems i n mathematics % An e x p l o r a t o r y s t u d y . Unpublished doctoral d i s s e r t a t i o n , Stanford University, 1967. K i l p a t r i c k , J . P r o b l e m - s o l v i n g and c r e a t i v e b e h a v i o r i n mathematics. I n J . W. W i l s o n a n d L . R. C a r r y ( E d s . ) , R e v i e w s o f r e c e n t r e s e a r c h i n mathematics education. Studies i n mathematics, V o l . 19. Stanford: S c h o o l M a t h e m a t i c s S t u d y G r o u p , 19o9« K i n g s l e y , H. L . & G a r r y , R. The n a t u r e and c o n d i t i o n s o f l e a r n i n g s (2nd. e d . ) E n g l e w o o d C l i f f s , New J e r s e y : Prentice Hall, 1957. K l e i n m u n t z , B. J . ( E d . ) , P r o b l e m s o l v i n g : New Y o r k : Wiley, 1966.  R e s e a r c h , method, and t h e o r y .  K r u t e t s k i i , V. A. M a t h e m a t i c a l a p t i t u d e s . I n J . K i l p a t r i c k and I . W i r s z u p ( E d s . ) , The s t r u c t u r e o f m a t h e m a t i c a l a b i l i t i e s . Soviet s t u d i e s i n t h e p s y c h o l o g y o f l e a r n i n g and t e a c h i n g mathematics. V o l . 2. Stanford: School Mathematics Study Group, 1969. L u c a s , . J . F. A n e x p l o r a t o r y s t u d y on t h e d i a g n o s t i c t e a c h i n g o f h e u r i s t i c problem s o l v i n g s t r a t e g i e s i n c a l c u l u s . Unpublished d o c t o r a l d i s s e r t a t i o n , The U n i v e r s i t y o f W i s c o n s i n , 1972. Luchins, 1942,  A. S. M e c h a n i z a t i o n £4. ( W h o l e N o . 248).  i n problem s o l v i n g .  P s y c h o l o g i c a l Monographs,  M a c P h e r s o n , E . D. A way o f l o o k i n g a t t h e c u r r i c u l u m . The J o u r n a l o f E d u c a t i o n o f t h e F a c u l t y o f E d u c a t i o n , V a n c o u v e r , 1970, 16 , 44-5"1° M a c P h e r s o n , E . D. U n p u b l i s h e d p a p e r o n h e u r i s t i c s , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , 1973. (Mimeographed) McGrew, W. C. A n e t h o l o g i c a l s t u d y Academic P r e s s , 1972. M i l l e r , G. A., of behavior.  of children's behavior  G a l a n t e r , E . & P r i b r a m , K. H. New Y o r k : H o l t , I960.  Plans  3  New Y o r k :  and t h e s t r u c t u r e  P a i g e , J . M. & S i m o n , H. A. C o g n i t i v e p r o c e s s e s i n s o l v i n g a l g e b r a word problems. I n B. J . K l e i n m u n t z ( E d . ) , P r o b l e m s o l v i n g : Research, method, and t h e o r y . New Y o r k : Wiley, 1966. P o l y a , G. How t o s o l v e i t . Anchor Books, 1957.  (2nd.  ed.)  G a r d e n C i t y , New Y o r k :  Doubleday  P o l y a , G. Mathematical discovery: On u n d e r s t a n d i n g , l e a r n i n g , a n d t e a c h i n g p r o b l e m s o l v i n g . V o l . I . New Y o r k : W i l e y , 19620 P o l y a , G. Mathematical discovery: On u n d e r s t a n d i n g , l e a r n i n g , and t e a c h i n g p r o b l e m s o l v i n g . V o l . I I , New Y o r k : W i l e y , 1965=  133 R o m b e r g , T. A . & W i l s o n , J . W. T h e d e v e l o p m e n t o f t e s t s . NLSMA R e p o r t s No. 7. J . W. W i l s o n , L . S. C o h e n & E . G. B e g l e ( E d s . ) , S t a n f o r d : S c h o o l M a t h e m a t i c s S t u d y G r o u p , 1969. Rosenfeld, I ; J . Mathematical a b i l i t y as a f u n c t i o n o f p e r c e p t u a l f i e l d dependency and c e r t a i n p e r s o n a l i t y v a r i a b l e s . Unpublished doctoral d i s s e r a t i o n , T h e U n i v e r s i t y o f O k l a h o m a , 1958. R o t h , B. The e f f e c t s o f o v e r t v e r b a l i z a t i o n on problem s o l v i n g . D i s s e r t a t i o n A b s t r a c t s . I966, 27_, 957-B. (Abstract) S a a r n i , C. I . P i a g e t i a n o p e r a t i o n s a n d f i e l d i n d e p e n d e n c e a s f a c t o r s i n c h i l d r e n ' s problem s o l v i n g performance. Paper presented a t t h e annual meeting o f t h e American Educational Research A s s o c i a t i o n , Chicago. A p r i l , 1972 S c h w i e g e r , R. D. A c o m p o n e n t a n a l y s i s o f m a t h e m a t i c a l p r o b l e m 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 , P u r d u e U n i v e r s i t y , 1974.  solving.  S p i t l e r , G. J . A n i n v e s t i g a t i o n o f v a r i o u s c o g n i t i v e s t y l e s a n d t h e i m p l i c a t i o n s f o r mathematics education. Unpublished doctoral 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» W a l b e r g , H. J . G e n e r a l i z e d r e g r e s s i o n m o d e l s i n e d u c a t i o n a l A m e r i c a n E d u c a t i o n a l R e s e a r c h J o u r n a l , 1971, 8, 71-93» W a l l a s , G.  1926.  The a r t o f t h o u g h t .  New Y o r k :  research.  H a r c o u r t , Brace and World,  W a l s h , R. J . 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 a n d s u c c e s s e x p e c t a n c i e s a s a f u n c t i o n o f t h e c o g n i t i v e s t y l e o f f i e l d independence v s . dependence, and e x p e r i m e n t a l l y i n d u c e d s u c c e s s - f a i l u r e e x p e c t a n c i e s . Dissertation A b s t r a c t s . 1974, 3_5_, 2 0 6 6 - A . (Abstract) Webb, N . A n e x p l o r a t i o n o f m a t h e m a t i c a l p r o b l e m - s o l v i n g p r o c e s s e s . Paper presented a t t h e annual meeting o f t h e 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 , W a s h i n g t o n , D. C , M a r c h , 1975» W i c k e l g r e n , W. A . How t o s o l v e p r o b l e m s : Elements and t h e o r y o f problems and p r o b l e m s o l v i n g . SanFrancisco: W. H. F r e e m a n , 1974* W i t k i n , H. A . E m b e d d e d f i g u r e s t e s t . P s y c h o l o g i s t s P r e s s , 1969.  Palo Alto,  California:  Consulting  W i t k i n , H. A., D y k , R. B., F a t e r s o n , H. J . , G o o d e n o u g h , D. R. & K a r p , S. A. P s y c h o l o g i c a l d i f f e r e n t i a t i o n . New Y o r k : W i l e y , 1962. W i t k i n , H. A., O l t m a n , P . K., R a s k i n , E . & K a r p , S. A. 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 . Palo Alto, California: Consulting Psychologists Press, 1971.  Appendix A  PROBLEMS USED I N P I L O T STUDY  A b o o k s h e l f w h o s e l e n g t h ( i n w h o l e i n c h e s ) l i e s b e t w e e n 30 a n d 5 0 i n c h e s holds e x a c t l y 5 p o e t r y books each a inches t h i c k , 3 h i s t o r y books each b inches t h i c k and 5 d i c t i o n a r i e s each c i n c h e s t h i c k ; (a, b, £, a r e u n e q u a l i n t e g e r s ) . I t could instead hold exactly 4 p o e t r y books, 5 h i s t o r y books, and 4 d i c t i o n a r i e s . I f instead, 7 p o e t r y b o o k s a n d 4 h i s t o r y b o o k s a r e p l a c e d o n i t , how many d i c t i o n a r i e s must t h e n be added t o f i l l t h e s h e l f e x a c t l y ? F i n d a l s o t h e t h i c k n e s s of each book and t h e l e n g t h o f t h e s h e l f . A l a d y gave t h e p o s t a g e stamp c l e r k a one d o l l a r b i l l and s a i d , " G i v e me some t w o - c e n t s t a m p s , t e n t i m e s a s many o n e - c e n t s t a m p s , a n d t h e balance i n f i v e s . " How c a n t h e c l e r k f u l f i l t h i s p u z z l i n g r e q u e s t ? A r e c t a n g u l a r l a w n i s t o b e f o r m e d so t h a t one s i d e o f a b a r n s e r v e s as one s i d e o f t h e r e c t a n g l e . Y o u h a v e 300 f e e t o f w i r e t o e n c l o s e t h e other three sides. What a r e t h e d i m e n s i o n s o f t h e r e c t a n g l e i f t h e a r e a i s t o b e o f maximum s i z e ? A g r o u p o f s t u d e n t s was s u r v e y e d t o a s c e r t a i n t h e i r p r e f e r e n c e s f o r s p o r t w a t c h i n g on t e l e v i s i o n . The t h r e e s p o r t s s e l e c t e d f o r t h e s u r v e y were f o o t b a l l , g o l f , and h o c k e y . The d a t a were c o l l e c t e d and o r g a n i z e d as f o l l o w s : 7 watched none o f t h e s p o r t s 10 w a t c h e d g o l f 8 watched hockey 6 watched f o o t b a l l 1 watched f o o t b a l l o n l y 4 watched f o o t b a l l and g o l f 5 watched g o l f and hockey 2 watched a l lt h r e e s p o r t s . Questions: a) b) c)  How many s t u d e n t s a n s w e r e d t h e s u r v e y ? How many s t u d e n t s w a t c h e d h o c k e y o n l y ? How many s t u d e n t s w a t c h e d h o c k e y a n d f o o t b a l l ?  F i n d t h e sum o f a l l  t h e odd i n t e g e r s ( p o s i t i v e ) l e s s than  100.  A c i r c u l a r t a b l e r e s t s i n a c o r n e r , t o u c h i n g b o t h w a l l s o f a room. A p o i n t o n t h e edge o f t h e t a b l e i s 8 i n c h e s f r o m one w a l l a n d 9 i n c h e s from t h e o t h e r w a l l . Find the diameter o f the t a b l e . F i n d t h e s o l u t i o n s e t o f X + Y = XY where X and Y a r e r e a l  numbers.  135 Find  a l l X such t h a t  ]X - 1 | - |X - 1 | = 2  12.  What i s t h e l o n g e s t p i e c e o f m e t a l r o d w h i c h c a n b e p l a c e d d i m e n s i o n 3 i n c h e s b y 4 i n c h e s b y 12 i n c h e s ?  i n a box o f  I m a g i n e a r o w o f 1000 l o c k e r s , a l l c l o s e d , a n d a l i n e o f 1000 men. S u p p o s e t h e f i r s t man g o e s a l o n g a n d o p e n s e v e r y l o c k e r . Suppose t h e s e c o n d man g o e s a l o n g a n d s h u t s e v e r y o t h e r l o c k e r . Suppose t h e t h i r d man g o e s a l o n g a n d c h a n g e s t h e s t a t e o f e v e r y t h i r d l o c k e r ( i f i t ' s open, he s h u t s i t , and v i c e v e r s a ) . S u p p o s e t h e f o u r t h man g o e s a l o n g and changes t h e s t a t e o f e v e r y f o u r t h l o c k e r , and so o n , u n t i l a l l t h e men h a v e p a s s e d b y a l l t h e l o c k e r s . Which l o c k e r s a r e open i n t h e end? What i s t h e maximum n u m b e r o f p e o p l e y o u c a n s e a t a t o n e t i m e u s i n g 100 c a r d t a b l e s p r o v i d e d y o u c a n o n l y s e a t o n e p e r s o n o n a s i d e o f a t a b l e a n d t h e t a b l e s a r e a r r a n g e d s o t h a t e a c h t i m e a new t a b l e i s added, a t l e a s t one o f i t s s i d e s must be p l a c e d a g a i n s t one s i d e o f a p r e v i o u s l y arranged t a b l e ? Suppose y o u have a g o l d mine and a r e h a u l i n g g o l d t o t h e s m e l t e r i n y o u r t r u c k which has a hole i n t h e back. I n order t o get t o t h e smelter y o u must d r i v e f r o m y o u r mine t o a r a i l r o a d t r a c k and t h e n t o t h e s m e l t e r w h i c h i s o n t h e same s i d e o f t h e t r a c k a s y o u r m i n e . S i n c e y o u r t r u c k has a hole i n t h e back y o u a r e i n t e r e s t e d i n f i n d i n g t h e s h o r t e s t r o u t e . What i s i t ? ^ ^  s  — H - f t i~pn i i'Ii i-v-rttttA r e a l e s t a t e agency o f f e r s y o u a c h o i c e o f two t : r i a n g u l a r p i e c e s o f land. One p i e c e h a s d i m e n s i o n s 30, 25, a n d 40 f e e t ; t h e o t h e r h a s d i m e n s i o n s 75, 90, 120 f e e t . T h e p r i c e o f t h e l a r g e r p i e c e i s 4 t i m e s the p r i c e o f t h e smaller piece. Which i s t h e b e t t e r buy? A b a r r e l o f h o n e y w e i g h s 50 p o u n d s . T h e same b a r r e l w i t h k e r o s e n e i n i t w e i g h s 35 p o u n d s . I f h o n e y i s t w i c e a s h e a v y a s k e r o s e n e , how much d o e s t h e empty b a r r e l w e i g h ? A f a r m e r h a s h e n s a n d r a b b i t s . T h e s e a n i m a l s h a v e 50 h e a d s a n d feet. How many h e n s a n d how many r a b b i t s h a s t h e f a r m e r ?  140  I n t h e f o l l o w i n g a d d i t i o n problem each l e t t e r stands f o r a d i f f e r e n t digit. I f a l e t t e r a p p e a r s m o r e t h a n o n c e , i t r e p r e s e n t s t h e same digit. What a r e t h e n u m b e r s i f t h e l e t t e r " o " s t a n d s f o r 6 ? MOM + POP LOVE  136 G i v e n a c i r c l e and a p o i n t i n i t s plane. O u t l i n e t h e major s t e p s yoi would use t o construct a tangent t o t h e c i r c l e through t h e p o i n t . T h r e e b o y s l i t a f i r e w o r k a n d i m m e d i a t e l y r a n o f f a t t h e same s p e e d i n d i f f e r e n t d i r e c t i o n s . T h e f i g u r e b e l o w s h o w s t h e i r p o s i t i o n when the firework exploded. Where d i d t h e f i r e w o r k e x p l o d e ?  ft  P r o v e t h a t n o t e r m i n t h e s e q u e n c e 11, square o f an i n t e g e r .  111,  1111,  H i l l ,  is the  G i v e n a r e g u l a r hexagon and a p o i n t i n i t s p l a n e . Draw a s t r a i g h t l i n e t h r o u g h t h e g i v e n p o i n t t h a t d i v i d e s t h e hexagon i n t o two p a r t s o f equal area. Can y o u do t h e p r o b l e m f o r any r e g u l a r p o l y g o n ? O t h e r c l o s e d geometric f i g u r e s ? F i n d t h e s e t o f p o s i t i v e i n t e g e r s w h i c h have l e s s t h a n 5 d i v i s o r s . A y a c h t i s m o o r e d a t A, 50 m e t e r s away f r o m a s t r a i g h t s e a w a l l CD. The c a p t a i n o f t h e y a c h t w i s h e s t o r o w t o t h e s e a w a l l t o c o l l e c t a p a s s e n g e r a n d t h e n t o a s p e e d - b o a t m o o r e d a t B, SO m e t e r s f r o m t h e w a l l . Where s h o u l d t h e p a s s e n g e r m e e t t h e c a p t a i n t o make t h i s r o u t e a s s h o r t as p o s s i b l e ? Jjj  ?0  c 7 - r r  There are 2 g l a s s e s : One c o n t a i n i n g 10 s p o o n s f u l o f w i n e a n d t h e o t h e r 10 o f w a t e r . A spoonful o f wine i s taken from t h e f i r s t g l a s s , p u t i n t h e second g l a s s , and mixed round. Then a s p o o n f u l o f t h e m i x t u r e i s t a k e n and p u t back i n t h e f i r s t g l a s s . I s t h e r e now m o r e w i n e i n t h e water than water i n t h e wine? G i v e n t h a t ABCD i s a parallelogram, AB p e r p e n d i c u l a r t o P C , DQ p e r p e n d i c u l a r t o BC, F i n d t h e l e n g t h o f DQ.  Appendix B PROBLEMS USED I N THE STUDY  Instructions  The p u r p o s e o f t h i s i n t e r v i e w t h e w a y s i n w h i c h s t u d e n t s who a r e solve mathematical problems. This h a v e a b s o l u t e l y n o t h i n g t o do. w i t h  i s t o o b t a i n some i n f o r m a t i o n o n completing grade eleven Algebra i s f o r my own i n f o r m a t i o n a n d w i l l your grade i n A l g e b r a ,  What y o u a r e a s k e d t o d o i s t o w o r k a s m a l l s e t o f p r o b l e m s , a n d t o t h i n k a l o u d as y o u a r e w o r k i n g on each p r o b l e m . T h i s means t h a t i n a d d i t i o n t o w r i t i n g y o u r steps on paper, y o u a r e b e i n g asked t o s a y e v e r y t h i n g t h a t you a r e t h i n k i n g about w h i l e y o u work each problem. T h i s i n c l u d e s a l l t h e l i t t l e t h i n g s t h a t pop i n t o y o u r mind — w h e t h e r you u s e them o r n o t . Y o u w i l l be t a p e - r e c o r d e d w h i l e y o u do t h i s . The reason f o r r e c o r d i n g y o u r work i s so t h a t I can g e t a c l e a r e r p i c t u r e o f w h a t t h o u g h t p r o c e s s e s y o u u s e a n d when y o u u s e t h e m w h i l e s o l v i n g a problem. There a r e s e v e r a l o t h e r i n s t r u c t i o n s I hope y o u w i l l working t h e problems: beginning  follow while  1)  Read each problem o u t l o u d b e f o r e  t o work i t ,  2)  W r i t e down e v e r y t h i n g y o u u s u a l l y w r i t e w h i l e s o l v i n g a p r o b l e m . T h i s i n c l u d e s s c r a t c h work, diagrams, e q u a t i o n s , c a l c u l a t i o n s , etc.; u s e a s much p a p e r a s y o u n e e d ; t a l k a s y o u w r i t e ,  3)  Do n o t e r a s e a n y t h i n g ; i f y o u d e c i d e n o t t o u s e s o m e t h i n g y o u ' v e a l r e a d y w r i t t e n d o w n j d r a w a l i n e t h r o u g h i t a n d t h e n w r i t e down the correction.  4)  I am m o r e i n t e r e s t e d i n how y o u go a b o u t s o l v i n g a p r o b l e m t h a n i n your solution. So p l e a s e do n o t a s k i f any o f y o u r work i s c o r r e c t u n t i l y o u have completed t h e i n t e r v i e w .  S i n c e o t h e r s t u d e n t s w i l l be p a r t i c i p a t i n g i n t h e s e i n t e r v i e w s , p l e a s e do n o t d i s c u s s t h e p r o b l e m s o r t h e i n t e r v i e w w i t h a n y o n e . This would i n v a l i d a t e t h e r e s u l t s f o r t h e whole s c h o o l . I f someone h a s q u e s t i o n s , t e l l h i m t o s e e me. Y o u r c o o p e r a t i o n w i l l be g r e a t l y a p p r e c i a t e d .  138 Mathematical World  Warm-up  Problems  Problems  A.  T h e sum o f t w o p o s i t i v e i n t e g e r s i s 23. T h e sum o f t h e f i r s t a n d t w i c e t h e s e c o n d i s 34» F i n d t h e numbers.  B.  T, D, H, J , a n d B a r e r e a l n u m b e r s w i t h t h e f o l l o w i n g r e l a t i o n s h i p s : T i s 20 l e s s t h a n D, D i s 50 g r e a t e r t h a n H, H i s 20 l e s s t h a n B, J i s 30 g r e a t e r t h a n T. Which number i s t h e l a r g e s t ? W h i c h i s s e c o n d , t h i r d , f o u r t h , and f i f t h largest?  Main  1.  integer  Problems The p o i n t A i s 50 u n i t s f r o m a s t r a i g h t l i n e CD, a n d B i s a p o i n t 80 u n i t s f r o m CD. F i n d t h e p o i n t X o n CD s o t h a t t h e d i s t a n c e from A t o X t o B i s as s m a l l as p o s s i b l e . B  £0  50  4 P  •/3o 2.  The sum o f a p o s i t i v e i n t e g e r a n d t w i c e a n o t h e r i s s u b t r a c t e d f r o m 100 a n d t h e d i f f e r e n c e i s d i v i s i b l e b y 5* The f i r s t i n t e g e r i s 10 t i m e s t h e s e c o n d . What a r e t h e n u m b e r s ? A r e c t a n g l e ABCD i s f o r m e d s o t h a t o n e o f i t s s i d e s CD i s o n a l i n e L . T h e sum o f t h e l e n g t h s o f t h e o t h e r s i d e s , A B , AD, a n d B C , i s 300 u n i t s . What a r e t h e d i m e n s i o n s o f t h e r e c t a n g l e i f t h e a r e a i s t o b e o f maximum s i z e ? Why d o e s y o u r s o l u t i o n g i v e t h e maximum area?  4.  What i s t h e g r e a t e s t d i s t a n c e b e t w e e n t w o p o i n t s i n a s o l i d o f d i m e n s i o n 3 u n i t s b y 4 u n i t s b y 12 u n i t s ?  5.  What i s t h e maximum p e r i m e t e r y o u c a n o b t a i n b y a r r a n g i n g o n e h u n d r e d o n e - i n c h s q u a r e s ? E a c h t i m e a new s q u a r e i s a d d e d a t l e a s t o n e o f i t s s i d e s must be p l a c e d a g a i n s t one s i d e o f a p r e v i o u s l y a r r a n g e d square. Why d o e s y o u r s o l u t i o n g i v e t h e maximum p e r i m e t e r ?  rectangular  139 Real World  Warm-up A.  Problems  Problems  J a b b a r s c o r e d 23 t i m e s i n a b a s k e t b a l l game. He s c o r e d 34 p o i n t s , two f o r e a c h f i e l d g o a l and one f o r e a c h f r e e t h r o w . How many f i e l d g o a l s d i d h e m a k e ? How many f r e e t h r o w s ?  B. . Tom, D i c k , H a r r y , J o e , a n d B i l l a r e l i n e d u p i n t h e s e p o s i t i o n s m i d w a y t h r o u g h a t r a c k meet: Tom i s 20 y a r d s b e h i n d D i c k , D i c k i s 50 y a r d s a h e a d o f H a r r y , H a r r y i s 20 y a r d s b e h i n d B i l l , J o e i s 30 y a r d s a h e a d o f Tom, A t t h i s p o i n t i n t h e r a c e , who i s w i n n i n g ? who i s s e c o n d , t h i r d , f o u r t h and f i f t h ? Main  1.  Problems A y a c h t i s m o o r e d a t A, 50 m e t e r s away f r o m a s t r a i g h t s e a w a l l CD. The c a p t a i n o f t h e y a c h t w i s h e s t o row t o t h e s e a w a l l t o c o l l e c t a p a s s e n g e r a n d t h e n t o a s p e e d b o a t m o o r e d a t B, 8 0 m e t e r s f r o m t h e wall. W h e r e s h o u l d t h e p a s s e n g e r m e e t t h e c a p t a i n t o make t h e r o u t e as s h o r t as p o s s i b l e ? .—^ 8  ft  ? 0  SO /////  kf  130  2.  A l a d y gave t h e p o s t a g e stamp c l e r k a one d o l l a r b i l l a n d s a i d , " G i v e me some t w o - c e n t s t a m p s , t e n t i m e s a s many o n e - c e n t s t a m p s , and t h e b a l a n c e i n f i v e s . " How c a n t h e c l e r k f u l f i l t h i s p u z z l i n g request?  3«  A r e c t a n g u l a r l a w n i s t o be formed s o t h a t one s i d e o f a b a r n s e r v e s as one s i d e o f t h e r e c t a n g l e . Y o u h a v e 300 f e e t o f w i r e t o e n c l o s e the other three sides. What a r e t h e d i m e n s i o n s o f t h e r e c t a n g l e i f t h e a r e a i s t o b e o f maximum s i z e ? Why d o e s y o u r s o l u t i o n g i v e t h e maximum a r e a ?  4.  What i s t h e l o n g e s t p i e c e o f m e t a l r o d w h i c h c a n b e p l a c e d i n a b o x o f d i m e n s i o n 3 i n c h e s b y 4 i n c h e s b y 12 i n c h e s ?  5«  What i s t h e maximum n u m b e r o f p e o p l e y o u c a n s e a t a t o n e t i m e u s i n g  140  100 s q u a r e c a r d t a b l e ? The t a b l e s a r e a r r a n g e d s o t h a t e a c h t i m e a new t a b l e i s a d d e d , a t l e a s t one o f i t s s i d e s m u s t be p l a c e d a g a i n s t one s i d e o f a p r e v i o u s l y a r r a n g e d t a b l e . Why d o e s y o u r s o l u t i o n g i v e t h e maximum s e a t i n g p l a n ?  Appendix  THE  C  CODING SYSTEM  T h i s c o d i n g system i s based on MacPherson's m o d e l o f m a t h e m a t i c a l problem s o l v i n g . o f s u b j e c t s who word  I t i s designed f o r coding problem  a r e a s k e d t o t h i n k a l o u d as t h e y s o l v e  mathematical  problems. The p r o b l e m  is  solving behavior  a s k e d t o do  s o l v i n g i n t e r v i e w i s a u d i o t a p e r e c o r d e d and t h e  all  of h i s t h i n k i n g aloud.  a l l h i s w r i t i n g and d i a g r a m s .  He  i s instructed to  Whenever t h e s u b j e c t f a l l s  subject  verbalize  s i l e n t , he i s  r e m i n d e d t o do h i s t h i n k i n g a l o u d . The p r o b l e m s  a r e p r e s e n t e d t o t h e s u b j e c t one  at a time.  Each  problem  i s t y p e w r i t t e n a t t h e t o p o f t h e page w i t h space u n d e r n e a t h f o r t h e s u b j e c t t o work.  Once t h e i n t e r v i e w i s c o m p l e t e , t h e s u b j e c t ' s  protocol  i s matched w i t h h i s w r i t t e n work by marking t h e a p p r o p r i a t e f o o t a g e from t h e t a p e on h i s w r i t t e n work.  The  c o d i n g i s done u s i n g b o t h t h e  subject's  w r i t t e n w o r k and t h e t a p e r e c o r d e d p r o t o c o l . A s u b j e c t ' s p r o t o c o l f o r each problem i s coded on t h e c o d i n g form (see n e x t two  pages).  solving behavior.  A m a t r i x i s used t o s e q u e n t i a l l y code  problem  A summary f o r m i s u s e d t o c o m p i l e i n f o r m a t i o n o b t a i n e d  f r o m t h e c o d i n g m a t r i x as w e l l as r e c o r d i n g o t h e r d a t a r e l a t e d t o t h e subject's problem The  solving behavior.  procedures from t h e coding form are d i v i d e d i n t o f i v e  core, h e u r i s t i c s ,  solution,  r e a s o n a b l e n e s s and c o n c e r n .  i d e n t i f i e s the procedure from the s u b j e c t ' s p r o t o c o l , i n t h e a p p r o p r i a t e row  and f i r s t  empty column.  As t h e  categories: coder  he e n t e r s a  check  I f the procedure i s  142 Reading problem Request d e f n . o f t e r r s Recall  same p r o b l e m  Recall related  problem  R e c a l l problem  type  Recall related  fact  Draw dlajnrani Modify Identify  diagram variable  S e t t i n g up e q u a t i o n s Algorithms-algebraic Algorithms-arithmetic Guessing Smoothing Analysis Templation Cases-all Cases-random Cases-systematic Cases-critical Cases-sequential Deduction Inverse  deduction  Invariation Analogy Symmetry Obtain  solution  Checking Checking  part  solution;  by s u b s t . i n e q u a t i o n : by r e t r a c i n g  steps  by r e a s n b l e / r e a l i s t i c uncodable Exp. Exp.  concern-method  concren-algorithm  Exp.  concern-equation  Exp. c o n c e r n - s o l u t i o n Work s t o p p e d - s o l n . Work stopped-no  soln.  THE  CODING  FORM  143 CODING  Number Solutioni  Problem#, Correct Inc orrect  FORM  SUMMARY  Tape#  Tape Reading.  Time spent on problem Errorsi  None  Arithmetic Algebraic Other  # of Cycles Heuristics used  Remarksi  # of times  # of Changes Core procedures used  # of times  144 f r o m t h e h e u r i s t i c o r c o r e c a t e g o r y ( e x c e p t r e c a l l o f same p r o b l e m , r e l a t e d p r o b l e m , o r p r o b l e m t y p e ; draw d i a g r a m ; m o d i f y d i a g r a m ) , he e n t e r s a 1,  The of  2,  o r 3,  depending upon t h e outcome:  1  Incomplete  2  Incorrect  3  Correct  outcome f r o m a c o r e p r o c e d u r e i s coded i n t e r m s o f t h e  the application of the procedure.  F o r example,  a 2 i n the  correctness  algorithm-  a r i t h m e t i c row i n d i c a t e s an e r r o r i n t h e u s e o f t h e a l g o r i t h m , w h e r e a s , 1 i n t h e s e t t i n g up e q u a t i o n r o w i n d i c a t e s t h e s u b j e c t h a s w r i t t e n incomplete equation. in  For the heuristic  h e u r i s t i c , not that the h e u r i s t i c i s i n c o r r e c t .  a f t e r the use F o r example,  t h e a n a l y s i s row i n d i c a t e s t h e o u t c o m e o f t h e p r o b l e m a f t e r t h e  of  a n a l y s i s i s i n c o r r e c t , not t h a t the subject committed  If  use  itself.  t w o o r more p r o c e d u r e s o c c u r s i m u l t a n e o u s l y t h e n a c h e c k  c a r r i e d on l o n g e r t h a n a column,  outcome e n t e r e d i n t h e l a s t is  2  an e r r o r i n t h e  n u m b e r i s e n t e r e d i n t h e same c o l u m n f o r e a c h p r o c e d u r e . is  of  a  in  heuristic  an  c a t e g o r y , t h e outcome i s coded  t e r m s o f t h e outcome o f t h e p r o b l e m o r s u b - p r o b l e m  the  a  or  I f a procedure  i t i s e n c l o s e d i n a box w i t h t h e  column e n c l o s e d .  F o r example,  a  subject  u s i n g random c a s e s and c o n s i d e r s t h r e e c a s e s a l o n g w i t h u s i n g .  a l g o r i t h m s and d r a w i n g d i a g r a m s . cases would Core  The  c o v e r more t h a n one  I n t h i s i n s t a n c e t h e " b l o c k " f o r random column.  core category includes behavior r e l a t e d t o the  use o f core p r o c e d u r e s .  subject's  A l l s u b j e c t s s t a r t w i t h r e a d i n g p r o b l e m and  check i s e n t e r e d i n t h e f i r s t  column.  Reading the problem i s also  each time t h e subject re-reads t h e problem o r p a r t o f the problem. he c o n t i n u e s t o r e - r e a d t h e p r o b l e m , w i t h no  other a c t i v i t y  a  checked I f  interspersed,  145 o n l y one  check i s e n t e r e d i n r e a d i n g problem.  terms i s checked i f a s u b j e c t  Request d e f i n i t i o n  asks the i n t e r v i e w e r t o define  of  a term  used i n the statement o f the problem. Recall  same p r o b l e m o r r e l a t e d p r o b l e m i s c h e c k e d i f t h e  subject  m e n t i o n s t h a t he h a s s e e n t h e p r o b l e m o r a s i m i l a r p r o b l e m b e f o r e .  I f  t h e s u b j e c t c l a s s i f i e s t h e p r o b l e m b y t y p e , s u c h as a work p r o b l e m a mixture problem, r e c a l l problem type i s checked. merely mentions a r e l a t e d mathematical t o p i c problem" or " I ' l l  have t o w r i t e  c a t e g o r i e s are not checked. is  I f the  or  subject  s u c h as " t h i s i s a g e o m e t r i c  an e q u a t i o n f o r t h i s " , t h e  I f a subject mentions  recall  a specific  fact  which  r e l a t e d t o t h e p r o b l e m , r e c a l l r e l a t e d f a c t i s c o d e d w i t h a 1,  o r 3, d e p e n d i n g on t h e c o r r e c t n e s s o f t h e f a c t .  F o r example  2,  a 2 i s  e n t e r e d i f a s u b j e c t m e n t i o n s t h e P y t h a g o r e a n Theorem as " t h e s i d e opposite the right two  sides",  a n g l e i s t w i c e as l o n g as t h e l o n g e s t o f t h e o t h e r  and r e c a l l s no o t h e r p r o p e r t i e s o f r i g h t  E a c h t i m e a s u b j e c t d r a w s a new  figure,  I f t h e s u b j e c t o n l y draws a few l i n e s category i s not checked. it  i n a n y way  is  checked. If  •  and i m m e d i a t e l y e r a s e s them, t h e  by d e l e t i n g o r adding a l i n e  i d e n t i f y v a r i a b l e i s coded.. an e q u a t i o n f o r t h i s  draw diagram i s checked^  Once a f i g u r e i s drawn,  a subject i d e n t i f i e s  a variable, The  triangles.  i f the subject  (or lines),  alters  modify diagram  either verbally or written,  s u b j e c t need not use t h e v a r i a b l e i n  c a t e g o r y t o be  checked.  I f a subject writes  an  e q u a t i o n and does n o t i d e n t i f y what t h e v a r i a b l e s r e p r e s e n t ,  identify  v a r i a b l e i s not coded.  subject  S e t t i n g up e q u a t i o n s i s coded i f t h e  w r i t e s any p a r t o f an e q u a t i o n . i n t h e e q u a t i o n , however, up e q u a t i o n s w i l l depend  A s u b j e c t need n o t i d e n t i f y t h e  i f he d o e s , t h e o u t c o m e a s s i g n e d t o on b o t h t h e e q u a t i o n s w r i t t e n and t h e  variables  setting variables  146 i d e n t i f i e d by t h e s u b j e c t .  F o r example,  i n t h e stamp problem (problem  #2, A p p e n d i x B ) , i f X i s t h e number o f 2 c e n t stamps, 1 c e n t stamps  a n d 5X t h e n u m b e r o f 5 c e n t s t a m p s  be coded 2 ) t h e e q u a t i o n equation  a l g e b r a i c i s coded.  (the equation  I f a subject  w i t h no change i n t e n d e d ,  ( i d e n t i f y v a r i a b l e would  10X + 2 X + 5 ( 5 X ) = 1 0 0 w o u l d b e c o d e d 3, t h e  10X + 2X + 5X = 1 0 0 w o u l d b e c o d e d 2.  a p p l i e d t o an e q u a t i o n  1CX t h e n u m b e r o f  I f an a l g o r i t h m i s  c o u l d be coded  copies  an e q u a t i o n  1, 2, o r 3),  algorithms-  o v e r on h i s paper  no code i s e n t e r e d .  A l g o r i t h m s - a r i t h m e t i c i s coded each t i m e t h e s u b j e c t uses o r a t t e m p t s t o u s e an a r i t h m e t i c a l g o r i t h m . more a l g o r i t h m s  I f t h e s u b j e c t uses two o r  c o n s e c u t i v e l y , such as t h e s u b t r a c t i o n o f two p a i r s o f  numbers, f o l l o w e d b y t h e m u l t i p l i c a t i o n o f t h e i r d i f f e r e n c e s , a r i t h m e t i c algorithms  i s o n l y coded once.  I f the algorithm i s performed  b y t h e s u b j e c t w i t h no d i f f i c u l t y , not to  coded.  G u e s s i n g i s coded each t i m e t h e s u b j e c t  t h e problem o r a subproblem  guessing,  s u c h a s o n e h a l f o f 500  provided  (such  mentally  i s 250,  i t i s  selects a solution  as an a l g o r i t h m o r e q u a t i o n ) b y  t h e s u b j e c t d o e s n o t i n d i c a t e some i n t e n t i o n o f f u r t h e r  trials.  Heuristics  This category  the  Smoothing i s coded each t i m e t h e s u b j e c t  subjects.  i n c l u d e s t h e h e u r i s t i c procedures used by  i r r e l e v a n t information i n t h e problem. 1  disregards  Smoothing i s coded t h e f i r s t  t i m e a s u b j e c t draws a diagram b u t n o t f o r s i m i l a r f i g u r e s drawn For  example, i n t h e r o d i n t h e box, problem  (#4,  after.  Appendix B) t h e subject  can smooth t h e b o x t o a r e c t a n g u l a r p a r a l l e l e p i p e d , however,  i f he  d r a w s a r e c t a n g l e , s m o o t h i n g i s coded as 2 and a c h e c k i s e n t e r e d i n  147  draws diagram.  I f a subject  considers  a s p e c i a l case such as  50 f e e t o f f e n c i n g i n a p r o b l e m r a t h e r t h a n 300,  using  smoothing i s coded.  A n a l y s i s i s coded whenever t h e s u b j e c t b r e a k s t h e problem subproblems  and a t t e m p t s t o s o l v e t h e s u b p r o b l e m s .  m e r e l y m e n t i o n s t h a t t h e p r o b l e m c a n be s u b p r o b l e m s , no  I f the  into  subject  s o l v e d i n t e r m s o f two  o r more  c o d e i s made.  Templation i s coded each t i m e t h e s u b j e c t mentions o r r e c a l l s category to  of content,  i . e . r e c a l l s part of the core, which i s r e l a t e d  t h e problem being  fact,  solved.  I f t h e s u b j e c t m e n t i o n s o n l y one  s u c h as r e c a l l i n g t h e P y t h a g o r e a n Theorem, r e c a l l  coded.  I n a problem which i n v o l v e s a r i g h t t r i a n g l e ,  m e n t i o n s some o f t h e t h e o r e m s , d e f i n i t i o n s , t r i a n g l e , t h e n t e m p l a t i o n i s coded. of the right t r i a n g l e 3 i s coded. the  I f the subject  subject  recalls  a 1 i s coded.  C a s e s - a l l i s coded i f t h e s u b j e c t  states there  are only a f i n i t e  u s e made o f t h e i n f o r m a t i o n o b t a i n e d  either of the  considers a l l  number o f c a s e s  number.  from previous  exist  Cases-random  t r i a l s or  apparant  cases.  I f t h e s u b j e c t ' s r e m a r k s i n d i c a t e t h a t he i s u s i n g i n f o r m a t i o n cases then cases-systematic  from  o r c a s e s - s e q u e n t i a l i s coded.  t h e d a t a a r e a p p l i e d i n a s e q u e n t i a l manner, s u c h as i n s o l v i n g t h e tion  12X + 5Y = 100 b y l e t t i n g X = 1, 2, 3,  a  a 2 i s coded and i f  c o d e d i f t h e s u b j e c t u s e s c a s e s i n a random f a s h i o n w i t h no  previous  right  any p r o p e r t i e s  o r m o r e c a s e s i n one  cases i n a problem where o n l y a f i n i t e  or the subject is  i f the  h e u r i s t i c s o f cases are coded each t i m e t h e s u b j e c t  e x p r e s s e s an i n t e n t i o n t o u s e o r u s e s two  possible  related fact i s  and u s e s any o f them i n s o l v i n g t h e p r o b l e m ,  I f the i n f o r m a t i o n used i s i n c o r r e c t ,  f o l l o w i n g manners:  specific  or properties of a  s u b j e c t does not use the. i n f o r m a t i o n obtained, The  a  . . ., o r f i n d i n g t h e  If equa-.  148  maximum o f X ( 3 0 0 - 2 X ) b y l e t t i n g X = 5 0 , 6 0 , 7 0 , . . .; is  coded.  cases-sequential  I f t h e subject i s using cases i n a non-sequential  or decreasing  manner o r u s i n g i n f o r m a t i o n f r o m t h e p r e v i o u s  increasing cases  such  as f i n d i n g t h e square r o o t o f 1 6 0 b y r e f i n i n g each case t o o b t a i n a value  c l o s e r t o t h e square r o o t , Cases^-systematic  critical  i s coded i f t h e s u b j e c t  when u s i n g c a s e s .  i s coded.  Cases-  s t a t e s a bound o r bounds f o r t h e v a r i a b l e s  F o r example, i n s o l v i n g t h e e q u a t i o n  12X + 5Y = 100  where X and Y a r e p o s i t i v e i n t e g e r s , i f t h e s u b j e c t i s u s i n g random o r systematic is  c a s e s and s t a t e s " t h e l a r g e s t v a l u e X c a n be i s 8" t h e n a 3  coded f o r  cases-critical.  D e d u c t i o n i n c l u d e s b o t h h y p o t h e t i c a l and d i r e c t d e d u c t i o n . is  coded whenever a s u b j e c t ' s r e m a r k s i n d i c a t e he h a s assumed a p r e m i s e  or i s using data obtained to  Deduction  from t h e problem as a p r e m i s e and i s a t t e m p t i n g  d e t e r m i n e a s many i m p l i c a t i o n s a s p o s s i b l e .  i f the subject  D e d u c t i o n i s n o t coded  states a single logical implication.  Inverse  deduction  is  c o d e d i f t h e s u b j e c t ' s comments i n d i c a t e h e i s i n t e n t i o n a l l y - a t t e m p t i n g  to  work t h e problem backwards.  can  find  provided be  a formula  F o r example, i f a s u b j e c t i n d i c a t e s he  f o r the general  3  h e c a n s o l v e aX  cubic equation  3  aX  + bX  2  + cX + d,  + c X + d = 0, t h e i n v e r s e d e d u c t i o n  would  coded d e p e n d i n g on t h e outcome. I n v a r i a t i o n i s c o d e d w h e n e v e r a s u b j e c t ' s comments a n d a c t i o n s  i n d i c a t e t h a t he i s i n t e n t i o n a l l y e x c l u d i n g o r f i x i n g constant) its  a v a r i a b l e , then attempting  (renaming a  t o s o l v e t h e new p r o b l e m a n d u s e  s o l u t i o n t o help him solve t h e o r i g i n a l problem.  I f a subject  f a i l s t o s o l v e t h e new p r o b l e m o r u s e i t s s o l u t i o n a f t e r i n d i c a t i n g a n i n t e n t t o do s o , a 1 i s coded f o r  invariation.  Analogy i s coded i f t h e s u b j e c t  recalls  an analogous  mathematical  s i t u a t i o n a n d i n t e n t i o n a l l y makes u s e o f i t s p r o p e r t i e s i n s o l v i n g  the problem.  I f a subject r e c a l l s  a non-mathematical  s i t u a t i o n such as  " t h i s i s s i m i l a r t o a b o a t m o v i n g u p a n d down o n a r i v e r " , problem t y p e i s coded.  The h e u r i s t i c  recall  o f symmetry i s coded i f t h e a c t i o n s  a n d comments o f t h e s u b j e c t - i n d i c a t e h e i s i n t e n t i o n a l l y m a k i n g u s e o f t h e i n h e r e n t o r c o n s t r u c t e d symmetry i n t h e p r o b l e m . Solution  The c a t e g o r y o f o b t a i n s o l u t i o n i s c h e c k e d e a c h t i m e t h e  subject obtains a s o l u t i o n t o t h e problem. Reasonableness  The r e a s o n a b l e n e s s c a t e g o r y p e r t a i n s t o c h e c k i n g p a r t  o f t h e s u b j e c t ' s work and c h e c k i n g h i s s o l u t i o n .  I f a subject's  actions  a n d comments i n d i c a t e h e i s c h e c k i n g p a r t o f h i s w o r k , c h e c k i n g p a r t is  coded  a n d t h e p r o c e d u r e o r p r o c e d u r e s h e u s e d a r e c o d e d i n t h e same  column.  I f he i n d i c a t e s he i s c h e c k i n g h i s s o l u t i o n ,  is  and t h e p r o c e d u r e o r p r o c e d u r e s he u s e d  coded  checking solution  a r e c o d e d i n t h e same  column.  T h e s e p r o c e d u r e s may i n c l u d e  system.  Four o t h e r c a t e g o r i e s are i n c l u d e d i n t h e c o d i n g system f o r  checking.  any p r o c e d u r e s f r o m t h e c o d i n g  Checking by s u b s t i t u t i n g i n equation i s checked i f t h e  s u b j e c t r e p l a c e s a v a r i a b l e i n an e q u a t i o n w i t h a v a l u e he b e l i e v e s t o be  a solution t o the equation.  Checking by r e t r a c i n g  steps i s checked  if  t h e s u b j e c t e x p l i c i t l y r e p e a t s an o p e r a t i o n o r s e r i e s o f o p e r a t i o n s  a f t e r he o b t a i n s a s o l u t i o n o r i n d i c a t e s he i s c h e c k i n g o v e r h i s work. C h e c k i n g by_ r e a s o n a b l e / r e a l i s t i c i s c o d e d i f t h e s u b j e c t i n d i c a t e s he h a s t e s t e d w h e t h e r problems  h i s s o l u t i o n i s reasonable e i t h e r i n terms o f t h e  o r i n terms o f t h e " r e a l world".  s o l u t i o n by r e a d i n g t h e problem, coded.  Uncodable  I f the subject  checks h i s  checking by reasonable/realistic i s  i s c h e c k e d i f t h e s u b j e c t ' s c o m m e n t s i n d i c a t e he h a s  c h e c k e d h i s w o r k b u t d o e s n o t i n d i c a t e how. right"  that  o r " L e t ' s see - t h a t ' s n o t r i g h t " .  e . g . "Oh!  That  c a n ' t be  Concern  The  t h e s u b j e c t may  c o n c e r n c a t e g o r y i n c l u d e s s e v e r a l k i n d s o f comments t h a t make a b o u t  t h e p r o c e d u r e s he i s u s i n g o r a b o u t h i s  solution.  E x p r e s s e s c o n c e r n about method i s checked i f t h e s u b j e c t  explicitly  e x p r e s s e s c o n c e r n a b o u t t h e p r o c e d u r e s he i s u s i n g .  the s u b j e c t might two  s a y , "I'm  not cure t h i s problem  e q u a t i o n s . " o r " T h i s may  not work, but I ' l l  can be  about  t o s o l v e i t a r e not coded.  t r y some n u m b e r s a n y w a y . "  s u c h a s "I'm  not sure t h i s problem  c o n c e r n about method.  or that  concern a concern  about  T h i s i n c l u d e s concern about t h e p r o c e s s  i n v o l v e d i n c a r r y i n g out the algorithm.  coded  Expresses  a l g o r i t h m i s checked whenever t h e s u b j e c t i n d i c a t e s  t h e a l g o r i t h m he h a s c h o s e n .  example,  s o l v e d by u s i n g  S t a t e m e n t s i n d i c a t i n g t h a t he d o e s n o t u n d e r s t a n d t h e p r o b l e m he d o e s n o t know how  For  However, an e x p r e s s i o n o f  c a n be  concern  s o l v e d w i t h an a l g o r i t h m " i s  E x p r e s s e s c o n c e r n about  e q u a t i o n i s coded  i f t h e s u b j e c t e x p r e s s e s c o n c e r n a b o u t t h e e q u a t i o n o r e q u a t i o n s he written or i s attempting to write. "I'm  not sure t h i s problem  c o n c e r n about method.  c a n be  An  e x p r e s s i o n of concern such  has  as  s o l v e d u s i n g e q u a t i o n s " i s coded  E x p r e s s e s c o n c e r n about  s o l u t i o n i s coded  whenever  t h e s u b j e c t ' s r e m a r k i n d i c a t e s t h a t he d o u b t s t h e c o r r e c t n e s s o f h i s solution.  Remarks t h a t t h e r e i s no  s o l u t i o n are not  coded.  Work s t o p p e d - s o l u t i o n i s c o d e d i f t h e s u b j e c t s t o p s w o r k o n problem w i t h a s o l u t i o n .  I f t h e s u b j e c t q u i t s w o r k i n g on t h e  the  problem  w i t h o u t a s o l u t i o n t h e n work stopped-no; s o l u t i o n i s coded. The  summary f o r m s u m m a r i z e s d a t a f r o m t h e c o d i n g f o r m a s w e l l  r e c o r d i n g o t h e r data r e l a t e d t o the subject's problem Most e n t r i e s on.the a r i t h m e t i c and coded  solving behavior.  summary f o r m a r e e a s i l y i d e n t i f i e d .  algebraic,  time t h e s u b j e c t ' s remarks  Errors,  are counted from the coding form.  e i t h e r 1 o r 2 i s c o u n t e d a s an e r r o r .  as  An  A c y c l e i s counted  both entry each  i n d i c a t e he i s a t t e m p t i n g t o r e - a t t a c k  the  problem.  A cycle i s also counted f o r t h e f i r s t  a t t a c k on t h e problem.  A c h a n g e i s c o u n t e d e a c h t i m e t h e s u b j e c t c h a n g e s h i s mode o f a t t a c k i n g the  problem  after a cycle.  A s u b j e c t c h a n g e s h i s mode o f a t t a c k i f h e  u s e s a p r o c e d u r e f r o m e i t h e r c o r e o r h e u r i s t i c s t h a t he d i d n o t u s e o n his  p r e v i o u s attempt a t s o l v i n g t h e problem. As i l l u s t r a t i o n s  o f how t h e c o d i n g s y s t e m o p e r a t e s , t h e w r i t t e n  p r o t o c o l s f o r two problems completed coding forms.  are given i n Chapter I I I together w i t h t h e  152  Appendix D HISTOGRAMS OF SELECTED VARIABLES  ;  16 14 M  +» o  12  CD  •r-3  w «H  o u o  10 8  • •  •  * «  * '#  #  •  •  #  *  #  *  * *  *  » #  *  »  *  *  * «  #  •  #  0  1 2  *  3  4  5  6 7 8  9 10 11 12 13 14 15 16 17 18  Number of Changes A Histogram Representing the D i s t r i b u t i o n of the Number of Changes Made by the Subjects i n the Sample (N=40)  |  16 ta +> o o>  14 12  •1-3  10 GO <H  o  8  u  6  •i  4  2  2  * •  * * #  *  0 1 2 3 4 5 6 7 8  «  •  «  9 10 11 12 13 14 15 16 17 18 19  Number of Heuristics A Histogram Representing the D i s t r i b u t i o n of the Number of Times Heuristics were used by the Subjects  (N=40)  29 30 31 32 33  16 CO  •p o <u  14 12 10  CO <H O  8  u  6  •g  4  Q)  3  2  *  *  * #  « *  0 1 2 3 4 5 6 7  *  «  •  8  9 10 11 12 13  -// 14  15 16 17  18  19 20  Number of Diagrams A Histogram: Representing the D i s t r i b u t i o n of, the Number of Timesdiagrams were Drawn by the Subjects &  (N=40)  25 26  // 33 34  16 14  to -p 12 o a> •<-> •§ CO  10  8  « «  o a>  • «  *  #  •  #  *  «  *  •  //-  -//0  1 2 3 4  5 6 7. 8 9 10 11 12 13 14 15 16 17 18  28  29 30  Number of Algebraic Algorithms A Histogram,Representing the Distribution, of the Number of Times Algebraic Algorithms were used by the Subjects (N=40)  55 5-6 57  16 w +> o  0)  •r-3  00 <H  o  <u  I 25  14 12 10  8  *  6 4 2  #  •  * #  * * *  «  * *  *  0 1 2 3  * *  « «  *  «  •  *  «  4 5 6 7 8 9 10 11 12 13 i4 15 16 17 18 Number of Templations  A Histogram Representing the D i s t r i b u t i o n of the Number of Times Templation was used by the Subjects (N=40)  16 14 CO  o  CD •r-j  •§ CO  o CD  •i 2S  12 10 8  6 4 2  « 0  «  1 2 3 4  *  5 6 ? 8 9 10 11 12 13 14 15 16 17 18 Number of Random Cases  A Histogram Representing the D i s t r i b u t i o n of the Number of Times Random Cases were used by the Subjects (N=40)  

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