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The effect of teaching heuristics on the ability of grade ten students to solve novel mathematical problems Pereira-Mendoza, Lionel 1975

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THE EFFECT OF TEACHING HEURISTICS ON THE ABILITY OF GRADE TEN STUDENTS TO SOLVE NOVEL MATHEMATICAL PROBLEMS by Lionel  Pereira-Mendoza  B. Sc.". Southampton U n i v e r s i t y , 1966 M. S c . , Southampton U n i v e r s i t y , 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION i n t h e Department of. Education We accept t h i s t h e s i s as conforming t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1975  In  presenting  this  an a d v a n c e d d e g r e e the I  Library  further  for  shall  agree  thesis  in  at  University  the  make  it  freely  that permission  s c h o l a r l y p u r p o s e s may  by  his  of  this  representatives. thesis  partial  for  financial  is  for  The  gain  of  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  of  Columbia,  British  Columbia  for  extensive by  the  understood  written permission.  Department  of  available  be g r a n t e d  It  fulfilment  shall  requirements  reference copying of  Head o f  that  not  the  I  agree  and  be a l l o w e d  that  study.  this  thesis  my D e p a r t m e n t  copying or  for  or  publication  without  my  ii  Abstract Research  Supervisor:  The  study  Dr.  Gail J.  Spitler  i n v e s t i g a t e d w h e t h e r kt was  t e a c h s t u d e n t s t o a p p l y a t l e a s t one examination problems,  o f 'cases o r a n a l o g y  p o s s i b l e to  of the h e u r i s t i c s  to novel  mathematical  and w h e t h e r l e a r n i n g t o - a p p l y h e u r i s t i c s  better accomplished  of  could  by h a v i n g s t u d e n t s f o c u s s o l e l y on  be  the  h e u r i s t i c s , r a t h e r t h a n a l s o i n v o l v e them with:..the simultaneous  l e a r n i n g of mathematical  content.  were c o n s i d e r e d i n the d e s i g n o f t h e s t u d y . f a c t o r considered employing different designed  o b j e c t i v e s : one  treatments  Two  The  designed  first t o meet  i n s t r u c t i o n a l treatment  to teach h e u r i s t i c s only  (H); another  factors  was  combined  i n s t r u c t i o n i n h e u r i s t i c s w i t h the teaching of'content and  a third  treatment  utilized  content only  The  l a t t e r treatment  an e x p e r i m e n t a l c o n t r o l .  The  second  employing  distinct  (HC);  (C), without  reference to h e u r i s t i c s .  factor  the  was  used  as  concerned  instructional vehicles (materials) for  the t r e a t m e n t s , namely a l g e b r a i c ( A ) , g e o m e t r i c  (G)  and  m a t h e m a t i c a l l y n e u t r a l (N) m a t e r i a l s . A s a m p l e o f g r a d e t e n b o y s was Nine  s e l e c t e d f o r the  s e l f - i n s t r u c t i o n a l b o o k l e t s were p r e p a r e d ,  to each of the treatment  A t t h e end  o f t h e t e n day  p e r i o d p e r day)  two  corresponding  x v e h i c l e combinations.  s u b j e c t s were r a n d o m l y a s s i g n e d t o one  The  of the nine  i n s t r u c t i o n a l p e r i o d (one  t r a n s f e r t e s t s , one  study.  groups. class  a l g e b r a i c and  o t h e r g e o m e t r i c , were a d m i n i s t e r e d t o a l l s u b j e c t s .  the  iii  Data from 1 8 9 s u b j e c t s ( 2 1 p e r group) were a n a l y s e d by ANOVA.  On the a l g e b r a i c t e s t the H t r e a t m e n t was  s i g n i f i c a n t l y b e t t e r than b o t h t h e HC and C t r e a t m e n t s , and. 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 between the HC and C treatments.  The c o n c l u s i o n was t h a t o n l y t h e H t r e a t m e n t  can be c l a i m e d t o be e f f e c t i v e i n t e a c h i n g s t u d e n t s t o apply at l e a s t one h e u r i s t i c t o a n o v e l a l g e b r a i c problem. g e o m e t r i c t e s t b o t h the H and HC t r e a t m e n t s were  On t h e  significantly  b e t t e r than the C t r e a t m e n t , and 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 between the H and HC t r e a t m e n t s .  The c o n c l u s i o n  was t h a t b o t h the H and HC t r e a t m e n t s were e f f e c t i v e i n t e a c h i n g s t u d e n t s t o apply at l e a s t one h e u r i s t i c t o a n o v e l geometric problem..  An a n a l y s i s o f the p a t t e r n o f  h e u r i s t i c a p p l i c a t i o n on t h e two t e s t s r e v e a l e d t h a t w h i l e b o t h h e u r i s t i c s were employed on the a l g e b r a i c t e s t t h e r e was l i t t l e evidence o f analogy b e i n g used on the g e o m e t r i c test.  The i n v e s t i g a t o r suggested t h a t the e f f e c t i v e n e s s o f  the HC t r e a t m e n t on t h e g e o m e t r i c t e s t may have been due t o the f a c t t h a t the HC treatment was more e f f e c t i v e i n t e a c h i n g e x a m i n a t i o n o f cases than t e a c h i n g a n a l o g y , and the g e o m e t r i c t e s t proved more amenable t o t h e a p p l i c a t i o n o f e x a m i n a t i o n o f cases. There were no s i g n i f i c a n t d i f f e r e n c e s i n v o l v i n g the vehicles.  Thus I t seemed r e a s o n a b l e t o conclude t h a t t h e  i n s t r u c t i o n a l v e h i c l e employed was not an i m p o r t a n t f a c t o r i n whether s t u d e n t s l e a r n e d t o apply h e u r i s t i c s .  iv  F i n a l l y , an e x p l o r a t o r y n o n - p a r a m e t r i c  ( x ) analysis  f o r the H treatment i n d i c a t e d t h a t s u b j e c t s who top  2  were i n the  t h i r d on e i t h e r m a t h e m a t i c a l or r e a d i n g achievement  both) s c o r e d s u b s t a n t i a l l y h i g h e r t h a n the bottom  third.  (or  TABLE OF CONTENTS  L I S T OF TABLES L I S T OF FIGURES ACKNOWLEDGEMENTS CHAPTER I  THE PROBLEM Background Purpose o f t h e Study The R o l e o f F r a m e w o r k s i n R e s e a r c h . . . . A Model  o f Mathematics  Definition  o f Terms  Three Q u e s t i o n s A r i s i n g  from t h e Model  Research Hypotheses II  RELATED LITERATURE Introduction The P r o b l e m S o l v i n g P r o c e s s Analogy E x a m i n a t i o n o f Cases Related Research  •  Discovery Versus Other Techniques Teaching H e u r i s t i c Summary  Instructional  Procedures  of the Related Literature  vi  CHAPTER III  Page 58  DESIGN AND PROCEDURE  58  Design Instructional Materials  . .'.  59  Booklets  6l  Instruments  62  G r a d i n g Scheme  63  Procedure  64  Pilot  64  Studies  Main Study  65  Subjects  69  Development o f t h e B o o k l e t s  70  Development o f T e s t s  82  A l g e b r a i c Instrument  82  Geometric  IV  '  Instrument..'  .  83  S t a t i s t i c a l Procedures  84  ANALYSIS OP THE DATA....  85  Grading  86  .. .  Data Used i n A n a l y s i s  86  T e s t i n g t h e Hypotheses  91  Plan f o r Analysis  93  A n a l y s i s o f Data...  97  Statistical  Conclusions  Interpretation of Results Discussion of Results  102  < .  103 104  'vii  CHAPTER V  Page 115  SUMMARY AND CONCLUSIONS  115  Summary Conclusions  116 119  Implications Limitations  ; ...  Recommendations f o r F u r t h e r Research BIBLIOGRAPHY APPENDICES...  121 124 138  . .• .  145  A  H e u r i s t i c s only  (Neutral Vehicle)  145  B  H e u r i s t i c s plus  content (Neutral V e h i c l e ) . . . . . .  C  Content o n l y  D  H e u r i s t i c s only  E  H e u r i s t i c s plus content (Algebraic V e h i c l e ) . . . .  216  F  Content o n l y  228  G  H e u r i s t i c s only  (Geometric V e h i c l e )  238  H  H e u r i s t i c s plus  c o n t e n t (Geometric V e h i c l e ) . . . .  249  I  Content o n l y  J  I n s t r u c t i o n s to Subjects  271  K  Algebraic  273  L M  Geometric Test G r a d i n g Scheme....;  166 184  (Neutral Vehicle)..  201  (Algebraic Vehicle)  (Algebraic Vehicle)  260  (Geometric V e h i c l e ) . . .  Test ;•  •.  278 283  viii  LIST OF TABLES Table  Page  I  Percentage Agreement on T e s t s  86  II  Summary o f Responses on the A l g e b r a i c Test  88  III  Summary o f Responses on the Geometric T e s t  89  IV  Mark A s s i g n e d f o r A n a l y s i s  90  V  The R e l a t i v e E f f i c i e n c e s and R a t i o s o f Confidence I n t e r v a l s f o r S c h e f f e , Tukey and Dunn  96  Means and S t a n d a r d D e v i a t i o n s A l g e b r a i c Test  f o r the 98  Means and S t a n d a r d D e v i a t i o n s  f o r the  VI VII  Geometric Test  98  VIII  O v e r a l l F - R a t i o s f o r the A l g e b r a i c Test  99  XIV  O v e r a l l F - R a t i o s f o r the Geometric T e s t  99  X  A l g e b r a i c Test - Simultaneous Confidence I n t e r v a l s f o r Hypotheses 1, 2 and 3 Geometric Test - Simultaneous Confidence  XI  I n t e r v a l s f o r Hypotheses  1, 2 and 3  100 100  XII  x  2  A n a l y s i s f o r the A l g e b r a i c Test  101  XIII  x  2  A n a l y s i s f o r t h e Geometric Test  101  XIV  D i s t r i b u t i o n s o f Data f o r S u b j e c t s Who Wrote Only One Test and A c t u a l Data Used i n the A n a l y s i s . I l l X Comparison o f A c t u a l Data and Only One Ii.;: Test Data 112  XV XVI XVII XVIII  2  X x  2  A n a l y s i s o f the Achievement o f t h e H e u r i s t i c s Only Group on t h e A l g e b r a i c Test  126  A n a l y s i s o f t h e Achievement o f t h e H e u r i s t i c s Only Group on t h e Geometric Test  127  2  A n a l y s i s o f t h e A p p l i c a t i o n o f Z e r o , One o r Two H e u r i s t i c s on t h e A l g e b r a i c Test  133  . LIST OF FIGURES Figure  Page  1  A Model o f Mathematics  5  2  E x p e r i m e n t a l Groups.....  3  Number o f C e l l s  27  4  T e s t i n g Arrangement f o r Treatment Groups  30  5  The B o o k l e t s  6  S t r u c t u r e o f the T e s t s  63  7  O r g a n i z a t i o n o f the Groups  65  8  E x p e r i m e n t a l Procedure.:  68  9  Topic 1  16/58  ,.  61/70  ' 71  10  Equivalence - Topic  1..  74  11  Topic 2  75  12  Example o f a R o t a t i o n  75  13-  Example o f a Mapping  76  14  Equivalence - Topic  15  Topic 3  16  Equivalence - Topic  17  Movements on a G r i d  18  V i s u a l Presentation of Results of S i g n i f i c a n c e  2  77 •  3  T e s t s f o r Hypotheses H  78 79 83  Q 1 3  HQ  2  and HQ^  102  19  Mean Scores o f Groups on-the A l g e b r a i c T e s t . . .  105  20  Mean Scores o f Groups on the Geometric T e s t . . .  106  21  O v e r a l l Means f o r A l g e b r a i c and Geometric Tests '  109  Figure 22  23  24  25.  26  27  28  Page A c h i e v e m e n t o f H i g h , M i d d l e and Low M a t h e m a t i c a l A c h i e v e m e n t S u b g r o u p s o f H e u r i s t i c s O n l y Group' Algebraic Test  128  A c h i e v e m e n t o f H i g h , M i d d l e and Low R e a d i n g A c h i e v e m e n t S u b g r o u p s o f H e u r i s t i c s O n l y Group Algebraic Test  128  A c h i e v e m e n t o f H i g h , M i d d l e and Low M a t h e m a t i c a l Achievement Subgroups o f H e u r i s t i c s • O n l y Group Geometric Test  129  A c h i e v e m e n t o f H i g h , M i d d l e and Low R e a d i n g A c h i e v e m e n t S u b g r o u p s o f H e u r i s t i c s O n l y Group Geometric Test  129  A c h i e v e m e n t o f S u b g r o u p s t h a t were H i g h o r on B o t h M a t h e m a t i c s and R e a d i n g Algebraic Test.... .....  Low 130  Achievement o f Subgroups t h a t were H i g h o r on B o t h M a t h e m a t i c s and R e a d i n g Geometric Test  Low  A p p l i c a t i o n o f Z e r o , One the A l g e b r a i c Test  on  o r Two  Heuristics  130 134  ACKNOWLEDGEMENTS I would l i k e t o thank my committee  f o r the help  and guidance g i v e n throughout the s t u d y . particularly  I would  l i k e t o thank Dr. S p i t l e r f o r h e r  h e l p and encouragement d u r i n g t h e l a t e r s t a g e s o f the  dissertation.  F i n a l l y , I would l i k e t o thank my w i f e , who managed t o s u r v i v e my c o n t i n u a l l y changing moods d u r i n g my graduate s t u d i e s .  Without h e r  c o n t i n u i n g encouragement I c o u l d n o t have s u c c e s s f u l l y completed m y . s t u d i e s .  CHAPTER I The  Problem  Background In  a w o r l d where t h e r e i s r a p i d growth i n t e c h n o l o g y  and knowledge, s o c i e t y i s f a c e d w i t h t h e c h a l l e n g e o f e d u c a t i n g i n d i v i d u a l s who w i l l be a b l e t o f u n c t i o n i n a c o n t i n u a l l y changing environment.  Educators  are faced w i t h  the problem o f i d e n t i f y i n g and t e a c h i n g the competencies needed t o d e a l w i t h the changing w o r l d .  Many e d u c a t o r s  have  d i s c u s s e d t h e c h a l l e n g e o f p r o v i d i n g an e d u c a t i o n t h a t w i l l meet t h e needs o f f u t u r e g e n e r a t i o n s , and one concept  that  was common t o many o f t h e i r w r i t i n g s i s t h a t ,cbf ' s t r a t e g i e s of i n q u i r y ' .  F o r example, when Downey (1965, p. 22)  d i s c u s s e d the i m p l i c a t i o n s o f t h e knowledge e x p l o s i o n on a secondary  school education he.stated:  For i t i s assumed t h a t i f s t u d e n t s a c q u i r e . a mastery and comprehension o f the broad s t r u c t u r e i n a s u b j e c t a r e a as w e l l as f a c i l i t y w i t h the s t r a t e g y o f i n q u i r y a p p r o p r i a t e t o t h e p a r t i c u l a r s u b j e c t d i s c i p l i n e , they w i l l be i n a good p o s i t i o n t o c o n t i n u e l e a r n i n g as f u t u r e e x p e r i e n c e s demand-. Bernard  (1972,  p. 284) advocated  t h a t s i n c e t h e s c h o o l cannot  a n t i c i p a t e the problems s t u d e n t s w i l l encounter  i n the  f u t u r e , approaches t o , r a t h e r than answers t o problems s h o u l d be o f s e r i o u s concern t o t e a c h e r s . suggested  Heathers (1965, p. 2)  that "problem-solving t h i n k i n g or inquiry i s  g e n e r a l l y c o n s i d e r e d t o be the core o f t h e e d u c a t i o n a l p r o c e s s , and the c h i e f mark o f t h e educated The  person".  g e n e r a l concern among e d u c a t o r s w i t h t h e r o l e o f  2  s t r a t e g i e s o f i n q u i r y i n the e d u c a t i o n a l process i s r e f l e c t e d i n the mathematics e d u c a t i o n l i t e r a t u r e . example, Rosenbloom  For  (1966, p. 1 3 0 — 1 3 1 ) i n d i c a t e d the  s i g n i f i c a n c e o f s t r a t e g i e s of i n q u i r y i n mathematics ( h e u r i s t i c s ) when he wrote: One of the major s o c i a l f u n c t i o n s of the mathematics c u r r i c u l u m i s to t e a c h one o f the p r i n c i p a l methods of a c q u i r i n g new knowledge. To f i n d out whether we are a c h i e v i n g t h i s o b j e c t i v e , we must see whether the student can s o l v e problems f o r which he has r e c e i v e d no s p e c i f i c i n s t r u c t i o n . One of the l e a d i n g advocates of the t e a c h i n g o f h e u r i s t i c s i n mathematics e d u c a t i o n i s George P o l y a . 1965)  has w r i t t e n many a r t i c l e s  Polya ( 1 9 5 7 ,  and t e x t s concerned w i t h the  t e a c h i n g of h e u r i s t i c s , and i n these he presented  specific  questions t h a t can be employed by students f o r a c q u i r i n g knowledge.  A few o f these q u e s t i o n s  include:  What i s the unknown? What are the data? are the c o n d i t i o n s ? . . .  What  Could 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?... ( P o l y a , 1 9 5 7 , pp. x v i - x v i i ) A s u c c i n c t summary of many educators' views on strategies of inquiry  ( h e u r i s t i c s ) i s embodied  f o l l o w i n g statement by Richards  1962,  (1968,  i n the  p. 4 6 ) .  In the most r a p i d l y changing s o c i e t y i n h i s t o r y , t e a c h e r s can no longer f o r e s e e what a student w i l l need t o know. A l l the t e a c h e r knows i s t h a t e d u c a t i o n w i l l be a l i f e l o n g a c t i v i t y , so the most important t h i n g s he can teach are methods o f a c q u i r i n g new knowledge. [ i t a l i c s not i n o r i g i n a l ]  new  3  Purpose of the Study The purpose o f t h i s study i s t o e x p l o r e one  component  o f problem s o l v i n g , namely whether i t i s p o s s i b l e t o t e a c h s t u d e n t s t o apply mathematical mathematical  problems.  d e s i g n of the s t u d y .  Two The  h e u r i s t i c s to novel  f a c t o r s are c o n s i d e r e d i n the  f i r s t f a c t o r c o n s i d e r s the e f f e c t  o f u t i l i z i n g t r e a t m e n t s d e s i g n e d t o meet d i f f e r e n t i n s t r u c t i o n a l o b j e c t i v e s : one treatment  designed to t e a c h .  h e u r i s t i c s o n l y and another t h a t combined i n s t r u c t i o n i n h e u r i s t i c s w i t h the t e a c h i n g o f c o n t e n t , on s t u d e n t s * a b i l i t y t o apply a h e u r i s t i c . the e f f e c t of employing a l g e b r a i c , geometric  The  second f a c t o r c o n s i d e r s  distinct instructional  settings,  and m a t h e m a t i c a l l y n e u t r a l on s t u d e n t s '  a b i l i t y t o apply a h e u r i s t i c . An e x p l i c i t statement  o f the r e s e a r c h hypotheses i s  I n c l u d e d at the c o n c l u s i o n o f t h i s c h a p t e r , a f t e r a d i s c u s s i o n o f the model and t e r m i n o l o g y employed %n the study. 'The Role o f Frameworks i n The  Research  i n i t i a l t a s k i n the development of a r e s e a r c h  study I n a p a r t i c u l a r a r e a i s the i d e n t i f i c a t i o n o f s i g n i f i c a n t q u e s t i o n s r e l a t e d t o that- a r e a .  E i t h e r a theory  or a model can p r o v i d e an I n v e s t i g a t o r w i t h a v a l u a b l e a i d i n the s e a r c h f o r and c l a s s i f i c a t i o n of these q u e s t i o n s . (1973, p. 77) suggested  t h a t t h e o r i e s and models are  Snow  4  extremely v a l u a b l e t o those p u r s u i n g r e s e a r c h on t e a c h i n g . Travers  (1969,  p. 18) n o t e d t h a t t h e e x t e n s i v e use o f models  i n s c i e n t i f i c work " d e r i v e s l a r g e l y from t h e i r v a l u e i n s u g g e s t i n g experiments  and s t u d i e s and t h e i r l o n g h i s t o r y as  a means o f a c h i e v i n g i m p o r t a n t knowledge". As w e l l as a c t i n g as a c a t a l y s t i n t h e i d e n t i f i c a t i o n o f s i g n i f i c a n t  research  q u e s t i o n s , models can a l s o serve o t h e r u s e f u l purposes. N u t h a l l and Snook (1973) suggested  t h a t models p r o v i d e a  framework f o r t h e i n t e r p r e t a t i o n and o r g a n i z a t i o n o f d a t a . Kaplan  (1964) i n d i c a t e d t h a t models can be u s e f u l i n  communication as w e l l as i n t h e o r g a n i z a t i o n o f d a t a . Many mathematics educators have a l l u d e d t o t h e need f o r a framework i n r e s e a r c h . 'For example, Romberg and Vere DeVault  (1967,  p. 95) noted t h a t t h e f i r s t s t e p towards  i d e n t i f y i n g needed r e s e a r c h i n t h e a r e a o f mathematics e d u c a t i o n i s t o o r g a n i z e a model o f t h e v a r i o u s c u r r i c u l u m components.  Becker  (1970, p. 21) made a p l e a f o r more  theory-related research.  P i n g r y (1967, p. 44) s t a t e d t h e '  need f o r a framework when he wrote: The proponents o f r e s e a r c h d i r e c t e d towards model b u i l d i n g and t h e o r y were o f t h e o p i n i o n t h a t r e a l p r o g r e s s i n r e s e a r c h i n mathematics e d u c a t i o n would n o t be made u n t i l such a time t h a t r e s e a r c h c o u l d be r e l a t e d t o a t h e o r y o f mathematics e d u c a t i o n , even i f t h e t h e o r y were a very crude one. The next s e c t i o n c o n t a i n s t h e model w i t h i n which t h i s study i s embedded.  5  A Model o f Mathematics MacPherson  (1970, 1971a) characterized  as h a v i n g t h r e e components.  a discipline  A p p l i c a t i o n , Core and D i s c o v e r y  S i n c e t h i s study i s concerned w i t h m a t h e m a t i c a l problem s o l v i n g t h e d i s c i p l i n e o f mathematics was chosen as an exemplar o f t h e model. Figure 1 A Model o f Mathematics  APPLICATION  DISCOVERY  lore  heuristics  MacPherson d e f i n e d these components as f o l l o w s . Application A p p l i c a t i o n incorporates the set o f acts of f a i t h ( l o r e ) by w h i c h systems d e v e l o p e d w i t h i n t h e d i s c i p l i n e a r e t i e d to the p h y s i c a l world.  S i n c e t h e r e i s no l o g i c a l  c o n n e c t i o n between systems i n t h e d i s c i p l i n e o f mathematics and t h e p h y s i c a l w o r l d , whenever one attempts t o s o l v e  6  p h y s i c a l w o r l d problems by m a t h e m a t i c a l means l o r e i s c a l l e d into play.  F o r example, c o n s i d e r t h e f o l l o w i n g problem:  John has t h r e e m a r b l e s . P a u l g i v e s John f o u r m a r b l e s . How many marbles does John now have? The  t r a n s l a t i o n o f t h i s problem i n t o t h e c o r r e s p o n d i n g  mathematical equation,  3 + 4 = ?  3  r e q u i r e s an a c t o f f a i t h ,  namely t h a t 3 + 4 = ? i s a v a l i d m a t h e m a t i c a l model f o r t h e p h y s i c a l world  problem.  Core The  core c o n s i s t s o f t h r e e components: f a c t s ,  a l g o r i t h m s and c a t e g o r i e s . F a c t s a r e p r o p o s i t i o n s about d a t a which a r e not under a c t i v e c o n s i d e r a t i o n as t o v a l i d i t y . 2 + 3 = 5; 2 ?< 6 = 12;  Examples o f t h e s e a r e :  5 < 10; t h e Fundamental  Theorem o f A r i t h m e t i c ; t h e axioms o f E u c l i d e a n Geometry..' Algorithms  a r e procedures f o r t h e a n s w e r i n g o f q u e s t i o n s  w h i c h , i f a p p l i e d c o r r e c t l y , w i l l guarantee 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 . The  Some t y p i c a l a l g o r i t h m s a r e :  decomposition algorithm f o r s u b t r a c t i o n ;  the E u c l i d e a n method f o r f i n d i n g t h e g r e a t e s t common d i v i s o r o f two numbers;  t h e Gaussian  e l i m i n a t i o n method f o r t h e s o l u t i o n o f a system of  equations.  Categories information.  a r e m a t h e m a t i c a l frameworks f o r o r g a n i z i n g F o r example:  ^  7  The n a t u r a l numbers; Non-Euclidean  geometries;  a l g e b r a ; geometry; t o p o l o g y . Discovery D i s c o v e r y i n c o r p o r a t e s the s e t o f techniques-, h e u r i s t i c s , by which m a t h e m a t i c a l  d i s c o v e r i e s are made  c o n c e r n i n g p h y s i c a l w o r l d or m a t h e m a t i c a l m a t e r i a l so d i s c o v e r e d may  problems.  Any  become p a r t of the core o f the  discipline. The term h e u r i s t i c has been u t i l i z e d by many mathematics e d u c a t o r s .  Kilpatrick  (1967  5  P- 19) d e f i n e d a  h e u r i s t i c as "any d e v i c e , t e c h n i q u e , r u l e o f thumb, e t c . t h a t improves p r o b l e m - s o l v i n g performance."  Wilson  (1967,  p. 3) r e f e r r e d t o a h e u r i s t i c as "a d e c i s i o n mechanism, a way of b e h a v i n g , which u s u a l l y l e a d s t o d e s i r e d outcomes, but w i t h no guarantee  of success."  P o l y a (1957, p. 112)  stated  t h a t the "aim o f h e u r i s t i c i s t o study the methods o f d i s c o v e r y and i n v e n t i o n . " P o l y a (1957, P- 113)  c o n t i n u e d by  d e f i n i n g the term h e u r i s t i c r e a s o n i n g : H e u r i s t i c r e a s o n i n g i s r e a s o n i n g not r e g a r d e d as f i n a l and s t r i c t but as p r o v i s i o n a l and p l a u s i b l e o n l y , whose purpose i s to d i s c o v e r the s o l u t i o n o f the p r e s e n t problem. One  s t r a n d t h a t l i n k s a l l these d e f i n i t i o n s I s t h a t a  h e u r i s t i c i s a f l e x i b l e procedure a solution.  t h a t does not  The i n v e s t i g a t o r f e e l s t h a t the  guarantee  following  d e f i n i t i o n , which i s employed i n t h i s s t u d y , i s i n a c c o r d t w i t h those s t a t e d p r e v i o u s l y :  8  A M a t h e m a t i c a l H e u r i s t i c i s a procedure which i s used f o r the purpose o f d i s c o v e r i n g m a t h e m a t i c a l r e l a t i o n s h i p s i n a n o v e l problem, and whose a p p l i c a t i o n does not guarantee s u c c e s s . The n o v e l problem i n which a h e u r i s t i c i s employed may be ~ '•••> t o t a l l y m a t h e m a t i c a l i n n a t u r e , o r be a s s o c i a t e d w i t h a p h y s i c a l world s i t u a t i o n . A l t h o u g h t h e term h e u r i s t i c i s i n g e n e r a l use by e d u c a t o r s t h e l i t e r a t u r e c o n t a i n s few s o u r c e s o f mathematical h e u r i s t i c s .  P o l y a ( 1 9 5 7 ) , Smith and Henderson  (1959), MacPherson (1971b). and Rousseau (1971) a l l have s u g g e s t i o n s c o n c e r n i n g s p e c i f i c t e c h n i q u e s t h a t can be employed i n s o l v i n g n o v e l problems, and t h e h e u r i s t i c s used i n t h i s experiment, namely analogy and e x a m i n a t i o n (enumeration) o f cases are common t o t h e s o u r c e s . The term analogy i s i n common use i n the E n g l i s h language.  Webster's  as "resemblance unlike."  dictionary  (1967, P- 32) d e f i n e d analogy  i n some p a r t i c u l a r s between t h i n g s o t h e r w i s e  P o l y a (1954, p. 32) d e f i n e d analogy as "a s o r t o f  s i m i l a r i t y , " w h i l e Dinsmore (1971, p. 17) d e f i n e d i t "as the p e r c e p t i o n o f a s i m i l a r i t y between t h e problem, t o be s o l v e d and one t h a t has a l r e a d y been s o l v e d . "  B a s i c a l l y a l l these  d e f i n i t i o n s d e s c r i b e analogy i n terms o f a p e r c e i v e d s i m i l a r i t y between t h i n g s . as f o l l o w s :  I n t h i s study analogy i s d e f i n e d  9  Analogy  Is. t h e h e u r i s t i c by which a p e r c e i v e d  s i m i l a r i t y between f a m i l i a r systems o r e x p e r i e n c e s Is u t i l i z e d i n order to raise, questions a novel mathematical  problem.  The f o l l o w i n g examples o f m a t h e m a t i c a l analogy  concerning  problems where  can be o f v a l u e are i n c l u d e d t o c l a r i f y t h e  definition. The student has d i s c u s s e d modulo 5 a r i t h m e t i c i n  (a)  some d e t a i l and i s then p r e s e n t e d w i t h modulo 6.  An analogy  between t h e new s i t u a t i o n and modulo 5 would.enable the student t o r a i s e questions.concerning the s t r u c t u r e of the new system.  F o r example, t h e analogy would r a i s e q u e s t i o n s  c o n c e r n i n g t h e c o m m u t a t i v l t y o f a d d i t i o n , t h e e x i s t e n c e o f an a d d i t i v e i d e n t i t y , as w e l l as many o t h e r f i e l d p r o p e r t i e s . Thus, t h e analogy w i t h modulo 5 p r o v i d e s t h e s t u d e n t w i t h a p o s s i b l e b a s i s from which t o a n a l y s e t h e new  problem.  (b) I n d i s c u s s i n g t h e h e u r i s t i c o f analogy  Polya  (1957j p. 37) n o t e d t h a t a r e c t a n g u l a r p a r a l l e l o g r a m I s analogous  to the three dimensional f i g u r e , the rectangular  p a r a l l e l e p i p e d , i n that- the r e l a t i o n s h i p between t h e s i d e s o f t h e p a r a l l e l o g r a m a r e l i k e those between t h e f a c e s of t h e parallelepiped.  Thus, when f a c e d w i t h t h e t h r e e d i m e n s i o n a l  f i g u r e f o r t h e f i r s t time employing  an analogy w i t h t h e  p a r a l l e l o g r a m would r a i s e q u e s t i o n s such as whether o p p o s i t e f a c e s a r e congruent, o r o p p o s i t e s o l i d angles a r e congruent.  10  Examining cases i s o f t e n r e f e r r e d t o i n t h e mathematics e d u c a t i o n l i t e r a t u r e .  Smith and Henderson (1959)  d i s c u s s e d t h e v a l u e o f enumeration o f c a s e s . i n f o r m u l a t i n g conjectures.  Leask  (1968,  p 8) d e f i n e d s i m p l e  enumeration  o f cases as c o n s i s t i n g " o f t h e p r e s e n t a t i o n o f many i n s t a n c e s o f t h e g e n e r a l i z a t i o n t o be d i s c o v e r e d . "  Leask  noted t h a t t h e s t u d e n t s can form hypotheses based on t h e cases and t h e n t e s t them t o determine t h e i r v a l i d i t y .  For  t h i s study e x a m i n a t i o n o f cases i s d e f i n e d a s : E x a m i n a t i o n o f Cases i s t h e h e u r i s t i c by which p a r t i c u l a r cases are examined i n o r d e r t o make d i s c o v e r i e s concerning a novel mathematical  problem.  The cases may be examined i n a s y s t e m a t i c o r random manner. F u r t h e r m o r e , t h i s e x a m i n a t i o n o f cases may be employed t o formulate a c o n j e c t u r e , or t o t e s t the v a l i d i t y o f a c o n j e c t u r e developed from t h e use o f e x a m i n a t i o n o f cases o r a n o t h e r h e u r i s t i c , such as analogy.  When v a l i d a t i n g a  c o n j e c t u r e by e x a m i n a t i o n o f cases one w i l l e i t h e r f i n d  cases  t h a t support t h e conjecture^';'or produce a counterexample necessitating i t s rejection.  The f o l l o w i n g examples a r e o f  m a t h e m a t i c a l problems where e x a m i n a t i o n o f cases can be o f value. (a)  A h i g h s c h o o l s t u d e n t has been p r e s e n t e d w i t h  the problem o f f i n d i n g a g e n e r a l f o r m u l a f o r t h e f o l l o w i n g series. 1+3  + 5 + 7 + 9 + . - . +  (2n-l)  11  The s t u d e n t can apply t h e h e u r i s t i c o f e x a m i n a t i o n o f cases by s u b s t i t u t i n g d i f f e r e n t v a l u e s f o r n. 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 =  16  1 + 3 + 5 + 7 + 9 =  25  T h i s s y s t e m a t i c e x a m i n a t i o n may suggest t h e f o l l o w i n g p a t t e r n , namely t h a t t h e sum o f n terms i s n . The 2  a p p l i c a t i o n o f t h e h e u r i s t i c o n l y enables the s t u d e n t t o make a r e a s o n a b l e guess t h a t t h e sum i s n , and does not 2  c o n s t i t u t e a mathematical proof o f the r e s u l t . (b)  An elementary s t u d e n t i s e x p l o r i n g t h e sum o f  the i n t e r i o r angles o f a t r i a n g l e .  By e x p e r i m e n t i n g w i t h  different t r i a n g l e s , selected either systematically or randomly, he i s a b l e t o conclude t h a t t h e sum o f t h e i n t e r i o r angles o f a t r i a n g l e i s 1 8 0 ° .  Again t h i s e x a m i n a t i o n o f  cases does not p r o v i d e a p r o o f , i n a m a t h e m a t i c a l sense, b u t o n l y r e a s o n a b l e evidence o f t h e r e s u l t . A g e n e r a l d i s c u s s i o n o f t h e s i g n i f i c a n c e o f these 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 i n c l u d e d i n the r e v i e w o f t h e l i t e r a t u r e .  (Chapter I I )  The n a t u r e o f h e u r i s t i c s i s t h a t they a r e a p p l i e d t o n o v e l •mathematical problems.  Problem,  l i k e analogy^ i s i n  common use i n t h e E n g l i s h language. Some r e s e a r c h s t u d i e s d e f i n e d problem i m p l i c i t l y i n terms o f t h e p a r t i c u l a r  12  problems i n v e s t i g a t e d i n t h e s t u d y , w h i l e o t h e r s  explicitly  s t a t e d what was meant by t h e word problem. Duncker (1945, p. 1) s u g g e s t e d t h a t "a problem a r i s e s when a l i v i n g c r e a t u r e has a g o a l but does not know' how t h i s g o a l i s t o be reached."  M a i e r ' s (1970) c o n c e p t i o n  the e x i s t e n c e  o f a problem i n v o l v e d  o f an o b s t a c l e , o r o b s t a c l e s between t h e  problem s o l v e r and the g o a l .  Polya  (1962, p.117) d i d not  d e f i n e p r o b l e m , p e r s e , but suggested what i t means t o have a problem: Thus t o have a problem means: t o s e a r c h consciously f o r some a c t i o n a p p r o p r i a t e t o a t t a i n a c l e a r l y c o n c e i v e d , but not i m m e d i a t e l y a t t a i n a b l e , aim. [italics i noriginal] I n each o f these d e f i n i t i o n s t h e r e i s e i t h e r explicitly  s t a t e d , o r i m p l i e d , one o f the e s s e n t i a l  components o f any d e f i n i t i o n o f a p r o b l e m , namely t h e existence reaching  o f a b a r r i e r t h a t p r e v e n t s t h e problem s o l v e r from the desired goal.  The e x i s t e n c e  o f such a b a r r i e r  means t h a t a problem must be more than j u s t the r o u t i n e a p p l i c a t i o n o f knowledge, s k i l l s o r a l g o r i t h m s . D a v i s (1973, p. 21) noted t h a t s i m p l e q u e s t i o n s invented  t h e F o r d ? ' c o u l d not be c o n s i d e r e d  average a d u l t .  F o r example, such as 'Who  problems f o r t h e  H i n t s on Problem S o l v i n g ( N a t i o n a l  Council  of Teachers o f Mathematics, 1969, p. 1) r e f e r r e d t o t h e nonr o u t i n e n a t u r e o f a m a t h e m a t i c a l problem.  13  In mathematics the u s u a l l y accepted concept of a p r o b l e m d i f f e r e n t i a t e s b e t w e e n s i t u a t i o n s s u c h as t h i s a s s i g n m e n t {.refers t o t h e a s s i g n i n g ^ o f : e x e r c i s e s from the textbook - the I n v e s t i g a t o r ] and t h o s e t h a t r e q u i r e c e r t a i n b e h a v i o r s b e y o n d t h e r o u t i n e a p p l i c a t i o n o f an - e s t a b l i s h e d p r o c e d u r e . A t r u e p r o b l e m i n m a t h e m a t i c s c a n be t h o u g h t o f as a s i t u a t i o n t h a t i s n o v e l , f o r t h e i n d i v i d u a l c . a l l e d upon t o s o l v e i t . The  sample d e f i n i t i o n s o f a problem  together with  cited  previously,  the t e r m i n o l o g y o f the model l e d t o the  following d e f i n i t i o n of A problem  problem.  i s a mathematical task  f o r which  the  l e a r n e r has no a v a i l a b l e : f a c t s , a l g o r i t h m s o r categories task  that  c a n be a p p l i e d d i r e c t l y  i n order to reach the d e s i r e d  This d e f i n i t i o n of problem but r a t h e r d e l i n e a t e s problem that this  formed  to the  goal.  i s n o t meant as a l l e m b r a c i n g ,  the i n t e r p r e t a t i o n o f the word  the basis  f o r developing the t e s t s f o r  study. D e f i n i t i o n o f Terms The  d i s c u s s i o n o f t h e m o d e l i n c l u d e s many d e f i n i t i o n s  t h a t are used throughout t h i s they are r e s t a t e d i n t h i s  study and,  f o r convenience  section.  A mathematical h e u r i s t i c i s a procedure which i s used f o r the purpose  of discovering  r e l a t i o n s h i p s i n a novel problem, a p p l i c a t i o n does n o t g u a r a n t e e  mathematical  and whose  success.  14 A n a l o g y i s t h e h e u r i s t i c by w h i c h a p e r c e i v e d s i m i l a r i t y between f a m i l i a r systems or  experiences,  i s u t i l i z e d i n order to r a i s e questions a novel mathematical Examination  discoveries  problem.  of cases  p a r t i c u l a r cases  i s t h e h e u r i s t i c by  concerning  no  a novel mathematical  available  c a t e g o r i e s t h a t can be i n order to reach One  applied directly  the  or  to the  task  the d e s i r e d g o a l .  t a s k t h a t c o u l d be  definitions  i s that  any  c a l l e d a problem would  c o n s i d e r e d as n o v e l f o r t h e p r o b l e m s o l v e r , and a p p l i c a t i o n of a h e u r i s t i c , or h e u r i s t i c s solution.  problem.  t a s k f o r which  f a c t s , algorithms  i m p l i c a t i o n of these  mathematical  which  a r e e x a m i n e d i n o r d e r t o make  A problem i s a mathematical l e a r n e r has  concerning  be  r e q u i r e the  i n attempting  a  For such problems, problem s o l v i n g i s d e f i n e d  follows: P r o b l e m s o l v i n g i s t h e s e a r c h f o r and of h e u r i s t i c s ,  f a c t s , a l g o r i t h m s and  application  categories that  a i d the l e a r n e r i n r e a c h i n g the d e s i r e d g o a l . The  model, t o g e t h e r w i t h the terms d e f i n e d i n t h i s  s e c t i o n provided the i n v e s t i g a t o r w i t h the w i t h i n which to develop  and  s t a t e the  framework  hypotheses.  as  15  Three q u e s t i o n s a r i s i n g from the model The s t r u c t u r e o f the model suggests c e r t a i n c o n c e r n i n g the t e a c h i n g o f h e u r i s t i c s . which d i r e c t l y r e l a t e d are d i s c u s s e d i n t h i s (1)  The t h r e e q u e s t i o n s  t o the f o r m a t i o n o f the hypotheses section..'  Is i t p o s s i b l e t o teach students to apply  h e u r i s t i c s t o n o v e l mathematical (2)  questions  problems?  Would i t be more e f f i c a c i o u s  to teach  heuristics  o n l y or combine i n s t r u c t i o n i n h e u r i s t i c s w i t h the t e a c h i n g of  content? (3)  Would t h e a b i l i t y t o l e a r n how t o a p p l y  heuristics  depend on the m a t e r i a l s s e l e c t e d f o r t h e i r i n t r o d u c t i o n ? The model s p e c i f i e s the e x i s t e n c e o f a s e t o f heuristics.  T h i s study s e l e c t e d the h e u r i s t i c s  of.analogy  and e x a m i n a t i o n o f cases and i n v e s t i g a t e d whether I t I s p o s s i b l e t o t e a c h s t u d e n t s t o apply a t l e a s t , one o f these h e u r i s t i c s to novel mathematical  problems.  The f i r s t  q u e s t i o n s l e d t o the s e l e c t i o n o f two t r e a t m e n t s , o n l y and h e u r i s t i c s p l u s c o n t e n t .  two  heuristics  The h e u r i s t i c s only-  treatment was d e s i g n e d t o t e a c h o n l y the h e u r i s t i c s , w h i l e the h e u r i s t i c s p l u s c o n t e n t treatment was d e s i g n e d t o combine i n s t r u c t i o n i n h e u r i s t i c s w i t h the teaching o f content.  A  t h i r d treatment u t i l i z e d content o n l y , w i t h r e f e r e n c e t o h e u r i s t i c s , and was used as an e x p e r i m e n t a l  control,  The core c o n t a i n s many c a t e g o r i e s such as a l g e b r a and  16  geometry.  Question. 3 concerns whether one c a t e g o r y i s  more e f f e c t i v e f o r i n t r o d u c i n g h e u r i s t i c s than S i n c e the secondary  another.  school' mathematics c u r r i c u l u m i s  p r i m a r i l y based on the c a t e g o r i e s o f a l g e b r a , and' geometry these two were u t i l i z e d as i n s t r u c t i o n a l v e h i c l e s f o r the heuristics.  Furthermore,  a s t u d e n t i s a l s o exposed t o  mathematics through a v a r i e t y o f e x p e r i e n c e s o u t s i d e a mathematics c o u r s e , such as s c i e n c e c o u r s e s , newspapers commercial  and  t r a n s a c t i o n s . Thus a t h i r d v e h i c l e , r e f e r r e d t o  as m a t h e m a t i c a l l y n e u t r a l was  employed.  This'resulted i n a  t o t a l - of n i n e groups, c o r r e s p o n d i n g t o the c e l l s i n f i g u r e Figure 2 E x p e r i m e n t a l Groups Vehicles  Neutral  Algebraic  Geometric  Heuristics  H - N  H - A  H - G  Heuristics plus content  HC - N  HC - A  HC - G  Content only  C..- N  C - A  C - G  2.  17  For the purpose o f . l a t e r d i s c u s s i o n have been i d e n t i f i e d by a two p a r t code.  the n i n e groups The H, HC and C  denote the t h r e e t r e a t m e n t s : h e u r i s t i c s o n l y (H). h e u r i s t i c s p l u s content  (HC) and content o n l y ( C ) .  The N, A and G  denote the t h r e e v e h i c l e s : m a t h e m a t i c a l l y n e u t r a l algebraic  (A) and geometric  (G).  (N),  F u r t h e r d e t a i l s on  groups can be found i n c h a p t e r I I I - D e s i g n and  these  Procedure.  The H, HC and C are used s e p a r a t e l y t o denote the t h r e e t r e a t m e n t s , i r r e s p e c t i v e o f the v e h i c l e , the group r e c e i v i n g the H treatment c o n s i s t s i n H - N, H - A and  I n each of hypotheses  Hypotheses 1 through 5 , the dependent  i s a measure o f a b i l i t y t o a p p l y at l e a s t one  the h e u r i s t i c s  of  (analogy or e x a m i n a t i o n o f cases) t o a  n o v e l mathematics  problem:  Hypothesis 1. 1  will  of a l l subjects  H G ,  Research  variable  F o r example,  A group taught h e u r i s t i c s o n l y  (H)  score h i g h e r than a group taught content o n l y ( C ) . H y p o t h e s i s 2_.  content  A group taught h e u r i s t i c s p l u s  (HC) w i l l s c o r e h i g h e r than a group taught  content  only ( C ) . Hypothesis  3_.  A group taught h e u r i s t i c s o n l y  w i l l score h i g h e r than a group taught h e u r i s t i c s p l u s content  (HC).  (H)  18  H y p o t h e s i s _.  The mean s c o r e o f a h e u r i s t i c s  (H - N, H - A, H - G) as compared corresponding content  o n l y group  only  t o t h e mean s c o r e o f t h e (C - N, C -A, C - G)  will  be I n d e p e n d e n t o f t h e v e h i c l e . H y p o t h e s i s 5_. content group  The mean s c o r e o f a h e u r i s t i c s  (HC - N, HC - A, HC - G) a s compared  mean s c o r e o f t h e c o r r e s p o n d i n g c o n t e n t o n l y g r o u p C - A, C - G) w i l l  be i n d e p e n d e n t o f t h e v e h i c l e s .  plus  to the (C - N,  group  CHAPTER I I Related  Literature  Introduction This review of the l i t e r a t u r e i s organized i n the f o l l o w i n g manner: process;  (1) l i t e r a t u r e on t h e problem s o l v i n g  (2) h i s t o r i c a l and e d u c a t i o n views o f analogy  and e x a m i n a t i o n  o f c a s e s ; and (3) r e l a t e d  research.  The Problem S o l v i n g Process Although  educators  have d i s c u s s e d 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 t h e r e have been few r e s e a r c h s t u d i e s t h a t have i n v e s t i g a t e d g e n e r a l h e u r i s t i c s .  Wittrock  (1964, p. 6 l ) had suggested  direct  that researchers should  t h e i r a t t e n t i o n t o t e a c h i n g c h i l d r e n how t o make d i s c o v e r i e s . He w r o t e : I n s t e a d o f a s k i n g i f l e a r n i n g by d i s c o v e r y i s o r i s n o t as e f f e c t i v e as some o t h e r way t o l e a r n , one ought now t o s t a r t t o ask by what sequence o f d i r e c t e d o r u n d i r e c t e d e x p e r i e n c e s can we t e a c h children to learn to discover. Shulman (1968, p. 89) observed  t h a t t h e r e was a l a c k o f  r e s e a r c h on g e n e r a l h e u r i s t i c s when he w r o t e : N o t a b l y absent a r e s t u d i e s which d e a l w i t h t h e q u e s t i o n o f whether g e n e r a l t e c h n i q u e s , s t r a t e g i e s , and h e u r i s t i c s o f d i s c o v e r y can be l e a r n e d - by d i s c o v e r y o r any o t h e r manner - which w i l l t r a n s f e r across g r o s s l y d i f f e r e n t kinds of t a s k s , One approach taken by e d u c a t o r s  and p s y c h o l o g i s t s  has been t o i n v e s t i g a t e t h e n a t u r e o f t h e problem s o l v i n g  20  process. problem  T h e i r e f f o r t s have r e s u l t e d i n a n a l y s e s o f t h e s o l v i n g process.  F o r example, Bloom and B r o d e r  (1950, p. 25) suggested t h a t t h e f o l l o w i n g f o u r components are i n v o l v e d i n the problem s o l v i n g p r o c e s s . 1. 2. 3.  U n d e r s t a n d i n g o f t h e n a t u r e o f t h e problem U n d e r s t a n d i n g o f the i d e a s c o n t a i n e d i n t h e problem G e n e r a l approach t o t h e s o l u t i o n o f problems  4.  A t t i t u d e towards t h e s o l u t i o n o f problems.  Garry and K i n g s l e y (1970, p. 464) suggested t h a t t h e r e are' t h r e e s t e p s i n problem  s o l v i n g : t h e s e a r c h phase, t h e  f u n c t i o n a l s o l u t i o n phase, and v e r i f i c a t i o n o f t h e f i n a l solution.  Gagne (1966, p. 138) i n c l u d e d t h e stages o f  d e f i n i n g t h e problem by d i s t i n g u i s h i n g e s s e n t i a l f e a t u r e s , s e a r c h i n g f o r and f o r m u l a t i n g h y p o t h e s e s , and v e r i f y i n g t h e s o l u t i o n i n h i s a n a l y s i s o f t h e problem s o l v i n g p r o c e s s . While these a n a l y s e s g i v e an o v e r a l l framework f o r examining t h e n a t u r e o f the. problem s o l v i n g p r o c e s s , they do not i n c l u d e s p e c i f i c t e c h n i q u e s that, can be a p p l i e d by t h e l e a r n e r t o m a k e . d i s c o v e r i e s about n o v e l problems.  For the  mathematics e d u c a t o r the works o f P o l y a (1954a, 1954b, 1957, 1962, 1965) p r o v i d e a major source o f s p e c i f i c q u e s t i o n s t h a t can be a p p l i e d t o n o v e l m a t h e m a t i c a l problems.  I n How To  S o l v e I t P o l y a (1957, pp. x v i - x v i i ) l i s t e d f o u r stages i n t h e problem  solving process.  21  UNDERSTANDING THE PROBLEM What i s t h e unknown? What are t h e data? ... DEVISING A PLAN Have you 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? ... Could 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? Could you s o l v e p a r t o f t h e problem? ... CARRYING OUT THE- PLAN Carry out your p l a n o f t h e s o l u t i o n , ... LOOKING BACK Can you check the r e s u l t s ? ... The major headings are g e n e r a l , b u t t h e v a r i o u s s u g g e s t i o n s under t h e s e headings c o n t a i n a s e r i e s o f s p e c i f i c q u e s t i o n s t h a t a problem s o l v e r can use when f a c e d w i t h a n o v e l problem.  The s i g n i f i c a n t c o n t r i b u t i o n o f P o l y a t o t h e study  o f h e u r i s t i c s i n mathematics .is e v i d e n c e d by t h e f a c t t h a t much o f t h e r e s e a r c h concerned w i t h m a t h e m a t i c a l h e u r i s t i c s I s based upon P o l y a ' s work.  F o r example, Ashton  (1962)  employed t h e q u e s t i o n s I n P o l y a ' s h e u r i s t i c p r o c e s s as t h e b a s i s o f h e r h e u r i s t i c t r e a t m e n t i n t h e study o f problem s o l v i n g by n i n t h grade a l g e b r a s t u d e n t s . of K i l p a t r i c k ' s  (1967)  A major component  study concerned t h e development o f  a system f o r i d e n t i f y i n g c h a r a c t e r i s t i c s o f t h e problem s o l v i n g process.  P o l y a ' s ' h e u r i s t i c q u e s t i o n s and  s u g g e s t i o n s formed t h e i n i t i a l b a s i s f o r d e v e l o p i n g t h e K i l p a t r l c k c a t e g o r i z a t i o n system.  22  Smith and Henderson (1959) have a l s o  developed  c a t e g o r i e s o f problem s o l v i n g thought p r o c e s s e s .  They  l i s t e d f i v e types o f evidence f o r p r o b a b l e i n f e r e n c e t h a t are  u s e f u l i n d e v e l o p i n g and t e s t i n g g e n e r a l i z a t i o n s .  These a r e t h e methods o f s i m p l e enumeration,  analogy,  e x t e n d i n g a p a t t e r n o:f t h o u g h t , hunches and t h e i r  relation  to p r o o f , and t e s t i n g t h e hunches (Smith and Henderson,  1959a p.  121).  Two a d d i t i o n a l sources were a v a i l a b l e t o t h e  i n v e s t i g a t o r , namely u n p u b l i s h e d a r t i c l e s by MacPherson (1971b) and Rousseau .(1971)•  I n s e l e c t i n g the h e u r i s t i c s of  analogy and e x a m i n a t i o n o f cases t h e i n v e s t i g a t o r chose two h e u r i s t i c s t h a t were common t o a l l f o u r sources  (Polya,  Smith and Henderson, MacPherson, Rousseau). Analogy and E x a m i n a t i o n o f Cases Analogy The h i s t o r i c a l s i g n i f i c a n c e o f analogy as a h e u r i s t i c cannot be o v e r e s t i m a t e d .  The c l a s s i c r e f e r e n c e t o t h e  v a l u e o f analogy can be a t t r i b u t e d t o H e n r i P o i n c a r e . I n d e v e l o p i n g t h e t h e o r y o f F u c h s i a n F u n c t i o n s P o i n c a r e was r e p o r t e d t o have s a i d (Hadamard, 1949  3  p , 13):  I want t o r e p r e s e n t t h e s e f u n c t i o n s by t h e q u o t i e n t o f two s e r i e s ; t h i s i d e a was p e r f e c t l y conscious' and d e l i b e r a t e ; t h e analogy w i t h e l l i p t i c f u n c t i o n s guided me. P o l y a (1954a, pp. 18,19) noted t h a t E u l e r employed an analogy t o c o n j e c t u r e a sum f o r t h e i n f i n i t e s e r i e s . £l/n . 2  23  The. v a l u e o f analogy d i s c i p l i n e o f mathematics.  I s not l i m i t e d t o the F o r example. M a x w e l l ( N l v e n ,  1.890. pp. 156 - 7) d i s c u s s e d t h e importance o f p h y s i c a l a n a l o g i e s i n the mathematical  s c i e n c e s , where one i s  i n v o l v e d I n a n a l o g i e s between p h y s i c a l laws and t h e laws o f number. analogy  Brumbaugh '(1968, p. 6) i n d i c a t e d how P l a t o used i n h i s arguments.  He w r o t e :  I f we can f i n d numbers f a l l i n g i n t h e same c l a s s e s as t h e e n t i t i e s we want t o s t u d y , t h e r e l a t i o n s o f these e n t i t i e s can be i l l u s t r a t e d and i n v e s t i g a t e d by o b s e r v i n g t h e c o r r e l a t e d r e l a t i o n s o f t h e analogous c l a s s e s o f numbers. D e l t h e i l (1971, P- 3 7 ) , a f t e r m e n t i o n i n g o f K e p l e r and P o i n c a r e  t h e works  concluded:  The e s t a b l i s h m e n t o f such comparisons i s t h e very essence o f t h e genius f o r d i s c o v e r y and the g o a l o f a mode o f i n t u i t i o n o f t h e f i r s t importance, a n a l o g i c a l i n t u i t i o n . The key h i s t o r i c a l s i g n i f i c a n c e o f analogy  as a  h e u r i s t i c has been summarized I n t h e f o l l o w i n g b r i e f b u t c o n c i s e q u o t a t i o n o f P o l y a (1954a, p. 1 7 ) , who s a i d o f analogy  t h a t i t "seems t o have a share- o f a l l d i s c o v e r i e s ,  but i n some I t has t h e l i o n ' s Many educators teaching.  share".  have d i s c u s s e d t h e r o l e o f analogy I n  I n i l l u s t r a t i n g t h e v a l u e o f analogy  s o l v i n g technique  as a problem  Smith and Henderson (1959, p. 124) w r o t e :  A n a l o g i e s between t h e two s u b j e c t s , [plane and s o l i d , geometry - t h e i n v e s t i g a t o r ] . f a c i l i t a t e l e a r n i n g , p r o v i d e a:\hierarchy o f r e l a t i o n s h i p s t h a t are more e a s i l y remembered, and suggest many c o n j e c t u r e s t h a t may l a t e r be proved o r disproved.  K l i n e ( 1 9 . 6 4 , p. 457) suggested  t h a t once the i d e a s o f Non-  E u c l i d e a n Geometries have been grasped t h e r e i s v a l u e i n " s e e k i n g the analogues o f theorems t h a t h e l d i n E u c l i d e a n . Geometry". o f analogy  Moser (1968, pp. 374 - 6) p o i n t e d t o t h e v a l u e i n d e v e l o p i n g r e l a t i o n s h i p s between t h e graphs o f  v a r i o u s c o n i e s and a p p r o p r i a t e a b s o l u t e v a l u e  equations.  For example, he c o n s i d e r e d the r e l a t i o n s h i p between the graphs o f y = x  2  and y =  |x/2| + |y/2| = 1.  | x | ; and  (x/2)  2  +  (y/2.)  2  = 1  and  F i n a l l y , B a l k (1971, p. 135) s t a t e d  that: I t i s w e l l known t h a t h e u r i s t i c methods, and above a l l analogy and i n d u c t i o n , p l a y an i m p o r t a n t r o l e i n any s p e c i a l i z e d f i e l d . B a l k (1971, pp> 135 - 6) c o n t i n u e d by c o n s i d e r i n g examples o f where analogy conjectures.  c o u l d be o f v a l u e i n the f o r m u l a t i o n ' o f  F o r example, he suggested how t h e p r o p o s i t i o n s  t h a t i f the sum o f t h e d i g i t s o f a number i s d i v i s i b l e by 3 (or 9) then the number i s d i v i s i b l e by 3 ( o r 9 ) , c o u l d l e a d t o the analogous p r o p o s i t i o n t h a t i f the,-;,sum o f the d i g i t s i s d i v i s i b l e by 27 t h e number i s d i v i s i b l e by 27.  The f a c t  t h a t t h i s p a r t i c u l a r c o n j e c t u r e i s i n v a l i d does n o t minimize  t h e v a l u e o f analogy  as a m a t h e m a t i c a l  heuristic.  These examples p r o v i d e an h i s t o r i c a l p e r s p e c t i v e o f analogy  as w e l l as an i n d i c a t i o n • o f t h e e d u c a t i o n a l  s i g n i f i c a n c e o f the h e u r i s t i c . conclude  I t seems r e a s o n a b l e t o  from t h i s l i t e r a t u r e t h a t analogy  worth t e a c h i n g .  Is a heuristic  25  E x a m i n a t i o n o f Cases D i r e c t h i s t o r i c a l evidence of mathematicians a p p l y i n g e x a m i n a t i o n o f cases i s p r a c t i c a l l y n o n - e x i s t e n t . The o n l y d i r e c t r e f e r e n c e i s t o Nichomachus (Heath, 1931, p. 68) c o n s i d e r i n g the s e r i e s o f odd numbers: 1,  3,  5,  7,  9,  11,  Nichomachus wrote t h a t 1 was two, 3 + 5  was  13,  ...  a cube; the sum o f the next  a cube; the sum  of the next t h r e e was  a cube;  and so on; and he noted t h a t such i n f o r m a t i o n c o u l d be u t i l i z e d t o deduce the f o r m u l a f o r the sum o f the f i r s t  n  cubes. I n d i r e c t e v i d e n c e o f the u t i l i z a t i o n o f e x a m i n a t i o n of  cases i s found i n Gauss's d i a r y .  There i s a note t o the  e f f e c t t h a t every p o s i t i v e i n t e g e r i s the sum o f t h r e e t r i a n g u l a r numbers ( B e l l , 1937, p. 228). The l a c k o f d i r e c t h i s t o r i c a l evidence o f mathematicians  a p p l y i n g the h e u r i s t i c o f e x a m i n a t i o n o f  cases c o u l d w e l l be a t t r i b u t e d t o the n a t u r e o f p u b l i s h e d mathematics.  Mathematical p u b l i c a t i o n s c o n s i s t of the  final  p r o d u c t s , u s u a l l y i n the form o f p r o o f s o f a m a t h e m a t i c a l endeavor.  S i n c e e x a m i n a t i o n o f cases would be most  a p p r o p r i a t e i n the e x p l o r a t o r y phase o f d i s c o v e r y , r e f e r e n c e to  the h e u r i s t i c would not appear i n the f i n a l  In p a r t i c u l a r , the cases may  literature.  have been the f i r s t  l e a d i n g t o a p r o o f by m a t h e m a t i c a l i n d u c t i o n .  steps  Hence, o n l y  26  the f i n a l form o f p r o o f would be p u b l i s h e d and not t h e mathematician's  i n v e s t i g a t i o n of d i f f e r e n t cases.  Hardy's  comment (1940, p. 114) on p r o o f by complete enumeration o f cases may r e f l e c t a p r e v a i l i n g a t t i t u d e t o r e f e r e n c e s t o the u t i l i z a t i o n o f t h e h e u r i s t i c : ... b u t I t I s j u s t the ' p r o o f by enumeration o f cases' (and cases which do n o t , at bottom, d i f f e r at a l l p r o f o u n d l y ) which a r e a l m a t h e m a t i c i a n tends t o d e s p i s e . Although there i s l i t t l e h i s t o r i c a l evidence o f the use o f e x a m i n a t i o n o f c a s e s , e d u c a t o r s c o n s i d e r t h e h e u r i s t i c t o be o f s i g n i f i c a n t v a l u e . (1959)  Smith and Henderson  i n c l u d e d enumeration o f cases i n t h e i r  f i v e types  o f e v i d e n c e f o r p r o b a b l e i n f e r e n c e t h a t s h o u l d be encouraged  i n t h e secondary s c h o o l .  K i n s e l l a (1970, p. 244)  suggested t h a t " t h e d i s c o v e r y o f r e l a t i o n s u s u a l l y I n v o l v e s the s t r a t e g y o f s t u d y i n g s p e c i a l cases i n an i n d u c t i v e manner u n t i l a p a t t e r n I s p e r c e i v e d " .  Szabo (1967, p. 840)  gave t h e f o l l o w i n g i m p r e s s i o n c o n c e r n i n g t h e h e u r i s t i c : We a l s o f e e l t h a t s t u d e n t s can l e a r n how [ i t a l i c s i n o r i g i n a l ] t o s e a r c h f o r p a t t e r n s and w i l l be a b l e t o use t h i s knowledge I n s o l v i n g new problems. Brown and W a l t e r (1972, pp. 42 - 44)  demonstrated  how s p e c i f i c and g e n e r a l cases can be employed t o f o r m u l a t e c o n j e c t u r e s and q u e s t i o n s .  They asked s t u d e n t s who were  f a m i l i a r w i t h the f o l l o w i n g formula f o r generating Pythagorean a=m  Triples 2  -n  2  ;  b = 2mn  ;  c = m  2  + n  2  27  'to use the f o r m u l a t o pose q u e s t i o n s and c o n j e c t u r e s c o n c e r n i n g Pythagorean  formulate  triples.  The  questions  raised include: How are a, b and c a f f e c t e d by the evenness and oddness o f m and n? What happens t o a, b and c i f m and n are c o n s e c u t i v e ? . . . The e x a m i n a t i o n o f s p e c i f i c t r i p l e s produced  conjectures,  which were o f t e n s t a t e d i n the form o f q u e s t i o n s such as: Is c always odd?; I f a i s odd, then c = b + 1; For &' f i x e d , are b and o u n i q u e ? . . . S c o t t (1971, pp. 38 - 45) showed how  the  examination  of the number o f c e l l s formed by t h e d i a g o n a l s o f d i f f e r e n t polygons patterns.  c o u l d l e a d t o an i n v e s t i g a t i o n o f d i f f e r e n t F o r example, what i s the p a t t e r n c o n n e c t i n g the  number of s i d e s o f a polygon and the number o f  cells  formed by the d i a g o n a l s . Figure 3 Number o f C e l l s  4  cells  11  cells  . 28.  It  seems r e a s o n a b l e t o c o n c l u d e t h a t  educators  c o n s i d e r b o t h a n a l o g y and e x a m i n a t i o n o f 'cases t o be valuable techniques f o r problem  solving.  The  historical  e v i d e n c e r e l a t e d t o analo.gy s u g g e s t s t h a t t h i s plays a s i g n i f i c a n t r o l e i n mathematical  heuristic  discovery, while  e d u c a t o r s have l o n g promoted t h e t e a c h i n g o f b o t h  analogy  and s y s t e m a t i c : . e x a m i n a t i o n o f c a s e s . Related  Research  At t h e p r e s e n t t i m e t h e r e i s l i t t l e teaching students to apply mathematical  r e s e a r c h on  h e u r i s t i c s , per se.  H o w e v e r , t e a c h i n g s t u d e n t s t o t r a n s f e r l e a r n i n g t o new s i t u a t i o n s h a s b e e n i n v e s t i g a t e d by b o t h e d u c a t o r s psychologists.  and  The f o l l o w i n g r e v i e w o f t h e r e s e a r c h i s  d i v i d e d i n t o two s e c t i o n s .  The f i r s t  section  reviews  s t u d i e s t h a t h a v e c o m p a r e d d i s c o v e r y , as a t e a c h i n g t e c h n i q u e , w i t h o t h e r forms o f i n s t r u c t i o n .  The  second  s e c t i o n c o n s i d e r s s t u d i e s where t h e m a j o r o b j e c t i v e was t o develop  students' a b i l i t y  procedures  t o apply general problem  ( i n c l u d i n g h e u r i s t i c s ) t o new  solving  mathematical  situations. Discovery Versus Other I n s t r u c t i o n a l Techniques . The s t u d i e s r e p o r t e d h e r e a l l i n c l u d e an  investigation  of the r e l a t i v e e f f e c t i v e n e s s of the i n s t r u c t i o n a l i n t e a c h i n g s t u d e n t s t o t r a n s f e r l e a r n i n g t o new  techniques  29  situations.  T h i s review, i s l i m i t e d t o s t u d i e s i n v o l v i n g  mathematical  content..  Kersh  (1962)  used a programmed b o o k l e t t o t e a c h  90  h i g h s c h o o l s t u d e n t s two r u l e s f o r a d d i t i o n o f s e r i e s . Kersh  (1962.  p. 66) d e s c r i b e d t h e r u l e s as f o l l o w s :  1. Odd Numbers r u l e . The sum o f any s e r i e s o f c o n s e c u t i v e odd numbers b e g i n n i n g w i t h 1 i s e q u a l t o t h e square o f the number o f f i g u r e s I n t h e series. ( F o r example, 1, 3, 5, 7, i s such a s e r i e s ; t h e r e are f o u r numbers, so 4 x 4 i s 16, the sum.) 2. Constant D i f f e r e n c e s r u l e . The sum o f any s e r i e s o f numbers i n which t h e d i f f e r e n c e between t h e numbers i s c o n s t a n t i s e q u a l t o o n e - h a l f t h e product o f the number o f f i g u r e s and t h e sum o f t h e f i r s t and l a s t numbers. (For example,' 2, 3, 4, 5 i s such a s e r i e s ; 2 and 5 I s 7; t h e r e are f o u r f i g u r e s , so 4 x 7 i s 28; h a l f 28 i s l 4 w h i c h I s the sum.) a  The s t u d e n t s were t o l d t h e r u l e s and g i v e n p r a c t i c e i n t h e i r application.  'On c o m p l e t i n g t h e i n i t i a l b o o k l e t t h e s t u d e n t s  were randomly a s s i g n e d t o one o f t h r e e treatment  groups,  namely d i r e c t e d l e a r n i n g ( D L ) , guided d i s c o v e r y (GD) and r o t e l e a r n i n g ( R L ) . The DL group r e c e i v e d an a d d i t i o n a l programmed b o o k l e t i n which t h e r u l e s were e x p l a i n e d . GD group was taught by a q u e s t i o n i n g t e c h n i q u e  The  I n which t h e  s t u d e n t s were g u i d e d towards an e x p l a n a t i o n o f t h e r u l e s . The RL group was g i v e n no a d d i t i o n a l i n s t r u c t i o n and was i n c o r p o r a t e d I n t h e experiment as a c o n t r o l group.  After  the i n s t r u c t i o n a l p e r i o d each group o f 30 was d i v i d e d i n t o t h r e e subgroups (10 p e r g r o u p ) . each treatment  D i f f e r e n t subgroups from  were t e s t e d a f t e r 3 d a y s , 2 weeks and 6 weeks."  30 The t e s t i n g arrangement i s summarized i n f i g u r e 4. Figure 4 . T e s t i n g Arrangement f o r Treatment  Tested a f t e r  Group  Subgroups  Groups  DL  GD  RL  10  10  10  3 days  10  10  10  2 weeks  10  10  10  6 weeks  The c r i t e r i o n t e s t c o n s i s t e d o f two word problems t h a t c o u l d be s o l v e d by a p p l y i n g t h e r u l e s i n the i n s t r u c t i o n a l u n i t s . The number o f s u b j e c t s who used the a p p r o p r i a t e r u l e i n the s o l u t i o n was the i n d e x o f t r a n s f e r .  With t h i s index Kersh  found s i g n i f i c a n t d i f f e r e n c e s between t r e a t m e n t s and between t e s t p e r i o d s on one o f the two t e s t problems.  He  s t a t e d t h a t perhaps the most s t r i k i n g r e s u l t , and c e r t a i n l y u n a n t i c i p a t e d , was t h a t the RL group "was  found t o be  c o n s i s t e n t l y s u p e r i o r i n every r e s p e c t t o t h e o t h e r t r e a t m e n t groups" ( K e r s h , 1962, p. 68).  I n terms o f the  i n d e x o f t r a n s f e r the RL group performed b e s t , the DL group worst and t h e GD group i n between. f o r two r e a s o n s .  This study i s important  F i r s t , s i n c e the a d d i t i o n r u l e s i n the  experiment were new t o a l l s t u d e n t s the r e s u l t s  indicate  t h a t a l l t h r e e t r e a t m e n t s r e g i s t e r e d some degree o f t r a n s f e r ( a b i l i t y t o apply the r u l e s t o a new p r o b l e m ) .  Second,  31  the r e s u l t s i n d i c a t e t h a t I t i s not n e c e s s a r y to t e a c h bv_ discovery  t o obtain  Ray  transfer.  (1961) compared d i s c o v e r y  teaching with  direct  i n s t r u c t i o n on c r i t e r i a of i n i t i a l l e a r n i n g , r e t e n t i o n t r a n s f e r to new  but  r e l a t e d problems.  He  used  two  e x p e r i m e n t a l groups, d i r e c t i n s t r u c t i o n (DI) and discovery  (DD),  directed  w h i l e a t h i r d group, which r e c e i v e d  i n s t r u c t i o n was  used as a c o n t r o l  (C).  Ray  and  no  also divided  sample i n t o t h r e e a b i l i t y groups based on h i g h , medium low  intelligence.  The  sample of 135  randomly a s s i g n e d to one  his and  grade 9 boys were  o f the s i x e x p e r i m e n t a l t r e a t m e n t  x i n t e l l i g e n c e groups (18 per g r o u p ) , or t o a c o n t r o l x i n t e l l i g e n c e group (9 per g r o u p ) .  The  i n s t r u c t i o n a l period  consisted  o f 47 minutes o f i n s t r u c t i o n on the  structure  of a micrometer and  how  t o use  physical  i t . The  students  were taught, i n groups of n i n e , w i t h t h r e e s t u d e n t s b e i n g selected  from each o f the  t e s t e d one and  ability levels.  Students were  week a f t e r /the i n s t r u c t i o n a l p e r i o d was  f u r t h e r r e t e n t i o n and  s i x weeks l a t e r .  The  completed  t r a n s f e r t e s t s were a d m i n i s t e r e d  transfer test involved  items  "that  attempted t o measure a p p l i c a t i o n of knowledge g a i n e d to but  r e l a t e d s i t u a t i o n s " (Ray,  1961,  p. 275).  He  found a  s i g n i f i c a n t d i f f e r e n c e between t r e a t m e n t s on b o t h .the (one  week) and  d e l a y e d ( s i x week) t r a n s f e r t e s t s , w i t h  r e s u l t s s t a t i s t i c a l l y favouring  the DD  treatment.  new  The  initial the  32  r e s u l t s , a l s o showed a s i g n i f i c a n t Intelligence ability those  d i f f e r e n c e on  the  f a c t o r with, the r e s u l t s , f a v o u r i n g the  students.  high  Ray's c o n c l u s i o n s a r e d i f f e r e n t  i n the Kersh  (1962) e x p e r i m e n t ,  from  i n t h a t Ray's' r e s u l t s  i n d i c a t e t h a t the d i s c o v e r y group p e r f o r m e d b e t t e r than programmed g r o u p  (direct  instruction).  K e r s h , t h e d a t a shows t h a t b o t h evidenced  some d e g r e e o f  Gagne" and techniques  However,  experimental  f o r t e a c h i n g s t u d e n t s how  i n t r o d u c t o r y programme w h i c h was  t o sum  example (R&E).  (D), guided  number  discovery  (GD)  each s t u d e n t .  The  Using  to  and  three  rule  10 b o y s w e r e  (11 p e r  group).  f o u r p r o b l e m s on s u m m a t i o n  s e r i e s , w i t h each problem i n v o l v i n g a s e r i e s to the students.  an  f o r a l l three  T h i r t y - t h r e e g r a d e 9 and  t r a n s f e r t e s t comprised  series.  to correspond  randomly a s s i g n e d to the t h r e e treatments The  instructional,  c o n s i s t i n g of  identical  u n i t s , f o l l o w e d by m a t e r i a l s d e s i g n e d  and  groups  Brown (1961) e m p l o y e d t h r e e  discovery  like  transfer.  T h r e e programmed u n i t s were d e v e l o p e d ,  treatments,  the  t e s t was  t h a t was  of new  administered i n d i v i d u a l l y  t h r e e measures o f e f f e c t i v e n e s s o f  programmes, namely time r e q u i r e d t o s o l v e the number o f h i n t s r e q u i r e d and  a weighted  and h i n t s , Gagne and B r o w n c o n c l u d e d was  t h e most e f f e c t i v e  R&E  treatment  the  problems,  c o m b i n a t i o n .of  t h a t t h e GD  i n teaching students  to  time  treatment  to t r a n s f e r ,  l e a s t e f f e c t i v e w i t h the D treatment  the  i n between.  33  Eldredge  (1965a, 1965b).  r e p o r t e d s t u d i e s t h a t were  based on Gagne and Brown's experiment.. 1  I n the f i r s t study  E l d r e d g e r e p l i c a t e d the. Gagne and Brown e x p e r i m e n t , 1  employing two o f the t h r e e t r e a t m e n t s , namely GD and  R&E.  The r e p l i c a t i o n d i f f e r e d from the o r i g i n a l experiment  in  t h a t E l d r e d g e used a mixed sample, two c l a s s e s each w i t h 13 boys and 13 g i r l s , and d i f f e r e n t h i n t s on the t r a n s f e r t e s t , s i n c e the h i n t s used i n the o r i g i n a l experiment not a v a i l a b l e .  One  o t h e r c l a s s the R&E  were  c l a s s r e c e i v e d the GD t r e a t m e n t and the treatment.  E l d r e d g e found o n l y p a r t i a l  support f o r the r e s u l t s of Gagne and Brown.  For example, the  a n a l y s i s o f the f o u r t r a n s f e r problems showed a s i g n i f i c a n t d i f f e r e n c e f a v o u r i n g the GD t r e a t m e n t on one another problem  problem,  f a v o u r e d the GD treatment and the o t h e r  two problems f a v o u r e d the R&E  t r e a t m e n t , a l t h o u g h none o f  the l a s t t h r e e d i f f e r e n c e s were s t a t i s t i c a l l y  significant.  The d i f f e r e n c e s between the Gagne and Brown and  Eldredge  r e s u l t s c o u l d be due t o many f a c t o r s , such as d i f f e r e n t samples and d i f f e r e n t In  hints.  a second study E l d r e d g e employed a m o d i f i e d  v e r s i o n o f the Gagne and Brown m a t e r i a l s t h a t had been developed by D e l l a - P i a n a and E d l r e d g e  (1965a, 1 9 6 5 b ) .  The  m o d i f i e d m a t e r i a l s c o n s i s t e d o f two programmes, a g u i d e d d i s c o v e r y programme (GD) and a r u l e and example programme (R&E).  The main d i f f e r e n c e s between these two programmes  were i n the sequencing o f the frames, and the c o n d i t i o n s  34  under which the,programmes prompt the students, f o r the f o r summing the' s e r i e s .  rule  A t o t a l of 97 grade 9 s t u d e n t s  (43 b o y s j 54 g i r l s ) were used i n the e x p e r i m e n t . As i n the o r i g i n a l e x p e r i m e n t s . ^ Gagne and Brown ( 1 9 6 1 ) , E l d r e d g e (1965a)  ) the t r a n s f e r t a s k c o n s i s t e d o f f o u r  problems on the summation o f s e r i e s . p e r i o d consisted of three a t r a n s f e r t e s t , and f o u r weeks l a t e r .  unfamiliar  The i n s t r u c t i o n a l  s e s s i o n s , immediately followed  then the t r a n s f e r t e s t was  U s i n g i n t e l l i g e n c e and  by  readminlstered  time t a k e n t o  complete the programme as c o v a r i a t e s , the r e s u l t s i n d i c a t e d t h a t the GD group performed s i g n i f i c a n t l y b e t t e r than R&E  group..  as one  A two-way a n a l y s i s o f v a r i a n c e  f a c t o r and  ability  using  treatments  ( h i g h , l o w ) as the o t h e r i n d i c a t e d  t h a t the h i g h a b i l i t y group performed s i g n i f i c a n t l y t h a n the low  a b i l i t y group, and  followed  was  found.  Identical  from the d a t a a n a l y s i s of the  ( f o u r week) t r a n s f e r t e s t .  better  again a s i g n i f i c a n t  d i f f e r e n c e between t r e a t m e n t s was conclusions  the  delayed  However, i f I n i t i a l performance  used as a c o v a r i a t e i n the a n a l y s i s the  differences  d i s a p p e a r e d , l e n d i n g credence t o the h y p o t h e s i s t h a t  the  d i f f e r e n c e s on the d e l a y e d t r a n s f e r t e s t were jjiust a r e f l e c t i o n on the i n i t i a l d i f f e r e n c e s produced by  the  programmes. I n the s t u d i e s r e v i e w e d t o t h i s p o i n t the  transfer  t e s t s c o n s i s t e d o f Items t h a t r e q u i r e d the a p p l i c a t i o n of knowledge a c q u i r e d  i n the I n s t r u c t i o n a l p e r i o d to problems  35  t h a t were new. t o the s t u d e n t but d i r e c t l y r e l a t e d t o the i n s t r u c t i o n a l m a t e r i a l s . " Two  o v e r a l l c o n c l u s i o n s can be  drawn from these s t u d i e s t h a t r e l a t e , t o the p r e s e n t  study.  F i r s t , the r e s u l t s o f the s t u d i e s r e v i e w e d show t h a t i t i s p o s s i b l e t o t e a c h . s t u d e n t s t o apply what has been l e a r n e d i n a mathematical  c o n t e x t t o new,  m a t h e m a t i c a l problems.  but d i r e c t l y  related  Second, some o f the s t u d i e s (Gagne  and Brown ( 1 9 6 1 ) , E l d r e d g e  (1965a,1965b)) i n d i c a t e  that  u s i n g s e l f - i n s t r u c t i o n a l m a t e r i a l s can produce t r a n s f e r .  The  p r e s e n t study can be c o n s i d e r e d an e x t e n s i o n of these experiments  i n t h a t i t r e q u i r e s s t u d e n t s t o apply what has  been l e a r n e d , the h e u r i s t i c s , t o t r a n s f e r t a s k s where the mathematical  content i s u n r e l a t e d t o t h a t used i n the  instructional materials.  Furthermore,  the p r e s e n t study  employs s e l f - i n s t r u c t i o n a l b o o k l e t s and the  evidence  i n d i c a t e s t h a t t h i s mode o f i n s t r u c t i o n can l e a d t o successful transfer. The s i n c e one  f o l l o w i n g s t u d i e s are o f p a r t i c u l a r  significance  (Worthen ( 1 9 6 5 ) ) i n c l u d e d an i n v e s t i g a t i o n o f  t r a n s f e r t o u n r e l a t e d problems,  and i n the o t h e r K e r s h  (1964)  s e a r c h e d f o r evidence o f h e u r i s t i c s i n the s t u d e n t s s o l u t i o n s t o the  problems.  Worthen (1965) conducted  an experiment  involving  two  e x p e r i m e n t a l t r e a t m e n t s , d i s c o v e r y (D) and e x p o s i t o r y ( E ) . The i n s t r u c t i o n a l content c o n s i s t e d o f a development o f  36  i n t e g e r s j a d d i t i o n and m u l t i p l i c a t i o n of i n t e g e r s , d i s t r i b u t i v e p r i n c i p l e and i n t e g r a l powers.  The  school  o f 432  with  students,  2 grade 5 c l a s s e s i n 8 s c h o o l s ,  with  s t u d e n t s (3 grade 6 c l a s s e s ) i n a n i n t h  used f o r c o n t r o l p u r p o s e s .  Eight  i n v o l v e d i n the s t u d y , each t e a c h i n g treatment D and  notation  sample c o n s i s t e d  14 grade 6 c l a s s e s and an a d d i t i o n a l 106  exponential  the  two  the o t h e r by t r e a t m e n t E.  t e a c h e r s were c l a s s e s , one  by  At the end  of  the  s i x week i n s t r u c t i o n a l p e r i o d ' a v a r i e t y o f c r i t e r i o n t e s t s were a d m i n i s t e r e d , i n c l u d i n g a t t i t u d e t e s t s , c o n t e n t t e s t s and  transfer tests.  The  t h r e e t e s t s germane t o the  present  study are the concept t r a n s f e r t e s t ( c T T ) , number sequences and  s e r i e s t e s t (nsTT) and  principle way  (apTT).  a t e s t i n v o l v i n g the  On the cTT  associative  d a t a Worthen employed a  a n a l y s i s o f c o v a r i a n c e w i t h main f a c t o r s t e a c h e r s  t r e a t m e n t s , and  covariates  solving, arithmetic  i n t e l l i g e n c e , arithmetic  computation and  the s c o r e on  problem  Worthen  concluded t h a t the D group performed better..than . (p He  <.08)  a l s o found s i g n i f i c a n t t e a c h e r e f f e c t s  interaction effects but  and  the  achievement t e s t on the i n s t r u c t i o n a l m a t e r i a l s .  the E group.  two-  (p <.001).  U s i n g the same  and  covariates  employing a one-way ANACOVA w i t h the D, E and  C groups  he found t h a t the D group performed s i g n i f i c a n t l y  better  t h a n the E group (p <.05), the D group performed s i g n i f i c a n t l y b e t t e r than the C group (p < . 0 1 ) was  ;  no s i g n i f i c a n t d i f f e r e n c e between the C and  and  there  E groups.  37  The  cTT i n c l u d e d new problems t h a t p e r t a i n e d  to the  i n s t r u c t i o n a l m a t e r i a l s and as such t h e t e s t items were extensions  of the m a t e r i a l s .  consistent w i t h the previous  Worthen's r e s u l t s , a r e s t u d i e s i n t h a t they show  e v i d e n c e o f t r a n s f e r to. new b u t d i r e c t l y r e l a t e d problems. The  nsTT and apTT's- c o n s i s t e d o f problems on-content t h a t •;  was  unrelated to the i n s t r u c t i o n a l m a t e r i a l s .  consisted o f questions  The nsTT  on number sequences and s e r i e s i n  which t h e s t u d e n t s had t o f i n d t h e p a t t e r n and r u l e .  The  two-way a n a l y s i s a g a i n showed s i g n i f i c a n t t e a c h e r and interaction effects.  This a n a l y s i s a l s o i n d i c a t e d t h a t t h e  D group performed 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 E group (p <.05).  I t appears t h a t Worthen used t h e words ' t r a n s f e r  of h e u r i s t i c s ' t o r e f e r t o t h e t r a n s f e r o f a b i l i t y t o make d i s c o v e r i e s i n new problem s i t u a t i o n s , i n t h i s case t o search  f o r and f i n d a p a t t e r n .  I n t h e apTT t h e problems  i n v o l v e d t h e a s s o c i a t i v e p r i n c i p l e , a p r i n c i p l e not taught during the i n s t r u c t i o n a l p e r i o d .  The s t u d e n t s were f a c e d •  w i t h problems i n which d i s c o v e r i n g t h i s p r i n c i p l e would provide  a shortcut to the s o l u t i o n .  F o r example, i n  f i n d i n g t h e sum o f 975 + (25 + 983) t h e a s s o c i a t i v e p r i n c i p l e would be o f g r e a t v a l u e .  Using a grading  scheme  i n which s t u d e n t s were g i v e n one p o i n t f o r a c o r r e c t answer and  an a d d i t i o n a l f o u r p o i n t s i f t h e i r work showed e v i d e n c e  o f t h e p r i n c i p l e , t h e D group performed b e t t e r than t h e E group (p <.025)-  significantly  Worthen's r e s u l t s  38  i n d i c a t e t h a t the d i s c o v e r y group l e a r n e d how to. search, f o r a p a t t e r n o r r u l e s i g n i f i c a n t l y b e t t e r than t h e e x p o s i t o r y group.  Thus, t h i s experiment p r o v i d e s • e v i d e n c e  t h a t an  i n s t r u c t i o n a l method, l e a r n i n g by d i s c o v e r y , r e s u l t e d i n t r a n s f e r o f h e u r i s t i c s t o new problems t h a t were u n r e l a t e d to the i n s t r u c t i o n a l m a t e r i a l s . K e r s h (1964) undertook a study t o compare h i g h l y d i r e c t e d (programmed) l e a r n i n g w i t h n o n - d i r e c t e d learning. primary  (discovery)  Three i n s t r u c t i o n a l u n i t s were designed  with the  o b j e c t i v e o o f teaching the d i s t r i b u t i v e p r i n c i p l e t o  f i f t h graders.  I n t h e f r e e d i s c o v e r y u n i t (D) each  student  was encouraged t o d i s c o v e r the p r i n c i p l e i n d e p e n d e n t l y  and  t h e n show t h e e x p e r i m e n t e r t h a t he had l e a r n e d t h e p r i n c i p l e . The e x p e r i m e n t e r worked w i t h t h e s t u d e n t s  i n d i v i d u a l l y , not  a l l o w i n g v e r b a l i z a t i o n of the p r i n c i p l e u n t i l a l l the students  demonstrated by example t h a t they had l e a r n e d t h e  distributive principle. gave guidance designed principle.  The programmed d i s c o v e r y u n i t (PD) t o a i d students  t o s e a r c h f o r the  V e r b a l i z e d a p r o v a l by the e x p e r i m e n t e r was used  to r e i n f o r c e the students.  A f i n a l programme, r e f e r r e d t o  as programmed guidance (PG) was developed t o t e a c h t h e same p r i n c i p l e , but w i t h o u t employing the d i s c o v e r y method. PG treatment  d i d not encourage t h e s e a r c h i n g  f o s t e r e d i n t h e D and PD programmes.  behaviours  A l l students  i n s t r u c t i o n u n t i l . t h e y met a p r e d e t e r m i n e d  The  received  criterion,  39  namely that-.they were i n i t i a l l y r e q u i r e d t o c o r r e c t l y answer f i v e o f s i x d i s t r i b u t i v e p r i n c i p l e q u e s t i o n s .  If  they f a i l e d they were r e t a u g h t , and the f i n a l r e s u l t  was  t h a t f o u r s t u d e n t s were dropped from the study f o r f a i l u r e to meet the c r i t e r i o n . l e a r n i n g , which was  The p o s t t e s t s i n c l u d e d a t e s t o f  a d m i n i s t e r e d w i t h i n 24 hours o f a  s t u d e n t c o m p l e t i n g the i n s t r u c t i o n a l m a t e r i a l s . was  new  This t e s t  a d m i n i s t e r e d i n d i v i d u a l l y t o each s t u d e n t , and  the  student's were r e q u i r e d t o " t h i n k out l o u d " , o r i n d i c a t e t h e i r t h i n k i n g by s c r a t c h work.  Kersh I d e n t i f i e d  problem  s o l v i n g b e h a v i o u r s such as 'search f o r p a t t e r n s ' , ' c h e c k i n g f o r e x c e p t i o n s t o p a t t e r n s ' , and  ' c h e c k i n g t o see i f  statements were t r u e o r f a l s e ' .  I n terms o f new  l e a r n i n g he  found s i g n i f i c a n t d i f f e r e n c e s between the t r e a t m e n t s 'search f o r p a t t e r n s ' . o f s t u d e n t s who  on  The D group had the g r e a t e s t number  evidenced t h i s behaviour  (27 of 3 0 ) ,  f o l l o w e d by the PG group (21 o f 30) and the l e a s t number i n the PD group (17 o f 30).  No s i g n i f i c a n t d i f f e r e n c e s were  found on the o t h e r c r i t e r i a o f new  learning.  For example,  on ' c h e c k i n g f o r e x c e p t i o n s t o p a t t e r n s ' 11 s t u d e n t s i n the D group used the t e c h n i q u e , 13 i n the PG group and, 12 i n the PD group.  Kersh's  study shows t h a t the D treatment was  more  e f f e c t i v e than the o t h e r s i n t e a c h i n g s t u d e n t s t o apply the problem  s o l v i n g h e u r i s t i c of searching f o r p a t t e r n s . To summarize, two major c o n c l u s i o n s can be drawn from  the s t u d i e s r e p o r t e d i n t h i s s e c t i o n t h a t have r e l e v a n c e f o r  40 the p r e s e n t s t u d y .  The s t u d i e s , i n d i c a t e t h a t (1) i t i s  p o s s i b l e to. t e a c h s t u d e n t s to. t r a n s f e r knowledge t o new but . d i r e c t l y r e l a t e d problems,  and (2) t h i s t r a n s f e r can be  f a c i l i t a t e d by s e l f - i n s t r u c t i o n a l m a t e r i a l s . Worthen's study to approach  (1965)  Furthermore,  i n d i c a t e s t h a t s t u d e n t s can be taught  problems t h a t are not d i r e c t l y r e l a t e d t o t h e  i n s t r u c t i o n a l ' m a t e r i a l s , a result that i s p a r t i c u l a r l y s i g n i f i c a n t i n l i g h t o f the f a c t t h a t t h e t r a n s f e r t e s t s t o be employed i n t h i s study w i l l be t o t a l l y u n r e l a t e d t o t h e instructional materials.  Kersh's  study a l s o  produced  evidence o f the t r a n s f e r o f the h e u r i s t i c o f s e a r c h i n g f o r a p a t t e r n , which i s a component o f e x a m i n a t i o n o f c a s e s . Teaching H e u r i s t i c Schaaf  Procedures  (1954)  d e s i g n e d a course which had as its;-.  major o b j e c t i v e the development o f s t u d e n t s ' a b i l i t y t o generalize.  A group o f n i n t h grade s t u d e n t s a t a  u n i v e r s i t y s c h o o l a c t e d as t h e e x p e r i m e n t a l group.  Data  from 127 n i n t h grade s t u d e n t s ( i n s i x c l a s s e s i n the P u b l i c S c h o o l System) were employed f o r c o n t r o l purposes. e x p e r i m e n t a l p e r i o d l a s t e d one y e a r .  The  The content s t u d i e d by  the e x p e r i m e n t a l group i n c l u d e d numbers and o p e r a t i o n s ; graphs and f o r m u l a s ; e q u a t i o n s and problem  solving; proportion  and i n d i r e c t measurement; and s t a t i s t i c s .  The i n s t r u c t i o n a l  procedure  f o r the e x p e r i m e n t a l course was d e s i g n e d t o  encourage s t u d e n t s t o extend p r e v i o u s  mathematical  41  l e a r n i n g , and to. d i s c o v e r new present experiences.  p r i n c i p l e s , from p a s t  and  Throughout the' .course, emphasis  was  p l a c e d on s t u d e n t d i s c o v e r y , w i t h the t e a c h e r a c t i n g as a guide In h e l p i n g the s t u d e n t s t o d i s c o v e r g e n e r a l i z a t i o n s . The methods o f g e n e r a l i z a t i o n t h a t were taught i n c l u d e d s i m p l e enumeration,  analogy, e x t e n s i o n o f form ( e x t r a p o l a t i o n  and i n t e r p o l a t i o n ) , d e d u c t i o n , v a r i a t i o n and i n v e r s e deduction.  A p r e t e s t o f the a b i l i t y t o g e n e r a l i z e i n d i c a t e d  t h a t the e x p e r i m e n t a l group was  s i g n i f i c a n t l y b e t t e r at  g e n e r a l i z i n g than the c o n t r o l group.  At the end of the  i n s t r u c t i o n a l y e a r the g e n e r a l i z a t i o n t e s t was  readministered.  A d d i t i o n a l I n f o r m a t i o n on the e x p e r i m e n t a l group  was  a v a i l a b l e from sources such as s t u d e n t n o t e b o o k s , own  Schaaf's  notes on the c l a s s r o o m meetings and o b s e r v e r r e p o r t s  (an o b s e r v e r v i s i t e d the c l a s s r o o m p e r i o d i c a l l y the e x p e r i m e n t a l p e r i o d ) .  Schaaf's  throughout  results indicate that  on some measures o f the a b i l i t y t o g e n e r a l i z e the e x p e r i m e n t a l group made s i g n i f i c a n t l y g r e a t e r g a i n s the c o n t r o l group.  than  For example, the e x p e r i m e n t a l group's  g a i n i n g e n e r a l i z i n g by e x t r a p o l a t i o n and i n s i t u a t i o n s were such g e n e r a l i z a t i o n was  interpolation justified  was  s i g n i f i c a n t l y g r e a t e r than the c o n t r o l group's g a i n .  On  o t h e r measures he found s i g n i f i c a n t s g a i n s by both  groups,  but no s i g n i f i c a n t d i f f e r e n c e between the g a i n . s c o r e s . suggested s c o r e s may  He  t h a t the l a c k o f s i g n i f i c a n c e between the g a i n have been due t o the e x p e r i m e n t a l group h a v i n g  42  an i n i t i a l l y  higher  s c o r e , g i v i n g them a more  restricted  r a n g e f o r i m p r o v e m e n t , o r t h a t t h e t e s t was n o t s e n s i t i v e enough i n m e a s u r i n g t h e g a i n . that exposure t o h e u r i s t i c s and  inverse deduction  ability  Schaaf's r e s u l t s  s u c h as a n a l o g y , , e n u m e r a t i o n  significantly  increased  t o g e n e r a l i z e t o new s i t u a t i o n s .  c e r t a i n measures t h i s  Indicate  students'  F u r t h e r m o r e , on  g a i n was s i g n i f i c a n t l y  greater  than  t h a t o f a c o n t r o l group. The  following four studies  ( 1 9 7 D , Lucas  (1972), G o l d b e r g  (Ashton  (1962), L i b e s k i n d  (1974)) a l l b a s e d  i n s t r u c t i o n a l p r o c e d u r e s on P o l y a ' s  heuristic  their  process.  They e m p l o y e d h i s q u e s t i o n s  and q u e s t i o n i n g t e c h n i q u e i n  the  ( s e e P o l y a , How t o S o l v e I t ,  instructional materials  1957) •  A s h t o n (1962) s e l e c t e d t e n g r a d e n i n e in five different  schools, f o rher study.  Each  contained  two c l a s s e s , b o t h b e i n g  teacher.  I n e a c h p a i r o f c l a s s e s one was an  ( h e u r i s t i c ) group and t h e o t h e r The  algebra classes, school  i n s t r u c t e d by t h e same experimental  a c o n t r o l (textbook)  i n s t r u c t i o n a l p e r i o d l a s t e d t e n weeks, d u r i n g  time whenever t h e h e u r i s t i c they  data?  which  g r o u p was f a c e d w i t h a p r o b l e m  were t o a s k t h e m s e l v e s , o r w e r e a s k e d q u e s t i o n s  Polya's  group.  w o r k , s u c h as "What i s t h e unknown? What a r e t h e c o n d i t i o n s ? " /  from  What a r e t h e  An e x a m i n a t i o n  o f Ashton's  procedure i n d i c a t e s that her i n s t r u c t i o n a l sessions  involved  43  more than j u s t P o l y a ' s q u e s t i o n s .  F o r example, s t u d e n t s i n  the h e u r i s t i c group were a l l o w e d t o w r i t e t h e i r own q u e s t i o n s t o c l a r i f y concepts.  F u r t h e r m o r e , b e f o r e anyone i n a  h e u r i s t i c c l a s s was p e r m i t t e d t o answer a q u e s t i o n a t l e a s t two o f the s t u d e n t s had t o r e s t a t e t h e q u e s t i o n , so t h a t by t h i s " t r a n s l a t i o n " the s t u d e n t s f a m i l i a r i z e d themselves w i t h a l g e b r a as a language.  D u r i n g the e x p e r i m e n t a l  the c o n t r o l c l a s s e s c o n t i n u e d w i t h t h e i r normal programme.  period classroom  At t h e end o f the t e n week i n s t r u c t i o n a l p e r i o d  a problem s o l v i n g t e s t c o n s i s t i n g o f word problems was administered.  Ashton found s i g n i f i c a n t r e s u l t s f a v o u r i n g  the h e u r i s t i c group.  Thus, h e r study p r o v i d e s evidence  that  the t e a c h i n g o f h e u r i s t i c p r o c e d u r e s based on the work o f P o l y a r e s u l t s i n s i g n i f i c a n t l y b e t t e r problem s o l v i n g performance. L i b e s k l n d (1971) employed P o l y a ' s h e u r i s t i c  approach  as t h e b a s i s f o r d e v e l o p i n g a h i g h s c h o o l u n i t on number theory.  I n o r d e r t o keep the p r e r e q u i s i t e s r e q u i r e d by the  s t u d e n t s t o a minimum t h e development and p r o o f o f theorems was r e s t r i c t e d t o the domain o f whole numbers.  The u n i t  was taught by L i b e s k l n d t o t e n s t u d e n t s , t h r e e o f whom had completed grade 9,  another  t h r e e had completed grade 10,  and t h e l a s t f o u r had completed grade 11.  The i n s t r u c t i o n a l  p e r i o d c o n s i s t e d o f 25 s e s s i o n s each o f 50 m i n u t e s , w i t h each s e s s i o n b e i n g immediately  f o l l o w e d by a 30 minute  44  study p e r i o d .  As p a r t of h i s e v a l u a t i o n o f the u n i t he  i n v e s t i g a t e d the q u e s t i o n o f whether s t u d e n t s some o f t h e p r o o f t e c h n i q u e s problems.  could  apply  developed i n the u n i t t o new  He c o n c l u d e d t h a t the s t u d e n t s were a b l e t o  t r a n s f e r t h e i r knowledge t o new problems.  This  conclusion  was not based on any s t a t i s t i c a l t e s t i n g , but he used as the j u s t i f i c a t i o n the f a c t t h a t t h e s t u d e n t s  were a b l e t o  s o l v e t e s t problems g i v e n t h r o u g h o u t . t h e e x p e r i m e n t a l  period.  For example, L i b e s k i n d n o t e d t h a t t h e s t u d e n t s were a b l e t o prove t h a t i f 2b d i v i d e s a and 5 d i v i d e s b then 10 d i v i d e s a (2b|a and 5|b t h e n 10|a).the s t u d e n t s  This was a new problem i n t h a t  had not seen i t b e f o r e , a l t h o u g h  the i n s t r u c t i o n a l  u n i t had i n c l u d e d p r o o f s o f g e n e r a l r e s u l t s such as i f a d i v i d e s b and b d i v i d e s c then a d i v i d e s c (a|b and b|c t h e n a|c).  As f u r t h e r e v i d e n c e o f t r a n s f e r he c i t e d the f a c t t h a t  the s t u d e n t s  were a b l e t o s o l v e a new problem i n v o l v i n g a  p r o o f t h a t t h e r e e x i s t s a prime g r e a t e r than 10 . had  The u n i t  i n c l u d e d a p r o o f o f the g e n e r a l r e s u l t t h a t t h e r e e x i s t s  no g r e a t e s t prime.  L i b e s k i n d ' s examples o f t r a n s f e r  'involved the a p p l i c a t i o n o f p r o o f t e c h n i q u e s  t o new problems  t h a t were c l o s e l y r e l a t e d t o t h e i n s t r u c t i o n a l m a t e r i a l s . Thus h i s r e s u l t s i n d i c a t e t h a t t r a i n i n g i n h e u r i s t i c s w i l l produce t r a n s f e r t o r e l a t e d problems.  Clearly this  r  c o n c l u s i o n i s l i m i t e d by t h e s m a l l sample. Goldberg's study theory.  (1974) a l s o used m a t e r i a l s on number  Two s e t s o f s e l f - i n s t r u c t i o n a l b o o k l e t s were  45  written.  Both s e t s c o n s i s t e d o f seven b o o k l e t s on  number t h e o r y , one s e t w i t h emphasis on h e u r i s t i c s and t h e other set without i n s t r u c t i o n i n h e u r i s t i c s . c o n t a i n i n g 238 undergraduates  Nine c l a s s e s  met f o r 12 s e s s i o n s (75  minutes p e r s e s s i o n ) w i t h two s e s s i o n s p e r week over a s i x week I n s t r u c t i o n a l p e r i o d .  The s t u d e n t s worked on  the b o o k l e t s f o r seven o f these s e s s i o n s .  The n i n e c l a s s e s  were randomly a s s i g n e d t o one o f t h r e e t r e a t m e n t s . r e i n f o r c e d h e u r i s t i c s treatment heuristics booklets.  (RH) t h e c l a s s e s used t h e  Furthermore,  d i s c u s s e d , o r new concepts  In the  when homework was  c o n s i d e r e d i n t h e s e s s i o n s when  b o o k l e t s were n o t b e i n g employed, t h e i n s t r u c t o r the h e u r i s t i c s as o f t e n as p o s s i b l e ,  utilized  In the non-reinforced  h e u r i s t i c s • t r e a t m e n t (H) t h e h e u r i s t i c s b o o k l e t s were used but t h e h e u r i s t i c s n o t r e i n f o r c e d i n c l a s s . h e u r i s t i c treatment  I n t h e non-  (C) t h e s t u d e n t s were g i v e n t h e non-  h e u r i s t i c b o o k l e t s and t h e i n s t r u c t o r d i d n o t employ t h e heuristics i n class.  The t e s t s t h a t were a d m i n i s t e r e d a t  the end o f t h e e x p e r i m e n t a l p e r i o d i n c l u d e d a t e s t o f u n d e r s t a n d i n g o f number t h e o r y  concepts  (concepts I ) and  a t e s t t o measure t h e a b i l i t y t o c o n s t r u c t p r o o f s ( p r o o f s I ) . F i v e weeks l a t e r d e l a y e d p o s t t e s t s o f concepts and p r o o f s ( p r o o f s I I ) were g i v e n .  (concepts I I ) .  P a r t o f these  delayed  t e s t s were p a r a l l e l t o t h e o r i g i n a l t e s t s , and p a r t d e a l t w i t h m a t e r i a l covered i n t h e i n t e r v e n i n g p e r i o d .  There  was 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 RH and H groups  46  a l t h o u g h t h e t r e n d i n d i c a t e d t h a t t h e RH b e t t e r than the H group. d a t a on m a t h e m a t i c a l thirds  group  performed  B a s e d on S t a n f o r d A c h i e v e m e n t  a c h i e v e m e n t t h e s a m p l e was  ( h i g h , medium, l o w )  and  divided  analysis indicated significant  high a b i l i t y  s t u d e n t s on t h e c o n c e p t s  f a v o u r e d s t u d e n t s i n t h e RH favoured t h i s  g r o u p on  r e s u l t s were not s t u d e n t s who  b e t t e r on u n d e r s t a n d i n g  The  Furthermore.,  concepts  A coding system  c h e c k l i s t was was  and  high  tended  t o p t h i r d on t h e p r o o f s p o s t t e s t . i n d i c a t i o n t h a t t h e s t u d e n t s who  to construct  r e c e i v e d the  The  seem t o i n d i c a t e treatment  b e n e f i t e d most f r o m t h e  The Goldberg  s t u d i e s by  Ashton  an  treatment were i n  additional information  O v e r a l l , Goldberg's  that high a b i l i t y  (1967)  and  results give  r e c e i v e d t h e RH  p a p e r g i v e s no  of h e u r i s t i c s .  H  the students i n the  u s e d h e u r i s t i c s more o f t e n t h a n t h e s t u d e n t s who  on t h e use  the  significantly  s i m i l a r to K i l p a t r i c k ' s  a p p l i e d t o t h e p r o o f s w r i t t e n by  The  also  ability  d e v i s e d t o i d e n t i f y h e u r i s t i c usage  the c o n t r o l group.  results  results  performed  p r o o f s b e t t e r t h a n t h e i r c o u n t e r p a r t s who treatment.  These  constructing proofs, although  r e c e i v e d the C treatment  A  differences for  tests.  treatment.  significant.  into  t h e s c o r e s on t h e p o s t t e s t s  w e r e d i v i d e d i n a s i m i l a r manner ( h i g h , medium, l o w ) . Chi-square  Test  results  students i n the  RH  instruction.  ( 1 9 6 2 ) , L i b e s k i n d (1971)  and  (1974) a l l b a s e d t h e i r i n s t r u c t i o n a l m a t e r i a l s on  Polya's h e u r i s t i c process.  These s t u d i e s i n d i c a t e  that  47  i n s t r u c t i o n in. h e u r i s t i c s , tends t o improve problem s o l v i n g performance when measured by t e s t s of. the a b i l i t y , t o a p p l y what had been l e a r n e d ( c o n t e n t ) , t o new b u t r e l a t e d problems. The next study d i f f e r s from these t h r e e i n t h a t Lucas 1974)  (1972,  s p e c i f i c a l l y i n v e s t i g a t e d whether s t u d e n t s c o u l d apply  h e u r i s t i c s t o new problems. Lucas (1972, 1974) i n v e s t i g a t e d the t e a c h i n g o f h e u r i s t i c s to u n i v e r s i t y students.  He taught c a l c u l u s t o two  c l a s s e s j one c l a s s (H) r e c e i v e d t h e e x p e r i m e n t a l programme c o n s i s t i n g o f an i n s t r u c t i o n a l approach t h a t exposed s t u d e n t s t o , and emphasized h e u r i s t i c s , w h i l e t h e o t h e r c l a s s (C) r e c e i v e d i n s t r u c t i o n i n the same content b u t w i t h no exposure to t h e h e u r i s t i c s .  The sample s e l e c t e d f o r study c o n s i s t e d  of 30 s t u d e n t s , 17 i n t h e H group and 13 i n t h e C group. The i n s t r u c t i o n a l p e r i o d l a s t e d e i g h t weeks.  At t h e end  of t h e i n s t r u c t i o n a l p e r i o d a problem s o l v i n g t e s t  consisting  of word problems was a d m i n i s t e r e d i n d i v i d u a l l y t o each student.  The s t u d e n t s  'thought  out l o u d ' w h i l e s o l v i n g  the p r o b l e m s , and Lucas made notes as w e l l as r e c o r d i n g the s e s s i o n s .  I t s h o u l d be n o t e d t h a t these problems  c o u l d be s o l v e d by c a l c u l u s techniques,, a l t h o u g h they were amenable t o s o l u t i o n by o t h e r methods, such as g r a p h i n g appropriate equations.  I n a n a l y s i n g t h e d a t a Lucas was  l o o k i n g f o r evidence o f a v a r i e t y o f h e u r i s t i c i n the students' s o l u t i o n s .  strategies  These s t r a t e g i e s i n c l u d e d  u s i n g mnemonic n o t a t i o n , u s i n g diagrams, c h e c k i n g t o see I f  48  the r e s u l t s were r e a s o n a b l e , and u s i n g analogy. h e u r i s t i c s were d e r i v e d from P o l y a ' s work.  These  He found  s i g n i f i c a n t d i f f e r e n c e s f a v o u r i n g the h e u r i s t i c s t r e a t m e n t on v a r i a b l e s such as mnemonic n o t a t i o n , p l a n n i n g t h e problem and use o f the r e s u l t s o f r e l a t e d problems.  Certain,  v a r i a b l e s are o f p a r t i c u l a r s i g n i f i c a n c e f o r the p r e s e n t d i s s e r t a t i o n and are d i s c u s s e d i n d e t a i l .  Uses method o f  a r e l a t e d problem r e f e r r e d t o t h e s t r a t e g y by which a r e l a t e d problem c o u l d be o f v a l u e i n "... p r o v i d i n g a method which e i t h e r may be a p p l i e d d i r e c t l y t o t h e g i v e n problem o r may s u p p l y a cue f o r d e v i s i n g an analogous p l a n of attack".  ( L u c a s , 1972, p.137)  I f a student s p e c i f i e d  a r e s u l t o f a n o t h e r problem then used the r e s u l t i n s o l v i n g the new problem he r e c e i v e d c r e d i t f o r e v i d e n c e o f t h e h e u r i s t i c , uses r e s u l t o f r e l a t e d problem.  Reasoning by  analogy i n v o l v e d the problem s o l v e r s e a r c h i n g f o r - s i m i l a r i t i e s between the g i v e n problem and a r e l a t e d s i t u a t i o n .  He  found s i g n i f i c a n t d i f f e r e n c e s between the H and C groups on b o t h uses methods o f r e l a t e d problems and uses r e s u l t s o f r e l a t e d problems, b o t h d i f f e r e n c e s f a v o u r i n g the H„group. Of the 30 s u b j e c t s t e s t e d t h e r e were o n l y f o u r who  exhibited  evidence o f r e a s o n i n g by analogy and a l l f o u r s u b j e c t s were i n the H group.  T h i s study i s i m p o r t a n t f o r two r e a s o n s .  F i r s t , Lucas l o o k e d f o r e v i d e n c e o f p a r t i c u l a r h e u r i s t i c s i n students' s o l u t i o n s .  Second, the c o n c l u s i o n i n v o l v i n g  the use o f r e l a t e d problems i n s o l v i n g new problems suggests  49  the p o s s i b i l i t y o f t e a c h i n g t h e h e u r i s t i c o f analogy as d e f i n e d i n t h i s study  (see page 14 ). Lucas n o t e d  l i m i t a t i o n s such as s m a l l sample s i z e and t h e f a c t t h a t students  were not randomly a s s i g n e d t o t r e a t m e n t s am any  attempt t o g e n e r a l i z e h i s r e s u l t s . do l e n d s u p p o r t t o t h e h y p o t h e s i s  However, h i s f i n d i n g s t h a t h e u r i s t i c s can be  taught. These l a s t f o u r s t u d i e s (Ashton ( 1 9 6 2 ) , L i b e s k i n d (197D,  Goldberg (1974) and Lucas (1972, 1 9 7 4 ) ) i n d i c a t e  t h a t t h e t e a c h i n g o f h e u r i s t i c s tends t o improve problem s o l v i n g performance.  F u r t h e r m o r e , Lucas's r e s u l t s , w h i l e  l i m i t e d , i n d i c a t e the p o s s i b i l i t y o f s t u d e n t s t r a n s f e r r i n g h e u r i s t i c s , not j u s t core m a t e r i a l t o new problems. The  remaining  s t u d i e s r e v i e w e d here have a l l employed  a v a r i e t y o f s t r a t e g i e s t o promote m a t h e m a t i c a l t r a n s f e r . P r i c e (1965) conducted a study  o f the e f f e c t o f  d i s c o v e r y on achievement and c r i t i c a l t h i n k i n g o f t e n t h grade s t u d e n t s .  He u t i l i z e d t h r e e groups:  the experimental  group (E) c o n t a i n i n g 22 s t u d e n t s ; t h e e x p e r i m e n t a l - t r a n s f e r group (T) c o n t a i n i n g 23 s t u d e n t s ; and a c o n t r o l group (C) of 18 s t u d e n t s .  The i n s t r u c t i o n a l p e r i o d l a s t e d 15 weeks,  d u r i n g which time t h e e x p e r i m e n t a l  group r e c e i v e d  mathematics i n s t r u c t i o n on t o p i c s t h a t i n c l u d e d s e t t h e o r y and p r o p e r t i e s o f number systems.  The programme f o r t h e E  group c o n s i s t e d o f q u e s t i o n and answer s e s s i o n s between t h e  50  t e a c h e r and t h e s t u d e n t s (a d i s c o v e r y o r i e n t e d approach). The T group, i n a d d i t i o n t o t h i s t e a c h i n g were g i v e n a s i n g l e study guide on i n f e r e n c e b e f o r e t h e l e s s o n s on e x t e n s i o n o f p a t t e r n .  (discussions)/  T h i s study m a t e r i a l was o f an  h i s t o r i c a l n a t u r e , and was f o l l o w e d by s p e c i f i c q u e s t i o n s t h a t the s t u d e n t was t o answer.  Price  (1965,  p. 72) s a i d  of t h i s p e r i o d of i n s t r u c t i o n : D u r i n g t h e c l a s s r o o m d i s c u s s i o n which took p l a c e r e g a r d i n g i n f e r e n c e t h e r e were many i n t e r e s t i n g o b s e r v a t i o n s made by t h e s t u d e n t s ,who had never b e f o r e r e a l l y stopped t o t h i n k about t h i s k i n d o f material. They d i s c o v e r e d t h a t from i n s u f f i c i e n t d a t a i n c o r r e c t o r many d i f f e r e n t i n f e r e n c e s may be made. D u r i n g t h e experiment  t h e c o n t r o l group r e c e i v e d t h e i r  normal c l a s s r o o m i n s t r u c t i o n .  Based on t h e r e s u l t s o f t h e  C a l i f o r n i a Test o f M a t h e m a t i c a l Achievement P r i c e  concluded  t h a t t h e r e were no s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s between t h e e x p e r i m e n t a l groups and c o n t r o l group on achievement.  However, b o t h e x p e r i m e n t a l groups d i d  s i g n i f i c a n t l y b e t t e r than t h e c o n t r o l on a Survey o f A l g e b r a A p t i t u d e , but o n l y t h e T group g a i n e d  significantly  on t h e W a t s o n - C r i t i c a l T h i n k i n g A p p r a i s a l T e s t . Post (1967) p r e s e n t e d a " S t r u c t u r e o f t h e ProblemS o l v i n g P r o c e s s " e x p e r i m e n t a l u n i t t o 74 grade seven mathematics s t u d e n t s .  C o r r e s p o n d i n g t o t h e s e 74 s t u d e n t s  were an e q u a l number i n a. c o n t r o l group, who had been matched on problem  solving a b i l i t y .  He a l s o d i v i d e d t h e  s e c t i o n s i n t o h i g h I n t e l l i g e n c e (I.Q. above 110) and low  51  intelligence  ( b e l o w 110).  The  solving process consisted (a)  R e c o g n i t i o n of the  phase; The  (c)  of the  existence  instructional material  period  on  not  categories:  (d)  (b)  phase.  experimental  days.  employed, d u r i n g  A  follow-  which time  the  where a p p r o p r i a t e .  introduction purely  Analysis  Verification  p r e s e n t e d to the  structure  employed f o r the  s o l v i n g p r o c e s s was  was  4  problem  of a problem;  three successive  o f s i x weeks was  teacher emphasized t h i s vehicle  following  P r e d i c t i o n p h a s e ; and  group i n t h r e e l e s s o n s up  o u t l i n e of his  of  the  mathematical.  The  problem For  example,  the  f o l l o w i n g p r o b l e m i s a n o n - m a t h e m a t i c a l example used  the  unit. A boy  dance i n the  comes home f r o m s c h o o l evening.  w i l l p r e v e n t him  He  has  solving structure. P o s t f o u n d no and  On  dance.  c o n t r o l groups g a i n  i n terms o f P o s t ' s  difference  f o r the  l a c k of s i g n i f i c a n c e . only  students to f u l l y  problem s o l v i n g process. was  only  partially  a  which  to solve  the  problem  between the  test  experimental  scores.  This i n v e s t i g a t o r suggests the  instructional period,  How  to  a mathematical problem s o l v i n g  significant  reasons f o r the  w a n t s t o go  c e r t a i n c h o r e s t o do  from going to the  boy's dilemma Is d i s c u s s e d  and  In  following  possible  F i r s t , the  short  t h r e e d a y s , gave l i t t l e assimilate Second, the  the  structure  instructional  mathematical i n nature, thus  time of  the  unit  requiring  52  the s t u d e n t s t o l e a r n b o t h the problem a non-mathematical  s i t u a t i o n and t h e n a p p l y i t t o a  m a t h e m a t i c a l problem Wills problem  (1967)  s o l v i n g process i n  solving  test.  e x p l i c i t l y c o n s i d e r e d t h e t r a n s f e r of  s o l v i n g a b i l i t y g a i n e d through, l e a r n i n g by  discovery.  H i s sample c o n s i s t e d o f 5 6 l h i g h s c h o o l s t u d e n t s  i n 24 c l a s s e s .  E i g h t c l a s s e s were randomly a s s i g n e d t o  two e x p e r i m e n t a l groups, Covert group (CG) and Overt group (OG), and one  c o n t r o l group ( C ) .  Over a p e r i o d o f  two  weeks the e x p e r i m e n t a l groups r e c e i v e d workbooks as w e l l as i n s t r u c t i o n from the t e a c h e r s .  Wills  (1967,  d e s c r i b e d the r o l e o f the t e a c h e r i n the two groups as  p.  155)  experimental  follows:  Covert Group Whenever a c r i t e r i o n problem a r i s e s i n the Covert group, the t e a c h e r f o l l o w s i t w i t h s i m p l e r problems, makes a t a b l e , and c o n t i n u e s a s k i n g q u e s t i o n s u n t i l the s t u d e n t s ' answers i n d i c a t e t h a t they have a c q u i r e d a g e n e r a l i z a t i o n . The t e a c h e r does not d i s c u s s why the e a s i e r problems are p r e s e n t e d . No e f f o r t i s made t o d i s c u s s the g e n e r a l method o f s o l v i n g such problems... I n the Covert group the t e a c h e r l e a d s the s t u d e n t t h r o u g h s e v e r a l s o l u t i o n s as does t h e i n s t r u c t i o n a l u n i t . . . Overt  Group  Whenever a c r i t e r i o n problem a r i s e s , the t e a c h e r t e l l s the s t u d e n t s t h a t an e f f e c t i v e way t o s o l v e a problem i s t o s o l v e s e v e r a l s i m p l e r problems o f the same type and t r y t o f i n d a p a t t e r n which can be a p p l i e d t o t h e more d i f f i c u l t problem. A l l t h r e e groups were g i v e n the same p r e t e s t and p o s t t e s t , and t h e f u n c t i o n o f the c o n t r o l group seems t o  53  have been t o c o n t r o l f o r growth o v e r t h e i n s t r u c t i o n a l period.  On t h e b a s i s o f t h e a n a l y s i s , which used t h e  p r e t e s t scores experimental  as a c o v a r i a t e . W i l l s c o n c l u d e d t h a t t h e  groups performed s i g n i f i c a n t l y b e t t e r than t h e  c o n t r o l , w h i l e 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 between the two e x p e r i m e n t a l two e x p e r i m e n t a l  groups.  The mean performance o f t h e  groups I n d i c a t e d a b e t t e r performance  from t h e o v e r t group.  The problem s o l v i n g t e s t was aimed  towards problems t h a t i n v o l v e d s e a r c h i n g Some o f t h e q u e s t i o n s patterns  involved asking the students to f i n d  i n problem s i t u a t i o n s t h a t were u n l i k e t h o s e i n  the i n s t r u c t i o n a l m a t e r i a l s . Involved  for a pattern.  patterns  F o r example, one problem  associated with Pascal's t r i a n g l e .  Other problems I n v o l v e d p a t t e r n s considered  extensions  I n s i t u a t i o n s t h a t can be  of the i n s t r u c t i o n a l m a t e r i a l s .  r e s u l t s i n d i c a t e that the discovery  approach r e s u l t e d i n  b e t t e r problem s o l v i n g performance on problems involved trying to find  Wills's  that  patterns.  Leask (1968) i n v e s t i g a t e d t h e t e a c h i n g o f s i m p l e enumeration as a s t r a t e g y f o r d i s c o v e r y .  The sample  c o n s i s t e d o f s i x grade t w e l v e c l a s s e s w i t h a t o t a l o f 158 students.  The s t u d e n t s had been randomly a s s i g n e d t o  classes at the beginning were a s s i g n e d  o f t h e y e a r , and t h r e e  t o each t r e a t m e n t , e x p o s i t o r y  enumeration ( S ) .  The l a t e r was d e s c r i b e d  classes  (E) and s i m p l e  as f o l l o w s :  54  T h i s c o n s i s t s of. t h e p r e s e n t a t i o n o f many i n s t a n c e s o f t h e g e n e r a l i z a t i o n t o be d i s c o v e r e d . The s t u d e n t s form hypotheses based on t h e examples and t e s t t h e s e t o determine which i s c o r r e c t . One counterexample i s s u f f i c i e n t t o w a r r a n t r e j e c t i o n of the hypothesis. (Leask, 1968, p. 8) In both treatments the experimental p e r i o d  consisted  o f seven one hour s e s s i o n s , w i t h t h e m a t e r i a l s c o n s i s t i n g o f the development  o f a r i t h m e t i c and g e o m e t r i c p r o g r e s s i o n s .  Both achievement the s t u d e n t s .  and t r a n s f e r t e s t s were a d m i n i s t e r e d t o  Leask (1968, p. 34) d e s c r i b e d t h e t r a n s f e r  t e s t as f o l l o w s : I t was a t w e n t y - i t e m t e s t based on m a t h e m a t i c a l m a t e r i a l s somewhat r e l a t e d t o t h e sequences b u t r e q u i r i n g a t r a n s f e r and e x t e n s i o n o f t h e 'knowledge a c q u i r e d . The items were comprised o f examples from which s t u d e n t s were r e q u i r e d t o generalize. Her r e s u l t s i n d i c a t e t h a t t h e S group performed  significantly  b e t t e r than t h e C group on t h e m a t h e m a t i c a l t r a n s f e r  test,  w h i l e 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 between t h e groups i n m a t h e m a t i c a l achievement.  An e x a m i n a t i o n o f t h e  r o l e of i n t e l l i g e n c e i n the conclusions i n d i c a t e s that the improvement i n t r a n s f e r was most s i g n i f i c a n t f o r t h e medium i n t e l l i g e n c e  range.  Jerman (1973) conducted a study o f m a t h e m a t i c a l t r a n s f e r i n t h e elementary s c h o o l .  He used two e x p e r i m e n t a l  programmes: t h e P r o d u c t i v e T h i n k i n g Programme o f C r u t c h f i e d and C o v i n g t o n , and a M o d i f i e d Wanted-Giveh Programme developed by Jerman.  The Jerman programme used  55 mathematical The  examples t o develop problem s o l v i n g  skills.  sample s e l e c t e d f o r t h e study was e i g h t c l a s s e s o f  f i f t h grade s t u d e n t s  (26l students), w i t h three classes  b e i n g a s s i g n e d t o each o f t h e e x p e r i m e n t a l programmes and the o t h e r two c l a s s e s r e c e i v e d t h e i r normal i n s t r u c t i o n and a c t e d as a c o n t r o l group.  classroom The e x p e r i m e n t a l  m a t e r i a l s were p r e s e n t e d i n s e l f - i n s t r u c t i o n a l b o o k l e t s , w i t h t h e t o t a l i n s t r u c t i o n a l p e r i o d b e i n g 16 days. t e s t s i n c l u d e d a word problem t e s t a d m i n i s t e r e d  The  immediately  a f t e r t h e i n s t r u c t i o n a l p e r i o d and a f o l l o w - u p t e s t g i v e n 7 weeks l a t e r .  Two NLSMA t e s t s , Working w i t h Numbers and  F i v e Dots were a l s o a d m i n i s t e r e d .  Jerman found no  s i g n i f i c a n t d i f f e r e n c e s between t h e t r e a t m e n t s .  However,  he n o t e d t h a t t h e word problems had proved more d i f f i c u l t than a n t i c i p a t e d .  I t appears t h a t t h i s d i f f i c u l t y was i n "  terms o f t h e computation  involved.  He proceeded t o  a n a l y s e t h e word problem d a t a a c c o r d i n g t o whether s u b j e c t s used a c o r r e c t procedure  to o b t a i n the s o l u t i o n .  That i s ,  whether t h e method would have r e s u l t e d i n a c o r r e c t s o l u t i o n i f no c o m p u t a t i o n a l e r r o r s were made.  With  this  new c r i t e r i o n Jerman found s i g n i f i c a n t d i f f e r e n c e s f a v o u r i n g the experimental treatments  on b o t h t h e immediate  word problem t e s t and t h e r e t e n t i o n t e s t g i v e n seven weeks later.  On t h e f i r s t t e s t t h e r e was a s i g n i f i c a n t s e x -  treatment  i n t e r a c t i o n f a v o u r i n g boys, a r e s u l t not repeated  56  on t h e . r e t e n t i o n t e s t . repeated the  may  The  f a c t t h a t t h i s r e s u l t was  h a v e b e e n due- to: t h e • i m p r o v e d p e r f o r m a n c e  students  on t h e d e l a y e d  t e s t , w h i c h was  v e r s i o n o f the o r i g i n a l word problem  literature i s reflected  design of the  study.  F i r s t , the  educators  c o n s i d e r analogy  important  mathematical  and  teach to  i t was  i n three facets of  of cases  ( f o r e x a m p l e , see  to select  these  to  be  Smith  (1971b)).  heuristics  to  students.  b o o k l e t s w e r e t o be s a m p l e and  control  ( f o r e x a m p l e , see Jerman  i n chapter  I I I ( p . 61 ) t h a t  employed i n o r d e r t o randomize the i n s t r u c t i o n a l  Gagne and  (1973)) p r o v i d e  instructional  sequence.  evidence  that u t i l i z i n g  m a t e r i a l s as t h e i n s t r u c t i o n a l  technique  f o r promoting  L i b e s k i n d (1971) and  that teaching involving  heuristic  (1965a) self-  medium i s a  transfer. studies s p e c i f i c a l l y  concerned w i t h the t e a c h i n g o f h e u r i s t i c s . e x a m p l e , see  the  Studies  Brown ( 1 9 6 l ) , E l d r e d g e  F i n a l l y , t h e r e w e r e a few  transfer  the  l i t e r a t u r e indicates that  (1954a), MacPherson  reasonable  Second, i t i s noted  feasible  a parallel  Literature  examination  heuristics  Henderson (1959), P o l y a  Therefore,  and  and  of  test.  Summary o f t h e R e l a t e d The  not  Goldberg  Studies (1974))  p r o c e d u r e s can  (for indicate  result  o f knowledge t o problems t h a t are r e l a t e d  to  in  the  57  instructional materials. Schaaf  (1954),  Wills  Other s t u d i e s ( f o r example, see  (1967),  Leask  (1968)  and Worthen (1965) )  p r o v i d e evidence t h a t s t u d e n t s can a p p l y g e n e r a l problem s o l v i n g t e c h n i q u e s t o new problems.  The p r e s e n t study can  be c o n s i d e r e d b o t h an e x t e n s i o n and r e f i n e m e n t o f c u r r e n t research.  I t i s a refinement i n that the i n s t r u c t i o n a l  m a t e r i a l s were r e s t r i c t e d t o t h e h e u r i s t i c s  o f analogy and  e x a m i n a t i o n o f c a s e s , and an e x t e n s i o n i n t h a t I t I n v e s t i g a t e d the t r a n s f e r o f h e u r i s t i c s content) t o novel mathematical  problems.  (as opposed to-  58  CHAPTER I I I Design and Procedure Design T h i s study was designed of h e u r i s t i c s .  to i n v e s t i g a t e the teaching  Two f a c t o r s were considered:', the f i r s t  f a c t o r concerned t h e employment o f t r e a t m e n t s  with different  o b j e c t i v e s ; t h e second f a c t o r w i t h employing d i s t i n c t instructional vehicles.  I n o r d e r t o i n v e s t i g a t e the  hypotheses r e l a t e d t o these  f a c t o r s a t o t a l of nine  were d e v i s e d , as d e s c r i b e d i n c h a p t e r I . Figure 2 Experimental  Groups  Vehicles  T r e a t m e n t s  Geometric  Neutral  Algebraic  Heuristics only  H - N  H - A  H - G .  Heuristics plus content  HC - N  HC - A  HC - G  Content only  C - N  C - A  •  C - G  groups  59 For the purpose o f l a t e r d i s c u s s i o n the n i n e groups have been i d e n t i f i e d by a two p a r t code.  The H, HC and C  denote the t h r e e t r e a t m e n t s : h e u r i s t i c s o n l y (H), h e u r i s t i c s p l u s content  (HC) and content o n l y ( C ) .  The N, A and G  denote the t h r e e v e h i c l e s : m a t h e m a t i c a l l y n e u t r a l (N), algebraic  (A) and geometric  (G).  The H, HC and C are used  s e p a r a t e l y t o denote the t h r e e t r e a t m e n t s , i r r e s p e c t i v e o f the v e h i c l e .  For example, the group r e c e i v i n g the H  treatment c o n s i s t s of a l l s u b j e c t s i n H - N, H - A and H - G. Employing  t h r e e t r e a t m e n t s and t h r e e v e h i c l e s  r e s u l t e d i n n i n e e x p e r i m e n t a l groups, one c o r r e s p o n d i n g t o each o f the c e l l s i n f i g u r e 2.  The f o l l o w i n g i s an  e x p l a n a t i o n o f the.nature o f the i n s t r u c t i o n a l m a t e r i a l s f o r these  groups.  Instructional Materials N e u t r a l V e h i c l e (H - N, HC - N, C - N). f o r t h e s e groups developed m a t h e m a t i c a l a p p l i e d t o the p h y s i c a l w o r l d .  The m a t e r i a l s  t o p i c s as they  The word n e u t r a l was  employed  to i n d i c a t e that these m a t e r i a l s , although mathematical  in  n a t u r e , were not a s s o c i a t e d w i t h any core c a t e g o r y o f the d i s c i p l i n e of mathematics, such as a l g e b r a or geometry. A l g e b r a i c V e h i c l e s (H - A, HC - A, C - A ) .  Both the  t o p i c s and treatment o f t o p i c s i n t h i s v e h i c l e can be c l a s s i f i e d as a l g e b r a i c .  The  s e l e c t i o n o f , and approach  to  60 t o p i c s e m p h a s i z e d t h e use o f s y m b o l s and o p e r a t i o n s  on  symbols. Geometric V e h i c l e  (H - G,.HC  - G, C - G ) .  Both the  '  t o p i c s and t r e a t m e n t o f t h e t o p i c s i n t h i s v e h i c l e c a n be classified  as g e o m e t r i c . . The e m p h a s i s  i n t h i s - v e h i c l e was  on t h e u s e o f g e o m e t r i c f i g u r e s , d i a g r a m m a t i c and g e o m e t r i c o p e r a t i o n s  representations  s u c h as f l i p p i n g a n d r o t a t i n g .  H e u r i s t i c s only,. (H - N, H - A, H - G ) .  The  materials  f o r t h e s e g r o u p s w e r e d e s i g n e d so t h a t t h e h e u r i s t i c s w e r e explicitly  s t a t e d and e m p h a s i z e d , and t h e o n l y  f o r the student  was l e a r n i n g t o a p p l y  Heuristics plus The m a t e r i a l s  content  objective  the h e u r i s t i c s .  (HC - N, HC - A, HC - G ) .  f o r t h e s e g r o u p s were d e s i g n e d so t h a t t h e  h e u r i s t i c s were e x p l i c i t l y on b o t h t h e h e u r i s t i c s a n d Content only  s t a t e d , b u t e m p h a s i s was  placed  content. The  materials  f o r t h e s e g r o u p s w e r e d e s i g n e d so t h a t t h e o n l y  objective  f o r the student content  only  (C - N, C - A, C - G ) .  was t h e l e a r n i n g o f t h e c o n t e n t .  g r o u p s were i n c l u d e d i n t h e d e s i g n  The as c o n t r o l  groups. A t o t a l of nine required.  distinct  i n s t r u c t i o n a l u n i t s were  F o r example, the m a t e r i a l s  received h e u r i s t i c s plus  content  f o r t h e group  i n s t r u c t i o n i n an  a l g e b r a i c s e t t i n g , HC - A, c o n s i s t e d o f an a l g e b r a i c w i t h emphasis content.  on b o t h h e u r i s t i c s and o t h e r  The m a t e r i a l s  f o r t h e C - G group  geometric u n i t designed to teach  that  content  unit  mathematical consisted of a  only.  61  Booklets B o o k l e t s were u s e d so t h a t s u b j e c t s c o u l d be r a n d o m l y a s s i g n e d t o one o f t h e n i n e e x p e r i m e n t a l to  c o n t r o l the i n s t r u c t i o n a l  were d e s i g n e d  A l l nine  as  booklets  so t h a t e a c h c o u l d be c o m p l e t e d i n t e n d a y s  class p e r i o d per day). topics.  sequence.  g r o u p s as w e l l  Each o f the b o o k l e t s d e a l t w i t h  T h r e e d a y s were a l l o t e d t o e a c h t o p i c and one  day f o r r e v i e w .  The v e h i c l e s c o m p r i s e d t h e f o l l o w i n g Figure The  (one  three additional  topics:  5  Booklets  Vehicles  Topic  1  Topic  2  Topic  3  Seating  Plans  Tiling  Electrical Networks  Geometric  Algebraic  Neutral  Permutations  Mappings  Arrangements  Flips  Binary Networks  & Turns  Networks  The o v e r a l l o r g a n i z a t i o n o f t h e b o o k l e t s was  as  follows: Topic Day  1_:  1 - M a t e r i a l presented  end o f t h e p e r i o d .  and a q u i z i n c l u d e d a t t h e  The q u i z t o be m a r k e d and  the next day, t o g e t h e r w i t h t h e  solutions.  returned  62  Day- 2 - F o l l o w e d Day  t h e i d e n t i c a l p a t t e r n to. day 1.  3 - M a t e r i a l presented  ;  b u t no q u i z .  Topic 2:. Days 4 - 6  f o l l o w e d the same - p a t t e r n as days 1 - 3 .  Topic 3_: Days 7 - 9 Day  f o l l o w e d the same p a t t e r n as days 1 - 3 -  10 was a l l o w e d f o r r e v i e w .  No new m a t e r i a l was  included. I n a d d i t i o n t o employing t h e same o r g a n i z a t i o n ^ e i t h e r i d e n t i c a l o r s i m i l a r e x e r c i s e s , w o r d i n g and examples were used i n a l l the b o o k l e t s .  T h i s meant t h a t t h e n i n e  I n s t r u c t i o n a l u n i t s were e q u i v a l e n t , v a r y i n g o n l y as the d i f f e r e n t o b j e c t i v e s and the n a t u r e s and g e o m e t r i c v e h i c l e r e q u i r e d .  of a n e u t r a l , a l g e b r a i c  A detailed discussion of  the development o f the m a t e r i a l s can be found l a t e r I n t h i s c h a p t e r , under the h e a d i n g , 'Development o f M a t e r i a l s ' . Sample pages from the b o o k l e t s  are i n c l u d e d i n appendices  A - I. Instruments Two t e s t s were d e v e l o p e d , one a l g e b r a i c and t h e other geometric.  The i n s t r u m e n t s  both followed  identical  f o r m a t , c o n s i s t i n g o f a n o v e l m a t h e m a t i c a l problem f o l l o w e d by an open ended q u e s t i o n .  63 F i g u r e '.6 S t r u c t u r e of. t h e T e s t s  Presentation o f the Mathematical  Problem  Q u e s t i o n : What c a n y o u find  out  about...?  The  t e s t items employed o p e r a t i o n s i n v e n t e d  for  the study.  Details  can be f o u n d i n t h i s  specifically  o f t h e development o f t h e i n s t r u m e n t s  c h a p t e r (see pages  82-3)-  G r a d i n g Scheme The  g r a d i n g scheme f o r t h e t e s t s  a s s i g n i n g a s u b j e c t ' s response  consisted of  t o one o f s i x c a t e g o r i e s .  They a r e : 0- -  No e v i d e n c e o f t h e u s e o f e i t h e r e x a m i n a t i o n o f c a s e s or  1  -  analogy.  The s u b j e c t h a s u s e d e x a m i n a t i o n o f c a s e s o n l y . has  2 .-  -  to look f o r a pattern.  The s u b j e c t h a s b o t h e x a m i n e d c a s e s a n d l o o k e d f o r a  3  not attempted  He  pattern.  The s u b j e c t h a s u s e d a n a l o g y o n l y .  There  i s no  e v i d e n c e t h a t he h a s t r i e d t o t e s t h i s g u e s s .  64 4  5  -  The s u b j e c t  -  h a s u s e d a n a l o g y and e x a m i n a t i o n o f  but  as d i s t i n c t  the  subject  The s u b j e c t has  techniques.  cases,  T h e r e i s no e v i d e n c e  that  h a s c o m b i n e d t h e two h e u r i s t i c s . has used b o t h h e u r i s t i c s .  F u r t h e r m o r e , he  c o m b i n e d t h e two i n s o l v i n g t h e p r o b l e m . Procedure  The  procedure i s reported  brief  summary o f t h e p i l o t  Pilot  Studies The  for  pilot  brief  i n two p a r t s .  First  a  s t u d i e s , and s e c o n d t h e m a i n  development o f the m a t e r i a l s s t u d i e s t o be u n d e r t a k e n .  study.  d i c t a t e d t h e need  The f o l l o w i n g i s a  statement o f the purposes o f the p i l o t s ,  summary o f t h e n a t u r e o f t h e m o d i f i c a t i o n s  and a  resulting  from  the p i l o t s . The  objectives  (1) t h e r e a d a b i l i t y the  of the p i l o t  s t u d i e s were t o d e t e r m i n e :  o f the m a t e r i a l s ;  concepts discussed  (2) t h e d i f f i c u l t y o f  i n the materials;  (3) i f t h e l e n g t h  o f an i n d i v i d u a l ; 3 d a y ' s m a t e r i a l was r e a s o n a b l e ; difficulty The suggestions  of the exercises results  a n d (4) t h e  and e x a m p l e s .  of the p i l o t  studies, together  from the i n v e s t i g a t o r ' s committee,  c e r t a i n minor m o d i f i c a t i o n s these m o d i f i c a t i o n s  included  necessitated  of the m a t e r i a l s .  F o r example,  removing e x e r c i s e s  p a r t i c u l a r d a y ' s w o r k r e q u i r e d t o o much t i m e ;  with  because a  inserting  a  65 paragraph to c l a r i f y  a c o n c e p t , s u c h as f l i p p i n g  a tile;  and.  a l l o w i n g a d d i t i o n a l space between e x e r c i s e s f o r s u b j e c t s write their  to  answers.  Main Study A s s i g n i n g Subjects, to Groups.  To  control for  f a c t o r s r e l a t e d t o p r o b l e m s o l v i n g , s u c h as Mathematical one  A c h i e v e m e n t , s u b j e c t s were randomly a s s i g n e d  o f the nine  g r o u p i n g s had c o u l d not  be  experimental  groups.  form.  The  to present  H-N  HC-N  C-N  10  the  instructional  7 of the  HC-A  H-A  i  Class  and  The  7.  Organization  2  of the y e a r ,  o v e r a l l o r g a n i z a t i o n I s shown  Figure  Class  class  take place w i t h i n a l l classrooms.  s o l u t i o n was  units i n booklet  1  the  a l t e r e d , t h i s random a s s i g n m e n t n e c e s s i t a t e d  sole p r a c t i c a l  Class  Since  been f i x e d at the b e g i n n i n g  that a l l treatments  In f i g u r e  I n t e l l i g e n c e and  C-A  Groups H-G  HC-G  C-G  to  66  A presentation c o n t r o l l i n g teacher  employing booklets  variability  i n s t r u c t i o n a l u n i t s ) w h i c h no training  c o u l d have t o t a l l y  It  should  be  amount o f  classroom  of  the  pre-experimental  eliminated.  noted t h a t randomly a s s i g n i n g  w i t h i n classrooms,  been i m p o s s i b l e  the e f f e c t  (in presenting  t o c l a s s r o o m s would have r e s u l t e d i n the nested  had  and  treatments  treatments  being  c o n s e q u e n t l y i t would have  to d i f f e r e n t i a t e between t r e a t m e n t  and  effects.  Instructions to Subjects.  The  day  p r i o r to  s t a r t i n g t h e i r r e s p e c t i v e t r e a t m e n t s a n o t e was e x p l a i n i n g t h a t the experiment  (see  instructions (a)  read  s u b j e c t s were about t o t a k e  appendix J ) .  The  subjects'  part  major p o i n t s i n  in  an  the  were: The  s u b j e c t s were i n f o r m e d t h a t t h e y were  i n v o l v e d i n an e x p e r i m e n t and  that booklets  w o u l d be  handed  out, (b)  The  s u b j e c t s were i n f o r m e d t h a t the  w o u l d be  c o l l e c t e d at the  w o u l d be  no  day.  on  f u r t h e r w o r k on  o f e a c h p e r i o d , and  the b o o k l e t s  They w e r e a l s o a s k e d n o t  the b o o k l e t s the  end  end  with  anyone.  By  of each experimental  the b o o k l e t s  m i n i m i z e any  outside  booklets  until  to d i s c u s s the  the  there next  content  c o l l e c t i n g the m a t e r i a l s s e s s i o n and  c l a s s , the  a s s i g n i n g no  i n v e s t i g a t o r hoped  d i s c u s s i o n among s u b j e c t s .  of at work to  67  Instructional Period.  On  the  first  i n s t r u c t i o n a l p e r i o d e a c h s u b j e c t was folder. first  The  f o l d e r contained  period.'  At  t h e end  of  presented  t h e p a g e s t o be  no  About h a l f w a y  i n s t r u c t i o n a l p e r i o d the they was  had  should  teacher- a d v i s e d On  s u g g e s t e d t h a t . s u b j e c t s who so.  I f a student  m a t e r i a l e a r l y he was  not  were added.  days on  the  which  had  subjects  d a y s on w h i c h  any  new  that there  o f the p e r i o d  not  s t a r t e d the  completed a p a r t i c u l a r  given  the  t h r o u g h each day's  twenty minutes remaining.  do  in  instructional  a q u i z , a b o u t t e n m i n u t e s f r o m t h e end  teacher  a  the n e x t day's pages  f o l l o w e d throughout the  quiz.  with  covered  p e r i o d , w i t h t h e o b v i o u s r e s t r i c t i o n on t h o s e t h e r e was  the  of the p e r i o d the b o o k l e t s  c o l l e c t e d , t h e q u i z m a r k e d , and T h i s p r o c e d u r e was  day  the quiz  day's  m a t e r i a l from  the  booklet. No  mark o r l e t t e r g r a d e was  q u i z s i n c e I t was effect  the r e s u l t s .  throughout the classes while  type  to a  not  day  c l a s s e s met  experimental  at v a r i o u s  i n progress.  s u c h as w h e t h e r one  than  the  teacher  to  times  teachers  directional the  concerning  suggested that a  investigator arrived.  p o s s i b l e to c o n t r o l the teacher  The  c o u l d w r i t e on  F o r more i n v o l v e d q u e s t i o n s  u n i t s the  until  attempt  e n a b l i n g the i n v e s t i g a t o r to v i s i t a l l  answer s u b j e c t s ' q u e r i e s , o t h e r  back o f a page.  continue  The  t h e e x p e r i m e n t was  questions,  subject's  that t h i s . f o r m of feedback might  a s u b j e c t ' s p e r f o r m a n c e , c o n f o u n d i n g any  interpret  did  felt  assigned  subject  Thus i t  input i n t o the  the  was  experimental  68  process. two  . At t h e c o n c l u s i o n o f t h e t e n i n s t r u c t i o n a l  t e s t s , were a d m i n s t e r e d  to. t h e  subjects.  A d m i n l s t r a t i o n o f t h e Tests.. f o l l o w i n g the l a s t i n s t r u c t i o n a l was  administered  On-'the day  p e r i o d the  t o a l l s u b j e c t s , and  administered  the next  experimental  p e r i o d s were c o n c e n t r a t e d  Since  day.  On  the  immediately  algebraic test  geometric  each of the t e s t i n the  between p e r i o d s t h i s  test  and  t h o s e who  the  afternoon.  arrangement  t h e p o s s i b i l i t y o f d i s c u s s i o n s b e t w e e n s u b j e c t s who completed the  test  days,  the s c h o o l a l l o w e d t h r e e minutes f o r s u b j e c t s  change c l a s s r o o m s  days,  to reduced had  would w r i t e i t l a t e r i n the  day . The  experimental  procedure i s summarized i n f i g u r e Figure  Experimental  Day  Procedure  Introductory Instructions  1  Days  8  7  2-11  Instructional Periods  Day  12  Algebraic Test  Day  13  Geometric Test  8.  69  Grading. one  E a c h s u b j e c t ' s r e s p o n s e was  o f s i x c a t e g o r i e s ( s e e p a g e s 63-4  c o n s i s t i n g o f t h e i n v e s t i g a t o r and  )••  two  A team o f other  education p r o f e s s o r s , scored the t e s t s . presented  assigned  to  judges,  mathematics  Each judge  was  w i t h a copy o f t h e g r a d i n g scheme ( s e e a p p e n d i x  t o g e t h e r w i t h sample responses involved i n this  study.  o b t a i n e d from s u b j e c t s  A s e r i e s of meetings to  data.  experimental  tests individually  class l i s t s .  No  j u d g e was  and  the t e s t s  No  try  a s s i g n e d t o one  disagreement.  agreed,  l i s t s were  o f two and  any then'  categories;  those  about  M e e t i n g s where t h e n h e l d t o  t o o b t a i n c o n s e n s u s on t h o s e  had been d i s a g r e e m e n t .  three  on  judges'  comments o r m a r k s o f The  on w h i c h a l l t h r e e j u d g e s  w h i c h t h e r e was  recorded h i s r e s u l t s  aware o f t h e o t h e r  f o r m were w r i t t e n on t h e t e s t s .  those  the  Then, each judge graded a l l the  c a t e g o r i z a t i o n s of the t e s t s .  c o l l a t e d and  not  discuss  t h e g r a d i n g scheme waso h e l d p r i o r t o t h e s c o r i n g o f experimental  M),  tests  about w h i c h  there  Throughout the d i s c u s s i o n s the  i n v e s t i g a t o r i n t r o d u c e d t e s t s on w h i c h a l l t h r e e j u d g e s a g r e e d i n o r d e r t o v e r i f y t h a t t h e d i s c u s s i o n s were  had  not  causing a l t e r a t i o n s i n the i n d i v i d u a l s ' p e r c e p t i o n s of  the  categories.  in  The  percentages  o f a g r e e m e n t can be  found  table I. Subjects The  s u b j e c t s c o n s i s t e d o f 294-grade t e n - b o y s f r o m  a l l male h i g h s c h o o l i n E a s t e r n  Canada.  The  choice of  an  an  70  all  male s a m p l e , w i t h i t s o b v i o u s l i m i t a t i o n s  g e n e r a l l z a b i l i t y . of the Since  the  s a m p l e was  on  r e s u l t s , r e q u i r e s ' some  t o be  d i v i d e d i n t o nine  explanation; groups,  for  experimental  the  i n v e s t i g a t o r w i s h e d t o o b t a i n as l a r g e a s a m p l e  possible.  The  contained  m o r t a l i t y ( C a m p b e l l and  the  b o y s ' s c h o o l was  one  Stanley,  o f the  -few  a l a r g e number ( i n e x c e s s o f 300)  allowing  1963,  p.  as  that  grade-ten  students. Development of Development o f the The figure  Materials  Booklets  t o p i c s used i n the  three  v e h i c l e s were l i s t e d i n  5 which, f o r convenience, i s repeated Figure The  below:  5  Booklets  Vehicles  Neutral Topic  1  Topic  2  Topic  3  Since  Seating  a l l the  Algebraic  Plans  Permutations  Tiling  Mappings  Electrical Networks.  booklets  Binary Networks  w e r e i n e x c e s s o f 100  complete m a t e r i a l s have not pages from the b o o k l e t s  been i n c l u d e d .  can be  Geometric Arrangements Flips  & Turns  Networks  pages,  the  However, sample  found i n appendices A - I.  5)  71  The  b o o k l e t s were d e s i g n e d  t o be  structurally  mathematically  equivalent.  types.  e q u i v a l e n c e b e t w e e n v e h i c l e s was  First,  s e l e c t i n g t o p i c s such  as s e a t i n g p l a n s  n e u t r a l ) , permutations ( g e o m e t r i c ) t h a t had Furthermore,  T h i s e q u i v a l e n c e was  ( a l g e b r a i c ) and  of  and two  obtained  by  (mathematically arrangements  i d e n t i c a l mathematical  w h e r e p o s s i b l e e x a m p l e s and  structures.  e x e r c i s e s were  t r a n s l a t e d Into the terminology a p p r o p r i a t e to a  particular  vehicle.  was  Second, e q u i v a l e n c e between t r e a t m e n t s  obtained by, wherever p o s s i b l e , employing e x e r c i s e s and booklets  examples i n d i f f e r e n t  contained approximately  e x e r c i s e s and  identical  treatments.  All  t h e same number o f  examples.  E q u i v a l e n c e between v e h i c l e s T h i s e q u i v a l e n c e i s b e s t d e s c r i b e d by c o n s i d e r i n g individual  topics.  <  Figure  9  Topic  1  Neutral Vehicle  -  Seating Plans  Algebraic Vehicle  -  Permutations  Geometric V e h i c l e  -  Arrangements  S e a t i n g P l a n s c o n s i d e r s d i f f e r e n t methods o f s e a t i n g people A,'B  i n a row,  and  C).  and  round a c i r c u l a r t a b l e  (See  appendices  A t y p i c a l question involves seating three  72  people  i n a row.  The t h r e e p e o p l e  a r e Mr. B l a d e ,  Mr. J o h n s o n a n d Mr. H a m i l t o n .  Three  chairs  The q u e s t i o n p o s e d t o t h e s t u d e n t s i s : How many d i f f e r e n t ways c a n y o u s e a t t h e t h r e e men? Permutations letters  c o n s i d e r s d i f f e r e n t ways o f p e r m u t i n g  and numbers, and t h e n p r o c e e d s  c i r c u l a r permutations, which of  c y c l i c permutations  t o t h e concept o f  i s an a d a p t i o n o f t h e c o n c e p t  (See a p p e n d i c e s  D, E a n d P ) .  The  question corresponding t o that given i n the seating plans booklet i s : C o n s i d e r t h e s e t c o n s i s t i n g o f b , j and h.  Using a l l  t h r e e l e t t e r s , how many d i f f e r e n t p o s s i b l e p e r m u t a t i o n s can y o u f i n d ? • Arrangements c o n s i d e r s d i f f e r e n t methods o f arranging geometric consider different and  I).  f i g u r e s i n a l i n e , and t h e n p r o c e e d s circular patterns  (See a p p e n d i c e s  to  G, H  The q u e s t i o n c o r r e s p o n d i n g t o t h o s e s t a t e d  previously i s : Consider the s e t c o n s i s t i n g o f a t r i a n g l e , circle.  U s i n g a l l t h r e e f i g u r e s how many  square and different  p o s s i b l e arrangements can you f i n d ? Seating people  i n a row and a r r a n g i n g  geometric  f i g u r e s i n a l i n e are equivalent.operations t o permuting  73  letters. and  Similarly,  arranging  equivalent first  s e a t i n g people round a c i r c u l a r  geometric figures i n a c i r c u l a r pattern are  to organizing  three  c i r c u l a r permutations.  t o p i c s are n e u t r a l , algebraic  m o d e l s o f t h e same m a t h e m a t i c a l s t r u c t u r e . exercises  to a given  vehicle.  nature o f t h i s rewording.  and g e o m e t r i c Corresponding  The q u e s t i o n s i n d i c a t e t h e  The t r a n s l a t i o n o f an e x e r c i s e o r  e x a m p l e f r o m v e h i c l e t o v e h i c l e means t h a t  each  topic  a p p r o x i m a t e l y t h e same number o f e x e r c i s e s  examples. inherent  The o n l y  d i f f e r e n c e s between v e h i c l e s  and  are those  i n t h e n a t u r e o f an a l g e b r a i c , g e o m e t r i c a n d  n e u t r a l a p p r o a c h t o t h e same m a t h e m a t i c a l Figure  Thus, t h e  and e x a m p l e s were r e w o r d e d i n t h e t e r m i n o l o g y  appropriate  contains  table  10 s u m m a r i z e s t h e t r a n s l a t i o n s .  structure.  74  F i g u r e 10 Equivalence  1  Geometric  (Permutations)  (Arrangements)  Sample q u e s t i o n  ! Sample q u e s t i o n  Plans)  Sample q u e s t i o n  Topic  Algebraic  Neutral (Seating  -  J  T r a n s l a t i o n •S  v  1  \  J  -Translation—^  M r . B l a d e , M r . J o h n s on  Consider the s e t  Consider the s e t  and M r . H a m i l t o n a r e  c o n s i s t i n g o f b,  consisting of a  t o be s e a t e d i n a  j and h.  triangle, and  row.  square  circle.  AGO  A A A, How many d i f f e r e n t  Using a l l three  jUsing a l l three  ways c a n y o u s e a t  l e t t e r s how many  j f i g u r e s how many  t h e t h r e e men?  different possible \ different possible permutations you  find?  can  i arrangements can j you find?  75  Figure Topic  11 2  •Neutral Vehicle Topic  Tiling  Algebraic Vehicle  -  Mappings  'Geometric V e h i c l e  -  Flips  2^  Tiling colours  -  considers  fitting  tiles  i n t o a p p r o p r i a t e l y shaped  positions  f o r the t i l e s ,  shapes  and  Different  as well;->,as movements a n d c o m b i n a t i o n s a tile  f r o m one p o s i t i o n  (See a p p e n d i c e s A, B a n d C ) .  t y p i c a l s i t u a t i o n Involves tile  of various  spaces.  o f movements r e q u i r e d t o t r a n s p o s e to another are surveyed  and Turns  fitting  A  an e q u i l a t e r a l t r i a n g u l a r  I n t o a c o n g r u e n t s p a c e a n d t h e n r o t a t i n g f r o m one  position  to another.  positions  The f o l l o w i n g a r e e x a m p l e s  and a r o t a t i o n . Figure  A.  Rotation  )  12  1  Position 1  Position 2  Mappings c o n s i d e r s and o p e r a t i o n s letters.  o f two  sets of three  and c o m b i n i n g o p e r a t i o n s  (See a p p e n d i c e s D, E and F ) .  corresponding  to that given  and f o u r that The  i n the t i l i n g  letters,  interchange situation  section i s :  76  F i g u r e 13 A  B  C  |  j  ^  B  C  A  n (Mapping ±  M-^ I s t h e a l g e b r a i c applying rotation  Position  1  Position  2  1)  equivalent of rotation  1 ( o r M^) t o p o s i t i o n  position previously  1.  Thus,  1, B o c c u p i e s t h e '  h e l d by A, C o c c u p i e s t h e p o s i t i o n  previously  h e l d by B, a n d A o c c u p i e s t h e p o s i t i o n  previously  h e l d by C.  labelled position Flips of  I n b o t h c a s e s t h e new a r r a n g e m e n t I s  2.  and T u r n s c o n s i d e r s f l i p s  and t u r n s  (rotations)  d i f f e r e n t p o l y g o n s and how t h e s e movements c a n be c o m b i n e d  t o p r o d u c e new p o s i t i o n s . to those I n the t i l i n g The  after  s i t u a t i o n that  tiling  section  The g e o m e t r i c f i g u r e s  section  (See a p p e n d i c e s G, H a n d I ) .  corresponds to that  Involves the rotation  t r i a n g l e , rather  presented i n the  o f an e q u i l a t e r a l  t h a n an e q u i l a t e r a l t r i a n g u l a r  To s u m m a r i z e , t o p i c (neutral, algebraic  correspond  2 consists  tile.  o f t h r e e models  and g e o m e t r i c ) o f t h e same m a t h e m a t i c a l  structure,  and c o r r e s p o n d i n g examples a r e e x e r c i s e s  translated  Into the terminology appropriate to a p a r t i c u l a r  vehicle.  F i g u r e 14 c o n t a i n s t h e d i f f e r e n t  models.  were  77  Figure Equivalence  14 -  Topic  Neutral  Algebraic  (Tiling)  (Mappings)  Sample  situation  Geometri c  Sample s i t u a t i o n '  Translation Rotation  2  (Flips  and T u r n s )  Sample  situation  Translation. Mapping 1  1  (of a t i l e ) .  (of  Rotation (of  letters)  1  a triangle)  I i i  A  B  A  B  A  B  >'  B  A  B  c  A  B  A  78 Figure Topic  15 3 Electrical  •Neutral V e h i c l e Topic  3  <  Algebraic  Vehicle  Binary  Geometric  Vehicle  Networks  The t i t l e s of the t o p i c s .  themselves r e f l e c t  Each c o n s i d e r s  Networks  Networks  the e q u i v a l e n t  combining switches  nature  t o form  n e t w o r k s , and t h e n d e v e l o p s t h e c o n c e p t o f e q u i v a l e n t networks. switches  I n t h e s e c t i o n s on e l e c t r i c a l are e l e c t r i c a l  whether current (See  and the development I s i n terms o f  f l o w s o r does n o t f l o w t h r o u g h t h e n e t w o r k ,  a p p e n d i c e s A, B and C ) .  Idea of switches the development or I I  Binary  switches  Involves  the  t h a t have two p o s s i b l e v a l u e s , 0 o r 1, centers  networks i n t e r p r e t  and  on w h e t h e r a n e t w o r k h a s t h e v a l u e  (See a p p e n d i c e s D, E and F ) .  are e i t h e r  networks the  F i n a l l y , geometric  a n e t w o r k as a c o m b i n a t i o n o f p a t h s t h a t  connected or disconnected,  and t h e  development  deals w i t h the c o n d i t i o n s under which a network I s , or i s not  complete  (See a p p e n d i c e s G, H and I ) .  As  before,  appropriate  e x e r c i s e s and e x a m p l e s w e r e t r a n s l a t e d i n t o  terminology  appropriate  summarizes  topic  3-  to a particular vehicle.  Figure  the 16  I  79  Figure  16  Equivalence  Neutral  ,  Topic  3  Geometric  Algebraic  (Networks)  ( E l e c t r i c a l Networks)! (Binary Networks)  Sample n e t w o r k  ,  Sample  B  A  H  network  -Translation'  -Trans l a t i o n - ^ A  Sample  network  B  A  B  0 I  A and B a r e s w i t c h e s  A and B a r e s w i t c h e s A  open  B closed Current not  flow  J does  1  A and B a r e s w i t c h e s  A has v a l u e  0  A  B has v a l u e  1  B connected  Network value I I  has  disconnected  Network i s not  complete  80 The  r e s u l t of designing  equivalence vehicles the  the m a t e r i a l s  to  between v e h i c l e s i s t h a t b o o k l e t s  contain corresponding  obtain  for different  t o p i c s t h a t are models  same m a t h e m a t i c a l s t r u c t u r e ; where p o s s i b l e  booklets  utilize  the b o o k l e t s exercises. result  exercises  contain approximately The  and  and  the  corresponding  examples;  nature  of a mathematically  g e o m e t r i c m o d e l and  and  same number o f  d i f f e r e n c e s between v e h i c l e s are  from the  algebraic  equivalent  of  those  that  neutral,  approach to the  same  mathematical structure. Equivalence The plus  between Treatments three  content  and  t r e a t m e n t s are h e u r i s t i c s o n l y , h e u r i s t i c s content  only.  Where a p p r o p r i a t e  i d e n t i c a l e x e r c i s e or example, or t r a n s l a t i o n of e x e r c i s e o r example are  utilized  For example, the  should  the  reader  seating plans  sections  the h e u r i s t i c s only  and  i n a l l three  examine the  (See  s o l u t i o n s to e x e r c i s e s o f t e n contain  the  treatments.  development  a p p e n d i c e s A,  h e u r i s t i c s plus  the  B and  content  of  C).  In  booklets  the  an e x p l a n a t i o n  of  how  e x a m i n a t i o n o f c a s e s o r a n a l o g y ils'.; e m p l o y e d i n f i n d i n g t h e solution. exercise the only  In the does n o t  solution. and  The  content  only booklets  i n c l u d e any  the  identical  mention of the h e u r i s t i c s i n  d i f f e r e n t i a t i o n between the h e u r i s t i c s  h e u r i s t i c s plus  content  booklets  d i f f e r e n t emphases i n t h e m a t e r i a l s .  For  occurs  in  example,  the the  8-1  h e u r i s t i c s p l u s c o n t e n t b o o k l e t s have f e w e r pages specifically cases  and  t o the h e u r i s t i c s ; the words e x a m i n a t i o n  analogy  b o o k l e t s but not  are u n d e r l i n e d i n the h e u r i s t i c s i n the o t h e r h e u r i s t i c s  t o emphasize content  o n l y , and  of  only  treatment;  are I n c l u d e d i n the h e u r i s t i c s p l u s content designed  devoted  exercises  booklets  these  exercises  ommitte.d f r o m t h e h e u r i s t i c s o n l y m a t e r i a l s . Thus two  c o n s t r a i n t s a r e p l a c e d on t h e d e s i g n o f  bookletsnamely vehicles i n the  and  attempts  the  t o o b t a i n .-..equivalence b e t w e e n  e q u i v a l e n c e between t r e a t m e n t s .  This  results  f o l l o w i n g e q u i v a l e n c e between a l l n i n e b o o k l e t s :  (1) a l l b o o k l e t s c o n t a i n t h r e e t o p i c s ;  (2)  corresponding  t o p i c s a r e m o d e l s o f t h e same m a t h e m a t i c a l where p o s s i b l e i d e n t i c a l e x a m p l e s and t r a n s l a t i o n s . o f these booklets  structure;  e x e r c i s e s , or  a r e u s e d i n a l l b o o k l e t s ; and  contain approximately  reasonable  assume t h a t a l l s u b j e c t s w o u l d n o t p r o g r e s s  allotted period.  To  t h a t most o f t h e new  segment he was at the q u i z .  t o accommodate  complete a day's work w i t h i n t h e  do t h i s  t h e m a t e r i a l was  concepts  h a l f o f a day's work.  are developed  organized  i n the  I f a subject completed  F o r s u b j e c t s who  finished early  provided.  I t was  so  first  this  I n a p o s i t i o n t o make a r e a s o n a b l e  a d d i t i o n a l m a t e r i a l was  to  a t t h e same r a t e .  I n p a r t i c u l a r , the d e s i g n o f the b o o k l e t s had would not  (4) a l l  t h e same number o f e x e r c i s e s .  I n d e v e l o p i n g t h e - b o o k l e t s I t was  subjects-who  (3)  attempt  no  decided  that  82 these  s u b j e c t s w o u l d be  a l l o w e d t o c o n t i n u e w i t h work  not  a s s o c i a t e d w i t h the booklets. Development o f the The  Tests  t e s t s , one  a l g e b r a i c and  c o n s i s t e d of novel problems. the instruments  was  geometric  Furthermore, the nature  s u c h as t o a l l o w t h e o p p o r t u n i t y  m a k i n g c o n j e c t u r e s by t o examine  the other  analogy  of for  as w e l l as p e r m i t t i n g s u b j e c t s  cases.  A l g e b r a i c Instrument.  Discussions with  personnel  f a m i l i a r w i t h t h e h i g h s c h o o l m a t h e m a t i c s programme, t o g e t h e r w i t h an currently  examination  employed i n the  ten students  of high school  schools  textbooks  i n d i c a t e d t h a t : (a) Grade  were f a m i l i a r w i t h a s t r u c t u r a l  mathematics.  approach  In p a r t i c u l a r , grade t e n students  I n v e s t i g a t e d s e l e c t e d f i e l d p r o p e r t i e s s u c h as  had commutatlvlty,  and  t h e e x i s t e n c e o f I n v e r s e s ; and  (b) Grade t e n  had  been exposed.to modulo systems  (clock arithmetic).  The two  algebraic test  mathematical  study,  What can y o u The  operations  the  students  c o n s i s t e d of the p r e s e n t a t i o n  operations  f o l l o w e d by  to  developed  specifically  of  f o r the  question:  f i n d out  about these  operations?  are:  Up-one m u l t i p l i c a t i o n '  a @ b = a x  (b+1)  (Modulo  7)  'Double a d d i t i o n '  a # b = 2x  (a+b)  (Modulo  7)  T  (See  appendix K f o r d e t a i l s )  83  The  f a c t t h a t s u b j e c t s , had  been exposed t o  a  s t r u c t u r a l a p p r o a c h i n t h e i r m a t h e m a t i c a l programme the  i n v e s t i g a t o r t h a t s u b j e c t s had  the necessary  m a t e r i a l w i t h w h i c h t o . make a n a l o g i e s , draw  development of t h i s  f o l l o w e d i n the The  design  or t r y cases  and  of the  i s s i m i l a r to  algebraic  you  that  a d e s c r i p t i o n of distance  between  question:  f i n d out  about s h o r t e s t r o u t e s  and  routes?  movement, i n v o l v i n g m o v i n g up i n figure  with  for  test.  c a l c u l a t i o n s of the  the  combining shortest  illustrated  procedure u t i l i z e d  instrument  a g r i d and  p o i n t s , f o l l o w e d by What can  The  s u b j e c t s were p r e s e n t e d  movements on  One  core  inferences. Geometric Instrument.  the  assured  and  across  the  grid i s  17. Figured? Movements on  a  Grid Distance A and  7  (See  f  appendix L f o r d e t a i l s )  between  B is 3  segments.  84  Statistical  Procedures.  I d e n t i c a l s t a t i s t i c a l a n a l y s e s were p e r f o r m e d the a l g e b r a i c effects  and g e o m e t r i c t e s t d a t a .  analysis  on b o t h  A two-way f i x e d  o f v a r i a n c e (ANOVA) was e m p l o y e d .  The  mo de1 i s : Y.  =  ii  +  a.  +  8  . + Y. • + e-  where u  =  grand  =  effect  o f treatment i  3.  =  effect  of vehicle  Y••  =  interaction effect  o f treatment i w i t h v e h i c l e  e^j^.  =  experimental error  o f person k  ct^ ( i = 1, only  3)  j  represents the treatment e f f e c t s ,  (a-j_)j h e u r i s t i c s p l u s c o n t e n t  ( j = 1,  g.  2,  mean  2,  3)  heuristics  ( a ) and-content  only  2  represents the vehicle  j  (ct^)-  e f f e c t s , mathematically  3  neutral  algebraic  (8-^),  Overall significance of  F-ratios  ( a t .05  were t e s t e d  Furthermore,  a n  ^ geometric (B^)-  w e r e computed a n d t e s t e d f o r  level).  the o v e r a l l F-ratios  hypotheses  (gj)  D e p e n d i n g on t h e s i g n i f i c a n c e  the contrasts  corresponding to the  by t h e S c h e f f e " M e t h o d ( K i r k ,  the i n v e s t i g a t o r  chose.to supplement c e r t a i n  o f t h e ANOVA t e s t p r o c e d u r e s w i t h a n o n - p a r a m e t r i c (_X ). 2  The d e t a i l s a n d j u s t i f i c a t i o n  found i n chapter IV - A n a l y s i s  1968).  method  f o r these decisions  o f the Data.  are  85  CHAPTER I V A n a l y s i s o f the Data The  r e s e a r c h hypotheses  Hypotheses 1 through hypotheses  3 refer  4 and 5 c o r r e s p o n d  are restated  t o the treatment to certain  In each o f t h e f i v e hypotheses, measure o f a b i l i t y  effect, while  interaction  effects.  t h e dependent v a r i a b l e  t o apply at l e a s t  (examination o f cases  below.  or analogy)  isa  one o f t h e h e u r i s t i c s  t o a novel mathematics  problem: Hypothesis will  will  2_.  A group taught  heuristics content  3_.  A group taught  o n l y (H)  only (C).  heuristics plus  s c o r e h i g h e r t h a n a group taught  Hypothesis will  A group taught  s c o r e h i g h e r than a group taught Hypothesis  (HC)  1.  content  heuristics  content  only (C),  o n l y (H)  score h i g h e r than a group taught h e u r i s t i c s  plus  content (HC). Hypothesis  4_.  The mean s c o r e o f a h e u r i s t i c s  only  group  (H - N, H - A, H - G) as c o m p a r e d t o t h e mean s c o r e o f t h e c o r r e s p o n d i n g c o n t e n t o n l y g r o u p . ( C - N, C - A, C - G) w i l l be  Independent o f t h e v e h i c l e Hypothesis  group  5_-  material).  The mean s c o r e o f a h e u r i s t i c s p l u s  content  (HC - N, HC - A, HC - G) a s c o m p a r e d t o t h e mean s c o r e o f  the corresponding content will  (Instructional  be i n d e p e n d e n t  o n l y group  of the vehicle  (C - N, C - A, C - G)  (Instructional  material),  86  Grading. different  Throughout the t e n i n s t r u c t i o n a l p e r i o d s  s u b j e c t s , were absent  from s c h o o l .  For a subject's  t e s t t o be g r a d e d he h a d t o be p r e s e n t f o r a t l e a s t n i n e o f the t e n i n s t r u c t i o n a l p e r i o d s .  Table I contains the t o t a l  number o f a l g e b r a i c and g e o m e t r i c t e s t s  graded, the  p e r c e n t a g e o f a g r e e m e n t p r i o r t o any d i s c u s s i o n judgesQ? and t h e f i n a l p e r c e n t a g e  among  agreement.  Table I P e r c e n t a g e Agreement on T e s t s  T o t a l ' number of  tests.  graded*  Algebraic Test  .  Geometric Test  Agreement  Agreement prior to  after  %  discussion  %  discussion  249  177  71  244  98  249  . 175  70  241  96  * 228 s u b j e c t s w r o t e b o t h  Data Used.in A n a l y s i s .  tests  A s u b s e t o f t h e t e s t s on w h i c h  t h e j u d g e s a g r e e d was u s e d i n t h e a n a l y s i s .  Subjects f o r ;  whom d a t a f r o m o n l y one t e s t was a v a i l a b l e were d i s c a r d e d ,  87  l e a v i n g data from 2 1 8 s u b j e c t s .  The r e s u l t i n g group  sizes  v a r i e d from 2 1 to 2 6 . In order t o e q u a l i z e the numbers i n a l l nine groups  a d d i t i o n a l data were randomly d i s c a r d e d so  t h a t there were 2 1 s u b j e c t s per group. There were two reasons f o r r e d u c i n g the t o t a l number o f s u b j e c t s i n c l u d e d i n the a n a l y s i s .  F i r s t , i n order to draw  c o n c l u s i o n s concerning both the a l g e b r a i c and geometric problems i t was a d v i s a b l e to r e s t r i c t  the a n a l y s i s to data  based on only those s u b j e c t s who wrote both t e s t s . the use o f a non-orthogonal  design (unequal c e l l  w i t h the a s s o c i a t e d l a c k of s t a t i s t i c a l  Second,  sizes),  independence  between the o v e r a l l F - t e s t s , i s u n d e s i r a b l e s i n c e i t makes it  difficult  to i n t e r p r e t  results.  Data from 1 8 9 s u b j e c t s were analysed ( 2 1 per group). Tables I I and I I I show the summary o f the scores on the a l g e b r a i c and geometric t e s t s ,  respectively.  R e c a t e g o r i z a t i o n of Data f o r A n a l y s i s .  For the purpose  of t e s t i n g the hypotheses, the data In t a b l e s I I and I I I were r e c a t e g o r i z e d a c c o r d i n g to the f o l l o w i n g 0  -  dichotomy:  No evidence o f e i t h e r the h e u r i s t i c o f examination o f cases or analogy.  1  -  Evidence o f at l e a s t one o f these heuristics.  The r e s u l t i n g f r e q u e n c i e s are shown i n t a b l e IV.  This  r e c a t e g o r l z a t i o n of the data was necessary s i n c e the  88  Table I I Summary o f R e s p o n s e s on t h e Algebraic  Test  Number o f S u b j e c t s i n a C a t e g o r y * Number o f ^ Group  o •H  c  -p co •H  rH .0 C -P O -P co Dh 0 to o  0  o  o —'  c E Sn u C D -P -P  PK Mj  oa co CD CO octi  rH C o  >>  50 O H <  cC oO O •H 0 -P to CO 3 •<!H H -P 3 o 0 a £ -H -p ,G CO -P o -Hp  cC oO o ^ •o.^ H 0 -P C O co 3 •UH T3 3 0  o , . , Sub,] e c t s p e r Group  0  c ^ -H ^ 6 -p o oo  PQ ^  1 o  i rH  1 C\J  1 1 on  H-N  10  1•1  1  4  --  5  21  HC- N  16  1  •-  1  --  3  21  C - N  17  -  1  1  -  '  2  21  6  2 2  3  2  2  6  6  21  HC- A  12  2  -  -  1  2  4  21  C - A  16  11  2  --  1  1  21  H-G  1 0 10  -  1  2  8  21  HC- G  13  --  1  2  ---  5  21  C-G  17  2  — -  1  1  21.  H-A  . -  1 •=r  1 1 LTv  •  — -  1  T o t a l number o f s u b j e c t s - 189 * D e t a i l s o f t h e c a t e g o r i e s , c a n be f o u n d on pages 6 3 - 4 .  89  Table I I I Summary o f R e s p o n s e s on t h e Geometric  Test  Number o f S u b j e c t s i n a C a t e g o r y *  Number o f Group  •  K) CJ •H  -P m •UH o o  0  rH CD C -P O -P CD CO  O  cu fCn D -P -P  o —»  >j rH O  P-1  >>  off  W CD M  •  bO O  rH Ctj  •H CD -P tt! ra 3 •H SH -P 3CD O ^ -H  -P ,C w -P o -H n  o  «j O  H  CM  00  -=t-  pq —-  m o  Subjects  ^  •H CKDl m 3 •H Hi CD CD C £ -H 40  per  Group  ^ S ' -P O O O  CP —•  H - N  6  6  8  -  -  1  21  HC- N  10  5  4  1  -  1  21  C - N  15  3  3  -  -  H - A  7  6  5  -  -  3  21  HC-. . A  8  6  5  -  -  2  21  C - A  14  5  2  -  -  -  21  H - G  9-  4  7  -  -  1  21  HC- G  9  6  5  -  -  1  2.1  C - G  12 .  6  3  —  —  —  21  21  T o t a l number, o f s u b j e c t s - 189 *Detalls of the categories  c a n be f o u n d on p a g e s  63-4  Table IV Mark A s s i g n e d ' f o r A n a l y s i s  Algebraic  Group  Test  Geometric  Test  0*  1**  0*  H - N  10  11  6  15  HC- N  16  5  10  11  C-N  17  4  15  6  1**  H - A  6  15  7  14  A  1.2  9  8  13  C - A  16  5  14  7  H - G  10  11  9  12  HC-  G  13  8  9  12  C - G  17  4  12  9  H  26  37  22  41  HC  41  22  27  36  C  50  13  41  22  HC-  *  -  No e v i d e n c e o f e i t h e r  **  -  Evidence of at l e a s t  heuristic. one o f t h e h e u r i s t i c s  o r i g i n a l 0 t h r o u g h 5 scheme does n o t r e p r e s e n t e i t h e r a n underlying that  continuum o r a t o t a l l y  ordinal scale,  form t h e d a t a were i n a p p r o p r i a t e  and i n  f o r a n a l y s i s by  ANOVA. The  i n t e r p r e t a t i o n d i f f i c u l t i e s were  further  c o n f o u n d e d by t h e u s e o f d i c h o t o m o u s d a t a .  H s u and'  (1969) e x a m i n e d t h e e f f e c t o f l i m i t e d s c a l e s 5 p o i n t ) on n o m i n a l a - l e v e l s .  ( 2 , 3, A and.  The r e s u l t s o f t h e i r s t u d y ,  w h i c h was r e s t r i c t e d t o e q u a l c e l l  sizes,  reasonable control  for a l l scales,  v. .  o f Type I e r r o r  W i t h a two p o i n t  scale  results  a o f .01 o r .10, b u t n o t f o r an a o f .05.  basis  concerning c e l l  o f Hsu and F e l d t ' s  desirability a cell  Indicate  an n o f 50 p l u s  seems i n d i c a t e d b y H s u a n d F e l d t ' s  picture  Feldt  (per  cell)  f o r a nominal Although the  s i z e i s n o t e n t i r e l y c l e a r , on t h e r e s u l t s , and t h e g e n e r a l  o f e q u a l n's, i t w o u l d a p p e a r a d v i s a b l e  t o use  s i z e l a r g e r t h a n 21. • S i n c e i t was n o t f e a s i b l e t o u s e a l a r g e r n , t h e  investigator  c h o s e t o s u p p l e m e n t c e r t a i n o f t h e ANOVA  procedures with  a non-parametricmethod  done f o r h y p o t h e s e s 1 - 3 was  not applicable  2  T h i s was  c o n c e r n i n g a main e f f e c t , b u t  t o t h e i n t e r a c t i o n h y p o t h e s e s 4 a n d 5. Testing  The  3  (x ) •  t h e Hypotheses  s t a t i s t i c a l model employed i n t h e a n a l y s i s o f  variance i s :  92  i j k  j  p  1  i  T  j  i  j  k  where ']i  =  g r a n d mean  a. l  =  effect  o f treatment  3.  =  effect  of vehicle  v..  =  interaction effect  e. .. ijk  =  experimental e r r o r o f person k ^ • ^  i  '  j  J  a^(i  o f treatment  = 1, 2, 3) r e p r e s e n t s t h e t r e a t m e n t  o n l y - (a-j^) h e u r i s t i c s p l u s c o n t e n t 5  3-(j  i with vehicle  (6^), algebraic  2  :  a-^ = a,->  :  a  HI  H  01  H2  H  02  H3  H  03  H  04l ^ll" 31  H  042  : Y  11- 31  H  043  : y  12  _ y  32  H  051  : y  21  _ y  31  H  052  : y  21- 31  = Y  V. 0 5 3  : y  22" 32  = y  r H5  below: Alternative  N u l l Hypothesis  H4  <  H  mathematically  hypotheses corresponding t o the  research hypotheses are stated  = a Y  = Y  H  _ Y  1 2  13  l  a  :  :  a  2  Hypothesis  > > a  2  >  :  32-  H  l4l  :  Y  1 1  -Y  3 1  ^Y  1 2  -Y32  13  _ Y  33  H  l42  :  Y  ll- 31^ 13- 33  = y  13  - Y  33  H  l43  :  Y  12- 32^ 13^ 33  = Y  22  _ y  32-  H  151  Y  21- 31^ 22- 32  Y  = Y  y  y  H  2  12  l l  H  ot ^  = 2  : :  effects:  (a-^)-  ( 8 2 ) - and g e o m e t r i c ( g ^ ) .  The s t a t i s t i c a l  Hypothesis  heuristics  ( a ) and c o n t e n t o n l y  = 1, 2 j 3) r e p r e s e n t s t h e v e h i c l e  neutral  effects:  j  23 23  _ y  - y  22 33  .H H  1 5 2  153  :  Y  Y  Y  y  Y  Y  Y  Y  Y  : T 1 ^ . 3 1 ^ 2 3 ^ 22 2  :  Y '2- 32^23" 33 Y  2  Y  Plan f o r Analysis. mathematics.  The  Chapter I c o n t a i n s a model of  form o f the model, w i t h the  components o f D i s c o v e r y and Core s t a t e r e s e a r c h hypotheses d i r e c t i o n a l hypotheses subsequent  1-3  l e d the i n v e s t i g a t o r directionally.  at the p r e s e n t time t h e r e i s l i t t l e  Although  ( C h a p t e r I I ) shows research that  supports or refutes d i r e c t i o n a l i t y . statement of the hypotheses s u c h as t h e HC  empirical  that  either  Furthermore, the  initial  would not c o n s i d e r a l t e r n a t i v e  treatment being  significantly  b e t t e r t h a n t h e H t r e a t m e n t , w h i c h have i m p o r t a n t implications.  to  a r e i m p l i e d by t h e m o d e l , t h e  review o f the l i t e r a t u r e  hypotheses  distinct  Thus f o r b o t h t h e s e r e a s o n s , l a c k  support f o r d i r e c t i o n a l i t y  and t h e  educational of  educational  importance o f o t h e r a l t e r n a t i v e hypotheses, the  investigator  t e s t e d n u l l hypotheses  non-  HQ^  HQ  d i r e c t i o n a l a l t e r n a t i v e s , H^,  H  ll'  : a  l ^ 2  '  a  ^12'  Since hypotheses permissible to test Dunn's m u l t i p l e first  testing  effect.  :  a  2  1-3  ^  and H ^ ^ . a g a i n s t  2  , H-j^t and H - ^ i >  a  3'  H  13'  :  a  l ^  a  2  were s p e c i f i e d a p r i o r i ,  appropriate non-orthogonal contrasts  comparison procedure  for a significant  H o w e v e r , due  ( K i r k , 1968)  chose  without  i m p o s e d by  0 - 1  to t e s t the hypotheses  t h e more c o n s e r v a t i v e a p o s t e r i o r i S c h e f f e p r o c e d u r e The  by  o v e r a l l main t r e a t m e n t  to the l i m i t a t i o n s  s c o r i n g , the i n v e s t i g a t o r  1968).  i t is  f o l l o w i n g i s a comparison  by (Kirk,  of the a p o s t e r i o r i  94 S c h e f f e ' and Tukey p r o c e d u r e s , and Dunn's M u l t i p l e procedure.  Since hypotheses  comparisons  o f row means t h e d i s c u s s i o n i s l i m i t e d t o  contrasts involving just  refer only to pairwise  two row means.  i> r e p r e s e n t any p o p u l a t i o n c o n t r a s t o f two means  Let and  1-3  Comparison  ip i t s e s t i m a t e .  Then c o n f i d e n c e I n t e r v a l s f o r t h e  c o n t r a s t s c a n be w r i t t e n . I n t h e f o r m : /v  A  *  -  /»  A  K< ±  <P < * + 5  ±  where *  =  x  -  x  and For the Scheffe  *L ^S  =  Procedure  / MS (/ win.  +  j  1  - ^ (1-1), F(I-1,N-IJ) n. , J 1-ct ' I ' '  with MS  w  I  n  -.—  w i t h i n c e l l s mean  square  number o f t r e a t m e n t  l e v e l s (row) levels  J  ••—  number o f t r e a t m e n t  N  --  t o t a l number o f s u b j e c t s  —  t o t a l number o f s u b j e c t s u s e d  i' i' n  X\  and  ,  (columns)  f o r computing  respectively  95 :  F o r t h e Tukey "  1 - a  q  Procedure  I, N - I J ,  MS ' ~~~w N  /  I  with -  Q-.-t-  ,  _  1 - a 4,1, N - • I J T  T  - - t h e 100(1 - a ) p e r c e n t i l e p o i n t I n ^ the S t u d e n t l z e d range  distribution  w i t h I and N - I J degrees o f freedom MS  w  , N . a n d I a r e t h e same as f o r S c h e f f e "  For  t h e Dunn M u l t i p l e  Comparison  A ?  D  t , D  a/2;  /MS / w  C, N - I J  (  v  Procedure  I  n.  +  I  )  n. , /  with C  - - Number o f c o m p a r i s o n s t o be made  t'D  „  „  -,- I s a c o e f f i c i e n t  a/2; C, N - I J / 0  T  table MS  w',  (Kirk,  obtained  f r o m Dunn's  1968, p . 551)  n.. and n . , a r e t h e same as f o r S c h e f f e 1  In  1'  the present  C = 3> a n d a g  .05.  study  Using  I = J.= 3,  this  information the r e l a t i v e  efficiencies  o f the three procedures  (Kirk,.1968,  p. 96) a n d t h e r a t i o s  confidence  Intervals (Kirk,  = n ^ , = 63, N =  were c a l c u l a t e d  of the lengths of the  1968, p . 9 7 ) .  l89  3  96 Table The  Relative Efficiencies for  V  and R a t i o s o f C o n f i d e n c e  S c h e f f | , Tukey a n d Dunn*  Ratio of  Relative  Confidence  Efficiency  ?2 T  Intervals  x 100 =  91.99%  x 100 =  96.01%  Intervals  • 96  I ^S 2  Pz  *=D Pz  k  A  ..  .98  ^S  ^S  A  i  2  22  x 100 =  95.81%  .98  *  * where  The  i  2  £  2  =  12.20  (MS /63)  =  11.22  (MS /63)  =  11.71  (MS /63)  w  w  Tir  following conclusions  c a n be d r a w n f r o m t h e  c o m p a r i s o n o f t h e S c h e f f e , . Tukey a n d Dunn's A  First,  as i n d i c a t e d by t h e l e n g t h s  A  o f ? , E,^, a n d £  Scheffe procedure produces the longest and  procedures.  A-  s  confidence  i s , t h e r e f o r e , t h e most c o n s e r v a t i v e . ; '  D  the  intervals ,  Second, t h e f a c t  97  the  relative  indicates  efficiencies  that  a l l three  a r e a l l i n e x c e s s o f 90 p e r c e n t p r o c e d u r e s have a p p r o x i m a t e l y t h e  same c o n t r o l o f Type I ( a - e r r o r ) pairwise  comparisons.  conservative  Scheffe  a n d Type I I ( 3 - e r r o r ) f o r  T h u s , t h e s e l e c t i o n , o f t h e most procedure f o r the a n a l y s i s of contrasts  does n o t a p p e a r t o be t o o c o s t l y I n t e r m s o f p o w e r . Analysis  o f Data.  T a b l e s V I and V I I c o n t a i n t h e  means and s t a n d a r d d e v i a t i o n s geometric data, The are  f o r t h e a l g e b r a i c and  respectively.  o v e r a l l F - r a t i o s were c o m p u t e d and t h e s e r e s u l t s  summarized i n T a b l e s V I I I and I X .  Indicate  that  there  The o v e r a l l F - r a t i o s  was no s i g n i f i c a n t i n t e r a c t i o n on  e i t h e r the a l g e b r a i c  or geometric t e s t .  Thus, t h e n u l l  h y p o t h e s e s c o r r e s p o n d i n g , t o h y p o t h e s e s 4 and 5 a r e a c c e p t e d , and  the i n v e s t i g a t o r concludes that  differences  that  there  a r e no  c o u l d be a t t r i b u t e d t o an i n t e r a c t i o n  between t r e a t m e n t s and v e h i c l e s . The  i n v e s t i g a t o r proceeded t o t e s t the three  treatment hypotheses H procedure.  Q 1  , H  Q 2  , and HQ^ u s i n g  T a b l e s X and XI c o n t a i n  the Scheffe  t h e .05 s i m u l t a n e o u s  confidence I n t e r v a l s . As  indicated previously  the Investigator  t e s t e d h y p o t h e s e s 1 - 3 by a n o n - p a r a m e t r i c The  results of this  and  XIII.  analysis  are contained  also  ( x ) method. 2  i n tables XII  98  Table V I Means a n d S t a n d a r d  Deviations  for the Algebraic  Test  Neutral.  Algebraic  Geometric  Total  .5238  .7143  .5283  .5873  (.5118)  (.4629)  (.5118)  (.4963)  .2381  .4286  .3810  .3492  (.4364)  (.5071)  (.4976)  (.4805)  .1905  .2381  .1905  .2064  (.4024)  (.4364)  (.40-24)  (.4071)  Heuristics only Heuristics plus content' Content only *  Standard deviations i n parentheses  Table V I I Means a n d S t a n d a r d  Deviations*  f o r the Geometric  Heuristics only Heuristics plus content  Neutral  Algebraic  Geometric  Total  .7143  .6667  .5714  .6508  (.4629)  (.4830)  (.5071)  (.4805)  .5238  .6190  .5714  .5714  (.5118)  (.4976)  (.5071)  (.4988)  .2857  .3333  .3810  .3333  (.4629)  (.4830)  (.4976)  (.4752)  f  Content only  Test  *  Standard deviations i n parentheses  99  Table Overall  VIII  F-Ratio's f o r t h e  Algebraic  Source  of  Test  Degrees o f  Mean  Variation  Freedom  Squares  Treatments  2  '2.3333  10.7825*  Vehicles  2  0.3333  1.5403  0.2171  Interaction  4  0.0714  0.3299  0.8577  Error  180'  „  Probability ( F >- F ) 0  0.00004  0.2164 *Significant  a t .05 l e v e l  Table IX Overall  F-Ratios f o r the  Geometric  Test  Probability  Source o f  Degrees o f  Mean  Variation  Freedom  Squares  Treatments  2  1.7196  7.1432*  0.0010  Vehicles  2  0.0212  0.0897  0.9159  Interaction  4  0.0?26  0.3846  0.8195  180  0.2407  Error  *Significant  Fc  (F » F f )  a t .05 l e v e l  100  Table X Algebraic  Test - Simultaneous  Confidence  Intervals  f o r H y p o t h e s e s 1, 2 and 3 D i f f e r e n c e between  s  means •HI .T .  •/^w' ' "  - Y" .. = 0.3819  0.2047  H2  Y . . - Y .. = 0.1429  0.2047  H3  Y ..  0.2047  lr  3  2  - Y ..  1  Ms • .  = 0.2381  2  "(-)"(3-D -  w  .05 C o n f i d e n c e  s  P(2,l80)  =  . 9Q q 5  n  *Slgnlfleant  Intervals (0,1772 , 0.5866)* •  (-0.0618, 0.3576) (0.0334 , 0 . 4 4 2 8 ) *  j/.2164 x 4^-x 3-05 63 a t .05  = 0.2047  level  Table XI Geometric  Test - Simultaneous  Confidence  Intervals  f o r H y p o t h e s e s 1, 2 and 3 Difference  / I s  means HI-  Y . . - Yy 1  H2 ' Y . ..' - Yy. 2  H3  Y  J  . = . 0 .3174  Confidence Intervals  w  •  (0.1015 , 0.5333)*  = 0.2381  0.2159  (0.'0222 , 0 . 4 5 4 0 ) *  0.0794  0.2159  (-0.1365,  **/MS (^)(3-Dw  -  0.2159  - Y,  1'  .05  between  . P(2,l80) 9 5  *Signifleant  7  /.2407 x 4 x 3.05 63  a t .05  level  0.2953) = 0.2159  101  Table XII •X A n a l y s i s 2  f o r the A l g e b r a i c Test .  Probability  Hypothesis  Groups  x  H 1  H vs C  19.0989  H 2  HC v s C  3.2044 .  .0734  H 3  H vs HC  7.1718  .0074*  *  S i g n i f i c a n t at  2  . 00001*  .05 l e v e l  Table X I I I X Analysis 2  Hypothesis  Groups  H 1  H vs C  H 2 H 3  o f the Geometric Data  .  x  2  Probability  11.4603  .0007*  HC v s C  6.2617  .0123*  H v s HC  ,,.8349  . 3609  * Significant  a t .05 l e v e l  102 Statistical  .Conclusions  Before discussing the conclusions important t o note that the analyses the Scheffe  Procedure and t h e x  statistical  results  Figure statistical  2  i n detail i t i s  o f h y p o t h e s e s 01 - 03 by  produced  identical  ( w i t h a = .05).  18 g i v e s a g r a p h i c a l p r e s e n t a t i o n o f t h e  conclusions. F i g u r e 18  Visual Presentation of Results o fSignificance Tests f o r Hypotheses H  Mean  n i  , H „ a n d H, ~ n  .3333  n  .5714  Group  GEOMETRIC  TEST Sig  - significant  a t .05 l e v e l  NS - n o t s i g n i f i c a n t  a t .05 l e v e l  .6508  103  T h e r e was  no s i g n i f i c a n t  overall interaction  effect  on e i t h e r t h e a l g e b r a i c o r g e o m e t r i c t e s t .  Thus, the  contrasts  5 were not  corresponding to hypotheses  t e s t e d , and n u l l h y p o t h e s e s Interpretation of The  H^^  4 and  and HQ_ w e r e a c c e p t e d .  Results.  H group s c o r e d s i g n i f i c a n t l y h i g h e r than the C  g r o u p on b o t h t h e a l g e b r a i c and g e o m e t r i c t e s t s 18).  S i n c e t h e C g r o u p was  heuristic  ability  present i n the p o p u l a t i o n p r i o r  investigator heuristics one  c o n c l u d e s t h a t by  focussing Instruction  on  i t i s p o s s i b l e to teach students to apply at  novel mathematical  least  o f e x a m i n a t i o n o f cases o r analogy  analyses of H  Q 2  and H Q ^  indicates  f o r t h e a l g e b r a i c and g e o m e t r i c t e s t s .  the a l g e b r a i c t e s t the H group p e r f o r m e d b e t t e r t h a n b o t h t h e H C and  C g r o u p s , w h i l e t h e r e was  no The  i n v e s t i g a t o r concludes that of the treatments t e s t e d , the treatment i s the only e f f e c t i v e s t u d e n t s t o a p p l y a t l e a s t one On  On  significantly  d i f f e r e n c e b e t w e e n t h e H and H C g r o u p s .  a l g e b r a i c problem.  to  problems.  statistical  different results  significant  might  independently of the t r e a t m e n t ^ the  of the h e u r i s t i c s  The  to  heuristic  f o r f a c t o r s s u c h as n o v e l t y t h a t  a f f e c t performance  figure  designed to c o n t r o l f o r possible  experimentation, f o r incidental learning of t e c h n i q u e s and  (see  H  treatment f o r teaching o f the h e u r i s t i c s  to a novel  the g e o m e t r i c t e s t b o t h t h e H and  HC  104 groups performed s i g n i f i c a n t l y b e t t e r than t h e C group, w h i l e no  significant  groups.  difference  The i n v e s t i g a t o r  treatments are e f f e c t i v e least  was f o u n d b e t w e e n t h e H and HC c o n c l u d e s t h a t b o t h t h e H a n d HC i n teaching students t o apply at  one o f t h e h e u r i s t i c s  t o a novel geometric  F u r t h e r m o r e , t h e r e i s no s t a t i s t i c a l  problem.  evidence t o support the  c o n c l u s i o n that they are d i f f e r e n t i a l l y  effective.  different  are discussed i nthe  next  c o n c l u s i o n s f o r t h e two t e s t s  section. T h e r e 'was., no s i g n i f i c a n t  on  either  the algebraic  the three treatments  for the algebraic  test,  overall  interaction  or the geometric t e s t .  that f o r a l l three vehicles of  The  effect  This  implies  (N, A, G) t h e p a t t e r n o f means  ( H , HC, C) i s e s s e n t i a l l y and e s s e n t i a l l y  t h e same  t h e same f o r t h e  g e o m e t r i c t e s t , a l t h o u g h t h e p a t t e r n s o f means a r e d i f f e r e n t f o r t h e two t e s t s  (see f i g u r e s  concludes that the vehicle  19 and 2 0 ) . The  investigator  does n o t seem t o be an i m p o r t a n t  c o n s i d e r a t i o n i n teaching students t o apply at least the is  heuristics.  Whatever t h e v e h i c l e ,  the c r i t i c a l  one o f determinant  the approach s e l e c t e d t o i n t r o d u c e the h e u r i s t i c s ,  heuristics  only, or heuristics  combined w i t h  namely  content.  Discussion of Results H y p o t h e s e s 1, 2 a n d 3 r e f e r be  attributed  to the treatments.  enables the i n v e s t i g a t o r  t o d i f f e r e n c e s that can The s t a t i s t i c a l  analysis  t o conclude t h a t i n terms o f average  Figure  19  Mean S c o r e s o f G r o u p s on the A l g e b r a i c  Test  75 70 65 60 .55 .50  Mean  .45  Scores .40 .35 .30 .25  . 20 .15  H  HC Treatments Overall N e u t r a l groups  only  A l g e b r a i c groups  only  Geometric  only  groups  Figure  20  Mean S c o r e s o f G r o u p s the  Geometric  on  Test  Me an Scores  .25 .20 .15 C  HC  H  Treatments Overall N e u t r a l groups  only  A l g e b r a i c groups  only  Geometric groups  only  107 test scores. It. i s possible, to. teach students, to apply at . least one of. the heuristics, of examination of cases or analogy to novel mathematical  problems.  In p a r t i c u l a r , the  heuristics•only experimental treatment i s successful i n teaching the students to apply the heuristics to both novel algebraic and geometric problems.  The results concerning the effectiveness  of the heuristics plus content treatment are different f o r the. two tests.  The analysis indicates the h e u r i s t i c s plus  content treatment i s successful i n teaching subjects to apply at least one of the heuristics to a novel geometric problem, but this is,; not true f o r the algebraic problem. It i s impossible to determine conclusions.  the reasons f o r these different  However, the following observations concerning  the two tests may, i n part., explain the different  statistical  results. An examination of tables II and I I I indicates a different pattern i n the application of h e u r i s t i c s on the two tests.  On the algebraic test 6'f the 72 subjects who  applied at least one h e u r i s t i c , 54 (75%) applied analogy, either alone (14 subjects) or i n conjunction with of cases (40 subjects).  examination-  On the geometric test 98 subjects  applied at least one h e u r i s t i c but only 10 (10%) of them used analogy, and nine of the ten used the h e u r i s t i c i n conjunction with examination of cases.  Subjects who had  exhibited the a b i l i t y to employ analogy i n the algebraic test could not, or did not, apply the h e u r i s t i c i n the geometric  108  test..  T h u s . I t a p p e a r s t h a t t h e g e o m e t r i c t e s t was  more  amenable to. t h e a p p l i c a t i o n o f e x a m i n a t i o n o f cases', t h a n t o the a p p l i c a t i o n o f analogy.  T h i s c o u l d b e due t o t h e  t h a t t h e g r a d e t e n s u b j e c t s were more f a m i l i a r systems, f a c i l i t a t i n g to  a geometric The  fact  with•algebraic  a n a l o g i e s i n an a l g e b r a i c as  opposed  setting.  i n v e s t i g a t o r concludes the d i f f e r e n t  concerning the e f f e c t i v e n e s s t r e a t m e n t on t h e t e s t s  results  of the h e u r i s t i c s p l u s  c o u l d be a t t r i b u t e d t o t h i s  content treatment  b e t t e r e q u i p p i n g s u b j e c t s to apply examination of cases to  than  a p p l y a n a l o g y , and t h a t t h i s m a n i f e s t s i t s e l f i n t h e  g e o m e t r i c t e s t where t h e p r o b l e m I s more a m e n a b l e t o t h e a p p l i c a t i o n of examination of cases. data supports t h i s  conclusion.  o n l y 18 s u b j e c t s f r o m t h e HC  An e x a m i n a t i o n o f t h e  First,  on t h e a l g e b r a i c  group employed a n a l o g y , w h i l e  s u b j e c t s from the H group a p p l i e d t h i s h e u r i s t i c . t h e 63 s u b j e c t s i n t h e HC no  evidence of a b i l i t y  t e s t b u t had One  test  g r o u p t h e r e w e r e 20 who  to apply e i t h e r h e u r i s t i c  Second, of had  shown  i n the  algebraic  a p p l i e d examination o f cases i n the geometric  f u r t h e r note  can be made c o n c e r n i n g t h e two (H, HC,  s c o r e d h i g h e r on t h e g e o m e t r i c t e s t t h a n on t h e  algebraic  test.  superior  r e s u l t s on t h e g e o m e t r i c t e s t . administered f i r s t  The  and s u b j e c t s may  f o r the  algebraic test  since both tests  C)  was  have d i s c u s s e d t h a t  t e s t p r i o r t o w r i t i n g the "geometric t e s t .  Furthermore,  f o l l o w t h e same g e n e r a l f o r m a t ,  test.  tests.  F i g u r e 21 shows t h a t a l l t h r e e t r e a t m e n t g r o u p s  T h e r e a r e many p o s s i b l e r e a s o n s  29  Involving  Figure Overall  21  Means f o r A l g e b r a i c  and. G e o m e t r i c  Tests  .70 .65 .60 .55 .50  Mean  .45  Scores .40 • 35 .30 .25 .20  • 15  H  HC Treatment Algebraic  test  Geometric  test  Groups  110  the p r e s e n t a t i o n  o f a p r o b l e m f o l l o w e d by t h e q u e s t i o n  "What c a n y o u f i n d o u t a b o u t . . . ? " t h e r e may h a v e b e e n a t e s t i n g e f f e c t (Campbell and S t a n l e y , Another p o s s i b i l i t y in  i s that there  1963  5  p. 5 ) .  i s something  inherent  t h e g e o m e t r i c p r o b l e m u s e d on t h e t e s t t h a t made i t  s i m p l e r t h a n t h e a l g e b r a i c p r o b l e m . . Any o f t h e s e a l t e r n a t i v e s , or combination o f a l t e r n a t i v e s could account f o r t h e b e t t e r scores  on t h e g e o m e t r i c The  original  test.  s t a t e m e n t s o f h y p o t h e s e s 1, 2 a n d 3  i n d i c a t e d an o r d e r i n g b e t w e e n t h e means o f t h e t h r e e treatment groups.  This  ordering i s :  Mean (H) > Mean (HC) > Mean ( C ) A l t h o u g h t h e a n a l y s i s does n o t r e s u l t i n s t a t i s t i c a l l y significant  differences fully  justifying this  ordering,  s u c h an o r d e r i n g e x i s t s on b o t h t e s t s ( f i g u r e 2 1 ) . Furthermore, t h i s vehicles.  ordering also holds  Figures  w i t h i n the three  19 a n d 20 show t h a t f o r t h e s a m p l e u s e d  Mean (H - V ) > Mean (HC - V ) >'::Mean (C - V ) ±  where  ±  = N, A, G, w i t h t h e s i n g l e e x c e p t i o n  Mean (H - G) = Mean (HC - G ) .  ±  that  Ill One, f u r t h e r a n a l y s i s been i n c l u d e d i n t h i s  a s s o c i a t e d w i t h the data  chapter.  Table I contains the  i n f o r m a t i o n on t h e number o f te:s.ts g r a d e d . o n l y one  test  There were  s u b j e c t s who  wrote  geometric).  Of t h e s e t h e j u d g e s a g r e e d on g r a d e s  (21 a l g e b r a i c and  a l g e b r a i c and 19 g e o m e t r i c s u b j e c t s . comparative the  Tables XIV  42  21 f o r 20  c o n t a i n s the  d a t a f o r t h e s e s u b j e c t s and t h e s u b j e c t s u s e d i n  analysis. Table  Distributions  XIV  o f D a t a f o r S u b j e c t s Who  and A c t u a l D a t a Used i n t h e  Wrote Only  H  (Data)  H  (One  (Data)  HC  (One  (One  1  0  1  26  27  22  41  4  5  3  4  41  22  27  36  3  1  3  1  50  13  41  22  5  2  5  3  117  72  90  99  12  8  11  8  Test)  C (Data) Test)  Total  (Data)  Total  (One  Test)  Geometry  0  Test)  HC  One  Analysis  Algebra  C  has  Test  112 The  number o f s u b j e c t s who w r o t e o n l y one t e s t was  too small t o undertake x treatments.between the  2  t e s t s , f o r i-each o f t h e  However, f o r each t e s t d a t a u s e d ,in t h e  a x  2  three  v a l u e was c o m p u t e d  a n a l y s i s ( T o t a l ( D a t a ) ) and  d a t a b a s e d on s u b j e c t s who w r o t e o n l y one t e s t  (Total(One  Test)). T a b l e XV X  C o m p a r i s o n s o f A c t u a l D a t a a n d O n l y One T e s t  2  Test  Groups Total  Algebraic  Total  (One  Total  (Data) (One  w i t h Yates's  0.0056  .940  0.3764  .539  Test)  correction for continuity  a n a l y s i s i n d i c a t e s that f o r both t e s t s t h e  d i s t r i b u t i o n o f scores (and  Probability  Test)  Vs Total  2  (Data)  Vs  Geometric  This  x  Data*  o f s u b j e c t s who w r o t e o n l y one t e s t  t h e j u d g e s a g r e e d on t h e g r a d e ) was e s s e n t i a l l y  same as t h e The  distribution  o f the  the  data used i n the a n a l y s i s .  f o l l o w i n g three p o i n t s are noted.  Since a l l  t h r e e have e d u c a t i o n a l i m p l i c a t i o n s o r suggest p o s s i b l e directions i n the  for further research  appropriate  they  are d i s c u s s e d  s e c t i o n s o f c h a p t e r V.  in  detail  113  (1) heuristic  Of  on t h e  heuristics  the  72  s u b j e c t s , who. u s e d a t l e a s t one .  a l g e b r a i c t e s t , 40  Of t h e  t r e a t m e n t 26  63  subjects  h e u r i s t i c t o the (3)  On  geometric  the  received  The  geometric  employed b o t h  22  received  e i t h e r h e u r i s t i c to  (35%)  d i d not  (H - A) was  performance of  c o n c e r n i n g the  a l s o second h i g h e s t  the  other on  results.  concerning  Conclusions  e f f e c t i v e n e s s of teaching  b e e n assumed t h a t  the  the  d i f f e r e n c e s are  due  H o w e v e r , b a s e d on  the  heuristics. to the  the  differences In  to d i f f e r e n t i a l m o t i v a t i o n a l  effects that  are  Independent of the  o f the p i l o t  studies  has  content.  study., treatments  e f f e c t s of the  booklets,  h e u r i s t i c s , rather  e m p h a s i s on h e u r i s t i c s c a n n o t be I n one  It  emphasis  with  e v i d e n c e a v a i l a b l e i n the  a l t e r n a t i v e hypotheses that  the  h a v e b e e n drawn  oh h e u r i s t i c s , e i t h e r a l o n e o r I n c o n j u n c t i o n  the  the  i n s t r u c t i o n by  b e t t e r than a l l the  comment i s I n c l u d e d  I n t e r p r e t a t i o n of the  due  either  test.  F i n a l l y , one  are  apply  problem.  the h e u r i s t i c s o n l y  H - A g r o u p was  the h e u r i s t i c s  apply  a l g e b r a i c t e s t the  algebraic vehicle groups.  who  ( 4 l % ) d i d not  algebraic problem, while  group t h a t  (55%)  in their solution.  (2) only  the  than  rejected.  d a t a were, o b t a i n e d  from  114  subjects, concerning  t h e i r a t t i t u d e towards the s e l f -  i n s t r u c t i o n a l booklets. s u b j e c t s were p l e a s e d were no d i s c e r n a b l e as was d i s c u s s e d designing  Their  comments I n d i c a t e t h a t most  t o complete the b o o k l e t s ,  d i f f e r e n c e s between groups.  i n chapter  the materials  and t h e r e Furthermore,  I I I e v e r y e f f o r t was made i n  t o make them as e q u i v a l e n t  as  possible. Thus, w h i l e into motivation  further research  n e e d s t o be u n d e r t a k e n  e f f e c t s associated with the teaching of  h e u r i s t i c s , the i n v e s t i g a t o r feels confident the d e s i g n  t h a t b a s e d on  o f t h e m a t e r i a l s , a n d s u b j e c t s ' comments o n t h e  b o o k l e t s , t h e s u c c e s s o f t h e H a n d HC t r e a t m e n t s i s due t o the  e m p h a s i s on h e u r i s t i c s a n d n o t d i f f e r e n t m o t i v a t i o n a l  effects  that are independent o f the h e u r i s t i c s .  CHAPTER V Summary a n d C o n c l u s i o n s The  study i n v e s t i g a t e s whether I t i s p o s s i b l e t o  t e a c h s t u d e n t s t o a p p l y a t l e a s t one o f t h e h e u r i s t i c s o f examination problems,  o f cases  o r analogy  to novel  mathematical  and whether l e a r n i n g t o a p p l y h e u r i s t i c s  b e t t e r accomplished  by h a v i n g t h e s t u d e n t s  c a n be  f o c u s s o l e l y on  t h e h e u r i s t i c s , r a t h e r t h a n a l s o i n v o l v e them w i t h t h e simultaneous  l e a r n i n g of mathematical  content.  were c o n s i d e r e d i n t h e d e s i g n o f t h e s t u d y . f a c t o r concerns  employing  different objectives: designed  treatments  Two  The  designed  only; another  first  t o meet  one I n s t r u c t i o n a l t r e a t m e n t  t o teach h e u r i s t i c s  factors  was  combined t h e  i n s t r u c t i o n i n h e u r i s t i c s w i t h t h e t e a c h i n g o f c o n t e n t ; and a t h i r d treatment to h e u r i s t i c s .  utilized  The s e c o n d  was u s e d as an  f a c t o r concerns  and m a t h e m a t i c a l l y  g r o u p s were u t i l i z e d  of the treatments  I n the study  x vehicle  employing  i n s t r u c t i o n a l p e r i o d l a s t e d t e n days a t t h e end o f which  neutral vehicles. corresponding t o each  combinations.  i n s t r u c t i o n a l b o o k l e t s were d e v e l o p e d  day)  reference  i n s t r u c t i o n a l v e h i c l e s f o r t h e t r e a t m e n t s , namely  a l g e b r a i c , geometric Nine  only, without  The l a t t e r t r e a t m e n t  experimental control. distinct  content  Self-  f o r each group.  The  (one c l a s s p e r i o d p e r  t w o t e s t s were a d m i n i s t e r e d , one  a l g e b r a i c and t h e o t h e r g e o m e t r i c .  Both t e s t s had the  116  same f o r m a t ,  c o n s i s t i n g of presenting students with a novel  mathematical  problem  f o l l o w e d by t h e q u e s t i o n :  What c a n y o u f i n d o u t a b o u t . . . ? Conclusions The  results  of the data analysis i n d i c a t e that i t i s  p o s s i b l e t o t e a c h s t u d e n t s t o a p p l y a t l e a s t one h e u r i s t i c to  a novel a l g e b r a i c o r geometric  problem.  the group t h a t r e c e i v e d t h e h e u r i s t i c s scored s i g n i f i c a n t l y  In particular,  only treatment  h i g h e r than t h e c o n t r o l group  (H)  (C) on  both t e s t s . The  e f f e c t i v e n e s s o f t h e combined  treatment, h e u r i s t i c s plus content t h e two t e s t s .  instructional  ( H C ) , was d i f f e r e n t f o r  On t h e a l g e b r a i c t e s t  t h e H group  performed  s i g n i f i c a n t l y b e t t e r t h a n b o t h t h e HC and C g r o u p s , no  statistically  significant  t h e HC a n d C g r o u p s . treatment students  d i f f e r e n c e was f o u n d  while  between  The c o n c l u s i o n i s t h a t o n l y t h e H  c a n be c l a i m e d t o be e f f e c t i v e  i n teaching  t o a p p l y a t l e a s t one h e u r i s t i c t o a n o v e l  a l g e b r a i c problem. indicates  On t h e g e o m e t r i c  t e s t the analysis  t h a t t h e H a n d HC g r o u p s b o t h  performed  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 C g r o u p , b u t they were n o t themselves  clearly  different  conclusion i s that both  i n their effectiveness.  The  t h e H and HC t r e a t m e n t s a r e  effective  i n t e a c h i n g s t u d e n t s t o a p p l y a t l e a s t one  heuristic  t o a novel geometric  problem,  and t h e e v i d e n c e  117  does n o t p e r m i t any  conclusion concerning t h e i r  relative  effectiveness. However, a l t h o u g h the a n a l y s i s general  c o n c l u s i o n t h a t t h e H . t r e a t m e n t i s more  t h a n t h e HC least  does n o t w a r r a n t  one  treatment  heuristic  efficacious  i n teaching students to apply at  to novel mathematical  e x a m i n a t i o n o f t h e o v e r a l l means ( f i g u r e in that direction.  The  problems, 21)  and  20).  same t r e n d a p p e a r s - i n t h e  C e r t a i n l y , i f one  i s solely  parallel (figures  interested i n  t e a c h i n g h e u r i s t i c s , t h e v - r e s u l t s on t h e a l g e b r a i c  test,  t o g e t h e r w i t h t h e t r e n d s r e f e r r e d t o above, i n d i c a t e the H.treatment i s the'most  tests  application  reveals that while both h e u r i s t i c s  e m p l o y e d on t h e a l g e b r a i c t e s t t h e r e was  little  o f a n a l o g y b e i n g u s e d on t h e g e o m e t r i c t e s t . suggests  t h e e f f e c t i v e n e s s o f t h e HC  geometric  t e s t may  treatment  i s more e f f e c t i v e  h a v e b e e n due  The  treatment  investigator  on  the  to the f a c t t h a t the  i n teaching examination the geometric  more amenable t o t h e a p p l i c a t i o n  of examination of  test  i s i m p o s s i b l e to a s c e r t a i n the reason f o r t h i s  c o n c e r n i n g t h e HC  treatment.  were  evidence  c a s e s t h a n t e a c h i n g a n a l o g y , and  It  that  efficacious.  An a n a l y s i s o f t h e p a t t e r n o f h e u r i s t i c on t h e two  an  shows a t r e n d  e x a m i n a t i o n o f t h e means w i t h i n t h e t h r e e v e h i c l e s 19  the  However, the  HC  of proved cases.  result  investigator  suggests  the f o l l o w i n g p o s s i b l e e x p l a n a t i o n .  o f cases  can be  Examination  used t o answer q u e s t i o n s , w h i l e  analogy  118 poses, r a t h e r , t h a n a n s w e r s q u e s t i o n s . heuristics  and  With  e m p h a s i s on  c o n t e n t s t u d e n t s : t e n d to. l e a r n to. apply, t h e  h e u r i s t i c t h a t a p p e a r s most, v a l u a b l e i n a n s w e r i n g namely the h e u r i s t i c  of examination  T h e r e were no vehicles.  both  significant  of  cases.  differences involving  Thus i t seems r e a s o n a b l e  to conclude  i n s t r u c t i o n a l v e h i c l e employed i s not In whether students  questions,  l e a r n to apply  that  an i m p o r t a n t  the  the  factor  heuristics.  The- f o l l o w i n g t h r e e p o i n t s a r e n o t e d .  Since a l l  t h r e e have e d u c a t i o n a l i m p l i c a t i o n s ,  or suggest p o s s i b l e  directions  are d i s c u s s e d i n  f o r f u r t h e r r e s e a r c h they  d e t a i l i n the a p p r o p r i a t e (1)  Of t h e 72  h e u r i s t i c on heuristics  26  s u b j e c t s who  ( k l % ) d i d not  the a l g e b r a i c problem,  w h i l e 22  to the geometric  (3)  On  40  geometric  r e c e i v e d the  heuristics  (35%)  d i d not  apply  to  either  problem.  (H - A) was  H - A g r o u p was  test.  both  the a l g e b r a i c t e s t the performance o f  algebraic vehicle The  employed  apply e i t h e r h e u r i s t i c  group t h a t r e c e i v e d the h e u r i s t i c s  groups.  (55%)  one  solution.  Of the. 63  only treatment  heuristic  sub j ects.who u s e d a t l e a s t  the a l g e b r a i c t e s t ,  In t h e i r  (2)  sections.  only treatment  b e t t e r than a l l the a l s o second  by  the the  other  h i g h e s t on  the  119  Implications• As a c o n s e q u e n c e of- t h i s  study  several suggestions  can be made c o n c e r n i n g t h e t e a c h i n g o f h e u r i s t i c s . c o n s i d e r i n g these study and  I m p l i c a t i o n s i t s h o u l d be n o t e d t h a t t h e  considered only the h e u r i s t i c s  analogy.  "heuristics" The  of examination  refers  t o these  two p a r t i c u l a r t e c h l q u e s .  a n a l y s i s I n d i c a t e s t h a t i t was p o s s i b l e t o t e a c h  mathematical secondary  problems.  objective ofthe skills  d e a l i n g w i t h u n f a m i l i a r problem s i t u a t i o n s , the that units designed  i n c o r p o r a t e d I n t o t h e secondary  curriculum. arises to  S i n c e an i m p o r t a n t  to novel  s c h o o l m a t h e m a t i c s programme I s t o d e v e l o p  I n v e s t i g a t o r suggests be  o f cases  Thus, I n t h e f o l l o w i n g d i s c u s s i o n t h e term  s t u d e n t s t o a p p l y a t l e a s t one o f t h e h e u r i s t i c s  In  When  I f this  heuristics  s c h o o l mathematics  suggestion i s accepted  the question  as t o t h e p o s s i b l e s e t t i n g s t h a t s h o u l d - b e e m p l o y e d  develop  factors  the h e u r i s t i c s .  The s t u d y  t h a t h e l p answer t h i s First,  treatment  Is likely  i n v e s t i g a t e d two  question.  the a n a l y s i s suggests  although i n this  that a h e u r i s t i c s  t o be s u p e r i o r t o a c o m b i n e d  s t u d y t h e s c o r e s were  b e t t e r o n l y on t h e a l g e b r a i c t e s t . trends  t o teach  only  treatment,  significantly  H o w e v e r , b a s e d on t h e  ( s e e f i g u r e s 19, 20 a n d 21) I t seems r e a s o n a b l e t o  conclude  t h a t u n i t s devoted  solely t o the teaching of  h e u r i s t i c s w i l l be more e f f e c t i v e t h a n u n i t s  t h a t combine  i n s t r u c t i o n i n h e u r i s t i c s with I n s t r u c t i o n i n content.  To  120 be  most e f f e c t i v e  the  teacher should devote p a r t of  mathematics course s o l e l y an  a r r a n g e m e n t does not  to  developing h e u r i s t i c s .  preclude, u s i n g h e u r i s t i c s  c o n j u n c t i o n w i t h - o t h e r segments o f c u r r i c u l u m , but instruction has  t o be  the  devoted s o l e l y  Second, the vehicle affect the  to  i n t r o d u c e the  d e v e l o p i n g the I t . has  does n o t  to  t h a t the  heuristics  a p p l y the  a p p e a r t o be  mathematics c u r r i c u l u m .  g r o u p , the  data indicates  number o f  s t u d e n t s i n the  added..-  q u e s t i o n as  ).  not Thus,  mathematically for  heuristics  However,  A l t h o u g h the  heuristics  heuristics  t h a n the  control  substantial  o n l y g r o u p who  n o v e l problems  t o w h e t h e r a l l s t u d e n t s can  i n h e u r i s t i c s , and  be  one  did  not  (see  These d a t a s u g g e s t i n v e s t i g a t i o n  recommendations f o r f u r t h e r  solely  on  t h a t t h e r e were a  h e u r i s t i c to the  c o n c l u s i o n s p. 113  To  instructional  does  critical  only group'performed s i g n i f i c a n t l y b e t t e r  instruction  some t i m e  heuristics.  geometric or  b e e n recommended t h a t u n i t s  c a u t i o n a r y n o t e s h o u l d be  the  the  heuristics.  i n c l u d e d i n the  apply e i t h e r  in  mathematics  most e f f e c t i v e  indicates  of•an a l g e b r a i c ,  vehicle  Such  heuristics.  students' a b i l i t y  selection  neutral  t o be  analysis  employed to the  the  evidence suggests that f o r  in heuristics  the .  benefit  of  from  this question i s explored  in  research.  s u m m a r i z e , i t i s recommended t h a t u n i t s  to teach students h e u r i s t i c s  be  designed  incorporated  into  121  the  secondary s c h o o l mathematics, c u r r i c u l u m .  the evidence vehicle  a v a i l a b l e from t h i s  study  Furthermore,  suggests•that the  s e l e c t e d does n o t a p p e a r t o be o f c r i t i c a l  importance.  Limitations This d i s s e r t a t i o n the  study  of heuristics.  heuristics restricts  focuses  on c e r t a i n  The l i m i t e d  components o f  selection of  employed and i n s t r u c t i o n a l v e h i c l e s the g e n e r a l i z a b i l i t y  utilized  o f the conclusions".  A  f u r t h e r l i m i t a t i o n was i m p o s e d by t h e s e l e c t i o n o f an a l l male sample.  Five limitations  section, resulting  from:  (1) c h o i c e o f s a m p l e ; (2) u s e o f  self-instructional booklets; (4)  are discussed i n t h i s  (3) s e l e c t i o n o f t e s t s ;  c h o i c e o f I n s t r u c t i o n a l v e h i c l e s ; a n d (5) s e l e c t i o n o f  heuristics.  The I n v e s t i g a t o r h a s I n c l u d e d i n t h i s  discussion the suggestions follow directly  f o r f u r t h e r research that'  from the l i m i t a t i o n s .  W i t h an a l l male s a m p l e i t I s p o s s i b l e t o make supportable  statements  male s t u d e n t s but  about t h e t e a c h i n g o f h e u r i s t i c s t o  i n t h e age r a n g e 15 - 17 y e a r s  (Grade t e n ) ,  l e a v e s open t h e q u e s t i o n o f how f e m a l e s t u d e n t s  respond t o s i m i l a r treatments.  will  Fennema (1974) r e v i e w e d t h e  l i t e r a t u r e on s e x a n d m a t h e m a t i c a l  achievement.  studies i n v o l v i n g high school students (NLSMA, Y - p o p u l a t i o n ) , f a v o u r e d  She  that favoured  females  (Easterday  cited males and  E a s t e r d a y ) , a n d t h a t f o u n d no s i g n i f i c a n t d i f f e r e n c e s '  122  between t h e sexes  ( B l u s h a n e_t. a l . ) .  Pennema (1974, p. 137)  s t a t e d t h a t no c o n c l u s i o n c a n be r e a c h e d  concerning the  r e l a t i o n s h i p between sex and mathematical high school students. ambivalent  concerning  mathematical  Since the research evidence i s t h e r o l e o f s e x as a f a c t o r i n  achievement o f h i g h s c h o o l s t u d e n t s , t h e  i n v e s t i g a t o r f e e l s that the study females,  achievement f o r  s h o u l d be r e p l i c a t e d  with  o r w i t h mixed groups i n a d e s i g n which permits t h e  assessment o f i n t e r a c t i o n e f f e c t s i n v o l v i n g sex. Using s e l f - I n s t r u c t i o n a l booklets r e s u l t e d i n c e r t a i n c o n s t r a i n t s on t h e l e a r n i n g s i t u a t i o n , teacher-student  and p e e r g r o u p i n t e r a c t i o n s , w h i c h a r e an  i n t e g r a l part o f the classroom from the study. lack of these  e n v i r o n m e n t , were e l i m i n a t e d  I t i s i m p o s s i b l e t o assess whether the  i n t e r a c t i o n s had a p o s i t i v e  on s t u d e n t s ' p e r f o r m a n c e . students  F o r example,  to progress  effect  I t i s p o s s i b l e t h a t by a l l o w i n g  a t t h e i r own r a t e , t h e s e l f -  i n s t r u c t i o n a l b o o k l e t s . may. h a v e h e l p e d apply the h e u r i s t i c s .  or negative  some s t u d e n t s  learn to  A l t e r n a t i v e l y , t h e l a c k o f any  i n t e r a c t i o n may have h i n d e r e d research c i t e d i n chapter  other students,  I I indicates that  Although the  self-  i n s t r u c t i o n a l m a t e r i a l s a r e a n e f f e c t i v e means o f i n s t r u c t i o n , f u r t h e r r e s e a r c h n e e d s t o be u n d e r t a k e n  t o determine  r o l e o f mode o f i n s t r u c t i o n o n s t u d e n t s ' a b i l i t y heuristics.  the  t o apply  An a d d i t i o n a l l i m i t a t i o n i m p o s e d by t h e u s e  o f b o o k l e t s i s t h a t a l l t h e i n s t r u c t i o n a l m a t e r i a l s were  123 p r e s e n t e d i n w r i t t e n form. the r e s u l t s o f t h i s  The r o l e o f r e a d i n g a b i l i t y i n  study i s d i s c u s s e d i n the  recommendations f o r f u r t h e r r e s e a r c h . The  nature o f the t e s t s i s o f c r i t i c a l  the study. problem,  Each-test  to apply the h e u r i s t i c s .  apply the h e u r i s t i c  similiarity  o f analogy  Although  a p p r o p r i a t e systems  to  mathematical might  F o r example,  s t u d e n t s must p e r c e i v e a  between t h e t e s t problem  e x p e r i e n c e d system. to  c o n s i s t s o f a n open e n d e d  and t h e s t u d e n t s ' p e r c e p t i o n o f t h e problem  affect their ability to  importance  a n d some p r e v i o u s l y  a l l s t u d e n t s had been exposed  ( f o r e x a m p l e , i t was p o s s i b l e t o  make a n a l o g i e s b e t w e e n t h e t e s t p r o b l e m s a n d t h e i n s t r u c t i o n a l m a t e r i a l s i n t h e b o o k l e t s ) t h e r e was no g u a r a n t e e analogy had  w o u l d be made.  I t i s p o s s i b l e that a student  learned the h e u r i s t i c  a similarity-  o f analogy  was u n a b l e  to perceive  t o apply the h e u r i s t i c .  the t e s t problems p l a y a s i g n i f i c a n t the i n v e s t i g a t o r feels  Because  role i n e l i c i t i n g  student  that a wide v a r i e t y o f  n o v e l t e s t p r o b l e m s must be u s e d t o o b t a i n a c l e a r of  the value of h e u r i s t i c s  such  of  cases i n s o l v i n g n o v e l  problems.  The  who  between t h e t e s t problems and h i s p r e v i o u s  e x p e r i e n c e , and h e n c e u n a b l e  response,  t h a t an  as a n a l o g y  picture  and e x a m i n a t i o n  i n s t r u c t i o n a l v e h i c l e s s e l e c t e d f o r the study  are a l g e b r a i c , geometric  and m a t h e m a t i c a l l y n e u t r a l , and t h e  t o p i c s were c h o s e n s o t h a t t h e v e h i c l e s a r e " e q u i v a l e n t " (See  chapter I I I ) .  This constraint l i m i t s  the s e l e c t i o n o f  124 topics  t h a t c o u l d be u s e d t o i n t r o d u c e t h e h e u r i s t i c s . -  investigator different  f e e l s that the study  topics  s h o u l d be r e p l i c a t e d  and v e h i c l e s .  analogy,  MacPherson Smith  (1959).  heuristics  Whether s t u d e n t s  c a n be t a u g h t  i s an open q u e s t i o n r e q u i r i n g  conclusions of this  section. (1)  Four major questions  t o apply  additional  other  research.  Research  study, together with the  l i m i t a t i o n s , suggest p o s s i b l e d i r e c t i o n s research.  (1957),  (1971), and Henderson and  Recommendations f o r F u r t h e r The  • -  o f cases,  were s e l e c t e d f r o m t h e w o r k s o f P o l y a  (1971b), Rousseau  with  '  F i n a l l y , o n l y two h e u r i s t i c s , e x a m i n a t i o n and  The.  f o r further  are discussed  In.this  They a r e : What f a c t o r s  affect  students'  ability  t o apply  heuristics? (2)  What r o l e  students' (3)  ability  Can s t u d e n t s  solve novel (4)  does k n o w l e d g e o f c o r e m a t e r i a l p l a y i n t o apply be t a u g h t  heuristics? t o combine h e u r i s t i c s t o  problems?  What I n s t r u c t i o n a l modes a r e . m o s t e f f e c t i v e i n  teaching  heuristics?  125  Factors E f f e c t i n g Heuristic A b i l i t y . a s u b s t a n t i a l number o f s t u d e n t s group  (H) d i d n o t  problems r a i s e s  on  59  i n the h e u r i s t i c s  to apply h e u r i s t i c s .  divided  each subgroup),  of these l e a s t one  2  These  reading  F o r each o f these measures the H group  i n t o t h r e e subgroups  Palrwise x  and  (approximately  namely h i g h , m i d d l e  a n a l y s e s were u n d e r t a k e n  factors  and  low  shown i n t a b l e s XVI  The  achievers.  to determine  and  r e s u l t s of t h i s  XVII.  d a t a f o r t h e M a t h e m a t i c s and  was  one-third i n  i s r e l a t e d to students' a b i l i t y  heuristic.  be  d a t a were  s u b j e c t s i n the H group.  d a t a I n c l u d e d m e a s u r e s on m a t h e m a t i c a l achievement.  might  Science  School Placement Test  o f t h e 63  only  test  t h e q u e s t i o n as t o what f a c t o r s  A s s o c i a t e s High  available  fact that  apply e i t h e r h e u r i s t i c to the  r e l a t e d t o the a b i l i t y Research  The  i f either to apply  analysis  are  F i g u r e s 22 - 27 show t h e Reading  subgroups.  at  126  T a b l e XVI X  2  A n a l y s i s o f the Achievement o f the H e u r i s t i c s  Only  Group on t h e A l g e b r a i c T e s t *  ?  Score  Mathematical Achievement  Achievement  Probability  2  H  4  15  H , M  0.8731  . 3501  M  8  12  H , L  7.5269  .0061  L  14  6  M , L  2.5252  .1120  ** Groups  x  2  Probability  0  1  H  5  15  H , M  2.1794  .1399  M  11  10  H , L  2.5336  .1114  L  10  8  M , L  0.0154  . 9014  Score 1  . 0  H i g h o r Low on BOTH measures  x  1  Score  Reading  Groups  0  HH  .3:i  LL  9  !  11  Groups HHjLL  X  2  Probability  2.9936  .0836  6  * with Yates's correction f o r continuity * not independent p a i r w i s e groupings.  127  Table XVII X  2  A n a l y s i s of the  Achievement of t h e H e u r i s t i c s  Group on t h e  Score  Mathematical Achievement  Achievement  Groups  Test*  X  Probability  0  1  H  5  14  H  ,M  0.0074  .9314  M  5  15  H  ,L  1.4170  .2339  L  10  10  M ,L  1.7067-  .1914  Score  Reading  Geometric  Only  Groups  X  Probability  0  1  H  4  16  H ,M  0.3728  .5415  M  7  14  H , L  2.5726  . 1087  L  9  9  M ,L  0.5305  .4664  Score  H i g h o r Low HH on BOTH LL measures  0  1  2  12  8  7  Groups  HH,LL  X  Probability  3.3116  .0688  * with Yates's correction f o r continuity ** n o t i n d e p e n d e n t p a i r w i s e g r o u p i n g s  128  F i g u r e 22 A c h i e v e m e n t o f H i g h , M i d d l e and.Low M a t h e m a t i c a l Subgroups  of Heuristics Algebraic  Achievement  O n l y Group  Test  20  Number o f  15 10  Students 5 0 0  Score  High Achievement  Subgroup  Middle Achievement  Subgroup  Low A c h i e v e m e n t S u b g r o u p .  F i g u r e 23 A c h i e v e m e n t o f H i g h , M i d d l e and Low R e a d i n g A c h i e v e m e n t Subgroups  of Heuristics Algebraic  O n l y Group  Test  20  Number o f Students  15 10 5 0 0  'Score  High Achievement  Subgroup  M i d d l e Achievement Low A c h i e v e m e n t  Subgroup  Subgroup  129  Figure  24  A c h i e v e m e n t o f H i g h , M i d d l e a n d Low M a t h e m a t i c a l A c h i e v e m e n t Subgroups  of Heuristics  O n l y Group  Geometric Test 20 Number o f Students  10 5 0  Score High Achievement  Subgroup  •  Middle Achievement  —  Low A c h i e v e m e n t  Subgroup  Subgroup  F i g u r e 25 A c h i e v e m e n t o f H i g h , M i d d l e a n d Low R e a d i n g A c h i e v e m e n t Subgroups  of Heuristics  O n l y Group  . Geometric Test 20 15  Number o f . 10  Students 5 0 \J  f~,  -L.  Score ' High_ A c h i e v e m e n t •  Subgroup  M i d d l e Achievement Subgroup Achievement Subgroup  L  o  w  F i g u r e 26 Achievement  o f S u b g r o u p s t h a t w e r e High.-',or Low  on.Both Mathematics Algebraic  and R e a d i n g Test  '15  Number o f Students  10 5 0 0  Score  H i g h on B o t h M a t h e m a t i c s Low on B o t h M a t h e m a t i c s Figure Achievement  a n d Reading -  and R e a d i n g  27  o f S u b g r o u p s t h a t w e r e H i g h o r Low  on B o t h M a t h e m a t i c s  and R e a d i n g  Geometric Test 15  Number o f Students  10 5 0 0  Score  High- on B o t h M a t h e m a t i c s a n d R e a d i n g Low on B o t h M a t h e m a t i c s  and R e a d i n g  131 The  d a t a - i n t a b l e s XVI  mathematical  and  XVII  a c h i e v e m e n t and  r e a d i n g achievement the  group p e r f o r m e d b e s t , the low group i n between.  The  t e s t where t h e m i d d l e  indicate, that f o r both  group w o r s t ,  mathematical  g r o u p s i s r e i n f o r c e d by  c o n s i s t i n g of students mathematical In low X  2  achievement  both  on b o t h value  and  who  reading  analyses  are l e s s than  are h i g h o r low  on  .1.  In the  who  probability  and  two  significance  However, the evidence  assigned  their  to  apply  for testing  the x  corresponding  to support  f a c t o r s s u c h as m a t h e m a t i c a l  particular,  are i n the no  value.  2  (on  or reading achievement) performed  Thus, t h e r e i s evidence  ability  others  the  associated  i n d i c a t e s t h a t the h i g h group  c o n s i d e r a b l y b e t t e r than  students'  or  involving  the a n a l y s i s i s only e x p l o r a t o r y  l e v e l was  e i t h e r mathematical  .05  are h i g h  four analyses  s u b g r o u p s , one  Since  compared t o  associated with  low  .1.  group.  both  c o m p a r i s o n s b e t w e e n h i g h and  region of  group  achievement. i n v o l v i n g students  than  geometric  data from subgroups  v a r i a b l e s , the p r o b a b i l i t i e s  i s less  middle  the h i g h achievement  s u p e r i o r p e r f o r m a n c e o f t h e h i g h g r o u p s as  the low  the  s i n g l e e x c e p t i o n i s on t h e  performed m a r g i n a l l y b e t t e r than The  and  high  to apply  and  low  the h y p o t h e s i s  be  a t l e a s t one  s e l f - i n s t r u c t i o n a l booklets  due  that  r e a d i n g achievement heuristic.  t h e r e l a t i o n s h i p b e t w e e n r e a d i n g and  a h e u r i s t i c may  group.  d e s i r a b l e t h a t f u r t h e r r e s e a r c h be  In  the  i n p a r t t o t h e use  i n the experiment.  effect  ability  of  It is  undertaken to i n v e s t i g a t e  132 the r e l a t i o n s h i p between these f a c t o r s apply h e u r i s t i c s .  . .  Knowledge o f Core M a t e r i a l . important  and t h e a b i l i t y t o  t o undertake  I t Is particularly  r e s e a r c h t h a t examines t h e  r e l a t i o n s h i p between s t u d e n t s ' core knowledge and t h e i r ability  t o a'pply h e u r i s t i c s .  paragraphs  The a n a l y s i s  Indicates that mathematical  a factor e f f e c t i n g the a b i l i t y  associated probability previous mathematical of  analogy  a c h i e v e m e n t may be  t o apply h e u r i s t i c s .  e x a m p l e , on t h e a l g e b r a i c t e s t t h e x h i g h and l o w m a t h e m a t i c a l  i n the previous  2  v a l u e between t h e  a c h i e v e m e n t g r o u p s h a d an  o f .0061. experience  The s p e c i a l r o l e o f In applying the h e u r i s t i c  h a s a l r e a d y b e e n d i s c u s s e d ( p . 1 2 3 ) . The  p e r f o r m a n c e o f t h e c o n t r o l g r o u p shows t h a t j u s t core m a t e r i a l (with which t e s t s ) , without  teaching  a n a l o g i e s c o u l d be made on t h e  s p e c i f i c reference to the heuristic^does  n o t r e s u l t i n much e v i d e n c e (six  For  o f students' a p p l y i n g  analogy  s t u d e n t s on t h e a l g e b r a i c t e s t , no s t u d e n t s on t h e  geometric  test).  The e v i d e n c e  group u s i n g analogy  of students i n the c o n t r o l  may,. _n---p'&rt j$.,jp&£lect a b i l i t y v  t o apply  the h e u r i s t i c p r e s e n t i n t h e sample p r i o r t o e x p e r i m e n t a t i o n . Research  s h o u l d be c a r r i e d o u t t o d e t e r m i n e  k n o w l e d g e o f c o r e m a t e r i a l on t h e a b i l i t y  the role of  to apply  heuristics. Combining H e u r i s t i c s the a l g e b r a i c and g e o m e t r i c  t o Solve Problems.  In both  t e s t s t h e s t u d e n t s were  given  133  t o t a l freedom i n approaching t h e problem.' instructions heuristics  T h e r e w e r e no  t o t h e students t o use e i t h e r , o r both o f t h e  i n t h e i r solutions,.although both  c o u l d be u s e d i n t h e t e s t data reveals  that  situations.  on t h e a l g e b r a i c  combined t h e h e u r i s t i c s  An e x a m i n a t i o n o f t h e  t e s t many  i n solving  heuristics  students  the problem  (See t a b l e  X V I I I and f i g u r e 28). Table Analysis  o f the Application  Heuristics  n~~,,~ G  r  o  u  p  1  2 * * *  Test  T> v. v, • 1 • ^ Probability  2  H  26  16  21  (H , HC)  HC  41  8  14  (H , O  C  50  8  5  *Not  o f Z e r o , One o r Two  on t h e A l g e b r a i c  Number o f Heuristics 0  (HC , c )  7.4287 20 .0918 5.1533  .0244 .00004 .0760  independent  Since t h i s significance  analysis  i s only exploratory,  no  l e v e l was a s s i g n e d f o r t e s t i n g t h e x  However, t h e r e s u l t s 0 - 1  XVIII  2  value.  are s i m i l a r to those obtained with the  d i c h o t o m y e m p l o y e d t o t e s t h y p o t h e s e s 1, 2 a n d 3-  T h a t i s , t h e H g r o u p p e r f o r m e d b e s t , t h e ,'.:c g r o u p w o r s t ,  Figure Application  28  o f Z e r o , One o r Two on t h e A l g e b r a i c  Heuristics  Test  50 ^5 40  35 Number o f Students  30 25 20 15 10  0  0  22-  Number o f H e u r i s t i c s Heuristics  o n l y group  Heuristic plus Content  (H)  c o n t e n t group  o n l y group  (C)  (HC)  135 and  t h e HC g r o u p i n b e t w e e n .  figure  The d a t a a r e r e p r e s e n t e d i n  28. In view o f these  f i n d i n g s the i n v e s t i g a t o r  t h a t f u r t h e r r e s e a r c h s h o u l d be u n d e r t a k e n best  to teach students  novel mathematical  to solve  The u s e o f b o o k l e t s  resulted  c o n s t r a i n t s on t h e l e a r n i n g s i t u a t i o n .  In  that booklets eliminated teacher-  and p e e r g r o u p I n t e r a c t i o n s f r o m t h e e x p e r i m e n t a l  p r o c e s s , a n d t h i s may h a v e h i n d e r e d their  how  problems.  p a r t i c u l a r i t Is noted student  to consider  t o combine h e u r i s t i c s  I n s t r u c t i o n a l Modes. in certain  feels  attempts  students i n  t o l e a r n how t o a p p l y h e u r i s t i c s .  more , t h e a n a l y s i s o f h i g h , m i d d l e achievement students to a student's  certain  tends  ability  and l o w r e a d i n g  t o suggest  t o apply  Further-  that reading Is related  a t l e a s t one h e u r i s t i c .  F u r t h e r r e s e a r c h s h o u l d employ o t h e r I n s t r u c t i o n a l modes f o r the t e a c h i n g o f h e u r i s t i c s . allow f o r student-teacher  Research  i n "normal"  An e x a m i n a t i o n algebraic group.  that I s , experiments  classroom  of the data i n d i c a t e  settings. t h a t on t h e  t e s t the H - A group s c o r e d h i g h e r than  Although  to  and p e e r g r o u p I n t e r a c t i o n s as  part o f the experimental procedure: s h o u l d be c o n d u c t e d  s h o u l d be d e s i g n e d  t h e r e was no s i g n i f i c a n t  overall  any o t h e r interaction,  f o r the purpose o f suggesting p o s s i b l e d i r e c t i o n s f o r  136  f u r t h e r r e s e a r c h , the I n t e r a c t i o n term this by  g r o u p was  investigated.  M a r a s c u i l o and  c a l c u l a t e d and formed.  The  study  and  =  1 2  - 421)  confirms  .048  t  w a s  J2  interval  .211  the e a r l i e r  significant  However, experiments  a significant  factor.  to the  interaction.  conclusion that In  this  differences involving  the  involving different vehicles  result  i n the vehicles''  T h i s r e m a i n s t o be  being  determined  by  research. Finally,  students'  i n view.of  the p o s s i b l e importance  core knowledge i n t e a c h i n g s t u d e n t s  h e u r i s t i c s , f u r t h e r r e s e a r c h • s h o u l d be grade l e v e l s . f a c t o r I t may  I f students' be  of mathematical  experience--  a n a l o g y , w h e r e he  he  1  c a n n o t be  a certain taught  to  T h i s c o u l d be p a r t i c u l a r l y  experience.  instruction i n heuristics  of mathematical  until  a s t u d e n t has  m a t u r i t y . ' I t may  to teach h e u r i s t i c s  breadth use  true  as  reached be  of  between  T h u s , i t may such  other  important  i s required to f i n d s i m i l a r i t i e s  c a n n o t be u n d e r t a k e n  designed  apply  c a r r i e d .out. a t  a s t u d e n t has  n o v e l p r o b l e m s and p r e v i o u s that e f f e c t i v e  to  of  c o r e k n o w l e d g e I s an  that u n t i l  certain heuristics.  level  suggested  y.  Scheffe confidence  i n s t r u c t i o n a l modes may  further  397  contribute significantly  t h e r e a r e no  vehicles.  the approach  intervaliis  does n o t  This simply  ( 1 9 7 0 , pp.  a 95 p e r c e n t  Y and  Levin  Using  associated with  be  analogy a  certain  t h a t a programme  should begin with  the  development of e x a m i n a t i o n o f cases i n the  elementary  grades, w i t h other h e u r i s t i c s being i n t r o d u c e d at stages.  later  138  BIBLIOGRAPHY A s h t o n , S i s t e r M. R.• H e u r i s t i c m e t h o d s i n p r o b l e m s o l v i n g i n n i n t h grade a l g e b r a . ( D o c t o r a l d i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y ) Ann A r b o r , M i c h i g a n : U n i v e r s i t y M i c r o f i l m s , 1962.  No.  62-02320.  B a l k , G. D. A p p l i c a t i o n o f h e u r i s t i c methods t o t h e of mathematics at s c h o o l . Educational Studies i n M a t h e m a t i c s , 1971,  3,  study  133-4(D.  B e c k e r , J . P. Research i n mathematics education: the r o l e o f t h e o r y and o f a p t i t u d e - t r e a t m e n t i n t e r a c t i o n s . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 1970, 1, 19-28. B e l l , E. T. Schuster,  Men o f m a t h e m a t i c s . 1937.  New  Y o r k : Simon  B e r n a r d , H. W. P s y c h o l o g y o f l e a r n i n g and New Y o r k : M c G r a w - H i l l , 1972.  and  teaching.  (3rd  ed  B l o o m , B. S., & B r o d e r , L. J . Problem-solving processes of college students. Supplementary E d u c a t i o n a l Monographs, C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s , 1950, No. 73. B r o w n , I . B., & W a l t e r , M. I . The r o l e s o f t h e s p e c i f i c and t h e g e n e r a l c a s e s i n p r o b l e m p o s i n g . Mathematics T e a c h i n g , 1972, 59, 52-4. B r u m b a u g h , R. S. P l a t o on t h e o n e ; t h e h y p o t h e s i s i n t h e parmenides. New Haven: Y a l e U n i v e r s i t y P r e s s , 1961. C a m p b e l l , D. T., & S t a n l e y , J . C. E x p e r i m e n t a l and q u a s i experimental designs f o r research. C h i c a g o : Rand M?Nally, 1963D a v i s , G. A. practice.  Psychology of problem s o l v i n g : theory New Y o r k : B a s i c B o o k s , 1973.  and  D e l l a - P i a n a , G. M., E l d r e d g e , G. M., & W o r t h e n , B. 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( D o c t o r a l d i s s e r t a t i o n , O h i o S t a t e ) Ann A r b o r , M i c h i g a n : U n i v e r s i t y M i c r o f i l m s , 1954. No. 59-02318. S c o t t , D. I n v e s t i g a t i o n o f a 'maths t o d a y ' p r o b l e m : an exercise i n pattern-spotting. Mathematics Teaching, 1971, 56, 38-45.  S h u l m a n , L. S. P s y c h o l o g i c a l c o n t r o v e r s i e s i n t h e t e a c h i n g of s c i e n c e and mathematics. S c i e n c e T e a c h e r , 1968, _35 34-38, 89.  S m i t h , E. P., & H e n d e r s o n , K. B. P r o o f . I n , The g r o w t h o f m a t h e m a t i c a l I d e a s g r a d e s K - 12. Y e a r b o o k o f t h e N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s , 1959 , 24, 111-57.  Snow, R. E. T h e o r y c o n s t r u c t i o n f o r r e s e a r c h on t e a c h i n g . 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 o f r e s e a r c h on teaching. C h i c a g o : Rand M ? N a l l y , 1973S z a b o , S. Some r e m a r k s on d i s c o v e r y . T e a c h e r , 1967, 60, 839-42  The M a t h e m a t i c s  Topics i n mathematics f o r elementary s c h o o l t e a c h e r s . B o o k l e t 17: H i n t s f o r p r o b l e m s o l v i n g . W a s h i n g t o n , D. The N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s , 1969. T r a v e r s , R. M. W. An I n t r o d u c t i o n t o e d u c a t i o n a l (3rd e d . ) New Y o r k : M a c M l l l a n , 196*9. W e b s t e r ' s s e v e n t h new c o l l e g i a t e L o n d o n : G. B e l l , 1967.  dictionary.  C:  research.  ( 7 t h ed.)  W i l l s I I I , H. T r a n s f e r o f p r o b l e m s o l v i n g a b i l i t y g a i n e d t h r o u g h l e a r n i n g by d i s c o v e r y . ( D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f I l l i n o i s ) Ann A r b o r , M i c h i g a n : U n i v e r s i t y M i c r o f i l m s , 1967. No. 67-11937. W i l s o n , J . W. G e n e r a l i t y o f h e u r i s t i c s a s an I n s t r u c t i o n a l variable. (Doctoral dissertation, Stanford University) Ann A r b o r , M i c h i g a n : U n i v e r s i t y M i c r o f i l m s , 1967. No. 67-17526.  144 W l t t r o c k , M. C. The l e a r n i n g by d i s c o v e r y E d u c a t i o n a l Resources Information Center . R e p r o d u c t i o n S e r v i c e , Post O f f i c e Drawer M a r y l a n d , 20014. Document R e f e r e n c e : ED  hypothesis. Document 0, B e t h e s d a , 003 340.  W o r t h e n , B. R. Discovery vs. expository sequencing In s i x weeks o f c l a s s r o o m I n s t r u c t i o n . I n D e l l a - P i a n a , G. M. _ t _ . _ a l . , S e q u e n c e c h a r a c t e r i s t i c s , o f t e x t m a t e r i a l s and t r a n s f e r l e a r n i n g . B u r e a u o f E d u c a t i o n a l R e s e a r c h , U n i v e r s i t y o f U t a h , P a r t I , 1965 ( a ) ; P a r t I I , 1965 (b).  145  APPENDIX A  Heuristics  only  - Neutral  H - N  Vehicle  146 INTRODUCTION The o n l y p u r p o s e o f t h i s b o o k l e t i s t o t e a c h y o u how t o make d i s c o v e r i e s a b o u t new m a t h e m a t i c a l t o p i c s . The b o o k l e t deals w i t h 3 t o p i c s , which are: 1)  How  2)  Tiling  3)  to organize  seating plans,  problems,  Electrical  Switching  Networks.  A t f i r s t g l a n c e t h e s e t o p i c s do n o t a p p e a r t o be mathematical i n nature. However, t h e t o p i c s have been c h o s e n to i n t r o d u c e you t o problem s o l v i n g t e c h n i q u e s f o r making m a t h e m a t i c a l d i s c o v e r i e s , and s e r v e no o t h e r p u r p o s e . When w o r k i n g t h r o u g h t h i s b o o k l e t remember t h a t y o u a r e t r y i n g t o l e a r n the problem s o l v i n g t e c h n i q u e s , not the content i t s e l f . The b o o k l e t w i l l t a k e 10 d a y s t o c o m p l e t e . 3 days are s p e n t on e a c h t o p i c , and 1 d a y a t t h e end i s u s e d f o r r e v i e w . A t t h e end o f most d a y s t h e r e i s a s h o r t q u i z . T h i s w i l l be m a r k e d and r e t u r n e d t h e n e x t d a y . You may k e e p t h e b o o k l e t when i t i s c o m p l e t e . Feel free t o w r i t e i n i t . Make a n y comments t h a t y o u t h i n k w i l l be h e l p f u l t o you.  - 1 -  147 SEATING PLANS First,  l e t us e x p l a i n  the n o t a t i o n  t h a t w i l l be used.  Notation  A A A A A  i n d i c a t e s t h a t a c h a i r i s emptyI n d i c a t e s t h a t a c h a i r i s o c c u p i e d . When a c h a i r i s o c c u p i e d , but we do not know the name o f the person s i t t i n g t h e r e ; we w i l l use one o f the l e t t e r s , x, y, o r z i n s i d e the triangle. i n d i c a t e s t h a t the c h a i r i s o c c u p i e d by the person and we do know t h e i r name. F o r example, suppose t h e r e a r e two lawyers a t t e n d i n g a meeting, Mr. Abel ( l a b e l l e d a) and Mr. Brock (labelled b). Then... i n d i c a t e s t h a t Mr. Abel I s o c c u p y i n g the c h a i r , i n d i c a t e s t h a t Mr. B r o c k i s o c c u p y i n g the c h a i r , I n d i c a t e s t h a t the c h a i r Is o c c u p i e d , but we do not know which o f t h e men i s . s i t t i n g there,  A A  i n d i c a t e s t h a t the l e f t c h a i r i s o c c u p i e d by Mr. Abel, and the r i g h t seat I s empty.  -  3 -  148  Exercise A meeting has been arranged between 4 d i r e c t o r s o f a f i s h p l a n t . They are Mr. Brace ( l a b e l l e d b ) , Mr. Drover ( d ) , Mr. Goodyear ( g ) , and Mrs. Murphy (m). The o r g a n i z e r s have p l a c e d 4 c h a i r s i n a row. Chair 1  Chair 2  A  Chair 3  A  A  Draw diagrams to r e p r e s e n t s e a t i n g arrangements. e.g.  C h a i r 1 - Mr.  Chair 4  A  each o f the  Drover; C h a i r s 2 and  following  3 -  occupied;  C h a i r 4 - empty. The  Now (i)  diagram would  A  A  A  A  draw the diagrams f o r the f o l l o w i n g : C h a i r 1 - Mr. C h a i r s 3 -nd  (ii)  be  Brace; 4 -  C h a i r 1 - empty; C h a i r 3 - Mr.  (iij) C h a i r 1 - Mr.  C h a i r 2 - Mrs.  Murphy;  occupied. C h a i r 2 - Mrs.  Brace;  Murphy;  C h a i r 4 - Mr.  Goodyear.  Drover; A l l o t h e r c h a i r s  occupied.  ( i v ) A l l 4 c h a i r s empty. (v)  C h a i r 1 - Mrs.  Murphy;  C h a i r 2 - Mr.  C h a i r 3 - Mr.  Goodyear;  C h a i r 4 - Mr.  ( v i ) C h a i r 1 - Mr.  Goodyear;  C h a i r s 2 and  Chair 4 -  occupied.  -4-  Brace; Drover. 3 - empty;  149  speaker's desks  audience  Above i s the p l a n o f the s m a l l meeting room i n a l a r g e company's head o f f i c e . Three branch managers, Mr. Blade, Mr. Johnson and Mr. Hamilton have been i n v i t e d t o t a l k t o a group o f head o f f i c e p e r s o n n e l . S i n c e the audience does n o t know the t h r e e men, you have t o p l a c e name p l a t e s on the desks. Exercise The t h r e e managers w i l l be s e a t e d a t the speaker's desks. How many d i f f e r e n t ways c o u l d you s e a t the t h r e e men? Draw diagrams t o r e p r e s e n t each o f the p o s s i b l e s e a t i n g p l a n s . e.g.  One p l a n would be  A A A where b stands f o r Mr. Blade, e t c . As i n the above example, i n w r i t i n g out your s o l u t i o n you need o n l y draw the c h a i r s . There i s no need t o draw the room and s p e a k e r s ' d e s k s . Now draw the o t h e r diagrams  - 8 -  below.  150  Solution There are 5 o t h e r Plan Plan  A A A »• A .A -A i.  Plan 3. Plan Plan Plan  5 6  AA AA AA AA AA AA  In your s o l u t i o n the p l a n s l i s t e d , a l t h o u g h they may have been i n a d i f f e r e n t order. For most people, l i s t i n g the v a r i o u s plans Is a m a t t e r of t r i a l and e r r o r . That i s , you t r y one p l a n , and then another, hoping t h a t i n the end you w i l l have l i s t e d a l l p o s s i b l e arrangements. I t i s now t h a t the f i r s t problem s o l v i n g t e c h n i q u e can be o f h e l p . T h i s problem s o l v i n g t e c h n i q u e i s c a l l e d examination o f c a s e s . The f i r s t stage of the technique I n v o l v e s the d e v e l o p i n g a s y s t e m a t i c method f o r w r i t i n g down a l l the d i f f e r e n t c a s e s . T h i s p a r t i c u l a r e x e r c i s e i s f a i r l y simple, s i n c e I t involves only 6 d i f f e r e n t seating plans. However, i t w i l l p r o v i d e a good example w i t h which to i n t r o d u c e the t e c h n i q u e . The next two pages d i s c u s s how a s y s t e m a t i c approach can be used t o l i s t the 6 plans g i v e n above.  - 9 -  151  The problem Is to l i s t a l l p o s s i b l e ways o f s e a t i n g 3 people In 3 c h a i r s . L a b e l the c h a i r s as f o l l o w s . Chair 1  Chair 2  Chair 3  S i n c e a l l 3 c h a i r s are to be o c c u p i e d , the f i r s t s t e p was to p l a c e one o f the 3 managers i n c h a i r 1. You c o u l d s t a r t w i t h any o f the 3, but i n t h i s example Mr. Blade was chosen. T h i s gave the f o l l o w i n g s i t u a t i o n .  A A A T h i s l e f t 2 people and 2 c h a i r s t o f i l l . be done i n 2 ways, l e a d i n g to plans 1 and 2.  A  Plan 1. Plan 2.  ^  This could  A  ^  ^  We have now c o n s i d e r e d a l l p o s s i b l e p l a n s Mr. Blade s i t t i n g i n c h a i r 1.  with  The next move was t o p l a c e one o f the o t h e r managers i n c h a i r 1. In t h i s example Mr. Johnson was chosen. This gave  A A A  to  Again t h i s l e f t p l a n s "3 and 4.  Plan 3.  2 people  and 2 c h a i r s to f i l l ,  leading  £\> fi\  Plan 4.  / \  /n\  F i n a l l y Mr. Hamilton was p l a c e d i n the f i r s t l e a d i n g t o the l a s t 2 p l a n s . Plan 5.  ^  Plan 6.  y/n\  A  chair,  A /i\  We have e l i m i n a t e d a l l the cases w i t h any o f the 3 men s i t t i n g In c h a i r 1. But t h e r e are o n l y 3 men i n t h e problem, and s i n c e the f i r s t c h a i r must be o c c u p i e d we have w r i t t e n down a l l the c a s e s . -10-  152  U s i n g a s y s t e m a t i c approach t o l i s t i n g cases ( i n t h i s example the cases were d i f f e r e n t s e a t i n g p l a n s ) has 2 major advantages:(1) You can be sure t h a t a l l p o s s i b l e cases have been listed. You do not have t o worry t h a t some might have been l e f t out. For example, i n t h e p r e v i o u s e x e r c i s e a l l the cases w i t h Mr. Blade i n c h a i r 1 were w r i t t e n down b e f o r e c o n s i d e r i n g anyone e l s e s i t t i n g t h e r e . (2) You can be sure t h a t no cases have been r e p e a t e d . T h i s may not be v e r y important when you have o n l y 5 cases t o c o n s i d e r , but i f you have 60 d i f f e r e n t p o s s i b i l i t i e s you do not want t o check t h a t t h e y a r e a l l d i f f e r e n t , and t h a t you have not r e p e a t e d some by m i s t a k e . In the p r e v i o u s e x e r c i s e the s y s t e m a t i c approach a s s u r e d us t h a t the 6 cases l i s t e d were a l l d i f f e r e n t and t h a t t h e r e were no o t h e r p o s s i b l e s e a t i n g p l a n s , A s y s t e m a t i c method w i l l h e l p when you want t o examine a l l c a s e s . As you go through t h i s b o o k l e t , you w i l l have more p r a c t i c e i n l i s t i n g cases by a s y s t e m a t i c method.  - 11 -  153 Since o n l y 3 days are b e i n g spent on s e a t i n g plans t h i s i s the l a s t day t h a t w i l l be spent on t h i s t o p i c . The next s e c t i o n w i l l i n t r o d u c e the second o f the problem s o l v i n g t e c h n i q u e s , namely analogy, and f u r t h e r develop examination o f c a s e s . I t may seem t h a t a g r e a t d e a l o f time i s b e i n g spent on t h i s p a r t i c u l a r s t e p o f examination of cases, namely l i s t i n g cases i n a s y s t e m a t i c manner. As you work through the o t h e r s e c t i o n s you w i l l see the v i t a l importance o f t h i s s t e p i n s o l v i n g problems you have not seen b e f o r e . In o r d e r to make the remainder o f today's l e s s o n more r e a l i s t i c the f o l l o w i n g problem w i l l be c o n s i d e r e d . However, as you attempt the- problems d_o not f o r g e t t h a t the o b j e c t i v e o f t h i s s e c t i o n i s to develop p r a c t i c e w i t h a s y s t e m a t i c approach to l i s t i n g cases and not w i t h the t o p i c of s e a t i n g plans. T h i s i s p a r t o f a map  The  of A f r i c a .  l e t t e r s stand f o r the f o l l o w i n g c o u n t r i e s . S i e r r a Leone (s) ;  Liberia  I v o r y Coast  (t)  (i)  ;  Togo ( t )  Upper V o l t a (u)  ;  Guinea  -40-  (gu)  ;  Ghana  ;  Mali  (g) (m)  154 As you a r e aware, many d e v e l o p i n g c o u n t r i e s have b o r d e r arguments w i t h t h e i r n e i g h b o r s . This i s true o f A f r i c a n c o u n t r i e s , so the n a t i o n s d e c i d e d t o s e t up The O r g a n i z a t i o n o f A f r i c a n S t a t e s t o h e l p the v a r i o u s n a t i o n s settle disputes. The O r g a n i z a t i o n a l s o h e l p s i n many o t h e r ways. C o n s i d e r the f o l l o w i n g h y p o t h e t i c a l problems t h a t c o u l d a r i s e between the c o u n t r i e s on the p r e v i o u s page. Note:The s o l u t i o n s a r e n o t i n c l u d e d . You w i l l r e c e i v e them when you have completed the b o o k l e t . Exercise In each o f the f o l l o w i n g l i s t a l l the s e a t i n g arrangements t h a t would s a t i s f y the c o n d i t i o n s . Before l i s t i n g the arrangements g i v e a_ b r i e f d e s c r i p t i o n o f the s y s t e m a t i c approach you i n t e n d to u s e . Spend some time t h i n k i n g about the approach b e f o r e s t a r t i n g t o l i s t the cases. T r y as many o f the problems as.you can. DO NOT WORRY I F YOU DO NOT COMPLETE ALL THE EXAMPLES. Do as many as you can i n the time. 1. A s p e c i a l meeting has been c a l l e d t o d i s c u s s the b o r d e r d i s p u t e s between the f o l l o w i n g 4 c o u n t r i e s . S i e r r a Leone (s) Liberia {£) Ivory  Coast  u)  Cj) Representatives as f o l l o w s . Ghana  o f the 4 c o u n t r i e s w i l l be seated A A A A  ZnA  Z*\  S i n c e these c o u n t r i e s a r e h a v i n g problems i t i s a d v i s a b l e not t o s e a t r e p r e s e n t a t i v e s from b o r d e r i n g s t a t e s together.  -41-  155  TilIng  Problems  Today you w i l l s t a r t the second t o p i c t o be d i s c u s s e d i n t h i s b o o k l e t , namely t i l i n g problems. The i d e a o f t i l i n g can be c o n s i d e r e d as a k i n d o f extended j i g s a w p u z z l e , where the q u e s t i o n asked i s how many ways a p a r t i c u l a r o b j e c t can be f i t t e d i n t o a g i v e n space. In the f i r s t s e c t i o n you were i n t r o d u c e d to the problem s o l v i n g t e c h n i q u e o f examination o f c a s e s . In t h i s s e c t i o n the second t e c h n i q u e , namely analogy, w i l l be used. Furthermore, you w i l l see how examination o f cases can be used to t e s t c o n j e c t u r e s (educated guesses)~~developed by analogy. As w i t h s e a t i n g p l a n s , the t o p i c o f t i l i n g i s not important. It i s j u s t used t o i n t r o d u c e a n a l o g y . On the p r e v i o u s page you saw a p i c t u r e o f a d e c o r a t o r w i t h a t i l e to f i t i n t o a w a l l . H i s problem i s how many d i f f e r e n t ways can he f i t the t i l e i n t o the w a l l . Let's c o n s i d e r the f o l l o w i n g problem. Problem A d e c o r a t o r i s h i r e d t o t i l e a bathroom. The owner o f the house has bought a s p e c i a l t r i a n g u l a r t i l e t o f i t i n t o the w a l l . T h i s t i l e i s i n f a c t an e q u i l a t e r a l t r i a n g l e . That i s , a t r i a n g l e w i t h a l l 3 s i d e s b e i n g o f e q u a l l e n g t h . The f r o n t o f the t i l e i s d i v i d e d i n t o 3 s e c t i o n s , each b e i n g a different colour. The back o f the t i l e i s covered w i t h cement, so t h a t i t w i l l s t i c k i n p l a c e . yellow blue red The d e c o r a t o r has completed the w a l l except f o r the space where the t i l e i s t o f i t . He i s now f a c e d w i t h a problem - how many d i f f e r e n t ways can the t i l e be f i t t e d Into the space?  -43-  «  156  Solution The t i l e  i s i n p o s i t i o n 2.  Exercise are  To save you r e f e r r i n g back the 3 p o s i t i o n s g i v e n below.  Ac  o  A  Position 2  Position 1  Now complete the f o l l o w i n g Initial Position (a)  f o r the t i l e  Position 3  table. Final Position  Rotation  1  240° c l o c k w i s e  (see below)  1  120° c l o c k w i s e  2  120° c o u n t e r c l o c k w i s e  3  120° c o u n t e r c l o c k w i s e  2  360° c l o c k w i s e  To f i n d the f i n a l p o s i t i o n i n (a) imagine the t i l e i s i n p o s i t i o n 1. Then t h i n k how i t moves when r o t a t e d t h r o u g h 240° i n a c l o c k w i s e d i r e c t i o n . You o b t a i n p o s i t i o n 3.  A ft  £  c  Initial Position  <r  7>  240° c l o c k w i s e  You would pla.ce a 3 i n the t a b l e .  -48-  A  Final Position  157 Solution Initial Position  Final Position  Rotation  1  240° c l o c k w i s e  3  1  120°  clockwise  2  2  120°  counterclockwise  1  3  120°  counterclockwise  2  clockwise  2  2  360°  From the r e s u l t s above I t would seem reasonable to conclude t h a t t h e r e were no o t h e r p o s i t i o n s f o r . the t i l e . However, can the d e c o r a t o r be sure? C l e a r l y with t h i s p a r t i c u l a r s i t u a t i o n i n v o l v i n g a t r i a n g u l a r t i l e d e c o r a t e d on 1 s i d e , -he c o u l d be r e a s o n a b l y sure. By r o t a t i n g the t i l e backward and forward he would soon conclude t h a t t h e r e were o n l y 3 p o s i t i o n s . However, i f he j u s t f i t s the t i l e i n t o the w a l l u s i n g a t r i a l and e r r o r approach t h i s would not h e l p him i n a more complex s i t u a t i o n . We w i l l now d i s c u s s a g e n e r a l approach t h a t can be used to a t t a c k t h i s problem. In d i s c u s s i n g how to approach t h i s problem from a s y s t e m a t i c p o i n t o f view we w i l l l a y the base f o r i n t r o d u c i n g the second problem s o l v i n g t e c h n i q u e , analogy. Imagine the d e c o r a t o r p i c k s up the t i l e and f i t s i t i n t o the w a l l . L e t the base p o s i t i o n , t h a t i s the f i r s t p o s i t i o n he t r i e s , be the one below.  Base  Position  The movement the d e c o r a t o r can use p o s i t i o n to another i s a r o t a t i o n . -  -49-  t o move from  one  158 Read  carefully • EXAMINATION OF CASES  Step 1  If possible,  f i x an o b j e c t as a s t a r t i n g  point.  Step 2  Apply any c o n d i t i o n s t h a t may the objects.  Step _3  Apply movements, o p e r a t i o n s , o r a r r a n g e o b j e c t s , a c c o r d i n g to the problem. Use a s y s t e m a t i c procedure to do t h i s .  be s t a t e d c o n c e r n i n g  Step k_ Look f o r a p a t t e r n . Let us see how these s t e p s a p p l y t o the two types o f problem we have been d e a l i n g with, namely s e a t i n g p l a n s and t i l i n g problems. Examination o f Cases as a p p l i e d t o t i l i n g in a starting  problems  Step 1  F i x the t i l e  Step 2  There were no c o n d i t i o n s o t h e r than the t i l e i n t o the w a l l .  Step 3  T i l e i s r o t a t e d i n a s y s t e m a t i c way. F i r s t looking at c l o c k w i s e , then c o u n t e r c l o c k w i s e r o t a t i o n s .  •Step k  The f i r s t 2 r o t a t i o n s g i v e you 3 d i f f e r e n t p o s i t i o n s . The base p o s i t i o n and those l a b e l l e d p o s i t i o n s 2 and 3. L o o k i n g a t the remainder o f the t a b l e you can see a p a t t e r n ; the same p o s i t i o n s o c c u r a g a i n i n a d e f i n i t e order. From t h i s you can conclude t h a t no new p o s i t i o n s will arise.  Examination o f Cases as a p p l i e d  position.  to s e a t i n g plans  Step 1  F i x a g i v e n person i n a s e a t .  Step 2  Apply c o n d i t i o n s . In some o f the problems people had to s i t i n p a r t i c u l a r s e a t s .  Step 3_  Other people were s e a t e d i n a s y s t e m a t i c  Step ii,  Had not been d i s c u s s e d and was seating plans.  If n e c e s s a r y the procedure was people i n d i f f e r e n t s e a t s .  -53-  must f i t  different way.  not a p p r o p r i a t e t o repeated f o r d i f f e r e n t  159  Read the f o l l o w i n g care f u l l y At the b e g i n n i n g o f the b o o k l e t i t was s t a t e d t h a t the aim was to i n t r o d u c e you to some problem s o l v i n g t e c h n i q u e s . T h i s page w i l l d e a l w i t h analogy. We have used analogy on two d i f f e r e n t o c c a s i o n s . I. When we were f i r s t f a c e d w i t h a. new problem. The problem I n v o l v i n g the 2 - s i d e d t i l e was i n t r o d u c e d . Analogy was used by l o o k i n g at what approach had beeri taken with the one s i d e d tile. T h i s l e d t o the Idea t h a t r o t a t i o n s might be o f v a l u e w i t h the new problem. II. When we i n v o l v i n g , the positions for from the base movement. We were p o s s i b l e p o s i t i o n with, enabled us t o r o t a t i o n s and  t r i e d to extend our knowledge o f movements 2-sided t i l e . We had found the 6 d i f f e r e n t the t i l e . We knew t h a t i t was p o s s i b l e to move p o s i t i o n to p o s i t i o n s 2 and 4 w i t h a s i n g l e t r i e d t o extend our knowledge by a s k i n g i f i t to move from the base p o s i t i o n to any o t h e r a s i n g l e movement. Again we used analogy which f i n d the o t h e r movements, which were d i f f e r e n t flips.  N o t i c e how i n I I what we had a l r e a d y found out about the movement o f a 2 - s i d e d t i l e l e d us t o use analogy t o f i n d out more. At the moment the i d e a o f analogy i s p r o b a b l y vague. Don't worry about t h i s . As you work through the remainder o f the b o o k l e t , and have more p r a c t i c e w i t h analogy, the i d e a w i l l become c l e a r e r .  -66-  160 Today we w i l l be' s t a r t i n g the. f i n a l s e c t i o n o f the booklet. The t o p i c t h a t w i l l be d i s c u s s e d i s t h a t o f e l e c t r i c a 1 s w i t c h i n g networks. You a r e f a m i l i a r w i t h switches from your everyday e x p e r i e n c e but you have p r o b a b l y never c o n s i d e r e d them as a t o p i c i n mathematics. As we develop the t o p i c you v ; i l l a l s o have p r a c t i c e i n a p p l y i n g the problem s o l v i n g t e c h n i q u e s o f analogy and examination o f c a s e s . Most s w i t c h e s , whatever t h e r e s i z e , shape, c o l o u r , e t c . , have one p r o p e r t y i n common, namely they a r e e i t h e r open o r closed. There i s no h a l f way s t a g e . You might l i k e t o t h i n k of a s w i t c h as an e l e c t r i c a l drawbridge. I f the b r i d g e i s up (the s w i t c h i s open) the t r a f f i c cannot move ( t h e c u r r e n t cannot f l o w ) . I f the b r i d g e i s down ( t h e s w i t c h i s c l o s e d ) the t r a f f i c moves (the c u r r e n t f l o w s ) . The p o s i t i o n o f the s w i t c h i s r e f e r r e d t o as the s t a t e o f the s w i t c h . Let A be a s w i t c h . in this section.  A  The f o l l o w i n g diagrams w i l l  be used  The s t a t e o f the s w i t c h i s unknown. We do n o t know i f the s w i t c h i s open o r c l o s e d , and hence we do n o t know i f the c u r r e n t w i l l f l o w . The s t a t e o f the s w i t c h i s open w i l l not f l o v i . The s t a t e o f the s w i t c h i s c l o s e d . current w i l l flow.  -  91 -  The  current  The  161 Exercise  o  ?  A  B  ,  op(A 0  B Under what c o n d i t i o n s w i l l the c u r r e n t flow through t h i s network? You should l o o k a t the d i f f e r e n t combinations o f s t a t e s and draw the networks. Try to l i s t  the cases  i n a s y s t e m a t i c manner,  -  100 -  162  When u s i n g examination o f cases to c o n s i d e r networks, the f o l l o w i n g r e s u l t s were o b t a i n e d .  A  B  These networks are The  current  flows  B  different  A  equivalent. i f A and  B are  closed.  (ii)  B  A  These networks are The  current  flows  equivalent. i f AjOr B,  o r both are  closed.  Exercise Can you t h i n k o f any mathematical o p e r a t i o n s you o f c o n n e c t i n g s w i t c h e s ? Hint:Look a t the u n d e r l i n e d words.  - 117  -  t h a t remind  163  Solution I f you have found o p e r a t i o n s t h a t remind you o f c o n n e c t i n g switches, t u r n t o the next page. I f not, the f o l l o w i n g comment may h e l p . In the networks on the p r e v i o u s page the switches have been c o m b i n e d ( j o i n e d ) . A l s o the o r d e r i n which they have been j o i n e d does not matter. That i s  A  B  i s equivalent to  -  118 -  B  A  164  Solution  continued  You c o u l d have chosen many o p e r a t i o n s t h a t remind you of connecting switches. F o r example, you c o u l d have thought o f any o f the f o l l o w i n g . Multiplication; v  Addition;  Union o f S e t s ; II  We must d e c i d e which analogy t o make. be used i s w i t h s e t t h e o r y .  Intersection o f Sets:  The analogy  Let us c o n s i d e r the d e f i n i t i o n o f Unicnand of Sets. I f X and Y a r e s e t s then  that w i l l Intersection  Xu Y i s the s e t o f elements i n X o r Y , or b o t h . Xfl Y i s the s e t o f elements i n X andY . If  we c o n s i d e r switches l a b e l l e d  A and B we have  C u r r e n t flows i f A o r B, o r both a r e c l o s e d .  B  Current flows i f A and B a r e c l o s e d .  A  B  Exercise Which network would you draw t o correspond Which one t o correspond t o A B ? r t  -  119 -  to A y B ?  Solution For two s w i t c h e s , A and B,  Ay B = B u A  Read the f o l l o w i n g care f u l l y Before c o n t i n u i n g to develop the analogy w i t h s e t theory, l e t us review the power o f analogy. We were I n i t i a l l y faced w i t h a s e t o f problems i n v o l v i n g s w i t c h i n g networks, a new t o p i c i n mathematics. By t r y i n g t o compare networks w i t h another mathematical system, a system w i t h which we a r e more f a m i l i a r , we a r e now l o o k i n g a t s e t .theory. I t i s hoped t h a t we w i l l be a b l e t o use many o f the p r o p e r t i e s a s s o c i a t e d w i t h set t h e o r y t o h e l p s o l v e problems w i t h networks. Hence, a n a l o g y has taken us from a t o t a l l y new t o p i c , s w i t c h i n g networks, i n t o a more f a m i l i a r t o p i c , s e t t h e o r y . As you work through the remainder o f t h i s s e c t i o n you w i l l see how s e t t h e o r y w i l l h e l p us w i t h network problems. As mentioned above we hope to use the p r o p e r t i e s o f set t h e o r y t o h e l p w i t h network problems. In p a r t i c u l a r , we would l i k e t o use the p r o p e r t i e s a s s o c i a t e d w i t h union and i n t e r s e c t i o n t o f i n d e q u i v a l e n t networks. In o r d e r t o do t h i s we w i l l have t o examine the v a r i o u s p r o p e r t i e s a s s o c i a t e d w i t h Union and I n t e r s e c t i o n .  - 121 -  APPENDIX.B  H e u r i s t i c s plus  content  HC  - Neutral  - N  Vehicle  167  INTRODUCTION T h e r e a r e two p u r p o s e s t o t h i s b o o k l e t . To t e a c h y o u how t o make d i s c o v e r i e s a b o u t new m a t h e m a t i c a l t o p i c s , and t o show y o u how m a t h e m a t i c a l i d e a s can h e l p y o u s o l v e p r o b l e m s taken from areas o t h e r than mathematics. The  booklet deals with 3 topics,  1) How  to organize seating plans,  2) T i l i n g 3)  which are:  problems,  Electrical  switching  networks.  A t f i r s t g l a n c e t h e s e t o p i c s may n o t a p p e a r t o be v e r y mathematical In n a t u r e . However, as y o u work t h r o u g h t h e b o o k l e t y o u w i l l see how t h e m a t h e m a t i c a l I d e a s developed t h r o u g h t h e use o f d i s c o v e r y t e c h n i q u e s w i l l h e l p you d e a l w i t h these t o p i c s . The b o o k l e t w i l l t a k e 10 d a y s t o c o m p l e t e . 3 days are s p e n t on e a c h t o p i c , and 1 day a t t h e end i s u s e d f o r r e v lev.. A t t h e end o f most days, t h e r e I s a s h o r t q u i z . T h i s w i l l be marked and r e t u r n e d t h e n e x t d a y . You may k e e p t h e b o o k l e t when i t i s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make any comments t h a t y o u t h i n k w i l l be h e l p f u l t o y o u .  -1-  168  SEATING PLANS You are probably wondering why t h i s t o p i c i s important. The f o l l o w i n g are two examples o f where s e a t i n g plans can p l a y , o r played, a r o l e . At the h i g h e s t l e v e l o f diplomacy, approximately one year was spent d i s c u s s i n g the s e a t i n g arrangements f o r the P a r i s Peace T a l k s on Vietnam. The f i n d i n g o f a s e a t i n g p l a n that was acceptable to a l l the p a r t i e s was e s s e n t i a l . This p l a n had to be found before the t a l k s c o u l d s t a r t . The o t h e r example i s o f a more p e r s o n a l n a t u r e . Some f r i e n d s ( o r maybe your parents) have to arrange the s e a t i n g p l a n f o r a wedding. You would p r e f e r to s i t next to Jane, not Tom. There are a l a r g e number o f guests and many of them have s i m i l a r requests concerning there own s e a t s . Your f r i e n d s are faced with the d i f f i c u l t task o f t r y i n g to organize a s e a t i n g p l a n t h a t w i l l keep a l l the guests happy. These are o n l y two o f many s i t u a t i o n s where s e a t i n g plans p l a y an important r o l e . For example, i n t e r n a t i o n a l conferences, c i v i c luncheons, board meeting, e t c . q u i t e o f t e n i n v o l v e s e t t i n g up s e a t i n g p l a n s . In many cases you are i n v i t i n g Important people who could be offended I f they were not s i t t i n g i n the ' c o r r e c t ' p l a c e . Before l o o k i n g at s e a t i n g problems the next page w i l l i n t r o d u c e the n o t a t i o n t h a t w i l l be used i n t h i s s e c t i o n .  -3-  169  First,  l e t us e x p l a i n  the n o t a t i o n  that w i l l be used.  Notation  A A  i n d i c a t e s that a c h a i r i s emptyi n d i c a t e s that a c h a i r i s o c c u p i e d . When a c h a i r i s occupied, but we do not know the name o f the person s i t t i n g there, we w i l l use one o f the l e t t e r s , x, y, o r z i n s i d e the triangle. i n d i c a t e s t h a t the c h a i r i s occupied by the person and we do know t h e i r name. For example, suppose there are two-lawyers a t t e n d i n g a meeting, Mr. Abel ( l a b e l l e d a) and Mr. Brock (labelled b). Then... i n d i c a t e s that Mr. Abel i s occupying the c h a i r . Indicates  that Mr. Brock i s o c c u p y i n g the c h a i r .  i n d i c a t e s t h a t the c h a i r i s occupied, but we do not know which o f the men i s s i t t i n g there  A A  i n d i c a t e s that the l e f t c h a i r i s occupied by Mr. Abel, and the r i g h t seat i s empty.  -  4  -  170  Exercise A meeting has been arranged between 4 d i r e c t o r s o f a f i s h p l a n t . They are Mr. Brace ( l a b e l l e d b), Mr. Drover ( d ) , Mr. Goodyear ( g ) , and Mrs. Murphy (m). The o r g a n i z e r s have placed 4 c h a i r s i n a row. Chair 1  Chair 2  A  A  Chair 3  A  Draw diagrams to r e p r e s e n t s e a t i n g arrangements. e.g.  C h a i r 1 - Mr.  Chair 4  A  each o f the f o l l o w i n g  Drover; C h a i r s 2 and  3 -  occupied;  C h a i r 4 - empty. The  Now (i)  diagram would  A  C h a i r 1 - Mr.  Brace; 4 -  C h a i r 1 - empty;  following:  C h a i r 2 - Mrs.  Murphy;  occupied. Chair 2 -  Mrs . Murphy;  C h a i r 4 - Mr.  Goodyear.  C h a i r 3 - Mr.  Brace;  C h a i r 1 - Mr.  Drover; A l l o t h e r c h a i r s  (iv) A l l (v)  A . A  draw the diagrams f o r the C h a i r s , 3 and  (ii)  A  be  occupied.  4 c h a i r s empty.  C h a i r 1 - Mrs. C h a i r 3 - Mr.  Murphy; Goodyear;  ( v i ) C h a i r 1 - Mr. Goodyear; Chair 4 - occupied.  C h a i r 2 - Mr.  Brace;  C h a i r -4 - Mr.  Drover.  C h a i r s 2 and  3 - empty;  -5-  171  A A A  speaker's desks  audience  Above i s the p l a n o f the small meeting room i n a l a r g e company's head o f f i c e . Three branch managers, Mr. Blade, Mr. Johnson and Mr. Hamilton have been i n v i t e d to t a l k to a group o f head o f f i c e p e r s o n n e l . Since the audience does not know the three men, you have t o p l a c e name p l a t e s on the desks. Exercise The t h r e e managers w i l l be seated a t the speaker's d e s k 3 .  How many d i f f e r e n t ways could you seat the three men? Draw diagrams to r e p r e s e n t each o f the p o s s i b l e s e a t i n g p l a n s . e.g.  One p l a n would be  where b stands f o r Mr. Blade, e t c . As i n the above example, i n w r i t i n g out your s o l u t i o n you need o n l y draw the c h a i r s . There i s no need t o draw the room and speakers' desks. Now draw the o t h e r diagrams  - 9 -  below.  172 Solution  Plan 1. Plan 2. Plan 3Plan Plan Plan 6.  AAA AAA AAA AAA AAA  In your s o l u t i o n to t h i s problem you probably had a l l the plans l i s t e d , although they may have been i n a d i f f e r e n t order. For most people, l i s t i n g the v a r i o u s plans i s a matter of t r i a l and e r r o r . That i s , you t r y one plan, and then another, hoping that i n the end you w i l l have l i s t e d a l l p o s s i b l e arrangements. I t i s now that the f i r s t problem s o l v i n g technique can be of h e l p . T h i s problem s o l v i n g technique i s c a l l e d examination of cases. The f i r s t stage of the technique i n v o l v e s the d e v e l o p i n g o f a s y s t e m a t i c method f o r w r i t i n g down a l l the d i f f e r e n t cases. T h i s p a r t i c u l a r e x e r c i s e i s f a i r l y simple, s i n c e i t involves only 6 d i f f e r e n t seating plans. However, i t w i l l provide a good example with which to Introduce the technique. The next two pages d i s c u s s how a s y s t e m a t i c approach can be used to l i s t the 6 plans given above.  - 10 -  173  The problem la to l i s t a l l possible ways of seating 3 people in 3 c h a i r s . Label the chairs as follows. Chair 1  Chair 2  Chair 3  A A A  Since a l l 3 chairs are to be occupied, the f i r s t step was to place one of the 3 managers in chair 1. You could start with any of the 3, but in this example Mr. Blade was chosen. This gave the following s i t u a t i o n .  A A A  This l e f t 2 people and 2 chairs to f i l l . be done in 2 ways, leading to plans 1 and 2. Plan 1. Plan 2.  y£\  /j±  / \  / n \  / \  This could  We have now considered a l l possible plans with Mr. Blade s i t t i n g in chair 1. The next move was to place one of the other managers in chair 1. In this example Mr. Johnson was chosen. This gave  A A A  Again this l e f t 2 people and 2 chairs to f i l l , to plans 3 and 4. Plan 3.  /\>  ^  Plan 4.  leading  £ ± / \  F i n a l l y Mr. Hamilton was placed in the f i r s t leading to the l a s t 2 plans. Plan 5.  / \  / \  A\  Plan 6.  yfi^  /b\  / \  chair,  We have eliminated a l l the cases with any of the 3 men s i t t i n g in chair 1. But there are only 3 men in the problem, and since the f i r s t chair must be occupied we have written down a l l the cases. - 11 -  174 Using a s y s t e m a t i c approach to l i s t i n g cases ( i n t h i s example the cases were d i f f e r e n t s e a t i n g plans) has 2 major advantages:(1) You can be sure that a l l p o s s i b l e cases have been listed. You do not have to worry that some might have been l e f t out. For example, i n the previous e x e r c i s e a l l the cases with Mr. Blade i n c h a i r 1 were w r i t t e n down before c o n s i d e r i n g anyone e l s e s i t t i n g t h e r e . (2) You can be sure that no cases have been repeated. This may not be very Important when you o n l y have 6 cases t o consider, but i f you have 6 0 d i f f e r e n t p o s s i b i l i t i e s you do not want to check t h a t they are a l l d i f f e r e n t , and t h a t you have not repeated some by mistake. In the previous e x e r c i s e the s y s t e m a t i c approach assured us that the 6 cases l i s t e d were a l l d i f f e r e n t and that there were no other p o s s i b l e s e a t i n g p l a n s . A s y s t e m a t i c method w i l l help when you want to examine a l l cases. As you go through t h i s booklet, you w i l l have more p r a c t i c e i n l i s t i n g cases by a s y s t e m a t i c method.  - 12 -  175  Since o n l y 3 days are being spent on s e a t i n g plans t h i s i s the l a s t day that w i l l be spent on t h i s t o p i c . As mentioned at the b e g i n n i n g o f t h i s s e c t i o n s e a t i n g plans o f t e n p l a y an important r o l e i n d i p l o m a t i c c i r c l e s . We w i l l c o n s i d e r problems that could a r i s e from d i p l o m a t i c s i t u a t i o n s . T h i s i s p a r t o f a map  The  l e t t e r s stand  f o r the  of  Africa.  following  countries. (t);  S i e r r a Leone ( s ) ;  Liberia  Ivory Coast ( i ) ;  Togo ( t )  Upper V o l t a  Guinea  (u)  ;  - 38  -  (gu)  Ghana ;  Mali  (g) (m)  176  As y o u e r e a w a r e , many d e v e l o p i n g c o u n t r i e s have b o r d e r disputes with t h e i r neighbors. T h i s ,1s t r u e o f A f r i c a n C o u n t r i e s , and vhen a r g u m e n t s o c c u r The O r g a n i z a t i o n o f A f r i c a n States t r i e s to help the countries involved s e t t l e their dispute. The O r g a n i z a t i o n a l s o h e l p s i n many o t h e r ways. could  Consider t h e f o l l o w i n g h y p o t h e t i c a l problems t h a t a r i s e b e t w e e n t h e c o u n t r i e s on t h e p r e v i o u s c a g e .  Note:The s o l u t i o n s a r e n o t i n c l u d e d . them when y o u have c o m p l e t e d t h e b o o k l e t .  You w i l l  receive  Exercise In each o f the f o l l o w i n g l i s t a l l t h e s e a t i n g arrangements that s a t i s f y the c o n d i t i o n s . Use a s y s t e m a t i c approach t o l i s t the cases ( s e a t i n g arrangements). T r y a s many p r o b l e m s a s y o u c a n . DO NOT WORRY I F YOU DO NOT COMPLETE A L L THE EXAMPLES. Do a s many a s y o u c a n I n the time. 1. A s p e c i a l m e e t i n g has been c a l l e d t o d i s c u s s t h e b o r d e r d i s p u t e s between the f o l l o w i n g f o u r c o u n t r i e s . S i e r r a Leone (S)  _  L i b e r i a (€) Ivory Coast Ghana as  (c)  (^)  Representatives follows:  o f the four countries w i l l  be  seated  A  Since these c o u n t r i e s a r e having problems i t i s a d v i s a b l e not t o seat r e p r e s e n t a t i v e s from b o r d e r i n g states together.  - 39 -  177 Tiling  Problems  Today you w i l l s t a r t the second t o p i c to be d i s c u s s e d i n t h i s booklet, namely t i l i n g problems. The Idea o f t i l i n g can be considered as a kind o f extended jigsaw puzzle, where the q u e s t i o n asked In how many ways a p a r t i c u l a r o b j e c t can be f i t t e d i n t o a given space. As you work through t h i s s e c t i o n you w i l l be i n t r o d u c e d to the second problem s o l v i n g technique, analogy. Furthermore you w i l l see how these techniques can be combined to help you solve t i l i n g problems. On the previous page you saw a p i c t u r e o f a d e c o r a t o r with a t i l e to f i t i n t o a w a l l . His problem i s how many d i f f e r e n t ways can he f i t the t i l e i n t o the w a l l . Lets c o n s i d e r the f o l l o w i n g problem. Problem A d e c o r a t o r i s h i r e d to t i l e , a bathroom. The owner o f the house has bought a s p e c i a l t r i a n g u l a r t i l e to f i t i n t o the wall. This t i l e i s i n f a c t an e q u a l a t e r a l t r i a n g l e . That i s , a t r i a n g l e with a l l 3 s i d e s being o f equal l e n g t h . The f r o n t of the t i l e i s d i v i d e d i n t o 3 s e c t i o n s , each being a d i f f e r e n t c o l o u r . The back o f the t i l e i s covered with cement so t h a t i t w i l l s t i c k i n place. y - yellow b - blue r - red The d e c o r a t o r has completed the w a l l except f o r the space where the t i l e i s t o f i t . He i s now faced w i t h a problem - how many d i f f e r e n t ways can the t i l e be f i t t e d i n t o the space?  - 41 -  178 Solution The t i l e  i s i n p o s i t i o n 2.  Exercise To save you r e f e r r i n g back the 3 p o s i t i o n s are g i v e n below.  A, *  6  Position  (a)  A  a  Position 3  Position 2  Now complete the f o l l o w i n g Initial Position  C  8  1  f o r the t i l e  table. Final Position  Rotation  1  240° c l o c k w i s e  (see below)  1  120° clockwise  2  120° c o u n t e r c l o c k w i s e  3  120° c o u n t e r c l o c k w i s e  2  360° clockwise  To f i n d the f i n a l p o s i t i o n i n (a) imagine the t i l e i s i n p o s i t i o n 1. Then t h i n k o f how i t moves when r o t a t e d through 240° i n a c l o c k w i s e d i r e c t i o n . You o b t a i n p o s i t i o n 3.  A.  8'  Initial Position  <^VJ>  240° clockwise  You would p l a c e a 3 i n the t a b l e .  - 46 -  .A  5 Final Position  179  Solution Initial Position  Final Position  Rotation  1  240° c l o c k w i s e  3  1  120°  clockwise  2  2  120°  counterclockwise  1  3  120°  counterclockwise  2  2  3o0°  clockwise  2  From the r e s u l t s above i t would seem reasonable to conclude that there were no o t h e r p o s i t i o n s f o r the t i l e . However, can the d e c o r a t o r be sure? C l e a r l y with t h i s p a r t i c u l a r s i t u a t i o n i n v o l v i n g a t r i a n g u l a r t i l e decorated on 1 s i d e , he could be r e a s o n a b l y s u r e . By r o t a t i n g the t i l e backward and forward he would soon conclude t h a t there were o n l y 3 p o s i t i o n s . However, i f he j u s t f i t s the t i l e i n t o the w a l l u s i n g a t r i a l and e r r o r approach t h i s would not help him i n a more complex s i t u a t i o n . We w i l l now d i s c u s s a g e n e r a l approach that can be used to a t t a c k t h i s problem. In d i s c u s s i n g how to approach t h i s problem from a s y s t e m a t i c p o i n t of view we w i l l l a y the b a s e . f o r i n t r o d u c i n g the second problem s o l v i n g technique, analogy. Imagine the d e c o r a t o r p i c k s up the t i l e and f i t s i t i n t o the w a l l . Let the base p o s i t i o n , that i s the f i r s t p o s i t i o n he t r i e s , be the one below.  . A . Base  Position  The movement the d e c o r a t o r can use to move from p o s i t i o n to another i s a r o t a t i o n .  -  kl  -  one  180  From the r e s u l t s I t seems reasonable to conclude that there are no other p o s i t i o n s f o r the t i l e . I f you look at the f i n a l p o s i t i o n s f o r the t i l e there i s a p a t t e r n i n the way they appear i n the t a b l e . The technique o f examination o f cases has been o f value In h e l p i n g the d e c o r a t o r be c e r t a i n that he had a l l p o s s i b l e positions for.the t i l e . By approaching the problem i n a systematic manner ( l i s t i n g the v a r i o u s r o t a t i o n s i n a systematic way) and observing that a p a t t e r n appeared i n the f i n a l p o s i t i o n s , he could be sure that he had l i s t e d a l l p o s s i b l e p o s i t i o n s f o r the t i l e . The components that form the technique o f examination of cases a r e : Step 1:  I f p o s s i b l e f i x an object as the s t a r t i n g  Step 2:  Apply any c o n d i t i o n s the o b j e c t s .  Step _3_:  Apply the movements, o p e r a t i o n s , o r arrange o b j e c t s , a c c o r d i n g to the problem. Use a systematic procedure to do t h i s .  Step 4_:  Look f o r a p a t t e r n .  that may be s t a t e d  point. concerning  You w i l l probably have noted that t h i s technique has been used i n both the s e a t i n g plans problems and those i n v o l v i n g the t i l e . A l l the steps may not be a p p r o p r i a t e t o a l l problems. In d e a l i n g with s e a t i n g plans we a p p l i e d step 2 , a p p l y i n g p a r t i c u l a r s e a t i n g c o n d i t i o n s , but d i d not use the l a s t step, l o o k i n g f o r a p a t t e r n . In the t i l i n g problem there are no c o n d i t i o n s t o apply, but we d i d look a t a p a t t e r n .  - 50 -  181  Today we w i l l be s t a r t i n g t h e f i n a l s e c t i o n o f t h e booklet. The t o p i c t h a t w i l l be d i s c u s s e d i s t h a t o f e l e c t r i c a l s w i t c h i n g networks. You a r e f a m i l i a r w i t h s w i t c h e s f r o m y o u r e v e r y d a y e x p e r i e n c e , b u t you h a v e p r o b a b l y n o t c o n s i d e r e d them as a t o p i c i n m a t h e m a t i c s . As we l e v e l o c t h e t o p i c you w i l l have an o p p o r t u n i t y o f u s i n g t h e p r o b l e m s o l v i n g t e c h n i q u e s o f a n a l o g y and e x a m i n a t i o n o f c a s e s . Most s w i t c h e s , whatever t h e i r s i z e , shape, c o l o u r , e t c . , have one p r o p e r t y i n common, n a m e l y t h e y a r e e i t h e r open o r closed. T h e r e i s no h a l f way s t a g e . You m i g h t l i k e t o t h i n k o f a s w i t c h as an e l e c t r i c a l d r a w b r i d g e . I f t h e b r i d g e i s up ( t h e s w i t c h i s open) t h e t r a f f i c c a n n o t move ( t h e c u r r e n t cannot f l o w ) . I f t h e b r i d g e i s down ( t h e s w i t c h i s c l o s e d ) t h e t r a f f i c moves ( t h e c u r r e n t f l o w s ) . The p o s i t i o n o f t h e s w i t c h i s r e f e r r e d t o as t h e s t a t e o f t h e s w i t c h . in  L e t A be a s w i t c h . this section.  A  rA  The  f o l l o w i n g diagrams w i l l  be  used  The s t a t e o f t h e s w i t c h i s unknown. We do n o t know i f t h e s w i t c h i s open o r c l o s e d , and h e n c e we do n o t know i f t h e c u r r e n t w i l l f l o w .  The s t a t e o f t h e w i l l not f l o w .  s w i t c h i s open.  The  switch i s closed.  s t a t e o f the  - 85  -  The  The  current  182  The  f o l l o w i n g r e s u l t s have been  obtained.  (i) - * - £ ] — E — -  A  These networks are The (ii)  +-0}  E  current flows  £ 0 "  B equivalent. i f A and  B are  i-m-i  closed.  1-0-1 B  L_Q]_J B  These networks are The  current flows  A  A equivalent. I f A _or_ B, or both are  closed.  Exercise you  Can you t h i n k of any mathematical operations of connecting switches.  Hint:  Look at the u n d e r l i n e d  words.  - 112  -  that remind  183  Solution  continued  You could have chosen many o p e r a t i o n s t h a t remind you of connecting switches. For example, you could have thought of any o f the f o l l o w i n g . Addition; -f  Multiplication; *  We must decide which analogy be used i s with s e t theory. Let of S e t s .  Union o f Sets; U  to make.  Intersection offsets:  The analogy that w i l l  us c o n s i d e r the d e f i n i t i o n o f Union and I n t e r s e c t i o n I f X and Y are s e t s then  XyY  i s the s e t o f elements i n X, o r Y, o r both.  XflY  i s the s e t o f elements In X and Y. I f we c o n s i d e r switches l a b e l l e d A and B we have  A  Current flows i f A, o r B, o r both are c l o s  B Current flows i f A and B are c l o s e d . B Exercise Which network would you draw to correspond Which one to correspond t o A^-j B?  - 114 -  to A y B ?  184  APPENDIX C  Content only  - Neutral  C-N  Vehicle  185  INTRODUCTION The  b o o k l e t d e a l s w i t h the  1)  How  to organize  2)  Tiling  3)  Electrical  following  three  topics:  seating plans,  problems, switching  networks.  A t f i r s t g l a n c e t h e s e t o p i c s do' n o t appear- t o be mathematical i n nature. However, as you work t h r o u g h t h e b o o k l e t , i n p a r t i c u l a r a s y o u t r y s e c t i o n s ( 2 ) and (3)> y o u w i l l see how m a t h e m a t i c a l i d e a s h e l p y o u s o l v e t h e p r o b l e m s . A t t h e end o f t h e b o o k l e t y o u s h o u l d be a b l e t o s o l v e p r o b l e m s w h i c h d e a l w i t h any o f the t h r e e t o p i c s l i s t e d above. The b o o k l e t w i l l t a k e 1 0 d a y s t o c o m p l e t e . 3 days are s p e n t on e a c h t o p i c , and 1 d a y a t t h e end i s u s e d f o r r e v i e w . A t t h e end o f most d a y s t h e r e I s a s h o r t q u i z . T h i s w i l l be m a r k e d and r e t u r n e d t h e n e x t d a y . You may k e e p t h e b o o k l e t when i t i s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make a n y comments t h a t y o u t h i n k w i l l be h e l p f u l t o y o u .  -1-  186  SEATING  PLANS  You a r e p r o b a b l y w o n d e r i n g why t h i s t o p i c i s I m p o r t a n t . The f o l l o w i n g a r e two e x a m p l e s o f w h e r e s e a t i n g p l a n s c a n play, or played, a r o l e . A t t h e h i g h e s t l e v e l o f d i p l o m a c y , a p p r o x i m a t e l y one year' was s p e n t d i s c u s s i n g t h e s e a t i n g a r r a n g e m e n t s f o r t h e P a r i s P e a c e T a l k s on V i e t n a m . The f i n d i n g o f a s e a t i n g p l a n t h a t was a c c e p t a b l e t o a l l t h e p a r t i e s was e s s e n t i a l . This p l a n had t o be f o u n d b e f o r e t h e t a l k s c o u l d s t a r t . The o t h e r e x a m p l e i s o f a more p e r s o n a l n a t u r e . Some f r i e n d s ( o r maybe y o u r p a r e n t s ) h a v e t o a r r a n g e t h e s e a t i n g p l a n f o r a wedding. You w o u l d p r e f e r t o s i t n e x t t o J a n e , n o t Tom. T h e r e a r e a l a r g e n u m b e r o f g u e s t s a n d many o f them h a v e s i m i l a r r e q u e s t s c o n c e r n i n g t h e r e own s e a t s . Your f r i e n d s are faced w i t h the d i f f i c u l t t a s k o f t r y i n g t o o r g a n i z e a s e a t i n g p l a n t h a t w i l l keep a l l the g u e s t s happy. T h e s e a r e o n l y two o f many s i t u a t i o n s w h e r e s e a t i n g p l a n s p l a y an i m p o r t a n t r o l e . For example, I n t e r n a t i o n a l conferences, c i v i c luncheons, board meeting, e t c . quite often i n v o l v e s e t t i n g up s e a t i n g p l a n s . I n many c a s e s y o u a r e i n v i t i n g i m p o r t a n t p e o p l e who c o u l d be o f f e n d e d I f t h e y w e r e not s i t t i n g In the ' c o r r e c t ' p l a c e . B e f o r e l o o k i n g a t s e a t i n g p r o b l e m s t h e n e x t page w i l l I n t r o d u c e t h e n o t a t i o n t h a t w i l l be u s e d i n t h i s s e c t i o n .  -3-  187  First,  l e t us e x p l a i n  the notation  that  w i l l be u s e d .  Notation  A A A A A  indicates  that  a chair  i s empty.  i n d i c a t e s t h a t a c h a i r i s o c c u p i e d . , When a c h a i r i s o c c u p i e d , b u t we do n o t know t h e name o f t h e p e r s o n s i t t i n g t h e r e , we w i l l u s e one o f t h e l e t t e r s , x, y, o r z i n s i d e t h e triangle. i n d i c a t e s t h a t the c h a i r i s o c c u p i e d by t h e p e r s o n a n d we do know t h e i r name. F o r e x a m p l e , s u p p o s e t h e r e a r e two l a w y e r s a t t e n d i n g a m e e t i n g , Mr. A b e l ( l a b e l l e d a) and Mr. B r o c k (labelled b). Then... indicates  that  Mr. A b e l  i s occupying the c h a i r ,  indicates  that  Mr. B r o c k i s o c c u p y i n g t h e c h a i r .  i n d i c a t e s t h a t t h e c h a i r i s o c c u p i e d , b u t we do n o t know w h i c h o f t h e men i s s i t t i n g t h e r e indicates that the l e f t chair i s occupied by Mr. A b e l , and t h e r i g h t s e a t i s empty.  - 4 -  •  188  Exercise A meeting has been arranged between 4 d i r e c t o r s o f a f i s h p l a n t . They are Mr. Brace ( l a b e l l e d b ) , Mr. Drover ( d ) , Mr. Goodyear ( g ) , and Mrs. Murphy (m). The o r g a n i z e r s have p l a c e d 4 c h a i r s i n a row. Chair 1  Chair 2  A  Chair 3  A  Draw diagrams to r e p r e s e n t s e a t i n g arrangements. Chair 1  e.g.  - Mr.  A  Chair 4  A  each o f the f o l l o w i n g  Drover; C h a i r s 2 and  3 -  occupied;  C h a i r 4 - empty. The  Now (i)  diagram would  A  draw the diagrams f o r the Chair 1  - Mr.  C h a i r s 3 and (ii)  (HI)  Chair 1  Brace; 4  -  - empty;  A  A  following:  Chair 2  - Mrs.  Murphy;  occupied. Chair 2 -  Mrs . Murphy;  C h a i r 3 - Mr.  Brace;  Chair 1  Drover; A l l o t h e r c h a i r s  (iv) A l l (v)  A  be  - Mr.  Chair 4  - Mrs.  C h a i r 3 - Mr.  Goodyear. occupied.  *  4 c h a i r s empty.  Chair 1  - Mr.  Murphy;  Chair  Goodyear;  ( v i ) C h a i r 1 - Mr. Goodyear; Chair 4 - occupied.  2 - Mr.  C h a i r 4 - Mr. C h a i r s 2 and  - 5 -  Brace; Drover. 3 - empty;  189  speaker's desks  audience  Above i s the p l a n o f the s m a l l meeting room i n a l a r g e company's head o f f i c e . Three branch managers, Mr. Blade, Mr. Johnson and Mr. H a m i l t o n have been i n v i t e d t o t a l k t o a group o f head o f f i c e p e r s o n n e l . S i n c e the audience does n o t know the t h r e e men, you have t o p l a c e name p l a t e s on the desks. Exercise The t h r e e managers w i l l be seated a t the speaker's desks. How many d i f f e r e n t ways c o u l d you seat the t h r e e men? Draw diagrams t o r e p r e s e n t each o f the p o s s i b l e s e a t i n g p l a n s e.g. b\  One p l a n would be /£\  /h>  where b s t a n d s f o r Mr. B l a d e ,  etc.  As I n t h e above example, i n w r i t i n g out y o u r s o l u t i o n you need o n l y draw the c h a i r s . There i s no need t o draw the room and s p e a k e r s ' d e s k s . Now draw the o t h e r diagrams  - 9 -  below.  190 Solution There a r e 5 o t h e r s e a t i n g p l a n s , g i v i n g a t o t a l o f 6 , Plan 1.  /b\  / j \  /h\  Plan 2 .  ^  ^  /b\  Plan 3. Plan 4.  P l a n s .  (given i n exercise)  A A A A A  Plan 6 . The important. plans.  o r d e r i n which you l i s t e d the plans i s not J u s t check t h a t you have l i s t e d a l l the d i f f e r e n t  I f i t d i d n o t m a t t e r where the men s a t t h e r e would be no problem i n c h o o s i n g a s e a t i n g p l a n . Any o f the 6 g i v e n above c o u l d be used. However, i n many problems there a r e r e s t r i c t i o n s as t o where people s i t . F o r example, you f i n d t h a t Mr. Blade and Mr. Hamilton do n o t l i k e each o t h e r . In t h i s case you t r y to keep them happy by c h o o s i n g a p l a n i n which they do n o t s i t next t o each o t h e r . Exercise still  How many o f the 6 arrangements can you choose and keep Mr. Blade and Mr. Hamilton happy? L i s t them below.  -10-  191  Since o n l y 3 days are b e i n g spent on s e a t i n g plans t h i s i s the l a s t day that w i l l be spent on t h i s t o p i c . As mentioned a t the b e g i n n i n g o f t h i s s e c t i o n s e a t i n g p l a n s o f t e n p l a y an important r o l e i n d i p l o m a t i c c i r c l e s . We w i l l c o n s i d e r problems t h a t c o u l d a r i s e from d i p l o m a t i c s i t u a t i o n s . This  The  letters  i s p a r t o f a map  stand  f o r the  of  Africa.  following  countries.  S i e r r a Leone ( s ) ;  Liberia  I v o r y Coast ( i ) ;  Togo ( t )  Upper V o l t a  Guinea  (u)  ;  -34-  (£); (gu)  ;  Ghana Mali  (g) (m)  192  As y o u a r e a w a r e , many d e v e l o p i n g c o u n t r i e s h a v e b o r d e r disputes with t h e i r neighbors. This i s true of A f r i c a n C o u n t r i e s , a n d when a r g u m e n t s o c c u r The O r g a n i z a t i o n o f African States t r i e s to help the countries involved s e t t l e their dispute. The O r g a n i z a t i o n a l s o h e l p s i n many o t h e r ways. Consider the f o l l o w i n g h y p o t h e t i c a l problems that c o u l d a r i s e b e t w e e n t h e c o u n t r i e s on t h e p r e v i o u s p a g e . Note:The s o l u t i o n s a r e n o t i n c l u d e d . You w i l l them a f t e r y o u have c o m p l e t e d t h e b o o k l e t .  receive  Exercise In each o f t h e f o l l o w i n g g i v e a l l p o s s i b l e s e a t i n g plans that w i l l s a t i s f y the c o n d i t i o n s . T r y a s many p r o b l e m s as y o u c a n . DO NOT WORRY IF YOU DO NOT COMPLETE A L L THE EXAMPLES. There i s no q u i z a t t h e end o f t o d a y s work. 1. A s p e c i a l m e e t i n g has been c a l l e d t o d i s c u s s t h e b o r d e r d i s p u t e s between the f o l l o w i n g c o u n t r i e s . S i e r r a Leone ( s ) Liberia  (*)  Ivory Coast ( i ) Ghana (g)  as  Representatives follows:  A  o f the four countries  will  be  seated  A A A  i J Since these c o u n t r i e s are having problems i t Is advisable not t o seat r e p r e s e n t a t i v e s from bordering s t a t e s together.  - 35 -  193  For example, s i n c e Ghana and the Ivory Coast b o r d e r each o t h e r , t h e i r r e p r e s e n t a t i v e s should not be seated next to each o t h e r . How many d i f f e r e n t s e a t i n g arrangements can you f i n d such t h a t no c o u n t r i e s w i t h a common b o r d e r s i t t o g e t h e r . e.g.  A A AA  i s an a c c e p t a b l e arrangement. 2. Can the 4 s t a t e s In 1 . be s e a t e d round a c i r c u l a r t a b l e so t h a t no c o u n t r i e s w i t h a common border s i t together? 3. Repeat problems 1 and 2 w i t h S i e r r a Guinea.  Leone r e p l a c e d by  Draw diagrams f o r the v a r i o u s s e a t i n g p l a n s . In the case o f the c i r c u l a r t a b l e draw n o n - e q u i v a l e n t diagrams. 4. How many d i f f e r e n t arrangements can you f i n d following f i v e countries are involved^.  i f the.  Togo ( t ) I v o r y Coast ( i ) Sierra  Leone ( s )  M a l i (m) Upper V o l t a (u) Again c o u n t r i e s w i t h a common b o r d e r cannot s i t t o g e t h e r . The t a b l e i s arranged as f o l l o w s .  5.  T r y problem 4 u s i n g a c i r c u l a r  -36-  table.  194  Tiling  Problems  Today you w i l l s t a r t the second t o p i c t o be d i s c u s s e d i n t h i s b o o k l e t , namely t i l i n g problems. The i d e a o f t i l i n g can be c o n s i d e r e d as a kind o f extended j i g s a w p u z z l e , where the q u e s t i o n asked i s how many ways a p a r t i c u l a r o b j e c t can be f i t t e d i n t o a g i v e n space. On the p r e v i o u s page you saw a p i c t u r e o f a d e c o r a t o r with a t i l e to f i t i n t o a w a l l . H i s problem I s how many d i f f e r e n t ways can he f i t the t i l e Into the w a l l . Let's c o n s i d e r the f o l l o w i n g problem. Problem A d e c o r a t o r I s h i r e d t o t i l e a bathroom. The owner o f the house has bought a s p e c i a l t r i a n g u l a r t i l e t o f i t i n t o the wall. T h i s t i l e i s i n f a c t an e q u i l a t e r a l t r i a n g l e . That i s , a t r i a n g l e w i t h a l l 3 s i d e s b e i n g o f e q u a l l e n g t h . The f r o n t of the t i l e i s d i v i d e d i n t o 3 s e c t i o n s , each b e i n g a d i f f e r e n t colour. The back o f the t i l e i s covered w i t h cement so t h a t i t w i l l stick i n place. -  yellow  -  blue  - red The d e c o r a t o r has completed the w a l l except f o r t h e space where the t i l e Is t o f i t . He i s now f a c e d w i t h a problem; how many d i f f e r e n t ways can the t i l e be f i t t e d i n t o the space'  -37-  Solution The t i l e  i s i n position  2.  Exercise are  To save you r e f e r r i n g back the 3 p o s i t i o n s g i v e n below.  B  A  Position 1  Position 2  Now complete the f o l l o w i n g Initial Position (a)  f o r the t i l e  Position 3  table. Final Position  Rotation  1  240° c l o c k w i s e  (see below)  1  120°  clockwise  2  120°  counterclockwise  3  120°  counterclockwise  2  360°  clockwise  To f i n d the f i n a l p o s i t i o n i n (a) imagine the t i l e i s i n position 1. Then t h i n k how i t moves when r o t a t e d t h r o u g h 240° i n a c l o c k w i s e d i r e c t i o n . You o b t a i n p o s i t i o n 3«  "  £  A  A  Initial Position  A  *  6*^  240° c l o c k w i s e  You would p l a c e a 3 i n t h e . t a b l e .  - 42 -  6  Final Position  196  Solution Initial Position  Rotation  Final Position  1  240° c l o c k w i s e  3  1  120°  clockwise  2  2  120°  counterclockwise  1  3  120°  counterclockwise  2  2  360°  clockwise  2  From these r e s u l t s I t seems t h a t no o t h e r p o s i t i o n s f o r the t i l e e x i s t . However, on page 39 we asked how the d e c o r a t o r c o u l d be r e a s o n a b l y sure t h a t no o t h e r p o s i t i o n s e x i s t . That i s , how he c o u l d be r e a s o n a b l y c e r t a i n t h a t he has a l l the p o s i t i o n s f o r the t i l e . The f o l l o w i n g approach w i l l work. Imagine the d e c o r a t o r p i c k s up the t i l e and f i t s i t i n t o the w a l l . L e t the f i r s t p o s i t i o n he t r i e s be p o s i t i o n 1.  A  Position 1 The movement p o s i t i o n t o another  the d e c o r a t o r can use t o move from one i s a rotation.  197 Today you w i l l be s t a r t i n g the f i n a l s e c t i o n o f the booklet. The t o p i c t h a t w i l l be d i s c u s s e d i s t h a t o f e l e c t r i c a l s w i t c h i n g networks. You a r e f a m i l i a r w i t h switches from your everyday e x p e r i e n c e , but you have p r o b a b l y never c o n s i d e r e d them as a t o p i c i n mathematics. Most switches, whatever t h e i r s i z e , shape c o l o u r , e t c . , have one p r o p e r t y i n common, namely they a r e e i t h e r open o r closed. There i s no h a l f way s t a g e . You might l i k e t o t h i n k of a s w i t c h as an e l e c t r i c a l drawbridge. I f the b r i d g e i s up (the s w i t c h i s open) the t r a f f i c cannot move ( t h e c u r r e n t cannot f l o w ) . I f the b r i d g e i s down (the s w i t c h i s c l o s e d ) the t r a f f i c moves ( t h e c u r r e n t f l o w s ) . The p o s i t i o n o f the s w i t c h i s r e f e r r e d t o as the s t a t e o f the s w i t c h . Let A be a s w i t c h . in this section. ^  I I LLJ A ?  ^—/?  ^ —  •  The f o l l o w i n g diagrams w i l l be  used  The s t a t e o f the s w i t c h i s unknown. We do n o t know i f the s w i t c h i s open o r c l o s e d , and hence we do not know i f the c u r r e n t w i l l f l o w .  The s t a t e o f the s w i t c h i s open. c u r r e n t w i l l n o t flow.  The  The s t a t e o f the s w i t c h i s c l o s e d . The current w i l l flow.  - 78 -  198  You a r e p r o b a b l y wondering why the t o p i c i s important. Before d i s c u s s i n g d i f f e r e n t networks l e t us c o n s i d e r the following situation. A f i r m i s i n v o l v e d i n manufacturing e l e c t r i c a l equipment. They a r e . I n v o l v e d i n t e n d e r i n g f o r jobs w i t h the government, l a r g e companies,' e t c . C l e a r l y the lower t h e i r b i d the more l i k e l y they a r e t o be g i v e n the j o b . They would l i k e t o keep the c o s t s as low as p o s s i b l e , and i n p a r t i c u l a r use as l i t t l e equipment t o do the j o b . Thus, i f they can d e s i g n a system c o n t a i n i n g o n l y 1 , 0 0 0 switches Instead o f 2 , 0 0 0 they can save money. T h i s b o o k l e t w i l l not c o n s i d e r networks i n v o l v i n g thousands o f s w i t c h e s . We w i l l d i s c u s s how t o reduce the number o f switches i n a network and s t i l l do the same j o b . Such a r e d u c t i o n would save the f i r m money and h e l p them reduce t h e i r b i d . Although we do not d i s c u s s l a r g e networks the technique used c o u l d be used i n s i t u a t i o n s i n v o l v i n g a l a r g e number o f s w i t c h e s .  - 79 -  199 Solution (i)  A closed;  B closed;  op(A)  open  B  op(A) A  There i s an unbroken path, so the (ii)  A open;  B closed;  op(A)  op(A)  -Y  current  flows  current  flows  closed  B  There i s an unbroken path, so the Exer'cise  B  A op(/ 0  .,  o B  Under what c o n d i t i o n s w i l l the c u r r e n t flow through t h i s network? You should l o o k a t d i f f e r e n t combinations o f s t a t e s and draw the networks.  -  87  -  200  Given 2 d i f f e r e n t switches there are 2 n o n - e q u i v a l e n t ways they can be connected. L e t A and B be the s w i t c h e s . (i) They can be-connected i n such a way t h a t the c u r r e n t o n l y flows i f both A and B are c l o s e d . e  'S-  9  <•  9  A  . B  Network 1 We examined t h i s the c u r r e n t o n l y are connected i n series. We w i l l  network i n a p r e v i o u s e x e r c i s e and found t h a t flows i f A and B are c l o s e d . I f 2 switches, t h i s manner we say t h a t they are connected i n denote network 1 by A  N  B  where o stands f o r i n t e r s e c t i o n . r e f e r r e d t o as A i n t e r s e c t i o n B . I t should be noted t h a t  The e x p r e s s i o n  AJ-JB  is  the f o l l o w i n g network  /  1  A  B  Network 2 has been shown t o be e q u i v a l e n t t o network 1 . The n o t a t i o n t h a t would be used f o r network 2 i s B f ) A . The l e t t e r s have been a l t e r e d t o i n d i c a t e t h a t the o r d e r o f the switches has changed. (ii) The switches can be connected so t h a t the c u r r e n t w i l l flow i f e i t h e r A , o r B, o r both are c l o s e d . ? e.g.  >  A >  9 B  I f 2 switches are connected i n t h i s manner we w i l l say they are connected i n p a r a l l e l . We w i l l denote t h i s network by A U B where \j stands f o r union, and the e x p r e s s i o n A u B w i l l be r e f e r r e d t o as A union B .  - 103 -  APPENDIX D  Heuristics  only - Algebraic  H -  A  Vehicle  202  INTRODUCTION The o n l y p u r p o s e o f t h i s b o o k l e t i s t o t e a c h y o u how t o make d i s c o v e r i e s a b o u t new m a t h e m a t i c a l t o p i c s . The booklet deals w i t h ' 3 t o p i c s , which a r e : 1)  Permutations,  2)  Mappings,  3)  Binary  Networks.  The t o p i c s h a v e b e e n c h o s e n t o i n t r o d u c e y o u t o problem s o l v i n g techniques f o rmaking mathematical d i s c o v e r i e s , and s e r v e n o o t h e r p u r p o s e . When w o r k i n g t h r o u g h t h i s b o o k l e t remember t h a t y o u a r e t r y i n g t o l e a r n t h e p r o b l e m s o l v i n g techniques, not the content i t s e l f . The b o o k l e t w i l l t a k e 10 d a y s t o c o m p l e t e . 3 days a r e s p e n t on e a c h t o p i c , and 1 d a y a t t h e end i s u s e d f o r r e v i e w . A t t h e e n d o f m o s t d a y s t h e r e i s a s h o r t q u i z . T h i s w i l l be marked a t r e t u r n e d t h e n e x t d a y . You may k e e p t h e b o o k l e t when I t i s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make a n y comments t h a t y o u t h i n k w i l l be h e l p f u l t o y o u .  - 1 -  203  PERMUTATIONS We w i l l c o n s i d e r d i f f e r e n t permutations o f l e t t e r s , numbers, e t c . That i s , d i f f e r e n t ways o f a r r a n g i n g o b j e c t s . For example, the f o l l o w i n g would be c o n s i d e r e d d i f f e r e n t permutations o f the numbers 1,2, and 3. 1  2  3  2  and 3 1 2  Before d i s c u s s i n g the t o p i c the n o t a t i o n t h a t w i l l be used.  in detail,  l e t us  introduce  Notation In the f o l l o w i n g the word o b j e c t could stand f o r a number, a l e t t e r , e t c . I t would depend on the p a r t i c u l a r exercise. w i l l be used as a p l a c e h o l d e r . When drawn l i k e t h i s i t w i l l show t h a t the p o s i t i o n i n d i c a t e d i s empty. I n d i c a t e s t h a t there i s an o b j e c t i n t h a t p l a c e , but we do not know what the o b j e c t i s . We w i l l use one o f the l e t t e r s x,y, o r z i n s i d e the placeholder. i n d i c a t e s t h a t there  i s a 1 In the p l a c e .  I n d i c a t e s t h a t the l e t t e r a Is i n t h a t place. An example  i s g i v e n on the next  page.  204 Exercise C o n s i d e r the s e t c o n s i s t i n g o f b , j j and U s i n g a l l three l e t t e r s , how permutations can you f i n d . e.g.  One  many d i f f e r e n t  permutation would be b  j  L i s t a l l the permutations,  - 8 -  h  h.  possible  205 Solution  -  There are 5 o t h e r permutations, g i v i n g a t o t a l o f 6 . Permutation 1.  b  j  h  Permutation 2.  b  h  j  Permutation 3.  j  h  b  Permutation 4.  j  b  h  Permutation 5«  h  j  b  Permutation 6 .  h  b  j  (given i n exercise)  In your s o l u t i o n t o t h i s problem you p r o b a b l y had a l l the permutations l i s t e d , a l t h o u g h they may have been i n a d i f f e r e n t o r d e r . For most people, l i s t i n g the v a r i o u s permutations i s a matter o f t r i a l and e r r o r . That I s , you t r y one permutation, and then another, hoping t h a t i n the end you w i l l have l i s t e d a l l p o s s i b l e combinations. It i s now t h a t the f i r s t problem s o l v i n g technique can be o f h e l p . T h i s problem s o l v i n g technique is. c a l l e d examination o f cases The f i r s t stage o f the technique i n v o l v e s the d e v e l o p i n g o f a s y s t e m a t i c method f o r w r i t i n g down the d i f f e r e n t c a s e s . T h i s p a r t i c u l a r e x e r c i s e i s f a i r l y simple s i n c e i t I n v o l v e s o n l y 6 d i f f e r e n t \permutatIons. However, i t w i l l p r o v i d e a good example w i t h which to i n t r o d u c e the t e c h n i q u e . The next two pages d i s c u s s how a s y s t e m a t i c approach can be used t o l i s t the 6 permutations g i v e n above.  - 9 -  206  We have e l i m i n a t e d a l l the cases w i t h any o f the 3 l e t t e r s i n p o s i t i o n 1. But there were o n l y 3 l e t t e r s i n the set b e i n g c o n s i d e r e d , so we must have w r i t t e n down a l l possible cases. Using a s y s t e m a t i c approach t o l i s t i n g cases ( i n t h i s example the cases are permutations) has two major advantages:(1) You can be sure t h a t a l l p o s s i b l e cases have been l i s t e d . You do not have to worry t h a t some cases might have been l e f t o u t . For example, i n the p r e v i o u s e x e r c i s e a l l the cases w i t h b i n p o s i t i o n 1 were w r i t t e n down b e f o r e c o n s i d e r i n g any o t h e r l e t t e r In t h a t p o s i t i o n . (2) You can be sure t h a t no cases have been r e p e a t e d . T h i s may not be v e r y important when you o n l y have 6 cases to c o n s i d e r , but i f you had 6 0 d i f f e r e n t p o s s i b i l i t i e s , you do not want t o check t h a t they a r e a l l d i f f e r e n t , and t h a t you have n o t repeated some by mistake. In the p r e v i o u s e x e r c i s e the s y s t e m a t i c approach assured us t h a t the 6 cases (permutations) l i s t e d were a l l d i f f e r e n t and t h a t t h e r e were no o t h e r p o s s i b l e permutations. A s y s t e m a t i c approach w i l l h e l p when you want t o examine a l l cases. As you go through t h i s b o o k l e t you w i l l have more p r a c t i c e l i s t i n g cases by a s y s t e m a t i c method.  -  11 -  207  Today we are going to d i s c u s s permutations i n v o l v i n g letters. Furthermore, these l e t t e r s w i l l be o r g a n i z e d i n a c i r c u l a r manner. You w i l l r e c a l l t h a t the aim o f t h i s b o o k l e t was to i n t r o d u c e some problem s o l v i n g t e c h n i q u e s . So f a r , o n l y an i n t r o d u c t i o n to examination o f cases has been g i v e n , and i n p a r t i c u l a r o n l y the f i r s t stage has been d i s c u s s e d , namely s y s t e m a t i c a l l y l i s t i n g cases. In d e a l i n g w i t h today's m a t e r i a l the value o f s y s t e m a t i c a l l y l i s t i n g cases can be seen more c l e a r l y . P a r t i c u l a r l y when the t o p i c o f e q u i v a l e n t permutations i s d i s c u s s e d , you w i l l see t h a t without a s y s t e m a t i c approach the danger o f r e p e a t i n g permutations without r e a l i z i n g i t i s v e r y h i g h . So the aim o f today's m a t e r i a l i s to g i v e you more p r a c t i c e w i t h s y s t e m a t i c a l l y l i s t i n g cases. Exercise Consider  the s e t c o n t a i n i n g the l e t t e r s a,b,  The 3 p l a c e s t h a t are to be manner. , .  filled  and  c.  are arranged i n a c i r c u l a r C i r c u l a r permutation i s re.corded as  A l l three p l a c e s are o c c u p i e d , b must be p l a c e d next to a. How many d i f f e r e n t permutations s a t i s f y t h i s c o n d i t i o n ? (No l e t t e r i s to be used more than once) Hint: F i x a i n one p l a c e f.nd then c o n s i d e r b. to a n o t h e r place,, e t c . Eg.  a i n PI, b Is P2,  (a j b j c )  c i n P3. w i l l be r e c o r d e d  - the p l a c e h o l d e r s  L i s t the c i r c u l a r  have been l e f t  Then move a as out.  permutations.  Note: Each c i r c u l a r p e r m u t a t i o n uses each l e t t e r once o n l y once.  - 21  -  and  208 Mappings T o d a y y o u w i l l s t a r t t h e s e c o n d t o p i c t o be d i s c u s s e d i n t h i s b o o k l e t , namely mappings. In the f i r s t s e c t i o n you were i n t r o d u c e d t o t h e p r o b l e m s o l v i n g t e c h n i q u e o f e x a m i n a t i o n of c a s e s . In t h i s s e c t i o n the second t e c h n i q u e , namely a n a l o g y , w i l l be u s e d . F u r t h e r m o r e , y o u w i l l s e e how e x a m i n a t i o n o f c a s e s c a n be u s e d t o t e s t c o n j e c t u r e s ( e d u c a t e d g u e s s e s ) developed by a n a l o g y . As w i t h p e r m u t a t i o n s , t h e t o p i c o f mappings i s n o t I m p o r t a n t . I t i s j u s t used t o i n t r o d u c e analogy. • L e t us s t a r t b y c o n s i d e r i n g t h e f o l l o w i n g  problem.  Problem The  set of objects that w i l l A  ,  B  ,  be e x a m i n e d c o n s i s t s o f  C  •  We s t a r t w i t h t h e o b j e c t s i n t h e o r d e r A  B  C  M a p p i n g 1 (M-j^) i s d e f i n e d a s  I B  That  1 C  follows:  I A  Position 2  i s , A i s mapped t o B, B i s mapped t o C, a n d C i s mapped t o A, A  apply  B  Position 1  B C to this  i s r e f e r r e d t o a s p o s i t i o n 1. When y o u p o s i t i o n " t h e l e t t e r s a r e moved t o p o s i t i o n  - 40 -  2.  Solution  B  C  1  i  i  i  A  B  C  I { C  A  209  M-  A  M.  When we a p p l y M-^ B maps t o C. C maps t o A.  Then a p p l y i n g  S i m i l a r l y f o r the o t h e r  When M-^ i s a p p l i e d  Position 3  B  again  columns.  twice we denote i t by  &  .  Exercise To save you r e f e r r i n g back the 3 p o s i t i o n s r e p e a t e d below:  are A  B  C  1  Position  A B C  C A  B  Now complete the f o l l o w i n g Initial Mapping Position  2  M  l  3  M  l  3  M  1  M-^ &  find  B  C  A  C  I  B  I  A  B  1  &  M  l ^  & M  x  s  e  e  & M  C  M.  A  table. Final Position  b e l o v ;  (a)  )  x  & M^  the p o s i t i o n  C  C ^  Position 3  B  To  B  Position 2  A  and mapping  i n (a) c o n s i d e r the f o l l o w i n g  Position 2 Mi Position  1  - 42 -  B i s mapped t o C and then t o A. C i s mapped t o A and then t o B, e t c , The f i n a l p o s i t i o n i s A B C, which i s p o s i t i o n 1. T h e r e f o r e you would p l a c e a 1 i n the t a b l e .  210  Read c a r e f u l l y EXAMINATION OF f i x an' o b j e c t  CASES  S t e p _1  I f possible,  as a s t a r t i n g p o i n t .  Step 2  A p p l y any c o n d i t i o n s the o b j e c t s .  Step 3  A p p l y movements, o p e r a t i o n s , o r a r r a n g e o b j e c t s , a c c o r d i n g t o the problem. Use a s y s t e m a t i c p r o c e d u r e t o do t h i s . .  S t e p j4  Look f o r a  t h a t may  be s t a t e d  concerning  pattern.  L e t u s s e e hov; t h e s e s t e p s a p p l y t o t h e two t y p e s o f p r o b l e m we h a v e b e e n d e a l i n g w i t h , n a m e l y p e r m u t a t i o n s a n d mappings. E x a m i n a t i o n o f Cases as a p p l i e d  t o mappings.  S t e p _1  F i x a p o s i t i o n as a s t a r t i n g p o i n t .  Step 2  T h e r e w e r e no c o n d i t i o n s  Step 3  The m a p p i n g i s a p p l i e d  Step 4  Step 2 Step 3  case.  i n a s y s t e m a t i c manner.  The f i r s t 2 a p p l i c a t i o n s (M-^ a n d M & M ^ g i v e y o u 3 d i f f e r e n t p o s i t i o n s . Looking a t the remainder o f the t a b l e you can see a p a t t e r n ; t h e same p o s i t i o n s occur again i n a d e f i n i t e order. From t h i s y o u c o n c l u d e t h a t n o new p o s i t i o n s w i l l a r i s e . 1  E x a m i n a t i o n o f Cases as a p p l i e d Step 1  i n this  F i x an o b j e c t  to permutations.  ( l e t t e r o r number) I n t h e s t a r t i n g p o s i t i o n .  Apply conditions. I n some o f t h e p r o b l e m s p a r t i c u l a r l e t t e r s o r numbers h a d t o be i n p a r t i c u l a r p o s i t i o n s . Other l e t t e r s ,  o r numbers,  S t e p 4^ Had n o t b e e n d i s c u s s e d permutations.  listed  i n a sj^stematic  a n d was n o t a p p r o p r i a t e  way.  to  I f n e c e s s a r y t h e p r o c e d u r e was r e p e a t e d f o r d i f f e r e n t l e t t e r s , o r numbers, i n d i f f e r e n t p o s i t i o n s .  - 47  -  211  Today you w i l l s t a r t t h e f i n a l s e c t i o n o f t h i s b o o k l e t . The t o p i c t h a t w i l l be d i s c u s s e d i s t h a t o f b i n a r y n e t w o r k s . As y o u w o r k t h r o u g h t h i s s e c t i o n remember t h a t t h e o n l y purpose o f t h i s b o o k l e t i s t o i n t r o d u c e you t o problem s o l v i n g t e c h n i q u e s f o r making m a t h e m a t i c a l d i s c o v e r i e s . Definition A binary  switch  L e t A be a b i n a r y section.  i s one w h i c h c a n h a v e two v a l u e s ,  switch.  The f o l l o w i n g w i l l  be u s e d  0 o r 1.  i n this  i n d i c a t e s t h a t we do n o t know i f t h e v a l u e o f t h e b i n a r y s w i t c h i s 0 o r 1.  0  indicates  that  the b i n a r y s w i t c h has the v a l u e  1  indicates  that  the b i n a r y  0,  s w i t c h has the value  A b i n a r y network i s any c o l l e c t i o n o f b i n a r y w i t h a s i n g l e i n p u t and o u t p u t .  switches  e.g. A  Input  Output B  Figure  1  D  w h e r e A, B, C a n d D a r e b i n a r y s w i t c h e s . I n t h e above network we do n o t know t h e v a l u e o f a n y o f t h e b i n a r y s w i t c h e s . T h e r e f o r e , e a c h one i s d r a w n w i t h a|? j .  - 83 -  When u s i n g e x a m i n a t i o n o f c a s e s t h e f o l l o w i n g have been o b t a i n e d .  results  (i) A  B  These n e t w o r k s The  networks  (ii)'  are  A  equivalent.  are complete  i f A and  B b o t h have the v a l u e  1.  .  A  B  B  0 — A  These n e t w o r k s The  B  networks  are  1  equivalent..  are complete  i f A or  B,  o r b o t h have t h e v a l u e  1.  Exercise Can y o u t h i n k o f a n y m a t h e m a t i c a l o p e r a t i o n s t h a t you o f c o n n e c t i n g b i n a r y s w i t c h e s ? Hint:-  Look a t the u n d e r l i n e d words.  -  102  -  remind  213 Solution I f you have found o p e r a t i o n s t h a t remind you o f connecting binary switches t u r n t o the n e x t page. I f not, t h e f o l l o w i n g comment may h e l p . I n t h e b i n a r y n e t w o r k s on t h e p r e v i o u s page t h e s w i t c h e s have been combined ( j o i n e d ) . A l s o the o r d e r i n which the s w i t c h e s have been j o i n e d does n o t m a t t e r . That i s  A  B  is  equivalent to  - 103  -  B  A  214 Solution  continued  You c o u l d h a v e c h o s e n many o p e r a t i o n s t h a t r e m i n d y o u of connecting binary switches. F o r example, you c o u l d have t h o u g h t o f any o f the f o l l o w i n g . Addition;  Multiplication; ^  We must d e c i d e w h i c h a n a l o g y be u s e d i s w i t h s e t t h e o r y .  Union of Sets; [j  t o make.  L e t us c o n s i d e r t h e d e f i n i t i o n of s e t s . I f X and Y a r e s e t s t h e n X (jY  i s the s e t o f elements  i n X,  XpY  i s the s e t of elements  i n X and  I f we  Is  The a n a l o g y  that  will  o f U n i o n and I n t e r s e c t i o n  or Y or both.  consider binary switches  A  Intersection of Sets:  Y. labelled  A a n d B we  c o m p l e t e i f A, o r B, o r b o t h value  have t h e  1.  B  is A  B  complete  i f A and B have t h e v a l u e  .  Exercise Which b i n a r y network would Which corresponds  correspond  t o A ^ B?  - 104  -  t o A ijB?  1.  have  215 Solution For Read t h e  two  switches,  following  A and  B,  A L> B — B ^ A  carefully  Before c o n t i n u i n g to develop the analogy w i t h s e t t h e o r y l e t us r e v i e w t h e power o f a n a l o g y . We w e r e i n i t i a l l y faced w i t h a set of problems i n v o l v i n g b i n a r y networks, a new t o p i c i n m a t h e m a t i c s . By t r y i n g t o compare n e t w o r k s w i t h a n o t h e r m a t h e m a t i c a l s y s t e m , a s y s t e m w i t h w h i c h we a r e more f a m i l i a r , we a r e now l o o k i n g a t s e t t h e o r y . I t i s hoped t h a t we w i l l be a b l e t o u s e many o f t h e p r o p e r t i e s a s s o c i a t e d w i t h set theory to help solve problems w i t h networks. H e n c e , a n a l o g y has t a k e n us f r o m a. t o t a l l y new topic, b i n a r y n e t w o r k s , i n t o a. more f a m i l i a r t o p i c , s e t t h e o r y ; As y o u w o r k t h r o u g h t h e r e m a i n d e r o f t h i s s e c t i o n y o u w i l l s e e how s e t t h e o r y w i l l h e l p us w i t h n e t w o r k p r o b l e m s . As m e n t i o n e d a b o v e we hope t o use t h e p r o p e r t i e s o f set theory to help w i t h network problems. In p a r t i c u l a r , we w o u l d l i k e t o u s e t h e p r o p e r t i e s a s s o c i a t e d w i t h u n i o n and i n t e r s e c t i o n t o f i n d e q u i v a l e n t n e t w o r k s . In o r d e r t o do t h i s we w i l l h a v e t o e x a m i n e t h e v a r i o u s p r o p e r t i e s a s s o c i a t e d w i t h U n i o n and I n t e r s e c t i o n .  -  106  -  APPENDIX E  Heuristics plus  content  HC  - Algebraic  - A  Vehicle  217 INTRODUCTION T h e r e a r e two p u r p o s e s t o t h i s b o o k l e t . how t o make d i s c o v e r i e s a b o u t new m a t h e m a t i c a l i n t r o d u c e ' y o u t o some nev; m a t h e m a t i c a l topics. deals with the f o l l o w i n g 3 t o p i c s : 1)  To t e a c h y o u t o p i c s , and t o The b o o k l e t  Permutations,  2)  Mappings,  3)  Binary  Networks.  The b o o k l e t w i l l t a k e 10 d a y s t o c o m p l e t e , 3 d a y s a r e s p e n t on e a c h t o p i c , a n d 1 d a y i s u s e d f o r r e v i e w . At the end o f m o s t d a y s t h e r e i s a s h o r t q u i z . . T h i s w i l l be m a r k e d and r e t u r n e d t h e n e x t d a y . You may k e e p t h e b o o k l e t when i t i s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make comments t h a t y o u t h i n k w i l l b e h e l p f u l t o you.  - 1 -  218 PERMUTATIONS We Y/Ill c o n s i d e r d i f f e r e n t numbers, e t c . That i s , d i f f e r e n t  permutations of l e t t e r s , ways o f a r r a n g i n g o b j e c t s .  For example, the f o l l o w i n g would be c o n s i d e r e d d i f f e r e n t p e r m u t a t i o n s o f the numbers 1,2, and 3« 1 2  3  2  and 3 1 2  B e f o r e d i s c u s s i n g the t o p i c i n d e t a i l , the n o t a t i o n t h a t w i l l be u s e d .  l e t us  introduce  Notation In the f o l l o w i n g the word o b j e c t c o u l d s t a n d f o r a number, a l e t t e r , e t c . I t would depend on the p a r t i c u l a r exercise. w i l l be used as a p l a c e h o l d e r . When • drawn l i k e t h i s i t w i l l show t h a t the p o s i t i o n i n d i c a t e d i s empty. i n d i c a t e s t h a t t h e r e i s an o b j e c t i n t h a t p l a c e , but we do not know what the o b j e c t i s . We w i l l use one o f the l e t t e r s x, y, o r z i n s i d e the placeholder. i n d i c a t e s that there  i s a 1 i n the p l a c e .  i n d i c a t e s t h a t the l e t t e r a i s i n t h a t place. An example .is g i v e n on the n e x t  - 2 -  page.  219  '• Exercise.  •  .  C o n s i d e r the s e t c o n s i s t i n g o f b , j , and U s i n g a l l t h r e e l e t t e r s , how p e r m u t a t i o n s can you f i n d . e.g.  One  List  many d i f f e r e n t p o s s i b l e  p e r m u t a t i o n would b  a l l the p e r m u t a t i o n s ,  h.  j  be h  _ ',  - 8 -  220  Solution There are 5 o t h e r permutations, g i v i n g a t o t a l 1.  b  j  h  Permutation 2 .  b  h  j  Permutation 3«  j  b  b  Permutation 4.  j  b  h  Permutation 5«  h  J  b  Permutation 6 .  h  b  j  Permutation  of  6.  (given i n exercise)  In your s o l u t i o n t o t h i s problem you p r o b a b l y had a l l the permutations l i s t e d , a l t h o u g h they may have been i n a d i f f e r e n t o r d e r . For most people, l i s t i n g the v a r i o u s permutations i s a matter o f t r i a l and e r r o r . That I s , you t r y one permutation, and then another, hoping t h a t i n the end you w i l l have l i s t e d a l l p o s s i b l e combinations. It i s now t h a t the f i r s t problem s o l v i n g technique can be o f h e l p . T h i s problem s o l v i n g technique i s c a l l e d examination o f c a s e s . The f i r s t stage o f the technique i n v o l v e s the d e v e l o p i n g o f a s y s t e m a t i c method f o r w r i t i n g down the d i f f e r e n t c a s e s . T h i s p a r t i c u l a r e x e r c i s e i s f a i r l y simple s i n c e i t involves only 6 d i f f e r e n t permutations. However, i t w i l l p r o v i d e a good example w i t h which to i n t r o d u c e the t e c h n i q u e . The next two pages d i s c u s s how a s y s t e m a t i c approach can be used to l i s t the 6 permutations g i v e n above.  - 9 -  221 T o d a y v:e a r e g o i n g t o d i s c u s s t h e idea, o f c i r c u l a r permutations. When d e a l i n g w i t h t h e i d e a o f e q u i v a l e n t c i r c u l a r p e r m u t a t i o n s y o u w i l l s e e how a s y s t e m a t i c a p p r o a c h e l i m i n a t e s the danger o f r e p e a t i n g a p a r t i c u l a r circular permutation without r e a l i z i n g i t . Exercise Consider  the s e t c o n s i s t i n g o f the l e t t e r s  a,b, a n d c .  The 3 p l a c e s t h a t a r e t o be f i l l e d a r e a r r a n g e d i n a c i r c u l a r manner. ^ ^ / >v C i r c u l a r permutation i s r e c o r d e d as  (A ' A , A )  A l l t h r e e p l a c e s are o c c u p i e d . permutations can you f i n d ? e.g. a  many d i f f e r e n t  circular  i n PI , b i n P2, and c i n P3 w i l l be r e c o r d e d as ( a , b , c )  List  How  - the p l a c e h o l d e r s have been l e f t  the c i r c u l a r permutations below:  Note: Each c i r c u l a r permutation uses each l e t t e r once and o n l y once.  - 20  -  out.  222  Ma pjxUvcs T o d a y y o u w i l l s t a r t t h e s e c o n d t o p i c t o be d i s c u s s e d i n t h i s b o o k l e t , namely mappings. I n the f i r s t s e c t i o n you were I n t r o d u c e d t o t h e p r o b l e m s o l v i n g t e c h n i q u e o f e x a m i n a t i o n of cases. In t h i s s e c t i o n the second technique, namely a n a l o g y , w i l l be u s e d . F u r t h e r m o r e , y o u w i l l s e e how t h e s e t e c h n i q u e s c a n be c o m b i n e d t o s o l v e m a p p i n g p r o b l e m s . Let  us s t a r t b y c o n s i d e r i n g  the f o l l o w i n g  problem.  Problem The  set of objects A  We  s t a r t with  that  be e x a m i n e d c o n s i s t s  of  , B , C  the objects A  will  B  i n the order  C  Mapping 1 (M ) i s d e f i n e d A B C 1  vl  Position as  1  follows:  I  C A Position 2 T h a t i s , A i s mapped t o B, B i s mapped t o C, a n d C i s mapped t o A, A B C i s r e f e r r e d t o a s p o s i t i o n 1. When y o u a p p l y t o p o s i t i o n 1 t h e l e t t e r s a r e moved t o p o s i t i o n 2,  - 38 -  223 Solution  M-  B  C  I  j When vie a p p l y C maps t o A.  A  4  Im-  B maps t o C.  position 3 Then a p p l y i n g M^ a g a i n  S i m i l a r l y f o r the o t h e r columns.  When M  i s a p p l i e d twice we denote i t by M^ &  1  .  Exercise To save you r e f e r r i n g back the 3 p o s i t i o n s are repeated below: A  B  • C  B  C  A  C  A  Position  1 A 1 4^ B  Position 2 Position 3  B  Now complete the f o l l o w i n g Initial Position  3  M  l  M  l  &  M  l(  B  |  C  C L  sf A  M, 1  table. Final Position  Mapping  2  and  (a)  below)  s e e  3 1  M_ & M & M ]  1  To f i n d  the p o s i t i o n  B  A  C A  C 4^  t B  I B C  x  i n (a) c o n s i d e r the f o l l o w i n g  Position 2 M, 4 1 J  Position  1  - ho  B i s mapped t o C and then t o A. C i s mapped t o A and then t o B, e t c The f i n a l p o s i t i o n i s A B C, which i s p o s i t i o n 1 . Therefore you would p l a c e a 1 i n the t a b l e .  224  Solution Initial Pos i t i o n  Final Position  Mappings  2  Base II  3  II  Base  II  2  II  3  II  Base  n  2  II  3  From the r e s u l t s i t seems r e a s o n a b l e to conclude t h a t t h e r e are no o t h e r p o s i t i o n s . I f you look: a t the f i n a l p o s i t i o n s there i s a p a t t e r n i n the way they appear i n the table. The technique o f examination o f cases has been o f v a l u e i n making sure t h a t you have a l l p o s s i b l e p o s i t i o n s f o r the l e t t e r s . By a p p r o a c h i n g the problem i n a s y s t e m a t i c manner ( l i s t i n g the v a r i o u s combined mappings i n a s y s t e m a t i c way) and o b s e r v i n g the p a t t e r n i n the f i n a l p o s i t i o n s , you could be sure t h a t a l l p o s s i b l e p o s i t i o n s have been l i s t e d . of  The components cases a r e :  t h a t form the t e c h n i q u e o f e x a m i n a t i o n  Step 1:  I f possible,  f i x an o b j e c t as the s t a r t i n g  Step 2:  A p p l y any c o n d i t i o n s t h a t may the o b j e c t s .  Step 3}  Apply the movements, o p e r a t i o n s , or arrange o b j e c t s , a c c o r d i n g t o the problem. Use a s y s t e m a t i c procedure, to do t h i s .  Step 4:  Look f o r a p a t t e r n .  _ 43 -  be s t a t e d  point.  concerning  225 Today you w i l l s t a r t the f i n a l s e c t i o n o f the b o o k l e t . The t o p i c t h a t w i l l be d i s c u s s e d i s t h a t o f b i n a r y networks. As we d e v e l o p the t o p i c you v / i l l have an o p p o r t u n i t y o f u s i n g the problem s o l v i n g t e c h n i q u e s o f a n a l o g y and e x a m i n a t i o n o f cases i n s o l v i n g problems. Definition A binary switch L e t A be a b i n a r y section.  i s one which can have two v a l u e s ,  switch,  The  0 o r 1.  f o l l o w i n g w i l l be used i n t h i s  i n d i c a t e s t h a t we do not know i f the v a l u e o f the b i n a r y s w i t c h i s 0 o r 1. i n d i c a t e s t h a t the b i n a r y s w i t c h has the v a l u e  1.  i n d i c a t e s t h a t the b i n a r y s w i t c h has the v a l u e 0, A b i n a r y network i s any c o l l e c t i o n o f b i n a r y w i t h a s i n g l e i n p u t and output.  switches  e.g. Output  Input B D  figure 1 where A,B, C and D a r e b i n a r y s w i t c h e s . In the above network we do n o t know the v a l u e o f a n y o f the b i n a r y s w i t c h e s . Therefore, each i s drawn w i t h a |? j .  - 72 -  • When u s i n g e x a m i n a t i o n o f cases have been o b t a i n e d . (i)  226 following results  -Gl B  A  B  These networks are The  the  networks are  (ii)  A  equivalent.  complete i f A and  B both have the v a l u e  1.  .  9  A  B  R  A  | 9 j. These networks are The  networks are  equivalent.  complete i f A en? B,  o r b o t h have the v a l u e  1,  Exercise you  Can you t h i n k o f any mathematical o p e r a t i o n s o f connecting b i n a r y switches?  Hint:-  Look a t the u n d e r l i n e d words,  - 90  -  t h a t remind  227 Exercise F i n d an e q u i v a l e n t  network f o r  1  r  B ?  •  op(B) Hint: The e x p r e s s i o n ( A  f o r t h i s network i s uOp(B)")  N  C  N  ( B  T  . A " )  We can r e w r i t e t h i s e x p r e s s i o n u s i n g the commutative p r o p e r t i e s Then use one o f the d i s t r i b u t i v e p r o p e r t i e s and c o n t i n u e .  - 105  -  APPENDIX F  Content only - A l g e b r a i c  C - A  Vehicle  229 INTRODUCTION The 1) 2)  b o o k l e t d e a l s w i t h the f o l l o w i n g three  topics:  Permutations, Mappings,  3) B i n a r y Networks. At t h e end o f t h e b o o k l e t y o u s h o u l d be a b l e t o s o l v e p r o b l e m s w h i c h d e a l v. 1th a n y o f t h e t h r e e t o p i c s l i s t e d a b o v e . 1  The b o o k l e t w i l l t a k e 1 0 d a y s t o c o m p l e t e . 3 days a r e s p e n t on e a c h t o p i c , and 1 d a y a t t h e end i s u s e d f o r r e v i e w . At t h e end o f most d a y s t h e r e i s a. s h o r t q u i z . T h i s w i l l be marked and r e t u r n e d t h e n e x t d a y . You may keep t h e b o o k l e t when i t i s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make a n y coinments t h a t y o u t h i n k w i l l be h e l p f u l t o y o u .  - 1 -  230  Exercise C o n s i d e r t h e s e t c o n s i s t i n g o f b , j , and U s i n g a l l t h r e e l e t t e r s , how p e r m u t a t i o n s can you f i n d . e.g.  One  p e r m u t a t i o n would b  List  a l l the  many d i f f e r e n t  j  permutations,  - 8 -  be h  h.  possible  231  Solution There are 3 o t h e r p e r m u t a t i o n s , g i v i n g a t o t a l P e r m u t a t i o n 1.  b  j  h  P e r m u t a t i o n 2.  h  J  b  P e r m u t a t i o n 3.  j  h  b  P e r m u t a t i o n 4.  h  b  j  P e r m u t a t i o n 5-  j  b  h  P e r m u t a t i o n 6.  b  h  j  of  6.  (given i n exercise)  The o r d e r i n w h i c h you l i s t e d t h e p e r m u t a t i o n s i n n o t important. J u s t c h e c k t h a t y o u have l i s t e d a l l t h e d i f f e r e n t permutations. Exercise b and  How many o f t h e 6 p e r m u t a t i o n s h are not next to each o t h e r ? List  them below..  -  9  -  can you  choose such  that  232 Today we w i l l be d e a l i n g w i t h t h e d e a o f c i r c u l e r p e r m u t a t i o n s . I n d i s c u s s i n g t h e s e p r o b l e m s you w i l l be i n t r o d u c e d to the i d e a o f e q u i v a l e n t c i r c u l a r p e r m u t a t i o n s . 1  Exercise C o n s i d e r the s e t c o n s i s t i n g o f the l e t t e r s The 3 p l a c e s t h a t a r e t o be manner.  f i l l e d are arranged  A l l three places are occupied. p e r m u t a t i o n s c a n you f i n d ? e.g.  a i n P I , b i s P2 and ( a , b , c )  List  the c i r c u l a r  in a  and  circular'  as  - t h e p l a c e h o l d e r s have b e e n l e f t  out.  below:  p e r m u t a t i o n uses each  - 15 -  c.  many d i f f e r e n t , c i r c u l a r  c i n P3 w i l l be r e c o r d e d  permutations  Note: Each c i r c u l a r o n l y once.  How  a,b,  l e t t e r once  and  233  Mappings Today you w i l l s t a r t the second i n t h i s b o o k l e t , namely mappings. Let  us s t a r t b y c o n s i d e r i n g  The  set of objects  topic  t o be d i s c u s s e d  the f o l l o w i n g  problem.  Problem  A We s t a r t w i t h  ,  ,  C  the objects A  Mapping  B  t o be e x a m i n e d  B  .  C  B  of  i n the order Position 1  1 (M-j^) i s d e f i n e d A  consists  as  follows:  C  4  I  \  B  C  A  Position 2  That  i s , A i s mapped t o B, B i s mapped t o C,and C i s mapped t o A. A B C i s r e f e r r e d t o as p o s i t i o n 1. When y o u a p p l y M^ t o t h i s p o s i t i o n t h e l e t t e r s a r e moved t o p o s i t i o n 2.  - 34 -  234  Solution A  B  C  I i 1 I I 4  M,  B  A  When we a p p l y M^ B maps t o C. C maps t o A.  Position 3 Then a p p l y i n g  again  S i m i l a r l y f o r the o t h e r columns.  When M^ i s a p p l i e d  twice we denote i t by M^ & M^.  Exercise are  To save you r e f e r r i n g back the 3 p o s i t i o n s repeated below:  A  B  C  Position 1  B  C  A  Position 2  A  B  i  I  Posltion 3  B  C  C  A  B  Now complete the f o l l o w i n g Initial Position  Mapping  2  M-j^ & M  3  M  3  M  1  M  x  and mapping  C A  M,  table Final Position  (see below)  (a)  l 1  & M  x  & M  1  & M  1  & M  x  x  To f i n d the p o s i t i o n i n (a) c o n s i d e r the f o l l o w i n g C  I B  A  I  M  B & C  M  Position 2  Position  E i s mapped to C and then t o A. C i s mapped t o A and then to B, e t c The f i n a l p o s i t i o n i s A E C, which i s p o s i t i o n 1. T h e r e f o r e you would place a 1 i n the t a b l e .  1 -  36 -  235 Today you w i l l s t a r t the f i n a l s e c t i o n o f the b o o k l e t . The t o p i c that w i l l be d i s c u s s e d i s that o f b i n a r y networks. As we develop the t o p i c you w i l l look at the Idea o f a complete network, and the idea o f e q u i v a l e n t networks. Definition A b i n a r y switch i s one which can have two values, 0 o r 1. Let A be a b i n a r y switch. section.  The f o l l o w i n g w i l l be used i n t h i s  i n d i c a t e s that we do not know i f the value of the b i n a r y switch i s 0 or 1. i n d i c a t e s t h a t the b i n a r y switch has the value 1. i n d i c a t e s that the b i n a r y switch has the value 0. A b i n a r y network i s any c o l l e c t i o n o f b i n a r y with a s i n g l e input and output.  switches  Input  figure 1 where A,B,C and D are b i n a r y switches. In the above network we do not know the value o f any o f the b i n a r y switches. Therefore, each one i s drawn with a|? j .  - 66 -  236  Consider the p a i r o f b i n a r y switches, A and B. (i)  They can be connected as f o l l o w s :  —CD  Network 1  B  We examined t h i s network i s a previous e x e r c i s e and found that i t was complete i f A and B had the value 1. I f 2 switches are connected i n t h i s manner we w i l l denote the network by Af]B where n stands f o r i n t e r s e c t i o n . r e f e r r e d to as A i n t e r s e c t i o n B.  The e x p r e s s i o n A ^ B i s  It should be noted that the f o l l o w i n g network  9 -CO-  Network 2  B  has been shown to be e q u i v a l e n t to network 1. The n o t a t i o n that would be used f o r network 2 i s B pA. The l e t t e r s have been a l t e r e d to i n d i c a t e that the o r d e r o f the switches has been changed. (ii)  They can be connected as f o l l o w s :  9 B  We have a l s o examined t h i s network i n a p r e v i o u s e x e r c i s e and found that I t i s complete i f e i t h e r A, o r B, or both, have the value 1. I f 2 switches are connected In t h i s manner we w i l l denote the network by  U  B  where u stands f o r union and the e x p r e s s i o n A {JB w i l l be r e f e r r e d to as A union B.  - 82 -  Exercise Here are some networks. e x p r e s s i o n s f o r them. e.g.)—rn—i  Write the c o r r e s p o n d i n g r—0A  corresponds t o  AyB  and corresponds to A<jC  and combining these two we have  corresponds t o (At>B) ^ ( A j j C )  Now  t r y the f o l l o w i n g , (ii)  (i) op(B) B  op(A) D  - 85 -  238  APPENDIX G  H e u r i s t i c s only  - Geometric  H -  G  Vehicle  239 INTRODUCTION The o n l y p u r p o s e o f t h i s b o o k l e t i s t o t e a c h y o u how t o make d i s c o v e r i e s a b o u t new m a t h e m a t i c a l t o p i c s . The booklet deals with 3 t o p i c s , which a r e : 1) A r r a n g e m e n t s , 2)  Flips  and t u r n s ,  3) N e t w o r k s . • The t o p i c s h a v e b e e n c h o s e n t o i n t r o d u c e y o u t o problem s o l v i n g techniques f o r making mathematical d i s c o v e r i e s , and s e r v e no o t h e r p u r p o s e . When w o r k i n g t h r o u g h t h i s b o o k l e t remember t h a t y o u a r e t r y i n g t o l e a r n t h e p r o b l e m s o l v i n g techniques, not the content itself. The b o o k l e t w i l l t a k e 10 d a y s t o c o m p l e t e . 3 days a r e s p e n t o n e a c h t o p i c , a n d 1 d a y a t t h e end i s u s e d f o r r e v i e w . A t t h e end o f most d a y s t h e r e i s a s h o r t q u i z . T h i s w i l l be marked and r e t u r n e d t h e next day. You may k e e p t h e b o o k l e t when I t i s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make a n y comments t h a t y o u t h i n k w i l l be h e l p f u l t o y o u .  - 1 -  240 Exercise C o n s i d e r t h e s e t c o n s i s t i n g o f a t r i a n g l e , s q u a r e and U s i n g a l l t h r e e f i g u r e s , how can you f i n d ?  e.g.  List  One  many d i f f e r e n t p o s s i b l e  arrangement would  a l l possible  arrangements.  - 8 -  be  circle.  arrangements  241 Solution There are 5 o t h e r arrangements, g i v i n g a t o t a l Arrangement  1.  Arrangement  2.  Arrangement  3-  Arrangement  4.  Arrangement  5•  Arrangement  6.  A-O  O  of 6 .  (given i n exerci  In your s o l u t i o n t o t h i s problem you p r o b a b l y had a l l the arrangements l i s t e d , a l t h o u g h they may have been i n a l i s t i n g the v a r i o u s d i f f e r e n t order. F o r most people That i s , you arrangements i s a m a t t e r o f t r i a l and e r r o r . t h e r , hoping t h a t i n the t r y one arrangement, and then ano end you w i l l have l i s t e d a l l poss i b l e c o m b i n a t i o n s . I t i s now t h a t the f i r s t problem s o l v i n g technique can be o f h e l p . T h i s problem s o l v i n g technique i s c a l l e d e x a m i n a t i o n o f cases The f i r s t stage o f the technique i n v o l v e s the d e v e l o p i n g o f a s y s t e m a t i c method f o r w r i t i n g d own the d i f f e r e n t c a s e s . T h i s p a r t i c u l a r e x e r c i s e i s f a i r l y simple s i n c e i t i n v o l v e s o n l y 6 d i f f e r e n t arrangements. However, i t w i l l p r o v i d e a good example w i t h which to i n t r o d u c e the t e c h n i q u e . The next two pages d i s c u s s how a s y s t e m a t i c approach can be used t o l i s t the 6 arrangements g i v e n above.  - 9 -  242 last  F i n a l l y the c i r c l e 2 arrangements.  was p l a c e d i n p o s i t i o n 1, g i v i n g the  Arrangement 5•  Arrangement o. We have e l i m i n a t e d a l l the cases w i t h any o f the 3 f i g u r e s i n p o s i t i o n 1. But t h e r e were o n l y 3 f i g u r e s i n the set b e i n g c o n s i d e r e d , so we must have w r i t t e n down a l l possible cases. U s i n g a s y s t e m a t i c approach t o l i s t i n g cases ( i n t h i s example the cases were arrangements) has two major advantages:(1) You can be sure t h a t a l l p o s s i b l e cases have been listed. You do not have t o worry t h a t some cases might have been l e f t o u t . For example, i n the p r e v i o u s e x e r c i s e a l l the cases w i t h the t r i a n g l e i n p o s i t i o n 1 were w r i t t e n down b e f o r e c o n s i d e r i n g any o t h e r f i g u r e i n t h a t p o s i t i o n . (2) You can be sure t h a t no cases have been r e p e a t e d . T h i s may n o t be v e r y Important when you o n l y have 6 cases to c o n s i d e r , but i f you had 6 0 d i f f e r e n t p o s s i b i l i t i e s you do not want t o check t h a t they a r e a l l d i f f e r e n t , and t h a t you have n o t repeated some by m i s t a k e . In the p r e v i o u s e x e r c i s e the s y s t e m a t i c approach assured us t h a t the 6 cases (arrangements) l i s t e d were a l l d i f f e r e n t and t h a t t h e r e were no o t h e r p o s s i b l e arrangements. A s y s t e m a t i c approach w i l l h e l p when you want t o examine a l l cases. As you go through t h i s b o o k l e t you w i l l have more p r a c t i c e l i s t i n g cases by a s y s t e m a t i c method.  • — 11 -  243 Flips  and t u r n s  T o d a y y o u w i l l s t a r t t h e s e c o n d t o p i c t o be d i s c u s s e d i n t h i s b o o k l e t , namely f l i p s and t u r n s . In the f i r s t s e c t i o n y o u were I n t r o d u c e d t o t h e p r o b l e m s o l v i n g t e c h n i q u e o f examination of cases. In t h i s s e c t i o n t h e second t e c h n i q u e , n a m e l y a n a l o g y , w i l l be u s e d . F u r t h e r m o r e , y o u w i l l s e e how e x a m i n a t i o n o f c a s e s c a n be u s e d t o t e s t c o n j e c t u r e s ( e d u c a t e d g u e s s e s ) d e v e l o p e d b y a n a l o g y . As w i t h a r r a n g e m e n t s , t h e t o p i c o f f l i p s and t u r n s i s n o t i m p o r t a n t . I t i s j u s t used to i n t r o d u c e analogy. Problem T h i s p r o b l e m w i l l be c o n c e r n e d w i t h a n e q u a l a t e r a l triangle. T h a t I s , a t r i a n g l e w i t h a l l 3 s i d e s t h e same length.  .A  How many d i f f e r e n t ways c a n t h i s t r i a n g l e be f i t t e d i n t o a s p a c e o f t h e same s i z e a n d s h a p e ? The t r i a n g l e c a n o n l y be r o t a t e d , a n d c a n n o t be t u r n e d o v e r .  fl  *  These a r e two d i f f e r e n t i n t o a s p a c e o f t h e same s h a p e  ways o f f i t t i n g and s i z e .  the t r i a n g l e  Exercise the  In what o t h e r p o s i t i o n space?  can the t r i a n g l e  A - 42 -  be f i t t e d  into  244 Solution  '' .  The t r i a n g l e i s i n p o s i t i o n 2. Exercise To save you r e f e r r i n g , back the 3 p o s i t i o n s t r i a n g l e a r e g i v e n below.  R  2> Position  1  Position  Now complete the f o l l o w i n g Initial Pos i t i o n (a)  2  f o r the  ft  Position  3  table. Final Pos i t I o n  Rotation.  1  240° c l o c k w i s e  1  120°  clockwise  2  120°  counterclockwise  3  120°  counterclockwise  2  360°  clockwise  (see below)  To f i n d the f i n a l p o s i t i o n i n (a) Imagine t h a t the t r i a n g l e i s in position 1. Then t h i n k hov; i t moves when r o t a t e d through 240° i n a c l o c k w i s e d i r e c t i o n . You o b t a i n p o s i t i o n 3 .  <3 Initial Position  0.7  240° c l o c k w i s e  You would p l a c e a 3 i n the t a b l e .  - 45 -  I Final Position  245 Solution Initial Pos i t l o n  Final Pos i t i o n  Rotation  1  240° c l o c k w i s e  3  1  120  clockwise  2  2  120  counterclockwise  1  3  120°  counterclockwise  2  3o0°  clockwise  2  2  From t h e r e s u l t s a b o v e i t w o u l d seem r e a s o n a b l e t o c o n c l u d e t h a t t h e r e a r e no o t h e r p o s i t i o n s f o r t h e t r i a n g l e . H o w e v e r , c a n y o u be s u r e ? Clearly with this particular situation involving a t r i a n g l e ( t h a t c a n o n l y be r o t a t e d ) y o u c a n be r e a s o n a b l y sure. By r o t a t i n g the t r i a n g l e backward and f o r w a r d y o u w o u l d s o o n c o n c l u d e t h a t t h e r e were o n l y 3 . p o s i t i o n s . However, i f y o u j u s t use a t r i a l and e r r o r a p p r o a c h i t w o u l d n o t h e l p i n a more c o m p l e x s i t u a t i o n . We w i l l now d i s c u s s a g e n e r a l a p p r o a c h t h a t c a n be used t o a t t a c k t h i s problem. I n d i s c u s s i n g how t o a p p r o a c h t h i s p r o b l e m f r o m a s y s t e m a t i c p o i n t o f v i e w we w i l l l a y t h e base f o r i n t r o d u c i n g the second problem s o l v i n g t e c h n i q u e , analogy. Imagine you s t a r t by p u t t i n g the t r i a n g l e a p p r o p r i a t e s p a c e u s i n g one p a r t i c u l a r p o s i t i o n . be r e f e r r e d t o a s t h e b a s e p o s i t i o n .  to  The another  i n the This w i l l  movement t h a t c a n be u s e d t o move f r o m one i s a rotation.  - 46 -  position  • Read  246  carefully EXAMINATION OF CASES  Step ]L  If possible,  Step 2  Apply any c o n d i t i o n s t h a t may the o b j e c t s .  Step 3  Apply movements, o p e r a t i o n s , o r arrange o b j e c t s , a c c o r d i n g to the problem. Use a s y s t e m a t i c procedure to do t h i s . Look f o r a p a t t e r n .  Step 4  f i x an o b j e c t as a s t a r t i n g  point.  be s t a t e d c o n c e r n i n g  Let us see how these s t e p s a p p l y t o the two types o f problem we have been d e a l i n g w i t h , namely arrangements, and f l i p s and t u r n s . Examination o_f Cases as a p p l i e d t o f l i p s in a starting  and  turns.  Step jL  F i x the t r i a n g l e  Step 2 *"  There were no c o n d i t i o n s o t h e r than the t r i a n g l e must f i t i n t o a space the same s i z e and shape.  Step 3  T r i a n g l e i s r o t a t e d i n a s y s t e m a t i c way. First at c l o c k w i s e , then c o u n t e r c l o c k w i s e r o t a t i o n s .  Step 4 "  The f i r s t two r o t a t i o n s g i v e you 3 d i f f e r e n t p o s i t i o n s . The base p o s i t i o n and those l a b e l l e d p o s i t i o n s 2 and 3L o o k i n g a t the remainder o f the t a b l e you can see a p a t t e r n ; the same p o s i t i o n s o c c u r a g a i n i n a d e f i n i t e order. From t h i s you can conclude t h a t no new p o s i t i o n s will arise.  Examination o f Cases as a p p l i e d  to  in a particular  position.  arrangements  Step 1  Fix a figure  Step 2 "~  Apply c o n d i t i o n s . In some problems d i f f e r e n t had t o be p l a c e d i n p a r t i c u l a r p o s i t i o n s .  position.  Other f i g u r e s were p l a c e d i n a s y s t e m a t i c Had not been d i s c u s s e d and was arrangements. I f n e c e s s a r y the procedure was figures in different positions.  - 50  -  looking  figures  way.  not a p p r o p r i a t e t o repeated f o r d i f f e r e n t  247  Exercise  You and  Under what c o n d i t i o n s w i l l should l o o k a t the d i f f e r e n t draw the networks.  Try to l i s t  t h i s network be complete? combinations o f c o n d i t i o n s  the cases i n a s y s t e m a t i c  -  96  -  manner.  When u s i n g e x a m i n a t i o n . o f cases to c o n s i d e r networks, the f o l l o w i n g r e s u l t s were o b t a i n e d . (i)  CO— A  They are  248  - c  B  These networks are  different  B  equivalent.  complete i f A and! B are  connected.  (ii) o  A  B  B  A  These networks are They are  equivalent.  complete i f A,  or B,  oj? both are  connected.  Exercise Can you t h i n k o f any mathematical o p e r a t i o n s you o f c o n n e c t i n g switches? Hint:-  Look a t the u n d e r l i n e d  words.  - 113  -  t h a t remind  APPENDIX H  Heuristics plus  content  HC-G  - Geometric  Vehicle  250  INTRODUCTION T h e r e a r e two p u r p o s e s t o t h i s b o o k l e t . how t o make d i s c o v e r i e s a b o u t new m a t h e m a t i c a l i n t r o d u c e y o u t o some new m a t h e m a t i c a l topics. deals with the f o l l o w i n g 3 topics:^ 1)  Arrangements,  2)  Flips  3)  Networks.  To t e a c h y o u t o p i c s , and t o The b o o k l e t  and t u r n s ,  The b o o k l e t w i l l t a k e 1 0 d a y s t o c o m p l e t e . 3 days a r e s p e n t on e a c h t o p i c , a n d 1 d a y a t t h e e n d i s u s e d f o r r e v i e w . At t h e end o f most days t h e r e i s a s h o r t q u i z . T h i s w i l l be marked and r e t u r n e d t h e n e x t d a y . You may k e e p t h e b o o k l e t when i t i s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make comments t h a t y o u t h i n k w i l l be h e l p f u l t o you.  -  1  -  251 ARRANGEMENTS We w i l l figures.  consider different  arrangements o f geometric  F o r e x a m p l e , t h e f o l l o w i n g w o u l d be c o n s i d e r e d d i f f e r e n t arrangements o f a t r i a n g l e and s q u a r e .  —A-  2  —A-—L_—  and  Before discussing the topic some n o t a t i o n t h a t c o u l d be u s e d .  i n detail,  l e t us I n t r o d u c e  Notation «  w i l l indicate i s empty.  r  x  — —  a particular  position  indicates that there i s a geometric figure i n t h a t p o s i t i o n , b u t we do n o t know what i t i s . We w i l l u s e one o f t h e l e t t e r s x, y, o r z f o r t h i s s i t u a t i o n . indicates position  An e x a m p l e  that  that  there  i s g i v e n on t h e n e x t  - 2 -  i sa circle  page.  i n the  252 Exe r c l s e C o n s i d e r t h e s e t c o n s i s t i n g o f a t r i a n g l e , s q u a r e and U s i n g a l l t h r e e f i g u r e s , how can you f i n d ? e.g.  List  One  many d i f f e r e n t p o s s i b l e  arrangement would  a l l possible  arrangements.  - 8 -  be  circle.  arrangements  last  F i n a l l y the c i r c l e 2 arrangements.  was p l a c e d  i n position  1, g i v i n g t h e  Arrangement ^  Arrangement We figures i set being possible  6.„  have e l i m i n a t e d a l l t h e c a s e s w i t h any o f t h e 3 n p o s i t i o n 1 . B u t t h e r e were o n l y 3 f i g u r e s i n t h e c o n s i d e r e d , s o we must h a v e w r i t t e n down a l l cases.  Using a s y s t e m a t i c approach t o l i s t n g cases ( i n t h i s e x a m p l e t h e c a s e s were a r r a n g e m e n t s ) h a s two m a j o r a d v a n t a g e s : (1) Y o u c a n be s u r e t h a t a l l p o s s i b l e c a s e s h a v e b e e n listed. Y o u do n o t h a v e t o w o r r y t h a t some c a s e s m i g h t h a v e been l e f t o u t . For example, i n the p r e v i o u s e x e r c i s e a l l t h e cases w i t h t h e t r i a n g l e i n p o s i t i o n 1 w e r e w r i t t e n down b e f o r e c o n s i d e r i n g any other f i g u r e i n t h a t p o s i t i o n . (2) Y o u c a n be s u r e t h a t n o c a s e s h a v e b e e n r e p e a t e d . T h i s may n o t be v e r y i m p o r t a n t when y o u o n l y h a v e 6 c a s e s t o c o n s d i e r , b u t i f y o u h a d 60 d i f f e r e n t p o s s i b i l i t i e s y o u do n o t w a n t t o c h e c k t h a t t h e y a r e a l l d i f f e r e n t , a n d t h a t y o u h a v e n o t r e p e a t e d some b y m i s t a k e . In the previous e x e r c i s e the s y s t e m a t i c approach a s s u r e d us t h a t t h e 6 c a s e s ( a r r a n g e m e n t s ) l i s t e d were a l l d i f f e r e n t and t h a t t h e r e were no o t h e r p o s s i b l e a r r a n g e m e n t s . A s y s t e m a t i c a p p r o a c h w i l l h e l p when y o u want t o e x a m i n e a l l cases. As y o u go t h r o u g h t h i s b o o k l e t y o u w i l l h a v e more p r a c t i c e l i s t i n g cases by a s y s t e m a t i c method.  - 11 -  254  Flips  and•turns  T o d a y y o u w i l l s t a r t t h e s e c o n d t o p i c t o be d i s c u s s e d i n t h i s b o o k l e t , n a m e l y f l i p s and t u r n s . In the f i r s t s e c t i o n y o u were i n t r o d u c e d t o t h e p r o b l e m s o l v i n g t e c h n i q u e of examination of cases. In t h i s s e c t i o n the second t e c h n i q u e , n a m e l y a n a l o g y , w i l l be u s e d . F u r t h e r m o r e , y o u w i l l s e e how t h e s e t e c h n i q u e s c a n be c o m b i n e d t o s o l v e p r o b l e m s . /  Problem T h i s p r o b l e m w i l l be c o n c e r n e d w i t h an e q u a l a t e r a l triangle. T h a t i s , a t r i a n g l e w i t h a l l 3 s i d e s t h e same length.  How many d i f f e r e n t ways c a n t h i s t r i a n g l e be i n t o a s p a c e t h e same s i z e and s h a p e ? The t r i a n g l e o n l y be r o t a t e d , a n d c a n n o t be t u r n e d o v e r .  T h e s e a r e two d i f f e r e n t ways o f f i t t i n g I n t o a s p a c e o f t h e same s i z e and s h a p e .  the  fitted can  triangle  Exercise the  I n what o t h e r p o s i t i o n space?  can the t r i a n g l e  - 39  -  be f i t t e d  into  255 Solution The  other  p o s i t i o n f o r the t r i a n g l e i s  -\Q>  i\L You w i l l n o t i c e t h a t a p o s i t i o n such as  since  t h e t r i a n g l e c a n o n l y be  rotated  A  Q  L  _\B  is  Impossible. By r o t a t i n g the t r i a n g l e you f i n d the 3 p o s i t i o n s t h a t have been g i v e n . H o w e v e r , c a n y o u be s u r e t h a t y o u h a v e a l l p o s s i b l e p o s i t i o n s t h a t c a n be o b t a i n e d by r o t a t i n g . S o o n v;e w i l l c o n s i d e r t h e way y o u c o u l d be s u r e t h a t you have a l l p o s s i b l e p o s i t i o n s . B e f o r e d o i n g t h i s t h e d i f f e r e n t p o s i t i o n s w i l l be l a b e l l e d . A  £  ft  1  Position  Position 2  Position 3  Exercise In each o f the f o l l o w i n g f i l l or p o s i t i o n s . (i)  A A  (iii)  <">  A  (iv)  6  cZ  i n the missing  i A  Position  Position - 40 -  vertices  256 From t h e r e s u l t s I t seems r e a s o n a b l e t o c o n c l u d e t h a t t h e r e a r e no o t h e r p o s i t i o n s f o r t h e t r i a n g l e . I f you l o o k at the f i n a l p o s i t i o n s f o r the t r i a n g l e there i s a p a t t e r n i n t h e way t h e y a p p e a r i n t h e t a b l e . The t e c h n i q u e o f e x a m i n a t i o n o f c a s e s h a s b e e n o f v a l u e i n h e l p i n g y o u be c e r t a i n t h a t a l l p o s s i b l e c a s e s h a v e been l i s t e d . By a p p r o a c h i n g t h e p r o b l e m i n a s y s t e m a t i c manner ( l i s t i n g t h e v a r i o u s r o t a t i o n s i n a s y s t e m a t i c way) and o b s e r v i n g t h a t a. p a t t e r n a p p e a r e d i n t h e f i n a l p o s i t i o n s , y o u c o u l d be s u r e t h a t a l l p o s s i b l e p o s i t i o n s had b e e n l i s t e d . The c o m p o n e n t s of cases are:  that  form the technique  of examination  S t e p 1:  I f possible,  f i x an o b j e c t  S t e p 2:  A p p l y any c o n d i t i o n s the o b j e c t s .  S t e p _3_:  A p p l y t h e movements, o p e r a t i o n s , o r a r r a n g e a c c o r d i n g to the problem. Use a s y s t e m a t i c t o do t h i s .  S t e p 4_:  Look f o r a  as t h e s t a r t i n g  t h a t may  be s t a t e d  point.  concerning objects, procedure  pattern.  You w i l l p r o b a b l y h a v e n o t e d t h a t t h i s t e c h n i q u e h a s b e e n used i n b o t h t h e p e r m u t a t i o n problems and t h o s e i n v o l v i n g the t r i a n g l e . A l l t h e s t e p s may n o t be a p p r o p r i a t e t o a l l problems. I n d e a l i n g w i t h p e r m u t a t i o n s (and c i r c u l a r p e r m u t a t i o n s ) we a p p l i e d s t e p 2 , a p p l y i n g p a r t i c u l a r c o n d i t i o n s on f i g u r e s , b u t d i d n o t u s e t h e l a s t s t e p , looking f o r a pattern. I n t h e t r i a n g l e p r o b l e m t h e r e were no c o n d i t i o n s ( o t h e r t h a n n o t b e i n g a l l o w e d t o t u r n i t o v e r ) t o a p p l y , b u t we d i d l o o k a t a p a t t e r n .  - 46 -  257 When u s i n g e x a m i n a t i o n o f cases to c o n s i d e r networks, the f o l l o w i n g r e s u l t s were o b t a i n e d .  (i)  B  These networks are They are  —E  -ED—  A  different  B  equivalent.  complete i f A and  B are  connected.  (ii)  '•  ? j—— ? B  These networks are They are  equivalent.  complete i f A, £r_B,  or b o t h are  connected.  Exercise you  Can you t h i n k o f any mathematical o p e r a t i o n s of connecting switches?  Hint:-  Look a t the u n d e r l i n e d  words.  - io5 -  t h a t remind  258  S o l u t i o n continued  You c o u l d have chosen many o p e r a t i o n s t h a t remind you o f c o n n e c t i n g s w i t c h e s . F o r example, you c o u l d have thought of any o f the f o l l o w i n g . Addition; 4-  Multiplication;  Union o f S e t s ;  X  Intersection o f Sets  U  We must d e c i d e which a n a l o g y t o make. be used Is w i t h s e t t h e o r y .  The analogy, t h a t  n  will  L e t us c o n s i d e r the d e f i n i t i o n o f Union and I n t e r s e c t i o n o f S e t s . I f X and Y a r e s e t s then X(jY  Is the s e t o f elements  Xj-jY  i s the s e t o f elements  i n X o r Y.or b o t h . i n X and Y.  I f we c o n s i d e r s w i t c h e s l a b e l l e d  A and B we have  Complete i f A o r B, o r b o t h a r e connected.  B  B  Complete i f A and B a r e connected,  Exercise Which network would you draw t o c o r r e s p o n d t o At/B? Which one t o c o r r e s p o n d t o A flB?  -  107 -  259 The properties networks. examples f may t a k e a  p r e v i o u s e x a m p l e h a s shown how t h e n o t a t i o n a n d o f s e t t h e o r y c a n be u s e d t o f i n d e q u i v a l e n t On t h e f o l l o w i n g pages y o u w i l l f i n d f u r t h e r o r y o u t o t r y . The e x e r c i s e s a r e NOT e a s y . They l o n g t i m e a n d i n v o l v e many s t e p s .  On one o f t h e two, p a g e s f o l l o w i n g t h e e x e r c i s e t h e r e i s a h i n t that should help you solve the problem. Only look a t t h e h i n t AFTER y o u h a v e t r i e d t h e p r o b l e m . You s h o u l d keep the pages w i t h t h e d i f f e r e n t p r o p e r t i e s handy s o t h a t y o u c a n r e f e r t o them (pages 1 1 4 - 6 ) . T h e r e i s no q u i z , a t t h e e n d o f t o d a y ' s w o r k . Do n o t w o r r y i f y o u do n o t c o m p l e t e a l l . t h e e x e r c i s e s . J u s t do a s many a s y o u c a n i n t h e t i m e a v a i l a b l e . Remember t h a t some o f t h e m i n v o l v e many s t e p s t o s i m p l i f y t h e s e t t h e o r e t i c e x p r e s s i o n . Exercise F i n d an e q u i v a l e n t  network f o r the f o l l o w i n g .  2J  •A B  B  Note:-  The c o r r e c t s e t t h e o r e t i c e x p r e s s i o n f o r t h e n e t w o r k i s g i v e n o n t h e n e x t page. Check t h e e x p r e s s i o n y o u o b t a i n before t r y i n g t o s i m p l i f yi t .  -  121  -  260  APPENDIX I  Content only - Geometric  C - G  Vehicle  261 INTRODUCTION The  booklet  deals with the f o l l o w i n g three  1)  Arrangements,  2)  Flips  3)  Networks.  topics:  and t u r n s ,  A t t h e e n d o f t h e b o o k l e t y o u s h o u l d be a b l e t o s o l v e p r o b l e m s w h i c h d e a l w i t h a n y o f t h e -three t o p i c s l i s t e d a b o v e . The b o o k l e t w i l l t a k e 10 d a y s t o c o m p l e t e . 3 d a y s a r e s p e n t on e a c h t o p i c , a n d 1 d a y a t t h e e n d i s u s e d f o r r e v i e w . At t h e end o f most days t h e r e i s a s h o r t q u i z . T h i s w i l l be marked and r e t u r n e d t h e n e x t d a y . You may k e e p t h e b o o k l e t when I t I s c o m p l e t e . Feel f r e e t o w r i t e i n i t . Make a n y comments t h a t y o u t h i n k w i l l be h e l p f u l t o y o u .  - 1 -  262  ARRANGEMENTS We w i l l figures.  consider different  arrangements o f geometric  F o r e x a m p l e , t h e f o l l o w i n g w o u l d be c o n s i d e r e d d i f f e r e n t arrangements o f a t r i a n g l e and s q u a r e .  HZI—"A~~  ~~Ar"Q~-  a n d  Before discussing some n o t a t i o n t h a t c o u l d  2  the topic be u s e d .  In detail,  l e t us I n t r o d u c e  Notation c — ~  x  *~—--  w i l l indicate i s empty.  a particular  position  indicates that there i s a geometric figure i n t h a t p o s i t i o n , b u t we d o n o t know what i t i s . We w i l l u s e one o f t h e l e t t e r s x, y, o r z f o r t h i s s i t u a t i o n . indicates position  An e x a m p l e  that  that  there  i s g i v e n on t h e n e x t  - 2 -  i sa circle  page.  i n the  Exercise  "  _  C o n s i d e r t h e s e t c o n s i s t i n g o f a t r i a n g l e , s q u a r e and U s i n g a l l t h r e e f i g u r e s , how many d i f f e r e n t p o s s i b l e can you f i n d ? e.g.  One arrangement would be  L i s t a l l p o s s i b l e arrangements.  - 8 -  circle.  arrangements  264 Solution There are 5 o t h e r arrangements, g i v i n g a t o t a l  (given i n exercise  Arrangement 1 .•  Arrangement  of 6 .  27  Arrangement 3  Arrangement 4.  Arrangement 5 •*  Arrangement  A—Qr  6.  The o r d e r i n which you l i s t e d the arrangements i s not important. J u s t check t h a t you have l i s t e d a l l the d i f f e r e n t arrangements. Exercise How many o f the 6 arrangements can you choose such t h a t the t r i a n g l e and c i r c l e are not next to each o t h e r . List  them below.  - 9 -  265 F l i p s and turns Today you w i l l s t a r t the second t o p i c t h i s b o o k l e t , namely t i l i n g problems. Let  to be d i s c u s s e d i n  us s t a r t by c o n s i d e r i n g the f o l l o w i n g  problem.  Problem T h i s problem w i l l be concerned w i t h an e q u a l a t e r a l t r i a n g l e . That i s , a t r i a n g l e w i t h a l l 3 s i d e s the same length. a  6  c  How many d i f f e r e n t ways can t h i s t r i a n g l e be f i t t e d i n t o a space the same s i z e and shape? The t r i a n g l e can o n l y be r o t a t e d , and cannot be t u r n e d o v e r .  These a r e two d i f f e r e n t ways o f f i t t i n g the t r i a n g l e i n t o a space o f the same s i z e and shape. Exercise the  In what o t h e r p o s i t i o n space?  can the t r i a n g l e  - 35 -  be f i t t e d  into  Solution  266  The  triangle  i s i n position  2.  Exercise To s a v e y o u r e f e r r i n g b a c k t r i a n g l e a r e g i v e n below. a B  the 3 positions  f o r the C  8 Position Nov; c o m p l e t e  1  the f o l l o w i n g  Initial Pos i t i o n (a)  Position  2  Final Pos1tion  1  240° c l o c k w i s e  1  120°  2  120° counterclockwise Q 120 counterclockwise 3o0^ c l o c k w i s e  2  3  table.  Rotation  3  Position  ( s e e below)  clockwise  To f i n d t h e f i n a l p o s i t i o n i n ( a ) i m a g i n e t h a t t h e t r i a n g l e i s i n p o s i t i o n 1. T h e n t h i n k how i t moves when r o t a t e d t h r o u g h 240° i n a c l o c k w i s e d i r e c t i o n . Y o u o b t a i n - p o s i t i o n 3-  .  ft  A  .  Initial Position  <5 Vo, 240  clockwise  You would p l a c e a 3 i n the t a b l e .  - 38 -  C  & Final Position  Solution Initial Position  Final Position  Rotation  1  240° c l o c k w i s e  3  1  120°  clockwise  2  2  120°  counterclockwise  1  3  120°  counterclockwise-  2  2  3600  clockwise  2  From these r e s u l t s i t seems t h a t no o t h e r p o s i t i o n s f o r the t r i a n g l e e x i s t . However, on page 36 we asked how you can be r e a s o n a b l y c e r t a i n t h a t you have a l l p o s s i b l e p o s i t i o n s f o r the t r i a n g l e . The f o l l o w i n g approach w i l l work. Let  the f i r s t  p o s i t i o n you t r y use be p o s i t i o n  1.  Position 1 The"movement t h a t can be used t o move the t r i a n g l e from one p o s i t i o n t o a n o t h e r i s a r o t a t i o n .  - 39 -  268 Today you w i l l s t a r t the f i n a l s e c t i o n o f t h e b o o k l e t . The t o p i c t h a t w i l l be d i s c u s s e d i s t h a t o f n e t w o r k s . As we develop the t o p i c you w i l l look a t the idea o f a complete network, and t h e i d e a o f e q u i v a l e n t n e t w o r k s . Definition A switch  c a n be e i t h e r c o n n e c t e d  L e t A be a s w i t c h . this section.  'A  r  A  A  or  disconnected.  The f o l l o w i n g d i a g r a m s w i l l  be u s e d i n  I n d i c a t e s t h a t we do n o t know i f t h e switch i s connected o r disconnected.  indicates that That i s , there  the switch i s disconnected i s a break i n the path.  indicates that That i s , there  the switch i s connected. i s no b r e a k i n t h e p a t h .  A network i s any c o l l e c t i o n i n p u t and o u t p u t p a t h .  o f switches  with a single  e.g. A Input  Output B Path  Path  D  where A,B,C a n d D a r e s w i t c h e s . I n t h e a b o v e n e t w o r k we do n o t know i f t h e s w i t c h e s a r e c o n n e c t e d o r d i s c o n n e c t e d . T h e r e f o r e , e a c h i s drawn w i t h a | ?| .  .  7  4  -  269  Exercise I n ..these e x a m p l e s t h e r e a r e p a i r s o f n e t w o r k s . I n e a c h case you are t o f i n d o u t i f the networks a r e e q u i v a l e n t .  (0 _ j 7 ] _ .  —H-  A  LU-  —4 9  B  A  B Network 1 (ii)  Network 2  -CD-  ?  A  -0  op(B)  A  9  A  L_{T].  B  B  -0  ' B  Netvjork  Network 1  - 93 -  2  270 C o n s i d e r t h e . p a i r o f s w i t c h e s , A and (i)  T h e y c a n be  connected  as  B.  follows:  B  Network 1  We e x a m i n e d t h i s n e t w o r k i s a p r e v i o u s e x e r c i s e and f o u n d t h a t i t was c o m p l e t e i f A and B a r e b o t h c o n n e c t e d . I f 2 switches a r e c o n n e c t e d i n t h i s manner we w i l l d e n o t e t h e n e t w o r k b y Afl B wheren stands f o r i n t e r s e c t i o n . The r e f e r r e d t o a s A i n t e r s e c t i o n B. If  s h o u l d be n o t e d  expression A f t B Is  t h a t the f o l l o w i n g  A  network  Network  2  h a s b e e n shown t o be e q u i v a l e n t t o n e t w o r k 1. The n o t a t i o n t h a t w o u l d be u s e d f o r n e t w o r k 2 i s Bf)A. The l e t t e r s h a v e been a l t e r e d t o I n d i c a t e t h a t t h e o r d e r o f the s w i t c h e s has been changed. (ii)  T h e y c a n be  c o n n e c t e d as  follows:  _A B We h a v e a l s o e x a m i n e d t h i s n e t w o r k i n a p r e v i o u s e x e r c i s e and f o u n d t h a t i t i s c o m p l e t e i f e i t h e r A, o r B, o r b o t h a r e connected. I f 2 s w i t c h e s a r e c o n n e c t e d i n t h i s m a n n e r we w i l l denote the network by Ap  B  w h e r e t f s t a n d s f o r u n i o n and r e f e r r e d t o a s A u n i o n B.  the e x p r e s s i o n A ^ B  - 97  -  will  be  271  APPENDIX J  Instructions  to  Subjects  272  I n s t r u c t i o n s t o students O v e r t h e n e x t few d a y s y o u w i l l be i n v o l v e d i n a n experiment. The f o l l o w i n g i s i n f o r m a t i o n on t h e e x p e r i m e n t . D u r i n g t h e n e x t 10 days, y o u w i l l s p e n d one m a t h e m a t i c s p e r i o d e a c h day w o r k i n g t h r o u g h b o o k l e t s . At t h e b e g i n n i n g o f e a c h day y o u w i l l be g i v e n t h e b o o k l e t t o w o r k w i t h . You w i l l h a v e a c e r t a i n number o f p a g e s t o be c o v e r e d e a c h d a y . The b o o k l e t c o n t a i n s e x e r c i s e s f o r y o u t o t r y , a n d I n most c a s e s the pages f o l l o w i n g each e x e r c i s e c o n t a i n I t ' s s o l u t i o n . P l e a s e t r y an e x e r c i s e b e f o r e l o o k i n g a t t h e s o l u t i o n . At t h e e n d o f most days t h e r e i s a s h o r t q u i s . The c o m p l e t e b o o k l e t w i l l be c o l l e c t e d a t t h e e n d o f t h e p e r i o d , t h e q u i z marked o v e r n i g h t , and r e t u r n e d t h e n e x t day. Once t h e b o o k l e t s h a v e b e e n c o l l e c t e d t h e r e w i l l f u r t h e r w o r k t o be done u n t i l t h e n e x t d a y . T h e r e a r e d i f f e r e n t b o o k l e t s b e i n g used. You w i l l p r o b a b l y have d i f f e r e n t one t h a n y o u r f r i e n d s . I t i s very important y o u DO NOT DISCUSS THE BOOKLET WITH ANYONE. PLEASE DO DISCUSS ANY OF THE WORK. for  be no many a that NOT  At t h e e n d o f * t h e t e n d a y s t h e r e w i l l be some t e s t s you t o t r y .  APPENDIX K  Algebraic  Test  Read t o S t u d e n t s  You  are to write  NO MATTER HOW L I T T L E  down ANYTHING y o u c a n f i n d o u t  OR UNIMPORTANT YOU MAY THINK I T I S  I f y o u h a v e any q u e s t i o n s a b o u t mod. 7 a s k .  Note t o t h e Teachers I will  t r y t o get round t o a l l c l a s s e s .  I f a s t u d e n t s a y s t o y o u do y o u want t h i s is  s h o u l d I do...) s a y y e s .  (that  275  Name  •  C l a s s 10 -  " ( P L E A S E PRINT) is  B e f o r e a t t e m p t i n g t h e p r o b l e m on t h e n e x t page t h i s t o remind, y o u o f m o d u l o a r i t h m e t i c , and what i t means.  Modulo 7  (Mod.  7)  - (Clock  7)  G i v e n any w h o l e number, t o c o n v e r t i t t o mod. d i v i d e I t by 7 and c o n s i d e r t h e r e m a i n d e r . For  =  +1  (2x7)  r e m a i n d e r a f t e r d i v i d i n g 1'5, by 15 13  The  =  (1x7)  +  that  say  that  7)  13 by 7 i s 6, and we  than 6 remain  >6  (Mod.  7)  unchanged.  example: 6  The  (Mod.  say  6  remainder a f t e r d i v i d i n g  Numbers l e s s  7 i s 1, and we » 1  13  For  will  example: 15  The  7 you  =  (0x7)  +  6  r e m a i n d e r a f t e r d i v i d i n g 6 by 5.  7 i s 6, and we ^5  (Mod.  say  that  7)  You w i l l n o t i c e t h a t c e r t a i n n u m b e r s , f o r e x a m p l e 13 and b o t h go t o 6 (mod. 7) We  say  6,  that 13=6  (Mod.  7)  I f y o u h a v e any p r o b l e m s w i t h mod. 7 p l e a s e a s k b e f o r e y o u go o n t o t h e p r o b l e m g i v e n on t h e n e x t p a g e .  -  1 -  Problem I am g o i n g t o d e f i n e 2 new m a t h e m a t i c a l o p e r a t i o n s Up-One M u l t i p l i c a t i o n The.symbol used f o r t h i s a @ b For  a x (b + 1)  (Mod.  7)  =  2 x (3 + 1)  (Mod.  7)  =  8  (Mod.  7)  =  1  (Mod.  7)  =  l x ( l + l )  (Mod.  7)  =  2  (Mod.  7)  =  example:  2 @ 3  1 @ 1  Double  Addition  The s y m b o l u s e d f o r t h i s a # b For  is @  i s *.  2 x (a + b)  (Mod.  =  2x(2+3)  (Mod. 7)  =  10  (Mod.  7)  =  3  (Mod.  7)  =  2 x  (Mod.  7)  =  2  (Mod.  7)  =  7)  example:  2*3  0*1  (0  + 1)  277  Name (PLEASE P R I N T )  C l a s s 10 QUESTION  What c a n y o u f i n d o u t a b o u t t h e s e I n c l u d e ALL y o u r  working.  operations?  APPENDIX L  Geometric  Test  Read t o S t u d e n t s The q u e s t i o n s h o u l d What c a n y o u f i n d combining You  read:  o u t about s h o r t e s t r o u t e s and  shortest routes?  a r e t o w r i t e down ANYTHING y o u c a n f i n d o u t  NO MATTER HOW  L I T T L E OR UNIMPORTANT  YOU MAY THINK I T I S  N o t e t o the, T e a c h e r I will  t r y t o get round t o a l l c l a s s e s .  I f a student is  says, t o y o u do y o u want t h i s  s h o u l d I .do...) s a y y e s .  (that  Class  10 -  280  Name (PLEASE PRINT) A t t a c h e d '-to t h i s s h e e t y o u w i l l f i n d a g r i d . The f o l l o w i n g r e f e r s t o a game t h a t c a n be p l a y e d on t h e g r i d . Rules 1) To move f r o m a p o i n t on t h e g r i d t o any p o i n t t h a t i s h i g h e r , o r a t t h e same l e v e l , y o u may move a c r o s s t h e g r i d i n e i t h e r d i r e c t i o n , a n d / o r up t h e g r i d - b u t y o u may NOT move down t h e g r i d . The f o l l o w i n g a r e e x a m p l e s o f r o u t e s t h a t c a n be u s e d t o go f r o m A t o B. 1  2) To move f r o m a p o i n t on t h e g r i d t o any p o i n t t h a t I s l o w e r , o r a t t h e same l e v e l , y o u may move a c r o s s t h e g r i d I n e i t h e r d i r e c t i o n , a n d / o r down t h e g r i d ; b u t y o u may NOT move up t h e g r i d . The f o l l o w i n g a r e e x a m p l e s o f r o u t e s t h a t c a n be u s e d t o go f r o m C t o B.  3) Y o u a r e l o o k i n g f o r t h e s h o r t e s t r o u t e s b e t w e e n p o i n t s The l e n g t h o f a r o u t e i s t h e number o f s e g m e n t s i t t a k e s For example t o go f r o m one p o i n t t o t h e o t h e r .  3  A t o t a l o f 3 segments  3  1  A t o t a l o f 7 segments  This path i s the shorter o f t h e two. S h o r t e s t r o u t e s c a n be c o m b i n e d . F o r example, the s h o r t e s t r o u t e f r o m A t o B can be combined w i t h t h e s h o r t e s t r o u t e f r o m B t o C. T h i s p r o d u c e s a r o u t e f r o m A t o C.  please  turn  over  281 Name (PLEASE PRINT)  Class  10 -  QUESTION What c a n y o u f i n d o u t a b o u t Include  ALL y o u r w o r k i n g .  combining s h o r t e s t  routes?  282  1  F  C  i  1  APPENDIX M G r a d i n g Scheme  284  GRADING SCHEME  The o b j e c t i v e o f t h e g r a d i n g scheme i s t o c a t e g o r i z e a t e s t a c c o r d i n g to the evidence that a student used the h e u r i s t i c s ( A n a l o g y and E x a m i n a t i o n o f C a s e s ) . Iti s important t o note t h a t you are l o o k i n g f o r evidence t h a t the s t u d e n t has t r i e d to_ a p p l y t h e h e u r i s t i c s i n a m e a n i n g f u l way, a n d n o t t h a t he h a s n e c e s s a r i l y u s e d them w e l l . The f o l l o w i n g a r e t h e c a t e g o r i e s t h a t w i l l be  used.  0  -  No e v i d e n c e  1  -  The s t u d e n t has u s e d E x a m i n a t i o n o f C a s e s ONLY. has NOT a t t e m p t e d t o l o o k f o r a p a t t e r n .  2  -  The s t u d e n t h a s e x a m i n e d c a s e s pattern.  3  -  The s t u d e n t has u s e d A n a l o g y ONLY. T h e r e i s NO e v i d e n c e t h a t he h a s t r i e d t o t e s t o u t h i s g u e s s e s .  4  -  The s t u d e n t h a s u s e d A n a l o g y a n d E x a m i n a t i o n o f C a s e s , b u t as DISTINCT t e c h n i q u e s . T h e r e i s NO e v i d e n c e t h a t t h e s t u d e n t has combined t h e two.  5  -  The s t u d e n t h a s u s e d b o t h t e c h n i q u e s ( A n a l o g y Examination o f Cases). F u r t h e r m o r e , he h a s c o m b i n e d t h e two i n s o l v i n g p r o b l e m s .  NOTE:  o f t h e use o f h e u r i s t i c s . He  AND l o o k e d f o r a  and  When a s s i g n i n g a s t u d e n t ' s r e s p o n s e t o a c a t e g o r y t h e r e must be WRITTEN EVIDENCE o f t h e a p p r o p r i a t e r e s p o n s e . F o r e x a m p l e , u n l e s s a s t u d e n t h a s w r i t t e n down some c a s e s , y o u c a n n o t assume t h a t he h a s e x a m i n e d c a s e s .  The f o l l o w i n g a r e e x a m p l e s o f t h e v a r i o u s c a t e g o r i e s f o r t h e ALGEBRAIC TEST. 0  -  No e v i d e n c e  of heuristics  a)  Blank  b)  E x p l a n a t i o n s o f how t h e o p e r a t i o n s w o r k , e.g.  Papers.  A v e r b a l d e s c r i p t i o n o f %.  Add one t o t h e s e c o n d number and t h e n m u l t i p l y by the f i r s t . D i v i d e y o u r r e s u l t by 7 and t h i s g i v e s you t h e answer. c)  A d e s c r i p t i o n o f Mod.  7.  e.g. A s t u d e n t may s a y t h a t mod. 7. i s l i k e s o m e t h i n g he d i d a t t h e b e g i n n i n g o f h i g h s c h o o l .  285 1  -  E x a m i n a t i o n o f Cases ONLY The s t u d e n t has happens. e.g.  tried different  A student would  1 § 3 = l x  (3 +  numbers t o s e e what  h a v e w r i t t e n down  1)  = 4 2 @ 4 = 2 x ( 4 + l ) =  10  =  3  3 @ l = 3 x ( l + l ) = 6 S i m i l a r l y , a p a p e r w o u l d be i n t h i s c a t e g o r y i f he cases f o r the o t h e r o p e r a t i o n , o r b o t h .  examined  NOTE:- T h e r e a r e two p o s s i b i l i t i e s i n t h i s e x a m i n a t i o n o f cases. The s t u d e n t c o u l d h a v e e x a m i n e d them s y s t e m a t i c a l l y or 'randomly'. I f he has e x a m i n e d them s y s t e m a t i c a l l y mark h i s r e s p o n s e 1A, i f r a n d o m l y I B . 2  -  E x a m i n a t i o n o f C a s e s and l o o k i n g  for a pattern  H e r e - a s t u d e n t w o u l d h a v e t r i e d d i f f e r e n t numbers attempted to look f o r a p a t t e r n In the r e s u l t s . e.g.  a) 1 @ 1 = 1 x (1 + 1) = 2  &  2 @ l = 2 x ( l + l ) = 4 3 % 1 = 3 x ( 1 + 1) = 6  m  and  , _ T h e r e f o r e the answer i s a  l  w  a  y  s  e v e n  -  NOTE: T h i s c o n c l u s i o n i s , i n f a c t , i n c o r r e c t . However, i t does show e v i d e n c e o f a m e a n i n g f u l a t t e m p t t o a p p l y e x a m i n a t i o n o f c a s e s and l o o k f o r a p a t t e r n . b) Any a t t e m p t t o e x a m i n e c a s e s and t h e n draw c o n c l u s i o n s a p a t t e r n i n the r e s u l t s . T h i s may be o f t h e f i e l d p r o p e r t i e s s u c h as a s s o c i a t i v i t y .  one  c) 2 @ 3 = 1 T h e r e f o r e t h e o r d e r makes a  difference.  3 @ 2 = 2 IMPORTANT: F o r a p a p e r t o f a l l I n t h i s c a t e g o r y a s t u d e n t must h a v e e x a m i n e d t h e c a s e s and t h e n c o n c l u d e d t h a t a pattern exists. The s t a r t i n g p o i n t MUST BE THE EXAMINATION OF CASES. F o r e x a m p l e , i n c) one w o u l d h a v e t o assume f r o m t h e c o n t e x t t h a t t h e s t a r t i n g p o i n t was t h e e x a m i n a t i o n o f cases. I f you f e e l t h a t the s t a r t i n g p o i n t i s the q u e s t i o n i t w o u l d be c a t e g o r i z e d u n d e r 5.  286 3  -  Analogy  H e r e a s t u d e n t w o u l d h a v e t r i e d t o make an a n a l o g y w i t h some o t h e r m a t h e m a t i c a l o p e r a t i o n o r p r o p e r t i e s . F o r a p a p e r t o f a l l i n t o t h i s c a t e g o r y t h e s t u d e n t must make t h e a n a l o g y , b u t NOT h a v e u s e d e x a m i n a t i o n o f c a s e s t o t e s t o r make t h e a n a l o g y . e.g. I n ( a ) a n d ( b ) t h e s t u d e n t must not.,suse c a s e s t o test h i s statements. a) The o p e r a t i o n i s c o m m u t a t i v e ,  like  addition.  b) The o p e r a t i o n i n v o l v e s t h e d i s t r i b u t i v e p r o p e r t y . c ) Use o f a n a l o g i e s w i t h o t h e r p r o p e r t i e s . d) Use o f a n a l o g y w i t h t h e b o o k l e t s . T h e r e h a s t o be a meaningful attempt at analogy. C l e a r l y a s t a t e m e n t s u c h as ' t h i s h a s s o m e t h i n g t o do w i t h p e r m u t a t i o n s ' i s n o t an a t t e m p t to apply analogy. However, such a s t a t e m e n t , t o g e t h e r w i t h an e x p l a n a t i o n as t o how i t has s o m e t h i n g t o do w i t h p e r m u t a t i o n s c o u l d be a c c e p t a b l e . A s t u d e n t w o u l d t r y t o r a i s e a q u e s t i o n by a n a l o g y . 4  -  Analogy  and E x a m i n a t i o n o f Cases  (DISTINCT)  Here a s t u d e n t w o u l d have used b o t h h e u r i s t i c s , b u t have made no a t t e m p t t o combine them. e.g. He h a s e x a m i n e d c a s e s He h a s made s t a t e m e n t s NOTE:  ( s e e 1) concerning properties  ( s e e 3)  The two h e u r i s t i c s h a v e b e e n u s e d , b u t t h e y h a v e b e e n applied independently.  He c o u l d h a v e t r i e d d i f f e r e n t c a s e s a n d t h e n made a statement ( i n a d i f f e r e n t p a r t o f t h e answer) c o n c e r n i n g the i d e n t i t y . T h i s w o u l d h a v e n'bthing t o do w i t h t h e c a s e s he e x a m i n e d . i.e.  2 @ 1 = 4 3 @ 1 = 6 5 % 5 = 4  @ i s commutative  T h i s c o m b i n a t i o n shows t h a t t h e s t u d e n t has u s e d e x a m i n a t i o n o f c a s e s a n d analogy. However, t h e cases examined have n o t h i n g t o do w i t h t h e s t a t e m e n t concerning commutativity.  287 5  -  Analaogy  a n d E x a m i n a t i o n o f C a s e s (COMBINED)  Here t h e s t u d e n t w o u l d have a s k e d a q u e s t i o n a n d used examination o f cases t o t e s t i t o u t . e.g. a)  I s % commutative?  No  2 @ 3 ¥ 3 § 2 b)  I s t h e r e an i d e n t i t y  f o r @?  1 @ 0 = 1 , 2 @ 0 = 2 , identity.  etc.  T h e r e f o r e , t h e r e i s an  NOTE: A l t h o u g h t h e c o n c l u s i o n c o n c e r n i n g t h e i d e n t i t y i s i n c o r r e c t , i t s t i l l shows t h e a s k i n g o f a q u e s t i o n , f o l l o w e d by an e x a m i n a t i o n o f c a s e s . The s t u d e n t ' s p r o b l e m l i e s i n h i s c o n c e p t i o n o f t h e concept o f an i d e n t i t y , n o t i n h i s u s e of the h e u r i s t i c . c)  Any o t h e r p a p e r Asks  question  than f o l l o w s the p a t t e r n . ^Tries  cases t o answer i t  This i s u s u a l l y i n t h e form o f a q u e s t i o n c o n c e r n i n g the field properties.  I n any p a p e r c o n t a i n i n g e v i d e n c e o f h e u r i s t i c s one w o u l d e x p e c t many s t a t e m e n t s t h a t a r e n o t a p p r o p r i a t e . . Y o u may f i n d e v i d e n c e o f 3 - A n a l o g y ONLY,, t o g e t h e r w i t h I r r e l e v a n t statements. I t i s a matter o f s h i f t i n g through the paper t o f i n d o u t what - i s a p p r o p r i a t e .  288 The f o l l o w i n g a r e e x a m p l e s o f t h e v a r i o u s c a t e g o r i e s f o r t h e GEOMETRIC TEST. 0  -  No e v i d e n c e o f h e u r i s t i c s  a)  Blank  papers.  b)  E x p l a n a t i o n s o f how t o combine r o u t e s . e.g. To combine r o u t e s y o u t a k e t h e l e n g t h o f t h e p a t h from A t o B s a y , and t h e n add t h e l e n g t h f r o m B to C say. c) A description o f the rules. Here y o u a r e l o o k i n g a t e x a m p l e s where t h e s t u d e n t h a s d e s c r i b e d t h e r u l e s i n h i s own w o r d s . 1  -  E x a m i n a t i o n o f C a s e s ONLY  The s t u d e n t w o u l d what h a p p e n s .  have t r i e d d i f f e r e n t p a t h s t o see  e.g. a)  He h a s c a l c u l a t e d t h e l e n g t h s o f d i f f e r e n t r o u t e s .  b)  He h a s c o m b i n e d d i f f e r e n t r o u t e s .  c ) He h a s e x a m i n e d d i f f e r e n t p a t h s b e t w e e n t h e same two p o i n t s . ( T h e r e a r e many p a t h s b e t w e e n A a n d B) NOTE: As w i t h t h e a l g e b r a i c t e s t , r e s p o n s e s c a t e g o r i z e d as e i t h e r 1A o r I B . 2  -  s h o u l d be  E x a m i n a t i o n o f Cases a n d l o o k i n g f o r a p a t t e r n  Here t h e s t u d e n t would have examined d i f f e r e n t and l o o k e d f o r a p a t t e r n .  paths  a) He h a s e x a m i n e d c a s e s a n d c o n c l u d e d t h a t t h e r e i s more t h a n one s h o r t e s t r o u t e b e t w e e n two p o i n t s . b) He h a s e x a m i n e d c a s e s a n d c o n c l u d e d t h a t c o m b i n i n g two s h o r t e s t r o u t e s does n o t a l w a y s g i v e a s h o r t e s t route. c) He h a s e x a m i n e d c a s e s a n d c o n c l u d e d t h a t t h e s h o r t e s t r o u t e f r o m X t o Y , s a y , I s t h e same as t h a t f r o m Y t o X. 3  -  Analogy  ONLY ..  H e r e a s t u d e n t w o u l d h a v e t r i e d t o make an a n a l o g y w i t h some o t h e r m a t h e m a t i c a l o p e r a t i o n o r p r o p e r t i e s . F o r a p a p e r t o f a l l i n t o t h i s c a t e g o r y t h e s t u d e n t must make t h e a n a l o g y , b u t NOT h a v e u s e d e x a m i n a t i o n o f c a s e s t o t e s t o u t o r make t h e a n a l o g y .  289 a) He makes s t a t e m e n t s l i n k i n g c o m b i n i n g r o u t e s w i t h networks. Statements concerning the f a c t that combining r o u t e s t o f o r m new r o u t e s i s l i k e c o m b i n i n g n e t w o r k s ( o r s w i t c h e s ) t o f o r m new n e t w o r k s . b) A n a l o g i e s b e t w e e n c o m b i n i n g p a t h s and. o p e r a t i o n s s u c h as a d d i t i o n , u n i o n , e t c . 4  -  -Analogy a n d E x a m i n a t i o n o f Cases  He c o u l d h a v e e x a m i n e d  different  He c o u l d h a v e made a n a l o g i e s  (DISTINCT)  r o u t e s ( s e e 1)  ( s e e 3)  These w o u l d a p p e a r i n d i f f e r e n t p a r t s o f t h e s t u d e n t s p a p e r w i t h no a t t e m p t t o u s e t h e h e u r i s t i c s t o g e t h e r . 5  -  Analogy  a n d E x a m i n a t i o n o f Cases  (COMBINED)  a) Does c o m b i n i n g s h o r t e s t r o u t e s g i v e a s h o r t e s t r o u t e ? NO - f o l l o w e d b y a c o u n t e r e x a m p l e  to prove t h i s statement.  YES - f o l l o w e d by s u p p o r t i n g e v i d e n c e . N o t e t h a t y e s i s incorrect. However, t h e p r o b l e m I s n o t w i t h t h e s t u d e n t s use o f t h e h e u r i s t i c s , b u t w i t h h i s c h o i c e o f e x a m p l e s . b)  I s the shortest  route unique?  NO - f o l l o w e d by c o u n t e r e x a m p l e . c) YES  Is finding  s h o r t e s t r o u t e symmetric?  - f o l l o w e d by e x a m p l e s . e.g. d)  L e n g t h AB = L e n g t h BA  '  Other papers f o l l o w i n g the p a t t e r n Ask q u e s t i o n  tTries  cases t o t e s t i t .  NOTE: When Jshe. p a p e r was h a n d e d o u t t h e s t u d e n t s w e r e t o l d that the question should read: What c a n y o u f i n d o u t a b o u t combining s h o r t e s t routes? I n c l u d e ALL y o u r w o r k i n g .  s h o r t e s t r o u t e s , and  

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