UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1975_A2 P04.pdf [ 11.14MB ]
Metadata
JSON: 831-1.0055064.json
JSON-LD: 831-1.0055064-ld.json
RDF/XML (Pretty): 831-1.0055064-rdf.xml
RDF/JSON: 831-1.0055064-rdf.json
Turtle: 831-1.0055064-turtle.txt
N-Triples: 831-1.0055064-rdf-ntriples.txt
Original Record: 831-1.0055064-source.json
Full Text
831-1.0055064-fulltext.txt
Citation
831-1.0055064.ris

Full Text

THE EFFECT OF TEACHING HEURISTICS ON THE ABILITY OF GRADE TEN STUDENTS TO SOLVE NOVEL MATHEMATICAL PROBLEMS by L i o n e l Pereira-Mendoza B. Sc.". Southampton U n i v e r s i t y , 1966 M. Sc., 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 the Department of. Education We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 i i A b s t r a c t R e s e a r c h S u p e r v i s o r : Dr. G a i l J . S p i t l e r The study i n v e s t i g a t e d whether kt was 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 the h e u r i s t i c s o f e x a m i n a t i o n o f 'cases or analogy t o n o v e l m a t h e m a t i c a l problems, 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 c o u l d be b e t t e r a c c o m p l i s h e d 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 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 s i m u l t a n e o u s l e a r n i n g o f m a t h e m a t i c a l c o n t e n t . Two f a c t o r s 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 . The f i r s t f a c t o r c o n s i d e r e d employing 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 o b j e c t i v e s : one i n s t r u c t i o n a l t r e a t m e n t was d e s i g n e d t o t e a c h h e u r i s t i c s o n l y (H); a n o t h e r combined the 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 (HC); and a t h i r d t r e a t m e n t u t i l i z e d c o n t e n t o n l y ( C ) , w i t h o u t r e f e r e n c e t o h e u r i s t i c s . The l a t t e r t r e a t m e n t was used as an e x p e r i m e n t a l c o n t r o l . The second f a c t o r concerned employing d i s t i n c t i n s t r u c t i o n a l v e h i c l e s ( m a t e r i a l s ) f o r 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 sample o f grade t e n boys was s e l e c t e d f o r the s t u d y . Nine 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 , c o r r e s p o n d i n g t o each o f the t r e a t m e n t x v e h i c l e c o m b i n a t i o n s . The s u b j e c t s were randomly a s s i g n e d t o one o f the n i n e groups. At t h e end o f the t e n day i n s t r u c t i o n a l p e r i o d (one c l a s s p e r i o d p er day) two t r a n s f e r t e s t s , one a l g e b r a i c and the 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 to a l l s u b j e c t s . i i i Data from 1 8 9 subjects ( 2 1 per group) were analysed by ANOVA. On the a l g e b r a i c t e s t the H treatment was s i g n i f i c a n t l y b e t t e r than both the HC and C treatments, and. 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 the HC and C treatments. The conclusion was that only the H treatment can be claimed to be e f f e c t i v e i n teaching students to apply at 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 problem. On the geometric t e s t both the H and HC treatments were s i g n i f i c a n t l y b e t t e r than the C treatment, and 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 the H and HC treatments. The conclusion was that both the H and HC treatments were e f f e c t i v e i n teaching students to apply at l e a s t one h e u r i s t i c to a novel geometric problem.. An a n a l y s i s of the p a t t e r n of h e u r i s t i c a p p l i c a t i o n on the two t e s t s revealed that while both 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 there was l i t t l e evidence of analogy being used on the geometric t e s t . The i n v e s t i g a t o r suggested that the e f f e c t i v e n e s s of the HC treatment on the geometric t e s t may have been due to the f a c t that the HC treatment was more e f f e c t i v e i n teaching examination of cases than teaching analogy, and the geometric t e s t proved more amenable to the a p p l i c a t i o n of examination of 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 v e h i c l e s . Thus I t seemed reasonable to conclude that the i n s t r u c t i o n a l v e h i c l e employed was not an important f a c t o r i n whether students learned to apply h e u r i s t i c s . i v F i n a l l y , an e x p l o r a t o r y non-parametric ( x 2 ) a n a l y s i s f o r the H treatment i n d i c a t e d that subjects who were i n the top t h i r d on e i t h e r mathematical or reading achievement (or both) scored s u b s t a n t i a l l y higher than the bottom t h i r d . TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ACKNOWLEDGEMENTS CHAPTER I THE PROBLEM Background Purpose o f the Study The Ro l e o f Frameworks i n Re s e a r c h . . . . A Model o f Mathematics D e f i n i t i o n o f Terms Three Q u e s t i o n s A r i s i n g from the Model Research Hypotheses I I RELATED LITERATURE I n t r o d u c t i o n The Problem 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 R e l a t e d Research • D i s c o v e r y Versus Other I n s t r u c t i o n a l Techniques T e a c h i n g H e u r i s t i c P r o c e d u r e s Summary o f the R e l a t e d L i t e r a t u r e v i CHAPTER Page I I I DESIGN AND PROCEDURE 5 8 Design 5 8 I n s t r u c t i o n a l M a t e r i a l s ' . .'. 59 B o o k l e t s 6 l I n s t r u m e n t s 62 G r a d i n g Scheme 6 3 P r o c e d u r e 6 4 P i l o t S t u d i e s 64 Main Study 6 5 S u b j e c t s 69 Development o f the 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 I n s t r u m e n t 82 Geometric Instrument..' . 83 S t a t i s t i c a l P r o c e d u r e s 84 IV ANALYSIS OP THE DATA.... 85 G r a d i n g . . . 86 Data Used i n A n a l y s i s 86 T e s t i n g the Hypotheses 91 P l a n f o r A n a l y s i s 9 3 A n a l y s i s o f Data... 9 7 S t a t i s t i c a l C o n c l u s i o n s < 1 0 2 I n t e r p r e t a t i o n o f R e s u l t s . 1 0 3 D i s c u s s i o n o f R e s u l t s 1 0 4 ' v i i CHAPTER Page V SUMMARY AND CONCLUSIONS 115 Summary 115 Conclusions 116 I m p l i c a t i o n s 119 L i m i t a t i o n s ; . . . 121 Recommendations f o r Further Research 124 BIBLIOGRAPHY . .• 138 APPENDICES... . 145 A H e u r i s t i c s only (Neutral V e h i c l e ) 145 B H e u r i s t i c s plus content (Neutral V e h i c l e ) . . . . . . 166 C Content only (Neutral V e h i c l e ) . . 184 D H e u r i s t i c s only ( A l g e b r a i c V e h i c l e ) 201 E H e u r i s t i c s plus content ( A l g e b r a i c V e h i c l e ) . . . . 216 F Content only ( A l g e b r a i c V e h i c l e ) 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 content (Geometric V e h i c l e ) . . . . 249 I Content only (Geometric V e h i c l e ) . . . 260 J I n s t r u c t i o n s to Subjects 271 K Al g e b r a i c Test 273 L Geometric Test 278 M Grading Scheme....; ;• • . 283 v i i i LIST OF TABLES Table Page I Percentage Agreement on Tests 86 I I Summary of Responses on the A l g e b r a i c Test 88 I I I Summary of Responses on the Geometric Test 89 IV Mark Assigned f o r An 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 Ratios of Confidence I n t e r v a l s f o r Scheffe, Tukey and Dunn 96 VI Means and Standard Deviations f o r the Al g e b r a i c Test 98 VII Means and Standard Deviations f o r the Geometric Test 9 8 V I I I O v e r a l l F-Ratios f o r the A l g e b r a i c Test 99 XIV O v e r a l l F-Ratios f o r the Geometric Test 99 X Al 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 3 100 XI Geometric Test - Simultaneous Confidence I n t e r v a l s f o r Hypotheses 1, 2 and 3 100 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 X I I I x 2 A n a l y s i s f o r the Geometric Test 101 XIV D i s t r i b u t i o n s of Data f o r Subjects Who Wrote Only One Test and Act u a l Data Used i n the A n a l y s i s . I l l XV X 2 Comparison of A c t u a l Data and Only One Ii.;: Test Data 112 XVI X 2 A n a l y s i s of the Achievement of the H e u r i s t i c s Only Group on the A l g e b r a i c Test 126 XVII x 2 A n a l y s i s of the Achievement of the H e u r i s t i c s Only Group on the Geometric Test 127 XVIII A n a l y s i s of the A p p l i c a t i o n of Zero, One or Two H e u r i s t i c s on the Alg e b r a i c Test 133 . LIST OF FIGURES Figure Page 1 A Model of Mathematics 5 2 Experimental Groups..... 16/58 3 Number of C e l l s 27 4 T e s t i n g Arrangement f o r Treatment Groups 30 5 The Booklets , . 61/70 6 S t r u c t u r e of the Tests 63 7 Organization of the Groups 65 8 Experimental Procedure.: 68 9 Topic 1 ' 71 10 Equivalence - Topic 1.. 74 11 Topic 2 75 12 Example of a Rotation 75 13- Example of a Mapping 76 14 Equivalence - Topic 2 77 15 Topic 3 • 78 16 Equivalence - Topic 3 79 17 Movements on a Gr i d 83 18 V i s u a l P r e s e n t a t i o n of Results of S i g n i f i c a n c e Tests f o r Hypotheses H Q 1 3 HQ 2 and HQ^ 102 19 Mean Scores of Groups on-the A l g e b r a i c Test... 105 20 Mean Scores of Groups on the Geometric Test... 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 F i g u r e Page 22 Achievement 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 Only Group' A l g e b r a i c T e s t 128 23 Achievement o f H i g h , M i d d l e and Low Reading Achievement Subgroups o f H e u r i s t i c s Only Group A l g e b r a i c Test 128 24 Achievement 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 25. Achievement o f H i g h , M i d d l e and Low Reading Achievement Subgroups o f H e u r i s t i c s Only Group Geometric T e s t 129 26 Achievement o f Subgroups t h a t were High o r Low on Both Mathematics and Reading A l g e b r a i c T e s t . . . . ..... 130 27 Achievement o f Subgroups t h a t were Hi g h or Low on Both Mathematics and Reading Geometric Test 130 28 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 the A l g e b r a i c Test 134 ACKNOWLEDGEMENTS I would l i k e to thank my committee f o r the help and guidance given throughout the study. I would p a r t i c u l a r l y l i k e to thank Dr. S p i t l e r f o r her help and encouragement during the l a t e r stages of the d i s s e r t a t i o n . F i n a l l y , I would l i k e to thank my w i f e , who managed to survive my c o n t i n u a l l y changing moods during my graduate s t u d i e s . Without her continui n g encouragement I could not have s u c c e s s f u l l y completed my.studies. CHAPTER I The Problem  Background In a world where there i s r a p i d growth i n technology and knowledge, s o c i e t y i s faced w i t h the challenge of educating i n d i v i d u a l s who w i l l be able to 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 of i d e n t i f y i n g and teaching the competencies needed to deal w i t h the changing world. Many educators have discussed the challenge of p r o v i d i n g an education that w i l l meet the needs of future generations, and one concept that was common to many of t h e i r w r i t i n g s i s that ,cbf ' s t r a t e g i e s of i n q u i r y ' . For example, when Downey (1965, p. 22) discussed the i m p l i c a t i o n s of the knowledge ex p l o s i o n on a secondary school education he.stated: For i t i s assumed that i f students acquire.a mastery and comprehension of the broad s t r u c t u r e i n a subject area as w e l l as f a c i l i t y w i t h the st r a t e g y of i n q u i r y appropriate to the p a r t i c u l a r subject 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 to continue l e a r n i n g as future experiences demand-. Bernard (1972, p. 284) advocated that since the school cannot a n t i c i p a t e the problems students 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 to problems should be of seriou s concern to teachers. Heathers (1965, p. 2) suggested that "problem-solving t h i n k i n g or i n q u i r y i s gen e r a l l y considered to be the core of the ed u c a t i o n a l process, and the c h i e f mark of the educated person". The general concern among educators w i t h the r o l e of 2 strategies of inquiry i n the educational process i s re f l e c t e d i n the mathematics education l i t e r a t u r e . For example, Rosenbloom (1966, p. 130—1 3 1 ) indicated the significance of strategies of inquiry i n mathematics (heuristics) when he wrote: One of the major s o c i a l functions of the mathematics curriculum i s to teach one of the p r i n c i p a l methods of acquiring new knowledge. To fi n d out whether we are achieving t h i s objective, we must see whether the student can solve problems for which he has received no s p e c i f i c i n s t r u c t i o n . One of the leading advocates of the teaching of h e u r i s t i c s i n mathematics education i s George Polya. Polya ( 1 9 5 7 , 1 9 6 2 , 1 9 6 5 ) has written many a r t i c l e s and texts concerned with the teaching of h e u r i s t i c s , and i n these he presented s p e c i f i c questions that can be employed by students f o r acquiring new knowledge. A few of these questions include: What i s the unknown? What are the data? What are the conditions?... Could you imagine a more accessible related problem? A more general problem? A more spe c i a l problem? An analogous problem?... (Polya, 1 9 5 7 , pp. xvi - x v i i ) A succinct summary of many educators' views on strategies of inquiry (heuristics) i s embodied i n the following statement by Richards ( 1 9 6 8 , p. 4 6 ) . In the most rapidly changing society i n hi s t o r y , teachers can no longer foresee what a student w i l l need to know. A l l the teacher knows i s that education 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 things he can teach are methods of  acquiring new knowledge. [ i t a l i c s not i n o r i g i n a l ] 3 Purpose of the Study The purpose of t h i s study i s to explore one component of problem s o l v i n g , namely whether i t i s p o s s i b l e to teach students to apply mathematical h e u r i s t i c s to novel mathematical problems. Two f a c t o r s are considered i n the design of the study. The f i r s t f a c t o r considers the e f f e c t of u t i l i z i n g treatments designed to 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 teach . h e u r i s t i c s only and another that 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, on students * a b i l i t y to apply a h e u r i s t i c . The second f a c t o r considers the e f f e c t of employing d i s t i n c t i n s t r u c t i o n a l s e t t i n g s , a l g e b r a i c , geometric and mathematically n e u t r a l on students' a b i l i t y to apply a h e u r i s t i c . An e x p l i c i t statement of the research hypotheses i s Included at the conclusion of t h i s chapter, a f t e r a d i s c u s s i o n of the model and terminology employed %n the study. 'The Role of Frameworks i n Research The i n i t i a l task i n the development of a research study In a p a r t i c u l a r area i s the i d e n t i f i c a t i o n of s i g n i f i c a n t questions r e l a t e d to that- area. E i t h e r a theory or a model can provide an I n v e s t i g a t o r w i t h a valuable a i d i n the search f o r and c l a s s i f i c a t i o n of these questions. Snow (1973, p. 77) suggested that t h e o r i e s and models are 4 extremely valuable to those pursuing research on teaching. Travers (1969, p. 18) noted that the extensive use of models i n s c i e n t i f i c work "derives l a r g e l y from t h e i r value i n suggesting experiments and studies and t h e i r long h i s t o r y as a means of achieving important 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 the i d e n t i f i c a t i o n of s i g n i f i c a n t research questions, models can also serve other u s e f u l purposes. N u t h a l l and Snook (1973) suggested that models provide a framework f o r the 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 of data. Kaplan (1964) i n d i c a t e d that models can be u s e f u l i n communication as w e l l as i n the o r g a n i z a t i o n of data. Many mathematics educators have a l l u d e d to the need f o r a framework i n research. 'For example, Romberg and Vere DeVault (1967, p. 95) noted that the f i r s t step towards i d e n t i f y i n g needed research i n the area of mathematics education i s to organize a model of the various 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 t h e o r y - r e l a t e d research. Pingry (1967, p. 44) s t a t e d t h e ' need f o r a framework when he wrote: The proponents of research d i r e c t e d towards model b u i l d i n g and theory were of the opinion that r e a l progress i n research i n mathematics education would not be made u n t i l such a time that research could be r e l a t e d to a theory of mathematics education, even i f the theory were a very crude one. The next s e c t i o n contains the model w i t h i n which t h i s study i s embedded. 5 A Model of Mathematics MacPherson ( 1 9 7 0 , 1971a) c h a r a c t e r i z e d a d i s c i p l i n e as having three components. A p p l i c a t i o n , Core and Discovery Since t h i s study i s concerned w i t h mathematical problem s o l v i n g the d i s c i p l i n e of mathematics was chosen as an exemplar of the model. Figure 1 A Model of Mathematics APPLICATION l o r e DISCOVERY h e u r i s t i c s MacPherson defined these components as f o l l o w s . A p p l i c a t i o n A p p l i c a t i o n incorporates the set of acts of f a i t h ( l o r e ) by which systems developed w i t h i n the d i s c i p l i n e are t i e d to the p h y s i c a l world. Since there i s no l o g i c a l connection between systems i n the d i s c i p l i n e of mathematics and the p h y s i c a l world, whenever one attempts to solve 6 p h y s i c a l world problems by mathematical means l o r e i s c a l l e d i n t o play. For example, consider the f o l l o w i n g problem: John has three marbles. Paul gives John four marbles. How many marbles does John now have? The t r a n s l a t i o n of t h i s problem i n t o the corresponding mathematical equation, 3 + 4 = ? 3 r e q u i r e s an act of f a i t h , namely that 3 + 4 = ? i s a v a l i d mathematical model f o r the p h y s i c a l world problem. Core The core c o n s i s t s of three components: f a c t s , algorithms and c a t e g o r i e s . Facts are p r o p o s i t i o n s about data which are not under a c t i v e c o n s i d e r a t i o n as to v a l i d i t y . Examples of these are: 2 + 3 = 5; 2 ?< 6 = 12; 5 < 10; the Fundamental Theorem of A r i t h m e t i c ; the axioms of Euclidean Geometry..' Algorithms are procedures f o r the answering of questions which, 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 of steps. Some t y p i c a l algorithms are: The decomposition a l g o r i t h m f o r s u b t r a c t i o n ; ^ the Euclidean method f o r f i n d i n g the greatest common d i v i s o r of two numbers; the Gaussian e l i m i n a t i o n method f o r the s o l u t i o n of a system of equations. Categories are mathematical frameworks f o r o r g a n i z i n g i n f o r m a t i o n . For example: 7 The n a t u r a l numbers; Non-Euclidean geometries; algebra; geometry; topology. Discovery Discovery incorporates the set of techniques-, h e u r i s t i c s , by which mathematical d i s c o v e r i e s are made concerning p h y s i c a l world or mathematical problems. Any m a t e r i a l so discovered may become part of the core of the d i s c i p l i n e . The term h e u r i s t i c has been u t i l i z e d by many mathematics educators. K i l p a t r i c k (1967 5 P- 19) defined a h e u r i s t i c as "any d e v i c e , technique, r u l e of thumb, e t c . that improves problem-solving performance." Wilson (1967, p. 3) r e f e r r e d to a h e u r i s t i c as "a d e c i s i o n mechanism, a way of behaving, which u s u a l l y leads to d e s i r e d outcomes, but w i t h no guarantee of success." Polya (1957, p. 112) s t a t e d that the "aim of h e u r i s t i c i s to study the methods of discovery and i n v e n t i o n . " Polya (1957, P- 113) continued by d e f i n i n g the term h e u r i s t i c reasoning: H e u r i s t i c reasoning i s reasoning not regarded 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 of the present problem. One strand that l i n k s a l l these d e f i n i t i o n s Is that a h e u r i s t i c i s a f l e x i b l e procedure that does not guarantee a s o l u t i o n . The i n v e s t i g a t o r f e e l s that the f o l l o w i n g d e f i n i t i o n , which i s employed i n t h i s study, i s i n accordt w i t h those s t a t e d p r e v i o u s l y : 8 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 d i s c o v e r i n g mathematical r e l a t i o n s h i p s i n a novel problem, and whose a p p l i c a t i o n does not guarantee success. The novel 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 mathematical i n nature, or be associated w i t h a p h y s i c a l world s i t u a t i o n . Although the term h e u r i s t i c i s i n general use by educators the l i t e r a t u r e contains few sources of mathematical h e u r i s t i c s . Polya (1957), Smith and Henderson (1959), MacPherson (1971b). and Rousseau (1971) a l l have suggestions concerning s p e c i f i c techniques that can be employed i n s o l v i n g novel problems, and the h e u r i s t i c s used i n t h i s experiment, namely analogy and examination (enumeration) of cases are common to the sources. The term analogy i s i n common use i n the E n g l i s h language. Webster's d i c t i o n a r y (1967, P- 32) defined analogy as "resemblance i n some p a r t i c u l a r s between things otherwise u n l i k e . " Polya (1954, p. 32) defined analogy as "a sort of s i m i l a r i t y , " while Dinsmore (1971, p. 17) defined i t "as the perception of a s i m i l a r i t y between the problem, to be solved and one that has already been solved." B a s i c a l l y a l l these d e f i n i t i o n s describe analogy i n terms of a perceived s i m i l a r i t y between t h i n g s . In t h i s study analogy i s defined as f o l l o w s : 9 Analogy Is. the h e u r i s t i c by which a perceived s i m i l a r i t y between f a m i l i a r systems or experiences Is u t i l i z e d i n order to rai s e , questions concerning a novel mathematical problem. The f o l l o w i n g examples of mathematical problems where analogy can be of value are i n c l u d e d to c l a r i f y the d e f i n i t i o n . (a) The student has discussed modulo 5 a r i t h m e t i c i n some d e t a i l and i s then presented with modulo 6. An analogy between the new s i t u a t i o n and modulo 5 would.enable the student to r a i s e questions.concerning the s t r u c t u r e of the new system. For example, the analogy would r a i s e questions concerning the commutativlty of a d d i t i o n , the existence of an a d d i t i v e i d e n t i t y , as w e l l as many other f i e l d p r o p e r t i e s . Thus, the analogy w i t h modulo 5 provides the student w i t h a p o s s i b l e b a s i s from which to analyse the new problem. (b) In d i s c u s s i n g the h e u r i s t i c of analogy Polya (1957j p. 37) noted that 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 Is analogous to the three dimensional f i g u r e , the r e c t a n g u l a r p a r a l l e l e p i p e d , i n that- the r e l a t i o n s h i p between the sides of the p a r a l l e l o g r a m are l i k e those between the faces of the p a r a l l e l e p i p e d . Thus, when faced w i t h the three dimensional f i g u r e f o r the f i r s t time employing an analogy w i t h the par a l l e l o g r a m would r a i s e questions such as whether opposite faces are congruent, or opposite s o l i d angles are congruent. 10 Examining cases i s of t e n r e f e r r e d to i n the mathematics education l i t e r a t u r e . Smith and Henderson (1959) discussed the value of enumeration of c a s e s . i n f o r m u l a t i n g conjectures. Leask (1968, p 8) defined simple enumeration of cases as c o n s i s t i n g "of the p r e s e n t a t i o n of many instances of the g e n e r a l i z a t i o n to be discovered." Leask noted that the students can form hypotheses based on the cases and then t e s t them to determine t h e i r v a l i d i t y . For t h i s study examination of cases i s defined as: Examination of Cases i s the 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 order to 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 systematic or random manner. Furthermore, t h i s examination of cases may be employed to formulate a conjecture, or to t e s t the v a l i d i t y of a conjecture developed from the use of examination of cases or another h e u r i s t i c , such as analogy. When v a l i d a t i n g a conjecture by examination of cases one w i l l e i t h e r f i n d cases that support the conjecture^';'or produce a counterexample n e c e s s i t a t i n g i t s r e j e c t i o n . The f o l l o w i n g examples are of mathematical problems where examination of cases can be of value. (a) A high school student has been presented w i t h the problem of f i n d i n g a general formula f o r the f o l l o w i n g s e r i e s . 1 + 3 + 5 + 7 + 9 + . - . + (2n-l) 11 The student can apply the h e u r i s t i c of examination of cases by s u b s t i t u t i n g d i f f e r e n t values f o r n. 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 This systematic examination may suggest the f o l l o w i n g p a t t e r n , namely that the sum of n terms i s n 2 . The a p p l i c a t i o n of the h e u r i s t i c only enables the student to make a reasonable guess that the sum i s n 2 , and does not c o n s t i t u t e a mathematical proof of the r e s u l t . (b) An elementary student i s e x p l o r i n g the sum of the i n t e r i o r angles of a t r i a n g l e . By experimenting with d i f f e r e n t t r i a n g l e s , s e l e c t e d e i t h e r s y s t e m a t i c a l l y or randomly, he i s able to conclude that the sum of the i n t e r i o r angles of a t r i a n g l e i s 180°. Again t h i s examination of cases does not provide a proof, i n a mathematical sense, but only reasonable evidence of the r e s u l t . A general d i s c u s s i o n of the s i g n i f i c a n c e of these h e u r i s t i c s i n mathematical problem s o l v i n g i s i n c l u d e d i n the review of the l i t e r a t u r e . (Chapter I I ) The nature of h e u r i s t i c s i s that they are a p p l i e d to novel •mathematical problems. Problem, l i k e analogy^ i s i n common use i n the E n g l i s h language. Some research s t u d i e s defined problem i m p l i c i t l y i n terms of the 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 the study, while others e x p l i c i t l y s t a t e d what was meant by the word problem. Duncker (1945, p. 1) suggested that "a problem a r i s e s when a l i v i n g creature has a goal but does not know' how t h i s goal i s to be reached." Maier's (1970) conception of a problem i n v o l v e d the existence of an o b s t a c l e , or obstacles between the problem s o l v e r and the goal. Polya (1962, p.117) d i d not define problem, per se, but suggested what i t means to have a problem: Thus to have a problem means: to search consciously f o r some a c t i o n appropriate to a t t a i n a c l e a r l y  conceived, but not immediately a t t a i n a b l e , aim. [ i t a l i c s i n o r i g i n a l ] In each of these d e f i n i t i o n s there i s e i t h e r e x p l i c i t l y s t a t e d , or i m p l i e d , one of the e s s e n t i a l components of any d e f i n i t i o n of a problem, namely the existence of a b a r r i e r that prevents the problem s o l v e r from reaching the de s i r e d g o a l . The existence of such a b a r r i e r means that a problem must be more than j u s t the rou t i n e a p p l i c a t i o n of knowledge, s k i l l s or algorithms. For example, Davis (1973, p. 21) noted that simple questions such as 'Who invented the Ford?' could not be considered problems f o r the average a d u l t . Hints on Problem S o l v i n g ( N a t i o n a l Council of Teachers of Mathematics, 1969, p. 1) r e f e r r e d to the non-rout i n e nature of a mathematical problem. 13 I n mathematics the u s u a l l y a c c e p t e d concept o f a problem d i f f e r e n t i a t e s between s i t u a t i o n s such as t h i s assignment {.refers t o the a s s i g n i n g ^ o f : e x e r c i s e s from the t e x t b o o k - the I n v e s t i g a t o r ] and those 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 beyond the 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 problem i n mathematics can be thought 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 the i n d i v i d u a l c.alled 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 c i t e d p r e v i o u s l y , t o g e t h e r w i t h the t e r m i n o l o g y o f the model l e d t o the f o l l o w i n g d e f i n i t i o n o f problem. A problem i s a m a t h e m a t i c a l t a s k 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 or c a t e g o r i e s t h a t can be a p p l i e d d i r e c t l y t o the t a s k i n o r d e r t o r e a c h the d e s i r e d g o a l . T h i s d e f i n i t i o n o f problem i s not meant as a l l embracing, but r a t h e r d e l i n e a t e s the i n t e r p r e t a t i o n o f the word problem t h a t formed the b a s i s f o r d e v e l o p i n g the t e s t s f o r t h i s s t u d y . 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 the model 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 study and, f o r convenience they are r e s t a t e d i n t h i s s e c t i o n . 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 . 14 Analogy i s the 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 , i s u t i l i z e d i n o r d e r t o r a i s e q u e s t i o n s c o n c e r n i n g a n o v e l m a t h e m a t i c a l problem. E x a m i n a t i o n o f cases i s the 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 c o n c e r n i n g a n o v e l m a t h e m a t i c a l problem. A problem i s a m a t h e m a t i c a l t a s k 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 c a t e g o r i e s t h a t can be a p p l i e d d i r e c t l y t o the t a s k i n o r d e r t o r e a c h the d e s i r e d g o a l . One i m p l i c a t i o n o f t h e s e d e f i n i t i o n s i s t h a t any m a t h e m a t i c a l t a s k t h a t c o u l d be c a l l e d a problem would be c o n s i d e r e d as n o v e l f o r the problem s o l v e r , and r e q u i r e t h e a p p l i c a t i o n o f a h e u r i s t i c , o r h e u r i s t i c s i n a t t e m p t i n g a s o l u t i o n . F o r such problems, problem s o l v i n g i s d e f i n e d as f o l l o w s : Problem s o l v i n g i s t h e s e a r c h f o r and a p p l i c a t i o n o f 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 c a t e g o r i e s t h a t a i d t h e 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 p r o v i d e d t h e i n v e s t i g a t o r w i t h t h e framework w i t h i n w h i c h t o develop and s t a t e the h y p o t h e s e s . 15 Three questions a r i s i n g from the model The s t r u c t u r e of the model suggests c e r t a i n questions concerning the teaching of h e u r i s t i c s . The three questions which d i r e c t l y r e l a t e d to the formation of the hypotheses are discussed i n t h i s section..' (1) Is i t p o s s i b l e to teach students to apply h e u r i s t i c s to novel mathematical problems? (2) Would i t be more e f f i c a c i o u s to teach h e u r i s t i c s only 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 with the teaching of content? (3) Would the a b i l i t y to l e a r n how to apply h e u r i s t i c s 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 existence of a set of h e u r i s t i c s . This study s e l e c t e d the h e u r i s t i c s of.analogy and examination of cases and i n v e s t i g a t e d whether I t Is p o s s i b l e to teach students to apply at l e a s t , one of these h e u r i s t i c s to novel mathematical problems. The f i r s t two questions l e d to the s e l e c t i o n of two treatments, h e u r i s t i c s only and h e u r i s t i c s plus content. The h e u r i s t i c s only-treatment was designed to teach only the h e u r i s t i c s , while the h e u r i s t i c s plus content treatment was designed to 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 the teaching o f content. A t h i r d treatment u t i l i z e d content only, with reference to h e u r i s t i c s , and was used as an experimental c o n t r o l , The core contains many categories such as algebra and 16 geometry. Question. 3 concerns whether one category 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 another. Since the secondary school' mathematics curriculum i s p r i m a r i l y based on the categories of algebra, 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 h e u r i s t i c s . Furthermore, a student i s also exposed to mathematics through a v a r i e t y of experiences outside a mathematics course, such as science courses, newspapers and commercial 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 to as mathematically n e u t r a l was employed. T h i s ' r e s u l t e d i n a t o t a l - of nine groups, corresponding to the c e l l s i n f i g u r e 2. Figure 2 Experimental Groups Veh i c l e s N e u t r a l A l g e b r a i c Geometric H e u r i s t i c s H - N H - A H - G H e u r i s t i c s plus content HC - N HC - A HC - G Content only C..- N C - A C - G 1 7 For the purpose o f . l a t e r d i s c u s s i o n the nine groups have been i d e n t i f i e d by a two part code. The H, HC and C denote the three treatments: h e u r i s t i c s only (H). h e u r i s t i c s plus content (HC) and content only (C). The N, A and G denote the three v e h i c l e s : mathematically n e u t r a l (N), a l g e b r a i c (A) and geometric (G). Further d e t a i l s on these groups can be found i n chapter I I I - Design and Procedure. The H, HC and C are used separately to denote the three treatments, i r r e s p e c t i v e of 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 subjects i n H - N, H - A and H G , Research Hypotheses In each of hypotheses 1 through 5 , the dependent v a r i a b l e i s a measure of a b i l i t y to apply at l e a s t one of the h e u r i s t i c s (analogy or examination of cases) to a novel mathematics problem: Hypothesis 11. A group taught h e u r i s t i c s only (H) w i l l score higher than a group taught content only (C). Hypothesis 2_. A group taught h e u r i s t i c s plus content (HC) w i l l score higher than a group taught content only (C). Hypothesis 3_. A group taught h e u r i s t i c s only (H) w i l l score higher than a group taught h e u r i s t i c s plus content (HC). 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 o n l y group (H - N, H - A, H - G) as compared t o the mean s c o r e o f the c o r r e s p o n d i n g c o n t e n t o n l y group (C - N, C -A, C - G) w i l l be Independent o f the v e h i c l e . H y p o t h e s i s 5_. The mean s c o r e o f a h e u r i s t i c s p l u s c o n t e n t group (HC - N, HC - A, HC - G) as compared t o the 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 group (C - N, C - A, C - G) w i l l be independent o f the v e h i c l e s . CHAPTER I I Related L i t e r a t u r e  I n t r o d u c t i o n 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: (1) l i t e r a t u r e on the problem s o l v i n g process; (2) h i s t o r i c a l and education views of analogy and examination of cases; and (3) r e l a t e d research. The Problem S o l v i n g Process Although educators have discussed the r o l e of h e u r i s t i c s i n problem s o l v i n g there have been few research studies that have i n v e s t i g a t e d general h e u r i s t i c s . Wittrock (1964, p. 6 l ) had suggested that researchers should d i r e c t t h e i r a t t e n t i o n to teaching c h i l d r e n how to make d i s c o v e r i e s . He wrote: Instead of asking i f l e a r n i n g by discovery i s or i s not as e f f e c t i v e as some other way to l e a r n , one ought now to s t a r t to ask by what sequence of d i r e c t e d or undirected experiences can we teach c h i l d r e n to l e a r n to discover. Shulman (1968, p. 89) observed that there was a l a c k of research on general h e u r i s t i c s when he wrote: Notably absent are studies which deal w i t h the question of whether general techniques, s t r a t e g i e s , and h e u r i s t i c s of discovery can be learned - by discovery or any other 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 educators and p s y c h o l o g i s t s has been to i n v e s t i g a t e the nature of the problem s o l v i n g 20 process. Their e f f o r t s have r e s u l t e d i n analyses of the problem s o l v i n g process. For example, Bloom and Broder (1950, p. 25) suggested that the f o l l o w i n g four components are i n v o l v e d i n the problem s o l v i n g process. 1. Understanding of the nature of the problem 2. Understanding of the ideas contained i n the problem 3. General approach to the s o l u t i o n of problems 4. A t t i t u d e towards the s o l u t i o n of problems. Garry and Kingsley (1970, p. 464) suggested that there are' three steps i n problem s o l v i n g : the search phase, the 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 of the f i n a l s o l u t i o n . Gagne (1966, p. 138) i n c l u d e d the stages of d e f i n i n g the 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 , searching f o r and for m u l a t i n g hypotheses, and v e r i f y i n g the s o l u t i o n i n h i s a n a l y s i s of the problem s o l v i n g process. While these analyses give an o v e r a l l framework f o r examining the nature of the. problem s o l v i n g process, they do not inclu d e s p e c i f i c techniques that, can be a p p l i e d by the l e a r n e r to make.discoveries about novel problems. For the mathematics educator the works of Polya (1954a, 1954b, 1957, 1962, 1965) provide a major source of s p e c i f i c questions that can be a p p l i e d to novel mathematical problems. In How To  Solve I t Polya (1957, pp. x v i - x v i i ) l i s t e d four stages i n the problem s o l v i n g process. 21 UNDERSTANDING THE PROBLEM What i s the unknown? What are the data? ... DEVISING A PLAN Have you seen i t before? Or have you seen the 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 general problem? A more s p e c i a l problem? An analogous problem? Could you solve part of the problem? ... CARRYING OUT THE- PLAN Carry out your plan of the 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 ge n e r a l , but the various suggestions under these headings contain a s e r i e s of s p e c i f i c questions that a problem s o l v e r can use when faced w i t h a novel 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 of Polya to the study of h e u r i s t i c s i n mathematics .is evidenced by the f a c t that much of the research concerned w i t h mathematical h e u r i s t i c s Is based upon Polya's work. For example, Ashton (1962) employed the questions In Polya's h e u r i s t i c process as the bas i s of her h e u r i s t i c treatment i n the study of problem s o l v i n g by n i n t h grade algebra students. A major component of K i l p a t r i c k ' s (1967) study concerned the development of 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 of the problem s o l v i n g process. P o l y a ' s ' h e u r i s t i c questions and suggestions formed the i n i t i a l b a s i s f o r developing the 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 also developed categories of problem s o l v i n g thought processes. They l i s t e d f i v e types of evidence f o r probable inference that are u s e f u l i n developing and t e s t i n g g e n e r a l i z a t i o n s . These are the methods of simple enumeration, analogy, extending a p a t t e r n o:f thought, hunches and t h e i r r e l a t i o n to proof, and t e s t i n g the 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 to the i n v e s t i g a t o r , namely unpublished a r t i c l e s by MacPherson (1971b) and Rousseau .(1971)• In s e l e c t i n g the h e u r i s t i c s of analogy and examination of cases the i n v e s t i g a t o r chose two h e u r i s t i c s that were common to a l l four sources (Polya, Smith and Henderson, MacPherson, Rousseau). Analogy and Examination of 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 of analogy as a h e u r i s t i c -cannot be overestimated. The c l a s s i c reference to the value of analogy can be a t t r i b u t e d to Henri Poincare. In developing the theory of Fuchsian Functions Poincare was reported to have s a i d (Hadamard, 1949 3 p , 13): I want to represent these functions by the quotient of two s e r i e s ; t h i s idea was p e r f e c t l y conscious' and d e l i b e r a t e ; the analogy with e l l i p t i c f u n c t i o n s guided me. Polya (1954a, pp. 18,19) noted that Euler employed an analogy to conjecture a sum f o r the i n f i n i t e series. £l/n 2. 23 The. value of analogy Is not l i m i t e d t o the d i s c i p l i n e o f mathematics. For example. Maxwell (Nlven, 1.890. pp. 156 - 7) discussed the importance of p h y s i c a l analogies i n the mathematical s c i e n c e s , where one i s in v o l v e d In analogies between p h y s i c a l laws and the laws of number. Brumbaugh '(1968, p. 6) i n d i c a t e d how P l a t o used analogy i n h i s arguments. He wrote: I f we can f i n d numbers f a l l i n g i n the same classes as the e n t i t i e s we want to study, the r e l a t i o n s of 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 observing the c o r r e l a t e d r e l a t i o n s of the analogous classes of numbers. D e l t h e i l (1971, P- 37), a f t e r mentioning the works of Kepler and Poincare concluded: The establishment of such comparisons i s the very essence of the genius f o r discovery and the goal of a mode of i n t u i t i o n of the 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 of analogy as a h e u r i s t i c has been summarized In the f o l l o w i n g b r i e f but concise quotation of Polya (1954a, p. 17), who s a i d of analogy that i t "seems to have a share- of a l l d i s c o v e r i e s , but i n some I t has the l i o n ' s share". Many educators have discussed the r o l e of analogy In teaching. In i l l u s t r a t i n g the value of analogy as a problem s o l v i n g technique Smith and Henderson (1959, p. 124) wrote: Analogies between the two subjects, [plane and s o l i d , geometry - the 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 , provide a:\hierarchy of r e l a t i o n s h i p s that are more e a s i l y remembered, and suggest many conjectures that may l a t e r be proved or disproved. K l i n e (19 .64 , p. 457) suggested that once the ideas o f Non-Euclidean Geometries have been grasped there i s value i n "seeking the analogues o f theorems that held i n Euclidean. Geometry". Moser (1968, pp. 374 - 6) pointed to the value of analogy i n developing r e l a t i o n s h i p s between the graphs of various conies and appropriate absolute value equations. For example, he considered the r e l a t i o n s h i p between the graphs of y = x 2 and y = | x | ; and ( x / 2 ) 2 + (y/2.) 2 = 1 and |x/2| + |y/2| = 1. F i n a l l y , Balk (1971, p. 135) s t a t e d t h a t : I t i s w e l l known that 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 , play an important r o l e i n any s p e c i a l i z e d f i e l d . Balk (1971, pp> 135 - 6) continued by c o n s i d e r i n g examples of where analogy could be of value i n the formulation'of conjectures. For example, he suggested how the p r o p o s i t i o n s that i f the sum of the d i g i t s of 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 (or 9), could lead to the analogous p r o p o s i t i o n that i f the,-;,sum of the d i g i t s i s d i v i s i b l e by 27 the number i s d i v i s i b l e by 27. The f a c t that t h i s p a r t i c u l a r conjecture i s i n v a l i d does not minimize the value of analogy as a mathematical h e u r i s t i c . These examples provide an h i s t o r i c a l p e r s p e c t i v e of analogy as w e l l as an i n d i c a t i o n • o f the educational s i g n i f i c a n c e of the h e u r i s t i c . I t seems reasonable to conclude from t h i s l i t e r a t u r e that analogy Is a h e u r i s t i c worth teaching. 25 Examination of Cases D i r e c t h i s t o r i c a l evidence of mathematicians applying examination of cases i s p r a c t i c a l l y non-existent. The only d i r e c t reference i s to Nichomachus (Heath, 1931, p. 68) c o n s i d e r i n g the s e r i e s of odd numbers: 1, 3, 5, 7, 9, 11, 13, ... Nichomachus wrote that 1 was a cube; the sum of the next two, 3 + 5 was a cube; the sum of the next three was a cube; and so on; and he noted that such i n f o r m a t i o n could be u t i l i z e d to deduce the formula f o r the sum of the f i r s t n cubes. I n d i r e c t evidence of the u t i l i z a t i o n of examination of cases i s found i n Gauss's d i a r y . There i s a note to the e f f e c t that every p o s i t i v e i n t e g e r i s the sum of three t r i a n g u l a r numbers ( B e l l , 1937, p. 228). The l a c k of d i r e c t h i s t o r i c a l evidence of mathematicians applying the h e u r i s t i c of examination of cases could w e l l be a t t r i b u t e d to the nature of published mathematics. Mathematical p u b l i c a t i o n s c o n s i s t of the f i n a l products, u s u a l l y i n the form of proofs of a mathematical endeavor. Since examination of cases would be most appropriate i n the exploratory phase of d i s c o v e r y , reference to the h e u r i s t i c would not appear i n the f i n a l l i t e r a t u r e . In p a r t i c u l a r , the cases may have been the f i r s t steps l e a d i n g to a proof by mathematical i n d u c t i o n . Hence, only 26 the f i n a l form of proof would be published and not the 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 proof by complete enumeration of 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 references to the u t i l i z a t i o n of the h e u r i s t i c : ... but I t Is j u s t the ' p r o o f by enumeration of cases' (and cases which do not , at bottom, d i f f e r at a l l profoundly) which a r e a l mathematician tends to despise. Although there i s l i t t l e h i s t o r i c a l evidence of the use of examination of cases, educators consider the h e u r i s t i c to be of s i g n i f i c a n t value. Smith and Henderson (1959) i n c l u d e d enumeration of cases i n t h e i r f i v e types of evidence f o r probable i n f e r e n c e that should be encouraged i n the secondary school. K i n s e l l a (1970, p. 244) suggested that "the discovery of r e l a t i o n s u s u a l l y Involves the strategy of studying 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 Is perceived". Szabo (1967, p. 840) gave the f o l l o w i n g impression concerning the h e u r i s t i c : We also f e e l that students can l e a r n how [ i t a l i c s i n o r i g i n a l ] to search f o r patterns and w i l l be able to use t h i s knowledge In s o l v i n g new problems. Brown and Walter (1972, pp. 42 - 44) demonstrated how s p e c i f i c and general cases can be employed to formulate conjectures and questions. They asked students 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 T r i p l e s a = m 2 - n 2 ; b = 2mn ; c = m2 + n 2 27 'to use the formula to pose questions and formulate conjectures concerning Pythagorean t r i p l e s . The questions r a i s e d i n c l u d e : How are a, b and c a f f e c t e d by the evenness and oddness of m and n? What happens to a, b and c i f m and n are consecutive?... The examination of s p e c i f i c t r i p l e s produced c o n j e c t u r e s , which were oft e n s t a t e d i n the form of questions 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 unique?... Scott (1971, pp. 38 - 45) showed how the examination of the number of c e l l s formed by the diagonals of d i f f e r e n t polygons could lead to an i n v e s t i g a t i o n of d i f f e r e n t p a t t e r n s . For example, what i s the p a t t e r n connecting the number of sides of a polygon and the number of c e l l s formed by the diagonals. Figure 3 Number of C e l l s 4 c e l l s 11 c e l l s . 28. 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 e d u c a t o r s c o n s i d e r b o t h analogy and e x a m i n a t i o n o f 'cases t o be v a l u a b l e t e c h n i q u e s f o r problem s o l v i n g . The h i s t o r i c a l e v i d e n c e r e l a t e d t o analo.gy suggests t h a t t h i s h e u r i s t i c p l a y s a s i g n i f i c a n t r o l e i n m a t h e m a t i c a l d i s c o v e r y , w h i l e e d u c a t o r s have l o n g promoted the t e a c h i n g o f both 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 . R e l a t e d Research At t h e p r e s e n t time t h e r e i s l i t t l e r e s e a r c h on t e a c h i n g s t u d e n t s t o a p p l y m a t h e m a t i c a l h e u r i s t i c s , per se. However, 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 has been 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 and p s y c h o l o g i s t s . 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 s e c t i o n r e v i e w s s t u d i e s t h a t have compared 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 the major o b j e c t i v e was t o develop s t u d e n t s ' a b i l i t y t o a p p l y g e n e r a l problem s o l v i n g p r o c e d u r e s ( i n c l u d i n g h e u r i s t i c s ) t o new m a t h e m a t i c a l s i t u a t i o n s . D i s c o v e r y V e r s u s 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 here a l l i n c l u d e an i n v e s t i g a t i o n of the r e l a t i v e e f f e c t i v e n e s s o f the i n s t r u c t i o n a l t e c h n i q u e s i n 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 29 s i t u a t i o n s . This review, i s l i m i t e d to studi e s i n v o l v i n g mathematical content.. Kersh (1962) used a programmed booklet to teach 90 high school students two r u l e s f o r a d d i t i o n of s e r i e s . Kersh (1962. p. 66) described the r u l e s as f o l l o w s : 1. Odd Numbers r u l e . The sum of any s e r i e s of consecutive odd numbers beginning w i t h 1 i s equal to the square of the number of f i g u r e s In the s e r i e s . (For example, 1, 3, 5, 7, i s such a s e r i e s ; there are four 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 of any s e r i e s of numbers i n which the d i f f e r e n c e between the numbers i s constant i s equal to one-half the product of the number of f i g u r e s and the sum of the f i r s t and l a s t numbers. (For example,' 2, 3, 4, 5 a i s such a s e r i e s ; 2 and 5 Is 7; there are four 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 Is the sum.) The students were t o l d the r u l e s and given p r a c t i c e i n t h e i r a p p l i c a t i o n . 'On completing the i n i t i a l booklet the students were randomly assigned to one of three treatment groups, namely d i r e c t e d l e a r n i n g (DL), guided discovery (GD) and rote l e a r n i n g (RL). The DL group r e c e i v e d an a d d i t i o n a l programmed booklet i n which the r u l e s were explained. The GD group was taught by a quest i o n i n g technique In which the students were guided towards an explanation of the r u l e s . The RL group was given no a d d i t i o n a l i n s t r u c t i o n and was inco r p o r a t e d In the experiment as a c o n t r o l group. 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 each group of 30 was d i v i d e d i n t o three subgroups (10 per group). D i f f e r e n t subgroups from each treatment were t e s t e d a f t e r 3 days, 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 .Testing Arrangement f o r Treatment Groups Sub-groups DL 10 10 10 Group GD 10 10 10 RL 10 10 10 Tested a f t e r 3 days 2 weeks 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 of two word problems that could be solved by applying the 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 of subjects who used the appropriate r u l e i n the s o l u t i o n was the index of 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 treatments and between t e s t periods on one of the two t e s t problems. He s t a t e d that 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 that the RL group "was found to be c o n s i s t e n t l y s u p e r i o r i n every respect to the other treatment groups" (Kersh, 1962, p. 68). In terms of the index of t r a n s f e r the RL group performed b e s t , the DL group worst and the GD group i n between. This study i s important f o r two reasons. F i r s t , since the a d d i t i o n r u l e s i n the experiment were new to a l l students the r e s u l t s i n d i c a t e that a l l three treatments r e g i s t e r e d some degree of t r a n s f e r ( a b i l i t y to apply the r u l e s to a new problem). Second, 31 the r e s u l t s i n d i c a t e that I t i s not necessary to teach bv_ discovery t o obtain t r a n s f e r . Ray (1961) compared discovery teaching w i t h d i r e c t 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 and t r a n s f e r to new but r e l a t e d problems. He used two experimental groups, d i r e c t i n s t r u c t i o n (DI) and d i r e c t e d discovery (DD), while a t h i r d group, which r e c e i v e d no i n s t r u c t i o n was used as a c o n t r o l (C). Ray also d i v i d e d h i s sample i n t o three a b i l i t y groups based on h i g h , medium and low i n t e l l i g e n c e . The sample of 135 grade 9 boys were randomly assigned to one of the s i x experimental treatment x i n t e l l i g e n c e groups (18 per group), or to a c o n t r o l x i n t e l l i g e n c e group (9 per group). 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 of 47 minutes of i n s t r u c t i o n on the p h y s i c a l s t r u c t u r e of a micrometer and how to use i t . The students were taught, i n groups of nine, w i t h three students being s e l e c t e d from each of the a b i l i t y l e v e l s . Students were te s t e d one 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 completed and f u r t h e r r e t e n t i o n and t r a n s f e r t e s t s were administered s i x weeks l a t e r . The t r a n s f e r t e s t i n v o l v e d items "that attempted to measure a p p l i c a t i o n of knowledge gained to new 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 treatments on both .the i n i t i a l (one week) and delayed ( s i x week) t r a n s f e r t e s t s , with the r e s u l t s s t a t i s t i c a l l y favouring the DD treatment. 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 d i f f e r e n c e on the I n t e l l i g e n c e 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 h i g h a b i l i t y s t u d e n t s . Ray's c o n c l u s i o n s are d i f f e r e n t from t h o s e i n the K e r s h (1962) e x p e r i m e n t , 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 performed b e t t e r t h a n the programmed group ( d i r e c t i n s t r u c t i o n ) . However, l i k e K e r s h , the d a t a shows t h a t b o t h e x p e r i m e n t a l groups e v i d e n c e d some degree o f t r a n s f e r . Gagne" and Brown (1961) employed t h r e e i n s t r u c t i o n a l , t e c h n i q u e s f o r t e a c h i n g s t u d e n t s how t o sum number s e r i e s . Three programmed u n i t s were d e v e l o p e d , c o n s i s t i n g o f an i n t r o d u c t o r y programme which was i d e n t i c a l f o r a l l t h r e e 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 to c o r r e s p o n d t o t h r e e t r e a t m e n t s , d i s c o v e r y (D), g u i d e d d i s c o v e r y (GD) and r u l e and example (R&E). T h i r t y - t h r e e grade 9 and 10 boys were randomly a s s i g n e d t o the t h r e e t r e a t m e n t s (11 p e r g r o u p ) . The t r a n s f e r t e s t comprised f o u r problems on summation o f 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 t h a t was new t o t h e s t u d e n t s . The t e s t 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 s t u d e n t . U s i n g t h r e e measures o f e f f e c t i v e n e s s o f the programmes, namely time r e q u i r e d t o s o l v e the p r o b l e m s , number o f h i n t s r e q u i r e d and a w e i g h t e d c o m b i n a t i o n .of time and h i n t s , Gagne and Brown c o n c l u d e d t h a t t h e GD t r e a t m e n t was the most 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 t r a n s f e r , the R&E t r e a t m e n t l e a s t e f f e c t i v e w i t h the D t r e a t m e n t i n between. 33 Eldredge (1965a, 1965b). reported s t u d i e s that were based on Gagne1 and Brown's experiment.. In the f i r s t study Eldredge r e p l i c a t e d the. Gagne1 and Brown experiment, employing two of the three treatments, 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 i n that Eldredge used a mixed sample, two classes 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 ince the h i n t s used i n the o r i g i n a l experiment were not a v a i l a b l e . One c l a s s r e c e i v e d the GD treatment and the other c l a s s the R&E treatment. Eldredge found only 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 of the four 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 favouring the GD treatment on one problem, another problem favoured the GD treatment and the other two problems favoured the R&E treatment, although none of the l a s t three 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 s i g n i f i c a n t . 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 could be due to 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 h i n t s . In a second study Eldredge employed a modified v e r s i o n of the Gagne and Brown m a t e r i a l s that had been developed by D e l l a - P i a n a and Edlredge (1965a, 1965b). The modified m a t e r i a l s c o n s i s t e d of two programmes, a guided discovery 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 of 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 r u l e f o r summing the' s e r i e s . A t o t a l of 97 grade 9 students (43 boysj 54 g i r l s ) were used i n the experiment. As i n the o r i g i n a l experiments.^ Gagne and Brown (1961), Eldredge (1965a) ) the t r a n s f e r task c o n s i s t e d of four u n f a m i l i a r problems on the summation of s e r i e s . 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 of three s e s s i o n s , immediately followed by a t r a n s f e r t e s t , and then the t r a n s f e r t e s t was readminlstered four weeks l a t e r . Using i n t e l l i g e n c e and time taken to 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 that the GD group performed s i g n i f i c a n t l y b e t t e r than the R&E group.. A two-way a n a l y s i s of variance using treatments as one f a c t o r and a b i l i t y (high,low) as the other i n d i c a t e d that the high a b i l i t y group performed s i g n i f i c a n t l y b e t t e r than the low a b i l i t y group, and again a s i g n i f i c a n t d i f f e r e n c e between treatments was found. I d e n t i c a l conclusions followed from the data a n a l y s i s of the delayed (four week) t r a n s f e r t e s t . However, i f I n i t i a l performance was used as a c o v a r i a t e i n the a n a l y s i s the d i f f e r e n c e s disappeared, le n d i n g credence to the hypothesis that the d i f f e r e n c e s on the delayed 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. In the s t u d i e s reviewed to t h i s point the t r a n s f e r t e s t s c o n s i s t e d of Items that r e q u i r e d the a p p l i c a t i o n of knowledge acquired 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 that were new. to the student but d i r e c t l y r e l a t e d to the i n s t r u c t i o n a l materials." Two o v e r a l l conclusions can be drawn from these studies that relate, to the present study. F i r s t , the r e s u l t s of the studies reviewed show that i t i s p o s s i b l e to teach.students to apply what has been learned i n a mathematical context to new, but d i r e c t l y r e l a t e d mathematical problems. Second, some of the s t u d i e s (Gagne and Brown (1961), Eldredge (1965a,1965b)) i n d i c a t e that using 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 present study can be considered an extension of these experiments i n that i t requires students to apply what has been l e a r n e d , the h e u r i s t i c s , to t r a n s f e r tasks where the mathematical content i s unrelated to that used 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 . Furthermore, the present study employs s e l f - i n s t r u c t i o n a l booklets and the evidence i n d i c a t e s that t h i s mode of i n s t r u c t i o n can lead to s u c c e s s f u l t r a n s f e r . The f o l l o w i n g studies are of p a r t i c u l a r s i g n i f i c a n c e since one (Worthen (1965)) i n c l u d e d an i n v e s t i g a t i o n of t r a n s f e r to unrelated problems, and i n the other Kersh (1964) searched f o r evidence of h e u r i s t i c s i n the students s o l u t i o n s to the problems. Worthen (1965) conducted an experiment i n v o l v i n g two experimental treatments, discovery (D) and expository (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 of a development of 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 , the d i s t r i b u t i v e p r i n c i p l e and exponential n o t a t i o n w i t h i n t e g r a l powers. The sample c o n s i s t e d of 432 students, 14 grade 6 c l a s s e s and 2 grade 5 classes i n 8 schools, with an a d d i t i o n a l 106 students (3 grade 6 c l a s s e s ) i n a n i n t h school used f o r c o n t r o l purposes. Eight teachers were i n v o l v e d i n the study, each teaching two c l a s s e s , one by treatment D and the other by treatment E. At the end of the s i x week i n s t r u c t i o n a l period'a v a r i e t y of c r i t e r i o n t e s t s were administered, i n c l u d i n g a t t i t u d e t e s t s , content t e s t s and t r a n s f e r t e s t s . The three t e s t s germane to the present study are the concept t r a n s f e r t e s t (cTT), number sequences and s e r i e s t e s t (nsTT) and a t e s t i n v o l v i n g the a s s o c i a t i v e p r i n c i p l e (apTT). On the cTT data Worthen employed a two-way a n a l y s i s of covariance w i t h main f a c t o r s teachers and treatments, and co v a r i a t e s i n t e l l i g e n c e , a r i t h m e t i c problem s o l v i n g , a r i t h m e t i c computation and the score on 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 . Worthen concluded that the D group performed better..than . (p <.08) the E group. He a l s o found s i g n i f i c a n t teacher e f f e c t s and i n t e r a c t i o n e f f e c t s (p <.001). Using the same co v a r i a t e s but employing a one-way ANACOVA wi t h the D, E and C groups he found that 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 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 <.01) ; and 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 the C and E groups. 37 The cTT included new problems that 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 the t e s t items were extensions of the m a t e r i a l s . Worthen's r e s u l t s , are cons i s t e n t w i t h the previous s t u d i e s i n that they show evidence of t r a n s f e r to. new but 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 of problems on-content that •; was unr 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 . The nsTT con s i s t e d of questions on number sequences and s e r i e s i n which the students had to f i n d the p a t t e r n and r u l e . The two-way a n a l y s i s again showed s i g n i f i c a n t teacher and i n t e r a c t i o n e f f e c t s . This a n a l y s i s a l s o i n d i c a t e d that 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 E group (p <.05). I t appears that Worthen used the words ' t r a n s f e r of h e u r i s t i c s ' to r e f e r to the t r a n s f e r of a b i l i t y to 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 to search f o r and f i n d a p a t t e r n . In the apTT the problems i n v o l v e d the 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 students were faced • wi 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 . For example, i n f i n d i n g the sum of 975 + (25 + 983) the a s s o c i a t i v e p r i n c i p l e would be of great value. Using a grading scheme i n which students were given one point f o r a corr e c t answer and an a d d i t i o n a l four p o i n t s i f t h e i r work showed evidence of the p r i n c i p l e , 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 E group (p <.025)- Worthen's r e s u l t s 38 i n d i c a t e that the discovery group learned how to. search, f o r a p a t t e r n or r u l e s i g n i f i c a n t l y b e t t e r than the exp o s i t o r y group. Thus, t h i s experiment provides•evidence that 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 of h e u r i s t i c s to new problems that were un 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 . Kersh (1964) undertook a study to 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 non-directed (discovery) l e a r n i n g . Three i n s t r u c t i o n a l u n i t s were designed w i t h the primary 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 to f i f t h graders. In the free discovery u n i t (D) each student was encouraged to discover the p r i n c i p l e independently and then show the experimenter that he had learned the p r i n c i p l e . The experimenter worked wi t h the students 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 that they had learned the d i s t r i b u t i v e p r i n c i p l e . The programmed discovery u n i t (PD) gave guidance designed to a i d students to search f o r the p r i n c i p l e . V e r b a l i z e d aproval by the experimenter 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 to as programmed guidance (PG) was developed to teach the same p r i n c i p l e , but without employing the discovery method. The PG treatment d i d not encourage the searching behaviours f o s t e r e d i n the D and PD programmes. A l l students r e c e i v e d i n s t r u c t i o n u n t i l . t h e y met a predetermined c r i t e r i o n , 39 namely that-.they were i n i t i a l l y r e q u i r e d to c o r r e c t l y answer f i v e of s i x d i s t r i b u t i v e p r i n c i p l e questions. I f they f a i l e d they were retaught, and the f i n a l r e s u l t was that four students 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 . The p o s t t e s t s i n c l u d e d a t e s t of new l e a r n i n g , which was administered w i t h i n 24 hours of a student completing the i n s t r u c t i o n a l m a t e r i a l s . This t e s t was administered i n d i v i d u a l l y to each student, and the student's were r e q u i r e d to " t h i n k out l o u d " , or 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 behaviours such as 'search f o r p a t t e r n s ' , 'checking f o r exceptions to p a t t e r n s ' , and 'checking to see i f statements were true or f a l s e ' . In terms of 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 treatments on 'search f o r p a t t e r n s ' . The D group had the greatest number of students who evidenced t h i s behaviour (27 of 30), followed by the PG group (21 of 30) and the l e a s t number i n the PD group (17 of 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 other c r i t e r i a of new l e a r n i n g . For example, on 'checking f o r exceptions to p a t t e r n s ' 11 students i n the D group used the technique, 13 i n the PG group and, 12 i n the PD group. Kersh's study shows that the D treatment was more e f f e c t i v e than the others i n teaching students to 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 conclusions can be drawn from the studies reported i n t h i s s e c t i o n that have relevance f o r 40 the present study. The studies, i n d i c a t e that (1) i t i s p o s s i b l e to. teach students to. t r a n s f e r knowledge to 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 . Furthermore, Worthen's study (1965) i n d i c a t e s that students can be taught to approach problems that are not d i r e c t l y 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 , a r e s u l t 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 of the f a c t t h a t the t r a n s f e r t e s t s to 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 to the i n s t r u c t i o n a l m a t e r i a l s . Kersh's study a l s o produced evidence of the t r a n s f e r of the h e u r i s t i c of searching f o r a p a t t e r n , which i s a component of examination of cases. Teaching H e u r i s t i c Procedures Schaaf (1954) designed a course which had as its;-. major o b j e c t i v e the development of students' a b i l i t y to g e n e r a l i z e . A group of n i n t h grade students at a u n i v e r s i t y school acted as the experimental group. Data from 127 n i n t h grade students ( i n s i x classes i n the P u b l i c School System) were employed f o r c o n t r o l purposes. The experimental p e r i o d l a s t e d one year. The content s t u d i e d by the experimental group i n c l u d e d numbers and operations; graphs and formulas; equations and problem s o l v i n g ; p r o p o r t i o n 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 experimental course was designed to encourage students to extend previous mathematical 41 l e a r n i n g , and to. discover new p r i n c i p l e s , from past and present experiences. Throughout the' .course, emphasis was placed on student d i s c o v e r y , with the teacher a c t i n g as a guide In h e l p i n g the students to discover g e n e r a l i z a t i o n s . The methods of g e n e r a l i z a t i o n that were taught i n c l u d e d simple enumeration, analogy, extension of 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 ) , deduction, v a r i a t i o n and inverse deduction. A p r e t e s t of the a b i l i t y to g e n e r a l i z e i n d i c a t e d that the experimental 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 year 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 Information on the experimental group was a v a i l a b l e from sources such as student notebooks, Schaaf's own notes on the classroom meetings and observer reports (an observer v i s i t e d the classroom p e r i o d i c a l l y throughout the experimental p e r i o d ) . Schaaf's r e s u l t s i n d i c a t e that on some measures of the a b i l i t y to g e n e r a l i z e the experimental group made s i g n i f i c a n t l y greater gains than the c o n t r o l group. For example, the experimental group's gain 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 t e r p o l a t i o n 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 j u s t i f i e d was s i g n i f i c a n t l y greater than the c o n t r o l group's gain. On other measures he found s i g n i f i c a n t s gains 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 gain.scores. He suggested that the l a c k of s i g n i f i c a n c e between the gain scores may have been due to the experimental group having 42 an i n i t i a l l y h i g h e r s c o r e , g i v i n g them a more r e s t r i c t e d range f o r improvement, o r t h a t t h e t e s t was not s e n s i t i v e enough i n measuring the g a i n . S c h a a f ' s r e s u l t s I n d i c a t e t h a t exposure t o h e u r i s t i c s such as analogy,, enumeration and i n v e r s e d e d u c t i o n s i g n i f i c a n t l y i n c r e a s e d s t u d e n t s ' a b i l i t y 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 . F u r t h e r m o r e , on c e r t a i n measures t h i s g a i n was s i g n i f i c a n t l y g r e a t e r than t h a t o f a c o n t r o l group. The f o l l o w i n g f o u r s t u d i e s (Ashton (1962), L i b e s k i n d (197D, Lucas (1972), G o l d b e r g (1974)) a l l based t h e i r 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 h e u r i s t i c p r o c e s s . They employed 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 i n s t r u c t i o n a l m a t e r i a l s (see P o l y a , How t o S o l v e I t , 1957) • Ashton (1962) s e l e c t e d t e n grade n i n e a l g e b r a c l a s s e s , i n f i v e d i f f e r e n t s c h o o l s , f o r h e r s t u d y . Each s c h o o l c o n t a i n e d two c l a s s e s , b o t h b e i n g i n s t r u c t e d by the same t e a c h e r . I n each p a i r o f c l a s s e s one was an e x p e r i m e n t a l ( h e u r i s t i c ) group and t h e o t h e r a c o n t r o l ( t e x t b o o k ) 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 t e n weeks, d u r i n g which time whenever the h e u r i s t i c group was f a c e d w i t h a problem they were t o ask t h e m s e l v e s , o r were asked q u e s t i o n s from P o l y a ' s work, such as "What i s t h e unknown? What are the data? What are the c o n d i t i o n s ? " / An e x a m i n a t i o n o f Ashton's p r o c e d u r e i n d i c a t e s t h a t h e r i n s t r u c t i o n a l s e s s i o n s i n v o l v e d 43 more than j u s t Polya's questions. For example, students i n the h e u r i s t i c group were allowed to w r i t e t h e i r own questions to c l a r i f y concepts. Furthermore, before anyone i n a h e u r i s t i c c l a s s was permitted to answer a question at l e a s t two of the students had to r e s t a t e the qu e s t i o n , so that by t h i s " t r a n s l a t i o n " the students f a m i l i a r i z e d themselves w i t h algebra as a language. During the experimental p e r i o d the c o n t r o l classes continued w i t h t h e i r normal classroom programme. At the end of the ten 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 of 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, her study provides evidence that the teaching of h e u r i s t i c procedures based on the work of Polya 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 Polya's h e u r i s t i c approach as the b a s i s f o r developing a high school u n i t on number theory. In order to 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 students to a minimum the development and proof of theorems was r e s t r i c t e d to the domain of whole numbers. The u n i t was taught by L i b e s k l n d to ten students, three of whom had completed grade 9, another three had completed grade 10, and the l a s t four 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 of 25 sessions each of 50 minutes, w i t h each session being immediately followed by a 30 minute 44 study p e r i o d . As part of h i s e v a l u a t i o n of the u n i t he i n v e s t i g a t e d the question of whether students could apply some of the proof techniques developed i n the u n i t to new problems. He concluded that the students were able to t r a n s f e r t h e i r knowledge to 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 that the students were able to solve t e s t problems given throughout.the experimental p e r i o d . For example, L i b e s k i n d noted that the students were able to prove that 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 then 10|a).- This was a new problem i n that the students had not seen i t before, although 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 proofs of general 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 then a|c). As f u r t h e r evidence of t r a n s f e r he c i t e d the f a c t that the students were able to solve a new problem i n v o l v i n g a proof that there e x i s t s a prime greater than 10 . The u n i t had i n c l u d e d a proof of the general r e s u l t that there e x i s t s no greatest prime. Libeskind's examples of t r a n s f e r 'involved the a p p l i c a t i o n of proof techniques to new problems that were c l o s e l y 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 . Thus h i s r e s u l t s i n d i c a t e that 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 to r e l a t e d problems. C l e a r l y t h i s r conclusion i s l i m i t e d by the small sample. Goldberg's study (1974) a l s o used m a t e r i a l s on number theory. Two sets of s e l f - i n s t r u c t i o n a l booklets were 45 w r i t t e n . Both sets c o n s i s t e d of seven booklets on number theory, one set wi t h emphasis on h e u r i s t i c s and the 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 . Nine c l a s s e s c o n t a i n i n g 238 undergraduates met f o r 12 sessions (75 minutes per session) w i t h two sessions per 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 students worked on the booklets f o r seven of these sessions. The nine c l a s s e s were randomly assigned to one of three treatments. In the r e i n f o r c e d h e u r i s t i c s treatment (RH) the classes used the h e u r i s t i c s b o o k l e t s . Furthermore, when homework was discussed, or new concepts considered i n the sessions when booklets were not being employed, the i n s t r u c t o r u t i l i z e d the h e u r i s t i c s as oft e n as p o s s i b l e , In the non-reinforced h e u r i s t i c s • t r e a t m e n t (H) the h e u r i s t i c s booklets were used but the h e u r i s t i c s not r e i n f o r c e d i n c l a s s . In the non-h e u r i s t i c treatment (C) the students were given the non-h e u r i s t i c booklets and the i n s t r u c t o r d i d not employ the h e u r i s t i c s i n c l a s s . The t e s t s that were administered at the end of the experimental p e r i o d i n c l u d e d a t e s t of understanding of number theory concepts (concepts I ) and a t e s t to measure the a b i l i t y to construct proofs (proofs I ) . Five weeks l a t e r delayed p o s t t e s t s of concepts (concepts I I ) . and proofs (proofs I I ) were given. Part of these delayed t e s t s were p a r a l l e l to the o r i g i n a l t e s t s , and part dealt w i t h m a t e r i a l covered i n the 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 the RH and H groups 46 a l t h o u g h the t r e n d i n d i c a t e d t h a t the RH group p e r f o r m e d b e t t e r than the H group. Based on S t a n f o r d Achievement T e s t d a t a on m a t h e m a t i c a l achievement the sample was d i v i d e d i n t o t h i r d s ( h i g h , medium, low) and the s c o r e s on the p o s t t e s t s were 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 ) . A C h i - s q u a r e a n a l y s i s i n d i c a t e d s i g n i f i c a n t d i f f e r e n c e s f o r h i g h a b i l i t y s t u d e n t s on the concepts t e s t s . These r e s u l t s f a v o u r e d s t u d e n t s i n the RH t r e a t m e n t . The r e s u l t s a l s o f a v o u r e d t h i s group on c o n s t r u c t i n g p r o o f s , a l t h o u g h the r e s u l t s were not s i g n i f i c a n t . Furthermore., h i g h a b i l i t y s t u d e n t s who r e c e i v e d the C t r e a t m e n t performed s i g n i f i c a n t l y b e t t e r on u n d e r s t a n d i n g concepts and tended t o c o n s t r u c t p r o o f s b e t t e r than t h e i r c o u n t e r p a r t s who r e c e i v e d the H t r e a t m e n t . A c o d i n g system s i m i l a r t o K i l p a t r i c k ' s (1967) c h e c k l i s t was 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 and was a p p l i e d t o the p r o o f s w r i t t e n by the s t u d e n t s i n the top t h i r d on the p r o o f s p o s t t e s t . The r e s u l t s g i v e an i n d i c a t i o n t h a t the s t u d e n t s who r e c e i v e d the RH t r e a t m e n t used h e u r i s t i c s more o f t e n than the s t u d e n t s who were i n the c o n t r o l group. The paper g i v e s no a d d i t i o n a l i n f o r m a t i o n on the use o f h e u r i s t i c s . O v e r a l l , Goldberg's r e s u l t s seem t o i n d i c a t e t h a t h i g h a b i l i t y s t u d e n t s i n the RH t r e a t m e n t b e n e f i t e d most from the i n s t r u c t i o n . The s t u d i e s by Ashton (1962), L i b e s k i n d (1971) and G o l d b e r g (1974) a l l based 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 P o l y a ' s h e u r i s t i c p r o c e s s . These s t u d i e s i n d i c a t e t h a t 47 i n s t r u c t i o n in. h e u r i s t i c s , tends to 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 , to apply what had been learned (content), to new but r e l a t e d problems. The next study d i f f e r s from these three i n that Lucas (1972, 1974) 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 students could apply h e u r i s t i c s to new problems. Lucas (1972, 1974) i n v e s t i g a t e d the teaching of 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 to two c l a s s e s j one c l a s s (H) r e c e i v e d the experimental programme c o n s i s t i n g of an i n s t r u c t i o n a l approach that exposed students t o , and emphasized h e u r i s t i c s , while the other c l a s s (C) re c e i v e d i n s t r u c t i o n i n the same content but with no exposure to the 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 students, 17 i n the H group and 13 i n the 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 the end of the 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 of word problems was administered i n d i v i d u a l l y to each student. The students 'thought out loud' while s o l v i n g the problems, and Lucas made notes as w e l l as re c o r d i n g the s e s s i o n s . I t should be noted that these problems could be solved by c a l c u l u s techniques,, although they were amenable to s o l u t i o n by other methods, such as graphing appropriate equations. In a n a l y s i n g the data Lucas was l o o k i n g f o r evidence of a v a r i e t y of h e u r i s t i c s t r a t e g i e s i n the students' s o l u t i o n s . These s t r a t e g i e s i n c l u d e d using mnemonic n o t a t i o n , using diagrams, checking to see I f 48 the r e s u l t s were reasonable, and using analogy. These h e u r i s t i c s were derived from Polya's work. He found s i g n i f i c a n t d i f f e r e n c e s favouring the h e u r i s t i c s treatment on v a r i a b l e s such as mnemonic n o t a t i o n , planning the problem and use of the r e s u l t s of r e l a t e d problems. Certain, v a r i a b l e s are of 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 present d i s s e r t a t i o n and are discussed i n d e t a i l . Uses method of a r e l a t e d problem r e f e r r e d to the strategy by which a r e l a t e d problem could be of value 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 to the given problem or may supply a cue f o r d e v i s i n g an analogous plan of attack". (Lucas, 1972, p.137) I f a student s p e c i f i e d a r e s u l t of another problem then used the r e s u l t i n s o l v i n g the new problem he received c r e d i t f o r evidence of the h e u r i s t i c , uses r e s u l t of 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 searching f o r - s i m i l a r i t i e s between the given 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 both uses methods of r e l a t e d problems and uses r e s u l t s of r e l a t e d problems, both d i f f e r e n c e s favouring the H„group. Of the 30 subjects t e s t e d there were only four who e x h i b i t e d evidence of reasoning by analogy and a l l four subjects were i n the H group. This study i s important f o r two reasons. F i r s t , Lucas looked f o r evidence of 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 conclusion i n v o l v i n g the use of 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 of teaching the h e u r i s t i c of analogy as defined i n t h i s study (see page 14 ). Lucas noted l i m i t a t i o n s such as small sample s i z e and the f a c t that students were not randomly assigned to treatments am any attempt to g e n e r a l i z e h i s r e s u l t s . However, h i s f i n d i n g s do lend support to the hypothesis that h e u r i s t i c s can be taught. These l a s t four studies (Ashton (1962), L i b e s k i n d ( 1 9 7 D , Goldberg (1974) and Lucas (1972, 1974)) i n d i c a t e that the teaching of h e u r i s t i c s tends to improve problem s o l v i n g performance. Furthermore, Lucas's r e s u l t s , while 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 of students 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 to new problems. The remaining s t u d i e s reviewed here have a l l employed a v a r i e t y of s t r a t e g i e s to promote mathematical t r a n s f e r . P r i c e (1965) conducted a study of the e f f e c t of discovery on achievement and c r i t i c a l t h i n k i n g of tenth grade students. He u t i l i z e d three groups: the experimental group (E) co n t a i n i n g 22 students; the ex p e r i m e n t a l - t r a n s f e r group (T) con t a i n i n g 23 students; and a c o n t r o l group (C) of 18 students. 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, during which time the experimental group re 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 that i n c l u d e d set theory and p r o p e r t i e s of number systems. The programme f o r the E group co n s i s t e d of question and answer sessions between the 50 teacher and the students (a discovery o r i e n t e d approach). The T group, i n a d d i t i o n to t h i s t eaching were given a s i n g l e study guide on inf e r e n c e before the lessons ( d i s c u s s i o n s ) / on extension of p a t t e r n . This study m a t e r i a l was of an h i s t o r i c a l nature, and was fol l o w e d by s p e c i f i c questions that the student was to answer. P r i c e (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 : During the classroom d i s c u s s i o n which took place regarding i n f e r e n c e there were many i n t e r e s t i n g observations made by the students ,who had never before r e a l l y stopped to th i n k about t h i s k i n d of m a t e r i a l . They discovered that from i n s u f f i c i e n t data i n c o r r e c t or many d i f f e r e n t inferences may be made. During the experiment the c o n t r o l group r e c e i v e d t h e i r normal classroom i n s t r u c t i o n . Based on the r e s u l t s of the C a l i f o r n i a Test of Mathematical Achievement P r i c e concluded that there 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 the experimental groups and c o n t r o l group on achievement. However, both experimental groups d i d s i g n i f i c a n t l y b e t t e r than the c o n t r o l on a Survey of Algebra A p t i t u d e , but only the T group gained s i g n i f i c a n t l y on the W a t s o n - C r i t i c a l Thinking A p p r a i s a l Test. Post (1967) presented a "S t r u c t u r e of the Problem-S o l v i n g Process" experimental u n i t to 74 grade seven mathematics students. Corresponding to these 74 students were an equal number i n a. c o n t r o l group, who had been matched on problem s o l v i n g a b i l i t y . He a l s o d i v i d e d the sect i o n s i n t o high I n t e l l i g e n c e (I.Q. above 110) and low 51 i n t e l l i g e n c e (below 110). The o u t l i n e o f h i s problem s o l v i n g p r o c e s s c o n s i s t e d o f the f o l l o w i n g 4 c a t e g o r i e s : (a) R e c o g n i t i o n of the e x i s t e n c e o f a p r oblem; (b) A n a l y s i s phase; (c) P r e d i c t i o n phase; and (d) V e r i f i c a t i o n phase. The i n s t r u c t i o n a l m a t e r i a l was p r e s e n t e d t o the e x p e r i m e n t a l group i n t h r e e l e s s o n s on t h r e e s u c c e s s i v e days. A f o l l o w -up p e r i o d o f s i x weeks was employed, d u r i n g w hich time the t e a c h e r emphasized t h i s s t r u c t u r e where a p p r o p r i a t e . The v e h i c l e employed f o r the i n t r o d u c t i o n o f the problem s o l v i n g p r o c e s s was not p u r e l y m a t h e m a t i c a l . F o r example, the f o l l o w i n g problem i s a n on-mathematical example used I n the u n i t . A boy comes home from s c h o o l and wants t o go t o a dance i n the e v e n i n g . He has c e r t a i n chores t o do w h i c h w i l l p r e v e n t him from g o i n g t o t h e dance. How t o s o l v e the boy's dilemma I s d i s c u s s e d i n terms o f P o s t ' s problem s o l v i n g s t r u c t u r e . On a m a t h e m a t i c a l problem s o l v i n g t e s t P o s t found no s i g n i f i c a n t d i f f e r e n c e between the e x p e r i m e n t a l and c o n t r o l groups g a i n s c o r e s . T h i s i n v e s t i g a t o r s u g g e s t s the f o l l o w i n g p o s s i b l e r easons f o r the l a c k o f s i g n i f i c a n c e . F i r s t , t h e s h o r t i n s t r u c t i o n a l p e r i o d , o n l y t h r e e days, gave l i t t l e time f o r t h e s t u d e n t s t o f u l l y a s s i m i l a t e the s t r u c t u r e o f the p roblem s o l v i n g p r o c e s s . Second, the i n s t r u c t i o n a l u n i t was o n l y p a r t i a l l y m a t h e m a t i c a l i n n a t u r e , thus r e q u i r i n g 52 the students to l e a r n both the problem s o l v i n g process i n a non-mathematical s i t u a t i o n and then apply i t to a mathematical problem s o l v i n g t e s t . W i l l s (1967) e x p l i c i t l y considered the t r a n s f e r of problem s o l v i n g a b i l i t y gained through, l e a r n i n g by discovery. His sample c o n s i s t e d of 56l high school students i n 24 c l a s s e s . Eight classes were randomly assigned to two experimental 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 of two weeks the experimental 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 teachers. W i l l s (1967, p. 155) described the r o l e of the teacher i n the two experimental groups as f o l l o w s : 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 teacher f o l l o w s i t w i t h simpler problems, makes a t a b l e , and continues asking questions u n t i l the students' answers i n d i c a t e that they have acquired a g e n e r a l i z a t i o n . The teacher does not discuss why the e a s i e r problems are presented. No e f f o r t i s made to discuss the general method of s o l v i n g such problems... In the Covert group the teacher leads the student through s e v e r a l s o l u t i o n s as does the 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 teacher t e l l s the students that an e f f e c t i v e way to solve a problem i s to solve s e v e r a l simpler problems of the same type and t r y to f i n d a p a t t e r n which can be a p p l i e d to the more d i f f i c u l t problem. A l l three groups were given the same p r e t e s t and p o s t t e s t , and the f u n c t i o n of the c o n t r o l group seems to 53 have been to c o n t r o l f o r growth over the i n s t r u c t i o n a l p e r i o d . On the ba s i s of the a n a l y s i s , which used the pr e t e s t scores as a c o v a r i a t e . W i l l s concluded that the experimental groups performed s i g n i f i c a n t l y b e t t e r than the c o n t r o l , w h i l e 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 the two experimental groups. The mean performance of the two experimental groups In d i c a t e d a b e t t e r performance from the overt group. The problem s o l v i n g t e s t was aimed towards problems that i n v o l v e d searching f o r a p a t t e r n . Some of the questions i n v o l v e d asking the students to f i n d patterns i n problem s i t u a t i o n s that were u n l i k e those 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 . For example, one problem Involved patterns a s s o c i a t e d w i t h Pascal's t r i a n g l e . Other problems Involved patterns In s i t u a t i o n s that can be considered extensions of the i n s t r u c t i o n a l m a t e r i a l s . W i l l s ' 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 that i n v o l v e d t r y i n g to f i n d p a t t e r n s . Leask (1968) i n v e s t i g a t e d the teaching of simple enumeration as a stra t e g y f o r discovery. The sample co n s i s t e d of s i x grade twelve classes w i t h a t o t a l of 158 students. The students had been randomly assigned to classes at the beginning of the year, and three classes were assigned to each treatment, expository (E) and simple enumeration (S). The l a t e r was described as f o l l o w s : 54 This c o n s i s t s of. the p r e s e n t a t i o n of many instances of the g e n e r a l i z a t i o n to be discovered. The students form hypotheses based on the examples and t e s t these to 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 to warrant 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 c o n s i s t e d of seven one hour s e s s i o n s , w i t h the m a t e r i a l s c o n s i s t i n g of the development of a r i t h m e t i c and geometric progressions. Both achievement and t r a n s f e r t e s t s were administered to the students. Leask (1968, p. 34) described the t r a n s f e r t e s t as f o l l o w s : I t was a twenty-item t e s t based on mathematical m a t e r i a l s somewhat r e l a t e d to the sequences but r e q u i r i n g a t r a n s f e r and extension of the 'knowledge acquired. The items were comprised of examples from which students were r e q u i r e d to ge n e r a l i z e . Her r e s u l t s i n d i c a t e that the S 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 on the mathematical t r a n s f e r t e s t , w hile 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 the groups i n mathematical achievement. An examination of the 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 the medium i n t e l l i g e n c e range. Jerman (1973) conducted a study of mathematical t r a n s f e r i n the elementary school. He used two experimental programmes: the Productive Thinking Programme of C r u t c h f i e d and Covington, and a Modif i e d Wanted-Giveh Programme developed by Jerman. The Jerman programme used 55 mathematical examples to develop problem s o l v i n g s k i l l s . The sample s e l e c t e d f o r the study was eight classes of f i f t h grade students (26l s t u d e n t s ) , w i t h three classes being assigned to each of the experimental programmes and the other two classes r e c e i v e d t h e i r normal classroom i n s t r u c t i o n and acted as a c o n t r o l group. The experimental m a t e r i a l s were presented 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 the t o t a l i n s t r u c t i o n a l p e r i o d being 16 days. The t e s t s i n c l u d e d a word problem t e s t administered immediately 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 and a follow-up t e s t given 7 weeks l a t e r . Two NLSMA t e s t s , Working w i t h Numbers and Five Dots were a l s o administered. 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 the treatments. However, he noted that the 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 that t h i s d i f f i c u l t y was i n " terms of the computation i n v o l v e d . He proceeded to analyse the word problem data according to whether subjects used a co r r e c t procedure to obtain the s o l u t i o n . That i s , whether the method would have r e s u l t e d i n a co r r e c t s o l u t i o n i f no computational e r r o r s were made. With t h i s 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 both the immediate word problem t e s t and the r e t e n t i o n t e s t given seven weeks l a t e r . On the f i r s t t e s t there was a s i g n i f i c a n t sex-treatment i n t e r a c t i o n f a vouring boys, a r e s u l t not repeated 56 on the . r e t e n t i o n t e s t . The f a c t t h a t t h i s r e s u l t was not r e p e a t e d may have been due- to: the • improved performance o f the s t u d e n t s on the d e l a y e d t e s t , w hich was a p a r a l l e l v e r s i o n o f the o r i g i n a l word problem t e s t . Summary of the R e l a t e d L i t e r a t u r e The l i t e r a t u r e i s r e f l e c t e d i n t h r e e f a c e t s o f the d e s i g n o f the s t u d y . F i r s t , the l i t e r a t u r e i n d i c a t e s t h a t e d u c a t o r s c o n s i d e r analogy and e x a m i n a t i o n o f cases t o be i m p o r t a n t m a t h e m a t i c a l h e u r i s t i c s ( f o r example, see Smith and Henderson (1959), P o l y a (1954a), MacPherson (1971b)). T h e r e f o r e , i t was r e a s o n a b l e t o s e l e c t these h e u r i s t i c s t o t e a c h t o s t u d e n t s . Second, i t i s n o t e d i n c h a p t e r I I I (p. 61 ) t h a t b o o k l e t s were t o be employed i n o r d e r t o randomize the sample and c o n t r o l the i n s t r u c t i o n a l sequence. S t u d i e s ( f o r example, see Gagne and Brown ( 1 9 6 l ) , E l d r e d g e (1965a) and Jerman (1973)) p r o v i d e e v i d e n c e t h a t u t i l i z 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 as t h e i n s t r u c t i o n a l medium i s a f e a s i b l e t e c h n i q u e f o r p r o m o t i n g t r a n s f e r . F i n a l l y , t h e r e were a few s t u d i e s 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 . S t u d i e s ( f o r example, see L i b e s k i n d (1971) and G o l d b e r g (1974)) i n d i c a t e t h a t t e a c h i n g i n v o l v i n g h e u r i s t i c p r o c e d u r e s can r e s u l t i n t r a n s f e r o f knowledge t o problems t h a t are r e l a t e d t o the 57 i n s t r u c t i o n a l m a t e r i a l s . Other s t u d i e s ( f o r example, see Schaaf (1954), W i l l s (1967), Leask (1968) and Worthen (1965) ) provide evidence that students can apply general problem s o l v i n g techniques to new problems. The present study can be considered both an extension and refinement of current 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 to the h e u r i s t i c s of analogy and examination of cases, and an extension i n that I t In v e s t i g a t e d the t r a n s f e r of h e u r i s t i c s (as opposed to-content) to novel mathematical problems. 58 CHAPTER I I I Design and Procedure  Design This study was designed to i n v e s t i g a t e the teaching of h e u r i s t i c s . Two f a c t o r s were considered:', the f i r s t f a c t o r concerned the employment of treatments w i t h d i f f e r e n t o b j e c t i v e s ; the second f a c t o r w i t h employing d i s t i n c t i n s t r u c t i o n a l v e h i c l e s . In order to i n v e s t i g a t e the hypotheses r e l a t e d to these f a c t o r s a t o t a l of nine groups were devised, as described i n chapter I. Figure 2 Experimental Groups Veh i c l e s N e u t r a l A l g e b r a i c Geometric T r H e u r i s t i c s only H - N H - A H - G . e a t m e n H e u r i s t i c s plus content HC - N HC - A HC - G t s Content only C - N C - A • C - G 59 For the purpose of l a t e r d i s c u s s i o n the nine groups have been i d e n t i f i e d by a two part code. The H, HC and C denote the three treatments: h e u r i s t i c s only (H), h e u r i s t i c s plus content (HC) and content only (C). The N, A and G denote the three v e h i c l e s : mathematically n e u t r a l (N), a l g e b r a i c (A) and geometric (G). The H, HC and C are used separately to denote the three treatments, i r r e s p e c t i v e of 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 subjects i n H - N, H - A and H - G. Employing three treatments and three v e h i c l e s r e s u l t e d i n nine experimental groups, one corresponding to each of 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 explanation of the.nature of 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. I n s t r u c t i o n a l M a t e r i a l s Ne u t r a l V e h i c l e (H - N, HC - N, C - N). The m a t e r i a l s f o r these groups developed mathematical t o p i c s as they a p p l i e d to the p h y s i c a l world. 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 i n nature, were not a s s o c i a t e d with any core category of the d i s c i p l i n e of mathematics, such as algebra or geometry. Alg 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 of 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 of, and approach to 60 t o p i c s emphasized the use o f symbols 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 the 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 g e o m e t r i c . . The emphasis i n t h i s - v e h i c l e was on the use o f g e o m e t r i c f i g u r e s , diagrammatic r e p r e s e n t a t i o n s and g e o m e t r i c o p e r a t i o n s such as f l i p p i n g and 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 m a t e r i a l s f o r t h e s e groups were d e s i g n e d so t h a t the h e u r i s t i c s were e x p l i c i t l y s t a t e d and emphasized, and the o n l y o b j e c t i v e f o r the s t u d e n t was l e a r n i n g t o a p p l y the h e u r i s t i c s . H e u r i s t i c s p l u s c o n t e n t (HC - N, HC - A, HC - G). The m a t e r i a l s f o r these groups were d e s i g n e d so t h a t the h e u r i s t i c s were e x p l i c i t l y s t a t e d , but emphasis was p l a c e d on b o t h t h e h e u r i s t i c s and c o n t e n t . Content o n l y (C - N, C - A, C - G). The m a t e r i a l s f o r t h e s e groups were d e s i g n e d so t h a t t h e o n l y o b j e c t i v e f o r the s t u d e n t was the l e a r n i n g o f the c o n t e n t . The co n t e n t o n l y groups were i n c l u d e d i n the d e s i g n as c o n t r o l groups. A t o t a l o f n i n e d i s t i n c t i n s t r u c t i o n a l u n i t s were r e q u i r e d . F o r example, the m a t e r i a l s f o r the group t h a t r e c e i v e d h e u r i s t i c s p l u s content 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 u n i t w i t h emphasis on b o t h h e u r i s t i c s and o t h e r m a t h e m a t i c a l c o n t e n t . The m a t e r i a l s f o r the C - G group c o n s i s t e d o f a g e o m e t r i c u n i t d e s i g n e d t o t e a c h c o n t e n t o n l y . 61 B o o k l e t s B o o k l e t s were used so t h a t s u b j e c t s c o u l d be randomly 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 groups as w e l l as t o c o n t r o l the i n s t r u c t i o n a l sequence. A l l n i n e b o o k l e t s were d e s i g n e d so t h a t each c o u l d be completed i n t e n days (one c l a s s p e r i o d p e r d a y ) . 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 t o p i c s . Three days were a l l o t e d t o each t o p i c and one a d d i t i o n a l day f o r r e v i e w . The v e h i c l e s c o m p r ised the f o l l o w i n g t o p i c s : F i g u r e 5 The B o o k l e t s V e h i c l e s N e u t r a l A l g e b r a i c Geometric T o p i c 1 S e a t i n g P l a n s P e r m u t a t i o n s Arrangements T o p i c 2 T i l i n g Mappings F l i p s & Turns T o p i c 3 E l e c t r i c a l Networks B i n a r y Networks Networks The o v e r a l l o r g a n i z a t i o n o f the b o o k l e t s was as f o l l o w s : T o p i c 1_: Day 1 - M a t e r i a l p r e s e n t e d and a q u i z i n c l u d e d at the end o f the p e r i o d . The q u i z t o be marked and r e t u r n e d the next day, t o g e t h e r w i t h the s o l u t i o n s . 62 Day- 2 - Followed the i d e n t i c a l p a t t e r n to. day ;1. Day 3 - M a t e r i a l presented but no q u i z . Topic 2:. Days 4 - 6 followed the same - p a t t e r n as days 1 - 3 . Topic 3_: Days 7 - 9 followed the same p a t t e r n as days 1 - 3 -Day 10 was allowed f o r review. No new m a t e r i a l was in c l u d e d . In a d d i t i o n to employing the 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 or s i m i l a r e x e r c i s e s , wording and examples were used i n a l l the book l e t s . This meant that the nine 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 only as the d i f f e r e n t o b j e c t i v e s and the natures of a n e u t r a l , a l g e b r a i c and geometric v e h i c l e r e q u i r e d . A d e t a i l e d d i s c u s s i o n of the development of the m a t e r i a l s can be found l a t e r In t h i s chapter, under the heading, 'Development of M a t e r i a l s ' . Sample pages from the booklets are in c l u d e d i n appendices A - I . Instruments Two t e s t s were developed, one a l g e b r a i c and the other geometric. The instruments both followed i d e n t i c a l format, c o n s i s t i n g of a novel mathematical problem f o l l o w e d by an open ended question. 63 F i g u r e '.6 S t r u c t u r e of. the T e s t s P r e s e n t a t i o n o f the M a t h e m a t i c a l Problem Q u e s t i o n : What can you f i n d out about...? The t e s t i tems employed o p e r a t i o n s i n v e n t e d s p e c i f i c a l l y f o r the s t u d y . D e t a i l s o f t h e development o f the i n s t r u m e n t s can be found i n t h i 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 the t e s t s c o n s i s t e d o f a s s i g n i n g a s u b j e c t ' s response 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 the use o f e i t h e r e x a m i n a t i o n o f cases o r a n a l o g y . 1 - The s u b j e c t has used e x a m i n a t i o n o f cases o n l y . He has not attempted t o l o o k f o r a p a t t e r n . 2 . - The s u b j e c t has b o t h examined cases and l o o k e d f o r a p a t t e r n . 3 - The s u b j e c t has used analogy o n l y . There i s no ev i d e n c e t h a t he has t r i e d t o t e s t h i s guess. 64 4 - The s u b j e c t has used analogy and e x a m i n a t i o n o f c a s e s , but as d i s t i n c t t e c h n i q u e s . There i s no e v i d e n c e t h a t the s u b j e c t has combined the two h e u r i s t i c s . 5 - The s u b j e c t 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 has combined the two i n s o l v i n g the problem. Pr o c e d u r e The p r o c e d u r e i s r e p o r t e d i n two p a r t s . F i r s t a b r i e f summary o f t h e p i l o t s t u d i e s , and second the main s t u d y . P i l o t S t u d i e s The development o f the m a t e r i a l s d i c t a t e d the need f o r p i l o t s t u d i e s t o be u n d e r t a k e n . The f o l l o w i n g i s a b r i e f statement o f the purposes o f the p i l o t s , and a summary o f the n a t u r e o f the m o d i f i c a t i o n s r e s u l t i n g from the p i l o t s . The o b j e c t i v e s o f 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 : (1) the r e a d a b i l i t y o f the m a t e r i a l s ; (2) the d i f f i c u l t y o f the concepts d i s c u s s e d i n the m a t e r i a l s ; (3) i f the l e n g t h of 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 ; and (4) the d i f f i c u l t y o f the e x e r c i s e s and examples. The r e s u l t s o f the p i l o t s t u d i e s , t o g e t h e r w i t h s u g g e s t i o n s from the i n v e s t i g a t o r ' s committee, n e c e s s i t a t e d c e r t a i n minor m o d i f i c a t i o n s o f the m a t e r i a l s . F o r example, the s e m o d i f i c a t i o n s i n c l u d e d removing e x e r c i s e s because a p a r t i c u l a r day's work r e q u i r e d t o o much t i m e ; i n s e r t i n g a 65 p a r a g r a p h t o c l a r i f y a c o n c e p t , such as f l i p p i n g a t i l e ; 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 t o w r i t e t h e i r answers. Main Study A s s i g n i n g Subjects, t o Groups. To c o n t r o l f o r f a c t o r s r e l a t e d t o problem s o l v i n g , such as I n t e l l i g e n c e and M a t h e m a t i c a l Achievement, s u b j e c t s were randomly a s s i g n e d t o one o f the n i n e e x p e r i m e n t a l groups. S i n c e t h e c l a s s g r o u p i n g s had been f i x e d a t t h e b e g i n n i n g o f the y e a r , and c o u l d not be a l t e r e d , t h i s random assignment n e c e s s i t a t e d t h a t a l l t r e a t m e n t s t a k e p l a c e w i t h i n a l l c l a s s r o o m s . The s o l e p r a c t i c a l s o l u t i o n was t o p r e s e n t the i n s t r u c t i o n a l u n i t s i n b o o k l e t form. The o v e r a l l o r g a n i z a t i o n I s shown I n f i g u r e 7. F i g u r e 7 O r g a n i z a t i o n o f the Groups H-N HC-N C-N H-A HC-A C-A H-G HC-G C-G C l a s s 1 C l a s s 2 i C l a s s 10 66 A p r e s e n t a t i o n employing b o o k l e t s had the e f f e c t o f c o n t r o l l i n g t e a c h e r v a r i a b i l i t y ( i n p r e s e n t i n g t h e i n s t r u c t i o n a l u n i t s ) which no amount of p r e - e x p e r i m e n t a l t r a i n i n g c o u l d have t o t a l l y e l i m i n a t e d . I t s h o u l d be n o t e d t h a t randomly a s s i g n i n g t r e a t m e n t s 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 t r e a t m e n t s b e i n g n e s t e d w i t h i n c l a s s r o o m s , and c o n s e q u e n t l y i t would have been i m p o s s i b l e t o d i f f e r e n t i a t e between t r e a t m e n t and c l a s s r o o m e f f e c t s . I n s t r u c t i o n s t o S u b j e c t s . The day p r i o r t o s u b j e c t s ' 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 note was r e a d e x p l a i n i n g t h a t the s u b j e c t s were about t o t a k e p a r t i n an experiment (see appendix J ) . The major p o i n t s i n t h e i n s t r u c t i o n s were: (a) The s u b j e c t s were i n f o r m e d t h a t they were i n v o l v e d i n an experiment and t h a t b o o k l e t s would 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 b o o k l e t s would be c o l l e c t e d at the end o f each p e r i o d , and t h e r e would be no f u r t h e r work on the b o o k l e t s u n t i l the next day. They were a l s o asked not t o d i s c u s s t h e c o n t e n t o f the b o o k l e t s w i t h anyone. By c o l l e c t i n g t h e m a t e r i a l s at the end o f each e x p e r i m e n t a l s e s s i o n and a s s i g n i n g no work on the b o o k l e t s o u t s i d e c l a s s , the i n v e s t i g a t o r hoped t o m i n i m i z e any d i s c u s s i o n among s u b j e c t s . 67 I n s t r u c t i o n a l P e r i o d . On the f i r s t day o f t h e i n s t r u c t i o n a l p e r i o d each s u b j e c t was p r e s e n t e d w i t h a f o l d e r . The f o l d e r c o n t a i n e d the pages t o be c o v e r e d i n the f i r s t p e r i o d . ' At the end o f the p e r i o d the b o o k l e t s were c o l l e c t e d , the q u i z marked, and the n e x t day's pages added. T h i s p r o c e d u r e was f o l l o w e d t h r o u g h o u t the i n s t r u c t i o n a l 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 days on which t h e r e was no q u i z . About h a l f w a y t h r o u g h each day's i n s t r u c t i o n a l p e r i o d the teacher- a d v i s e d the s u b j e c t s t h a t they had twenty minutes r e m a i n i n g . On days on which t h e r e was a q u i z , about t e n minutes from th e end o f the p e r i o d the t e a c h e r s u g g e s t e d t h a t . s u b j e c t s who had not s t a r t e d t h e q u i z s h o u l d do so. I f a s t u d e n t completed a p a r t i c u l a r day's m a t e r i a l e a r l y he was not g i v e n any new m a t e r i a l from the b o o k l e t . No mark o r l e t t e r grade was a s s i g n e d t o a s u b j e c t ' s q u i z s i n c e I t was f e l t t h a t t h i s . f o r m o f feedback might e f f e c t 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 attempt t o i n t e r p r e t the r e s u l t s . The c l a s s e s met at v a r i o u s times t h roughout the day e n a b l i n g the i n v e s t i g a t o r t o v i s i t a l l c l a s s e s w h i l e the experiment was i n p r o g r e s s . The t e a c h e r s d i d not answer s u b j e c t s ' q u e r i e s , o t h e r than d i r e c t i o n a l t y pe q u e s t i o n s , such as whether one c o u l d w r i t e on the back o f a page. F o r more i n v o l v e d q u e s t i o n s c o n c e r n i n g t h e e x p e r i m e n t a l u n i t s the t e a c h e r s u g g e s t e d t h a t a s u b j e c t c o n t i n u e u n t i l the i n v e s t i g a t o r a r r i v e d . Thus i t was p o s s i b l e t o c o n t r o l t h e t e a c h e r i n p u t i n t o t h e e x p e r i m e n t a l 68 p r o c e s s . . At the 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 days, two t e s t s , were a d m i n s t e r e d to. the s u b j e c t s . Adminls t r a t i o n o f the Tests.. On-'the day i m m e d i a t e l y 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 p e r i o d the a l g e b r a i c t e s t was 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 , and the g e o m e t r i c t e s t a d m i n i s t e r e d the next day. On each o f the t e s t days, the e x p e r i m e n t a l p e r i o d s were c o n c e n t r a t e d i n t h e a f t e r n o o n . S i n c e 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 t o change c l a s s r o o m s between p e r i o d s t h i s arrangement reduced the p o s s i b i l i t y o f d i s c u s s i o n s between s u b j e c t s who had completed the t e s t and t h o s e who would w r i t e i t l a t e r i n the day . The e x p e r i m e n t a l p r o c e d u r e i s summarized i n f i g u r e 8. F i g u r e 8 E x p e r i m e n t a l P r o c e d u r e Day 1 Days 2 - 1 1 Day 12 Day 13 I n t r o d u c t o r y I n s t r u c t i o n s 7 I n s t r u c t i o n a l P e r i o d s A l g e b r a i c Test Geometric T e s t 69 G r a d i n g . Each s u b j e c t ' s response was a s s i g n e d t o one o f s i x c a t e g o r i e s (see pages 63-4 )•• A team o f j u d g e s , c o n s i s t i n g o f the i n v e s t i g a t o r and two o t h e r mathematics e d u c a t i o n p r o f e s s o r s , s c o r e d t h e t e s t s . Each judge was p r e s e n t e d w i t h a copy o f the g r a d i n g scheme (see appendix M), t o g e t h e r w i t h sample responses o b t a i n e d from s u b j e c t s not i n v o l v e d i n t h i s s t u d y . A s e r i e s o f meetings t o d i s c u s s the 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 the e x p e r i m e n t a l d a t a . Then, each judge graded a l l t h e e x p e r i m e n t a l t e s t s i n d i v i d u a l l y and r e c o r d e d h i s r e s u l t s on c l a s s l i s t s . No judge was aware o f t h e o t h e r judges' c a t e g o r i z a t i o n s o f the t e s t s . No comments o r marks o f any form were w r i t t e n on t h e t e s t s . The t h r e e l i s t s were then' c o l l a t e d and the t e s t s a s s i g n e d t o one o f two c a t e g o r i e s ; t h o s e on w h i c h a l l t h r e e judges a g r e e d , and those about which t h e r e was disagreement. Meetings where th e n h e l d t o t r y t o o b t a i n consensus on t h o s e t e s t s about w h i c h t h e r e had been disagreement. Throughout th e 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 judges had agreed i n o r d e r t o v e r i f y t h a t the d i s c u s s i o n s were not c a u s i n g 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 o f the c a t e g o r i e s . The p e r c e n t a g e s o f agreement can be found i n t a b l e I . S u b j e c t s The s u b j e c t s c o n s i s t e d o f 294-grade ten-boys from an 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 c h o i c e o f an 70 a l l male sample, w i t h i t s obvious l i m i t a t i o n s on the g e n e r a l l z a b i l i t y . of the r e s u l t s , r e q u i r e s ' some e x p l a n a t i o n ; S i n c e the sample was t o be d i v i d e d i n t o n i n e g r o u p s , a l l o w i n g f o r e x p e r i m e n t a l m o r t a l i t y (Campbell and S t a n l e y , 1963, p. 5) 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 sample as p o s s i b l e . The boys' s c h o o l was one o f the -few t h a t c o n t a i n e d a l a r g e number ( i n excess o f 300) g r a d e - t e n s t u d e n t s . Development of M a t e r i a l s  Development o f the B o o k l e t s The t o p i c s used i n t h e t h r e e v e h i c l e s were l i s t e d i n f i g u r e 5 w h i c h , f o r c o n v e n i e n c e , i s r e p e a t e d below: F i g u r e 5 The B o o k l e t s V e h i c l e s N e u t r a l A l g e b r a i c Geometric T o p i c 1 S e a t i n g P l a n s P e r m u t a t i o n s Arrangements T o p i c 2 T i l i n g Mappings F l i p s & Turns T o p i c 3 E l e c t r i c a l Networks. B i n a r y Networks Networks S i n c e a l l the b o o k l e t s were i n excess o f 100 pages, the complete m a t e r i a l s have not been i n c l u d e d . However, sample pages from the b o o k l e t s can be found i n appendices A - I . 71 The b o o k l e t s were d e s i g n e d t o be s t r u c t u r a l l y and m a t h e m a t i c a l l y e q u i v a l e n t . T h i s e q u i v a l e n c e was o f two t y p e s . F i r s t , e q u i v a l e n c e between v e h i c l e s was o b t a i n e d by 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 ( m a t h e m a t i c a l l y n e u t r a l ) , p e r m u t a t i o n s ( a l g e b r a i c ) and arrangements ( g e o m e t r i c ) t h a t had i d e n t i c a l m a t h e m a t i c a l s t r u c t u r e s . F u r t h e r m o r e , where p o s s i b l e examples and e x e r c i s e s were t r a n s l a t e d I n t o the t e r m i n o l o g y a p p r o p r i a t e t o a p a r t i c u l a r v e h i c l e . Second, e q u i v a l e n c e between t r e a t m e n t s was o b t a i n e d by, wherever p o s s i b l e , employing i d e n t i c a l e x e r c i s e s and examples i n d i f f e r e n t t r e a t m e n t s . A l l b o o k l e t s c o n t a i n e d 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 and 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 i n d i v i d u a l t o p i c s . F i g u r e 9 T o p i c 1 <N e u t r a l V e h i c l e - S e a t i n g P l a n s A l g e b r a i c V e h i c l e - P e r m u t a t i o n s 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 p e o p l e i n a row, and round a c i r c u l a r t a b l e (See appendices A,'B and C). A t y p i c a l q u e s t i o n i n v o l v e s s e a t i n g t h r e e 72 p e o p l e i n a row. The t h r e e p e o p l e are Mr. B l a d e , Mr. Johnson and Mr. H a m i l t o n . Three c h a i r s The q u e s t i o n posed 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 can you s e a t the t h r e e men? P e r m u t a t i o n s 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 l e t t e r s and numbers, and the n proceeds t o t h e concept o f c i r c u l a r p e r m u t a t i o n s , which i s an a d a p t i o n o f t h e concept o f c y c l i c p e r m u t a t i o n s (See appendices D, E and P ) . 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 a t g i v e n i n the s e a t i n g p l a n s b o o k l e t 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. U s i n g 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 you 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 a r r a n g i n g g e o m e t r i c f i g u r e s i n a l i n e , and then proceeds t o c o n s i d e r d i f f e r e n t c i r c u l a r p a t t e r n s (See appendices G, H and I ) . 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 p r e v i o u s l y i s : C o n s i d e r 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 , square and c i r c l e . 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 arrangements can you f i n d ? S e a t i n g p e o p l e i n a row and a r r a n g i n g g e o m e t r i c f i g u r e s i n a l i n e are e q u i v a l e n t . o p e r a t i o n s t o p e r m u t i n g 73 l e t t e r s . S i m i l a r l y , s e a t i n g p e o p l e round a c i r c u l a r t a b l e and a r r a n g i n g g e o m e t r i c f i g u r e s i n a c i r c u l a r p a t t e r n a r e e q u i v a l e n t t o o r g a n i z i n g c i r c u l a r p e r m u t a t i o n s . Thus, the f i r s t t h r e e t o p i c s are n e u t r a l , a l g e b r a i c and g e o m e t r i c models o f the same m a t h e m a t i c a l s t r u c t u r e . C o r r e s p o n d i n g e x e r c i s e s and examples were reworded i n t h e t e r m i n o l o g y a p p r o p r i a t e t o a g i v e n v e h i c l e . The q u e s t i o n s i n d i c a t e t h e n a t u r e o f t h i s r e w o r d i n g . 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 example from v e h i c l e t o v e h i c l e means t h a t each t o p i c c o n t a i n s a p p r o x i m a t e l y the same number o f e x e r c i s e s and examples. 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 are t h o s e i n h e r e n t i n the 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 and n e u t r a l approach t o 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 . F i g u r e 10 summarizes the t r a n s l a t i o n s . 74 F i g u r e 10 E q u i v a l e n c e - T o p i c 1 N e u t r a l ( S e a t i n g P l a n s ) A l g e b r a i c ( P e r m u t a t i o n s ) Geometric (Arrangements) Sample q u e s t i o n T r a n s l a t i o n Sample q u e s t i o n ! Sample q u e s t i o n J v \ J •S1 - T r a n s l a t i o n — ^ Mr.Blade,Mr.Johns on and Mr.Hamilton are t o be s e a t e d i n a row. A A A, How many d i f f e r e n t ways can you s e a t the t h r e e men? 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 h. C o n s i d e r 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 , square and c i r c l e . AGO U s i n g a l l t h r e e j U s i n g a l l t h r e e l e t t e r s how many j 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 \ 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 i arrangements can you f i n d ? j you f i n d ? 75 F i g u r e 11 T o p i c 2 • N e u t r a l V e h i c l e - T i l i n g T o p i c 2 ^ A l g e b r a i c V e h i c l e - Mappings 'Geometric V e h i c l e - F l i p s and Turns T i l i n g c o n s i d e r s f i t t i n g t i l e s o f v a r i o u s shapes and c o l o u r s i n t o a p p r o p r i a t e l y shaped s p a c e s . D i f f e r e n t p o s i t i o n s f o r the t i l e s , as well;->,as movements and c o m b i n a t i o n s o f movements r e q u i r e d t o t r a n s p o s e a t i l e from one p o s i t i o n t o a n o t h e r are s u r v e y e d (See appendices A, B and C). A t y p i c a l s i t u a t i o n I n v o l v e s f i t t i n g an e q u i l a t e r a l t r i a n g u l a r t i l e I n t o a congruent space and th e n r o t a t i n g from one p o s i t i o n t o a n o t h e r . The f o l l o w i n g are examples o f two p o s i t i o n s and a r o t a t i o n . F i g u r e 12 A. R o t a t i o n 1 ) P o s i t i o n 1 P o s i t i o n 2 Mappings c o n s i d e r s s e t s o f t h r e e and f o u r l e t t e r s , and o p e r a t i o n s and combining o p e r a t i o n s t h a t i n t e r c h a n g e l e t t e r s . (See appendices D, E and F ) . The s i t u a t i o n c o r r e s p o n d i n g t o t h a t g i v e n i n the t i l i n g s e c t i o n i s : 76 F i g u r e 13 A B C P o s i t i o n 1 | j ^ n±(Mapping 1) B C A P o s i t i o n 2 M-^  I s the a l g e b r a i c e q u i v a l e n t o f r o t a t i o n 1. Thus, a f t e r a p p l y i n g r o t a t i o n 1 ( o r M^) t o p o s i t i o n 1, B o c c u p i e s t h e ' p o s i t i o n p r e v i o u s l y 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 p r e v i o u s l y h e l d by B, and A o c c u p i e s the p o s i t i o n p r e v i o u s l y h e l d by C. I n b o t h cases t h e new arrangement I s l a b e l l e d p o s i t i o n 2. F l i p s and Turns c o n s i d e r s f l i p s and t u r n s ( r o t a t i o n s ) o f 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 can be combined to produce new p o s i t i o n s . The g e o m e t r i c f i g u r e s c o r r e s p o n d t o those I n the t i l i n g s e c t i o n (See appendices G, H and I ) . The s i t u a t i o n t h a t c o r r e s p o n d s t o t h a t p r e s e n t e d i n t h e t i l i n g s e c t i o n I n v o l v e s t h e r o t a t i o n o f an e q u i l a t e r a l t r i a n g l e , r a t h e r 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 t i l e . To summarize, t o p i c 2 c o n s i s t s o f t h r e e models ( n e u t r a l , a l g e b r a i c and g e o m e t r i c ) o f the same m a t h e m a t i c a l s t r u c t u r e , and c o r r e s p o n d i n g examples are e x e r c i s e s were t r a n s l a t e d I n t o the t e r m i n o l o g y a p p r o p r i a t e t o a p a r t i c u l a r v e h i c l e . F i g u r e 14 c o n t a i n s the d i f f e r e n t models. 77 F i g u r e 14 E q u i v a l e n c e - T o p i c 2 N e u t r a l ( T i l i n g ) A l g e b r a i c (Mappings) Geometri c ( F l i p s and Turns) Sample s i t u a t i o n T r a n s l a t i o n Sample s i t u a t i o n ' Sample s i t u a t i o n T r a n s l a t i o n . R o t a t i o n 1 ( o f a t i l e ) . A B Mapping 1 (o f l e t t e r s ) R o t a t i o n 1 (o f a t r i a n g l e ) B A I i i A B >' B c A A B B A 78 F i g u r e 15 T o p i c 3 < •Neutral V e h i c l e E l e c t r i c a l Networks T o p i c 3 A l g e b r a i c V e h i c l e B i n a r y Networks Geometric V e h i c l e Networks The t i t l e s t h emselves r e f l e c t the e q u i v a l e n t n a t u r e o f the t o p i c s . Each c o n s i d e r s combining s w i t c h e s 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 concept o f e q u i v a l e n t n e t w o r k s . I n t h e s e c t i o n s on e l e c t r i c a l networks t h e s w i t c h e s are e l e c t r i c a l and the development I s i n terms o f whether c u r r e n t f l o w s o r does not f l o w t h r o u g h the network, (See appendices A, B and C). B i n a r y s w i t c h e s I n v o l v e s the Id e a o f s w i t c h e s t h a t have two p o s s i b l e v a l u e s , 0 o r 1, and the development c e n t e r s on whether a network has t h e v a l u e I o r I I (See appendices D, E and F ) . F i n a l l y , g e o m e t r i c  networks i n t e r p r e t a network as a c o m b i n a t i o n o f p a t h s t h a t are e i t h e r connected o r d i s c o n n e c t e d , and the development d e a l s w i t h the c o n d i t i o n s under w h i c h a network I s , o r i s not complete (See appendices G, H and I ) . As b e f o r e , a p p r o p r i a t e e x e r c i s e s and examples were t r a n s l a t e d i n t o t h e t e r m i n o l o g y a p p r o p r i a t e t o a p a r t i c u l a r v e h i c l e . F i g u r e 16 summarizes t o p i c 3-79 F i g u r e 16 E q u i v a l e n c e T o p i c 3 N e u t r a l , A l g e b r a i c ( E l e c t r i c a l Networks)! ( B i n a r y Networks) Geometric (Networks) Sample network , Sample network Sample network -Trans l a t i o n - ^ A B A and B are s w i t c h e s A open B c l o s e d C u r r e n t does not f l o w A B - T r a n s l a t i o n ' A H 0 B I A and B are s w i t c h e s 1 A and B are s w i t c h e s A has v a l u e 0 J B has v a l u e 1 Network has v a l u e I I A d i s c o n n e c t e d B connected Network i s not complete 80 The r e s u l t o f d e s i g n i n g the m a t e r i a l s t o o b t a i n e q u i v a l e n c e between v e h i c l e s i s t h a t b o o k l e t s f o r d i f f e r e n t v e h i c l e s c o n t a i n c o r r e s p o n d i n g t o p i c s t h a t are models o f the 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 c o r r e s p o n d i n g b o o k l e t s u t i l i z e e q u i v a l e n t e x e r c i s e s and examples; and the b o o k l e t s c o n t a i n 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 . The d i f f e r e n c e s between v e h i c l e s are t h o s e t h a t r e s u l t from the n a t u r e o f a m a t h e m a t i c a l l y n e u t r a l , a l g e b r a i c and g e o m e t r i c model and approach t o 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 . E q u i v a l e n c e between Treatments The t h r e e 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 p l u s c o n t e n t and c o n t e n t o n l y . Where a p p r o p r i a t e the i d e n t i c a l e x e r c i s e o r example, or t r a n s l a t i o n o f t h e e x e r c i s e o r example are u t i l i z e d i n a l l t h r e e t r e a t m e n t s . F o r example, the r e a d e r s h o u l d examine the development o f the s e a t i n g p l a n s s e c t i o n s (See appendices A, B and C). I n the h e u r i s t i c 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 b o o k l e t s t h e s o l u t i o n s t o e x e r c i s e s o f t e n c o n t a i n an e x p l a n a t i o n o f how e x a m i n a t i o n of cases o r analogy ils'.; employed i n f i n d i n g t h e s o l u t i o n . I n the c o n t e n t o n l y b o o k l e t s the i d e n t i c a l e x e r c i s e does not i n c l u d e any mention o f the h e u r i s t i c s i n the s o l u t i o n . The 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 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 b o o k l e t s o c c u r s i n the d i f f e r e n t emphases i n t h e m a t e r i a l s . F o r example, 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 fewer pages d e v o t e d s p e c i f i c a l l y 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 o f cases and analogy are u n d e r l i n e d i n the h e u r i s t i c s o n l y b o o k l e t s but not i n the o t h e r h e u r i s t i c s t r e a t m e n t ; e x e r c i s e s 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 c o n t e n t b o o k l e t s d e s i g n e d t o emphasize c o n t e n t o n l y , and t h e s e e x e r c i s e s ommitte.d from the 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 are p l a c e d on the d e s i g n o f the b o o k l e t s n a m e l y attempts t o o b t a i n .-..equivalence between v e h i c l e s and e q u i v a l e n c e between t r e a t m e n t s . T h i s r e s u l t s i n the 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) c o r r e s p o n d i n g t o p i c s are models o f the same m a t h e m a t i c a l s t r u c t u r e ; (3) where p o s s i b l e i d e n t i c a l examples and e x e r c i s e s , o r t r a n s l a t i o n s . o f these are used i n a l l b o o k l e t s ; and (4) a l l b o o k l e t s c o n t a i n 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 . I n d e v e l o p i n g the -booklets I t was r e a s o n a b l e t o assume t h a t a l l s u b j e c t s would not p r o g r e s s 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 t o accommodate subj e c t s - w h o would not complete a day's work w i t h i n t h e a l l o t t e d p e r i o d . To do t h i s the m a t e r i a l was o r g a n i z e d so t h a t most o f the new concepts are d e v e l o p e d i n t h e f i r s t h a l f o f a day's work. I f a s u b j e c t completed t h i s segment he was 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 attempt at the q u i z . F o r s u b j e c t s who f i n i s h e d e a r l y no a d d i t i o n a l m a t e r i a l was p r o v i d e d . I t was d e c i d e d t h a t 82 t h e s e s u b j e c t s would 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 b o o k l e t s . Development o f the T e s t s The t e s t s , one a l g e b r a i c and the o t h e r g e o m e t r i c c o n s i s t e d o f n o v e l problems. F u r t h e r m o r e , t h e n a t u r e o f the i n s t r u m e n t s was such as t o a l l o w t h e o p p o r t u n i t y f o r making c o n j e c t u r e s by analogy as w e l l as p e r m i t t i n g s u b j e c t s t o examine c a s e s . A l g e b r a i c I n s t r u m e n t . D i s c u s s i o n s w i t h p e r s o n n e l f a m i l i a r w i t h the h i g h s c h o o l mathematics programme, t o g e t h e r w i t h an e x a m i n a t i o n o f h i g h s c h o o l t e x t b o o k s c u r r e n t l y employed i n the s c h o o l s i n d i c a t e d t h a t : (a) Grade t e n s t u d e n t s were f a m i l i a r w i t h a s t r u c t u r a l approach t o mathematics. I n p a r t i c u l a r , grade t e n s t u d e n t s had 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 such as c o m m u t a t l v l t y , and the 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 s t u d e n t s had been exposed.to modulo systems ( c l o c k a r i t h m e t i c ) . The a l g e b r a i c t e s t c o n s i s t e d o f the p r e s e n t a t i o n o f two m a t h e m a t i c a l o p e r a t i o n s d e v e l o p e d s p e c i f i c a l l y f o r the s t u d y , f o l l o w e d by the q u e s t i o n : What can you f i n d out about t h e s e o p e r a t i o n s ? The o p e r a t i o n s a r e : TUp-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 = 2 x ( a + b ) (Modulo 7) (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 approach i n t h e i r m a t h e m a t i c a l programme a s s u r e d 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 n e c e s s a r y core 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 , o r t r y cases and draw i n f e r e n c e s . Geometric I n s t r u m e n t . The p r o c e d u r e u t i l i z e d f o r the development o f t h i s i n s t r u m e n t i s s i m i l a r t o t h a t f o l l o w e d i n the d e s i g n o f the a l g e b r a i c t e s t . The s u b j e c t s were p r e s e n t e d w i t h a d e s c r i p t i o n o f movements on a g r i d and c a l c u l a t i o n s o f the d i s t a n c e between p o i n t s , f o l l o w e d by the q u e s t i o n : What can you f i n d out about s h o r t e s t r o u t e s and c ombining s h o r t e s t r o u t e s ? One movement, i n v o l v i n g moving up and a c r o s s the g r i d i s i l l u s t r a t e d i n f i g u r e 17. F i g u r e d ? Movements on a G r i d f 7 D i s t a n c e between A and B i s 3 segments. (See appendix L f o r d e t a i l s ) 84 S t a t i s t i c a l 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 on b o t h the a l g e b r a i c and g e o m e t r i c t e s t d a t a . A two-way f i x e d e f f e c t s a n a l y s i s of v a r i a n c e (ANOVA) was employed. The mo de1 i s : Y . = ii + a . + 8 . + Y. • + e-where u = grand mean = e f f e c t o f t r e a t m e n t i 3. = e f f e c t o f v e h i c l e j Y•• = i n t e r a c t i o n e f f e c t o f t r e a t m e n t i w i t h v e h i c l e j e ^ j ^ . = e x p e r i m e n t a l e r r o r o f p e r s o n k ct^ ( i = 1, 2, 3) r e p r e s e n t s the t r e a t m e n t e f f e c t s , h e u r i s t i c s o n l y (a-j_)j h e u r i s t i c s p l u s c o n t e n t ( a 2 ) and-content o n l y (ct^)-g. ( j = 1, 2, 3) r e p r e s e n t s the v e h i c l e e f f e c t s , m a t h e m a t i c a l l y 3 n e u t r a l ( 8 - ^ ) , a l g e b r a i c ( g j ) a n ^ g e o m e t r i c ( B ^ ) -O v e r a l l F - r a t i o s were computed and t e s t e d f o r s i g n i f i c a n c e ( a t .05 l e v e l ) . Depending on the s i g n i f i c a n c e o f the o v e r a l l F - r a t i o s t h e c o n t r a s t s c o r r e s p o n d i n g t o the hypotheses were t e s t e d by t h e Scheffe" Method ( K i r k , 1968). F u r t h e r m o r e , 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 the 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 method (_X2). The d e t a i l s and j u s t i f i c a t i o n f o r these d e c i s i o n s are found i n c h a p t e r IV - A n a l y s i s o f the Data. 85 CHAPTER IV A n a l y s i s o f the Data The r e s e a r c h hypotheses a re r e s t a t e d below. Hypotheses 1 t h r o u g h 3 r e f e r t o the t r e a t m e n t e f f e c t , w h i l e hypotheses 4 and 5 c o r r e s p o n d t o c e r t a i n i n t e r a c t i o n e f f e c t s . I n each o f the f i v e h y p o t h e s e s , the dependent v a r i a b l e 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 o f t h e h e u r i s t i c s ( e x a m i n a t i o n o f cases o r ana l o g y ) t o a n o v e l mathematics problem: H y p o t h e s i s 1. A group t a u g h t h e u r i s t i c s o n l y (H) w i l l s c o r e h i g h e r than a group taught c o n t e n t o n l y ( C ) . H y p o t h e s i s 2_. A group t a u g h t h e u r i s t i c s p l u s c o n t e n t (HC) w i l l s c o r e h i g h e r t h a n a group taught c o n t e n t o n l y ( C ) , H y p o t h e s i s 3_. A group taught h e u r i s t i c s o n l y (H) w i l l s c o r e h i g h e r t h a n a group t a u g h t h e u r i s t i c s p l u s c o n t e n t (HC). Hy p o t h e s i s 4_. The mean s c o r e o f a h e u r i s t i c s o n l y group (H - N, H - A, H - G) as compared t o the mean s c o r e o f the c o r r e s p o n d i n g c o n t e n t o n l y group.(C - N, C - A, C - G) w i l l be Independent o f t h e v e h i c l e ( I n s t r u c t i o n a l m a t e r i a l ) . H y p o t h e s i s 5_- The mean s c o r e o f a h e u r i s t i c s p l u s c o n t e n t group (HC - N, HC - A, HC - G) as compared t o the mean s c o r e o f the c o r r e s p o n d i n g c o n t e n t o n l y group (C - N, C - A, C - G) w i l l be independent o f the v e h i c l e ( I n s t r u c t i o n a l m a t e r i a l ) , 86 G r a d i n g . 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 d i f f e r e n t s u b j e c t s , were absent from s c h o o l . F o r a s u b j e c t ' s t e s t t o be graded he had t o be p r e s e n t f o r at 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 c o n t a i n s 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 g r aded, the pe r c e n t a g e o f agreement p r i o r t o any d i s c u s s i o n among judgesQ? and the f i n a l p e r c e n t a g e agreement. Table I Perc e n t a g e Agreement on T e s t s T o t a l ' number o f t e s t s . g r a ded* Agreement p r i o r t o d i s c u s s i o n % Agreement a f t e r d i s c u s s i o n % A l g e b r a i c Test . 249 177 71 244 98 Geometric Test 249 . 175 70 241 96 * 228 s u b j e c t s wrote b o t h t e s t s Data U s e d . i n A n a l y s i s . A s u b s e t o f the t e s t s on wh i c h the judges agreed was used i n the a n a l y s i s . S u b j e c t s f o r ; whom d a t a from 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 , 8 7 leaving data from 2 1 8 subjects. The r e s u l t i n g group sizes varied from 2 1 to 2 6 . In order to equalize the numbers i n a l l nine groups additional data were randomly discarded so that there were 2 1 subjects per group. There were two reasons for reducing the t o t a l number of subjects included i n the analysis. F i r s t , i n order to draw conclusions concerning both the algebraic and geometric problems i t was advisable to r e s t r i c t the analysis to data based on only those subjects who wrote both tests. Second, the use of a non-orthogonal design (unequal c e l l s i z e s ) , with the associated lack of s t a t i s t i c a l independence between the o v e r a l l F-tests, i s undesirable since i t makes i t d i f f i c u l t to interpret r e s u l t s . Data from 1 8 9 subjects were analysed ( 2 1 per group). Tables II and III show the summary of the scores on the algebraic and geometric te s t s , respectively. Recategorization of Data for Analysis. For the purpose of t e s t i n g the hypotheses, the data In tables II and III were recategorized according to the following dichotomy: 0 - No evidence of either the h e u r i s t i c of examination of cases or analogy. 1 - Evidence of at least one of these h e u r i s t i c s . The r e s u l t i n g frequencies are shown i n table IV. This recategorlzation of the data was necessary since the 88 Table I I Summary o f Responses on the A l g e b r a i c Test Number of Subj e c t s i n a C a t e g o r y * ^ E CO CO o , . , Group u o.^ Sub,] e c t s p e r Group c co co Sn O o ^ CD •H 0 •H 0 o -P -P to -P CO •H c -P rH CO 3 co 3 -p Kj C •<H •H co rH .0 PM o !H -P U T3 •H C -P 3 o 3 0 O -P oa >> 0 a 0 c 50 £ -H ^ -H 0 co Dh co O -p 0 CD H ,G CO ^ 6 to o CO -P -H -p o o cti o p o o o —' o < 1 • PQ ^ 1 1 o i rH 1 C\J 1 on 1 •=r 1 LTv 10 •1 1 4 - 5 16 1 • - 1 - 3 17 - 1 1 2 6 2 3 2 2 6 12 2 - 1 2 4 16 . 1 2 - 1 1 10 - 1 2 - 8 13 - 1 2 - 5 17 2 — 1 — 1 Number o f H - N 1 -  21 HC- N - 21 C - N   - '  21 H - A 2  6 21 HC- A  21 C - A   21 H - G 0 - - 21 HC- G   - 21 C - G  - - 1 21. 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 , can be found on pages 6 3 - 4 . 89 Table I I I Summary of Responses on the Geometric Test Number o f Subj e c t s i n a C a t e g o r y * Number o f Group • u S u b j e c t s p e r Group c m fn o —» o ^ K) CD •H CD •H CD CJ -P >j -P tt! 4 0 K l • H -P rH ra 3 m 3 -P •H •H m rH CD P-1 O SH -P •H C -P 3 O Hi CD U O -P off >> CD CD C bO ^ -H £ -H o W • O -P CD CD rH ,C w ^ S CO O M Ctj -P -H ' -P O o «j o n O O o O pq —- CP —• 0 H CM 00 -=t-H - N 6 6 8 - - 1 21 HC- N 10 5 4 1 - 1 21 C - N 15 3 3 - - 21 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 T o t a l number, o f s u b j e c t s - 189 * D e t a l l s o f t h e c a t e g o r i e s can be found on pages 63-4 Table IV Mark Assigned' f o r A n a l y s i s Group A l g e b r a i c T e s t Geometric Test 0* 1** 0* 1** H - N 10 11 6 15 HC- N 16 5 10 11 C - N 17 4 15 6 H - A 6 15 7 14 HC- 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 * - No e v i d e n c e o f e i t h e r h e u r i s t i c . ** - E v i d e n c e o f at l e a s t one o f the 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 not r e p r e s e n t e i t h e r an u n d e r l y i n g continuum o r a t o t a l l y o r d i n a l s c a l e , and i n t h a t form the d a t a were i n a p p r o p r i a t e 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 f u r t h e r confounded by t h e use o f dichotomous d a t a . Hsu and' F e l d t (1969) examined the e f f e c t o f l i m i t e d s c a l e s (2, 3, A and. 5 p o i n t ) on n o m i n a l a - l e v e l s . The r e s u l t s o f t h e i r s t u d y , which was r e s t r i c t e d t o e q u a l c e l l s i z e s , I n d i c a t e r e a s o n a b l e c o n t r o l o f Type I e r r o r f o r a l l s c a l e s , v. . W i t h a two p o i n t s c a l e an n o f 50 p l u s (p e r c e l l ) seems i n d i c a t e d by Hsu and F e l d t ' s r e s u l t s f o r a n o m i n a l a o f .01 o r .10, but not f o r an a o f .05. A l t h o u g h t h e p i c t u r e c o n c e r n i n g c e l l s i z e i s not e n t i r e l y c l e a r , on the b a s i s o f Hsu and F e l d t ' s r e s u l t s , and the g e n e r a l d e s i r a b i l i t y o f e q u a l n's, i t would appear a d v i s a b l e t o use a c e l l s i z e l a r g e r than 21. • S i n c e i t was not f e a s i b l e t o use a l a r g e r n, the i n v e s t i g a t o r chose t o supplement c e r t a i n o f the ANOVA pr 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 m e t h o d ( x 2 ) • T h i s was done f o r hypotheses 1 - 3 3 c o n c e r n i n g a main e f f e c t , but was not a p p l i c a b l e t o the i n t e r a c t i o n hypotheses 4 and 5. T e s t i n g the Hypotheses The s t a t i s t i c a l model employed i n the a n a l y s i s o f v a r i a n c e i s : 92 i j k 1 p j T i j i j k where ']i = grand mean a. = e f f e c t o f t r e a t m e n t i ' l 3. = e f f e c t o f v e h i c l e j J v.. = i n t e r a c t i o n e f f e c t o f t r e a t m e n t i w i t h v e h i c l e j e. .. = e x p e r i m e n t a l e r r o r o f p e r s o n k i j k ^ • ^ a ^ ( i = 1, 2, 3) r e p r e s e n t s the t r e a t m e n t e f f e c t s : h e u r i s t i c s only- (a-j^) 5 h e u r i s t i c s p l u s c o n t e n t ( a 2 ) and c o n t e n t o n l y (a-^)-3 - ( j = 1, 2j 3) r e p r e s e n t s t h e v e h i c l e e f f e c t s : m a t h e m a t i c a l l y n e u t r a l ( 6 ^ ) , a l g e b r a i c (82)- and g e o m e t r i c ( g ^ ) . The s t a t i s t i c a l hypotheses c o r r e s p o n d i n g t o the r e s e a r c h hypotheses are s t a t e d below: H y p o t h e s i s N u l l H y p o t h e s i s A l t e r n a t i v e H y p o t h e s i s HI H 0 1 : a-^  = a,-> H l l : a l > > a 2 H2 H 0 2 : a 2 = ot ^  H 1 2 : a 2 > H3 H 0 3 : = a 2 H 1 3 : H 0 4 l : ^ l l " Y 3 1 = Y 1 2 _ Y 3 2 - H l 4 l : Y 1 1 - Y 3 1 ^ Y 1 2 - Y 3 2 H4 H 0 4 2 : Y 1 1 - Y 3 1 = Y 1 3 _ Y 3 3 H l 4 2 : Y l l - Y 3 1 ^ Y 1 3 - y 3 3 H 0 4 3 : y 1 2 _ y 3 2 = y 1 3 - Y 3 3 H l 4 3 : Y 1 2 - Y 3 2 ^ Y 1 3 ^ Y 3 3 r H 0 5 1 : y 2 1 _ y 3 1 = Y 2 2 _ y 3 2 - H 1 5 1 : Y 2 1 - Y 3 1 ^ Y 2 2 - Y 3 2 H5 < H 0 5 2 : y 2 1 - y 3 1 = Y 2 3 _ y 2 2 . H 1 5 2 : T 2 1 ^ . 3 1 ^ 2 3 ^ 2 2 V. H 0 5 3 : y 2 2 " y 3 2 = y 2 3 - y 3 3 H 1 5 3 : Y 2'2- Y32^23" Y33 P l a n f o r A n a l y s i s . Chapter I c o n t a i n s a model of mathematics. The form o f the model, w i t h the d i s t i n c t components o f D i s c o v e r y and Core l e d the i n v e s t i g a t o r t o s t a t e r e s e a r c h hypotheses 1 - 3 d i r e c t i o n a l l y . A l t h o u g h d i r e c t i o n a l hypotheses are i m p l i e d by the model, the subsequent r e v i e w o f the l i t e r a t u r e ( C h a pter I I ) shows t h a t a t the p r e s e n t time t h e r e i s l i t t l e r e s e a r c h t h a t e i t h e r s u p p o r t s o r r e f u t e s d i r e c t i o n a l i t y . F u r t h e r m o r e , the i n i t i a l s t atement o f the hypotheses would not c o n s i d e r a l t e r n a t i v e hypotheses such as the HC t r e a t m e n t b e i n g s i g n i f i c a n t l y b e t t e r than the H t r e a t m e n t , which have i m p o r t a n t e d u c a t i o n a l i m p l i c a t i o n s . Thus f o r b o t h t h e s e r e a s o n s , l a c k o f e m p i r i c a l s u pport f o r d i r e c t i o n a l i t y and t h e e d u c a t i o n a l i mportance o f o t h e r a l t e r n a t i v e h y p o t h e s e s , the i n v e s t i g a t o r t e s t e d n u l l hypotheses HQ^ H Q 2 and H ^ ^ . a g a i n s t non-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-j^t and H-^i > H l l ' : a l ^ a2 ' ^12' : a2 ^ a 3 ' H 1 3 ' : a l ^ a2 S i n c e hypotheses 1 - 3 were s p e c i f i e d a p r i o r i , i t i s p e r m i s s i b l e t o t e s t a p p r o p r i a t e n o n - o r t h o g o n a l c o n t r a s t s by Dunn's m u l t i p l e comparison p r o c e d u r e ( K i r k , 1968) w i t h o u t f i r s t t e s t i n g f o r a s i g n i f i c a n t o v e r a l l main t r e a t m e n t e f f e c t . However, due t o the l i m i t a t i o n s imposed by 0 - 1 s c o r i n g , the i n v e s t i g a t o r chose t o t e s t the hypotheses by the 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 ( K i r k , 1968). The f o l l o w i n g i s a comparison o f the a p o s t e r i o r i 94 Scheffe' and Tukey p r o c e d u r e s , and Dunn's M u l t i p l e Comparison p r o c e d u r e . S i n c e hypotheses 1 - 3 r e f e r o n l y t o p a i r w i s e 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 c o n t r a s t s i n v o l v i n g j u s t two row means. Le t 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 and 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 the c o n t r a s t s can be w r i t t e n . I n the form: A /v A / » * - K±< <P < * + 5 ± where * = x - x and For the S c h e f f e P r o c e d u r e *L = / MS (- + - ^ (1-1), F ( I - 1 , N - I J ) S^ / w i n . n. , J 1-ct ' j 1 I ' ' w i t h MS w -- w i t h i n c e l l s mean square I . — number o f t r e a t m e n t l e v e l s (row) J ••— number o f t r e a t m e n t l e v e l s (columns) N -- t o t a l number o f s u b j e c t s n i ' n i ' — t o t a l number o f s u b j e c t s used f o r computing X\ and , r e s p e c t i v e l y 9:5 F o r the Tukey Procedure MS " ' ~~~w 1 - a q I , N - I J , N / I w i t h - Q-.-t- ,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 1 - a 4,1, N - • I J ^ the S t u d e n t l z e d range d i s t r i b u t i o n w i t h I and N - I J degrees o f freedom MS , N . and I are the same as f o r Scheffe" w Fo r the Dunn M u l t i p l e Comparison P r o c e d u r e A ?D t , D a / 2 ; C, N - I J /MS ( I + I ) / w v n. n. , / w i t h C - - Number o f comparisons t o be made t'D / 0 „ „ -,-T I s a c o e f f i c i e n t o b t a i n e d from Dunn's a/2; C, N - I J t a b l e ( K i r k , 1968, p. 551) MS , n.. and n., are the same as f o r S c h e f f e w' 1 1' I n t he p r e s e n t study I = J.= 3, = n^, = 63, N = l 8 9 3 C = 3> and a g .05. U s i n g t h i s i n f o r m a t i o n the r e l a t i v e e f f i c i e n c i e s o f the t h r e e p r o c e d u r e s were c a l c u l a t e d ( K i r k , . 1 9 6 8 , p. 96) and t h e r a t i o s o f the l e n g t h s o f the c o n f i d e n c e I n t e r v a l s ( K i r k , 1968, p.97). 96 T a b l e V The R e l a t i v e E f f i c i e n c i e s and R a t i o s o f Co n f i d e n c e I n t e r v a l s f o r S c h e f f | , Tukey and Dunn* R e l a t i v e R a t i o o f E f f i c i e n c y C o n f i d e n c e I n t e r v a l s ? 2 T I2 ^S x 100 = 91.99% • 96 Pz *=D Pz ^S x 100 = 96.01% k . . A ^S .98 i 2 2 2 x 100 = 95.81% A * .98 * where i 2 = 12.20 (MS w/63) £ 2 = 11.22 (MS w/63) = 11.71 (MSTir/63) The f o l l o w i n g c o n c l u s i o n s can be drawn from t h e comparison o f the S c h e f f e , . Tukey and Dunn's p r o c e d u r e s . A A- A F i r s t , as i n d i c a t e d by the l e n g t h s o f ? s , E,^, and £ D the S c h e f f e p r o c e d u r e produces the l o n g e s t c o n f i d e n c e i n t e r v a l s , and i s , t h e r e f o r e , the most conservative.;' Second, t h e f a c t 97 t h e r e l a t i v e e f f i c i e n c i e s a r e a l l i n excess o f 90 p e r c e n t i n d i c a t e s t h a t a l l t h r e e 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 ) and Type I I ( 3 - e r r o r ) f o r p a i r w i s e comparisons. Thus, t h e s e l e c t i o n , o f t h e most c o n s e r v a t i v e S c h e f f e p r o c e d u r e f o r the a n a l y s i s o f c o n t r a s t s does not appear t o be t o o c o s t l y I n terms o f power. A n a l y s i s o f Data. T a b l e s VI and V I I c o n t a i n the means and s t a n d a r d d e v i a t i o n s f o r the a l g e b r a i c and ge o m e t r i c d a t a , r e s p e c t i v e l y . The o v e r a l l F - r a t i o s were computed and the s e r e s u l t s are summarized i n T a b l e s V I I I and IX. The o v e r a l l F - r a t i o s I n d i c a t e t h a t t h e r e 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 o r g e o m e t r i c t e s t . Thus, the n u l l h ypotheses c o r r e s p o n d i n g , t o hypotheses 4 and 5 are a c c e p t e d , and t h e i n v e s t i g a t o r c o n c l u d e s t h a t t h e r e are no d i f f e r e n c e s t h a t 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 t h e t h r e e t r e a t m e n t hypotheses H Q 1 , H Q 2 , and HQ^ u s i n g t h e S c h e f f e p r o c e d u r e . T a b l e s X and XI c o n t a i n the .05 s i m u l t a n e o u s c o n f i d e n c e I n t e r v a l s . As i n d i c a t e d p r e v i o u s l y the I n v e s t i g a t o r a l s o t e s t e d hypotheses 1 - 3 by a n o n - p a r a m e t r i c ( x 2 ) method. The r e s u l t s o f t h i s a n a l y s i s are c o n t a i n e d i n t a b l e s X I I and X I I I . 98 Table V I Means and S t a n d a r d D e v i a t i o n s f o r the A l g e b r a i c T e s t N e u t r a l . A l g e b r a i c Geometric T o t a l H e u r i s t i c s o n l y .5238 (.5118) .7143 (.4629) .5283 (.5118) .5873 (.4963) H e u r i s t i c s p l u s c o n t e n t ' .2381 (.4364) .4286 (.5071) .3810 (.4976) .3492 (.4805) Content o n l y .1905 (.4024) .2381 (.4364) .1905 (.40-24) .2064 (.4071) * S t a n d a r d d e v i a t i o n s i n p a r e n t h e s e s T a b l e V I I Means and S t a n d a r d D e v i a t i o n s * f o r the Geometric Test N e u t r a l A l g e b r a i c Geometric T o t a l H e u r i s t i c s o n l y .7143 (.4629) .6667 (.4830) .5714 (.5071) .6508 (.4805) H e u r i s t i c s p l u s c o n t e n t .5238 (.5118) .6190 (.4976) .5714 (.5071) .5714 (.4988) f Content o n l y .2857 (.4629) .3333 (.4830) .3810 (.4976) .3333 (.4752) * S t a n d a r d d e v i a t i o n s i n p a r e n t h e s e s 99 T a b l e V I I I O v e r a l l F-Ratio's f o r the A l g e b r a i c Test Source o f Degrees o f Mean „ P r o b a b i l i t y V a r i a t i o n Freedom Squares (F >- F 0 ) Treatments 2 '2.3333 10.7825* 0.00004 V e h i c l e s 2 0.3333 1.5403 0.2171 I n t e r a c t i o n 4 0.0714 0.3299 0.8577 E r r o r 180' 0.2164 * S i g n i f i c a n t at .05 l e v e l Table IX O v e r a l l F - R a t i o s f o r the Geometric Test Source o f V a r i a t i o n Treatments V e h i c l e s I n t e r a c t i o n E r r o r Degrees o f Freedom 2 2 4 180 Mean Squares 1.7196 0.0212 0.0?26 0.2407 Fc P r o b a b i l i t y (F » F f ) 7.1432* 0.0010 0.0897 0.9159 0.3846 0.8195 * S i g n i f i c a n t at .05 l e v e l 100 T a b l e X A l g e b r a i c Test - Simu l t a n e o u s C o n f i d e n c e I n t e r v a l s f o r Hypotheses 1, 2 and 3 D i f f e r e n c e between s s .05 C o n f i d e n c e means •/^w' ' " I n t e r v a l s •HI .Tlr. - Y"3.. = 0.3819 0.2047 (0,1772 , 0.5866)* H2 Y 2.. - Y .. = 0.1429 0.2047 • (-0.0618, 0.3576) H3 Y1.. - Y 2.. = 0.2381 0.2047 (0.0334 , 0.4428)* Ms "(-)"(3-D Q q P ( 2 , l 8 0 ) = /.2164 x 4 x 3-05 = 0.2047 • . w n - . 9 5 j ^-63 * S l g n l f l e a n t at .05 l e v e l T a b le XI Geometric Test - Simultaneous C o n f i d e n c e I n t e r v a l s f o r Hypotheses 1, 2 and 3 D i f f e r e n c e between means HI- Y 1. . - Yy . = . 0 .3174 H2 ' Y 2. ..' - Yy. = 0.2381 H3 Y 1' - Y, 0.0794 / I s -J w 0.2159 0.2159 0.2159 **/MS w(^)(3-D- . 9 5 P ( 2 , l 8 0 ) .05 C o n f i d e n c e I n t e r v a l s • (0.1015 , 0.5333)* (0.'0222 , 0.4540)* (-0.1365, 0.2953) /.2407 x 4 x 3.05 = 0.2159 7 63 * S i g n i f l e a n t at .05 l e v e l 101 Table X I I •X2 A n a l y s i s f o r the A l g e b r a i c Test . H y p o t h e s i s Groups x 2 P r o b a b i l i t y H 1 H vs C 19.0989 . 00001* H 2 HC vs C 3.2044 . .0734 H 3 H vs HC 7.1718 .0074* * S i g n i f i c a n t at .05 l e v e l Table X I I I X 2 A n a l y s i s o f the Geometric Data H y p o t h e s i s Groups . x 2 P r o b a b i l i t y H 1 H vs C 11.4603 .0007* H 2 HC vs C 6.2617 .0123* H 3 H vs HC ,,.8349 . 3609 * S i g n i f i c a n t at .05 l e v e l 102 S t a t i s t i c a l .Conclusions B e f o r e d i s c u s s i n g the c o n c l u s i o n s i n d e t a i l i t i s im p o r t a n t t o note t h a t the a n a l y s e s o f hypotheses 01 - 03 by the S c h e f f e Procedure and the x 2 produced i d e n t i c a l s t a t i s t i c a l r e s u l t s ( w i t h a = .05). F i g u r e 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 the s t a t i s t i c a l c o n c l u s i o n s . F i g u r e 18 V i s u a l P r e s e n t a t i o n o f R e s u l t s o f S i g n i f i c a n c e T e s t s f o r Hypotheses H n i , H n„ and H,n~ Mean .3333 .5714 .6508 Group GEOMETRIC TEST S i g - s i g n i f i c a n t at .05 l e v e l NS - not s i g n i f i c a n t at .05 l e v e l 103 There was no s i g n i f i c a n t o v e r a l l i n t e r a c t i o n e f f e c t on e i t h e r the 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 c o n t r a s t s c o r r e s p o n d i n g t o hypotheses 4 and 5 were not t e s t e d , and n u l l hypotheses H^^ and HQ_ were a c c e p t e d . I n t e r p r e t a t i o n o f R e s u l t s . The 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 group 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 (see f i g u r e 18). S i n c e the C group was d e s i g n e d t o c o n t r o l f o r p o s s i b l e h e u r i s t i c a b i l i t y p r e s e n t i n the p o p u l a t i o n p r i o r t o e x p e r i m e n t a t i o n , f o r i n c i d e n t a l l e a r n i n g o f h e u r i s t i c t e c h n i q u e s and f o r f a c t o r s such as n o v e l t y t h a t might a f f e c t performance i n d e p e n d e n t l y o f the t r e a t m e n t ^ the i n v e s t i g a t o r c o ncludes t h a t by f o c u s s i n g I n s t r u c t i o n on h e u r i s t i c s 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 the h e u r i s t i c s o f e x a m i n a t i o n o f cases o r analogy t o n o v e l m a t h e m a t i c a l problems. The s t a t i s t i c a l a n a l y s e s o f H Q 2 and HQ^ i n d i c a t e s d i f f e r e n t r e s u l t s f o r the a l g e b r a i c and g e o m e t r i c t e s t s . On the a l g e b r a i c t e s t the 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 the HC and C g r o u p s , 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 H and HC groups. The i n v e s t i g a t o r c o ncludes t h a t o f the t r e a t m e n t s t e s t e d , the H t r e a t m e n t i s the o n l y e f f e c t i v e t r e a t m e n t f o r t e a c h i n g s t u d e n t s t o a p p l y at l e a s t one o f the h e u r i s t i c s t o a n o v e l a l g e b r a i c problem. On the g e o m e t r i c t e s t b o t h the 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 t h a n the C group, w h i l e no s i g n i f i c a n t d i f f e r e n c e was found between the H and HC groups. The i n v e s t i g a t o r c o n c l u d e s t h a t b o t h t h e H and HC t r e a t m e n t s a re 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 a p p l y a t l e a s t one o f the h e u r i s t i c s t o a n o v e l g eometric problem. 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 e v i d e n c e t o support the c o n c l u s i o n t h a t they are d i f f e r e n t i a l l y e f f e c t i v e . The d i f f e r e n t c o n c l u s i o n s f o r the two t e s t s are d i s c u s s e d i n the next s e c t i o n . There 'was., no s i g n i f i c a n t o v e r a l l i n t e r a c t i o n e f f e c t on e i t h e r the a l g e b r a i c or the ge o m e t r i c t e s t . T h i s i m p l i e s t h a t f o r a l l t h r e e v e h i c l e s (N, A, G) the p a t t e r n o f means of the t h r e e t r e a t m e n t s (H, HC, C) i s e s s e n t i a l l y t h e same f o r the a l g e b r a i c t e s t , and e s s e n t i a l l y the same f o r the geo m e t r i c t e s t , a l t h o u g h the p a t t e r n s o f means are d i f f e r e n t f o r the two t e s t s (see f i g u r e s 19 and 2 0 ) . The i n v e s t i g a t o r c o n c l u d e s t h a t t h e v e h i c l e does not 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 t e a c h i n g s t u d e n t s t o a p p l y at l e a s t one o f the h e u r i s t i c s . Whatever the v e h i c l e , the c r i t i c a l d e t e r m i n a n t i s 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 , namely h e u r i s t i c s o n l y , o r h e u r i s t i c s combined w i t h c o n t e n t . D i s c u s s i o n o f R e s u l t s Hypotheses 1, 2 and 3 r e f e r t o d i f f e r e n c e s t h a t can be a t t r i b u t e d t o the t r e a t m e n t s . The s t a t i s t i c a l a n a l y s i s e n a b l e s the i n v e s t i g a t o r t o co n c l u d e t h a t i n terms o f average F i g u r e 19 Mean Scores of Groups on the A l g e b r a i c Test Mean Scores 75 70 65 60 .55 .50 .45 .40 .35 .30 .25 . 20 .15 HC Treatments O v e r a l l N e u t r a l groups o n l y A l g e b r a i c groups o n l y Geometric groups o n l y H F i g u r e 20 Mean Scores o f Groups on the Geometric T e s t Me an Scores .25 .20 .15 C HC H Treatments O v e r a l l N e u t r a l groups o n l y A l g e b r a i c groups o n l y Geometric groups o n l y 107 test scores. It. is possible, to. teach students, to apply at . least one of. the heuristics, of examination of cases or analogy to novel mathematical problems. In particular, the heuristics•only experimental treatment is successful in 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 for the. two tests. The analysis indicates the heuristics plus content treatment is successful in teaching subjects to apply at least one of the heuristics to a novel geometric problem, but this is,; not true for the algebraic problem. It is impossible to determine the reasons for these different conclusions. However, the following observations concerning the two tests may, in part., explain the different s t a t i s t i c a l results. An examination of tables II and III indicates a different pattern in the application of heuristics on the two tests. On the algebraic test 6'f the 72 subjects who applied at least one heuristic, 54 (75%) applied analogy, either alone (14 subjects) or in conjunction with examination-of cases (40 subjects). On the geometric test 98 subjects applied at least one heuristic but only 10 (10%) of them used analogy, and nine of the ten used the heuristic in conjunction with examination of cases. Subjects who had exhibited the ability to employ analogy in the algebraic test could not, or did not, apply the heuristic in the geometric 108 t e s t . . Thus. I t appears t h a t t h e g e o m e t r i c t e s t was more amenable to. the 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 a n a l o g y . T h i s c o u l d be due t o t h e f a c t t h a t the grade t e n s u b j e c t s were more f a m i l i a r w i t h • a l g e b r a i c systems, f a c i l i t a t i n g a n a l o g i e s i n an a l g e b r a i c as opposed to a g e o m e t r i c s e t t i n g . The i n v e s t i g a t o r c o ncludes the d i f f e r e n t r e s u l t s c o n c e r n i n g the e f f e c t i v e n e s s o f the h e u r i s t i c s p l u s c o n t e n t t r e a t m e n t on the t e s t s c o u l d be a t t r i b u t e d t o t h i s t r e a t m e n t b e t t e r e q u i p p i n g s u b j e c t s t o a p p l y e x a m i n a t i o n o f cases than t o 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 the g e o m e t r i c t e s t where the problem I s more amenable t o the 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 c a s e s . An e x a m i n a t i o n o f the d a t a s u p p o r t s t h i s c o n c l u s i o n . F i r s t , on the a l g e b r a i c t e s t o n l y 18 s u b j e c t s from the HC group employed a n a l o g y , w h i l e 29 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 . Second, o f the 63 s u b j e c t s i n t h e HC group t h e r e were 20 who had shown no e v i d e n c e o f a b i l i t y t o a p p l y e i t h e r h e u r i s t i c i n the a l g e b r a i c t e s t but had a p p l i e d e x a m i n a t i o n o f cases i n the g e o m e t r i c t e s t . One f u r t h e r note can be made c o n c e r n i n g the two t e s t s . 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 groups (H, HC, C) s c o r e d h i g h e r on the g e o m e t r i c t e s t than on the a l g e b r a i c t e s t . There are many p o s s i b l e reasons f o r the s u p e r i o r r e s u l t s on the g e o m e t r i c t e s t . The a l g e b r a i c t e s t was a d m i n i s t e r e d f i r s t and s u b j e c t s may 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 . F u r t h e r m o r e , s i n c e b o t h t e s t s f o l l o w the same g e n e r a l f o r m a t , I n v o l v i n g F i g u r e 21 O v e r a l l Means f o r A l g e b r a i c and. Geometric T e s t s Mean Scores .70 .65 .60 .55 .50 .45 .40 • 35 .30 .25 .20 • 15 HC Treatment Groups A l g e b r a i c t e s t H Geometric t e s t 110 the p r e s e n t a t i o n o f a problem f o l l o w e d by the q u e s t i o n "What can you f i n d out about . . . ? " t h e r e may have been a t e s t i n g e f f e c t (Campbell and S t a n l e y , 1963 5 p. 5). Another p o s s i b i l i t y i s t h a t t h e r e i s something i n h e r e n t i n the g e o m e t r i c problem used on the t e s t t h a t made i t s i m p l e r t h a n the a l g e b r a i c problem. . Any o f t h e s e a l t e r n a t i v e s , o r c o m b i n a t i o n o f a l t e r n a t i v e s c o u l d account f o r the b e t t e r s c o r e s on the g e o m e t r i c t e s t . The o r i g i n a l s t a t ements o f hypotheses 1, 2 and 3 i n d i c a t e d an o r d e r i n g between t h e means o f the t h r e e t r e a t m e n t groups. T h i s o r d e r i n g i s : Mean (H) > Mean (HC) > Mean (C) A l t h o u g h the a n a l y s i s does not r e s u l t i n 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 f u l l y j u s t i f y i n g t h i s o r d e r i n g , such 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 21). F u r t h e r m o r e , t h i s o r d e r i n g a l s o h o l d s w i t h i n t h e t h r e e v e h i c l e s . F i g u r e s 19 and 20 show t h a t f o r the sample used Mean (H - V ± ) > Mean (HC - V ± ) >'::Mean (C - V ± ) where = N, A, G, w i t h the s i n g l e e x c e p t i o n t h a t Mean (H - G) = Mean (HC - G). I l l One, f u r t h e r a n a l y s i s a s s o c i a t e d w i t h the d a t a has been i n c l u d e d i n t h i s c h a p t e r . Table I c o n t a i n s the i n f o r m a t i o n on the number o f te:s.ts graded. There were 42 s u b j e c t s who wrote o n l y one t e s t (21 a l g e b r a i c and 21 g e o m e t r i c ) . Of t h e s e the judges agreed on grades f o r 20 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 . T a b l e s XIV c o n t a i n s the comparative d a t a f o r these s u b j e c t s and the s u b j e c t s used i n the a n a l y s i s . T a ble 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 A l g e b r a Geometry 0 1 0 1 H (Data) 26 27 22 41 H (One T e s t ) 4 5 3 4 HC (Data) 41 22 27 36 HC (One T e s t ) 3 1 3 1 C (Data) 50 13 41 22 C (One T e s t ) 5 2 5 3 T o t a l (Data) 117 72 90 99 T o t a l (One T e s t ) 12 8 11 8 112 The number o f s u b j e c t s who wrote o n l y one t e s t was too s m a l l t o under t a k e x 2 t e s t s , f o r i-each o f the t h r e e treatments.- However, f o r each t e s t a x 2 v a l u e was computed between the d a t a used ,in the a n a l y s i s ( T o t a l ( D a t a ) ) and d a t a based on s u b j e c t s who wrote o n l y one t e s t ( T o t a l ( O n e T e s t ) ) . T a b l e XV X 2 Comparisons o f A c t u a l Data and Only One Test Data* Test Groups x 2 P r o b a b i l i t y T o t a l (Data) A l g e b r a i c Vs 0.0056 .940 T o t a l (One T e s t ) T o t a l (Data) Geometric Vs 0.3764 .539 T o t a l (One T e s t ) w i t h Y a t e s ' s c o r r e c t i o n f o r c o n t i n u i t y T h i s a n a l y s i s i n d i c a t e s t h a t f o r b o t h t e s t s t he d i s t r i b u t i o n o f s c o r e s o f s u b j e c t s who wrote o n l y one t e s t (and the judges agreed on the grade) was e s s e n t i a l l y the same as the d i s t r i b u t i o n o f the d a t a used i n the a n a l y s i s . The f o l l o w i n g t h r e e p o i n t s are n o t e d . S i n c e 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 d i r e c t i o n s f o r f u r t h e r r e s e a r c h they are d i s c u s s e d i n d e t a i l i n the a p p r o p r i a t e s e c t i o n s o f c h a p t e r V. 113 (1) Of the 72 s u b j e c t s , who. used at l e a s t one . h e u r i s t i c on t h e a l g e b r a i c t e s t , 40 (55%) employed b o t h h e u r i s t i c s i n t h e i r s o l u t i o n . (2) Of t h e 63 s u b j e c t s who r e c e i v e d the h e u r i s t i c s o n l y t r e a t m e n t 26 ( 4 l % ) d i d not a p p l y e i t h e r h e u r i s t i c t o t h e a l g e b r a i c p r o b l e m , w h i l e 22 (35%) d i d not a p p l y e i t h e r h e u r i s t i c t o the g e o m e t r i c problem. (3) On t h e a l g e b r a i c t e s t the performance o f the group t h a t r e c e i v e d the h e u r i s t i c s o n l y i n s t r u c t i o n by t h e a l g e b r a i c v e h i c l e (H - A) was b e t t e r t h a n a l l the o t h e r groups. The H - A group was a l s o second h i g h e s t on the g e o m e t r i c t e s t . F i n a l l y , one comment i s I n c l u d e d c o n c e r n i n g the I n t e r p r e t a t i o n o f the r e s u l t s . C o n c l u s i o n s have been drawn c o n c e r n i n g the e f f e c t i v e n e s s o f t e a c h i n g h e u r i s t i c s . I t has been assumed t h a t the d i f f e r e n c e s are due t o the emphasis 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 w i t h c o n t e n t . However, based on the e v i d e n c e a v a i l a b l e i n the study., a l t e r n a t i v e hypotheses t h a t the d i f f e r e n c e s I n t r e a t m e n t s are due t o d i f f e r e n t i a l m o t i v a t i o n a l e f f e c t s o f the b o o k l e t s , e f f e c t s t h a t are Independent o f the h e u r i s t i c s , r a t h e r than the emphasis on h e u r i s t i c s cannot be r e j e c t e d . I n one o f the p i l o t s t u d i e s d a t a were, o b t a i n e d from 114 s u b j e c t s , c o n c e r n i n g 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 b o o k l e t s . T h e i r comments I n d i c a t e t h a t most s u b j e c t s were p l e a s e d t o complete the b o o k l e t s , and t h e r e were no d i s c e r n a b l e d i f f e r e n c e s between groups. F u r t h e r m o r e , as was d i s c u s s e d i n c h a p t e r I I I e v e r y e f f o r t was made i n d e s i g n i n g the m a t e r i a l s t o make them as e q u i v a l e n t as p o s s i b l e . Thus, w h i l e f u r t h e r r e s e a r c h needs t o be un d e r t a k e n i n t o m o t i v a t i o n e f f e c t s a s s o c i a t e d 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 , the i n v e s t i g a t o r f e e l s c o n f i d e n t t h a t based on the d e s i g n o f the m a t e r i a l s , and s u b j e c t s ' comments on the b o o k l e t s , the success o f the H and HC t r e a t m e n t s i s due t o the emphasis on h e u r i s t i c s and not d i f f e r e n t m o t i v a t i o n a l e f f e c t s t h a t are independent o f the h e u r i s t i c s . CHAPTER V Summary and C o n c l u s i o n s The s t u d y 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 app l y at l e a s t one o f the h e u r i s t i c s o f e x a m i n a t i o n o f cases o r analogy t o n o v e l m a t h e m a t i c a l p r o b l e m s , 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 can be b e t t e r a c c o m p l i s h e d by h a v i n g the s t u d e n t s f o c u s s o l e l y on the h e u r i s t i c s , r a t h e r than a l s o i n v o l v e them w i t h the s i m u l t a n e o u s l e a r n i n g of m a t h e m a t i c a l c o n t e n t . Two f a c t o r s were c o n s i d e r e d i n the d e s i g n o f the s t u d y . The f i r s t f a c t o r concerns employing 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 o b j e c t i v e s : one I n s t r u c t i o n a l t r e a t m e n t was d e s i g n e d t o t e a c h h e u r i s t i c s o n l y ; a n o t h e r combined the 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 ; and a t h i r d t r e a t m e n t u t i l i z e d c o n t e n t o n l y , w i t h o u t r e f e r e n c e t o h e u r i s t i c s . The l a t t e r t r e a t m e n t was used as an e x p e r i m e n t a l c o n t r o l . The second f a c t o r concerns e m p l o y i n g d i s t i n c t 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 t r e a t m e n t s , namely a l g e b r a i c , g e o m e t r i c and m a t h e m a t i c a l l y n e u t r a l v e h i c l e s . Nine groups were u t i l i z e d I n the study corresponding t o each o f the t r e a t m e n t s x v e h i c l e c o m b i n a t i o n s . 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 d e v e l o p e d f o r each 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 t e n days (one c l a s s p e r i o d p e r day) a t the end o f which two 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 the 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 o f p r e s e n t i n g s t u d e n t s w i t h 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 the q u e s t i o n : What can you f i n d out about...? C o n c l u s i o n s The r e s u l t s o f the d a t a a n a l y s i s i n d i c a t e 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 app l y at l e a s t one h e u r i s t i c to a n o v e l a l g e b r a i c o r g e o m e t r i c problem. I n p a r t i c u l a r , t h e 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 o n l y t r e a t m e n t (H) 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 o n t r o l group (C) on b o t h t e s t s . The e f f e c t i v e n e s s o f the combined i n s t r u c t i o n a l t r e a t m e n t , h e u r i s t i c s p l u s c o n t e n t (HC), was d i f f e r e n t f o r the two t e s t s . On 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 s i g n i f i c a n t l y b e t t e r than b o t h the HC and C gr o u p s , w h i l e 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 was found between the HC and C groups. The c o n c l u s i o n i s t h a t o n l y the 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 a p p l y 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. On the g e o m e t r i c t e s t the a n a l y s i s i n d i c a t e s t h a t the H and HC groups 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 the C group, but they were not themselves c l e a r l y d i f f e r e n t i n t h e i r e f f e c t i v e n e s s . The c o n c l u s i o n i s t h a t b o t h the H and HC t r e a t m e n t s are 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 app l y at l e a s t one h e u r i s t i c t o a n o v e l g e o m e t r i c problem, and the e v i d e n c e 117 does not p e r m i t any c o n c l u s i o n c o n c e r n i n g t h e i r r e l a t i v e e f f e c t i v e n e s s . However, a l t h o u g h the a n a l y s i s does not w a r r a n t the g e n e r a l c o n c l u s i o n t h a t the H.treatment i s more e f f i c a c i o u s t han the HC t r e a t m e n t 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 h e u r i s t i c t o n o v e l m a t h e m a t i c a l p r o b l e m s , an e x a m i n a t i o n o f the o v e r a l l means ( f i g u r e 21) shows a t r e n d i n t h a t d i r e c t i o n . The same t r e n d a p p e a r s - i n the p a r a l l e l e x a m i n a t i o n o f the means w i t h i n the t h r e e v e h i c l e s ( f i g u r e s 19 and 20). C e r t a i n l y , i f one i s s o l e l y i n t e r e s t e d 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 the a l g e b r a i c t e s t , t o g e t h e r w i t h the 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 t h a t the H.treatment i s the'most e f f i c a c i o u s . 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 the two t e s t s r e v e a l s 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 e v i d e n c e o f analogy b e i n g used on the g e o m e t r i c t e s t . The i n v e s t i g a t o r s u g g e s t s 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 the 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 t r e a t m e n t i s 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 t h a n 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 p r o v e d more amenable t o the 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 c a s e s . I t i s i m p o s s i b l e t o a s c e r t a i n the r e a s o n f o r t h i s r e s u l t c o n c e r n i n g the HC t r e a t m e n t . However, the i n v e s t i g a t o r s u g g ests t h e 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 . E x a m i n a t i o n o f cases can be 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 , than answers q u e s t i o n s . W i t h emphasis on b o t h h e u r i s t i c s and c o n t e n t students: t e n d to. l e a r n to. apply, the h e u r i s t i c t h a t appears most, v a l u a b l e i n a n s w e r i n g q u e s t i o n s , namely 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 c a s e s . 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 v e h i c l e s . Thus i t seems r e a s o n a b l e to conclude t h a t the i n s t r u c t i o n a l v e h i c l e employed i s 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 t o a p p l y h e u r i s t i c s . The- f o l l o w i n g t h r e e p o i n t s are n o t e d . S i n c e 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 d i r e c t i o n s f o r f u r t h e r r e s e a r c h they are d i s c u s s e d i n d e t a i l i n the a p p r o p r i a t e s e c t i o n s . (1) Of the 72 sub j ects.who used a t l e a s t one h e u r i s t i c on the a l g e b r a i c t e s t , 40 (55%) employed b o t h h e u r i s t i c s I n t h e i r s o l u t i o n . (2) Of the. 63 s u b j e c t s who r e c e i v e d the h e u r i s t i c s o n l y t r e a t m e n t 26 ( k l % ) d i d not a p p l y e i t h e r h e u r i s t i c t o the a l g e b r a i c p r o b l e m , w h i l e 22 (35%) d i d not a p p l y e i t h e r h e u r i s t i c t o the g e o m e t r i c problem. (3) On the a l g e b r a i c t e s t the performance o f the group t h a t r e c e i v e d the h e u r i s t i c s o n l y t r e a t m e n t by the a l g e b r a i c v e h i c l e (H - A) was b e t t e r than a l l t h e o t h e r groups. The H - A group was a l s o second h i g h e s t on the g e o m e t r i c t e s t . 119 I m p l i c a t i o n s • As a consequence of- t h i s study s e v e r a l s u g g e s t i o n s can be made 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 . When c o n s i d e r i n g these 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 the study c o n s i d e r e d o n l y the h e u r i s t i c s o f e x a m i n a t i o n o f cases and a n a l o g y . Thus, I n the f o l l o w i n g d i s c u s s i o n t h e term " h e u r i s t i c s " r e f e r s t o these two p a r t i c u l a r t e c h l q u e s . The 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 s t u d e n t s t o apply at l e a s t one o f the h e u r i s t i c s t o n o v e l m a t h e m a t i c a l problems. S i n c e an i m p o r t a n t o b j e c t i v e o f the secondary s c h o o l mathematics programme I s to develop s k i l l s I n 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 , t h e I n v e s t i g a t o r suggests t h a t u n i t s d e s i g n e d t o t e a c h h e u r i s t i c s be i n c o r p o r a t e d I n t o the secondary s c h o o l mathematics c u r r i c u l u m . I f t h i s s u g g e s t i o n i s a c c e p t e d the q u e s t i o n a r i s e s as t o the p o s s i b l e s e t t i n g s t h a t should- be employed to develop the h e u r i s t i c s . The study i n v e s t i g a t e d two f a c t o r s t h a t h e l p answer t h i s q u e s t i o n . F i r s t , the a n a l y s i s s u g g e s t s t h a t a h e u r i s t i c s o n l y t r e a t m e n t I s l i k e l y t o be s u p e r i o r t o a combined t r e a t m e n t , a l t h o u g h i n t h i s s tudy the s c o r e s were s i g n i f i c a n t l y b e t t e r o n l y on the a l g e b r a i c t e s t . However, based on the t r e n d s (see f i g u r e s 19, 20 and 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 s o l e l y t o the t e a c h i n g o f 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 than 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 w i t h I n s t r u c t i o n i n c o n t e n t . To 120 be most e f f e c t i v e the t e a c h e r s h o u l d devote p a r t o f the . mathematics course s o l e l y t o d e v e l o p i n g h e u r i s t i c s . Such an arrangement does not preclude, u s i n g h e u r i s t i c s i n c o n j u n c t i o n w i t h - o t h e r segments o f the mathematics c u r r i c u l u m , but the e v i d e n c e suggests t h a t f o r the i n s t r u c t i o n i n h e u r i s t i c s t o be most e f f e c t i v e some time has t o be devoted s o l e l y t o h e u r i s t i c s . Second, the a n a l y s i s i n d i c a t e s t h a t th e i n s t r u c t i o n a l v e h i c l e employed t o i n t r o d u c e the h e u r i s t i c s does not a f f e c t the s t u d e n t s ' a b i l i t y t o a p p l y the h e u r i s t i c s . Thus, the s e l e c t i o n o f • a n a l g e b r a i c , g e o m e t r i c o r m a t h e m a t i c a l l y n e u t r a l v e h i c l e does not appear t o be c r i t i c a l f o r d e v e l o p i n g the h e u r i s t i c s . It. has been recommended t h a t u n i t s on h e u r i s t i c s be i n c l u d e d i n the mathematics c u r r i c u l u m . However, one c a u t i o n a r y n ote s h o u l d be added..- A l t h o u g h the h e u r i s t i c s o n l y 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 the c o n t r o l g roup, the d a t a i n d i c a t e s t h a t t h e r e were a s u b s t a n t i a l number o f s t u d e n t s i n the h e u r i s t i c s o n l y group who d i d not a pply e i t h e r h e u r i s t i c t o t h e n o v e l problems (see c o n c l u s i o n s p. 113 ). These d a t a suggest i n v e s t i g a t i o n o f the q u e s t i o n as t o whether a l l s t u d e n t s can b e n e f i t from i n s t r u c t i o n i n h e u r i s t i c s , and t h i s q u e s t i o n i s e x p l o r e d i n recommendations f o r f u r t h e r r e s e a r c h . To summarize, i t i s recommended t h a t u n i t s d e s i g n e d s o l e l y t o t e a c h s t u d e n t s h e u r i s t i c s be i n c o r p o r a t e d i n t o 121 the secondary s c h o o l mathematics, c u r r i c u l u m . F u r t h e r m o r e , the e v i d e n c e a v a i l a b l e from t h i s s tudy s u g g e s t s • t h a t the v e h i c l e s e l e c t e d does not appear t o be o f c r i t i c a l i m p o r t a n c e . L i m i t a t i o n s T h i s d i s s e r t a t i o n f o c u s e s on c e r t a i n components o f the study o f h e u r i s t i c s . The l i m i t e d s e l e c t i o n o f h e u r i s t i c s employed and i n s t r u c t i o n a l v e h i c l e s u t i l i z e d r e s t r i c t s the g e n e r a l i z a b i l i t y o f the conclusions". A f u r t h e r l i m i t a t i o n was imposed by the s e l e c t i o n o f an a l l male sample. F i v e l i m i t a t i o n s are d i s c u s s e d i n t h i s s e c t i o n , r e s u l t i n g from: (1) c h o i c e o f sample; (2) use 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 ; (3) s e l e c t i o n o f t e s t s ; (4) 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 ; and (5) s e l e c t i o n o f h e u r i s t i c s . The I n v e s t i g a t o r has I n c l u d e d i n t h i s d i s c u s s i o n the s u g g e s t i o n s f o r f u r t h e r r e s e a r c h t h a t ' f o l l o w d i r e c t l y from the l i m i t a t i o n s . W i t h an a l l male sample i t I s p o s s i b l e t o make s u p p o r t a b l e statements about the t e a c h i n g o f h e u r i s t i c s t o male s t u d e n t s i n the age range 15 - 17 y e a r s (Grade t e n ) , but l e a v e s open the q u e s t i o n o f how female s t u d e n t s w i l l r e spond t o s i m i l a r t r e a t m e n t s . Fennema (1974) r e v i e w e d the l i t e r a t u r e on sex and m a t h e m a t i c a l achievement. She c i t e d s t u d i e s i n v o l v i n g h i g h s c h o o l s t u d e n t s t h a t f a v o u r e d males (NLSMA, Y - p o p u l a t i o n ) , f a v o u r e d females ( E a s t e r d a y and E a s t e r d a y ) , and t h a t found 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 the 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 can be rea c h e d c o n c e r n i n g t h e r e l a t i o n s h i p between sex and m a t h e m a t i c a l achievement f o r h i g h s c h o o l s t u d e n t s . S i n c e the r e s e a r c h e v i d e n c e i s am b i v a l e n t c o n c e r n i n g the r o l e o f sex as a f a c t o r i n ma t h e m a t i c a l achievement o f h i g h s c h o o l s t u d e n t s , the i n v e s t i g a t o r f e e l s t h a t the study s h o u l d be r e p l i c a t e d w i t h f e m a l e s , o r w i t h mixed groups i n a d e s i g n which p e r m i t s the 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. U s i n g 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 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 , F o r example, t e a c h e r - s t u d e n t and peer group i n t e r a c t i o n s , which a re an i n t e g r a l p a r t o f t h e c l a s s r o o m environment, were e l i m i n a t e d from the stu d y . I t i s i m p o s s i b l e t o a s s e s s whether the l a c k o f these i n t e r a c t i o n s had a p o s i t i v e or n e g a t i v e e f f e c t on s t u d e n t s ' performance. 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 s t u d e n t s t o p r o g r e s s a t t h e i r own r a t e , 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 . may. have h e l p e d some s t u d e n t s l e a r n t o ap p l y the h e u r i s t i c s . A l t e r n a t i v e l y , the 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 o t h e r s t u d e n t s , A l t h o u g h the r e s e a r c h c i t e d i n c h a p t e r I I i n d i c a t e s t h a t 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 are an 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 needs t o be unde r t a k e n t o determine the r o l e o f mode o f i n s t r u c t i o n on s t u d e n t s ' a b i l i t y t o a p p l y h e u r i s t i c s . An a d d i t i o n a l l i m i t a t i o n imposed by the use o f b o o k l e t s i s t h a t a l l the 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 o l e o f r e a d i n g a b i l i t y i n the r e s u l t s o f t h i s s tudy 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 n a t u r e o f the t e s t s i s o f c r i t i c a l i m portance t o the s t u d y . E a c h - t e s t c o n s i s t s o f an open ended m a t h e m a t i c a l p r o b l e m , and t h e s t u d e n t s ' p e r c e p t i o n o f the problem might a f f e c t t h e i r a b i l i t y t o a p p l y the h e u r i s t i c s . F o r example, to a p p l y the h e u r i s t i c o f analogy s t u d e n t s must p e r c e i v e a s i m i l i a r i t y between the t e s t problem and some p r e v i o u s l y e x p e r i e n c e d system. A l t h o u g h a l l s t u d e n t s had been exposed to a p p r o p r i a t e systems ( f o r example, i t was p o s s i b l e t o make a n a l o g i e s between the t e s t problems and the i n s t r u c t i o n a l m a t e r i a l s i n the b o o k l e t s ) t h e r e was no guarantee t h a t an analogy would be made. I t i s p o s s i b l e t h a t a s t u d e n t who had l e a r n e d the h e u r i s t i c o f analogy was unable t o p e r c e i v e a s i m i l a r i t y - between the 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 hence unable t o a p p l y the h e u r i s t i c . Because the t e s t problems p l a y a s i g n i f i c a n t r o l e i n e l i c i t i n g s t u d e n t r e s p o n s e , the i n v e s t i g a t o r f e e l s t h a t a wide v a r i e t y o f n o v e l t e s t problems must be used t o o b t a i n a c l e a r p i c t u r e o f the v a l u e o f h e u r i s t i c s such as analogy and e x a m i n a t i o n o f cases i n s o l v i n g n o v e l problems. The 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 st u d y are a l g e b r a i c , g e o m e t r i c and m a t h e m a t i c a l l y n e u t r a l , and the t o p i c s were chosen so t h a t the v e h i c l e s are " e q u i v a l e n t " (See c h a p t e r I I I ) . T h i s c o n s t r a i n t l i m i t s the s e l e c t i o n o f 124 t o p i c s t h a t c o u l d be used t o i n t r o d u c e the h e u r i s t i c s . - The. i n v e s t i g a t o r f e e l s t h a t the study s h o u l d be r e p l i c a t e d w i t h d i f f e r e n t t o p i c s and v e h i c l e s . ' • -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 o f cases, and a n a l o g y , were s e l e c t e d from the works o f P o l y a (1957), MacPherson (1971b), Rousseau (1971), and Henderson and Smith (1959). Whether s t u d e n t s can be taught t o app l y o t h e r h e u r i s t i c s i s an open q u e s t i o n r e q u i r i n g a d d i t i o n a l r e s e a r c h . Recommendations f o r F u r t h e r R esearch The c o n c l u s i o n s o f t h i s s t u d y , t o g e t h e r w i t h 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 f o r f u r t h e r r e s e a r c h . Four major q u e s t i o n s are d i s c u s s e d I n . t h i s s e c t i o n . They a r e : (1) What f a c t o r s a f f e c t s t u d e n t s ' a b i l i t y t o a p p l y h e u r i s t i c s ? (2) What r o l e does knowledge o f core m a t e r i a l p l a y i n s t u d e n t s ' a b i l i t y t o a p p l y h e u r i s t i c s ? (3) Can s t u d e n t s be ta u g h t t o combine h e u r i s t i c s t o s o l v e n o v e l problems? (4) What I n s t r u c t i o n a l modes are.most e f f e c t i v e i n t e a c h i n g h e u r i s t i c s ? 125 F a c t o r s E f f e c t i n g H e u r i s t i c A b i l i t y . The f a c t t h a t a s u b s t a n t i a l number o f s t u d e n t s i n t h e h e u r i s t i c s o n l y group (H) d i d not a p p l y e i t h e r h e u r i s t i c t o the t e s t problems r a i s e s the q u e s t i o n as t o what f a c t o r s might be r e l a t e d t o the a b i l i t y t o a p p l y h e u r i s t i c s . S c i e n c e Research A s s o c i a t e s High S c h o o l Placement Test d a t a were a v a i l a b l e on 59 o f the 63 s u b j e c t s i n the H group. These d a t a I n c l u d e d measures on m a t h e m a t i c a l and r e a d i n g achievement. F o r each o f these measures the H group was d i v i d e d i n t o t h r e e subgroups ( a p p r o x i m a t e l y o n e - t h i r d i n each s u b g r o u p ) , namely h i g h , middle and low a c h i e v e r s . P a l r w i s e x 2 a n a l y s e s were undertaken t o determine i f e i t h e r o f these f a c t o r s i s r e l a t e d to s t u d e n t s ' a b i l i t y t o a p p l y a t l e a s t one h e u r i s t i c . The r e s u l t s o f t h i s a n a l y s i s are shown i n t a b l e s XVI and X V I I . F i g u r e s 22 - 27 show the d a t a f o r the Mathematics and Reading subgroups. 1 2 6 Table 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 the A l g e b r a i c T e s t * ? Score 0 1 Groups x 2 P r o b a b i l i t y H 4 1 5 H , M 0 . 8 7 3 1 . 3 5 0 1 M a t h e m a t i c a l M 8 1 2 H , L 7 . 5 2 6 9 . 0 0 6 1 Achievement L 14 6 M , L 2 . 5 2 5 2 . 1 1 2 0 Score ** 0 1 Groups x 2 P r o b a b i l i t y H 5 1 5 H , M 2 . 1 7 9 4 . 1 3 9 9 Reading M 1 1 1 0 H , L 2 . 5 3 3 6 .1114 Achievement L 1 0 8 M , L 0 . 0 1 5 4 . 9 0 1 4 Score . 0 1 Groups X 2 P r o b a b i l i t y H igh o r Low HH .3 !:i 1 1 HHjLL 2 . 9 9 3 6 . 0 8 3 6 on BOTH measures LL 9 6 * w i t h Y a t e s ' s c o r r e c t i o n f o r c o n t i n u i t y * not independent p a i r w i s e g r o u p i n g s . 127 Table XVII X 2 A n a l y s i s o f the Achievement of the H e u r i s t i c s Only Group on the Geometric T e s t * Score 0 1 Groups X P r o b a b i l i t y H 5 14 H , M 0.0074 .9314 M a t h e m a t i c a l M 5 15 H , L 1.4170 .2339 Achievement L 10 10 M , L 1.7067- .1914 Score 0 1 Groups X P r o b a b i l i t y H 4 16 H , M 0.3728 .5415 Reading M 7 14 H , L 2.5726 . 1087 Achievement L 9 9 M , L 0.5305 .4664 Score 0 1 Groups X P r o b a b i l i t y H igh o r Low HH 2 12 on BOTH HH,LL 3.3116 .0688 measures LL 8 7 * w i t h Y a t e s ' s c o r r e c t i o n f o r c o n t i n u i t y ** not independent 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 Achievement 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 Only Group Number o f Stud e n t s 20 15 10 5 0 A l g e b r a i c Test 0 Score High Achievement Subgroup M i d d l e Achievement Subgroup Low Achievement Subgroup . F i g u r e 23 Achievement o f H i g h , M i d d l e and Low Reading Achievement Subgroups o f H e u r i s t i c s Only Group Number o f Stud e n t s A l g e b r a i c T est 20 15 10 5 0 0 'Score High Achievement Subgroup M i d d l e Achievement Subgroup Low Achievement Subgroup 129 F i g u r e 24 Achievement 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 Only Group Geometric T e s t 20 Number o f 10 S t u d e n t s 5 0 Score High Achievement Subgroup • M i d d l e Achievement Subgroup — Low Achievement Subgroup F i g u r e 25 Achievement o f H i g h , M i d d l e and Low Reading Achievement Subgroups of H e u r i s t i c s Only Group . Geometric Test 20 15 Number o f . 10 S t u d e n t s 5 0 \J f~, -L. Score ' High_ Achievement Subgroup • M i d d l e Achievement Subgroup L o w Achievement Subgroup F i g u r e 26 Achievement o f Subgroups t h a t were High.-',or Low on.Both Mathematics and Reading Number o f St u d e n t s '15 10 5 0 A l g e b r a i c Test 0 Score H i g h on Both Mathematics and - Reading Low on Both Mathematics and Reading F i g u r e 27 Achievement o f Subgroups t h a t were High o r Low on Both Mathematics and Reading Number o f Stud e n t s Geometric T e s t 15 10 5 0 0 Score High- on Both Mathematics and Reading Low on Both Mathematics and Reading 131 The d a t a - i n t a b l e s XVI and X V I I i n d i c a t e , t h a t f o r b o t h m a t h e m a t i c a l achievement and r e a d i n g achievement the h i g h group performed b e s t , the low group w o r s t , and the m i d d l e group i n between. The s i n g l e e x c e p t i o n i s on the g e o m e t r i c t e s t where the m i d d l e m a t h e m a t i c a l achievement group performed m a r g i n a l l y b e t t e r than the h i g h achievement group. The s u p e r i o r performance o f the h i g h groups as compared t o the low groups i s r e i n f o r c e d by d a t a from subgroups c o n s i s t i n g o f s t u d e n t s who are h i g h o r low on b o t h m a t h e m a t i c a l and r e a d i n g achievement. I n b o t h a n a l y s e s i n v o l v i n g s t u d e n t s who are h i g h o r low on b o t h v a r i a b l e s , the p r o b a b i l i t i e s a s s o c i a t e d w i t h the X 2 v a l u e are l e s s t h a n .1. I n the f o u r a n a l y s e s i n v o l v i n g comparisons between h i g h and low subgroups, one a s s o c i a t e d p r o b a b i l i t y i s l e s s than .05 and two o t h e r s are i n the r e g i o n o f .1. S i n c e the a n a l y s i s i s o n l y e x p l o r a t o r y no s i g n i f i c a n c e l e v e l was a s s i g n e d f o r t e s t i n g the x 2 v a l u e . However, the e v i d e n c e i n d i c a t e s t h a t t h e h i g h group (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) performed c o n s i d e r a b l y b e t t e r than t h e i r c o r r e s p o n d i n g low group. Thus, t h e r e i s e v i d e n c e t o s u p p o r t the h y p o t h e s i s t h a t f a c t o r s such as m a t h e m a t i c a l and r e a d i n g achievement e f f e c t s t u d e n t s ' a b i l i t y t o a p p l y a t l e a s t one h e u r i s t i c . I n p a r t i c u l a r , the r e l a t i o n s h i p between r e a d i n g and the a b i l i t y t o a p p l y a h e u r i s t i c may be due i n p a r t t o the use 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 i n the e x p e r i m e n t . I t i s 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 u n d e r t a k e n t o i n v e s t i g a t e 132 t h e r e l a t i o n s h i p between th e s e f a c t o r s and the a b i l i t y t o apply h e u r i s t i c s . . . Knowledge o f Core M a t e r i a l . I t I s p a r t i c u l a r l y i m p o r t a n t t o unde r t a k e 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 a b i l i t y t o a'pply h e u r i s t i c s . The a n a l y s i s i n the p r e v i o u s paragraphs I n d i c a t e s t h a t m a t h e m a t i c a l achievement may be a f a c t o r e f f e c t i n g the a b i l i t y t o a p p l y h e u r i s t i c s . F o r example, on the a l g e b r a i c t e s t the x 2 v a l u e between t h e h i g h and low m a t h e m a t i c a l achievement groups had an a s s o c i a t e d p r o b a b i l i t y o f .0061. The s p e c i a l r o l e o f p r e v i o u s m a t h e m a t i c a l e x p e r i e n c e I n a p p l y i n g the h e u r i s t i c o f analogy has a l r e a d y been d i s c u s s e d (p. 123). The performance o f the c o n t r o l group shows t h a t j u s t t e a c h i n g core m a t e r i a l ( w i t h which a n a l o g i e s c o u l d be made on the t e s t s ) , w i t h o u t s p e c i f i c r e f e r e n c e t o t h e h e u r i s t i c ^ d o e s not r e s u l t i n much ev i d e n c e o f students' a p p l y i n g analogy ( s i x s t u d e n t s on the a l g e b r a i c t e s t , no s t u d e n t s on the ge o m e t r i c t e s t ) . The e v i d e n c e o f s t u d e n t s i n the c o n t r o l group u s i n g analogy may,. _n---p'&rt j$.,vjp&£lect a b i l i t y t o a p p l y 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 out t o determine the r o l e o f knowledge o f core m a t e r i a l on the a b i l i t y t o a p p l y h e u r i s t i c s . Combining H e u r i s t i c s t o S o l v e Problems. I n b o t h the 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 s t u d e n t s were g i v e n 133 t o t a l freedom i n a p p r o a c h i n g the problem.' There were no i n s t r u c t i o n s t o t h e s t u d e n t s t o use e i t h e r , o r b o t h o f t h e h e u r i s t i c s i n t h e i r s o l u t i o n s , . a l t h o u g h b o t h h e u r i s t i c s c o u l d be used i n the t e s t s i t u a t i o n s . An e x a m i n a t i o n o f t h e d a t a r e v e a l s t h a t on the a l g e b r a i c t e s t many s t u d e n t s combined the h e u r i s t i c s i n s o l v i n g the problem (See t a b l e X V I I I and f i g u r e 28). Table X V I I I A n a l y s i s o f the A p p l i c a t i o n o f Zer o , One o r Two H e u r i s t i c s on the A l g e b r a i c Test n~~,,~ Number o f 2 * T> v. v, • 1 • ^ G r o u p H e u r i s t i c s * * P r o b a b i l i t y 0 1 2 H 26 16 21 (H , HC) 7.4287 .0244 HC 41 8 14 (H , O 20 .0918 .00004 C 50 8 5 (HC , c) 5.1533 .0760 *Not independent S i n c e t h i s a n a l y s i s i s o n l y e x p l o r a t o r y , no s i g n i f i c a n c e 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 he x 2 v a l u e . However, the r e s u l t s a r e s i m i l a r t o t h o s e o b t a i n e d w i t h t h e 0 - 1 dichotomy employed t o t e s t hypotheses 1, 2 and 3-That i s , t h e H group performed b e s t , the ,'.:c group w o r s t , F i g u r e 28 A p p l i c a t i o n o f Zero, One o r Two H e u r i s t i c s on the A l g e b r a i c T e s t Number o f Stud e n t s 50 5^ 40 35 30 25 20 15 10 0 0 2 2 -Number o f H e u r i s t i c s H e u r i s t i c s o n l y group (H) H e u r i s t i c p l u s c o n t e n t group (HC) Content o n l y group (C) 135 and the HC group i n between. The d a t a are r e p r e s e n t e d i n f i g u r e 28. I n view o f the s e f i n d i n g s the i n v e s t i g a t o r f e e l s 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 und e r t a k e n t o c o n s i d e r how bes t t o t e a c h s t u d e n t s to combine h e u r i s t i c s t o s o l v e n o v e l m a t h e m a t i c a l problems. I n s t r u c t i o n a l Modes. The use o f b o o k l e t s 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 the l e a r n i n g s i t u a t i o n . I n p a r t i c u l a r i t I s n o t e d t h a t b o o k l e t s e l i m i n a t e d t e a c h e r -s t u d e n t and p e e r group I n t e r a c t i o n s from the e x p e r i m e n t a l p r o c e s s , and t h i s may have h i n d e r e d c e r t a i n s t u d e n t s i n t h e i r attempts t o l e a r n how t o apply h e u r i s t i c s . F u r t h e r -more , the a n a l y s i s o f h i g h , m i d d l e and low r e a d i n g achievement s t u d e n t s tends t o suggest t h a t r e a d i n g I s r e l a t e d to a s t u d e n t ' s a b i l i t y t o apply at 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 . Research s h o u l d be d e s i g n e d t o a l l o w f o r s t u d e n t - t e a c h e r and peer group I n t e r a c t i o n s as p a r t o f the e x p e r i m e n t a l p r o c e d u r e : t h a t I s , exp e r i m e n t s s h o u l d be conducted i n "normal" c l a s s r o o m s e t t i n g s . An e x a m i n a t i o n o f the d a t a i n d i c a t e t h a t on the a l g e b r a i c t e s t the H - A group s c o r e d h i g h e r than any o t h e r group. A l t h o u g h t h e r e was no s i g n i f i c a n t o v e r a l l i n t e r a c t i o n , f o r the purpose o f s u g g e s t i n g 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 a s s o c i a t e d w i t h t h i s group was i n v e s t i g a t e d . U s i n g the approach s u g g e s t e d by M a r a s c u i l o and L e v i n (1970, pp. 397 - 421) y.J2 w a s c a l c u l a t e d and a 95 p e r c e n t S c h e f f e c o n f i d e n c e i n t e r v a l formed. The i n t e r v a l i i s Y 1 2 = .048 t .211 and does not c o n t r i b u t e s i g n i f i c a n t l y t o the i n t e r a c t i o n . T h i s s i m p l y c o n f i r m s the e a r l i e r c o n c l u s i o n t h a t I n t h i s study t h e r e are 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 v e h i c l e s . However, experiments i n v o l v i n g d i f f e r e n t v e h i c l e s and i n s t r u c t i o n a l modes may r e s u l t i n the v e h i c l e s ' ' b e i n g a s i g n i f i c a n t f a c t o r . T h i s remains t o be d e t e r m i n e d by f u r t h e r r e s e a r c h . F i n a l l y , i n v i e w . o f the p o s s i b l e importance o f s t u d e n t s ' core knowledge 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 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 c a r r i e d .out. a t o t h e r grade l e v e l s . I f s t u d e n t s ' core knowledge I s an i m p o r t a n t f a c t o r I t may be t h a t u n t i l a s t u d e n t has a c e r t a i n b r e a d t h o f m a t h e m a t i c a l experience-- 1 he cannot be t a u g h t t o use c e r t a i n h e u r i s t i c s . T h i s c o u l d be p a r t i c u l a r l y t r u e o f a n a l o g y , where he i s r e q u i r e d t o f i n d s i m i l a r i t i e s between n o v e l problems and p r e v i o u s e x p e r i e n c e . Thus, i t may be t h a t e f f e c t i v e i n s t r u c t i o n i n h e u r i s t i c s such as analogy cannot be u n d e r t a k e n u n t i l a s t u d e n t has r e a c h e d a c e r t a i n l e v e l of m a t h e m a t i c a l m a t u r i t y . ' I t may be t h a t a programme de s i g n e d t o t e a c h h e u r i s t i c s s h o u l d b e g i n w i t h t h e development of e x a m i n a t i o n o f cases i n the e l e m e n t a r y g r a d e s , w i t h o t h e r h e u r i s t i c s b e i n g i n t r o d u c e d a t l a t e r s t a g e s . 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 methods i n problem 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 the study o f mathematics at s c h o o l . E d u c a t i o n a l S t u d i e s i n  M a t h e m a t i c s , 1971, 3, 133-4(D. B e c k e r , J . P. Research i n mathematics e d u c a t i o n : the r o l e o f t h e o r y and o f a p t i t u d e - t r e a t m e n t i n t e r a c t i o n s . J o u r n a l f o r Research i n Mathematics E d u c a t i o n , 1970, 1, 19-28. B e l l , E. T. Men o f mathematics. New Y ork: Simon and S c h u s t e r , 1937. 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 t e a c h i n g . (3rd ed New Y ork: M c G r a w - H i l l , 1972. Bloom, B. S., & B r o d e r , L. J . P r o b l e m - s o l v i n g p r o c e s s e s o f c o l l e g e s t u d e n t s . Supplementary E d u c a t i o n a l Monographs, Chicago: U n i v e r s i t y o f Chicago P r e s s , 1950, No. 73. Brown, I . B., & W a l t e r , M. I . The r o l e s o f the s p e c i f i c and the g e n e r a l cases i n problem p o s i n g . Mathematics  T e a c h i n g , 1972, 59, 52-4. Brumbaugh, R. S. P l a t o on the one; the h y p o t h e s i s i n the  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 - e x p e r i m e n t a l d e s i g n s f o r r e s e a r c h . C h i c a g o : Rand M ? N a l l y , 1963-D a v i s , G. A. P s y c h o l o g y o f problem s o l v i n g : t h e o r y and p r a c t i c e . New York: B a s i c Books, 1973. D e l l a - P i a n a , G. M., E l d r e d g e , G. M., & Worthen, B. R. • Sequence 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 and t r a n s f e r  l e a r n i n g . P a r t I: E x p e r i m e n t s i n d i s c o v e r y l e a r n i n g . Bureau 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 Utah, 1965. D e l l a - P i a n a , G. M., E l d r e d g e , G. M., & Worthen, B. R. Sequence 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 and t r a n s f e r . l e a r n i n g . P a r t I I : A p pendices : Tasks and t e s t s . Bureau 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 Utah, 1965. 139 D e l t h e i l , R. Analogy I n mathematics. I n P. Le L i o n n a i s ( E d . ) , Great c u r r e n t s o f m a t h e m a t i c a l t h o u g h t . New Yo r k : Dover, 1971. D e s s a r t , D. J . , & P r a n d s e n , H. Research on t e a c h i n g s e c o n d a r y - s c h o o l mathematics. I n R. M. W. T r a v e r s ( E d . ) , Second handbook o f r e s e a r c h on t e a c h i n g . C h i c a g o : Rand M P N a l l y , 1973-Dinsmore, L. A. An i n v e s t i g a t i o n o f the h e u r i s t i c s used by s e l e c t e d grade e l e v e n academic a l g e b r a s t u d e n t s i n the s o l u t i o n o f mathematics problems. U n p u b l i s h e d master's t h e s i s , U n i v e r s i t y o f B r i t i s h C olumbia, 1 9 7 1 -Downey, L. W. The secondary phase o f e d u c a t i o n . T o r o n t o : B l a i s d e l l , 19637 Duncker, K. On problem s o l v i n g ( T r a n s . S. L e e s ) . P s y c h o l o g i c a l Monographs, 1945, 5_8, (5, Whole number 270). E l d r e d g e , G. M. D i s c o v e r y v s . e x p o s i t o r y s e q u e n c i n g I n a programed u n i t on summing number s e r i e s . I n D e l l a - P l a n a , G. M., E l d r e d g e , G. M., & Worthen B. R., Sequence  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 . Bureau 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 Utah, P a r t I , 1965 ( a ) ; P a r t I I , 1965 ( b ) . Fennema, E. H. Mathematics l e a r n i n g and the s e x e s : a r e v i e w . J o u r n a l f o r Research i n Mathematics E d u c a t i o n , 1974, 5, 1 2 6 - 3 9 . Gagne, R. M. Human problem s o l v i n g : I n t e r n a l and e x t e r n a l e v e n t s . I n B. K l e i n m u n t z ( E d . ) , Problem s o l v i n g : r e s e a r c h , method, and t h e o r y . New York: W i l e y , 1966"! Gagne, R. M., & Brown, L. T. Some f a c t o r s i n the progr a m i n g o f c o n c e p t u a l l e a r n i n g . J o u r n a l o f E x p e r i m e n t a l - P s y c h o l o g y , 1961, 62 s 3 1 3-21. G a r r y , R. , & Ki n g s l e y , H. L. The n a t u r e and ( c o n d i t i o n s o f  l e a r n i n g . ( 3 r d ed.) New J e r s e y : P r e n t i c e - H a l l , 1970. G o l d b e r g , D. J . The e f f e c t s o f t r a i n i n g i n h e u r i s t i c methods on the a b i l i t y t o w r i t e p r o o f s i n number t h e o r y . Paper p r e s e n t e d at t h e annual m e e t i n g o f the N a t i o n a l C o u n c i l o f Teachers o f Mathem a t i c s , A t l a n t i c C i t y , A p r i l , 1 9 7 4 . Hadamard, J . An essay on. the p s y c h o l o g y o o f i n v e n t i o n i n the m a t h e m a t i c a l f i e l d . New J e r s e y : P r i n c e t o n U n i v e r s i t y P r e s s , 1949. 140 Hardy, G. H. A m a t h e m a t i c i a n ' s a p o l o g y . Cambridge: Cambridge U n i v e r s i t y P r e s s , 1940.. H e a t h , S i r T. L. A manual o f greek mathematics. London: O x f o r d U n i v e r s i t y P r e s s , 1931 . H e a t h e r s , G. Four p r o c e s s g o a l s o f e d u c a t i o n . C e n t e r f o r Research and Development on E d u c a t i o n a l D i f f e r e n c e s , H a r v a r d Graduate S c h o o l , 1965-Henderson, K. B. Research on t e a c h i n g secondary s c h o o l mathematics. I n N. L. Gage ( E d . ) , Handbook o f r e s e a r c h  on t e a c h i n g . Chicago: Rand M9Nally~^ 1 9 6 3 . Hsu, T s e - C h i , & F e l d t , L. S. The e f f e c t o f l i m i t a t i o n s o f c r i t e r i o n s c o r e v a l u e s on the s i g n i f i c a n c e l e v e l s o f the F - t e s t . American E d u c a t i o n a l Research J o u r n a l , 1 9 6 9 , 6., 5 1 5 - 2 7 . Jerman, M. I n d i v i d u a l i z e d i n s t r u c t i o n i n problem s o l v i n g • i n e l ementary s c h o o l mathematics. J o u r n a l f o r Research i n Mathematics E d u c a t i o n , 1 9 7 3 , 4 , 6-19• K a p l a n , A. The conduct o f i n q u i r y . San F r a n c i s c o : C h a n d l e r , 1964. K e r s h , B. Y.• The m o t i v a t i o n a e f f e c t s o f l e a r n i n g by d i r e c t e d d i s c o v e r y . J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , 1 9 6 2 , 5 3 , 6 5 - 7 1 . K e r s h , B. Y. D i r e c t e d d i s c o v e r y v s . programed i n s t r u c t i o n : a t e s t o f a t h e o r e t i c a l p o s i t i o n i n v o l v i n g e d u c a t i o n a l  t e c h n o l o g y . Oregon S t a t e System o f H i g h e r E d u c a t i o n , Monmouth, Oregon, 1964. K i l p a t r i c k , J . A n a l y z i n g the s o l u t i o n o f word problems i n mathematics: an e x p l o r a t o r y s t u d y . ( 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 , 1967 . No. 6 8 - 0 6 4 4 2 . K i l p a t r i c k , J . Problem s o l v i n g i n mathematics. Review o f E d u c a t i o n a l R e s e a r c h , 1 9 6 9 , 3 9 , 5 2 5 - 3 3 . K i n s e l l a , J . J . Problem s o l v i n g . I n , The t e a c h i n g o f secondary s c h o o l mathematics. Yearbook o f t h e N a t i o n a l  C o u n c i l o f Teachers o f Ma t h e m a t i c s , 1 9 7 2 , 3 3 , 241 - 6 6 . 1 4 1 K i r k , R. E. E x p e r i m e n t a l d e s i g n : p r o c e d u r e s f o r the b e h a v i o r a l s c i e n c e s . Belmont, C a l i f o r n i a : B r o o k s / C o l e , 1968. K l i n e , M. Mathematics I n w e s t e r n c u l t u r e . New York: O x f o r d U n i v e r s i t y P r e s s , 1 9 6 4 . L e a s k , I . C. The e f f e c t i v e n e s s 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 . U n p u b l i s h e d master's t h e s i s , U n i v e r s i t y of. B r i t i s h C olumbia, 1968. L i b e s k l n d , S. A development o f a u n i t on number t h e o r y f o r  use i n h i g h s c h o o l , based on a h e u r i s t i c approach. ( 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 W i s c o n s i n ) Ann Arbor::, 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 , 1971. No. 71-25201. L u c a s , J . F. An e x p l o r a t o r y study on the d i a g n o s t i c  t e a c h i n g o f h e u r i s t i c problem s o l v i n g s t r a t e g i e s i n  c a l c u l u s . ( 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 W i s c o n s i n ) 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 , 1972. No. 72-15368. L u c a s , J . F. The t e a c h i n g o f h e u r i s t i c p r o b l e m - s o l v i n g s t r a t e g i e s • i n e l ementary c a l c u l u s . J o u r n a l f o r Research i n Mathematics E d u c a t i o n , 1974, 5, 3 6 - 4 6 . MacPherson, E. D. A way o f l o o k i n g at the c u r r i c u l u m . The J o u r n a l o f E d u c a t i o n o f the F a c u l t y o f E d u c a t i o n o f  the U n i v e r s i t y o f B r i t i s h Columbia, 1970, 16, 44-51. MacPherson, E. D. U n p u b l i s h e d papers on the model, U n i v e r s i t y o f B r i t i s h C o lumbia, 1969-71 ( a ) . MacPherson, E. D. U n p u b l i s h e d papers on h e u r i s t i c s , U n i v e r s i t y o f B r i t i s h Columbia, 1969-71"(b). M a i e r , N. R. F. Problem s o l v i n g and c r e a t i v i t y i n i n d i v i d u a l s and groups. Belmont, C a l i f o r n i a : B r o o k s / C o l e , 1970. M a r a s c u i l o , L. A., & L e v i n , J . R. A p p r o p r i a t e p o s t hoc comparisons f o r i n t e r a c t i o n and n e s t e d hypotheses I n a n a l y s i s o f v a r i a n c e d e s i g n s : the e l i m i n a t i o n o f type IV e r r o r s . American E d u c a t i o n a l Research J o u r n a l , 1970, 7, 3 9 7 - 4 2 4 . M a x w e l l , J . C . The s c i e n t i f i c papers o f James C l e r k M a x w e l l , Volume I (W. D. N i v e n ( E d . ) ) , New York: Dover, 1890. 1 4 2 Moser, J . M. Mathematics by analogy.. The Mathematics  Teacher, 1 9 . 6 8 , 6 l , 3 7 4 - 6 . N u t h a l l , G., & Snook, I . Contemporary models o f t e a c h i n g . I n R. M. W.'Travers ( E d . ) , Second handbook o f r e s e a r c h on  t e a c h i n g . Chicago: Rand M 9 N a l l y , 1973. P i n g r y , R. Summary o f d i s c u s s i o n . J o u r n a l o f R e s e a r c h and  Development i n E d u c a t i o n , 1 9 6 7 , 1 , 4 4 - 7 . P o l y a , G. Mathematics and p l a u s i b l e r e a s o n i n g . Volume I . I n d u c t i o n and analogy i n mathematics. New J e r s e y : P r i n c e t o n U n i v e r s i t y Press"^ 1954 . ( (a) P o l y a , G. Mathematics and p l a u s i b l e r e a s o n i n g . Volume I I . P a t t e r n s o f p l a u s i b l e i n f e r e n c e . New J e r s e y : P r i n c e t o n U n i v e r s i t y P r e s s , 1954"^ (b) . P o l y a , G. How t o s o l v e i t . New Y o r k : Doubleday Anchor Books, 1 9 5 7 . P o l y a , G. M a t h e m a t i c a l disco.very. Volume I . New York: W i l e y , 196~2: P o l y a , G. M a t h e m a t i c a l d i s c o v e r y . Volume I I . New Y ork: W i l e y , 19 b"!T P o s t , T. R.• The e f f e c t s o f the p r e s e n t a t i o n o f a s t r u c t u r e  o f the p r o b l e m - s o l v i n g p r o c e s s upon p r o b l e m - s o l v i n g  a b i l i t y i n s e v e n t h grade mathematics• ( D o c t o r a l d i s s e r t a t i o n , I n d i a n a 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 , 1967. No. 6 8 - 0 4 7 4 6 . P r i c e , J . S. D i s c o v e r y : i t s e f f e c t on the achievement and  c r i t i c a l t h i n k i n g o f grade t e n g e n e r a l mathematics  s t u d e n t s . ( D o c t o r a l d i s s e r t a t i o n , Wayne S t a t e ) Ann A r b o r : U n i v e r s i t y M i c r o f i l m s , 1965. No. 6 6 - 0 1 2 4 6 . Ray, W. E. P u p i l d i s c o v e r y v s . d i r e c t e d i n s t r u c t i o n . J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 1 9 6 1 , 29., 2 7 1-80. R i c h a r d s , D. K. C u r r i c u l u m development at t h e l o c a l l e v e l . The B u l l e t i n o f the N a t i o n a l A s s o c i a t i o n o f Secondary  S c h o o l P r i n c i p a l s , 1968, 52, 3 8 - 4 7 . Romberg, T. A., & Vere D e v a u l t , M . Mathematics c u r r i c u l u m : needed r e s e a r c h . J o u r n a l of Research and Development i n E d u c a t i o n , 1967, 1 , . 9 5 - 1 1 2 . . 0 143 Rosenbloom, P. Problem making and problem s o l v i n g . I n The Conference Board o f the M a t h e m a t i c a l S c i e n c e s ( E d s . ) , The r o l e o f a x l o m a t i c s and problem s o l v i n g I n  mathematics. B o s t o n : G i n n , 1966. Rousseau, L. U n p u b l i s h e d papers on h e u r i s t i c s , U n i v e r s i t y o f B r i t i s h C olumbia, 1969-71. S c h a a f , 0. F. Student d i s c o v e r y o f a l g e b r a i c p r i n c i p l e s as a means o f d e v e l o p i n g a b i l i t y t o g e n e r a l i z e . ( D o c t o r a l d i s s e r t a t i o n , Ohio 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 today' problem: an e x e r c i s e i n p a t t e r n - s p o t t i n g . Mathematics T e a c h i n g , 1971, 56, 38-45. Shulman, 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 the t e a c h i n g o f s c i e n c e and mathematics. S c i e n c e Teacher, 1968, _35 34-38, 89. S m i t h , E. P., & Henderson, K. B. P r o o f . I n , The growth o f m a t h e m a t i c a l Ideas grades K - 12. Yearbook o f the  N a t i o n a l C o u n c i l o f Teachers o f Mathem a t i c s , 1959 , 24, 111-57. Snow, R. E. Theory 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 . ) , Second handbookoof r e s e a r c h on  t e a c h i n g . Chicago: Rand M ? N a l l y , 1973-Szabo, S. Some remarks on d i s c o v e r y . The Mathematics  Teacher, 1967, 60, 839-42 T o p i c s 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 problem s o l v i n g . Washington, D. C : The N a t i o n a l C o u n c i l o f Teachers o f Ma 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 r e s e a r c h . (3rd ed.) New York: M a c M l l l a n , 196*9. Webster's seventh new c o l l e g i a t e d i c t i o n a r y . (7th ed.) London: G. B e l l , 1967. W i l l s I I I , H. T r a n s f e r o f problem 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 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 as an I n s t r u c t i o n a l v a r i a b l e . ( 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 , 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 h y p o t h e s i s . E d u c a t i o n a l Resources I n f o r m a t i o n Center Document . R e p r o d u c t i o n S e r v i c e , P o s t O f f i c e Drawer 0, B e t h e s d a , M a r y l a n d , 20014. Document R e f e r e n c e : ED 003 340. Worthen, B. R. D i s c o v e r y v s . e x p o s i t o r y s e q u e n c i n g I n s i x weeks of 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 . , Sequence 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 . Bureau 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 H e u r i s t i c s o n l y - N e u t r a l V e h i c l e H - N 146 INTRODUCTION The o n l y purpose o f t h i s b o o k l e t i s t o t e a c h you how t o make d i s c o v e r i e s about 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 d e a l s w i t h 3 t o p i c s , w h i c h a r e : At 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 m a t h e m a t i c a l i n n a t u r e . However, the t o p i c s have been chosen 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 , and s e r v e no o t h e r purpose. 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 you a r e t r y i n g t o l e a r n the p r oblem s o l v i n g t e c h n i q u e s , n o t the c o n t e n t i t s e l f . The b o o k l e t w i l l t a k e 10 days t o c o m p l e t e . 3 days a r e s p e n t on each t o p i c , and 1 day a t the end i s used f o r r e v i e w . At the 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 the n e x t day. You may keep the b o o k l e t when i t i s c o m p l e t e . F e e l f r e e t o w r i t e i n i t . Make any comments t h a t you t h i n k w i l l be h e l p f u l t o you. 1) 2) 3) How t o o r g a n i z e s e a t i n g p l a n s , T i l i n g p r oblems, E l e c t r i c a l S w i t c h i n g N e t w o r k s . - 1 -147 SEATING PLANS F i r s t , l e t us explain the notation that w i l l be used. Notation A A A A A A A indicates that a ch a i r i s empty-Indicates that a ch a i r i s occupied. When a chai r i s occupied, but we do not know the name of the person s i t t i n g there; we w i l l use one of the l e t t e r s , x, y, or z inside the t r i a n g l e . indicates that the cha 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 attending a meeting, Mr. Abel ( l a b e l l e d a) and Mr. Brock ( l a b e l l e d b). Then... indicates that Mr. Abel Is occupying the ch a i r , indicates that Mr. Brock i s occupying the chair, Indicates that the cha i r Is occupied, but we do not know which of the men i s . s i t t i n g there, indicates 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 Is empty. - 3 -148 Exercise A meeting has been arranged between 4 d i r e c t o r s of a f i s h plant. They are Mr. Brace ( l a b e l l e d b), Mr. Drover (d), Mr. Goodyear (g), and Mrs. Murphy (m). The organizers have placed 4 chairs i n a row. Chair 1 Chair 2 Chair 3 Chair 4 A A A A Draw diagrams to represent each of the following seating arrangements. e.g. Chair 1 - Mr. Drover; Chairs 2 and 3 - occupied; Chair 4 - empty. The diagram would be A A A A Now draw the diagrams f o r the following: ( i ) Chair 1 - Mr. Brace; Chair 2 - Mrs. Murphy; Chairs 3 -nd 4 - occupied. ( i i ) Chair 1 - empty; Chair 2 - Mrs. Murphy; Chair 3 - Mr. Brace; Chair 4 - Mr. Goodyear. (iij) Chair 1 - Mr. Drover; A l l other chairs occupied. (iv) A l l 4 chairs empty. (v) Chair 1 - Mrs. Murphy; Chair 2 - Mr. Brace; Chair 3 - Mr. Goodyear; Chair 4 - Mr. Drover. (vi) Chair 1 - Mr. Goodyear; Chairs 2 and 3 - empty; Chair 4 - occupied. -4-149 speaker's desks audience Above i s the plan of the small meeting room i n a large 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 of head o f f i c e personnel. Since the audience does not know the three men, you have to place name plates on the desks. Exercise The three managers w i l l be seated at the speaker's desks. How many d i f f e r e n t ways could you seat the three men? Draw diagrams to represent each of the possible seating plans. e.g. One plan 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 only draw the c h a i r s . There i s no need to draw the room and speakers' desks. Now draw the other diagrams below. - 8 -150 Solution There are 5 other Plan i . A A A Plan A A A Plan 3. A A A Plan »• A A A Plan 5. A A A Plan 6- A A A In your s o l u t i o n 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 various plans Is 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 possible arrangements. It i s now that the f i r s t problem so l v i n g technique can be of help. This problem so 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 Involves the developing a systematic  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. This p a r t i c u l a r exercise i s f a i r l y simple, since It 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 discuss how a systematic approach can be used to l i s t the 6 plans given above. - 9 -151 The problem Is 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 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 i n chair 1. You could s t a r t with any of the 3, but i n t h i s 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 . This could be done i n 2 ways, leading to plans 1 and 2. Plan 1. A A Plan 2. ^ ^ ^ We have now considered a l l possible plans with Mr. Blade s i t t i n g i n c h a i r 1. The next move was to place one of the other managers i n c h a i r 1. In t h i s example Mr. Johnson was chosen. This gave A A A Again t h i s l e f t 2 people and 2 chairs to f i l l , leading to plans "3 and 4. Plan 3. £ \ > Plan 4. fi\ / \ / n \ F i n a l l y Mr. Hamilton was placed i n the f i r s t c h a i r , leading to the l a s t 2 plans. Plan 5. ^ A A Plan 6. y/n\ /i\ 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 i n the problem, and since the f i r s t c h air must be occupied we have written down a l l the cases. -10-152 Using a systematic 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 seating plans) has 2 major advantages:-(1) You can be sure that a l l possible cases have been  l i s t e d . You do not have to worry that some might have been l e f t out. For example, i n the previous exercise a l l the cases with Mr. Blade i n c h a i r 1 were written down before considering anyone else s i t t i n g there. (2) You can be sure that no cases have been repeated. This may not be very important when you have only 5 cases to consider, 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 to check that they are a l l d i f f e r e n t , and that you have not repeated some by mistake. In the previous exercise the systematic 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 possible seating plans, A systematic 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 prac t i c e i n l i s t i n g cases by a systematic method. - 11 -153 Since only 3 days are being spent on seating 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 . The next section w i l l introduce the second of the problem so l v i n g techniques, namely analogy, and further develop examination of cases. It may seem that a great deal of time i s being spent on t h i s p a r t i c u l a r step of examination of cases, namely l i s t i n g cases i n a systematic manner. As you work through the other sections you w i l l see the v i t a l importance of t h i s step i n s o l v i n g problems you have not seen before. In order to make the remainder of today's lesson more r e a l i s t i c the following problem w i l l be considered. However, as you attempt the- problems d_o not forget that  the objective of t h i s section i s to develop practice with a systematic approach to l i s t i n g cases and not with the  topic of seating plans. This i s part of a map of A f r i c a . The l e t t e r s stand f o r the following countries. S i e r r a Leone (s) ; L i b e r i a (t) ; Ghana (g) Ivory Coast ( i ) ; Togo (t) ; Mali (m) Upper Volta (u) ; Guinea (gu) -40-154 As you are aware, many developing countries have border arguments with t h e i r neighbors. This i s true of Afri c a n countries, so the nations decided to set up The Organization of A f r i c a n States to help the various nations s e t t l e disputes. The Organization also helps i n many other ways. Consider the following hypothetical problems that could a r i s e between the countries on the previous page. Note:- The solutions are not included. You w i l l receive them when you have completed the booklet. Exercise In each of the following l i s t a l l the seating arrangements that would s a t i s f y the conditions. Before l i s t i n g the arrangements give a_ b r i e f d e s c r i p t i o n of the systematic approach you intend to use. Spend some time thinking about the approach before s t a r t i n g to l i s t the cases. Try as many of the problems as.you can. DO NOT WORRY IF 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 to discuss the border disputes between the following 4 countries. S i e r r a Leone (s) L i b e r i a {£) Ivory Coast u ) Ghana C j ) Representatives of the 4 countries w i l l be seated as follows. A A A A ZnA Z * \ Since these countries are having problems i t i s advisable not to seat representatives from bordering states together. -41-155 T i l I n g Problems Today you w i l l s t a r t the second topic to be discussed i n t h i s booklet, namely t i l i n g problems. The idea of t i l i n g can be considered as a kind of extended jigsaw puzzle, where the question asked i s how many ways a p a r t i c u l a r object can be f i t t e d into a given space. In the f i r s t section you were introduced to the problem solv i n g technique of examination of cases. In t h i s section the second technique, namely analogy, w i l l be used. Furthermore, you w i l l see how examination of cases can be used to tes t conjectures (educated guesses)~~developed by analogy. As with seating plans, the topic of t i l i n g i s not important. It i s j u s t used to introduce analogy. On the previous page you saw a picture of a decorator with a t i l e to f i t into a wall. 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 wa l l . Let's consider the following problem. Problem A decorator i s hired to t i l e a bathroom. The owner of 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 into the w a l l . This 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 with a l l 3 sides being of equal length. The front of the t i l e i s divided into 3 sections, each being a d i f f e r e n t colour. The back of the t i l e i s covered with cement, so that i t w i l l s t i c k i n place. yellow blue red The decorator has completed the wall except f o r the space where the t i l e i s to f i t . He i s now faced with 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 To save you r e f e r r i n g back the 3 positions f o r the t i l e are given below. Ac A P o s i t i o n 1 Now complete the following t a b l e . o P o s i t i o n 2 Pos i t i o n 3 (a) I n i t i a l P o sition Rotation F i n a l P o s i t i o n 1 240° clockwise (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 po s i t i o n 1. Then think how i t moves when rotated through 240° i n a clockwise d i r e c t i o n . You obtain p o s i t i o n 3. ft £ Ac <r 7 > A I n i t i a l P o s i t i o n 240° clockwise F i n a l P o s i t i o n You would pla.ce a 3 i n the ta b l e . -48-157 Solution I n i t i a l P o sition Rotation F i n a l Position 1 240° clockwise 3 1 1 2 0 ° clockwise 2 2 1 2 0 ° counterclockwise 1 3 1 2 0 ° counterclockwise 2 2 3 6 0 ° clockwise 2 From the r e s u l t s above It would seem reasonable to conclude that there were no other positions for. the t i l e . However, can the decorator 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 side, -he could be reasonably sure. By r o t a t i n g the t i l e backward and forward he would soon conclude that there were only 3 p o s i t i o n s . However, i f he just f i t s the t i l e into the wall using 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 will now discuss a general approach that can be used to attack t h i s problem. In discussing how to approach t h i s problem from a systematic point of view we w i l l l a y the base f o r introducing the second problem sol v i n g technique, analogy. Imagine the decorator picks up the t i l e and f i t s i t into 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 tries, be the one below. Base Posit i o n The movement the decorator can use to move from one p o s i t i o n to another i s a rotation.--49-158 Read c a r e f u l l y • EXAMINATION OF CASES Step 1 I f possible, f i x an object as a s t a r t i n g point. Step 2 Apply any conditions that may be stated concerning the objects. Step _3 Apply movements, operations, or arrange objects, according to the problem. Use a systematic procedure to do t h i s . Step k_ Look f o r a pattern. Let us see how these steps apply to the two types of problem we have been dealing with, namely seating plans and t i l i n g problems. Examination of Cases as applied to t i l i n g problems  Step 1 Fix the t i l e i n a s t a r t i n g p o s i t i o n . Step 2 There were no conditions other than the t i l e must f i t into the w a l l . Step 3 T i l e i s rotated i n a systematic way. F i r s t looking at clockwise, then counterclockwise r o t a t i o n s . •Step k The f i r s t 2 rotations give 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 positions 2 and 3. Looking at the remainder of the table you can see a pattern; the same positions occur again i n a d e f i n i t e order. From t h i s you can conclude that no new positions w i l l a r i s e . Examination of Cases as applied to seating plans  Step 1 Fix a given person i n a seat. Step 2 Apply conditions. In some of the problems d i f f e r e n t people had to s i t i n p a r t i c u l a r seats. Step 3_ Other people were seated i n a systematic way. Step ii, Had not been discussed and was not appropriate to seating plans. If necessary the procedure was repeated f o r d i f f e r e n t people i n d i f f e r e n t seats. -53-159 Read the following care f u l l y At the beginning of the booklet i t was stated that the aim was to introduce you to some problem sol v i n g techniques. This page w i l l deal with analogy. We have used analogy on two d i f f e r e n t occasions. I. When we were f i r s t faced with a. new problem. The problem Involving the 2-sided t i l e was introduced. Analogy was used by looking at what approach had beeri taken with the one sided t i l e . This led to the Idea that rotations might be of value with the new problem. II. When we t r i e d to extend our knowledge of movements involving, the 2-sided t i l e . We had found the 6 d i f f e r e n t positions f o r the t i l e . We knew that i t was possible to move from the base p o s i t i o n to positions 2 and 4 with a sing l e movement. We t r i e d to extend our knowledge by asking i f i t were possible to move from the base p o s i t i o n to any other p o s i t i o n with, a single movement. Again we used analogy which enabled us to f i n d the other movements, which were d i f f e r e n t rotations and f l i p s . Notice how i n II what we had already found out about the movement of a 2-sided t i l e led us to use analogy to f i n d out more. At the moment the idea of analogy i s probably vague. Don't worry about t h i s . As you work through the remainder of the booklet, and have more practice with analogy, the idea 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 section of the booklet. The topic that w i l l be discussed i s that of e l e c t r i c a 1 switching networks. You are f a m i l i a r with switches from your everyday experience but you have probably never considered them as a topic i n mathematics. As we develop the topic you v ; i l l also have practice i n applying the problem so l v i n g techniques of analogy and examination of cases. Most switches, whatever there s i z e , shape, colour, etc., have one property i n common, namely they are e i t h e r open or closed. There i s no h a l f way stage. You might l i k e to think of a switch as an e l e c t r i c a l drawbridge. I f the bridge i s up (the switch i s open) the t r a f f i c cannot move (the current cannot flow). I f the bridge i s down (the switch i s closed) the t r a f f i c moves (the current flows). The p o s i t i o n of the switch i s referred to as the state of the switch. Let A be a switch. The following diagrams w i l l be used i n t h i s section. A The state of the switch i s unknown. We do not know i f the switch i s open or closed, and hence we do not know i f the current w i l l flow. The state of the switch i s open w i l l not flovi. The current The state of the switch i s closed. current w i l l flow. The - 91 -161 Exercise o A ? , B op(A 0 B Under what conditions w i l l the current flow through t h i s network? You should look at the d i f f e r e n t combinations of states and draw the networks. Try to l i s t the cases i n a systematic manner, - 100 -162 When using examination of cases to consider d i f f e r e n t networks, the following r e s u l t s were obtained. A B B A These networks are equivalent. The current flows i f A and B are closed. ( i i ) B A These networks are equivalent. The current flows i f AjOr B, or both are closed. Exercise Can you think of any mathematical operations that remind you of connecting switches? Hint:- Look at the underlined words. - 117 -163 Solution I f you have found operations that remind you of connecting switches, turn to the next page. I f not, the following comment may help. In the networks on the previous page the switches have been combined(joined). Also the order i n which they have been joined does not matter. That i s i s equivalent to A B B A - 118 -164 Solution continued You could have chosen many operations that remind you of connecting switches. For example, you could have thought of any of the following. We must decide which analogy to make. The analogy that w i l l be used i s with set theory. Let us consider the d e f i n i t i o n of Unicnand Intersection of Sets. I f X and Y are sets then Xu Y i s the set of elements i n X orY , or both. Xfl Y i s the set of elements i n X andY . Addition; M u l t i p l i c a t i o n ; Union of Sets; Intersection v II of Sets: If we consider switches l a b e l l e d A and B we have Current flows i f A or B, or both are closed. B Current flows i f A and B are closed. A B Exercise Which network would you draw to correspond to AyB? Which one to correspond to A r tB? - 119 -Solution For two switches, A and B, Ay B=Bu A Read the following care f u l l y Before continuing to develop the analogy with set theory, l e t us review the power of analogy. We were I n i t i a l l y faced with a set of problems involving switching networks, a new topic in mathematics. By t r y i n g to compare networks with another mathematical system, a system with which we are more f a m i l i a r , we are now looking at set .theory. It i s hoped that we w i l l be able to use many of the properties associated with set theory to help solve problems with networks. Hence, analogy has taken us from a t o t a l l y new topic, switching networks, into a more f a m i l i a r topic, set theory. As you work through the remainder of thi s section you w i l l see how set theory w i l l help us with network problems. As mentioned above we hope to use the properties of set theory to help with network problems. In p a r t i c u l a r , we would l i k e to use the properties associated with union and i n t e r s e c t i o n to f i n d equivalent networks. In order to do t h i s we w i l l have to examine the various properties associated with Union and Intersection. - 121 -APPENDIX.B H e u r i s t i c s p l u s c o n t e n t - N e u t r a l V e h i c l e HC - N 167 INTRODUCTION There are two purposes to t h i s b o o k l e t . To t e a c h you how t o make d i s c o v e r i e s about new m a t h e m a t i c a l t o p i c s , and t o show you how m a t h e m a t i c a l i d e a s can h e l p you s o l v e problems ta k e n from area s o t h e r than mathematics. The b o o k l e t d e a l s w i t h 3 t o p i c s , which a r e : 1) How t o o r g a n i z e s e a t i n g p l a n s , 2) T i l i n g problems, 3) E l e c t r i c a l s w i t c h i n g n e t w o r k s . At f i r s t g l a n c e t h e s e t o p i c s may not appear t o be v e r y m a t h e m a t i c a l In n a t u r e . However, as you work th r o u g h the b o o k l e t you w i l l see how the m a t h e m a t i c a l Ideas developed t h r o u g h the 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 take 10 days t o complete. 3 days are spent on each t o p i c , and 1 day a t the end i s used f o r r e v lev.. At the end o f most days, t h e r e Is 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 the n e x t day. You may keep the b o o k l e t when i t i s complete. F e e l f r e e t o w r i t e i n i t . Make any comments t h a t you t h i n k w i l l be h e l p f u l t o you. -1-168 SEATING PLANS You are probably wondering why t h i s topic i s important. The following are two examples of where seating plans can play, or played, a r o l e . At the highest l e v e l of diplomacy, approximately one year was spent discussing the seating arrangements for the Paris Peace Talks on Vietnam. The f i n d i n g of a seating plan that was acceptable to a l l the parties was e s s e n t i a l . This plan had to be found before the talks could s t a r t . The other example i s of a more personal nature. Some friends (or maybe your parents) have to arrange the seating plan f o r a wedding. You would prefer to s i t next to Jane, not Tom. There are a large number of guests and many of them have s i m i l a r requests concerning there own seats. Your friends are faced with the d i f f i c u l t task of t r y i n g to organize a seating plan that w i l l keep a l l the guests happy. These are only two of many situations where seating plans play an important r o l e . For example, international conferences, c i v i c luncheons, board meeting, etc. quite often involve s e t t i n g up seating plans. 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 'correct' place. Before looking at seating problems the next page w i l l introduce the notation that w i l l be used i n t h i s section. -3 -169 F i r s t , l e t us explain the notation that w i l l be used. Notation A A indicates that a chair i s empty-indicates that a chair i s occupied. When a chair i s occupied, but we do not know the name of the person s i t t i n g there, we w i l l use one of the l e t t e r s , x, y, or z inside the t r i a n g l e . A A indicates that the chair i s occupied by the person and we do know t h e i r name. For example, suppose there are two-lawyers attending a meeting, Mr. Abel ( l a b e l l e d a) and Mr. Brock ( l a b e l l e d b). Then... indicates that Mr. Abel i s occupying the chair. Indicates that Mr. Brock i s occupying the chair. indicates that the chair i s occupied, but we do not know which of the men i s s i t t i n g there indicates that the l e f t chair i s occupied by Mr. Abel, and the ri 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 of a f i s h plant. They are Mr. Brace ( l a b e l l e d b), Mr. Drover (d), Mr. Goodyear (g), and Mrs. Murphy (m). The organizers have placed 4 chairs in a row. Chair 1 Chair 2 Chair 3 Chair 4 A A A A Draw diagrams to represent each of the following seating arrangements. e.g. Chair 1 - Mr. Drover; Chairs 2 and 3 - occupied; Chair 4 - empty. The diagram would be A A A . A Now draw the diagrams for the following: ( i ) Chair 1 - Mr. Brace; Chair 2 - Mrs. Murphy; Chairs , 3 and 4 - occupied. ( i i ) Chair 1 - empty; Chair 2 - Mrs . Murphy; Chair 3 - Mr. Brace; Chair 4 - Mr. Goodyear. Chair 1 - Mr. Drover; A l l other chairs occupied. (iv) A l l 4 chairs empty. (v) Chair 1 - Mrs. Murphy; Chair 2 - Mr. Brace; Chair 3 - Mr. Goodyear; Chair -4 - Mr. Drover. (vi) Chair 1 - Mr. Goodyear; Chairs 2 and 3 - empty; Chair 4 - occupied. -5 -171 A A A speaker's desks audience Above i s the plan of the small meeting room i n a large company's head o f f i c e . Three branch managers, Mr. Blade, Mr. Johnson and Mr. Hamilton have been in v i t e d to t a l k to a group of head o f f i c e personnel. Since the audience does not know the three men, you have to place name plates on the desks. Exercise The three managers w i l l be seated at 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 represent each of the possible seating plans. where b stands for Mr. Blade, etc. As i n the above example, i n writing out your solution you need only draw the chairs. There i s no need to draw the room and speakers' desks. e.g. One plan would be Now draw the other diagrams below. - 9 -172 Solution Plan 1. Plan Plan Plan Plan Plan 6. 2. 3-A A A A A A A A A A A A A A A In your solution to this problem you probably had a l l the plans l i s t e d , although they may have been in a d i f f e r e n t order. For most people, l i s t i n g the various plans i s a matter of t r i a l and error. 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 possible arrangements. It i s now that the f i r s t problem solving technique can be of help. This problem solving technique i s c a l l e d examination of cases. The f i r s t stage of the technique involves the developing of a systematic method for writing down a l l the d i f f e r e n t cases. This p a r t i c u l a r exercise i s f a i r l y simple, since 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 discuss how a systematic 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 chairs. 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 tuat ion. A A A This le f t 2 people and 2 chairs to f i l l . This could be done in 2 ways, leading to plans 1 and 2. Plan 1. /j± / \ Plan 2. y £ \ / n \ / \ We have now considered a l l possible plans with Mr. Blade s i t t ing 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 le f t 2 people and 2 chairs to f i l l , leading to plans 3 and 4. Plan 3. / \ > ^ £ ± Plan 4. / \ F inal ly Mr. Hamilton was placed in the f i r s t chair , leading to the last 2 plans. Plan 5. / \ / \ A\ Plan 6. yfi^ / b \ / \ We have eliminated a l l the cases with any of the 3 men s i t t ing 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 systematic approach to l i s t i n g cases ( i n this example the cases were d i f f e r e n t seating plans) has 2 major advantages:-(1) You can be sure that a l l possible cases have been l i s t e d . You do not have to worry that some might have been l e f t out. For example, in the previous exercise a l l the cases with Mr. Blade i n chair 1 were written down before considering anyone else s i t t i n g there. (2) You can be sure that no cases have been repeated. This may not be very Important when you only have 6 cases to 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 that they are a l l d i f f e r e n t , and that you have not repeated some by mistake. In the previous exercise the systematic 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 possible seating plans. A systematic method w i l l help when you want to examine a l l cases. As you go through this booklet, you w i l l have more practice i n l i s t i n g cases by a systematic method. - 12 -175 Since only 3 days are being spent on seating 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 topic. As mentioned at the beginning of this section seating plans often play an important r o l e i n diplomatic c i r c l e s . We w i l l consider problems that could a r i s e from diplomatic s i t u a t i o n s . This i s part of a map of A f r i c a . The l e t t e r s stand f o r the following countries. S i e r r a Leone ( s ) ; L i b e r i a (t); Ghana (g) Ivory Coast ( i ) ; Togo (t) ; Mali (m) Upper Volta (u) ; Guinea (gu) - 38 -176 As you ere 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 d i s p u t e s w i t h t h e i r n e i g h b o r s . 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 arguments 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 S t a t e s t r i e s t o h e l p the c o u n t r i e s i n v o l v e d s e t t l e t h e i r d i s p u t e . 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 cage. Note:- The s o l u t i o n s are not 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 . E x e r c i s e 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 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 a r r a n g e m e n t s ) . T r y as many 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 t i m e . 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 f o u r c o u n t r i e s . S i e r r a Leone (S) _ R e p r e s e n t a t i v e s o f the f o u r c o u n t r i e s w i l l be s e a t e d as f o l l o w s : A L i b e r i a (€) I v o r y Coast (c) Ghana (^) S i n c e these c o u n t r i e s a re 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 t o g e t h e r . - 39 -177 T i l i n g Problems Today you w i l l start the second topic to be discussed in this booklet, namely t i l i n g problems. The Idea of t i l i n g can be considered as a kind of extended jigsaw puzzle, where the question asked In how many ways a p a r t i c u l a r object can be f i t t e d into a given space. As you work through this section you w i l l be introduced to the second problem solving 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 picture of a decorator with a t i l e to f i t into a wall. His problem i s how many di f f e r e n t ways can he f i t the t i l e into the wall. Lets consider the following problem. Problem A decorator is hired to t i l e , a bathroom. The owner of the house has bought a special triangular t i l e to f i t into the wall. This t i l e i s in fact an equalateral t r i a n g l e . That i s , a triangle with a l l 3 sides being of equal length. The front of the t i l e i s divided into 3 sections, each being a d i f f e r e n t colour. The back of the t i l e i s covered with cement so that i t w i l l s t i c k in place. y - yellow b - blue r - red The decorator has completed the wall except for the space where the t i l e i s to f i t . He is now faced with 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? - 41 -178 Solution The t i l e i s i n position 2. Exercise To save you r e f e r r i n g back the 3 positions f o r the t i l e are given below. * 8 C 6 A, A a Position 1 Position 2 Now complete the following table. Position 3 I n i t i a l Position Rotation F i n a l Position 1 240° clockwise (see below) 1 120° clockwise 2 120° counterclockwise 3 120° counterclockwise 2 360° clockwise (a) To f i n d the f i n a l p o s i t i o n in (a) imagine the t i l e i s i n po s i t i o n 1. Then think of how i t moves when rotated through 240° i n a clockwise d i r e c t i o n . You obtain p o s i t i o n 3. A. 8' In i t i a l Position <^VJ> .A 240° clockwise You would place a 3 i n the table. 5 F i n a l Position - 46 -179 Solution I n i t i a l Position Rotation Fi n a l Position 1 240° clockwise 3 1 1 2 0 ° clockwise 2 2 1 2 0 ° counterclockwise 1 3 1 2 0 ° counterclockwise 2 2 3 o 0 ° clockwise 2 From the r e s u l t s above i t would seem reasonable to conclude that there were no other positions for the t i l e . However, can the decorator 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 involving a t r i a n g u l a r t i l e decorated on 1 side, he could be reasonably sure. By r o t a t i n g the t i l e backward and forward he would soon conclude that there were only 3 positions. However, i f he just f i t s the t i l e into the wall using a t r i a l and error approach th i s would not help him in a more complex s i t u a t i o n . We will now discuss a general approach that can be used to attack t h i s problem. In discussing how to approach thi s problem from a systematic point of view we w i l l lay the base.for introducing the second problem solving technique, analogy. Imagine the decorator picks up the t i l e and f i t s i t into the wall. Let the base position, that is the f i r s t p o s i t i o n he tries, be the one below. . A . Base Position The movement the decorator can use to move from one po s i t i o n to another i s a r o t a t i o n . - kl -180 From the results It seems reasonable to conclude that there are no other positions for the t i l e . If you look at the f i n a l positions for the t i l e there is a pattern in the way they appear in the table. The technique of examination of cases has been of value In helping the decorator be certain that he had a l l possible positions for.the t i l e . By approaching the problem in a systematic manner ( l i s t i n g the various rotations in a systematic way) and observing that a pattern appeared in the f i n a l positions, he could be sure that he had l i s t e d a l l possible positions for the t i l e . The components that form the technique of examination of cases are: Step 1: If possible f i x an object as the s t a r t i n g point. Step 2: Apply any conditions that may be stated concerning the objects. Step _3_: Apply the movements, operations, or arrange objects, according to the problem. Use a systematic procedure to do t h i s . Step 4_: Look for a pattern. You w i l l probably have noted that this technique has been used in both the seating plans problems and those involving the t i l e . A l l the steps may not be appropriate to a l l problems. In dealing with seating plans we applied step 2 , applying p a r t i c u l a r seating conditions, but did not use the l a s t step, looking for a pattern. In the t i l i n g problem there are no conditions to apply, but we did look at a pattern. - 50 -181 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 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 e l e c t r i c a l s w i t c h i n g networks. You are f a m i l i a r w i t h s w i t c h e s from your e v e r y d a y e x p e r i e n c e , but you have p r o b a b l y not c o n s i d e r e d them as a t o p i c i n mathematics. As we l e v e l o c the 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 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 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, namely they are e i t h e r open or c l o s e d . 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 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 the b r i d g e i s up ( t h e s w i t c h i s open) the t r a f f i c cannot move (the 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 . L e t A be a s w i t c h . The f o l l o w i n g diagrams w i l l be used i n t h i s s e c t i o n . A The s t a t e o f the s w i t c h i s unknown. We do not 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 . r The s t a t e o f the s w i t c h i s open. The c u r r e n t w i l l not f l o w . A The s t a t e o f the s w i t c h i s c l o s e d . The - 85 -182 The following results have been obtained. (i) - * - £ ] — E — -A E + - 0 } B £ 0 " A These networks are equivalent. The current flows i f A and B are closed. ( i i ) i-m-i L_Q]_J B 1 - 0 - 1 B A These networks are equivalent. The current flows I f A _or_ B, or both are closed. Exercise Can you think of any mathematical operations that remind you of connecting switches. Hint: Look at the underlined words. - 112 -183 Solution continued You could have chosen many operations that remind you of connecting switches. For example, you could have thought of any of the following. Addition; M u l t i p l i c a t i o n ; Union of Sets; Intersection -f * U o f f s e t s : We must decide which analogy to make. The analogy that w i l l be used is with set theory. Let us consider the d e f i n i t i o n of Union and Intersection of Sets. If X and Y are sets then X y Y i s the set of elements in X, or Y, or both. XflY i s the set of elements In X and Y. If we consider switches la b e l l e d A and B we have A Current flows i f A, or B, or both are clos B Current flows i f A and B are closed. B Exercise Which network would you draw to correspond to AyB? Which one to correspond to A^ -j B? - 114 -184 APPENDIX C Content o n l y - N e u t r a l V e h i c l e C - N 185 INTRODUCTION The 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 t h r e e t o p i c s : 1) How t o o r g a n i z e s e a t i n g p l a n s , 2) T i l i n g p r o b l e m s , 3 ) E l e c t r i c a l s w i t c h i n g n e t w o r k s . At 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 m a t h e m a t i c a l i n n a t u r e . 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 as you t r y s e c t i o n s ( 2 ) and (3)> you 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 you s o l v e t h e p r o b l e m s . At t h e end o f t h e b o o k l e t you s h o u l d be a b l e t o s o l v e problems 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 days t o c o m p l e t e . 3 days a r e s p e n t on each t o p i c , and 1 day a t the end i s used f o r r e v i e w . At the 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 the n e x t day. You may keep the b o o k l e t when i t i s c o m p l e t e . F e e l f r e e t o w r i t e i n i t . Make any comments t h a t you t h i n k w i l l be h e l p f u l t o you. -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 examples o f where s e a t i n g p l a n s can p l a y , o r p l a y e d , a r o l e . At the 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 the s e a t i n g a rrangements f o r t h e P a r i s Peace 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 the p a r t i e s was e s s e n t i a l . T h i s p l a n had t o be found b e f o r e 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 y o u r p a r e n t s ) have t o a r r a n g e 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 t o s i t n e x t t o J a n e , n o t Tom. There a r e a l a r g e number o f g u e s t s and many o f them have 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 a r e f a c e d 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 t h e g u e s t s happy. These a r e 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 p l a n s p l a y an i m p o r t a n t r o l e . F o r example, I n t e r n a t i o n a l c o n f e r e n c e s , c i v i c l u n c h e o n s , b o a r d m e e t i n g , 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 . I n many ca s e s you 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 were n o t s i t t i n g I n t h e ' 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 problems t h e n e x t 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 . -3-187 F i r s t , 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. N o t a t i o n A A i n d i c a t e s t h a t a c h a i r i s empty. i n d i c a t e s t h a t a c h a i r i s occupied., When a c h a i r i s o c c u p i e d , b u t we do not know the name o f the p e r s o n 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 t r i a n g l e . A A A 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 and we do know t h e i r name. F o r example, suppose 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 ( l a b e l l e d b ) . Then... i n d i c a t e s t h a t Mr. A b e l 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 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 i n d i c a t e s t h a t t h e l e f t c h a i r i s o c c u p i e d by Mr. A b e l , and the 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 of a f i s h plant. They are Mr. Brace ( l a b e l l e d b), Mr. Drover (d), Mr. Goodyear (g), and Mrs. Murphy (m). The organizers have placed 4 chairs i n a row. Chair 1 Chair 2 Chair 3 Chair 4 A A A A Draw diagrams to represent each of the following seating arrangements. e.g. Chair 1 - Mr. Drover; Chairs 2 and 3 - occupied; Chair 4 - empty. The diagram would be A A A A Now draw the diagrams f o r the following: ( i ) Chair 1 - Mr. Brace; Chair 2 - Mrs. Murphy; Chairs 3 and 4 - occupied. ( i i ) Chair 1 - empty; Chair 2 - Mrs . Murphy; Chair 3 - Mr. Brace; Chair 4 - Mr. Goodyear. (HI) Chair 1 - Mr. Drover; A l l other chairs occupied. (iv) A l l 4 chairs empty. * (v) Chair 1 - Mrs. Murphy; Chair 2 - Mr. Brace; Chair 3 - Mr. Goodyear; Chair 4 - Mr. Drover. (vi) Chair 1 - Mr. Goodyear; Chairs 2 and 3 - empty; Chair 4 - occupied. - 5 -189 speaker's desks audience Above i s the plan of the small meeting room i n a large 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 of head o f f i c e personnel. Since the audience does not know the three men, you have to place name plates on the desks. Exercise The three managers w i l l be seated at the speaker's desks. How many d i f f e r e n t ways could you seat the three men? Draw diagrams to represent each of the possible seating plans e.g. One plan would be b\ / £ \ /h> where b stands f o r Mr. Blade, etc. As In the above example, i n w r i t i n g out your s o l u t i o n you need only draw the ch a i r s . There i s no need to draw the room and speakers' desks. Now draw the other diagrams below. - 9 -190 Solution There are 5 other seating plans, gi v i n g a t o t a l of 6 , Plan 1. / b \ / j \ / h \ (given i n exercise) Plan 2. ^ ^ / b \ Plan 3. Plan 4. A A P l a n s . A A A Plan 6 . The order i n which you l i s t e d the plans i s not important. Just check that you have l i s t e d a l l the d i f f e r e n t plans. I f i t did not matter where the men sat there would be no problem i n choosing a seating plan. Any of the 6 given above could be used. However, i n many problems there are r e s t r i c t i o n s as to where people s i t . For example, you f i n d that Mr. Blade and Mr. Hamilton do not l i k e each other. In th i s case you t r y to keep them happy by choosing a plan i n which they do not s i t next to each other. Exercise How many of the 6 arrangements can you choose and s t i l l keep Mr. Blade and Mr. Hamilton happy? L i s t them below. -10-1 9 1 Since only 3 days are being spent on seating 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 beginning of t h i s section seating plans often play an important role i n diplomatic c i r c l e s . We w i l l consider problems that could a r i s e from diplomatic s i t u a t i o n s . This i s part of a map of A f r i c a . The l e t t e r s stand f o r the following countries. S i e r r a Leone ( s ) ; L i b e r i a (£); Ghana (g) Ivory Coast ( i ) ; Togo (t) ; Mali (m) Upper Volta (u) ; Guinea (gu) -34-192 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 d i s p u t e s w i t h t h e i r n e i g h b o r s . T h i s i s t r u e o f A f r i c a n C o u n t r i e s , and when arguments 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 S t a t e s t r i e s t o h e l p the c o u n t r i e s i n v o l v e d s e t t l e t h e i r d i s p u t e . 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 t h e 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 a f t e r you have completed the b o o k l e t . E x e r c i s e In e a c h 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 p l a n s t h a t w i l l s a t i s f y t h e c o n d i t i o n s . T r y as many problems as you can. DO NOT WORRY IF YOU DO NOT COMPLETE ALL THE EXAMPLES. There i s no q u i z a t the 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 ) L i b e r i a (*) I v o r y Coast ( i ) Ghana (g) R e p r e s e n t a t i v e s o f the f o u r c o u n t r i e s w i l l be s e a t e d as f o l l o w s : A A A A i J S i n c e t h e s e 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 n o t t o s e a t r e p r e s e n t a t i v e s f r o m b o r d e r i n g s t a t e s t o g e t h e r . - 35 -1 9 3 For example, since Ghana and the Ivory Coast border each other, t h e i r representatives should not be seated next to each other. How many d i f f e r e n t seating arrangements can you f i n d such that no countries with a common border s i t together. e.g. A A A A i s an acceptable arrangement. 2. Can the 4 states In 1 . be seated round a c i r c u l a r table so that no countries with a common border s i t together? 3. Repeat problems 1 and 2 with S i e r r a Leone replaced by Guinea. Draw diagrams f o r the various seating plans. In the case of the c i r c u l a r table draw non-equivalent diagrams. 4. How many d i f f e r e n t arrangements can you f i n d i f the. following f i v e countries are involved^. Togo (t) Ivory Coast ( i ) Si e r r a Leone (s) Mali (m) Upper Volta (u) Again countries with a common border cannot s i t together. The table i s arranged as follows. 5. Try problem 4 using a c i r c u l a r t a b l e . -36-194 T i l i n g Problems Today you w i l l s t a r t the second topic to be discussed i n th i s booklet, namely t i l i n g problems. The idea of t i l i n g can be considered as a kind of extended jigsaw puzzle, where the question asked i s how many ways a p a r t i c u l a r object can be f i t t e d into a given space. On the previous page you saw a picture of a decorator with a t i l e to f i t into a wall. His problem Is how many d i f f e r e n t ways can he f i t the t i l e Into the wall. Let's consider the following problem. Problem A decorator Is hired to t i l e a bathroom. The owner of 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 into the wall. This t i l e i s i n fact 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 with a l l 3 sides being of equal length. The front of the t i l e i s divided into 3 sections, each being a d i f f e r e n t colour. The back of the t i l e i s covered with cement so that i t w i l l s t i c k i n place. - yellow - blue - red The decorator has completed the wall except f o r the space where the t i l e Is to f i t . He i s now faced with 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' -37-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 f o r the t i l e are given below. A B P o s i t i o n 1 P o s i t i o n 2 Now complete the following table. P o s i t i o n 3 (a) I n i t i a l P o sition Rotation F i n a l P o s i t i o n 1 240° clockwise (see below) 1 1 2 0 ° clockwise 2 1 2 0 ° counterclockwise 3 1 2 0 ° counterclockwise 2 3 6 0 ° clockwise To f i n d the f i n a l p o s i t i o n in (a) imagine the t i l e i s i n -p o s i t i o n 1 . Then think how i t moves when rotated through 240° i n a clockwise d i r e c t i o n . You obtain p o s i t i o n 3« " A * £ A 6*^ A 6 I n i t i a l P o s i t i o n 240° clockwise F i n a l P o s i t i o n You would place a 3 i n the.table. - 42 -196 Solution I n i t i a l P o sition Rotation F i n a l P o s i t i o n 1 240° clockwise 3 1 120° clockwise 2 2 120° counterclockwise 1 3 120° counterclockwise 2 2 3 6 0 ° clockwise 2 From these r e s u l t s It seems that no other positions f o r the t i l e e x i s t . However, on page 39 we asked how the decorator could be reasonably sure that no other positions e x i s t . That i s , how he could be reasonably c e r t a i n that he has a l l the positions f o r the t i l e . The following approach w i l l work. Imagine the decorator picks up the t i l e and f i t s i t into the wa l l . Let 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 P o s i t i o n 1 The movement the decorator can use to move from one p o s i t i o n to another i s a r o t a t i o n . 197 Today you w i l l be s t a r t i n g the f i n a l section of the booklet. The topic that w i l l be discussed i s that of e l e c t r i c a l switching networks. You are f a m i l i a r with switches from your everyday experience, but you have probably never considered them as a topic i n mathematics. Most switches, whatever t h e i r s i z e , shape colour, etc., have one property i n common, namely they are ei t h e r open or closed. There i s no h a l f way stage. You might l i k e to think of a switch as an e l e c t r i c a l drawbridge. If the bridge i s up (the switch i s open) the t r a f f i c cannot move (the current cannot flow). I f the bridge i s down (the switch i s closed) the t r a f f i c moves (the current flows). The po s i t i o n of the switch i s referred to as the state of the switch. Let A be a switch. The following diagrams w i l l be used in t h i s s e c t i on. ^ I ? I The state of the switch i s unknown. We do LLJ not know i f the switch i s open or closed, and A hence we do not know i f the current w i l l flow. ^—/? • The state of the switch i s open. The current w i l l not flow. ^ — The state of the switch i s closed. The current w i l l flow. - 78 -198 You are probably wondering why the topic i s important. Before discussing d i f f e r e n t networks l e t us consider the following s i t u a t i o n . A firm i s involved in manufacturing e l e c t r i c a l equipment. They are.Involved i n tendering f o r jobs with the government, large companies,' etc. C l e a r l y the lower t h e i r bid the more l i k e l y they are to be given the job. They would l i k e to keep the costs as low as possible, and i n p a r t i c u l a r use as l i t t l e equipment to do the job. Thus, i f they can design a system containing only 1 , 0 0 0 switches Instead of 2 , 0 0 0 they can save money. This booklet w i l l not consider networks in v o l v i n g thousands of switches. We w i l l discuss how to reduce the number of switches i n a network and s t i l l do the same job. Such a reduction would save the firm money and help them reduce t h e i r b i d . Although we do not discuss large networks the technique used could be used i n s i t u a t i o n s i n v o l v i n g a large number of switches. - 79 -199 Solution ( i ) A closed; B closed; op(A) open op(A) B A There i s an unbroken path, so the current flows ( i i ) A open; B closed; op(A) closed op(A) B -Y There i s an unbroken path, so the current flows Exer'cise A B op(/ ., o 0 B Under what conditions w i l l the current flow through t h i s network? You should look at d i f f e r e n t combinations of states and draw the networks. - 87 -2 0 0 Given 2 d i f f e r e n t switches there are 2 non-equivalent ways they can be connected. Let A and B be the switches. ( i ) They can be-connected i n such a way that the current only flows i f both A and B are closed. e'S- <• 9 9 A . B Network 1 We examined th i s network in a previous exercise and found that the current only flows i f A and B are closed. I f 2 switches, are connected i n t h i s manner we say that they are connected i n s e r i e s . We w i l l 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 . The expression A J - J B i s referred to as A i n t e r s e c t i o n B . It should be noted that the following network 1 / B A Network 2 has been shown to be equivalent to network 1 . The notation that would be used f o r network 2 i s Bf)A. The l e t t e r s have been a l t e r e d to indicate that the order of the switches has changed. ( i i ) The switches can be connected so that the current w i l l flow i f e i t h e r A , or B, or both are closed. e.g. ? A 9 > > B I f 2 switches are connected in 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 for union, and the expression A u B w i l l be referred to as A union B . - 103 -APPENDIX D H e u r i s t i c s o n l y - A l g e b r a i c V e h i c l e H - A 202 INTRODUCTION The o n l y purpose o f t h i s b o o k l e t i s t o t e a c h you how t o make d i s c o v e r i e s about 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 d e a l s w i t h ' 3 t o p i c s , w h i c h a r e : 1) P e r m u t a t i o n s , 2 ) Mappings, 3) B i n a r y N e t w o r k s . The t o p i c s have been chosen 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 , 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 you a r e t r y i n g t o l e a r n t h e problem s o l v i n g  t e c h n i q u e s , n o t the c o n t e n t i t s e l f . The b o o k l e t w i l l t a k e 10 days t o c o m p l e t e . 3 days a r e sp e n t on each t o p i c , and 1 day a t the end i s used f o r r e v i e w . A t the 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 a t r e t u r n e d the n e x t day. 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 . F e e l f r e e t o w r i t e i n i t . Make any comments t h a t you t h i n k w i l l be h e l p f u l t o you. - 1 -203 PERMUTATIONS We w i l l consider d i f f e r e n t permutations of l e t t e r s , numbers, etc. That i s , d i f f e r e n t ways of arranging objects. For example, the following would be considered 2 d i f f e r e n t permutations of the numbers 1,2, and 3. 1 2 3 and 3 1 2 Before discussing the topic i n d e t a i l , l e t us introduce the notation that w i l l be used. Notation In the following the word object could stand f o r a number, a l e t t e r , etc. It would depend on the p a r t i c u l a r exercise. w i l l be used as a placeholder. When drawn l i k e t h i s i t w i l l show that the p o s i t i o n indicated i s empty. Indicates that there i s an object i n that place, but we do not know what the object i s . We w i l l use one of the l e t t e r s x,y, or z inside the placeholder. Indicates that the l e t t e r a Is i n that place. indicates that there i s a 1 In the place. An example i s given on the next page. 204 Exercise Consider the set c o n s i s t i n g of b , j j and h. Using a l l three l e t t e r s , how many d i f f e r e n t possible permutations can you f i n d . e.g. One permutation would be b j h L i s t a l l the permutations, - 8 -205 Solution -There are 5 other permutations, g i v i n g a t o t a l of 6 . Permutation 1. b j h (given i n exercise) 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 In your s o l u t i o n to t h i s problem you probably had a l l the permutations 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 various permutations i s a matter of t r i a l and e r r o r . That Is, you t r y one permutation, and then another, hoping that in the end you w i l l have l i s t e d a l l possible combinations. It i s now that the f i r s t problem sol v i n g technique can be of help. This problem sol v i n g technique is. c a l l e d examination of cases The f i r s t stage of the technique involves the developing of a systematic method fo r wr i t i n g down the d i f f e r e n t cases. This p a r t i c u l a r exercise i s f a i r l y simple since i t Involves only 6 d i f f e r e n t \permutatIons. However, i t w i l l provide a good example with which to introduce the technique. The next two pages discuss how a systematic approach can be used to l i s t the 6 permutations given above. - 9 -206 We have eliminated a l l the cases with any of the 3 l e t t e r s i n pos i t i o n 1. But there were only 3 l e t t e r s i n the set being considered, so we must have written down a l l possible cases. Using a systematic approach to 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 that a l l possible cases have been  l i s t e d . You do not have to worry that some cases might have been l e f t out. For example, i n the previous exercise a l l the cases with b i n p o s i t i o n 1 were written down before considering any other l e t t e r In that p o s i t i o n . (2) You can be sure that no cases have been repeated. This may not be very important when you only have 6 cases to consider, 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 to check that they are a l l d i f f e r e n t , and that you have not repeated some by mistake. In the previous exercise the systematic approach assured us that the 6 cases (permutations) 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 possible permutations. A systematic approach 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 practice l i s t i n g cases by a systematic method. - 11 -207 Today we are going to discuss permutations i n v o l v i n g l e t t e r s . Furthermore, these l e t t e r s w i l l be organized i n a c i r c u l a r manner. You w i l l r e c a l l that the aim of t h i s booklet was to introduce some problem solving techniques. So far, only an introduction to examination of cases has been given, and in p a r t i c u l a r only the f i r s t stage has been discussed, namely systematically l i s t i n g cases. In dealing with today's material the value of systematically 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 topic of equivalent permutations i s discussed, you w i l l see that without a systematic approach the danger of repeating permutations without r e a l i z i n g i t i s very high. So the aim of today's material i s to give you more practice with systematically l i s t i n g cases. Exercise Consider the set containing the l e t t e r s a,b, and c. The 3 places that are to be f i l l e d are arranged i n a c i r c u l a r manner. , . C i r c u l a r permutation i s re.corded as A l l three places are occupied, b must be placed 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 condition? (No l e t t e r i s to be used more than once) Hint: Fix a i n one place f.nd then consider b. Then move a to another place,, etc. Eg. a i n PI, b Is P2, c i n P3. w i l l be recorded as (a j b j c ) - the placeholders have been l e f t out. L i s t the c i r c u l a r permutations. Note: Each c i r c u l a r permutation uses each l e t t e r once and only once. - 21 -208 Mappings 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 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 the problem 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 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 a n a l o g y , w i l l be used. F u r t h e r m o r e , you w i l l see how e x a m i n a t i o n o f  cases can be used 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 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 , the 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 a n a l o g y . • L e t us s t a r t by c o n s i d e r i n g t h e f o l l o w i n g p roblem. Problem The s e t o f o b j e c t s t h a t w i l l be examined c o n s i s t s o f A , B , C • We s t a r t w i t h the o b j e c t s i n t h e o r d e r A B C P o s i t i o n 1 Mapping 1 (M-j^ ) i s d e f i n e d as f o l l o w s : B I 1 I B C A P o s i t i o n 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 you a p p l y t o t h i s p o s i t i o n " the l e t t e r s a r e moved t o p o s i t i o n 2. - 40 -Solution A B C I { 1 B C A i i i C A B 209 M-M. Posi t i o n 3 When we apply M-^  B maps to C. Then applying again C maps to A. S i m i l a r l y f o r the other columns. When M-^  i s applied twice we denote i t by & . Exercise To save you r e f e r r i n g back the 3 positions and mapping are repeated below: A B C P o s i t i o n 1 A B C B C A Po s i t i o n 2 ^ C A B Po s i t i o n 3 B C A Now complete the following table. M. I n i t i a l P o s i t i o n Mapping F i n a l P o s i t i o n 2 M l & M l ^ s e e b e l o v ; ) 3 M l 3 M1 & Mx & Mx 1 M-^  & & M^  (a) To f i n d the p o s i t i o n i n (a) consider the fol l o w i n g P o s i t i o n 2 B I C A C I B A B C Mi P o s i t i o n 1 B i s mapped to C and then to A. C i s mapped to A and then to B, etc, 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 place a 1 i n the table. - 42 -2 1 0 Read c a r e f u l l y EXAMINATION OF CASES Step _1 I f p o s s i b l e , f i x an' o b j e c t as a s t a r t i n g p o i n t . S t e p 2 A p p l y any c o n d i t i o n s t h a t may be s t a t e d c o n c e r n i n g the o b j e c t s . S tep 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 ep j4 Look f o r a p a t t e r n . L e t us see hov; t h e s e s t e p s a p p l y t o the two t y p e s o f problem we have been d e a l i n g w i t h , namely p e r m u t a t i o n s and 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 . S t e p 2 There were no c o n d i t i o n s i n t h i s c a s e . Step 3 The mapping i s a p p l i e d i n a s y s t e m a t i c manner. Step 4 The f i r s t 2 a p p l i c a t i o n s (M-^ and M 1 & M^ g i v e you 3 d i f f e r e n t p o s i t i o n s . L o o k i n g a t the r e m a i n d e r 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 o r d e r . From t h i s you c o n c l u d e t h a t no new p o s i t i o n s w i l l a r i s e . E x a m i n a t i o n o f Cases as a p p l i e d t o p e r m u t a t i o n s . S t e p 1 F i x an o b j e c t ( l e t t e r o r number) I n the s t a r t i n g p o s i t i o n . S t e p 2 A p p l y c o n d i t i o n s . I n some o f the problems p a r t i c u l a r l e t t e r s o r numbers had t o be i n p a r t i c u l a r p o s i t i o n s . S t e p 3 O t h e r l e t t e r s , o r numbers, l i s t e d i n a s j ^ s t e m a t i c way. S t e p 4^ Had n o t been d i s c u s s e d and was not a p p r o p r i a t e t o p e r m u t a t i o n s . I f n e c e s s a r y the 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 the 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 you work t h r o u g h t h i s s e c t i o n remember t h a t the 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 pr o b l e m 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 . D e f i n i t i o n A b i n a r y s w i t c h i s one w h i c h can have two v a l u e s , 0 o r 1 . L e t A be a b i n a r y s w i t c h . The f o l l o w i n g w i l l be used i n t h i s s e c t i o n . i n d i c a t e s t h a t we do n o t 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 0 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 n e t w o r k i s any c o l l e c t i o n o f b i n a r y s w i t c h e s w i t h a s i n g l e i n p u t and o u t p u t . e.g. I n p u t A B F i g u r e 1 D Output where A, B, C and D a r e b i n a r y s w i t c h e s . I n the above n e t w o r k we do n o t know the v a l u e o f any o f the 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 ch one i s drawn 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 cases the f o l l o w i n g r e s u l t s have been o b t a i n e d . ( i ) A B B A These net w o r k s a r e e q u i v a l e n t . The n e t w o r k s a r e complete i f A and B b o t h have the v a l u e 1. ( i i ) ' . A B B 0 — 1 A These net w o r k s a r e e q u i v a l e n t . . The networks a r e complete i f A or B, o r b o t h have th e v a l u e 1. E x e r c i s e Can you t h i n k o f any 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 remind 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 ? H i n t : - Look a t the u n d e r l i n e d words. - 102 -213 S o l u t i o n 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 b i n a r y s w i t c h e s t u r n t o the n e x t page. I f n o t , the f o l l o w i n g comment may h e l p . In the b i n a r y networks on the p r e v i o u s page the 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 w h i c h the s w i t c h e s have been j o i n e d does not m a t t e r . That i s i s e q u i v a l e n t t o A B B A - 103 -214 S o l u t i o n c o n t i n u e d 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 b i n a r y s w i t c h e s . F o r example, you c o u l d have tho u g h t o f any o f the f o l l o w i n g . A d d i t i o n ; M u l t i p l i c a t i o n ; U n i o n o f S e t s ; I n t e r s e c t i o n ^ [j o f S e t s : We must d e c i d e w h i c h a n a l o g y t o make. The a n a l o g y t h a t w i l l be used i s w i t h s e t t h e o r y . 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 U n i o n 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 t h e n X (jY i s the s e t o f e l e m e n t s i n X, o r Y o r b o t h . X p Y i s the s e t o f e l e m e n t s i n X and Y. I f we c o n s i d e r b i n a r y s w i t c h e s l a b e l l e d A and B we have A I s c o mplete i f A, o r B, o r b o t h have the v a l u e 1. B A B . E x e r c i s e i s c o m plete i f A and B have the v a l u e 1. Which b i n a r y n e t w o r k would c o r r e s p o n d t o A i j B ? Which c o r r e s p o n d s t o A ^ B? - 104 -215 S o l u t i o n F o r two s w i t c h e s , A and B, A L> B — B ^ A Read the f o l l o w i n g c a r e f u l l y B e f o r e c o n t i n u i n g t o d e v e l o p t h e a n a l o g y w i t h s e t t h e o r y l e t us r e v i e w the power o f a n a l o g y . We were i n i t i a l l y f a c e d w i t h a s e t o f problems i n v o l v i n g b i n a r y n e t w o r k s , 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 system, 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 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 s e t t h e o r y t o h e l p s o l v e problems w i t h n e t w o r k s . Hence, a n a l o g y has t a k e n us from a. t o t a l l y new t o p i c , 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 you work t h r o u g h the r e m a i n d e r 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 n e t w o r k p r o b l e m s . As mentioned above we hope t o use the p r o p e r t i e s o f s e t t h e o r y t o h e l p w i t h n e t w o r k p r o b l e m s . 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 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 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 U n i o n and I n t e r s e c t i o n . - 106 -APPENDIX E H e u r i s t i c s p l u s c o n t e n t - A l g e b r a i c V e h i c l e HC - A 217 INTRODUCTION There are two purposes t o t h i s b o o k l e t . To t e a c h you how t o make d i s c o v e r i e s about new m a t h e m a t i c a l t o p i c s , and t o 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 t o p i c s . The 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 3 t o p i c s : 1) P e r m u t a t i o n s , 2) Mappings, 3) B i n a r y N e t w o r k s . The b o o k l e t w i l l t a k e 10 days t o c o m p l e t e , 3 days a r e sp e n t on each t o p i c , and 1 day i s used 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 quiz.. T h i s w i l l be marked and r e t u r n e d the n e x t day. 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 . F e e l f r e e t o w r i t e i n i t . Make comments t h a t you t h i n k w i l l be h e l p f u l t o you. - 1 -218 PERMUTATIONS We Y/Ill consider d i f f e r e n t permutations of l e t t e r s , numbers, e t c . That i s , d i f f e r e n t ways of arranging o b j e c t s . For example, the f o l l o w i n g would be considered 2 d i f f e r e n t permutations of the numbers 1,2, and 3« 1 2 3 and 3 1 2 Before d i s c u s s i n g the topi c i n d e t a i l , l e t us introduce the n o t a t i o n that w i l l be used. Notation In the f o l l o w i n g the word object could stand f o r a number, a l e t t e r , e t c . It would depend on the p a r t i c u l a r e x e r c i s e . w i l l be used as a placeholder. When • drawn l i k e t h i s i t w i l l show that 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 that there i s an object i n that place, but we do not know what the object i s . We w i l l use one of the l e t t e r s x, y, or 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 place. i n d i c a t e s that the l e t t e r a i s i n that place. An example .is given on the next page. - 2 -'• 219 E x e r c i s e . • . Consider the set c o n s i s t i n g of b , j , and h. Using a l l three 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 permutations can you f i n d . e.g. One permutation would be b j h L i s t a l l the permutations, _ ' , - 8 -220 Solution There are 5 other permutations, giv i n g a t o t a l of 6 . Permutation 1 . b j h (given i n exercise) 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 In your s o l u t i o n to t h i s problem you probably had a l l the permutations 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 various permutations i s a matter of t r i a l and e r r o r . That Is, you t r y one permutation, and then another, hoping that i n the end you w i l l have l i s t e d a l l possible combinations. It i s now that the f i r s t problem s o l v i n g technique can be of help. This problem sol 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 involves the developing of a systematic method f o r wr i t i n g down the d i f f e r e n t cases. This p a r t i c u l a r exercise i s f a i r l y simple since i t involves only 6 d i f f e r e n t permutations. However, i t w i l l provide a good example with which to introduce the technique. The next two pages discuss how a systematic approach can be used to l i s t the 6 permutations given above. - 9 -221 Today v:e a r e g o i n g t o d i s c u s s the idea, o f c i r c u l a r p e r m u t a t i o n s . When d e a l i n g w i t h 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 you w i l l see 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 c i r c u l a r p e r m u t a t i o n w i t h o u t r e a l i z i n g i t . E x e r c i s e 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 a,b, and 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 p e r m u t a t i o n i s r e c o r d e d as (A ' A , A ) A l l three places are occupied. How many d i f f e r e n t c i r c u l a r permutations can you find? e.g. a i n PI , b i n P2, and c i n P3 w i l l be recorded as ( a , b , c ) - the placeholders have been l e f t out. L i s 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 only once. - 20 -222 Ma pjxUvcs 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 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 the problem 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 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 a n a l o g y , w i l l be used. F u r t h e r m o r e , you w i l l see how t h e s e t e c h n i q u e s can be combined t o s o l v e mapping p r o b l e m s . L e t 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 p r oblem. P r o b l e m The s e t o f o b j e c t s t h a t w i l l be examined c o n s i s t s o f A , B , C We s t a r t w i t h the o b j e c t s i n the o r d e r A B C P o s i t i o n 1 Mapping 1 (M 1) i s d e f i n e d as f o l l o w s : A B C v l I C A P o s i t i o n 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 you a p p l y t o p o s i t i o n 1 the 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 B C A j I 4 M-Im-position 3 When vie apply B maps to C. Then applying M^  again C maps to A. S i m i l a r l y for the other columns. When M1 i s applied twice we denote i t by M^  & . Exercise To save you r e f e r r i n g back the 3 positions and are repeated below: A B • C Posi t i o n 1 A B C B C A Pos i t i o n 2 1 | L M, 4^  sf C A B Positi o n 3 B C A 1 Now complete the following table. I n i t i a l P o s i t i o n Mapping F i n a l P o s i t i o n 2 M l & M l ( s e e below) 3 M l 3 1 M]_ & M1 & Mx (a) To f i n d the p o s i t i o n i n (a) consider the following Pos i t i o n 2 41 B C A C 4^  t B A I M, B C J P o s i t i o n 1 - ho B i s mapped to C and then to A. C i s mapped to A and then to B, etc The f i n a l p o s i t i o n i s A B C, which i s pos i t i o n 1 . Therefore you would place a 1 i n the table. 224 Solution I n i t i a l Pos i t ion Mappings F i n a l Position Base 2 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 reasonable to conclude that there are no other p o s i t i o n s . I f you look: at the f i n a l positions there i s a pattern i n the way they appear i n the table. The technique of examination of cases has been of value i n making sure that you have a l l possible positions f o r the l e t t e r s . By approaching the problem i n a systematic manner ( l i s t i n g the various combined mappings i n a systematic way) and observing the pattern i n the f i n a l p ositions, you could be sure that a l l possible positions have been l i s t e d . The components that form the technique of examination of cases are: Step 1: I f possible, f i x an object as the s t a r t i n g point. Step 2: Apply any conditions that may be stated concerning the objects. Step 3} Apply the movements, operations, or arrange objects, according to the problem. Use a systematic procedure, to do t h i s . Step 4: Look f o r a pattern. _ 43 -225 Today you w i l l s t a r t the f i n a l section of the booklet. The topic that w i l l be discussed i s that of binary networks. As we develop the topic you v / i l l have an opportunity of using the problem s o l v i n g techniques of analogy and examination of cases i n so l v i n g problems. D e f i n i t i o n A binary switch i s one which can have two values, 0 or 1. Let A be a binary switch, section. The following w i l l be used i n t h i s indicates that we do not know i f the value of the binary switch i s 0 or 1. indicates that the binary switch has the value 1. indicates that the binary switch has the value 0, A binary network i s any c o l l e c t i o n of binary switches with a single input and output. e.g. Input B D f i g u r e 1 Output where A,B, C and D are binary switches. In the above network we do not know the value of a n y o f the binary switches. Therefore, each i s drawn with a |? j . - 72 -226 • When using examination of cases the following r e s u l t s have been obtained. ( i ) - G l B B A A These networks are equivalent. The networks are complete i f A and B both have the value 1. ( i i ) . 9 A | 9 j. B R A These networks are equivalent. The networks are complete i f A en? B, or both have the value 1, Exercise Can you think of any mathematical operations that remind you of connecting binary switches? Hint:- Look at the underlined words, - 90 -227 Exercise Find an equivalent network f o r op(B) 1 r • B ? Hint: The expression f o r t h i s network i s ( A uOp ( B ) " ) N C N ( B T . A " ) We can rewrite t h i s expression using the commutative properties Then use one of the d i s t r i b u t i v e properties and continue. - 105 -APPENDIX F Content o n l y - A l g e b r a i c V e h i c l e C - A 229 INTRODUCTION The 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 t h r e e t o p i c s : 1) P e r m u t a t i o n s , 2) Mappings, 3) B i n a r y Networks. At the end o f the b o o k l e t you s h o u l d be a b l e t o s o l v e  problems which d e a l v.11th 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 take 1 0 days t o complete. 3 days are spent on each t o p i c , and 1 day a t the end i s used f o r r e v i e w . At the 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 the next day. You may keep the b o o k l e t when i t i s complete. F e e l f r e e t o w r i t e i n i t . Make any coinments t h a t you t h i n k w i l l be h e l p f u l t o you. - 1 -230 E x e r c i s e 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 h. U s i n g 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 you f i n d . e.g. One p e r m u t a t i o n would be b j h L i s t a l l the p e r m u t a t i o n s , - 8 -2 3 1 S o l u t i o n 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 o f 6. ( g i v e n i n e x e r c i s e ) 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 The o r d e r i n which you l i s t e d the p e r m u t a t i o n s i n not i m p o r t a n t . 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 p e r m u t a t i o n s . E x e r c i s e How many o f the 6 p e r m u t a t i o n s can you choose such t h a t b and h are not next t o each o t h e r ? L i s t them below.. - 9 -232 Today we w i l l be d e a l i n g w i t h the 1dea o f c i r c u l e r  p e r m u t a t i o n s . In d i s c u s s i n g these problems 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 . E x e r c i s e 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 a,b, and c. The 3 p l a c e s t h a t are to be f i l l e d a re a r r a n g e d i n a c i r c u l a r ' manner. A l l t h r e e p l a c e s are o c c u p i e d . How many d i f f e r e n t , c i r c u l a r p e r m u t a t i o n s can you f i n d ? e.g. a i n P I , b i s P2 and c i n P3 w i l l be r e c o r d e d as ( a , b , c ) - the p l a c e h o l d e r s have been l e f t o u t . L i s t the c i r c u l a r p e r m u t a t i o n s below: 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 and o n l y once. - 15 -233 Mappings 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 mappings. Let 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 The s e t o f o b j e c t s t o be examined c o n s i s t s o f A , B , C . We s t a r t w i t h the o b j e c t s i n the o r d e r A B C P o s i t i o n 1 Mapping 1 (M-j^ ) i s d e f i n e d as f o l l o w s : A B C 4 I \ B C A P o s i t i o n 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 you a p p l y M^ t o t h i s p o s i t i o n the l e t t e r s a r e moved t o p o s i t i o n 2. - 34 -2 3 4 Solution A B C I i 1 I I 4 A M, B Position 3 When we apply M^  B maps to C. Then applying again C maps to A. S i m i l a r l y for the other columns. When M^  i s applied twice we denote i t by M^  & M^. Exercise To save you r e f e r r i n g back the 3 positions and mapping are repeated below: M, A B C Position 1 A B C B C A Position 2 i I C A B Posltion 3 B C A Now complete the following table I n i t i a l Position Mapping Fi n a l Position 2 M-j^  & Mx (see below) 3 M l 3 M1 & Mx & Mx 1 M1 & M1 & Mx (a) To find the position in (a) consider the following Position 2 C I B M A I B & M C Position 1 - 36 -E i s mapped to C and then to A. C i s mapped to A and then to B, etc The f i n a l position i s A E C, which i s position 1. Therefore you would place a 1 in the table. 235 Today you w i l l s t a r t the f i n a l section of the booklet. The topic that w i l l be discussed i s that of binary networks. As we develop the topic you w i l l look at the Idea of a complete network, and the idea of equivalent networks. D e f i n i t i o n A binary switch i s one which can have two values, 0 or 1. Let A be a binary switch. The following w i l l be used in this indicates that we do not know i f the value of the binary switch is 0 or 1. section. indicates that the binary switch has the value 1. indicates that the binary switch has the value 0. A binary network i s any c o l l e c t i o n of binary switches with a single input and output. Input figure 1 where A,B,C and D are binary switches. In the above network we do not know the value of any of the binary switches. Therefore, each one is drawn with a|? j . - 66 -236 Consider the pair of binary switches, A and B. (i) They can be connected as follows: —CD B Network 1 We examined this network i s a previous exercise and found that i t was complete i f A and B had the value 1. If 2 switches are connected i n this manner we w i l l denote the network by Af]B where n stands for in t e r s e c t i o n . The expression A ^ B i s referred to as A interse c t i o n B. It should be noted that the following network 9 -CO- Network 2 B has been shown to be equivalent to network 1. The notation that would be used for network 2 i s B pA. The l e t t e r s have been altered to indicate that the order of the switches has been changed. ( i i ) They can be connected as follows: 9 B We have also examined this network i n a previous exercise and found that It i s complete i f eithe r A, or B, or both, have the value 1. I f 2 switches are connected In this manner we w i l l denote the network by U B where u stands for union and the expression A {JB w i l l be referred to as A union B. - 82 -Exercise Here are some networks. Write the corresponding expressions for them. e . g . ) — r n — i r — 0 -A corresponds to AyB and corresponds to A<jC and combining these two we have corresponds to (At>B) ^  (AjjC) Now t r y the following, ( i ) B op(B) op(A) ( i i ) - 85 -D 238 APPENDIX G H e u r i s t i c s o n l y - Geometric V e h i c l e H - G 239 INTRODUCTION The o n l y purpose o f t h i s b o o k l e t i s t o t e a c h you how t o make d i s c o v e r i e s about 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 d e a l s w i t h 3 t o p i c s , w h i c h a r e : 1) Arrangements, 2) F l i p s and t u r n s , 3) Networks. • The t o p i c s have been chosen 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 , 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 you a r e t r y i n g t o l e a r n t h e problem s o l v i n g  t e c h n i q u e s , n o t the c o n t e n t i t s e l f . The b o o k l e t w i l l t a k e 10 days t o c o m p l e t e . 3 days a r e sp e n t on each t o p i c , and 1 day a t the end i s used f o r r e v i e w . At the 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 the n e x t day. You may keep the b o o k l e t when I t i s c o m p l e t e . F e e l f r e e t o w r i t e i n i t . Make any comments t h a t you t h i n k w i l l be h e l p f u l t o you. - 1 -240 E x e r c i s e C o n s i d e r 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 , square and c i r c l e . 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 arrangements 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 a r r a n g e m e n t s . - 8 -241 Solution There are 5 other arrangements, g i v i n g a t o t a l of 6 . Arrangement 1. Arrangement 2. O (given i n exerci A - O Arrangement 3-Arrangement 4. Arrangement 5 • Arrangement 6 . In your s o l u t i o n to t h i s the arrangements l i s t e d , although d i f f e r e n t order. For most people arrangements i s a matter of t r i a l 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 now that the f i r s t problem s o l v i n This problem s o l v i n g technique i s The f i r s t stage of the technique a systematic method f o r writing d problem you probably had a l l they may have been i n a l i s t i n g the various and e r r o r . That i s , you ther, hoping that i n the i b l e combinations. It i s g technique can be of help. c a l l e d examination of cases involves the developing of own the d i f f e r e n t cases. This p a r t i c u l a r exercise i s f a i r l y simple since i t involves only 6 d i f f e r e n t arrangements. However, i t w i l l provide a good example with which to introduce the technique. The next two pages discuss how a systematic approach can be used to l i s t the 6 arrangements given above. - 9 -242 F i n a l l y the c i r c l e was placed i n po s i t i o n 1, gi v i n g the l a s t 2 arrangements. Arrangement 5• Arrangement o. We have eliminated a l l the cases with any of the 3 figures i n p o s i t i o n 1. But there were only 3 figures i n the set being considered, so we must have written down a l l possible cases. Using a systematic approach to 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 that a l l possible cases have been  l i s t e d . You do not have to worry that some cases might have been l e f t out. For example, i n the previous exercise a l l the cases with the t r i a n g l e i n p o s i t i o n 1 were written down before considering any other figure i n that p o s i t i o n . (2) You can be sure that no cases have been repeated. This may not be very Important when you only have 6 cases to consider, 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 to check that they are a l l d i f f e r e n t , and that you have not repeated some by mistake. In the previous exercise the systematic approach assured us that the 6 cases (arrangements) 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 possible arrangements. A systematic approach 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 practice l i s t i n g cases by a systematic method. • — 11 -243 F l i p s and t u r n s 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 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 you were I n t r o d u c e d t o the problem 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 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. F u r t h e r m o r e , you w i l l see how e x a m i n a t i o n o f c a s e s can be used 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 guesses) d e v e l o p e d by a n a l o g y . As w i t h a r r a n g e m e n t s , the 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 t o i n t r o d u c e a n a l o g y . P r o b lem T h i s p r oblem 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 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 l e n g t h . . A 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 o f 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 . fl * 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 t h e same shape and s i z e . E x e r c i s e I n what o t h e r p o s i t i o n can t h e t r i a n g l e be f i t t e d i n t o the space? A - 42 -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 ref e r r i n g , back the 3 positions f o r the t r i a n g l e are given below. 2> R ft P o s i t i o n 1 P o s i t i o n 2 Now complete the following t a b l e . P o s i t i o n 3 (a) I n i t i a l Pos i t i o n Rotation. F i n a l Pos i t I o n 1 240° clockwise (see below) 1 1 2 0 ° clockwise 2 1 2 0 ° counterclockwise 3 1 2 0 ° counterclockwise 2 3 6 0 ° clockwise To f i n d the f i n a l p o s i t i o n in (a) Imagine that the t r i a n g l e i s in p o s i t i o n 1 . Then think hov; i t moves when rotated through 240° i n a clockwise d i r e c t i o n . You obtain p o s i t i o n 3 . <3 0 . 7 I n i t i a l P o s i t i o n 240° clockwise You would place a 3 i n the ta b l e . I F i n a l P o s i t i o n - 45 -245 S o l u t i o n I n i t i a l Pos i t l o n R o t a t i o n F i n a l Pos i t i o n 1 240° c l o c k w i s e 3 1 120 c l o c k w i s e 2 2 120 c o u n t e r c l o c k w i s e 1 3 120° c o u n t e r c l o c k w i s e 2 2 3o0° c l o c k w i s e 2 From the r e s u l t s above i t would 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 the t r i a n g l e . However, can you be s u r e ? C l e a r l y w i t h 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 l e ( t h a t can o n l y be r o t a t e d ) you can 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 r i a n g l e backward and f o r w a r d you would soon 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 you 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 would n o t h e l p 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 a p p r o a c h t h a t can be used t o a t t a c k t h i s p r oblem. 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 from 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 the base f o r i n t r o d u c i n g the second p r o b l e m s o l v i n g t e c h n i q u e , a n a l o g y . Imagine you s t a r t by p u t t i n g the t r i a n g l e i n the a p p r o p r i a t e space u s i n g one p a r t i c u l a r p o s i t i o n . T h i s w i l l be r e f e r r e d t o as t h e base p o s i t i o n . The movement t h a t can be used t o move f r o m 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 . - 46 -• 246 Read c a r e f u l l y EXAMINATION OF CASES Step ]L I f possible, f i x an object as a s t a r t i n g point. Step 2 Apply any conditions that may be stated concerning the objects. Step 3 Apply movements, operations, or arrange objects, according to the problem. Use a systematic procedure to do t h i s . Step 4 Look for a pattern. Let us see how these steps apply to the two types of problem we have been dealing with, namely arrangements, and f l i p s and turns. Examination o_f Cases as applied to f l i p s and turns. Step jL F i x the t r i a n g l e i n a s t a r t i n g p o s i t i o n . Step 2 There were no conditions other than the t r i a n g l e must *" f i t into a space the same size and shape. Step 3 Triangle i s rotated i n a systematic way. F i r s t looking at clockwise, then counterclockwise r o t a t i o n s . Step 4 The f i r s t two rotations give 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 positions 2 and 3-Looking at the remainder of the table you can see a pattern; the same positions occur again in a d e f i n i t e order. From t h i s you can conclude that no new positions w i l l a r i s e . Examination of Cases as applied to arrangements Step 1 Fix a figu r e i n a p a r t i c u l a r p o s i t i o n . Step 2 Apply conditions. In some problems d i f f e r e n t f igures "~ had to be placed i n p a r t i c u l a r p o s i t i o n s . Other figures were placed i n a systematic way. Had not been discussed and was not appropriate to arrangements. I f necessary the procedure was repeated f o r d i f f e r e n t figures i n d i f f e r e n t p o s i t i o n s . - 50 -2 4 7 Exercise Under what conditions w i l l t h i s network be complete? You should look at the d i f f e r e n t combinations of conditions and draw the networks. Try to l i s t the cases i n a systematic manner. - 96 -When using examination.of cases to consider d i f f e r e n t networks, the following r e s u l t s were obtained. ( i ) 248 CO— - c A B B These networks are equivalent. They are complete i f A and! B are connected. ( i i ) o A B B A These networks are equivalent. They are complete i f A, or B, oj? both are connected. Exercise Can you think of any mathematical operations that remind you of connecting switches? Hint:- Look at the underlined words. - 113 -APPENDIX H H e u r i s t i c s p l u s c o n t e n t - Geometric V e h i c l e H C - G 250 INTRODUCTION There are two purposes t o t h i s b o o k l e t . To t e a c h you how t o make d i s c o v e r i e s about new m a t h e m a t i c a l t o p i c s , and t o i n t r o d u c e you t o some 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 d e a l s w i t h the f o l l o w i n g 3 t o p i c s : ^ 1) Arrangements, 2) F l i p s and t u r n s , 3 ) N e t w o r k s . The b o o k l e t w i l l t a k e 1 0 days t o c o m p l e t e . 3 days a r e spent on each t o p i c , and 1 day a t the end i s used f o r r e v i e w . At the 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 the n e x t day. You may keep the b o o k l e t when i t i s c o m p l e t e . F e e l f r e e t o w r i t e i n i t . Make comments t h a t you 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 c o n s i d e r d i f f e r e n t a r r a ngements o f g e o m e t r i c f i g u r e s . F o r example, the f o l l o w i n g would be c o n s i d e r e d 2 d i f f e r e n t a r r a n g e m e n t s o f a t r i a n g l e and s q u a r e . — A - a n d —A-—L_— B e f o r e d i s c u s s i n g t he t o p i c i n d e t a i l , l e t us I n t r o d u c e some n o t a t i o n t h a t c o u l d be used. N o t a t i o n « r w i l l i n d i c a t e t h a t a p a r t i c u l a r p o s i t i o n i s empty. i n d i c a t e s t h a t t h e r e i s a g e o m e t r i c f i g u r e i n t h a t p o s i t i o n , but we do n o t know what x — — i 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 f o r t h i s s i t u a t i o n . i n d i c a t e s t h a t t h e r e i s a c i r c l e i n t h e p o s i t i o n An example i s g i v e n on the n e x t page. - 2 -252 Exe r c l s e C o n s i d e r 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 , s q u are and c i r c l e . 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 a r r a n g e m e n t s 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 a r r a n g e m e n t s . - 8 -F i n a l l y the c i r c l e 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 l a s t 2 a r r a n g e m e n t s . We have e l i m i n a t e d a l l the c a s e s w i t h any o f t h e 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 s e t 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 p o s s i b l e c a s e s . U s i n g a s y s t e m a t i c a p p r o a c h t o l i s t n g c a s e s ( i n t h i s example the ca s e s were arrangements) has two major a d v a n t a g e s : -( 1 ) You can 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 have been l i s t e d . You do n o t have t o w o r r y t h a t some ca s e s might have been l e f t o u t . F o r 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 ca s e s 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 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 s u r e t h a t no c a s e s have been 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 you o n l y have 6 c a s e s t o c o n s d i 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 n ot 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 n o t r e p e a t e d some by m i s t a k e . I n 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 a p p r o a c h a s s u r e d us t h a t the 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 you want t o examine a l l c a s e s . As you go t h r o u g h 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 c a s e s by a s y s t e m a t i c method. Arrangement ^ Arrangement 6 . „ - 11 -254 F l i p s a n d • t u r n s / 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 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 you were i n t r o d u c e d t o the problem 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 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 a n a l o g y , w i l l be used. F u r t h e r m o r e , you w i l l see how t h e s e t e c h n i q u e s can be combined t o s o l v e p r o b l e m s . T h i s p r oblem 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 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 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 t h e t r i a n g l e I n t o a space o f the same s i z e and shape. E x e r c i s e I n what o t h e r p o s i t i o n can the t r i a n g l e be f i t t e d i n t o the space? Problem l e n g t h . - 39 -255 S o l u t i o n The o t h e r p o s i t i o n f o r the t r i a n g l e i s i\L -\Q> You w i l l n o t i c e t h a t s i n c e the t r i a n g l e can o n l y be r o t a t e d a p o s i t i o n such as A QL _\B i s I m p o s s i b l e . 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 . However, can you 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 t h a t can be o b t a i n e d by r o t a t i n g . Soon v;e w i l l c o n s i d e r the way you 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 the 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 £ P o s i t i o n 1 P o s i t i o n 2 P o s i t i o n 3 E x e r c i s e I n each o f the f o l l o w i n g f i l l i n t h e m i s s i n g v e r t i c e s o r p o s i t i o n s . ( i ) A <"> A A ( i i i ) 6 (iv) c Z i A P o s i t i o n P o s i t i o n - 40 -256 From the 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 the t r i a n g l e . I f you l o o k a t t h e 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 t h e r e i s a p a t t e r n i n the way t h e y appear i n the 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 cases has been o f v a l u e i n h e l p i n g you be c e r t a i n t h a t a l l p o s s i b l e c ases have been l i s t e d . 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 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 appeared i n the f i n a l p o s i t i o n s , you 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 been l i s t e d . The components t h a t f o rm 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 a r e : S t e p 1: I f p o s s i b l e , f i x an o b j e c t as the s t a r t i n g p o i n t . S t e p 2: A p p l y any c o n d i t i o n s t h a t may be s t a t e d c o n c e r n i n g the o b j e c t s . S t e p _3_: A p p l y the 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 4_: Look f o r a p a t t e r n . You w i l l p r o b a b l y have n o t e d t h a t t h i s t e c h n i q u e has been used i n b o t h the 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 the s t e p s may not be a p p r o p r i a t e t o a l l p r o b l e m s . 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 use t h e l a s t s t e p , l o o k i n g f o r a p a t t e r n . In the t r i a n g l e p r o b lem 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 using examination of cases to consider d i f f e r e n t networks, the following r e s u l t s were obtained. (i) -ED— — E A B B These networks are equivalent. They are complete i f A and B are connected. ( i i ) ' • ? j—— ? B These networks are equivalent. They are complete i f A, £r_B, or both are connected. Exercise Can you think of any mathematical operations that remind you of connecting switches? Hint:- Look at the underlined words. - io5 -Solution continued 258 You could have chosen many operations that remind you of connecting switches. For example, you could have thought of any of the following. Addition; M u l t i p l i c a t i o n ; Union of Sets; 4- X U I n t e r s e c t i o n of Sets n We must decide which analogy to make. The analogy, that w i l l be used Is with set theory. Let us consider the d e f i n i t i o n of Union and Intersection of Sets. I f X and Y are sets then X(jY Is the set of elements i n X or Y.or both. Xj-jY i s the set of elements i n X and Y. I f we consider switches l a b e l l e d A and B we have Complete i f A or B, or both are connected. B B Complete i f A and B are connected, Exercise Which network would you draw to correspond to At/B? Which one to correspond to A flB? - 107 -259 The p r e v i o u s example has shown how the n o t a t i o n and p r o p e r t i e s o f s e t t h e o r y can be used t o f i n d e q u i v a l e n t n e t w o r k s . On the f o l l o w i n g pages you w i l l f i n d f u r t h e r examples f o r you t o t r y . The e x e r c i s e s a re NOT e a s y . They may t a k e a l o n g time and i n v o l v e many s t e p s . On one o f the two, pages f o l l o w i n g the e x e r c i s e t h e r e i s a h i n t t h a t s h o u l d h e l p you s o l v e the problem. O n l y l o o k a t the h i n t AFTER you have t r i e d the problem. You s h o u l d keep the pages w i t h the d i f f e r e n t p r o p e r t i e s handy so t h a t you can r e f e r t o them (pages 1 1 4 - 6 ) . There i s no q u i z , a t the end o f t o d a y ' s work. Do not wo r r y i f you do n o t complete a l l . t h e e x e r c i s e s . J u s t do as many as you can i n the time a v a i l a b l e . Remember t h a t some o f them i n v o l v e many s t e p s t o s i m p l i f y the s e t t h e o r e t i c e x p r e s s i o n . E x e r c i s e F i n d an e q u i v a l e n t n e t w o r k f o r the f o l l o w i n g . 2J •A B B N o t e : - 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 the netw o r k i s g i v e n o n the n e x t page. Check the e x p r e s s i o n you o b t a i n b e f o r e t r y i n g t o s i m p l i f y i t . - 121 -260 APPENDIX I Content o n l y - Geometric V e h i c l e C - G 261 INTRODUCTION The 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 t h r e e t o p i c s : 1) Arrangements, 2) F l i p s and t u r n s , 3) N e t w o r k s . A t the end o f the b o o k l e t you s h o u l d be a b l e t o s o l v e  problems w h i c h d e a l w i t h any o f the -three 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 10 days t o c o m p l e t e . 3 days a r e sp e n t on each t o p i c , and 1 day a t the end i s used f o r r e v i e w . At the 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 day. You may keep the b o o k l e t when I t I s c o m p l e t e . F e e l f r e e t o w r i t e i n i t . Make any comments t h a t you t h i n k w i l l be h e l p f u l t o you. - 1 -262 ARRANGEMENTS We w i l l c o n s i d e r d i f f e r e n t a r r a n g e m e n t s o f g e o m e t r i c f i g u r e s . F o r example, the f o l l o w i n g would be c o n s i d e r e d 2 d i f f e r e n t a r r a n g e m e n t s o f a t r i a n g l e and s q u a r e . HZI—"A~~ a n d ~ ~ A r " Q ~ -B e f o r e d i s c u s s i n g t he t o p i c I n d e t a i l , l e t us I n t r o d u c e some n o t a t i o n t h a t c o u l d be u s e d . N o t a t i o n c — ~ w i l l i n d i c a t e t h a t a p a r t i c u l a r p o s i t i o n i s empty. i n d i c a t e s t h a t t h e r e i s a g e o m e t r i c f i g u r e i n t h a t p o s i t i o n , b u t we do n o t know what x *~—-- i 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 f o r t h i s s i t u a t i o n . i n d i c a t e s t h a t t h e r e i s a c i r c l e i n t h e p o s i t i o n An example i s g i v e n on the n e x t page. - 2 -E x e r c i s e " _ Consider the set c o n s i s t i n g o f a t r i a n g l e , square and c i r c l e . U s ing a l l three 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 arrangements 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 -264 Solution There are 5 other arrangements, givi n g a t o t a l of 6 . Arrangement 1 .• Arrangement 27 (given i n exercise Arrangement 3 Arrangement 4. Arrangement 5 •* Arrangement 6 . A—Qr The order i n which you l i s t e d the arrangements i s not important. Just check that 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 of the 6 arrangements can you choose such that the t r i a n g l e and c i r c l e are not next to each other. L i s t them below. - 9 -265 F l i p s and turns Today you w i l l s t a r t the second topic to be discussed i n this booklet, namely t i l i n g problems. Let us s t a r t by considering the following problem. This problem w i l l be concerned with an equalateral 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 sides the same length. a 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 into a space the same size and shape? The t r i a n g l e can only be rotated, and cannot be turned over. These are two d i f f e r e n t ways of f i t t i n g the t r i a n g l e into a space of the same siz e and shape. Exercise In what other p o s i t i o n can the t r i a n g l e be f i t t e d into the space? Problem 6 c - 35 -S o l u t i o n The t r i a n g l e i s i n p o s i t i o n 2. E x e r c i s e To save you r e f e r r i n g back the 3 p o s i t i o n s f o r the t r i a n g l e a r e g i v e n below. a B C 266 8 P o s i t i o n 1 P o s i t i o n 2 Nov; complete the f o l l o w i n g t a b l e . P o s i t i o n 3 (a) I n i t i a l Pos i t i o n R o t a t i o n F i n a l P o s 1 t i o n 1 240° c l o c k w i s e ( see below) 1 1 2 0 ° c l o c k w i s e 2 1 2 0 ° c o u n t e r c l o c k w i s e 3 Q 120 c o u n t e r c l o c k w i s e 2 3 o 0 ^ c l o c k w i s e To f i n d t he 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. 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-ft . C A . <5 Vo, I n i t i a l P o s i t i o n 240 clockwise You would place a 3 i n the t a b l e . & F i n a l P o s i t i o n - 38 -Solution I n i t i a l P o sition Rotation F i n a l P o s i t i o n 1 240° clockwise 3 1 1 2 0 ° clockwise 2 2 1 2 0 ° counterclockwise 1 3 1 2 0 ° counterclockwise- 2 2 3 6 0 0 clockwise 2 From these r e s u l t s i t seems that no other positions 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 reasonably c e r t a i n that you have a l l possible positions f o r the t r i a n g l e . The following 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. P o s i t i o n 1 The"movement that can be used to move the t r i a n g l e from one p o s i t i o n to another 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 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 n e t w o r k s . As we d e v e l o p the t o p i c you w i l l l o o k a t the i d e a o f a complete  network, and the i d e a o f e q u i v a l e n t n e t w o r k s . D e f i n i t i o n A s w i t c h can be e i t h e r c o n n e c t e d o r d i s c o n n e c t e d . L e t A be a s w i t c h . The f o l l o w i n g diagrams w i l l be used i n t h i s s e c t i o n . ' A I n d i c a t e s t h a t we do not know i f the s w i t c h i s c o n n e c t e d o r d i s c o n n e c t e d . r A i n d i c a t e s t h a t the s w i t c h i s d i s c o n n e c t e d That i s , t h e r e i s a b r e a k i n the p a t h . i n d i c a t e s t h a t the s w i t c h i s c o n n e c t e d . A That i s , t h e r e i s no b r e a k i n the p a t h . A n e t w o r k i s any c o l l e c t i o n o f s w i t c h e s w i t h a s i n g l e i n p u t and o u t p u t p a t h . e.g. I n p u t P a t h A B D Output P a t h where A,B,C and D a r e s w i t c h e s . I n the above n e t w o r k we do not 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 E x e r c i s e In ..these examples t h e r e a r e p a i r s o f n e t w o r k s . In each case you are t o f i n d out i f the n e t w o r k s a r e e q u i v a l e n t . (0 _ j 7 ] _ . A B Network 1 ( i i ) ? A 9 A -CD-op(B) B - 0 ' B — H -A —4 9 LU-B Network 2 -0 A L _ { T ] . B Network 1 Netvjork 2 - 93 -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 B. ( i ) They can be c o n n e c t e d as f o l l o w s : B Network 1 We examined 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 found t h a t i t was complete i f A and B a r e b o t h c o n n e c t e d . I f 2 s w i t c h e s are c o n n e c t e d i n t h i s manner we w i l l denote the n e t w o r k by Afl B wheren s t a n d s f o r i n t e r s e c t i o n . The e x p r e s s i o n A f t B I s 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 f s h o u l d be n o t e d t h a t the f o l l o w i n g n e t w o r k A Network 2 has been 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 would be used f o r n e t w o r k 2 i s Bf)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 t h e o r d e r o f the s w i t c h e s has been changed. ( i i ) They can be c o n n e c t e d as f o l l o w s : _A B We have a l s o examined 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 found t h a t i t i s complete i f e i t h e r A, o r B, o r b o t h a r e c o n n e c t e d . 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 manner we w i l l denote t h e n e t w o r k by A p B wheretf s t a n d s f o r u n i o n and t h e e x p r e s s i o n A ^ B w i l l be r e f e r r e d t o as A u n i o n B. - 97 -271 APPENDIX J I n s t r u c t i o n s t o S u b j e c t s 272 I n s t r u c t i o n s t o s t u d e n t s Over the next few days you w i l l be i n v o l v e d i n an experi m e n t . The f o l l o w i n g i s i n f o r m a t i o n on the e x p e r i m e n t . D u r i n g the next 10 days, you w i l l spend one mathematics p e r i o d each day w o r k i n g t h r o u g h b o o k l e t s . At the b e g i n n i n g o f each day you w i l l be g i v e n the b o o k l e t t o work w i t h . You w i l l have a c e r t a i n number o f pages t o be co v e r e d each day. 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 you t o t r y , and I n most cases 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 at the s o l u t i o n . At the end o f most days t h e r e i s a s h o r t q u i s . The complete b o o k l e t w i l l be c o l l e c t e d at the end o f the p e r i o d , the 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 next day. Once the b o o k l e t s have been c o l l e c t e d t h e r e w i l l be no f u r t h e r work t o be done u n t i l t he n e x t day. There are many  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 a d i f f e r e n t one than y o u r f r i e n d s . I t i s v e r y i m p o r t a n t t h a t you DO NOT DISCUSS THE BOOKLET WITH ANYONE. PLEASE DO NOT DISCUSS ANY OF THE WORK. At t h e end of * the t e n days t h e r e w i l l be some t e s t s f o r you t o t r y . APPENDIX K A l g e b r a i c Test Read t o S t u d e n t s You are t o w r i t e down ANYTHING you can f i n d out NO MATTER HOW LITTLE OR UNIMPORTANT YOU MAY THINK IT IS I f you have any q u e s t i o n s about mod. 7 ask. Note t o t h e Teachers I w i l l 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 says t o you do you want t h i s ( t h a t i s s h o u l d I do...) say y e s . 275 Name • C l a s s 10 -"(PLEASE PRINT) B e f o r e a t t e m p t i n g the problem on t h e n e x t page t h i s i s t o remind, you o f modulo a r i t h m e t i c , and what i t means. Modulo 7 (Mod. 7) - ( C l o c k 7) Given any whole number, t o c o n v e r t i t t o mod. 7 you w i l l d i v i d e I t by 7 and c o n s i d e r the remainder. Fo r example: 15 = (2x7) + 1 The remainder a f t e r d i v i d i n g 1'5, by 7 i s 1, and we say t h a t 15 » 1 (Mod. 7) 13 = (1x7) + 6 The remainder a f t e r d i v i d i n g 13 by 7 i s 6, and we say t h a t 13 >6 (Mod. 7) Numbers l e s s than 6 remain unchanged. For example: 6 = (0x7) + 6 The remainder a f t e r d i v i d i n g 6 by 7 i s 6, and we say t h a t 5. ^5 (Mod. 7) You w i l l n o t i c e t h a t c e r t a i n numbers, f o r example 13 and 6, b o t h go t o 6 (mod. 7) We say t h a t 1 3 = 6 (Mod. 7) I f you have any problems w i t h mod. 7 p l e a s e ask b e f o r e you go onto the problem g i v e n on the n e x t page. - 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 i s @ a @ b = a x (b + 1) (Mod. 7) F o r example: 2 @ 3 = 2 x (3 + 1) (Mod. 7) = 8 (Mod. 7) = 1 (Mod. 7) 1 @ 1 = l x ( l + l ) (Mod. 7) = 2 (Mod. 7) Double A d d i t i o n The symbol used f o r t h i s i s *. a # b = 2 x (a + b) (Mod. 7) F o r example: 2 * 3 = 2 x ( 2 + 3 ) (Mod. 7) = 10 (Mod. 7) = 3 (Mod. 7) 0 * 1 = 2 x (0 + 1) (Mod. 7) = 2 (Mod. 7) 277 Name (PLEASE PRINT) C l a s s 10 -QUESTION What can you f i n d out about th e s e o p e r a t i o n s ? I n c l u d e ALL your w o r k i n g . APPENDIX L Geometric Test Read t o Stu d e n t s The q u e s t i o n s h o u l d r e a d : What can you f i n d out about s h o r t e s t r o u t e s and combining s h o r t e s t r o u t e s ? You are t o w r i t e down ANYTHING you can f i n d out NO MATTER HOW LITTLE OR UNIMPORTANT YOU MAY THINK IT IS Note t o the, Teacher I w i l l 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 says, t o you do you want t h i s ( t h a t i s s h o u l d I .do...) say y e s . 280 Name C l a s s 10 -(PLEASE PRINT) A t t a c h e d '-to t h i s sheet you 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 can be p l a y e d on t h e g r i d . R ules 1) To move from a p o i n t on the 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 at the same l e v e l , you may move a c r o s s the g r i d i n e i t h e r d i r e c t i o n , and/or up the g r i d - but you may NOT move down t h e g r i d . The f o l l o w i n g are examples o f r o u t e s t h a t can be used t o go from A t o B. 1 2) To move from a p o i n t on the 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 at the same l e v e l , you may move a c r o s s the g r i d I n e i t h e r d i r e c t i o n , and/or down the g r i d ; but you may NOT move up the g r i d . The f o l l o w i n g are examples o f r o u t e s t h a t can be used t o go from C to B. 3) You are l o o k i n g f o r the s h o r t e s t r o u t e s between p o i n t s The l e n g t h o f a r o u t e i s the number o f segments i t t a k e s t o go from one p o i n t t o t h e o t h e r . For example 3 1 3 A t o t a l o f 7 segments A t o t a l o f 3 segments T h i s p a t h i s the s h o r t e r o f t h e two. S h o r t e s t r o u t e s can be combined. F or example, the s h o r t e s t r o u t e from 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 from B to C. T h i s produces a r o u t e from A t o C. p l e a s e t u r n o v e r 281 Name (PLEASE PRINT) C l a s s 10 -QUESTION What can you f i n d out about combining s h o r t e s t r o u t e s ? I n c l u d e ALL your w o r k i n g . 282 1 F C 1 i 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 the 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 t o the e v i d e n c e t h a t a s t u d e n t used the h e u r i s t i c s (Analogy and E x a m i n a t i o n o f C a s e s ) . I t i s i m p o r t a n t t o note t h a t you are l o o k i n g f o r e v i d e n c e t h a t the s t u d e n t has t r i e d to_ ap 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, and not t h a t he has n e c e s s a r i l y used them w e l l . The f o l l o w i n g are the 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 o f the use o f h e u r i s t i c s . 1 - The s t u d e n t has used E x a m i n a t i o n o f Cases ONLY. He has NOT attempted 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 has examined cases AND l o o k e d f o r a p a t t e r n . 3 - The s t u d e n t has used Analogy ONLY. There i s NO e v i d e n c e t h a t he has t r i e d t o t e s t out h i s gue s s e s . 4 - The s t u d e n t has used Analogy and E x a m i n a t i o n o f Cases, but as DISTINCT t e c h n i q u e s . There i s NO ev i d e n c e t h a t the s t u d e n t has combined the two. 5 - The s t u d e n t has used b o t h t e c h n i q u e s (Analogy and E x a m i n a t i o n o f C a s e s ) . F u r t h e r m o r e , he has combined the two i n s o l v i n g problems. NOTE: When a s s i g n i n g a s t u d e n t ' s response t o a c a t e g o r y t h e r e must be WRITTEN EVIDENCE o f the a p p r o p r i a t e r e s p o n s e . F o r example, u n l e s s a s t u d e n t has w r i t t e n down some c a s e s , you cannot assume t h a t he has examined c a s e s . The f o l l o w i n g are examples o f the v a r i o u s c a t e g o r i e s f o r the ALGEBRAIC TEST. 0 - No e v i d e n c e o f h e u r i s t i c s a) B l a n k Papers. 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 work, e.g. A v e r b a l d e s c r i p t i o n o f %. Add one t o the second number and then m u l t i p l y by the f i r s t . D i v i d e your r e s u l t by 7 and t h i s g i v e s you the 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 say t h a t mod. 7. i s l i k e something he d i d at the 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 t r i e d d i f f e r e n t numbers t o see what happens. e.g. A s t u d e n t would have w r i t t e n down 1 § 3 = l x (3 + 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 paper would be i n t h i s c a t e g o r y i f he examined 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 . NOTE:- There are 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 c a s e s . The s t u d e n t c o u l d have examined them s y s t e m a t i c a l l y o r 'randomly'. I f he has examined them s y s t e m a t i c a l l y mark h i s response 1A, i f randomly IB. 2 - E x a m i n a t i o n o f Cases and l o o k i n g f o r a p a t t e r n Here-a s t u d e n t would have t r i e d d i f f e r e n t numbers and attempted t o l o o k f o r a p a t t e r n I n the r e s u l t s . e.g. a) 1 @ 1 = 1 x (1 + 1) = 2 m, _ & T h e r e f o r e the answer i s 2 @ l = 2 x ( l + l ) = 4 3 % 1 = 3 x (1 + 1) = 6 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 attempt t o a p p l y e x a m i n a t i o n o f cases and l o o k f o r a p a t t e r n . b) Any attempt t o examine cases 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 one o f t h e f i e l d p r o p e r t i e s such as a s s o c i a t i v i t y . c) 2 @ 3 = 1 T h e r e f o r e the o r d e r makes a d i f f e r e n c e . 3 @ 2 = 2 IMPORTANT: For a paper 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 have examined the cases and t h e n c o n c l u d e d t h a t a p a t t e r n e x i s t s . The s t a r t i n g p o i n t MUST BE THE EXAMINATION OF CASES. F o r example, i n c) one would have t o assume from the c o n t e x t t h a t the s t a r t i n g p o i n t was the e x a m i n a t i o n o f c a s e s . 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 would be c a t e g o r i z e d under 5. 286 3 - Analogy Here a s t u d e n t would have t r i e d t o make an analogy 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 paper t o f a l l i n t o t h i s c a t e g o r y the s t u d e n t must make the a n a l o g y , but NOT have used e x a m i n a t i o n o f cases t o t e s t o r make the an a l o g y . e.g. I n (a) and (b) the s t u d e n t must not.,suse cases t o t e s t h i s s t a t e m e n t s . a) The o p e r a t i o n i s commutative, l i k e a d d i t i o n . b) The o p e r a t i o n i n v o l v e s the 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 analogy w i t h the b o o k l e t s . There has t o be a m e a n i n g f u l attempt at an a l o g y . C l e a r l y a statement such as ' t h i s has something t o do w i t h p e r m u t a t i o n s ' i s not an attempt to a p p l y a n a l o g y . 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 something 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 would t r y t o r a i s e a q u e s t i o n by an 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 would have used b o t h h e u r i s t i c s , but have made no attempt t o combine them. e.g. He has examined cases (see 1) He has made statements c o n c e r n i n g p r o p e r t i e s (see 3) NOTE: The two h e u r i s t i c s have been used, but they have been a p p l i e d i n d e p e n d e n t l y . He c o u l d have t r i e d d i f f e r e n t cases and the n made a statement ( i n a d i f f e r e n t p a r t o f the answer) c o n c e r n i n g the i d e n t i t y . T h i s would have n'bthing t o do w i t h the cases he examined. T h i s c o m b i n a t i o n shows t h a t the s t u d e n t has used e x a m i n a t i o n o f cases and an a l o g y . However, the cases examined have n o t h i n g t o do w i t h the statement c o n c e r n i n g c o m m u t a t i v i t y . @ i s commutative i . e . 2 @ 1 = 4 3 @ 1 = 6 5 % 5 = 4 287 5 - Analaogy and E x a m i n a t i o n o f Cases (COMBINED) Here the s t u d e n t would have asked a q u e s t i o n and used e x a m i n a t i o n 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 , e t c . T h e r e f o r e , t h e r e i s an i d e n t i t y . NOTE: A l t h o u g h the c o n c l u s i o n c o n c e r n i n g the 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 the 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 cases. The s t u d e n t ' s problem l i e s i n h i s c o n c e p t i o n o f the concept o f an i d e n t i t y , not i n h i s use of the h e u r i s t i c . c) Any o t h e r paper than f o l l o w s the p a t t e r n . Asks q u e s t i o n ^ T r i e s cases t o answer i t Th i s i s u s u a l l y i n the form o f a q u e s t i o n c o n c e r n i n g the f i e l d p r o p e r t i e s . I n any paper 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 would expect many statements t h a t a re not a p p r o p r i a t e . . You may f i n d e v i d e n c e o f 3 - Analogy ONLY,, t o g e t h e r w i t h I r r e l e v a n t s t a t e m e n t s . I t i s a m a t t e r o f s h i f t i n g t h r o u g h the paper t o f i n d out 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 are examples o f the v a r i o u s c a t e g o r i e s f o r the 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) B l a n k p a p e r s . 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 you ta k e the l e n g t h o f the p a t h from A t o B s a y , and the n add t h e l e n g t h from B t o C say. c) A d e s c r i p t i o n o f the r u l e s . Here you are l o o k i n g at examples where the s t u d e n t has d e s c r i b e d the r u l e s i n h i s own words. 1 - E x a m i n a t i o n o f Cases ONLY The s t u d e n t would have t r i e d d i f f e r e n t paths t o see what happens. e.g. a) He has c a l c u l a t e d the 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 has combined d i f f e r e n t r o u t e s . c) He has examined d i f f e r e n t paths between the same two p o i n t s . (There are many paths between A and B) NOTE: As w i t h the a l g e b r a i c t e s t , r e sponses s h o u l d be c a t e g o r i z e d as e i t h e r 1A o r IB. 2 - E x a m i n a t i o n o f Cases and l o o k i n g f o r a p a t t e r n Here the s t u d e n t would have examined d i f f e r e n t p a t h s and l o o k e d f o r a p a t t e r n . a) He has examined cases and c o n c l u d e d t h a t t h e r e i s more than one s h o r t e s t r o u t e between two p o i n t s . b) He has examined cases and c o n c l u d e d t h a t combining two s h o r t e s t r o u t e s does not always g i v e a s h o r t e s t r o u t e . c) He has examined cases and c o n c l u d e d t h a t the s h o r t e s t r o u t e from X t o Y,say, I s t h e same as t h a t from Y to X. 3 - Analogy ONLY .  Here a s t u d e n t would have t r i e d t o make an analogy 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 paper t o f a l l i n t o t h i s c a t e g o r y the s t u d e n t must make the a n a l o g y , but NOT have used e x a m i n a t i o n o f cases t o t e s t out o r make the an a l o g y . 289 a) He makes statements l i n k i n g combining r o u t e s w i t h n etworks. Statements c o n c e r n i n g the f a c t t h a t combining r o u t e s t o form new r o u t e s i s l i k e combining networks (or s w i t c h e s ) t o form new ne t w o r k s . b) A n a l o g i e s between combining paths and. o p e r a t i o n s such as a d d i t i o n , u n i o n , e t c . 4 - -Analogy and E x a m i n a t i o n o f Cases (DISTINCT) He c o u l d have examined d i f f e r e n t r o u t e s (see 1) He c o u l d have made a n a l o g i e s (see 3) These would appear i n d i f f e r e n t p a r t s o f the s t u d e n t s paper w i t h no attempt t o use the h e u r i s t i c s t o g e t h e r . 5 - Analogy and E x a m i n a t i o n o f Cases (COMBINED) a) Does combining 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 by a counterexample to prove t h i s s t a t e m e n t . 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 . Note t h a t yes i s i n c o r r e c t . However, the problem I s not w i t h the s t u d e n t s use o f the h e u r i s t i c s , but w i t h h i s c h o i c e o f examples. b) I s the s h o r t e s t r o u t e unique? NO - f o l l o w e d by counterexample. c) I s f i n d i n g s h o r t e s t r o u t e symmetric? YES - f o l l o w e d by examples. e.g. Length AB = Length BA ' d) 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. paper was handed out the s t u d e n t s were t o l d t h a t the q u e s t i o n s h o u l d r e a d : What can you f i n d out about s h o r t e s t r o u t e s , and combining s h o r t e s t r o u t e s ? I n c l u d e ALL y o u r w o r k i n g . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0055064/manifest

Comment

Related Items