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

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THE EFFECT OF PROBLEM CONTEXT UPON THE PROBLEM SOLVING PROCESSES USED BY FIELD DEPENDENT AND INDEPENDENT STUDENTS: A CLINICAL STUDY by R i c k N. B l a k e B.A., C h i c o S t a t e C o l l e g e , 1964 M.S., U n i v e r s i t y o f N o t r e Dame, 1969 M.A., Wa s h i n g t o n S t a t e U n i v e r s i t y , 1972 A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF EDUCATION i n t h e F a c u l t y o f E d u c a t i o n We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA F e b r u a r y , 1976 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 ree tha t 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 tudy . 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 pu rpo se 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 tha t 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 Eduction The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date March 2. 1976 i i Abstract Research Supervisor: Dr. G a i l J . S p i t l e r I t was the purpose of t h i s study to analyze the processes students used i n s o l v i n g mathematical word problems and to determine the e f f e c t of problem context on these processes. A concomitant purpose was to determine whether students who d i f f e r i n t h e i r degree of f i e l d independence, d i f f e r i n the processes they use i n s o l v i n g mathematical problems. Forty subjects of both sexes, who were completing a grade eleven academic mathematics program were randomly selected from 14 Algebra I I cl a s s e s . The subjects were of average a b i l i t y f o r students on t h i s program (IQ range 115-125). The subjects were teste d i n d i v i d u a l l y , using Witkin's Embedded Figures Test. They were matched on t h i s v a r i a b l e and randomly assigned to one of two groups. One group was given problems using a r e a l world s e t t i n g , while the other group was given the same problems using a mathematical s e t t i n g . The subjects were i n d i v i d u a l l y interviewed and asked to think aloud as they solved f i v e mathematical word problems. To analyze the problem s o l v i n g procedures the subjects' tape recorded protocols of the interviews were coded by means of a system based on a model of mathematical problem s o l v i n g by MacPherson. The coding system had two parts: a coding matrix used to se q u e n t i a l l y code problem s o l v i n g behavior, and a summary sheet f o r compiling information obtained from the coding matrix as w e l l as other data r e l a t e d to the subjects' problem s o l v i n g behavior. . The coding system had intercoder r e l i a b i l i t y of .80 and intracoder r e l i a b i l i t y of .86. i i i Problem context proved to be unrelated to the h e u r i s t i c s used. Both the t o t a l number of h e u r i s t i c s used and the number of d i f f e r e n t h e u r i s t i c s used were not influenced by problem s e t t i n g . Subjects working problems i n the math world s e t t i n g had a s l i g h t l y more d i f f i c u l t time understanding the problems, but performed as w e l l as the other group. Within the IQ range, 115 to 125, f i e l d independence had a marked e f f e c t on the use of h e u r i s t i c s and on the number of correct solutions obtained. The f i e l d independent subjects used a greater v a r i e t y of h e u r i s t i c s ( r = .33) i n attacking and s o l v i n g problems. They were more w i l l i n g to change t h e i r mode of attack ( r = .27) and they obtained a greater number of correct solutions ( r = .30) than t h e i r f i e l d dependent counterparts. Both t o t a l number of h e u r i s t i c s as w e l l as number of d i f f e r e n t h e u r i s t i c s , accounted f o r a s i g n i f i c a n t (P <^.0l) amount of the variance i n number of correct s o l u t i o n s . In p a r t i c u l a r , h e u r i s t i c s accounted f o r an a d d i t i o n a l 2lf0 of variance not accounted f o r by core procedures (algorithms, diagramming, equations, and guessing). The h e u r i s t i c s used by the subjects i n t h i s study added to t h e i r a b i l i t y to solve problems beyond t h e i r mathematical core knowledge. The number of times a subject attempted to solve a problem was found to be unrelated to obtaining a correct s o l u t i o n , while changing one's mode of attack i n s o l v i n g a problem was s i g n i f i c a n t l y (P <^.0l) r e l a t e d to obtaining a correct s o l u t i o n . When the subjects were grouped by problem context, both groups exhibited the same general pattern of problem s o l v i n g behavior. The r e a l world students expressed concern f o r solutions obtained using h e u r i s t i c s while those students i n the math world s e t t i n g expressed none. However, expression of concern f o r s o l u t i o n was unrelated t o i v the correctness of the s o l u t i o n . When grouped by f i e l d independence a d i f f e r e n c e was observed i n the o v e r a l l pattern of sequential moves i n the problem s o l v i n g process. These d i f f e r e n c e s among the groups were not tested f o r s t a t i s t i c a l s i g n i f i c a n c e . The f i e l d independent student moved more f r e e l y across a l l procedures coded. He was more concerned with h i s work, and c o n t i n u a l l y checked both the procedures he was using as w e l l as h i s s o l u t i o n . The f i e l d independent student was more w i l l i n g to check h i s work, us u a l l y by r e t r a c i n g h i s steps, whereas the f i e l d dependent student u s u a l l y reread the problem. •v: TABLE OF CONTENTS Page L I S T OF TABLES vi'i LIST OF FIGURES v i i i ACKNOWIiEDGEMENT x-C h a p t e r I . THE PROBLEM 1 P u r p o s e o f t h e S t u d y 2 A n a l y z i n g t h e P r o b l e m S o l v i n g P r o c e s s 2 The Need f o r a M o d e l o f P r o b l e m S o l v i n g 4 MacPherson's M o d e l f o r M a t h e m a t i c a l P r o b l e m S o l v i n g 5 D e f i n i t i o n o f Terms 20 The P r o b l e m 22 G e n e r a l H y p o t h e s e s 23 S i g n i f i c a n c e o f t h e S t u d y 24 A s s u m p t i o n s and L i m i t a t i o n s 25 I I . REVIEW OF RELATED LITERATURE 27 M o d e l s o f P r o b l e m S o l v i n g 27 F i e l d Independence 42 I I I . PROCEDURES 51 S u b j e c t s 51 P i l o t S t u d y 52 The C o d i n g System 55 The I n t e r v i e w P r o c e d u r e 61 The C o d i n g P r o c e d u r e 67 v i C h a p t e r Page I V . ANALYSIS AND RESULTS 73 R e s e a r c h H y p o t h e s e s 73 Method o f A n a l y s i s 74 R e s u l t s o f A n a l y s i s - P r o b l e m C o n t e x t 80 R e s u l t s o f A n a l y s i s - F i e l d Independence 93 R e s u l t s o f A n a l y s i s - C o r e and H e u r i s t i c P r o c e d u r e s .... 100 V. CONCLUSIONS 114 Summary o f t h e E x p e r i m e n t a l S t u d y « » 114 Summary and C o n c l u s i o n s f o r t h e Mo d e l and C o d i n g System .. 119 I m p l i c a t i o n s f o r E d u c a t i o n 122 L i m i t a t i o n s o f t h e S t u d y 125 I m p l i c a t i o n s f o r R e s e a r c h 127 LITERATURE CITED 130 A p p e n d i x A. PROBLEMS USED I N PILOT STUDY 134 B. PROBLEMS USED I N THE STUDY 137 C. THE CODING SYSTEM 141 D. HISTOGRAMS OF SELECTED VARIABLES 152 v i i L I S T OF TABLES T a b l e Page 1. Types and Numbers o f D i s a g r e e m e n t s Between Two D i f f e r e n t C o d e r s 70 2. Types and Numbers o f D i s a g r e e m e n t Between Code and Recede 71 3. Means and S t a n d a r d D e v i a t i o n s f o r Me a s u r e s o f H y p o t h e s i z e d V a r i a b l e s 80 4. R e s u l t s o f R e g r e s s i o n A n a l y s i s w i t h P r o b l e m C o n t e x t as Independent V a r i a b l e . 82 5« R e s u l t s o f R e g r e s s i o n A n a l y s i s w i t h F i e l d Independence as I n d e p e n d e n t V a r i a b l e . 94 6. I n t e r c o r r e l a t i o n s Among Measures o f H y p o t h e s i z e d V a r i a b l e s 105 7. R e s u l t s o f R e g r e s s i o n A n a l y s i s w i t h C o r r e c t S o l u t i o n as Dependent V a r i a b l e 107 8. Means and S t a n d a r d D e v i a t i o n s f o r P r o c e d u r e s D e r i v e d f r o m C o d i n g System 108 9. I n t e r c o r r e l a t i o n s Between P a i r s o f V a r i a b l e s f r o m C o d i n g System 109 10. R e g r e s s i o n A n a l y s i s on C o r r e c t S o l u t i o n w i t h C o r e E n t e r e d B e f o r e H e u r i s t i c s I l l 11. R e g r e s s i o n A n a l y s i s on C o r r e c t S o l u t i o n w i t h H e u r i s t i c s E n t e r e d B e f o r e Core 112 v i i i LIST OF FIGURES Figure Page 1. Three Facets of the D i s c i p l i n e of Mathematics 6 2. The H e u r i s t i c s from MacPherson's Model 8 3. A Model of the Events Involved i n Mathematical Problem Solving 19 4. The Coding Form 56 5. Example of Coding System: Student #36, Problem #3 62 6. Example of Coding System: Student #24, Problem #4 63 7. Examples of Coder E r r o r 69 8. The Process Matrix with I d e n t i f i e d Areas 77 9. Process Matrix Categories 78 10. Process Matrix: Frequency D i s t r i b u t i o n of Change of Problem Solving Procedures f o r 100 Math World Problems .... 83 11. Process Matrix: Frequency D i s t r i b u t i o n of Change of Problem Solving Procedures f o r 100 Real World Problems .... 84 12. Percent of Procedures Used i n Each Category - Math World .... 85 13. Percent of Procedures Used i n Each Category - Real World .... 86 14. Process Matrix: Frequency D i s t r i b u t i o n of Change of Problem So l v i n g Procedures, Counted Once per Problem, f o r 100 Math World Problems 88 15. Process Matrix: Frequency D i s t r i b u t i o n of Change of Problem S o l v i n g Procedures, Counted Once per Problem, f o r 100 Real World Problems 89 16. Row Percentage of Moves: Process Matrix f o r Math World Problems 91 17. Row Percentage of Moves: Process Matrix f o r Real World Problems 92 18. Process Matrix: Frequency D i s t r i b u t i o n of Change of Problem Solving Procedures f o r F i e l d Independent Students 96 ix Figure Page 19- Process Matrix: Frequency D i s t r i b u t i o n of Change of Problem S o l v i n g Procedures f o r F i e l d Dependent Students ... 97 20. Percent of Procedures Used i n Each Category - F i e l d Independence 98 21. Percent of Procedures Used i n Each Category - F i e l d Dependence 99 22. Process Matrix: Frequency D i s t r i b u t i o n of Changes of Problem Solving Procedures, Counted Once per Problem, f o r F i e l d Independent Students 101 23. Process Matrix: Frequency D i s t r i b u t i o n of Change of Problem Solving Procedures, Counted Once per Problem, f o r F i e l d Dependent Students 102 24. Row Percentage- of Moves: Process Matrix f o r F i e l d Independent Students 103 25. Row Percentage of Moves: Process Matrix f o r F i e l d Dependent Students 104 26. R e l a t i v e Contributions of Core and H e u r i s t i c s to Correct S o l u t i o n 113 X ACKNOWLEDGEMENT I wish to thank the members of my d i s s e r t a t i o n committee, Dr. Douglas Owens, Dr. James S h e r r i l l , Dr. Peter Olle y , and Dr. Robert Conry f o r t h e i r assistance. I wish p a r t i c u l a r l y t o thank my chairman, Dr. G a i l S p i t l e r , f o r her patience and guidance i n the completion of my d i s s e r t a t i o n . I wish to thank, too, Dr. E r i c MacPherson f o r h i s i n s p i r a t i o n and counsel. Chapter I THE PROBLEM The I n t e r n a t i o n a l Commission of Mathematics, which met i n Moscow i n the summer of I966, chose three t o p i c s of p a r t i c u l a r importance f o r di s c u s s i o n . These t o p i c s were: the u n i v e r s i t y programs i n mathematics f o r p h y s i c i s t s , the use of the axiomatic method i n the teaching of mathe-matics i n the secondary school, and the r o l e of problems i n developing students' mathematical a c t i v i t y . The report to the commission by the Conference Board of Mathematical Sciences (Begle, I966) i n d i c a t e d that the cre a t i o n of problem sets should be a body of problem material which w i l l provide the environment f o r imaginative and cr e a t i v e t h i n k i n g . The Cambridge Conference on School Mathematics (Goals f o r School Mathematics, 1963) has noted that "the composition of problem sequence i s one of the l a r g e s t and one of the most urgent tasks i n c u r r i c u l a r development [p. 28]." Even with the growing emphasis on s o l v i n g more complex and challenging problems, l i t t l e has been done i n developing methods that the teacher may use to improve the problem s o l v i n g s k i l l s of h i s students. In f a c t Dessart and Frandsen (l?72) have suggested that the "major question of how problems are solved. . . [ i s one]. . .of the to p i c s f o r which a more inten s i v e search f o r fundamental p r i n c i p l e s might begin i n hopes of b u i l d i n g foundations to support v a l i d research [p. 1191]." In view of the l i m i t e d knowledge of how students solve problems, precursors to any 2 large scale study i n the teaching of problem s o l v i n g should be c l i n i c a l studies of i n d i v i d u a l subjects ( K i l p a t r i c k , 1969,p. 179)• Purpose of the Study The question of how mathematical problem s o l v i n g may be improved has att r a c t e d considerable a t t e n t i o n among mathematics educators. In most problem s o l v i n g tasks, there i s no simple r e l a t i o n s h i p between the s o l u t i o n obtained by an i n d i v i d u a l and the process used to achieve i t . Measures such as the correctness of a s o l u t i o n or the time needed t o achieve a correct s o l u t i o n do not i n d i c a t e the p a r t i c u l a r procedures the i n d i v i d u a l used i n reaching a s o l u t i o n . In order to study the problem s o l v i n g process techniques must be used which allow the subject t o generate an observable sequence of behavior ( K i l p a t r i c k , 1967, p« 4)« I t i s a purpose of t h i s study to analyze an i n d i v i d u a l ' s behavior during the s o l u t i o n of a mathematical problem and to determine the e f f e c t of varying the problem context on t h i s behavior. The experimenter w i l l also t r y to determine i f students d i f f e r i n g i n t h e i r degree of f i e l d , independence, d i f f e r i n t h e i r behavior while s o l v i n g mathematical problems. F i e l d independence i s a construct which provides information about i n d i v i d u a l d i f f e r e n c e s . S p i t l e r (1970, pp. 53-57) contends that f i e l d independent students are more able to deal with complex geometric problems and with problems which deal with patterned s t i m u l i than f i e l d dependent students. This would seem t o imply that the field-independent-dependent construct i s an important v a r i a b l e i n mathematical problem s o l v i n g . Analyzing the Problem Solving Process A search of the l i t e r a t u r e reveals that many techniques have been used to study the problem s o l v i n g process. One of the p r i n c i p a l objectives which must be sought by the i n v e s t i g a t o r i s that of obtaining and 3 maintaining observable behavior. I t i s not enough to t r y to detect which problem s o l v i n g methods are used by studying the r e s u l t s of the students' work on t h e i r f i n a l papers. The i n v e s t i g a t o r must t r y to e l i c i t an observable response from the student even though t h i s type of behavior may be d i f f i c u l t f o r the student working a novel problem. Psychologists have used numerous devices and techniques i n attempts to encourage t h e i r subjects to exte r n a l i z e t h e i r thought processes. Comprehensive reviews can be found i n Lucas (1972) and K i l p a t r i c k (1967). In view of the general purpose of t h i s study a method w e l l s u i t e d f o r e x t e r n a l i z i n g the problem s o l v i n g procedure was suggested by K i l p a t r i c k (1967), who observed: There i s one method of g e t t i n g a subject to produce s e q u e n t i a l l y - l i n k e d , observable behavior that requires n e i t h e r s k i l l i n s e l f - o b s e r v a t i o n nor the manipulation pf mechanical devices: have the subject think aloud as he works [ p. 6 ] . Thinking aloud requires only that the subject give an account of h i s mental a c t i v i t y as best he can. However, t h i s method may have c e r t a i n l i m i t a t i o n s . Thoughts may come and go too q u i c k l y to be v e r b a l i z e d . Subjects may also tend to remain s i l e n t during moments of deepest thought. More serious, however, i s the p o s s i b i l i t y that a subject may solve a problem i n a d i f f e r e n t manner when asked to v e r b a l i z e h i s thoughts than when he i s l e f t alone to solve i t s i l e n t l y ( K i l p a t r i c k , 1967, p» 7)» Two research studies have examined the e f f e c t of r e q u i r i n g subjects to think aloud when s o l v i n g problems. Flaherty (1973) found no s i g n i f i c a n t d i f f e r e n c e with respect to problem s o l v i n g t e s t scores between subjects who were required to think aloud and those subjects who were not required to v e r b a l i z e o v e r t l y . However, there was a s i g n i f i c a n t d i f f e r e n c e between the two groups i n the area of computational e r r o r s . Flaherty i n d i c a t e d t h a t : 4 Perhaps the tendency of overt v e r b a l i z a t i o n subjects to make more computational errors than the non-verbaliza-t i o n subjects may be a t t r i b u t e d t o t h e i r being somewhat d i s t r a c t e d by the requirement to think aloud [ p. 1767 ] Roth (1966) found there was no s i g n i f i c a n t d i f f e r e n c e i n the number of correct s o l u t i o n s or the time required to f i n d a s o l u t i o n between subjects who were required to t h i n k aloud while s o l v i n g reasoning problems and subjects not required to think aloud. Information-processors have made frequent use of the t h i n k i n g aloud procedure. Paige and Simon (1966) used the protocols of subjects asked t o think aloud to i d e n t i f y some of the important processes that a person must use i n order to do algebra word problems s u c c e s s f u l l y . The method of t h i n k i n g aloud i s both productive and easy to use, only r e q u i r i n g the subject to work on the s o l u t i o n to a mathematical problem and to t e l l about i t as he goes along. I f the method i s used i n t e l l i g e n t l y and conscientiously, keeping i n mind i t s l i m i t a t i o n s , i t can provide information about the d e t a i l e d process of thought. The problem i s not so much to c o l l e c t the data as i t i s to know what to do with them . ( M i l l e r , Galanter, and Pribram, i960, p. 193). The Need f o r a Model of Problem S o l v i n g I t i s apparent that a framework f o r c l a s s i f y i n g and analyzing the data i s necessary. In f a c t , before undertaking a study i n problem s o l v i n g , the i n v e s t i g a t o r must decide which procedures to examine and which to ignore. This choice of f a c t o r s w i l l not only determine the usefulness of the study, but also i t s l i m i t a t i o n s . In order to minimize the chances of overlooking or not being able to account f o r a procedure used by a subject, the i n v e s t i g a t o r must be aware of the model he i s using. Begle and Wilson (1970) described a model as "providing an o r g a n i z a t i o n a l framework; i t represents a 5 c a t e g o r i z a t i o n system with some stated rules and r e l a t i o n s h i p s f o r using the system [ p. 372]." The functions of a model should be to make c l e a r to others what one has i n mind and so a i d communication, to d i s t i n g u i s h between d e f i n i t i o n and empirical propositions, and to provide a means of data organization and i n t e r p r e t a t i o n (Kaplan, 1964, ch. 7) . With more emphasis being placed on the r o l e of problems i n both the curriculum and i n research, i t i s necessary t o know more about the problem s o l v i n g process than the usual three or four "steps" suggested by models i n t h i s area. The f o l l o w i n g statement made by Johnson (1944) which summarized the knowledge of problem s o l v i n g i n the f o r t i e s i s s t i l l relevent today: Problem s o l v i n g begins with the i n i t i a l o r i e n t a t i o n and ends with the c l o s i n g judgment, but between these bounds almost anything can happen, i n any sequence [p. 203]. The writings of George Polya (1957, 1962, I965) have added a great deal t o the knowledge of problem s o l v i n g i n mathematics. Polya describes mathematical problem s o l v i n g i n terms of "methods and rules of discovery and i n vention" which he c a l l s h e u r i s t i c s (1957, P» 112). However, i t i s s t i l l "between these bounds" where a more c a r e f u l l y constructed model i s needed. The procedures given i n the model must be broad enough to account f o r a l l the observed processes and yet not so broad that no r e a l d i s t i n c t i o n can be made. The model used i n t h i s study appears to be such a model. The model i s p r i m a r i l y the work of E r i c MacPherson, presently Dean of the School of Education, U n i v e r s i t y of Manitoba. MacPherson 1s Model f o r Mathematical Problem So l v i n g For the purpose of d i s c u s s i o n from a problem s o l v i n g point of view i t i s convenient t o view the d i s c i p l i n e of mathematics as having three f a c e t s : A p p l i c a t i o n , Core and Discovery (MacPherson, 1970, 1973). 6 APPLICATION l o r e CORE fa c t s categories algorithms notation DISCOVERY h e u r i s t i c s FIGURE 1 THREE FACETS OF THE DISCIPLINE OF MATHEMATICS Core. The core i s that part of the d i s c i p l i n e which i s not under ac t i v e question at the moment. I t consists of categories, notation, f a c t s and algorithms. The categories are the conventionally agreed upon groupings of f a c t s and algorithms. In mathematics such groupings are broadly termed algebra, geometry, analysis, topology, and more p a r t i c u l a r l y , number systems, integers and the l i k e . Associated with these categories i s a set of f a c t s which are propositions about the data not under a c t i v e question; and algorithms are the procedures f o r answering questions, which, when applied c o r r e c t l y , guarantee a s o l u t i o n i n a f i n i t e number of steps. Associated with each of these i s u s u a l l y found well-defined notation (MacPherson, 1973). Lore. Lore i s the set of acts of f a i t h by which mathematical systems are t i e d to the r e a l world. The e f f i c a c y of l o r e i s not determined by the l o g i c a l consistency of any algorithms used, but by the j u s t i f i c a t i o n of the acts of f a i t h (MacPherson, 1970, 1973). Discovery. A d i s c i p l i n e i n v a r i a b l y has a set of procedures f o r adding t o i t s core. Such procedures are termed h e u r i s t i c s (MacPherson, 1970, 1973). That i s : a h e u r i s t i c i s a general, non-core strategy which i s used f o r the purpose of di s c o v e r i n g some order of mathematical g e n e r a l i z a t i o n i n a novel s i t u a t i o n . Operationally, h e u r i s t i c s d i f f e r from algorithms i n that h e u r i s t i c s are characterized by an uncertainty as t o the choice of the i n i t i a l data and the e f f i c a c y of the procedure, whereas algorithms have a guarantee of producing a s o l u t i o n when applied c o r r e c t l y . • I t seems u s e f u l to consider four h i e r a r c h i c a l categories of h e u r i s t i Low, Cases, Middle, and General. Lower h e u r i s t i c s lead more d i r e c t l y to the use of algorithms and i n general do not create new problems, whereas the higher h e u r i s t i c s u s u a l l y create new problems and then lead to use of the lower h e u r i s t i c s or core procedures (MacPherson, 1973)<-The h e u r i s t i c s from MacPherson*s model are l i s t e d i n Figure 2 and defined as follows: Smoothing^" - The h e u r i s t i c of smoothing i s used when the problem i s a l t e r e d i n order to obtain some isomorphism between the new problem and a mathematical system. For example, i n the problems What i s the longest piece of metal rod which can be placed i n a box of dimension 3 inches by 4 inches by 12 inches? One might " i d e a l i z e " or smooth the box t o a rectangular p a r a l l e l e p i p e d and the metal rod to a l i n e segment. Another example, i n s o l v i n g the problem: The point A i s 50 u n i t s from a s t r a i g h t l i n e CD, and B i s a point 80 u n i t s from CD. Find the point X on CD so that the distance from A to X to B i s as small as p o s s i b l e . The d e f i n i t i o n s f o r the twelve h e u r i s t i c s i n the model were determined i n consultation with MacPherson. Low Cases M i d d l e 1. S m o o t h i n g 2. A n a l y s i s , a l l 1. Cases ^ ^ ^ r a n d o m someN^^ s e q u e n t i a l s y s t e m a t i c c r i t i c a l 2. T e m p l a t i o n . • d i r e c t 1. D e d u c t i o n ^ " ^ h y p o t h e t i c a l 2. I n v e r s e d e d u c t i o n f i x a t i o n 3. I n v a r i a t i o n e x c l u s i o n 4. A n a l o g y 5. Symmetry 6. P r e s e r v a t i o n G e n e r a l 1. V a r i a t i o n 2. E x t e n s i o n FIGURE 2 THE HEURISTICS FROM MACPHERSON'S MODEL 9 A SO 130 One could d i v i d e each of the measurements by 10, c r e a t i n g a s i m i l a r f i g u r e with dimensions of 5, 13, and 8. This may s i m p l i f y the arithmetic involved i n obtaining a s o l u t i o n . Once a s o l u t i o n f o r the s i m p l i f i e d problem i s found, the s o l u t i o n t o the o r i g i n a l problem can be found by m u l t i p l y i n g the simpler s o l u t i o n by ten. Analysis - The h e u r i s t i c of a n a l y s i s i s used when the problem i s separated or broken up i n t o subproblems. The solutions obtained from the subproblems are then used i n s o l v i n g the o r i g i n a l problem. One s o l u t i o n f o r the problem: What i s the maximum perimeter you can obtain by arranging one hundred one inch squares by f o l l o w i n g the r u l e that each time a new square i s added at l e a s t one of i t s sides must be placed against one side of a previously arranged square. i s to arrange the squares i n a s t r a i g h t row. When determining the perimeter, one might consider the end squares separate from the others, thus c r e a t i n g two subproblems. One f i n d i n g the perimeter f o r the two end squares i n v o l v i n g three sides and a subproblem f o r f i n d i n g the perimeter of the remaining squares Involving two s i d e s . 10 Cases - A case i s renaming the variables i n a problem as constants. Using the h e u r i s t i c s of cases i s to consider two or more cases i n one of the f ollowing manners: (a) To consider a l l possible cases, u s u a l l y i f there i s only a small number. (b) To consider cases at random. (c) To consider cases i n some systematic fashion, such as s e q u e n t i a l l y or to determine the c r i t i c a l cases. and then to recognize a pattern; that i s , to recognize the c h a r a c t e r i s t i c s shared by the data c o l l e c t e d from the cases considered and the p r o p e r t i e s and procedures from core. Consider t h i s example of the use of systematic cases: Find the sum of the f i r s t one hundred odd n a t u r a l numbers. The following cases ( i n v o l v i n g the number of n a t u r a l numbers i n the sum) could be considered i n sequence: 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 There i s a r e l a t i o n s h i p between the number of terms i n the sum and the square root of the term on the r i g h t of the equation. Hence, the s o l u t i o n to the problem i s 100 squared. One can also use sequential cases when a l l of the p o s s i b l e s o l u t i o n s are recognized. For example: A lady gave the postage stamp cle r k a one d o l l a r b i l l and said, "Give me some two-oent stamps, ten times as many one-cent stamps, and the balance i n f i v e s . " How can the c l e r k f u l f i l l t h i s p u z z l i n g request? In answering t h i s question, one could write the equation 12X + 5Y = 100, f o r w h i c h t h e r e a r e e i g h t p o s s i b l e v a l u e s f o r X. A l l c a s e s c o u l d be c o n s i d e r e d , o r s y s t e m a t i c c a s e s f o r X = 1, 2, 3,*>*>8 c o u l d be u s e d u n t i l a s o l u t i o n i s o b t a i n e d . S e q u e n t i a l c a s e s can be u s e d i n f i n d i n g a*** where a / 0 i f one i s f a m i l i a r w i t h t h e use o f p o s i t i v e e x p o n e n t s . T h a t i s : F i n d t h e v a l u e o f a ^ where a / 0. The f o l l o w i n g c a s e s c o u l d be c o n s i d e r e d i n sequence u s i n g a b a s e o f 2: 2 4 = 16 As each exponent d e c r e a s e s by 1, t h e v a l u e on t h e r i g h t o f t h e e q u a t i o n i s o n e - h a l f t h e p r e c e d i n g v a l u e . I f t h e p a t t e r n i s t o h o l d , t h e n 2^ = 1. O t h e r b a s e s can be c o n s i d e r e d , e i t h e r a t random o r s y s t e m a t i c a l l y o b t a i n i n g a s i m i l a r r e s u l t . Hence a^ = 1 f o r a / 0. T e m p i a t i o n - The h e u r i s t i c o f t e m p l a t i o n i s r e c a l l i n g a c a t e g o r y o f c o n t e n t w h i c h i s r e l a t e d t o t h e p r o b l e m b e i n g s o l v e d . T h i s i n c l u d e s t h e r e c a l l o f s u c h t h i n g s as a l g o r i t h m s , p r o b l e m t y p e s , p r o c e d u r e s , t h e o r e m s , and p r o p e r t i e s r e l a t e d t o t h e p r o b l e m o r p r o b l e m a r e a . The p u r p o s e o f t e m p l a t i n g i s t o r e c a l l r e l a t e d c o r e m a t e r i a l w h i c h can be u s e d i n s o l v i n g t h e p r o b l e m . F o r example: A y a c h t i s moored a t A, 50 metres away f r o m a s t r a i g h t s e a w a l l , CD. The c a p t a i n o f t h e y a c h t w i s h e s t o row t o t h e s e a w a l l t o c o l l e c t a p a s s e n g e r and t h e n t o a speed-boat moored a t B, 80 metres f r o m t h e w a l l . Where s h o u l d t h e p a s s e n g e r meet t h e c a p t a i n t o make t h e r o u t e as s h o r t as p o s s i b l e ? 12 One might pick a point X between C and D„then looking at the two t r i a n g l e s , consider the theorems r e l a t e d to r i g h t t r i a n g l e s to see i f any information about the p o s i t i o n of the point X can be obtained. Deduction - The h e u r i s t i c of deduction can be s u b - c l a s s i f i e d i n t o two h e u r i s t i c s , d i r e c t and hy p o t h e t i c a l . To use d i r e c t deduction i s to ask what consequences can be implied from a given premise or set of premises. Hypothetical deduction i s asking what consequences can be implied from a premise or set of premises assumed by the problem s o l v e r . In e i t h e r instance, an attempt must be made to answer the questions posed. For an example of d i r e c t deduction, assume the fo l l o w i n g data, or set of premises are given: Given t r i a n g l e XYZ with A the midpoint of XY and B the midpoint of YZ, l e t C and D be points on XZ so that q u a d r i l a t e r a l ABCD can be formed by f o l d i n g along AD so that X i s a point on CD, f o l d on BC so that Z i s a point on CD, and f o l d on AB. What consequences can be i m p l i e d b y t h e s e t o f p r e m i s e s ? ABCD i s a r e c t a n g l e . The sum o f a n g l e s X, Y, and Z i s 180 d e g r e e s . The a r e a o f q u a d r i l a t e r a l ABCD i s e q u a l t o o n e - h a l f t h e a r e a o f t r i a n g l e XYZ. These a r e examples o f some o f t h e i m p l i c a t i o n s w h i c h c o u l d be made. H y p o t h e t i c a l d e d u c t i o n may be u s e d i n s o l v i n g t h e p r e v i o u s y a c h t p r o b l e m . I f X i s t h e p o i n t on t h e s e a w a l l where t h e c a p t a i n p i c k s up t h e p a s s e n g e r , t h e n two r i g h t t r i a n g l e s a r e f o r m e d . A s o l u t i o n t o t h e p r o b l e m can be o b t a i n e d b y m i n i m i z i n g t h e sum o f t h e l e n g t h s o f t h e i r h y p o t e n u s e . I f one assumes t h e h y p o t e n u s e o f a r i g h t t r i a n g l e i s t w i c e SO ( c A / / / \ 8 p ( ^ i_r I P t h e l o n g e s t s i d e ( w h i c h i t i s n o t ) , what c a n be c o n c l u d e d i n t h i s p r o b l e m ? I f CX i s l e s s t h a n 50m, t h e n t h e l e n g t h o f h y p o t e n u s e AX i s 100m and XB i s 2 ( 1 3 0 - CX)m. The sum o f AX and BX i s i n c r e a s e d as t h e p o i n t X i s moved c l o s e r t o C. I f XD i s l e s s t h a n 80m, t h e l e n g t h o f XB w i l l be 160m and AX w i l l have l e n g t h 2 ( 1 3 0 - XD)m. Hence t h e sum o f AX and XB w i l l i n c r e a s e as t h e p o i n t X i s moved t o w a r d s t h e p o i n t D. So one c o u l d c o n c l u d e t h e minimum o f t h e sum, AX + XB w i l l o c c u r when X i s 50m f r o m C. I n v e r s e D e d u c t i o n - The h e u r i s t i c o f i n v e r s e d e d u c t i o n i s u s e d when one assumes ( o r one i s g i v e n ) a c o n c l u s i o n , and a s k s what p r e m i s e s o r a n t e c e d e n t s i m p l y i t . I n v e r s e d e d u c t i o n means ' w o r k i n g t h e p r o b l e m backwards'.. F o r example: 14 What i s the longest piece of metal rod which can be placed i n a box of dimension 3 inches by 4 inches by 12 inches? The longest piece of metal rod which w i l l f i t i n t o the box has the length AD. Now AD i s the hypotenuse of r i g h t g £ t r i a n g l e CDA, and i f the length of DC and AC were known, then the Pythagorean Theorem could be applied to f i n d AD. CD i s known to be 4 inches long, so the length of AC must be found. But AC i s the hypotenuse of r i g h t t r i a n g l e BCA and the lengths of AB and BC are known to be 3 said 12 inches r e s p e c t i v e l y . So by applying the Pythagorean Theorem to t r i a n g l e BCA, AC can be found. I n v a r i a t i o n - The h e u r i s t i c of i n v a r i a t i o n can be s u b c l a s s i f i e d i n t o two h e u r i s t i c s , f i x a t i o n and exclusion. In using i n v a r i a t i o n e i t h e r a v a r i a b l e i s renamed as a constant ( f i x a t i o n ) , or a v a r i a b l e i s excluded i n the problem (e x c l u s i o n ) . Then an attempt i s made to solve the new problem and i t s s o l u t i o n studied i n order to gain some i n s i g h t i n t o the given problem. The h e u r i s t i c of i n v a r i a t i o n - f i x a t i o n can be used i n s o l v i n g the f o l l o w i n g problem: Derive a formula to f i n d the roots of the general cubic 3 2 aX + bX + cX + d, where a, b, c, and d are r e a l numbers. For t h i s problem, one or more of the constants i s f i x e d at zero and the roots of the new fun c t i o n are found. Then the s o l u t i o n of t h i s new problem i s studied i n hopes of gaining some i n s i g h t i n t o s o l v i n g the o r i g i n a l problem. The h e u r i s t i c of i n v a r i a t i o n - e x c l u s i o n can be used i n s o l v i n g the foll o w i n g geometry problem: Construct a t r i a n g l e ABC given the length of the side BC, the measure of angle A, and the length of the a l t i t u d e h from A. In t h i s problem, one might exclude the measure of angle A and formulate the new problem of constructing a t r i a n g l e ABC given BC and the a l t i t u d e h from A. Again the new problem may be solved by some method and i t s s o l u t i o n studied t o gain some i n s i g h t i n t o the o r i g i n a l problem. It. f K B c Analogy - I f there i s a "quasi" isomorphism between the problem and a mathematical system p r e v i o u s l y studied then the h e u r i s t i c of analogy can be used. The h e u r i s t i c of analogy involves asking questions and considering properties based on the recogni t i o n of the isomorphism. Analogy can be used to solve the fo l l o w i n g problem by r e c a l l i n g an analogous s i t u a t i o n from two-dimensional geometry f o r which the s o l u t i o n i s known: What i s the greatest distance between two points of a rectangular s o l i d of dimensions 3 by 4 by 12? I f one considers the s i m i l a r i t y between t h i s problem and that of f i n d i n g the length of a diagonal of a rectangle of dimension a by b, then, a + b , the length of the diagonal 3/ 3 3 3 / of t h i s rectangular s o l i d could be d 3 + 4 + 1 2 , y ; 0 2 ,2 1 0 2 3 + 4 + 1 2 , or \J J> + 4 3 + 12 3 . Symmetry - To use the h e u r i s t i c of symmetry i s to make use of the inherent or constructed symmetry i n a problem. For example: A yacht i s moored at A , 50 meters away from a s t r a i g h t sea wall, CD. The captain of the yacht wishes to row to the sea wa l l to c o l l e c t a passenger and then row to a speed-boat moored at B, 80 meters from the w a l l . Where should the passenger meet the captain to make the route as short as possible? c IN N \ , 6 p 130 \ N N N j 3' One might take advantage of the f a c t that once the captain has moved from point A to the sea w a l l CD at X, the distance from X to B i s the same as the distance from X to a point B' (where B' i s the r e f l e c t i o n of B through the l i n e CD.) Since the shortest distance between two points i s a s t r a i g h t path, the point X i s e a s i l y found. In the fo l l o w i n g example, the problem i s stated symmetrically i n terms of the sets A and B: Let A and B be sets such that f o r any set, C, A AC = B/\ C and A U C =. BU C. Prove A = B. The r o l e of A and B can be interchanged without changing the problem. 17 So i f one shows A i s a subset of B, then by the symmetry, B w i l l be a subset of A. Since two sets are equal i f and only i f they are subsets of each other, A = B. Preservation - The h e u r i s t i c of preservation i s used when a conscious e f f o r t i s made to preserve the properties of a mathematical system when i t s domain i s extended (isomorphically) to form a new system. For example i n the extension of the r e a l numbers to the complex, one wishes to preserve the f i e l d properties of the r e a l number system. The h e u r i s t i c of preservation i s used i f the commutative, a s s o c i a t i v e and d i s t r i b u t i v e properties are taken i n t o account when d e f i n i n g a d d i t i o n and m u l t i p l i c a t i o n f o r the complex numbers so as to preserve the f i e l d p r o p e r t i e s . The h e u r i s t i c of preservation can be used i n s o l v i n g the f o l l o w i n g problem: Find a d e f i n i t i o n of a^ where a ^ 0. X Y X—Y One of the properties of exponents i s b /b = b , f o r X / Y. I f a conscious e f f o r t i s made to preserve t h i s property when d e f i n i n g a^, then the h e u r i s t i c of preservation has been used. V a r i a t i o n - In using the h e u r i s t i c of v a r i a t i o n the c o n s t r a i n t s on one of the v a r i a b l e s of the problem are relaxed i n order to study the e f f e c t on the other v a r i a b l e s . As an example from geometry, l e t X and Y be the midpoints of AC and AB r e s p e c t i v e l y i n t r i a n g l e ABC. Then one can conclude that XY i s p a r a l l e l to BC and the length X . of XY i s -g- the length of BC. >v The h e u r i s t i c of v a r i a t i o n i s C R J used i f one relaxes the c o n d i t i o n that X and Y are midpoints and studies the e f f e c t on the r e l a t i o n s h i p s between XY and BC. 18 Extension - The h e u r i s t i c of extension i s posing a problem that i s an extension or a g e n e r a l i z a t i o n of the given problem. Given the problem, "Find a n a t u r a l number which i s both a square and a cube.", one uses the h e u r i s t i c of extension when posing the question, 'can a l l n a t u r a l numbers which are both a square and a cube be characterized?'. For the purposes of t h i s study, the sequence of events used while so l v i n g a mathematical problem i s characterized by Figure 3. A b r i e f d e s c r i p t i o n of the model foll o w s : Willingness: The subject's willingness to accept the problem and once he has s t a r t e d to solve i t , h i s w i l l i n g n e s s to continue u n t i l he has a s o l u t i o n . Sieve: The mathematical content and processes (core) which are f a m i l i a r to the problem s o l v e r . Upon presentation of the problem, the problem i s e i t h e r accepted or rejected (willingness box). I f the problem i s accepted, then the i n d i v i d u a l immediately t r i e s t o r e c a l l whatever content he knows that i s r e l a t e d to the problem (sieve) i n hopes of recognizing a pattern which w i l l enable him to obtain a solution.. I f the e x i t i s from the sieve to pattern, one of three things can happen depending upon the confidence ( p r o b a b i l i t y ) the i n d i v i d u a l has i n the pattern he has obtained: l ) Keeping the pattern i n mind, he could return to the sieve and t r y to obtain a b e t t e r f i t of the pattern to the core and problem. 2) I f the i n d i v i d u a l has l i t t l e confidence i n the pattern he could return to the w i l l i n g n e s s box and decide to e i t h e r continue working on the problem or to q u i t without a s o l u t i o n . 3) He could e x i t to the use of core algorithms. Two confidence l e v e l s or p r o b a b i l i t i e s (indicated by dotted l i n e ) are assigned to the sequence of procedures used f o r every problem, one concerning the f i t of the pattern and the other concerning the accuracy of the algorithm. These F I G U R E 3 A MODEL OF THE E V E N T S I N V O L V E D I N M A T H E M A T I C A L PROBLEM S O L V I N G 20 p r o b a b i l i t i e s determine the p o s s i b l e routes from the concern box. S i m i l a r l y with the a p p l i c a t i o n of an algorithm, a reasonableness i s attached to the s o l u t i o n . I f there i s a high p r o b a b i l i t y attached to the pattern and yet, there i s some concern f o r the use of an algorithm, the. problem solver may return d i r e c t l y to the algorithm box. I f however, there i s some concern f o r a p a r t i c u l a r pattern then he may return to the sieve ( v i a the willingness box) and t r y to obtain a b e t t e r f i t . I f the outcome of the a p p l i c a t i o n of the algorithm i s reasonable, then the i n d i v i d u a l w i l l e x i t with a s o l u t i o n . However, i f no e x i t i s made from the sieve to pattern, then he must e i t h e r return to the w i l l i n g n e s s box or to the use of h e u r i s t i c s . The lower h e u r i s t i c s such as smoothing and analysis appear to be c l o s e l y r e l a t e d to the sieve so there may be a considerable movement between the sieve and h e u r i s t i c s . The use of the lower h e u r i s t i c s may also lead more d i r e c t l y to the sieve and pattern whereas the use of higher h e u r i s t i c s u s u a l l y creates new problems and then leads to the use of the lower h e u r i s t i c s . D e f i n i t i o n of Terms Certain terms occur throughout the study and are defined here. The term degree of f i e l d independence r e f e r s to the r e l a t i v e p o s i t i o n of the student's score i n the d i s t r i b u t i o n of the experimental sample scores on the Embedded Figures Test (Witkin, 1969)• See Chapter I I f o r f u r t h e r d i s c u s s i o n of the construct of f i e l d independence. Conventionally a student i s f i e l d independent i f h i s score i s above the sample median on the Embedded Figures Test (EFT) and f i e l d dependent i f h i s score i s below the median. To t e s t one of the hypotheses i n t h i s study, a subject i s f i e l d independent i f h i s score on the EFT i s i n the top t h i r d of the scores f o r the sample and f i e l d dependent i f h i s score i s i n the bottom one-third. The term problem r e f e r s t o " a s i t u a t i o n i n which one must give a response (that i s , when he seeks s a t i s f a c t i o n ) and has no h a b i t u a l response which w i l l give t h i s s a t i s f a c t i o n [ Cronbach, 1948, p. 32]." This d e f i n i t i o n implies that what i s a problem f o r one student may not be a problem f o r another. I t i s not the posing of a question that makes i t a problem, but the wi l l i n g n e s s of the i n d i v i d u a l t o accept i t as something he must t r y t o solve. Furthermore, a question such as: What, i s the greatest distance between two points i n a rectangular s o l i d of dimension 3 inches by 4 inches by 12 inches? i s not a problem t o someone who i s f a m i l i a r with three-dimensional geometry and r e c a l l s a formula f o r f i n d i n g the length of the diagonal of a rectangular p a r a l l e l e p i p e d . However, i t may be a problem f o r the student who has studied only plane geometry. The term problem context r e f e r s to the s e t t i n g i n which the problem i s stated, mathematical vs. r e a l or p h y s i c a l world. . The vocabulary used i n the statement of the problem should be f a m i l i a r t o a student who i s taking a grade eleven mathematics. The terms "mathematical" and " r e a l " r e f e r t o the s i t u a t i o n and the vocabulary used i n the problem. A problem stated i n each context i s given below: Mathematical World or Math World: What i s the greatest distance between two points i n a rectangular s o l i d of dimension 3 u n i t s by 4 u n i t s by 12 u n i t s ? Real World: What i s the longest piece of metal rod which can be placed i n a box of dimension 3 inches by 4 inches by 12 inches? 22 The term algorithm r e f e r s to a systematic procedure which, i f c a r r i e d out c o r r e c t l y , guarantees a s o l u t i o n i n a f i n i t e number of steps. Algorithms vary i n complexity from very simple such as the a d d i t i o n of two d i g i t whole numbers without regrouping, to complex and involved procedures, l i k e using the e l i m i n a t i o n method to f i n d the s o l u t i o n of a system of l i n e a r equations. An example of an algorithm from the grade eleven curriculum i s the procedure used to f i n d the roots of the quadratic 2 4x - l6x - 9 by completing the square. The term h e u r i s t i c r e f e r s to the twelve procedures defined i n the model. A h e u r i s t i c d i f f e r s from an algorithm i n that h e u r i s t i c s have a low p r o b a b i l i t y of guaranteeing any success i n s o l v i n g the problem. For example, the renaming of a v a r i a b l e as a constant i n order t o study i t s e f f e c t s on the other v a r i a b l e s i n the problem does not guarantee that one w i l l f i n d a s o l u t i o n , but i t may give some i n s i g h t i n t o the problem. The Problem In t h i s study, the w r i t e r attempted to determine, by use of a c l i n i c a l procedure, whether there were any s i g n i f i c a n t d i f f e r e n c e s i n the problem sol v i n g behavior among subjects working problems i n e i t h e r a mathematical or r e a l world s e t t i n g . An attempt was also made to i n v e s t i g a t e the r e l a t i o n s h i p between these problem s o l v i n g behaviors, i n terms of h e u r i s t i c and nonheuristic procedures, and how these behaviors r e l a t e to f i e l d independence. The present study has as i t s c e n t r a l theme, the use of h e u r i s t i c s i n mathematical problem s o l v i n g . L i t t l e i s known about how students attack and solve mathematical problems and what part, i f any, h e u r i s t i c s play i n t h i s procedure. In order to gain some i n s i g h t i n t o t h i s area, a study which i s exploratory i n nature was designed. The general aims of the study were: 1. To develop a workable system, based on the model of MacPherson, f o r coding the audio tape recorded protocols of subjects asked to think aloud as they solve mathematical world problems. 2. To i n v e s t i g a t e the r e l a t i o n s h i p of the problem s o l v i n g behaviors derived from the coding system, to one another. 3. To i n v e s t i g a t e the e f f e c t of problem context and f i e l d independence on problem s o l v i n g behaviors. In l i g h t of these aims, the f o l l o w i n g questions were asked: 1. What procedures from core and what h e u r i s t i c s are used by students i n attacking and s o l v i n g mathematical word problems? 2. Does the context of the problem influence h e u r i s t i c usage? 3. What e f f e c t does field-dependence-independence have on the problem s o l v i n g process? 4. Do selected groups of i n d i v i d u a l s e x h i b i t patterns of h e u r i s t i c usage when s o l v i n g mathematical word problems? General Hypotheses • Some of the hypotheses that were t e s t e d i n order to help the i n v e s t i g a t o r gain some i n s i g h t t o the answers f o r the above questions are as foll o w s : HI: Problem context w i l l not a f f e c t the t o t a l number of h e u r i s t i c s used by a student. H2: Problem context w i l l not a f f e c t the number of d i f f e r e n t h e u r i s t i c s used by a student. H3* Problem context w i l l not a f f e c t the number of correct solutions obtained by a student. H4: F i e l d independence w i l l not a f f e c t the number of times h e u r i s t i c s 2k are used by a student. H"5: F i e l d independence w i l l not a f f e c t the number of d i f f e r e n t h e u r i s t i c s used by a student. H6: F i e l d independence w i l l not a f f e c t the number of correct solutions obtained by a student. H7: The number of times h e u r i s t i c s are used by a student i s unrelated to the number of correct solutions he obtains. H8: The number of d i f f e r e n t h e u r i s t i c s used by a student i s unrelated t o the number of correct solutions he obtains. H9: The sequence of h e u r i s t i c s used by the subjects w i l l not be a f f e c t e d by problem context. H10: The sequence of h e u r i s t i c s used by the subjects w i l l not be af f e c t e d by f i e l d independence. S i g n i f i c a n c e of the Study K i l p a t r i c k (1969) stated i n a recent review of mathematical problem, solving, ...the researcher...who chooses t o i n v e s t i g a t e problem so l v i n g i n mathematics i s probably best advised t o undertake c l i n i c a l studies of i n d i v i d u a l subjects... because our ignorance i n t h i s area demands c l i n i c a l studies as precursors to l a r g e r e f f o r t s [p. 179]• The s i g n i f i c a n c e of t h i s study i s apparent when one considers that so few attempts i n research have been made to analyze i n d i v i d u a l behavior during the problem s o l v i n g process. This behavior w i l l be analyzed i n terms of a problem s o l v i n g model developed by MacPherson. U n t i l more i s known about the way these h e u r i s t i c s are employed l i t t l e can be gained by way of i n s i g h t i n t o an adequate teaching methodology to enhance problem s o l v i n g s k i l l s . This study i s also s i g n i f i c a n t i n that i t makes use of complex 25 problems, u s u a l l y not found i n today's textbooks. With a greater emphasis being placed on challenging problems i n the curriculum, there i s a need f o r research which makes use of such problems. In K i l p a t r i c k ' s words (1967): . The mathematics educator, i n p a r t i c u l a r , sometimes complains that, the kind, of complex, challenging problems that are the most d i f f i c u l t to l e a r n how to solve r a r e l y appear i n the research l i t e r a t u r e . He questions how much one can e x t r a -polate from f i n d i n g s based on l e v e r p u l l i n g and card s o r t i n g to the processes that underlie the search f o r an elegant geometric proof or the production of an equation d e s c r i b i n g a p h y s i c a l s i t u a t i o n . The analogies may be d i r e c t or they may be exceedingly subtle and complicated. We have no way of knowing because complex problems have so seldom been used i n research [p. l j . F i n a l l y , the most important c o n t r i b u t i o n of t h i s study may be that of adding to the current s e l e c t i o n , as small as i t may be, of t h e o r e t i c a l models of mathematics problem s o l v i n g . This model can be used as a framework f o r extending research i n mathematical problem s o l v i n g and perhaps as a mode f o r teaching mathematics. Assumptions and L i m i t a t i o n s Five basic assumptions were made i n beginning t h i s study. I t was assumed that the h e u r i s t i c s from MacPherson's model could be observed and used t o c l a s s i f y problem s o l v i n g procedures. I t was assumed the problems used i n t h i s study would e l i c i t the use of h e u r i s t i c s . Also, i t was assumed that the students would be w i l l i n g to t r y to solve the f i v e problems assigned to them, and that they would v e r b a l i z e t h e i r thoughts. I t was also assumed that students i n grade eleven were mature enough, both emotionally and mathematically, to p a r t i c i p a t e i n t h i s c l i n i c a l study. During the p i l o t study these assumptions were examined and found t o be tenable. The students i n the p i l o t study were w i l l i n g to work on a l l assigned problems and to v e r b a l i z e t h e i r thoughts. In modifying the coding system i t was found that the h e u r i s t i c s could be observed and used to c l a s s i f y problem s o l v i n g procedures. The l i m i t a t i o n s of t h i s study are a t t r i b u t a b l e t o : (a) The problems used. The problems were not selected from a s p e c i f i c category, such as algebra or geometry, but were selected from those used i n the p i l o t study using the f o l l o w i n g four c r i t e r i a : 1. They were problems not u s u a l l y found i n the school curriculum 2. They could be solved by about h a l f the students 3. They could be solved i n a number of d i f f e r e n t ways 4. They e l i c i t e d the use of h e u r i s t i c s A set of d i f f e r e n t problems could have produced d i f f e r e n t data. (b) The sample of subjects used. The subjects were selected from the middle IQ range of academic grade eleven mathematics students. The data are based on a r e l a t i v e l y homogeneous group of students. I t i s quite p o s s i b l e that, i f the study had included a greater v a r i e t y of i n d i v i d u a l s , a greater v a r i e t y of h e u r i s t i c s might have been observed. (c) The method of data c o l l e c t i o n . Having the subjects think aloud as they solve mathematical problems may cause them to commit errors they normally would not; i n f a c t they may solve the problem i n a d i f f e r e n t manner when asked to v e r b a l i z e than when l e f t alone to solve i t . Chapter I I REVIEW OF RELATED LITERATURE This chapter w i l l examine some of the l i t e r a t u r e relevant to the study. Rather than attempt a comprehensive survey of recent problem so l v i n g research (see Kleinmuntz, 1966; K i l p a t r i c k , 1969; Jerman, 197l) , the d i s c u s s i o n w i l l be confined to two relevant t o p i c s . The f i r s t concerns the problem s o l v i n g process. A s e r i e s of models of the problem s o l v i n g process w i l l be discussed with the r e l a t e d research. The second t o p i c deals with the consistent or i n c o n s i s t e n t problem s o l v i n g behaviors exhibited by subjects. Recent research i n which the f i e l d independence construct i s used as an aptitude v a r i a b l e i n a problem s o l v i n g s e t t i n g w i l l be c i t e d . Models of Problem S o l v i n g Over the years, scholars have sought to shed l i g h t upon the problem s o l v i n g process by s p e c i f y i n g the sequence of behaviors through which one might proceed i n s o l v i n g a problem (Davis, 1973, p. 15)« One example i s given by Johnson (1944) i n which he i d e n t i f i e s three processes or groups of processes which, he says r e g u l a r l y occur during problem s o l v i n g : ( l ) o r i e n t a t i o n to the problem, (2) producing relevant material, and (3) judging. Dewey (1933) gives a s l i g h t l y d i f f e r e n t a n a l y s i s . As he defines r e f l e c t i v e thinking, there i s l i t t l e d i f f e r e n c e between i t and problem s o l v i n g . Hence, hi s analysis of r e f l e c t i v e t h i n k i n g can be taken as an analysis of the process of problem s o l v i n g (Henderson and Pingry, 1953)* 28 Dewey (1933, pp. 107-116) o u t l i n e s f i v e phases of r e f l e c t i v e t h i n k i n g : 1. "A f e l t d i f f i c u l t y " , a question f o r which the answer must be sought 2. Location and d e f i n i t i o n of the d i f f i c u l t y 3 . "The i d e n t i f i c a t i o n of various hypotheses...to i n i t i a t e and guide observation and other operations i n c o l l e c t i o n of f a c t u a l m a t e r i a l [p. 107]." 4. Elaboration of each hypothesis by reasoning and the t e s t i n g of the hypothesis 5« A c t i n g on the basi s of the p a r t i c u l a r hypothesis s e l e c t e d i n step four to see whether the r e s u l t s t h e o r e t i c a l l y i n d i c a t e d a c t u a l l y occur. Dewey notes these f i v e phases or stages of thought do not follow one another i n a set order. Each step i n t h i n k i n g does something to the formation of a suggestion t o change i t i n t o an i d e a or hypothesis. Each step also does something to promote the l o c a t i o n and d e f i n i t i o n of the problem. Each improvement i n the idea leads to new observations that y i e l d new f a c t s t o help judge the relevance of f a c t s already at hand. The elaboration of an hypothesis does not wait u n t i l the problem has been defined, but i t may come at any intermediate time. As we l l , any evaluation need not be the f i n a l step i n the process. I t may be introductory t o new observations or suggestions, according to what happens as a consequence of i t (Dewey, 1933, p. H 5 ) . Other examples of the suggested "steps" i n problem s o l v i n g follow: Burt (1928) Gray (1935) 1. Occurrence of a p e r p l e x i t y 1. S e n s i t i v i t y to the problem 2. C l a r i f i c a t i o n of the per-p l e x i t y 2. Knowledge of the problem con-d i t i o n s , recognition of s i g -n i f i c a n t information 3 . Appearance of suggested solutions 4« Deducing implications of suggested solutions 5. V e r i f y i n g action or ob-ser v a t i o n . Humphrey (1948) Directed t h i n k i n g i n v o l v e s : 1. A problem s i t u a t i o n 2. Motivating f a c t o r s 3 . T r i a l and e r r o r 4. Use of as s o c i a t i o n and images 5. A f l a s h of i n s i g h t (the place of 3, 4» snd 5 v a r i e s with the problem) 6. Some applications i n action . Bloom and Brader (1950) "problem s o l v i n g c h a r a c t e r i s t i c s " are: 1. Understanding of the nature of the problem 2. Understanding of the ideas contained i n the problem 3 . General approach t o the s o l u t i o n of the problem 4. A t t i t u d e towards the s o l -u t i o n of the problem. 3 . Suggested s o l u t i o n or hy-pothesis 4. Subjective evaluation, does the proposed s o l u t i o n work? 5. Objective t e s t 6. Conclusion or g e n e r a l i z a t i o n . Burock (1950) 1. Clear formulation of the prob-lem 2. Preliminary survey of a l l concepts of the material 3. Analysis i n t o major v a r i a b l e s 4. Locating a c r u c i a l aspect of the problem 5. A p p l i c a t i o n of past experience 6. Varied t r i a l s 7. Control 8. E l i m i n a t i o n of sources of error 9. V i s u a l i z a t i o n . Kingsley and Garry (1957) 1. A d i f f i c u l t y i s f e l t 2. The problem i s c l a r i f i e d and defined 3. A search f o r clues i s made 4. Various suggestions appear and are t r i e d out 5. A suggested s o l u t i o n i s accepted 6. The s o l u t i o n i s tested. Both Brownell (1942, p. 432) and K i l p a t r i c k (1967, p. 20) caution against the tendency to misuse conceptual frameworks such as those above 30 by assuming that i n r e a l i t y , problem s o l v i n g occurs i n w e l l defined sequential stages. In the 1930's students were taught t o use the technique of formal analysis to solve arithmetic problems. Students were encouraged to ask themselves a sequence of questions, such as " ( l ) What i s the given? (2) What i s to be found? (3) What i s to be done? (4) What i s a close estimate of the answer? [Burch, 1953, p. 47]." These questions were to be asked before any computational work was done. Brownell (1942) i n d i c a t e s that formal analysis, "represents a l o g i c a l pattern of t h i n k i n g which may or may not characterize expert t h i n k i n g on the part of adults, but c e r t a i n l y has not yet been shown t o characterize good t h i n k i n g on the part of c h i l d r e n [p. 432]." Formal analysis has been shown to be i n e f f e c t i v e and troublesome f o r the student. To study the e f f e c t of formal a n a l y s i s , Burch (1953), conducted a study using 305 elementary school c h i l d r e n who had been t r a i n e d through the use of formal analysis to solve arithmetic problems. He found that students g e n e r a l l y attained higher scores on the t e s t which d i d not require formal analysis than on one which d i d . In an attempt to determine whether students used formal analysis even i f not required to do so, Burch selected 51 students from the sample and asked them to think aloud as they solved a s e r i e s of arithmetic problems. Out of approximately f i v e hundred problems attempted during the interviews, formal steps were used i n only two instances. Furthermore, when students were questioned f o l l o w i n g the interview, each of them i n d i c a t e d that he never used formal analysis except when required to do so. Many reported that when they t r i e d to use formal a n a l y s i s , they became confused. Burch noted that "one explanation of the inadequacy of formal analysis may be that i t fragments the problem i n t o i s o l a t e d parts [p. 47]." One of the best known models of problem s o l v i n g was given by-Wallas (1926). Wallas' stages of problem s o l v i n g are: ( l ) Preparation, c l a r i f y i n g and d e f i n i n g the problem, (2) incubation, unconscious mental a c t i v i t y , (3) i n s p i r a t i o n , s o l u t i o n appears suddenly, and (4) v e r i f i c a t i o n , checking the s o l u t i o n . The stages of incubation and i n s p i r a t i o n , by d e f i n i t i o n , are unobservable mental processes. However, both Polya (1945, 1962, 1965) and Hadamard (1945) r e a d i l y acknowledge the unconscious, preconscious or sometimes "fringe-conscious" a c t i v i t y leading to the s o l u t i o n of a d i f f i c u l t problem. Green (1966) observed that the Wallas model may characterize the research e f f o r t s of s c i e n t i s t s working on d i f f i c u l t problems, but i t f a i l s to describe the high school student t r y i n g to solve an algebra problem. Wickelgren's (1974) model of mathematical problem s o l v i n g has been influenced by current work i n a r t i f i c i a l i n t e l l i g e n c e and computer simulation of human problem s o l v i n g done by Newell and Simon. The p r i n c i p a l aim of the Wickelgren model i s to present the elementary p r i n c i p l e s necessary to solve mathematical problems of e i t h e r the "to f i n d " or the "to prove" character, but not problems of d e f i n i n g "mathematically i n t e r e s t i n g " axiom systems (Wickelgren, 1974, P» 2 ) . The procedures i n Wickelgren's model include inference, t r i a l and error, state evaluation, subgoal, c o n t r a d i c t i o n , working backwards, and r e c a l l of r e l a t e d problems. These procedures are t o be used by the problem so l v e r only i f he can't solve the problem by the d i r e c t use of algorithms. A d e s c r i p t i o n of Wickelgren's model follows: Inference: To draw inferences from e x p l i c i t l y and i m p l i c i t l y presented information that s a t i s f y one or both of the f o l l o w i n g two c r i t e r i a : (a) the inferences have frequently been made i n the past from the same type of information; (b) the i n -ferences are concerned with properties (variables, terms, expressions, and so on) that appear i n the goal, the given, 32 or inferences from the goal and the givens [p. 23]. Drawing inferences i s the f i r s t problem s o l v i n g procedure employed i n attempting to solve a problem. The goal or the givens are e s s e n t i a l l y expanded by br i n g i n g to bear a l l of the knowledge the problem s o l v e r has concerning the problem. Random T r i a l and E r r o r : to apply the allowable operations to the given i n a random fashion [p. 46]• Systematic T r i a l and E r r o r : to produce a mutually exclusive and exhaustive l i s t i n g of a l l sequences of actions up to some maximum length [p. 47 !• C l a s s i f i c a t o r y T r i a l and E r r o r : to organize sequences of actions i n t o classes that are equivalent with respect to the s o l u t i o n of the problem [p. 473» In t h i s case i f one sequence of actions within a c l a s s w i l l solve (not solve) the problem, then a l l the other sequences of actions within the same c l a s s w i l l probably also solve (not solve) the problem. State Evaluation and H i l l Climbing: This method has two parts: (a) d e f i n i n g an 'evaluation function' over a l l states (the set of a l l the expressions that e x i s t i n the world of the problem at a given time) i n c l u d i n g the goal state and (b) choosing actions at any given state to achieve a next state with an evaluation c l o s e r t o that of the goal. P i c k i n g an action on the basi s of such a l o c a l evaluation of i t s consequences i s known as ' h i l l climbing' [p. 67]. An example of state evaluation and h i l l climbing i s given i n the following example: You have a p i l e of 24 coins. Twenty-three of these coins have the same weight, and one i s heavier. Your task i s to determine which coin i s heavier and to do so i n the minimum number of weighings. You are given a beam balance ( s c a l e ) , which w i l l compare the weight of any two sets of coins out of the t o t a l set of 24 coins. A s u i t a b l e evaluation function f o r s o l v i n g t h i s problem would be the number of coins whose c l a s s i f i c a t i o n as heavy or l i g h t i s known. At the beginning of the prob-lem, the value of the function i s zero, since none of the 24 coins i s known to be e i t h e r heavy or l i g h t . In the goal state, the heavy-light c l a s s i f i c a t i o n of a l l 24 coins i s known, so the value of the function i s 24. Thus, a h i l l c l i m b i n g a p p r o a c h w o u l d choose an a c t i o n a t e a c h node t h a t m a x i m i z e d t h e number o f c o i n s whose h e a v y - l i g h t c l a s s i f i c a t i o n i s known [ p . 7l]« S u b g o a l : t o a n a l y z e a p r o b l e m i n t o s u b p r o b l e m s o r t o b r e a k i t i n t o p a r t s [ p . 9l]« The p u r p o s e o f t h i s p r o c e d u r e i s t o r e p l a c e a s i n g l e d i f f i c u l t p r o b l e m w i t h two o r more s i m p l e r p r o b l e m s . C o n t r a d i c t i o n ; P r o v i n g t h e g o a l c o u l d n o t p o s s i b l y be o b t a i n e d f r o m t h e g i v e n s [ p . 109]. The method o f c o n t r a d i c t i o n c a n be a p p l i e d i n t h e f o l l o w i n g f o u r ways: I n d i r e c t P r o o f : t o assume t h e c o n t r a r y i s t r u e and show t h a t t h e c o n t r a r y s t a t e m e n t i n c o m b i n a t i o n w i t h t h e g i v e n s , r e s u l t s i n a c o n t r a d i c t i o n [ p . I l l ] , M u l t i p l e C h o i c e - S m a l l S e a r c h Space: i n p r o b l e m s i n v o l -v i n g a s m a l l s e t o f a l t e r n a t i v e g o a l s , t o s y s t e m a t i c a l l y a p p l y t h e method o f c o n t r a d i c t i o n t o e v e r y a l t e r n a t i v e g o a l [ p . 115]. • C l a s s i f i c a t o r y C o n t r a d i c t i o n - L a r g e S e a r c h S p a c e : t o d e v i s e an e f f e c t i v e s e a r c h p r o c e d u r e t h a t c o n t r a d i c t s l a r g e c l a s s e s o f a l t e r n a t i v e g o a l s s i m u l t a n e o u s l y [ p . 1 2 6 ] . C l a s s i f i c a t o r y C o n t r a d i c t i o n - I n f i n i t e S e a r c h S p a c e : t o d e v i s e an e f f e c t i v e s e a r c h p r o c e d u r e t h a t c o n t r a d i c t s i n f i n i t e l y l a r g e c l a s s e s on t h e b a s i s o f some common p r o p e r t y [ p . 133]. W o r k i n g Backwards: t o guess a p r e c e d i n g s t a t e m e n t o r s t a t e m e n t s t h a t , t a k e n t o g e t h e r , w o u l d i m p l y t h e g o a l s t a t e m e n t [ p . 1 3 8 ] . W i c k e l g r e n d e s c r i b e s f o u r f u n d a m e n t a l t y p e s o f r e l a t i o n s h i p s between p r o b l e m s t h a t c a n be u s e d b y t h e p r o b l e m s o l v e r . E q u i v a l e n t P r o b l e m s : t o r e c o g n i z e t h a t p r o b l e m s d i f f e r o n l y w i t h r e s p e c t t o t h e names a t t a c h e d t o d i f f e r e n t e l e m e n t s , b u t whose r e l a t i o n s and o p e r a t i o n s a r e i d e n t i c a l [ p . 156]. S i m i l a r P r o b l e m s : t o r e c o g n i z e t h a t two p r o b l e m s s h a r e common e l e m e n t s , and t h e n t o r e c a l l t h e methods u s e d t o s o l v e t h e s i m i l a r p r o b l e m [ p . 153]. S i m p l e r P r o b l e m s : t o pose and s o l v e o r r e c a l l a p r o b l e m w h i c h i s s i m p l e r o r a s p e c i a l c a s e o f t h e more complex p r o b l e m [ p . 157], 34 More Complex Problems t Posing a problem that i s more complex than the given problem and i n which the given problem i s embedded. Then solve the more complex problem [p. 166]. In MacPherson's (1973) terms, Wickelgren's model consists of procedures from both core and h e u r i s t i c s . His d e f i n i t i o n s and examples of random t r i a l and error, subgoals, working backwards, and complex problems correspond to the h e u r i s t i c s of random cases, analysis, inverse deduction, and extension, r e s p e c t i v e l y . Rather than procedures f o r s o l v i n g problems, state evaluation and h i l l climbing, systematic and c l a s s i f i c a t o r y t r i a l and error, and the methods of co n t r a d i c t i o n appear t o be o v e r a l l methods or plans of attacking p a r t i c u l a r types of problems. This model may have some i m p l i c a t i o n s f o r teaching problem s o l v i n g but as a framework f o r c l a s s i f y i n g data i t i s quite r e s t r i c t e d . Some of the procedures, such as subgoals and working backwards, are u s e f u l i n s o l v i n g many kinds of problems and should be included i n a c l a s s i f i c a t o r y scheme. However, many of Wickelgren's procedures appear to be too s p e c i f i c to be used to characterize problem s o l v i n g i n any general sense. A framework f o r c l a s s i f y i n g data should be f l e x i b l e enough to accommodate broad s t r a t e g i e s of problem s o l v i n g as w e l l as narrower t a c t i c s used t o r e a l i z e these s t r a t e g i e s ( K i l p a t r i c k , 1967). Schwieger (1974) describes a model of mathematical problem s o l v i n g based on the i d e n t i f i c a t i o n and d e s c r i p t i o n of eight b a s i c components generated from the l i t e r a t u r e . These basic components are: C l a s s i f y : to recognize pertinent c h a r a c t e r i s t i c s and at t r i b u t e s of mathematical problems or expressions and to s p e c i f y the c l a s s or classes t o which they belong [p. 38]• Deduce: to r e l a t e a set of statements so that acceptance of the statements and t h e i r i n t e r r e l a t i o n s h i p s dictates' a p a r t i c u l a r conclusion [p. 4l]« Estimate: to use a v a i l a b l e mathematical information t o make a judgement of measurement or of a r e s u l t of c a l c u l a -t i o n [p. 44]• 35 Generate Pattern: to put known or a v a i l a b l e mathematical data i n t o a systematic arrangement [p. 47]• Hypothesize: to recognize or to generate c o n d i t i o n a l r e l a t i o n s h i p s between mathematical statements [p. 50]• Translate: to substitute f o r one mathematical form, an equivalent representation [p. 53]. T r i a l and E r r o r : to apply knowledge to a mathematical problem i n an unorganized manner [p. 56]. V e r i f y : to apply data to a hypothesis i n t e s t i n g i t s v a l i d i t y [p. 59], Schwieger also i d e n t i f i e d the f o l l o w i n g hierarchy among the basic components (p. 80).. A s o l i d arrow i n d i c a t e s that the a b i l i t y at the t a i l of the arrow i s a p r e r e q u i s i t e t o the a b i l i t y at the head of the arrow. The dashed arrows i n d i c a t e some of the more common t a s k - s p e c i f i c _ ^ - _ — ^ » ' C l a s s i f y T r i a l of E r r o r Estimate \ Translate Generate Patterns x * . A. X Hypothesize N X Deduce V e r i f y * " " p r e r e q u i s i t e s , ( i . e . " a b i l i t y to hypothesize may depend on p r i o r generation of p a t t e r n s . " ) . A problem so l v e r may return to a co-nponent higher on the diagram at any time. Schwieger (pp. 36-39) claims "These eight components are considered t o be basic t o any t h i n k i n g i n mathematical problem solving, and any mathematical problem s o l v i n g can be explained i n terms of them.", and that, i n f a c t , these basic components are independent of each other. However, there i s very l i t t l e evidence to support t h i s claim of independence. Schwieger reported the analysis of only two problems coded using t h i s model. 36 I f the model i s to be used to analyze problem solving, some of the components appear to be not defined c l e a r l y enough f o r r e l i a b l e coding. This i s e s p e c i a l l y true f o r the components of deduce, hypothesize, and estimate. The current i n t e r e s t i n the use of " h e u r i s t i c s " i n mathematical problem s o l v i n g i s due p r i n c i p a l l y to Polya (l957t 1962, 1965). Polya's model includes a v a r i e t y of procedures, both general and s p e c i f i c , f o r s o l v i n g mathematical problems i n a number of content areas. A po r t i o n of Polya's model i s presented i n terms of a l i s t o f questions one asks himself as he t r i e s to solve a problem. He postulates that these correspond to mental a c t i o n . Polya's l i s t includes d i f f e r e n t forms of questioning geared to d e f i n i n g and approaching d i f f i c u l t and u n f a m i l i a r mathematical tasks. Polya's (1957) mathematical c h e c k l i s t i n c l u d e s : Understanding the Problem What i s the unknown? What are the data? What i s the condition? Is i t pos s i b l e to s a t i s f y the condition? I s the condition s u f f i c i e n t to determine the unknown? Or i s i t i n s u f f i c i e n t ? Or redundant? Or contradictory? Draw a f i g u r e . Introduce s u i t a b l e notation. Separate the various parts of the condition. Can you write them down? 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? Do you know a r e l a t e d problem? Do you know a theorem that could be useful? Look at the unknown! And t r y to think of a f a m i l i a r problem having the same or a s i m i l a r unknown. Here i s a problem r e l a t e d t o yours and solved before. Could you use i t ? Could you use i t s r e s u l t ? Could you use i t s method? Should you introduce some a u x i l i a r y element i n order t o make i t s use possible? Could you restate the problem? Could you r e s t a t e i t s t i l l d i f f e r e n t l y ? Go back to d e f i n i t i o n s . I f you cannot solve the proposed problem t r y t o solve f i r s t some r e l a t e d problem. Could you imagine a more ac 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 a part of the problem? Keep only a part of the condition, drop the other part; how f a r i s the unknown then determined, how can i t vary? Could you 37 derive something u s e f u l from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or the data, or both i f necessary, so that the new unknown and the new data are nearer to each other? Did you use a l l the data? Did you use the whole condition? Have you taken i n t o account a l l e s s e n t i a l notions involved i n the problem? Carrying Out the Plan Carrying out your plan of the s o l u t i o n , check each step. Can you see c l e a r l y that the step i s correct? Can you prove that i t i s correct? Looking Back Can you check the r e s u l t ? Can you check the argument? Can you derive the r e s u l t d i f f e r e n t l y ? Can you see i t at a glance? Can you use the r e s u l t , or the method, f o r some other problem [p. x v i , x v i i ] . The model consists of four phases, but i t i s not t o be implied that i n r e a l i t y these phases always occur i n sequence, or that a given problem so l v e r w i l l e x h i b it behavior characterized by every phase during the s o l u t i o n of a problem. This model i s expanded, (Polya, 1957) i n a "Short D i c t i o n a r y of H e u r i s t i c s " and i n two of Polya*s other books (Polya, 1962, 1965). The model includes procedures from drawing f i g u r e s t o g e n e r a l i z a t i o n . Most of these are e i t h e r expansions of the questions from the l i s t or are explained i n terms of combinations of questions from the l i s t . These procedures include g e n e r a l i z a t i o n , v a r i a t i o n of the problem, and working backwards. D i f f i c u l t i e s i n attempting to use Polya's c h e c k l i s t t o analyze the protocols of f i f t y - s i x eighth grade students of above average a b i l i t y were i n d i c a t e d by K i l p a t r i c k (1967) who stated: Attempts to apply the c h e c k l i s t to several protocols from the p i l o t study demonstrated c l e a r l y that whatever merits Polya's l i s t has f o r teaching problem solving, i t i s of l i m i t e d use-fulness, as i t stands, f o r c h a r a c t e r i z i n g the behavior of these subjects. Many of the categories were unoccupied: subjects seem-i n g l y d i d not exhibit behavior even remotely resembling actions suggested by the h e u r i s t i c questions. For example, no subject 38 asked themselves aloud whether they were using a l l of the e s s e n t i a l notions of the problem. Furthermore, the categories were, not defined c l e a r l y enough f o r r e l i a b l e coding [p. 44] • K i l p a t r i c k (I967) used a modified c h e c k l i s t and "Process-sequence" system based on Polya's model to analyze the protocols of the subjects i n h i s study. Non-sequential behavior, such as "draw f i g u r e " , " r e c a l l s same or r e l a t e d problem", "uses successive approximation", etc., was checked on the c h e c k l i s t the f i r s t time i t was observed, and repeated occurrences were not recorded. The sequential behavior was t r a n s l a t e d i n t o coding symbols and recorded i n t h e i r order of occurrence. These procedures were: R Reading and t r y i n g to understand the problem D Deduction from condition E S e t t i n g up equation T T r i a l and er r o r C Checking s o l u t i o n The three symbols D, E, and T were followed by a number from 1 t o 5 that represented the outcome of the process. For example, DI stands f o r an incomplete deduction. A f u l l explanation o f t h i s coding system can be found i n K i l p a t r i c k ' s (1967, Appendix F) d i s s e r t a t i o n . When K i l p a t r i c k applied t h i s coding system to the students' tape recorded protocols, only a few of the procedures from Polya's model were observed. For instance, few students v a r i e d the conditions of the problem or attempted to derive a s o l u t i o n by another method. Most of the subjects drew f i g u r e s while s o l v i n g the problems, but the frequency with which f i g u r e s were drawn was unrelated t o the other procedures or to success i n s o l v i n g the problems. Both t r i a l and e r r o r (P < . 0 5 ) and successive approximation (P < .01) were co r r e l a t e d with successful problem s o l v i n g . Checking, as i d e n t i f i e d by the coding system, was also r e l a t e d (P < .05) to the number of correct s o l u t i o n s . K i l p a t r i c k also compared the r e s u l t s of the coding system f o r each subject with h i s performance on a battery of t e s t s from the National Longitudinal Study of Mathematical A b i l i t i e s (NLSMA) f i l e s . He found that subjects who attempted to set up and use equations were s i g n i f i c a n t l y superior to those students who d i d not use equations on measures of mathematics achievement, general reasoning, word fluency, q u a n t i t a t i v e a b i l i t y , and r e f l e c t i v e - i m p u l s i v e s t y l e . Of those subjects who d i d not use equations, those that used t r i a l and er r o r were higher than those students who d i d not use t r i a l and er r o r i n achievement and qu a n t i t a t i v e a b i l i t y . The subjects who used the l e a s t t r i a l and er r o r and d i d not use equations had the most trouble with word problems, spent the l e a s t amount of time on them, and got the fewest number of correct s o l u t i o n s . Using a modified v e r s i o n of K i l p a t r i c k ' s coding system based on Polya's model, Webb (1975) analyzed the problem s o l v i n g a b i l i t y of f o r t y second year high school algebra students. The students were interviewed i n d i v i d u a l l y and asked t o think aloud as they solved eight mathematical problems. The protocols were recorded and coded at a l a t e r date. Each student was given a t o t a l score from zero t o f i v e f o r each of the problems. The score was based on the sum of subscores f o r approach, plan, and r e s u l t . Using regression a n a l y s i s , Webb found that the mathematical achievement component accounted f o r 50/£ of the variance i n the t o t a l score and the h e u r i s t i c strategy components accounted f o r an a d d i t i o n a l 13% of the variance. Kantowski (1975) analyzed the problem s o l v i n g a b i l i t y of eight above average a b i l i t y n i n t h grade algebra students as they learned to solve problems i n geometry. Her study was comprised of four phases: a pretest, a readiness i n s t r u c t i o n phase i n the use of selected h e u r i s t i c s of Polya's, an i n s t r u c t i o n i n geometry phase using teaching s t r a t e g i e s 40 based on the same h e u r i s t i c s , and a p o s t t e s t . During each of the phases, the subjects were asked to think aloud as they solved problems and t h e i r protocols were recorded and then analyzed using a modified v e r s i o n of K i l p a t r i c k * s coding scheme. A score was assigned to each problem based on the procedures used by the subject as w e l l as the s o l u t i o n . Kantowski's study was a c l i n i c a l exploratory study to determine the procedures used by students as they learned t o solve problems i n geometry. The objective of the study was to seek r e g u l a r i t i e s that would generate hypotheses f o r f u r t h e r experimental study and not to state conclusions. Kantowski found that the increased use of h e u r i s t i c s and the development of problem s o l v i n g a b i l i t y were p o s i t i v e l y c o r r e l a t e d . The use of "looking back" s t r a t e g i e s d i d not increase as problem s o l v i n g a b i l i t y developed nor d i d i t appear to be r e l a t e d to success i n problem s o l v i n g . Kantowski noted the l e v e l of r i g o r required i n the use of these s t r a t e g i e s may be beyond students who are j u s t beginning to study i n a content area. Polya's model has been modified and used i n research t o analyze problem s o l v i n g . MacPherson's model of h e u r i s t i c s may be considered as a refinement of P o l y a r s model. Each h e u r i s t i c can be accounted f o r by a combination of procedures from Polya's work. For example, the h e u r i s t i c of i n v a r i a t i o n - f i x a t i o n can be described i n terms of Polya's l i s t . The h e u r i s t i c of f i x a t i o n i s the renaming of a v a r i a b l e as a constant and then attempting to solve the new problem and study i t ' s s o l u t i o n i n order t o gain some i n s i g h t i n t o the given problem. In terms of Polya's model, the f o l l o w i n g questions would have to be asked: I f you cannot solve the proposed problem t r y to solve some r e l a t e d problem f i r s t . Could you change the unknown or the data (rename a v a r i a b l e as a constant)? Carry out your plan (on the new problem) of the s o l u t i o n . Can you use the r e s u l t , or the method, f o r some other problem ( i n t h i s case the o r i g i n a l one)? One of the major advantages of using MacPherson's model to analyze problem s o l v i n g may be the r e l a t i v e l y small number of h e u r i s t i c s (twelve), as compared to Polya's c h e c k l i s t of t h i r t y - s i x questions. Models of problem s o l v i n g vary i n complexity from the 3 or 4 steps of "formal a n a l y s i s " t o those of Wickelgren, Schwieger, and Polya who attempt d e t a i l e d d e s c r i p t i o n s of procedures used i n mathematical problem s o l v i n g . Of these models, Polya's has been used i n recent research to analyze problem s o l v i n g . The model was modified by K i l p a t r i c k , with other modifications by Webb and Kantowski, and used at d i f f e r e n t grade l e v e l s . Wickelgren's model seems inappropriate to use f o r analyzing problem s o l v i n g . His model includes some procedures s i m i l a r to Polya's but many of them are o v e r a l l plans f o r attacking problems rather than problem s o l v i n g procedures. Many of the procedures i n Schwieger's model appear to overlap and are not c l e a r l y defined enough t o use i n analyzing problem s o l v i n g . This review of the l i t e r a t u r e has r a i s e d several important questions i n terms of the u s a b i l i t y of models of mathematical problem s o l v i n g f o r analyzing problem s o l v i n g . These questions are: 1. Are the procedures from the model used by the subjects? 2. Do the procedures from the model describe the problem s o l v i n g process, i . e . , are the procedures i n the model broad enough to account f o r a l l of the processes used by the subject and yet not so broad that no r e a l d i s t i n c t i o n can be made? 3. Are the procedures from the model defined c l e a r l y enough t o be coded r e l i a b l y ? 42 In terms of the u s a b i l i t y of MacPherson's model to analyze mathematical problem solving, an attempt was made to answer each of these questions. F i e l d Independence In the l a s t decade, psychologists have attempted to revive and extend the study of dominant patterns or modes of cognitive behavior. One speaks of these dominant patterns or modes as cognitive s t y l e ( S p i t l e r , 1970, p. 1) and c l a s s i f i e s together i n d i v i d u a l s who.typically use the same s t y l e . The study of cognitive s t y l e s , which began i n observations of how i n d i v i d u a l s perceive and categorize information, has gradually broadened to include the operation of these s t y l e s i n i n t e l l e c t u a l tasks such as problem-solving ( K i l p a t r i c k , 1967, pp. 20-21). Modern experimental work on cognitive s t y l e began with Witkin and hi s colleagues (Witkin, Dyk, Faterson, Goodenough, and Karp, 1962),, : who found that people d i f f e r i n the way they o r i e n t themselves i n space. When a subject i s seated i n a t i l t e d c h a i r and the room i s t i l t e d independently of the chair, the c o n f l i c t i n g v i s u a l cues and b o d i l y sensations often make i t d i f f i c u l t f o r him to b r i n g e i t h e r h i s body or the room i n t o a v e r t i c a l p o s i t i o n . Subjects seem to make consistent errors on t h i s task and t h e i r score i s c o r r e l a t e d with t h e i r performance on the Embedded Figures Test (EFT), i n which they are asked to f i n d a p a r t i c u l a r simple f i g u r e within a l a r g e r complex design (Witkin, Ch. 4)« A subject whose score i s above the median f o r a given sample i s s a i d to be f i e l d independent and those whose score i s below the median are s a i d t o be f i e l d dependent. F i e l d independence-dependence i s an index of perceptual components. F i e l d independence represents the a b i l i t y to. overcome an embedding context and perceive an item as d i s t i n c t from i t s background. 44 Research i n d i c a t e s (Witkin, 1962, Walsh, 1974) that the construct of f i e l d independence i s not s i g n i f i c a n t l y c o r r e l a t e d with IQ l e v e l . Witkin found s i g n i f i c a n t c o r r e l a t i o n between the construct of f i e l d independence and a group of subtests of WISC (Wechsler I n t e l l i g e n c e Scale f o r Child r e n ) ; Block Design, Object Assembly and P i c t u r e Completion; however, there was n o n - s i g n i f i c a n t c o r r e l a t i o n between the construct and v e r b a l comprehension and arithmetic subtest scores of WISC. Hence " . . . i n t e l l i g e n c e t e s t scores cannot be i n t e r p r e t e d t o mean field-independent c h i l d r e n are of generally superior i n t e l l i g e n c e [Witkin, 1962, p. 70]." Research has shown that f i e l d independence i s r e l a t e d to mathematical achievement. In a study of 100 grade nine boys, Rosenfeld (1958) examined the r e l a t i o n s h i p of mathematical a b i l i t y as measured on scores of the Progressive Achievement Test and performance on the EFT. S i g n i f i c a n t C o r r e l a t i o n s of - . 5 6 , - . 3 2 , - . 6 4 , (P < . 0 5 ) , were found r e l a t i n g f i e l d independent scores on the EFT (used time as subject's score: f i e l d independent below the median, f i e l d dependent above the median) and the t o t a l mathematics score, mathematical reasoning score, and the mathematical fundamentals score, r e s p e c t i v e l y . The poor mathematics students were: more f i e l d dependent than the good mathematics students. Few studies have i n v e s t i g a t e d the r e l a t i o n s h i p between f i e l d indepen-dence and problem s o l v i n g i n mathematics. However, the f i e l d independent construct has been used i n studying problem s o l v i n g i n other areas. Saarni (1972) i n v e s t i g a t e d d i f f e r e n c e s i n problem s o l v i n g as a f u n c t i o n of cognitive s t y l e . She proposed that Piaget's development of l o g i c a l t h i n k i n g would provide an o v e r a l l framework f o r understanding problem s o l v i n g performance and that Witkin's f i e l d independence would provide information about i n d i v i d u a l d i f f e r e n c e s i n problem s o l v i n g behavior within each Piagetian developmental l e v e l . S i x t y - f o u r students, eight male and e i g h t f e m a l e , p e r grade were randomly s e l e c t e d f r o m g r a d e s s i x t h r o u g h n i n e . Two P r o d u c t i v e T h i n k i n g p r o b l e m s , "The M i s s i n g J e w e l " and "The O l d B l a c k House", c o n s t i t u t e d t h e p r o b l e m s o l v i n g t a s k s . The s u b j e c t s ' p e r f o r m a n c e on each p r o b l e m was s c o r e d i n f o u r c a t e g o r i e s : ( a ) number o f r e l e v a n t c l u e s c i t e d ( p u z z l i n g f a c t s i n s e c o n d p r o b l e m ) , (b) number o f c o r r e c t a n a l y t i c c h o i c e s made, ( c ) number o f p l a u s i b l e i d e a s g e n e r a t e d f o r s o l u t i o n , and ( d ) s c o r e o f 1 t o 5 f o r speed and adequacy o f a t t a i n m e n t o f t h e c o r r e c t s o l u t i o n . The n e s t i n g o f f i e l d i n d e p e n d e n c e w i t h i n P i a g e t i a n l e v e l s d i d n o t y i e l d any s i g n i f i c a n t (P <.05) d i f f e r e n c e s i n p r o b l e m s o l v i n g p e r f o r m a n c e . S a a r n i c o n c l u d e d t h a t : The c o n s t r u c t f i e l d i n d e p e n d e n c e a p p e a r s t o have d o u b t f u l i m p l i c a t i o n s f o r complex p r o b l e m s o l v i n g p e r f o r m a n c e . The a n a l y s i s i n d i c a t e t h a t f i e l d i n d e p e n d e n c e w i t h i n e ach P i a g e t i a n l e v e l does n o t a f f e c t complex, m u l t i - s t e p p r o b l e m s o l v i n g p e r -formance as m a n i f e s t e d i n t h e P r o d u c t i v e T h i n k i n g p r o b l e m s . T h i s does n o t i n v a l i d a t e t h e r o l e f i e l d i n d e p e n d e n c e m i g h t have i n d e t e r m i n i n g p e r f o r m a n c e on p r o b l e m s w h i c h a r e more p e r c e p t u a l l y bound a n d / o r r e l a t i v e l y n o n - v e r b a l [p.2 2 ] , F a r r (1968) c o n d u c t e d a s t u d y t o d e t e r m i n e w h e t h e r f i e l d i n d e p e n d e n c e and p r o b l e m d i f f i c u l t y were r e l a t e d t o p r o b l e m s o l v i n g p e r f o r m a n c e on o r g a n i z e d and d i s o r g a n i z e d r e o r g a n i z a t i o n - t y p e p r o b l e m s . Two h u n d r e d n i n e t y e i g h t s t u d e n t s between t h e ages o f 18 and 24, e n r o l l e d i n u n d e r -g r a d u a t e e d u c a t i o n c o u r s e s were u s e d i n t h e s t u d y . The s t u d e n t s were asked t o s o l v e anagram problems r e p r e s e n t i n g v e r b a l , and match s t i c k p r o b l e m s r e p r e s e n t i n g n o n - v e r b a l , r e o r g a n i z a t i o n - t y p e p r o b l e m s . E a c h t y p e o f p r o b l e m was p r e s e n t e d i n o r g a n i z e d and d i s o r g a n i z e d f o r m s , w i t h e a s y and d i f f i c u l t p r o b l e m s i n each f o r m . She f o u n d t h a t when m a t h e m a t i c s a p t i t u d e was h e l d c o n s t a n t , f i e l d i n d e p e n d e n t s t u d e n t s g e n e r a l l y r e c e i v e d s i g n i f i c a n t l y h i g h e r s c o r e s t h a n f i e l d dependent s t u d e n t s on n o n - v e r b a l b u t n o t on v e r b a l p r o b l e m s , r e g a r d l e s s o f p r o b l e m o r g a n i z a t i o n o r d i f f i c u l t y . However, both Walsh (1974) and Cooperman (1974) found f i e l d independent students performed s i g n i f i c a n t l y b e t t e r than f i e l d dependent students i n s o l v i n g anagram problems of moderate d i f f i c u l t y . The students i n Walsh's study were 12 and 13 years o l d while those i n Cooperman's study were 10 years o l d . The r e s u l t s of research i n v o l v i n g f i e l d independence and problem s o l v i n g i n a non-mathematical s e t t i n g are mixed. Both Saarni and Farr found n o n - s i g n i f i c a n t d i f f e r e n c e s between f i e l d independence and verbal problem s o l v i n g . However, studies by Walsh and Cooperman found f i e l d independent students performed s i g n i f i c a n t l y b e t t e r than f i e l d dependent subjects on verbal problems. The r e s u l t s of Farr's study also i n d i c a t e that with mathematical a b i l i t y held constant, f i e l d independence i s s i g n i f i c a n t l y r e l a t e d to problem s o l v i n g i n a non-verbal s e t t i n g . Two research studies have examined the r e l a t i o n s h i p of f i e l d indepen-dence and a subject's a b i l i t y to change h i s mode of attack i n s o l v i n g non-mathematical problems. The E i n s t e l l u n g problems of Luchins (1942) have been used t o study the e f f e c t s of the " s e t " of a subject upon h i s problem s o l v i n g behavior. A set i s defined as "the tendency of an i n d i v i d u a l t o perseverate i n a given mode of attack [Buetzkow, 1951, P« 219]." The problems require an i n d i v i d u a l to obtain a given quantity of l i q u i d by using various combinations of three j a r s . A, B, and C, with given volumes. The i n i t i a l f i v e problems which a l l have the same so l u t i o n , A + B - 2C, e s t a b l i s h a " s e t " . A measure of the strength of the " s e t " i s contained i n the succeeding two problems which can be solved by a simple d i r e c t procedure, A - C, or by the complex manner used t o solve the f i r s t f i v e problems. F i n a l l y , the terminal problem which can only be solved by the simpler method, i s used as a measure of a b i l i t y to overcome the " s e t " . Guetzkow (1951) c a r r i e d out a study using Luchins' E i n s t e l l u n g problems. He divided h i s subjects i n t o two groups according t o t h e i r set-breaking a b i l i t y . The set breakers were those who adopted the set by using the (A + B - 2 C ) method of s o l u t i o n , rather than the (A - C) method on the c r i t i c a l problems, but who were able to break the set on the terminal problem. The non-set breakers were those subjects who were unable to break the set on the terminal problem. He found the set-breakers d i d s i g n i f i c a n t l y b e t t e r than the non-set-breakers on Thurstone's Gottschaldt 2 Figures Test. Guetzkow also found s i g n i f i c a n t r e l a t i o n s h i p between Thurstone's Gottschaldt Figures Test and the time required to s o l v e the terminal problem. Goodman ( i960) , conducted a s i m i l a r study using college students. These students received the EFT, Thurstone's Gottschaldt Figures Test, and the E i n s t e l l u n g t e s t . No s i g n i f i c a n t d i f f e r e n c e i n performance of e i t h e r of the perceptual t e s t s were found between students who solved the c r i t i c a l E i n s t e l l u n g problems by the short method and those who solved them by the long method. However, s i g n i f i c a n t c o r r e l a t i o n s were found between both the EFT and Thurstone's Gottschaldt Figures Test and the time required t o solve the terminal E i n s t e l l u n g problem. These r e s u l t s , confirming and extending the f i n d i n g s of Guetzkow, i n d i c a t e that set-breaking a b i l i t y i n the E i n s t e l l u n g s i t u a t i o n i s re l a t e d t o f i e l d independence. A mu l t i v a r i a t e design was used by Dodson (1972) to t e s t the relevance Witkin's Embedded Figures Test was adapted from Thurstone's Gottschaldt Figures Test. Color was added to Gottschaldt's black and white o u t l i n e complex f i g u r e s to make them more d i f f i c u l t . Studies c a r r i e d on i n the 1950's have a l l shown s i g n i f i c a n t c o r r e l a t i o n s between these two t e s t s (See Witkin 1962, p. 40) . of 77 concomitant v a r i a b l e s f o r the a b i l i t y to solve i n s i g h t f u l mathematics problems and to determine which of these v a r i a b l e s discriminate best among a b i l i t y groups. I n s i g h t f u l problems are problems which cannot be solved by simple r e c a l l from memory or standard computational algorithms, nor does the s o l u t i o n depend on a s p e c i a l t r i c k (Dodson, 1973. p» 3 ) . A random sample of 1123 grade eleven students was se l e c t e d from those students p a r t i c i p a t i n g i n NLSMA who were c u r r e n t l y e n r o l l e d i n mathematics. A l l of the data f o r each subject were obtained from NLSMA. The students were placed i n one of s i x a b i l i t y groups depending upon t h e i r score on a t e s t of i n s i g h t f u l mathematics problem s o l v i n g . The 77 d e s c r i p t o r v a r i a b l e s were c l a s s i f i e d i n t o f i v e major categories: ( l ) mathematics aptitude and achievement v a r i a b l e s , (2) psychological v a r i a b l e s (e. g. a t t i t u d e s , anxiety, and cognitive f a c t o r s ) , (3) teacher background v a r i a b l e s , (4) school and community v a r i a b l e s and (5) mathematics curriculum v a r i a b l e s . Dodson found the mathematical achievement v a r i a b l e s to be the strongest d i s c r i m i n a t o r s among a b i l i t y groups and that the cognit i v e v a r i a b l e s were second strongest. Of the cognitive v a r i a b l e s , Dodson found the best i n s i g h t f u l mathematics problem solvers tended to have the highest scores on the reasoning t e s t s - verbal and l o g i c a l reasoning as w e l l as the numerical reasoning t e s t . F i e l d independence was found t o discriminate as w e l l (P < . 0 0 1 ) among the a b i l i t y groups as d i d the poorest reasoning v a r i a b l e s and was included i n the composite l i s t of the strongest c h a r a c t e r i s t i c s of a successful i n s i g h t f u l mathematics problem solver (Dodson, 1973. P» 122). Analyzing the problem s o l v i n g procedures of 40 second year high school algebra students, Webb (1974) supported the f i n d i n g of Dodson that mathematical achievement i s the strongest component i n accounting f o r problem s o l v i n g a b i l i t y . Using regression analysis, Webb found that mathematical achievement accounts f o r 50% of the variance i n the t o t a l scores from h i s problem s o l v i n g inventory. With mathematical achievement, ve r b a l reasoning and negative anxiety entered i n the regression equation f i r s t , f i e l d independence d i d not add s i g n i f i c a n t l y to the amount of variance on the t o t a l score already accounted f o r . Research fin d i n g s i n v o l v i n g the f i e l d independent construct and problem s o l v i n g i n non-mathematical settings are mixed, but i n general favor the f i e l d independent subject. F i e l d independent students are more able to overcome a problem s o l v i n g set than t h e i r f i e l d dependent counterparts. This would seem to be an important c h a r a c t e r i s t i c of a good problem so l v e r i n mathematics, that i s , when s o l v i n g a problem i f a mode of attack i s not leading t o a so l u t i o n , rather than continue to use t h i s mode, change the procedures being used and t r y a d i f f e r e n t method of attacking the problem. F i e l d independence has been shown t o r e l a t e s i g n i f i c a n t l y to mathematical reasoning and i n s i g h t f u l problem s o l v i n g . The c h a r a c t e r i s t i c s associated with a f i e l d independent student are: ( l ) a b i l i t y to r e s i s t d i s t r a c t i o n , (2) a b i l i t y to i d e n t i f y the c r i t i c a l elements of a problem, (3) a b i l i t y to separate parts from the whole and recombine them to form a new whole, (4) a b i l i t y to remain independent of i r r e l e v a n t elements, and (5) a b i l i t y to overcome a problem set (Witkin, 1962; Romberg and Wilson, 1969)» In conclusion, the l i t e r a t u r e from two major areas has influenced the design of the present study. The l i t e r a t u r e on models of problem s o l v i n g has r a i s e d three important questions i n terms of the u s a b i l i t y of models of mathematical problem s o l v i n g f o r analyzing problem s o l v i n g behavior. 50 1. Are the procedures from the model used by subjects i n s o l v i n g mathematical problems? 2. Can the procedures from the model be coded r e l i a b l y ? 3. Do the procedures from the model describe the problem s o l v i n g process? In terms of the u s a b i l i t y of MacPherson's model f o r analyzing problem s o l v i n g an attempt was made to answer each of these questions. The c h a r a c t e r i s t i c s associated with a f i e l d independent student i n d i c a t e that these students are b e t t e r problem solvers than the f i e l d dependent students. In terms of the outcome of t h i s study, the f i e l d independent student i s expected t o use a greater v a r i e t y of h e u r i s t i c s from MacPherson's model, to be more w i l l i n g t o change procedures i f he i s not being, successful i n s o l v i n g a problem, and to obtain more correct solutions than the f i e l d dependent student. Chapter I I I PROCEDURES Two major considerations i n analyzing problem s o l v i n g behavior are the choice of subjects and the choice of problems. I f the subjects are too widely diverse i n t h e i r background and a b i l i t i e s , there i s l i t t l e hope of observing any common patterns of t h i n k i n g . The choice of the problem material used must be made with the subjects i n mind. I f the problems are too hard there w i l l be l i t t l e problem s o l v i n g behavior observed. On the other hand, i f the problems are too easy, they may not e l i c i t any problem s o l v i n g behavior at a l l ( K i l p a t r i c k , 1967, p. 32) . Subjects The subjects selected f o r t h i s study were a l l completing the eleventh grade academic mathematics program. The sample consisted of boys and g i r l s , of average a b i l i t y f o r those students e n r o l l e d i n t h i s program, who were p a r t i c i p a t i n g i n a one semester algebra course. A popular modern algebra and trigonometry text was used f o r t h i s course. At the time of the problem s o l v i n g interviews the students were studying the chapter r e l a t e d to the quadratic f u n c t i o n . One of the aims of the study was t o assess the h e u r i s t i c s used by students i n s o l v i n g word problems. Since a large p o r t i o n of the content of second year algebra deals with word problems, students who have completed the course should have a s u f f i c i e n t background f o r d e a l i n g with word problems. An average IQ range was selected i n order to cont r o l the IQ va r i a b l e and hopefully s t i l l obtain a f a i r l y wide range of EFT scores. A p i l o t study had shown that students i n t h i s a b i l i t y range were not unduly threatened by the task of s o l v i n g d i f f i c u l t mathematical problems i n an interview s i t u a t i o n . To determine those students of average a b i l i t y i n the academic mathematical program, a sample of 150 students was randomly drawn from the Algebra I I classes of the schools involved i n the study. The mean IQ score on. the C a l i f o r n i a Test of Mental Maturity f o r t h i s sample was 119.8 with a standard d e v i a t i o n of 10.4. A student was defined to be of average a b i l i t y i f h i s IQ was w i t h i n one-half standard d e v i a t i o n of the mean. A t o t a l of 40 students, i n the IQ range 115 to 125, was randomly selected from 14 Algebra I I classes i n three senior secondary schools (grades 11 and 12) i n the greater Vancouver area. Three students were selected from each of twelve classes and two students from each of the other two c l a s s e s . The classes were taught by ten d i f f e r e n t teachers, no teacher having more than two c l a s s e s . The mean IQ f o r the students i n the study was 119*5 with a standard d e v i a t i o n of 10.2. Ages of the students at the time of the interview ranged from 16 years, 4 months, to 17 years, 4 months, with a median age of 16 years, 11 months. The sample contained 22 boys and 18 g i r l s . P i l o t Study The p i l o t study was conducted from December, 1973 to March, 1974, i n a senior high school i n the same general area as the schools used i n the main study. The i n i t i a l phase of the p i l o t study involved the use of written problem sets to a i d i n the s e l e c t i o n of the problems to be used i n the study. This phase was followed by a s e r i e s of p i l o t 53 interviews. The major aims of the p i l o t study were: 1. Determine i f eleventh grade students of average a b i l i t y could respond w e l l to the interview procedure 2. Select the problems to be used i n the study 3. Modify K i l p a t r i c k ' s coding system or develop a new coding system using MacPherson*s model f o r problem s o l v i n g 4» Develop an interview format 5. P r a c t i c e the interviewing and coding techniques Twelve mathematical problems selected from the areas of number theory, geometry, and algebra were used i n the i n i t i a l t e s t i n g of the p i l o t study. The problems were stated i n both the r e a l world and math world s e t t i n g . These problems, selected on the bas i s of t h e i r p o t e n t i a l to e l i c i t the use of c e r t a i n h e u r i s t i c s , were those that the experimenter f e l t could be solved by approximately h a l f of the students p a r t i c i p a t i n g i n the study. The problems were then randomly d i v i d e d i n t o sets of three i n the same s e t t i n g and a<±irinistered as w r i t t e n examinations through December, 1973, and January, 1974, to 90 students who would complete the Algebra I I course i n February. Throughout the. period, 11 new problems were added to the l i s t and many others were deleted because they proved to be too d i f f i c u l t . The students were asked to work on a problem as long as they wished and not to erase anything. Each student was asked to record the amount of time he spent on each problem. Appendix A contains the problems used i n the p i l o t study. F i f t e e n problems were selected from t h i s l i s t as p o s s i b l e candidates f o r the main study. Over the next three months, twelve students were interviewed using from 3 to 8 of the 15 problems. Each interview was designed to generate as much observable problem s o l v i n g behavior as p o s s i b l e . A room was provided at the high school f o r the interview sessions. Subjects were t o l d that the session would be tape recorded and they should think aloud while working on the problems. The students were also i n s t r u c t e d not to erase anything and i f a diagram was to be modified, t o draw a new one. A l l i n s t r u c t i o n s , except the problem s t a t e -ments, were communicated v e r b a l l y by the interviewer. The problems were typewritten and presented to the students one at a time. I t was determined that grade eleven students of average a b i l i t y are not unduly threatened by the task of s o l v i n g d i f f i c u l t mathematical problems i n the interview s i t u a t i o n . Also, they are able to v e r b a l i z e t h e i r thoughts, e s p e c i a l l y i f they have some success and understand what the interviewer expects. The interview format was rev i s e d to take t h i s i n t o account. The need f o r several p r a c t i c e problems t o acclimate the subjects to s o l v i n g problems aloud became evident i n the p i l o t study when the students involved became accustomed to t h i n k i n g aloud only a f t e r attempting one or two problems. The l i s t of problems was reduced to 5 and two p r a c t i c e problems were added (see Appendix B). This allowed the subject s u f f i c i e n t time to work on the problems and s t i l l complete the interview i n one session. Most students were w i l l i n g t o work f o r up t o 2-|- hours i f given a short re s t period between problems. Four of the problems were selected from the p i l o t study because they d i d e l i c i t the use of d i f f e r e n t h e u r i s t i c s and the f i f t h problem was added i n an e f f o r t to obtain some use of the h e u r i s t i c of symmetry (problem #1, Appendix B). However, a l l f i v e problems could be solved i n a number of d i f f e r e n t ways. A f i n a l four students were interviewed using the 7 problems, two students doing problems i n each of the se t t i n g s , as a check on the problems The protocols were also analyzed using the The Coding System One of the main reasons f o r spacing the p i l o t interview sessions over the three month period was t o allow ample time to evaluate and modify the coding system. The i n i t i a l coding forms were quite s i m i l a r to K i l p a t r i c k ' s (1967, pp. 50-53). Both the c h e c k l i s t and the process sequence had been modified to take i n t o account the procedures from MacPherson's. model. A f t e r each interview period during the p i l o t study, the i n v e s t i g a t o r used both the tape recorded protocols as w e l l as a l l w r i t t e n work to help analyze the subjects' problem s o l v i n g behavior. The problem s o l v i n g behaviors were analyzed i n terms of process-sequence coding symbols and other events were recorded by checking o f f appropriate categories on the c h e c k l i s t . During the attempts to apply the coding system to protocols from the p i l o t study, d i f f i c u l t i e s and shortcomings became apparent. Many of the categories on the c h e c k l i s t were not used. In some instances, the process sequence became very long and complex with some procedures nested i n others. More importantly, i t became evident that the p o s i t i o n of the procedure i n the sequence of the problem s o l v i n g process was more important than i t s frequency of occurance. The f i n a l coding system, used i n the analysis of t h i s study, combined both the check l i s t and coding sequence i n t o a complete sequential coding system. The use of a matrix provided the researcher with a convenient device f o r recording the procedures obtained from the tape recorded protocols. The coding matrix, (see Figure 4), accounts f o r d i f f e r e n t and the interview method, modified coding system. Reading problem Request defn. o f ter-13 R e c a l l same problem R e c a l l r e l a t e d problem R e c a l l problem type R e c a l l r e l a t e d f a c t Draw diagram Modify diagram I d e n t i f y v a r i a b l e S e t t i n g up equations A l g o r i t h m s - a l g e b r a i c Algorithms-arithmetic Guessing Smoothing A n a l y s i s Templation C a s e s - a l l Cases-random Cases-systematic C a s e s - c r i t i c a l Cases-sequential Deduction Inverse deduction I n v a r i a t i o n Analogy Symmetry Obtain s o l u t i o n Checking part Checking s o l u t i o n by subst. i n equation by r e t r a c i n g steps by r e a s n b l e / r e a l i s t i c uncodable Exp. concern-method Exp. concren-algorithm Exp. concern-equation Exp. concern-solution Work stopped-soln. Work stopped-no s o l n . FIGURE -* THE CODING FORM (Reduced by 33%) procedures used simultaneously; procedures which are nested i n others, such as the use of d i f f e r e n t algorithms and diagrams, which might take place while a subject i s using the h e u r i s t i c of random cases; and the sequential order of a l l procedures. At a glance one can t e l l the methods used by a subject, the order i n which he used them, and how s u c c e s s f u l he was. To take i n t o account the sequence of events used i n s o l v i n g a problem, see Figure 3, P' 19. the procedures were grouped i n the coding matrix as follo w s : Preparation Reading problem Request d e f i n i t i o n of terms Sieve R e c a l l same problem R e c a l l r e l a t e d problem R e c a l l problem type R e c a l l r e l a t e d f a c t Draw diagram Modify diagram I d e n t i f y v a r i a b l e S e t t i n g up equations Algorithms-algebraic Algorithms-arithmetic Guessing H e u r i s t i c s Smoothing Analysis Templation C a s e s - a l l Cases-random Cases-systematic C a s e s - c r i t i c a l Cases-sequential Deduction Inverse deduction I n v a r i a t i o n Analogy Symmetry S o l u t i o n Obtain s o l u t i o n Reasonableness Checking part Checking s o l u t i o n by s u b s t i t u t i n g i n equations by r e t r a c i n g steps by r e a s o n a b l e / r e a l i s t i c uncodable Concern Express concern about method Express concern about algorithm Express concern about equation Express concern about s o l u t i o n Work Stopped Work stopped-solution Work stopped-no s o l u t i o n As the coder i d e n t i f i e s these processes from a subject's protocol, he enters a check ) i n the appropriate row and f i r s t empty column. I f the process i s from the h e u r i s t i c category or the sieve (except r e c a l l of same problem r e l a t e d problem, problem type) he enters e i t h e r a 1, 2, or 3, depending upon the outcome: 1 Incomplete 2 Incorrect 3 Correct For example, a 2 i n the algorithm-algebraic row i n d i c a t e s an e r r o r in. the use of the algorithm, whereas, a 2 f o r one of the h e u r i s t i c s , a n alysis, f o r example, i n d i c a t e s the outcome of the problem a f t e r the use of analysis i s i n c o r r e c t , not that the subject committed an e r r o r i n the h e u r i s t i c i t s e l f . I f two or more procedures occur simultaneously then a check or number i s entered i n the same column f o r each one. I f a procedure i s 59 c a r r i e d on longer than a column, i t i s enclosed i n a box with the outcome entered i n the l a s t column enclosed. For example, suppose a subject i s using random cases and considers three cases along with using algorithm, checking part of h i s work and r e c a l l i n g r e l a t e d f a c t . In t h i s instance the "block" f o r random cases would cover ten columns. See the example i n Figure 5« Most of the time the subject, i n checking h i s work or s o l u t i o n , did so by s u b s t i t u t i n g i n an equation, r e t r a c i n g steps or checking to see i f h i s r e s u l t was reasonable, or r e a l i s t i c . I f a subject was checking h i s s o l u t i o n , a check would go i n the appropriate column i n the checking s o l u t i o n row and a check or numbers i n the row which i n d i c a t e d the procedure he used. According to MacPherson's model, a p r o b a b i l i t y (amount of concern) i s attached to the outcome from the pattern box, a p p l i c a t i o n of an algorithm, and the s o l u t i o n obtained by the problem s o l v e r . To d i s c e r n some f e e l i n g f o r these p r o b a b i l i t i e s the concern expressed by a subject was coded i n one of the four categories. As an i l l u s t r a t i o n of the coding system, two sample protocols follow, with the coded forms given i n Figures 5 and 6. Problem #3. A rectangular lawn i s to be formed so that one side of a barn serves as one side of the rectangle. You have 300 f e e t of wire to enclose the other three sides. What are the dimensions of the rectangle i f the area i s to be a maximum size? Why does your s o l u t i o n give the maximum area? Protocol Column # (Student #36, problem #3) A rectangular lawn i s t o be formed so that one side of a barn serves as one side of the rectangle. You have 300 feet of wire to enclose the other three sides. What are the dimensions of the rectangle i f the area i s to be of maximum size? Why does your s o l u t i o n give the maximum area? I don't know. O.K., I've got a rectangular 1 lawn...is to be formed so that one side of the barn serves as one side of the lawn. O.K., 60 l e t ' s draw a rectangle...O.K. barn i s here. Covers 2 that whole side. You have 300 feet of wire to enclose the other three s i d e s . O.K. Let's see - a rectangular lawn i s to be formed so t h a t . . . I don't know 3 how b i g t h i s rectangle has t o be, but I have to get the most s i z e out of i t . That's 300 f e e t . O.K. -Let's see, no equation f o r t h i s one. I ' l l scrap the equation. O.K., Now, i f i t ' s going to need more 4 area f o r my barn, I think I'd use somewhat of a square, or as close t o a square as po s s i b l e . O.K.,. 5 How b i g i s my barn? Let's imagine a 50 foot length 6 across there, 50 down here - r i g h t , O.K., 50 and 50 from there i s 250 so 125 feet long. Now am I using 7 as much area as I could? Now, l e t ' s see, area i s a 125 times 50. O.K., so 0 i s 5 i s 5 i s 2, 2 over 8 there i s 12, i s 1 over the 6, 625O square f e e t . O.K., 9 i f I had a 100 there, so 100, 100, 100, O.K., so 10 100 times 100 i s 0, 0, 0 i s 10,000 square f e e t . How am 11 I going t o get a rectangular lawn? How rectangular i s rectangular? Now, that's the most I can have f o r a square. Now, 0 ,0 ,1 that's r i g h t . That would be 12 the maximum f o r a square. I want a rectangle. O.K., I ' l l give my barn 90 f e e t , 0,K., 90 and 90 i s 180, 13 O.K., that's 120, that's 60 times 60, 60 times 90 14 i s 0, nope, wrong. A...I don't know how I'm going 15 to do t h i s . . . 16 Problem #4: What i s the longest piece of metal rod which can be placed i n a box of dimensions 3 inches by 4 inches by 12 inches? P r o t o c o l Column # (Student #24, problem #4) What i s the longest piece of metal rod which can be placed i n a box of dimensions 3 1 inches by 4 inches by 12 inches? I t ' s not hard t o see the longest piece of metal rod w i l l be from opposite corners, from the bottom opposite corner of one to the top opposite corner of the other. Now the question i s 2 how long that i s ? I know i t s longer than 12. Let's see i f I can work out some kind of t r i a n g l e here. The two 3 ends of the boxes l i k e the ends of the box are 3 by 4, now that would mean that, A, I'm t r y i n g to form 2 t r i -angles here to f i n d out, so I can f i n d out, adjacent, or complementary, or corresponding parts so I can get that long rod as a corresponding part i n t h i s box. Now, i f I 4 had a 3 by 4 inch box, I'm assuming t h i s box i s rectan-gular i n shape. That, I'm j u s t p u t t i n g i n the angles 5 here. I f there i s r i g h t angles i n the box, side-angle 6 side, I forgot how to do that, oh yeah, I want to f i n d 7 out i n that box, I want to f i n d out r i g h t on the two ends, a l i n e going r i g h t through the middle to the oppo-s i t e sides so that one side would connect with the l i n e I have put as the imaginary rod and therefore form a t r i a n g l e , w ell, I sa i d the a l t i t u d e of the box i s 4 inches, and the width i s 3, therefore, I formed a 61 t r i a n g l e with two legs of 3 inches and 4 inches and a hypotenuse that I do not know, and since i t i s a r i g h t angle t r i a n g l e , since I'm assuming i t s a rectangle, a, 4 squared i s equal to the hypotenuse squared, I ' l l say H squared, but should be C squared that the o l d equation. So that's 9 plus 16 i s equal to 25 and that i s 5 squared. Therefore, 5 i s equal to the length of that l i n e , 5 inches. Now, I've found out that one side of my imaginary t r i a n g l e i s 5 inches. Now I know the other side i s 12 inches because of the t h i n g . Again, I am assuming I have a r i g h t angle t r i a n g l e , because of the angles of the box. Therefore, a l l I would do i s 5 squared plus 12 squared i s equal t o C squared. 5 squared i s 25 and 12 squared i s 144 i s equal t o 169. Now, I know I69 i s 13 squared. Therefore, 13 inches must be the length of that rod. Let me think. 13 inches I'm saying. That means I've formed a t r i a n g l e from one part of that box. I ' l l say i t s 13 inches. I t must be because, l i k e I t r i e d to form a whole t r i a n g l e and t r y i n g to f i n d the sides of the t r i a n g l e so that I could assume what the l a s t side i s by mathematical e q u a l i t i e s . . . S o 25, 5 squared, that.one side plus 12 squared which i s 144 i s equal to 169. And 169 i s 13 squared. Therefore the answer i s 13, 13 inches. A coding form summary sheet was also used to summarize the major parts from the coding form and to count errors and procedures used. The number of cycles- end changes was also noted here. A cycle occurs each time the subject attempts to solve the problem. These are determined from the subject's tape recorded protocol f o r the problem e i t h e r by the subject himself, i n d i c a t i n g that he i s going to s t a r t the problem over or from the sequence of procedures used. Usually the subject w i l l read the problem each time he c y c l e s . I f a subject a l t e r s h i s method of attack, e i t h e r by changing the core procedures or h e u r i s t i c s he i s using or by changing h i s o v e r a l l plan, then a change occurs. A f u l l explanation of the coding system i s given i n Appendix C. The Interview Procedure 8 9 10 11 12 13 14, 15 The Embedded Figures Test(EFT), was administered during the l a s t h a l f of A p r i l , 1974» and the problem s o l v i n g interviews were held 62 Reading problem Request defn. of terr.s R e c a l l same problem R e c a l l r e l a t e d problem R e c a l l problem type R e c a l l r e l a t e d f a c t Draw diagram Modify diagram I d e n t i f y v a r i a b l e S e t t i n g up equations Algorithms-algebraic Algorithms-arithmetic Guessing Smoothing A n a l y s i s Templation C a s e s - a l l Cases-random Cases-systematic C a s e s - c r i t i c a l Cases-sequential Deduction Inverse deduction I n v a r i a t i o n Analogy Symmetry Obtain s o l u t i o n Checking p a r t Checking solution] by subst. i n equation by r e t r a c i n g steps by r e a s n b l e / r e a l i s t i c uncodable Exp. concern-method Exp. concren-aleorithm Exp. concern-equation Exp. concern-solution Work stopped-soln. Work stopped-no s o l n . J V1 % 3 i i i j 3 j i 3 1 3 1 1 I 3 y y y I i i 1 1 V yi 1 F I G U R E 5 j E X A M P L E OF C O D I N G S Y S T E M STUDENT #36, P R O B L E M # 3 ( R e d u c e d b y 33?6) 63 Reading problem Request defn. of ter~r.3 R e c a l l same problem R e c a l l r e l a t e d problem R e c a l l problem type R e c a l l r e l a t e d f a c t Draw diagram Modify diagram I d e n t i f y v a r i a b l e S e t t i n g up equations Algorithms-algebraic Algorithms-arithmetic Guessing Smoothing A n a l y s i s Templation C a s e s - a l l Cases-random Cases-systematic C a s e s - c r i t i c a l Cases-sequential Deduction Inverse deduction I n v a r i a t i o n Analogy Symmetry Obtain s o l u t i o n Checking p a r t Checking solution] by subst. i n equation by r e t r a c i n g steps by r e a s n b l e / r e a l i s t l c uncodable Exp. concern-method Exp. concren-algorithm Exp. concern-equation Exp. concern-solution Work stopped-soln Work stopped-no s o l n i J 3 ! t 3 3 i | 3 1 i i 3-! V 1 \/ | i | — FIGURE 6 EXAMPLE OF CODING SYSTEM STUDENT # 2k, PROBLEM # k (Reduced by 33>) during May and June, 1974, i n the senior secondary schools the subjects attended. Each school made a room a v a i l a b l e f o r both t e s t i n g periods. A l l EFT's and interviews were conducted by the experimenter. The subjects were t o l d that at a l a t e r date they would spend about 2|- hours s o l v i n g mathematical problems and t h i n k i n g aloud while they worked on them. I t was emphasized that the purpose was to l e a r n more about how grade eleven students solved problems and was not to make an i n d i v i d u a l diagnosis or evaluation. I t was also emphasized to each i n d i v i d u a l that the outcome of the interview had nothing to do with h i s grade i n Algebra I I nor would the information be made a v a i l a b l e to h i s mathematics teacher. The subjects were t o l d that the interviews would be audio tape recorded to a s s i s t the interviewer i n determining what procedures were used. The EFT was administered i n d i v i d u a l l y to the subjects i n a 50 minute period. The t e s t m a t e r i a l consists of a set of 12 cards with complex fi g u r e s and a set of 8 cards with simple f i g u r e s . The task i s to f i n d a given simple f i g u r e which i s embedded i n a complex pattern. The subject i s asked to describe the complex f i g u r e i n any way he wishes. Then the simple f i g u r e i s shown t o the subject and he i s asked to f i n d i t i n the complex design. When he i n d i c a t e s he has found i t , the interviewer stops timing and the subject o u t l i n e s the f i g u r e with a s t y l u s . I f he i s correct, the time i s noted and the next problem i s presented. I f he i s i n c o r r e c t the subject i s t o l d he i s wrong and may continue to look f o r the sample f i g u r e . A maximum of 180 seconds i s allowed f o r each of the 12 problems (See Witkin, Oltman, Raskin, and Karp, 1971, pp. 16, 17). A f t e r a l l 40 subjects had taken the EFT, the scores were rank ordered and paired. Each subject from a p a i r was randomly assigned to one of two groups: those working problems i n the r e a l world s e t t i n g and those working problems i n the math world s e t t i n g . The problem sets along with the i n s t r u c t i o n s that were given t o the students can be found i n Appendix B. The problems were ordered so that each of them appeared i n p o s i t i o n s 1 to 5 exactly 4 times and each problem was preceded by the same problem 4 times and followed the same problem 4 times. This gave a t o t a l of twenty d i f f e r e n t orderings of the f i v e problems. The orderings were randomly assigned to the twenty p a i r s of subjects. Each student was given the same two warm up problems (except f o r s e t t i n g ) i n the same order. The problem s o l v i n g interviews were held from May 6, to June 7, 1974* The subjects were dismissed from classes f o r e i t h e r the morning or afternoon. A l l subjects from the same mathematics c l a s s were i n t e r -viewed i n consecutive sessions i n order to minimize d i s c u s s i o n between the subjects. No interviews were held on Fr i d a y afternoons. The interviews were tape recorded with the; microphone i n f u l l view of the subject. The subject was asked to do h i s t h i n k i n g aloud, to say everything that came to mind. He was also i n s t r u c t e d to v e r b a l i z e a l l w r i t i n g and diagrams. Whenever the subject f e l l s i l e n t f o r ten seconds or more, the interviewer would ask, "What are you t h i n k i n g now?" or "Can you t e l l me what you are t h i n k i n g about?" or some s i m i l a r probe. The subjects were given the sheet of d i r e c t i o n s and asked t o read them. The importance of t h i n k i n g aloud was stressed. The subject was t o l d that he was free to work the problems any way he saw f i t . I t was i n d i c a t e d that the only kind of question the interviewer would answer would be to define any terms used i n the statement of the problem that were u n f a m i l i a r to the subject* I t was also stated that any di s c u s s i o n of the problem with the interviewer, i n c l u d i n g the correctness of any solutions obtained by the subjects, would take place a f t e r the interview was over. Once a subject s t a r t e d work on a problem he was asked to continue to work on the problem and not to return to i t at a l a t e r time. A f t e r the i n s t r u c t i o n s were c a r e f u l l y presented to the subject, he was given the f i r s t p r a c t i c e problem t o solve. I f the subject had tro u b l e expressing himself or f e l l s i l e n t too often during the time he was working on t h i s problem, the interviewer would remind the subject of the purpose of the interview and give some i n s t r u c t i o n s and examples. For example, the interviewer might work part of the problem t h i n k i n g aloud, or i f the subject had been w r i t i n g equations, the interviewer would go back over the work and v e r b a l i z e what the subject might have been t h i n k i n g . By the end of the second problem, a l l the subjects were t a l k i n g most of the time and i n d i c a t i n g what they were doing. As soon as the subject had f i n i s h e d p r a c t i c i n g on the two sample problems and had discussed any remaining questions on format, he was given the f i r s t of the f i v e problems. Each problem was typewritten at the top of the page with space underneath f o r the subject's work. Extra paper was also a v a i l a b l e . While the subject was engaged i n working the given problem, the interviewer observed the student and i f the subject f e l l s i l e n t , he was encouraged t o v o c a l i z e as much of h i s t h i n k i n g as p o s s i b l e . When each problem was completed, the subject had the option of e i t h e r stopping temporarily t o re s t or proceding t o the next problem. Most subjects p r e f e r r e d to work continuously on the problem set. Once the interview session was completed and the subject had l e f t , the interviewer played the tape recording back, matching the pr o t o c o l 67 with the written work. This was done so the two could be combined when the coding took place. The Coding Procedure As a f i n a l check on the coding system and f o r p r a c t i c e coding, the w r i t e r coded 40 problems chosen from the main study. These included a l l f i v e problems taken from both the r e a l world and math world s e t t i n g s . At the time i t was f e l t that the coding system was indeed accounting f o r the procedures used by the subjects i n the various problems. The f i v e problems were randomly ordered as follows: 3, 1, 4, 5, 2. Then the protocols of a l l 40 subjects f o r problem number 3 were randomly assigned. These protocols were then coded by the w r i t e r using the f i n a l coding form. The procedure was repeated f o r the other 4 problems, g i v i n g a t o t a l of 200 p r o t o c o l s . Each problem was coded on a separate coding form. The coding was done using the tape recorded protocols and the wr i t t e n work of each subject. To assess the r e l i a b i l i t y of the coding, both i n t e r c o d e r r e l i a b i l i t y and intracoder r e l i a b i l i t y of the coding were used. A second coder had been t r a i n e d during the p i l o t study. She and the w r i t e r spent nine hours working together coding sixteen problems. The second coder then coded another 14 problems alone. Like the w r i t e r , she was very f a m i l i a r with the h e u r i s t i c s used i n the study and had had several years experience teaching secondary mathematics. For each of the 5 problems, 4 protocols were selected at random. These were then coded by both the w r i t e r and the second coder. The r e s u l t s of these codings were compared t o the w r i t e r ' s o r i g i n a l codings of the same pro t o c o l s . Four types of coding e r r o r s were i d e n t i f i e d . These are i l l u s t r a t e d i n Figure 7 and described here: 1) A Blank occurs when a procedure i s coded by one coder and not the other. The assumption i s that t h i s i s the only e r r o r . That i s , i n the example coder A missed the procedure of i d e n t i f y i n g a v a r i a b l e which was coded by coder B. 2) C a l l i n g a procedure by a d i f f e r e n t name occurs when a procedure i s coded d i f f e r e n t l y by the two coders. 3) Coding procedures i n d i f f e r e n t order occurs when the order of two or more successive procedures i s reversed by one of the coders. I t i s assumed that the only er r o r made i s the r e v e r s a l of the procedures. In the example, r e c a l l of r e l a t e d f a c t and templation were reversed. 4) The l a s t error, cases, occurs when one coder i n d i c a t e s the use of the h e u r i s t i c s of cases, stops, and then codes i t s t a r t i n g again and the other coder codes i t as a continuous process. A summary of the procedures coded d i f f e r e n t l y , under each of the categories, by the two coders, i s found i n Table 1 and the code-recode by the w r i t e r i s found i n Table 2. As a measure of r e l i a b i l i t y , the percentage of items coded i d e n t i c a l l y was c a l c u l a t e d using a formula derived from McGrew (1971» p. 24).., The formula used i n t h i s study i s a more conservative measure than that of McGrew because i t takes i n t o account the number of items coded i d e n t i c a l l y by both coders only once. Let the number of items coded by A which agree with those coded by B equal a, and the remainder equal a'. Then a + a' = t o t a l coded by A. 69 CODER "A" R e c a l l r e l a t e d f a c t 3 3 Draw diagram / Modify diagram I d e n t i f y v a r i a b l e S e t t i n g up equations 3 1 3 Algorithms-algebraic z i Algorithms-arithmetic 3 3 3 3 3 3-Guessing T \ Smoothing — 3 Templation 3 C a s e s - a l l Cases-random / / V 3 Cases-systematic Checking part .C i. V c • ! • <D — t J O n o in a 8 ! 0) -p o c . o u U -p c/2 >> a  u u i I > O I •+-> . 1 O H <H O ti * H | ( D O -•3 0> | U W •H CD ft ! w to *o p O i fl> t> <H -P CODER MB" H X5 ! U o iH n w R e c a l l r e l a t e d f a c t .3. | j 3 Draw diagram T i V •/ Modify diagram i 1 I d e n t i f y v a r i a b l e V ! i S e t t i n g up equations 3 3 1 Algorithms-algebraic % Algorithms-arithmetic 3 3 3 3 3 3 Guessing Smoothing 3 Analysis -V Templatior i 2 3 3 Cases-al! L Case3-randor n V i V V 3 Cases-systemati( Checking par t / FIGURE 7 EXAMPLES OF CODER ERROR (Reduced by 33$) 70 TABLE 1 TYPES AND NUMBERS OF DISAGREEMENTS BETWEEN TWO DIFFERENT CODERS FIRST CODER SECOND CODER BLANKS Reading Problem -2 Modify Diagram -3 Ide n t i f y Variable -2 Set t i n g up Equations -4 Guessing -3 Templation -2 Analysis -1 Concern -2 Reading Problem -3 R e c a l l Related Fact -2 Modify Diagram -4 Ide n t i f y Variable -2 Sett i n g up Equations -1 Guessing Smoothing -2 Templation -1 Checking Part -2 Concern -3 CODING PROCESS DIFFERENTLY Templation S e t t i n g up Equations Diagram Checking Part Checking Part Templation Arithmetic Algorithms R e c a l l Related Fact -2 Algebraic Algorithms -1 Modify Diagram -1 Concern -2 Cases(All) -1 Cases(Random) -1 Algebraic Algorithms -2 ORDER REVERSED Templation - R e c a l l Related Fact -2 Modify Diagram - Smoothing - Cases(Start) -1 CASES Continuous - Start-Stop -3 71 TABLE 2 TYPES AND NUMBERS OF DISAGREEMENTS BETWEEN CODE AND RECODE FIRST CODE RECODE BLANKS Reading Problem -3 Modify Diagram -2 Identi f y Variable -1 Set t i n g up Equations -2 Arithmetic Algorithms -1 Guessing -1 Checking Part -2 Concern -1 Reading Problem -1 Rec a l l Problem Type -1 Re c a l l Related Fact -4 Identify Variable -2 Se t t i n g up Equations -1 Arithmetic Algorithms -3 Guessing -2 Templation -1 Checking Part -2 Concern -3 CODING PROCESS DIFFERENTLY Templation Obtain Solution Checking Part R e c a l l Related Fact Checking Part R e c a l l Related Fact -2 Checking Part -1 Arithmetic Algorithms -1 Templation. -1 Concern -1 ORDER REVERSED Smoothing - Diagram -1 Rec a l l Related Fact - Templation -1 CASES Continuous - Start-Stop -1 72 S i m i l a r l y , a + b* = t o t a l coded by B. Then a + a' + b 1 = t o t a l number of d i f f e r e n t items coded by both A and B, c a l l t h i s sum T. Then a/T = y, the percentage of items coded i d e n t i c a l l y by both coders i s the measure of r e l i a b i l i t y . For i n t ercoder r e l i a b i l i t y : a = 294, a' = 40, b' = 35, T = 369 The percent of items coded i d e n t i c a l l y by two d i f f e r e n t coders i s 80. For intracoder r e l i a b i l i t y : a = 307, a' = 27, b' = 2 1 , T = 355 The percent of items coded i d e n t i c a l l y on the code/recode by the same coder i s 86. These r e l i a b i l i t i e s were f e l t to be s u f f i c i e n t l y high. There are several important f a c t o r s which may have contributed to the r e l i a b i l i t i e s being high. Both coders were very f a m i l i a r with the second year algebra course, both having several years teaching experience at t h i s l e v e l . Both coders were very f a m i l i a r with the coding system and with the h e u r i s t i c s . In ad d i t i o n a great d e a l of time was spent on the coding i t s e l f . MacPherson's model appears to be a r e l i a b l e model to use f o r analyzing mathematical problem s o l v i n g . I t can be used e f f e c t i v e l y by a person who: ( l ) knows and understands the h e u r i s t i c s well, (2) knows the core and i s f a m i l i a r with the background of the subjects, and (3) i s f a m i l i a r with t h i s coding system. C h a p t e r IV ANALYSIS AND RESULTS I n t h i s s t u d y two phases were i n v o l v e d i n t h e a n a l y s i s o f t h e d a t a . The f i r s t phase c o n s i s t e d o f t h e a n a l y s i s o f t h e h y p o t h e s e s s t a t e d i n C h a p t e r I . The h y p o t h e s e s a r e r e s t a t e d b e l o w . H y p o t h e s e s 1-8 were t e s t e d s t a t i s t i c a l l y . A r e g r e s s i o n a n a l y s i s p r o c e d u r e was d e s i g n e d t o t e s t h y p o t h e s e s 1-6. H y p o t h e s e s 7 and 8 were t e s t e d u s i n g P e a r s o n product-moment c o r r e l a t i o n c o e f f i c i e n t s . H y p o t h e s e s 9 and 10 were n o t t e s t e d s t a t i s t i c a l l y . A F l a n d e r s (See Amidon an d Hough, 1967) t y p e i n t e r a c t i o n m a t r i x ( a p r o c e s s m a t r i x ) was u s e d t o g a i n some i n s i g h t i n t o t h e sequence o f p r o c e d u r e s u s e d b y t h e s t u d e n t s i n t h i s s t u d y . The second phase i n v o l v e d p a s t hoc a n a l y s i s o f t h e d a t a o b t a i n e d f r o m t h e c o d i n g system. P r o b l e m s o l v i n g b e h a v i o r s were examined i n t e r m s o f t h e i r i n t e r c o r r e l a t i o n s as w e l l as w i t h t h e u s e o f r e g r e s s i o n a n a l y s i s . , R e s e a r c h H y p o t h e s e s H I : P r o b l e m c o n t e x t w i l l n o t c o n t r i b u t e t o t h e number o f t i m e s h e u r i s t i c s a r e u s e d by a s t u d e n t . H2: P r o b l e m c o n t e x t w i l l n o t c o n t r i b u t e t o t h e number o f d i f f e r e n t h e u r i s t i c s u s e d b y a s t u d e n t . H3: P r o b l e m c o n t e x t w i l l n o t c o n t r i b u t e t o t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d b y a s t u d e n t . H4: F i e l d i n d e p e n d e n c e w i l l n o t c o n t r i b u t e t o t h e number o f t i m e s h e u r i s t i c s a r e u s e d by a s t u d e n t . H5: F i e l d i n d e p e n d e n c e w i l l n o t c o n t r i b u t e t o t h e number o f d i f f e r e n t h e u r i s t i c s u s e d b y a s t u d e n t . H6: F i e l d i n d e p e n d e n c e w i l l n o t c o n t r i b u t e t o t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d by a s t u d e n t . 74 H7s There i s no c o r r e l a t i o n between t h e number o f t i m e s h e u r i s t i c s a r e u s e d b y a s t u d e n t and t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d . HS: T h e r e i s no c o r r e l a t i o n between t h e number o f d i f f e r e n t h e u r i s t i c s u s e d b y a s t u d e n t and t h e number o f c o r r e c t s o l u t i o n s he o b t a i n e d . H9: P r o b l e m c o n t e x t w i l l n o t o b s e r v a b l y a f f e c t t h e sequence o f p r o c e d u r e s u s e d b y a s u b j e c t . H10: F i e l d i n d e p e n d e n c e w i l l n o t o b s e r v a b l y a f f e c t t h e sequence o f p r o c e d u r e s u s e d b y a s u b j e c t . Method o f A n a l y s i s The r e g r e s s i o n a n a l y s i s a p p r o a c h employed f o r t h i s s t u d y has been d e s c r i b e d b y many w r i t e r s ( e . g . K e r l i n g e r and Pe d h a z u r , 1973)* W a l b e r g (1971) d e s c r i b e s t h r e e a d v a n t a g e s t h a t r e g r e s s i o n a n a l y s i s has o v e r t h e c o n v e n t i o n a l a n a l y s i s o f v a r i a n c e : ( l ) t h e use o f c o n t i n u o u s v a r i a b l e s , (2) l e s s d a t a p r o c e s s i n g t i m e , and (3) d i r e c t c o m p r e h e n s i v e e s t i m a t e s o f t h e magnitude and s i g n i f i c a n c e o f t h e i n d e p e n d e n t v a r i a b l e s e f f e c t s on t h e dependent v a r i a b l e . The f i r s t and t h i r d o f t h e s e a r e e s p e c i a l l y i m p o r t a n t t o t h i s s t u d y . W i t h t h e u s e o f c o n t i n u o u s v a r i a b l e s and a s m a l l sample s i z e (N=40), p r e c i s i o n i n g r o u p i n g s c o r e s i n t o two o r t h r e e l e v e l s c o u l d be l o s t and t h e sample s i z e i s t o o s m a l l t o m a i n t a i n c e l l s i z e s i f more l e v e l s were added. I n an e x p l o r a t o r y s t u d y s u c h as t h i s one, t h e t h i r d advantage appears t o have d i r e c t a p p l i c a t i o n , t h a t i s , t h e u s e o f t h e m u l t i p l e r e g r e s s i o n c o e f f i c i e n t : when s q u a r e d i t r e v e a l s d i r e c t l y how much v a r i a n c e i n t h e dependent v a r i a b l e i s a s s o c i a t e d w i t h o r a c c o u n t e d f o r by t h e i n d e p e n d e n t v a r i a b l e s . A s e p a r a t e r e g r e s s i o n e q u a t i o n was d e f i n e d f o r t h e a n a l y s i s o f ea c h o f t h e h y p o t h e s e s 1 t o 6. The r e g r e s s i o n a n a l y s e s were p e r f o r m e d u s i n g The T r i a n g u l a r R e g r e s s i o n Package (TRIP) ( B j e r r i n g and S e a g r a v e s , 1974) a v a i l a b l e a t t h e Computing C e n t e r o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . 75 A l t h o u g h i n some c a s e s t h e r e l a t i o n s h i p between t h e v a r i a b l e s was l o g a r i t h m i c i n n a t u r e , t h e range o f t h e i n d e p e n d e n t v a r i a b l e s was so s m a l l t h a t v e r y l i t t l e s t a t i s t i c a l d i f f e r e n c e ( F - p r o b a b i l i t i e s a g r e e d i n f i r s t two d e c i m a l p l a c e s ) was n o t e d between a l o g a r i t h m i c and l i n e a r m odel. Hence a l i n e a r r e l a t i o n s h i p was assumed between t h e v a r i a b l e s . D e t e r i n i n i n g t h e a p p r o p r i a t e s i g n i f i c a n c e l e v e l f o r m e a n i n g f u l i n t e r p r e t a t i o n must be c o n s i d e r e d c a r e f u l l y . T h i s i s an e x p l o r a t o r y s t u d y u s i n g MacPherson's model t o d e t e r m i n e i f any d i f f e r e n c e e x i s t s between gro u p s o f i n d i v i d u a l s i n methods o f a t t a c k i n g and s o l v i n g mathemat-i c a l p r o b l e m s . As an e x p l o r a t o r y s t u d y , i f d i f f e r e n c e s do e x i s t , a s i g n i f i c a n c e l e v e l w h i c h i s n o t t o o s t r i n g e n t s h o u l d be c h o s e n o t h e r w i s e i m p o r t a n t i m p l i c a t i o n s may be l o s t . A s i g n i f i c a n c e l e v e l o f .10 has been chosen t o t e s t t h e h y p o t h e s e s o f t h i s s t u d y . The p r o b a b i l i t y o f m a king a Type 1 e r r o r by r e j e c t i n g t h e n u l l h y p o t h e s i s w i l l be g i v e n f o r e a c h s t a t i s t i c a l t e s t . B e s i d e s t h e h y p o t h e s i z e d v a r i a b l e s , two v a r i a b l e s o f i n t e r e s t t o t h e w r i t e r were o b t a i n e d f r o m t h e p r o t o c o l s o f t h e s u b j e c t s . These a r e t h e number o f t i m e s a s u b j e c t a t t e m p t e d t o b e g i n a p r o b l e m a g a i n ( c y c l e s ) and t h e number o f t i m e s he changed h i s method o f a t t a c k . I t a p p e a r e d f r o m t h e p r o t o c o l s t h a t some s u b j e c t s w o u l d a t t e m p t a p r o b l e m a number o f t i m e s and c o n t i n u e t o do t h e same t h i n g e ach t i m e w h i l e o t h e r s t r i e d a v a r i e t y o f methods. Many o f t h e s e changes i n v o l v e d t h e use o f h e u r i s t i c s w h i l e o t h e r s were more d i r e c t l y r e l a t e d t o c o r e . I n an a t t e m p t t o g a i n some i n s i g h t i n t o t h e sequence o f p r o c e d u r e s u s e d b y t h e s u b j e c t , a p r o c e s s m a t r i x was d e v e l o p e d f r o m t h e c o d i n g s y s t e m s . T h i s p r o c e s s m a t r i x c o n t a i n s s i x m a j o r c a t e g o r i e s f o r a n a l y z i n g t h e p r o c e d u r e s . They a r e : ( l ) C o r e , (2) H e u r i s t i c s , (3) S o l u t i o n , (4) C h e c k i n g , (5) C o n c e r n , and (6) Work s t o p p e d (See Figures)-The d a t a a r e t a k e n f r o m t h e c o d i n g s h e e t s and e n t e r e d i n t h e p r o c e s s m a t r i x ( F i g u r e s g and 9) i n t h e a p p r o p r i a t e row and column. F o r example: i f a s t u d e n t r e a d s t h e p r o b l e m and t h e n draws a d i a g r a m , one i s added t o t h e e n t r y i n t h e f i r s t row, f o u r t h column. A f t e r d r a w i n g t h e d i a g r a m , i f t h e s t u d e n t u s e s t e m p l a t i o n , one i s added t o t h e e n t r y i n t h e f o u r t h row, t h i r t e e n t h column. T h i s s y s t e m does n o t a c c o u n t f o r t h e p r o c e d u r e s w h i c h a r e n e s t e d i n o t h e r s , s u c h as t h e use o f a l g o r i t h m s o r d i a g r a m s t h e s u b j e c t may be u s i n g w h i l e he i s u s i n g random o r s y s t e m a t i c c a s e s . A l s o , when a s u b j e c t i s c h e c k i n g e i t h e r p a r t o f h i s work o r h i s s o l u t i o n , t h e method he i s u s i n g i s n o t n o t e d i n t h e m a t r i x . D i f f e r e n t p a r t s o f t h e m a t r i x i n d i c a t e d i f f e r e n t k i n d s o f p r o b l e m -s o l v i n g p r o c e d u r e s . These a r e a s a r e i d e n t i f i e d i n F i g u r e 8 and d e s c r i b e d as f o l l o w s : AREA A. T h i s a r e a i n d i c a t e s t h e p r o b l e m s o l v i n g p r o c e d u r e s r e l a t e d t o t h e c o r e . A l l p r o c e d u r e s a r e fr o m c o r e t o c o r e . AREA B. The c e l l s i n t h i s a r e a i n c l u d e a l l o f t h e c o r e p r o c e d u r e s w h i c h a r e f o l l o w e d b y t h e u s e o f a h e u r i s t i c . AREA C. T h i s g roup o f c e l l s i n c l u d e s a l l o f t h e c o r e f o l l o w e d b y t h e use o f a c h e c k i n g p r o c e d u r e . AREA D. T h i s a r e a i n d i c a t e s t h e c o n c e r n w h i c h i s shown a f t e r t h e u s e o f a c o r e p r o c e d u r e . AREA E. T h i s a r e a c o n s i s t s o f t h e h e u r i s t i c s w h i c h a r e f o l l o w e d b y t h e use o f a c o r e p r o c e d u r e . AREA F. T h i s a r e a i n d i c a t e s t h e use o f a h e u r i s t i c f o l l o w e d b y a h e u r i s t i c . AREA G. These c e l l s i n c l u d e t h e c h e c k i n g p r o c e d u r e s w h i c h f o l l o w t h e u s e o f a h e u r i s t i c . AREA H. T h i s g roup o f c e l l s i n d i c a t e s t h e c o n c e r n shown a f t e r t h e use o f a h e u r i s t i c . AREA I . T h i s a r e a r e p r e s e n t s t h e c o r e p r o c e d u r e s u s e d a f t e r a s o l u t i o n was o b t a i n e d . AREA J . T h i s shows a l l o f t h e h e u r i s t i c s u s e d a f t e r a s o l u t i o n was 77 1 2 3 4 5 6 ? 8 9 10 11 12 13 14 15 16 17 18 19 20 22 23 24 25 26 27 1 2 3 4 5 6 A B C D 7 8 9 10 11 12 13 14 15 16 F G H 17 18 19 20 21 I J K L 22 23 M N 0 P 24 25 26 Q R S T 27 FIGURE 8 THE PROCESS MATRIX WITH IDENIFIED AREAS 78 FIGURE 9 PROCESS MATRIX CATEGORIES CATEGORIES Core 1. Reading the problem 2. R e c a l l problem (includes: R e c a l l same problem R e c a l l r e l a t e d problem R e c a l l problem type) 3. Re c a l l r e l a t e d fact 4. Draws diagram 5. Modify diagram 6. Id e n t i f y v a r i a b l e 7. S e t t i n g up equations 8. Algorithms-algebraic 9. Algorithms-arithmetic 10. Guessing Heu r i s t i c s 11. Smoothing 12. Analysis 13. Templation 14. A l l cases 15. Random cases 16. Systematic cases (includes: Systematic cases Sequential cases) 17. C r i t i c a l cases 18. Deduction 19. Inverse deduction 20. V a r i a t i o n Solution 21. Obtain s o l u t i o n Checking 22. Checking part 23. Checking s o l u t i o n Concern 24. Express concern about method 25. Express concern about algorithm 26. Express concern about equation 27. Express concern about s o l u t i o n Work Stopped 28. Work stopped-solution 29. Work stopped-no solu t i o n 79 o b t a i n e d . AREA K. The c e l l s i n t h i s a r e a i n d i c a t e t h e o b t a i n i n g o f a s o l u t i o n f o l l o w e d b y a c h e c k i n g p r o c e d u r e . AREA L. T h i s a r e a i n d i c a t e s t h e c o n c e r n shown f o l l o w i n g a s o l u t i o n . AREA M. These c e l l s r e p r e s e n t t h e use o f a c h e c k i n g p r o c e d u r e f o l l o w e d b y t h e u s e o f c o r e . AREA N, T h i s i n d i c a t e s t h e use o f h e u r i s t i c s f o l l o w i n g a c h e c k i n g p r o c e d u r e . AREA 0. The c e l l s i n t h i s a r e a r e p r e s e n t t h e u s e o f two c o n s e c u t i v e c h e c k i n g p r o c e d u r e s . AREA P. T h i s a r e a i n d i c a t e s t h e c o n c e r n shown f o l l o w i n g a c h e c k i n g p r o c e d u r e . AREA Q. These c e l l s show t h e use o f a c o r e p r o c e d u r e f o l l o w i n g an e x p r e s s i o n o f c o n c e r n . AREA R. The c e l l s i n t h i s a r e a show t h e h e u r i s t i c s u s e d f o l l o w i n g some t y p e o f c o n c e r n . AREA S. These c e l l s i n d i c a t e c o n c e r n f o l l o w e d b y a c h e c k i n g p r o c e d u r e . AREA T. These c e l l s i n c l u d e t h e c o n s e c u t i v e e x p r e s s i o n o f c o n c e r n . The m a t r i x i n d i c a t e s t h e amount and t o some e x t e n t a p a t t e r n o f t h e p r o c e d u r e s u s e d t o s o l v e m a t h e m a t i c a l p r o b l e m s a c c o r d i n g t o t h e c a t e g o r i e s i n t h e p r o c e s s m a t r i x . The m a t r i x a l s o i n d i c a t e s t h e number o f t i m e s t h a t s u b j e c t s change c o r e and h e u r i s t i c s and t h e amount o f p r o c e d u r e s u s e d w i t h i n e a c h c a t e g o r y . S t u d e n t s w o r k i n g p r o b l e m s i n t h e m a t h e m a t i c a l w o r l d s e t t i n g were coded 1 and t h o s e w o r k i n g p r o b l e m s i n t h e r e a l w o r l d s e t t i n g were coded 2. " T o t a l H e u r i s t i c s " i s t h e t o t a l number o f a l l h e u r i s t i c s coded f o r a l l f i v e p r o b l e m s . " D i f f e r e n t H e u r i s t i c s " i s t h e number o f d i f f e r e n t h e u r i s t i c s c o ded f o r a l l f i v e p r o b l e m s and t h e c o r r e c t s o l u t i o n i s t h e t o t a l number o f c o r r e c t s o l u t i o n s f o r t h e f i v e p r o b l e m s . The means and s t a n d a r d d e v i a t i o n s of; t h e h y p o t h e s i z e d v a r i a b l e s . a r e g i v e n i n T a b l e 3. The r e a d e r s h o u l d be aware o f t h e d i s t r i b u t i o n o f some v a r i a b l e s i n i n t e r p r e t i n g t h e r e s u l t s f r o m t h e r e g r e s s i o n a n a l y s i s . A p p e n d i x D c o n t a i n s t h e h i s t o g r a m s f o r s e v e r a l s e l e c t e d v a r i a b l e s . Extreme s c o r e s may have i n f l u e n c e d t h e 80 TABLE 3 MEANS AND STANDARD DEVIATIONS FOR MEASURES OF HYPOTHESIZED VARIABLES (N = 40) V a r i a b l e Mean S t a n d a r d D e v i a t i o n F i e l d Independence 143.53 20.03 T o t a l H e u r i s t i c s 8.60 7.90 D i f f e r e n t H e u r i s t i c s 3.00 1.70 C y c l e s 10.37 4.02 Changes 1.85 2.30 C o r r e c t S o l u t i o n 1.15 .89 s t a t i s t i c a l r e s u l t s , b u t be c a u s e o f t h e s m a l l sample s i z e (N = 40) t h e s e s c o r e s were n o t d e l e t e d f r o m t h e s t a t i s t i c a l a n a l y s i s . T a b l e 6 shows t h e i n t e r c o r r e l a t i o n s between t h e s e v a r i a b l e s . I t was n o t e d i n C h a p t e r I I I t h a t one o f t h e r e a s o n s f o r c h o o s i n g t h e s e p a r t i c u l a r p r o b l e m s was t h a t t h e y c o u l d be s o l v e d b y about h a l f o f t h e s u b j e c t s . The mean f o r t h e c o r r e c t number o f s o l u t i o n s i s 1,15 w h i c h i m p l i e s t h e p r o b l e m s may have been more d i f f i c u l t t h a n t h e p i l o t s t u d y i n d i c a t e d . O n l y p r o b l e m s 2 and 5 were answered c o r r e c t l y b y about h a l f o f t h e s u b j e c t s . The d i s c u s s i o n o f t h e r e s u l t s o f t h e a n a l y s i s i s d i v i d e d i n t o t h r e e a r e a s . The i n f l u e n c e o f p r o b l e m c o n t e x t i s examined f i r s t a l o n g w i t h t h e r e l a t e d h y p o t h e s e s . The sec o n d a r e a d e a l s w i t h f i e l d i n d e p e n d e n c e and i t s r e l a t e d h y p o t h e s e s . I n t h e t h i r d a r e a , t h e r e s u l t s o f b e h a v i o r s d e r i v e d f r o m t h e c o d i n g s y s t e m a r e d i s c u s s e d . The s e c t i o n d e a l s w i t h t h e i n f l u e n c e o f b o t h c o r e and h e u r i s t i c p r o c e d u r e s on p r o b l e m s o l v i n g . R e s u l t s o f A n a l y s i s - P r o b l e m C o n t e x t P r o b l e m c o n t e x t a p p e a r s t o have had l i t t l e e f f e c t on t h e p r o b l e m s o l v i n g p r o c e d u r e s u s e d b y t h e s u b j e c t s i n t h i s s t u d y . The r e s u l t s o f t h e 81 r e g r e s s i o n a n a l y s i s p e r f o r m e d w i t h p r o b l e m c o n t e x t as t h e i n d e p e n d e n t v a r i a b l e a r e g i v e n i n T a b l e 4. The r e s u l t s o f h y p o t h e s e s 1, 2, 3, and 9 a r e g i v e n as f o l l o w s : H y p o t h e s i s 1: No s i g n i f i c a n t d i f f e r e n c e (P = .5195) was f o u n d i n t h e t o t a l number o f d i f f e r e n t h e u r i s t i c s u s e d b y s t u d e n t s s o l v i n g p r o b l e m s i n a r e a l w o r l d s e t t i n g and s t u d e n t s s o l v i n g p r o b l e m s i n a m a t h e m a t i c a l s e t t i n g . H y p o t h e s i s 2: No s i g n i f i c a n t d i f f e r e n c e (P = .1556) was f o u n d i n t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d b y s t u d e n t s s o l v i n g p r o b l e m s i n a r e a l w o r l d s e t t i n g and s t u d e n t s s o l v i n g p r o b l e m s i n a math w o r l d s e t t i n g . 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 d i f f e r e n c e f o r t h e h y p o t h e s i s t e s t e d , t h e .1556 p r o b a b i l i t y l e v e l may be s u f f i c i e n t t o i n d i c a t e a t r e n d . The mean f o r c o r r e c t s o l u t i o n s f o r t h e r e a l w o r l d s u b j e c t s i s 1.35 compared t o .95 f o r t h e math w o r l d s u b j e c t s . The r e s u l t s o f p o s t hoc a n a l y s e s u s i n g l i n e a r r e g r e s s i o n i n d i c a t e d t h a t p r o b l e m c o n t e x t d i d n o t a f f e c t t h e number o f c y c l e s o r changes made b y a s t u d e n t . No s i g n i f i c a n t d i f f e r e n c e (P = .6232) w a s f o u n d between t h e number o f t i m e s a s t u d e n t i s w i l l i n g t o a t t a c k a p r o b l e m ( c y c l e s ) and t h e p r o b l e m c o n t e x t . N o r was t h e r e a s i g n i f i c a n t d i f f e r e n c e (P = .4225) between t h e number o f t i m e s a s t u d e n t c h a n g e s h i s method o f a t t a c k i n g a p r o b l e m and t h e p r o b l e m c o n t e x t . The p r o c e s s m a t r i c e s f o r h y p o t h e s i s 9 a r e g i v e n i n F i g u r e s 10 and 11 w i t h t h e p e r c e n t o f p r o c e d u r e s u s e d i n each c a t e g o r y g i v e n i n F i g u r e s 12 and 13. H y p o t h e s i s 9 was n o t t e s t e d f o r s t a t i s t i c a l s i g n i f i c a n c e . I n e v a l u a t i n g h y p o t h e s i s 9, t h e f o l l o w i n g m a j o r a r e a s f r o m t h e p r o c e s s m a t r i c e s were c o n s i d e r e d : 1. Moves fr o m c o r e t o c o r e o r t o h e u r i s t i c s ( A r e a s A and B) 2. Moves fr o m h e u r i s t i c s t o c o r e o r t o h e u r i s t i c s ( A r e a s E and F) TABLE 4 RESULTS OF REGRESSION ANALYSIS WITH PROBLEM CONTEXT AS INDEPENDENT VARIABLE (N=40) DEPENDENT VARIABLE SOURCE OF VARIATION F - VALUE TO ENTER/REMOVE 2 F - PROB. R WITH DEPENDENT VARIABLE 3RSQ TOTAL HEURISTICS PROBLEM ,4478 .5145 .1077 .0116 CONTEXT DIFFERENT HEURISTICS .9762 .3312 .1581 .0250 CYCLES .2533 .6232 .0802 .0064 CHANGES .6722 .4225 .1319 .0174 CORRECT SOLUTION 2.06l .1556 .2267 .0514 Each of These is a Separate Simple Regression. P r o b a b i l i t y of Making a Type 1 Error by Rejecting Null Hypothesis, that i s , Claiming S t a t i s t i c a l Significance. -^ The Proportion of Variance i n the Dependent Variable Accounted for by Problem Context. 83 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23- 24 25 26 27 29 29 1 27 18 71 12 27 87 7 20 26 12 9 2 7 14 8 2 1 7 6 7 2 3 8 3 9 2 5 4 4 10 5 3 4 i i 2 2 2 4 39 13 19 7 3 4 U 8 7 1 9 5 3 1 2 4 4 1 5 l 5 6 7 8 9 10 4 10 44 19 10 9 1 4 4 5 2 8 8 4 3 l 1 2 4 2 4 26 4 3 48 61 3 3 12 62 5 2 3 43 3 2 21 6 7 5 7 2 2 2 5 4 8 1 6 6 1 2 1 3 2 1 1 4 8 31 32 1 1 6 24 4 2 1 2 3 4 1 4 ,3 1 3 3 l l 2 3 2 1 1 11 2 3 2 1 2 2 1 12 1 1 13 13 3 8 2 6 6 1 4 8 X 4 5 2 1 2 1 3 i 5 14 15 3 1 1 1 1 1 1 1 22 2 3 l 2 3 16 1 1 1 1 6 1 1 1 1 17 1 18 1 1 2 i 19 1 20 1 21 21 3 5 2 5 1 4 4 2 27 l 18 22 J 22 23 20 14 3 5 2 2 1 8 3 7 3 2 6 1 4 1 1 3 2 3 4 1 2 3 9 5 2 1? 24 25 2 6 3 l 7 1 1 2 1 2 1 1 4 1 1 1 3 1 • l 1 x * 1 1 27 9 4 1 1 4 1 1 1 2 8 1 2 FIGURE 10 PROCESS MATRIX»FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR 100 MATH WORLD PROBLEMS (N=20) 84 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 1 21 3 12 64 6 31 31 2 15 35 9 7 15 2 2 6 15 2 4 10 3 2 1 2 1 1 3 2 2 1 6 9 3 2 8 2 1 1 1 4 24" 1 9 17 4 4 l 13 15 10 1 5 3 3 2 1 1 1 1 5 2 2 2 1 1 6 5 1 2 1 1 l 3 6 8 1 1 1 41 1 1 2 1 l 7 19 3 5 36 56 2 5 1 2 2 1 17 1 7 8 8 1 l 8 64 6 5 1 4 16 14 5 1 1 1 1 9 12 2 4 2 1 67 6 3 6 7 1 1 42 11 2 4 3 5 1 l 10 6 3 2 1 3 24 1 3 1 35 9 1 3 -11 5 2 3 4 3 3 1 9 4 1 12 1 1 7 13 10 1 5 1 1 3 11 4 3 1 4 1 1 1 1 2 2 2 14 15 8 1 2 2 6 2 3 2 2 1 2 4 2 1 18 3 1 1 2 5 16 1 1 1 2 1 1 2 5 1 1 ' 17 1 18 1 1 1 19 1 20 21 19 2 2 2 2 2 4 1 3 49 2 12 22 22 18 3 4 14 5 9 3 1 2 5 3 2 1 1 1 1 1 1 4 23 15 1 4 1 1 3 3 4 1 3 8 3 4 1 1 9 22 2 24 4 1 1 2 2 1 1 1 1 4 1 3 25 1 1 2 2 1 26 2 3 3 1 1 27 9 2 1 1 1 3 2 5 11 2 FIGURE 11 PROCESS MATRIX*FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR 100 REAL WORLD PROBLEMS (N=20) 85 1 2 3 4 5 6 7 8 9 10 11 12 131** 1516 1718 19 20 22 23 2U 25 26 27 1 2 3 if 5 62 7 5 2 7 8 9 10 11 12 13 14 15 16 2 1 . 1 17 18 19 20 21 2 1 2 1 22 23 5 1 1 24 25 26 3 1 1 27 FIGURE 12 PERCENT OF PROCEDURES USED IN EACH CATEGORY - MATH WORLD (N=20) 86 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 23 24 25 26 27 54 7 3 3 7, 3 1 • 2 1 4 1 6: 2 1 1 3 1 1 FIGURE 13 PERCENT OF PROCEDURES USED IN EACH CATEGORY - REAL WORLD (N=20) 87 3. Moves c o n c e r n e d w i t h c h e c k i n g ( A r e a s C, G, M, and N) 4. Moves c o n c e r n e d w i t h e x p r e s s i o n o f c o n c e r n ( A r e a s D, H, Q, and R ) . I n c o m p a r i n g t h e s e a r e a s , f o r t h e two g r o u p s (math w o r l d and r e a l w o r l d ) f t h e r e s e a r c h e r c o n s i d e r e d t h e p e r c e n t a g e o f moves i n each a r e a f o r e a c h g r o u p as w e l l as c o m p a r i n g t h e number and t y p e s o f moves f o r each p r o c e d u r e . That i s , f o r a g i v e n p r o c e d u r e , i s t h e r e a p r o c e d u r e ( o r p r o c e d u r e s ) w h i c h s t u d e n t s have a t e n d e n c y t o use n e x t ? H y p o t h e s i s 9: The p a t t e r n o f p r o c e d u r e s u s e d by s u b j e c t s i n t h e two g r o u p s , r e a l w o r l d and math w o r l d , was n o t o b s e r v a b l y d i f f e r e n t . A l t h o u g h t h e s u b j e c t s w o r k i n g p r o b l e m s i n t h e math w o r l d s e t t i n g u s e d a h i g h e r p e r c e n t o f c o r e r e l a t e d p r o c e d u r e s as compared t o t h e s u b j e c t s i n t h e r e a l w o r l d s e t t i n g (See A r e a A and p e r c e n t a g e s o f c o r e r e l a t e d a r e a s i n F i g u r e s 12 and 13)> t h e s e d i f f e r e n c e s a r e n o t g r e a t . Two e x c e p t i o n s i n v o l v e t h e number o f t i m e s s u b j e c t s move f r o m r e a d i n g ( # l ) t o s e t t i n g up e q u a t i o n s (#7) and f r o m s e t t i n g up e q u a t i o n s t o r e a d i n g t h e p r o b l e m . I n b o t h o f t h e s e i n s t a n c e s t h e math w o r l d s u b j e c t s made more moves t h a n t h e r e a l w o r l d subjects, . . . 4 4 t o 19 and 87 t o 37 r e s p e c t i v e l y . •These d i f f e r e n c e s come m a i n l y f r o m problem: number 2, where t h e number o f moves f r o m e q u a t i o n s t o r e a d i n g and r e a d i n g t o e q u a t i o n s f o r t h e math w o r l d and r e a l w o r l d i s 31 t o 13 and 66 t o 23 r e s p e c t i v e l y . W i t h o v e r h a l f o f t h e moves i n A r e a A, t h e d i v i s i o n s i n t h e p r o c e s s m a t r i x may n o t have been f i n e enough t o d e t e c t any d i f f e r e n c e s between t h e two g r o u p s . A l s o a few s t u d e n t s may have c o n t r i b u t e d t h e m a j o r i t y o f moves i n some a r e a s . R a t h e r t h a n c o u n t i n g t h e t o t a l number o f moves f o r e a c h s u b j e c t , a s e c o n d p r o c e s s m a t r i x was o b t a i n e d where a move f o r each s u b j e c t was o n l y c o u n t e d once f o r each p r o b l e m ( s e e F i g u r e s 14 and 15). The e n t r i e s i n t h e s e m a t r i c e s i n d i c a t e t h e number o f p r o b l e m s i n w h i c h a p a r t i c u l a r move was made. F i v e a r e a s from t h e c o r e were i d e n t i f i e d : 88 M >-) Q < < o w w EC K S < K CD 2 O M E-i to < tO tD W W CD CD o to w CO < a O H 2 S W << O o p CO to w to o o M F-i £ W S3 to 6-< 2 o M F-t o to CD C3 I—I M o o w w o o to to w o o oc re W o O o 2 O M o w ( X w o o o READING 16 11 55 34 32 5 6 13 2 5 14 2 4 RECALL 2 3 14 2 6 2 1 1 DIAGRAM 24 8 18 6 20 8 6 4 4 5 1 1 EQUATION 21 4 1 51 4 2 2 4 2 15 6 GUESS 6 5 1 1 1 9 1 3 SMOOTHING 4 2 6 3 1 9 4 1 TEMPLATION 10 1 6 4 4 2 1 3 1 1 2 RANDOM CASES 7 1 2 8 3 2 1 2 4 2 2 SYSTEMATIC CASES 1 1 1 1 1 1 1 1 OBTAIN SOLUTION 19 2 4 2 4 1 3 38 2 11 CHECKING PART" 13 3 13 3 1 2 5 3 1 1 3 1 CHECKING SOLUTION 12 1 5 3 4 3 6 1 4 2 6 CONCERN PROCESS 6 2 3 1 1 2 7 1 CONCERN SOLUTION 9 3 1 3 2 5 FIGURE 14 PROCESS MATRIXiFREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM, FOR 100 MATH WORLD PROBLEMS (N=20) 89 2 M P < 2 S O < M « E H 175 < W W M C? co w •3 K K Q W O 2 M « E H O O 2 CO o M E H PH H W E H CO W CO «< o o D CO W CO o o M E H < JE W E-* ss CO E H < o 2 M is."* 6 w o 2 o M E H 3 o CO e> M & O w >-r-o to CO w o o a : p< a : W o 2 O o 2 O M E H o w 2 ( X w o 2 o o READING 21 16 65 54 25 11 8 2 13 8 3 6 RECALL 8 3 11 8 5 2 4 1 2 2 DIAGRAM 35 14 22 9 12 7 12 5 3 4 5 1 1 EQUATION 33 9 3 47 7 11 6 5 7 1 GUESS 9 7 3 2 1 3 SMOOTHING 2 4 1 2 TEMPLATION 12 1. 7 10 8 4 5 2 ^ l 1 RANDOM CASES 3 1 1 2 1 1 2 3 1 2 SYSTEMATIC CASES 1 1 1 i 2 1 OBTAIN SOLUTION 19 3 6 1 4 4 2 25 1 14 CHECKING PART" 14 3 4 6 2 6 3 3 3 CHECKING SOLUTION 12 2 3 1 4 1 4 8 CONCERN PROCESS 11 1 1 2 3 1 5 2 CONCERN SOLUTION 9 4 l 4 1 l 1 2 2 FIGURE 15 PROCESS MATRIXiFREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM* FOR 100 REAL WORLD PROBLEMS (N=20) 90 r e a d i n g , r e c a l l r e l a t e d f a c t , w o r k i n g w i t h a d i a g r a m ( i n c l u d e s d i a g r a m and m o d i f y d i a g r a m ) , w r i t i n g e q u a t i o n s ( i n c l u d e s i d e n t i f y v a r i a b l e and s e t t i n g up e q u a t i o n s ) , and g u e s s i n g . F o u r o f t h e h e u r i s t i c s : s m o o t h i n g , t e m p l a t i o n , random c a s e s , and s y s t e m a t i c c a s e s ) were u s e d a c r o s s t h e f i v e p r o b l e m s and a r e i d e n t i f i e d i n t h e m a t r i x . The o t h e r a r e a s i n c l u d e d were o b t a i n i n g a s o l u t i o n , c h e c k i n g p a r t , c h e c k i n g s o l u t i o n , c o n c e r n f o r p r o c e s s ( i n c l u d e s c o n c e r n f o r method, a l g o r i t h m , and e q u a t i o n ) , and c o n c e r n f o r s o l u t i o n . The p e r c e n t a g e o f moves was o b t a i n e d f o r e a c h o f t h e s e p r o c e d u r e s a c r o s s each row ( s e e F i g u r e s 16 and 17)• F o r example i n F i g u r e 16, i n 2 8 % o f t h e p r o b l e m s , w o r k i n g w i t h a d i a g r a m f o l l o w e d r e a d i n g t h e p r o b l e m . The s u b j e c t s i n t h e r e a l w o r l d s e t t i n g u s e d t h e h e u r i s t i c o f sm o o t h i n g a c r o s s a l m o s t a l l o f t h e p r o c e d u r e s i n t h e p r o c e s s m a t r i x w h i l e t h e s u b j e c t s i n t h e math w o r l d s e t t i n g u s e d i t m a i n l y i n c o n n e c t i o n w i t h diagramming. The math w o r l d s u b j e c t s made a g r e a t e r p e r c e n t a g e o f moves t o r e a d i n g f r o m a l m o s t e v e r y p r o c e s s . K i l p a t r i c k (1967. P« 64) i n d i c a t e d t h e number o f t i m e s a s u b j e c t r e a d s t h e p r o b l e m i s a measure o f h i s d i f f i c u l t y i n u n d e r s t a n d i n g t h e p r o b l e m . I t appears as t h o u g h s u b j e c t s i n b o t h p r o b l e m s e t t i n g s had d i f f i c u l t y i n u n d e r s t a n d i n g t h e p r o b l e m s , w i t h t h o s e i n t h e math w o r l d s e t t i n g b e i n g t h e h a r d e s t t o u n d e r s t a n d . A l s o , t h e r e a l w o r l d s u b j e c t s seemed t o be more c o n f i d e n t i n t h e s o l u t i o n s t h e y o b t a i n e d u s i n g t h e h e u r i s t i c s , o b t a i n i n g 21 s o l u t i o n s u s i n g random and s y s t e m a t i c c a s e s compared t o 25 f o r t h e math w o r l d s u b j e c t s . The math w o r l d s u b j e c t s e x p r e s s e d c o n c e r n f o r s o l u t i o n s o b t a i n e d u s i n g b o t h o f t h e s e p r o c e d u r e s w h i l e t h e r e a l w o r l d s u b j e c t s e x p r e s s e d none. B o t h g r o u p s o f s u b j e c t s moved t o and f r o m t h e use o f t e m p l a t i o n and random c a s e s , u s i n g a l l t h e p r o c e d u r e s i n t h e p r o c e s s m a t r i x . B o t h o f t h e s e g r o u p s s t o p p e d work w i t h a s o l u t i o n u s i n g a p p r o x i m a t e l y t h e same p r o c e d u r e s . I n g e n e r a l , t h e o v e r a l l p a t t e r n o f moves ( e x c e p t f o r s m o o t h i n g ) f o r b o t h g r o u p s o f s u b j e c t s i s t h e same. 91 C5 P < S cc o ce 2 O M < c? GO to w CC Q W O o o 2 M g a o s 00 E - « CO w CO o o p cc to w CO o o M w CO 2 o M CC i-3 < o CH CO o e> 2 2 M M O O w w 6 o to CO w o o CC P-« 2 CC w o 2 O o 2; O M o CO 2 cc w o 2 O O R E A D I N G 9 7 2 8 23 11 5 3 1 1 3 4 3 R E C A L L 1 7 7 2 4 1 7 1 1 4 9 2 4 4 D I A G R A M 27 1 1 1 7 7 9 5 9 4 2 3 4 I 1 1 E Q U A T I O N 2 6 7 2 3 8 6 9 5 4 6 1 G U E S S 3 6 2 8 1 2 8 4 1 2 S M O O T H I N G 2 2 4 4 11 2 2 -T E M P L A T I O N 23 2 13 1 9 15 8 9 4 4 2 2 R A N D O M C A S E S 1 8 6 6 1 2 6 6 1 2 18 6 1 2 S Y S T E M A T I C C A S E S 1 4 1 4 1 4 1 4 29 14 O B T A I N S O L U T I O N 2 4 4 8 5 5 3 32 1 1 8 C H E C K I N G PART" 32 7 9 1 4 5 1 4 7 7 7 C H E C K I N G S O L U T I O N 3 5 6 9 3 12 3 12 2 4 C O N C E R N P R O C E S S 4 2 4 4 8 1 2 4 1 9 8 C O N C E R N S O L U T I O N 2 8 12 3 1 2 3 3 3 6 1 9 3 6 F I G U R E 16 ROW P E R C E N T A G E O F M O V E S i P R O C E S S M A T R I X F O R M A T H W O R L D P R O B L E M S (N=20) 92 o 2 M Q < o 2 O M En K Q W CO CO w o n EH O O s oo 2 O M En Cm S w EH CO o 2 w n to O CO E H CO w t—i E H E H CO o t-H o £D w <; O o \A CO u PH CO O < M PL, W o E H C5 o < *~ 2 M M K « o W W W Q O O o o < w w 2 2 5H O O CO o o o o READING 8 6 28 17 16 5 3 7 1 3 7 1 2 RECALL 6 9 44 6 19 6 3 3 DIAGRAM . 23 8 17 6 19 8 6 4 4 5 1 1 EQUATION 19 5 49 4 2 2 4 2 14 5 GUESS 22 19 4 4 4 • 33 4 11 SMOOTHING 13 7 20 10 3 30 13 3 TEMPLATION 29 3 17 11 11 6 3 9 3 3 6 RANDOM CASES 21 3 6 24 9 6 3 6 11 6 6 SYSTEMATIC CASES 12 12 12 12 12 12 12 12 OBTAIN SOLUTION 22 2 5 2 5 1 3 44 2 13 ! CHECKING PART" 25 6 25 6 2 4 9 • 6 2 2 6 2 CHECKING SOLUTION 26 2 11 6 9 6 13 2 9 4 13 CONCERN PROCESS 26 9 13 4 4 9 30 6 CONCERN SOLUTION 39 13 4 13 9 22 FIGURE 17 ROW PERCENTAGE OF MOVES «PROCESS MATRIX FOR REAL WORLD PROBLEMS (N=20) R e s u l t s o f A n a l y s i s - F i e l d Independence W i t h IQ r e s t r i c t e d t o one s t a n d a r d d e v i a t i o n , 115 t o 125., t h e f i e l d i n d e p e n d e n t - d e p e n d e n t v a r i a b l e had a s i g n i f i c a n t e f f e c t on many o f t h e v a r i a b l e s c o n s i d e r e d i n t h i s s t u d y . The r e s u l t s o f t h e r e g r e s s i o n a n a l y s i s p e r f o r m e d w i t h f i e l d i n d e p e n d e n c e as t h e i n d e p e n d e n t v a r i a b l e a r e g i v e n i n T a b l e 5. The r e s u l t s o f h y p o t h e s e s 4, 5, 6, and 10 a r e g i v e n b e l o w : H y p o t h e s i s l+: A s i g n i f i c a n t d i f f e r e n c e (P=.0110) was f o u n d i n t h e t o t a l ^•number o f h e u r i s t i c s u s e d b y f i e l d i n d e p e n d e n t s t u d e n t s o v e r f i e l d dependent s t u d e n t s . By i t s e l f t h e f i e l d i n d e p e n d e n c e v a r i a b l e a c c o u n t s f o r 16% o f t h e v a r i a n c e o f t h e t o t a l number o f h e u r i s t i c s . H y p o t h e s i s 5: A s i g n i f i c a n t d i f f e r e n c e (P=.0385) was f o u n d i n t h e number o f d i f f e r e n t h e u r i s t i c s u s e d b y f i e l d i n d e p e n d e n t s t u d e n t s o v e r f i e l d dependent s t u d e n t s . By i t s e l f t h e f i e l d i n d e p e n d e n c e v a r i a b l e a c c o u n t s f o r 11% o f t h e v a r i a n c e o f t h e number o f d i f f e r e n t h e u r i s t i c s u s e d . H y p o t h e s i s 6: .A s i g n i f i c a n t d i f f e r e n c e (P=.0575) was f o u n d i n t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d b y f i e l d i n d e p e n d e n t s t u d e n t s o v e r f i e l d dependent s t u d e n t s . By i t s e l f t h e f i e l d i n d e p e n d e n t v a r i a b l e a c c o u n t e d f o r 9% o f t h e v a r i a n c e o f t h e number o f c o r r e c t s o l u t i o n s . P o s t hoc a n a l y s e s r e v e a l e d t h a t t h e r e was no s i g n i f i c a n t d i f f e r e n c e (P=.3263) between t h e number o f t i m e s a s u b j e c t i s w i l l i n g t o a t t a c k a p r o b l e m ( c y c l e s ) and f i e l d i n d e p e n d e n c e . However, f i e l d i n d e p e n d e n c e d i d have a s i g n i f i c a n t e f f e c t (P=.0892) on t h e number o f changes he makes i n a t t a c k i n g a p r o b l e m . F i e l d i n d e p e n d e n c e a c c o u n t s f o r 7% o f t h e v a r i a n c e i n t h e number o f changes. TABLE 5 RESULTS OF -""REGRESSION ANALYSIS WITH FIELD INDEPENDENCE AS INDEPENDENT VARIABLE (N=40) DEPENDENT VARIABLE SOURCE OF VARIATION F - VALUE TO ENTER/REMOVE 2 F - PROB. R WITH DEPENDENT VARIABLE R^SQ TOTAL HEURISTICS FIELD INDEPENDENCE 7.084 .0110 .3964 .1571 DIFFERENT HEURISTICS 4.495 .0385 .3252 .1058 CYCLES .9957 .3263 .1597 .0255 CHANGES 2.970 .0892 .2693 .0725 CORRECT SOLUTION 3.745 .0575 .2995 .0879 Each of These is a Separate Simple Regression. Probability of Making a Type 1 Error by Rejecting Null Hypothesis, that i s , Claiming S t a t i s t i c a l Signifance. 'The Proportion of Variance i n the Dependent Variable Accounted for by Field Independence. 95 I n t e s t i n g h y p o t h e s i s 10, t h e s u b j e c t s were d i v i d e d i n t o t h i r d s a c c o r d i n g t o t h e i r s c o r e on t h e EFT. The t o p t h i r d (14 s u b j e c t s ) formed t h e f i e l d i n d e p e n d e n t g r o u p and t h e b o t t o m t h i r d (14 s u b j e c t s ) formed t h e f i e l d dependent g r o u p . Those s u b j e c t s i n t h e m i d d l e range o f s c o r e s were e x c l u d e d f r o m t h i s a n a l y s i s . T h i s was done i n o r d e r t o o b t a i n a g r e a t e r d i f f e r e n c e i n t h e d e g r e e o f f i e l d i n d e p e n d e n c e between s u b j e c t s c l a s s i f i e d as f i e l d i n d e p e n d e n t and t h o s e c l a s s i f i e d as f i e l d d e pendent. H y p o t h e s i s 10 was n o t t e s t e d f o r s t a t i s t i c a l s i g n i f i c a n c e . I f f i e l d i n d e p e n d e n c e does e f f e c t t h e p a t t e r n o f p r o c e d u r e s u s e d , t h e d i f f e r e n c e may be e a s i e r t o o b s e r v e u s i n g t h e s e two g r o u p s o f s u b j e c t s r a t h e r t h a n t h e e n t i r e sample. To t e s t h y p o t h e s i s 10 a p a i r o f p r o c e s s m a t r i c e s were formed u s i n g t h e r e s u l t s f r o m t h e c o d i n g forms f o r t h e t o p 14 f i e l d i n d e p e n d e n t s u b j e c t s f o r one m a t r i x ( F i g u r e 18) and t h e b o t t o m 14 f i e l d dependent s u b j e c t s f o r t h e o t h e r m a t r i x ( F i g u r e 19)• The p e r c e n t a g e o f moves i n e a c h a r e a i s g i v e n i n F i g u r e s 20 and 21. I n e v a l u a t i n g h y p o t h e s i s 10, t h e f o l l o w i n g m a j o r a r e a s f r o m t h e p r o c e s s m a t r i x were c o n s i d e r e d * 1. Moves f r o m c o r e t o c o r e o r t o h e u r i s t i c s ( A r e a s A and B) 2. Moves f r o m h e u r i s t i c s t o c o r e o r t o h e u r i s t i c s ( A r e a s E and F ) 3. Moves c o n c e r n e d w i t h c h e c k i n g ( A r e a s C, G, M and N) 4. Moves c o n c e r n e d w i t h e x p r e s s i o n o f c o n c e r n ( A r e a s D, H, Q and R ) * I n c o m p a r i n g t h e s e a r e a s f o r t h e two g r o u p s ( f i e l d i n d e p e n d e n t and f i e l d d e p e n d e n t ) t h e r e s e a r c h e r c o n s i d e r e d t h e p e r c e n t a g e o f moves i n e a c h a r e a f o r each g r o u p as w e l l as c o m p a r i n g t h e number and t y p e s o f moves f o r each p r o c e d u r e . That i s , f o r a g i v e n p r o c e d u r e i s t h e r e a p r o c e d u r e ( o r p r o c e d u r e s ) w h i c h s t u d e n t s have a t e n d e n c y t o use n e x t ? H y p o t h e s i s 10: The p a t t e r n o f p r o c e d u r e s u s e d b y t h e t o p 14 f i e l d i n d e p e n d e n t s u b j e c t s i s o b s e r v a b l y d i f f e r e n t t h a n t h a t o f t h e b o t t o m 96 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 1 9 3 11 hz 5 16 41 3 9 16 8 10 1 2 9 4 5 4 2 1 2 3 3 1 5 1 5 6 2 6 2 8 2 1 2 2 1 1 4 17 12 10 4 2 2 13 7 8 5 4 3 5 4 1 2 5 1 3 2 1 l 1 5 2 3 3 1 1 6 1 3 1 1 27 3 2 l l 1 7 25 3 2 30 39 2 6 4 3 l l 7 1 7 1 8 10 2 7 33 1 3 3 l l 8 12 3 2 1 3 9 6 7 1 1 1 65 8 3 7 5 3 29 7 1 4 6 3 2 10 5 3 1 1 1 2 18 1 1 2U 4 1 3 1 -11 2 2 5 1 5 1 1 4 4 1 12 3 13 5 4 3 2 2 1 1 10 •7. 3 3 6 1 2 1 1 1 4 2 4 14 15 6 1 2 2 3 2. 2 2 1 1 2 1 1 17 1 2 1 1 3 2 16 1 1 2 1 8 2 1 1 1 l 17 18 1 1 1 21 2 1 19 1 20 1 21 13 1 7 3 7 2 6 3 3 4 3 4 3 1 1 3 3 l 22 13 1 7 3 7 2 6 3 3 It 3 4 3 1 1 3 3 l 23 5 1 4 2 l 1 1 3 1 2 6 l 1 1 4 1 1 4 14 24 4 1 1 2 l 1 1 3 1 4 1 3 -25 1 2 1 5 1 1 26 4 3 2 1 2 1 27 7 4 1 3 1 4 1 8 1 4 FIGURE 18 PROCESS MATRIX«FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR FIELD INDEPENDENT STUDENTS (N=l4) 97 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 23 29 1 25 9 45 4 23 52 3 15 32 1 6 11 1 2 6 10 2 1 6 4 8 2 1 3 4 1 1 1 4 3 1 4 4 1 3 1 4 25 1 6 4 1 3 3 2 5 1 1 2 1 2 1 3 5 2 1 3 4 1 1 1 i 6 11 1 22 1 1 l 1 7 24 3 1 3 28 50 2 4 1 2 l 7 1 2 1 8 13 1 1 7 57 6 6 1 1 11 16 2 2 1 9 12 2 1 1 2 22 3 1 4 3 1 22 5 1 1 3 1 10 6 3 3 16 1 2 1 1 29 2 1 •11 2 2 1 2 12 1 2 1 13 9 3 1 3 2 2 2 1 1 2 1 l 1 14 15 3 l 3 l 1 1 1 12 2 2 A t 16 1 1 1 1 2 1 17 1 18* 19 20 21 16 1 2 3 2 2 2 2 23 10 __16_ 22 23 17 16 1 1 1 6 2 2 4 2 1 1 1 3 1 3 1 1 1 2 1 4 10 t 1 24 2 2 1 1 1 25 1 A 26 1 1 3 27 6 l 1 1 3 2 11 2 FIGURE 19 PROCESS MATRIX 1FREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES FOR FIELD DEPENDENT STUDENTS (N=l4) 98. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19.20 22 23 24 25 26 27 1 2 3 4 5 6 49 8 5 3 7 8 9 10 11 12 13 14 15 16 9 3 1 1 17 18 19 20 21 2 1 3 1 22 23 5 2 1 1 24 25 26 3 1 2 27 FIGURE 20 P E R C E N T OF PROCEDURES U S E D I N E A C H CATEGORY - F I E L D I N D E P E N D E N C E ( N = l 4 ) 99 1 2 3 * 5 6 7 8 9 10 11 12 13 14 15 lfS 17 18 19 20 22 23 24 25 26 27 1 2 3 4 5 6 64 5 5 2 7 8 9 10 11 12 13 14 15 16 4 2 1 17 18 19 20 21 3 3 1 22 23 6 1 1 24 25 26 2 27 FIGURE 21 PERCENT OF PROCEDURES USED IN EACH CATEGORY - FIELD DEPENDENCE (N=l4) 100 14 f i e l d dependent s u b j e c t s . The f i e l d dependent s u b j e c t s a p p e a r t o be more e o r i e n t e d w i t h a h i g h e r p e r c e n t o f p r o c e d u r e s i n A r e a A. The f i e l d i n d e p e n d e n t s u b j e c t s a r e more c o n s i s t e n t l y c o n c e r n e d w i t h t h e i r work ( s e e A r e a s D, H, Q, R, S, T ) . I n t h e use o f h e u r i s t i c s , t h e f i e l d i n d e p e n d e n t s u b j e c t s make g r e a t e r use o f s y s t e m a t i c c a s e s (#16) and g i v e what l i t t l e e v i d e n c e t h e r e i s o f t h e h i g h e r h e u r i s t i c s . B o t h g r o u p s appear t o use b o t h t e m p l a t i o n (#13) and random c a s e s (#15) a c r o s s a l l p r o c e d u r e s . A g a i n , a s e c o n d p r o c e s s m a t r i x was o b t a i n e d f o r e a c h g r o u p where t h e e n t r i e s i n t h e m a t r i x i n d i c a t e t h e number o f p r o b l e m s i n w h i c h a p a r t i c u l a r move i s made. (See F i g u r e s 22 and 23). The row p e r c e n t a g e s f o r t h e s e m a t r i c e s a r e g i v e n i n F i g u r e s 24 and 25. The f i e l d dependent s u b j e c t s a p p e a r t o have had g r e a t d i f f i c u l t y i n u n d e r s t a n d i n g t h e p r o b l e m s w i t h 31 t o 57 p e r c e n t o f t h e i r moves t o r e a d i n g f r o m a m a j o r i t y o f t h e p r o c e d u r e s . The m a j o r i t y o f t h e i r moves a r e t o d i a g r a m s , e q u a t i o n s o r t e m p l a t i o n . H a l f o f t h e i r r e s p o n s e s t o c o n c e r n i s t o r e r e a d t h e p r o b l e m h o p i n g t o g a i n some i n s i g h t . The f i e l d i n d e p e n d e n t s u b j e c t moves more f r e e l y t o a g r e a t e r v a r i e t y o f p r o c e d u r e s and t o t h e f o u r h e u r i s t i c s . He i s more c o n c e r n e d w i t h h i s work, c o n t i n u a l l y c h e c k i n g b o t h h i s work and s o l u t i o n . When c o n c e r n e d about t h e p r o c e s s he u s e s o r h i s s o l u t i o n , he i s more w i l l i n g t o check h i s p r o c e d u r e s o r s o l u t i o n s . I n g e n e r a l , t h e f i e l d i n d e p e n d e n t s u b j e c t moves between a l l t h e p r o c e d u r e s i n t h e m a t r i x w h i l e t h e f i e l d dependent s u b j e c t spends a g r e a t d e a l o f t i m e i n t h e c o r e a r e a s . R e s u l t s o f A n a l y s i s - Gore and H e u r i s t i c P r o c e d u r e s The c o r r e l a t i o n s f o r h y p o t h e s e s 7 and 8 a r e g i v e n i n T a b l e 6. An F - T e s t was u s e d t o d e t e r m i n e i f t h e s e c o r r e l a t i o n s were s i g n i f i c a n t l y d i f f e r e n t f r o m 0. 101 CO w CO •< EH CO o CC W < 2 CO o o -< M o 2 2 M o E H C J O M E H < 2 2 •-H < M «< s M H CC E-t 0 0 E H •3 o w Q < C 5 < CO O PH t—« o E H O < <*: c w O 2 {0 w W w i—i s W •< >H CC Q w o 0"} E H CC CO o 2 O M E H O CO e> 2 o w •T* f— o CO CO W o o CC PH 2 CC w o 2 O o 2 O M E H t=> •H O CO 2 CC W o 2 O O READING 6 9 36 28 14 3 7 9 1 8 4 4 RECALL 3 1 6 9 2 6 2 1 2 1 1 DIAGRAM 13 12 14 6 9 7 8 4 3 4 5 1 1 EQUATION 13 6 1 39 5 6 3 2 6 7 GUESS 5 3 2 1 1 4 1 3 SMOOTHING 6 1 1 1 4 4 1 TEMPLATION 5 2 3 7 2 3 5 1 1 1 1 RANDOM CASES 5 1 2 5 2 2 1 2 4 1 3 1 SYSTEMATIC CASES 1 1 1 2 3 1 ; OBTAIN SOLUTION 11 2 5 1 2 4 1 3 23 3 9 CHECKING PART" 9 1 6 8 3 3 4 3 3 2 3 : CHECKING SOLUTION 5 1 6 2 3 2 5 1 4 2 4 CONCERN PROCESS 8 1 3 2 1 2 2 10 2 CONCERN SOLUTION 7 4 3 1 4 1 7 1 J ... . FIGURE 22 PROCESS MATRIX tFREQUENCY DISTRIBUTION OF CHANGE OF PROBLEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM, FOR FIELD INDEPENDENT STUDENTS (N=l4) 102 2 M p t-H •H < 2 o M E H << to to w o 2 r-H. X E H O 2 O to w to <; r-H O E H PL. P O S K P W O s to w E H < 2 to o ON w M to ON to F H to M •< E H w £< o K o E> O o hH o PH to re o M e» CM to E H •< 2 2 2 2 l - l M re I X w iii & w w E H O O o o w w o 2 O to o o o READING 20 9 40 34 30 2 5. 9 1 5 9 3 6 RECALL 4 1 2 7 4 1 3 DIAGRAM 24 6 6 6 8 3 2 1 1 3 1 EQUATION 26 4 l 32 4 1 3 2 7 3 1 GUESS 6 3 1 1 1 2 1 SMOOTHING 3 2 1 2 • TEMPLATION . 9 4 3 2 1 2 1 1 RANDOM CASES 3 4 1 1 2 2 SYSTEMATIC CASES 1 1 1 1 OBTAIN SOLUTION 16 1 4 2 2 2 19 8 CHECKING PART- 11 1 5 1 1 3 1 1 1 CHECKING SOLUTION 11 1 1 1 1 1 3 2 3 CONCERN PROCESS 4 1 1 1 CONCERN SOLUTION 6 1 1 1 2 2 FIGURE 23 PROCESS MATRIXiFREQUENCY DISTRIBUTION OF CHANGE OF PR03LEM SOLVING PROCEDURES, COUNTED ONCE PER PROBLEM, FOR FIELD DEPENDENT STUDENTS (N=l4) 103 CO o ff—< o w M to CO UT CO M E H UT M CO d CC >-} o & W < o o r-H 2 CO o P-, CO CC o O < M CM to 2 2 M o a e» o O M EH < 2 2 2 2 r-H •X M n l-H CC CC M HH CC E H CO •4 o w w w n < o < CO O PH p E H o o o o o < •—\ w O t :" r — w w 2 w w M s W < >-r" o O CC CC P w o 00 E H CC to o o o O READING 5 7 28 22 11 2 5 7 1 6 3 3 RECALL 9 3 18 26 6 15 6 3 6 3 3 DIAGRAM 14 13 16 7 10 8 9 4 3 4 6 1 i EQUATION 15 7 1 44 6 7 3 2 7 8 GUESS 25 15 10 5 5 20 5 15 SMOOTHING 33 6 6 6 22 22 6 TEMPLATION 14 6 11 9 20 6 9 14 3 3 3 3 RANDOM CASES 17 3 7 17 7 7 3 7 14 3 10 3 SYSTEMATIC CASES 11 11 11 22 33 11 OBTAIN SOLUTION 17 3 8 2 3 6 2 5 36 5 14 ; CHECKING PART" 20 2 13 17 7 7 9 7 7 4 7 CHECKING SOLUTION 14 3 17 6 9 6 14 3 11 6 11 CONCERN PROCESS 26 3 10 6 3 6 6 32 6 CONCERN SOLUTION 25 14 11 4 14 4 2 5 1 4 FIGURE 24 ROW PERCENTAGE OF MOVES »PROCESS MATRIX FOR FIELD INDEPENDENT STUDENTS (N.= l4) 104 READING RECALL DIAGRAM EQUATION GUESS SMOOTHING TEMPLATION RANDOM CASES SYSTEMATIC CASES OBTAIN SOLUTION CHECKING PART CHECKING SOLUTION CONCERN PROCESS CONCERN; SOLUTION CO M0 w M to o CO F H CO M E H ID w E H CO o ce >-J o E> w < O o HH 2 CO o PH CO a: o o O << M PH to 2 2 M o E H o e» s O H E H < 2 2 2 2 M << s: M r-H re re M re e-t CO EH ••3 o w W w Q < •< CO O PH o E-- o O o o < o < n? w O f-0 w IV! 2 W w i — 1 c 2 W *H o O r e K Q w o CO E-i re CO o o o o 12 5 23 20 17 1 3 5 1 3 5 2 3 18 5 9 32 18 5 14 39 10 10 10 13 5 3 2 2 5 2 31 5 1 38 5 1 4 2 8 4 1 40 20 7 7 7 13 7 38 25 13 25 • 39 17 13 9 4 9 4 4 23 31 8 8 15 15 25 25 25 25 30 2 7 4 4 4 35 15 , 44 4 20 4 4 12 4 4 4 46 4 4 4 4 4 13 8 57 14 14 14 46 8 8 8 15 15 FIGURE 25 . ROW PERCENTAGE OF MOVESiPROCESS MATRIX FOR FIELD DEPENDENT STUDENTS (N=l4) T A B L E 6 I N T E R C O R R E L A T I O N S AMOUNG MEASURES OF H Y P O T H E S I Z E D V A R I A B L E S (N=40) 2 3 4 5 6 7 8 1. P R O B L E M C O N T E X T 3 , ,01 .11 .16 -.08 .13 .23 -.02 2.. F I E L D I N D E P E N D E N C E .40* .33* .16 .27 .30 .23 3. T O T A L H E U R I S T I C S .84 .48 .80 .49 .08 4. D I F F E R E N T H E U R I S T I C S .43 .75 .47 .19 5. C Y C L E S .52 .14 .12 6. CHANGES .51** .20 7. CORRECT S O L U T I O N .18 8.. O F F E R E D S O L U T I O N is Equivalent to the Point-Blserial Correlation Coefficent. # ## otomous Variable. R c r i t = # 3 l 2 1 @ P=.05 R c r i t = , ^ ° 2 6 @ p = = * 0 1 o H y p o t h e s i s 7: A s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n (P=.0013) was f o u n d between t h e t o t a l number o f h e u r i s t i c s u s e d b y a s t u d e n t and t h e number o f c o r r e c t s o l u t i o n s he o b t a i n e d . H y p o t h e s i s 8: A s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n (P=.0022) was f o u n d between t h e number o f d i f f e r e n t h e u r i s t i c s u s e d b y a s t u d e n t and t h e number o f c o r r e c t s o l u t i o n s he o b t a i n e d . P o s t hoc a n a l y s i s was p e r f o r m e d , u s i n g l i n e a r r e g r e s s i o n , t o d e t e r m i n e t h e amount o f v a r i a n c e i n t h e number o f c o r r e c t s o l u t i o n s w h i c h c o u l d be a c c o u n t e d f o r b y t h e h e u r i s t i c s , c y c l e s and c h a n g e s . These r e s u l t s a r e g i v e n i n T a b l e 7» The t o t a l number o f h e u r i s t i c s a c c o u n t s f o r 22$ o f t h e v a r i a n c e i n c o r r e c t s o l u t i o n s w h i l e t h e number o f d i f f e r e n t h e u r i s t i c s u s e d a c c o u n t e d f o r 2 2 % . The number o f c y c l e s , ( t h e number o f t i m e s a s u b j e c t a t t e m p t s t o s o l v e a p r o b l e m ) a c c o u n t s f o r l e s s t h a n 2% o f t h e v a r i a n c e , w h i l e t h e change o f p r o c e d u r e s a c c o u n t s f o r 2 6 % o f t h e v a r i a n c e o f c o r r e c t s o l u t i o n s . T a b l e 8 l i s t s t h e means and s t a n d a r d d e v i a t i o n s and T a b l e 9 l i s t s t h e i n t e r c o r r e l a t i o n s between p a i r s o f v a r i a b l e s d e r i v e d f r o m t h e c o d i n g s y s t e m f o r t h e t o t a l sample o f f o r t y s u b j e c t s . I t s h o u l d be n o t e d t h a t a c o r r e l a t i o n c o e f f i c i e n t , w h i c h i s s t a t i s t i c a l l y s i g n i f i c a n t , p r o v i d e s a d d i t i o n a l i n s i g h t i n t o t h e d a t a , b u t d o e s n o t a l l o w c a u s a l i n f e r e n c e t o be made. Most o f t h e s i g n i f i c a n t c o r r e l a t i o n s have o b v i o u s i n t e r p r e t a t i o n s o The l a r g e number o f s i g n i f i c a n t c o r r e l a t i o n s w i t h c y c l e s and changes r e f l e c t t h e f a c t t h a t s u b j e c t s who a r e a t t e m p t i n g t o b e g i n p r o b l e m s a g a i n o r change t h e i r p r o c e d u r e s a r e p e r f o r m i n g more p r o c e s s e s . As w o u l d be e x p e c t e d , t h e c o r e a r e a s a r e s i g n i f i c a n t l y c o r r e l a t e d . Those p r o c e d u r e s w h i c h a r e a l g e b r a i c s u c h a s , i d e n t i f y i n g v a r i a b l e s , s e t t i n g up e q u a t i o n s TABLE 7 RESULTS OF REGRESSION ANALYSIS WITH CORRECT SOLUTION AS DEPENDENT VARIABLE (N=40) DEPENDENT SOURCE OF F - VALUE TO *F - PR OB'. R WITH -*RSQ VARIABLE VARIATION ENTER/REMOVE DEPENDENT VARIABLE CORRECT TOTAL HEURISTICS 11.84 .0015 .4873 .2375 SOLUTION DIFFERENT HEURISTICS 10.53 .0025 .4658 .2170 CHANGES 13.33 .0009 .5096 .2597 CYCLES .7424 .3986 .1386 .0192 •^Each of These is a Separate Simple Regression. 2 Probability of Making a Type 1 Error by Rejecting Null Hypothesis, that i s Claiming S t a t i s t i c a l Significance. 3 ^The Proportion) of Variance in the Dependent Variable Accounted for by Independent Variable., 108 TABLE 8 MEANS PROCEDURES AND STANDARD DEVIATIONS FOR DERIVED FROM CODING SYSTEM (N=40) Variable Mean Standard Deviation Reading 17.85 6.93 Recall Fact 3.25 2.68 Draw Diagram: 8.12 6.80 Modify Diagram. 2.32 2.29 Identify Variable 2-80 2.33 Set up Equations 9.08 6.85 Algebraic Algorithms 8.28 10.53 Arithmetic- Algorithms 31.10 35.68 Guessing 4.90 2.63 Smoothing 1.08 1.99 Templation. 3.75 3.54 Random Cases 2.35 2.51 Systematic Cases .95 1.95 Obtain Solution 6.15 2.69 Checking Part 5.98 4.44 Checking Solution; 4.42 3.42 Concern Process 2.32 3.12 Concern Solution 2.52 3.13 Cycles 10.38 4.12 Changes ; 1.85 2.30 Stops without Solution 1.12 1.22 Correct Solution. 1.15 .89 Note- Appendix D Contains Histograms for Some Selected Variables. 109 1. READING PR03LEM .20 2 . RECALL RELATED FACT 3 . DRAW DIAGRAM 4. KCDIFY DIAGRAM 5.IDENTIFY VARIABLE 6.SETTING UP EQUATIONS 7.ALG0RITKKS-ALGE3RAIC 8 .ALGORXTHIS-ARITHKETIC 9.GUESSING 10.SMOOTHING 11 .TEMPLATION 12.CASES-RANDOM 13 .CASES-SYSTEMATIC l4.OBTAIN SOLUTION 1 5 . CHECKING PART 16. CHECKING SOLUTION 17. CONCERN-PROCESS 13.CONCERN-SOLUTION 1 9 . CYCLES 2 0 . CKANGES 21. WORK STOP-NO SOLN. 2 2 . CORRECT SOLUTION .26 .25 «» .51 . 4 8 . 5 * * * .50 . 4 4 * .24 .28 .39 .13 .09 .56* -.01 .18 -.02 .04 - . 0 9 .23 .46* .05 .59** .30 . 8 0 .01 -.11 - . 1 9 .18 .18 .44* - . 0 2 .10 - . 0 8 .14 -.19 .13 • 50 .18 .30 - . 0 6 - . 13 .62* .19 .02 .07 • .49 « 2 0 » • .38 .39 .3^* .47** .25 .17 - 0 * .07 - .12 .19 .79 .17 .03 • ** .36 «W .14 .08 .22 .43 .21 » .32 .11 . 0 0 .89 .21 . 5 6 # .08 .62* .14 .26 « .37 .18 .28 .21 .15 .59 .4l*< .36* - . 0 5 * .57 .58* .30 .17 .05 .60 .02 . 4 5 " • • .50 .13 .29 .15 .22 .12 .62 .05 .38* • .39 - . 0 6 .22 « .31 - . 0 2 .14 .67 .22 .20 .40* .27 . ** .42 .21 • . *» .42 -.03 .48** ,** .56 .29 .52 «* .52 .21 .40* , .60 .61 .30 .18 .75 - .02 .26 .20 « . 3 * .45 . 1 0 .20 .12 .23 .63 .16 .04 .18 . 3 * .56 .33 .59* .21 .26 .36 .51 .18 .60 .60 .43* .70* • 32 1 .49 * .32 # . 5 8 ,68* .62* .74 .25 . * .60 .49* . .08 .25 .38* - . 0 3 .62** - . 0 8 .21 .02 .45** .20 ,15 - .08 .09 .13 ,84** - .27 .18 .66 .36* -.18 -.18 .26 .62 - . 2 1 . 8 1 * * - 3 0 . 4 5 " - 5 3 ' „ 5 ? " - 0 2 .01 - . 2 8 ,49 ,14 - . 1 2 ®° - .12 - . 2 0 ** .16 ,52 - . 13 .06 .41 .24 ,14 - . 13 - . 16 .47 - . 0 3 .44 .16 ,58 .30 f ,22 .22 .21 .15 - . 0 ? .14 .51 -.18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 c r i t R .,.= . 4 0 2 6 @ P = . 0 1 ^ c r i t T A B L E 9 I N T E R C O R R E L A T I O N B E T W E E N P A I R S OF V A R I A B L E S FROM C O D I N G S Y S T E M ( N = 4 0 ) 110 and a l g e b r a i c a l g o r i t h m s a r e a l l c o r r e l a t e d . As w e l l , t h o s e f r o m t h e g e o m e t r i c a r e a s u c h as d r a w i n g d i a g r a m s , m o d i f y i n g d i a g r a m s and a r i t h m e t i c a l g o r i t h m s ( m a i n l y f r o m t h e use o f f o r m u l a s ) a r e c o r r e l a t e d . C h e c k i n g p a r t o f work i s p o s i t i v e l y c o r r e l a t e d w i t h most o f t h e p r o c e s s v a r i a b l e s , w h e r e a s , c h e c k i n g s o l u t i o n s i s s i g n i f i c a n t l y c o r r e l a t e d w i t h t h o s e p r o c e d u r e s p e r t a i n i n g t o s o l u t i o n s o r w h i c h were u s e d t o o b t a i n most o f t h e s o l u t i o n s . However, e x p r e s s i n g c o n c e r n f o r s o l u t i o n i s u n r e l a t e d t o o b t a i n i n g a c o r r e c t s o l u t i o n . F o u r p r o c e d u r e s f r o m t h e c o d i n g f o r m a c c o u n t e d f o r 8 6 % o f t h e s o l u t i o n s o b t a i n e d : a l g e b r a i c a l g o r i t h m (10%), a r i t h m e t i c a l g o r i t h m (31%), g u e s s i n g ( 2 8 % ) , and random c a s e s (17%), ( w i t h s y s t e m a t i c c a s e s , 22%). Of t h e s e , g u e s s i n g and a l g e b r a i c a l g o r i t h m s a r e u n r e l a t e d t o c o r r e c t s o l u t i o n , whereas b o t h a r i t h m e t i c a l g o r i t h m s and random c a s e s a r e s i g n i f i c a n t l y c o r r e l a t e d w i t h c o r r e c t s o l u t i o n . A d e t a i l e d a n a l y s i s o f t h e c o d i n g forms shows t h a t more t h a n h a l f o f t h e c o r r e c t s o l u t i o n s (26) were o b t a i n e d b y random c a s e s f o l l o w e d b y a r i t h m e t i c a l g o r i t h m s (14) w i t h g u e s s i n g a c c o u n t i n g f o r o n l y 6 c o r r e c t s o l u t i o n s and a l g e b r a i c a l g o r i t h m s f o r 1. B o t h a r i t h m e t i c a l g o r i t h m s and d i a g r a m s were u s e d c o n s i d e r a b l y w i t h b o t h random and s y s t e m a t i c c a s e s as w e l l as w i t h o t h e r c o r e p r o c e d u r e s . To g a i n some f u r t h e r i n s i g h t i n t o t h e r e l a t i o n s h i p between t h e s e v a r i a b l e s , a r e g r e s s i o n a n a l y s i s was p e r f o r m e d u s i n g c o r r e c t s o l u t i o n as t h e dependent v a r i a b l e . The v a r i a b l e s f r o m t h e c o r e a r e a were e n t e r e d i n t o t h e r e g r e s s i o n . e q u a t i o n f i r s t , . f o l l o w e d b y the. h e u r i s t i c s t r a t e g i e s ( v a r i a b l e s w i t h i n c o r e . a n d h e u r i s t i c s were a l l o w e d t o e n t e r t h e r e g r e s s i o n e q u a t i o n f r e e ) . The c o r e p r o c e d u r e s a c c o u n t f o r 40% o f t h e v a r i a n c e i n c o r r e c t s o l u t i o n , i n a d d i t i o n h e u r i s t i c s a c c o u n t e d f o r a n o t h e r 21% w i t h s y s t e m a t i c c a s e s a c c o u n t i n g f o r 13% o f t h i s ( s e e T a b l e 10). When t h e I l l TABLE 10 REGRESSION ANALYSIS ON CORRECT SOLUTION WITH CORE ENTERED BEFORE HEURISTICS (N=40) SOURCE OF MULTIPLE ? ARSQ F - VALUE TO F-PROB1 VARIANCE R RSQ ENTER/REMOVE CORE Arith.Algorithms .4742 .2249 .2249 11.0246 .0021 Draw Diagram .5242 .2748 .0499 2.5477 .1151 Reading .5597 .3133 .0384 2.0153 .1608 Recall Fact .5829 .3398 .0266 1.4085 .2418 Guessing .5981 .3577 .0178 .9442 .3402 Modify Diagram. .6322 .3997 .0420 2.3098 .1343 Alg. Algorithms .6340 .4019 .0022 .1194 .7292 Set up Equations .6343 .4024 .0005 .0248 .8496 Iden. Variables .6348 .4026 .0002 .0088 .8877 HEURISTICS Systematic. Cases .7552 .5307 .1281 11.3247 .0023 Random Cases .7712 .5948 .0641 1.6888 .2018 Templation .7778 .6058 .0110 .6986 .4154 Smoothing .7800 .6084 .0026 .2226 .6453 Probability of Making a Type 1 Error by Rejecting Null Hypothesis,i.e. Claming S t a t i s t i c a l Significance. 2 The Amount of Variance in Correct Solution Accounted for by the Independent Variable. 112 TABLE 11 REGRESSION ANALYSIS ON CORRECT SOLUTION WITH HEURISTICS ENTERED BEFORE CORE (N=40) SOURCE OF VARIANCE MULTIPLE ? R RSQ ARSQ F - VALUE TO ENTER/REMOVE F-PROB HEURISTICS Random Cases . 5 8 3 0 .3399 .3399 I 9 . 5 6 5 6 . 0 0 0 1 Smoothing . 6 1 3 3 . 3 7 6 0 . 0 3 6 1 2.1427 . 1484 Systematic Cases . 6 3 6 5 . 4 0 5 1 . 0 2 9 1 1.7580 . 1 9 0 3 Templation .6366 . 4 0 5 2 . 0 0 0 1 .OO96 .8852 CORE -Draw Diagram .6596 .4351 . 0 2 9 8 1.7946 .1862 Reading .6952 . 4 8 3 3 .0482 3 .0808 . 0 8 4 9 Arith. Algorithms . 7 0 9 3 . 5 0 3 1 .0198 1 .2731 . 2 6 7 0 Modify Diagram- . 7 3 0 5 . 5 3 3 6 . 0 3 0 6 2 . 0 3 1 5 . 1 6 0 6 Guessing .7481 .5596 . 0 2 6 0 1 . 7 7 0 0 .1905 Recall Fact .7658 .5865 .0269 1.8846 .1772 Set up Equations .7783 . 6 0 5 8 . 0 1 9 3 1 . 3 7 0 6 . 2 5 0 4 Alg. Algorithms .7793 . 6 0 7 4 . 0 0 1 6 .1089 .7398 Iden. Variable .7800 .6084 . 0 0 1 0 . 0 6 6 6 .7867 1 •""Probability of Making a Type 1 Error by Rejecting Null Hypothesis,i.e. Claming S t a t i s t i c a l Significance. 2 The Amount of Variance in Correct Solution Accounted for by. the Independent Variable. The a r e a w i t h i n t h e r e c t a n g l e r e p r e s e n t s t h e t o t a l v a r i a n c e o f c o r r e c t s o l u t i o n . C i r c l e s C and H r e p r e s e n t t h e p e r c e n t a g e s o f v a r i a n c e o f c o r r e c t s o l u t i o n w h i c h a r e c o n t r i b u t e d b y c o r e and h e u r i s t i c s , r e s p e c t i v e l y . FIGURE 26 RELATIVE CONTRIBUTIONS OF CORE AND HEURISTICS TO CORRECT SOLUTION o r d e r was r e v e r s e d i n t h e r e g r e s s i o n e q u a t i o n ( s e e T a b l e 11), t h e h e u r i s t i c s a c c o u n t e d f o r 41$ o f t h e v a r i a n c e . The c o r e p r o c e d u r e s added an a d d i t i o n a l 20$. I t a p p e a r s as t h o u g h f o r t h e s e s u b j e c t s , t h e i n f l u e n c e o f c o r e and h e u r i s t i c s p r o c e d u r e s i s e q u i v a l e n t ( s e e F i g u r e 26) CHAPTER V CONCLUSIONS T h i s s t u d y was u n d e r t a k e n t o g a i n some i n s i g h t i n t o m a t h e m a t i c a l p r o b l e m s o l v i n g . C o n f i n i n g a t t e n t i o n t o m a t h e m a t i c a l word p r o b l e m s , an at t e m p t was made t o u n d e r s t a n d s o m e t h i n g o f t h e s o l u t i o n t e c h n i q u e s u s e d b y a group o f s e n i o r h i g h s c h o o l s t u d e n t s . S p e c i f i c a l l y , what c o r e p r o c e d u r e s and h e u r i s t i c s do t h e y u s e ? What e f f e c t does p r o b l e m c o n t e x t have on t h e s e p r o c e d u r e s ? Does f i e l d i n d e p e n d e n c e i n f l u e n c e h e u r i s t i c u s a g e ? What p a t t e r n s o f h e u r i s t i c usage do s p e c i f i c g r o u p s o f i n d i v i d u a l s e x h i b i t ? To h e l p answer t h e s e q u e s t i o n s , a c o d i n g s y s t e m was d e v e l o p e d i n a f o r m w h i c h a l l owed c o m p a r i s o n a c r o s s i n d i v i d u a l s n o t o n l y o f t h e p r o c e d u r e s t h e y u s e d , b u t a l s o o f t h e sequence i n w h i c h t h e y were a p p l i e d . The c o d i n g s y s t e m a p p e a r s t o be b o t h u s e f u l and u s a b l e . I t ' s a p p l i c a t i o n t o t h e d a t a i n t h i s s t u d y has y i e l d e d many i n t e r e s t i n g r e s u l t s and i m p l i -c a t i o n s f o r f u r t h e r r e s e a r c h o Summary o f t h e E x p e r i m e n t a l S t u d y F o r t y s t u d e n t s who were j u s t c o m p l e t i n g t h e grade e l e v e n academic program i n m a t h e m a t i c s were r a n d o m l y s e l e c t e d f r o m f o u r t e e n A l g e b r a I I c l a s s e s i n t h r e e s e n i o r s e c o n d a r y s c h o o l s . The s u b j e c t s were o f av e r a g e m e n t a l a b i l i t y (IQ r a n g e : 115 t o 125) f o r t h i s p o p u l a t i o n . The s u b j e c t s were g i v e n t h e Embedded F i g u r e s T e s t i n d i v i d u a l l y and t h e s c o r e s r a n k - o r d e r e d and p a i r e d . The s u b j e c t s f r o m e a c h p a i r were r a n d o m l y a s s i g n e d t o one o f two g r o u p s , one gr o u p w o r k i n g p r o b l e m s i n a m a t h e m a t i c a l s e t t i n g , t h e o t h e r i n a r e a l w o r l d s e t t i n g . They were t h e n i n t e r v i e w e d 115 i n d i v i d u a l l y and a s k e d t o t h i n k a l o u d as t h e y s o l v e d a s e l e c t i o n o f m a t h e m a t i c a l word p r o b l e m s . A c o d i n g system was d e v e l o p e d b a s e d on t h e m o d e l o f h e u r i s t i c s o f MacPherson and was u s e d t o a n a l y z e t h e t a p e r e c o r d e d p r o t o c o l s o f t h e s u b j e c t s . The c o d i n g s y s t e m n o t o n l y i n d i c a t e d t h e p r o c e d u r e s u s e d b y t h e s u b j e c t s , b u t t h e sequence i n w h i c h t h e y were a p p l i e d . When t h e c o d i n g s y s t e m was a p p l i e d t o t h e s u b j e c t s ' t a p e r e c o r d e d p r o t o c o l s , o n l y a few o f t h e h e u r i s t i c s d e s c r i b e d b y M a c P h e r s o n were o b s e r v e d s u f f i c i e n t l y o f t e n t o be u s e d i n t h e s e q u e n t i a l a n a l y s i s . Most o f t h e s u b j e c t s were v e r y c o r e o r i e n t e d and made l i m i t e d u s e o f t h e ... h e u r i s t i c s . The d a t a c o l l e c t e d f r o m t h e c o d i n g f o r m s a l o n g w i t h t h e u s e o f r e g r e s s i o n a n a l y s i s were u s e d i n a n s w e r i n g t h e f i r s t t h r e e q u e s t i o n s -p o s e d i n C h a p t e r I . The i n f o r m a t i o n g a i n e d f r o m t h e p r o c e s s m a t r i c e s was u s e d i n a n s w e r i n g t h e l a s t q u e s t i o n . C o n s i d e r t h e f o u r q u e s t i o n s i n o r d e r : 1. What p r o c e d u r e s f r o m c o r e and what h e u r i s t i c s a r e u s e d b y s t u d e n t s i n a t t a c k i n g and s o l v i n g m a t h e m a t i c a l word p r o b l e m s ? A l l o f t h e c o r e p r o c e d u r e s i d e n t i f i e d i n t h e c o d i n g s y s t e m were u s e d b y t h e f o r t y s u b j e c t s . As w o u l d be e x p e c t e d , most s u b j e c t s a t t e m p t e d t o use a l g e b r a i c o p e r a t i o n s t o s o l v e t h e p r o b l e m s b y i d e n t i f y i n g v a r i a b l e s and a t t e m p t i n g t o s e t up some e q u a t i o n s . The h e u r i s t i c s o f t e m p l a t i o n and random c a s e s w e r e u s e d b y t h r e e -f o u r t h s o f t h e s u b j e c t s on a t l e a s t one p r o b l e m . O v e r h a l f o f t h e s t u d e n t s u s e d e a c h o f t h e s e h e u r i s t i c s more t h a n o n c e . S m o o t h i n g , a n a l y s i s , and s e q u e n t i a l c a s e s were u s e d by o n e - f o u r t h o f t h e s t u d e n t s , w i t h a n a l y s i s u s e d o n l y on p r o b l e m number 5« Some e v i d e n c e was o b t a i n e d o f t h e use o f c r i t i c a l c a s e s , a l l c a s e s , h y p o t h e t i c a l d e d u c t i o n , i n v e r s e 116 d e d u c t i o n ( u s e d by f o u r s t u d e n t s on p r o b l e m number 4), and v a r i a t i o n . The o t h e r h e u r i s t i c s were n o t o b s e r v e d . Of t h e f o r t y - s e v e n c o r r e c t s o l u t i o n s o b t a i n e d , 26 were t h r o u g h t h e u s e o f random o r s y s t e m a t i c c a s e s , a r i t h m e t i c a l g o r i t h m s m a i n l y a c c o u n t e d f o r 14, g u e s s i n g f o r 6, and a l g e b r a i c a l g o r i t h m s f o r 1. The use o f d i a g r a m s was r e l a t e d t o t h e number o f c o r r e c t s o l u t i o n s as w e l l as many o f t h e h e u r i s t i c s . F o u r o f t h e p r o b l e m s were g e o m e t r i c i n n a t u r e , and most o f t h e s m o o t h i n g and t e m p l a t i o n was r e l a t e d t o g e o m e t r i c concepts« Many o f t h e s t u d e n t s u s e d d i a g r a m s w i t h b o t h random and s y s t e m a t i c c a s e s and drew a s e p a r a t e d i a g r a m f o r e a c h c a s e . B o t h t h e t o t a l number o f h e u r i s t i c s u s e d b y a s u b j e c t as w e l l as t h e number o f d i f f e r e n t h e u r i s t i c s u s e d , a c c o u n t f o r a s i g n i f i c a n t amount o f t h e v a r i a n c e i n t h e number o f c o r r e c t s o l u t i o n s . I n p a r t i c u l a r , .." h e u r i s t i c s a c c o u n t e d f o r an a d d i t i o n a l 21% o f t h e v a r i a n c e n o t a c c o u n t e d f o r b y c o r e p r o c e d u r e s w i t h s y s t e m a t i c c a s e s a c c o u n t i n g f o r a s t a t i s t i c a l l y s i g n i f i c a n t amount o f t h i s (13$). Hence, t h e h e u r i s t i c s u s e d b y t h e s u b j e c t s i n t h i s s t u d y added t o t h e i r a b i l i t y t o s o l v e p r o b l e m s beyond t h e i r m a t h e m a t i c a l c o r e knowledge» S t u d e n t s who u s e d a w i d e r a n g e o f h e u r i s t i c s were on t h e a v e r a g e , b e t t e r p r o b l e m s o l v e r s . The s u b j e c t s who u s e d a w i d e r r a n g e o f h e u r i s t i c s u s e d them more e f f e c t i v e l y . These s t u d e n t s were more w i l l i n g t o change t h e i r mode o f a t t a c k o r t h e p r o c e d u r e s t h e y were u s i n g i f t h e y were n o t h a v i n g s u c c e s s i n o b t a i n i n g a s o l u t i o n . C h a n g i n g one's mode o f a t t a c k i n s o l v i n g a p r o b l e m was s i g n i f i c a n t l y r e l a t e d t o o b t a i n i n g a c o r r e c t s o l u t i o n . 2. Does t h e c o n t e x t o f t h e p r o b l e m i n f l u e n c e h e u r i s t i c u s a g e ? P r o b l e m c o n t e x t , as i d e n t i f i e d i n t h i s s t u d y , p r o v e d t o be u n r e l a t e d t o t h e h e u r i s t i c s u s e d . B o t h t h e t o t a l number o f h e u r i s t i c s u s e d and t h e number o f d i f f e r e n t h e u r i s t i c s u s e d were n o t i n f l u e n c e d b y t h e p r o b l e m s e t t i n g . A l t h o u g h i n e a c h c a s e , t h e means o f t h e v a r i a b l e s f a v o r t h e r e a l ' w o r l d s e t t i n g , t h e d i f f e r e n c e s a r e n o t g r e a t . S u b j e c t s w o r k i n g p r o b l e m s i n t h e m a t h e m a t i c a l w o r l d s e t t i n g had a s l i g h t l y h a r d e r t i m e u n d e r s t a n d i n g t h e p r o b l e m s , b u t p e r f o r m e d as w e l l as t h e o t h e r g r o u p . 3. What e f f e c t does f i e l d - d e p e n d e n c e - i n d e p e n d e n c e have on t h e p r o b l e m s o l v i n g p r o c e s s ? The s u b j e c t s u s e d i n t h i s s t u d y were chosen f r o m t h e m i d d l e IQ ran g e o f s t u d e n t s t a k i n g A l g e b r a I I . W i t h i n t h i s IQ r a n g e , f i e l d i n d e p e n d e n c e had a marked e f f e c t on t h e use o f h e u r i s t i c s and on t h e number o f c o r r e c t s o l u t i o n s o b t a i n e d . F i e l d dependent s t u d e n t s a p p e a r t o be v e r y o r i e n t e d t o t h e u s e o f c o r e p r o c e d u r e s . . They e x h i b i t some u s e o f t h e l o w e r h e u r i s t i c s , b u t - i n g e n e r a l t h e y a t t a c k e d most p r o b l e m s f r o m an a l g e b r a i c p o i n t o f v i e w . Most o f them drew a d i a g r a m f o r a p r o b l e m , b u t t h i s was u s e d i n o r d e r t o h e l p s e t up e q u a t i o n s . Once t h e y had a t t e m p t e d t o s o l v e a p r o b l e m w i t h o u t s u c c e s s , most c o n t i n u e d i n t h e same v e i n u s i n g e x a c t l y t h e same p r o c e d u r e s as b e f o r e . E v i d e n c e o f t h i s i s e x e m p l i f i e d i n t h e s o l v i n g o f t h e stamp p r o b l e m ( p r o b l e m #2) i n w h i c h a s i n g l e e q u a t i o n w i t h two unknowns can be w r i t t e n , and h e n c e , t h e r e a p p e a r s t o be a need f o r a second e q u a t i o n . The f i e l d d ependent s t u d e n t s c o n t i n u e d t o do so u n t i l t h e y o b t a i n e d one w h i c h was i n c o r r e c t o r dependent on t h e f i r s t e q u a t i o n . The f i e l d i n d e p e n d e n t s t u d e n t s soon r e a l i z e d a second e q u a t i o n c o u l d n o t be o b t a i n e d and changed t h e i r mode o f a t t a c k u s u a l l y t o c a s e s . What l i t t l e e v i d e n c e t h e r e was o f t h e use o f t h e h i g h e r h e u r i s t i c s came f r o m t h e f i e l d i n d e p e n d e n t s t u d e n t s . These s t u d e n t s u s e d a g r e a t e r v a r i e t y o f b o t h c o r e and h e u r i s t i c s i n a t t a c k i n g and s o l v i n g p r o b l e m s . They were much more w i l l i n g t o change t h e i r mode o f lid a t t a c k . They o b t a i n e d a g r e a t e r number o f c o r r e c t s o l u t i o n s . From t h e v i e w p o i n t o f p r o c e d u r e s u s e d , as w e l l as c o r r e c t s o l u t i o n s , t h e f i e l d i n d e p e n d e n t s t u d e n t i s a b e t t e r p r o b l e m s o l v e r . 4. Do s e l e c t e d g r o u p s o f i n d i v i d u a l s e x h i b i t p a t t e r n s o f h e u r i s t i c  usage when s o l v i n g m a t h e m a t i c a l word p r o b l e m s ? Two d i f f e r e n t c r i t e r i a were u s e d t o g r o u p s u b j e c t s i n an a t t e m p t t o answer t h i s q u e s t i o n , b y p r o b l e m c o n t e x t and b y d e g r e e o f f i e l d i n d e p e n d e n c e . S t u d e n t s g r o u p e d b y p r o b l e m c o n t e x t e x h i b i t e d t h e same g e n e r a l p a t t e r n s o f p r o b l e m s o l v i n g p r o c e d u r e s , b u t when gro u p e d by f i e l d i n d e p e n d e n c e showed an o b s e r v a b l e d i f f e r e n c e . These d i f f e r e n c e s were n o t t e s t e d f o r s t a t i s t i c a l s i g n i f i c a n c e . When g r o u p e d b y p r o b l e m c o n t e x t , b o t h g r o u p s e x h i b i t e d t h e same g e n e r a l p a t t e r n o f p r o b l e m s o l v i n g moves i n v o l v i n g c o r e p r o c e d u r e s and h e u r i s t i c s e x c e p t f o r s m o o t h i n g . S u b j e c t s i n t h e r e a l w o r l d s e t t i n g moved t o s m o o t h i n g f r o m a v a r i e t y o f p r o c e d u r e s w h e r e a s t h o s e i n t h e math w o r l d s e t t i n g u s e d s m o o t h i n g m a i n l y t h r o u g h d i a g r a m m i n g . B o t h g r o u p s o f s t u d e n t s moved t o and f r o m t h e u s e o f t e m p l a t i o n and random c a s e s t o a l l o f t h e c o r e p r o c e d u r e s . B o t h t h e r e a l w o r l d and math w o r l d g r o u p s o b t a i n e d s o l u t i o n s u s i n g t h e same p r o c e d u r e s and s t o p p e d work on t h e p r o b l e m s f o l l o w i n g t h e same moves. The s t u d e n t s who e n c o u n t e r e d r e a l w o r l d p r o b l e m s e x p r e s s e d c o n c e r n f o r s o l u t i o n s o b t a i n e d u s i n g h e u r i s t i c s w h i l e t h o s e s t u d e n t s i n t h e math w o r l d s e t t i n g e x p r e s s e d none. However, e x p r e s s i o n o f c o n c e r n f o r s o l u t i o n was f o u n d t o be u n r e l a t e d t o c o r r e c t s o l u t i o n . When grouped b y f i e l d i n d e p e n d e n c e a d i f f e r e n c e was f o u n d i n t h e o v e r a l l p a t t e r n o f s e q u e n t i a l moves i n t h e p r o b l e m s o l v i n g p r o c e s s . F i e l d dependent s t u d e n t s had g r e a t e r d i f f i c u l t y i n u n d e r s t a n d i n g t h e p r o b l e m s . I n g e n e r a l t h e y were v e r y c o r e o r i e n t e d , m a k i n g t h e m a j o r i t y o f t h e i r moves f r o m c o r e p r o c e d u r e s t o c o r e p r o c e d u r e s . The c o r e p r o c e d u r e s u s e d most o f t e n were w r i t i n g an e q u a t i o n and some use o f d i a g r a m s . F i e l d dependent s t u d e n t s e x h i b i t e d some u s e o f random c a s e s , b u t mainly-u s e d t h e h e u r i s t i c o f t e m p l a t i o n . I n g e n e r a l , t h e y were p o o r e r a t t e m p l a t i n g t h a n t h e f i e l d i n d e p e n d e n t s t u d e n t . The f i e l d i n d e p e n d e n t s t u d e n t moved more f r e e l y between t h e c o r e and h e u r i s t i c p r o c e d u r e s . He u s e d f o u r o f t h e h e u r i s t i c s : s m o o t h i n g , t e m p l a t i o n , random and s y s t e m a t i c c a s e s , a c r o s s a l l t h e p r o c e d u r e s i n , t h e p r o c e s s m a t r i x . He was more c o n c e r n e d w i t h h i s work b e c a u s e he was c o n t i n u a l l y c h e c k i n g b o t h t h e p r o c e d u r e s he was u s i n g as w e l l as h i s s o l u t i o n . When c o n c e r n e d w i t h h i s s o l u t i o n t h e f i e l d i n d e p e n d e n t s t u d e n t was more w i l l i n g t o check i t , u s u a l l y b y r e t r a c i n g h i s s t e p s , whereas t h e f i e l d dependent s t u d e n t r e r e a d t h e p r o b l e m . P e r h a p s t h e f i e l d . dependent s t u d e n t was n o t s u r e o f what t h e p r o b l e m was a s k i n g s I n g e n e r a l , t h e f i e l d i n d e p e n d e n t s t u d e n t was a b e t t e r p r o b l e m s o l v e r . He u s e d a g r e a t e r v a r i e t y o f h e u r i s t i c s and moved more f r e e l y a c r o s s a l l t h e c o r e and h e u r i s t i c p r o c e d u r e s . The f i e l d dependent s t u d e n t was v e r y c o r e o r i e n t e d i n t h e p r o c e d u r e s he u s e d . Summary and C o n c l u s i o n s f o r The M o d e l and C o d i n g System The h e u r i s t i c s d e f i n e d i n t h i s model a c c o u n t f o r t h e n o n - c o r e p r o c e d u r e s u s e d b y t h e f o r t y s u b j e c t s i n t h e s t u d y . They a r e b r o a d enough t o a c c o u n t f o r a l l o f t h e o b s e r v e d p r o c e s s e s and y e t , n o t so b r o a d t h a t no r e a l d i s t i n c t i o n c a n be made. The h e u r i s t i c s a p p e a r t o be t h e n a t u r a l k i n d s o f p r o c e d u r e s t h a t w o u l d be u s e d i n a t t a c k i n g and s o l v i n g p r o b l e m s . The u s e o f c a s e s p r o v e d t o be an i m p o r t a n t p r o c e d u r e i n o b t a i n i n g s o l u t i o n s . One o f t h e s t r o n g a d v a n t a g e s o f t h i s model o v e r t h o s e d i s c u s s e d i n C h a p t e r I I ' , i s t h e d i s t i n c t i o n between t h e d i f f e r e n t k i n d s o f " t r i a l and e r r o r " o r c a s e s . O ver t h r e e - f o u r t h s o f t h e s t u d e n t s i n v o l v e d i n t h i s s t u d y u s e d c a s e s o f some k i n d i n an a t t e m p t t o o b t a i n a s o l u t i o n t o a p r o b l e m . I t i s i m p o r t a n t t o d i s t i n g u i s h t h e t y p e o f sequence t h e y a r e u s i n g . I s t h e s t u d e n t j u s t t r y i n g some numbers i n a c o m p l e t e l y random f a s h i o n i n hopes o f h i t t i n g on t h e s o l u t i o n o r i s he a t t e m p t i n g t o f i n d a s o l u t i o n i n some k i n d o f s y s t e m a t i c f a s h i o n ? I s he p e r h a p s u s i n g t h e i n f o r m a t i o n o b t a i n e d f r o m one c a s e t o "improve** t h e n e x t o r a p p l y i n g t h e d a t a f r o m t h e p r o b l e m i n a s e q u e n t i a l f a s h i o n ? The s t u d e n t m i g h t d e t e r m i n e t h a t t h e r e can be o n l y a s m a l l f i n i t e number o f p o s s i b l e s o l u t i o n s ( a s i n p r o b l e m # 2) and hence, c o n s i d e r a l l p o s s i b l e c a s e s . A l l o f t h e s e t y p e s o f c a s e s have t h e i r p l a c e i n p r o b l e m s o l v i n g . They a r e d i f f e r e n t p r o c e d u r e s and s h o u l d be c o n s i d e r e d as s u c h . As w o u l d be e x p e c t e d , t h e s u b j e c t s i n t h i s s t u d y were v e r y o r i e n t e d t o u s i n g c o r e p r o c e d u r e s . A l t h o u g h t h e y d i d e x h i b i t u s e o f t h e l o w e r h e u r i s t i c s , t h e y s p e n t t h e m a j o r i t y o f t h e i r t i m e and e f f o r t i n a t t e m p t i n g t o s o l v e t h e p r o b l e m u s i n g o n l y c o r e m a t e r i a l . E v e n t h o u g h f e w e r c o r r e c t s o l u t i o n s were o b t a i n e d b y t h e s e s t u d e n t s t h a n t h o s e i n t h e p i l o t s t u d i e s , t h e s t u d e n t s were s t i l l m o t i v a t e d and a p p e a r e d t o do t h e i r b e s t i n w o r k i n g a l l t h e p r o b l e m s . The i n e f f e c t i v e u s e o f t h e h e u r i s t i c o f t e m p l a t i o n , w h i c h was t h e most u s e d h e u r i s t i c , r e v e a l e d t h a t s t u d e n t s do n o t know what c o r e m a t e r i a l t h e y can b r i n g t o b e a r on a p a r t i c u l a r p r o b l e m . They can r e c a l l g e n e r a l c a t e g o r i e s o f i n f o r m a t i o n ; sometimes c o r r e c t l y , sometimes n o t ; b u t cann o t r e c a l l s p e c i f i c d e t a i l s . I n p a r t i c u l a r , o n l y f o u r s t u d e n t s c o u l d r e c a l l the. P y t h a g o r e a n Theorem. Many o f t h e o t h e r s u b j e c t s knew t h e r e was some r e l a t i o n s h i p among t h e s i d e s o f a r i g h t t r i a n g l e , w h i l e o t h e r s d i d n o t even comment on t h e r e l a t i o n s h i p . The way t e m p l a t i o n i s d e f i n e d , i t 121 g i v e s no e v i d e n c e o f w h e t h e r t h e s u b j e c t i s l o o k i n g a t t h e g i v e n i n f o r m a t i o n i n t h e p r o b l e m and t h e n t r y i n g t o roam o v e r some c o n t e n t a r e a t o h e l p b r i d g e t h e gap t o t h e c o n c l u s i o n o r w o r k i n g t h e o t h e r way, l o o k i n g a t t h e c o n c l u s i o n o f t h e p r o b l e m and t r y i n g t o d e t e r m i n e w h i c h c o n t e n t w i l l h e l p h i m a c h i e v e i t . T h i s d i s t i n c t i o n may r e v e a l some i m p o r t a n t c h a r a c t e r i s t i c o f t h e p r o b l e m s o l v e r . Does he a t t a c k t h e p r o b l e m i n a f o r w a r d manner o n l y , o r f r o m t h e c o n c l u s i o n , w o r k i n g backwards ( v e r y l i t t l e e v i d e n c e was o b t a i n e d o f t h i s ) o r i s i t a c o m b i n a t i o n o f b o t h ? As c o r e o r i e n t e d as s t u d e n t s a r e . t e m p l a t i o n a p p e a r s t o be an i m p o r t a n t h e u r i s t i c and s h o u l d be r e d e f i n e d as f o l l o w s : D i r e c t T e m p l a t i o n - The h e u r i s t i c o f d i r e c t t e m p l a t i o n i s u s e d when a p r e m i s e o r s e t o f p r e m i s e s i s g i v e n o r assumed. The u s e o f t h e h e u r i s t i c i n v o l v e s r e c a l l i n g a c a t e g o r y o f c o n t e n t w h i c h i s r e l a t e d t o t h e p r e m i s e s g i v e n . T h i s i n c l u d e s t h e r e c a l l o f such t h i n g s as a l g o r i t h m s , p r o b l e m t y p e s , p r o c e d u r e s , t h e o r e m s , and p r o p e r t i e s r e l a t e d t o t h e p r e m i s e s . The p u r p o s e o f d i r e c t t e m p l a t i o n i s t o recall c o r e m a t e r i a l w h i c h i s r e l a t e d t o t h e p r e m i s e s i n o r d e r t o h e l p b r i d g e t h e gap t o t h e c o n c l u s i o n b y e x p a n d i n g knowledge about t h e p r e m i s e s . I n v e r s e . T e m p l a t i o n - The h e u r i s t i c o f i n v e r s e t e m p l a t i o n i s u s e d when a c o n c l u s i o n i s g i v e n o r assumed. The u s e o f t h e h e u r i s t i c i n v o l v e s r e c a l l i n g a c a t e g o r y o f c o n t e n t w h i c h i s r e l a t e d t o t h e c o n c l u s i o n o f t h e p r o b l e m . T h i s i n c l u d e s t h e r e c a l l o f s u c h t h i n g s as a l g o r i t h m s , p r o b l e m t y p e s , p r o c e d u r e s , t h e o r e m s , and p r o p e r t i e s r e l a t e d t o t h e c o n c l u s i o n . The p u r p o s e o f i n v e r s e t e m p l a t i o n i s t o r e c a l l c o r e m a t e r i a l w h i c h i s r e l a t e d t o t h e c o n c l u s i o n i n o r d e r t o e x p a n d knowledge about t h e c o n c l u s i o n and d e t e r m i n e c o n t e n t t h a t w i l l h e l p a c h i e v e i t . V e r y l i t t l e e v i d e n c e was o b t a i n e d t o s u p p o r t a n y o r d e r i n g o f t h e h e u r i s t i c s ( s e e F i g u r e 2, p. 8 ) . Few o f t h e h e u r i s t i c s i n t h e m i d d l e and 122 g e n e r a l c a t e g o r i e s were o b s e r v e d . T h e r e a p p e a r e d t o be c o n s i d e r a b l e movement between t h e s i e v e ( c o r e ) and t h e l o w e r h e u r i s t i c s . From t h o s e h i g h e r h e u r i s t i c s w h i c h were o b s e r v e d , s t u d e n t s u s i n g them d i d move t o a l o w e r h e u r i s t i c o r b a c k t o c o r e . S u b j e c t s d i d move f r o m b o t h t h e s i e v e and h e u r i s t i c s t o o b t a i n i n g a s o l u t i o n ( F i g u r e 3 , p. 1 Q)<> A Some e v i d e n c e was o b t a i n e d w h i c h r e l a t e d t o e x p r e s s i o n o f c o n c e r n , b u t t h i s e x p r e s s i o n was u n r e l a t e d t o o b t a i n i n g a s o l u t i o n . C o n c e r n f o r p r o c e s s was r e l a t e d t o b o t h c o r e and h e u r i s t i c p r o c e d u r e s . The c o d i n g s y s t e m p r o v e d t o be r e l i a b l e and c a p a b l e o f i d e n t i f y i n g i n d i v i d u a l d i f f e r e n c e s . The sys t e m n o t o n l y i d e n t i f i e s t h e p r o c e d u r e s u s e d , b u t a l s o i n d i c a t e s t h e sequence i n w h i c h t h e y a r e a p p l i e d . I n o r d e r t o be a b l e t o i d e n t i f y p r o b l e m s o l v i n g s t y l e s , t h e sequence i n w h i c h p r o c e d u r e s a r e u s e d w i l l c e r t a i n l y be a p r i n c i p a l s o u r c e o f i n f o r m a t i o n . I m p l i c a t i o n s f o r E d u c a t i o n The f i n d i n g s o f t h i s s t u d y i n d i c a t e t h a t s t u d e n t s who a r e c o m p l e t i n g an A l g e b r a I I c o u r s e have some f a c i l i t y w i t h t h e h e u r i s t i c s d e f i n e d i n MacPherson's model. I n p a r t i c u l a r , some e v i d e n c e was o b t a i n e d t o s u g g e s t t h a t s t u d e n t s who u s e t h e h e u r i s t i c s o f random and s y s t e m a t i c c a s e s , when f a c e d w i t h a d i f f i c u l t m a t h e m a t i c a l p r o b l e m , t e n d t o be s u c c e s s f u l i n s o l v i n g t h e p r o b l e m . Many o f t h e s t u d e n t s who f a i l e d t o us e c a s e s may have known how t o u s e t h i s t e c h n i q u e , b u t a v o i d e d i t b e c a u s e t h e y t h o u g h t i t was i n a p p r o p r i a t e . I n f a c t , most o f t h e s t u d e n t s who u s e d c a s e s d i d so as a l a s t r e s o r t w i t h some i m p l y i n g t h a t , a l t h o u g h t h i s t e c h n i q u e was n o t a c c e p t a b l e t o use as a p r o c e d u r e i n t h e c l a s s r o o m , t h e y w o u l d a t l e a s t " t r y some numbers" t o see what w o u l d happen. Many m a t h e m a t i c s t e a c h e r s who a r e e a g e r t o have t h e i r s t u d e n t s l e a r n t h e 123 " r i g h t " way o f s o l v i n g p r o b l e m s , p e n a l i z e t h e s t u d e n t s who r e s o r t t o t h e u s e o f c a s e s . P o l y a (1962, p. 26) s u g g e s t s t h a t t h e t e a c h e r s h o u l d n o t d i s c o u r a g e h i s s t u d e n t s f r o m u s i n g t r i a l and e r r o r ( c a s e s ) - b u t t o t h e c o n t r a r y , he s h o u l d e n c o u r a g e t h e i n t e l l i g e n t use o f t h i s f u n d a m e n t a l p r o c e d u r e . Many o f t h e good p r o b l e m s o l v e r s i n t h i s s t u d y d i d u s e b o t h random and s y s t e m a t i c c a s e s e f f e c t i v e l y . Some o f t h e s t u d e n t s d i d n o t u n d e r s t a n d how t o use t h e method. When t r y i n g t o use c a s e s , t h e s e s t u d e n t s g a i n e d v e r y l i t t l e i n f o r m a t i o n about t h e p r o b l e m . I t w o u l d be advantageous t o b o t h m a t h e m a t i c s t e a c h e r s and t h e i r s t u d e n t s i f more c o n s i d e r a t i o n were g i v e n t o t h e use o f c a s e s as an a c c e p t a b l e p r o b l e m s o l v i n g p r o c e d u r e . The h e u r i s t i c o f t e m p l a t i o n was u s e d b y t h r e e - f o u r t h s o f t h e s t u d e n t s and y e t had no e f f e c t on t h e s o l u t i o n s o b t a i n e d . A r e a s o n f o r t h i s may be t h a t s t u d e n t s d o n ' t have t o t e m p l a t e f o r t h e p r o b l e m s t h e y a r e a s k e d t o do i n t h e c l a s s r o o m . I n most t e x t b o o k s t h e m a t e r i a l r e q u i r e d f o r a p r o b l e m i s g i v e n i n t h e two o r t h r e e pages i m m e d i a t e l y p r e c e e d i n g t h e e x e r c i s e . I n g e n e r a l , s t u d e n t s d o n ' t have t o r e c a l l _ _..v m a t e r i a l f r o m t h e p r e v i o u s y e a r o r even t h e p r e v i o u s c h a p t e r . Most e x e r c i s e s i n v o l v e t h e d i r e c t a p p l i c a t i o n o f t h e c o n t e n t s o f a few pages f r o m t h e t e x t b o o k . The s t u d e n t s i n t h i s . s t u d y had v e r y p o o r r e c a l l o f t h e c o r e m a t e r i a l t h e y had s t u d i e d and v e r y l i t t l e i d e a o f what c o r e m a t e r i a l t h e y c o u l d u s e i n a p r o b l e m . T h i s i s n o t s u r p r i s i n g s i n c e most s t u d e n t s a r e t o l d what t o do by b o t h t h e t e x t and t h e t e a c h e r . I f t h e c r e a t i o n .: -o f new p r o b l e m s e t s i s as i m p o r t a n t as s u g g e s t e d i n t h e o p e n i n g r emarks o f t h i s p a p e r , t h e n n o t o n l y a r e c u r r i c u l u m d e v e l o p e r s g o i n g t o have t o d e v o t e more t i m e t o t h e c r e a t i o n o f i m a g i n a t i v e and c r e a t i v e p r o b l e m s , b u t a l s o t o t h e p l a c e m e n t o f t h e s e p r o b l e m s i n t h e c u r r i c u l u m . P r o b l e m s s h o u l d n o t a l w a y s be p l a c e d i n t h e e x e r c i s e s i m m e d i a t e l y f o l l o w i n g t h e 124 p r e s e n t a t i o n o f t h e m a t e r i a l needed t o s o l v e them, b u t t h e y s h o u l d be i n c l u d e d i n e x e r c i s e s a t a l a t e r d a t e . A n o t h e r o f t h e f i n d i n g s o f t h i s s t u d y i s t h a t t h o s e s t u d e n t s who change t h e i r mode o f a t t a c k a r e more s u c c e s s f u l i n o b t a i n i n g a s o l u t i o n . Most t e a c h e r s do t e l l t h e i r s t u d e n t s t o t r y a n o t h e r method i f t h e y a r e h a v i n g d i f f i c u l t y i n s o l v i n g a p r o b l e m , b u t i f s t u d e n t s a r e g o i n g t o be a b l e t o change t h e i r mode o f a t t a c k , t h e y have t o be a b l e t o r e c a l l t h e c o r e m a t e r i a l t h e y have s t u d i e d ( t e m p l a t e ) o r be a b l e t o a t t a c k t h e p r o b l e m b y u s i n g v a r i o u s h e u r i s t i c s . H e n d e r s o n and P i n g r y (l953f p. 238) s t a t e t h a t p r a c t i c e i n s o l v i n g p r o b l e m s and a c o n s c i o u s awareness o f t h e p r o b l e m s o l v i n g p r o c e s s w i l l i m p r o v e p r o b l e m s o l v i n g p e r f o r m a n c e . P o l y a p r o p o s e s a program f o r t e a c h i n g p r o b l e m s o l v i n g w h i c h c o n s i s t s o f two a s p e c t s : abundant e x p e r i e n c e i n s o l v i n g p r o b l e m s md s e r i o u s s t u d y o f t h e s o l u t i o n p r o c e s s . He ( P o l y a , 1962, p. v ) e x p r e s s e s t h e need f o r t h e f i r s t o f t h e s e as f o l l o w s : " S o l v i n g p r o b l e m s i s a p r a c t i c a l a r t , l i k e swimming o r s k i i n g o r p l a y i n g t h e p i a n o ; y o u can l e a r n i t o n l y b y i m i t a t i o n and p r a c t i c e . " P o l y a a l s o warns t h a t i n p r o b l e m s o l v i n g , i m i t a t i o n and p r a c t i c e a r e n o t s u f f i c i e n t . Not o n l y must p r o b l e m s be s o l v e d , b u t t h e l e a r n e r s ' a t t e n t i o n must be d i r e c t e d t o t h e methods u s e d . These must be g e n e r a l enough so t h a t t h e y become a v a i l a b l e f o r u s e i n s o l v i n g s i m i l a r p r o b l e m s i n t h e f u t u r e . The h e u r i s t i c s i n MacPherson's model a r e o f a g e n e r a l n a t u r e . They appear t o be t h e n a t u r a l k i n d s o f p r o c e d u r e s t h a t w o u l d be u s e d i n a t t a c k i n g and s o l v i n g m a t h e m a t i c a l p r o b l e m s . I f t h e programs o f Henderson, P i n g r y and P o l y a do i n d e e d have m e r i t f o r t e a c h i n g p r o b l e m s o l v i n g , t h e n p e r h a p s t h e h e u r i s t i c s f r o m t h i s model s h o u l d be i n c l u d e d . P r o b l e m s o l v i n g , as c o n s i d e r e d i n t h i s s t u d y w i t h s t u d e n t s o f a v e r a g e a b i l i t y , i s r e l a t e d t o f i e l d i n d e p e n d e n c e . Not o n l y does t h e f i e l d 125 dependent s t u d e n t o b t a i n f e w e r c o r r e c t s o l u t i o n s , he i s a l s o l e s s w i l l i n g t o change h i s method o f a t t a c k once he s t a r t s t o s o l v e a p r o b l e m . S p i t l e r (1970) d i s c u s s e d f i v e p r o c e d u r e s w h i c h may be u s e d b y t h e t e a c h e r t o h e l p s t u d e n t s l e a r n t o s o l v e p r o b l e m s u s i n g d i f f e r e n t modes o f a t t a c k . 1. N e a r t h e end o f e a c h s e t o f p r o b l e m s w h i c h a r e d e s i g n e d t o p r o v i d e p r a c t i c e i n a g i v e n p r o c e d u r e , i n c l u d e s e v e r a l p r o b l e m s w h i c h r e q u i r e o t h e r p r o c e d u r e s . 2. P r o b l e m s w h i c h c a n be o r must be done u s i n g d i f f e r e n t p r o c e d u r e s s h o u l d be i n t e r s p e r s e d t h r o u g h o u t a s e t o f p r o b l e m s . 3. The t e a c h e r s h o u l d r e g u l a r l y d i s c u s s t h e i n f l u e n c e s o f p a t t e r n e d r e s p o n s e s w i t h h i s s t u d e n t s . 4. S u p e r f l u o u s i n f o r m a t i o n s h o u l d be i n c l u d e d i n p r o b l e m s t h r o u g h o u t t h e e x e r c i s e s . 5. When p r a c t i c i n g complex p r o b l e m s o l v i n g , p r o b l e m s w h i c h t h e s t u d e n t r e c o g n i z e s t h a t he c o u l d s o l v e b y a l e s s complex p r o c e d u r e , s h o u l d be i n c l u d e d . S p i t l e r a d d i t i o n a l l y recommends t h a t f i e l d d e p e ndent s t u d e n t s be g i v e n p r o b l e m s e t s w h i c h r e q u i r e many d i f f e r e n t methods o f s o l u t i o n . The recommendations i n t h i s s e c t i o n r e q u i r e t h a t t h e m a t h e m a t i c s . t e a c h e r c o n t i n u a l l y a s s i g n p r o b l e m s w h i c h r e q u i r e d i f f e r e n t p r o c e d u r e s t o o b t a i n a s o l u t i o n and t h a t t h e s e p r o c e d u r e s be d i s c u s s e d w i t h t h e s t u d e n t s . A l s o t h e f i e l d i n d e p e n d e n t - d e p e n d e n t c o n s t r u c t may be u s e f u l i n g r o u p i n g s t u d e n t s f o r t e a c h i n g m a t h e m a t i c a l p r o b l e m s o l v i n g b e c a u s e t h e p r o b l e m s o l v i n g b e h a v i o r o f s t u d e n t s i n t h e s e two group s i n d i c a t e t h e y may have a need f o r d i f f e r e n t t e a c h i n g s t r a t e g i e s . L i m i t a t i o n s o f t h e S t u d y There a r e s e v e r a l l i m i t a t i o n s o f t h e p r e s e n t s t u d y . .The f i r s t t h r e e l i m i t a t i o n s were n o t e d i n C h a p t e r I as i n c o m i n g l i m i t a t i o n s o f t h e 126 s t u d y . These l i m i t a t i o n s a r e r e s t a t e d h e r e a l o n g w i t h o t h e r l i m i t a t i o n s w h i c h were d e t e r m i n e d as t h e s t u d y was c a r r i e d o u t . A l l o f t h e s u b j e c t s were o f av e r a g e a b i l i t y f o r s u b j e c t s on an academic m a t h e m a t i c s program, t h e r e f o r e , r e g u l a r i t i e s o b s e r v e d g i v e no e v i d e n c e t h a t above a v e r a g e o r weak s t u d e n t s w o u l d e x h i b i t t h e same p r o b l e m s o l v i n g b e h a v i o r . The s u b j e c t s were r a n d o m l y s e l e c t e d f r o m t h e academic g r a d e e l e v e n program ( A l g e b r a I I ) and so any g e n e r a l i z a t i o n s made b a s e d upon t h e s e r e s u l t s a r e l i m i t e d t o t h i s o r a s i m i l a r p o p u l a t i o n o f s t u d e n t s . An o b v i o u s l i m i t a t i o n i s t h e s e l e c t i o n o f p r o b l e m s u s e d i n t h e s t u d y . The d a t a f o r e a c h s t u d e n t were c o l l e c t e d f r o m one s e s s i o n i n w h i c h t h e s t u d e n t was a s k e d t o s o l v e f i v e p r o b l e m s . An i n d i v i d u a l may have e x h i b i t e d d i f f e r e n t p r o b l e m s o l v i n g c h a r a c t e r i s t i c s i f a s k e d t o a t t e m p t a d i f f e r e n t s e t o f p r o b l e m s . The p r o b l e m s u s e d i n t h i s s t u d y were n o t s e l e c t e d f r o m any p a r t i c u l a r c o r e a r e a n o r were t h e y o f any p a r t i c u l a r t y p e , a l t h o u g h f o u r o f them i n v o l v e a maximum o r minimum s o l u t i o n . T h e r e were a l s o l i m i t a t i o n s r e s u l t i n g f r o m t h e use o f t h e " t h i n k a l o u d " method i n d e t e r n i i n i n g t h e p r o c e d u r e s u s e d b y t h e s t u d e n t s i n s o l v i n g p r o b l e m s . The method i t s e l f may be u n r e l i a b l e s i n c e an i n d i v i d u a l m i g h t r e m a i n s i l e n t d u r i n g moments o f d e e p e s t t h o u g h t . M o r e o v e r , a v e r b a l i z e d s o l u t i o n c o u l d be e s s e n t i a l l y d i f f e r e n t f r o m one a f f e c t e d s i l e n t l y . The p r e s e n c e o f an o b s e r v e r m i g h t i n h i b i t a p r o b l e m s o l v e r i n s u c h a way t h a t he m i g h t n o t a t t e m p t s o l u t i o n s w h i c h he f e l t w o u l d be c o n s i d e r e d f o o l i s h b y someone e l s e . These l i m i t a t i o n s were n o t e d b y b o t h K i l p a t r i c k (1967) and K a n t o w s k i (1975). T h e r e were a l s o l i m i t a t i o n s i n t h e c o d i n g system u s e d . The s u b j e c t s i n t h e s t u d y were v e r y o r i e n t e d t o u s i n g c o r e p r o c e d u r e s and s p e n t a 127 g r e a t d e a l o f t i m e a t t e m p t i n g t o u n d e r s t a n d t h e p r o b l e m s . P e r h a p s i f t h e c o r e p r o c e d u r e s u s e d i n t h e c o d i n g s y s t e m had i n c l u d e d a f i n e r d i v i s i o n i n t h e r e a d i n g and p r e p a r a t i o n c a t e g o r i e s , some i m p o r t a n t p r o b l e m s o l v i n g c h a r a c t e r i s t i c s may have been i d e n t i f i e d . F i n a l l y , t h e a n a l y s i s o f t h e s e q u e n t i a l d a t a was o f l i m i t e d u s e . Sequences were o n l y a n a l y z e d i n p a i r s o f c o n s e c u t i v e moves and t h i s a n a l y s i s o n l y i n v o l v e d a n o n - s t a t i s t i c a l c o m p a r i s o n o f p e r c e n t a g e s . I f p r o b l e m s o l v i n g s t y l e s a r e t o be i d e n t i f i e d , t h e sequence o f p r o c e d u r e s w i l l most c e r t a i n l y o f f e r many i m p o r t a n t c l u e s . T h e r e i s a l s o a l i m i t a t i o n i n t e r m s o f t h e i n t e r p r e t a t i o n o f t h e r e s u l t s f r o m t h e r e g r e s s i o n a n a l y s i s . E x t r e m e s c o r e s on some o f t h e v a r i a b l e s may have i n f l u e n c e d t h e s t a t i s t i c a l r e s u l t s . However, b e c a u s e o f t h e s m a l l sample s i z e (N = 4 0 ) , t h e s e s c o r e s were n o t d e l e t e d f r o m t h e s t a t i s t i c a l a n a l y s i s . A p p e n d i x D c o n t a i n s t h e h i s t o g r a m s f o r s e v e r a l o f t h e s e v a r i a b l e s . I m p l i c a t i o n s f o r R e s e a r c h One p r a c t i c a l d i f f i c u l t y i n u n d e r t a k i n g s t u d i e s i n w h i c h t h e p r o b l e m s o l v i n g p r o c e s s i s i n v e s t i g a t e d i s t h a t o f d e t e r m i n i n g t h e p r o c e d u r e s u s e d . The u s e o f t h e " t h i n k a l o u d " method as a d a t a g a t h e r i n g t e c h n i q u e i s g a i n i n g i n p o p u l a r i t y . However, w i t h t h e e x c e p t i o n o f t h e s t u d i e s b y R o t h (1966) and F l a h e r t y (1973), l i t t l e r e s e a r c h has been done t o d e t e r m i n e t h e r e l i a b i l i t y o f t h i s method. Do s t u d e n t s a t t a c k p r o b l e m s d i f f e r e n t l y when a s k e d t o v e r b a l i z e t h e i r t h o u g h t s t h a n when l e f t a l o n e t o s o l v e t h e pr o b l e m ? T h i s and o t h e r q u e s t i o n s r e l a t e d t o t h e " t h i n k a l o u d " method n e e d f u r t h e r i n v e s t i g a t i o n . K r u t e t s k i i (1969) b e l i e v e s t h a t t h e 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 s t u d e n t s w i t h above a v e r a g e o r l o w m a t h e m a t i c a l a p t i t u d e d i f f e r f r o m 128 t h o s e s t u d e n t s w i t h a v e r a g e m a t h e m a t i c a l a p t i t u d e s . T h e r e f o r e , t h i s s t u d y s h o u l d be r e p l i c a t e d u s i n g s t u d e n t s o f h i g h e r and l o w e r a b i l i t y . I t i s p o s s i b l e t h a t t h e p r o b l e m s s e l e c t e d f o r t h e l o w e r a b i l i t y s t u d e n t s w o u l d have t o be d i f f e r e n t . S i n c e t h e p r o c e d u r e s u s e d i n s o l v i n g p r o b l e m s i n d i f f e r e n t c o n t e n t a r e a s may v a r y , t h i s s t u d y s h o u l d be r e p e a t e d u s i n g c o n t e n t a r e a s s u c h as a l g e b r a , geometry, number t h e o r y , and c a l c u l u s t o d e t e r m i n e i f r e g u l a r i t i e s e x i s t i n h e u r i s t i c usage a c r o s s a r e a s o f c o n t e n t as w e l l as t o s u g g e s t h y p o t h e s e s s p e c i f i c t o a c o n t e n t a r e a . S i n c e t h e p r o c e d u r e s u s e d i n p r o b l e m s o l v i n g may d i f f e r as s t u d e n t s g a i n more e x p e r i e n c e i n p r o b l e m s o l v i n g , e x p l o r a t o r y s t u d i e s s u c h as t h i s one s h o u l d be done u s i n g s t u d e n t s a t v a r i o u s s t a g e s o f m a t h e m a t i c a l s o p h i s t i c a t i o n . I n p a r t i c u l a r , t h i s s t u d y g i v e s v e r y l i t t l e i n f o r m a t i o n on t h e u s e o f t h e h i g h e r h e u r i s t i c s , t h e r e f o r e , i t s h o u l d be r e p l i c a t e d u s i n g s t u d e n t s who a r e v e r y m a t h e m a t i c a l l y m a t u r e . A l s o t h e s t u d y may be r e p l i c a t e d u s i n g open ended p r o b l e m s t o d e t e r m i n e i f t h e y w i l l e l i c i t t h e u s e o f h i g h e r h e u r i s t i c s . One i n t e r e s t i n g a s p e c t o f MacPherson's model i s t h e o r d e r i n g o f t h e h e u r i s t i c s ( s e e F i g u r e 2 t p. 8) i n t o f o u r c a t e g o r i e s . I n o r d e r t o d e t e r m i n e i f t h e o r d e r i n g o f t h e h e u r i s t i c s i n MacPherson's model does i n d e e d h o l d , a s t u d y w i l l have t o be c a r r i e d o u t u s i n g s u b j e c t s who m i g h t d e m o n s t r a t e t h e use o f t h e s e h e u r i s t i c s . A l s o , a m o d i f i c a t i o n o f t h e model, a d d i t i o n o r d e l e t i o n o f some h e u r i s t i c s , c o u l d be c a r r i e d o u t i f t h e d a t a o b t a i n e d f r o m s u c h a s t u d y so i n d i c a t e , The c o d i n g s y s t e m p r o v e d t o be v e r y r e l i a b l e when u s e d b y c o d e r s who were b o t h v e r y f a m i l i a r w i t h t h e h e u r i s t i c s and t h e c o d i n g s y s t e m . The system was d e s i g n e d p r i m a r i l y w i t h t h e p r o b l e m s u s e d i n t h i s s t u d y i n m i nd and hence needs much r e f i n e m e n t , e s p e c i a l l y i n t h e c o r e a r e a . The c o d i n g s y s t e m s h o u l d be a p p l i e d t o new s t u d e n t s and new s e t s o f p r o b l e m s t o see i f t h e d i f f e r e n c e s o b s e r v e d i n t h e p r e s e n t s t u d y a r e m a i n t a i n e d . I f t h e s y s t e m i s u s e d w i t h s t u d e n t s who a r e more m a t h e m a t i c a l l y s o p h i s t i c a t e d t h a n t h e ones i n t h i s s t u d y , some o f t h e c a t e g o r i e s m i g h t show g r e a t e r f r e q u e n c y o f usag e . The c o d i n g s y s t e m a l l o w s f o r a l l o f t h e p r o c e d u r e s u s e d t o be coded i n sequence. Some a t t e m p t was made t o a n a l y z e t h e s e q u e n t i a l d a t a , b u t t h e i n f o r m a t i o n o b t a i n e d was l i m i t e d . I f any p r o b l e m s o l v i n g s t y l e s a r e g o i n g t o be i d e n t i f i e d , t h e sequence i n w h i c h p r o c e d u r e s a r e u s e d w i l l be a m a j o r s o u r c e o f i n f o r m a t i o n o More r e s e a r c h i s needed i n d e v e l o p i n g and r e f i n i n g methods o f a n a l y z i n g s e q u e n t i a l i n f o r m a t i o n . T h i s s t u d y d i d n o t i n c l u d e an i n s t r u c t i o n a l phase be c a u s e i t was d e s i r e d t o o b s e r v e what h e u r i s t i c s were b e i n g u s e d b y s t u d e n t s w i t h o u t any s p e c i a l i n s t r u c t i o n . R e s e a r c h i s needed w h i c h i n c l u d e s i n s t r u c t i o n on t h e use o f h e u r i s t i c s t o d e t e r m i n e i f s u c h i n s t r u c t i o n w i l l i n c r e a s e t h e e f f e c t i v e n e s s o f t h e h e u r i s t i c s beyond t h a t a c c o u n t e d f o r b y t h e c o r e p r o c e d u r e s . R e s e a r c h i s a l s o needed t o d e t e r m i n e t h e e f f e c t o f i n s t r u c t i o n on t h e p r o b l e m s o l v i n g p r o c e s s e s u s e d b y f i e l d i n d e p e n d e n t and f i e l d dependent s t u d e n t s . The s t u d e n t s ' knowledge o f b o t h c o r e and h e u r i s t i c s , no m a t t e r how l i m i t e d , s h o u l d be t a k e n i n t o a c c o u n t i n d e s i g n i n g e x p e r i m e n t a l s t u d i e s w h i c h i n c l u d e an i n s t r u c t i o n a l p h a s e . E x p l o r a t o r y s t u d i e s s u c h as L u c a s (1972) and K a n t o w s k i (1975) where a n a l y s i s o f t h e p r o b l e m s o l v i n g p r o c e s s was u n d e r t a k e n d u r i n g t h e s t u d y i n d i c a t e t h a t i n s t r u c t i o n i n h e u r i s t i c s n o t o n l y a f f e c t s p r o b l e m s o l v i n g p e r f o r m a n c e , b u t a l s o has a p o s i t i v e e f f e c t on t h e p r o c e d u r e s u s e d b y t h e s t u d e n t s . A p l a n n e d program o f p r o c e s s r e s e a r c h , where t h e r o l e o f h e u r i s t i c s i n p r o b l e m s o l v i n g i s s y s t e m a t i c a l l y i n v e s t i g a t e d i s needed. LITERATURE CITED Amidon, E. J . & Hough, J . B. ( E d s . ) I n t e r a c t i o n a n a l y s i s ; T h e o r y , r e s e a r c h and a p p l i c a t i o n . R e a d i n g , Mass. : A d d i s o n - W e s l e y , 1967• B e g l e , E. G., (Ed.) The r o l e o f a x i o m a t i c s and p r o b l e m s o l v i n g i n m a t h e m a t i c s . R e p o r t , C o n f e r e n c e B o a r d o f t h e M a t h e m a t i c a l S c i e n c e s , B o s t o n : G i n n , 1966. B e g l e , E. G. & W i l s o n , J . W. E v a l u a t i o n o f m a t h e m a t i c s p r o g r a m s . I n E.G.. B e g l e ( E d . ) , M a t h e m a t i c s e d u c a t i o n . Y e a r b o o k , N a t i o n a l S o c i e t y f o r t h e S t u d y o f E d u c a t i o n , 1967. 2 2 , P a r t I . 367-407. B j e r r i n g , J . H. & S e a g r a v e s , P. T r i a n g u l a r r e g r e s s i o n p a c k a g e (UBC  T R I P j . 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 , V a n c o u v e r , J u n e 1974. 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 . S u p p l e m e n t a r y E d u c a t i o n a l Monographs, No. 73. C h i c a g o ; U n i v e r s i t y o f C h i c a g o P r e s s , 1950. B r o w n e l l , W. A. P r o b l e m s o l v i n g . I n N. B. H e n e r y ( E d . ) , The p s y c h o l o g y  o f l e a r n i n g . Y e a r b o o k , N a t i o n a l S o c i e t y f o r t h e S t u d y o f E d u c a t i o n , 1942, 4 i , P a r t I I . 415-443. B u r a c k , B. The n a t u r e and e f f i c a c y o f methods o f a t t a c k on r e a s o n i n g p r o b l e m s . P s y c h o l o g i c a l Monographs, 1953, 64. (7, Whole No. 287)» B u r c h , R. L. F o r m a l a n a l y s i s as a p r o b l e m - s o l v i n g p r o c e d u r e . J o u r n a l o f E d u c a t i o n . 1953, 136, 44-47. B u r t , C. L . F a c t o r s o f t h e mind. London: U n i v e r s i t y o f London P r e s s , 1928. Cooperman, M. L. F i e l d dependence and c h i l d r e n ' s p r o b l e m s o l v i n g u n d e r v a r y i n g c o n t i n g e n c i e s o f p r e d e t e r m i n e d f e e d b a c k . D i s s e r t a t i o n  A b s t r a c t s . 1974, 3_£, 4243-A ( A b s t r a c t ) C r o n b a c k , L . J . The meanings o f p r o b l e m s i n a r i t h m e t i c . S u p p l e m e n t a r y E d u c a t i o n a l Monographs. No. 60, C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s , 1948. D a v i s , G. A. P s y c h o l o g y o f p r o b l e m s o l v i n g . New Y o r k : B ^ s i c B o o k s , 1973. D e s s a r t , D. J . & F r a n d s e n , H. R e s e a r c h on t e a c h i n g s e c o n d a r y - s c h o o l m a t h e m a t i c s . 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 c N a l l y , 1973. Dewey, J . How we t h i n k . (2nd. ed.) B o s t o n : H e a t h , 1933. 131 Dodson, J . W. 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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 : On u n d e r s t a n d i n g , l e a r n i n g , and  t e a c h i n g p r o b l e m s o l v i n g . V o l . I . New Y o r k : W i l e y , 19620 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 : On u n d e r s t a n d i n g , l e a r n i n g , and  t e a c h i n g p r o b l e m s o l v i n g . V o l . I I , New Y o r k : W i l e y , 1965= 133 Romberg, T. A. & W i l s o n , J . W. The development o f t e s t s . NLSMA R e p o r t s No. 7. J . W. W i l s o n , L. S. Cohen & E. G. B e g l e ( E d s . ) , S t a n f o r d : S c h o o l M a t h e m a t i c s S t u d y G r o u p , 1969. R o s e n f e l d , I ; J . M a t h e m a t i c a l a b i l i t y as a f u n c t i o n o f p e r c e p t u a l f i e l d -dependency and c e r t a i n p e r s o n a l i t y v a r i a b l e s . U n p u b l i s h e d d o c t o r a l d i s s e r a t i o n , The U n i v e r s i t y o f Oklahoma, 1958. R o t h , B. The e f f e c t s o f o v e r t v e r b a l i z a t i o n on p r o b l e m s o l v i n g . D i s s e r t a t i o n A b s t r a c t s . I966, 27_, 957-B. ( A b s t r a c t ) S a a r n i , C. I . P i a g e t i a n o p e r a t i o n s and f i e l d i n d e p e n d e n c e as f a c t o r s i n c h i l d r e n ' s p r o b l e m s o l v i n g p e r f o r m a n c e . P a p e r p r e s e n t e d a t t h e a n n u a l m e e t i n g o f t h e A m e r i c a n E d u c a t i o n a l R e s e a r c h A s s o c i a t i o n , C h i c a g o . A p r i l , 1972 S c h w i e g e r , R. D. A component a n a l y s i s o f m a t h e m a t i c a l p r o b l e m s o l v i n g . U n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , Purdue U n i v e r s i t y , 1974. S p i t l e r , G. J . An i n v e s t i g a t i o n o f v a r i o u s c o g n i t i v e s t y l e s and t h e i m p l i c a t i o n s f o r m a t h e m a t i c s e d u c a t i o n . U n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , Wayne S t a t e U n i v e r s i t y , 1970» W a l b e r g , H. J . G e n e r a l i z e d r e g r e s s i o n models i n e d u c a t i o n a l r e s e a r c h . A m e r i c a n E d u c a t i o n a l R e s e a r c h J o u r n a l , 1971, 8, 71-93» W a l l a s , G. The a r t o f t h o u g h t . New Y o r k : H a r c o u r t , B r a c e and W o r l d , 1926. W a l s h , R. J . P r o b l e m s o l v i n g p e r f o r m a n c e and s u c c e s s e x p e c t a n c i e s as a f u n c t i o n o f t h e c o g n i t i v e s t y l e o f f i e l d i n d e p e n d e n c e v s . dependence, and e x p e r i m e n t a l l y i n d u c e d s u c c e s s - f a i l u r e e x p e c t a n c i e s . D i s s e r t a t i o n  A b s t r a c t s . 1974, 3_5_, 2066-A. ( A b s t r a c t ) Webb, N. An e x p l o r a t i o n o f m a t h e m a t i c a l p r o b l e m - s o l v i n g p r o c e s s e s . P a p e r p r e s e n t e d a t t h e a n n u a l m e e t i n g o f t h e A m e r i c a n E d u c a t i o n a l R e s e a r c h A s s o c i a t i o n , W a s h i n g t o n , D. C , Ma r c h , 1975» W i c k e l g r e n , W. A. How t o s o l v e p r o b l e m s : E l e m e n t s and t h e o r y o f p r o b l e m s  and p r o b l e m s o l v i n g . San F r a n c i s c o : W. H. Freeman, 1974* W i t k i n , H. A. Embedded f i g u r e s t e s t . P a l o A l t o , C a l i f o r n i a : C o n s u l t i n g P s y c h o l o g i s t s P r e s s , 1969. W i t k i n , H. A., Dyk, R. B., F a t e r s o n , H. J . , Goodenough, D. R. & K a r p , S. A. P s y c h o l o g i c a l d i f f e r e n t i a t i o n . New Y o r k : W i l e y , 1962. W i t k i n , H. A., Oltman, P. K., R a s k i n , E. & K a r p , S. A. A manual f o r t h e embedded f i g u r e s t e s t . P a l o A l t o , C a l i f o r n i a : C o n s u l t i n g P s y c h o l o g i s t s P r e s s , 1971. A p p e n d i x A PROBLEMS USED I N PILOT STUDY A b o o k s h e l f whose l e n g t h ( i n whole i n c h e s ) l i e s between 30 and 50 i n c h e s h o l d s e x a c t l y 5 p o e t r y books each a i n c h e s t h i c k , 3 h i s t o r y b ooks e a c h b i n c h e s t h i c k and 5 d i c t i o n a r i e s e ach c i n c h e s t h i c k ; ( a , b, £, a r e u n e q u a l i n t e g e r s ) . I t c o u l d i n s t e a d h o l d e x a c t l y 4 p o e t r y b o o k s, 5 h i s t o r y b o o k s , and 4 d i c t i o n a r i e s . I f i n s t e a d , 7 p o e t r y books and 4 h i s t o r y books a r e p l a c e d on i t , how many d i c t i o n a r i e s must t h e n be added t o f i l l t h e s h e l f e x a c t l y ? F i n d a l s o t h e t h i c k n e s s o f e a ch book and t h e l e n g t h o f t h e s h e l f . A l a d y gave t h e p o s t a g e stamp c l e r k a one d o l l a r b i l l and s a i d , " G i v e me some t w o - c e n t stamps, t e n t i m e s as many o n e - c e n t stamps, and t h e b a l a n c e i n f i v e s . " How c a n t h e c l e r k f u l f i l t h i s p u z z l i n g r e q u e s t ? A r e c t a n g u l a r l a w n i s t o be formed so t h a t one s i d e o f a b a r n s e r v e s as one s i d e o f t h e r e c t a n g l e . Y o u have 300 f e e t o f w i r e t o e n c l o s e t h e o t h e r t h r e e s i d e s . What a r e t h e d i m e n s i o n s o f t h e r e c t a n g l e i f t h e a r e a i s t o be o f maximum s i z e ? A group o f s t u d e n t s was s u r v e y e d t o a s c e r t a i n t h e i r p r e f e r e n c e s f o r s p o r t w a t c h i n g on t e l e v i s i o n . The t h r e e s p o r t s s e l e c t e d f o r t h e s u r v e y were f o o t b a l l , g o l f , and h o c k e y . The d a t a were c o l l e c t e d and o r g a n i z e d as f o l l o w s : 7 watched none o f t h e s p o r t s 10 watched g o l f 8 w a t c h e d h o c k e y 6 w a t c h e d f o o t b a l l 1 watched f o o t b a l l o n l y 4 w a t c h e d f o o t b a l l and g o l f 5 watched g o l f and h o c k e y 2 watched a l l t h r e e s p o r t s . Q u e s t i o n s : a) How many s t u d e n t s answered t h e s u r v e y ? b ) How many s t u d e n t s watched h o c k e y o n l y ? c ) How many s t u d e n t s w a t c h e d h o c k e y and f o o t b a l l ? F i n d t h e sum o f a l l t h e odd i n t e g e r s ( p o s i t i v e ) l e s s t h a n 100. A c i r c u l a r t a b l e r e s t s i n a c o r n e r , t o u c h i n g b o t h w a l l s o f a room. A p o i n t on t h e edge o f t h e t a b l e i s 8 i n c h e s from one w a l l and 9 i n c h e s f r o m t h e o t h e r w a l l . F i n d t h e d i a m e t e r o f t h e t a b l e . F i n d t h e s o l u t i o n s e t o f X + Y = XY where X and Y a r e r e a l numbers. 135 F i n d a l l X s u c h t h a t ] X - 1 | 2 - | X - 1 | = 12. What i s t h e l o n g e s t p i e c e o f m e t a l r o d w h i c h c a n be p l a c e d i n a b o x o f d i m e n s i o n 3 i n c h e s b y 4 i n c h e s b y 12 i n c h e s ? I magine a row o f 1000 l o c k e r s , a l l c l o s e d , and a l i n e o f 1000 men. Suppose t h e f i r s t man goes a l o n g and opens e v e r y l o c k e r . Suppose t h e s e c o n d man goes a l o n g and s h u t s e v e r y o t h e r l o c k e r . Suppose t h e t h i r d man goes a l o n g and changes t h e s t a t e o f e v e r y t h i r d l o c k e r ( i f i t ' s open, he s h u t s i t , and v i c e v e r s a ) . Suppose t h e f o u r t h man goes a l o n g and changes t h e s t a t e o f e v e r y f o u r t h l o c k e r , and so on, u n t i l a l l t h e men have p a s s e d by a l l t h e l o c k e r s . Which l o c k e r s a r e open i n t h e end? What i s t h e maximum number o f p e o p l e y o u can s e a t a t one t i m e u s i n g 100 c a r d t a b l e s p r o v i d e d y o u can o n l y s e a t one p e r s o n on a s i d e o f a t a b l e and t h e t a b l e s a r e a r r a n g e d so t h a t e a c h t i m e a new t a b l e i s added, a t l e a s t one o f i t s s i d e s must be p l a c e d a g a i n s t one s i d e o f a p r e v i o u s l y a r r a n g e d t a b l e ? Suppose y o u have a g o l d mine and a r e h a u l i n g g o l d t o t h e s m e l t e r i n y o u r t r u c k w h i c h has a h o l e i n t h e b a c k . I n o r d e r t o g e t t o t h e s m e l t e r y o u must d r i v e f r o m y o u r mine t o a r a i l r o a d t r a c k and t h e n t o t h e s m e l t e r w h i c h i s on t h e same s i d e o f t h e t r a c k as y o u r m ine. S i n c e y o u r t r u c k has a h o l e i n t h e b a c k y o u a r e i n t e r e s t e d i n f i n d i n g t h e s h o r t e s t r o u t e . What i s i t ? ^ ^ s —H-ft i~pn i i'Ii i-v-rtttt-A r e a l e s t a t e agency o f f e r s y o u a c h o i c e o f two t : r i a n g u l a r p i e c e s o f l a n d . One p i e c e h a s d i m e n s i o n s 30, 25, and 40 f e e t ; t h e o t h e r h a s d i m e n s i o n s 75, 90, 120 f e e t . The p r i c e o f t h e l a r g e r p i e c e i s 4 t i m e s t h e p r i c e o f t h e s m a l l e r p i e c e . Which i s t h e b e t t e r b uy? A b a r r e l o f honey w e i g h s 50 pounds. The same b a r r e l w i t h k e r o s e n e i n i t w e i g h s 35 pounds. I f honey i s t w i c e as h e a v y as k e r o s e n e , how much does t h e empty b a r r e l w e i g h ? A f a r m e r has hens and r a b b i t s . These a n i m a l s have 50 heads and 140 f e e t . How many hens and how many r a b b i t s has t h e f a r m e r ? I n t h e f o l l o w i n g a d d i t i o n p r o b l e m each l e t t e r s t a n d s f o r a d i f f e r e n t d i g i t . I f a l e t t e r a p p e a r s more t h a n once, i t r e p r e s e n t s t h e same d i g i t . What a r e t h e numbers i f t h e l e t t e r "o" s t a n d s f o r 6? MOM + POP LOVE 136 G i v e n a c i r c l e and a p o i n t i n i t s p l a n e . O u t l i n e t h e m a j o r s t e p s yoi w o u l d u s e t o c o n s t r u c t a t a n g e n t t o t h e c i r c l e t h r o u g h t h e p o i n t . T h r e e boys l i t a f i r e w o r k and i m m e d i a t e l y r a n o f f a t t h e same s p e e d i n d i f f e r e n t d i r e c t i o n s . The f i g u r e b e l o w shows t h e i r p o s i t i o n when t h e f i r e w o r k e x p l o d e d . Where d i d t h e f i r e w o r k e x p l o d e ? ft P r o v e t h a t no t e r m i n t h e sequence 11, 111, 1111, H i l l , s q u a r e o f an i n t e g e r . i s t h e G i v e n a r e g u l a r hexagon and a p o i n t i n i t s p l a n e . Draw a s t r a i g h t l i n e t h r o u g h t h e g i v e n p o i n t t h a t d i v i d e s t h e hexagon i n t o two p a r t s o f e q u a l a r e a . Can y o u do t h e p r o b l e m f o r any r e g u l a r p o l y g o n ? O t h e r c l o s e d g e o m e t r i c f i g u r e s ? F i n d t h e s e t o f p o s i t i v e i n t e g e r s w h i c h have l e s s t h a n 5 d i v i s o r s . A y a c h t i s moored a t A, 50 m e t e r s away f r o m a s t r a i g h t s e a w a l l CD. The c a p t a i n o f t h e y a c h t w i s h e s t o row t o t h e s e a w a l l t o c o l l e c t a p a s s e n g e r and t h e n t o a speed - b o a t moored a t B, SO m e t e r s f r o m t h e w a l l . Where s h o u l d t h e p a s s e n g e r meet t h e c a p t a i n t o make t h i s r o u t e as s h o r t as p o s s i b l e ? J j j ? 0 c 7 - r r T h e r e a r e 2 g l a s s e s : One c o n t a i n i n g 10 s p o o n s f u l o f w i n e and t h e o t h e r 10 o f w a t e r . A s p o o n f u l o f wine i s t a k e n f r o m t h e f i r s t g l a s s , p u t i n t h e second g l a s s , and mi x e d r o u n d . Then a s p o o n f u l o f t h e m i x t u r e i s t a k e n and p u t back i n t h e f i r s t g l a s s . I s t h e r e now more wine i n t h e w a t e r t h a n w a t e r i n t h e w i n e ? a G i v e n t h a t ABCD i s p a r a l l e l o g r a m , AB p e r p e n d i c u l a r t o PC, DQ p e r p e n d i c u l a r t o BC, F i n d t h e l e n g t h o f DQ. A p p e n d i x B PROBLEMS USED I N THE STUDY I n s t r u c t i o n s The p u r p o s e o f t h i s i n t e r v i e w i s t o o b t a i n some i n f o r m a t i o n on t h e ways i n w h i c h s t u d e n t s who a r e c o m p l e t i n g g r a d e e l e v e n A l g e b r a s o l v e m a t h e m a t i c a l p r o b l e m s . T h i s i s f o r my own i n f o r m a t i o n and w i l l have a b s o l u t e l y n o t h i n g t o do. w i t h y o u r g r a d e i n A l g e b r a , What y o u a r e a s k e d t o do i s t o work a s m a l l s e t o f p r o b l e m s , and t o t h i n k a l o u d as y o u a r e w o r k i n g on e a c h p r o b l e m . T h i s means t h a t i n a d d i t i o n t o w r i t i n g y o u r s t e p s on p a p e r , y o u a r e b e i n g a s k e d t o s a y  e v e r y t h i n g t h a t y o u a r e t h i n k i n g about w h i l e y o u wo r k e a c h p r o b l e m . T h i s i n c l u d e s a l l t h e l i t t l e t h i n g s t h a t pop i n t o y o u r mind — w h e t h e r y o u u s e them o r n o t . Y o u w i l l be t a p e - r e c o r d e d w h i l e y o u do t h i s . The r e a s o n f o r r e c o r d i n g y o u r work i s so t h a t I can g e t a c l e a r e r p i c t u r e o f what t h o u g h t p r o c e s s e s y o u u s e and when y o u u s e them w h i l e s o l v i n g a p r o b l e m . T h e r e a r e s e v e r a l o t h e r i n s t r u c t i o n s I hope y o u w i l l f o l l o w w h i l e w o r k i n g t h e p r o b l e m s : 1) Read each p r o b l e m o u t l o u d b e f o r e b e g i n n i n g t o work i t , 2) W r i t e down e v e r y t h i n g y o u u s u a l l y w r i t e w h i l e s o l v i n g a p r o b l e m . T h i s i n c l u d e s s c r a t c h work, d i a g r a m s , e q u a t i o n s , c a l c u l a t i o n s , e t c . ; u s e as much p a p e r as y o u need; t a l k a s y o u w r i t e , 3 ) Do n o t e r a s e a n y t h i n g ; i f y o u d e c i d e n o t t o u s e s o m e t h i n g you've a l r e a d y w r i t t e n downj draw a l i n e t h r o u g h i t and t h e n w r i t e down t h e c o r r e c t i o n . 4 ) I am more i n t e r e s t e d i n how y o u go about s o l v i n g a p r o b l e m t h a n i n y o u r s o l u t i o n . So p l e a s e do n o t a s k i f any o f y o u r work i s c o r r e c t u n t i l y o u have c o m p l e t e d t h e i n t e r v i e w . S i n c e o t h e r s t u d e n t s w i l l be p a r t i c i p a t i n g i n t h e s e i n t e r v i e w s , p l e a s e do n o t d i s c u s s t h e p r o b l e m s o r t h e i n t e r v i e w w i t h anyone. T h i s w o u l d i n v a l i d a t e t h e r e s u l t s f o r t h e whole s c h o o l . I f someone has q u e s t i o n s , t e l l him t o see me. Y o u r c o o p e r a t i o n w i l l be g r e a t l y a p p r e c i a t e d . 138 M a t h e m a t i c a l W o r l d P r o b l e m s Warm-up P r o b l e m s A. The sum o f two p o s i t i v e i n t e g e r s i s 23. The sum o f t h e f i r s t i n t e g e r and t w i c e t h e se c o n d i s 34» F i n d t h e numbers. B. T, D, H, J , and B a r e r e a l numbers w i t h t h e f o l l o w i n g r e l a t i o n s h i p s : T i s 20 l e s s t h a n D, D i s 50 g r e a t e r t h a n H, H i s 20 l e s s t h a n B, J i s 30 g r e a t e r t h a n T. Which number i s t h e l a r g e s t ? Which i s se c o n d , t h i r d , f o u r t h , and f i f t h l a r g e s t ? M a i n P r o b l e m s 1. 2. 4. 5. The p o i n t A i s 50 u n i t s f r o m a s t r a i g h t l i n e CD, and B i s a p o i n t 80 u n i t s f r o m CD. F i n d t h e p o i n t X on CD so t h a t t h e d i s t a n c e f r o m A t o X t o B i s as s m a l l as p o s s i b l e . B 50 £0 P •/3o 4 The sum o f a p o s i t i v e i n t e g e r and t w i c e a n o t h e r i s s u b t r a c t e d f r o m 100 and t h e d i f f e r e n c e i s d i v i s i b l e b y 5* The f i r s t i n t e g e r i s 10 t i m e s t h e s e c o n d . What a r e t h e numbers? A r e c t a n g l e ABCD i s formed so t h a t one o f i t s s i d e s CD i s on a l i n e L. The sum o f t h e l e n g t h s o f t h e o t h e r s i d e s , AB, AD, and BC, i s 300 u n i t s . What a r e t h e d i m e n s i o n s o f t h e r e c t a n g l e i f t h e a r e a i s t o be o f maximum s i z e ? Why does y o u r s o l u t i o n g i v e t h e maximum a r e a ? What i s t h e g r e a t e s t d i s t a n c e between two p o i n t s i n a r e c t a n g u l a r s o l i d o f d i m e n s i o n 3 u n i t s by 4 u n i t s b y 12 u n i t s ? What i s t h e maximum p e r i m e t e r y o u can o b t a i n b y a r r a n g i n g one hundred o n e - i n c h s q u a r e s ? E a c h t i m e a new s q u a r e i s added a t l e a s t one o f i t s s i d e s must be p l a c e d a g a i n s t one s i d e o f a p r e v i o u s l y a r r a n g e d s q u a r e . Why does y o u r s o l u t i o n g i v e t h e maximum p e r i m e t e r ? 139 R e a l W o r l d P r o b l e m s Warm-up P r o b l e m s A. J a b b a r s c o r e d 23 t i m e s i n a b a s k e t b a l l game. He s c o r e d 34 p o i n t s , two f o r e a c h f i e l d g o a l and one f o r e a c h f r e e t h r o w . How many f i e l d g o a l s d i d he make? How many f r e e t h r o w s ? B. . Tom, D i c k , H a r r y , J o e , and B i l l a r e l i n e d up i n t h e s e p o s i t i o n s midway t h r o u g h a t r a c k meet: Tom i s 20 y a r d s b e h i n d D i c k , D i c k i s 50 y a r d s ahead o f H a r r y , H a r r y i s 20 y a r d s b e h i n d B i l l , J o e i s 30 y a r d s ahead o f Tom, A t t h i s p o i n t i n t h e r a c e , who i s w i n n i n g ? who i s second, t h i r d , f o u r t h and f i f t h ? M a i n P r o b l e m s 1. 2. A y a c h t i s moored a t A, 50 m e t e r s away f r o m a s t r a i g h t s e a w a l l CD. The c a p t a i n o f t h e y a c h t w i s h e s t o row t o t h e s e a w a l l t o c o l l e c t a p a s s e n g e r and t h e n t o a speedboat moored a t B, 80 m e t e r s f r o m t h e w a l l . Where s h o u l d t h e p a s s e n g e r meet t h e c a p t a i n t o make t h e r o u t e as s h o r t as p o s s i b l e ? . — ^ 8 ft SO ///// kf 130 ? 0 A l a d y gave t h e p o s t a g e stamp c l e r k a one d o l l a r b i l l and s a i d , " G i v e me some t w o - c e n t stamps, t e n t i m e s as many o n e - c e n t stamps, and t h e b a l a n c e i n f i v e s . " How c a n t h e c l e r k f u l f i l t h i s p u z z l i n g r e q u e s t ? 3« A r e c t a n g u l a r l a w n i s t o be fo r m e d so t h a t one s i d e o f a b a r n s e r v e s as one s i d e o f t h e r e c t a n g l e . Y o u have 300 f e e t o f w i r e t o e n c l o s e t h e o t h e r t h r e e s i d e s . What a r e t h e d i m e n s i o n s o f t h e r e c t a n g l e i f t h e a r e a i s t o be o f maximum s i z e ? Why does y o u r s o l u t i o n g i v e t h e maximum a r e a ? 4. What i s t h e l o n g e s t p i e c e o f m e t a l r o d w h i c h c a n be p l a c e d i n a box o f d i m e n s i o n 3 i n c h e s b y 4 i n c h e s by 12 i n c h e s ? 5« What i s t h e maximum number o f p e o p l e y o u can s e a t a t one t i m e u s i n g 140 100 s q u a r e c a r d t a b l e ? The t a b l e s a r e a r r a n g e d so t h a t e a c h t i m e a new t a b l e i s added, a t l e a s t one o f i t s s i d e s must be p l a c e d a g a i n s t one s i d e o f a p r e v i o u s l y a r r a n g e d t a b l e . Why does y o u r s o l u t i o n g i v e t h e maximum s e a t i n g p l a n ? A p p e n d i x C THE CODING SYSTEM T h i s c o d i n g s y s t e m i s b a s e d on MacPherson's m o d e l o f m a t h e m a t i c a l p r o b l e m s o l v i n g . I t i s d e s i g n e d f o r c o d i n g p r o b l e m s o l v i n g b e h a v i o r o f s u b j e c t s who a r e a s k e d t o t h i n k a l o u d as t h e y s o l v e m a t h e m a t i c a l word p r o b l e m s . The p r o b l e m s o l v i n g i n t e r v i e w i s a u d i o t a p e r e c o r d e d and t h e s u b j e c t i s a s k e d t o do a l l o f h i s t h i n k i n g a l o u d . He i s i n s t r u c t e d t o v e r b a l i z e a l l h i s w r i t i n g and d i a g r a m s . Whenever t h e s u b j e c t f a l l s s i l e n t , he i s r e m i n d e d t o do h i s t h i n k i n g a l o u d . The p r o b l e m s a r e p r e s e n t e d t o t h e s u b j e c t one a t a t i m e . E a c h p r o b l e m i s t y p e w r i t t e n a t t h e t o p o f t h e page w i t h s p a c e u n d e r n e a t h f o r t h e s u b j e c t t o work. Once t h e i n t e r v i e w i s c o m p l e t e , t h e s u b j e c t ' s p r o t o c o l i s matched w i t h h i s w r i t t e n work b y m a r k i n g t h e a p p r o p r i a t e f o o t a g e f r o m t h e t a p e on h i s w r i t t e n work. The c o d i n g i s done u s i n g b o t h t h e s u b j e c t ' s w r i t t e n work and t h e t a p e r e c o r d e d p r o t o c o l . A s u b j e c t ' s p r o t o c o l f o r e a c h p r o b l e m i s c o d e d o n t h e c o d i n g f o r m ( s e e n e x t two p a g e s ) . A m a t r i x i s u s e d t o s e q u e n t i a l l y code p r o b l e m s o l v i n g b e h a v i o r . A summary f o r m i s u s e d t o c o m p i l e i n f o r m a t i o n o b t a i n e d f r o m t h e c o d i n g m a t r i x as w e l l as r e c o r d i n g o t h e r d a t a r e l a t e d t o t h e s u b j e c t ' s p r o b l e m s o l v i n g b e h a v i o r . The p r o c e d u r e s f r o m t h e c o d i n g f o r m a r e d i v i d e d i n t o f i v e c a t e g o r i e s : c o r e , h e u r i s t i c s , s o l u t i o n , r e a s o n a b l e n e s s and c o n c e r n . As t h e c o d e r i d e n t i f i e s t h e p r o c e d u r e f r o m t h e s u b j e c t ' s p r o t o c o l , he e n t e r s a c h e c k i n t h e a p p r o p r i a t e row and f i r s t empty column. I f t h e p r o c e d u r e i s 142 Reading problem Request defn. o f t e r r s R e c a l l same problem R e c a l l r e l a t e d problem R e c a l l problem type R e c a l l r e l a t e d f a c t Draw dlajnrani Modify diagram I d e n t i f y v a r i a b l e S e t t i n g up equations Algorithms-algebraic Algorithms-arithmetic Guessing Smoothing A n a l y s i s Templation C a s e s - a l l Cases-random Cases-systematic C a s e s - c r i t i c a l Cases-sequential Deduction Inverse deduction I n v a r i a t i o n Analogy Symmetry Obtain s o l u t i o n Checking p a r t Checking s o l u t i o n ; by subst. i n equation: by r e t r a c i n g steps by r e a s n b l e / r e a l i s t i c uncodable Exp. concern-method Exp. concren-algorithm Exp. concern-equation Exp. concern-solution Work stopped-soln. Work stopped-no s o l n . T H E C O D I NG FORM 143 C O D I N G FORM SUMMARY Number Problem#, Solutioni Correct Inc orrect None # of Cycles Heuristics used # of times Tape# Tape Reading. Time spent on problem Errorsi Arithmetic Algebraic Other # of Changes Core procedures used # of times Remarksi 144 f r o m t h e h e u r i s t i c o r c o r e c a t e g o r y ( e x c e p t r e c a l l o f same p r o b l e m , r e l a t e d p r o b l e m , o r p r o b l e m t y p e ; draw d i a g r a m ; m o d i f y d i a g r a m ) , he e n t e r s a 1, 2, o r 3, d e p e n d i n g upon t h e outcome: 1 I n c o m p l e t e 2 I n c o r r e c t 3 C o r r e c t The outcome f r o m a c o r e p r o c e d u r e i s coded i n t e r m s o f t h e c o r r e c t n e s s o f t h e a p p l i c a t i o n o f t h e p r o c e d u r e . F o r example, a 2 i n t h e a l g o r i t h m -a r i t h m e t i c row i n d i c a t e s an e r r o r i n t h e use o f t h e a l g o r i t h m , whereas, a 1 i n t h e s e t t i n g up e q u a t i o n row i n d i c a t e s t h e s u b j e c t has w r i t t e n an i n c o m p l e t e e q u a t i o n . F o r t h e h e u r i s t i c c a t e g o r y , t h e outcome i s coded i n t e r m s o f t h e outcome o f t h e p r o b l e m o r s u b - p r o b l e m a f t e r t h e u s e o f t h e h e u r i s t i c , n o t t h a t t h e h e u r i s t i c i s i n c o r r e c t . F o r example, a 2 i n t h e a n a l y s i s row i n d i c a t e s t h e outcome o f t h e p r o b l e m a f t e r t h e use o f a n a l y s i s i s i n c o r r e c t , n o t t h a t t h e s u b j e c t c o m m i t t e d an e r r o r i n t h e h e u r i s t i c i t s e l f . I f two o r more p r o c e d u r e s o c c u r s i m u l t a n e o u s l y t h e n a c h e c k o r number i s e n t e r e d i n t h e same column f o r e a c h p r o c e d u r e . I f a p r o c e d u r e i s c a r r i e d on l o n g e r t h a n a column, i t i s e n c l o s e d i n a b o x w i t h t h e outcome e n t e r e d i n t h e l a s t column e n c l o s e d . F o r example, a s u b j e c t i s u s i n g random c a s e s and c o n s i d e r s t h r e e c a s e s a l o n g w i t h u s i n g . a l g o r i t h m s and d r a w i n g d i a g r a m s . I n t h i s i n s t a n c e t h e " b l o c k " f o r random c a s e s w o u l d c o v e r more t h a n one column. Core The c o r e c a t e g o r y i n c l u d e s b e h a v i o r r e l a t e d t o t h e s u b j e c t ' s u s e o f c o r e p r o c e d u r e s . A l l s u b j e c t s s t a r t w i t h r e a d i n g p r o b l e m and a check i s e n t e r e d i n t h e f i r s t column. R e a d i n g t h e p r o b l e m i s a l s o c h e c k e d e a c h t i m e t h e s u b j e c t r e - r e a d s t h e p r o b l e m o r p a r t o f t h e p r o b l e m . I f he c o n t i n u e s t o r e - r e a d t h e p r o b l e m , w i t h no o t h e r a c t i v i t y i n t e r s p e r s e d , 145 o n l y one check i s e n t e r e d i n r e a d i n g p r o b l e m . Request d e f i n i t i o n o f  t e r m s i s c h e c k e d i f a s u b j e c t a s k s t h e i n t e r v i e w e r t o d e f i n e a t e r m u s e d i n t h e s t a t e m e n t o f t h e p r o b l e m . R e c a l l same p r o b l e m o r r e l a t e d p r o b l e m i s c h e c k e d i f t h e s u b j e c t m e n t i o n s t h a t he has s e e n t h e p r o b l e m o r a s i m i l a r p r o b l e m b e f o r e . I f t h e s u b j e c t c l a s s i f i e s t h e p r o b l e m b y t y p e , s u c h as a work p r o b l e m o r a m i x t u r e p r o b l e m , r e c a l l p r o b l e m t y p e i s c h e c k e d . I f t h e s u b j e c t m e r e l y m e n t i o n s a r e l a t e d m a t h e m a t i c a l t o p i c s u c h as " t h i s i s a g e o m e t r i c p r o b l e m " o r " I ' l l have t o w r i t e an e q u a t i o n f o r t h i s " , t h e r e c a l l c a t e g o r i e s a r e n o t c h e c k e d . I f a s u b j e c t m e n t i o n s a s p e c i f i c f a c t w h i c h i s r e l a t e d t o t h e p r o b l e m , r e c a l l r e l a t e d f a c t i s coded w i t h a 1, 2, o r 3, d e p e n d i n g on t h e c o r r e c t n e s s o f t h e f a c t . F o r example a 2 i s e n t e r e d i f a s u b j e c t m e n t i o n s t h e P y t h a g o r e a n Theorem as " t h e s i d e • o p p o s i t e t h e r i g h t a n g l e i s t w i c e as l o n g as t h e l o n g e s t o f t h e o t h e r two s i d e s " , and r e c a l l s no o t h e r p r o p e r t i e s o f r i g h t t r i a n g l e s . E a c h t i m e a s u b j e c t draws a new f i g u r e , draw d i a g r a m i s checked^ I f t h e s u b j e c t o n l y draws a few l i n e s and i m m e d i a t e l y e r a s e s them, t h e c a t e g o r y i s n o t c h e c k e d . Once a f i g u r e i s drawn, i f t h e s u b j e c t a l t e r s i t i n any way b y d e l e t i n g o r a d d i n g a l i n e ( o r l i n e s ) , m o d i f y d i a g r a m i s c h e c k e d . I f a s u b j e c t i d e n t i f i e s a v a r i a b l e , e i t h e r v e r b a l l y o r w r i t t e n , i d e n t i f y v a r i a b l e i s coded.. The s u b j e c t need n o t use t h e v a r i a b l e i n an e q u a t i o n f o r t h i s c a t e g o r y t o be c h e c k e d . I f a s u b j e c t w r i t e s an e q u a t i o n and does n o t i d e n t i f y what t h e v a r i a b l e s r e p r e s e n t , i d e n t i f y v a r i a b l e i s n o t coded. S e t t i n g up e q u a t i o n s i s coded i f t h e s u b j e c t w r i t e s any p a r t o f an e q u a t i o n . A s u b j e c t need n o t i d e n t i f y t h e v a r i a b l e s i n t h e e q u a t i o n , however, i f he d o e s , t h e outcome a s s i g n e d t o s e t t i n g up e q u a t i o n s w i l l depend on b o t h t h e e q u a t i o n s w r i t t e n and t h e v a r i a b l e s 146 i d e n t i f i e d by t h e s u b j e c t . F o r example, i n t h e stamp p r o b l e m ( p r o b l e m #2, A p p e n d i x B ) , i f X i s t h e number o f 2 c e n t stamps, 1CX t h e number o f 1 c e n t stamps and 5X t h e number o f 5 c e n t stamps ( i d e n t i f y v a r i a b l e w o u l d be coded 2 ) t h e e q u a t i o n 10X + 2X + 5(5X) = 100 w o u l d be coded 3, t h e e q u a t i o n 10X + 2X + 5X = 100 w o u l d be coded 2. I f an a l g o r i t h m i s a p p l i e d t o an e q u a t i o n ( t h e e q u a t i o n c o u l d be coded 1, 2, o r 3), a l g o r i t h m s - a l g e b r a i c i s coded. I f a s u b j e c t c o p i e s an e q u a t i o n o v e r on h i s p a p e r w i t h no change i n t e n d e d , no code i s e n t e r e d . A l g o r i t h m s - a r i t h m e t i c i s coded each t i m e t h e s u b j e c t u s e s o r a t t e m p t s t o use an a r i t h m e t i c a l g o r i t h m . I f t h e s u b j e c t u s e s two o r more a l g o r i t h m s c o n s e c u t i v e l y , s u c h as t h e s u b t r a c t i o n o f two p a i r s o f numbers, f o l l o w e d by t h e m u l t i p l i c a t i o n o f t h e i r d i f f e r e n c e s , a r i t h m e t i c -a l g o r i t h m s i s o n l y coded once. I f t h e a l g o r i t h m i s p e r f o r m e d m e n t a l l y b y t h e s u b j e c t w i t h no d i f f i c u l t y , s u c h as one h a l f o f 500 i s 250, i t i s n o t coded. G u e s s i n g i s coded e a c h t i m e t h e s u b j e c t s e l e c t s a s o l u t i o n t o t h e p r o b l e m o r a s u b p r o b l e m ( s u c h as an a l g o r i t h m o r e q u a t i o n ) b y g u e s s i n g , p r o v i d e d t h e s u b j e c t does n o t i n d i c a t e some i n t e n t i o n o f f u r t h e r t r i a l s . H e u r i s t i c s T h i s c a t e g o r y i n c l u d e s t h e h e u r i s t i c p r o c e d u r e s u s e d b y t h e s u b j e c t s . S m o o t h i n g i s coded each t i m e t h e s u b j e c t d i s r e g a r d s i r r e l e v a n t 1 i n f o r m a t i o n i n t h e p r o b l e m . S m o o t h i n g i s coded t h e f i r s t t i m e a s u b j e c t draws a d i a g r a m b u t n o t f o r s i m i l a r f i g u r e s drawn a f t e r . F o r example, i n t h e r o d i n t h e box, p r o b l e m (#4, A p p e n d i x B) t h e s u b j e c t can smooth t h e b o x t o a r e c t a n g u l a r p a r a l l e l e p i p e d , however, i f he draws a r e c t a n g l e , s m o o t h i n g i s coded as 2 and a c h e c k i s e n t e r e d i n 147 draws d i a g r a m . I f a s u b j e c t c o n s i d e r s a s p e c i a l c a s e s u c h as u s i n g 50 f e e t o f f e n c i n g i n a p r o b l e m r a t h e r t h a n 300, s m o o t h i n g i s coded. A n a l y s i s i s coded whenever t h e s u b j e c t b r e a k s t h e p r o b l e m i n t o s u b problems and a t t e m p t s t o s o l v e t h e s u b p r o b l e m s . I f t h e s u b j e c t m e r e l y m e n t i o n s t h a t t h e p r o b l e m c a n be s o l v e d i n t e r m s o f two o r more sub p r o b l e m s , no code i s made. T e m p l a t i o n i s coded e a c h t i m e t h e s u b j e c t m e n t i o n s o r r e c a l l s a c a t e g o r y o f c o n t e n t , i . e . r e c a l l s p a r t o f t h e c o r e , w h i c h i s r e l a t e d t o t h e p r o b l e m b e i n g s o l v e d . I f t h e s u b j e c t m e n t i o n s o n l y one s p e c i f i c f a c t , s u c h as r e c a l l i n g t h e P y t h a g o r e a n Theorem, r e c a l l r e l a t e d f a c t i s coded. I n a p r o b l e m w h i c h i n v o l v e s a r i g h t t r i a n g l e , i f t h e s u b j e c t m e n t i o n s some o f t h e t h e o r e m s , d e f i n i t i o n s , o r p r o p e r t i e s o f a r i g h t t r i a n g l e , t h e n t e m p l a t i o n i s coded. I f t h e s u b j e c t r e c a l l s any p r o p e r t i e s o f t h e r i g h t t r i a n g l e and u s e s any o f them i n s o l v i n g t h e p r o b l e m , a 3 i s coded. I f t h e i n f o r m a t i o n u s e d i s i n c o r r e c t , a 2 i s coded and i f t h e s u b j e c t does n o t use t h e . i n f o r m a t i o n o b t a i n e d , a 1 i s coded. The h e u r i s t i c s o f c a s e s a r e coded each t i m e t h e s u b j e c t e i t h e r e x p r e s s e s an i n t e n t i o n t o use o r u s e s two o r more c a s e s i n one o f t h e f o l l o w i n g manners: C a s e s - a l l i s coded i f t h e s u b j e c t c o n s i d e r s a l l p o s s i b l e c a s e s i n a p r o b l e m where o n l y a f i n i t e number o f c a s e s e x i s t o r t h e s u b j e c t s t a t e s t h e r e a r e o n l y a f i n i t e number. Cases-random i s coded i f t h e s u b j e c t u s e s c a s e s i n a random f a s h i o n w i t h no a p p a r a n t use made o f t h e i n f o r m a t i o n o b t a i n e d f r o m p r e v i o u s t r i a l s o r c a s e s . I f t h e s u b j e c t ' s r emarks i n d i c a t e t h a t he i s u s i n g i n f o r m a t i o n f r o m p r e v i o u s c a s e s t h e n c a s e s - s y s t e m a t i c o r c a s e s - s e q u e n t i a l i s coded. I f t h e d a t a a r e a p p l i e d i n a s e q u e n t i a l manner, s u c h as i n s o l v i n g t h e equa-. t i o n 12X + 5Y = 100 by l e t t i n g X = 1, 2, 3, . . ., o r f i n d i n g t h e 148 maximum o f X ( 3 0 0 - 2 X ) b y l e t t i n g X = 50, 6 0 , 7 0 , . . .; c a s e s - s e q u e n t i a l i s coded. I f t h e s u b j e c t i s u s i n g c a s e s i n a n o n - s e q u e n t i a l i n c r e a s i n g o r d e c r e a s i n g manner o r u s i n g i n f o r m a t i o n f r o m t h e p r e v i o u s c a s e s s u c h as f i n d i n g t h e s q u a r e r o o t o f 160 b y r e f i n i n g e ach c a s e t o o b t a i n a v a l u e c l o s e r t o t h e s q u a r e r o o t , C a s e s ^ - s y s t e m a t i c i s coded. C a s e s - c r i t i c a l i s coded i f t h e s u b j e c t s t a t e s a bound o r bounds f o r t h e v a r i a b l e s when u s i n g c a s e s . F o r example, i n s o l v i n g t h e e q u a t i o n 12X + 5Y = 1 0 0 where X and Y a r e p o s i t i v e i n t e g e r s , i f t h e s u b j e c t i s u s i n g random o r s y s t e m a t i c c a s e s and s t a t e s " t h e l a r g e s t v a l u e X can be i s 8" t h e n a 3 i s coded f o r c a s e s - c r i t i c a l . D e d u c t i o n i n c l u d e s b o t h h y p o t h e t i c a l and d i r e c t d e d u c t i o n . D e d u c t i o n i s coded whenever a s u b j e c t ' s r e m a r k s i n d i c a t e he has assumed a p r e m i s e o r i s u s i n g d a t a o b t a i n e d f r o m t h e p r o b l e m as a p r e m i s e and i s a t t e m p t i n g t o d e t e r m i n e as many i m p l i c a t i o n s as p o s s i b l e . D e d u c t i o n i s n o t coded i f t h e s u b j e c t s t a t e s a s i n g l e l o g i c a l i m p l i c a t i o n . I n v e r s e d e d u c t i o n i s coded i f t h e s u b j e c t ' s comments i n d i c a t e he i s i n t e n t i o n a l l y - a t t e m p t i n g t o work t h e p r o b l e m b a c k w a r d s . F o r example, i f a s u b j e c t i n d i c a t e s he 3 2 c a n f i n d a f o r m u l a f o r t h e g e n e r a l c u b i c e q u a t i o n aX + bX + cX + d, 3 p r o v i d e d he can s o l v e aX + cX + d = 0, t h e i n v e r s e d e d u c t i o n w o u l d be coded d e p e n d i n g on t h e outcome. I n v a r i a t i o n i s coded whenever a s u b j e c t ' s comments and a c t i o n s i n d i c a t e t h a t he i s i n t e n t i o n a l l y e x c l u d i n g o r f i x i n g ( r e n a m i n g a c o n s t a n t ) a v a r i a b l e , t h e n a t t e m p t i n g t o s o l v e t h e new p r o b l e m and use i t s s o l u t i o n t o h e l p h im s o l v e t h e o r i g i n a l p r o b l e m . I f a s u b j e c t f a i l s t o s o l v e t h e new p r o b l e m o r use i t s s o l u t i o n a f t e r i n d i c a t i n g an i n t e n t t o do s o , a 1 i s coded f o r i n v a r i a t i o n . A n a l o g y i s coded i f t h e s u b j e c t r e c a l l s an a n a l o g o u s m a t h e m a t i c a l s i t u a t i o n and i n t e n t i o n a l l y makes u se o f i t s p r o p e r t i e s i n s o l v i n g t h e p r o b l e m . I f a s u b j e c t r e c a l l s a n o n - m a t h e m a t i c a l s i t u a t i o n s u c h as " t h i s i s s i m i l a r t o a b o a t moving up and down on a r i v e r " , r e c a l l p r o b l e m t y p e i s coded. The h e u r i s t i c o f symmetry i s coded i f t h e a c t i o n s and comments o f t h e s u b j e c t - i n d i c a t e he i s i n t e n t i o n a l l y m a k i n g u s e o f t h e i n h e r e n t o r c o n s t r u c t e d symmetry i n t h e p r o b l e m . S o l u t i o n The c a t e g o r y o f o b t a i n s o l u t i o n i s c h e c k e d e a c h t i m e t h e s u b j e c t o b t a i n s a s o l u t i o n t o t h e p r o b l e m . R e a s o n a b l e n e s s The r e a s o n a b l e n e s s c a t e g o r y p e r t a i n s t o c h e c k i n g p a r t o f t h e s u b j e c t ' s work and c h e c k i n g h i s s o l u t i o n . I f a s u b j e c t ' s a c t i o n s and comments i n d i c a t e he i s c h e c k i n g p a r t o f h i s work, c h e c k i n g p a r t i s coded and t h e p r o c e d u r e o r p r o c e d u r e s he u s e d a r e coded i n t h e same column. I f he i n d i c a t e s he i s c h e c k i n g h i s s o l u t i o n , c h e c k i n g s o l u t i o n i s coded and t h e p r o c e d u r e o r p r o c e d u r e s he u s e d a r e coded i n t h e same column. These p r o c e d u r e s may i n c l u d e any p r o c e d u r e s f r o m t h e c o d i n g s y s t e m . F o u r o t h e r c a t e g o r i e s a r e i n c l u d e d i n t h e c o d i n g s y s t e m f o r c h e c k i n g . C h e c k i n g b y s u b s t i t u t i n g i n e q u a t i o n i s c h e c k e d i f t h e s u b j e c t r e p l a c e s a v a r i a b l e i n an e q u a t i o n w i t h a v a l u e he b e l i e v e s t o be a s o l u t i o n t o t h e e q u a t i o n . C h e c k i n g b y r e t r a c i n g s t e p s i s c h e c k e d i f t h e s u b j e c t e x p l i c i t l y r e p e a t s an o p e r a t i o n o r s e r i e s o f o p e r a t i o n s a f t e r he o b t a i n s a s o l u t i o n o r i n d i c a t e s he i s c h e c k i n g o v e r h i s work. C h e c k i n g by_ r e a s o n a b l e / r e a l i s t i c i s coded i f t h e s u b j e c t i n d i c a t e s t h a t he has t e s t e d w h e t h e r h i s s o l u t i o n i s r e a s o n a b l e e i t h e r i n t e r m s o f t h e pr o b l e m s o r i n t e r m s o f t h e " r e a l w o r l d " . I f t h e s u b j e c t c h e c k s h i s s o l u t i o n b y r e a d i n g t h e p r o b l e m , c h e c k i n g b y r e a s o n a b l e / r e a l i s t i c i s coded. U n c o d a b l e i s c h e c k e d i f t h e s u b j e c t ' s comments i n d i c a t e he has c h e c k e d h i s work b u t does n o t i n d i c a t e how. e.g. "Oh! T h a t c a n ' t be r i g h t " o r " L e t ' s see - t h a t ' s n o t r i g h t " . C o n c e r n The c o n c e r n c a t e g o r y i n c l u d e s s e v e r a l k i n d s o f comments t h a t t h e s u b j e c t may make about t h e p r o c e d u r e s he i s u s i n g o r about h i s s o l u t i o n . E x p r e s s e s c o n c e r n about method i s c h e c k e d i f t h e s u b j e c t e x p l i c i t l y e x p r e s s e s c o n c e r n about t h e p r o c e d u r e s he i s u s i n g . F o r example, t h e s u b j e c t m i g h t s a y , "I'm n o t c u r e t h i s p r o b l e m can be s o l v e d b y u s i n g two e q u a t i o n s . " o r " T h i s may n o t work, b u t I ' l l t r y some numbers anyway." S t a t e m e n t s i n d i c a t i n g t h a t he does n o t u n d e r s t a n d t h e p r o b l e m o r t h a t he does n o t know how t o s o l v e i t a r e n o t coded. E x p r e s s e s c o n c e r n  about a l g o r i t h m i s c h e c k e d whenever t h e s u b j e c t i n d i c a t e s a c o n c e r n about t h e a l g o r i t h m he has c h o s e n . T h i s i n c l u d e s c o n c e r n about t h e p r o c e s s i n v o l v e d i n c a r r y i n g o u t t h e a l g o r i t h m . However, an e x p r e s s i o n o f c o n c e r n s u c h as "I'm n o t s u r e t h i s p r o b l e m c a n be s o l v e d w i t h an a l g o r i t h m " i s coded c o n c e r n about method. E x p r e s s e s c o n c e r n about e q u a t i o n i s coded i f t h e s u b j e c t e x p r e s s e s c o n c e r n about t h e e q u a t i o n o r e q u a t i o n s he has w r i t t e n o r i s a t t e m p t i n g t o w r i t e . An e x p r e s s i o n o f c o n c e r n s u c h as "I'm n o t s u r e t h i s p r o b l e m c a n be s o l v e d u s i n g e q u a t i o n s " i s coded c o n c e r n about method. E x p r e s s e s c o n c e r n about s o l u t i o n i s coded whenever t h e s u b j e c t ' s remark i n d i c a t e s t h a t he d o u b t s t h e c o r r e c t n e s s o f h i s s o l u t i o n . Remarks t h a t t h e r e i s no s o l u t i o n a r e n o t coded. Work s t o p p e d - s o l u t i o n i s coded i f t h e s u b j e c t s t o p s work on t h e p r o b l e m w i t h a s o l u t i o n . I f t h e s u b j e c t q u i t s w o r k i n g on t h e p r o b l e m w i t h o u t a s o l u t i o n t h e n work stopped-no; s o l u t i o n i s coded. The summary f o r m summarizes d a t a f r o m t h e c o d i n g f o r m as w e l l as r e c o r d i n g o t h e r d a t a r e l a t e d t o t h e s u b j e c t ' s p r o b l e m s o l v i n g b e h a v i o r . Most e n t r i e s o n .the summary f o r m a r e e a s i l y i d e n t i f i e d . E r r o r s , b o t h a r i t h m e t i c and a l g e b r a i c , a r e c o u n t e d f r o m t h e c o d i n g f o r m . An e n t r y coded e i t h e r 1 o r 2 i s c o u n t e d as an e r r o r . A c y c l e i s c o u n t e d each t i m e t h e s u b j e c t ' s r emarks i n d i c a t e he i s a t t e m p t i n g t o r e - a t t a c k t h e p r o b l e m . A c y c l e i s a l s o c o u n t e d f o r t h e f i r s t a t t a c k on t h e p r o b l e m . A change i s c o u n t e d each t i m e t h e s u b j e c t changes h i s mode o f a t t a c k i n g t h e p r o b l e m a f t e r a c y c l e . A s u b j e c t changes h i s mode o f a t t a c k i f he u s e s a p r o c e d u r e f r o m e i t h e r c o r e o r h e u r i s t i c s t h a t he d i d n o t use on h i s p r e v i o u s a t t e m p t a t s o l v i n g t h e p r o b l e m . As i l l u s t r a t i o n s o f how t h e c o d i n g s y s t e m o p e r a t e s , t h e w r i t t e n p r o t o c o l s f o r two p r o b l e m s a r e g i v e n i n C h a p t e r I I I t o g e t h e r w i t h t h e c o m p l e t e d c o d i n g f o r m s . 152 Appendix D HISTOGRAMS OF SELECTED VARIABLES ; | M +» o CD •r-3 w «H o u o 16 14 12 10 8 • • • * * » « '# # # • • # * # * * » * * * * * * « # • # * 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Number of Changes A Histogram Representing the Distribution of the Number of Changes Made by the Subjects in the Sample (N=40) 16 ta +> o o> •1-3 GO <H o u • i 2 14 12 10 8 6 4 2 * • * * # * 0 1 2 3 4 5 « • « 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Number of Heuristics 29 30 31 32 33 A Histogram Representing the Distribution of the Number of Times Heuristics were used by the Subjects (N=40) 16 CO •p o <u CO <H O u Q) •g 3 14 12 10 8 6 4 2 * * # « * * * « • 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Diagrams -// // 25 26 33 34 A Histogram: Representing the Distribution of, the Number of Timesdiagrams &were Drawn by the Subjects (N=40) to -p o a> •<-> •§ CO o a> 16 14 12 10 8 « « • « # * • # * « * • 0 1 2 3 4 5 6 7. 8 9 10 11 12 13 14 15 16 17 18 Number of Algebraic Algorithms -//- //-2 8 29 30 55 5-6 57 A Histogram,Representing the Distribution, of the Number of Times Algebraic Algorithms were used by the Subjects (N=40) w +> o 0) •r-3 00 <H o <u I 25 16 14 12 10 8 6 4 2 * # « * • * * « « * * « • * * * * « # * * 0 1 2 3 4 5 6 7 8 9 10 11 12 13 i4 15 16 17 18 Number of Templations A Histogram Representing the Distribution of the Number of Times Templation was used by the Subjects (N=40) 16 CO o CD • r - j •§ CO o CD • i 2S 14 12 10 8 6 4 2 « « * 0 1 2 3 4 5 6 ? 8 9 10 11 12 13 14 15 16 17 18 Number of Random Cases A Histogram Representing the Distribution of the Number of Times Random Cases were used by the Subjects (N=40) 

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