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The operational curricula of mathematics 8 teachers in British Columbia Dirks, Michael Karel 1986

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THE OPERATIONAL CURRICULA OF MATHEMATICS 8 TEACHERS IN BRITISH COLUMBIA by MICHAEL KAREL DIRKS B. Sc., U n i v e r s i t y o-f Washington, 1968 M. Sc. Nat. Sc., S e a t t l e U n i v e r s i t y , 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION i n THE FACULTY OF GRADUATE STUDIES (Mathematics Education) We accept t h i s t h e s i s as con-forming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January 1.986 Q Michael Karel D i r k s , 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. MICHAEL K. D I RK-S Department of MATHEMATICS EDUCAT The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 4 A P R I L 1 9 8 6 Date DE-6 n/a'n i i Abstract Research Supervisor: Dr. D. F. R o b i t a i l l e The purpose of t h i s study was to d e s c r i b e the mathematics c u r r i c u l a as a c t u a l l y implemented by a sample of Mathematics 8 teachers i n B r i t i s h Columbia. A survey of previous research i n d i c a t e d that knowledge about the mathematics subject matter which teachers present to t h e i r students and the i n t e r p r e t a t i o n s which teachers give to that subject matter i s sparse i n s p i t e of the importance such knowledge might have for the c u r r i c u l u m r e v i s i o n process, textbook s e l e c t i o n , the i d e n t i f i c a t i o n of i n s e r v i c e education needs, and the i n t e r p r e t a t i o n of student achievement r e s u l t s . The Mathematics 8 c u r r i c u l u m was d i v i d e d i n t o three content areas: a r i t h m e t i c , a l g e b r a , and geometry. Within these content areas a t o t a l of 16 t o p i c s were i d e n t i f i e d as among the basic t o p i c s of the formal Mathematics 8. course. Four v a r i a b l e s were i d e n t i f i e d as r e p r e s e n t i n g important aspects of a mathematics c u r r i c u l u m . The f i r s t of these, content emphasis, was d e f i n e d as a f u n c t i o n of the amount of time a teacher spent on each content area. The other three v a r i a b l e s , mode of content r e p r e s e n t a t i o n , r u l e - o r i e n t e d n e s s of i n s t r u c t i o n , and d i v e r s i t y of i n s t r u c t i o n , were d e f i n e d as f u n c t i o n s of the content-s p e c i f i c methods teachers used to i n t e r p r e t the t o p i c s to t h e i r students. C l a s s achievement l e v e l and the primary textbook were i d e n t i f i e d as having strong p o t e n t i a l r e l a t i o n s h i p s with a teacher's o p e r a t i o n a l c u r r i c u l u m . These were used as background I l l v a r i a b l e s i n t h i s study. The data f o r t h i s study were c o l l e c t e d as p a r t of the Second I n t e r n a t i o n a l Mathematics Study d u r i n g the 1980/1981 s c h o o l y e a r . The sample c o n s i s t e d of 93 t e a c h e r s who submitted f i v e T o p i c - S p e c i f i c Q u e s t i o n n a i r e s throughout the s c h o o l year r e g a r d i n g what they taught to one of t h e i r Mathematics 8 c l a s s e s . Each c l a s s took a 40 item p r e t e s t a t the b e g i n n i n g of the s c h o o l y e a r . The 27 c l a s s e s with the h i 9 h e s * s c l a s s means were d e s i g n a t e d as "high achievement c l a s s e s " f o r t h i s study while the 27 c l a s s e s w i t h the lowest c l a s s means were d e s i g n a t e d as "low achievement c l a s s e s . " Among the f i n d i n g s of t h i s study were: (1) Wide v a r i a t i o n e x i s t e d i n the emphasis given by t e a c h e r s to the t h r e e content areas with 60% g i v i n g at l e a s t one area l i g h t or very l i g h t emphasis. (2) The median p r o p o r t i o n of c l a s s time a l l o c a t e d f o r geometry was s l i g h t l y h i g h e r than f o r a l g e b r a or a r i t h m e t i c . However, te a c h e r s showed the mos.t v a r i a t i o n f o r t h i s c o n t e n t area spending between 0% and 66% of t h e i r c o u r s e s on geometry. (3) In low achievement c l a s s e s somewhat more time was spent on a r i t h m e t i c and somewhat l e s s time on geometry than i n h i g h achievement c l a s s e s . (4) Teachers u s i n g a t e x t which p l a c e d more emphasis on a p a r t i c u l a r c o n t e n t a r e a tended to spend more time on t h a t c o n t e n t area i n t h e i r c l a s s e s . (5) The mode of r e p r e s e n t a t i o n of mathematical content was i v s l i g h t l y more abstract than perceptual in general. (6) The median mode of content repesentation varied s u b s t a n t i a l l y among t o p i c s . (7) Teachers of low achievement classes tended to present mathematics in a s l i g h t l y more abstract and r u l e -oriented way than teachers of high achievement classes. (8) A weak p o s i t i v e association was found between the l e v e l of d i v e r s i t y in the textbook used and the l e v e l of d i v e r s i t y in the operational c u r r i c u l a of teachers using that textbook. V Table of Contents A b s t r a c t i i L i s t of T a b l e s v i i i L i s t of F i g u r e s ix Acknowledgement x i i i Chapter I INTRODUCTION 1 1. PURPOSE AND RATIONALE 2 2. HISTORICAL ORIENTATIONS TO THE TERM CURRICULUM 4 3. LEVELS OF CURRICULUM 6 4. THE COMPONENTS OF A CURRICULUM: CONTENT AND METHOD 8 5. TYPES OF MATHEMATICAL CONTENT 12 6. MATHEMATICS CURRICULUM VARIABLES 13 6.1 Content Emphasis 14 6.2 Mode Of Content R e p r e s e n t a t i o n 15 6.3 R u l e - O r i e n t e d n e s s Of I n s t r u c t i o n 18 6.4 D i v e r s i t y Of I n s t r u c t i o n 20 7. THE CONTEXTUAL VARIABLE: CLASS ACHIEVEMENT LEVEL 21 8. RESEARCH QUESTIONS 22 Chapter II REVIEW OF THE LITERATURE 23 1. THE TEACHING OF SECONDARY SCHOOL MATHEMATICS 23 1.1 Pre-NACOME Research In North America v . . . . 2 5 1.2 The I n t e r n a t i o n a l Study Of Achievement In Mathematics 29 1.3 The NACOME Study 32 1.4 The N a t i o n a l Sc ience Foundat ion S t u d i e s 33 1.5 S t u d i e s Conducted W i t h i n B r i t i s h Columbia 36 2. THE USE OF QUESTIONNAIRES IN RESEARCH 39 2.1 Quest ionnaire• , Research: T h e o r e t i c a l V a l i d i t y P r i n c i p l e s 41 2.2 Ques t ionnaire ; Research: E m p i r i c a l V a l i d i t y S t u d i e s 45 Chapter III RESEARCH DESIGN AND PROCEDURES 52 1. THE B. C . SIMS PROJECT 52 1.1 D e s c r i p t i o n Of SIMS I n s t r u m e n t a t i o n 54 1.2 Sample S e l e c t ion 56 v i 1.3 P a r t i c i p a t i o n Rate And Instrument Return Rate 56 1.4 Representativeness Of The Achieved Mathematics 8 Sample 58 1.5 SIMS V a l i d a t i o n : Instrument Development And Research 61 2. MATHEMATICS 8 CONTENT INCLUDED IN THE STUDY 65 2.1 The Content Areas 65 2.2 The S p e c i f i c Mathematics Topics 67 3. MEASUREMENT OF THE CURRICULUM VARIABLES 69 3.1 Content Emphasis 69 3.2 Mode Of Content Representation 71 3.3 Rule-Orientedness Of I n s t r u c t i o n 74 3.4 D i v e r s i t y Of I n s t r u c t i o n 78 4. THE CONTEXTUAL VARIABLE: CLASS ACHIEVEMENT 80 5. DATA ANALYSIS 81 Chapter IV THE RESULTS OF THE STUDY 83 1. DESCRIPTION OF THE GRAPHICAL DISPLAYS 83 1.1 The Stem-and-Leaf P l o t 83 1 .2 The Boxplot 87 2. CONTENT EMPHASIS 91 2.1 The Emphasis Given To A r i t h m e t i c 91 2.2 The Emphasis Given To Algebra 95 2.3 The Emphasis Given To Geometry 97 2.4 Comparisons Among The Content Areas 100 2.5 Content Emphasis Of Teachers And Textbooks 103 3. MODE OF CONTENT REPRESENTATION 108 3.1 Mode Of Content Representation For A r i t h m e t i c ....110 3.2 Mode Of Content Representation For Algebra 113 3.3 Mode Of Content Representation For Geometry 116 3.4 Comparisons Among The Topics And Content Areas ...118 3.5 Achievement Level Comparisons 122 3.6 Content Representation Of Teachers And Textbooks .124 4. RULE-ORIENTEDNESS OF INSTRUCTION 127 4.1 Rule-Orientedness In Teaching A r i t h m e t i c 128 4.2 Rule-Orientedness In Teaching Algebra 130 4.3 Rule-Orientedness In Teaching Geometry 133 4.4 Comparisons In Rule-Orientedness For Topics And Content Areas 135 4.5 Achievement Level Comparisons 138 5. DIVERSITY OF INSTRUCTION 140 5.1 D i v e r s i t y In Teaching A r i t h m e t i c 143 5.2 D i v e r s i t y In Teaching Algebra 146 5.3 D i v e r s i t y In Teaching Geometry 149 VI 1 5.4 Comparisons In D i v e r s i t y Between Content Areas And T o p i c s 152 5.5 Achievement L e v e l Comparisons 155 5.6 D i v e r s i t y Scores Of Teachers And Textbooks 156 Chapter V CONCLUSIONS 160 1 . SUMMARY OF THE RESULTS 160 2. IMPLICATIONS FOR PRACTICE 173 3. SIGNIFICANCE OF THE STUDY 177 4. LIMITATIONS OF THE STUDY 178 5. SUGGESTIONS FOR FUTURE RESEARCH 180 BIBLIOGRAPHY 183 APPENDIX A - TOPICS FROM THE SIMS TOPIC SPECIFIC QUESTIONNAIRES USED IN THIS STUDY 195 APPENDIX B - SIMS CORE PRETEST FOR MATHEMATICS 8 213 v i i i L i s t of Tables 3- 1 . Teacher Workload i n Mathematics 59 3- 2. Years of Teaching Experience 60 3- 3. Defined Values f o r Rule-Orientedness 76 4- 1 . Percent of Teachers S c o r i n g Within Each L e v e l of Content Emphasis f o r Each Content Area 102 4--- 2. Percent of the Commonly Used Textbooks Devoted to Each-Content Area 105 4- 3. Mode of Content Representation Scores 109 4- 4. Pro p o r t i o n of Abstr a c t TSQ Methods to T o t a l TSQ Methods in the Textbooks 125 4- 5. Rule-Orientedness.of I n s t r u c t i o n Scores 128 4 - 6 . D i v e r s i t y of I n s t r u c t i o n Scores •" 141 4 - 7 . Number of TSQ Methods Contained in the Commonly Used Textbooks f o r S e l e c t e d Topics 157 ix L i s t of Figures 1- 1 . F i v e i n t e r p r e t a t i o n s of the concept of i n t e g e r s 11 1-2. The concrete r e p r e s e n t a t i o n of q u a d r a t i c expressions using blocks 16 3- 1 . C o n t e n t - s p e c i f i c methods for teaching the concept of dec imals 73 3- 2. C o n t e n t - s p e c i f i c methods for teaching the Pythagorean theorem 77 4- 1 . D i s t r i b u t i o n of a r i t h m e t i c emphasis scores ( i l l u s t r a t i v e stem-and-leaf p l o t ) 84 4- 2. D i s t r i b u t i o n of mode of r e p r e s e n t a t i o n scores f o r a r i t h m e t i c for low and high achievement c l a s s e s 85 4- 3. D i s t r i b u t i o n of o v e r a l l d i v e r s i t y scores f o r a r i t h m e t i c 87 4- 4. The d i s t r i b u t i o n of a r i t h m e t i c emphasis scores ( i l l u s t r a t i v e boxplot) 90 4- 5. D i s t r i b u t i o n of a r i t h m e t i c emphasis scores 91 4- 6. D i s t r i b u t i o n of a r i t h m e t i c emphasis scores f o r low and high achievement l e v e l c l a s s e s 93 4- 7. D i s t r i b u t i o n of algebra emphasis scores 95 4- 8. D i s t r i b u t i o n s of algebra emphasis scores f o r low and high achievement c l a s s e s 97 X 4- 9. D i s t r i b u t i o n of geometry emphasis scores 98 4-10. D i s t r i b u t i o n s of geometry emphasis scores f o r low and high achievement l e v e l c l a s s e s 100 4-11. D i s t r i b u t i o n of content emphasis scores 101 4-12. D i s t r i b u t i o n s of content emphasis scores f o r low and high achievement l e v e l c l a s s e s ....104 4-13. D i s t r i b u t i o n s of content emphasis scores f o r c l a s s e s using each of the commonly used textbooks 107 4-14. D i s t r i b u t i o n of content r e p r e s e n t a t i o n scores averaged over a l l t o p i c s 110 4-15. D i s t r i b u t i o n s of the content r e p r e s e n t a t i o n scores f o r the a r i t h m e t i c scores 111 4-16. D i s t r i b u t i o n of o v e r a l l content r e p r e s e n t a t i o n scores f o r a r i t h m e t i c 113 4-17. D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores for the a l g e b r a i c t o p i c s 114 4-18. D i s t r i b u t i o n of o v e r a l l content r e p r e s e n t a t i o n scores f o r algebra 115 4-19. D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores for the geometric t o p i c s 117 4-20. D i s t r i b u t i o n of o v e r a l l content r e p r e s e n t a t i o n scores f o r geometry 118 4-21 . D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores for 14 t o p i c s . 119 4-22. D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores for each content area and across a l l t o p i c s 120 x i 4-23. D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores f o r low and high achievement l e v e l c l a s s e s 123 4-24. D i s t r i b u t i o n s of Content Representation Scores f o r • Class e s Using Each of the Commonly Used Textbooks. ..126 4-25. D i s t r i b u t i o n of r u l e - o r i e n t e d n e s s scores averaged over e i g h t t o p i c s 129 4-26. D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores for a l g e b r a i c t o p i c s 131 4-27. D i s t r i b u t i o n of r u l e - o r i e n t e d n e s s scores f o r a l g e b r a . 1 32 4-28. D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores f o r geometric t o p i c s 134 4-29. D i s t r i b u t i o n of r u l e - o r i e n t e d n e s s scores f o r geometry. 1 35 4-30. D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores f o r eig h t t o p i c s 1 36 4-31. D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores for each content area and o v e r a l l ...137 4-32. D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores for low and high achievement l e v e l c l a s s e s 139 4-33. D i s t r i b u t i o n of d i v e r s i t y scores averaged over a l l t o p i c s 142 4-34. D i s t r i b u t i o n of d i v e r s i t y scores f o r a r i t h m e t i c 143 4-35. Boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores f o r a r i t h m e t i c t o p i c s 145 4-36. D i s t r i b u t i o n of d i v e r s i t y scores for algebra ...146 x i i 4-37. Boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores for a l g e b r a i c t o p i c s 148 4-38. D i s t r i b u t i o n of d i v e r s i t y scores for geometry 149 4-39. Boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores for geometric t o p i c s 151 4-40. D i s t r i b u t i o n of d i v e r s i t y scores for each content area and across a l l t o p i c s 153 4-41 . D i s t r i b u t i o n s of d i v e r s i t y scores for the 16 t o p i c s . 154 4-42. D i s t r i b u t i o n s of d i v e r s i t y scores f o r low and high achievement l e v e l c l a s e s 156 4-43. D i s t r i b u t i o n s of average d i v e r s i t y scores for the groups of t o p i c s for users of each textbook 158 XI 1 1 Acknowledgement I wish to thank David R o b i t a i l l e , James S h e r r i l l , and Walter S z e t e l a who comprised my s u p e r v i s o r y committee. They provided encouragement and d i r e c t i o n throughout the research process and kept me on-task while I wrote t h i s d i s s e r t a t i o n 700 kilometers from campus. My wife Suree D i r k s was u n f a i l i n g i n her help and understanding during my seven years as a d o c t o r a l student. I thank her and my two sons W i l l i a m and Robert most of a l l . 1 I. INTRODUCTION The p r a c t i c e of c r i t i c i z i n g the p u b l i c school, both i t s programs and i t s products, has been a popular a c t i v i t y since the i n c e p t i o n of that i n s t i t u t i o n i n i t s c u r r e n t form in North America over a century ago. The i n t e r n a t i o n a l "modern mathematics movement" of 1955-1975 1 was preceded by p a r t i c u l a r l y strong c r i t i c i s m of school c u r r i c u l u m m a t e r i a l s , teacher competence, and student achievement wi t h i n that subject area (e.g., Bestor, 195.3; Lynd, 1953; Smith, M. , 1949). There i s now evidence that c r i t i c i s m of school mathematics as w e l l as science i s once again i n c r e a s i n g ( K e i t e l , 1982; U s i s k i n , 1985). While a c o n s i d e r a b l e body of knowledge about student mathematics achievement, both recent s t a t u s and trends over time, 2 now e x i s t s , l i t t l e i s known about many aspects of mathematics classroom p r a c t i c e (NACOME, 1975). As M i l e s (1981) has noted: . . . i t seems c l e a r that much more d i r e c t l y d e s c r i p t i v e data are needed on matters of the most s t r a i g h t f o r w a r d s o r t [ i n c l u d i n g ] the a c t u a l i n s t r u c t i o n a l modes being used by 1 As Howson (1982, p. 205) has noted, i t i s not p o s s i b l e to give f i x e d dates for t h i s movement which a c t u a l l y encompasses many cu r r i c u l u m development and. other a c t i v i t i e s with widely d i f f e r i n g aims and o r i e n t a t i o n s to both mathematics and education (Suydam & Osborne, 1977). The dates given here, however, do not d i f f e r widely from the v a r i o u s ones' u s u a l l y used in the l i t e r a t u r e (e.g., Suydam & Osborne, 1977; S t e i n e r , 1980; Howson, K e i t e l , & K i l p a t r i c k , 1981; Stebbins, 1978). 2 ( R o b i t a i l l e & S h e r r i l l , 1977) and ( R o b i t a i l l e , 1981) are examples of s t u d i e s which have documented l e v e l s of student achievement in mathematics wit h i n B r i t i s h Columbia. O'Shea (1979) s t u d i e d achievement trends w i t h i n the same j u r i s d i c t i o n . The N a t i o n a l Assessment of E d u c a t i o n a l Progress (NAEP) and the Assessment of Performance Unit (APU) have conducted s i m i l a r survey research on a n a t i o n a l l e v e l i n the United States and the United Kingdom r e s p e c t i v e l y . 2 tea c h e r s . . . . In most cases we do not have r e l i a b l e , c a r e f u l l y sampled s t u d i e s that would t e l l us, simply, what i s r e a l l y going on. (pp. 110-111, emphasis i n o r i g i n a l ) In p a r t i c u l a r , few s t u d i e s have been conducted to i n v e s t i g a t e the c u r r i c u l u m that has been implemented by teachers in t h e i r classrooms even though i t i s widely recognized that what has been p r e s c r i b e d as an o f f i c i a l c u r r i c u l u m may "bear l i t t l e r e l a t i o n s h i p to what a c t u a l l y goes on i n the classroom" (Theisen, 1981, p. 7). Where s t u d i e s have been conducted, i t i s not c l e a r that the most important v a r i a b l e s have been i d e n t i f i e d (Young, 1979). Such a lack of knowledge may e x p l a i n why the modern mathematics movement was appraised as r e v o l u t i o n a r y i n i t s impact on school p r a c t i c e at one time (NCTM, 1961; S e y f e r t , 1968), but was l a t e r c r i t i c i z e d as " r e l a t i v e l y i n s i g n i f i c a n t " (Howson et a l . , 1981, p. 238) and f i n a l l y as only a "minor p e r t u r b a t i o n " (Wheeler, 1982, p. 23). If the p e r i o d i c c r i t i c i s m s of school mathematics programs are to be assessed and i f c u r r i c u l u m development i s to be c a r r i e d out e f f e c t i v e l y , a means of d e s c r i b i n g mathematics c u r r i c u l a as implemented by teachers i s needed as w e l l as such d e s c r i p t i o n s themselves. 1. PURPOSE AND RATIONALE T h i s study had four major purposes: (1) to develop and j u s t i f y a framework for the d e s c r i p t i o n of a school mathematics cu r r i c u l u m ; (2) to use the v a r i a b l e s d e f i n e d as part of t h i s framework to d e s c r i b e the o p e r a t i o n a l c u r r i c u l a , c u r r i c u l u m - i n -3 use or implemented c u r r i c u l u m 3 of a sample of Mathematics 8 teachers i n B r i t i s h Columbia during the 1980-81 school year (the year f o r which the necessary data are a v a i l a b l e ) ; (3) to evaluate p a r t i a l l y the congruence of the o p e r a t i o n a l c u r r i c u l a of these teachers with the formal c u r r i c u l u m of the c u r r i c u l u m guide and adopted textbooks as well as the i d e a l c u r r i c u l u m of mathematics educators; (4) to generate hypotheses about the o p e r a t i o n a l c u r r i c u l a of Mathematics 8 teachers i n B. C. There i s a need for theory b u i l d i n g i n the general area of c u r r i c u l u m . In p a r t i c u l a r , no adequate theory of mathematics c u r r i c u l u m or i n s t r u c t i o n i s a v a i l a b l e f o r confirmatory, h y p o t h e s i s - t e s t i n g i n v e s t i g a t i o n s or to plan c u r r i c u l u m m a t e r i a l s or classroom a c t i v i t i e s ( B a u e r s f e l d , 1979). T h i s need for theory b u i l d i n g was expressed by Mann (1975) as f o l l o w s : I b e l i e v e i t i s w e l l known that there are'no comprehensive t h e o r i e s about c u r r i c u l u m phenomena. But even such rudiments of theory as a l i m i t e d set of explanatory p r o p o s i t i o n s about s e l e c t e d c u r r i c u l u m phenomena, or d i s c i p l i n e d e f f o r t s to suggest an approach to c o n c e p t u a l i z i n g the events to which a theory might p e r t a i n , are q u i t e l i m i t e d i n number. (p. 158) Recognizing the l i m i t e d development of c u r r i c u l u m theory g e n e r a l l y and mathematics c u r r i c u l u m theory s p e c i f i c a l l y , i t became apparent that a reasonable t h e o r e t i c a l goal of t h i s study The terms o p e r a t i o n a l c u r r i c u l u m , c u r r i c u l u m - i n - u s e , and implemented c u r r i c u l u m are used interchangeably in t h i s study to r e f e r to the a c t u a l c u r r i c u l u m of a teacher i n a classroom. 4 would be the e x p l i c a t i o n of a number of mathematics c u r r i c u l u m v a r i a b l e s rather than the development of a complete theory of mathematics c u r r i c u l u m . The c o n t r i b u t i o n s which t h i s study makes to the theory of mathematics c u r r i c u l u m have t h e i r foundations i n the w r i t i n g s and research of Cooney (1976, 1980a, 1980b) and Goodlad (1979, 1983). 2. HISTORICAL ORIENTATIONS TO THE TERM CURRICULUM The d e f i n i t i o n and scope of the term "curriculum" continues to be an u n s e t t l e d issue w i t h i n the f i e l d of education. While many v a r i a n t s are found in the l i t e r a t u r e , two basic formulations can be i d e n t i f i e d , each having a h i s t o r y which can be t r a c e d to the f i r s t quarter of t h i s century. One d e f i n i t i o n employs an ends-means model in which c u r r i c u l u m c o n s t i t u t e s the planned ends of the e d u c a t i o n a l process with i n s t r u c t i o n as the means. Johnson's d e f i n i t i o n i s t y p i c a l of those w i t h i n t h i s category: . . . i t i s here s t i p u l a t e d that c u r r i c u l u m i s a s t r u c t u r e d s e r i e s of intended l e a r n i n g outcomes. Curriculum p r e s c r i b e s (or at l e a s t a n t i c i p a t e s ) the r e s u l t s of i n s t r u c t i o n . (1967, p. 130) In t h i s d e f i n i t i o n , c u r r i c u l u m precedes i n s t r u c t i o n . I n s t r u c t i o n i s the instrumental process by which the c u r r i c u l u m or intended l e a r n i n g outcomes are t r a n s m i t t e d to students. This formulation of the concept of c u r r i c u l u m has i t s o r i g i n s i n the work of T a y l o r , B o b b i t t , and Thorndike (Howson et a l . , 1981, p. 85). The second common o r i e n t a t i o n to the term c u r r i c u l u m 5 emphasizes the a c t u a l experiences of persons in an e d u c a t i o n a l s e t t i n g (Brubaker, 1982). Stenhouse o f f e r e d t h i s c h a r a c t e r i z a t i o n of what might be c a l l e d the e x p e r i e n t i a l formulat i o n : ...the c u r r i c u l u m i s not the i n t e n t i o n or p r e s c r i p t i o n but what happens in r e a l s i t u a t i o n s . I t i s not the a s p i r a t i o n , but the achievement. The problem of s p e c i f y i n g the c u r r i c u l u m i s one of p e r c e i v i n g , understanding and d e s c r i b i n g what i s a c t u a l l y going on i n school and classroom. (1975, p. 2) T h i s a s s o c i a t i o n of c u r r i c u l u m with the l i v e d experience of the classroom i s rooted in the w r i t i n g and p r a c t i c e of Dewey (Howson et a l . , 1981, p. 84). The recent debate regarding the extent to which modern mathematics has been implemented in t y p i c a l school programs has i l l u s t r a t e d that i f the term c u r r i c u l u m i s to be u s e f u l , i t needs to encompass both i n t e n t i o n and r e a l i t y . In essence i t seems to me that c u r r i c u l u m study i s concerned with the r e l a t i o n s h i p between the two views of c u r r i c u l u m — a s i n t e n t i o n and as r e a l i t y . I b e l i e v e that our e d u c a t i o n a l r e a l i t i e s seldom conform to our e d u c a t i o n a l i n t e n t i o n s . We cannot put our p o l i c i e s i n t o p r a c t i c e . . . The c e n t r a l problem of c u r r i c u l u m study i s the gap between ideas and a s p i r a t i o n s and our attempts to o p e r a t i o n a l i z e them. (Stenhouse, 1975, pp. 2-3) A t t e n t i o n to both the l e a r n i n g s intended by such documents as c u r r i c u l u m guides and embodied in textbooks, as w e l l as the r e a l i t i e s of the p r e s e n t a t i o n s of teachers and the a c t i v i t i e s and a s s i m i l a t i o n s of students has c h a r a c t e r i z e d the work of Goodlad (e.g., Goodlad, 1979, 1983; Goodlad & K l e i n , 1970). His notion that c u r r i c u l u m should be p e r c e i v e d at s e v e r a l l e v e l s i n c o r p o r a t e s a concern f o r both i n t e n t i o n and r e a l i t y as c e n t r a l 6 to c u r r i c u l u m . I t was used as a ba s i c conceptual framework f o r t h i s study. 3. LEVELS OF CURRICULUM The m u l t i - l e v e l conception of c u r r i c u l u m used i n the present study i s s i m i l a r to the s e v e r a l v e r s i o n s which have been proposed by Goodlad (1979) and used in h i s research. He i d e n t i f i e d four l e v e l s : the i d e a l c u r r i c u l u m , the formal c u r r i c u l u m , the o p e r a t i o n a l c u r r i c u l u m , and the e x p e r i e n t i a l c u r r i c u l u m . The i d e a l or e x p e r t / p r o f e s s i o n a l l e v e l of the mathematics cu r r i c u l u m r e f e r s to a course of study proposed or produced by mathematics educators or other e d u c a t i o n a l experts. Exemplars would include d e t a i l e d recommendations f o r the content and methods of a p a r t i c u l a r course but they c o u l d be more general i n nature. One example i s the N a t i o n a l C o u n c i l of Teachers of Mathematics (NCTM) Agenda for A c t i o n (See S h u f e l t & Smart, 1983) which s p e c i f i e d a broad o u t l i n e f o r an i d e a l c u r r i c u l u m in mathematics. Ideal c u r r i c u l a are of t e n s p e c i f i e d in mathematics education methods t e x t s . In t h i s context i d e a l r e f e r s to ideas and academia and not to a best or p e r f e c t c u r r i c u l u m . The second l e v e l , the formal c u r r i c u l u m , r e f e r s to c u r r i c u l a which have been f o r m a l l y or o f f i c i a l l y adopted w i t h i n some l e g a l j u r i s d i c t i o n such as a province or school d i s t r i c t . Such c u r r i c u l a are represented by the contents of c u r r i c u l u m guides, approved textbooks, or other m a t e r i a l s . The t h i r d l e v e l , the o p e r a t i o n a l c u r r i c u l u m , r e f e r s to a 7 course of study as a c t u a l l y presented i n the classroom by a teacher. The focus at t h i s l e v e l i s on the content and the i n t e r p r e t a t i o n s given to that content by teachers. These p r e s e n t a t i o n s might mirror the contents of the textbook or might d i f f e r from i t i n some way. U n l i k e the previous two l e v e l s there are u s u a l l y no w r i t t e n records of an o p e r a t i o n a l c u r r i c u l u m . I t was t h i s l e v e l of c u r r i c u l u m that was i n v e s t i g a t e d in t h i s study. The f o u r t h l e v e l , the e x p e r i e n t i a l c u r r i c u l u m , r e f e r s to the course of study a c t u a l l y r e c e i v e d by an i n d i v i d u a l student. In a classroom of t h i r t y students there could be t h i r t y d i s t i n c t e x p e r i e n t i a l ' c u r r i c u l a with c e r t a i n commonalities. As with the o p e r a t i o n a l c u r r i c u l u m , w r i t t e n documentation of an e x p e r i e n t i a l c u r r i c u l u m i s unusual. Achievement t e s t s provide measures of some of the e f f e c t s of such a c u r r i c u l u m . Interview p r o t o c o l s have the p o t e n t i a l of p r o v i d i n g a more comprehensive view of the cu r r i c u l u m as r e c e i v e d and i n t e r p r e t e d by the student (e.g., Erlwanger, 1975). No c l a i m i s made that the c a t e g o r i z a t i o n of c u r r i c u l a i n t o the four l e v e l s presented here provides a complete model for a theory of c u r r i c u l u m . I t does, however, provide a way of making d i s t i n c t i o n s between courses of study and of rec o g n i z i n g c u r r i c u l a that otherwise might not be i d e n t i f i e d as such. There are obvious connections among these four l e v e l s of cu r r i c u l u m . For example, the formal c u r r i c u l u m within a j u r i s d i c t i o n exerts a strong i n f l u e n c e on the o p e r a t i o n a l l e v e l . Indeed i t has been claimed that t h i s i n f l u e n c e i s e s p e c i a l l y 8 strong in the case of mathematics (e.g., Goodlad, 1983). Likewise, the content covered and the i n t e r p r e t a t i o n s given to that content i n a teacher's o p e r a t i o n a l c u r r i c u l u m w i l l probably e s t a b l i s h l i m i t s on the e x p e r i e n t i a l c u r r i c u l a of most c l a s s members. The problem of i d e n t i f y i n g i n f l u e n c e s on any cu r r i c u l u m and measuring t h e i r s t r e n g t h i s complex. Much of Goodlad's research has been i n t h i s a r e a . 4 4 . THE COMPONENTS OF A CURRICULUM: CONTENT AND METHOD It was noted above that there i s no agreement as to whether the term c u r r i c u l u m should r e f e r to intended l e a r n i n g outcomes, the r e a l i t y of e d u c a t i o n a l experience, or both. Likewise, there i s no consensus within education as to the s p e c i f i c elements which might c o n s t i t u t e a course of study or a c u r r i c u l u m so conceived. Howson (1979, p. 134) has argued that the i d e n t i f i c a t i o n of a mathematics c u r r i c u l u m with a s y l l a b u s or t o p i c o u t l i n e has impeded c u r r i c u l u m development. He i d e n t i f i e d aims f o r education, and mathematics education, as well as content, methods, and assessment procedures as being c e n t r a l components of a (mathematics c u r r i c u l u m . Goodlad and h i s a s s o c i a t e s (1979, p.. 66) o f f e r e d an even more extensive l i s t of what they c a l l " c urriculum commonplaces": goals and o b j e c t i v e s , m a t e r i a l s , content,; l e a r n i n g a c t i v i t i e s , teaching s t r a t e g i e s , ft A growing body o f , r e s e a r c h i s emerging for each l e v e l of the c u r r i c u l u m in which attempts to i n v e s t i g a t e i n f l u e n c e s have been made. Stebbins (1978) and Quick (1978), f o r example, have both examined the i n f l u e n c e s of the i d e a l c u r r i c u l u m on the formal c u r r i c u l u m in the context of the c u r r i c u l u m reform movement of 1955-1975. 9 e v a l u a t i o n , grouping p a t t e r n s , the use of time, and the use of i space. Huebner (1976), on-the-other-hand, contended that the term c u r r i c u l u m should have a narrower, more focused set of r e f e r e n t s . He argued that because c u r r i c u l u m has become concerned with so many f a c e t s of education i t has l o s t i t s coherence, focus, and e f f e c t i v e n e s s (p. 156). He f u r t h e r a s s e r t e d that "the nature of the student and the f u n c t i o n of the teacher, examinations and school o r g a n i z a t i o n s " (p. 159), for example, should not be among the elements of > the c u r r i c u l u m . Rather, the c e n t r a l components in h i s view are: (1) i d e n t i f i c a t i o n of those segments of c u l t u r e . . . that can become the content of the course of study; (2) i d e n t i f i c a t i o n of the te c h n o l o g i e s by which t h i s content can be made a c c e s s i b l e or made present to p a r t i c u l a r i n d i v i d u a l s . (p. 160) Huebner p r e s c r i b e d , then, that content and method should be the c e n t r a l f o c i of a c u r r i c u l u m and the c e n t r a l concerns of cu r r i c u l u m study. The p o s i t i o n that c u r r i c u l u m should focus on content and method was adopted in t h i s study f o r two reasons. F i r s t , these components were included e x p l i c i t l y or i m p l i c i t l y i n , and were basic t o, every formulation of the concept of cu r r i c u l u m which was examined as part of t h i s study. Secondly, the c u r r i c u l u m commonplaces i d e n t i f i e d by Goodlad as well as the cu r r i c u l u m components i d e n t i f i e d by other authors can, i n gener a l , be de s c r i b e d as aspects of or important to content, 10 method or t h e i r i n t e r r e l a t i o n s h i p s . The instrumental d e f i n i t i o n of method given by Huebner above was not, however, considered as adequate for t h i s study. Confrey (1981), in an essay on mathematics and c u r r i c u l u m , expands on Huebner's two c a t e g o r i e s or, more p r e c i s e l y , notes t h e i r i n t e r r e l a t i o n s h i p . I w i l l add a t h i r d c o n s i d e r a t i o n which I think has been addressed inadequately and o f t e n , in f a c t , ignored by c u r r i c u l u m t h e o r i s t s ; that i s , the i n t e g r a l r e l a t i o n s h i p between i d e n t i f y i n g e d u c a t i o n a l content and d e c i d i n g how to make i t a v a i l a b l e to young people. It i s the p a r t i c u l a r s of t h i s r e l a t i o n s h i p which I think are subject-matter s p e c i f i c , i f not to a large extent concept s p e c i f i c , and hence must be undertaken with respect to one's subject matter. (p. 243) I t was the notion of method impl i e d by the " t h i r d c o n s i d e r a t i o n " above that was used to c o n c e p t u a l i z e c o n t e n t - s p e c i f i c methods in t h i s study. Thus, c o n t e n t - s p e c i f i c methods in t h i s study were taken to r e f e r to the ways content can be made meaningful to students such as the ways mathematical concepts can be i n t e r p r e t e d . Teaching methods or i n s t r u c t i o n a l methods or t e c h n o l o g i e s u n r e l a t e d to s p e c i f i c content such as overhead p r o j e c t o r s , programmed i n s t r u c t i o n , advance o r g a n i z e r s , or c l a r i t y were not c o n s i d e r e d to be among the c o n t e n t - s p e c i f i c methods of a c u r r i c u l u m . As an example of c o n t e n t - s p e c i f i c methods, consider F i g u r e 1-1 which comes from a q u e s t i o n n a i r e used in the Second I n t e r n a t i o n a l Mathematics Study (SIMS), a p r o j e c t of the I n t e r n a t i o n a l A s s o c i a t i o n f o r the E v a l u a t i o n of E d u c a t i o n a l Achievement (IEA). T h i s study i s d i s c u s s e d i n more d e t a i l in 11 Chapter 2 and Chapter 3. In the Fi g u r e 1-1, f i v e i n t e r p r e t a t i o n s of the mathematical concept of i n t e g e r s are shown. In t h i s study, each of these i n t e r p r e t a t i o n s was con s i d e r e d as a c o n t e n t - s p e c i f i c method of making the concept of i n t e g e r s a v a i l a b l e to students. 2 0 . E x t e n d i n g t h e number r a y t o t h e n u u i e r l i n e : I e x t e n d e d t h e number n y (0 and p o s i t i v e n u m b e r s ) t o t h e l e f t by I n t r o d u c i n g d i r e c t i o n as w e l l a s n a g n i t i l d e . Ex: - 4 - 3 - 2 - 1 0 1 2 3 4 - 3 Deans 3 u n i t s t o t h e l e f t ot* 0 . 2 1 . E x t e n d i n g t h e number s y s t e a t o f i n d s o l u t i o n s t o e q u a t i o n s : I d i s c u s s e d t h e n e e d t o e x t e n d t h e p o s i t i v e i n t e g e r s In o r d e r t o f i n d a s o l u t i o n t o e q u a t i o n s l i k e + 7 - 5 . 2 2 . U s i n g v e c t o r s o r d i r e c t e d s e g -ments o n t h e n u n i i e r l i n e : 1 d e f i n e d an I n t e g e r as a s e t o f v e c t o r s ( d i r e c t e d l i n e s e g -s e n t s ) on t h e n u n b e r l i n e . E x : - 2 c a n be r e p r e s e n t e d b y any o f : - 1 0 - 5 4 5 10 E x : +2 c a n be r e p r e s e n t e d by any o f : TO 2 3 . D e f i n i n g I n t e g e r s as e q u i v a -l e n c e c l a s s e s o f w h o l e n u m b e r s : I d e v e l o p e d t h e I n t e g e r s as e q u i v a l e n c e c l a s s e s o f o r d e r e d p a i r s o f w h o l e n u m b e r s . E x : { ( 0 , 2 ) . ( 1 . 3 ) , ( 2 , 4 ) . . . . ) -o r { ( a , b ) c WXW: b =• a + 2 } « 2 4 . U s i n g e x a m p l e s o f p h y s i c a l s i t u a t i o n s : I d e v e l o p e d I n t e g e r s by r e f e r r i n g t o d i f f e r e n t p h y s i c a l s i t u a t i o n s w h i c h c a n be d e s c r i b e d wi th I n t e g e r s . ' E x : t h e r m o m e t e r , e l e v a t i o n , money ( c r e d i t / d e b i t ) , s p o r t s ( s c o r i n g ) , t i m e ( b e f o r e / a f t e r ) , e t c . F i g u r e 1- 1 - Fiv e i n t e r p r e t a t i o n s of the concept of i n t e g e r s . 12 5. TYPES OF MATHEMATICAL CONTENT Within t h i s study, mathematical content or subject matter was c a t e g o r i z e d i n t o four types: f a c t s , concepts, o p e r a t i o n s , and p r i n c i p l e s . T h i s c l a s s i f i c a t i o n scheme was borrowed from Begle (1979) and i s s i m i l a r to that proposed by Cooney, Davis, and Henderson (1975). Although Begle d i d not d e f i n e the terms " f a c t " and "concept", he d i d provide examples. He c i t e d "two plus three equals f i v e " and " 7 X 8 = 56" as f a c t s , r e f e r r i n g to the f i r s t as an " a r b i t r a r y f a c t " and the second as "deducible." Cooney et a l . (1975, p. 64) used the term " s i n g u l a r statements" for f a c t s and d e s c r i b e d them as "statements about j u s t one o b j e c t " such as "2 i s the only even prime number." According to Cooney et a l . (1975, p. 61) "a concept i s knowledge of what something i s . " Skemp (1971) a s s o c i a t e d concepts with the processes of c l a s s i f i c a t i o n and a b s t r a c t i o n and s t a t e d t h a t : [An a b s t r a c t i o n ] i s something l e a r n t which enables us to c l a s s i f y ; i t i s the d e f i n i n g property of a c l a s s . To d i s t i n g u i s h between a b s t r a c t i n g as an a c t i v i t y , and an a b s t r a c t i o n as i t s end product, we s h a l l h e r e a f t e r c a l l the l a t t e r a concept. (p. 22, emphasis i n o r i g i n a l ) Begle (1979, p. 7) l i s t e d r e c t a n g u l a r ajrray, f r a c t i o n , and congruence among examples of mathematical concepts. Begle's other two types of mathematical content were of a higher order than f a c t s and concepts and he was able to provide def i n i t i o n s . 1 3 An o p e r a t i o n i s a f u n c t i o n which assigns mathematical o b j e c t s to mathematical o b j e c t s . Examples a r e : . . . c o u n t i n g , a d d i n g two numbers, [and] measuring the length of a l i n e segment.... A p r i n c i p l e i s a r e l a t i o n s h i p between two or more mathematical o b j e c t s : f a c t s , concepts, o p e r a t i o n s , other p r i n c i p l e s . Any p r i n c i p l e can be expressed as a mathematical theorem or axiom and every theorem or axiom expresses a p r i n c i p l e (except f o r those which express a f a c t by s t a t i n g the e x i s t e n c e of a p a r t i c u l a r kind of mathematical o b j e c t ) . (1979, p. 7) In adopting Begle's c l a s s i f i c a t i o n of mathematical content, i t was decided to use the word t o p i c to r e f e r to any p a r t i c u l a r f a c t , concept, o p e r a t i o n , or p r i n c i p l e . The term content area was used to r e f e r to major groupings of mathematical t o p i c s . 6. MATHEMATICS CURRICULUM VARIABLES From the ba s i c c u r r i c u l u m components of content and method, four g l o b a l v a r i a b l e s were co n s t r u c t e d in t h i s study to c h a r a c t e r i z e mathematics c u r r i c u l a and, i n p a r t i c u l a r , o p e r a t i o n a l mathematics c u r r i c u l a . These v a r i a b l e s were among those o r i g i n a l l y suggested by Cooney (1980a) and McKnight (1980), and in c l u d e d : content emphasis, mode of content r e p r e s e n t a t i o n , r u l e - o r i e n t e d n e s s of i n s t r u c t i o n , and d i v e r s i t y of i n s t r u c t i o n . 1 4 6 . 1 C o n t e n t E m p h a s i s I n t h i s s t u d y , t h e m a i n c o m p o n e n t s o f a c u r r i c u l u m w e r e s p e c i f i e d t o be c o n t e n t a n d c o n t e n t - s p e c i f i c m e t h o d s . . A l t h o u g h c u r r i c u l u m t h e o r i s t s d i f f e r a s t o w h a t t h e c o n c e p t o f c u r r i c u l u m s h o u l d i n c l u d e b e s i d e s c o n t e n t , no o n e d i s p u t e s t h a t c o n t e n t i s a f u n d a m e n t a l p a r t o f a c u r r i c u l u m . T h e r e f o r e , a n e c e s s a r y , t h o u g h n o t s u f f i c i e n t , way o f d e s c r i b i n g a c u r r i c u l u m i s t o l i s t t h e t o p i c s o r c o n t e n t a r e a s i n c l u d e d i n t h a t c u r r i c u l u m . T h u s , a n o p e r a t i o n a l c u r r i c u l u m i n w h i c h g e o m e t r y , f o r e x a m p l e , i s p r e s e n t e d t o s t u d e n t s d i f f e r s f r o m a n o p e r a t i o n a l c u r r i c u l u m i n w h i c h g e o m e t r y i s o m i t t e d . Some m e a s u r e o f w h a t c o n t e n t i s i n c l u d e d , t h e n , i s n e e d e d t o d e s c r i b e a c u r r i c u l u m . T h r e e c o n t e n t a r e a s w e r e i d e n t i f i e d f o r i n v e s t i g a t i o n : a r i t h m e t i c , a l g e b r a , a n d g e o m e t r y . T h e b a s i s f o r s e l e c t i n g t h e s e p a r t i c u l a r c o l l e c t i o n s o f t o p i c s i s d i s c u s s e d i n C h a p t e r 3. S i n c e i t was h y p o t h e s i z e d t h a t a l m o s t e v e r y t e a c h e r w o u l d i n c l u d e e a c h o f t h e s e c o n t e n t a r e a s i n h i s o r h e r c u r r i c u l u m t o some d e g r e e , i t was n e c e s s a r y t o d e f i n e a v a r i a b l e w h i c h w o u l d q u a n t i f y t h e a m o u n t o f c o v e r a g e e a c h c o n t e n t a r e a r e c e i v e d i n t h e o p e r a t i o n a l c u r r i c u l a . F o r t h i s r e a s o n t h e v a r i a b l e " c o n t e n t e m p h a s i s " was i n c o r p o r a t e d i n t h e s t u d y . T h e way i n w h i c h t h i s v a r i a b l e was d e f i n e d a n d m e a s u r e d i s d i s c u s s e d i n C h a p t e r 3. 1 5 6.2 Mode Of Content Representation The t h e o r e t i c a l and e m p i r i c a l work of Bruner and Dienes i n the area of mathematical concept formation and mathematics i n s t r u c t i o n has had a s i g n i f i c a n t e f f e c t on research conducted i n these areas and, to some extent, on school c u r r i c u l u m m a t e r i a l s over the l a s t 20 years (Resnick & Ford, 1981). A c e n t r a l c o n s t r u c t in the t h e o r i e s of i n s t r u c t i o n which each of these researchers developed was the idea of the mode in which content i s represented. • Bruner i d e n t i f i e d three modes: e n a c t i v e , i c o n i c , and symbolic. Any domain of knowledge (or any problem w i t h i n that domain of knowledge) can be represented i n three ways: by a set of a c t i o n s a p p r o p r i a t e f o r a c h i e v i n g a c e r t a i n r e s u l t (enactive r e p r e s e n t a t i o n ) ; by a set of summary images or graphics that stand f o r a concept without d e f i n i n g i t f u l l y ( i c o n i c r e p r e s e n t a t i o n ) ; and by a set of symbolic or l o g i c a l p r o p o s i t i o n s drawn from a symbolic system that i s governed by r u l e s or laws for forming and transforming p r o p o s i t i o n s (symbolic r e p r e s e n t a t i o n ) . (Bruner, 1966, pp. 44-45) Bruner a s s e r t e d that i n teaching mathematics i t i s necessary to represent concepts f i r s t c o n c r e t e l y (the e n a c t i v e mode), then using diagrams or some other semi-concrete means of p r e s e n t a t i o n (the i c o n i c mode), and f i n a l l y using the c o n v e n t i o n a l or some other mathematical symbolism (the symbolic mode). Figure 1-2 i l l u s t r a t e s t h i s idea a p p l i e d to q u a d r a t i c e x p r e s s i o n s . The f i g u r e shows blocks which represent q u a d r a t i c q u a n t i t i e s along with the corresponding symbolism. The blocks themselves are concrete, while a p i c t u r e of the blocks i s semi-co n c r e t e . Bruner d e s c r i b e d the teaching sequence as f o l l o w s : 16 The object was to begin with an enactive r e p r e s e n t a t i o n of q u a d r a t i c s — something that could l i t e r a l l y be "done" or b u i l t - - a n d to move from there to an i c o n i c r e p r e s e n t a t i o n , however r e s t r i c t e d . Along the way, n o t a t i o n was developed and, by the use of v a r i a t i o n and c o n t r a s t , converted i n t o a p r o p e r l y symbolic system. ( 1966, pp.. 64-65) i • 2 it x 2 x* *4x «-4 Figure 1- 2 - The concrete r e p r e s e n t a t i o n of q u a d r a t i c expressions using blocks. Dienes (1973; See a l s o Dienes, 1960 and 1964) i d e n t i f i e d s i x stages in the process of l e a r n i n g mathematics. Within t h i s process, he a l s o a s s e r t e d the n e c e s s i t y of r e p r e s e n t i n g content using each of the:modes of r e p r e s e n t a t i o n i d e n t i f i e d by Bruner and in the same order. Dienes a p p l i e d h i s teaching sequence to a r i t h m e t i c concepts and a l s o to such advanced t o p i c s as q u a d r a t i c s , logarithms, v e c t o r s , and f u n c t i o n s . Research has e s t a b l i s h e d a strong case f o r the use of concrete and semi-concrete r e p r e s e n t a t i o n s of mathematical content before that content i s represented s y m b o l i c a l l y (Suydam & Higgins, 1977) and t h i s sequence i s t y p i c a l l y discussed in 1 7 in d e t a i l in elementary methods t e x t s (e.g., Heimer & Trueblood, 1977). The t h e o r i e s of mathematics i n s t r u c t i o n formulated by Bruner and Dienes were based l a r g e l y on the developmental psychology of Piaget and h i s c l a s s i f i c a t i o n of human development i n t o four stages: sensori-motor, p r e o p e r a t i o n a l , the stage of concrete o p e r a t i o n s , and the stage of formal operations (Ginsburg & Opper, 1979). While, according to P i a g e t i a n theory, a c h i l d of between 11 and 15 should enter the stage of formal o p e r a t i o n s , and presumably be able to l e a r n mathematics using symbolic r e p r e s e n t a t i o n s alone, research has shown that t h i s stage i s f r e q u e n t l y not reached u n t i l l a t e r , i f at a l l (Ginsburg & Opper, 1979, p. 201). Research r e s u l t s i n mathematics education have been c o n s i s t e n t with t h i s general p s y c h o l o g i c a l f i n d i n g i n that the use of concrete and semi-concrete r e p r e s e n t a t i o n s of content has been shown to be b e n e f i c i a l to ol d e r c h i l d r e n and adolescents as well as to younger c h i l d r e n (Suydam & Higgins, 1977, p. 38). Skemp speculated that some form of p r o g r e s s i o n from the concrete to the a b s t r a c t may be r e q u i r e d i n l e a r n i n g mathematical ideas r e g a r d l e s s of age. But i t may well be the case that we a l l have to go, perhaps more r a p i d l y than the growing c h i l d , through s i m i l a r stages i n each new t o p i c which we e n c o u n t e r — that the mode of t h i n k i n g a v a i l a b l e i s p a r t l y a f u n c t i o n of the degree to which the concepts have been developed i n the primary system. One can h a r d l y be expected to r e f l e c t on concepts which have not yet been formed, however w e l l developed one's r e f l e c t i v e system. So the " i n t u i t i v e - b e f o r e - r e f l e c t i v e " order may be p a r t i a l l y true for each new f i e l d of mathematical study. (Skemp, 1971, p. 66) 1 8 Bruner, i n f a c t , a s s e r t e d that even when i t might seem p o s s i b l e to omit enactive and i c o n i c r e p r e s e n t a t i o n s of content, i t c o uld w e l l be unwise to do so: For when the l e a r n e r has a well-developed symbolic system, i t may be p o s s i b l e to by-pass the f i r s t two stages. But one does so with the r i s k that the l e a r n e r may not possess the imagery to f a l l back on when h i s symbolic transformations f a i l to achieve a goal i n problem s o l v i n g . (1966, p. 49) Because of the prominence of the concept of mode of r e p r e s e n t a t i o n in t h e o r e t i c a l d i s c u s s i o n s of mathematics l e a r n i n g and i n s t r u c t i o n as w e l l as the supporting evidence of research s t u d i e s , the v a r i a b l e "mode of content r e p r e s e n t a t i o n " was i d e n t i f i e d as an important c u r r i c u l u m v a r i a b l e and inc o r p o r a t e d in t h i s study. 6.3 Rule-Orientedness Of I n s t r u c t i o n Rules i n mathematics are standard procedures or a s s o c i a t i o n s . They f i g u r e prominently in every branch of the subject (Beatley, 1954; Gordon, Achiman, & Melman, 1981). The school mathematics c u r r i c u l u m a f f o r d s numerous examples of r u l e s i n connection with o p e r a t i o n s : f o r example, the r u l e of signs fo r i nteger m u l t i p l i c a t i o n , the " i n v e r t and m u l t i p l y " r u l e f o r the d i v i s i o n of f r a c t i o n s and the "transpose and change s i g n s " r u l e f o r the s o l v i n g of l i n e a r equations. A l s o , d e f i n i t i o n s of concepts can be considered as r u l e s of a s s o c i a t i o n . Despite the conspicuousness of r u l e s i n mathematics, an emphasis on r u l e s in teaching to the e x c l u s i o n or under-emphasis 19 of other approaches to content has been d e c r i e d as rote i n s t r u c t i o n . Skemp, for example, argues v i g o r o u s l y against what he c a l l s the use of " r u l e s without reasons" in teaching (Skemp, 1971, p. 17; Skemp, 1977, p. 20). He a s s e r t s that concepts must be introduced through examples rather than by d e f i n i t i o n s or r u l e s . Concepts of a higher order than those which a person al r e a d y has cannot be communicated to him by a d e f i n i t i o n , but only by arranging for him to encounter a s u i t a b l e c o l l e c t i o n of examples. (1971, p. 32) Teaching i n which r u l e s are emphasized has f r e q u e n t l y been c o n t r a s t e d with teaching for understanding. An emphasis on the understanding 5 of mathematical ideas was one of the s t a t e d goals of the modern mathematics movement (Willoughby, 1968, p. 15; School Mathematics Study Group (SMSG), 1961, p. v ) . According to Callahan and Glennon (1975): "The 'new math' was intended to be more c o n c e p t u a l l y meaningful to the l e a r n e r s ; r o t e , meaningless l e a r n i n g was to be de-emphasized" (p. 6). P r i c e (1975) noted that a concern f o r the promotion of understanding in mathematics i s not a new phenomenon: The problem i s c e r t a i n l y not new and indeed r e a c t i o n s to the widespread l e a r n i n g of mathematics without; some degree of understanding, that i s "parrot f a s h i o n " , "by r o t e " or "mechanically", have been going on f o r over a century (p. 34) 5 The nature of mathematical understanding i s a complex t o p i c . See, f o r example, Backhouse, 1982; Byers & H e r s c o y i c s , 1977; Skemp, 1977, 1982. 20 Because of the importance of r u l e s i n mathematics content and because of the s t r e s s placed on a v o i d i n g over-emphasis or premature i n t r o d u c t i o n of r u l e s by mathematics educators, the v a r i a b l e " r u l e - o r i e n t e d n e s s " was i d e n t i f i e d as an important c u r r i c u l u m v a r i a b l e and in c o r p o r a t e d i n t h i s study. 6.4 D i v e r s i t y Of I n s t r u c t i o n Cooney (1980a) and McNight (1980) argued that the d i v e r s i t y or v a r i e t y of content p r e s e n t a t i o n s which teachers employ i s an important v a r i a b l e in t h e i r o p e r a t i o n a l c u r r i c u l a . McNight (1980) noted the prominent place of d i v e r s i t y or m u l t i p l e embodiments i n the' l e a r n i n g and i n s t r u c t i o n a l theory of mathematics formulated by Dienes (See Dienes, 1960, p. 44) as d i d Resnick and Ford (1981, pp. 116-123). Cooney (1980a) implied a connection between d i v e r s i t y of p r e s e n t a t i o n s and teacher f l e x i b i l i t y : One might expect d i f f e r e n t student outcomes for teachers with high v a r i a b i l i t y than f o r those with medium or low v a r i a b i l i t y . There i s evidence in the l i t e r a t u r e to suggest that teachers who are more " f l e x i b l e " are more e f f e c t i v e . One aspect of being f l e x i b l e i s to be able to i d e n t i f y and u t i l i z e a number of i n s t r u c t i o n a l approaches. (p. 8) Without n e c e s s a r i l y a c c e p t i n g Cooney's a s s o c i a t i o n between d i v e r s i t y and f l e x i b i l i t y , i t would seem that d i v e r s i t y of approach i s a c u r r i c u l u m v a r i a b l e which warrants i n v e s t i g a t i o n . In t h i s study d i v e r s i t y w i t h i n the o p e r a t i o n a l c u r r i c u l u m was d e f i n e d i n terms of the number of ways teachers used to i n t e r p r e t and present mathematical ideas. 21 7. THE CONTEXTUAL VARIABLE; CLASS ACHIEVEMENT LEVEL A mathematics c u r r i c u l u m i s , in g e n e r a l , subject to many i n f l u e n c e s . These include p s y c h o l o g i c a l and s o c i o l o g i c a l f a c t o r s as w e l l as changes w i t h i n mathematics i t s e l f (Howson et a l . , 1981; R o b i t a i l l e & D i r k s , 1982). At the o p e r a t i o n a l l e v e l , i n f l u e n c e s both i n t e r n a l and e x t e r n a l to the classroom context may a f f e c t teachers' c u r r i c u l u m d e c i s i o n s . The content s e l e c t e d and the c o n t e n t - s p e c i f i c methods employed may be a f f e c t e d , for example, by the nature of the students in the c l a s s (Cooney, 1981). One of the recommendations of the B r i t i s h Columbia Curriculum Guide(Curriculum Development Branch, 1978) i s that teachers gear the depth of t h e i r courses and the approaches used to meet the needs of t h e i r students. Student needs in l e a r n i n g mathematics are r e l a t e d , at l e a s t in p a r t , to t h e i r mathematical a b i l i t y and p r i o r achievement in the s u b j e c t . I t was hypothesized i n t h i s study that d i f f e r e n c e s might e x i s t i n the four c u r r i c u l u m v a r i a b l e s between c l a s s e s of high achievement and c l a s s e s of low achievement. One might expect, for example, to f i n d more s t r e s s on a r i t h m e t i c and lower l e v e l s of a b s t r a c t i o n in low achievement c l a s s e s as compared to high achievement c l a s s e s . Because i t seemed reasonable to expect such d i f f e r e n c e s , i t was decided to analyze the four c u r r i c u l u m v a r i a b l e s s e p a r a t e l y for d i f f e r e n t c l a s s achievement l e v e l s . 22 8. RESEARCH QUESTIONS « The o p e r a t i o n a l c u r r i c u l a of B. C. Mathematics 8 teachers are explored i n t h i s study through the use of the c u r r i c u l u m and con t e x t u a l v a r i a b l e s d i s c u s s e d above. The f o l l o w i n g four research questions have been formulated to guide t h i s i n q u i r y : (1) What patterns of content emphasis are present in the sample c l a s s e s and how much v a r i a t i o n e x i s t s ? (2) For each of the other three c u r r i c u l u m v a r i a b l e s what l e v e l s are most common and how much v a r i a t i o n e x i s t s fo r s i n g l e t o p i c s , f o r each content area, and o v e r a l l ? (3) Are there any d i f f e r e n c e s i n the d i s t r i b u t i o n s of each c u r r i c u l u m v a r i a b l e between the low and high achievement c l a s s e s ? (4) To what degree do the d e s c r i p t i o n s of the o p e r a t i o n a l c u r r i c u l a provided by t h i s study c o i n c i d e with or d i f f e r from the formal or i d e a l c u r r i c u l a ? 23 I I . REVIEW OF THE LITERATURE The l i t e r a t u r e review i s d i v i d e d i n t o two major s e c t i o n s . In the f i r s t s e c t i o n the North American l i t e r a t u r e on the o p e r a t i o n a l c u r r i c u l a of secondary school mathematics teachers i s d i s c u s s e d . This l i t e r a t u r e c o n s i s t s p r i m a r i l y of d o c t o r a l s t u d i e s and components of l a r g e - s c a l e e v a l u a t i o n p r o j e c t s which have had as one of t h e i r s t a t e d purposes the d e s c r i p t i o n of the content which secondary teachers incorporated i n t h e i r courses and, more r a r e l y , the c o n t e n t - s p e c i f i c approaches which teachers employed duri n g i n s t r u c t i o n . While s e v e r a l of these s t u d i e s have u t i l i z e d i nterview and o b s e r v a t i o n a l data (e.g., Stake & Easley, 1978a, 1978b), most, i n c l u d i n g the Second I n t e r n a t i o n a l Mathematics Study, have r e l i e d h e a v i l y upon teacher s e l f - r e p o r t data gathered v i a q u e s t i o n n a i r e s . In the second s e c t i o n of the chapter the methodological i s s u e s surrounding q u e s t i o n n a i r e s which are r e l e v a n t to the SIMS p r o j e c t and the present study are reviewed. 1. THE TEACHING OF SECONDARY SCHOOL MATHEMATICS Within the l a s t decade the lack of comprehensive data on the b e l i e f s , planning procedures and classroom p r a c t i c e s of teachers of mathematics has been recognized. For example, in the United S t a t e s the authors of the N a t i o n a l Advisory Committee on Mathematics Education (NACOME) report (1975) a s s e r t e d : The q u e s t i o n "What goes on i n the ord i n a r y classroom in the United S t a t e s ? " i s s u r e l y an important one, but i n attempting to survey the st a t u s of mathematical education at "benchmark 1975," one i s immediately confronted by the 24 f a c t that a major gap i n e x i s t i n g data occurs here. A p p a l l i n g l y l i t t l e i s known about teaching in any l a r g e f r a c t i o n of U.S. classrooms. (p. 68) The authors of .the NACOME report noted the recent trend toward student assessment programs. They were concerned about the danger of formulating and the d i f f i c u l t y of r e f u t i n g cause-and-e f f e c t e xplanations of u n s a t i s f a c t o r y achievement r e s u l t s given "the vacuum of data on classroom processes" (p. 68). S i m i l a r l y , L a n i e r (1978) a s s e r t e d that " d e s c r i p t i v e analyses of teachers planning f o r and i n s t r u c t i n g groups of l e a r n e r s i n classrooms are o b v i o u s l y absent in mathematics education" (p. 7). As i t i s used w i t h i n the l i t e r a t u r e of e d u c a t i o n a l research, the term "classroom processes" i n c l u d e s more than the curr i c u l u m - i n - u s e in the classroom. Classroom management and teaching v a r i a b l e s , for example, have been d e f i n e d to d e s c r i b e aspects of the classroom process. Research in t h i s area has advanced c o n s i d e r a b l y s i n c e the NACOME report was w r i t t e n (See Rosenshine, 1982). However, i f one focuses on knowledge of the o p e r a t i o n a l c u r r i c u l u m — t h e content taught and how that content i s approached—one f i n d s that the s i t u a t i o n at "benchmark 1985" wit h i n North America i s only m a r g i n a l l y advanced beyond the s i t u a t i o n d e s c r i b e d ten years e a r l i e r w i t h i n the United S t a t e s . No attempt w i l l be made i n t h i s l i t e r a t u r e review to summarize the r e s e a r c h on teaching or any other area w i t h i n the domain of classroom processes except f o r those s t u d i e s which i n v e s t i g a t e d the o p e r a t i o n a l Curriculum i n some way. 25 1.1 Pre-NACOME Research In North America Few research p r o j e c t s conducted p r i o r to the NACOME study had, as a primary purpose, the d e s c r i p t i o n of the mathematics content teachers included i n t h e i r courses or the content-s p e c i f i c approaches they employed. Most of the i n v e s t i g a t i o n s in t h i s area were conducted i n connection with the academic year i n s t i t u t e s and the summer i n s t i t u t e s sponsored by the N a t i o n a l Science Foundation (NSF) i n the USA to upgrade the academic background of secondary mathematics and science t e a c h e r s . In the main, these s t u d i e s i n v e s t i g a t e d the degree to which p a r t i c i p a n t s in NSF i n s t i t u t e s introduced "modern" content i n t o the courses they taught. Connellan (1962), f o r example, concluded that p a r t i c i p a n t s i n Colorado tended to d i s c u s s such "modern" t o p i c s as set theory, the r e a l number system, and non-Eu c l i d e a n geometry in t h e i r courses more f r e q u e n t l y than a matched group of c o n t r o l teachers. S i m i l a r l y , Bradberry (1967) reported that over 70% of teachers p a r t i c i p a t i n g in NSF programs in the Southeastern region of the USA who responded to her q u e s t i o n n a i r e agreed that they had " r e v i s e d the course content they taught to include more up-to-date subject matter" (p. 2114A). Corbet (1976) reported that NSF p a r t i c i p a n t s i n Kansas were more l i k e l y to introduce such t o p i c s as l o g i c and s e t s , number theory, matrices, and transformation geometry than randomly s e l e c t e d Kansas mathematics teachers. F i e l d s (1970), Martinen (1968), Roye (1968), Wiersma (1962), and Wilson (1967) each a l s o concluded that NSF p a r t i c i p a n t s a l t e r e d t h e i r c u r r i c u l a by i n t r o d u c i n g "modern" content f o l l o w i n g t h e i r 26 experience in i n s t i t u t e s . These s t u d i e s d i d not, however, i n v e s t i g a t e the r e l a t i v e emphasis "modern" t o p i c s were given compared to " t r a d i t i o n a l " or other t o p i c s . A l s o , i n none of these s t u d i e s was the mathematics content s p e c i f i e d at the l e v e l of p a r t i c u l a r concepts, o p e r a t i o n s , and p r i n c i p l e s . While each of the s t u d i e s c i t e d above focused on the content implemented by NSF p a r t i c i p a n t s , rather than on content-s p e c i f i c or more general methods, s e v e r a l s t u d i e s d i d make some mention of methodology or approach. Corbet (1976), for example, s t a t e d t h a t : i t c o uld not be concluded from the data that the teaching of mathematics content courses had any e f f e c t upon the teaching methods of NSF p a r t i c i p a n t s , (p. 5206A) In reaching a more negative c o n c l u s i o n , Connellan (1962) a s s e r t e d t h a t : the Academic Year I n s t i t u t e i s not doing as much as i t should in p o i n t i n g out how t r a d i t i o n a l t o p i c s can be t r e a t e d from a modern point of view. (p. 541) Roye (1968), on-the-other-hand, concluded from h i s study that not only d i d NSF p a r t i c i p a n t s tend to teach new concepts in t h e i r courses, but a l s o that they used "new c u r r i c u l a r approaches and greater depth and d e t a i l i n the subject or course taught" (p. 503A). Taken together these s t u d i e s provide scant information about the approaches employed in teaching mathematical ideas. Another group of American d o c t o r a l s t u d i e s i s more d i r e c t l y r e l e v a n t to the subject of t h i s l i t e r a t u r e review in that one of 27 t h e i r s t a t e d purposes was to d e s c r i b e the content being taught w i t h i n some j u r i s d i c t i o n (e.g., Alspaugh, 1966; Crawford, 1967; Dunson, 1970; Rudnick, 1963; S h e t l e r , 1959). As with the previous group of s t u d i e s , however, an o v e r r i d i n g concern was the determination of what p r o p o r t i o n of courses taught could be c l a s s i f i e d as "modern" as c o n t r a s t e d with " t r a d i t i o n a l . " Thus, teachers and p r i n c i p a l s were surveyed f o r t i t l e s of courses taught and textbooks used and f o r t h e i r e v a l u a t i o n s of how modern or t r a d i t i o n a l t h e i r courses were. Alspaugh (1966) c o l l e c t e d more d e t a i l e d c u r r i c u l u m information than most of these r e s e a r c h e r s . He asked a sample of secondary teachers from M i s s o u r i whether or not each of a l i s t of t o p i c s was i n c l u d e d i n the course they taught. On t h i s b a s i s , rather than by course t i t l e or the o p i n i o n s of teachers, he rated the courses as modern or t r a d i t i o n a l . He concluded that 50%' of the Algebra 1 courses and 7% of the Geometry courses in h i s sample had a "modern" c u r r i c u l u m . S e v e r a l of these s t u d i e s a l s o addressed questions of methodology. However, in each case "method" was conceived at the l e v e l of general teaching s t r a t e g i e s or procedures. Thus, Alspaugh (1966) and Woods (1973) both concluded that a "show and t e l l " method of i n s t r u c t i o n predominated i n which d i s c u s s i o n of homework was followed by teacher explanation of new ideas which was i n turn followed by the supervised study of new homework. Woods (1973) a l s o i n v e s t i g a t e d the use of team teaching, audio-v i s u a l m a t e r i a l s , and the grouping of students. In none of these s t u d i e s were teachers' methods of i n t e r p r e t i n g concepts or 28 of teaching p r i n c i p l e s or o p e r a t i o n s the subject of i n v e s t i g a t i o n . A few research p r o j e c t s can be i d e n t i f i e d i n which some aspect of the mathematics o p e r a t i o n a l c u r r i c u l u m in secondary schools was st u d i e d , but f o r which the primary focus was not the implementation of "modern mathematics." Neatrour (1969) and Smith, G. A. (1972) are of i n t e r e s t not only f o r t h i s reason but a l s o because they looked at the e a r l i e r secondary grades i n c o n t r a s t to most of the s t u d i e s reviewed here. Neatrour (1969) sought to determine the geometry content included in the middle school c u r r i c u l a in 19 American s t a t e s . To do t h i s he i n v e s t i g a t e d the 16 textbook s e r i e s i n use and a l s o c o l l e c t e d q u e s t i o n n a i r e data using a "scope and sequence form" from 156 teachers. He reported that, at the Grade 8 l e v e l , textbooks devoted an average of 32% of t h e i r pages to geometry content and that 79% of the teachers surveyed i n c l u d e d at l e a s t 26 of 34 s e l e c t e d t o p i c s . He d i d not report the p r o p o r t i o n of i n s t r u c t i o n a l time devoted to geometric t o p i c s however. Among h i s f i n d i n g s was a tendency f o r more geometric t o p i c s to be taught where a p o l i c y of m u l t i p l e textbook adoption e x i s t e d . Smith, G. A. (1972), using q u e s t i o n n a i r e s , addressed the f o l l o w i n g questions i n h i s study of Los Angeles j u n i o r high school teachers: What t o p i c a l content has r e c e n t l y been in use i n seventh grade mathematics classrooms, which of these t o p i c s do teachers see as m e r i t i n g s p e c i a l emphasis, and what t o p i c s do teachers d e s i r e to have included in the c u r r i c u l u m at t h i s l e v e l ? (p. 560A) 29 Among Smith's f i n d i n g s was the c e n t r a l place of the basic operations on whole numbers, decimals, and f r a c t i o n s at t h i s l e v e l as we l l as the common i n c l u s i o n of t o p i c s r e l a t e d to percent, exponential n o t a t i o n , and geometry. Smith a l s o noted a trend toward combining t r a d i t i o n a l t o p i c s with modern approaches in i n s t r u c t i o n although i t i s not c l e a r e x a c t l y what was meant by t h i s . The i n t e r p r e t a t i o n s given by teachers to concepts in t h e i r o p e r a t i o n a l c u r r i c u l a were not i n v e s t i g a t e d i n any of these s t u d i e s ; n e i t h e r were the c o n t e n t - s p e c i f i c methods used to teach f a c t s , o p e r a t i o n s , and p r i n c i p l e s . One g l o b a l c u r r i c u l u m v a r i a b l e was a c e n t r a l focus of many of these s t u d i e s . T h i s v a r i a b l e d e a l t with how up-to-date school programs were and g e n e r a l l y took on only two values — t r a d i t i o n a l or modern. The four c u r r i c u l u m v a r i a b l e s d e f i n e d i n t h i s study were not i n v e s t i g a t e d by any of these re s e a r c h e r s . 1.2 The I n t e r n a t i o n a l Study Of Achievement In Mathematics The f i r s t l a r g e - s c a l e c r o s s - n a t i o n a l e v a l u a t i o n p r o j e c t conducted by IEA d e a l t with secondary school mathematics at two l e v e l s , students 13 years of age and p r e - u n i v e r s i t y s t u d e n t s . 1 1 At age 13 two groups were d e f i n e d and t e s t e d : Population 1a c o n s i s t i n g of a l l p u p i l s 13 years o l d on the t e s t i n g date and Population 1b c o n s i s t i n g of a l l p u p i l s at the grade l e v e l which contained the ma j o r i t y of 13 year o l d s . The p r e - u n i v e r s i t y l e v e l c o n s i s t e d of the grade(s) p r i o r to u n i v e r s i t y from which u n i v e r s i t y students were normally drawn. Population 3a c o n s i s t e d of p u p i l s who were studying mathematics as an i n t e g r a l part of t h e i r program; Population 3b c o n s i s t e d of p u p i l s who were studying; mathematics as a complemantary part of t h e i r program. (Husen, 1967a, p. 46) 30 The o v e r a l l aim of t h i s p r o j e c t , which w i l l be r e f e r r e d to as the F i r s t Mathematics Study, ...was to r e l a t e c e r t a i n s o c i a l , economic and pedagogic c h a r a c t e r i s t i c s of the d i f f e r e n t systems to the outcomes of i n s t r u c t i o n in terms of student achievement and a t t i t u d e s . (Husen, 1975, p. 127) While there were s p e c i f i c reasons f o r the choice of mathematics for IEA's f i r s t l a r g e - s c a l e study, the study was r e a l l y one of e d u c a t i o n a l systems i n general and not school mathematics. IEA regarded mathematics p r i m a r i l y as a surrogate f o r general school achievement and only s e c o n d a r i l y as mathematics per se. I t s analyses were t h e r e f o r e mainly aimed at p r o v i d i n g information f o r p o l i c y makers and hence the chapter on school and system o r g a n i z a t i o n and s o c i a l f a c t o r s . I t could be argued that more a t t e n t i o n should have been given to data of i n t e r e s t to mathematics c u r r i c u l u m w r i t e r s and tea c h e r s . . . ( P o s t l e t h w a i t e , 1972, p. 102) The place of the c u r r i c u l u m i n the F i r s t Mathematics Study and the i m p l i c a t i o n s of t h i s p r o j e c t f o r the c u r r i c u l u m f i e l d have been the su b j e c t s of debate. Bloom (1974) has a s s e r t e d that t h i s and subsequent " f i r s t round" IEA s t u d i e s have documented d i f f e r e n c e s in the " o p p o r t u n i t y - t o - l e a r n " (OTL) c u r r i c u l u m content i n the schools of va r i o u s j u r i s d i c t i o n s and have demonstrated the importance of the OTL v a r i a b l e as a 31 p r e d i c t o r of achievement. 2 Freudenthal (1975), on-the-other-hand, has claimed that c h i e f among the d e f e c t s of the f i r s t round of IEA stu d i e s has been the neglect of the c u r r i c u l u m as a f a c t o r f o r accounting f o r achievement d i f f e r e n c e s between and withi n c o u n t r i e s . 3 C o n t e n t - s p e c i f i c methods were not i n v e s t i g a t e d as such i n the F i r s t Mathematics Study, but an attempt was made to i n v e s t i g a t e the prominence of " i n q u i r y - c e n t e r e d methods" (Husen, 1967b, p. 148) as g e n e r a l l y evidenced in the l e a r n i n g environment of each classroom. To determine the degree of i n q u i r y in each classroom, students were asked to respond to items such as: My mathematics teacher wants p u p i l s to solve problems only by the procedures he teaches. My mathematics teacher r e q u i r e s the p u p i l s not only to master the steps in s o l v i n g problems, but a l s o to understand the reasoning i n v o l v e d . (Husen, 1967a, p. 116) 2 OTL v a r i a b l e s within e d u c a t i o n a l research are, in gene r a l , measures of content i n c l u s i o n w i t h i n i n s t r u c t i o n a l programs. In the F i r s t Mathematics Study OTL was o p e r a t i o n a l l y d e f i n e d for each teacher and each achievement t e s t item based on teacher assessments of the percentage of students who had "had an opportunity to le a r n t h i s type of problem" (Husen, 1967b, p. 167). In the SIMS p r o j e c t teachers were a l s o asked i f the mathematics needed to answer each t e s t item had been taught. 3 Freudenthal (1975) was concerned that OTL might be pe r c e i v e d as an i n d i c a t o r of the importance of mathematics in the o v e r a l l c u r r i c u l u m of each j u r i s d i c t i o n p a r t i c i p a t i n g in the IEA p r o j e c t . He f e l t that OTL r e a l l y measured only the opportunity students had to l e a r n the m a t e r i a l necessary for a p a r t i c u l a r set of t e s t items and that these items might poorly represent the a c t u a l mathematics c u r r i c u l a of many c o u n t r i e s . S p e c i f i c a l l y , he asserted that one-half to two-thirds of the mathematics program of the Netherlands was not represented by the IEA t e s t items (p. 139). 32 Students were not asked about the methods used i n teaching s p e c i f i c mathematical t o p i c s . The report focused on the importance of the i n q u i r y v a r i a b l e as a p r e d i c t o r of achievement w i t h i n and between c o u n t r i e s and provided only summary measures of t h i s v a r i a b l e at the country l e v e l . In the F i r s t Mathematics Study, then, an input-output model of school achievement was used. The o p e r a t i o n a l c u r r i c u l a of teachers were t r e a t e d l a r g e l y as part of the black box of the sc h o o l i n g process and was not the d i r e c t object of i n v e s t i g a t i o n (See K i l p a t r i c k , 1972). 1.3 The NACOME Study C l a r i f i c a t i o n of classroom p r a c t i c e s was part of NACOME's terms of re f e r e n c e . However, only gross i n d i c a t o r s of the o p e r a t i o n a l c u r r i c u l u m were measured i n the survey conducted under the auspices of that group." Examples i n c l u d e d such broad t o p i c areas as p r o b a b i l i t y , s t a t i s t i c s , the metric system, and r e l a t i o n s and f u n c t i o n s , the co p y r i g h t date and number of t e x t s used by teachers i n the sample, the t o t a l time devoted to mathematics i n s t r u c t i o n , and general information on teaching methods (NACOME,;. 1975, pp. 68-78; P r i c e , K e l l e y , & K e l l e y , 1977). KA second l i m i t a t i o n of the NACOME Report was that while i t s d i s c u s s i o n / and c o n c l u s i o n s covered grades K-12, the NACOME was appointed by the Conference Board of the Mathematical Sciences i n May 1974 and was " d i r e c t e d to prepare an overview and a n a l y s i s of U. S. s c h o o l - l e v e l mathematical e d u c a t i o n - - i t s o b j e c t i v e s , c u r r e n t p r a c t i c e s , and attainments." (NACOME, 1975, p. i i i ) 33 e m p i r i c a l survey was r e s t r i c t e d to Grades 2 and 5. In f a c t , the d i s c u s s i o n of the secondary school c u r r i c u l u m i n the NACOME report i s based on the l i s t i n g of course t i t l e s and enrollments provided by a 1972-73 survey as we l l as i n f e r e n c e s drawn from the content of items used by the Na t i o n a l Assessment of Edu c a t i o n a l Progress (NAEP) in t e s t i n g 13 and 17 year o l d s . At the j u n i o r secondary school l e v e l "NACOME found no f i r s t h a n d survey data that i n d i c a t e r e l a t i v e emphasis of new and t r a d i t i o n a l . . . t o p i c s " (NACOME, 1975, p. 9) and r e l i e d s o l e l y on a content a n a l y s i s of items i n the NAEP t e s t b a t t e r y as well as s e v e r a l standardized t e s t b a t t e r i e s . As a b a s i s f o r infer e n c e s regarding what i s a c t u a l l y being taught in classrooms such data are o b v i o u s l y weak e s p e c i a l l y s i n c e "the four most widely used b a t t e r i e s appear to be measuring q u i t e d i f f e r e n t kinds of school programs" (NACOME, 1975, p. 10). The NACOME Report's v a l i d i t y for d e s c r i b i n g a c t u a l classroom c u r r i c u l a was probably l i m i t e d to the grade l e v e l s at which e m p i r i c a l data were a c t u a l l y c o l l e c t e d . 1.4 The N a t i o n a l Science Foundation Studies In 1978 the N a t i o n a l Science Foundation p u b l i s h e d s e v e r a l volumes d e t a i l i n g the r e s u l t s of a study commissioned to i n v e s t i g a t e the l a s t i n g outcomes of the c u r r i c u l u m reform e f f o r t s which NSF had sponsored in science and mathematics education during the previous two decades (NSF, 1978, p r e f a c e ) . The NSF study c o n s i s t e d of three separate sub-studies: 34 (1) Three l i t e r a t u r e reviews of research and h i s t o r i c a l data c o v e r i n g 1955-1975—one each fo r science education (Helgeson, B l o s s e r , & Howe, 1977), mathematics education (Suydam & Osborne, 1977), and s o c i a l s t u d i e s education (Wiley & Race, 1978), (2) A n a t i o n a l survey of course o f f e r i n g s , enrollments and p r a c t i c e s (Weiss, 1978), (3) Eleven ethnographic case s t u d i e s conducted at school s i t e s throughout the U. S. (Stake & Easley, 1978a, 1978b). The l i t e r a t u r e review in mathematics education addressed the q u e s t i o n : What were and are c u r r e n t p r a c t i c e s in mathematics education for c u r r i c u l u m , i n s t r u c t i o n , teacher education, performance of l e a r n e r s , and needs assessments during the twenty-year p e r i o d beginning i n 1955? (Suydam & Osborne, 1977, p. 3) The researchers who conducted t h i s extensive l i t e r a t u r e review a l s o sought to determine the extent to which information about teaching p r a c t i c e s had been u t i l i z e d by decision-makers during t h i s p e r i o d . The f i n d i n g s of what teachers teach and how they teach i t are not impressive: I t comes as a s u r p r i s e to most people that there are a c t u a l l y r e l a t i v e l y few s t u d i e s which d e s c r i b e the a c t u a l classroom s i t u a t i o n . . . . In most s t u d i e s i n the classroom, the s e t t i n g i s d e s c r i b e d only g e n e r a l l y . Comparisons are made with the " t r a d i t i o n a l " o r - " u s u a l " classroom, as i f everyone knew p r e c i s e l y what that was. (p. 54) 35 The s p e c i f i c f i n d i n g s that Suydam and Osborne c i t e d d e a l t with classroom management and general methods rather than with what content was taught and how i t was taught. Examples are the p r o p o r t i o n of teacher t a l k i n the average classroom, the amount of time spent on managerial d u t i e s , the pace of i n s t r u c t i o n , and the types of i n s t r u c t i o n a l m a t e r i a l s u t i l i z e d (pp. 54-56). The second part of the NSF study, a n a t i o n a l survey, was conducted by the Research T r i a n g l e I n s t i t u t e and reported by Weiss (1978). I t d i d not provide s i g n i f i c a n t new information regarding the o p e r a t i o n a l c u r r i c u l u m . At the l e v e l of the formal c u r r i c u l u m data were c o l l e c t e d regarding courses o f f e r e d and t h e i r enrollments, textbooks used and, s p e c i f i c a l l y , NSF m a t e r i a l s used. Teachers were asked about t h e i r use of general methods such as l e c t u r e s , d i s c u s s i o n s and l e a r n i n g c o n t r a c t s , and about a u d i o - v i s u a l and other equipment such as overhead p r o j e c t o r s , games and p u z z l e s , and c a l c u l a t o r s . Teachers were a l s o asked for the names of the courses they taught and the textbooks they used. Teachers were not aske'd anything about the content they taught except course t i t l e s . The t h i r d component of the NSF p r o j e c t , eleven ethnographic case s t u d i e s , was d i r e c t e d by Robert Stake and Jack Easley of the U n i v e r s i t y of I l l i n o i s . In the s y n t h e s i s of these case s t u d i e s , Stake and Easley (1978b) contended that s o c i a l i z a t i o n of students was an o v e r r i d i n g goal i n the classrooms observed and that subject matter i t s e l f was used f o r aims of c l a s s 36 c o n t r o l . 5 They noted that "subject matter that d i d not f i t these aims got r e j e c t e d , neglected, or changed i n t o 'something that worked'" (p. 19:5). The observers d i d not, however, c o l l e c t d e t a i l e d information on the approaches or s t r a t e g i e s used to teach mathematics. In f a c t , no uniform system of observations was employed across s i t e s . While s e v e r a l i l l u m i n a t i n g examples of mathematics content being used f o r s o c i a l i z a t i o n were reported (Stake & Easley, 1978b, chapter 16), the contention that mathematics content i t s e l f i s s e l e c t e d or d i s t o r t e d so that i t serves as a v e h i c l e f o r s o c i a l i z a t i o n was not c o n v i n c i n g l y supported or elaborated by t h i s study (House & T a y l o r , 1978, pp. 11-13). I t remains an i n t e r e s t i n g hypothesis regarding the o p e r a t i o n a l c u r r i c u l u m , p a r t i c u l a r l y s i n c e r u l e s are a conspicuous part of mathematics as noted above, but one r e q u i r i n g more s p e c i f i c , comprehensive data f o r c o n f i r m a t i o n . 1.5 Studies Conducted Within B r i t i s h Columbia Both the 1977 and 1981 B r i t i s h Columbia P r o v i n c i a l Mathematics Assessments i n c l u d e d a component in which information was gathered from teachers regarding v a r i o u s aspects of t h e i r backgrounds, b e l i e f s , and classroom p r a c t i c e s . One of the volumes produced as a r e s u l t of the 1977 Assessment d e a l t e x c l u s i v e l y with what the authors termed " i n s t r u c t i o n a l p r a c t i c e s " ( R o b i t a i l l e & S h e r r i l l , 1977). The survey reported 5 By " s o c i a l i z a t i o n of students" Stake and Easley appear to mean the process whereby students come to accept the s o c i a l norms of the school such as p e r s i s t e n t l y t r y i n g one's best. (See Stake and E a s l e y , 1978b, pp. 16:5 and 16:13.) 37 in that volume was s i m i l a r to the NACOME study but represented an advance as a comprehensive source of knowledge regarding mathematics teaching p r a c t i c e s in s e v e r a l r e s p e c t s : (1) A more r e p r e s e n t a t i v e sample was s p e c i f i e d and a higher r e t u r n r a t e achieved, (2) Data were c o l l e c t e d at seven grade l e v e l s : 1, 3, 5, 7, 8, 10, and 12, (3) Questions were asked regarding the i n c l u s i o n of more s p e c i f i c t o p i c s . In reference to the o p e r a t i o n a l c u r r i c u l u m at the secondary l e v e l , teachers were asked to c a t e g o r i z e l e a r n i n g outcomes on a f i v e point s c a l e of "not important" to "very important." R o b i t a i l l e (1980) summarized the r e s u l t s : On the assessment q u e s t i o n n a i r e teachers were asked to rank order a l i s t of some 20 o b j e c t i v e s s e l e c t e d from the t o t a l l i s t of o b j e c t i v e s f o r t h e i r grade, as p u b l i s h e d in the o f f i c i a l c u r r i c u l u m guide. Four of the f i v e o b j e c t i v e s rated most important by grade 8 teachers d e a l t with the computational s k i l l s of a r i t h m e t i c ; the f i f t h d e a l t with the a b i l i t y to solve problems i n v o l v i n g percentages. A l l but one of the geometry o b j e c t i v e s were ranked among the lowest t h i r d of the e n t i r e l i s t , and the only o b j e c t i v e in the l i s t which e x p l i c i t l y mentioned the term " s e t s " was ranked l a s t , i . e . , l e a s t important. A s i m i l a r r e s u l t was found among teachers of grade 10 mathematics. The three highest-ranked o b j e c t i v e s at that l e v e l concerned computational s k i l l s . A l l of the o b j e c t i v e s d e a l i n g i n any way with "new mathematics" were given low rankings of importance r e l a t i v e to the others, (p. 302) The o p e r a t i o n a l c u r r i c u l u m in B. C. has only been p a r t l y 38 s p e c i f i e d by t h i s survey, however. While teachers were asked to rank order l e a r n i n g o b j e c t i v e s which involved s p e c i f i c content, they were not asked i f they included corresponding i n s r u c t i o n or other l e a r n i n g a c t i v i t i e s i n t h e i r c u r r i c u l a . F urther, while the l e a r n i n g o b j e c t i v e s encompassed many of the concepts, o p e r a t i o n s , and p r i n c i p l e s which might be taught at t h i s l e v e l , the c o n t e n t - s p e c i f i c methods used by teachers were not i n v e s t i g a t e d . A second study which i n v e s t i g a t e d the o p e r a t i o n a l mathematics c u r r i c u l u m of B r i t i s h Columbia teachers was the B. C. component of the IEA SIMS p r o j e c t which was conducted d u r i n g the 1980-81 school year. T h i s p r o j e c t , i n which Grades 8 and 12 were i n v e s t i g a t e d , i s of note both f o r i t s l o n g i t u d i n a l design and f o r the scope and d e t a i l of the data c o l l e c t e d . At each l e v e l approximately 100 teachers were asked what content they taught both at the l e v e l of major t o p i c areas and at the l e v e l of s p e c i f i c concepts, o p e r a t i o n s , and p r i n c i p l e s , as w e l l as the amount of time spent on that content. At the Grade 8 l e v e l .teachers were a l s o asked what c o n t e n t - s p e c i f i c methods they used in t h e i r p r e s e n t a t i o n s . The f i n d i n g s , which are repor t e d in R o b i t a i l l e , O'Shea, and Di r k s (1982), included the frequency with which teachers used v a r i o u s i n t e r p r e t a t i o n s pf such mathematical ideas as i n t e g e r s and the Pythagorean theorem and the number of c l a s s p eriods a l l o c a t e d to content areas and s p e c i f i c t o p i c s . As the p r i n c i p a l author noted i n the preface to the re p o r t , however, many other s t u d i e s might be conducted using t h i s r i c h data source s i n c e only a l i m i t e d amount of data 39 a n a l y s i s had as yet been p o s s i b l e . For example, s i n c e each of the teacher q u e s t i o n n a i r e s was analyzed s e p a r a t e l y , i t was not p o s s i b l e to i n v e s t i g a t e each teacher's c u r r i c u l u m - i n - u s e comprehensively in terms of the cu r r i c u l u m v a r i a b l e s d i s c u s s e d i n Chapter 1 of t h i s d i s s e r t a t i o n . One r e - a n a l y s i s of the B. C. SIMS data has been completed. Tarn (1983) was i n t e r e s t e d i n the mode of r e p r e s e n t a t i o n Mathematics 8 teachers used i n pr e s e n t i n g content and used the methods of Exp l o r a t o r y Data A n a l y s i s in her study. She concluded that teachers p r e f e r r e d a b s t r a c t approaches over concrete approaches f o r most t o p i c s . An i n s p e c t i o n of the box p l o t s included i n her report i n d i c a t e s c o n s i d e r a b l e v a r i a t i o n among teachers i n t h e i r p r e f e r e n c e s , however. P o s s i b l e a s s o c i a t i o n s between mode of r e p r e s e n t a t i o n and other f a c t o r s such as c l a s s achievement were not explored i n her study. 2. THE USE OF QUESTIONNAIRES IN RESEARCH The o p e r a t i o n a l c u r r i c u l u m of Mathematics 8 teachers in B. C. was i n v e s t i g a t e d in t h i s study by r e a n a l y z i n g data c o l l e c t e d as part of the B. C. SIMS p r o j e c t . These data i n c l u d e d q u e s t i o n n a i r e s e l f - r e p o r t s of content taught, time a l l o c a t e d to content, and c o n t e n t - s p e c i f i c methods used in tea c h i n g . A r e l i a n c e on s e l f - r e p o r t s was r e q u i r e d s i n c e a reasonably comprehensive study of even a s i n g l e teacher's o p e r a t i o n a l c u r r i c u l u m would r e q u i r e nearly a f u l l school year. D i r e c t observations would have l i m i t e d the study to two or three teachers at most, p r e c l u d i n g the survey scope necessary to 40 address the research q u e s t i o n s . S e l f - r e p o r t s from m u l t i p l e i n t e r v i e w s c o u l d have been conducted with perhaps ei g h t or ten teachers. T h i s sample s i z e , however, would s t i l l have been inadequate p a r t i c u l a r l y f o r e x p l o r i n g a s s o c i a t i o n s between the c u r r i c u l u m and c l a s s achievement l e v e l . Furthermore, the l i t e r a t u r e does not suggest gr e a t e r v a l i d i t y f o r intervi e w data in c o n t r a s t to q u e s t i o n n a i r e data for questions of the type asked in t h i s study (Conger, Conger, & Riccobono, 1976). By using q u e s t i o n n a i r e s , d e t a i l e d c u r r i c u l u m information was c o l l e c t e d at f i v e p o i n t s in time from nearly 100 teachers over the course of a.school year. Surveys w i t h i n e d u c a t i o n a l research and d e s c r i p t i v e surveys using q u e s t i o n n a i r e data have been c r i t i c i z e d by some commentators. 6 Mouly (1978), in f a c t , has a s s e r t e d that "probably no instrument of research has been more subject to censure than the q u e s t i o n n a i r e " (p. 188). 7 In a d d i t i o n , the v a l i d i t y of teacher r e p o r t s of t h e i r own behavior has been c a l l e d i n t o question in one review (Hook & Rosenshine, 1979). 6 Sieber (1968) has documented a perv a s i v e c r i t i c a l stance by the authors of e d u c a t i o n a l research textbooks and other commentators from the beginning of t h i s century to survey methods. He a s c r i b e d t h i s a t t i t u d e to two f a c t o r s : (1) the i d e n t i f i c a t i o n of the q u e s t i o n n a i r e with the "school survey," which i s a s e r v i c e rather than a research o p e r a t i o n ; and (2) the predominance of p s y c h o l o g i c a l approaches in ed u c a t i o n a l research. (p. 275) 7 While there may be a bias a g a i n s t survey research w i t h i n education ( H e f r i o t t , 1969, p. 1400), t h i s methodology i s the dominant research form wi t h i n the s o c i a l sciences (Orenstein & P h i l l i p s , 1978, p. 170) having achieved t h i s s t a t u s s i n c e the end of the Second World War ( S z a l a i & P e t r e l l a , 1977, p. i x ) . 41 Because of these c r i t i c i s m s , the t h e o r e t i c a l l i t e r a t u r e on the v a l i d i t y of q u e s t i o n n a i r e surveys i s d i s c u s s e d below. Also reviewed i s the research l i t e r a t u r e concerning q u e s t i o n n a i r e v a l i d i t y which i s relevant to the instrumentation and design of the B. C. SIMS p r o j e c t . The v a l i d i t y s t u d i e s which were conducted as part of the SIMS p r o j e c t are d i s c u s s e d in Chapter 3. 8 2.1 Questionnaire Research: T h e o r e t i c a l V a l i d i t y P r i n c i p l e s A q u e s t i o n n a i r e item i s v a l i d i n s o f a r as i t e l i c i t s from a respondent the information intended by the i n v e s t i g a t o r . T h i s i m p l i e s that response d i f f e r e n c e s between i n d i v i d u a l s represent true d i f f e r e n c e s of o p i n i o n , behavior, or other c h a r a c t e r i s t i c s of the respondents who are being s t u d i e d (Berdie & Anderson, 1974). The r e s u l t s of a q u e s t i o n n a i r e survey are v a l i d only i f i n d i v i d u a l items have e l i c i t e d the intended information and an adequate return r a t e of q u e s t i o n n a i r e s has been achieved. Several t h r e a t s to v a l i d i t y i n research using q u e s t i o n n a i r e s have been i d e n t i f i e d . If a q u e s t i o n n a i r e item does not communicate the meaning intended by the framer of the item, the v a l i d i t y of the response i s in doubt. Further, i f the 8 Questions of v a l i d i t y occur i n every area of educational r e s e a r c h . They are not r e s t r i c t e d to surveys using q u e s t i o n n a i r e s . Standardized t e s t s , for example, may not be v a l i d i n d i c a t o r s of student understanding, achievement, or a b i l i t y (Davis & S i l v e r , 1982; K r u t e t s k i i , 1976. p. 13). Rigorous, r e p l i c a b l e l a b o r a t o r y experiments may lack e c o l o g i c a l v a l i d i t y (Cole, Hood, & McDermott., 1979, p. 2). Observational research may produce i n v a l i d measures (Rowan, Bossert, & Dwyer, 1983, p. 25) and may s u f f e r from observer s u b j e c t i v i t y and hence lack v a l i d i t y (House & T a y l o r , 1978, pp. 11-13). 42 respondent i s e i t h e r unable or u n w i l l i n g to provide the information s o l i c i t e d , v a l i d i t y i s threatened. Even i f the information r e c e i v e d using q u e s t i o n n a i r e s i s v a l i d , the v a l i d i t y of the survey i t s e l f i s i n doubt i f , as noted above, the response rate has been low si n c e nonrespondents may d i f f e r from respondents in some systematic way (Mouly, 1978, pp. 189-190.) The l i t e r a t u r e d e a l i n g with q u e s t i o n n a i r e v a l i d i t y i s not exten s i v e . Because the response rate to surveys conducted by mail has t y p i c a l l y been low, 9 much of t h i s l i t e r a t u r e has focused on s t r a t e g i e s f o r maximizing the return of q u e s t i o n n a i r e s rather than other issues concerning v a l i d i t y . R e f l e c t i n g the preoccupation with response' r a t e , Nisbet and E n t w i s t l e (1970, p. 44) a s s e r t e d t h a t : "the percentage response i s the most important c o n s i d e r a t i o n in e v a l u a t i n g a qu e s t i o n n a i r e study." S i m i l a r l y , Mouly (1978, p. 189) and H e r r i o t t (1969, p. 1402) l i s t the problem of nonreturns as the primary l i m i t a t i o n of q u e s t i o n n a i r e surveys. An examination of e d u c a t i o n a l research t e x t s (e.g., Ary, Jacobs, & Razavich, 1979; Best, 1981; BOrg & G a l l , 1979; Cohen & Manion, 1979; Mouly, 1978; Nisbet & E n t w i s t l e , 1970) and those few r e f e r e n c e s devoted p r i m a r i l y to q u e s t i o n n a i r e c o n s t r u c t i o n and survey design (e.g., Berdie & Anderson, 1974; Dillman, 1978; Hyman, 1955) shows c o n s i d e r a b l e emphasis on p r i n c i p l e s r e l a t e d to i n c r e a s i n g q u e s t i o n n a i r e r e t u r n s . Dillman (1978), f o r example, reviewed 16 g u i d e l i n e s that had been o f f e r e d i n the 9 Rates below 50 percent have of t e n been considered acceptable (Dillman, 1978 p. 2; H e r r i o t t , 1969, p. 6). 43 past to improve response rate and then e x p l i c a t e d h i s own comprehensive system. Most authors of survey references do provide to some degree, however, c r i t e r i a f o r designing surveys and c o n s t r u c t i n g q u e s t i o n n a i r e items so that the information gathered w i l l correspond to a c t u a l a t t i t u d e s , p e r c e p t i o n s , or behaviors. The c r i t e r i a given t y p i c a l l y d e al with both the issues of respondent a b i l i t y and w i l l i n g n e s s to provide information, although these c a t e g o r i e s are not always e x p l i c i t l y noted. Respondent a b i l i t y to answer questions i s a f u n c t i o n of both the knowledge of the respondents themselves and the questions being asked. Several p r i n c i p l e s and cautions are provided in the l i t e r a t u r e which dea l s with question c o n s t r u c t i o n . Borg and G a l l (1979, p. 297), f o r example, recommended the use of c l e a r l y w r i t t e n , b r i e f l y s t a t e d items which are focused on a s i n g l e idea and which avoid t e c h n i c a l language. Berdie and Anderson (1974, pp. 36-48) advocated the use of items which are unambiguous, s e l f - e x p l a n a t o r y and which communicate something s p e c i f i c and r e q u i r e a minimum of " p u z z l i n g out." In a d d i t i o n , they noted that adequate response options must be provided. Lack of ambiguity r e q u i r e s that each item be meaningful to the respondents, that only one b a s i c meaning be a s c r i b e d by a l l respondents, and that t h i s be the meaning intended by the researcher (Berdie & Anderson, 1974; Orenstein & P h i l l i p s , 1978). For items i n v o l v i n g s e l f - r e p o r t s of s t a t u s or behavior, s p e c i f i c i t y i s p a r t i c u l a r l y important (Orenstein & P h i l l i p s , 1978, pp. 218-219). Questionnaire items 44 which r e q u i r e a high degree of in f e r e n c e and i n t e r p r e t a t i o n by respondents would be p a r t i c u l a r l y suspect a c c o r d i n g to the foregoing standards. Obviously, a respondent i s unable to provide a v a l i d response to a q u e s t i o n n a i r e item i f he or she does not possess the information being s o l i c i t e d . This would be the case i f f a c t u a l knowledge has been fo r g o t t e n by the respondent or i f he or she has no opini o n about a p a r t i c u l a r i s s u e . A respondent would a l s o be unable to answer questions about h i s or her behavior or environment i f the questions d e a l t with phenomena beyond h i s or her a b i l i t y to p e r c e i v e . A teacher, f o r example, might be able to pe r c e i v e c e r t a i n aspects of h i s or her behavior i n the classroom and of the classroom l e a r n i n g environment but might be unable to pe r c e i v e other aspects (Fraser, 1982). Regarding w i l l i n g n e s s of respondents to answer questions a c c u r a t e l y , apart from a basic w i l l i n g n e s s to p a r t i c i p a t e and return the q u e s t i o n n a i r e , the authors of the t h e o r e t i c a l l i t e r a t u r e are q u i t e c o n s i s t e n t i n admonishing that biased, loaded, or emotional language w i l l cause inacurate responses to qu e s t i o n n a i r e items. The comments of Borg and G a l l (1979) are t y p i c a l : . . . i t i s very important that an e f f o r t be made to avoid biased or le a d i n g q u e s t i o n s . If the subject i s given h i n t s as to the type of answer you would most p r e f e r , there i s some tendency to give you what you want. (p. 297) Berdie and Anderson (1974) note f u r t h e r that many respondents may be u n w i l l i n g to respond to h y p o t h e t i c a l questions as well as what they term "why" questions, i . e . , questions which are 45 w r i t t e n to s o l i c i t reasons f o r a t t i t u d e s or behavior. Dillman (1978), in d i s c u s s i n g h i s " T o t a l Design Method" f o r survey research, o f f e r s two other t h e o r e t i c a l g u i d e l i n e s f o r maximizing respondent w i l l i n g n e s s to answer questions a c c u r a t e l y . F i r s t , "contamination by oth e r s " should be avoided. For example, teachers or students may c o l l a b o r a t e in completing q u e s t i o n n a i r e s i f the s i t u a t i o n permits and produce r e s u l t s that are not v a l i d f o r a l l respondents. Secondly, Dillman (1978) advocates the a v a i l a b i l i t y of c o n s u l t a t i o n and follow-up s e r v i c e s to p a r t i c i p a n t s of surveys to increase t h e i r i n t e r e s t and motivation and thereby increase the v a l i d i t y of responses as well as the rate of q u e s t i o n n a i r e r e t u r n s . 2.2 Questionnaire Research: E m p i r i c a l V a l i d i t y Studies The number of e m p i r i c a l s t u d i e s which have i n v e s t i g a t e d the v a l i d i t y of q u e s t i o n n a i r e data i s not large and those s t u d i e s which have been conducted have not produced a comprehensive set of p r i n c i p l e s f o r the c o n s t r u c t i o n of q u e s t i o n n a i r e items. In f a c t , c o n t r a d i c t o r y c o n c l u s i o n s have been reached in these s t u d i e s as to the general v a l i d i t y of surveys using q u e s t i o n n a i r e s (Berdie & Anderson, 1974; Walsh, 1968). T y p i c a l l y q u e s t i o n n a i r e s have been used i n e d u c a t i o n a l research when other methodologies such as observations or int e r v i e w s have not been f e a s i b l e due to high c o s t , or when a concern has e x i s t e d that other methods might have r e a c t i v e e f f e c t s such as the presence of an observer changing the patterns of behavior of teachers or students. Thus, i t i s probably not s u r p r i s i n g that 46 Berdie and Anderson (1974, p. 20) have noted that "owing to the nature of q u e s t i o n n a i r e s , the ways to check the r e l i a b i l i t y and v a l i d i t y of q u e s t i o n n a i r e items are l i m i t e d . " In those cases where v a l i d i t y s t u d i e s have been conducted they have u s u a l l y been adjuncts to s t u d i e s designed f o r other purposes. For t h i s reason Berdie and Anderson (1974) cautioned against a c c e p t i n g g e n e r a l i z a t i o n s about q u e s t i o n n a i r e v a l i d i t y that go beyond the s p e c i f i c instruments which were used i n the s t u d i e s . In s e v e r a l s t u d i e s the v a l i d i t y of teachers' responses to q u e s t i o n n a i r e items has been the subject of i n v e s t i g a t i o n . The r e s u l t s l a r g e l y support the p r i n c i p l e s which were d i s c u s s e d i n the previous s e c t i o n of t h i s chapter. In p a r t i c u l a r , the responses to q u e s t i o n n a i r e s have shown the l e a s t v a l i d i t y when there has been reason to doubt e i t h e r the a b i l i t y 'or w i l l i n g n e s s of the s u b j e c t s to respond. As part of the N a t i o n a l L o n g i t u d i n a l Study of the C l a s s of 1972, Conger, Conger, and Riccobono (1976) reviewed the l i t e r a t u r e on the v a l i d i t y and r e l i a b i l i t y of survey research q u e s t i o n n a i r e s . They were concerned i n p a r t i c u l a r with how data c o l l e c t i o n procedures, item c h a r a c t e r i s t i c s , respondent c h a r a c t e r i s t i c s , and i n t e r a c t i o n s among these f a c t o r s might i n f l u e n c e r e l i a b i l i t y and v a l i d i t y . On the b a s i s of t h e i r l i t e r a t u r e review they concluded: Demographic c h a r a c t e r i s t i c s and f a c t u a l information about present behavior y i e l d the highest v a l i d i t y (and 47 r e l i a b i l i t y ) c o e f f i c i e n t s , 1 0 r e s p e c t i v e l y . F a c t u a l information on past behavior and e v a l u a t i v e or judgmental behavior y i e l d s the l e a s t s t a b l e data, with the l a t t e r r e p r e s e n t i n g the lowest response s t a b i l i t y . Furthermore, v a l i d i t y of r e ports of past behavior may be moderated by the importance of the accomplishment. That i s , past events which have low ambiguity and are s i g n i f i c a n t to the respondent in terms of accomplishment...tend to have high r a t e s of v a l i d i t y , (p. 10) Further in t h e i r report Conger et a l . (1976) s t a t e d even more emphatically t h a t : The l i t e r a t u r e and r e l i a b i l i t y study are u n e q u i v o c a l l y c o n s i s t e n t in the f i n d i n g that contemporaneous, o b j e c t i v e , f a c t u a l l y o r i e n t e d items are more r e l i a b l e than s u b j e c t i v e , temporally remote, or ambiguous items. The v a l i d i t y items are s i m i l a r l y c o n s i s t e n t , (p. 31) T h i s a s s e r t i o n , based on a review of e m p i r i c a l research, provides g u i d e l i n e s which are q u i t e s i m i l a r to some of the t y p i c a l t h e o r e t i c a l admonitions d i s c u s s e d above, namely, that questions should be s p e c i f i c and unambiguous. What i s of p a r t i c u l a r i n t e r e s t i s the c o n c l u s i o n by Conger et a l . (1976), c i t e d above, that q u e s t i o n n a i r e s can be used for b e h a v i o r a l s e l f - r e p o r t s , in cases where the items are designed to s o l i c i t i nformation about behavior which i s both reasonably recent and reasonably important to the respondent. 1 0 Conger, Conger, and Riccobono (1976) do not e x p l i c i t l y d e f i n e high, moderate, and low l e v e l s of r e l i a b i l i t y and v a l i d i t y in t h e i r d i s c u s s i o n s perhaps due to "the v a r i e t y of i n d i c e s used to summarize the r e s u l t s " of s t u d i e s i n v e s t i g a t i n g the r e l i a b i l i t y and v a l i d i t y of survey data (p. 4). However, they do report c o r r e l a t i o n c o e f f i c i e n t s , and at one point i n d i c a t e that "moderate to moderately high c o e f f i c i e n t s " are in the 0.70 < r < 0.88 range (p. 10). 48 Hook and Rosenshine (1979) reviewed nine s t u d i e s i n which q u e s t i o n n a i r e v a l i d i t y was i n v e s t i g a t e d i n con j u n c t i o n with research on teac h i n g . The c o n c l u s i o n s they reached were mixed and o f f e r e d only l i m i t e d support f o r the use of q u e s t i o n n a i r e s in t h i s area of research. . . . i f a teacher answers a q u e s t i o n n a i r e on a v a r i e t y of s p e c i f i c a c t i v i t i e s , we cannot assume that these r e p o r t s correspond to a c t u a l p r a c t i c e . . . . o n e i s not advised to accept teacher reports of s p e c i f i c behaviors as p a r t i c u l a r l y a ccurate. No s l u r i s intended; teachers do not have p r a c t i c e i n e s t i m a t i n g t h e i r behavior and then checking a g a i n s t a c t u a l performance. There appears to be some value i n teacher r e p o r t s when behaviors are grouped i n t o dimensions, but one has no way of knowing, a p r i o r i , which dimensions w i l l c o r r e l a t e with a c t u a l p r a c t i c e . F i n a l l y , based on the two a v a i l a b l e s t u d i e s on t h i s t o p i c , teacher r e p o r t s used to c l a s s i f y teachers on a continuum such as t r a d i t i o n a l or i n f o r m a l , appear to be trustworthy. (pp. 9-10, emphasis i n o r i g i n a l ) In the s t u d i e s reviewed by Hook and Rosenshine, teachers' responses to q u e s t i o n n a i r e items were inaccurate when observers coded t h e i r p e rceptions of the teachers' behaviors using the Flanders I n t e r a c t i o n A n a l y s i s System, when the q u e s t i o n n a i r e items contained a bias towards p r e f e r r e d responses or when both c o n d i t i o n s were, present. In s t u d i e s reported by Ehman (1970), Johnson, D. P. (1969), and S t e e l e , House, and Kerins (1971), the Flanders system was used, and i n each case i t was found that teachers could/not a c c u r a t e l y estimate the percent of c l a s s time spent i n teacher t a l k when t h e i r q u e s t i o n n a i r e responses were compared to the frequency count data of observers. I t may be, however, that the teachers i n these s t u d i e s were unable to 49 respond a c c u r a t e l y because they could not p e r c e i v e the " m u l t i p l i c i t y of molecular events," as Walberg and H a e r t e l (1980, p. 232) c h a r a c t e r i z e d them, which made up the c l a s s d i s c u s s i o n and l e c t u r e time. It should be noted that in research on teaching the term " s p e c i f i c behavior" has u s u a l l y been used to r e f e r to the very short events which are coded using i n t e r a c t i o n a n a l y s i s systems and which may exceed the a b i l i t y of teachers to p e r c e i v e or r e c a l l . Thus, the q u e s t i o n n a i r e items used i n research on teaching which are s p e c i f i c i n t h i s sense of the word have tended to produce i n v a l i d responses, while items which are s p e c i f i c i n the more usual sense of the word have tended to produce v a l i d responses in other s t u d i e s . In s e v e r a l of the s t u d i e s reviewed by Hook and Rosenshine the w i l l i n g n e s s of the respondents to provide accurate answers can be questioned because of item b i a s . For example, in the study conducted by Goodlad and K l e i n (1970) teachers tended to over estimate t h e i r use of innovative teaching techniques. Squire and Applebee (1966) found that teachers over; estimated t h e i r use of S o c r a t i c q u e s t i o n i n g and underestimated the time they spent l e c t u r i n g . In both cases i t can be argued that c e r t a i n response options were perceived by teachers as p r e f e r r e d by the r e s e a r c h e r s . In other studies the responses of teachers to q u e s t i o n n a i r e items about t h e i r c u r r i c u l a and classroom p r a c t i c e s have been c o n s i s t e n t with the judgement of observers. For example, in a study i n v o l v i n g 37 teachers, Bennett (1976) found agreement 5 0 between observers and teachers on the teaching s t y l e used in the classroom. Teaching s t y l e was measured by asking s p e c i f i c questions such as "Do you put an a c t u a l mark or grade on p u p i l s ' work?" of both teachers and observers (Bennett, 1976, p 168). Kazarian (1978) and M a r l i a v e , F i s h e r , and F i l b y (1977) found that teachers were able to estimate and w i l l i n g to report the amount of time a l l o c a t e d to i n s t r u c t i o n a l a c t i v i t i e s at l e v e l s judged acceptable by the r e s e a r c h e r s . Kazarian (1978) used agreement of teachers' q u e s t i o n n a i r e responses with teachers' interview responses as the c r i t e r i o n f o r q u e s t i o n n a i r e v a l i d i t y . M a r l i a v e et a l . (1977), however, v a l i d a t e d the responses of teachers to q u e s t i o n n a i r e items by comparing them to classroom o b s e r v a t i o n s . The focus of Hardebeck's (1974) d o c t o r a l study was the v a l i d i t y of t e a c h e r s ' ; q u e s t i o n n a i r e r e p o r t s of i n d i v i d u a l i z a t i o n of i n s t r u c t i o n . Classroom observations were used to v a l i d a t e s e l f - r e p o r t s . While teachers who were observed to do l i t t l e i n d i v i d u a l i z i n g d i d tend to over-report t h e i r i n d i v i d u a l i z i n g behavior, the r e l a t i o n s h i p between observations and s e l f - r e p o r t s was strong enough "so as to permit d e s c r i b i n g s e l f - r e p o r t e d teacher p r a c t i c e s of f i v e aspects of i n d i v i d u a l i z a t i o n of i n s t r u c t i o n as being high, medium, or low" (p. 126A). In the s t u d i e s reviewed above, q u e s t i o n n a i r e items sometimes e l i c i t e d accurate information from respondents according to c r i t e r i a e s t a b l i s h e d by the researcher. Sometimes they d i d not. As Berdie and Anderson (1974) noted: 51 ...the c o n t r a d i c t o r y r e p o r t s concerning q u e s t i o n n a i r e methods are not s u r p r i s i n g , as they are based on r e s u l t s from d i f f e r e n t q u e s t i o n n a i r e s used for d i f f e r e n t reasons with d i f f e r e n t people at d i f f e r e n t times. (p. 12) Questionnaire data obtained from teachers showed the l e a s t v a l i d i t y in those s t u d i e s in which they were asked to evaluate the percent of c l a s s time they t a l k e d (e.g., S t e e l e et a l . , 1971), t h e i r degree of openness (e.g., Ehman, 1970), t h e i r use of i n n o v a t i v e methods (e.g., Goodlad & K l e i n , 1970), and the l i k e . In these cases e i t h e r the a b i l i t y of teachers to perc e i v e the behaviors r e q u i r e d to answer the questions or t h e i r w i l l i n g n e s s to appear out of touch with the l a t e s t e d u c a t i o n a l fashion can be questioned. In cases where teachers were asked n e u t r a l questions about s p e c i f i c , yet p e r c e i v a b l e classroom p r a c t i c e s , the v a l i d i t y of q u e s t i o n n a i r e resonses was much be t t e r (e.g., Conger et a l . , 1976; Bennett, 1976; Mar l i a v e et a l . , 1977). These r e s u l t s provide support f o r the v a l i d i t y of the type of items i n c o r p o r a t e d in the q u e s t i o n n a i r e s used in the B. C. SIMS: p r o j e c t . The l i t e r a t u r e a l s o supports the p o s i t i o n that any study in which q u e s t i o n n a i r e s are used needs to have a component i n which item v a l i d i t y i s i n v e s t i g a t e d . Several v a l i d i t y s t u d i e s were conducted i n co n j u n c t i o n with the SIMS p r o j e c t . These are discus s e d i n Chapter 3. 52 I I I . RESEARCH DESIGN AND PROCEDURES The methodology chapter has been d i v i d e d i n t o f i v e major s e c t i o n s . In the f i r s t s e c t i o n , aspects of the B. C. SIMS p r o j e c t , the data source f o r t h i s study, are d i s c u s s e d . P a r t i c u l a r a t t e n t i o n i s given to the basic design of that study, the nature of the sample, and a d e s c r i p t i o n of the instruments used as we l l as t h e i r development and v a l i d a t i o n . The next three s e c t i o n s contain a d i s c u s s i o n of the content areas and t o p i c s , the c u r r i c u l u m v a r i a b l e s , and the c o n t e x t u a l v a r i a b l e r e s p e c t i v e l y which were incorporated in t h i s study. In the f i n a l s e c t i o n the data a n a l y s i s s t r a t e g y , u t i l i z i n g techniques of E x p l o r a t o r y Data A n a l y s i s (Tukey, 1977), i s presented. 1. THE B. C. SIMS PROJECT As noted e a r l i e r , B r i t i s h Columbia was one of the p a r t i c i p a n t s 1 in the SIMS p r o j e c t , the second survey of school mathematics organized by IEA. The B. C. SIMS r e p o r t , ( R o b i t a i l l e , O'Shea, & D i r k s , 1982), d e s c r i b e d the context of B. C. involvement: B r i t i s h Columbia's p a r t i c i p a t i o n i n the Second I n t e r n a t i o n a l Study of Mathematics was sponsored by the Learning A s s i s t a n c e Branch of the B. C. M i n i s t r y of Education. The p r o j e c t was undertaken as an adjunct to the 1981 Mathematics Assessment which was conducted during the same school year. P a r t i c i p a t i o n in the i n t e r n a t i o n a l study provided an opportunity to 1 Over 20 j u r i s d i c t i o n s took part i n the SIMS p r o j e c t . Most of these j u r i s d i c t i o n s were c o u n t r i e s , but there were exceptions such as B r i t i s h Columbia, Ontario, Flemish-speaking Belgium, and French-speaking Belgium. 53 a c q u i r e i m p o r t a n t i n f o r m a t i o n a b o u t t h e t e a c h i n g o f m a t h e m a t i c s a n d a b o u t s t u d e n t s ' p e r f o r m a n c e w h i c h d i d n o t f a l l w i t h i n t h e t e r m s o f r e f e r e n c e o f t h e M a t h e m a t i c s A s s e s s m e n t . ( p . 1) T h e c u r r i c u l u m w a s a k e y c o n c e p t i n t h e f o r m u l a t i o n o f a f r a m e w o r k f o r t h e SIMS p r o j e c t . A s was s t a t e d i n t h e B . C . r e p o r t : T h e i n t e r n a t i o n a l s t u d y [ w a s ] a b r o a d l y - b a s e d , c o m p a r a t i v e i n v e s t i g a t i o n o f t h e m a t h e m a t i c s c u r r i c u l u m a s p r e s c r i b e d , a s t a u g h t , a n d a s l e a r n e d . F o r t h e p u r p o s e s o f t h e s t u d y , t h e m a t h e m a t i c s c u r r i c u l u m may b e v i e w e d a s c o n s i s t i n g o f t h r e e c o m p o n e n t s o r d i m e n s i o n s : t h e i n t e n d e d c u r r i c u l u m , t h e i m p l e m e n t e d c u r r i c u l u m , a n d t h e a t t a i n e d c u r r i c u l u m , ( p . 5~^  e m p h a s i s i n o r i g i n a l ) T h e c u r r i c u l u m f r a m e w o r k u s e d i n t h e S I M S p r o j e c t w a s v e r y s i m i l a r t o t h e o n e a d o p t e d i n t h i s s t u d y . T h e t h r e e c o m p o n e n t s m e n t i o n e d c o r r e s p o n d t o t h e f o r m a l , o p e r a t i o n a l , a n d e x p e r i e n t i a l l e v e l s o f c u r r i c u l u m r e s p e c t i v e l y . A s n o t e d i n C h a p t e r 2 , t h e B . C . S I M S p r o j e c t i n v o l v e d s u b s t u d i e s a t b o t h t h e G r a d e 8 l e v e l ( r e f e r r e d t o a s P o p u l a t i o n A ) a n d t h e G r a d e 12 l e v e l ( r e f e r r e d t o a s P o p u l a t i o n B ) . T h e s p e c i f i c c o u r s e s s u r v e y e d w e r e M a t h e m a t i c s 8 a n d A l g e b r a 1 2 . T h e d i s c u s s i o n w h i c h f o l l o w s i s r e s t r i c t e d t o t h e G r a d e 8 o r P o p u l a t i o n A p h a s e o f t h i s p r o j e c t . 54 1.1 D e s c r i p t i o n Of SIMS Instrumentation The instrumentation used i n the B. C. SIMS p r o j e c t i n c l u d e d p r e t e s t s and p o s t t e s t s , teacher and student a t t i t u d e s c a l e s , a teacher background q u e s t i o n n a i r e , and teacher "Classroom Process" q u e s t i o n n a i r e s . R o b i t a i l l e , O'Shea, and D i r k s (1982) c h a r a c t e r i z e d the l a t t e r q u e s t i o n n a i r e s as: "unique instruments designed to c o l l e c t h i g h l y s p e c i f i c information from teachers regarding the methods they used i n teaching s p e c i f i c t o p i c s in the c u r r i c u l u m " (p. 7). Six classroom process instruments were used i n the B. C. SIMS for Population A. One of these, the General Classroom P r a c t i c e s Questionnaire (GCPQ), s o l i c i t e d g eneral information about classroom o r g a n i z a t i o n and management and the use of m a t e r i a l s . The other f i v e instruments, the Topic S p e c i f i c Q u e stionnaires (TSQs), were each d i r e c t e d at one of the f o l l o w i n g areas: • Common and decimal f r a c t i o n s (the F r a c t i o n TSQ) • Ratio, p r o p o r t i o n , and percent (the Ra t i o TSQ) • Algebra (formulas and equations) and i n t e g e r s (the Algebra TSQ) • Geometry (the Geometry TSQ) • Measurement (the Measurement TSQ) The B. C. SIMS report c a t e g o r i z e d the aspects of classroom process d e a l t with by these f i v e q u e s t i o n n a i r e s as f o l l o w s : 55 • resources such as textbooks, worksheets, and games used in teaching; • s p e c i f i c subtopics taught; • i n t e r p r e t a t i o n s of s p e c i f i c concepts such as ir • c o n t e n t - s p e c i f i c methods and s t r a t e g i e s such as the procedures used f o r teaching s u b t r a c t i o n of i n t e g e r s ; • f a c t o r s teachers p e r c e i v e d as i n f l u e n c i n g t h e i r choice of s p e c i f i c concept i n t e r p r e t a t i o n s , methods, and s t r a t e g i e s ; • time a l l o c a t e d to an e n t i r e t o p i c and to i n d i v i d u a l s u b t o p i c s ; • teacher' opinions regarding i s s u e s such as the need to j u s t i f y for students the r u l e s f o r m u l t i p l i c a t i o n of i n t e g e r s , or the place of c a l c u l a t o r s i n teaching decimals. (pp. 28-29) Measures for the four c u r r i c u l u m v a r i a b l e s i n v e s t i g a t e d i n t h i s study, which were d i s c u s s e d i n Chapter 1 and which are d e f i n e d o p e r a t i o n a l l y l a t e r i n t h i s chapter, were obtained using TSQ data. Measures of content emphasis came from TSQ questions about time a l l o c a t i o n s . An example taken from the F r a c t i o n TSQ i s : "How many t o t a l c l a s s p e r i o d s d i d you spend on teaching f r a c t i o n s ? (Combine p a r t i a l lessons where n e c e s s a r y . ) " Measures of the other c u r r i c u l u m v a r i a b l e s , content r e p r e s e n t a t i o n l e v e l , r u l e - o r i e n t e d n e s s of i n s t r u c t i o n , and d i v e r s i t y of i n s t r u c t i o n , were obtained using TSQ data about the methods teachers used in i n t e r p r e t i n g s p e c i f i c concepts as w e l l as teaching p r i n c i p l e s and o p e r a t i o n s (the t h i r d and f o u r t h c a t e g o r i e s l i s t e d above). The c o n t e n t - s p e c i f i c methods for each t o p i c are l i s t e d i n Appendix A. C l a s s achievement, the c o n t e x t u a l v a r i a b l e i n v e s t i g a t e d i n t h i s study, was d e f i n e d using the SIMS Core p r e t e s t data as d e s c r i b e d l a t e r in t h i s chapter. The t e s t which was used i s provided i n Appendix B. 56 1.2 Sample S e l e c t i o n Each of the j u r i s d i c t i o n s which p a r t i c i p a t e d i n the SIMS p r o j e c t had some l a t i t u d e in d e f i n i n g the s i z e and composition of the sample of c l a s s e s s e l e c t e d for i n v e s t i g a t i o n ( R o b i t a i l l e , O'Shea, & D i r k s , 1982, p. 8). The B. C. SIMS popul a t i o n s were de f i n e d and the samples s e l e c t e d as f o l l o w s : For the B. C. sample, Population A was d e f i n e d to include a l l students e n r o l l e d i n re g u l a r Grade 8 c l a s s e s i n the p u b l i c schools in the province as of September 1980. Excluded by t h i s d e f i n i t i o n were the approximately 5% of the age cohort e n r o l l e d i n independent schools as w e l l as those f o l l o w i n g programs where the l e v e l of m a t e r i a l covered was s i g n i f i c a n t l y below that p r e s c r i b e d f o r the Mathematics 8 course.... In order to achieve a sample s i z e of approximately 100 Grade 8 and Algebra 12 c l a s s e s s t r a t i f i e d a ccording to geographic zone of the province and by school s i z e , i n i t i a l samples of 125 c l a s s e s at each l e v e l were drawn. In most cases t h i s r e s u l t e d i n the s e l e c t i o n of no more than one c l a s s per s c h o o l . Of the 125 c l a s s e s , 105 were s e l e c t e d f o r i n i t i a l contact and the remainder reserved f o r use when needed. In the s p r i n g of 1980, l e t t e r s were sent from the M i n i s t r y of Education to a l l of the p r i n c i p a l s of the schools s e l e c t e d , s o l i c i t i n g t h e i r c ooperation in the study and asking them to s e l e c t a Mathematics 8 or Algebra 12 teacher or teachers at random from among the teachers a v a i l a b l e . In cases where i t was not p o s s i b l e to make a random s e l e c t i o n , the p r i n c i p a l s were asked to e x e r c i s e t h e i r best judgement about which teacher or teachers to s e l e c t . 1.3 P a r t i c i p a t i o n Rate And Instrument Return Rate / In 78 of the 105 c l a s s e s s e l e c t e d in the Mathematics 8 sample, a l l of the student and teacher instruments were returned to the t e c h n i c a l agency which administered the B.C. SIMS 57 p r o j e c t . 2 In 13 of the remaining 27 c l a s s e s s e l e c t e d , some of the t e s t s and q u e s t i o n n a i r e s were r e t u r n e d . 3 Twelve c l a s s e s which were s e l e c t e d f o r p a r t i c i p a t i o n were subsequently excluded. A f t e r the sample had been drawn and the m a t e r i a l s for the study d i s t r i b u t e d to the schools, a problem developed i n the Grade 8 sample which r e s u l t e d i n the l o s s of 12 of the 105 Mathematics 8 c l a s s e s in the sample. In the schools in which these 12 c l a s s e s were l o c a t e d , a l l of the Mathematics 8 c l a s s e s e s t a b l i s h e d at the beginning of the school year were disbanded at the end of the f i r s t semester, and new c l a s s e s were set up f o r the second h a l f of the Mathematics 8 course. Since the students d i d not stay together but were d i s t r i b u t e d among s e v e r a l c l a s s e s and teachers, i t was not p o s s i b l e to include them i n the study. U n f o r t u n a t e l y , t h i s problem was not i d e n t i f i e d u n t i l i t was too l a t e i n the course of the study to obtain replacements for those c l a s s e s . ( R o b i t a i l l e , O'Shea, & D i r k s , 1982, pp. 9-10) Thus, out of the 105 c l a s s e s s e l e c t e d , 93 were s u i t a b l e f o r p a r t i c i p a t i o n . Of these e l i g i b l e c l a s s e s complete instrument returns were r e c e i v e d from 84%, p a r t i a l r e t u r n s from 14%, and no returns from 2%.* The r e t u r n rate of the TSQs f o r the 93 e l i g i b l e c l a s s e s was 2 B. C. Research administered the t e c h n i c a l aspects of the B. C. SIMS p r o j e c t . This i n c l u d e d the p r i n t i n g and d i s t r i b u t i o n of the t e s t s and q u e s t i o n n a i r e as well as follow-up a c t i v i t i e s to maximize r e t u r n r a t e . 3 In the B. C. SIMS report the number of p a r t i a l r e t u r n s i s given as 11 and the number of n o n - p a r t i c i p a n t s i s given as 15 (p. 10). A f t e r the report was w r i t t e n , however, data were re c e i v e d from one of the c l a s s e s which had been c a t e g o r i z e d as a non-p a r t i c i p a n t . A l s o , one c l a s s was apparently miscategorized i n the report s i n c e complete returns were e v e n t u a l l y r e c e i v e d from 78 rather than 79 c l a s s e s . a Labor d i s p u t e s by school support s t a f f , a p o s t a l s t r i k e , and teacher i l l n e s s and t r a n s f e r account for most of the unreturned q u e s t i o n n a i r e s . ( R o b i t a i l l e , O'Shea, & D i r k s , 1982, p. 10) 58 89%; 416 TSQs were returned out of the 465 d i s t r i b u t e d . In a d d i t i o n to the two n o n - p a r t i c i p a t i n g teachers, three other teachers f a i l e d to complete a s i n g l e TSQ. Thus, f i v e teachers accounted f o r 25 of the 49 q u e s t i o n n a i r e s which were not returned. The remaining 78 teachers returned a l l 390 of the TSQs they had been g i v e n . 5 1.4 Representativeness Of The Achieved Mathematics 8 Sample One goal of the B. C. SIMS p r o j e c t was to make in f e r e n c e s regarding a target p o p u l a t i o n c o n s i s t i n g of teachers of Mathematics 8 in B. C. T h i s target p o p u l a t i o n was not i d e n t i c a l to the set of a l l teachers who taught Mathematics 8. In p a r t i c u l a r , no attempt was made to secure r e p r e s e n t a t i v e p a r t i c i p a t i o n of teachers whose primary teaching load was outside of mathematics. The percent of teacher workload i n mathematics f o r the Mathematics 8 B. C. SIMS sample and f o r the sample of Mathematics 8 teachers who p a r t i c i p a t e d i n the 1981 B. C. Mathematics Assessment i s shown in Table 1. While almost one quarter of the B. C. Assessment sample had 25% or l e s s of t h e i r workload in mathematics, the corresponding f i g u r e for the B. C. SIMS p r o j e c t was only 6%. R o b i t a i l l e , O'Shea, and D i r k s (1982) d e s c r i b e d the B. C. SIMS sample as f o l l o w s : 5 On 14 of these 390 q u e s t i o n n a i r e s i t was i n d i c a t e d that the p a r t i c u l a r content d e a l t with by the q u e s t i o n n a i r e was not part of the c u r r i c u l u m as implemented by the teacher. E i g h t teachers omitted one of the TSQ t o p i c s and three teachers omitted two TSQ t o p i c s from t h e i r courses. 5 9 Table 3- 1 - Teacher Workload i n Mathematics Percent of 1981 Mathematics B.C. SIMS Workload Assessment Teachers Teachers R e l a t i v e frequency (percent) Cumulative frequency (percent) R e l a t i v e frequency (percent) Cumulative frequency (percent) o - 2 5 2 3 . 9 2 3 . 9 5 . 6 5 . 6 2 6 - 50 1 7 . 6 4 1 . 5 1 4 . 6 2 0 . 2 5 1 - 7 5 1 2 . 6 5 4 . 1 1 6 . 9 3 7 . 1 7 6 - 100 4 5 . 9 1 0 0 . 0 6 2 . 9 1 0 0 . 0 Teachers s e l e c t e d to p a r t i c i p a t e i n the i n t e r n a t i o n a l study at the Population A l e v e l were more l i k e l y to be mathematics s p e c i a l i s t s and were more experienced than the general p o p u l a t i o n of teachers of secondary mathematics. Teachers of Mathematics 8 who were in the IEA sample had an average of 14 years of teaching experience, compared to 9 years f o r the po p u l a t i o n of teachers of secondary mathematics. There were a l s o i n d i c a t i o n s that the IEA teachers spent a greater p r o p o r t i o n of t h e i r teaching time conducting mathematics c l a s s e s than d i d the popula t i o n of teachers of secondary mathematics. T h i s l a t t e r f i n d i n g i s not s u r p r i s i n g s i n c e p r i n c i p a l s were asked to s e l e c t a teacher of mathematics to p a r t i c i p a t e in the study. It i s u n l i k e l y , in such circumstances, that they would have considered s e l e c t i n g a teacher who taught s e v e r a l subject areas, or whose s p e c i a l i z a t i o n was i n a f i e l d other than mathematics. The comparison of teaching experience in the quote above i s somewhat mis l e a d i n g s i n c e the mean of 14 years for the IEA sample r e f e r s to t o t a l teaching experience while the f i g u r e of 9 years a c t u a l l y r e f e r s to experience teaching mathematics only. U n f o r t u n a t e l y , i d e n t i c a l questions were not asked of the Mathematics 8 teachers in the SIMS and Assessment surveys so that a c t u a l comparisons of teaching experience are not p o s s i b l e . 60 Tab le 3- 2 - Years of Teach ing E x p e r i e n c e 1981 Mathematics B.C. SIMS Assessment Sample Years of experience Teaching Mathematics Teaching a l l subjects Teaching Mathematics 8 Relative frequency (percent) Cumulative frequency (percent) Relative Cumulative frequency frequency (percent) (percent) Relative Cumulative frequency frequency (percent) (percent) 1- 2 18.5 18.5 3.4 3.4 18.0 18.0 3- 5 20.3 38.8 11.2 14.2 16.8 34.8 6-10 23.2 62.0 24.7 39.3 36.0 70.8 11-15 14.3 76.3 28.1 67.4 18.0 88.8 over 15 23.7 100.0 32.6 100.0 11.2 100.0 The percent of t eachers wi th v a r i o u s l e v e l s of mathematics t e a c h i n g exper ience who p a r t i c i p a t e d in the Assessment as w e l l as l e v e l s of t o t a l t each ing exper i ence and Mathematics 8 t e a c h i n g exper ience for the B. C . SIMS sample are shown in T a b l e 3-2 . While these data do not a l l ow any exact comparisons , i t can be noted that the two columns of c u m u l a t i v e f requenc ie s for the B. C . SIMS teachers d i f f e r w i d e l y . A p p a r e n t l y the average number of years these t eachers taught Mathematics 8 was c o n s i d e r a b l y l e s s than t h e i r average number of years of t e a c h i n g . I t i s a l s o p o s s i b l e , perhaps l i k e l y , that the average number of years these t eachers had taught mathematics in genera l was l e s s than t h e i r average of 14 years of t o t a l t each ing 61 experience. Thus, while the teachers in the B. C. SIMS sample were probably more experienced than the- p o p u l a t i o n of teachers who taught Mathematics 8 during the 1980-81 school year, f i v e years i s perhaps an overestimate of the d i f f e r e n c e between the average teaching experience of the two groups. 1.5 SIMS V a l i d a t i o n : Instrument Development And Research As noted in SIMS B u l l e t i n Number 5 (IEA, 1980), concerns regarding the v a l i d i t y of the T o p i c - S p e c i f i c Questionnaires "have been c e n t r a l to the development of the instruments" (p. 30). B u l l e t i n Number 5 o u t l i n e s the s e v e r a l phases of the v a l i d a t i o n process for these survey instruments. In order for the T o p i c - S p e c i f i c Q u e s t i o n n a i r e s to have basic content v a l i d i t y i t i s necessary that the q u e s t i o n n a i r e items r e f l e c t the content which might be taught and the t o p i c -s p e c i f i c methods which teachers might use. At t h i s l e v e l v a l i d i t y can be e s t a b l i s h e d in part by expert opinion (Moser & Kalton, 1971, p. 356). The SIMS T o p i c - S p e c i f i c Questionnaires were i n i t i a l l y c o n s t r u c t e d at the U n i v e r s i t y of Georgia in 1978 and were r e v i s e d on s e v e r a l occasions by a Working Group composed of prominent members of the mathematics education community. They were reviewed f u r t h e r by the SIMS I n t e r n a t i o n a l Mathematics Committee. At the second stage of the v a l i d a t i o n process, the reactions' of classroom teachers to the SIMS q u e s t i o n n a i r e s were obtained through l i m i t e d p i l o t t e s t i n g over a seven month p e r i o d . S p e c i f i c a l l y , 62 . . . s e v e r a l experienced researchers in mathematics education volunteered to conduct in-depth interviews with classroom teachers concerning (a) the c l a r i t y and i n t e n t i o n of the items, (b) the coverage of the instruments with respect to content and method, and (c) the time demands of the instruments. (IEA, 1980, p. 30) On the b a s i s of teacher r e a c t i o n s the q u e s t i o n n a i r e s were subsequently r e v i s e d and presented to the meeting of the IEA General Assembly in P a r i s i n September 1979. The o b j e c t i v e of the t h i r d stage of the SIMS v a l i d a t i o n scheme was to assess the conformity of teacher s e l f - r e p o r t s of t h e i r o p e r a t i o n a l c u r r i c u l a with the assessments of observers. The nature of the TSQ items made t h i s a d i f f i c u l t task s i n c e a s i n g l e c u r r i c u l u m question on one of the TSQs c o u l d take days or weeks to v e r i f y through assessments- by observers. For example, i f a teacher reported spending 20 days teaching i n t e g e r s , then at l e a s t 20 days would . be r e q u i r e d to v a l i d a t e the s i n g l e Algebra TSQ question s o l i c i t i n g t h i s i n f o r m a t i o n . Because of the time r e q u i r e d to v a l i d a t e the TSQs through d i r e c t o b s e r v a t i o n , i t i s not s u r p r i s i n g that only two "small s c a l e " v a l i d i t y s t u d i e s of the SIMS T o p i c - S p e c i f i c Questionnaires have been reported. The f i r s t o b s e r v a t i o n a l study was conducted at the U n i v e r s i t y of Georgia and c o n s i s t e d of comparing the responses to two of the q u e s t i o n n a i r e s ( F r a c t i o n s and Ratio) by teachers and by observers of the teachers' c l a s s e s . Rather high agreement was found, but the study was too small to provide f i r m c o n c l u s i o n s . (IEA, 1980, p. 30) 63 The c r i t e r i a f o r a s s e r t i n g "rather high agreement" were not reported. F l e x e r (1980) conducted an o b s e r v a t i o n a l study of three e i g h t h grade mathematics classrooms i n I l l i n o i s using the Integer and the Ratio q u e s t i o n n a i r e s . She s p e c i f i c a l l y sought to i n v e s t i g a t e the v a l i d i t y of these instruments. Classroom o b s e r v a t i o n s were made over periods of between 3.5 and 9 weeks depending on the length of the i n s t r u c t i o n a l u n i t . A f t e r a t o p i c corresponding to the content of a q u e s t i o n n a i r e had been taught by a p a r t i c i p a t i n g teacher, that teacher completed the a p p r o p r i a t e instrument i n conformity with the SIMS research design. F l e x e r found c o r r e l a t i o n s between observations and q u e s t i o n n a i r e responses of 0.83 on average. In r e p o r t i n g her research F l e x e r noted that d e s p i t e these encouraging r e s u l t s the teachers in her study expressed concern over the length and complexity of the classroom process instruments. The q u e s t i o n n a i r e s were l a t e r shortened and s i m p l i f i e d i n format by the Working Group. A f u r t h e r v a l i d i t y study was conducted in B. C. by t h i s r esearcher. F i v e Mathematics 8 teachers who had p a r t i c i p a t e d i n the SIMS p r o j e c t were interviewed i n June 1981 a f t e r the year long data c o l l e c t i o n p e r i o d . The interviews of 80 to 100 minutes each had two components. In the f i r s t part the teachers were asked e i g h t general questions regarding the appropriateness of the T o p i c - S p e c i f i c Questionnaires f o r t h e i r courses. They were a l s o asked i f the items were c l e a r and fr e e from apparent b i a s and i f the response c a t e g o r i e s provided were, in f a c t , 64 adequate i n t h e i r o p i n i o n . These teachers reported that the q u e s t i o n n a i r e s were c l e a r and h i g h l y r e l e v a n t to t h e i r courses. In p a r t i c u l a r , none of the teachers reported using content-s p e c i f i c methods other than those l i s t e d on the q u e s t i o n n a i r e s . T h i s i s not s u r p r i s i n g c o n s i d e r i n g the extensive p i l o t i n g of these instruments which had taken plac e i n B. C. The only c r i t i c i s m of the instruments was d i r e c t e d at those questions which asked them why they d i d or d i d not use each of the c o n t e n t - s p e c i f i c methods i n t h e i r t e a c h i n g . Teachers who p a r t i c i p a t e d i n the F l e x e r study a l s o s i n g l e d out these questions f o r c r i t i c i s m and t h i s i s c o n s i s t e n t with Berdie and Anderson's (1974) observation that "why" questions are in general not w e l l s u i t e d to data c o l l e c t i o n using q u e s t i o n n a i r e s . In the second part of the i n t e r v i e w s , teachers were asked questions from the T o p i c - s p e c i f i c Q u e s t i o n n a i r e which they had most r e c e n t l y completed. For the three teachers who had taught the q u e s t i o n n a i r e m a t e r i a l w i t h i n three weeks of the i n t e r v i e w s , i t was found that on average 9 percent of the subtopics which had been reported as taught on the questionnaires\were reported as hot taught during the in t e r v i e w s or v i c e versa. S i m i l a r l y , 16 percent of the c o n t e n t - s p e c i f i c methods which had been reported as u t i l i z e d on the q u e s t i o n n a i r e s were reported as not u t i l i z e d during the interviews or v i c e v e r s a . 6 T/ime a l l o c a t i o n s 6 Each of these three teachers was asked questions from only one of the TSQs they had completed, s p e c i f i c a l l y Algebra, Geometry, and Measurement. O v e r a l l , 75 questions were asked about subtopics and 99 questions about t o p i c - s p e c i f i c methods to the three teachers interviewed. 65 reported during the interviews d i f f e r e d by an average of 13 percent from those which had been reported on the q u e s t i o n n a i r e s . These f i n d i n g s were considered as supportive of the assumption that teachers had taken reasonable care i n completing the T o p i c - S p e c i f i c Q u e s t i o n n a i r e s . It should be noted that each B. C. teacher i n the SIMS p r o j e c t p a r t i c i p a t e d i n a one day o r i e n t a t i o n workshop. A l s o , contacts were made by mail and telephone when teachers d i d not return i n d i v i d u a l q u e s t i o n n a i r e s soon a f t e r the probable t o p i c completion dates which they had provided i n September 1980. These f a c t o r s may well have enhanced teacher care i n completing the q u e s t i o n n a i r e s in B. C. 2. MATHEMATICS 8 CONTENT INCLUDED IN THE STUDY 2.1 The Content Areas This study was designed to i n v e s t i g a t e a s u b s t a n t i a l p o r t i o n of the content which might be taught in Mathematics 8 c l a s s e s . . Three broad content a r e a s — a r i t h m e t i c , a l g e b r a , and geometry—were s e l e c t e d f o r study. These areas were i d e n t i f i e d both because they represent t y p i c a l c l a s s i f i c a t i o n s of school mathematics content (e.g., Begle, 1979, pp. 14-15) and because of the prominence each area i s given i n the formal B. C. cu r r i c u l u m at t h i s l e v e l . For the purposes of t h i s study the formal B. C. c u r r i c u l u m at the Grade 8 l e v e l was d e f i n e d to c o n s i s t of those mathematical t o p i c s which were e i t h e r : 66 ( 1 ) e x p l i c i t l y s p e c i f i e d i n t h e P r o v i n c i a l M a t h e m a t i c s C u r r i c u l u m G u i d e , or ( 2 ) c o n t a i n e d w i t h i n a c h a p t e r of any of the t h r e e p r e s c r i b e d t e x t s f o r M a t h e m a t i c s 8 w h i c h a l s o c o n t a i n e d some t o p i c s w h i c h were s p e c i f i e d i n t h e C u r r i c u l u m G u i d e . (3) c o n t a i n e d i n t h e f i r s t t w o - t h i r d s of any of the p r e s c r i b e d t e x t s . The s e c o n d and t h i r d c r i t e r i a were i n c l u d e d b e c a u s e i t i s assumed t h a t p r e s c r i b e d t e x t s c a r r y messages t o t e a c h e r s as t o what s h o u l d be t a u g h t which may be as s t r o n g as t h e s p e c i f i c a t i o n s of t h e C u r r i c u l u m G u i d e . The " f i r s t t w o - t h i r d s " s t i p u l a t i o n of t h e t h i r d c r i t e r i o n i s somewhat a r b i t r a r y . I t r e f l e c t s an a s s u m p t i o n t h a t t e a c h e r s t e n d t o view t h e l a t t e r p o r t i o n s of a t e x t b o o k as o p t i o n a l . The t h r e e c o n t e n t a r e a s w hich were i n v e s t i g a t e d i n t h i s s t u d y each c o n t a i n a s u b s t a n t i a l amount of c o n t e n t r e l e v a n t t o t h e f o r m a l c u r r i c u l u m of M a t h e m a t i c s 8 as d e f i n e d above, b o t h i n terms of t o p i c s l i s t e d i n t h e C u r r i c u l u m G u i d e and i n terms of p r e s c r i b e d t e x t b o o k e m p h a s i s . The s c o p e of t h e terms a r i t h m e t i c , a l g e b r a , and geometry as us e d h e r e was d e t e r m i n e d not o n l y by t h e m a t h e m a t i c a l t o p i c s w h i c h f a l l under t h e s e c a t e g o r i e s and appear w i t h i n t h e f o r m a l c u r r i c u l u m f o r M a t h e m a t i c s 8 but a l s o by t h e i n c l u s i o n of p a r t i c u l a r t o p i c s on t h e q u e s t i o n n a i r e s used f o r t h e B. C. SIMS p r o j e c t . S i n c e the d a t a from t h a t p r o j e c t were r e - a n a l y z e d f o r 67 t h i s study, a few t o p i c s which might be expected to be included wi t h i n the three content areas at t h i s grade l e v e l were omitted. For example, the t o p i c s of square roots and s c i e n t i f i c n o t a t i o n were not incorporated i n the SIMS q u e s t i o n n a i r e and thus c o u l d not be i n c l u d e d in a r i t h m e t i c i n t h i s study though they are part of the formal B.C. c u r r i c u l u m . 2.2 The S p e c i f i c Mathematics Topics Within each of the three content areas f i v e or s i x concepts, operations, and p r i n c i p l e s were s e l e c t e d f o r i n v e s t i g a t i o n . The t o p i c s s e l e c t e d were both important t o p i c s wi t h i n the formal c u r r i c u l u m of Mathematics 8 and included in the B. C. SIMS instrumentation. To be considered as important content i n the formal c u r r i c u l u m each t o p i c had to s a t i s f y one or both of the f o l l o w i n g c r i t e r i a : ( 1 ) the t o p i c was l i s t e d as "core" content i n the Curriculum Guide at t h i s l e v e l , or (2) the t o p i c was contained in the f i r s t two-thirds of each of the three p r e s c r i b e d t e x t s . The t o p i c s which were included in t h i s study are l i s t e d below a c c o r d i n g to content area. The SIMS TSQ which contained items about the c o n t e n t - s p e c i f i c methods teachers used i n pr e s e n t i n g these t o p i c s i s i n d i c a t e d i n parentheses. Each t o p i c s a t i s f i e d one or both c r i t e r i a f o r important content as s p e c i f i e d above. 68 A r i t h m e t i c the concept of f r a c t i o n s ( F r a c t i o n TSQ) the a d d i t i o n of f r a c t i o n s ( F r a c t i o n TSQ) the concept of decimals ( F r a c t i o n TSQ) operations with decimals ( F r a c t i o n TSQ) the concept of p r o p o r t i o n s (Ratio TSQ) Algebra the concept of i n t e g e r s (Algebra TSQ) the a d d i t i o n of i n t e g e r s (Algebra TSQ) the s u b t r a c t i o n of i n t e g e r s (Algebra TSQ) the m u l t i p l i c a t i o n of i n t e g e r s (Algebra TSQ) the concept of formulas (Algebra TSQ) s o l v i n g l i n e a r equations (Algebra TSQ) Geometry the t r i a n g l e angle sum theorem (Geometry TSQ) the Pythagorean theorem (Geometry TSQ) the concept of n (Measurement TSQ) the area of a p a r a l l e l o g r a m (Measurement TSQ) the volume of a rectangular prism (Measurement TSQ) For each of the mathematical t o p i c s l i s t e d above the teaching methods incorporated in the SIMS TSQs were assumed to i n c l u d e the most common a l t e r n a t i v e s which teachers might use i n t h e i r p r e s e n t a t i o n s . T h i s assumption i s warranted as the q u e s t i o n n a i r e s were developed and r e f i n e d by a panel of mathematics educators, as noted e a r l i e r , and were e x t e n s i v e l y 69 p i l o t e d , a process which involved c o n s i d e r a b l e input from teachers i n B. C. and elsewhere. 3. MEASUREMENT OF THE CURRICULUM VARIABLES 3.1 Content Emphasis The emphasis given to each of the three content areas: a r i t h m e t i c , algebra, and geometry was measured i n terms of the number of c l a s s periods spent on that content. Since there was v a r i a t i o n in the number of periods teachers had a v a i l a b l e for t h e i r courses, the measure of emphasis which each content area r e c e i v e d was defined in t h i s study as the p r o p o r t i o n of time a l l o c a t e d to a p a r t i c u l a r t o p i c r e l a t i v e to the t o t a l time a l l o c a t e d f o r a l l three areas. Thus, f o r each teacher and each content area the l e v e l of content emphasis c o u l d take on values between zero and one i n c l u s i v e . The sum of content, emphasis measures f o r the three areas was one for each teacher. In order to f a c i l i t a t e the d i s c u s s i o n of the f i n d i n g s of t h i s study, f i v e l e v e l s of content emphasis were d e f i n e d p r i o r to data a n a l y s i s : very heavy emphasis, heavy emphasis, moderate emphasis, l i g h t emphasis, and very l i g h t emphasis. The l e t t e r C was used to represent the content emphasis v a r i a b l e . The f o l l o w i n g values f o r C were a s s o c i a t e d with the f i v e s p e c i f i e d l e v e l s : 70 0.66 < C < 1.00: very heavy emphasis 0.50 < C ^ 0.66: heavy emphasis 0.25 < C ^ 0.50: moderate emphasis 0.17 < C < 0.25: l i g h t emphasis 0.00 ^ C < 0.17: very l i g h t emphasis For each teacher there were a c t u a l l y three separate content emphasis v a r i a b l e s , one f o r each content area. For example, suppose a teacher a l l o c a t e d 36 p e r i o d s to a r i t h m e t i c , 36 periods to a l g e b r a , and 18 periods to geometry. Then the emphasis score for a r i t h m e t i c was 0.40, the emphasis score for algebra was a l s o 0.40, and the emphasis score for geometry was 0.20. The i n t e r v a l s s p e c i f i e d above were c o n s t r u c t e d by noting that equal emphasis on a l l three content areas by a teacher would r e s u l t i n three content emphasis values of 1/3 f o r that teacher. Very heavy content emphasis was d e f i n e d as twice the equal emphasis value or g r e a t e r , while heavy emphasis was d e f i n e d as between one-and-one-half and two times the equal emphasis va l u e . Given that one area r e c e i v e d very heavy emphasis by a teacher according to the foregoing, the sum of the other two content emphasis values for that teacher could not exceed 1/3. The c r i t e r i o n f o r very l i g h t emphasis was s t i p u l a t e d as a value l e s s than one-half of t h i s remaining amount or l e s s than 1/6. S i m i l a r l y , the i n t e r v a l a s s o c i a t e d with l i g h t emphasis was formulated i n r e l a t i o n to that d e f i n e d f o r heavy emphasis, i . e . , 1/2(1 - 1/2) = 1/4 and 1/2(1 - 2/3) 1/6 were taken as c u t o f f values. The remaining i n t e r v a l 71 spanning the equal emphasis value of 1/3 was d e f i n e d as i n d i c a t i n g moderate emphasis. 3.2 Mode Of Content Representation As part of the development of the SIMS Classroom Process Q u e s t i o n n a i r e s , Cooney (1980a) c l a s s i f i e d the content s p e c i f i c methods or approaches to most of the mathematical t o p i c s which were contained i n those instruments as e i t h e r p e r c e p t u a l or a b s t r a c t . In p a r t i c u l a r , the approaches a s s o c i a t e d with 14 of the 16 t o p i c s i n v e s t i g a t e d i n t h i s study, the exceptions being p r o p o r t i o n s and s o l v i n g l i n e a r equations, were c l a s s i f i e d by Cooney, and reviewed by members of the Classroom Process Questionnaire Working Group, who used the f o l l o w i n g d e f i n i t i o n s : A perceptual treatment of the content r e l i e s on concrete m a t e r i a l s , diagrams or p i c t u r e s or d e r i v e s i t s meaning from the environment, e x p e r i e n t i a l a c t i v i t i e s or some s o r t of perceptual a c t i v i t y . An a b s t r a c t treatment of the content r e l i e s on explanations which are symbolic i n nature and d e r i v e s i t s meaning from other mathematical content. (Cooney, 1980a, p. 20) In t h i s study Cooney's d e f i n i t i o n s as s t a t e d above were used to c l a s s i f y the c o n t e n t - s p e c i f i c methods f o r teaching a r i t h m e t i c and a l g e b r a i c t o p i c s i n t o a b s t r a c t and p e r c e p t u a l c a t e g o r i e s . For geometric t o p i c s Cooney's d e f i n i t i o n s were modified so that the use of a diagram or p i c t u r e i n and of i t s e l f was not s u f f i c i e n t for a teaching method to be c l a s i f i e d as p e r c e p t u a l . Rather, a method used f o r teaching a geometric t o p i c was c l a s s i f i e d as perceptual only when an a c t u a l p h y s i c a l a c t i v i t y was involved or a diagram was used which suggested some 72 p h y s i c a l a c t i v i t y . Otherwise, the method was c l a s s i f i e d as a b s t r a c t . In q u a n t i f y i n g t h i s v a r i a b l e for each t o p i c and content area, the measure of i n t e r e s t was the r a t i o of a b s t r a c t to t o t a l methods. Teachers were asked in the SIMS Classroom Process q u e s t i o n n a i r e s whether they emphasized, used without emphasis, or d i d not use the approaches l i s t e d f o r each mathematical t o p i c . In order that emphasized approaches r e c e i v e more weight than those which were used only, the emphasized approaches were given a double weight in t h i s r a t i o . Thus, L, the l e v e l of a b s t r a c t i o n used by a teacher i n p r e s e n t i n g a p a r t i c u l a r t o p i c , c o u l d take on values between zero (no a b s t r a c t approaches)' and one ( a l l approaches a b s t r a c t ) i n c l u s i v e . As with the content emphasis s c a l e , f i v e l e v e l s of content r e p r e s e n t a t i o n were def i n e d p r i o r to data a n a l y s i s to f a c i l i t a t e the d i s c u s s i o n of the f i n d i n g s of t h i s study: h i g h l y a b s t r a c t , somewhat a b s t r a c t , balanced, somewhat p e r c e p t u a l , and h i g h l y p e r c e p t u a l . The f o l l o w i n g values f o r L were a s s o c i a t e d with the f i v e s p e c i f i e d l e v e l s : 0.80 < L < 1.00 0.60 < L < 0.80 0.40 < L < 0.60 0.20 < L < 0.40 0.00 < L < 0.20 h i g h l y a b s t r a c t somewhat a b s t r a c t balanced somewhat pe r c e p t u a l h i g h l y p e r c e p t u a l Since there seemed to be no compelling reason for doing otherwise, the f i v e i n t e r v a l s were chosen of equal l e n g t h . 73 Decimals The inteApA.eXatioM g i v e n belov) may be includid in youA in&tAuctioncU piogiam. CHECK the \tipon6e. code which deicUbei the' tn.zcUme.nt ot each topic in youx claii. Response coves: /. Emphasized lu&ed cu a pximcuiy explanation, xeienxed to ex-tensively ox l*.equently) I. Uicd but not emphasized i. Hot ated 51. A decimal as the coordinate of a point on the number line. 1 .28 .28 < .8 2 3 52. A decimal as another way of writing a fraction. 0.17 17 TOO" 0.8 8 TO" 53. A decimal as part of a region. 0. 38 54. A decimal as an extension of place value. 55. A decimal as a series 0.243 » 2 + 4 • 3 To" TOO" TOM 56. A decimal as a comparison CXI unit rod ' l i n n 0.5 I I I UJ 0.45 Figure 3- 1 - C o n t e n t - s p e c i f i c methods for teaching the concept of decimals. F i g u r e 3-1 i s taken from the F r a c t i o n TSQ. I t shows the s i x teaching approaches included in that instrument for the concept of decimals. Of these, options 51, 53, and 56 were considered as p e r c e p t u a l , the others as a b s t r a c t . As an example of the computation of L, suppose that a p a r t i c u l a r teacher emphasized the a b s t r a c t approach given by option 54 and used 74 without emphasis the a b s t r a c t approach given by o p t i o n 52 as w e l l as the perceptual approach given by option 53. For t h i s teacher the value of L f o r t h i s t o p i c would be computed as f o l l o w s : Emphasized 54 ( a b s t r a c t ) = 2 Used 52 ( a b s t r a c t ) = 1 Used 53 (perceptual) = 1 L = 2+2 = 0.75 2+1 + 1 Thus, according to the foregoing d e f i n i t i o n t h i s teacher's i n s t r u c t i o n was somewhat a b s t r a c t f o r the concept of decimals. 3.3 Rule-Orientedness Of I n s t r u c t i o n If a concept i s d e f i n e d f o r m a l l y but no f u r t h e r i n t e r p r e t a t i o n provided or i f an operation or p r i n c i p l e i s s t a t e d as a r u l e with no i n t e r p r e t a t i o n or j u s t i f i c a t i o n , then the mathematical idea i n question i s being presented in what might be c a l l e d a r u l e - o r i e n t e d way as d i s c u s s e d e a r l i e r . For e i g h t of the 16 mathematical t o p i c s i n c l u d e d in t h i s i n v e s t i g a t i o n Cooney (1980a) i d e n t i f i e d one of the a l t e r n a t e approaches as c l e a r l y a r u l e , e.g., the r u l e of signs f o r i n t e g e r m u l t i p l i c a t i o n . These t o p i c s were: operations with decimals, a d d i t i o n of i n t e g e r s , s u b t r a c t i o n of i n t e g e r s , m u l t i p l i c a t i o n of i n t e g e r s , the Pythagorean theorem, the concept of ir, the area of a p a r a l l e l o g r a m , and the volume of a r e c t a n g u l a r prism. 75 Conceivably, f o r some t o p i c s teachers might: (1) Present r u l e s without any conceptual development, (2) Present r u l e s together with one or more conceptual approaches, or (3) Develop a mathematical concept without an e x p l i c i t statement of r u l e s . Cooney (1980a, p. 29) in d i s c u s s i n g t h i s v a r i a b l e hypothesized that a h i g h l y r u l e - o r i e n t e d teacher might e f f e c t i v e l y promote computational s k i l l s in students but at the expense of higher order outcomes such as problem s o l v i n g . A l t e r n a t e l y , one might expect that some teachers would be more r u l e - o r i e n t e d for review areas than f o r new content. Table 3-3 d e f i n e s values f o r the r u l e - o r i e n t e d n e s s , R, of a teacher on a p a r t i c u l a r t o p i c . This measure was used to q u a n t i f y the r u l e - o r i e n t e d n e s s of a teacher's o p e r a t i o n a l c u r r i c u l u m . The value of R i s a f u n c t i o n of (1) whether the p r e s e n t a t i o n of r u l e s was emphasized, used without emphasis, or not used i n i n s t r u c t i o n , and (2) whether other approaches were emphasized, used without emphasis, or not used in i n s t r u c t i o n . The higher the value of R the stronger the emphasis on r u l e s in i n s t r u c t i o n . As with the content emphasis and l e v e l of r e p r e s e n t a t i o n v a r i a b l e s , f i v e l e v e l s of r u l e - o r i e n t e d n e s s were d e f i n e d p r i o r to data a n a l y s i s : h i g h l y r u l e - o r i e n t e d , somewhat r u l e - o r i e n t e d , balanced, n o n - r u l e - o r i e n t e d , h i g h l y n o n - r u l e - o r i e n t e d . The 76 Table 3- 3 - Defined Values f o r Rule-Orientedness Rule approach Emphasized Used Not Used Any other approach Emphasized 0.50 0.25 0.00 Used 0.75 0.50 0.25 Not Used 1.00 0.75 f o l l o w i n g values f o r R were a s s o c i a t e d with the f i v e s p e c i f i e d l e v e l s : 0.80 < R < 1.00: h i g h l y r u l e - o r i e n t e d 0.60 < R < 0.80: somewhat r u l e - o r i e n t e d 0.40 < R < 0.60: balanced 0.20 < R < 0.40: somewhat n o n - r u l e - o r i e n t e d 0.00 < R < 0.20: h i g h l y n o n - r u l e - o r i e n t e d For a p a r t i c u l a r teacher and t o p i c , R takes on values of 1.00, 0.75, 0.50, 0.25, and 0.00 only, each of the above c a t e g o r i e s c o n t a i n i n g e x a c t l y one of these f i v e v a l u e s . Related measures found by aggregating across t o p i c s and/or teachers can, however, a t t a i n other values and hence the n e c e s s i t y of i n t e r v a l s . For example, i f a teacher had r u l e - o r i e n t e d n e s s scores of 0.75 on s i x t o p i c s and scores of 0.25 on two t o p i c s , then that teacher's o v e r a l l measure for R f o r those e i g h t t o p i c s would be 0.625, the a r i t h m e t i c average. F i g u r e 3-2 i s taken from the Geometry TSQ. I t shows the seven teaching approaches in that instrument f o r the Pythagorean theorem. Option 69 s p e c i f i e s the p r e s e n t a t i o n of a r u l e and was 77 SivtAol u x t h o d i j o * c e n d i u i g { A c P u l u n g c r t e o j i r h e o t e n <uit g i u e r t btX.au. CHECX till w p o n a t c o d e u m v i d i rftactibu A t f i m t m e j i c o j t a e h mttiiod in u o i n cjaii. ttSPOHSE CUVIS: I. Hi id 44 » p u m a v i i / « . £ < i o d e x p l a n a t i o n U i e r f o u t n o t a l a p u n u . i i / m t o j u o j e x p l a n a t i o n 3 . NaC lUtr i 6 7 . I p r e s e n t e d my s t u d e n t s w i t h a v a r i e t y o f r i g h t t r i a n g l e s a n d h a d t h e n m e a s u r e a n d r e c o r d t h e l e n g t h s o f t h e l e g s a n d h y p o -t e n u s e . The p a t t e r n was d i s -c u s s e d a n d t h e n we s t a t e d t h e p r o p e r t y . E x : l e a 1 e q h y p o t e n u s e 3 5 3 2 * 4 2 9 2 • U 2 • 5 ' 1 5 2 a 2 * b 2 l 6 8 . ! u s e d d i a g r a m s l i k e t h e f o l l o w i n g t o show t h a t , i n r i g h t t r i a n g l e a 2 * b 2 • c 2 5 9 . I g a v e n y s t u d e n t s t h e f o r m u l a a 2 • b 2 • c 2 a n d h a d t h e m u s e i t I n w o r k i n g e x a m p l e s . 7 1 . 1 p r e s e n t e d a n I n f o r m a l a r e a a r g u m e n t u s i n g p h y s i c a l , e . g . g e o b o a r d s , o r p i c t o r a l m o d e l s E x : I s h o w e d t h a t t h e two s q u a r e s h a d e q u a l a r e a . a ^ tt b a b 7 2 . 3 I p r e s e n t e d a f o r m a l d e d u c t i v e " a l g e b r a i c " a r g u m e n t . E x : U s i n g s i m i l a r r i g h t t r i a n g l e s , p r o p o r t i o n s c a n b e s e t up t o y i e l d " .2 7 3 . 1 I p r e s e n t e d a f o r m a l d e d u c t i v e a r g u m e n t u s i n g a r e a . E x : T h i s f i g u r e i s s o m e t i m e s u s e d t o p r e s e n t a f o r m a l p r o o f . 1 7 0 . The t h e o r e m was p r e s e n t e d i n t h e c o n t e x t o f a h i s t o r i c a l a c c o u n t o f P y t h a g o r a s a n d E u c l i d . F i g u r e 3- 2 - C o n t e n t - s p e c i f i c methods f o r teaching the Pythagorean theorem. designated as the r u l e approach. To i l l u s t r a t e the determination of R, suppose that a p a r t i c u l a r teacher emphasized t h i s approach but a l s o used without emphasis the approach given by o p t i o n 71. The appr o p r i a t e value of R f o r t h i s teacher's p r e s e n t a t i o n s of t h i s t o p i c would be 0.75. Thus, according to 78 the foregoing d e f i n i t i o n t h i s teacher's i n s t r u c t i o n was somewhat r u l e - o r i e n t e d for the Pythagorean theorem. 3.4 D i v e r s i t y Of I n s t r u c t i o n In t h i s study the number of approaches which a teacher employed i n pr e s e n t i n g a mathematical concept, o p e r a t i o n , or p r i n c i p l e was used as the b a s i s f o r q u a n t i f y i n g the d i v e r s i t y with which mathematical t o p i c s were approached wi t h i n the o p e r a t i o n a l c u r r i c u l a . For each of the 16 mathematical t o p i c s included i n t h i s study the measure of the d i v e r s i t y employed by a teacher i n p r e s e n t i n g a p a r t i c u l a r t o p i c , D, was taken to be the number of approaches emphasized plus one-half m u l t i p l i e d by the number of approaches used without emphasis. U n l i k e the other three c u r r i c u l u m v a r i a b l e s i n v e s t i g a t e d in t h i s study, the d i v e r s i t y measure d i d not take on values of from 0.00 to 1.00 i n c l u s i v e . If a teacher i n d i c a t e d i n c l u s i o n of one of the t o p i c s i n v e s t i g a t e d i n t h i s study in h i s or her o p e r a t i o n a l c u r r i c u l u m , the r e l a t e d d i v e r s i t y measure was at l e a s t 0.50. T h i s value would occur i n .the case of use without emphasis of a s i n g l e approach to a t o p i c and was the minimum value f o r D. If a t o p i c was not taught no d i v e r s i t y measure was computed. The maximum value f o r t h i s v a r i a b l e was dependent upon the number of approaches given for a p a r t i c u l a r t o p i c in the Topic S p e c i f i c Q u estionnaires which v a r i e d between three and ten. Thus, the maximum value f o r D v a r i e d between 3.0 and 10.0. While the d i v e r s i t y measure c o u l d have been standardized across t o p i c s by making i t a p r o p o r t i o n of the number of approaches 79 l i s t e d i n the q u e s t i o n n a i r e s for each t o p i c , the r e s u l t i n g s c a l e would not have adequately r e f l e c t e d a c t u a l d i v e r s i t y i n i n s t r u c t i o n . For example, i f p r o p o r t i o n s were used, the emphasis of f i v e of the ten l i s t e d f r a c t i o n i n t e r p r e t a t i o n s , and e x c l u s i o n of the others, would have r e s u l t e d in the same d i v e r s i t y measure as the emphasis of two of the four l i s t e d p r o p o r t i o n i n t e r p r e t a t i o n s , and e x c l u s i o n of the o t h e r s . Assuming, as was done in t h i s study, that the a l t e r n a t i v e approaches to t o p i c s given in the Topic S p e c i f i c Q u e s t i o n n a i r e s include n e a r l y a l l of a c t u a l approaches used by teachers, the f i r s t example should have r e s u l t e d in a higher d i v e r s i t y value than the second. Using the s c a l e as d e f i n e d , the r e s u l t i n g values f o r D i n these examples are 5.0 and 2.0 r e s p e c t i v e l y . As with the other v a r i a b l e s , l e v e l s of d i v e r s i t y were def i n e d p r i o r to data a n a l y s i s to f a c i l i t a t e d i s c u s s i o n . These l e v e l s were as f o l l o w s : D > 3.00: high d i v e r s i t y 1.50 < D < 3.00: moderate d i v e r s i t y 0.50 < D < 1.50: low d i v e r s i t y Thus, i f a teacher emphasized three or more approaches to a mathematical idea, the i n s t r u c t i o n was c h a r a c t e r i z e d as h i g h l y d i v e r s i f i e d f o r that t o p i c i n t h i s study. A l t e r n a t e l y , the emphasis of two approaches and the use of two others r e s u l t e d i n the same c h a r a c t e r i z a t i o n . At the other extreme, i f a teacher .emphasized only one approach to an idea in i n s t r u c t i o n and used at most one other approach without emphasis, the i n s t r u c t i o n was 80 c h a r a c t e r i z e d as showing a low l e v e l of d i v e r s i t y for that t o p i c . Despite the reasonableness of these c a t e g o r i e s as d e f i n e d , i t should be kept in mind that the d i f f e r e n c e s i n the number of approaches given for the v a r i o u s t o p i c s as noted above make comparisons of d i v e r s i t y between teachers or groups of teachers f o r the same t o p i c or group of t o p i c s l e s s problematic than comparisons between t o p i c s across teachers. 4. THE CONTEXTUAL VARIABLE: CLASS ACHIEVEMENT Each of the c u r r i c u l u m v a r i a b l e s i n v e s t i g a t e d in t h i s study: content emphasis, content r e p r e s e n t a t i o n l e v e l , r u l e -orientedness of i n s t r u c t i o n , and d i v e r s i t y of i n s t r u c t i o n , was examined with reference to c l a s s achievement. U n l i k e the case of the c u r r i c u l u m v a r i a b l e s , however, the c a t e g o r i e s f o r c l a s s achdievement were i d e n t i f i e d a f t e r rather than before a p r e l i m i n a r y a n a l y s i s of the data and on the b a s i s of n a t u r a l l y o c c u r r i n g v a r i a t i o n . C l a s s means on the 40 item SIMS p r e t e s t were used to designate each c l a s s i n the study as low achievement, middle achievement, or high achievement. 7 The low achievement group c o n s i s t e d of the 29 lowest s c o r i n g c l a s s e s . T h e i r c l a s s means v a r i e d between 10.57 and 15.83 and had a mean value of 14.04. The high achievement group c o n s i s t e d of the 29 highest s c o r i n g 7 Each student took the Core P r e t e s t at the; beginning of the course and an i d e n t i c a l Core Posstest plus, one of four r o t a t e d t e s t forms at the end of the course. 81 c l a s s e s . T h e i r c l a s s means v a r i e d between 19.81 and 30.71 and had a mean value of 23.48. 5. DATA ANALYSIS The r a t i o n a l e for t h i s study r e s t s on the assumption that the o p e r a t i o n a l c u r r i c u l a of secondary mathematics teachers are important e d u c a t i o n a l phenomena which warrant d i s c i p l i n e d i n q u i r y . Furthermore, i t was assumed that no adequate, g l o b a l theory of mathematics c u r r i c u l u m and i n s t r u c t i o n i s c u r r e n t l y a v a i l a b l e thus implying that d e f i n i t i v e , hypothesis t e s t i n g s t u d i e s are premature at t h i s p o i n t . By c o n c e p t u a l i z i n g c u r r i c u l u m as i n v o l v i n g mathematical concepts, o p e r a t i o n s , and p r i n c i p l e s and the c o n t e n t - s p e c i f i c methods which are used in p r e s e n t i n g these ideas, i t was p o s s i b l e to d e f i n e q u a n t i t a t i v e d e s c r i p t o r s of mathematics c u r r i c u l a . In pursuing a q u a n t i t a t i v e approach to d e s c r i b i n g curriculum-in-use and the r e l a t i o n s h i p s between c u r r i c u l u m - i n - u s e and c l a s s achievement l e v e l , methods of E x p l o r a t o r y Data A n a l y s i s (EDA) were u t i l i z e d . In t h i s s e c t i o n the reasons for using EDA w i l l be b r i e f l y o u t l i n e d and those EDA techniques employed in t h i s i n v e s t i g a t i o n are i d e n t i f i e d . EDA i s a body of s t a t i s t i c a l techniques developed by John Tukey (Eric k s o n & Nosanchuk, 1977, p. v) who c h a r a c t e r i z e d i t as i n v o l v i n g : . . . l o o k i n g at data to see what i t seems to say. It concentrates on simple a r i t h m e t i c and easy-to-draw p i c t u r e s . It regards whatever appearances we have recognized as p a r t i a l d e s c r i p t i o n s , and t r i e s to look beneath them for new i n s i g h t s . I t s concern i s with 82 appearance, not with c o n f i r m a t i o n . (Tukey, 1977, p. v) Leinhardt and Leinhardt (1980) have noted the relevance of EDA to e d u c a t i o n a l research: ...EDA i s e s p e c i a l l y important to e d u c a t i o n a l research, where many of the v a r i a b l e s s t u d i e d and data c o l l e c t e d are not brought i n t o analyses because w e l l -v e r i f i e d , s u b s t a n t i v e theory demands t h e i r i n c l u s i o n . Rather, v a r i a b l e s are o f t e n i n c l u d e d i n a study because i n v e s t i g a t o r s " f e e l " they ought to be, because they are "convenient" to use, t h e i r measures have been recorded i n some assumedly "reasonable" manner. Nor do the data always d e r i v e from s c i e n t i f i c a l l y designed random experiments. It i s p r e c i s e l y in such ad hoc e m p i r i c a l research that EDA can be used to i t s g r e a t e s t advantage because i t i s here that an open mind i s an absolute n e c e s s i t y : The a n a l y s t r a r e l y has the support of t h e o r e t i c a l l y based e x p e c t a t i o n s , and the r e a l task c o n f r o n t i n g the data a n a l y s t i s to e x p l o r e — t o search f o r ideas that make sense of the data. (p. 87) In t h i s study two of the b a s i c techniques of EDA were employed: stem-and-leaf p l o t s and box-and-whisker p l o t s , with an emphasis on the l a t t e r form of data d i s p l a y . The f o l l o w i n g were produced and are presented i n Chapter 4. (1) P l o t s of content emphasis for a l l teachers and s e p a r a t e l y for the teachers of low and high achievement c l a s s e s . (2) P l o t s of r u l e - o r i e n t e d n e s s , l e v e l of r e p r e s e n t a t i o n , and d i v e r s i t y f o r the t o p i c s for which each of these v a r i a b l e s are d e f i n e d , across each of the content areas, and o v e r a l l . P l o t s were made for a l l teachers and s e p a r a t e l y f o r teachers of low and high achievement c l a s s e s . 83 IV. THE RESULTS OF THE STUDY 1. DESCRIPTION OF THE GRAPHICAL DISPLAYS The r e s u l t s of t h i s study are presented in part using the stem-and-leaf p l o t s and the boxplots of E x p l o r a t o r y Data A n a l y s i s . In a stem-and-leaf p l o t a l l of the values i n a given data set are r e t a i n e d in a d i s p l a y which i s s i m i l a r to a r o t a t e d histogram. In a boxplot the d i s t r i b u t i o n of the data i s shown using f i v e summary s t a t i s t i c s as well as any o u t l i e r s which may occur in the data. The f o l l o w i n g d i s c u s s i o n e x p l a i n s these g r a p h i c a l techniques and the a s s o c i a t e d terminology i n the context of the c u r r i c u l u m v a r i a b l e s examined i n t h i s study. 1.1 The Stem-and-Leaf P l o t To c o n s t r u c t a stem-and-leaf p l o t each data value i s f i r s t s p l i t at the l a s t p a i r of adjacent d i g i t s . For example, a value of 22.9 i n a data set would appear as 22|9 and every other value in the set would be s i m i l a r l y s p l i t between the ones d i g i t and the tenths d i g i t s . The emphasis which a teacher gave to a p a r t i c u l a r content area, C, was def i n e d n u m e r i c a l l y as the p r o p o r t i o n of time a l l o c a t e d to that content area r e l a t i v e to the time a l l o c a t e d to a l l three content areas. The p o s s i b l e values f o r t h i s v a r i a b l e were thus between 0.0 and 1.0 i n c l u s i v e . In c o n s t r u c t i n g the stem-and-leaf p l o t s for t h i s v a r i a b l e each content emphasis score was f i r s t rounded to the hundredths d i g i t . For the content area of aithm e t i c these scores ranged from a low of 0.00 84 to a high of 0.66. On a stem-and-leaf p l o t these two values appear as 0|0 and 6|6. The l e a d i n g d i g i t s of the data values form the "stem" of the p l o t . To c o n s t r u c t the d i s p l a y these values are w r i t t e n in a v e r t i c a l column which i s followed by a v e r t i c a l l i n e . The d i s p l a y i s completed by w r i t i n g down the t r a i l i n g d i g i t (the " l e a f " ) of each data value on the l i n e corresponding to i t s lea d i n g d i g i t . The leaves are w r i t t e n i n numerical order on each l i n e . 0 5689 00023455667788889 00011233444555566777888889 0011122333344567 1 1 477 6 0 1 2 3 4 5 6 7 8 9 1 0 1|5 represents 0.15 N=70 0 1 1 2 2 3 4 5 5 6 6 7 8 9 1 0 0 56 89 000234 55667788889 00011233444555566777888889 0011122333344567 1 1 477 6 Figu r e 4- 1 - D i s t r i b u t i o n of a r i t h m e t i c emphasis scores ( i l l u s t r a t i v e stem-and-leaf p l o t ) . In F i g u r e 4-1 the content emphasis scores f o r a r i t h m e t i c are shown. The p l o t at the l e f t i s a standard stem-and-leaf d i s p l a y . The p l o t at the r i g h t i s a modified d i s p l a y showing the d i s t r i b u t i o n of content emphasis scores f o r a r i t h m e t i c i n t o the f i v e s p e c i f i e d l e v e l s : 85 0.66 < C ^ 1.00 very heavy emphasis 0.50 < C ^ 0.66 heavy emphasis 0.25 ^ C ^ 0.50 moderate emphasis 0.17 < C < 0.25 l i g h t emphasis 0.00 <• C < 0.17 very l i g h t emphasis Note that i t i s necessary to write s e v e r a l of the l e a d i n g d i g i t s twice i n the stem s i n c e , f o r example, 0.16 and 0.18 are i n d i f f e r e n t l e v e l s . M o d i f i e d stem-and-leaf p l o t s w i l l be used throughout t h i s chapter in place of the standard p l o t s s i n c e they provide a d d i t i o n a l information while s t i l l showing the basic d i s t r i b u t i o n of s c o r e s . h i g h l y p e r c e p t u a l 0 1 somewhat perc e p t u a l 2 3 4 3 balanced 9972 86222220 0 4 5 6 377899 33588 somewhat a b s t r a c t 8511 75322210 6 7 8 123337799 2 h i g h l y a b s t r a c t 995 N. =28 8 9 10 2 012 00 N H = 28 3|3 represents 0.33 low high achievement achievement Fig u r e 4- 2 - D i s t r i b u t i o n of mode of r e p r e s e n t a t i o n scores for a r i t h m e t i c f o r low and high achievement c l a s s e s . To compare the d i s t r i b u t i o n s of scores for low and high 86 achievement c l a s s e s on one of the c u r r i c u l u m v a r i a b l e s , b a c k - t o -back s t e m - a n d - l e a f p l o t s are used . Us ing t h i s d i s p l a y , the l eaves for low achievement c l a s s e s appear to the l e f t of the stem whi l e the l eaves for h i g h achievement c l a s s e s appear to the r i g h t of the stem. To i l l u s t r a t e t h i s p l o t , the mode of r e p r e s e n t a t i o n scores for a r i t h m e t i c content are shown for c l a s s e s of low and h igh achievement in F i g u r e 4-2 . The p l o t i s a m o d i f i e d v e r s i o n showing the d i s t r i b u t i o n of mode of r e p r e s e n t a t i o n scores us ing the f i v e s p e c i f i e d l e v e l s : 0.80 < L < 1.00 h i g h l y a b s t r a c t 0.60 < L <, 0.80 somewhat a b s t r a c t 0.40 < L 5 0.60 ba lanced 0.20 ^ L < 0.40 somewhat p e r c e p t u a l 0.00 < L < 0.20 h i g h l y p e r c e p t u a l In c o n s t r u c t i n g s t e m - a n d - l e a f p l o t s for the d i v e r s i t y of i n s t r u c t i o n v a r i a b l e , i t was necessary to spread out the data by u s i n g two l i n e s for each stem. On one l i n e the d i g i t s 0-4 which o c c u r r e d as l eaves were w r i t t e n ; on the other l i n e the d i g i t s 5-9 were w r i t t e n . In F i g u r e 4-3 the d i s t r i b u t i o n of o v e r a l l d i v e r s i t y s cores for a r i t h m e t i c i s shown. Each o v e r a l l score r e p r e s e n t s the average number of t e a c h i n g methods a teacher used in p r e s e n t i n g each a r i t h m e t i c t o p i c . Note t h a t 1|3 r e p r e s e n t s 1.3 and not 0.13 as i t would i n the p l o t s for the other v a r i a b l e s . The p l o t i s a m o d i f i e d v e r s i o n of the s tandard 87 low d i v e r s i t y moderate d i v e r s i t y high d i v e r s i t y 1|3 represents 1.3 33 5 7889 0000111122223333333444 555555666667778899999 00000111122223444 5566667779999 022 5 N=85 Figur e 4- 3 - D i s t r i b u t i o n of o v e r a l l d i v e r s i t y scores f o r a r i t h m e t i c d i s p l a y showing the d i s t r i b u t i o n of scores i n t o the three s p e c i f i e d l e v e l s : D > 3.00 high d i v e r s i t y 1.50 < D < 3.00 moderate d i v e r s i t y 0.50 £ D < 1.50 low d i v e r s i t y 1 .2 The Boxplot The c o n s t r u c t i o n of a boxplot from a set of ordered data i s e a s i l y c a r r i e d out by s o r t i n g and counting. The d e s c r i p t i o n of t h i s process as w e l l as the i n t e r p r e t a t i o n and comparison of boxplots, however, r e q u i r e s the use of some t e c h n i c a l terminology. The d e f i n i t i o n s which follow presuppose that a given set of N observations i s arranged i n t o ascending order. The p o s i t i o n of a data value r e f e r s to i t s place within t h i s o r d e r i n g . 88 Lower Extreme: the l e a s t data value. Upper Extreme: the g r e a t e s t data value. Upward Rank: the p o s i t i o n of a data value counting upward from the lower extreme. Downward Rank: the p o s i t i o n of a data value counting downward from the upper extreme. Depth: the smaller of the upward and downward ranks of a given data value. Median: the data value whose depth i s (n+1)/2. If the depth of the median i s not an i n t e g e r , the median i s determined by i n t e r p o l a t i n g between the two data values whose depths a-re nearest the depth of the median. Lower Fourth ( F L ) and Upper Fourth (F ): the data values whose upward and downward ranks r e s p e c t i v e l y are given by the f o l l o w i n g equation: depth of fourths = [depth of median] + 1 2 where [X] stands for the l a r g e s t i n t e g e r not exceeding X. If the depth of the fourths i s not an i n t e g e r , the lower and upper f o u r t h s are determined by i n t e r p o l a t i n g between the ; data values whose depths are nearest the depth of the f o u r t h s . Fourth-spread or F-spread (Dp): the number determined by s u b t r a c t i n g the lower f o u r t h from the upper/ f o u r t h . Lower O u t l i e r C u t o f f : the value of F^ - 1.50Dp. 89 Upper O u t l i e r C u t o f f : the value of F u + 1.50Dp.1 O u t l i e r : any data value which i s l e s s than the lower o u t l i e r c u t o f f or g r e a t e r than the upper o u t l i e r c u t o f f . The c o n s t r u c t i o n of a boxplot w i l l be i l l u s t r a t e d using the data given i n the stem-and-leaf p l o t of Figure 4-1. Since n = 70, the median has a depth of 35.5. I t s value i s 0.35. The depth of the fourths i s 18. The lower and upper f o u r t h s are 0.28, and 0.41 r e s p e c t i v e l y . The p l o t i s begun by drawing a r e c t a n g l e or box using the f o u r t h s to determine two s i d e s . The box thus shows the l o c a t i o n of the c e n t r a l 50% of the data. The F-spread, 0.13, i s the length of the box. The median i s i n d i c a t e d by a segment w i t h i n the box. The o u t l i e r c u t o f f s are determined next. In t h i s case the values are 0.28 - (1.5)x(0.13) rounded to 0.09, and 0.41 + (1.5)x(0.13), rounded to 0.61. Two t a i l s or "whiskers" are drawn from the box. The lower t a i l i s drawn to the g r e a t e s t data value not l e s s than the lower c u t o f f , in t h i s case 0.15. The upper t a i l i s drawn to the g r e a t e s t data value not exceeding the upper c u t o f f , i n t h i s case 0.57. There are two o u t l i e r s , 0.00 and 0.66. These are i n d i c a t e d by Xs on the p l o t . If e i t h e r o u t l i e r value had occurred more than once in the data set, the number of occurrences would have been i n d i c a t e d in 1 For normally d i s t r i b u t e d data s l i g h t l y l e s s than 0.7% of the o b s e r v a t i o n s would be o u t l i e r s using these standard d e f i n i t i o n s for o u t l i e r c u t o f f s . ( H o a g l i n , 1983, p.40) 90 0,80 0.60 M o o r o r R r P R f s c NT AT 1 ON 0,20 P E P C f P T U A L 0 > 0 0 , Figure 4 - 4 - The d i s t r i b u t i o n of a r i t h m e t i c emphasis scores ( i l l u s t r a t i v e b o x p l o t ) . parentheses a f t e r the X. The completed boxplot i s shewn in F i g u r e 4 - 4 . 91 2. CONTENT EMPHASIS 2.1 The Emphasis Given To A r i t h m e t i c F i g u r e 4-5 i s a stem-and-leaf p l o t which shows the pr o p o r t i o n of time each teacher spent on a r i t h m e t i c in h i s or her c l a s s . It i s a modified v e r s i o n of F i g u r e 4-1. The d i s t r i b u t i o n i s c l o s e to normal i n form and has a median value of 0.35. The lower f o u r t h i s 0.28, the upper f o u r t h i s 0.41, and the fourth-spread i s 0.13. Thus, about h a l f of the teachers spent between 28% and 41% of t h e i r time on t h i s review area of Mathematics 8. very l i g h t emphasis l i g h t emphasis moderate emphasis heavy emphasis very heavy emphasis 0 1 1 2 2 3 4 5 5 6 6 7 8 9 10 0 56 89 000234 55667788889 00011233444555566777888889 0011122333344567 1 1 477 6 N=70 Fig u r e 4- 5 - D i s t r i b u t i o n of a r i t h m e t i c emphasis scores. Three of the teachers i n t h i s study had content emphasis scores for a r i t h m e t i c which were i n the very l i g h t emphasis category. One of these teachers spent no time at a l l on t h i s 92 area. An a d d i t i o n a l e i g h t teachers had scores w i t h i n the l i g h t emphasis category. A l t o g e t h e r 11 teachers devoted l e s s than 25% of t h e i r o p e r a t i o n a l c u r r i c u l u m to a r i t h m e t i c . No teacher gave a r i t h m e t i c very heavy emphasis. Six teachers, however, d i d give a r i t h m e t i c heavy emphasis by spending over 50% of t h e i r i n s t r u c t i o n a l c l a s s time on that area. The remaining 53 t e a c h e r s , 2 76% of the t o t a l , gave a r i t h m e t i c moderate emphasis. These teachers thus a l l o c a t e d between one-quarter and one-half of t h e i r i n s t r u c t i o n a l time to a r i t h m e t i c content. Since student assessment r e s u l t s have not always been s a t i s f a c t o r y in a r i t h m e t i c for Grade 8 students (e.g., R o b i t a i l l e , 1981,pp. 134-142), one can argue that a r i t h m e t i c should continue to be taught to students at t h i s grade l e v e l . However, i t i s not c l e a r that i n s t r u c t i o n i n these t o p i c s for a l l students i s necessary or d e s i r a b l e . One might expect that d i f f e r e n c e s in the amount of time teachers give to a r i t h m e t i c would be r e l a t e d to the achievement l e v e l of the c l a s s with low achievement c l a s s e s r e c e i v i n g more review of a r i t h m e t i c content than high achievement c l a s s e s . There was, in f a c t , some tendency i n t h i s d i r e c t i o n in the o p e r a t i o n a l c u r r i c u l a which were i n v e s t i g a t e d . The d i s t r i b u t i o n s of a r i t h m e t i c emphasis scores f o r the low 2 The data f o r t h i s v a r i a b l e c o n s i s t of 70 scores. There were more missing data f o r t h i s v a r i a b l e than for the other c u r r i c u l u m v a r i a b l e s . T h i s was due to the f a c t that a teacher c o u l d not have a score on t h i s v a r i a b l e unless he or she returned a l l f i v e TSQs and i n each case provided a response for the time a l l o c a t i o n items. 93 and high achievement c l a s s e s are shown s e p a r a t e l y i n Figure 4-6. Although only four teachers of low achievement c l a s s e s spent under 30% of t h e i r i n s t r u c t i o n a l time on a r i t h m e t i c , ten teachers of high achievement c l a s s e s d i d so. While three teachers of low achievement c l a s s e s devoted over 50% of t h e i r courses to a r i t h m e t i c , no teacher of a high achievement c l a s s spent that much time on that area. very l i g h t emphasis l i g h t emphasis moderate emphasis heavy emphasis very heavy emphasis 8877554443321 6330 71 6 NL=24 low achievement c l a s s e s 0 1 1 2 2 3 4 5 5 6 6 7 8 9 1 0 8 00234 5669 0016678 1134457 N =24 H high achievement c l a s s e s Figure 4- 6 - D i s t r i b u t i o n of a r i t h m e t i c emphasis scores for low and high achievement l e v e l c l a s s e s . The expected d i f f e r e n c e s i n a r i t h m e t i c emphasis for low and high achievement l e v e l c l a s s e s were found when extreme cases were considered. Otherwise, the d i s t r i b u t i o n s were s i m i l a r . In 75% of each group of c l a s s e s the emphasis of a r i t h m e t i c was moderate. The median values of emphasis f o r a r i t h m e t i c were 0.35 and 0.31 for the low and high /achievement c l a s s e s 94 r e s p e c t i v e l y . Thus, while more time was spent on a r i t h m e t i c content i n the median low achievement l e v e l c l a s s than in the median high achievement l e v e l c l a s s , the d i f f e r e n c e was s l i g h t . 3 These r e s u l t s provide some cause for concern, p a r t i c u l a r l y with regard to the amount of time that was a l l o c a t e d to a r i t h m e t i c i n most high achievement l e v e l c l a s s e s . I t i s arguable that too much time was spent teaching a r i t h m e t i c i n these c l a s s e s . It i s not c l e a r , however, on what b a s i s these teachers decided to a l l o c a t e t h i s much time to review m a t e r i a l . It i s p o s s i b l e that they were not aware of the achievement l e v e l of t h e i r c l a s s e s . " It i s a l s o p o s s i b l e that the i n c l u s i o n of a r i t h m e t i c t o p i c s i n the a u t h o r i z e d t e x t s was an i n f l u e n t i a l f a c t o r in teachers' d e c i s i o n s regarding content s e l e c t i o n . A t h i r d p o s s i b i l i t y i s that these teachers b e l i e v e d that an extensive review of a r i t h m e t i c would f u r t h e r enhance performance and r e t e n t i o n f o r t h e i r high achievement l e v e l c l a s s e s . In any event, the f i n d i n g that i n most high achievement l e v e l c l a s s e s a r i t h m e t i c r e c e i v e d the same l e v e l of moderate emphasis that i t r e c e i v e d in most low achievement l e v e l c l a s s e s c o u l d i n d i c a t e that there i s a need to s p e c i f y options in the formal c u r r i c u l u m for c l a s s e s of low and high achievement in t h i s area and to 3 For a l l c l a s s e s the c o r r e l a t i o n between the c l a s s Core P r e t e s t mean and the content emphasis score f o r a r i t h m e t i c was -0.23. This i n d i c a t e s a weak tendency for more a r i t h m e t i c i n s t r u c t i o n in lower achievement c l a s s e s . " Subsequent a n a l y s i s of the B.C. SIMS data has shown a strong p o s i t i v e a s s o c i a t i o n between these teachers' p e r c e p t i o n of the achievement l e v e l of t h e i r c l a s s e s and achievement l e v e l based on core p r e t e s t scores. Further d i s c u s s i o n of t h i s issue i s provided i n Chapter 9 5 provide the necessary i n s t r u c t i o n a l m a t e r i a l s . 2.2 The Emphasis Given To Algebra F i g u r e 4-7 shows the p r o p o r t i o n of time each teacher in t h i s study a l l o c a t e d to algebra content. T h i s d i s t r i b u t i o n has a median of 0.29, a lower f o u r t h of 0.24 and an upper f o u r t h of 0.35. Each of these values i s somewhat lower than the corresponding value for a r i t h m e t i c i n d i c a t i n g l e s s emphasis in the implemented c u r r i c u l u m on a l g e b r a than a r i t h m e t i c . The fourth-spread i s 0.11 i n d i c a t i n g j u s t s l i g h t l y l e s s than the l e v e l of v a r i a t i o n which was present i n the a r i t h m e t i c emphasis scores f o r the middle h a l f of the d i s t r i b u t i o n . very l i g h t emphasis 0 1 66 l i g h t emphasis 1 2 889 0122222333334 moderate emphasis 2 3 4 5 566677777788888899 0000111122234455567889 12233336788 0 heavy emphasis 5 6 very heavy emphasis 6 7 8 9 10 N=70 Fig u r e 4- 7 - D i s t r i b u t i o n of algebra emphasis scores. Most teachers, 52 out of 70 or 74%, gave moderate emphasis to a lgebra in t h e i r o p e r a t i o n a l c u r r i c u l a . No teacher gave t h i s 96 content area heavy or very heavy emphasis. On-the-other-hand, 16 teachers gave algebra l i g h t emphasis and two teachers gave i t very l i g h t emphasis. Algebra was, however, part of every o p e r a t i o n a l c u r r i c u l u m r e c e i v i n g no l e s s than 16% of the i n s t r u c t i o n a l time. I t i s not c l e a r that one should expect s u b s t a n t i a l d i f f e r e n c e s in the amount of time given to algebra i n high achievement c l a s s e s as compared to low achievement c l a s s e s . One might expect that those teachers of high achievement l e v e l c l a s s e s who spent r e l a t i v e l y l i t t l e time reviewing a r i t h m e t i c would have spent a cor r e s p o n d i n g l y larger, amount of time on algeb r a , geometry or other content such as p r o b a b i l i t y . F i g u r e 4-8 shows the d i s t r i b u t i o n s of algebra emphasis scores f o r the achievement groups s e p a r a t e l y . Although the d i s t r i b u t i o n s are not i d e n t i c a l , they are c e r t a i n l y s i m i l a r . The median p r o p o r t i o n of time given to algebra i n low achievement c l a s s e s was 0.31 compared to 0.29 in high achievement c l a s s e s . In both cases the lower f o u r t h i s 0.27. In f i v e c l a s s e s i n each group, algebra r e c e i v e d l i g h t emphasis. The major d i f f e r e n c e between the two achievement d i s t r i b u t i o n s i n v o l v e s the number of c l a s s e s in which over 40% of the i n s t r u c t i o n a l time was devoted to a l g e b r a . For low achievement c l a s s e s t h i s number was e i g h t , f o r high achievement c l a s s e s i t was two. As a r e s u l t , t h e upper f o u r t h f o r low achievement c l a s s e s i s 0.43 compared to 0.32 f o r high achievement c l a s s e s . Thus, the hypothesis that high achievement c l a s s e s might show a stronger tendency to emphasize algebra was 97 very l i g h t emphasis 0 6 l i g h t emphasis 32222 2 01 34 moderate emphasis 998776 76220 88633322 2 3 4 5 6778888 0001112355 17 heavy emphasis 5 6 very heavy emphasis 6 7 8 9 1 0 low achievement c l a s s e s high achievement c l a s s e s F i g u r e 4- 8 - D i s t r i b u t i o n s of algebra emphasis scores for low and high achievement c l a s s e s . not borne out. The median, upper f o u r t h , and lower f o u r t h were a l l higher f o r the low achievement c l a s s scores than f o r the high achievement c l a s s s c o r e s . 5 2.3 The Emphasis Given To Geometry Fi g u r e 4-9 shows the p r o p o r t i o n of time each teacher spent teaching geometry content. T h i s d i s t r i b u t i o n has a median of 0.36 and lower and upper fourths of 0.28 and 0.44 r e s p e c t i v e l y . Thus, about h a l f of the teachers spent between 28% and 44% of t h e i r time on geometry, an area that i n c l u d e d t o p i c s which are 5 For a l l c l a s s e s the c o r r e l a t i o n between the c l a s s Core p r e t e s t mean and the content emphasis score for algebra was -0.15. This i n d i c a t e s a very weak tendency f o r more al g e b r a i n s t r u c t i o n in lower achievement c l a s s e s . 98 not part of the formal c u r r i c u l u m in p r i o r grades. F i v e of the teachers gave very l i g h t emphasis to geometry. Of these, two spent no time at a l l teaching geometry. An a d d i t i o n a l 10 teachers gave l i g h t emphasis to t h i s content area. None of the content emphasis scores f o r geometry were w i t h i n the very heavy emphasis category. Seven scores, however, were w i t h i n the heavy emphasis category. very l i g h t emphasis 0 1 0077 1 l i g h t emphasis 1 2 7889 233444 moderate emphasis 2 3 4 5 57888999 003335555666677778899 111233344456677779 0 heavy emphasis 5 6 1 2355 46 very heavy emphasis 6 7 8 9 10 N=70 Fig u r e 4- 9 - D i s t r i b u t i o n of geometry emphasis scores. As with the other two content areas, most of the geometry emphasis scores were in the moderate category; 48 of the 70 scores (69%). T h i s value i s somewhat below the corresponding values f o r a r i t h m e t i c and a l g e b r a . As with algebra, i t was not c l e a r whether to expect teachers of low achievement c l a s s e s to spend more or l e s s time on geometry than teachers of high achievement c l a s s e s . One 99 might expect l e s s emphasis on geometry i n low achievement c l a s s e s due to a greater s t r e s s on a r i t h m e t i c . A l t e r n a t e l y , one might expect more emphasis on geometry i n low achievement c l a s s e s due to the p o s s i b i l i t i e s f o r student e x p l o r a t i o n and the use of concrete m a t e r i a l s in teaching t h i s content area. In f a c t , teachers of low achievement c l a s s e s tended to spend l e s s time on geometry than teachers of high achievement c l a s s e s as shown in Figure 4-10. Both d i s t r i b u t i o n s appear approximately normal and appear to have s i m i l a r spreads but d i f f e r i n g c e n t r a l values. The median value for the low achievement c l a s s e s i s 0.32 compared to 0.40 f o r the high achievement c l a s s e s . The lower fourths of the two d i s t r i b u t i o n s are 0.23 and 0.30; the upper fourths are 0.39 and 0.47. While the content emphasis i n nine low achievement c l a s s e s was l i g h t or very l i g h t , t h i s was true in only two high achievement c l a s s e s . ' While geometry r e c e i v e d heavy emphasis in four high achievement c l a s s e s , i t r e c e i v e d t h i s degree of emphasis i n only one low achievement c l a s s . A l l of these d e s c r i p t i v e s t a t i s t i c s r e i n f o r c e the d i f f e r i n g v i s u a l f e a t u r e s of the two d i s t r i b u t i o n s . 6 The greater emphasis which geometry r e c e i v e d i n high achievement c l a s s e s compared to the emphasis i t r e c e i v e d i n low achievement c l a s s e s may represent an u n d e s i r a b l e s t a t e of a f f a i r s . Geometry approached i n an informal and e x p e r i e n t i a l 6 For a l l c l a s s e s the c o r r e l a t i o n between the c l a s s Core p r e t e s t mean and the content emphasis score f o r geometry was +0.28. This i n d i c a t e s a weak tendency for more geometry i n s t r u c t i o n i n higher achievement c l a s s e s . 1 00 very l i g h t emphasis 70 0 7 l i g h t emphasis 987 4432 8 2 moderate 885 9877665 7442 2 3 4 5 899 0035668 11334779 heavy emphasis 5 5 6 123 4 very heavy emphasis 6 7 8 9 0 low achievement c l a s s e s high achievement c l a s s e s Figure 4-10 - D i s t r i b u t i o n s of geometry emphasis scores f o r low and high achievement l e v e l c l a s s e s . manner i s probably j u s t as important, i f not more important, f o r the low as f o r the high achievement student. I f so, s t r a t e g i e s which c o u l d reduce the time needed to review and extend a r i t h m e t i c content with low achievement c l a s s e s , such as a great e r use of c a l c u l a t o r s , need to rec e i v e more c o n s i d e r a t i o n . 2.4 Comparisons Among The Content Areas F i g u r e 4-11 shows the d i s t r i b u t i o n s of emphasis scores f o r the three content areas. Boxplots have been used to f a c i l i t a t e comparisons. As noted above, the median time a l l o c a t i o n s were: a r i t h m e t i c , 0.35; algebra 0.29; and geometry, 0.36. While the alg e b r a d i s t r i b u t i o n has the lowest median value, i t a l s o has the lowest spread i n d i c a t i n g that somewhat more u n i f o r m i t y in emphasis occured for al g e b r a than for the other two areas. In 101 c o n t r a s t , the geometry d i s t r i b u t i o n shows the g r e a t e s t spread. The F-spread values for a r i t h m e t i c , a l g e b r a , and geometry are 0.13, 0.11, and 0.16 r e s p e c t i v e l y . Despite these d i f f e r e n c e s , however, the o v e r a l l p a t t e r n s of emphasis for the three content areas are s t r i k i n g l y s i m i l a r . 1.00 0.80 0.60 o.uo 0.20 T 0.00 X (2) A R I T H M C T I C A L Q C B R A G C O M C T R Y F i g u r e 4-11 - D i s t r i b u t i o n of content emphasis scores In Table 4-1 are the percent of teachers whose content emphasis scores f e l l i n t o each of the f i v e c a t e g o r i e s f o r each content area. For each area the m a j o r i t y of teachers p r o v i d e d a moderate l e v e l of emphasis. No teacher gave a content area very 1 02 heavy emphasis and only 5% of a l l scores were wit h i n the very l i g h t range. The corresponding values f o r heavy and l i g h t emphasis were 6% and 16% r e s p e c t i v e l y . Despite the preponderance of moderate emphasis scores, 60% of a l l teachers surveyed gave at l e a s t one area l i g h t or very l i g h t emphasis. Table 4- 1 - Percent of Teachers Scoring Within Each L e v e l of Content Emphasis for Each Content Area Content Area Content Emphasis D i s t r i b u t i o n Median (% of scores i n Category) Score Very L i g h t Moderate Heavy Very L i g h t Heavy A r i t h m e t i c 4.3 11.4 75.7 8.6 0.0 0.35 Algebra 2.9 22.9 74.3 0.0 0.0 0.29 Geometry 7.1 14.3 68.6 10.0 0.0 0.36 F i g u r e 4-12 shows the d i s t r i b u t i o n s of content emphasis scores for low and high achievement c l a s s e s for each content area. As noted above, geometry r e c e i v e d greater emphasis in high achievement c l a s s e s than in low achievement c l a s s e s . Both a r i t h m e t i c and algebra r e c e i v e d greater emphasis in low achievement c l a s s e s . The f a c t that the d i s t r i b u t i o n of algebra emphasis scores f o r high achievement c l a s s e s has r e l a t i v e l y short t a i l s as well as a r e l a t i v e l y small F-spread of 0.05 means that the scores show l e s s v a r i a t i o n from the median than cases where d i s t r i b u t i o n s have longer t a i l s and l a r g e r F-spreads. With the exception of the three o u t l i e r s , a l l of the scores are rather t i g h t l y bunched. In f a c t , none of these three values would be o u t l i e r s w i t h i n any of the other d i s t r i b u t i o n s . 103 T h i s shows f u r t h e r the r e l a t i v e u n i f o r m i t y among the teachers of high achievement c l a s s e s in t h e i r time a l l o c a t i o n s to a l g e b r a . In c o n t r a s t , the d i s t r i b u t i o n of a r i t h m e t i c emphasis scores f o r low achievement c l a s s e s while having a small F-spread has r e l a t i v e l y long t a i l s . T h i s means that while the middle h a l f of the teachers i n t h i s group emphasized a r i t h m e t i c rather uniformly, there was r e l a t i v e l y high d i v e r s i t y among the other h a l f of the teachers in t h i s group. Each of the other four d i s t r i b u t i o n s shown i n F i g u r e 4-12 has an F-spread of e i t h e r 0.16 or 0.17. Thus, there was much more v a r i a t i o n among the middle h a l f of the teachers i n these cases than i n the two alr e a d y d i s c u s s e d . The geometry emphasis d i s t r i b u t i o n s f o r both low and high achievement c l a s s e s show the g r e a t e s t o v e r a l l v a r i a t i o n s i n c e the F-spreads are r e a l a t i v e l y l a r g e and the t a i l s at both ends are c o n s i d e r a b l y longer than i s the case for the other d i s t r i b u t i o n s . Thus, even when c l a s s achievement i s taken i n t o c o n s i d e r a t i o n , one f i n d s that the l e a s t u n i f o r m i t y occured regarding how much time should be spent teaching geometry. 2.5 Content Emphasis Of Teachers And Textbooks I t has been a s s e r t e d f a i r l y f r e q u e n t l y that school mathematics i n s t r u c t i o n i s textbook o r i e n t e d i n that a text i s u s u a l l y used and c l o s e l y followed (e.g., Begle, 1973; Fey, 1979). One aspect of t h i s study was to explore the strengt h of the l i n k between t h i s component of the formal c u r r i c u l u m and the o p e r a t i o n a l c u r r i c u l u m of the classroom. S p e c i f i c a l l y , were 1 04 1.00 9.PO 0.60 O.l»0 0.20 0.00 I I 1 T T LOW H I R H A R I T H M C T I C LOW H I C H A L G t D t A LOW H I C H ccoucTny Figure 4-12 - D i s t r i b u t i o n s of content emphasis scores for low and high achievement l e v e l c l a s s e s . d i f f e r i n g emphases in textbooks r e f l e c t e d in the i n s t r u c t i o n of teachers using those books? The vB.C. Mathematics Curriculum Guide a u t h o r i z e s the use of three textbooks f o r Mathematics 8. In p r a c t i c e , two of these books are used with about the same degree of frequency by teachers while the t h i r d book i s seldom used/as the b a s i c t e x t ( R o b i t a i l l e , 19S1, p. 244). The two widely used books are Mathematics II (Sobel & Maletsky, 1971) and School Mathematics 2 105 (Fleenor, E i c h o l z , & O'Daffer, 1975). Table 4-2 shows the percent of each textbook devoted to a r i t h m e t i c , a l g e b r a , geometry, and other c o n t e n t . 7 Although "other content" i n c l u d e s a s i z e a b l e p r o p o r t i o n of the content of each textbook, i t c o n s i s t s p r i m a r i l y of the m a t e r i a l at the end of each book. Moreover, most of t h i s m a t e r i a l i s i n areas such as trigonometry and p r o b a b i l i t y which are not part of the formal c u r r i c u l u m at t h i s grade l e v e l . An i n s p e c t i o n of Table 4-2 shows that the two Table 4- 2 - Percent of the Commonly Used Textbooks Devoted to Each Content Area. Content Area Textbook A r i t h m e t i c Algebra Geometry Other School Mathematics 2 28. 1 16.0 22.0 33.9 Mathematics II 18.2 16.3 39.0 26.5 t e x t s d i f f e r markedly i n the emphasis given to a r i t h m e t i c and geometry while they provide n e a r l y i d e n t i c a l emphasis to al g e b r a . A r i t h m e t i c r e c e i v e s 54% more emphasis in School Mathematics 2 than i n Mathematics II while geometry r e c e i v e s 77% more emphasis in Mathematics II than i n the other t e x t . In c o n t r a s t to these l a r g e d i f f e r e n c e s , algebra r e c e i v e s j u s t 2% more emphasis i n Mathematics I I . These percents were determined by f i r s t c a t e g o r i z i n g each page of a.text which contained mathematical content according to the content area with which i t d e a l t . The t o t a l number of pages devoted to a r i t h m e t i c i n a t e x t , f o r example, compared to the t o t a l number of pages i n that text which contained mathematical content was then used as the percent of that te x t devoted to a r i t h m e t i c . 106 Figure 4-13 d i s p l a y s the d i s t r i b u t i o n s of content emphasis scores s e p a r a t e l y according to the basic textbook used in each c l a s s . The median values for the algebra emphasis d i s t r i b u t i o n s are almost i d e n t i c a l at 0.30 and 0.28. The medians for the a r i t h m e t i c d i s t r i b u t i o n s d i f f e r much more. For c l a s s e s which used School Mathematics 2 the median for a r i t h m e t i c i s 0.40 while for c l a s s e s which used Mathematics II the median i s 0.29. This d i f f e r e n c e i s c o n s i s t e n t with the d i f f e r e n c e i n emphasis in the books themselves. The d i f f e r e n c e between the medians of the two d i s t r i b u t i o n s of geometry emphasis scores i s s i m i l a r l y c o n s i s t e n t with the d i f f e r e n c e i n emphasis of geometry in the two t e x t s . For c l a s s e s which used School Mathematics 2 the median f o r geometry i s 0.28 while for c l a s s e s which used Mathematics II the median i s 0.41. These r e s u l t s can be i n t e r p r e t e d as supporting the hypothesis that the content of the formal c u r r i c u l u m as embodied by a textbook has an observable i n f l u e n c e on the o p e r a t i o n a l c u r r i c u l a of teachers. The two books which were used as basic t e x t s contained v i r t u a l l y the same number of pages of algebra m a t e r i a l and the d i s t r i b u t i o n s of a l g e b r a emphasis scores f o r the groups of teachers using each book were very s i m i l a r . Likewise, the d i f f e r e n c e s in emphasis of a r i t h m e t i c and geometry content i n the books were a s s o c i a t e d with c o n s i s t e n t d i f f e r e n c e s between the o p e r a t i o n a l c u r r i c u l a of the teachers using the books. The a s s o c i a t i o n which was found between textbook emphasis of content and teacher emphasis of content can a l s o be 107 ».on o.no 0.60 o.fco 0.20 0..00 T 1 X j_ 1 T T SM2 Ml I APITHUCTIC SM2 Ml ALGCBRA SM2 M|| QCOMCTKT F i g u r e 4 - 1 3 - D i s t r i b u t i o n s o f c o n t e n t e m p h a s i s s c o r e s f o r c l a s s e s u s i n g e a c h o f t h e c o m m o n l y u s e d t e x t b o o k s . i n t e r p r e t e d a s s u p p o r t i n g t h e v a l i d i t y o f t h e s e l f - r e p o r t s o f t h e t e a c h e r s i n t h i s s t u d y . I f o n e t a k e s a s a g i v e n t h a t t h e o p e r a t i o n a l c u r r i c u l a w i l l b e s t r o n g l y i n f l u e n c e d by t h e c o v e r a g e g i v e n t o c o n t e n t i n t h e t e x t b o o k s , t h e n t h e s e l f -r e p o r t s o f t e a c h e r s a r e v a l i d o n l y i n s o f a r a s t h e r e i s a c o r r e s p o n d e n c e b e t w e e n t h e c o n t e n t e m p h a s i s t e a c h e r s r e p o r t e d a n d t h e e m p h a s i s g i v e n t o c o n t e n t i n t h e b o o k s t h e y u s e d . F o r t h e t e a c h e r s who p a r t i c i p a t e d i n t h i s s t u d y s u c h a c o r r e s p o n d e n c e d i d e x i s t . 108 3. MODE OF CONTENT REPRESENTATION The content r e p r e s e n t a t i o n v a r i a b l e was d e f i n e d so that a score of 0.0 i n d i c a t e s the use of only p e r c e p t u a l approaches to content i n teaching a t o p i c while, at the other extreme, a score of 1.0 i n d i c a t e s a r e l i a n c e on only a b s t r a c t approaches. In Table 4-3 the percentage of teachers whose mode of re p r e s e n t a t i o n scores are i n each of three c a t e g o r i e s 8 i s given f o r each t o p i c , each content area and o v e r a l l . The content area and o v e r a l l scores were obtained by averaging the app r o p r i a t e t o p i c scores f o r each teacher. The t a b l e a l s o c o n t a i n s median content r e p r e s e n t a t i o n scores as w e l l as the p r o p o r t i o n of the teaching methods contained in the TSQs which were c l a s s i f i e d as a b s t r a c t . The median content r e p r e s e n t a t i o n score a c r o s s a l l t o p i c s i s 0.57 i n d i c a t i n g that o v e r a l l teachers used a b s t r a c t approaches to t h e i r course content somewhat more f r e q u e n t l y than p e r c e p t u a l approaches. S l i g h t l y over o n e - t h i r d of the o v e r a l l scores are with i n the a b s t r a c t r e p r e s e n t a t i o n category, with 63% in the balanced category. Only 2% of the scores are below 0.40 and thus c l a s s i f i e d as i n d i c a t i n g a perc e p t u a l o r i e n t a t i o n to content. The d i s t r i b u t i o n of o v e r a l l content r e p r e s e n t a t i o n scores i s shown in Figure 4-14. I t i s nearly normal i n form and con t a i n s two scores i n the h i g h l y abstact category. These 8 The somewhat perceptual and h i g h l y p e r c e p t u a l c a t e g o r i e s of content r e p r e s e n t a t i o n are combined i n t o a s i n g l e perceptual category f o r t h i s t a b l e . The somewhat a b s t r a c t and h i g h l y a b s t a c t c a t e g o r i e s are s i m i l a r l y combined. The o r i g i n a l c a t e g o r i z a t i o n i s used in the stem-and-leaf p l o t s which f o l l o w . 1 0 9 T a b l e 4- 3 - Mode of C o n t e n t R e p r e s e n t a t i o n S c o r e s T o » t c IM M I * M o o t MT R C P K f MMTATIOM PROPORTION o r HID IAN ( t f o r ICORCB IM C A T t O O R V ) ABSTRACT MXTHOOS Pit PMC AC MT A T 1 OM IM T S Q s S c o r n P t A C t P T U A l •ALAMCCD A B S T R A C T FaACTIONS 32.1 55.1 12.8 0.60 O . M FRACTION tooIT ION M . I 32.0 to.o 0.3» o.*o OCCIMALS 0.0 13.2 76.8 0.50 0.71 DCCIMAL O r r R A T l o w s 0.0 1.2 98.6 0.6? 1.00 ARITHMETIC T o x i c a 2.* 36. J 6 1.2 0.52 0 . 6 2 iNTCaCRS 90.0 10.0 0.0 o.to 0.25 IMTISCM AOOITIOM 65.0 32.5 2.5 0.33 0.33 IMTCKR SUBTRACTION 1.3 27.5 7 1.2 0 . 6 7 0.67 INTTMCR MULTIPLICATION 1 •* 1 3 . o 6).2 0.75 0.80 FORMULAS 5.9 •2.9 53.2 O.ko 0.67 A t o c a R A i c TOPICS 0.0 M . o 1 6.0 0.52 0.5* AMOLC SUM TNCORCM 90.• 8.2 !.• 0.29 0.0 PVTNAQORCAN T M t O R C M 6 1 . • 21 .• 17 . 1 0.50 0.33 • 10.1 21.7 6 8 . 1 0.«3 0.63 ARC A OT A PORALLCLOOP.AM 7.: »5.9 76.B 0.63 * 0.71 V O L u a r o r A P« l EM 27.2 '5.9 56.5 0.33 0.67 _GCOMCTRIC TOPICS 25.9 •6.9 27.2 0.»5 0.5* ALL TOPICS 2.} 6 ) . 2 3*.5 0.»9 0.57 R t e a c h e r s u s e d a b s t r a c t methods o v e r f o u r t i m e s as f r e q u e n t l y as p e r c e p t u a l methods. No s c o r e s i n t h e h i g h l y p e r c e p t u a l c a t e g o r y a r e c o n t a i n e d i n the d i s t r i b u t i o n . An i n s p e c t i o n of s c o r e s a t t h e t o p i c l e v e l shows t h a t s e v e r a l t o p i c s such as i n t e g e r s and t h e a n g l e sum th e o r e m were t y p i c a l l y p r e s e n t e d u s i n g p e r c e p t u a l modes of r e p r e s e n t a t i o n a l m o s t e x c l u s i v e l y . O t h e r t o p i c s , such as t h e c o n c e p t o f rr and d e c i m a l o p e r a t i o n s , were p r e s e n t e d v e r y a b s t r a c t l y . These r e s u l t s a r e examined i n more d e t a i l i n the s e c t i o n s t h a t f o l l o w . B o x p l o t s a r e used so t h a t t h e d i s t r i b u t i o n s of s c o r e s can be compared. 1 10 h i g h l y p e r c e p t u a l 0 somewhat perc e p t u a l 2 3 26. balanced 4 5 6 12223344577789 00000111122222233334455666677788889 000000 somewhat a b s t r a c t 6 7 8 1111122333455555678899 001238 h i g h l y a b s t r a c t 8 9 0 19 N=87 Figur e 4-14 - D i s t r i b u t i o n of content r e p r e s e n t a t i o n scores averaged over a l l t o p i c s . 3.1 Mode Of Content Representation For A r i t h m e t i c In F i g u r e 4-15 boxplots of the d i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores f o r the four a r i t h m e t i c t o p i c s f o r which t h i s v a r i a b l e was d e f i n e d are shown. 9 One of the p l o t s does not appear to be a boxplot at a l l because 71 of the 81 scores are 1.00 causing a degenerate p l o t of Xs. There are, however, two ba s i c p a t t e r n s of content r e p r e s e n t a t i o n f o r the four a r i t h m e t i c t o p i c s . O v e r a l l , teachers r e l i e d more h e a v i l y on pe r c e p t u a l methods than a b s t r a c t methods f o r f r a c t i o n s and a d d i t i o n of f r a c t i o n s . For each of these t o p i c s 75% of the content r e p r e s e n t a t i o n scores are at or below 0.50, the value i n d i c a t i n g an exact 9 In Appendix A the teaching methods given i n the TSQs f o r the 16 t o p i c s examined i n t h i s study are given. A l s o i n c l u d e d i s a l i s t i n g of which methods were co n s i d e r e d p e r c e p t u a l and which were considered a b s t r a c t f o r the 14 t o p i c s f o r which the mode of r e p r e s e n t a t i o n v a r i a b l e was d e f i n e d . 111 1 . 0 0 o.Po 0.60 0,'rO 0.20 c o o * (10) * 00 X K • ( 7 1 ) * (2) * <5> C O N C E P T or ITACTIONS ADDITION Or TBMCTIONS C O N C E P T O P E R A T I O N S o r W I T H D E C I M A L S D E C I M A L S Figure 4-15 - D i s t r i b u t i o n s of the content r e p r e s e n t a t i o n scores f o r the a r i t h m e t i c s c o r e s . balance between the two types of r e p r e s e n t a t i o n . In each case over 30% of the scores are within the p e r c e p t u a l c a t e g o r i e s and 20% or l e s s are w i t h i n the abstact c a t e g o r i e s . The tendency toward the use of p e r c e p t u a l methods by teachers was strongest f o r a d d i t i o n of f r a c t i o n s . The l e a s t consensus occured for t h i s t o p i c , however, as the F-spread i s the l a r g e s t and o u t l i e r scores of both 0.00 and 1.00 are present in the d i s t r i b u t i o n . Decimals and o p e r a t i o n s with decimals were both t r e a t e d i n an a b s t r a c t manner in most cases. For both t o p i c s 75% or more 1 12 of the scores are w i t h i n one of the two a b s t r a c t c a t e g o r i e s . The tendency toward an abstact r e p r e s e n t a t i o n of content was strongest f o r operations with decimals with 86% of the scores wi t h i n the h i g h l y a b s t r a c t range. The a r i t h m e t i c content i s review m a t e r i a l at t h i s grade l e v e l . The a b s t r a c t treatment given by teachers to the two decimal t o p i c s r e f l e c t s t h i s f a c t . Many teachers, on-the-other-hand, app a r e n t l y f e l t that students s t i l l needed p e r c e p t u a l r e p r e s e n t a t i o n s of the f r a c t i o n m a t e r i a l in s p i t e of i t s review nature. Although t r a d i t i o n a l l y f r a c t i o n s are introduced e a r l i e r than decimals i n the elementary grades, teachers may b e l i e v e that the d i f f i c u l t y many students have with f r a c t i o n s r e q u i r e s more frequent enactive and i c o n i c r e p r e s e n t a t i o n s than i s the case for decimals even at the Grade 8 l e v e l . The mean of the content r e p r e s e n t a t i o n scores f o r the f i v e a r i t h m e t i c topics' f o r each teacher was taken as the o v e r a l l content r e p r e s e n t a t i o n score f o r a r i t h m e t i c f o r that teacher. i The d i s t r i b u t i o n of these scores i s shown i n F i g u r e 4-16. Out of 85 scores, 31 are in the balanced category and 38 are i n the somewhat a b s t r a c t category. Only two scores are in the p e r c e p t u a l c a t e g o r i e s , while 14 scores are i n the h i g h l y a b s t r a c t category. The median of the d i s t r i b u t i o n i s 0.64; the lower and upper fourths are 0.55 and 0.72. Although more pe r c e p t u a l methods were used by most teachers for two of the four a r i t h m e t i c t o p i c s , the other two t o p i c s were d e a l t with i n an a b s t r a c t way very uniformly by teachers. The e f f e c t of t h i s i s that the; average scores are almost a l l i n the 1 13 h i g h l y p e r c e p t u a l 0 somewhat p e r c e p t u a l 2 3 33 balanced 4 5 6 12377789999 022222335567788888 00 somewhat a b s t r a c t 6 7 8 1112223334456778888999 011122222234577 0 h i g h l y a b s t r a c t 8 9 10 23599999 0123 00 N=85 Fi g u r e 4-16 - D i s t r i b u t i o n of o v e r a l l content r e p r e s e n t a t i o n scores f o r a r i t h m e t i c . balanced or the two a b s t r a c t c a t e g o r i e s . Thus, across a l l a r i t h m e t i c t o p i c s over o n e - t h i r d of the teachers i n the sample took a balanced approach between p e r c e p t u a l and a b s t r a c t methods. Almost a l l of the other teachers represented content more f r e q u e n t l y i n an a b s t r a c t manner than i n a p e r c e p t u a l manner. 3.2 Mode Of Content Representation For Algebra Boxplots of the d i s t r i b u t i o n s of the content r e p r e s e n t a t i o n scores f o r the f i v e a l g e b r a i c t o p i c s are shown i n Figure 4-17. As with the a r i t h m e t i c t o p i c s , two p a t t e r n s are e v i d e n t . The concept of i n t e g e r s and the a d d i t i o n of i n t e g e r s were taught by most teachers using predominantly p e r c e p t u a l methods. This tendency was e s p e c i a l l y strong f o r the concept of i n t e g e r s ; a l l but 10% of the scores f o r t h i s t o p i c were below 0.40. For each of these two t o p i c s over 18% of the teachers used no a b s t r a c t methods at a l l . 1 1 4 1.00 * (10) 0.80 I 0.60 O.Uo o.co 0.00 C O K C C P T or A O O I T I O N or i*TEccns S U B T R A C T I O N o r U U L T I P L I C A T I O r ! o r l - N T C C C R S concert or r J R I J D L A S Figure 4-17 - D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores f o r the a l g e b r a i c t o p i c s . The other two operations with i n t e g e r s which were i n v e s t i g a t e d as w e l l as the concept of formulas were presented by most teachers using more a b s t r a c t than p e r c e p t u a l methods. No more than 10% of the scores for each of these d i s t r i b u t i o n s were below 0.50. Teachers were g e n e r a l l y not c o n s i s t e n t i n the type of r e p r e s e n t a t i o n with which the four i n t e g e r t o p i c s were 1 15 presented. T y p i c a l l y , the concept of i n t e g e r s and the ope r a t i o n of a d d i t i o n of i n t e g e r s were taught with a p e r c e p t u a l o r i e n t a t i o n . Apparently, most teachers b e l i e v e d that the oper a t i o n s of s u b t r a c t i o n and m u l t i p l i c a t i o n of i n t e g e r s c o u l d then be presented i n a l a r g e l y a b s t r a c t manner, as that was the usual approach. For each teacher the mean of the content r e p r e s e n t a t i o n scores f o r the f i v e a l g e b r a i c t o p i c s was taken as the o v e r a l l content r e p r e s e n t a t i o n score f o r al g e b r a f o r that teacher. The d i s t r i b u t i o n of these scores i s shown i n F i g u r e 4-18. h i g h l y p e r c e p t u a l somewhat p e r c e p t u a l 2 3 balanced somewhat a b s t r a c t h i g h l y a b s t r a c t 4 5 6 6 7 8 8 9 10 022344444555566777777889 000000000022233444445555666667777788888899 00 23333444588 5 N=81 F i g u r e 4-18 - D i s t r i b u t i o n of o v e r a l l content r e p r e s e n t a t i o n scores f o r a l g e b r a . Out of 81 scores, 68 are in the balanced category and 12 are i n the somewhat a b s t r a c t category. The remaining score i s in the h i g h l y a b s t r a c t category. The median f o r the d i s t r i b u t i o n i s 0.54; the lower and upper fourths are 0.47 and 0.58. Thus, a l a r g e m a j o r i t y of teachers, 84%, presented 1 16 a l g e b r a i c content using a roughly equal balance of p e r c e p t u a l and a b s t r a c t methods. A l l of the other teachers r e l i e d more h e a v i l y on a b s t r a c t than on p e r c e p t u a l methods. While most teachers d i d present algebra content using a balance of both types of methods, i t should be emphasized that t h i s was not true at the l e v e l of i n d i v i d u a l t o p i c s . As was noted above, two t o p i c s tended to be presented p e r c e p t u a l l y , three a b s t r a c t l y . In f a c t , only 25% of the content r e p r e s e n t a t i o n scores f o r the f i v e a l g e b r a i c t o p i c s are in the balanced category. The e f f e c t of a p e r c e p t u a l r e p r e s e n t a t i o n of some t o p i c s by teachers and an a b s t r a c t r e p r e s e n t a t i o n of others was a balanced o v e r a l l r e p r e s e n t a t i o n f o r most teachers, however. 3.3 Mode Of Content Representation For Geometry Boxplots f o r the d i s t r i b u t i o n s of the content r e p r e s e n t a t i o n scores f o r the f i v e geometric t o p i c s are shown i n F i g u r e 4-19. As with the a r i t h m e t i c and a l g e b r a i c t o p i c s there were s u b s t a n t i a l d i f f e r e n c e s among the t y p i c a l l e v e l s of a b s t r a c t i o n with which each geometric t o p i c was presented. At one extreme, the angle sum theorem f o r t r i a n g l e s and the Pythagorean theorem were u s u a l l y taught p e r c e p t u a l l y ; 90% and 61% of the scores f o r these t o p i c s r e s p e c t i v e l y are below 0.40. At the other extreme, the concept of n and the area of a p a r a l l e l o g r a m were u s u a l l y taught a b s t r a c t l y ; 68% and 77% of the scores f o r these t o p i c s r e s p e c t i v e l y are over 0.60. The f i f t h geometric t o p i c , the 1 1 7 1.00 * (5) 0.B0 0.60 O.fco 0.20 0.00 X (2) X S U M Or T H C P V T M A f S O R E A N CONCEPT ANGLrS I H A T H C O R C M OT T R I A N G L E IT Ant* o r A P A R A L L E L O C R A U V O L U M E or A R C C T A N G U L A R P R I S M F i g u r e 4-19 - D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores for the geometric t o p i c s . volume of a rec t a n g u l a r prism, was a l s o d e a l t with a b s t r a c t l y by a ma j o r i t y of teachers. In c o n t r a s t to the previous two t o p i c s , however, t h i s t o p i c was a l s o taught with an emphasis on per c e p t u a l methods by a s u b s t a n t i a l number of teachers. For each teacher the mean of the content r e p r e s e n t a t i o n scores f o r the f i v e geometric t o p i c s was taken as the o v e r a l l content r e p r e s e n t a t i o n score f o r geometry for that teacher. The d i s t r i b u t i o n of these scores i s shown i n Fi g u r e 4-20. Thi s d i s t r i b u t i o n appears roughly normal. Out of 81 scores 118 h i g h l y p e r c e p t u a l 0 06 0 somewhat p e r c e p t u a l 2 3 0589 2333333366689 balanced 4 5 6 0013666677888 00022334456666788889999 000 somewhat a b s t r a c t 6 7 8 1222333558899 1558 h i g h l y a b s t r a c t 8 9 10 27 0 00 N=81 F i g u r e 4-20 - D i s t r i b u t i o n of o v e r a l l content r e p r e s e n t a t i o n scores f o r geometry. 39 are i n the balanced category, while 17 are i n the somewhat perce p t u a l and 17 are i n the somewhat a b s t r a c t c a t e g o r i e s . Three scores are in the h i g h l y p e r c e p t u a l category and f i v e i n the h i g h l y a b s t r a c t category. Thus, s l i g h t l y l e s s than h a l f of the teachers r e l i e d about e q u a l l y on p e r c e p t u a l and a b s t r a c t methods i n teaching geometry. The remaining teachers were almost, evenly s p l i t between those who put more emphasis on pe r c e p t u a l methods and those who put more emphasis on a b s t r a c t methods. 3.4 Comparisons Among The Topics And Content Areas S u b s t a n t i a l d i f f e r e n c e s e x i s t among the 15 t o p i c s regarding the mode of content r e p r e s e n t a t i o n employed by teachers. / In F i g u r e 4-21 the d i s t r i b u t i o n s f o r t h i s v a r i a b l e are shown With t o p i c s i d e n t i f i e d by content area. At one extreme, an a r i t h m e t i c t o p i c has a median l e v e l of content r e p r e s e n t a t i o n of 0.00, while at the other extreme a geometry t o p i c has a median 1 19 l e v e l of 1.00. Since t o p i c s from a l l three content areas have both low and high median values, there does not appear to be any strong a s s o c i a t i o n between the content area and the median value of the d i s t r i b u t i o n when t o p i c s are co n s i d e r e d i n d i v i d u a l l y . firo A L G A ^ I AU A L O A R I A L T Ar»i A L C G E O A L G C E O C T O CEO F i g u r e 4-21 - D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores f o r 14 t o p i c s . When the content r e p r e s e n t a t i o n scores are averaged f o r each teacher both f o r t o p i c s w i t h i n a content area and a c r o s s 1 20 a l l t o p i c s , the d i s t r i b u t i o n s of sc o r e s vary much l e s s than was true f o r the d i s t r i b u t i o n s at the t o p i c l e v e l . These d i s t r i b u t i o n s are shown in Figure 4-22. In c o n t r a s t to the very 1.00 0.90 0.80 0.70 0.60 0.50 o.fco 0.30 0.20 0.10 0.00 T A L L T O P I C S A R I T H M E T I C A L S C B R A aCOMCTRY F i g u r e 4-22 - D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores f o r each content area and acr o s s a l l t o p i c s . n o t i c e a b l e d i f f e r e n c e s which e x i s t e d among the d i s t r i b u t i o n s at the t o p i c l e v e l , the d i s t r i b u t i o n s f o r the three content area scores and the o v e r a l l score are more n e a r l y s i m i l a r . The algebra and geometry d i s t r i b u t i o n s , f o r example, both have 121 i d e n t i c a l median values of 0.54. Thus, f o r these content areas the median teacher used j u s t s l i g h t l y more a b s t r a c t than p e r c e p t u a l r e p r e s e n t a t i o n s of content. The median value f o r a l l t o p i c s was somewhat higher at 0.57. While teachers t y p i c a l l y balanced p e r c e p t u a l . and a b s t r a c t approaches to algebra and geometry content, they d e a l t with a r i t h m e t i c content i n a s l i g h t l y more a b s t r a c t manner. The median score f o r the a r i t h m e t i c content d i s t r i b u t i o n i s 0.64, a value i n the somewhat a b s t r a c t category. The upper and lower f o u r t h s are a l s o higher f o r the a r i t h m e t i c d i s t r i b u t i o n . Since the a r i t h m e t i c content was l a r g e l y review m a t e r i a l , d i f f e r e n c e s are not s u r p r i s i n g . Another d i f f e r e n c e that can be noted among the d i s t r i b u t i o n s i s the degree- of v a r i a t i o n i n the sc o r e s . In p a r t i c u l a r , i n comparing the middle 50% of the d i s t r i b u t i o n s the l a r g e s t v a r i a t i o n occurs f o r geometry with an F-spread of 0.22. The F-spreads of the a r i t h m e t i c , algebra and o v e r a l l d i s t r i b u t i o n s are 0.17, 0.11, and 0.12 r e s p e c t i v e l y . The t a i l s of the geometry d i s t r i b u t i o n extend f u r t h e r than those of the other d i s t r i b u t i o n s and the geometry d i s t r i b u t i o n c o n t a i n s the l a r g e s t number of o u t l i e r s . These r e s u l t s provide f u r t h e r i n d i c a t i o n s that these teachers showed the l e a s t u n i f o r m i t y i n t h e i r mode of content r e p r e s e n t a t i o n f o r geometry. A f u r t h e r comparison of the way i n which teachers represented content i n the three areas can be gained by a r e i n s p e c t i o n of Table 4-3. Almost no teachers d e a l t with a r i t h m e t i c or algebra p e r c e p t u a l l y . A r i t h m e t i c was presented 122 a b s t r a c t l y by a m a j o r i t y of teachers, while a l a r g e m i n o r i t y balanced p e r c e p t u a l and a b s t r a c t p r e s e n t a t i o n s . A l a r g e m a j o r i t y of teachers used a balanced approach f o r algebra content. While almost h a l f of the teachers a l s o balanced p e r c e p t u a l and a b s t r a c t approaches to geometry, the remainder were about evenly s p l i t between those whose pre f e r e n c e was f o r p e r c e p t u a l methods and those whose preference was f o r a b s t r a c t methods. 3.5 Achievement L e v e l Comparisons In F i g u r e 4-23 separate d i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores are shown for low and high achievement l e v e l c l a s s e s over a l l t o p i c s and f o r the content areas. I t might be expected that p e r c e p t u a l methods would be used more f r e q u e n t l y i n low achievement c l a s s e s ; such was not the case, however. The median f o r the o v e r a l l d i s t r i b u t i o n i s 0.58 f o r low achievement c l a s s s and 0.54 f o r high achievement c l a s s e s . The lower and upper f o u r t h s are a l s o s l i g h t l y higher f o r the low achievement c l a s s e s . An i n s p e c t i o n of the d i s t r i b u t i o n s of scores f o r the content areas does not show a c o n s i s t e n t tendency for more a b s t r a c t p r e s e n t a t i o n s i n low achievement c l a s s e s . For algebra and geometry the median content r e p r e s e n t a t i o n score i s s l i g h t l y higher f o r low achievement c l a s s e s i n d i c a t i n g a more a b s t r a c t p r e s e n t a t i o n . For a r i t h m e t i c , however, the median score i s s l i g h t l y higher for the high achievement c l a s s e s . Although the f i n d i n g t h a t , o v e r a l l , teachers presented 123 X I 1 T T T T X X A L L . T O P I C S LOW M « C H -A R I T H W C T I C LOW HI«H A L O C B R A LOW H I OH OF. O U T T R Y LOW H I OH Figure 4-23 - D i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores f o r low and high achievement l e v e l c l a s s e s . content i n a s l i g h t l y more a b s t r a c t way to low achievement c l a s s e s than to high achievement c l a s s e s was not r e p l i c a t e d f o r a l l content areas, i t i s c o n s i s t e n t with the f i n d i n g s of another recent study. Crosswhite et a l . (1985) using SIMS data c o l l e c t e d in the United States reported that " i n s t r u c t i o n tended to be more symbolic with remedial c l a s s e s than with other types of c l a s s e s and tended to be more symbolic when reviewing content than when c o v e r i n g new subject matter." (p. 24) 124 Although the d i f f e r e n c e s found i n the way content was represented to low and high achievement c l a s s e s were not l a r g e , t h i s r e s u l t i s d i s t u r b i n g . One can speculate, f o r example, that while the low achievement student might p r o f i t at l e a s t as much as the high achievement student from p e r c e p t u a l methods, d i f f e r e n c e s i n the motivation and behavior p a t t e r n s of these types of students might m i t i g a t e a g a i n s t a stronger p e r c e p t u a l o r i e n t a t i o n i n low achievement c l a s s e s . 3.6 Content Representation Of Teachers And Textbooks The two major textbooks used i n Mathematics 8 classrooms s t r e s s a b s t r a c t approaches to content somewhat more of t e n than p e r c e p t u a l approaches. Table 4-4 shows the p r o p o r t i o n of a b s t r a c t TSQ methods to t o t a l TSQ methods that appear in each t e x t f o r the three content areas and o v e r a l l . Except f o r the treatment of geometry in School Mathematics 2 more a b s t r a c t methods are i n c l u d e d i n the t e x t s than p e r c e p t u a l methods i n each case. The treatment of geometry i n School Mathematics 2 i n c l u d e s an equal number of a b s t r a c t and p e r c e p t u a l methods. O v e r a l l , Mathematics II c o n t a i n s the g r e a t e r p r o p o r t i o n of a b s t r a c t methods. In that te x t 62% of the methods are a b s t r a c t compared to 55% f o r School Mathematics 2. The treatment of a l g e b r a and geometry content i s l i k e w i s e more a b s t r a c t in Mathematics I I . Only f o r a r i t h m e t i c content does School  Mathematics 2 c o n t a i n a s l i g h t l y g reater p r o p o r t i o n of a b s t r a c t methods than the other t e x t . Figure 4-24 shows the d i s t r i b u t i o n s of content 125 Table 4- 4 Pro p o r t i o n of A b s t r a c t TSQ Methods to T o t a l TSQ Methods i n the Textbooks Mathematics II School Mathematics 2 A r i t h m e t i c Algebra Geometry A l l t o p i c s 0.60 0.60 0.67 0.62 0.63 0.54 0.50 0.55 r e p r e s e n t a t i o n scores f o r each content area and o v e r a l l s e p a r a t e l y f o r users of the two textbooks. In each case the median teacher used more a b s t r a c t than p e r c e p t u a l methods and f o r users of both t e x t s p r e s e n t a t i o n s were more a b s t r a c t f o r a r i t h m e t i c than for the other content areas. For users of Mathematics II the median content r e p r e s e n t a t i o n scores are 0.68, 0.53, and 0.51 f o r a r i t h m e t i c , a l g e b r a , and geometry r e s p e c t i v e l y . For School Mathematics 2 the corresponding values are 0.63, 0.55, and 0.56. The way content was t y p i c a l l y represented by teachers i n i n s t r u c t i o n and the way i t was represented i n the textbooks are c o n s i s t e n t i n that in both instances s l i g h t l y more a b s t r a c t methods occurred than p e r c e p t u a l methods. The median content r e p r e s e n t a t i o n score across a l l t o p i c s f o r users of School Mathematics 2 i s 0.57 compared with a p r o p o r t i o n of 0.55 a b s t r a c t methods in the textbook i t s e l f . S i m i l a r l y , the median content r e p r e s e n t a t i o n score f o r users of Mathematics 11 i s 0.55 and the p r o p o r t i o n of a b s t r a c t methods in that textbook i s 0.62. Although median content r e p r e s e n t a t i o n scores and the two 1 2 6 1.00 0.90 O.fiO 0.70 0.60 0.50 o.ko 0.30 0.20 0.10 0.00 I T 1 IT: A L L T O P I C S M l I 8 U 2 A R I T H M E T I C M i l 8M2 A L Q E B R A Ml I SM2 G E O M E T R Y Ml I S U 2 F i g u r e 4-24 - D i s t r i b u t i o n s of Content R e p r e s e n t a t i o n Scores for C l a s s e s U s i n g Each of the Commonly Used Textbooks . textbooks a l l / showed a greater s t r e s s on a b s t r a c t than p e r c e p t u a l methods, the d i f f e r e n c e s between the content . r e p r e s e n t a t i o n ; s c o r e s of the users of the two textbooks were not c o n s i s t e n t wi th the d i f f e r e n c e s between the textbooks themselves . Thus , wh i l e Mathematics II c o n t a i n e d a s l i g h t l y more a b s t r a c t treatment of content than the other t e x t , the users of Mathematics II d i s p l a y e d a s l i g h t l y l e s s a b s t r a c t o r i e n t a t i o n i n t h e i r c la s sroom p r e s e n t a t i o n s than the users of the other t e x t . T h i s i n c o n s i s t e n c y might be a r e s u l t of the 127 small d i f f e r e n c e s i n v o l v e d . A l t e r n a t e l y , i t might be the case that the contents of a textbook i n f l u e n c e teachers' c h o i c e s of the t o p i c s they w i l l teach and the emphasis they w i l l give to content areas more s t r o n g l y than t h e i r c h o i c e s of how they w i l l represent content during i n s t r u c t i o n . 4. RULE-ORIENTEDNESS OF INSTRUCTION Table 4-5 shows the percentage of teachers whose r u l e -orientedness scores are i n each of three c a t e g o r i e s 1 0 f o r each t o p i c f o r which t h i s v a r i a b l e was d e f i n e d , f o r the content areas of algebra and geometry, and o v e r a l l . Median scores are a l s o included i n the t a b l e . The content area and o v e r a l l scores were determined f o r each teacher by averaging the scores of the ap p r o p r i a t e t o p i c s f o r that teacher. The median o v e r a l l r u l e - o r i e n t e d n e s s score i s 0.47. Thus, teachers t y p i c a l l y e x p l a i n e d concepts, o p e r a t i o n s , and p r i n c i p l e s j u s t s l i g h t l y l e s s o f t e n by s t a t i n g a computational r u l e , a d e f i n i t i o n or a theorem followed by examples than by using a p h y s i c a l i n t e r p r e t a t i o n , the i n v e s t i g a t i o n of a p a t t e r n or some other n o n - r u l e - o r i e n t e d method. J u s t over h a l f of the o v e r a l l s cores, 52%, are in the balanced category while 26% are in the n o n - r u l e - o r i e n t e d category and 22% are i n the r u l e -o r i e n t e d category. Thus, almost a q u a r t e r of the teachers 1 0 For t h i s t a b l e the h i g h l y n o n - r u l e - o r i e n t e d and somewhat non-r u l e - o r i e n t e d c a t e g o r i e s have been combined i n t o a s i n g l e non-r u l e - o r i e n t e d category. S i m i l a r l y , the h i g h l y r u l e - o r i e n t e d and somewhat r u l e - o r i e n t e d c a t e g o r i e s have been combined i n t o a s i n g l e r u l e - o r i e n t e d category. 128 p l a c e d strong emphasis on r u l e s i n e x p l a i n i n g mathematical ideas while over a quarter placed weak emphasis on r u l e s . Table 4- 5 - Rule-Orientedness of I n s t r u c t i o n Scores Topic Area Rule-Orientedness d i s t r i b u t i o n (% of Scores in Category) ^ _ Median Rule Non-Rule Balanced Rule O r i e n t e d -O r i e n t e d O r i e n t e d ness Score Decimal Operations 7.2 21 .4 71.4 0.75 Integer A d d i t i o n 47.6 45.0 7.6 0.50 Integer S u b t r a c t i o n 22.5 40.0 37.5 0.50 Inteqer M u l t i p l i c a t i o n 67.9 18.5 13.5 0.25 A l g e b r a i c Topics 48.8 37.2 14.0 0.42 Pythagorean Theorem 52.8 25.0 22.3 0.25 ir 27.8 40.3 32.0 0.50 Area of a P a r a l l e l o g r a m 29.1 38.9 31.9 0.50 Volume of a Prism 25.7 21 .6 52.7 0.75 Geometric Topics 26.8 40.2 32.9 0.50 A l l T o p i c s 26.4 51 .7 21 .8 0.47 The d i s t r i b u t i o n of o v e r a l l r u l e - o r i e n t e d n e s s scores i s shown i n the stem-and-leaf p l o t of F i g u r e 4-25. Out of 87 scores, only a s i n g l e value i s i n the h i g h l y n o n - r u l e - o r i e n t e d category and no scores are i n the h i g h l y r u l e - o r i e n t e d category. The other scores are d i s t r i b u t e d n e a r l y normally from a low of 0.20 to a high of 0.78. Thus, while teachers showed c o n s i d e r a b l e v a r i a t i o n i n the amount of s t r e s s they put on r u l e s i n t h e i r implemented c u r r i c u l a , v i r t u a l l y none r e l i e d almost t o t a l l y on r u l e s or excluded r u l e s a l t o g e t h e r . 4.1 Rule-Orientedness In Teaching A r i t h m e t i c The r u l e - o r i e n t e d n e s s v a r i a b l e was d e f i n e d f o r a s i n g l e a r i t h m e t i c t o p i c , namely decimal o p e r a t i o n s . Of the three methods l i s t e d for t h i s t o p i c i n the F r a c t i o n TSQ, the o p t i o n : "Related operations with decimals to o p e r a t i o n s with whole 129 h i g h l y n o n - r u l e - o r i e n t e d 0 0 somewhat n o n - r u l e - o r i e n t e d 2 3 012288888 1144445688889 balanced 4 5 6 0011111124444456667777 00000033333566666778999 somewhat r u l e - o r i e n t e d 6 7 8 1333466668999 125558 h i g h l y r u l e - o r i e n t e d 8 9 10 N=87 Fig u r e 4-25 - D i s t r i b u t i o n of r u l e - o r i e n t e d n e s s scores averaged over e i g h t t o p i c s . numbers, teaching r u l e s f o r p l a c i n g the decimal p o i n t " was c l a s s i f i e d as the r u l e a p p r o a c h . 1 1 T h i s t o p i c was taught by 52% of the teachers i n a somewhat r u l e - o r i e n t e d way and by 19% of the teachers i n a h i g h l y r u l e -o r i e n t e d way. Twenty-one percent of the teachers r e l i e d on the r u l e approach and other methods e q u a l l y , while an a d d i t i o n a l 7% taught t h i s t o p i c i n a somewhat or h i g h l y n o n - r u l e - o r i e n t e d manner. An i n s p e c t i o n of Table 4-5 shows that f o r no other t o p i c was the t o t a l percent of scores i n the two r u l e - o r i e n t e d c a t e g o r i e s so high. On t h i s b a s i s , i t can be s t a t e d that the a r i t h m e t i c t o p i c decimal op e r a t i o n s was t r e a t e d i n a more r u l e -o r i e n t e d way than any of the a l g e b r a i c or geometric t o p i c s . While these comparative f i g u r e s are probably to be expected given the review nature of a r i t h m e t i c at t h i s grade l e v e l , i t i s 1 1 In Appendix B the r u l e o p t i o n i s l i s t e d f o r each of the e i g h t t o p i c s . 1 30 not n e c e s s a r i l y the case that a strong r e l i a n c e on r u l e s i n teaching operations with decimals i s d e s i r a b l e . On a SIMS Test item which r e q u i r e d that students be able to estimate the answer to a m u l t i p l i c a t i o n of decimals problem the r e s u l t s were poor " i n d i c a t i n g that students may be a p p l y i n g a mechanical process r a t h e r than d e a l i n g with q u a n t i t i e s with understanding" ( R o b i t a i l l e , O'Shea, & D i r k s , 1982, p. 98). Perhaps i f i n s t r u c t i o n had been l e s s s t r o n g l y r u l e - o r i e n t e d , student understanding and achievement might have been hi g h e r . 4.2 Rule-Orientedness In Teaching Algebra F i g u r e 4-26 shows boxplots of the d i s t r i b u t i o n s of r u l e -orientedness scores for the three a l g e b r a i c t o p i c s . No two p l o t s are p a r t i c u l a r l y s i m i l a r o v e r a l l , although the median value f o r both a d d i t i o n and s u b t r a c t i o n of i n t e g e r s i s 0.50. The p l o t s show, however, that f o r a d d i t i o n most teachers who d i d not s t r e s s r u l e s and other methods e q u a l l y , tended to emphasize non-rule methods. For s u b t r a c t i o n the tendency was to s t r e s s r u l e s more f r e q u e n t l y . M u l t i p l i c a t i o n of i n t e g e r s was the t h i r d a l g e b r a i c t o p i c f o r which teachers' s t r e s s on r u l e s was i n v e s t i g a t e d . Of the four methods given i n the ^algebra TSQ for t h i s t o p i c , the o p t i o n : "No development—students were given r u l e s " was c l a s s i f i e d as the r u l e approach. An i n s p e c t i o n of Table 4-5 shows that the m a j o r i t y of teachers put more s t r e s s on methods other than r u l e s f o r m u l t i p l i c a t i o n of i n t e g e r s . The wording of the r u l e o p t i o n which seems to preclude the use of other methods 131 1.00 0.80 0.60 0.«I0 0.20 0.00 X(2) X(6) A O O I I T I O N or i m r e c R S SUBTRACT ION of INTEGERS MULTIPLICATION U J T E f i T R S F i g u r e 4-26 - D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s s c o r e s f o r a l g e b r a i c t o p i c s . i s unfortunate, however, and c a s t s some doubt on the v a l i d i t y of the r e s u l t s for t h i s t o p i c . Only 13% of the scores are i n the two r u l e - o r i e n t e d c a t e g o r i e s and only 19% are in the balanced category. Over two-thirds of the scores, 68%, are i n the two n o n - r u l e - o r i e n t e d c a t e g o r i e s . The mean of the r u l e - o r i e n t e d n e s s scores f o r the three a l g e b r a i c t o p i c s for each teacher was taken as the o v e r a l l r u l e -o r i e n t e d n e s s score for a l g e b r a for that teacher. The d i s t r i b u t i o n of tese scores i s shown in Figure 4-27. 1 32 h i g h l y n o n - r u l e - o r i e n t e d 0 888 7777777777 somewhat n o n - r u l e - o r i e n t e d 2 3 5555555 3333333333333333333333 balanced 4 5 6 222222222222 00000000000008888888 somewhat r u l e - o r i e n t e d 6 7 8 77777 555555 h i g h l y r u l e - o r i e n t e d 8 9 10 3 N=86 Figure 4-27 - D i s t r i b u t i o n of r u l e - o r i e n t e d n e s s scores f o r a l g e b r a . Out of 86 scores, 32 are i n the balanced category. Nearly as many scores, 29, are in the somewhat n o n - r u l e - o r i e n t e d category, while 13 scores are i n the h i g h l y n o n - r u l e - o r i e n t e d category. Of the remaining 12 scores, 11 are in the somewhat r u l e - o r i e n t e d category and only 1 i s i n the h i g h l y r u l e - o r i e n t e d category. The median of the d i s t r i b u t i o n i s 0.42; the lower and upper f o u r t h s are 0.33 and 0.50. Thus, when a l l three operations with i n t e g e r s are considered, almost h a l f of the teachers, 49%, s t r e s s e d non-rule-o r i e n t e d methods more than r u l e - o r i e n t e d methods i n t h e i r implemented c u r r i c u l a . Somewhat fewer, 37%, put equal s t r e s s on r u l e s and other c o n t e n t - s p e c i f i c methods. Only 14% of the sample put more emphasis on r u l e s than on a l t e r n a t i v e methods. Given that these operations are probably new to students at t h i s grade l e v e l , t h i s s t r e s s on methods other than r u l e s i s probably d e s i r a b l e . Students, i n f a c t , showed t h e i r g r e a t e s t improvement 133 between the Core P r e t e s t and Core P o s t t e s t on a m u l t i p l i c a t i o n of i n t e g e r s item ( R o b i t a i l l e , O'Shea, and D i r k s , 1982, p. 96). 4.3 Rule-Orientedness In Teaching Geometry F i g u r e 4-28 shows boxplots of the d i s t r i b u t i o n s of r u l e -o r i e n t e d n e s s scores f o r the four geometric t o p i c s . In each case the lower t a i l extends to 0.00 and the lower f o u r t h i s 0.25. Each d i s t r i b u t i o n except the one f o r the Pythagorean theorem has an upper f o u r t h of 0.75 and an upper t a i l which extends to 1.00. These p l o t s are very s i m i l a r to each other and show g r a p h i c a l l y the wide v a r i a t i o n between teachers i n t h e i r r e l a t i v e emphasis of r u l e s f o r each t o p i c . Although the d i s t r i b u t i o n s are s i m i l a r , there are d i f f e r e n c e s i n the median v a l u e s . For the Pythagorean theorem the median i s 0.25, f o r the volume of a re c t a n g u l a r prism i t i s 0.75. Thus, the g r e a t e s t d i f f e r e n c e i n the emphasis given to r u l e s occured between the Pythagorean theorem and the volume of a r e c t a n g u l a r prism. Most teachers tended not to emphasize the statement of the r u l e i t s e l f in teaching the former t o p i c but i n s t e a d tended to emphasize methods which provided j u s t i f i c a t i o n or i n t e r p r e t a t i o n . When teaching the l a t t e r t o p i c , however, the more common tendency was to put s t r e s s on the r u l e i t s e l f . Based on poor performance by students on a SIMS t e s t item i n v o l v i n g volume, one can speculate that teaching the c a l c u l a t i o n of volumes as an e x e r c i s e i n s u b s t i t u t i n g values i n t o a formula does not ensure student understanding of vol u m e t r i c concepts ( R o b i t a i l l e , O'Shea, and D i r k s , 1982, p. 1 34 1.00 0.80 0.60 o.fco 0.20 0.00 »>VTHA«0RCA« T M f O n C M C O N C C P T or tr A R C A- O r A R A R A L U C L O C n A U V O I U U C or A N C C T A M C U L A n F i g u r e 4-28 - D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores f o r geometric t o p i c s . 108). The mean of t h e : r u l e - o r i e n t e d n e s s scores for the four geometric t o p i c s for each teacher was taken as the o v e r a l l -rule-o r i e n t e d n e s s score for geometry for that teacher. The d i s t r i b u t i o n of these scores i s shown in Figure 4-29. Out of 81 scores, 33 are i n the balanced category, MO are in the somewhat n o n - r u l e - o r i e n t e d category, 11 are i n t h e / h i g h l y n o n - r u l e - o r i e n t e d category, 23 are in the somewhat r u l e - o r i e n t e d category, and 4 are in the h i g h l y r u l e - o r i e n t e d category. The median for/ the d i s t r i b u t i o n i s 0.50; the lower and upper f o u r t h s 135 h i g h l y n o n - r u l e - o r i e n t e d 0 1 06688 339999 somewhat n o n - r u l e - o r i e n t e d 2 3 55555 1 1 1 88 balanced 4 5 6 2444444 00000000000000000666666666 somewhat r u l e - o r i e n t e d 6 7 8 3333377999999 5555555555 h i g h l y r u l e - o r i e n t e d 8 9 10 1338 N=81 Fi g u r e 4-29 - D i s t r i b u t i o n of r u l e - o r i e n t e d n e s s scores f o r geometry. are 0.38 and 0.67. Thus, when a l l four geometric t o p i c s are considered, wide v a r i a t i o n among teachers i n t h e i r s t r e s s of r u l e s i s s t i l l present j u s t as i t was at the i n d i v i d u a l t o p i c l e v e l . The percent of scores i n the two n o n - r u l e - o r i e n t e d c a t e g o r i e s , the balanced category and the two r u l e - o r i e n t e d c a t e g o r i e s are 26%, 41%, and 33% r e s p e c t i v e l y . Thus, the l a r g e s t number of teachers put equal emphasis on r u l e s and other c o n t e n t - s p e c i f i c methods. Of the remaining teachers, s l i g h t l y more s t r e s s e d r u l e s than s t r e s s e d a l t e r n a t i v e approaches to content. 4.4 Comparisons In Rule-Orientedness For Topics And Content  Areas In F i g u r e 4-30 the d i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores f o r the e i g h t t o p i c s f o r which t h i s v a r i a b l e was d e f i n e d are shown i d e n t i f i e d by content area. These d i s t r i b u t i o n s are 136 not i d e n t i c a l . In p a r t i c u l a r , two have medians of 0.25, four have medians of 0.50, and two have medians of 0.75. A l s o , the F-spread of four d i s t r i b u t i o n s i s 0.25, while the F-sread of the other four d i s t r i b u t i o n s i s 0.50. There does not appear to be any strong a s s o c i a t i o n between the content area and the median value of F-spread of the d i s t r i b u t i o n . 1 .00 0.80 o.6o 0.40 0.20 0.00 T * (3) X T * (2) - 7 - - r -~ L - • X (6) A L O G f O A L G G C O C E O A L G C f O A n I F i g u r e 4-30 - D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores f o r e i g h t t o p i c s . In s p i t e of the d i f f e r e n c e s among these d i s t r i b u t i o n s , they 137 are not as d i f f e r e n t from each other as was the case f o r the d i s t r i b u t i o n s of content r e p r e s e n t a t i o n scores at the t o p i c l e v e l . In p a r t i c u l a r , i n each of the r u l e - o r i e n t e d n e s s d i s t r i b u t i o n s c o n s i d e r a b l e v a r i a t i o n between scores e x i s t s . In four of the e i g h t d i s t r i b u t i o n s , the d i f f e r e n c e between the o u t l i e r c u t o f f s i s 1.00 and in three others i t i s 0.75. 0.?? *>-l- A L C C B B A G r i w c T P y T O P i r c F i g u r e 4—31 - D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s s c o r e s for each content area and o v e r a l l . When the r u l e - o r i e n t e d n e s s scores are averaged f o r each 1 38 teacher both f o r t o p i c s w i t h i n a content area and across a l l t o p i c s , the r e s u l t i n g d i s t i b u t i o n s s t i l l show c o n s i d e r a b l e v a r i a t i o n between scores. These d i s t r i b u t i o n s are shown i n Figu r e 4-31. An i n s p e c t i o n of these d i s t r i b u t i o n s shows that geometry was t r e a t e d i n a more r u l e - o r i e n t e d manner than was the case f o r a l g e b r a . The lower f o u r t h , median, and upper f o u r t h of the algebra d i s t r i b u t i o n are 0.33, 0.42 and 0.50. The corresponding values f o r the geometry d i s t r i b u t i o n are 0.38, 0.50 and 0.67. The F-spreads of the algebra and geometry d i s t r i b u t i o n s are 0.17 and 0.29 showing more v a r i a t i o n i n the s t r e s s given to r u l e s i n geometry than i n a l g e b r a . 4.5 Achievement Level Comparisons In F i g u r e 4-32 boxplots are used to compare the d i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores s e p a r a t e l y f o r low and high achievement c l a s s e s . P l o t s are given f o r algebra scores, geometry sc o r e s , and o v e r a l l s c o r e s . The p l o t s show the s i m i l a r emphasis which r u l e s r e c e i v e d i n both types of c l a s s e s . For a l l t o p i c s the s t r e s s p l a c e d on r u l e s was s l i g h t l y g reater i n low achievement c l a s s e s . The median of the d i s t r i b u t i o n f o r low achievement c l a s s e s i s 0.46 compared with 0.44 f o r high achievement c l a s s e s . For Algebra content, however, the median score i s higher f o r high achievement c l a s s e s than f o r low achievement c l a s s e s . For a l g e b r a content the r u l e - o r i e n t e d n e s s scores are more f r e q u e n t l y i n the two r u l e - o r i e n t e d c a t e g o r i e s f o r low 1 39 1.00 0.90 0.B0 0.70 0.60 0.50 o.uo 0.30 0..20 0.10 o.oo T 1 low high a l l t o p i c s low h i g h algebra low high geometry Figure 4-32 - D i s t r i b u t i o n s of r u l e - o r i e n t e d n e s s scores f o r low and high achievement l e v e l c l a s s e s . achievement c l a s s e s than for high achievement c l a s s e s . While there are s i x scores above 0.60 f o r low achievement c l a s s e s , there i s a s i n g l e score above t h i s value f o r high achievement c l a s s e s , the scores are a l s o most f r e q u e n t l y in the non-rule-o r i e n t e d c a t e g o r i e s for low as compared to high achievement c l a s s e s . The d i f f e r e n c e i s s m a l l , however, with 16 scores below 0.40 f o r low achievement c l a s s e s compared to 14 scores for high achievement c l a s s e s . Thus, the most n o t i c e a b l e c o n t r a s t between 1 40 the two groups i s the grea t e r frequency with which r u l e s were s t r e s s e d i n low achievement c l a s s e s . For ' geometry content the two d i s t r i b u t i o n s have i d e n t i c a l medians of 0.50. The d i s t r i b u t i o n s d i f f e r , however, i n the extent to which scores vary from t h i s c e n t r a l value. Considerably more v a r i a t i o n from the median occurs f o r the d i s t r i b u t i o n of scores f o r low achievement c l a s s e s . The F-spread f o r that d i s t r i b u t i o n i s 0.42 while the corresponding value for the d i s t r i b u t i o n of scores f o r high achievement c l a s s e s i s only 0.18. For low achievement c l a s s e s , e i g h t scores are in e i t h e r the h i g h l y r u l e - o r i e n t e d or the h i g h l y non-rule-o r i e n t e d c a t e g o r i e s . For high achievement c l a s s e s the corresponding number of scores i s two. Thus, i t was more l i k e l y i n low achievement c l a s s e s than i n high achievement c l a s s e s that r u l e s would e i t h e r r e c e i v e a great deal of s t r e s s or very l i t t l e s t r e s s . 5. DIVERSITY OF INSTRUCTION Table 4-6 shows the percentage of teachers whose d i v e r s i t y of i n s t r u c t i o n scores are i n the low, moderate and high d i v e r s i t y c a t e g o r i e s f o r each t o p i c , each content area, and o v e r a l l . The t a b l e a l s o shows median d i v e r s i t y scores as w e l l as the number of teaching methods l i s t e d i n the TSQs f o r each t o p i c . The content area scores represent the average number of methods used by teachers f o r those t o p i c s i n each content area. S i m i l a r l y , the o v e r a l l scores represent the average number of methods used by teachers a c r o s s a l l 16 t o p i c s . The median d i v e r s i t y score over a l l t o p i c s i s 2.5. ' Such a Table 4- 6 - D i v e r s i t y of I n s t r u c t i o n Scores TOPIC OR ARIA MO. or DIVERSITY SCORE DISTRIBUTION MEDIAN TSQ (% or SCORES IN CATCOORY) DIVERSITY METHOOS Low MODERATE HIGH SCORE FRACTIONS. 10 1.3 11.5 87.2 5.0 FRACTION ADDITION e 40.0 29.3 30.7 2.0 DECIMALS 6 1.2 23.2 75.6 3.3 DECIMAL OPERATIONS 3 80.7 18.1 1.2 1.5 PROPORT1ONS u 17.1 50.0 32.9 2.5 ARITHMETIC TOPICS (6.2) 3.5 55.3 41.2 2.6 INTEGERS 5 17.5 43.8 38.8 2.5 INTEGER AOOITION 3 33.8 55.0 11.3 2.0 INTEGER SUBTRACTION 6 17.5 20.0 62.5 3.0 INTEGER MULTIPLICATION 5 20.0 56.0 24.0 2.0 LINEAR EQUATIONS 5 7.4 34.6 58.0 3.0 FORMULAS 5 37.8 36.5 25.7 2.0 AL GE RRAIc Topics (4.8) 1.2 80.2 18.5 2.5 ANCLE SUM THEOREM 8 35.6 45.2 19.2 2.0 PYTHAGOREAN THEOREM 7 28.6 5^ .2 17.1 2.0 7 31.9 44.9 23.2 2.0 AREA or A PARALLELOGRAM B 20.3 26.1 53.6 3.0 VOLUME or A PRISM 3 ei .2 18.8 0.0 1.5 GEOMETRIC TOPICS (6.6) 14.8 71.6 13.6 2.0 A L L TOPICS (Q1) 1.1 78.2 20.7 2.5 score c o u l d be a t t a i n e d by the emphasis of two methods and the use without emphasis of one method, by the emphasis of one method and the use without emphasis of three methods, or by s e v e r a l other combinations of emphasizing and using content s p e c i f i c methods. The d i s t r i b u t i o n of o v e r a l l d i v e r s i t y scores i s shown in the stem-and-leaf p l o t of Figure 4-33. The d i s t r i b u t i o n has three o u t l i e r s c o n s i s t i n g of one e x c e p t i o n a l l y low score of 1.2, the only o v e r a l l d i v e r s i t y score in the low 142 category, and two r e l a t i v e l y high scores of 4.0 and 5.3. The l a t t e r score could be a t t a i n e d through the emphasis of over f i v e methods f o r each t o p i c . low d i v e r s i t y high d i v e r s i t y moderate d i v e r s i t y 1 2 2 88999 0000000001111111222223333333334444444 55555556667777777888888999 000002334 6777777 0 3 N=87 Fi g u r e 4-33 - D i s t r i b u t i o n of d i v e r s i t y scores averaged over a l l t o p i c s . Almost f o u r - f i f t h s of the o v e r a l l d i v e r s i t y scores were between 1.8 and 2.9 i n c l u s i v e and were thus i n the moderate category. A l l of the other scores (except f o r the one lower o u t l i e r ) were 3.0 or higher and were thus i n the high d i v e r s i t y category. Almost a l l of the teachers t y p i c a l l y , then, taught the concepts, o p e r a t i o n s , and p r i n c i p l e s i n t h e i r c u r r i c u l a through the ; use of s e v e r a l approaches on average. About one-f i f t h of the sample employed enough methods to be considered h i g h l y d i v e r s e i n t h e i r i n s t r u c t i o n . 143 5.1 D i v e r s i t y In Teaching A r i t h m e t i c The mean of the d i v e r s i t y scores f o r the f i v e a r i t h m e t i c t o p i c s for each teacher was taken as the d i v e r s i t y score f o r a r i t h m e t i c f o r that teacher. The d i s t i b u t i o n of these scores i s shown in F i g u r e 4-34. low d i v e r s i t y moderate d i v e r s i t y high d i v e r s i t y 33 5 7889 0000111122223333333444 555555666667778899999 00000111122223444 5566667779999 022 5 N=85 Figure 4-34 - D i s t r i b u t i o n of d i v e r s i t y scores f o r a r i t h m e t i c . Out of 85 scores, 47 are i n the moderate d i v e r s i t y category, 3 are i n the low d i v e r s i t y category and 35 are in the high d i v e r s i t y category. The median of the d i s t r i b u t i o n i s 2.7; the lower and upper f o u r t h s are 2.3. and 3.3. The only o u t l i e r i s the extremely high score of 6.1. Thus, i n teaching a r i t h m e t i c t o p i c s , somewhat over one-half of the teachers chose to present a moderate number of content-s p e c i f i c methods on average. S l i g h t l y over t w o - f i f t h s of the teachers showed a greater degree of d i v e r s i t y i n t h e i r 1 44 i n s t r u c t i o n with scores i n the high range. Very few teachers presented so few methods that t h e i r a r i t h m e t i c d i v e r s i t y scores were i n the low range. Figure 4-35 shows boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores f o r the a r i t h m e t i c t o p i c s . The most d i v e r s i t y was shown f o r the concept of f r a c t i o n s . The mean f o r the d i s t r i b u t i o n of scores f o r t h i s t o p i c i s 5.0; the lower and upper fourths are 4.0 and 5.5. A l l of these values are w i t h i n the high d i v e r s i t y range. A l t o g e t h e r , 87% of the teachers showed a high degree of d i v e r s i t y when teaching t h i s t o p i c . T h i s was the l a r g e s t percentage of scores i n t h i s category f o r a l l 16 t o p i c s . Teachers a l s o , i n gene r a l , showed high d i v e r s i t y when teaching the concept of decimals. The mean f o r that t o p i c i s 3.3 and the lower and upper fourths are 3.0 and 4.0. For t h i s t o p i c , 76% of the scores are w i t h i n the high d i v e r s i t y range. The median d i v e r s i t y scores f o r two t o p i c s , the a d d i t i o n of f r a c t i o n s and the concept of p r o p o r t i o n s / are w i t h i n the moderate range being 2.0 and 2.5 r e s p e c t i v e l y . The lower and upper fourths f o r a d d i t i o n of f r a c t i o n s are 1.5 and 3.0. For the concept of p r o p o r t i o n s the corresponding values are 2.0 and 3.0. The f i f t h t o p i c , o perations with decimals, was taught with much l e s s d i v e r s i t y than the other four t o p i c s . The median score f o r t h i s t o p i c i s 1.5; the lower and upper fourths are 1.0 and 1.5. A l l of these values are w i t h i n the low d i v e r s i t y range. In a l l , 81% of the teachers showed low d i v e r s i t y i n 145 10.0 B.O f .0 2.0 0.0 * (2) T C O N C E P T C O N C C P T O P C M . A T I O M S C O N C E P T or A O O I T I O N O T W I T H or TRACtlOHS O f O r R I U A L S O C C I V A L S P R O P O R T I O N S F fl ACT I O N S F i g u r e 4-35 - Boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores for a r i t h m e t i c t o p i c s . t e a c h i n g operations w i t h decimals. For the f i v e a r i t h m e t i c t o p i c s there was a strong p o s i t i v e a s s o c i a t i o n between t h e number of methods u s e d i n p r e s e n t i n g a t o p i c and the number of a v a i l a b l e methods. The c o r r e l a t i o n between the median d i v e r s i t y score and the number of methods l i s t e d in the TSQs i s 0.74. Teachers d i d not fo l l o w t h i s trend in t e a c h i n g the a d d i t i o n o f f r a c t i o n s , however. W h i l e e i g h t methods were l i s t e d i n t h e F r a c t i o n TSQ f o r t h i s t o p i c , the median d i v e r s i t y score was only 2.0. One ex p l a n a t i o n for the 1 46 f a c t that teachers showed the most d i v e r s i t y i n p r e s e n t i n g the concepts of f r a c t i o n s , decimals and p r o p o r t i o n s and the l e a s t d i v e r s i t y i n p r e s e n t i n g the o p e r a t i o n of a d d i t i o n of f r a c t i o n s and o p e r a t i o n s with decimals i s th a t , perhaps, these teachers f e l t that while a r i t h m e t i c concepts should be presented with c o n s i d e r a b l e d i v e r s i t y , a r i t h m e t i c operations are b e t t e r taught using fewer methods even i f many are a v a i l a b l e . 5.2 D i v e r s i t y In Teaching Algebra The mean of the d i v e r s i t y scores f o r the s i x a l g e b r a i c t o p i c s f o r each teacher was taken as the d i v e r s i t y score of alge b r a f o r that teacher. The d i s t r i b u t i o n of these scores i s shown i n F i g u r e 4-36. low d i v e r s i t y moderate d i v e r s i t y high d i v e r s i t y 677888999 000000011111222223333333334 55555556666777788888888888999 0000223333 55667 N=8 1 Fi g u r e 4-36 - D i s t r i b u t i o n of d i v e r s i t y scores f o r a l g e b r a . Out of 81 scores, 65 are i n the moderate d i v e r s i t y range, one i s i n the low d i v e r s i t y range, and 15 are i n the high d i v e r s i t y range. The median of the d i s t r i b u t i o n i s 2.5; the lower and upper fo u r t h s are 2.1 and 2.8. These r e s u l t s are 147 s i m i l a r to those that were obtained f o r a r i t h m e t i c . For algebra even a l a r g e r percentage of scores, 80%, are wi t h i n the moderate d i v e r s i t y range and a cor r e s p o n d i n g l y smaller percentage of scor e s , 19%, are w i t h i n the high d i v e r s i t y range. As with a r i t h m e t i c , very few teachers showed low o v e r a l l d i v e r s i t y f o r al g e b r a . F i g u r e 4-37 shows boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores f o r the a l g e b r a i c t o p i c s . These d i s t r i b u t i o n s d i f f e r l e s s markedly from each other than was the case f o r the a r i t h m e t i c t o p i c s . While the medians f o r a r i t h m e t i c t o p i c s ranged from 1.5 to 5.0, f o r a l g e b r a i c t o p i c s the range i s form 2.0 to 3.0. The g r e a t e s t mean d i v e r s i t y f o r a l g e b r a i c t o p i c s occurred for s u b t r a c t i o n of i n t e g e r s and s o l v i n g l i n e a r equations. In both cases the median d i v e r s i t y score i s 3.0. The lower f o u r t h i n both cases i s 2.5. The upper f o u r t h for s u b t r a c t i o n of in t e g e r s i s 3.8, while f o r s o l v i n g l i n e a r equations i t i s 3.5. Thus, in terms of d i v e r s i t y these t o p i c s were t r e a t e d i n a very s i m i l a r f a s h i o n . The concept of i n t e g e r s has the t h i r d highest median d i v e r s i t y value f o r t h i s group of t o p i c s at 2.5. The lower and upper fo u r t h s are 2.0 and 3.0. Teachers tended to show the l e a s t d i v e r s i t y w i t h i n algebra when they taught a d d i t i o n and m u l t i p l i c a t i o n of i n t e g e r s and the concept of formulas. In each case the median d i v e r s i t y score i s 2.0. For m u l t i p l i c a t i o n of i n t e g e r s the middle 50% of the scores vary l i t t l e from t h i s value with a lower f o u r t h of 2.0 1 4 8 10.0 Co 6.0 2.0 0.0 T X X *(2) X (5) T CZZ) * — - * » X (3) 1 1 C O N C E P T S U B T H A i C T I O N C O N C E P T OF A D D I T I O N Or! M U L T I P L I C A T I O N OF S O L V I N G I N T E G E R S Or I N T E G E R S O r r O R M U L A S L I N E A R I N T E G E R S I N T E G E R S E Q U A T I O N S F i g u r e 4-37 - Boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores for a l g e b r a i c t o p i c s . and an upper f o u r t h of 2.5. For a d d i t i o n of i n t e g e r s the corresponding values are 1.5 and 2.5 while for the concept of formulas they are 1.5 and 3 .0 . The a s s o c i a t i o n between the number of teaching methods a v a i l a b l e to teachers for each t o p i c and the number which they a c t u a l l y used was not as strong f o r algebra as i t was f o r a r i t h m e t i c . The c o r r e l a t i o n between the number of TSQ methods and the median d i v e r s i t y score for each t o p i c was 0.59 compared to 0.74 f o r a r i t h m e t i c . Also, teachers d i d not c o n s i s t e n t l y use 149 more approaches for a l g e b r a i c concepts than for a l g e b r a i c o p e r a t i o n s as had been the case f o r a r i t h m e t i c . 5.3 D i v e r s i t y In Teaching Geometry As with the other content areas, the mean of the d i v e r s i t y scores f o r the f i v e geometric t o p i c s f o r each teacher was taken as the d i v e r s i t y score "for geometry for that teacher. The d i s t r i b u t i o n of these scores i s shown i n F i g u r e 4-38. low d i v e r s i t y moderate d i v e r s i t y high d i v e r s i t y 5 233344 55555 6666777777778888899999 0000000111112233333334444 55566778889 033 566779 03 N=81 F i g u r e 4-38 - D i s t r i b u t i o n of d i v e r s i t y scores f o r geometry. Out of 81 scores f o r t h i s v a r i a b l e , 58 are i n the moderate d i v e r s i t y range. T h i s i s s i m i l a r to the r e s u l t s f o r a r i t h m e t i c and a l g e b r a . U n l i k e the other two content areas f o r which the remaining d i v e r s i t y scores were almost a l l in the high d i v e r s i t y range, however, the remaining d i v e r s i t y scores f o r geometry are almost e q u a l l y s p l i t between the low and high d i v e r s i t y c a t e g o r i e s . Of the 23 scores, 12 are i n the low d i v e r s i t y category and 11 are i n the high d i v e r s i t y category. The median of the geometry d i s t r i b u t i o n i s 2.0. T h i s score 1 50 c o u l d be a t t a i n e d by the emphasis of j u s t two c o n t e n t - s p e c i f i c methods f o r each t o p i c . The lower and upper fourths of the d i s t r i b u t i o n are 1.7 and 2.5. Each of these values i s l e s s than the corresponding values f o r both a r i t h m e t i c and a l g e b r a . Thus, teachers showed c o n s i d e r a b l y l e s s d i v e r s i t y i n teaching geometric t o p i c s than was the case f o r the other two content areas. T h i s i s in s p i t e of the f a c t that more c o n t e n t - s p e c i f i c methods appear to be a v a i l a b l e f o r the geometric t o p i c s than f o r the t o p i c s i n the other content areas. An average of 6.6 methods were l i s t e d i n the TSQs for each geometric t o p i c compared to 6.2 and 4.8 methods f o r the a r i t h m e t i c and a l g e b r a i c t o p i c s r e s p e c t i v e l y . F i g u r e 4-39 shows boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores f o r the geometric t o p i c s . The f i r s t three of these p l o t s appear almost i d e n t i c a l . The medians, lower f o u r t h s , and upper fourths are 2.0, 1.5 and 2.5 i n each case. For each of these t o p i c s , the t r i a n g l e angle sum theorem, the Pythagorean theorem, and the concept of it, low d i v e r s i t y scores occurred more f r e q u e n t l y than high d i v e r s i t y scores. T h i s i s of i n t e r e s t because of the r e l a t i v e l y l a r g e number of teaching options a v a i l a b l e f o r the t o p i c s . The Geometry TSQ l i s t e d e i g h t methods for the t r i a n g l e angle sum theorem and seven methods f o r the Pythagorean theorem. The Measurement TSQ l i s t e d seven methods for the concept of it . Very few teachers were f a m i l i a r with or chose to use s e v e r a l of the content s p e c i f i c methods f o r each of these t o p i c s . During the in t e r v i e w s which t h i s researcher conducted, teachers I n d i c a t e d 151 10.0 8.0 6.0 *».o 2.0 0.0 X X X T T s u u p r PYTHACOHEAN CONCEPT ADEA o r VOLUME o r A TMC ANGLES THEOREM Or A RECTANGULAR. PRISM 'IN j*- TT PARALLCLOCnA». TP I ANGLE Figure 4 - 3 9 - Boxplots of the d i s t r i b u t i o n s of d i v e r s i t y scores f o r geometric t o p i c s . that they were not f a m i l i a r with s e v e r a l of the te a c h i n g methods given in the T S Q s for geometry but intended to u s e them in t h e i r teaching in the f u t u r e . The f a c t that for these three t o p i c s the d i v e r s i t y scores tend to be low in s p i t e of a r e l a t i v e l y l a r g e number of options l i s t e d in the TSQs f u r t h e r supports the contention that teachers completed the q u e s t i o n n a i r e s in a c o n s c i e n t i o u s manner. Of the remaining two geometric t o p i c s , one was approached 152 with more d i v e r s i t y and one with l e s s d i v e r s i t y than the three t o p i c s d i s c u s s e d above. The area of a p a r a l l e l o g r a m was presented with the g r e a t e s t d i v e r s i t y of the geometric t o p i c s . The median of the d i s t r i b u t i o n of scores f o r t h i s t o p i c i s 3.0, the only median of a geometric t o p i c i n the high d i v e r s i t y range. The lower and upper fourths are 2.0 and 4.0. For the volume of a r e c t a n g u l a r prism, i n c o n t r a s t , the median i s 1.5 and the lower and upper f o u r t h s are 1.0 and 1.5, a l l values i n the low d i v e r s i t y range. As noted, the number of methods teachers used f o r geometric t o p i c s was l e s s than might be expected based on the number of methods a v a i l a b l e and the number of methods used f o r teaching t o p i c s w i t h i n other content areas. The c o r r e l a t i o n between a v a i l a b l e methods and the median d i v e r s i t y scores f o r the geometric t o p i c s was nonetheless q u i t e high at 0.70. 5.4 Comparisons In D i v e r s i t y Between Content Areas And Topics The d i s t r i b u t i o n s of d i v e r s i t y scores f o r the three content areas and o v e r a l l t o p i c s are shown i n F i g u r e 4-40. The boxplots show the s i m i l a r i t y between the four d i s t r i b u t i o n s . The most n o t i c e a b l e d i f f e r e n c e among them i s the lower median and lower f o u r t h s f o r geometry. As noted e a r l i e r , the median d i v e r s i t y score f o r geometry i s 2.0 while the median values f o r a r i t h m e t i c and a l g e b r a are 2.7 and 2.5 r e s p e c t i v e l y . For the d i s t r i b u t i o n of o v e r a l l d i v e r s i t y scores the median i s 2.5. One p o s s i b l e i m p l i c a t i o n of t h i s r e s u l t i s that teachers may need more i n s e r v i c e t r a i n i n g i n the area of geometry in order to l e a r n 1 53 10.0 . P.O . 6.0 . * «*.o . n : X 2.0 . 0.0 . A L L A R I T H M E T I C A L B t P R A C C O M C T f c Y TOP I C S F i g u r e 4-40 - D i s t r i b u t i o n of d i v e r s i t y s c o r e s f o r e a c h c o n t e n t a r e a a nd a c r o s s a l l t o p i c s . more c o n t e n t - s p e c i f i c t e a c h i n g methods t h a n t h e y do i n t h e a r e a s o f a r i t h m e t i c a n d a l g e b r a . In F i g u r e 4-41 t h e d i s t r i b u t i o n s of d i v e r s i t y s c o r e s f o r t h e 16 t o p i c s a r e shown i d e n t i f i e d by c o n t e n t a r e a w i t h t h e number of methods l i s t e d f o r e a c h i n t h e TSQs i n c l u d e d i n p a r e n t h e s e s . T e a c h e r s a p p a r e n t l y t e n d e d t o p r e s e n t a r i t h m e t i c t o p i c s w i t h a g r e a t e r d i v e r s i t y o f methods t h a n a l g e b r a i c o r g e o m e t r i c t o p i c s . T h i s c o u l d r e f l e c t a g r e a t e r d e g r e e of f a m i l i a r i t y w i t h a r i t h m e t i c t o p i c s on t h e ; p a r t o f t h e t e a c h e r s . T X x * (2) T 10.0 8 . 0 6.0 4 . 0 2 . 0 0 . 0 1 x (*> X A n I A L G ( i l U G t O A i l A L G A L C A R I (3) (3) (7) (e) (0) (5) (6) (6) S C O A L G O r O A L G A R I A L C G t O A R I (3) (5) (7) (5) CO (5) (C) (io) F i g u r e 4-41 - D i s t r i b u t i o n s of d i v e r s i t y scores for the 16 t o p i c s . F i g u r e 4-41 a l s o shows that, with the exception of three geometric t o p i c s , teachers tended to use more methods in t h e i r teaching when more a l t e r n a t i v e s were a c t u a l l y a v a i l a b l e to them. The c o r r e l a t i o n between the median d i v e r s i t y score f o r each of the 16 t o p i c s and the number of methods l i s t e d in the TSQs for that t o p i c i s 0.59. 155 5.5 Achievement L e v e l Comparisons In F i g u r e 4-42 boxplots are used to compare the d i s t r i b u t i o n s of d i v e r s i t y scores of low and high achievement c l a s s e s f o r each content area and over a l l t o p i c s . These p l o t s show that there was no apparent a s s o c i a t i o n between c l a s s achievement l e v e l and the number of methods teachers used i n p r e s e n t i n g mathematical t o p i c s . The median values f o r each of the four p a i r s of d i s t r i b u t i o n s d i f f e r from each other by no more than 0.1. • Teachers of low achievement c l a s s e s , f o r example, have a median d i v e r s i t y score of 2.1 f o r geometry while the corresponding value f o r teachers of high achievement c l a s s e s i s 2.2. Greater d i f f e r e n c e s between the d i s t r i b u t i o n s e x i s t for v a r i a t i o n as measured by F-spread, but these d i f f e r e n c e s are not c o n s i s t e n t f o r a l l content areas. O v e r a l l , teachers of low achievement c l a s s e s show more u n i f o r m i t y in t h e i r d i v e r s i t y scores than teachers of high achievement c l a s s e s . The F-spread of the d i s t r i b u t i o n of o v e r a l l d i v e r s i t y scores f o r low achievement c l a s s e s i s 0.7 compared to 0.9 f o r high achievement c l a s s e s . For a r i t h m e t i c content the F-spreads f o r low and high achievement c l a s s e s are 1.0 and 1.4 r e s p e c t i v e l y . For geometry, however, the F-spread i s gr e a t e r f o r the d i s t r i b u t i o n of high achievement l e v e l c l a s s e s and f o r algebra the F-spreads of the two d i s t r i b u t i o n s are the same. Thus, i t would appear that teachers d i d not i n general base t h e i r d e c i s i o n s as to how many methods would be optimal in p r e s e n t i n g the t o p i c s i n t h e i r c u r r i c u l a on the achievement I O . O 9.0 R .O 7.0 rt.O 5.0 fc.o 3.o 2.0 1.0 0.0 T J . T I T 5 L O W HI OH ALL TOPICS LOW HIOH ARITHMETIC LOW HI OH ALGEBRA LOW HIOH GEOMETRY Figure 4-42 - Distributions of diversity scores for low and high achievement level classes. level of their classes. Alternately, they may have felt that the same number of content-specific methods was appropriate for both low and high achievement classes. 5.6 Diversity Scores Of Teachers And Textbooks When the content emphasis scores of teachers using each of the two major textbooks were analyzed separately, an association was found between the relative emphasis of the content areas in the textbooks and the emphasis given by teachers to content 157 areas i n t h e i r implemented c u r r i c u l a . A s i m i l a r , though weaker, p o s i t i v e a s s o c i a t i o n was found between the number of methods inc l u d e d i n each textbook f o r groups of t o p i c s and the average number of methods employed by teachers using each textbook. Table 4-7 shows the number of c o n t e n t - s p e c i f i c methods l i s t e d i n the TSQs for the 16 t o p i c s which appeared i n the two Mathematics 8 textbooks. The t a b l e a l s o shows the number of methods contained i n each textbook for two groups of t o p i c s . Topic Group 1 c o n s i s t s of those t o p i c s f o r which more TSQ methods were found i n Mathematics II than i n School Mathematics 2 while Topic Group 2 c o n s i s t s of those t o p i c s f o r which more methods were found i n School Mathematics 2. 1 2 Table 4- 7 - Number of TSQ Methods Contained i n the Commonly Used Textbooks f o r S e l e c t e d Topics Textbook A l l T o pics Topic Group 1 Topic Group 2 Mathematics II 31 16 4 School Mathematics 2 36 1 1 1 1 In F i g u r e 4-43 boxplots are used to compare the average d i v e r s i t y scores of the teachers using each textbook f o r the three groups of t o p i c s . The two d i s t r i b u t i o n s of d i v e r s i t y scores f o r a l l t o p i c s are n e a r l y i d e n t i c a l . The teachers using School Mathematics 2 had the higher median score, however, 2.5 compared to 2.3. T h i s corresponds to a somewhat greater number of t o t a l teaching methods i n School Mathematics 2 for the 16 1 2 Topic Group 1 inc l u d e s i n t e g e r s u b t r a c t i o n , s o l v i n g l i n e a r equations, decimals and the Pythagorean theorem. Topic Group 2 c o n s i s t s of f r a c t i o n s , decimal o p e r a t i o n s , the concept of it, and the volume of a rectangular prism. 1 58 t o p i c s . 10.0 1.0 e.o 7.0 6.0 5.0 **.o 3.0 2.0 1.0 0.0 T ^ T SW2 MM A L L T O P I C S S M 2 M i l T O P I C 8 R 0 U P 1 T SM2 Ml I T O P I C O R O U P 2 F i g u r e 4-4 3 - D i s t r i b u t i o n s of average d i v e r s i t y scores; f o r the ^groups of t o p i c s for users of each textbook. For Topic Group 1, the median score of those teachers using Mathematics II i s 2.9 compared to 2.8 for teachers using School  Mathematics 2. The upper f o u r t h s f o r the two d i s t r i b u t i o n s are 3.4 and 3.1 r e s p e c t i v e l y . These f i g u r e s are c o n s i s t e n t with the f a c t that 16 methods are contained in the former textbook f o r t h i s group of t o p i c s , compared to 11 methods contained in the 1 59 l a t t e r t e x t . For Topic Group 2, only four teaching methods are contained i n Mathematics II compared to 11 methods in School Mathematics  2. For t h i s group of t o p i c s , the teachers using School  Mathematics 2 showed more d i v e r s i t y in t h e i r i n s t r u c t i o n than the teachers using the other t e x t . The median d i v e r s i t y score for the teachers using School Mathematics 2 i s 2.6 compared to a median score of 2.3 f o r the teachers using the other book. The lower and upper fourths were a l s o higher f o r the teachers using School Mathematics 2. Although d i f f e r e n c e s between the d i s t r i b u t i o n s of d i v e r s i t y scores f o r users of the two textbooks were not l a r g e , the p a t t e r n was c o n s i s t e n t . When more methods were contained i n one textbook f o r a group of t o p i c s , teachers tended to use more methods in teaching those t o p i c s . 160 V. CONCLUSIONS 1. SUMMARY OF THE RESULTS The implemented c u r r i c u l a of Mathematics 8 teachers which were examined i n t h i s study had, almost u n i v e r s a l l y , c e r t a i n commonalities with respect to content. Nearly a l l of these c u r r i c u l a i n c l u d e d a r i t h m e t i c , a l g e b r a , and geometry. Only three teachers reported o m i t t i n g one of these content areas a l t o g e t h e r . Furthermore, the mean r e l a t i v e emphasis given each area was n e a r l y the same with a l g e b r a r e c e i v i n g somewhat l e s s c l a s s time than a r i t h m e t i c or geometry. On average, teachers devoted 29% of t h e i r time to a l g e b r a i c t o p i c s , as compared with 35% f o r a r i t h m e t i c t o p i c s and 36% f o r geometric t o p i c s . Thus, not only were each of the content areas almost always • part of the implemented c u r r i c u l a , t h e i r average degrees of emphasis wi t h i n these c u r r i c u l a were n e a r l y equal'. Despite these s i m i l a r i t i e s , i t cannot be s a i d that there was a common c u r r i c u l u m i n terms of content emphasis i n the c l a s s e s which were s t u d i e d . The range of; emphasis scores w i t h i n each content area was l a r g e with the most d i v e r s i t y o c c u r r i n g f o r geometry. Two teachers, f o r example, spent only 7% of t h e i r c l a s s time on geometry while a t h i r d devoted 66% to that content area. C o n s i d e r i n g only the middle 50% of the content emphasis scores i n each case, teachers d i f f e r e d by 13% i n the percent of c l a s s time given to a r i t h m e t i c , by 9% i n the percent of c l a s s time given to alge b r a , and by 16% i n the percent of c l a s s time given to geometry. I n c l u d i n g a l l non-zero values, the range of 161 content emphasis scores expressed as percents was 51% f o r a r i t h m e t i c , 34% for al g e b r a , and 59% f o r geometry. Thus, there was f a r from a common c u r r i c u l u m i n the courses s t u d i e d with wide v a r i a t i o n in the emphasis given to the three content areas. The p r o p o r t i o n of scores w i t h i n each category of content emphasis f u r t h e r demonstrates both the s i m i l a r i t i e s and d i f f e r e n c e s i n the implemented c u r r i c u l a as measured by t h i s v a r i a b l e . For a r i t h m e t i c , a l g e b r a , and geometry 76%, 74%, and 69% of the r e s p e c t i v e emphasis scores were w i t h i n the moderate range. Thus, f o r each content area most teachers provided a moderate degree of content emphasis. On-the-other-hand, 60% of the teachers i n t h i s study gave l i g h t or very l i g h t emphasis to at l e a s t one content area. Thus, most teachers d i d not give moderate emphasis to a l l content areas. The d i f f e r e n c e s in content emphasis among teachers were the strongest f o r geometry. Although t h i s content area had the highest mean emphasis score, 0.36, i t i s a l s o the case that a l a r g e r percentage of teachers, 31%, gave t h i s area l i g h t or very l i g h t emphasis than was true f o r the other content areas. Due to the s e q u e n t i a l nature of mathematics and the s p i r a l approach i n c o r p o r a t e d in many textbooks and suggested by the B. C. Curriculum Guide, t h i s l e v e l of v a r i a t i o n in content emphasis may not be d e s i r a b l e . E s p e c i a l l y with respect to geometry t h i s d i v e r s i t y w i t h i n the Mathematics 8 c u r r i c u l u m may make assumptions made by Mathematics 9 teachers regarding p r i o r l e a r n i n g of content q u i t e problematic. O v e r a l l and f o r each content area, teachers showed a s l i g h t 162 preference for a b s t r a c t as compared with p e r c e p t u a l teaching methods. The median o v e r a l l mode of r e p r e s e n t a t i o n score was 0.57. For a r i t h m e t i c , a l g e b r a , and geometry the median scores were 0.64, 0.54, and 0.54 r e s p e c t i v e l y . S l i g h t l y over o n e - t h i r d of the teachers i n t h i s study c o u l d be c l a s s i f i e d as a b s t r a c t i n t h e i r p r e s e n t a t i o n s to students based on t h e i r o v e r a l l content r e p r e s e n t a t i o n scores. Almost a l l of the other teachers c o u l d be c l a s s i f i e d as showing a balance between a b s t r a c t and pe r c e p t u a l teaching methods. Of the three content areas, a r i t h m e t i c was most u s u a l l y d e a l t with a b s t r a c t l y by te a c h e r s . About o n e - t h i r d of the teachers balanced t h e i r i n s t r u c t i o n between p e r c e p t u a l and a b s t r a c t methods f o r a r i t h m e t i c while almost a l l of the other teachers showed a c l e a r preference f o r a b s t r a c t methods f o r t h i s review content area. Most teachers, 84% of the sample, balanced t h e i r i n s t r u c t i o n f o r algebra content. The remainder favored a b s t r a c t methods. Teachers showed the l e a s t u n i f o r m i t y i n t h e i r mode of content r e p r e s e n t a t i o n f o r geometry. While 48% balanced a b s t r a c t and perceptual methods i n teaching geometry to students, 27% c l e a r l y favored a b s t r a c t methods while almost as many, 25%, c l e a r l y favored p e r c e p t u a l methods. Geometry was the only content area f o r which an a p p r e c i a b l e number of teachers put a d e f i n i t e s t r e s s on p e r c e p t u a l approaches to content across a l l t o p i c s . Wide v a r i a t i o n e x i s t e d among t o p i c s i n the mode of 163 r e p r e s e n t a t i o n t y p i c a l l y used during i n s t r u c t i o n . Some t o p i c s were most of t e n taught using p e r c e p t u a l methods while other t o p i c s were most of t e n taught using a b s t r a c t methods. For four t o p i c s the median r e p r e s e n t a t i o n score was below 0.40 and hence w i t h i n the pe r c e p t u a l range. Two of the t o p i c s are a l g e b r a i c , two geometric. These t o p i c s were i n t e g e r s , i n t e g e r a d d i t i o n , the angle sum theorem, and the Pythagorean theorem. The median r e p r e s e n t a t i o n score was not i n the pe r c e p t u a l range f o r any a r i t h m e t i c t o p i c . The m a j o r i t y of t o p i c s , e i g h t out of the 14 f o r which t h i s v a r i a b l e was d e f i n e d , were u s u a l l y taught a b s t r a c t l y . For each of these t o p i c s the median r e p r e s e n t a t i o n score was above 0.60. The a r i t h m e t i c t o p i c s which were taught with t h i s l e v e l of a b s t r a c t i o n were decimals and decimal o p e r a t i o n s . The corresponding a l g e b r a i c t o p i c s were in t e g e r s u b t r a c t i o n , i n t e g e r m u l t i p l i c a t i o n , and formulas while the corresponding geometric t o p i c s were the concept of IT, the area of a p a r a l l e l o g r a m , and the volume of a prism. For only two t o p i c s was the median r e p r e s e n t a t i o n score w i t h i n the balanced range. These t o p i c s , f r a c t i o n s and f r a c t i o n a d d i t i o n , each i n v o l v e d a r i t h m e t i c content. v An i n s p e c t i o n of the mode of r e p r e s e n t a t i o n scores f o r the three content areas as a whole would l e a d one to the c o n c l u s i o n that the two areas which contained mostly new m a t e r i a l , algebra and geometry, were u s u a l l y taught with a nea r l y equal balance between pe r c e p t u a l and a b s t r a c t methods while the review area, a r i t h m e t i c , was taught with an emphasis on a b s t r a c t methods. 164 While true i n a general sense, i t i s a l s o true that t h i s p a t t e r n was not evident at the l e v e l of s p e c i f i c t o p i c s . Four t o p i c s were taught q u i t e p e r c e p t u a l l y by most teachers while e i g h t t o p i c s were taught q u i t e a b s t r a c t l y by most te a c h e r s . O v e r a l l , the teachers i n t h i s study tended to show a s l i g h t p r e f e r ence f o r n o n - r u l e - o r i e n t e d approaches to content over r u l e - o r i e n t e d approaches. Averaged over the e i g h t t o p i c s f o r which t h i s v a r i a b l e was d e f i n e d , the median r u l e - o r i e n t e d n e s s score was 0.47. F i f t y - t w o percent of the o v e r a l l r u l e -o r ientedness scores were i n the balanced category, while 26% were i n the two n o n - r u l e - o r i e n t e d c a t e g o r i e s , and 22% were in the two r u l e - o r i e n t e d c a t e g o r i e s . Thus, over a q u a r t e r of the teachers i n t h i s study showed a c l e a r tendency to emphasize approaches to content other than the statement of r u l e s while almost as many showed a c l e a r tendency to emphasize statements of mathematical r u l e s . Rules were emphasized s t r o n g l y f o r the one a r i t h m e t i c t o p i c f o r which t h i s v a r i a b l e was d e f i n e d . The median r u l e -o rientedness score f o r t h i s t o p i c , decimal o p e r a t i o n s , was 0.75. Over 70% of the teachers i n t h i s study put more emphasis on the a c t u a l r u l e s f o r decimal p o i n t placement dur i n g i n s t r u c t i o n than on approaches which provide reasons f o r the placement of the decimal p o i n t . O v e r a l l , teachers put more emphasis on r u l e s when teaching geometric t o p i c s than when teaching a l g e b r a i c t o p i c s . For a l g e b r a i c content the median score was 0.42, with 49% of the scores i n the n o n - r u l e - o r i e n t e d c a t e g o r i e s , 37% i n the balanced 1 65 category, and 14% i n the r u l e - o r i e n t e d c a t e g o r i e s . For geometric content the median score was 0.50 with 27% of the scores i n the n o n - r u l e - o r i e n t e d c a t e g o r i e s , 40% i n the balanced category, and 33% i n the r u l e - o r i e n t e d category. Thus, while o n e - t h i r d of the teachers emphasized r u l e s such as statements of d e f i n i t i o n s and theorems when teaching geometry, l e s s than h a l f as many teachers put a s i m i l a r heavy emphasis on a l g e b r a i c r u l e s such as the r u l e s f o r signed numbers. The d i f f e r e n c e s between the r u l e - o r i e n t e d n e s s scores measured at the l e v e l of i n d i v i d u a l t o p i c s and the r u l e -orientedness scores measured at the l e v e l of the content areas were smaller than the corresponding d i f f e r e n c e s f o r the mode of content r e p r e s e n t a t i o n v a r i a b l e . For four of the e i g h t t o p i c s : i n t e g e r a d d i t i o n , i n t e g e r s u b t r a c t i o n , the concept of IT, and the area of a p a r a l l e l o g r a m the median r u l e - o r i e n t e d n e s s score was 0.50. T h i s compares with medians of 0.42 and 0.50 f o r a l g e b r a i c and geometric content, r e s p e c t i v e l y , as noted above. Teachers put r e l a t i v e l y heavy emphasis on r u l e s f o r decimal operations and the area of a p a r a l l e l o g r a m . In each case the median r u l e - o r i e n t e d n e s s score was 0.75. In c o n t r a s t , teachers put r e l a t i v e l y l i g h t emphasis on r u l e s f o r inte g e r m u l t i p l i c a t i o n and the Pythagorean theorem. In each case the median r u l e - o r i e n t e d n e s s score was 0.25. Across a l l 16 t o p i c s , almost 80% of the teachers showed moderate d i v e r s i t y i n t h e i r use of teaching methods while over 20% showed high d i v e r s i t y . Only one percent of the sample showed low d i v e r s i t y as d e f i n e d i n t h i s study. 166 Teachers showed almost i d e n t i c a l median l e v e l s of d i v e r s i t y i n teaching a r i t h m e t i c and a l g e b r a . For a r i t h m e t i c content the median d i v e r s i t y score was 2.6, while f o r a l g e b r a i c content i t was 2.5. More d i v e r s i t y scores were in the high d i v e r s i t y category f o r a r i t h m e t i c as compared with a l g e b r a , 41% compared with 19%. In both cases n e a r l y a l l of the other scores were i n the moderate d i v e r s i t y category. Teachers showed s u b s t a n t i a l l y l e s s d i v e r s i t y i n t h e i r t e a c h i n g of geometry than i n t h e i r teaching of the other two content areas. The median d i v e r s i t y score f o r geometric content was 2.0. F i f t e e n percent of the scores were i n the low d i v e r s i t y category, 72% were i n the moderate category and 14% were i n the high category. As was the case f o r the mode of content r e p r e s e n t a t i o n and r u l e - o r i e n t e d n e s s v a r i a b l e s , there were marked d i f f e r e n c e s among the median scores f o r the d i v e r s i t y v a r i a b l e at the t o p i c l e v e l . For two t o p i c s , decimal operations and the volume of a prism, the median score was i n the low d i v e r s i t y category. In both cases over 80% of the teachers showed low d i v e r s i t y when teaching the t o p i c . Teachers showed high d i v e r s i t y i n teaching f i v e t o p i c s , two from a r i t h m e t i c , two from a l g e b r a , and one from geometry. These t o p i c s were f r a c t i o n s , decimals, i n t e g e r s u b t r a c t i o n , l i n e a r equations, and the area of a p a r a l l e l o g r a m . The m o s t / d i v e r s i t y was shown by teachers i n approaching f r a c t i o n s . The median d i v e r s i t y score f o r that t o p i c was 5.0. For the other four t o p i c s the median score was between 3.0 and 3.3 i n c l u s i v e . 167 Teachers showed moderate d i v e r s i t y i n t h e i r approach to the remaining nine t o p i c s . In each case the median score was between 2.0 and 2.5 i n c l u s i v e . D i f f e r e n c e s were found between the implemented c u r r i c u l a i n c l a s s e s i n which the o v e r a l l student achievement l e v e l was low compared with c l a s s e s in which the o v e r a l l student achievement l e v e l was h i g h . These d i f f e r e n c e s , however, were i n general q u i t e s m a l l . T h i s i s s u r p r i s i n g because the d i f f e r e n c e s i n mathematical achievement between the two groups of c l a s s e s were q u i t e l a r g e . A l s o , the B. C. Curriculum Guide (1978) s t a t e s that teachers should provide " d i f f e r e n c e s i n approach, depth and ra t e of l e a r n i n g " through the use of the m u l t i p l e - t e x t adoption ( p . 3 ) 1 . Thus, one might have expected gre a t e r d i f f e r e n c e s i n i n s t r u c t i o n between the two groups of c l a s s e s than were a c t u a l l y found to occur. In g e n e r a l , students i n low achievement c l a s s e s were taught l e s s geometry, more a r i t h m e t i c , and s l i g h t l y more algebra than t h e i r c o u n t e r p a r t s i n high achievement c l a s s e s . The c o r r e l a t i o n s between c l a s s achievement as measured by the SIMS Core P r e t e s t and the content emphasis scores f o r the three content areas were +0.28 f o r geometry, -0.23 f o r a r i t h m e t i c , and -0.15 f o r a l g e b r a . The d i f f e r e n c e s i n content emphasis f o r a r i t h m e t i c and geometry were p a r t i c u l a r l y n o t i c e a b l e when scores o u t s i d e of the 1 Response by teachers to the GCPQ i n d i c a t e d that very few classrooms in t h i s study were organized so that i n d i v i d u a l i z a t i o n occured w i t h i n classrooms. 168 moderate emphasis category were c o n s i d e r e d . The only three c l a s s e s i n which a r i t h m e t i c r e c e i v e d heavy emphasis were low achievement c l a s s e s while s i x of the nine c l a s s e s i n which a r i t h m e t i c r e c e i v e d l i g h t or very l i g h t emphasis were high achievement c l a s s e s . For geometry t h i s p a t t e r n was reversed. Four out of f i v e c l a s s e s i n which geometry r e c e i v e d heavy emphasis were high achievement c l a s s e s while nine out of 11 c l a s s e s i n which geometry r e c e i v e d l i g h t or very l i g h t emphasis were low achievement c l a s s e s . Across a l l content, teachers taught somewhat more a b s t r a c t l y i n low achievement than i n high achievement c l a s s e s . The median mode of content r e p r e s e n t a t i o n score was 0.58 f o r low achievement c l a s s e s compared with 0.54 for high achievement c l a s s e s . Although t h i s i s a small d i f f e r e n c e , t h i s f i n d i n g i s of i n t e r e s t f o r two reasons. F i r s t , i t i s c o n s i s t e n t with recent r e s e a r c h conducted in the United States as noted p r e v i o u s l y . Second, t h i s f i n d i n g i s counter to what might be expected based on the i d e a l c u r r i c u l u m ; o f mathematics educators in which the slower student might be expected to r e q u i r e more concrete r e p r e s e n t a t i o n s of content than the more able student. O v e r a l l , there was a s l i g h t tendency fo r r u l e s to r e c e i v e heavier emphasis i n low achievement c l a s s e s than i n high achievement l e v e l c l a s s e s . Rules r e c e i v e d more emphasis i n low achievement l e v e l c l a s s e s f o r geometry content considered s e p a r a t e l y . Although the median r u l e - o r i e n t e d n e s s score f o r low achievement c l a s s e s was not greater than the corresponding value f o r high achievement l e v e l c l a s s e s f o r algebra content, s i x out 169 of seven scores which were in the two r u l e - o r i e n t e d c a t e g o r i e s for a l g e b r a were scores f o r low achievement c l a s s e s . For the s i n g l e a r i t h m e t i c t o p i c f o r which the r u l e - o r i e n t e d n e s s v a r i a b l e was d e f i n e d no achievement l e v e l d i f f e r e n c e s e x i s t e d . T h i s t o p i c was u s u a l l y taught with a heavy emphasis on r u l e s i n both low and high achievement c l a s s e s . In general students i n low achievement c l a s s e s tended to be taught more a r i t h m e t i c and a l g e b r a and l e s s geometry than students in high achievement c l a s s e s . These students a l s o tended to be taught somewhat more a b s t r a c t l y and with a very s l i g h t l y g r e a t e r emphasis on r u l e s . C l a s s e s of both achievement l e v e l s tended to be taught with the same number of content s p e c i f i c teaching methods. A f a i r l y strong r e l a t i o n s h i p was found between the emphasis which a content area r e c e i v e s i n a textbook and the emphasis which that content area r e c e i v e d i n the implemented c u r r i c u l u m of a teacher using that textbook. School Mathematics 2 c o n t a i n s 54% more a r i t h m e t i c content than Mathematics II and the median content emphasis score f o r a r i t h m e t i c was 38% higher f o r teachers using the former book. S i m i l a r l y , Mathematics II c o n t a i n s 77% more geometry content than School Mathematics 2 and the median content emphasis score f o r geometry was 46,% higher f o r teachers using the former book. Algebra r e c e i v e s approximately the same amount of emphasis in both textbooks and the median content emphasis scores f o r teachers using the two books were n e a r l y equal. There was, then, a c l o s e a s s o c i a t i o n between the content 170 emphasis of the formal B.C. Mathematics 8 c u r r i c u l u m as i n d i c a t e d by the number of pages the textbooks devoted to v a r i o u s content areas and the emphasis the teachers i n t h i s study gave to those content areas. Thus, i t would seem that a necessary c o n d i t i o n f o r i n f l u e n c i n g teachers to emphasize a content area more h e a v i l y i s to s e l e c t a textbook which pro v i d e s an a p p r o p r i a t e amount of emphasis for that content area. I t should be noted, however, that the presence of a content area i n a textbook i n and of i t s e l f i s no guarantee that teachers w i l l teach that content. P r o b a b i l i t y and s t a t i s t i c s content, f o r example, i s included i n both textbooks. However, t h i s m a t e r i a l i s c ontained i n chapters near the end of the textbooks and t h i s content area i s not i n c l u d e d in the B. C. Curriculum Guide (1978) for Mathematics 8. The teachers i n t h i s study were asked i n the GCPQ how much time they would spend teaching p r o b a b i l i t y and s t a t i s t i c s and i n d i c a t e d that they would devote a median of only 4% of t h e i r i n s t r u c t i o n a l time to t h i s content area. Most commonly, teachers reported that they would spend no time at a l l teaching p r o b a b i l i t y and s t a t i s t i c s . Thus, the presence of p r o b a b i l i t y and s t a t i s t i c s chapters near the end of the commonly used t e x t s d i d not i n f l u e n c e most teachers to teach t h i s m a t e r i a l . The a s s o c i a t i o n between the other c u r r i c u l u m v a r i a b l e s examined in t h i s study and the contents of the two commonly used textbooks were weaker than was the case f o r the content emphasis v a r i a b l e . T h i s i s probably true at l e a s t i n part because the two t e x t s d i f f e r much l e s s with respect to these other v a r i a b l e s 171 than they d i f f e r with respect to content emphasis. The teachers in t h i s study represented mathematical content to t h e i r students more f r e q u e n t l y i n an a b s t r a c t mode than in a pe r c e p t u a l mode. The median content r e p r e s e n t a t i o n score across a l l t o p i c s was 0.57. The two textbooks a l s o represent content more f r e q u e n t l y i n an a b s t r a c t than i n a pe r c e p t u a l manner. In Mathematics II 62% of the content s p e c i f i c methods are a b s t r a c t while i n School Mathematics 2 55% of the methods are a b s t r a c t . Thus, there was a c o n s i s t e n c y between the formal c u r r i c u l u m as embodied i n the textbooks and the implemented c u r r i c u l a of teachers. The d i f f e r e n c e i n a b s t r a c t i o n between the two textbooks was not a s s o c i a t e d with a s i m i l a r d i f f e r e n c e between the implemented c u r r i c u l a of the users of the textbooks, however. The users of the l e s s a b s t r a c t t e x t , School  Mathematics 2 were a c t u a l l y s l i g h t l y more a b s t r a c t in t h e i r p r e s e n t a t i o n s to students than were the users of the other t e x t . Because of these mixed f i n d i n g s , i t i s d i f f i c u l t to hypothesize what e f f e c t , i f any, a change i n the l e v e l of a b s t r a c t i o n of the textbooks might have on the implemented c u r r i c u l a . I t i s p o s s i b l e that while teachers are i n f l u e n c e d i n the amount of time they spend on a content area by the amount of coverage given to that area by the textbook they are using, they are not s i m i l a r l y i n f l u e n c e d i n the content s p e c i f i c methods they choose to use to represent that content by the methods contained i n the textbook. A l t e r n a t e l y , i t i s p o s s i b l e that a r e l a t i o n s h i p may e x i s t and that the adoption of a more p e r c e p t u a l l y o r i e n t e d textbook might r e s u l t i n : a greater 172 u t i l i z a t i o n of p e r c e p t u a l methods of r e p r e s e n t i n g content by t e a c h e r s . A weak p o s i t i v e a s s o c i a t i o n was found between the number of methods teachers employed i n teaching the 16 t o p i c s examined in t h i s study and the number of methods contained i n the textbook they used. O v e r a l l , Mathematics II c o n t a i n s 16% more teaching methods than School Mathematics 2 f o r these t o p i c s and the median d i v e r s i t y of i n s t r u c t i o n score f o r teachers using the former te x t was 9% higher than f o r teachers using the l a t t e r t e x t . For four t o p i c s (Topic Group 2), School Mathematics 2 p r o v i d e s more i n s t r u c t i o n a l options f o r teaching c o n t a i n i n g almost three times as many teaching methods as the other t e x t . For those t o p i c s , teachers using School Mathematics 2 had a median d i v e r s i t y score 13% higher than users of the other t e x t . These f i n d i n g s provide some evidence that teachers may be i n f l u e n c e d i n the number of approaches they use f o r the mathematical t o p i c s they teach by the number of approaches contained i n the textbook they use. The r e l a t i v e l y small d i f f e r e n c e i n methods used by teachers f o r Topic Group 2 i n s p i t e of the l a r g e d i f f e r e n c e i n the number of methods contained in the two t e x t s i n d i c a t e s , however, that t h i s i n f l u e n c e may be s m a l l . 1 73 2. IMPLICATIONS FOR PRACTICE Many of the a s s e r t i o n s which are made in t h i s s e c t i o n go beyond the data presented in t h i s study. Inferences are drawn not only from the r e s u l t s of the study i t s e l f but a l s o from the author's experience as a teacher as w e l l as h i s i n t e r p r e t a t i o n of the i d e a l c u r r i c u l a contained i n the mathematics education l i t e r a t u r e . T h i s has been done because the s e c t i o n deals s p e c i f i c a l l y with i m p l i c a t i o n s f o r p r a c t i c e and r e q u i r e s a d d i t i o n a l knowledge beyond that gained from t h i s study. I t i s a n t i c i p a t e d that some readers w i l l reach c o n c l u s i o n s d i f f e r e n t from those of the author based on t h i s study as w e l l as t h e i r own experiences and thereby a r r i v e at i m p l i c a t i o n s f o r p r a c t i c e d i f f e r e n t from those o u t l i n e d below. One important f i n d i n g of t h i s study i s that f o r a l l of the c u r r i c u l u m v a r i a b l e s which were examined c o n s i d e r a b l e v a r i a t i o n e x i s t e d among teachers. In p a r t i c u l a r , teachers d i d not follow a s i n g l e p a t t e r n of content emphasis in t h e i r courses with the widest v a r i a t i o n e x i s t i n g f o r geometry. Whether or not the degree of v a r i a t i o n observed i s d e s i r a b l e i s a- matter f o r d i s c u s s i o n . C e r t a i n l y , when teachers plan t h e i r courses on the b a s i s of an assumed c u r r i c u l a r background of students, such a l a c k of u n i f o r m i t y may be problematic. If so, a more d e t a i l e d and p r e s c r i p t i v e c u r r i c u l u m guide might be needed although the r e l a t i o n s h i p between the contents of such a document and teacher p r a c t i c e i s not known. A s i n g l e text adoption might a l s o i n c r e a s e u n i f o r m i t y i n implemented c u r r i c u l a . A l t e r n a t e l y , a method of d e c i d i n g which of s e v e r a l t e x t s i s a p p r o p r i a t e for any 174 particular class might provide a more rational basis for having different mathematics c u r r i c u l a at the Grade 8 l e v e l . Although mathematics deals inherently with abstractions, mathematical ideas, at least those included in the formal Mathematics 8 curriculum, can be represented quite concretely. P a r t i c u l a r l y when students are introduced to mathematical concepts, operations, and p r i n c i p l e s for the f i r s t time i t may be advisable for perceptual representations to predominate over abstract representations. Although the majority of teachers balanced their approaches to algebraic and geometric content, a s i g n i f i c a n t number r e l i e d heavily on abstract methods. Also, a majority of teachers dealt with three of the algebraic topics and three of the geometric topics in an abstract manner. Inservice training to make teachers aware of the mode of representation variable and to expose them to more perceptual teaching methods for selected topics might increase the number of perceptual methods they present to students. The results of this study indicate (to me) that ; overall teachers are probably putting a reasonable emphasision rules during instruction. However the median rule-orientedness scores varied considerably among topics. Rules should probably be more heavily emphasized for review topics than for new materials. Inservice training to make teachers aware of the rule-orientedness variable might increase the probability that teachers would follow a more consistent pattern in emphasizing rules. Overall, teachers seemed to provide their students with a 175 reasonable d i v e r s i t y of approaches to the t o p i c s they taught. However, a number of t o p i c s were taught with low d i v e r s i t y by a s i g n i f i c a n t number of teachers and i n s e v e r a l cases by a ma j o r i t y of teachers. The median d i v e r s i t y scores were p a r t i c u l a r l y low f o r geometric t o p i c s . I n s e r v i c e t r a i n i n g to make teachers aware of a d d i t i o n a l methods f o r teaching the t o p i c s which were f r e q u e n t l y d e a l t with with low d i v e r s i t y may improve t h i s s i t u a t i o n Some evidence of a s s o c i a t i o n s between textbook contents and the implemented c u r r i c u l a of teachers was provided by t h i s study. A f a i r l y strong p o s i t i v e a s s o c i a t i o n was found between the way content areas were emphasized i n textbooks and the content emphasis scores of teachers using those textbooks. A l s o , there was evidence that teachers used g r e a t e r d i v e r s i t y in teaching t o p i c s when more content s p e c i f i c teaching methods were contained i n the book they were using. A l s o , the o v e r a l l l e v e l of a b s t r a c t i o n that teachers used i n p r e s e n t i n g mathematical t o p i c s was n e a r l y the same as the l e v e l of a b s t r a c t i o n with which those t o p i c s were d e a l t with i n the two textbooks. While the d i f f e r e n c e s i n l e v e l of a b s t r a c t i o n i n the textbooks was not a s s o c i a t e d with s i m i l a r d i f f e r e n c e s i n the i n s t r u c t i o n of teachers using those textbooks, t h i s i s probably true because of the s i m i l a r i t y of the two textbooks i n t h e i r mode of r e p r e s e n t i n g content. The a s s o c i a t i o n s found between the formal c u r r i c u l u m of textbooks and the implemented c u r r i c u l u m of teachers do not n e c e s s i t a t e a cause and e f f e c t r e l a t i o n s h i p . However, i t can be 176 hypothesized that such a r e l a t i o n s h i p does e x i s t and that the contents of a textbook i n terms of the four c u r r i c u l u m v a r i a b l e s w i l l i n f l u e n c e teachers' implemented c u r r i c u l a . Thus, i t can be speculated that i f i t i s d e s i r e d that teachers put more emphasis on geometry, f o r example, or use more p e r c e p t u a l teaching methods, then a textbook c o n s i s t e n t with these goals should be s e l e c t e d as part of the formal c u r r i c u l u m . I t was found i n t h i s study that the r e l a t i o n s h i p between a teacher's implemented c u r r i c u l u m and the achievement l e v e l of the c l a s s to be taught i s not a strong one. I t can be a s s e r t e d that a c l a s s with a l a r g e number of low achievement students r e q u i r e s more pe r c e p t u a l approaches to content than a c l a s s with a small number of low achievement students. The r e s u l t s of t h i s study do not support the hypothesis that t h i s occurs i n p r a c t i c e but rather t h a t , i f anything, the opposite occurs. While low achievement students may r e q u i r e more p e r c e p t u a l methods than high achievement students, they may a l s o r e q u i r e that a stronger emphasis be put on the r u l e s of mathematics. Very weak evidence was found that some d i f f e r e n t i a t i o n of t h i s s o r t by c l a s s achievement does occur. The c l e a r e s t r e l a t i o n s h i p between c l a s s achievement and the implemented c u r r i c u l u m e x i s t e d f o r the content emphasis v a r i a b l e . In gen e r a l , low achievement c l a s s e s r e c e i v e d more a r i t h m e t i c and algebra i n s t r u c t i o n and l e s s geometry i n s t r u c t i o n than high achievement c l a s s e s . While more s t r e s s on a r i t h m e t i c in low achievement c l a s s e s i s probably a p p r o p r i a t e , i t may be that t h i s i s happening too much at the expense of geometry. 1 77 A l s o , while the median content emphasis score f o r low achievement c l a s s e s i s higher than the corresponding score f o r high achievement c l a s s e s , both d i s t r i b u t i o n s of scores c o n t a i n c o n s i d e r a b l e v a r i a t i o n . Thus, there appear to be many low achievement c l a s s e s i n which a r i t h m e t i c i s emphasized too l i t t l e and many high achievement c l a s s e s i n which t h i s content area i s emphasized too much. In s p i t e of the d i f f e r e n c e s in the implemented c u r r i c u l a of low and high achievement c l a s s e s noted above, a major f i n d i n g of t h i s study i s that two c l a s s e s of markedly d i f f e r e n t achievement l e v e l s may have n e a r l y i d e n t i c a l c u r r i c u l a or may, i n f a c t , d i f f e r i n ways opposite to what might be thought d e s i r a b l e . While the B. C. Curriculum Guide does i n d i c a t e that teachers should organize t h e i r c l a s s e s to meet t h e i r students' needs, i t may be that teachers w i l l r e q u i r e more a s s i s t a n c e to r e a l i z e t h i s g o a l . More s p e c i f i c a t i o n and d i f f e r e n t i a t i o n by achievement may be r e q u i r e d w i t h i n the formal c u r r i c u l a i f one i s to expect s u b s t a n t i a l c u r r i c u l a r d i f f e r e n c e s f o r c l a s s e s of d i f f e r e n t achievement l e v e l s . 3. SIGNIFICANCE OF THE STUDY Thi s study i s s i g n i f i c a n t f o r s e v e r a l reasons. F i r s t , i t represents an i n i t i a l attempt to d e s c r i b e implemented c u r r i c u l a using the four mathematics c u r r i c u l u m v a r i a b l e s which were in c o r p o r a t e d i n t h i s study and to d i s p l a y and analyze the r e s u l t s using b a s i c techniques of E x p l o r a t o r y Data A n a l y s i s . T h i s represents a more d e t a i l e d method of i n v e s t i g a t i n g the c u r r i c u l a r aspect of teaching p r a c t i c e than has been the case 178 p r e v i o u s l y . Besides the methodological c o n t r i b u t i o n made by t h i s study, t h i s r e s e a r c h provides r e s u l t s which may be u s e f u l i n planning c u r r i c u l u m r e v i s i o n s t r a t e g i e s and i n e s t a b l i s h i n g a benchmark from which c u r r i c u l u m change can be measured. Information over time s i m i l a r to that presented here should provide a sounder b a s i s f o r a s s e s s i n g the c u r r i c u l u m that i s reaching students and for comparing t h i s c u r r i c u l u m to those of the past. 4. LIMITATIONS OF THE STUDY A l i m i t a t i o n of t h i s study i s that i t u t i l i z e d i nstrumentation which was not s p e c i f i c a l l y c o n s t r u c t e d to c o l l e c t data about the c u r r i c u l u m v a r i a b l e s which were examined in the study. The TSQ and GCPQ instruments were designed as part of the SIMS p r o j e c t to provide data about the implemented c u r r i c u l a w i t h i n v a r i o u s i n t e r n a t i o n a l j u r i s d i c t i o n s and the teaching methods which were contained i n the TSQs for each mathematical t o p i c were q u i t e comprehensive. However, the wording of some of the content s p e c i f i c methods given i n TSQ items was such that i t was not p o s s i b l e to i n v e s t i g a t e each c u r r i c u l u m v a r i a b l e f o r every t o p i c . For example, f o r the angle sum t o p i c one item contained the r u l e approach "I t o l d my students that the sum of the measures of the angles of a t r i a n g l e i s 180 degrees." U n f o r t u n a t e l y , t h i s item i n c l u d e d a d d i t i o n a l wording so that i t was not p o s s i b l e to say that a teacher who u t i l i z e d the teaching method contained i n that item was, s t r i c t l y speaking, p r e s e n t i n g the mathematical idea i n 179 quest i o n by j u s t g i v i n g students a r u l e . Because of s i m i l a r problems of wording i t was only p o s s i b l e to d e f i n e the r u l e -orientedness v a r i a b l e f o r h a l f of the t o p i c s which were s t u d i e d . The f a c t that t h i s study i n v o l v e d the r e - a n a l y s i s of data c o l l e c t e d f o r another study meant that s e v e r a l t o p i c s which were part of the formal B.C. Mathematics 8 c u r r i c u l u m c o u l d not be examined. Square r o o t s , s c i e n t i f i c n o t a t i o n , and the t r a n s l a t i o n of E n g l i s h phrases i n t o mathematical expressions are examples of t o p i c s which c o u l d not be i n c l u d e d in t h i s study because they were not i n c l u d e d in the TSQs. A f u r t h e r p o s s i b l e l i m i t a t i o n of t h i s study i s r e l a t e d to teachers' assessments of the achievement l e v e l s of t h e i r c l a s s e s . In t h i s study a s s o c i a t i o n s between implemented c u r r i c u l a and c l a s s achievement were examined. A s s o c i a t i o n s between implemented c u r r i c u l a and teacher p e r c e p t i o n of c l a s s achievement were not examined. If teachers' perceptions d i f f e r e d markedly from a c t u a l c l a s s achievement, some r e s u l t s of t h i s study may be m i s l e a d i n g . To determine the congruence between c l a s s achievement and teacher p e r c e p t i o n , data which were c o l l e c t e d as part of the B.C. SIMS p r o j e c t were used. Each teacher was asked to estimate the number of students i n h i s or her c l a s s who f i t i n t o each of three c a t e g o r i e s of mathematical achievement f o r the p r o v i n c e : 1) top t h i r d , 2) middle t h i r d , and 3) bottom t h i r d . A f o u r t h category, "unable to judge," was a l s o i n c l u d e d . A measure of teacher p e r c e p t i o n of c l a s s achievement was determined f o r each teacher by s u b t r a c t i n g the number of 180 students reported i n category three from the number of students reported i n category one and d i v i d i n g t h i s q u a n t i t y by the sum of the number of students i n c a t e g o r i e s one, two, and three. The r e s u l t i n g q u a n t i t y c o u l d take on values from -1 to +1 with -1 i n d i c a t i n g that the teacher p e r c e i v e d every student as low achievement (bottom t h i r d ) and +1 i n d i c a t i n g that the teacher p e r c e i v e d every student as high achievement (top t h i r d ) . T h i s measure of teacher p e r c e p t i o n of c l a s s achievement was then c o r r e l a t e d with the measure of a c t u a l c l a s s achievement which was used i n t h i s study, the Core P r e t e s t s c o r e . The r e s u l t i n g c o r r e l a t i o n was +0.65. While one must s t i l l use ca u t i o n in equating teacher p e r c e p t i o n of c l a s s achievement with a c t u a l c l a s s achievement when i n t e r p r e t i n g the r e s u l t s of t h i s study, the rather high p o s i t i v e c o r r e l a t i o n obtained i n d i c a t e s that the teachers i n t h i s study d i d , in g e n e r a l , assess the achievement l e v e l of t h e i r c l a s s e s rather a c c u r a t e l y . 5. SUGGESTIONS FOR FUTURE RESEARCH Th i s study provides a survey of the implemented Mathematics 8 c u r r i c u l u m i n B r i t i s h Columbia as i t e x i s t e d i n the 1980/1981 school year. In order to assess changes i n the implemented c u r r i c u l u m over time r e p l i c a t i o n s of t h i s study c o u l d be undertaken on a p e r i o d i c b a s i s . The data c o l l e c t i o n instruments c o u l d be r e f i n e d to more a c c u r a t e l y measure the c u r r i c u l u m v a r i a b l e s and the q u e s t i o n n a i r e s c o u l d be expanded to include a d d i t i o n a l t o p i c s and content areas such as p r o b a b i l i t y and s t a t i s t i c s . A l s o , instrumentation c o u l d be developed f o r other 181 grade l e v e l s . Because of the survey nature of t h i s research i t was only p o s s i b l e to examine a s s o c i a t i o n s and not to e s t a b l i s h c a u s a l r e l a t i o n s h i p s . Future research could examine how manipulations of the formal c u r r i c u l u m might i n f l u e n c e the c u r r i c u l u m as implemented i n the classroom. For example, c u r r i c u l u m m a t e r i a l s c o u l d be s e l e c t e d or produced as part of a c u r r i c u l u m r e v i s i o n and randomly assigned to some B.C. classrooms. If these new m a t e r i a l s c o n t a i n a d i f f e r e n t p a t t e r n of content emphasis than the o l d m a t e r i a l s , then the implemented c u r r i c u l a of c l a s s e s using both the o l d and new m a t e r i a l s c o u l d be examined. In t h i s way the s t r e n g t h of the l i n k between the content i n the formal and implemented c u r r i c u l a c o u l d be i n v e s t i g a t e d . In a s i m i l a r way the i n f l u e n c e of the mode of content r e p r e s e n t a t i o n , r u l e -orientedness and d i v e r s i t y w i t h i n the formal c u r r i c u l u m on the implemented c u r r i c u l u m c o u l d be i n v e s t i g a t e d . Wide v a r i a t i o n among the implemented c u r r i c u l a which were examined i n t h i s study e x i s t e d f o r each of the c u r r i c u l u m v a r i a b l e s . I t i s not c l e a r why such v a r i a t i o n s e x i s t . Further research on teacher decision-making i n c u r r i c u l u m implementation and s e l e c t i o n i s needed. In t h i s regard the r e l a t i o n s h i p between the degree of s p e c i f i c i t y w i t h i n the formal c u r r i c u l u m and the amount of v a r i a t i o n i n content emphasis should be i n v e s t i g a t e d . The formal c u r r i c u l u m i n B r i t i s h Columbia f o r Mathematics 8 allows teachers c o n s i d e r a b l e f l e x i b i l i t y in planning t h e i r courses. In other j u r i s d i c t i o n s the formal c u r r i c u l u m i s more p r e c i s e l y s p e c i f i e d . The degree to which 182 this s p e c i f i c i t y tends to produce a more uniform implemented cu r r i c u l a i s unknown and warrants further investigation. The evidence of this study supports the hypothesis that class achievement level i s not an especially i n f l u e n t i a l factor in teacher decision making regarding curriculum for Mathematics 8 courses. In particular no strong associations were observed between mode of content representation, rule-orientedness of instruction, and d i v e r s i t y of instruction on the one hand and class achievement level on the other hand. Since i t would seem that some differences in curriculum might be desirable for classes of d i f f e r i n g achievement lev e l s , this hypothesis warrants further investigation. Although the teachers who participated in t h i s study in general used both perceptual and abstract teaching methods, i t is not clear i f they usually or ever sequenced their presentations in accordance with the enactive, iconic, symbolic model of Bruner (1966) and Dienes (1960, 1964, 1973). This is another area requiring further research. 183 BIBLIOGRAPHY Alspaugh, J . W. 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Item numbers are given f o r the methods which were c l a s s i f i e d as pe r c e p t u a l and abstact f o r each t o p i c . Topic 1 . 2. 3. 4. 6. 7. 8. Perceptual A b s t r a c t Methods Methods 21,22,23,27,28,30 24,25,26,29 31,32,33,37,38 34,35,36 51 ,53,56 52,54,55 59 57,58 20,22,24 21 ,23 25,27 26 28,32 29,30,31,33 36 34,37,38 45,47,48 44,46 . 59,61,62,63,66 60,64 67,68,71 69,72,73 48,50,53,54 49,51,52 56,57,59 55,58,60,61,62 64,65 63 F r a c t i o n s A d d i t i o n of F r a c t i o n s Dec imals Operations with decimals Integers A d d i t i o n of i n t e g e r s S u b t r a c t i o n of in t e g e r s 9. M u l t i p l i c a t i o n of i n t e g e r s 10. Formulas 12. Angle sum theorem 13. Pythagorean theorem 14. Concept of ir 15. Area of a p a r a l l e l o g r a m 16. Volume of a prism Topics 5 and 11 are the concept of p r o p o r t i o n s and s o l v i n g l i n e a r equations r e s p e c t i v e l y . The mode of r e p r e s e n t a t i o n v a r i a b l e was not d e f i n e d f o r these two t o p i c s . 196 The r u l e - o r i e n t e d n e s s v a r i a b l e was d e f i n e d for the f o l l o w i n g e i g h t t o p i c s . An item number i s given f o r the method which was considered the r u l e approach f o r each t o p i c . Topic Rule Method 4. Operations with decimals 58 7. A d d i t i o n of i n t e g e r s 26 8. S u b t r a c t i o n of i n t e g e r s 33 9. M u l t i p l i c a t i o n of i n t e g e r s 38 12. Pythagorean theorem 69 14. Concept of IT 49 15. Area of a p a r a l l e l o g r a m 55 196a TOPICS 1-16., SIMS TOPIC SPECIFIC QUESTIONNAIRE Leaves 197-212 not filmed; permission not obtained. APPENDIX B - SIMS CORE PRETEST FOR MATHEMATICS 8 Leaves 213-28 not filmed; permission not obtained. 197 Topic 1 (Arithmetic) - Concept of Fractions 21. Fract ions as part of regions: 27. Fract ions as r a t i o s : 22. Fract ions as part of a c o l l e c t i o n : 3 « . n , O O O A * A A A A i 23. Fract ions as the coordinates of points on the number l i n e : -I I V 24. Fract ions as quotients: 3 neans "3 divided by 4* 28. Fract ions as measurements: th i s container holds 0 t h i s s t i c k Is 3 aa 4" 25. Fract ions as decimals: 3 - 0.75 4" 29. Fract ions as operators: 30. Fract ions as comparisons 26. F ract ions as repeated add i t i on of a unit f r a c t i o n . 3 • 1 • 1 • 1 4" T 4" ? 1 I - • • ) 1 2 u n i t rod 1 r o d J 2 rod I 198 Topic 2 ( A r i t h m e t i c ) - A d d i t i o n of F r a c t i o n s 31. The sum of two fractions as t)it union of two regions 35. 32. The sum of '.M fractions as combine vior of fractional parts of a collection Lc. 2 • 1 as (Mote: the collection consists of 20 dots) 33. The sum of two fractions on the number line Ex. 2 * 3 as: . * * T x 34. _t 2 3 £ I The sun of fractions as the sua of two quotients Ex. 2 * 3 as (2 • 3) • (3 • 4) Sine* 2 • 3 » 3 t 12 And 3 • 4 • 9 • 12 C8 • 1 1 * (9 • 121 - ( 3 * 9 ) * 12, • 17 • 12 The sum of two fractions as the sum of two decimals. Ex. 3 • 2 0.75 • 0.40 1.15 l 36. The sum of two fractions using fractions as repeated addition of the unit fractions Ex. 2 * 4 •f i • i\ *r i * '\ * i • }\ ~ r f ( T ? r r • I + I * I * I * I * I i r r r i i > 6 37. The sum of two fractions as a combination of two measurements 38. The sum of two fractions 4S joining two segments Ex. 2 • 3 as T T 199 Topic 3 ( A r i t h m e t i c ) - Concept of Decimals 51. A decimal as the coordinate of a point on the number line. t .2a 1 2 1 T I" 52. A decimal as another May of writing a fraction. 0.17 - 17 0.8-8 TCO* TO" S3. A decimal as part of a region. m a. 3a i I 3 54. A decimal as an extension of place value. 55. A decimal as a series 0.243 - 2 • 4 • 3 TO" TOO" TOOT 56. A decimal as a comparison 1 1 1 1 i 11 i -m unit rod 1 1 1 1 1 " 0.6 n u n 0.45 200 Topic 4 ( A r i t h m e t i c ) - Operations with Decimals 57. Related operations with decimals to operations with f rac t ions . Ex. 0.7 x 0.6 -58. But 0.7 So o.: 7 and 0.6 • 6 TO" TO" x 0.6 - 7 x 6 TS VS - 42 VSS Therefore 3.7 x 0.6 » 0.42 1 2 3 Related operations with decimals to operations with whole numbers, teaching rules for placing the decimal point. Ex. 1.38 x 5.2 • Since 13S x 52 m~ 690 7T76-1.38 x 5.2 • 7.176 59. Used concrete materials to I l l u s t r a t e operations • -with decimals. Ex. 3.47 • 2.13 -Using rods I demonstrated that 3.47 fi and 2.13 f i makes 5.60 m 1 2 3 1 2 3 201 Topic 5 ( A r i t h m e t i c ) - Concept of P r o p o r t i o n s 26. Proportions as equivalent ratios: Ci. 12 heartbeats per 10 seconds Is the same as 72 beats per aln. ) * i 27. Proportions as equivalent comparisons: Ex. 9 red cars to 12 blue ones Is the same as 3 to 4 . 28. Proportions as equivalent fractions: Exs. 1) 1/3 - 4/12 11) 3 3x2 3x3 3 x 4 2 2x2 2 x 3 2 x 4 3 6 9 12 1 ? Z S 29. Proportions as equivalent quotients: Ex. 3:4 and 9:12 Since 3 t 4 - 0.75 and 9 • 12 • 0.75. the quotients are equal; jo 3:4 and 9:12 are equivalent. I 2 3 202 Topic 6 (Algebra) - Concept of Integers 20. Extending the number ray U the number l i ne : I extended UM number ray (0 «nd positive numbers) U the l e f t by Introducing direction as M i l as magnitude. U: - 4 - 1 - 2 - 1 0 1 2 ) 4 -3 I U I I I 3 units to the left ef 0. 21. Citending the nunber system U find solutions to equations: I discussed Lhe need to extend the positive integers in order to find « solution to equations l i ke • 7 • S. 22. Using vectors or directed seg-ments on the number l ine: 1 defined «n Integer *s < set of vectors (directed line seg-ments) on the number l ine. Ex: -2 can be represented by any of: -IT 5 ro Ex: +2 can be represented by any of: 23. Def in ing Integers as equiva -lence c l a s s e s o f whole numbers: I developed the Integers as equivalence c l a s s e s of ordered p a i r s of whole numbers. Ex: ( ( 0 . 2 ) . ( 1 . 3 ) . ( 2 . 4 ) . . . . ) - "2 or ((a.b) c WXW: b - a • 2) - "2 I 24. Using examples of phys ica l s i t u a t i o n s : 1 developed integers by r e f e r r i n g to d i f f e r e n t phys ica l s i t u a t i o n s which can be described wi th Integers . Ex: thermometer, e l e v a t i o n , money ( c r e d i t / d e b i t ) , sports ( s c o r i n g ) , time ( b e f o r e / a f t e r ) , e t c . -to1 I I I I I I I m" •A 1 - ' i . i i i I . I 5 10 2 0 3 T o p i c 7 ( A l g e b r a ) - A d d i t i o n of I n t e g e r s 25. Addition on the number line: I used the number line to add integers. 26. Add i t ion by r u l e s : I used r u l e s to add in tegers . Ex: I f both addends have the same s i g n , the sum Is found by adding t h e i r numerical (absolute) values and ad jo in ing the common s i g n . 27. use o f phys ica l s i t u a t i o n s : I used phys ica l s i t u a t i o n s to add Integers . Ex: In c l imb ing out of the Dead Sea V a l l e y , the car s t a r t e d at an e l e v a t i o n of -643 feet and cl imbed 432 feet to an e l e v a t i o n of f e e t . i 2 3 204 T o p i c 8 (Algebra) - S u b t r a c t i o n of In tegers 28. Subtract ion as add i t ion of opposites: 1 used the number l i n e to subtract Integers by s t a r t i n g at the minuend and gotng the number of uni ts Indicated by the subtrahend but In the d i r e c t i o n opposite o f i t s s i gn . 31. Subtract ion as a number o f u n i t s : I extended the meaning o f subt ract ion of whole numbers ( I . e . y - x means the number of un i ts from x to y) to Integers. Ex: *4 - *3 means the nuni>er.of un i ts from "3 to 4. 29. Subtract ion as Inverse of add i t i on : I used the inverse r e l a t i o n between addi t ion and subtract ion to subtract Integers. Ex: *4 - "3 -Solve +4 - • "3 32. Subtract ion as d is tance: I used the nunber l i n e to subtract Integers by f ind ing tiie number of units (or d istance) from the subtrahend to the minuend. 30. Subtract ion by r u l e s i I used ru les to subtract Integers. Ex: To subtract an in teger , add i t s opposite. 33. Subtract ion as "what must be added"*. I Interpreted subtract ion to mean "what must be added" to the subtrahend to g*t the minuend. Ex: 44 - " 3 • means "what must'be added to " 3 to get V . 205 T o p i c 9 (Algebra) - M u l t i p l i c a t i o n of In tegers 34. Development by use of repeated add i t ion : I developed the concept of m u l t i p l i c a t i o n by appealing to repeated add i t i on , e . g . , 4 x 3 • *3 • "3 • "3 • *3 • '12 35. Development by the extension of propert ies o f the whole number system: I developed the concept of m u l t i p l i c a t i o n of Integers by using 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 to j u s t i f y the products, e . g . , " 4 x " 3 • B u t O " ("4 • + 4 ) x " 3 • h x "3) • T4 x -3 ) - ("4 x "3) • "12 Hence "4 x " 3 - *12 36. Development by use of physical s i tuat ions ; I developed the concept of m u l t i p l i c a t i o n of Integers by appealing to physical s i tuat ions that might i l l u s t r a t e the product of pos i t i ve and negative numbers, e . g . , A r e f r i ge ra to r Is cool ing at a rate of 4* per minute. I ts thermometer 1s at 0*. What wHT be i t s temperature 4 minutes from now? 37. Development by use of patterns: I developed the concept of m u l t i p l i c a t i o n of integers by appealing to patterns of products, e . g . , *4 x "3 - "12 *3 x "3 - "9 2 x "3 - "6 *1 x "3 - "3 0 x "3 • 0 "1 x "3 - 3 "2 x "3 - 6 38. No development — students were given ru les : I d id not develop the facts for m u l t i p l i c a -t i on of integers by using any of the above methods. I instead gave them rules s im i la r to the fa l lowing . I f the signs are a l i k e , the answer i s p o s i t i v e . " I f they are d i f fe rent the answer i s negative. I f one factor i s zero, the answer i s zero. 206 T o p i c 10 (Algebra) - Concept of Formulas 4 7 . H i v i n g s t u d e n t s c o l l e c t d a t a o n r e l a t e d v a r i a b l e s a n d f o r m u l a t e t h e r e l a t i o n s h i p b e t w e e n t h e v a r i a b l e s : Ex. o n e r e v o l u t i o n ' \ » \ I 1 S . S c o 4 4 . P r e s e n t i n g f o r m u l a s and e x p l a i n i n g t h e m e a n i n g o f L i e t a r s i n . t h e f o r m u l a s : E x : F o r m u l a : A • ^ oh A s t a n d s f o r t h e a r e a o f a t r i a n g l e b_ s t a n d s f o r t h e b a s e al a t r i a n g l e h_ s t a n d s f o r t h e h e i g h t ol a t r i a n g l e 4 5 . H a v i n g t h e s t u d e n t s . I n s p e c t g r a p h s a n d f i n d f o r m u l a s t o e x p r e s s t h e r e l a t i o n -s h i p s p o r t r a y e d b y t h e g r a p h : R a t i o : 1 L 5 Ex: 4 6 . -4-4-1 2 3 4 5 6 7 8 A • 2 X L P r o v i d i n g d a t a f r o m w h i c h f o r m u l a s o r e q u a t i o n s a r e d e v e l o p e d : X r 4 0 1 3 2 5 3 7 4 9 5 11 H e n c e y • 2 x + 1 3 . 1 2 , , one r e v o l u t i o n / x 4 0 . 9 c a R a t i o : 4 0 . 9 -rr 3 . 1 5 H e n c e e . 3.1 S o C • 3 . 1 d 4 8 . H a v i n g s t u d e n t s c r e a t e new f o r m u l a s b a s e d on k n o w n , s i m p l e r f o r m u l a s : E x . C r e a t e f o r m u l a f o r s u r f a c e a r e a o f a c y l i n d e r b a s e d o n f o r m u l a s f o r a r e a o f t h e r e c t a n g l e and t h e c i r c l e . S o , s u r f a c e a r e a • 2 i r h + 2 » r 5A • 2 » r ( h • r ) 207 Topic 11 (Algebra) - S o l v i n g L i n e a r Equations Using propert ies of e q u a l i t y w i t h operat ions wi th numbers: Ex : 7x • S • 40 7x • 5 - 5 • 40 - 5 (Subtract 5 from both s ides ) 7x • 35 (a r i thmet ic f a c t ) 7x . 35 •7 "7 (d iv ide both s ides by 7) x • 5 41. Using a r i t h m e t i c a l reasoning : Ex: Given 7x • 5 • 40 What number Increased by 5 i s 40 ( _ • 5 • 40)7 Since the number i s 35, then 7 times what number gives 35 (7 x • 35)7 The s o l u t i o n Is 5. 4? . Using t r i a l and e r r o r : Ex: Given 7x • 5 • 40 Try x - 4. But 7(4) + 5 - 3 3 . So t ry x • 5 , as x needs to be l a r g e r . 7(5) + 5 - 4 0 . So, x - 5. 40. Using inverse operat ions with numbers: Ex: 7x • 5 - 40 7x • 5 - "5 - 40 • "5 (add the Inverse of 5 to both s ides ) 7x - 35 \ . (7x) - \ . 35 (mu l t ip l y both s ides by the r e c i p r o c a l of 7) 43. Given 7x • S • 40 Example Rules - - c o l l e c t a l l constant terms on one s ide o f the equation and a l l v a r i a b l e terms on the other . 7x • 40 - 5 — combine l i k e terms. 7x - 35 — d i v i d e by the c o e f f i c i e n t of x x • 5 208 Topic 12 (Geometry) - Angle Sum Theorem 59. My students measured the angles of a t r i ang le and added the •easures to discover that the SUM of the Measures i s 180 . I 60. I drew a Une through a vertex p a r a l l e l to the opposite s ide and used a l ternate I n t e r i o r angles to show that the sun of the angles of a t r i a n g l e 1s 180°. Ex: ln the f igure j l • )4 and )3 • J 5 , So )1 • ,2 • • )4 + )2 + *S • 180° 61. My students cut the angles o f f a t r iang le and arranged thea on a s t ra ight l i n e . 62. I t o l d ny students that the sua of the neasures of t h e angles of a t r iang le 1s 180° and had then v e r i f y i t by M e a s u r i n g the angles and adding the measures. 63. 1 had my students v e r i f y the re la t ionsh ip by paper f o l d i n g . • f l f o l d . f o l d 64. I used the fact that (as i l l u s -trated 1n the f igure ) in t r a v e l i n g AB, BC, CA, a complete revo lu t ion (360°) Is swept. Using t h i s and angle supplements. )2 • ) 3 • 180° 65. Using t e s s e l l a t i o n s perhaps from the real wor ld , 1 I d e n t i f i e d three angles at a point (C) congruent with three angles In a t r i ang le (ABC) embedded In the t e s s e l l a t i o n . 66. A ru le r and compass const ruct ion was used to show the r e l a t i o n s h i p . *A - )1 )B - )2 K - )3 209 T o p i c 13 ( G e o m e t r y ) - P y t h a g o r e a n T h e o r e m 67. 1 presented my students with • va r ie ty of r i gh t t r i ang les and had them measure and record the lengths of the legs and hypo-tenuse. The pattern was d i s -cussed and then we stated the property. Ex: leg l eg hypotenuse 3 4 5 9 12 15 3 2 * 4 2 - 5 2 9 2 • 12 2 - 1 5 2 2 ^ J2 2 . . a • b • c l 2 3 68. I used diagrams l i k e the fo l lowing to show that . In a r i g h t t r i ang le a 2 • b 2 - c 2 69. I gave my students the formula ? 2 2 a • b » c and had them use i t In working examples. 1 70. The theorem was presented In the context of a h i s t o r i c a l account of Pythagoras and E u c l i d . 71. I presented an informal area argument using p h y s i c a l , e . g . geoboards, or p l c t o r a l models. Ex: I showed that the two squares had equal area. '4 72. I presented a formal deductive " a l g e b r a i c " argument. Ex: Using s i m i l a r r i gh t t r i a n g l e s , proportions can be set up to y i e l d 73. I presented a formal deductive argument using area . Ex: This f igure Is sometimes used to present a formal proof. o E< \ .6 210 Topic 14 (Geometry) - Concept of TT 48. I had my students measure and f ind the r a t i o of the circum-ference to the diameter of a number of c i r c u l a r o b j e c t ! , -and approximate c for any c i r c l e . 3 l * > 49. 1 to ld my students that , • " or 3.14. I J 1 50. Ny students estimated the value of i using Buf fon ' l Needle Problem. S3. I had my students use regular polygons Inscribed i n a C i r c l e to obtain successive approximations of i . Using square ABCO, • * 2.75 Using the octagon, i * ? aitd so on, to show that » approaches 3.14 as the nuniber of sides of the polygons increases. I i ) SI . 1 presented a chart re la t ing the values of the circumference to that of the diameter of various c i r c l e s l i k e the fo l lowing : c 44 28 37 d 14.0 8 . 9 1 1 . 8 I asked the students to f ind the r a t i o of the circumference to the diameter for each c i r c l e and generalized that | ' 3.14. 1 52. 1 t o l d my students that i i s an i r r a t i o n a l number obtained • as the r e s u l t of d iv id ing the circumference of any c i r c l e by I ts d laaeter . 54. I introduced • as the area of a c i r c l e of radius 1 . Ex. Using successive approxi -mations to the area of the uni t c i r c l e , I showed that : area of c i r c l e OR < 4 2 < < 4 68 area of 88 ?5 " c i r c l e ' 7S Using a f i ne r -gr id, I showed that : OK 2.72 < i < 3.S2 Using s t i l l a f iner g r i d . 1 showed t U t : 2B8 area of 344 TOO ' c i r c l e " TOO 2.88 < i < 3.44 and so on. 21 1 Topic 15 (Geometry) - Area of a Parallelogram 55. I presented the f o w l * A • b i h <nd demonstrated how to apply It by wans 7 A • 4 ca a K 7 I 1 6.8 ca> l 1 presented a pera l le lograa on a grid (or a geoboard) l i k e the one below (parallelogram ABCD), and had the students relate the number of square units inside ADCO to the base and al t i tude of the parallelogram. 58. I derived the formula A • b i h by cougar Ing the are* of the paral lelogram to that of a re la ted rectangle of equal dimensions. 50. I oave the student a p a r a l l e l o -Crao l i k e the one below, and asked them to cut o f f t r i a n g l e FOC and to use t h i s to form a rectangle (AF'FO). The students then re la ted the formula for the area of the rectangle to the area of the paral lelogram. A< 57. I presented a parallelogram on a gr id (or a geoboard) l i k e the one shown above and had the students count the square units Inside tr iangles ABE and CDF. Then I had then re la te the area of ABCD to that of rectangle BEFC based on the congruence of A ABE and A 8 C F . 1 60. I part i t ioned the parallelogram .by a diagonal Into two congruent t r iang les . 61. Then the area of A ABD i s >i bh and the area of the p a r a l l e l o -gram is then bh. I par t i t ioned the paral le logram ABCD into A ABE, & CDF and r e c t a n g l e ATCF so t h a t the area of t h e p a r a l l e l o g r a m 1s obtained by adding the areas of the two t r iang les and the rectangle. 62. I obtained the area of the parallelogram by subt ract ing the areas of A A8G and A OCH from the area of the rectangle g811d. G -—A 0 I x 1 212 T o p i c 16 ( G e o m e t r y ) - V o l u m e o f a P r i s m 63. I presented the formula Y • 1 i V i k or V • (area of base) x height and demonstrated how to apply I t by means of examples. 64. 1 presented a physical model of a r ight rectangular prism (box) with Its faces marked o f f In square un i ts , as i l l u s t r a t e d below. I had students generate the formula by re la t ing the number of cubic units contained in the prism to the dimensions of the box, giving hints only 1f necessary. 65. I provided my students with unit cubes and asked them to bui ld rectangular prisms of speci f ied dimensions. I asked them to relate the number of unit cubes used to the given dimensions, giving hints only i f necessary. 2 1 3 APPENDIX B - SIMS CORE PRETEST FOR MATHEMATICS 8 1 . 2 m e t r e s + 3 m i l l i m e t r e s i s e q u a l t o A. 2 . 0 0 0 3 m e t r e s 3 . I f 5x + 4 = 4x - 3 1 , then . x i s e q u a l t o A . - 3 5 B. 2 . 0 0 3 m e t r e s B . -27 C. 2 . 0 3 m e t r e s 0 . 2 . 3 m e t r e s E. 5 m e t r e s C . 3 0 . 27 E . 35 2 . A . 0 . 2 0 % B. 2% C. 5", 0 . 20": E. 2 5 v Four 1 - l i t r e bowls o f i c e cream were s e t o u t a t a p a r t y . A f t e r t h e p a r t y , 1 bowl was e m p t y , 2 were h a l f f u l l , and 1 was t h r e e q u a r t e r s f u l l . How many l i t r e s o f i c e cream had been EATEN? A. 3 j 4 C . 2 E. None of t h e s e 214 8.8 a 6.9 m Which o f t h e f o l l o w i n g i s the c l o s e s t a p p r o x i m a t i o n to t h e a r e a o f the r e c t a n g l e w i t h measurements g i v e n ? A . 48 m 2 B. 54 m2 C . 56 m 2 0 . 63 m 2 E. 72 m 2 6 . I 1 I I 1 1 1 1 1 1 1 1 vm\ 1 i ivmtmiiumn n^mnm . : — ; — ; ; — i 1 1 1 j 1 s q u a r e u n i t T h e a r e a o f t h e s h a d e d f i g u r e , t o t h e n e a r e s t s q u a r e u n i t , i s A . 23 s q u a r e u n i t s B . 20 s q u a r e u n i t s C . 18 s q u a r e u n i t s D . 15 s q u a r e u n i t s E . 12 s q u a r e u n i t s 2 1 5 S T 1 ' P Q R U V i M N O Z X w The d i a g r a m shows a c a r d b o a r d cube w h i c h has been c u t a l o n g some edges and f o l d e d o u t f l a t . I f i t i s f o l d e d t o a g a i n make t h e c u b e , w h i c h two c o r n e r s w i l l t o u c h c o r n e r P? A c o r n e r s Q and S B c o r n e r s T and Y £ c o r n e r s W and Y 0 c o r n e r s T and V E c o r n e r s U and Y 8. The l e n g t h o f A B i s 1 u n i t . Which i s the b e s t e s t i m a t e f o r the l e n g t h o f PIT? A 2 u n i t s 8 6 u n i t s C 10 u n i t s D 14 u n i t s p V E 18 u n i t s 216 9. On t h e above s c a l e t h e r e a d i n g i n d i c a t e d by t h e a r r o w i s between A. 51 and 52 B. 57 and 58 C 60 and 62 D. 62 and 64 E. 64 and 66 1 0 . A s o l i d p l a s t i c cube w i t h edges 1 c e n t i m e t r e l o n g weighs 1 g r a m . How much w i l l a s o l i d cube o f t h e same p l a s t i c weigh i f each edge i s 2 c e n t i m e t r e s l o n g . A. 8 grams B- 4 grams C- 3 grams D- 2 grams E. 1 gram 2 1 7 11. On a number l i n e two p o i n t s A and B a r e g i v e n . The c o o r d i n a t e of A i s - 3 and t h e c o o r d i n a t e o f fl i s +7. What i s t h e c o o r d i n a t e of the p o i n t C, i f 5 2 1 t h e m i d p o i n t o f the l i n e segment AC? 1 3 . I f P = LW and i f P then W 1s e q u a l t o = 12 and L = 3 , A. - 1 3 8. 4 C. +2 D. +12 E. +17 B. 3 C. 4 0 . 12 E. 36 1 2 . A p a i n t e r i s t o mix g r e e n and y e l l o w p a i n t i n t h e r a t i o o f 4 to 7 t o o b t a i n t h e c o l o u r he w a n t s . I f he has 28 l i t r e s o f g r e e n p a i n t , how many l i t r e s o f y e l l o w p a i n t s h o u l d be added? 14. A model b o a t i s b u i l t t o s c a l e so t h a t ^ i t i s y j j as l o n g as t h e o r i g i n a l b o a t . I f t h e w i d t h o f t h e o r i g i n a l b o a t i s 4 m e t r e s , the w i d t h o f the model s h o u l d b e , B. 16 C. 28 A. 0 . 1 metre D. 49 E. 196 B. 0 . 4 metres C 1 metre D. 4 metres E. 40 metres 218 The value of 0.2131 x 0.02958 1s 17. Which of the indicated angles is A C U T E approximately A. 0.6 B. 0.06 C. 0.006 D. 0.0006 E. 0.00006 0. E. -rn. 4x 18. If j2 - 0, then x is equal to (-2) x (-3) is equal to A. -6 B. -5 C. -1 0. 5 A. . 0 B. 3 C. 8 0. 12 E. 16 2 1 9 1 9 . The l e n g t h o f t h e c i r c u m f e r e n c e o f t h e c i r c l e w i t h c e n t r e a t 0 i s 24 and t h e l e n g t h o f a r c RS 1s 4 . What i s t h e measure i n d e g r e e s o f t h e c e n t r a l a n g l e ROS? A . 24 B . 30 C. 45 D. 60 E. 90 2 0 . In a d i s c u s - t h r o w i n g c o m p e t i t i o n , t h e w i n n i n g throw was 6 1 . 6 0 m e t r e s . The s e c o n d p l a c e throw was 5 9 . 7 2 m e t r e s . How much l o n g e r was t h e w i n n i n g throw t h a n the second p l a c e t h r o w ? A. 1 .12 metres B. 1 . 8 8 metres C. 1 . 9 2 metres 2 . 1 2 metres E. 121.32 metres 220 In the above diagram, triangles ABC and DEF are congruent, with BC = EF. What is the measure of angle EGC? A. 20° B. 40° C. 60° 0. 80° E. 100° s is equal to A. 75 B. 70 C. 65 D. 60 E. 40 221 20 a A s q u a r e i s removed f r o m t h e r e c t a n g l e as shown. What i s t h e a r e a o f t h e r e m a i n i n g p a r t ? A. 316 m2 B. 300 m2 C 284 m2 2 0. 80 m 2 E. 16 m C l o t h i s s o l d by t h e s q u a r e m e t r e . I f 6 s q u a r e metres o f c l o t h c o s t $ 4 . 8 0 , t h e c o s t o f 16 s q u a r e m e t r e s w i l l be A . $ 1 2 . 8 0 B- $ 1 4 . 4 0 C $ 2 8 . 8 0 D- $ 5 2 . 8 0 E. $128.00 222 T h e a i r t e m p e r a t u r e a t t h e f o o t o f a m o u n t a i n i s 31 d e g r e e s . On t o p o f t h e m o u n t a i n t h e t e m p e r a t u r e i s - 7 d e g r e e s . How much w a r m e r i s t h e a i r a t t h e f o o t o f t h e m o u n t a i n ? A . - 3 8 d e g r e e s B. - 2 4 d e g r e e s C . 7 d e g r e e s D. 24 d e g r e e s E. 38 d e g r e e s 0 . 4 0 x 6 . 3 8 i s e q u a l t o A . . 2 5 5 2 B . 2 . 4 5 2 C 2 . 5 5 2 D. 2 4 . 5 2 E. 2 5 . 5 2 223 A shopkeeper has x kg of tea 1n stock. He sells 15 kg and then receives a new lot weighing Zy kg. What weight of tea does he now have? x - 15 - Zy B. x + 15 + 2y C x - 15 + Zy x + 15 - Zy None of these In the figure the l i t t l e squares are all the same size and the area of the whole rectangle is equal to 1. The area of the shaded part is equal to A. B. C D-2 T5" 3 2 5" 3 E. 1 7 224 2 9 . The d i s t a n c e between two towns i s u s u a l l y 3 1 . m e a s u r e d i n A. B. C. 0. E. mi 1 1 i m e t r e s c e n t i m e t r e s d e c i m e t r e s m e t r e s k i l o m e t r e s | + | i s e q u a l t o 8. C. 0. 5 50 _ 5 _ 40 16 TF 31 40" 3 0 . 0 . 0 0 0 4 6 i s e q u a l t o A . 46 x 1 0 " 3 3 2 . 1 2Q ^ s e q u a 1 t o B. C 0. E. 4 . 6 x 10~-0 . 4 6 x 103 4 . 6 x 10* 46 x 1 0 s 8. C 0. E. 7 . 0 3 7 . 1 5 7 . 2 3 7 . 3 7 . 6 225 3 3 . I n a s c h o o l o f 8 0 0 p u p i l s , 3 0 0 a r e b o y s . T h e r a t i o o f t h e n u m b e r o f b o y s t o t h e n u m b e r o f g i r l s i s A. 3 : 8 B. 5 : 8 C. 3 : 11 D. 5 : 3 E. 3 : 5 34. What is 20 as a percent of 80? ' A. 4% B. 20* C 25% D. 40% E. None of these 226 The s e n t e n c e " a number x d e c r e a s e d 6y 6 i s l e s s than 12" can be w r i t t e n as t h e i n e q u a l i t y A . x - 6 > 12 B. x - 6 >. 12 C x - 6 < 12 D. 6 - x > 12 E . 6 - x < 12 30 i s 75% o f what number? A. 40 B. 90 C. 105 D. 225 E . 2250 227 37. Which of the points A, B, C, D, E on this number line corresponds to b 5* A B C D E t « I i • • I * » I • m t t i | > I ) A. point A B. point B C point C 0- point 0 E. point E 38. 20% of 125 is equal to A. 6.25 B- 12.50 C 15 D- 25 E. 50 228 39. .» *t -x - i . I l X l * * • ? What are the coordinates of point P? A. (-3,4) 3. (-4,-3) C (3,4) 0. (4,-3) E. (-4,3) 40 Triangles PQR and STU are similar, long is 50? How A. 5 B. 10 C 12.5 0. 15 E. 25 

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