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Relationships between classroom processes and student performance in mathematics : an analysis of cross-sectional… Taylor, Alan Richard 1987

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RELATIONSHIPS BETWEEN CLASSROOM PROCESSES AND STUDENT PERFORMANCE IN MATHEMATICS An A n a l y s i s o f C r o s s - s e c t i o n a l D a t a From t h e 1985 P r o v i n c i a l A s s e s s m e n t o f M a t h e m a t i c s By A l a n R i c h a r d T a y l o r B . S c , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1967 M.Ed., W e s t e r n W a s h i n g t o n U n i v e r s i t y , 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION i n THE FACULTY OF GRADUATE STUDIES ( D e p a r t m e n t o f M a t h e m a t i c s a n d S c i e n c e E d u c a t i o n ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA Septenber, 1987 © Alan Richard Taylor, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of /*f/0 T l C S &0#C>4 T/Qfl/ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date £/9?~7 i  DE-6(3/81) A B S T R A C T i i The purpose of t h i s i n v e s t i g a t i o n was t o examine, through the use of survey data, r e l a t i o n s h i p s between i n p u t s of s c h o o l i n g and outcomes, as measured by student achievement i n mathematics. The i n p u t s of s c h o o l i n g were comprised of a number of v a r i a b l e s grouped under each of the f o l l o w i n g c a t e g o r i e s : s t u d e n t s 1 and te a c h e r s ' backgrounds; s t u d e n t s 1 and teac h e r s ' p e r c e p t i o n s of mathematics; classroom o r g a n i z a t i o n and problem-s o l v i n g processes. Outcome measures i n c l u d e d student achievement on t e s t t o t a l , problem s o l v i n g and a p p l i c a t i o n s . A r e l a t e d q u e stion i n v o l v e d e x p l o r a t i o n of the appro p r i a t e n e s s of u s i n g c r o s s - s e c t i o n a l survey data t o make d e c i s i o n s based on the r e l a t i o n s h i p s found among the i n p u t and output v a r i a b l e s . To address t h i s q u e s t i o n , r e s u l t s from a subsequent l o n g i t u d i n a l study, which u t i l i z e d the same instruments, were examined f i r s t w i t h p o s t - t e s t data and second w i t h the i n c l u s i o n of p r e - t e s t data as c o v a r i a t e s . Data c o l l e c t e d from teachers and students of Grade 7 i n the 1985 B r i t i s h Columbia Assessment of Mathematics were re-analysed i n order t o l i n k responses t o Teacher Q u e s t i o n n a i r e s w i t h the students* r e s u l t s i n teac h e r s ' r e s p e c t i v e classrooms. Responses were r e c e i v e d from students i n 1816 classrooms across the pr o v i n c e and from 1073 teachers of Grade 7 mathematics. The data underwent s e v e r a l stages of a n a l y s i s . F o l l o w i n g the numerical coding of v a r i a b l e s and the aggregation of student data t o c l a s s l e v e l , Pearson product-moment c o r r e l a t i o n s were c a l c u l a t e d between p a i r s of v a r i a b l e s . F a c t o r a n a l y s i s and i i i m u l tiple regression techniques were u t i l i z e d at subsequent stages of the a n a l y s i s . A number of s i g n i f i c a n t r e l a t i o n s h i p s were found between teacher and student behaviors, and student achievement. Among the v a r i a b l e s found to be most strongly r e l a t e d to achievement were teachers 1 a t t i t u d e s toward problem so l v i n g , the number and v a r i e t y of approaches and methods used by teachers, student perceptions of mathematics, and socio-economic status. Results also show that student background, students' and teachers' perceptions of mathematics, classroom organization and problem-solving processes a l l account f o r measurable variances i n student achievement. The amount of variance accounted f o r , however, was higher f o r achievement on a p p l i c a t i o n items, measuring lower cognitive l e v e l s of behavior, than on problem-so l v i n g items which measured cognitive behavior at the c r i t i c a l t hinking l e v e l . Through examination of the standardized beta weights from the c r o s s - s e c t i o n a l and l o n g i t u d i n a l models, i t was found that p r e d i c t i o n of change i n achievement based on corresponding change i n classroom process v a r i a b l e s was s i m i l a r f o r both models. Differences, however, were found f o r v a r i a b l e s i n the other categories. i v TABLE OF CONTENTS Page ABSTRACT i i REFERENCES v i i TABLES V i i FIGURES i x APPENDICES i x ACKNOWLEDGEMENTS X CHAPTER 1. STATEMENT OF THE PROBLEM 1 1.1 BACKGROUND 1 1.2 PURPOSE OF THE STUDY 3 1.3 ASSUMPTIONS OF THE STUDY 3 1.4 SIGNIFICANCE OF THE STUDY 5 1.5 SUMMARY 6 CHAPTER 2. REVIEW OF THE LITERATURE 8 2.1 RELATED LITERATURE ON THE EFFECTS OF SCHOOLING 8 E a r l y S t u d i e s 8 More Recent F i n d i n g s 9 F a c t o r s A f f e c t i n g Student Outcomes 10 2.2 FACTORS AFFECTING THE LEARNING OF MATHEMATICS 12 Teacher Background and Behaviors 14 Classroom Processes 15 Student A t t i t u d e s 17 2.3 SIMILAR STUDIES OF A CROSS-SECTIONAL NATURE 20 The F i r s t I n t e r n a t i o n a l Mathematics Study 20 The N a t i o n a l Assessment of E d u c a t i o n a l Progress (NAEP) - A R e p l i c a t i o n Study 23 2.4 SUMMARY 25 V CHAPTER 3. RESEARCH DESIGN AND METHODOLOGY 26 3.1 A MODEL OF INPUTS AND OUTCOMES OF SCHOOLING 26 3.2 POPULATION AND SAMPLING PLAN 29 The 1985 Mathematics Assessment 29 The 1987 V a l i d a t i o n Study 29 3.3 OVERVIEW OF THE METHOD OF STUDY 3 0 3.4 INSTRUMENTATION 31 Boo k l e t s f o r Students 31 Students 1 Background Items 3 6 Students' P e r c e p t i o n s of Mathematics 37 Teacher Q u e s t i o n n a i r e 37 3.5 DESCRIPTION AND DEFINITION OF THE VARIABLES 38 3.6 DATA COLLECTION 41 1985 Data C o l l e c t i o n 41 C a l c u l a t i o n s f o r Dependent V a r i a b l e s 42 C a l c u l a t i o n s f o r Independent V a r i a b l e s 42 1986-87 Data C o l l e c t i o n 43 3.7 DATA ANALYSIS PROCEDURES 43 1985 Assessment Data 43 C o r r e l a t i o n a l A n a l y s i s 44 F a c t o r A n a l y s i s 44 M u l t i p l e R e g r e s s i o n 44 1987 V a l i d a t i o n Study 46 3.8 SUMMARY 47 CHAPTER 4. FINDINGS 48 4.1 PREPARATION OF THE DATA 48 4.2 DESCRIPTIVE ANALYSIS OF THE INDEPENDENT VARIABLES 50 Student Background V a r i a b l e s 51 Teacher Background 54 Classroom O r g a n i z a t i o n 56 Problem-Solving Processes 59 Teachers' P e r c e p t i o n s o f Mathematics 65 Students' P e r c e p t i o n s o f Mathematics 69 v i 4.3 CORRELATIONAL ANALYSIS 73 Student Background and Achievement 74 Teacher Background and Student Achievement 76 Classroom O r g a n i z a t i o n and Achievement 78 Problem-Solving Processes and Student Achievement 81 Teachers' P e r c e p t i o n s o f Mathematics and Student Achievement 83 Students' P e r c e p t i o n s of Mathematics and Ach i evement 85 I n t e r p r e t a t i o n o f C o r r e l a t i o n C o e f f i c i e n t s 86 4.4 FACTOR ANALYSIS 87 Student Background F a c t o r s 87 Teacher Background F a c t o r s 89 Classroom O r g a n i z a t i o n F a c t o r s 90 Problem-Solving Processes 93 Teachers' P e r c e p t i o n s of Mathematical T o p i c s 95 Students' P e r c e p t i o n s o f Mathematical T o p i c s 95 Summary of the F a c t o r Analyses 95 4.5 MULTIPLE REGRESSION ANALYSIS 96 Student Background 97 Teacher Background 98 Classroom O r g a n i z a t i o n 99 Problem-Solving Process 100 Teachers' P e r c e p t i o n s o f Mathematics 101 Students' P e r c e p t i o n s of Mathematics 102 4.6 THE PROVINCIAL MODELS 103 Problem S o l v i n g 104 T e s t T o t a l 105 A p p l i c a t i o n s 106 4.7 THE SURREY MODEL 108 1987 P o s t - T e s t Model 108 1987 L o n g i t u d i n a l Model 110 CHAPTER 5. SUMMARY AND CONCLUSIONS 113 5.1 SIGNIFICANT FINDINGS AND CONCLUSIONS 114 The F i n a l Models 129 5.3 IMPLICATIONS FOR DECISION MAKERS 134 v i i 5.4 LIMITATIONS OF THE STUDY 13 6 E s t i m a t i o n of C l a s s Means 137 D e f i n i t i o n of V a r i a b l e s 138 Impacts o f Independent V a r i a b l e s on the Achievement of I n d i v i d u a l Students 138 The C l a s s as a U n i t of A n a l y s i s 138 5.5 IMPLICATIONS FOR FURTHER RESEARCH 139 REFERENCES 141 TABLES 1. Domains and Item Assignments 34 2. Summary S t a t i s t i c s f o r Grade 7 T e s t B o o k l e t s 36 3. Independent V a r i a b l e s and T h e i r Sources 39 4. Frequency D i s t r i b u t i o n o f C l a s s S i z e s 50 5. Mother Tongue 51 6. Time Spent on Homework 52 7. E d u c a t i o n a l L e v e l o f Parents 53 8. Time Spent on Classroom A c t i v i t i e s 58 9. Problem-Solving S t r a t e g i e s Taught 60 10. Sources of Problem-Solving E x e r c i s e s 61 11. A c t i v i t i e s Used t o M o t i v a t e Students 62 12. Problem Types Assigned t o Students 63 13. Classroom Features t o Promote Problem S o l v i n g 64 14. Number of Problem-Solving A c t i v i t i e s and Sources Used 64 15. Teachers' P e r c e p t i o n s o f Mathematical T o p i c s 66 16. R e l i a b i l i t y Analyses of Teachers' P e r c e p t i o n S c a l e s 68 17. D i s t r i b u t i o n s of Index Numbers f o r Teachers' P e r c e p t i o n s of Mathematics 69 18. Students' P e r c e p t i o n s of Mathematical T o p i c s 70 19. R e l i a b i l i t y Analyses o f Students' P e r c e p t i o n S c a l e s 72 20. D i s t r i b u t i o n s of Index Numbers f o r Students' P e r c e p t i o n s of Mathematics 72 V l l l 21. C o r r e l a t i o n s Among Student Background V a r i a b l e s and Achievement 75 22. C o r r e l a t i o n s Among Teacher Background V a r i a b l e s and Student Achievement 77 23. C o r r e l a t i o n s Among Classroom O r g a n i z a t i o n V a r i a b l e s and Achievement 79 24. C o r r e l a t i o n s Among Problem-Solving Process V a r i a b l e s and Achievement 82 25. C o r r e l a t i o n s Among Teachers' P e r c e p t i o n s and Student Achievement 84 26. C o r r e l a t i o n s Among Students' P e r c e p t i o n s and T h e i r Achievement 85 27. P r i n c i p a l Components of Student Background V a r i a b l e s 88 28. Rotated F a c t o r M a t r i x o f Student Background V a r i a b l e s 88 29. P r i n c i p a l Components of Teacher Background V a r i a b l e s 89 30. Rotated F a c t o r M a t r i x o f Teacher Background V a r i a b l e s 90 31. P r i n c i p a l Components of Classroom O r g a n i z a t i o n V a r i a b l e s 91 32. Rotated F a c t o r M a t r i x o f Classroom O r g a n i z a t i o n V a r i a b l e s 92 33. P r i n c i p a l Components of Problem-Solving Process V a r i a b l e s 93 34. Rotated F a c t o r M a t r i x o f Problem-Solving Process V a r i a b l e s 94 35. Numbers of V a r i a b l e s and F a c t o r s 96 36. Student Background F a c t o r s Regressed on C r i t e r i o n V a r i a b l e s 98 37. Classroom O r g a n i z a t i o n F a c t o r s Regressed on C r i t e r i o n V a r i a b l e s 99 38. Problem-Solving Process F a c t o r s Regressed on C r i t e r i o n V a r i a b l e s 100 39. Teachers' P e r c e p t i o n s Regressed on C r i t e r i o n V a r i a b l e s 102 i x 40. Students* Perceptions Regressed on C r i t e r i o n V a r i a b l e s 103 41. P r o v i n c i a l Regression Model f o r Problem S o l v i n g 104 42. P r o v i n c i a l Regression Model f o r Test T o t a l 105 43. P r o v i n c i a l Regression Model f o r A p p l i c a t i o n s 107 44. Post-Test Regression Model 109 45. L o n g i t u d i n a l Regression Model f o r Test T o t a l 111 46. C o r r e l a t i o n s Among Problem-Solving Process V a r i a b l e s 119 47. S i g n i f i c a n t R e l a t i o n s h i p s With Achievement on Problem S o l v i n g and A p p l i c a t i o n s 125 48. Variances i n Achievement Accounted For 129 FIGURES FIGURE 1 Factors a f f e c t i n g outcomes of s c h o o l i n g . 11 FIGURE 2 B l o c k i n g e f f e c t s on school achievement. 23 FIGURE 3 A model of inputs and outcomes of s c h o o l i n g . 27 FIGURE 4 E f f e c t s of sc h o o l i n g on achievement i n problem s o l v i n g , t e s t t o t a l and a p p l i c a t i o n s . 131 FIGURE 5 E f f e c t s of sc h o o l i n g on mathematics achievement based on c r o s s - s e c t i o n a l and l o n g i t u d i n a l data. 133 APPENDICES A. B r i t i s h Columbia Mathematics Assessment Test Booklets 153 R, S, and T B. Teacher's Guide Questionnaire 232 C. Coding of V a r i a b l e s 251 X ACKNOWLEDGEMENTS I would l i k e to thank my committee members who played a key r o l e i n the development of t h i s d i s s e r t a t i o n . Dr. David R o b i t a i l l e , as my research supervisor, provided i n s p i r a t i o n and a commitment to a l l aspects of the task at hand. His high standards and willingness to allow scope i n my work caused me to pursue objectives with corresponding d i l i g e n c e . Dr. James S h e r r i l l , with h i s meticulous reviews, focussed my att e n t i o n on the importance of d e t a i l and accuracy i n research. Dr. John Anderson played an important r o l e i n responding to my methods of s t a t i s t i c a l analyses and Dr. Douglas Owens served an important function as a reviewer with a ph i l o s o p h i c a l perspective. My family contributed s i g n i f i c a n t l y to the development of t h i s d i s s e r t a t i o n as well. They provided support and encouragement at a l l steps along the way. Brenda, my wife, and daughters Charlene and Cindy share with others i n helping t h i s p u b l i c a t i o n to come into being. I would also l i k e to acknowledge some of my friends who also provided encouragement and support. Among them was Dr. Nand Kishor who was always w i l l i n g to discuss issues which dealt with s t a t i s t i c s and computers. 1 CHAPTER 1 STATEMENT OF THE PROBLEM 1.1 BACKGROUND A major purpose of c r o s s - s e c t i o n a l s t u d i e s i n mathematics education i s the c o l l e c t i o n of data t o p r o v i d e d i r e c t i o n f o r d e c i s i o n making i n a number of areas r e l e v a n t t o the t e a c h i n g and l e a r n i n g of mathematics. These areas o f i n t e r e s t i n c l u d e d i r e c t i o n f o r resource a l l o c a t i o n , c u r r i c u l u m r e v i s i o n , p r e -s e r v i c e and i n - s e r v i c e t r a i n i n g , f u r t h e r r e s e a r c h , and maintenance o f st r e n g t h s and improvement of weaknesses as demonstrated by l e v e l s of achievement by students. During the l a s t twenty years s e v e r a l l a r g e , q u a n t i t a t i v e s t u d i e s of mathematics education have been conducted i n a number of j u r i s d i c t i o n s . Vast amounts of data r e l e v a n t t o students' achievement and a t t i t u d e s , t e a c h e r s ' backgrounds and a t t i t u d e s , and classroom processes were c o l l e c t e d . Among these s t u d i e s were the F i r s t and Second I n t e r n a t i o n a l S t u d i e s of Achievement i n Mathematics [Husen, 1967; McKnight, Travers & Dossey, 1985; R o b i t a i l l e , 1985; R o b i t a i l l e & Garden ( i n p r e s s ) ] , the N a t i o n a l Assessments of E d u c a t i o n a l Progress i n the U n i t e d S t a t e s (Carpenter, C o r b i t t , Kay, L i n d q u i s t & Reys, 1980; Carpenter, L i n d q u i s t , Matthews & S i l v e r , 1983), the N a t i o n a l Science Foundation Surveys (Fey, 1979) and a number of Canadian s t u d i e s i n c l u d i n g t h e 1977, 1981 and 1985 P r o v i n c i a l Assessments of Mathematics i n B r i t i s h Columbia ( R o b i t a i l l e & S h e r r i l l , 1977; R o b i t a i l l e , 1981; R o b i t a i l l e & O'Shea, 1985). 2 Each of these s t u d i e s r e p o r t e d i n f o r m a t i o n p r o v i d i n g d i r e c t i o n f o r d e c i s i o n making i n most of the areas noted e a r l i e r . Although data were c o l l e c t e d on i n p u t s of the e d u c a t i o n a l system, as r e f l e c t e d by classroom process v a r i a b l e s , t e a c h e r s ' a t t i t u d e s and t e a c h e r s ' backgrounds, and on outputs as measured by students' achievement, analyses of the r e l a t i o n s h i p s between them were not r e p o r t e d . N e v e r t h e l e s s , they were of i n t e r e s t t o the i n v e s t i g a t o r s i n each study. For example, R o b i t a i l l e and O'Shea (1985) r e p o r t e d t h a t one of the questions of i n t e r e s t i n the 1985 P r o v i n c i a l Assessment of Mathematics was, "How are achievement l e v e l s r e l a t e d t o c e r t a i n aspects of students' backgrounds, t h e i r a t t i t u d e s and o p i n i o n s , and those of t h e i r t e a c h e r s ? " (p. 3). An answer t o t h i s q u e s t i o n , however, r e q u i r e d f u r t h e r a n a l y s i s of the data. Among the reasons why these and other c r o s s - s e c t i o n a l s t u d i e s have not r e p o r t e d r e l a t i o n s h i p s between i n p u t s and outputs, i n c l u d e problems a s s o c i a t e d w i t h t h r e e areas of concern: l e v e l s of data aggregation, l i n k a g e s between teachers and classrooms, and l a c k of p r e - t e s t data. I n the present study, r e l a t i o n s h i p s between e d u c a t i o n a l i n p u t s and student outcomes f o r Grade 7 mathematics i n the p r o v i n c e of B r i t i s h Columbia were examined through f u r t h e r a n a l y s i s of data from the 1985 P r o v i n c i a l Assessment of Mathematics, the design of which prov i d e d an o p p o r t u n i t y t o address two of the t h r e e concerns s t a t e d e a r l i e r . F i r s t , data c o u l d be aggregated t o the classroom l e v e l . Second, r e s u l t s from teachers who responded t o q u e s t i o n n a i r e s on classroom processes, p e r c e p t i o n s of mathematics, and background - 3 c h a r a c t e r i s t i c s c o u l d be l i n k e d t o achievement r e s u l t s of students i n t h e i r c l a s s e s . I n order t o address the t h i r d i s s u e , l a c k of a p r e - t e s t , the 1985 Assessment was r e p l i c a t e d w i t h a sample of Grade 7 students and teachers from a l a r g e suburban sc h o o l d i s t r i c t d u r i n g the 1986-87 sc h o o l year. A p r e - t e s t and a p o s t - t e s t were ad m i n i s t e r e d and p r e - t e s t scores were t r e a t e d as c o v a r i a t e s . A comparison was then made between the r e s u l t s u s i n g p o s t - t e s t data o n l y and those found when p r e - t e s t data were a l s o i n c l u d e d i n the a n a l y s i s . 1.2 PURPOSE OF THE STUDY The o b j e c t i v e s of the present study were t h r e e f o l d : f i r s t , t o i d e n t i f y through the a n a l y s i s of p r o v i n c i a l survey data, those classroom processes, and teacher c h a r a c t e r i s t i c s and behaviors which are r e l a t e d i n a s i g n i f i c a n t way t o student achievement i n mathematics; second, t o t e s t a t h e o r e t i c a l model i n which r e l a t i o n s h i p s between i n p u t s and outcomes of s c h o o l i n g are hypothesized; and t h i r d , t o use r e s u l t s from a subsequent v a l i d a t i o n study t o compare f i n d i n g s from survey r e s e a r c h based on both the presence and absence of p r e - t e s t data. R e s u l t s were a l s o expected t o pr o v i d e d i r e c t i o n f o r f u t u r e r e s e a r c h i n the a n a l y s i s of c r o s s - s e c t i o n a l data. 1.3 ASSUMPTIONS OF THE STUDY D i r e c t i o n f o r the de t e r m i n a t i o n of procedures, data c o l l e c t i o n and a n a l y s i s was provided by two b a s i c assumptions of 4 the present study. The f i r s t assumption was that what students learn in mathematics i s , in part, a function of their attitudes toward i t and the teacher behaviors which occur during the course of instruction. Second, was the assumption that student learning in the classroom can be measured by the aggregation of individual results to obtain class-level data. In making these assumptions some evidence existed to suggest that they were plausible. The f i r s t assumption i s one generally held by many educators. However, Willms and Cuttance (1985) reported that studies prior to the mid 1970s, for the most part, found l i t t l e evidence to support the notion that teachers or schools made a difference in student learning. They stated that evidence from a number of these early studies suggested home and student background accounted for most of the variance in student learning, and that instructional factors had no significant effect. Willms and Cuttance proceeded to report that those findings were subsequently challenged by a number of more recent studies on effective schooling, in which instructional effects were partialled out from the others. The more recent studies concluded that instructional factors do make a difference in student learning. Some teachers appear to be more effective than others and since evidence from recent studies exists to suggest that i s the case, the f i r s t assumption underlying the study was proposed. The second assumption, that data aggregated to classroom level can be used to typify student behaviors, has been adopted in numerous other studies. For example, the large quantitative 5 studies referenced earlier used aggregated data for analysis and reporting. C r i t i c s of this practice argue that within class variances in student achievement are not accounted for when the data are aggregated (Willms & Cuttance, 1985), resulting in lower magnitudes for correlation coefficients. The classroom, however, was used as a unit of analysis in the present study since the benefits in this case outweighed the disadvantages. For example, the classroom i s a functional unit with which to compare teacher behavior and second, student level results would not be meaningful in this study due to a multiple matrix-sampling design employed in the 1985 Provincial Assessment of Mathematics. 1.4 SIGNIFICANCE OF THE STUDY A number of questions of significance to decision makers were addressed in the current study. Answers suggested direction for the alteration of various aspects of classroom process and for planning both pre-service and in-service a c t i v i t i e s , and identified areas for further research. The questions were as follows: 1. What relationships exist among teacher background characteristics and student background characteristics; and between these variables and student achievement in mathematics? 2. What relationships exist among types of classroom organizations and structures; and between these variables and students 1 achievement in mathematics? 3. What relationships exist between different approaches to the teaching of problem solving and students' achievements' in mathematics? 6 4. What r e l a t i o n s h i p s e x i s t among teachers 1 perceptions of mathematics and students• perceptions of mathematics; and between these perceptions and students' achievement i n mathematics? 5. What dif f e r e n c e s , i f any, e x i s t i n the strengths of the r e l a t i o n s h i p s i n questions 1 to 4 when achievement i s measured at d i f f e r e n t cognitive behavior l e v e l s ? 6. How much variance i n student achievement i n mathematics i s accounted f o r by the e f f e c t s of teacher and student background, classroom organization and processes, and teachers' and students' perceptions of mathematics? 7. What di f f e r e n c e s occur i n the r e s u l t s found through the analysis of cr o s s - s e c t i o n a l data a f t e r l o n g i t u d i n a l data are included i n the analysis? 1.5 SUMMARY The importance of t h i s study i s based on a need to c o l l e c t meaningful information f o r de c i s i o n making. I t i s e s s e n t i a l that r e l a t i o n s h i p s between the inputs and the outcomes of schooling be established i n order that guidelines can be determined f o r e f f e c t i v e classroom organization, use of appropriate teaching s t r a t e g i e s and the e f f i c i e n t a l l o c a t i o n of resources. A need f o r c o l l e c t i o n of t h i s information was supported by Randhawa and Fu (1973). They concluded, a f t e r a survey of l i t e r a t u r e on the e f f e c t s of input v a r i a b l e s , that the learning environment of a classroom can be a p r e d i c t o r of achievement. This recognition brings with i t the r e s p o n s i b i l i t y f o r assessing the environmental v a r i a b l e s and examining t h e i r r e l a t i o n s h i p s with achievement. The next chapter deals with a review of pertinent l i t e r a t u r e and the t h e o r e t i c a l perspectives of the study. A conceptual framework, based on the l i t e r a t u r e review, i s developed i n Chapter 3 i n which i n t e r r e l a t i o n s h i p s among factors a f f e c t i n g student outcomes are proposed. Descriptions of the instruments and procedures, d e f i n i t i o n s of the v a r i a b l e s and methods of analysis are also dealt with i n Chapter 3. Chapters 4 and 5 discuss r e s u l t s of the analyses and a r r i v e at conclusions. 8 CHAPTER 2 REVIEW OP THE LITERATURE Examinations of the r e l a t i o n s h i p s among a number of input and outcome v a r i a b l e s have been conducted by researchers i n attempts to determine the e f f e c t s of schooling. Included among these v a r i a b l e s are teacher c h a r a c t e r i s t i c s , teacher behaviors, student c h a r a c t e r i s t i c s , student behaviors and student learning outcomes (Centra & Potter, 1980). Many of these studies used a c r o s s - s e c t i o n a l design i n which data were c o l l e c t e d at a given point i n time. Others were of a l o n g i t u d i n a l nature i n which time was included as a v a r i a b l e . This review i s organized on the basis of the r e l a t i o n s h i p s among v a r i a b l e s which were examined i n these studies. I t also i d e n t i f i e s a number of issues dealt with i n the analyses of data. 2 . 1 RELATED LITERATURE ON THE EFFECTS OF SCHOOLING Ear l y Studies A review of the l i t e r a t u r e on teacher ef f e c t i v e n e s s i n the two decades leading up to the early 1970s could lead one to the conclusion that teaching does not make a d i f f e r e n c e i n student learning. For example, Gage (1960) reported that i n more than IC 000 studies on teacher effectiveness, the combined l i t e r a t u r e was overwhelming i n i t s inconsistencies across f i n d i n g s . In "Equality of Educational Opportunity" (Coleman, Campbell, Hobson, McPartland, Mood, Weinfeld & York, 1966) r e s u l t s were reported i n which the r e l a t i o n s h i p s between over 400 input v a r i a b l e s and achievement outcome measures were examined. I t was concluded that v a r i a b l e s which could be manipulated by school p o l i c i e s , such as teacher s a l a r i e s , type of curriculum, and per p u p i l expenditures had l i t t l e e f f e c t on p u p i l achievement. A number of subsequent reviews of t h i s e a r l y work also reached the conclusion that teachers do not make a d i f f e r e n c e . (Fey, 1969; Jenks, Smith, Ackland, Bane, Cohen, Gentis, Heyns & Michelson, 1972; Dunkin and Biddle, 1974.) C r i t i c s of these e a r l i e r studies, however, claim that t h e i r f a i l u r e s to uncover important input e f f e c t s were a t t r i b u t e d to a number of weaknesses i n design and a n a l y s i s . For example, i t was contended that the d i f f i c u l t y of separating the e f f e c t s due to school resources and those due to family background (Bowles and Levin, 1968) , the use of achievement t e s t s which were not l i n k e d d i r e c t l y to the curriculum (Postlethwaite, 1975) and several methodological problems, dealing with use of aggregated data (Burstein, 1980) , l e d to inconclusive f i n d i n g s . Other researchers argued that r e l a t i o n s h i p s could not be found between teachers' behaviors and students' achievement i n e a r l y studies because of the type of data c o l l e c t e d . Dunkin and Biddle (1974), f o r example, contended that a basic problem was that teachers were seldom observed. The same c r i t i c i s m was d i r e c t e d at the Coleman et. a l . (1966) report. For example, Good, Grouws & Ebmeier (1983) claimed that i t dealt with input and output v a r i a b l e s but not classroom processes. More Recent Findings Willms and Cuttance (1985) claimed that as a r e s u l t of the c r i t i c a l reviews of e a r l i e r studies, "a new l i t e r a t u r e i s 10 emerging t h a t emphasizes w i t h i n school processes t h a t l i n k p u p i l i n p u t s t o s c h o o l i n g outcomes" (p. 290). These more recent s t u d i e s examined the types of l e a r n i n g environments w i t h i n classrooms and schools (McLaughlin, 1978; R u t t e r , Maughan, Mortimore & Ouston, 1979); teacher behaviors and s t y l e s (Brophy, 1982a, b; Evertson, Anderson, Anderson & Brophy, 1980) and the a l l o c a t i o n of t e a c h i n g resources w i t h i n the s c h o o l ( B i d w e l l and Kasarda, 1980). F u r t h e r work done by Anderson (1982) and Moos (1979) attempted t o d e f i n e s c h o o l c l i m a t e and s t u d i e d p u p i l and t e a c h e r i n t e r a c t i o n s . Based on t h i s l i t e r a t u r e i t i s i n c r e a s i n g l y c l e a r t h a t what c h i l d r e n l e a r n from t h e i r classroom experiences i s a f u n c t i o n of what i s done d u r i n g c l a s s time. A number of other s t u d i e s of i n p u t - o u t p u t r e l a t i o n s h i p s found t h a t the e f f e c t s of i n p u t s such as socio-economic s t a t u s (SES) , teacher c h a r a c t e r i s t i c s , time spent on s p e c i f i c t a s k s and e f f e c t s of p u p i l m o t i v a t i o n are r e l a t e d t o academic outcomes. Reviewers of t h i s l i t e r a t u r e (Murnane, 1981; C l a r k , L o t t o and McCarthy, 1980; B r i d g e , Judd and Mooch, 1979; R u t t e r , 1983) conclude t h a t s c h o o l s do make a d i f f e r e n c e . For example, Murnane (1981, p. 27) contends t h a t , " c h i l d r e n l e a r n more when they are taught by t a l e n t e d , h i g h l y motivated teachers who b e l i e v e t h a t t h e i r p u p i l s can l e a r n and who s t r u c t u r e the school day so t h a t p u p i l s spend l a r g e amounts of time 'on t a s k ' working a t b a s i c s k i l l development." Factors A f f e c t i n g Student Outcomes I t can be concluded, based on the preceding g e n e r a l review of r e s e a r c h evidence on f a c t o r s a f f e c t i n g students' performance, t h a t v a r i a b l e s r e l a t e d t o home background and students' 11 c h a r a c t e r i s t i c s have the most s i g n i f i c a n t impact on achievement l e v e l s . However, findings of recent studies have also shown that school-related factors have some e f f e c t as w e l l . A model depi c t i n g these r e l a t i o n s h i p s i s shown i n Figure l . s Home Background C h a r a c t e r i s t i c s of Students \ School-Related Factors Student Outcomes Figure 1. Factors a f f e c t i n g outcomes of schooling. The main influences on student outcomes f a l l under three categories i n the model (shown i n Figure 1) : Home Background, C h a r a c t e r i s t i c s of Students and School-Related Factors. Interactions among these categories are also i l l u s t r a t e d i n the model. Influences on academic performance have been the subject of extensive research, but none of the research has been able to provide a comprehensive and accurate account of them or of the impacts they have on students with d i f f e r e n t l e v e l s of a b i l i t y and motivation. The e f f e c t i v e schools research continues to struggle with the complexity of the i n t e r a c t i o n s , the appropriateness of instruments to measure them and acceptable methods of an a l y s i s . 2.2 FACTORS AFFECTING THE LEARNING OF MATHEMATICS 12 The l i t e r a t u r e r e l a t e d s p e c i f i c a l l y to the teaching and lear n i n g of mathematics follows a pattern s i m i l a r to that reported e a r l i e r . In reporting on the Missouri Mathematics Effe c t i v e n e s s Project, Good, Grouws & Ebmeier (1983) made the following observation: When we began our program of research i n the e a r l y 1970s there was very l i t t l e u s e f u l or r e l i a b l e information a v a i l a b l e f o r describing the r e l a t i o n s h i p between classroom processes (e.g., teacher behavior) and classroom products (e.g., student achievement). What knowledge existed i n 1970 about the e f f e c t s of classroom processes on student achievement was weak and contradictory. A f t e r a decade of extensive research on classroom processes (much of t h i s research supported by the National I n s t i t u t e of Education), there i s now much pertinent information about t h i s r e l a t i o n s h i p (p. 1). This observation i s supported by evidence from several, large scale, c o r r e l a t i o n a l studies i n which the data i l l u s t r a t e d that i t was p o s s i b l e to i d e n t i f y some teachers who c o n s i s t e n t l y produced higher achievement i n students than expected (e.g., Brophy and Evertson, 1974; Good and Grouws, 1975). I t was also possible, through design and analysis, to i d e n t i f y i n s t r u c t i o n a l patterns that d i f f e r e n t i a t e d these teachers from those who were l e s s successful according to an operational d e f i n i t i o n of effectiveness (e.g., B e r l i n e r and Tikunoff, 1976; Brophy and Evertson, 1976; Rosenshine, 1979). A number of field-based experimental studies supported these patterns of e f f e c t i v e i n s t r u c t i o n a l behavior suggested by the c o r r e l a t i o n a l studies (e.g. Brophy, 1979; Good and Grouws, 1977; S t a l l i n g s , 1980). 13 In a study of fourth-grade mathematics i n s t r u c t i o n , Good and Grouws (1977) i d e n t i f i e d nine e f f e c t i v e and nine l e s s e f f e c t i v e teachers from a sample of over one hundred. They found that effectiveness was strongly associated with the following behavioral c l u s t e r s : c l a r i t y of i n s t r u c t i o n , a task-focused environment, a non-evaluative and relaxed learning environment, higher achievement expectations, r e l a t i v e l y few behavioral problems and teaching the c l a s s as a u n i t . In another process-product study focusing on mathematics, Evertson et. a l . (1980), found that more e f f e c t i v e teachers, i n contrast to l e s s e f f e c t i v e ones at the Grade 7 and 8 l e v e l s , demonstrated the following c h a r a c t e r i s t i c s : they spent more time on content presentations and discussions with l e s s time on seat work, held higher expectations of students and exhibited stronger management s k i l l s . In l i g h t of these findings, a case that teachers do make a d i f f e r e n c e i n the learning of mathematics can be made. For example, Good, et. a l . (1983), i n t h e i r report on the Missouri Mathematics Effectiveness Project referenced e a r l i e r , confirm t h i s p o s i t i o n with the following statement: Our research provides compelling evidence that teachers make a d i f f e r e n c e i n student learning and o f f e r s some useful information about how more or l e s s e f f e c t i v e teachers d i f f e r i n t h e i r behavior and i n t h e i r e f f e c t s on student achievement (p. 13). The s i z e of these teacher e f f e c t s , however, va r i e s considerably across studies. Good (1979, p. 54) f o r example, reported that i n a study by Inman, i n s t r u c t i o n a l v a r i a b l e s accounted f o r 26 percent of adjusted variance i n minority 14 students' achievement scores but f o r only 12 percent of the adjusted variance f o r majority students. Since the major purpose of the present study i s to examine r e l a t i o n s h i p s between a number of input v a r i a b l e s such as students' background and a t t i t u d e , background of teachers and t h e i r a t t i t u d e s , classroom processes and organization; and output v a r i a b l e s as defined by student achievement i n mathematics, t h i s review proceeds to examine findings from other studies on the e f f e c t s of these p a r t i c u l a r v a r i a b l e s . Teacher Background and Behaviors In general, studies on the r e l a t i o n s h i p s between background c h a r a c t e r i s t i c s of teachers and student outcomes have shown l i t t l e e f f e c t s . For instance, Rutter et. a l . , (1979), and M c D i l l and Rigsby (1973) found that preparation time, the keeping of records, and salary l e v e l had no r e l a t i o n s h i p with students' achievement or a s p i r a t i o n s . However, the same researchers d i d f i n d that the r e l a t i o n s h i p between students* achievement and teachers with more than a bachelor's degree was s i g n i f i c a n t l y and p o s i t i v e l y c o r r e l a t e d . Brophy (1982b), i n a discussion of teacher c h a r a c t e r i s t i c s or behaviors associated with student achievement gains, l i s t e d the following eight categories based on research findings of the seventies: teacher expectations; teacher e f f i c i e n c y ; student opportunity to learn; classroom management and organization; curriculum pacing; a c t i v e teaching; teaching to mastery; and a supportive learning environment. Rosenshine and Furst (1971), i d e n t i f i e d the following v a r i a b l e s as strong c o r r e l a t e s of student achievement: c l a r i t y , v a r i a b i l i t y , enthusiasm, task 15 o r i e n t a t i o n and opportunity to le a r n . Further evidence of p o s i t i v e c o r r e l a t i o n s between teacher v a r i a b i l i t y and student achievement were found by Kolb (1977) and Cooney, Davis and Henderson (1975). In a major c r o s s - s e c t i o n a l study i n v o l v i n g more than 20 000 secondary students and t h e i r teachers, M c D i l l and Rigsby (1973) found a p o s i t i v e r e l a t i o n s h i p between the educational background of teachers and student achievement. This r e l a t i o n s h i p was confirmed i n a study by the New York State Department (1976) i n which both teacher education and experience showed p o s i t i v e c o r r e l a t i o n s with student achievement. The importance of t r a i n i n g and i n - s e r v i c e was stressed by Ward (1979), i n presenting the r e s u l t s of a survey undertaken from the Schools Council, i n which he reported that the main handicap of elementary teachers was lack of t r a i n i n g i n mathematics education. He suggested that because of the l i n e a r i t y of the subject, mathematics can s u f f e r most from poor teachers. Based on the r e s u l t s of these studies i t appears that teacher background v a r i a b l e s have some e f f e c t on student learning. Although findings are not consistent across a l l studies, v a r i a b l e s r e l a t e d to the behaviors and professional preparation of teachers have p o s i t i v e r e l a t i o n s h i p s with student achievement. Classroom Processes In a comprehensive study of classroom processes and i n s t r u c t i o n a l p r a c t i c e s i n Grade 1 and Grade 3 classrooms, S t a l l i n g s (1976) reported that out of a poss i b l e 340 16 c o r r e l a t i o n s between mathematics achievement and classroom processes, 108 were s i g n i f i c a n t l y r e l a t e d at the 0.05 l e v e l . The classroom process v a r i a b l e which c o r r e l a t e d the highest with achievement was the amount of time spent by students on mathematical a c t i v i t i e s . Emphases placed on d r i l l , follow-up to homework, and i n s t r u c t i o n with small groups were among the other v a r i a b l e s p o s i t i v e l y r e l a t e d to student achievement. Further evidence of the importance of time spent on mathematical a c t i v i t i e s has been provided i n numerous studies. (Husen, 1967; S t a l l i n g s and Kaskowitz, 1974; Wiley and Harnischfeger, 1974; McDonald and E l i a s , 1976; Ebmeier and Good, 1979). In an analysis of r e s u l t s from the 1977-1978 National Assessment of Educational Progress i n Mathematics, Welch, Anderson & H a r r i s (1982) reported that the amount of mathematics studied accounted f o r 34 percent of the variance i n achievement f o r 17-year olds. An important d i s t i n c t i o n among types of time-r e l a t e d a c t i v i t i e s was made by B e r l i n e r (1978) i n which he defined three v a r i a b l e s : a l l o c a t e d time, engaged time, and academic learn i n g time. He found that while a l l of these v a r i a b l e s were p o s i t i v e l y associated with achievement, a l l o c a t e d time accounted f o r l e s s variance than the others. In a study by Wiener (1979) i t was found that the most e f f e c t i v e teachers of mathematics at the Grade 2 and 3 l e v e l s used small group teaching to present new s k i l l s f o r the a c c e l e r a t i o n of students and to provide general review for students i n need of remedial assistance. This f i n d i n g was extended by r e s u l t s of a study of Grade 4 students by Sindelar, Rosenberg, Wilson & Bursuck (1984) i n which i t was found that work i n small groups promoted students' engaged time, and that 17 engaged time was r e l a t e d t o h i g h e r achievement l e v e l s . S t u d i e s by Moody, B a u s e l l & J e n k i n s , (1974) and F i s h e r , B e r l i n e r , F i l b y , M a r l i a v e , Cahen & Dishaw (1980), a l s o r e p o r t e d t h a t s m a l l group s i z e s had c o n s i d e r a b l e i n f l u e n c e on l e a r n i n g . R e s u l t s from these s t u d i e s suggest t h a t time spent on s p e c i f i c a c t i v i t i e s i n mathematics i s p o s i t i v e l y c o r r e l a t e d t o student achievement i n the s u b j e c t . For example, time spent on f o l l o w - u p t o homework ( S t a l l i n g s , 1976) and time i n which students were engaged i n mathematical a c t i v i t i e s ( B e r l i n e r , 1978) r e l a t e d t o student achievement. Co n s i d e r a b l e evidence a l s o e x i s t e d t o show t h a t working w i t h s m a l l groups had a p o s i t i v e e f f e c t on student l e a r n i n g (Moody e t . a l . , 1974; S t a l l i n g s , 1976; Weiner, 1979; F i s h e r e t . a l . , 1980; S i n d e l a r e t . a l . , 1984). Student A t t i t u d e s I n a d d i t i o n t o the l e a r n i n g of p r i n c i p l e s , f a c t s and methods, important outcomes of s c h o o l i n g a l s o i n c l u d e a t t i t u d e s , v a l u e s and a p p r e c i a t i o n . The l a t t e r t h r e e outcomes r e l a t e t o o b j e c t i v e s from the a f f e c t i v e domain, which c o n s t i t u t e s a major goal o f the c u r r i c u l u m . A number of methods have been used t o measure students' a t t i t u d e towards mathematics. These i n c l u d e o b s e r v a t i o n s , i n t e r v i e w s , sentence completion t e s t s and a t t i t u d e s c a l e s . The most popular of these methods i s the use of a t t i t u d e s c a l e s such as the Thurstone or L i k e r t (Aiken, 1972). L i t t l e r e s e a r c h evidence i s a v a i l a b l e t o support the b e l i e f t h a t f a v o r a b l e a t t i t u d e s towards mathematics l e a d t o h i g h e r achievement. For example, Jackson (1968), i n a review of 18 research i n t h i s area found few studies that reported s i g n i f i c a n t r e l a t i o n s h i p s between a t t i t u d e and achievement. Caezza (1970) and Van de Walle (1973), f o r example, found no s i g n i f i c a n t r e l a t i o n s h i p s i n studies at the elementary l e v e l . However, low, p o s i t i v e c o r r e l a t i o n s between students' and teachers' a t t i t u d e s toward mathematics and students' performance i n mathematics have been found i n a number of other studies (Torrance, 1966; Wess, 1970; P h i l l i p s , 1973). Lester (1980), however, a t t r i b u t e s inconclusive findings to the elusiveness of a t t i t u d e to be defined as a v a r i a b l e and to the lack of r e l i a b l e instruments. Several s p e c i f i c research p r o j e c t s follow i n which s i g n i f i c a n t , p o s i t i v e c o r r e l a t i o n s between a t t i t u d e and performance were found. Robinson (1973), i n a study of the problem-solving behaviors of s i x t h graders found that good problem solvers had higher self-esteem than poor ones. Evertson et. a l . (1980) found s i g n i f i c a n t c o r r e l a t i o n s with achievement among j u n i o r high school mathematics students. Newman (1984), i n a l o n g i t u d i n a l study i n v o l v i n g an a n a l y s i s of students' achievement and s e l f - p e r c e p t i o n i n mathematics i n Grades 2, 5 and 10 found that between Grades 2 and 5 mathematics achievement i s c a u s a l l y r e l a t e d to s e l f - r a t i n g s of a b i l i t y and that between grades 5 and 10 the strength of the r e l a t i o n s h i p diminishes. In a summary of some studies r e l a t i n g a t t i t u d e to achievement, Hart (1977) commented that even where s i g n i f i c a n t c o r r e l a t i o n occurs, i t i s d i f f i c u l t to determine whether a t t i t u d e a f f e c t s achievement or v i c e versa. The d i r e c t i o n of 19 the r e l a t i o n s h i p between these v a r i a b l e s was also questioned by Neal (1969). Begle (1979), i n reviewing studies on the a t t i t u d e s of mathematics teachers, found that teacher enjoyment of mathematics had a p o s i t i v e e f f e c t on p u p i l s ' achievement. He also found that a f f e c t i v e v a r i a b l e s of teachers have a stronger e f f e c t than background v a r i a b l e s such as gender or marital status. Further evidence of r e l a t i o n s h i p s between teacher a t t i t u d e s and student achievement was found by Edmonds and Frederickson (1978) i n a study of Grade 6 students and s t a f f i n 812 elementary schools. The work of Kyles and Sumner (1977) investigated not only general a t t i t u d e s toward mathematics, but also d e t a i l s of students' perceptions of d i f f e r e n t t o p i c s and a c t i v i t i e s within the subject. They confirmed that student perceptions d i f f e r e d among t o p i c s . S i m i l a r r e s u l t s were reported i n the primary survey conducted by the Assessment of Performance Unit (1980). Although findings were not consistent across these studies, p o s i t i v e c o r r e l a t i o n s between a t t i t u d e and achievement were found by several of the researchers (Robinson, 1973; Edmonds and Frederickson, 1978; Begle, 1979; Evertson et. a l . , 1980; Newman, 1984). Of further i n t e r e s t are the d i r e c t i o n of r e l a t i o n s h i p s between a t t i t u d e and achievement (Neal, 1969 ; Hart, 1977) and student perceptions of d i f f e r e n t t o p i c s within the curriculum (Kyles and Sumner, 1977). 20 2.3 SIMILAR STUDIES OF A CROSS-SECTIONAL NATURE In much of the e a r l i e r work with c r o s s - s e c t i o n a l survey data, researchers reported on cause-and-effeet r e l a t i o n s h i p s using measures of a s s o c i a t i o n such as regression and c o r r e l a t i o n c o e f f i c i e n t s . These s t a t i s t i c s were used to estimate the e f f e c t that would occur i n one v a r i a b l e given a measured change i n another. C r i t i c s of t h i s approach such as Willms and Cuttance (1985) suggest that the genuine r e l a t i o n s h i p of school and teacher v a r i a b l e s to student achievement can be revealed accurately only through the analysis of l o n g i t u d i n a l data. They claim that only by c o n t r o l l i n g f o r the influence of a b i l i t y , family, and other non-school fac t o r s can cause-and-effeet be studied. While l o n g i t u d i n a l data may be necessary to report casual r e l a t i o n s h i p s , i t i s assumed that c r o s s - s e c t i o n a l data can be used to measure the strengths of a s s o c i a t i o n between v a r i a b l e s under examination and the extent to which variances i n student achievement can be a t t r i b u t e d to those v a r i a b l e s . A d e s c r i p t i o n of the methods of a n a l y s i s used i n two studies of a s i m i l a r nature to the present one, which used c r o s s - s e c t i o n a l data to examine r e l a t i o n s h i p s between inputs of schooling and student achievement, follows. The F i r s t International Mathematics study One of the f i r s t major i n t e r n a t i o n a l studies of the e f f e c t s of schooling on the learning of mathematics was the F i r s t I nternational Mathematics Study (Husen, 1967), conducted i n 1964 under the auspices of the International A s s o c i a t i o n f o r the Evaluation of Educational Achievement. A t o t a l of twelve countries p a r t i c i p a t e d i n the study which involved students i n the following four populations: 13-year olds; grade l e v e l containing most 13-year olds; mathematics students i n t h e i r f i n a l secondary year; and non-mathematics students i n t h e i r f i n a l secondary year. Husen (1967) reported that the main obstacle faced i n the anal y s i s of r e s u l t s was the lack of consistent measurement of a number of independent v a r i a b l e s with o p e r a t i o n a l l y f e a s i b l e i n d i c e s . D i f f e r e n t i n t e r p r e t a t i o n s were often applied to va r i a b l e s among countries due to d i f f e r e n t c u l t u r e s and educational systems. In the 13-year o ld group t o t a l achievement scores were co r r e l a t e d with f o r t y - f i v e independent v a r i a b l e s which characterized the school, the teacher and the student. Among the school v a r i a b l e s , s i z e was found to be c o r r e l a t e d at the 0.12 l e v e l . The teacher v a r i a b l e s which c o r r e l a t e d the highest with achievement were teacher t r a i n i n g and teacher ratings of students 1 opportunity to learn. These c o r r e l a t i o n s were reported as 0.08 and 0.19 r e s p e c t i v e l y . Students' c h a r a c t e r i s t i c s which were found to c o r r e l a t e p o s i t i v e l y included fathers' education (0.18), students' i n t e r e s t i n mathematics (0.30) and students' plans and a s p i r a t i o n s (0.18 to 0.22). At the next stage of analysis several v a r i a b l e s were dropped due to overlapping or unsuitable coding. There remained twenty-six independent v a r i a b l e s grouped under the following four headings: parental v a r i a b l e s , teacher v a r i a b l e s , school v a r i a b l e s and student v a r i a b l e s . A stepwise regression analysis was conducted next with v a r i a b l e s under each heading regressed on t o t a l s c o r e . The average amount of v a r i a n c e a c r o s s c o u n t r i e s i n the achievement of 13-year o l d s accounted f o r by v a r i a b l e s i n each separate heading f o l l o w s : p a r e n t a l v a r i a b l e s , 4.4 percent; t e a c h e r v a r i a b l e s , 1.3 percent; s c h o o l v a r i a b l e s , 1.3 percent; and student v a r i a b l e s , 7.5 percent. A d i f f e r e n t approach t o the a n a l y s i s of r e s u l t s from t h r e e subsequent I n t e r n a t i o n a l S t u d i e s i n s c i e n c e e d u c a t i o n , r e a d i n g comprehension and l i t e r a t u r e was r e p o r t e d by Coleman (1975). He r e p o r t e d on a number of methodological i s s u e s which were faced i n the a n a l y s i s of r e s u l t s from these s t u d i e s . The dependent v a r i a b l e was achievement i n each of t h r e e t o p i c areas. Independent v a r i a b l e s on the other hand, were c l u s t e r e d i n t o t h r e e B l o c k s : Block 1- Home Background; Block 2- Type of School and Program; and Block 3- School I n s t r u c t i o n . The purpose of s e p a r a t i n g v a r i a b l e s i n t o these b l o c k s a p r i o r i was t o b r i n g some order i n t o the r e g r e s s i o n analyses. The e x p e c t a t i o n s , a c c o r d i n g t o Coleman, were t h a t Block 1 v a r i a b l e s were more important than Block 3 v a r i a b l e s and t h a t s c h o o l v a r i a b l e s (Block 2) would show l i t t l e e f f e c t . Using t h i s r a t i o n a l e , v a r i a b l e s were entered i n t o the r e g r e s s i o n a n a l y s i s i n order of Block number. E f f e c t s were then i d e n t i f i e d as increments t o e x p l a i n e d v a r i a n c e s f o r each preceding Block. The diagram shown i n F i g u r e 2, i n d i c a t e s the c a u s a l reasoning behind the use of t h i s sequence of b l o c k s . Block 1 Block 2 Block 3 Block 4 Figure 2. Blocking e f f e c t s on school achievement (Coleman, 1975, p.361). The National Assessment of Educational Progress (NAEP) - A Rep l i c a t i o n study In a study conducted to assess the dependence of high school mathematics achievement on a number of input factors, Horn and Walberg (1984) regressed the achievement and i n t e r e s t scores of a sample of 17-year olds on each other and on fourteen other v a r i a b l e s . The purposes of the study were to investigate the dependencies of students' i n t e r e s t and achievement i n mathematics on a lar g e r set of v a r i a b l e s through r e p l i c a t i o n of the e a r l i e r NAEP study. Data from the 1977-78 NAEP were used i n the analyses. Horn and Walberg (1984) reported that a sample of 1480 17-year olds was drawn, using a s t r a t i f i e d , three-stage national p r o b a b i l i t y design: the primary sampling units were representative of a l l the regions and community s i z e s i n the United States. At the second stage, sc h o o l s w i t h i n each primary sampling u n i t were sampled from a l i s t of a l l p u b l i c and p r i v a t e s c h o o l s . At the t h i r d stage a random sample of a g e - e l i g i b l e students was s e l e c t e d w i t h i n each s c h o o l . Students i n the sample were ad m i n i s t e r e d the NAEP b o o k l e t e n t i t l e d "Number 1". A w e i g h t i n g procedure was used a t the a n a l y s i s stage t o e s t i m a t e n a t i o n a l s t a t i s t i c s by compensating f o r oversampling of s e l e c t e d groups. Achievement was measured by a t e s t c o n s i s t i n g o f 55 items, w i t h a r e l i a b i l i t y of 0.92, u s i n g Cronbach's alpha c o e f f i c i e n t . The t e s t was comprised of f i v e content areas and f o u r c o g n i t i v e l e v e l s of behavior. The independent v a r i a b l e s under study i n c l u d e d the f o l l o w i n g : i n s t r u c t i o n (3 l e v e l s ) , number of courses, most advanced course, home environment, TV, homework, SES, sex and e t h n i c i t y . These v a r i a b l e s were l i m i t e d i n number by the items contained i n the NAEP b o o k l e t and were s e l e c t e d on the b a s i s of e a r l i e r f i n d i n g s of r e l a t i o n s h i p s between background, i n s t r u c t i o n and achievement. U n i v a r i a t e analyses showed h i g h p o s i t i v e c o r r e l a t i o n s between achievement and both the number of courses (0.62) and the most advanced course taken (0.63). Moderate c o r r e l a t i o n s , r anging between 0.38 and 0.41 were found between achievement and t r a d i t i o n a l i n s t r u c t i o n , home environment and SES. C o r r e l a t i o n s of 0.23 and 0.21 were r e p o r t e d between achievement and both frequency of c o u r s e - r e l a t e d a c t i v i t i e s and homework r e s p e c t i v e l y , whereas t e l e v i s i o n exposure c o r r e l a t e d n e g a t i v e l y . M u l t i p l e r e g r e s s i o n techniques were used t o determine the e f f e c t s of v a r i a b l e s on the outcome measures. A reduced model f o r achievement accounted f o r 57 percent of the v a r i a n c e and showed that each of 12 v a r i a b l e s was s t a t i s t i c a l l y s i g n i f i c a n t when the others were c o n t r o l l e d . Findings from the Horn and Walberg (1984) study showed that mathematics achievement i s a function of the l e v e l and amount of mathematics coursework completed i n high school. I t i s also influenced by student i n t e r e s t i n mathematics, t r a d i t i o n a l i n s t r u c t i o n , education of parents and q u a l i t y of the home environment. 2.4 SUMMARY This chapter has discussed a number of issues faced by researchers i n examining the r e l a t i o n s h i p s between inputs and outcomes of schooling. I t reported on the l i m i t e d findings of e a r l y studies and the more s i g n i f i c a n t ones of subsequent research i n the 1970s and 1980s. Results from numerous studies on the r e l a t i o n s h i p s between student achievement and v a r i a b l e s r e l a t e d to student and teacher backgrounds, student and teacher a t t i t u d e s , and classroom processes were then reported. In the next chapter, a conceptual model, based on the l i t e r a t u r e review j u s t completed, i s presented. A d e s c r i p t i o n of instruments and procedures, d e f i n i t i o n of v a r i a b l e s and methods of analysis are included. ) 26 CHAPTER 3 RESEARCH DESIGN AND METHODOLOGY This chapter contains a discussion of a model of the re l a t i o n s h i p s between inputs and outcomes of schooling. I t also describes the population and the sampling plan, i d e n t i f i e s the relevant v a r i a b l e s and the methods by which they were measured, provides a d e s c r i p t i o n of the instruments and t h e i r development, and describes the data c o l l e c t i o n and a n a l y t i c a l procedures. 3.1 A MODEL OF INPUTS AND OUTCOMES OF SCHOOLING In the preceding chapter the impact of schooling on students' achievement i n mathematics was discussed. I t was reported that e a r l y studies found students' a b i l i t y and family background accounted f o r almost a l l s t a t i s t i c a l l y s i g n i f i c a n t variance i n student achievement (e.g., Coleman, et. a l . , 1966; Fey, 1969; Jencks et. a l . , 1972; Dunkin and Biddle, 1974). However, subsequent studies i n the l a t e 1970s and 1980s found that, a f t e r c o n t r o l l i n g f o r student a b i l i t y and background, schools d i d make a di f f e r e n c e (e.g., Willms and Cuttance, 1985; Anderson, 1982; Murnane, 1981; Moos, 1979). A further search of the l i t e r a t u r e , which focused on those fact o r s of schooling that have an e f f e c t on students' at t i t u d e s and achievement i n mathematics, uncovered a number of vari a b l e s r e l a t e d to these outcomes. I n s t r u c t i o n a l processes, classroom organization, and teacher background and a t t i t u d e s were among the inputs of schooling shown to have some such e f f e c t (e.g., Brophy, 1982a,b; Good, et. a l . , 1983; Evertson, et. a l . , 1980). The e f f e c t s of input v a r i a b l e s , however, are complex and i n t e r r e l a t e d . For example, Neal (1969) and Hart (1977) pointed out that, even where s i g n i f i c a n t c o r r e l a t i o n s occurred, i t was d i f f i c u l t to determine whether a given v a r i a b l e a f f e c t e d student a t t i t u d e and achievement or v i c e versa. This suggested that a model f o r i n v e s t i g a t i o n should provide f o r the i n t e r a c t i o n of some of these v a r i a b l e s . Based on the preceding review of the l i t e r a t u r e , the conceptual model shown i n Figure 3 i s presented f o r i n v e s t i g a t i o n . Student Background ± Student Perceptions of Mathematics Achievement Classroom Processes Teacher Perceptions of Mathematics Legend. One way r e l a t i o n s h i p s of i n t e r e s t Interactions of i n t e r e s t ^ > Relationships not under study <•* — a * Figure 3 . A model of inputs and outcomes of schooling. A basic assumption of the model i s that students' outcomes, as represented by achievement i n mathematics, are functions of students 1 backgrounds and perceptions, teachers' backgrounds and perceptions, and classroom processes. For the purposes of t h i s 28 study, students' and teachers' perceptions are l i m i t e d to what t h e i r a t t i t u d e s and opinions toward mathematics are. These fa c t o r s , however, involve complex sets of i n t e r a c t i o n s . For example, i n exploring a number of causal models f o r achievement, Parkerson, S c h i l l e r , Lomax & Walberg (1984) concluded that r e c i p r o c a l paths of causal influence should be taken into account i n attempting to obtain a better understanding of classroom learning. The model portrayed i n Figure 3 c l a s s i f i e s student and teacher background v a r i a b l e s i n a t h e o r e t i c a l s t ructure used to explain variance i n student achievement. As shown i n the diagram, the impacts of these p a r t i c u l a r f a c t o r s are expected to be i n one d i r e c t i o n . They are r e l a t i v e l y f i x e d and hence students' achievement i s not expected to have an e f f e c t on them. Other inputs, such as classroom processes, and students' and teachers' perceptions, are expected to i n t e r a c t with student achievement. Two of these r e l a t i o n s h i p s were examined to determine which had the greater e f f e c t : teachers' perceptions on students' perceptions or v i c e versa. Although classroom processes and teachers' perceptions may be i n t e r r e l a t e d , these c o r r e l a t i o n s were not examined. Neither were r e l a t i o n s h i p s between students' background and t h e i r perceptions or teachers 1 background and classroom processes. Hence, dotted l i n e s connecting these v a r i a b l e s are shown i n the model. 3.2 POPULATION AND SAMPLING PLAN 29 The 1985 Mathematics Assessment The population f o r the present study consisted of a l l students and teachers of Grade 7 i n B r i t i s h Columbia p u b l i c and funded independent schools as of May, 1985. Based on s t a t i s t i c s provided by the M i n i s t r y of Education, the population of students at that l e v e l as of February, 1985 was 35 890. A t o t a l of 33 888 Grade 7 students wrote t e s t booklets f o r a return rate of 94 percent. Since teachers completed at most one questionnaire, inclu d i n g cases where they taught more than one c l a s s of Grade 7 mathematics, fewer teachers than classes p a r t i c i p a t e d . A t o t a l of 1073 teacher questionnaires were returned. The 1987 V a l i d a t i o n study The sample f o r the 1987 v a l i d a t i o n study consisted of a l l students and teachers at the Grade 7 l e v e l i n Surrey School D i s t r i c t , a large suburban d i s t r i c t located i n a metropolitan area of about 1.5 m i l l i o n people. The May, 1987 administration involved 2146 students from 104 classrooms. Since the study was part of the school d i s t r i c t ' s plan to r e p l i c a t e the 1985 Mathematics Assessment during the 198 6-87 school year, a l l students and teachers of Grade 7 i n the d i s t r i c t were involved. Students were required to complete t e s t booklets, c o n s i s t i n g of achievement items, background questions and an a t t i t u d e scale i n September, 1986 and again i n May, 1987. During the September administration teachers were asked to complete an a t t i t u d e scale which formed part of the more comprehensive questionnaire from the 1985 Assessment. Teachers were asked to complete the e n t i r e questionnaire during the May administration. 3.3 OVERVIEW OP THE METHOD OP STUDY The independent v a r i a b l e s were selected from those v a r i a b l e s which previous research had shown to be r e l a t e d to academic achievement, and which were included i n the 1985 Assessment. The pool of independent v a r i a b l e s included factors d escribing students 1 home backgrounds, teachers• backgrounds and experience, classroom organization, techniques of i n s t r u c t i o n , and students' and teachers' perceptions of mathematics. The dependent v a r i a b l e s , on the other hand, were students' achievement i n problem solving, mathematical a p p l i c a t i o n s and t e s t t o t a l . In examining the r e l a t i o n s h i p s among independent and dependent v a r i a b l e s , the un i t s of analyses were teacher and c l a s s . Responses of those students whose teachers completed the Teacher Questionnaire were re-scored from the p r o v i n c i a l data tape and aggregated to the classroom l e v e l . Class means and variances were then link e d to responses of teachers and c o r r e l a t i o n s between independent and dependent v a r i a b l e s determined. Since a major goal of the current study was to determine a model to explain variance i n student outcomes as measures of achievement, regression equations were determined with each of the three dependent v a r i a b l e s f o r each of the input categories. 31 F i n a l models were then determined by regressing a l l v a r i a b l e s on each dependent v a r i a b l e . Pre-test data were not a v a i l a b l e from the p r o v i n c i a l study and hence there was no control at that stage of the a n a l y s i s f o r the e n t r y - l e v e l knowledge of students. Consequently regression weights a r r i v e d at through multiple regression techniques were estimates which may have contained bias or e r r o r a t t r i b u t e d to d i f f e r e n c e s i n i n i t i a l students* c h a r a c t e r i s t i c s . To determine the extent of bias which may have occurred, r e s u l t s from the 1986-7 V a l i d a t i o n Study were compared both with and without pre-t e s t data as a covariate. Further analysis of the 1986^87 data, using a cross-lagged panel c o r r e l a t i o n between teachers• perceptions and perceptions of students', provided an i n d i c a t i o n of the extent to which teachers' perceptions a f f e c t students' perceptions. 3.4 INSTRUMENTATION Booklets f o r students In the 1985 Assessment four booklets, e n t i t l e d Q, R, S, and T, were administered randomly to Grade 7s' on a one-per-student basis. The Q booklet, which contained open-ended t e s t items, was administered to a sample of four percent of the population whereas the others were evenly d i s t r i b u t e d to the balance, using a matrix-sampling design. Each of booklets R, S, and T consisted of 50 d i f f e r e n t multiple-choice achievement items, one of three a t t i t u d e scales, and a common set of background items. The achievement items measured the following seven domains: Number and O p e r a t i o n , Geometry , Measurement , A l g e b r a i c T o p i c s , P r o b l e m S o l v i n g , P r o b a b i l i t y and S t a t i s t i c s , and C a l c u l a t o r s and Computers . The f i r s t f i v e o f t h e s e domains measured c o n t e n t i n t h e B r i t i s h C o l u m b i a c u r r i c u l u m , whereas t h e l a t t e r two were n o n - c u r r i c u l a r . F o r t h e p u r p o s e s o f t h e p r e s e n t s t u d y , t h e l a t t e r two domains were d e l e t e d and two new domains e n t i t l e d T e s t T o t a l and A p p l i c a t i o n s were e s t a b l i s h e d . The T e s t T o t a l domain was c o m p r i s e d o f t h e sum o f a l l i t e m s c o n t a i n e d i n the f i v e c u r r i c u l a r domains , whereas t h e A p p l i c a t i o n s domain c o n s i s t e d o f r o u t i n e s t o r y prob lems drawn from t h e f i r s t f o u r . Ach ievement i t ems were d i s t r i b u t e d e v e n l y a c r o s s b o o k l e t s by c o n t e n t a r e a and an a t t empt was made t o a l l o c a t e e q u i v a l e n t r a n g e s o f d i f f i c u l t y . (A copy each o f b o o k l e t s R , S and T i s c o n t a i n e d i n A p p e n d i x A . ) S i n c e t h e p r e s e n t s t u d y examined r e l a t i o n s h i p s between i n p u t s o f s c h o o l i n g and ach ievement on t h e p r e s c r i b e d c u r r i c u l u m , t e s t i t ems from t h e two n o n - c u r r i c u l a r domains , p r o b a b i l i t y and s t a t i s t i c s , and c a l c u l a t o r s and c o m p u t e r s , were n o t i n c l u d e d i n t h e a n a l y s i s . Form Q r e s u l t s were a l s o d e l e t e d from t h e a n a l y s i s s i n c e t h e y i n v o l v e d d i f f e r e n t i t e m s . A l l ach ievement i t ems were o f m u l t i p l e c h o i c e format w i t h f i v e o p t i o n s , i n c l u d i n g "I d o n ' t know." S t u d e n t s answered on o p t i c a l s c a n s h e e t s . T a b l e 1 l i s t s t h e i t ems c o n t a i n e d i n e a c h domain i n c l u d e d i n t h e s t u d y . As r e p o r t e d e a r l i e r , Domain 7, e n t i t l e d A p p l i c a t i o n s , was c r e a t e d from i tems c o n t a i n e d i n t h e f i r s t f o u r c a t e g o r i e s . T h i s domain was n o t p a r t o f t h e o r i g i n a l t a b l e o f s p e c i f i c a t i o n s and therefore no attempt had been made to d i s t r i b u t e items contained i n i t evenly across forms. The a t t i t u d e scales contained i n the booklets were as follows: Mathematics i n School, Booklet R; Gender i n Mathematics, Booklet S; Calculators and Computers, Booklet T. Only r e s u l t s from the Mathematics i n School scale were examined i n the current study. In t h i s scale students were asked t h e i r perceptions of the importance, d i f f i c u l t y i n learning and enjoyment i n learning associated with ten major t o p i c s i n the mathematics curriculum. 34 Table 1 Domains and Item Assignments Domain Item Numbers (booklet and no.) 1. Number and Operation R: 1, 2, 3, 4, 8, 9, 14, 15, 16 37, 40, 41, 42, 43 S: 1, 2, 3, 4, 11, 12, 13, 19, 25, 26, 27, 48, 49, 50 T: 1, 2, 3, 4, 9, 10, 11, 15, 16, 17, 39, 47, 48, 49 2. Geometry R: 12, 13, 19, 20, 26, 27, 31, 32, 38, 39, 44, 45 S: 5, 6, 9, 10, 20, 21, 36, 37, 39, 40, 41, 42 T: 12, 13, 14, 18, 19, 27, 28, 29, 40, 41, 42, 43 3 . Measurement R: 21, 28, 48, 49, 50 S: 14, 15, 16, 22, 38 T: 35, 36, 37, 38, 44 4. Algebraic Topics R: 5, 29, 30, 33, 34, 35, 36, 46 S: 23, 24, 30, 31, 32, 33, 43, 44 T: 5, 6, 23, 24, 25, 26, 45, 46 5. Problem Solving R: 17, 18, 25 S: 34, 35, 47 T: 8, 20, 50 6. Test T o t a l R: a l l items l i s t e d above S: a l l items l i s t e d above T: a l l items l i s t e d above 7. Applications R: 14, 30 S: 2, 3, 4, 24, 39, 44 T: 10, 26, 38, 39 Test items from non-curricular domains are not included i n t h i s t a b l e . In generating the 1985 achievement items, a contract team from the U n i v e r s i t y of B r i t i s h Columbia developed some and selected others from instruments used i n t e s t i n g programs from several d i f f e r e n t j u r i s d i c t i o n s . Items were selected to r e f l e c t a pre-determined table of s p e c i f i c a t i o n s and on the basis of t h e i r psychometric properties. Sources f o r these items included the 1977 and 1981 P r o v i n c i a l Assessments of Mathematics, the National Assessment of Educational Progress i n the United States, the Second International Study of Mathematics, and a number of surveys conducted i n England, New Zealand, and i n other Canadian provinces. Items included i n the pool were chosen to measure a wide range of a b i l i t y and d i f f e r e n t l e v e l s of c o g n i t i v e behavior. For example, a l l questions i n the Problem-Solving domain consisted of items which measured students' a b i l i t y to apply p r i o r knowledge i n unfamiliar s i t u a t i o n s . To complete the table of s p e c i f i c a t i o n s a number of new items needed to be developed and some, selected from other j u r i s d i c t i o n s , required modification. These items were p i l o t e d with Grade 8 students i n the F a l l of 1984. R o b i t a i l l e and O'Shea (1985, p.14), reported that items had to meet the following c r i t e r i a before they were considered f o r addi t i o n to the pool. Standard item s t a t i s t i c s were computed f o r each option of each item. On the basis of these r e s u l t s , items which showed any of the following c h a r a c t e r i s t i c s were e i t h e r eliminated or modified p r i o r to being considered f o r possible use i n the Assessment: • more than 95% or fewer than 10% of the students c o r r e c t l y answered the question; • not a l l d i s t r a c t o r s a t t r a c t e d respondents; • the b i s e r i a l c o r r e l a t i o n between the correct answer and t o t a l t e s t score was l e s s than 0.20; 36 • the b i s e r i a l c o r r e l a t i o n between the corre c t answer and t o t a l t e s t score was le s s than the b i s e r i a l c o r r e l a t i o n between a d i s t r a c t o r and the t o t a l t e s t score. The t a b l e of s p e c i f i c a t i o n s and the items chosen to appear on f i n a l forms were reviewed f o r content v a l i d i t y by an Advisory Committee, composed of mathematics educators. A summary of the psychometric properties of the set of t e s t items contained on each form i s shown i n Table 2 below (Anderson, 1986). Table 2 Summary S t a t i s t i c s f or Grade 7 Test Booklets Mean Percent Form Correct Standard Dev. Cronbach's Alpha R 55.6 9.0 0.87 S 51.0 8.3 0.83 T 55.2 8.3 0.84 These data show that the forms were of s i m i l a r l e v e l s of d i f f i c u l t y with means ranging between 51.0 and 55.6 percent. Students' r e s u l t s were d i s t r i b u t e d i n a s i m i l a r way about each mean as r e f l e c t e d by standard deviations ranging between 8.3 and 9.0. Values of Cronbach's c o e f f i c i e n t alpha i n d i c a t e that each instrument was r e l i a b l e . Based on these r e s u l t s i t can be stated that the.forms were r e l a t i v e l y p a r a l l e l i n nature. Students 1 Background Items The student background items gave information i n the following areas: age, gender, f i r s t language, mathematics program, homework, parents' education, and f a m i l i a r i t y with metric u n i t s of measure. The majority of these items had been used i n previous p r o v i n c i a l assessments. Students' Perceptions of Mathematics Based on findings of Kyles and Sumner (1977) and Whitaker (1982), i t was assumed that students' a t t i t u d e s toward mathematics were composites of t h e i r a t t i t u d e s toward d i f f e r e n t t o p i c s i n the mathematics curriculum. As a r e s u l t the present study examined students' responses to each of several items contained i n Scale R, e n t i t l e d "Mathematics i n School." The "Mathematics i n School" scale consisted of ten items which measured students' perceptions of the importance, d i f f i c u l t y and enjoyment of the following t o p i c s : basic operations with f r a c t i o n s ; basic operations with decimals; working with percents; learning about estimation; memorizing basic f a c t s ; s o l v i n g equations; s o l v i n g word problems; learning about the metric system; working with perimeter and area; and doing geometry. Students responded on a f i v e - p o i n t Likert-type sc a l e . The scale was adapted from one developed f o r use i n the Second International Study of Mathematics. Teacher Quest ionnaire The Teacher Questionnaire was comprised of four major sections, which are l i s t e d as follows: Background Information, Mathematics i n School, Problem Solving, and Calculators and Computers. For the purpose of the current study, however, information examined was l i m i t e d to the f i r s t three sections l i s t e d . A copy of the questionnaire i s included i n Appendix B. 38 Items i n the background section included questions on teachers' preparation f o r teaching mathematics, p r o f e s s i o n a l development a c t i v i t i e s , frequency and length of mathematics periods, and a number of s p e c i f i c a c t i v i t i e s i n the classroom. Items i n the "Mathematics i n School" s e c t i o n d e a l t with the same top i c s as those i n the student booklet. In the problem-solving sec t i o n questions were included on teachers• a t t i t u d e s toward problem solving, s t r a t e g i e s taught to students and a c t i v i t i e s used to f a c i l i t a t e the learning of problem s o l v i n g . 3 .5 DESCRIPTION AND DEFINITION OF THE VARIABLES Dependent v a r i a b l e s were selected as i n d i c a t o r s of mathematical achievement i n three categories of i n t e r e s t : Problem Solving, Applications and Test T o t a l . The Problem-Solving v a r i a b l e measured student achievement on items designed to require higher order thinking s k i l l s whereas the A p p l i c a t i o n v a r i a b l e assessed student achievement on routine story problems. O v e r a l l achievement i n mathematics was measured by the Test To t a l v a r i a b l e . Categories of independent v a r i a b l e s i n the present study included the following: Student Background, Teacher Background, Students' Perceptions of Mathematics, Teachers' Perceptions of Mathematics, Classroom Organization and Problem-Solving Processes. A number of items were used to measure va r i a b l e s within each category. Since teachers' and students' perceptions of major t o p i c s i n the curriculum involved three d i f f e r e n t scales—importance, d i f f i c u l t y and enjoyment—the sum of each scale was used as a v a r i a b l e f o r further a n a l y s i s a f t e r a review of t h e i r psychometric properties. A l i s t i n g of each independent v a r i a b l e and i t s source, i s shown i n Table 3. Table 3 Independent V a r i a b l e s and T h e i r Sources Variable Source and Item Number A. Student Background Background Information Student Booklet 1. Language f i r s t spoken 2. Language spoken at home now 3. Time spent on mathematics homework 4. Father's l e v e l of education 5. Mother's l e v e l of education 1 2 7 9 10 B. Teacher Background 1. Years experience 2. Preference to teach mathematics 3. Proportion of teaching load 4. Attendance at conferences 5. Attendance at workshops 6. Mathematics courses completed 7. Mathematics education courses completed C. Student Perceptions of Mathematics Background Information Teacher Questionnaire 1 2 3 5 6 8 Scale R Student Booklet Perceptions of importance, d i f f i c u l t y and enjoyment of the following t o p i c s : 1. Adding, subtracting, multiplying, and d i v i d i n g f r a c t i o n s 2. Adding, subtracting, multiplying, and d i v i d i n g decimals 3. Working with percents 4. Learning about estimation 5. Memorizing basic f a c t s 6. Solving equations 7. Solving word problems 8. Learning about the metric system 9. Working with perimeter and area 10. Doing geometry 2 3 4 5 6 7 8 9 10 40 D. Teacher Perceptions of Scale R Mathematics Teacher Questionnaire Perceptions of importance, d i f f i c u l t y to teach and enjoyment i n teaching t o p i c s l i s t e d under "Student Perceptions of Mathematics" E. Classroom Organization Background Information Teacher Questionnaire 1. Type of course 2. Frequency of t e s t i n g 3. Number of classes per week 4. Length of period 5. Time on homework a c t i v i t i e s 6. Questioning 7. Seatwork 8. Working i n small groups 9. Working at sta t i o n s 10. Time on computational d r i l l 11. Giving l e c t u r e s t y l e i n s t r u c t i o n G. Problem-Solving Processes Scale S Teacher Questionnaire 1. Perception of student enjoyment 1 2. Expectation of performance 2 3 . S a t i s f a c t i o n to teach 5 4. Ease of teaching 7 5. Uses of d i f f e r e n t s t r a t e g i e s 8 6. Inservice involvement 9 7. Uses of materials resources 10 8. A c t i v i t i e s f o r motivation 11 9. Frequency of teaching 12 10. Problem types used 14 11. Organization of room 15 10 11 12 13 14 15 16 17 18 19 20 ' In order to examine common dimensions or constructs underlying items, a fa c t o r analysis of v a r i a b l e s within each of the categories shown i n Table 3 (with the exception of Student and Teacher Perceptions), was subsequently undertaken using data from the 1985 Assessment. A f t e r analysis they were combined into r e l a t e d c l u s t e r s . To d i s t i n g u i s h these new combinations of va r i a b l e s from the o r i g i n a l ones they are r e f e r r e d to as f a c t o r s , rather than v a r i a b l e s . Since the sums of each of the ten-item scales under Students' and Teachers' Perceptions produced only 3 v a r i a b l e s f o r each of these categories, a f a c t o r a n alysis was not conducted. However, f o r purposes of ease of de s c r i p t i o n , these sums are r e f e r r e d to as fa c t o r s when r e s u l t s are discussed. 3.6 DATA COLLECTION A d e s c r i p t i o n of the procedures used f o r the d i s t r i b u t i o n , administration and c o l l e c t i o n of t e s t booklets and teacher questionnaires i s presented i n t h i s section. Discussion of the c r i t e r i a used f o r the aggregation of data i s also included. 1985 Data C o l l e c t i o n In the 1985 P r o v i n c i a l Assessment of Mathematics, booklets f o r students were packaged by school, at a c e n t r a l l o c a t i o n , i n numbers which exceeded reported enrollments by 10 percent. Since a matrix sampling design was used, booklets were interleaved at the packaging stage. Administration i n s t r u c t i o n s to teachers d i r e c t e d them to d i s t r i b u t e the booklets to students i n the order they appeared i n each package. A l e t t e r to p r i n c i p a l s i n s t r u c t e d them to assign one cl a s s code number to each teacher. In cases where a teacher taught more than one c l a s s of Grade 7 mathematics, the code f o r that teacher was assigned to the f i r s t c l a s s met during the week or i n the time table c y c l e . In responding to the questionnaire, the teacher was inst r u c t e d to answer c l a s s - s p e c i f i c questions r e l a t i v e to the assigned c l a s s . Both students and teachers responded to questions on an o p t i c a l scan sheet. Teachers and t h e i r students used the same cl a s s code numbers to allow f o r linkages between t h e i r responses to be made. C a l c u l a t i o n s for Dependent V a r i a b l e s Means and variances f o r each achievement v a r i a b l e were aggregated to the c l a s s l e v e l . Since the assessment booklets were power t e s t s , these s t a t i s t i c s were c a l c u l a t e d using the number of items responded to by each student. The "I don't know" response was included i n the determination of these reported scores. Inclusion of that response i s consistent with past p r a c t i c e f o r p r o v i n c i a l assessments i n B r i t i s h Columbia and with the National Assessment of Educational Progress i n the United States. Since students responded to only one i n three items, c l a s s s t a t i s t i c s are reported as estimates. C a l c u l a t i o n s f o r Independent V a r i a b l e s Options f o r a l l non-achievement items contained i n the instruments were l a b e l l e d a l p h a b e t i c a l l y . In order to c a l c u l a t e values f o r responses, the options were re-coded numerically. The weightings assigned to each option are reported i n Appendix C. Due to the matrix-sampling design that was used, a l l students d i d not have an opportunity to respond to each item. As a r e s u l t , missing responses were not used i n the c a l c u l a t i o n s . A c l a s s mean was ca l c u l a t e d f o r each item by summing the weightings of options selected and d i v i d i n g by the number of respondents. For teachers, options f o r s i n g l e -response items were given the weighting assigned to i t . However, options f o r multiple-response items, included i n the Teacher Questionnaire, were assigned a "1" i f they were chosen or a "0" i f they were not. Teachers' scores f o r each multiple response item was a r r i v e d at by summing the " l " s . 1986-87 Data C o l l e c t i o n S i m i l a r procedures to those used i n the 1985 assessment were followed f o r the packaging and d i s t r i b u t i o n of materials used i n the 1986-87 V a l i d a t i o n Study. Since the booklets were administered during September and again the following May, arrangements were made to c o l l e c t a l l booklets and questionnaires a f t e r each s i t t i n g . To ensure that teachers and students used the same c l a s s codes f o r each session, they were assigned c e n t r a l l y . Two versions of the Teacher Questionnaire were used. In September i t consisted of only the "Mathematics i n School" scale. In May, teachers were asked to complete the f u l l questionnaire used i n the 1985 p r o v i n c i a l assessment. 3.7 DATA ANALYSIS PROCEDURES 1985 Assessment Data Class means and variances were computed f o r each v a r i a b l e r e l a t i n g to students' background, perceptions, and achievement by domain. Teacher v a r i a b l e s on background, perceptions and classroom processes were coded numerically as reported i n Appendix C. A l l c a l c u l a t i o n s were based on the number of respondents to each item. 44 C o r r e l a t i o n a l A n a l y s i s At the second stage of an a l y s i s , an index of the degree of as s o c i a t i o n among the independent and dependent v a r i a b l e s was determined. Pearson c o r r e l a t i o n c o e f f i c i e n t s were used as a measure of t h i s a s s o c i a t i o n . The r e s u l t i n g c o r r e l a t i o n matrices provided measures of the s t a t i s t i c a l i n t e r r e l a t i o n s h i p s among v a r i a b l e s . Factor A n a l y s i s At the next stage, f a c t o r analysis was used to i d e n t i f y whether patterns of r e l a t i o n s h i p s existed within groups of independent v a r i a b l e s . Results of t h i s procedure provided information on the common, underlying dimensions on which v a r i a b l e s were located. As a by-product of t h i s process the number of v a r i a b l e s to be investigated i n the following stage of analysis was reduced. The v a r i a b l e s under each of the category headings of Student Background, Teacher Background, Classroom Organization and Problem-Solving Processes were clustered, based on fac t o r loadings c a l c u l a t e d i n the f a c t o r a n a l y s i s . Resulting combinations of v a r i a b l e s were r e f e r r e d to as fact o r s and discussed according to composition. Since the number of va r i a b l e s under Teachers 1 and Students 1 Perceptions had already been reduced by c a l c u l a t i n g the t o t a l f o r each of the three subscales, they were not fac t o r analyzed. M u l t i p l e Regression Subsequently the set of factors within each category was regressed on each of the three dependent or c r i t e r i o n v a r i a b l e s . This r e s u l t e d i n production of a family of regression equations f o r each dependent v a r i a b l e . A step-wise regression analysis was conducted during t h i s stage to determine which f a c t o r s were the best pre d i c t o r s of success i n each equation. A l l fact o r s were then regressed again on each dependent v a r i a b l e , without reference to category, to a r r i v e at the f i n a l models. The f u n c t i o n a l r e l a t i o n s h i p between student achievement and the independent v a r i a b l e s could be described as follows: Student Achievement = f (SB, TB, SP, TP, CO,PS) where: SB = Student Background TB = Teacher Background SP = Student Perceptions TP = Teacher Perceptions CO = Classroom Organization PS = Problem-Solving Processes A general l i n e a r multiple regression model des c r i b i n g t h i s r e l a t i o n s h i p i s shown below: /3j — unstandardized regression c o e f f i c i e n t XT--j = value of the j t n independent • . t h v a r i a b l e on the 1 " t r i a l £ J = r e s i d u a l or error term where: J 4 observed score of the i t h c l a s s on Y 1, M classrooms 1,..., 6 independent v a r i a b l e s constant term 46 1987 V a l i d a t i o n Study Analysis of the data from the 1987 V a l i d a t i o n Study involved two approaches. The f i r s t employed the same method used on the p r o v i n c i a l data, with post-test c l a s s achievement means as dependent v a r i a b l e s . The second involved an analysis of covariance, which combined regression analysis with analysis of variance. I t c o n t r o l l e d f o r the variance i n achievement contributed by students 1 learning which took place p r i o r to t h e i r a r r i v a l i n these classes. In t h i s process, c l a s s pre-test means f o r achievement i n mathematics were treated as covariates i n the regression equations. Although c o r r e l a t i o n c o e f f i c i e n t s provided measures of the extent to which teachers' and students' perceptions were s i m i l a r , the question of which had a greater impact on the other remained unanswered i n the analysis of r e s u l t s from the 1985 P r o v i n c i a l Assessment. To address t h i s issue, the V a l i d a t i o n Study employed time as a t h i r d v a r i a b l e , using a cross-lagged panel c o r r e l a t i o n (Campbell & Stanley, 1963). To determine which v a r i a b l e had the greater e f f e c t on the other, c o r r e l a t i o n s between teachers 1 perceptions at Time 1 and students' perceptions at Time 2; and teachers • perceptions at Time 2 and students' perceptions at Time 1 were compared. I f one of the cross-lagged c o r r e l a t i o n s was s i g n i f i c a n t l y more p o s i t i v e than the other, t h i s would provide evidence of which v a r i a b l e had the greater e f f e c t on the other. 3 . 8 SUMMARY This chapter has described the population and sampling plan, the instruments and v a r i a b l e s , and data c o l l e c t i o n and analysis procedures. The next chapter presents the r e s u l t s of the a n a l y s i s and i n Chapter 5, the r e l a t i o n s h i p s between these r e s u l t s and the questions under study are discussed. 48 CHAPTER 4 FINDINGS Analyses of the data involved four d i s t i n c t phases: preliminary analyses, c o r r e l a t i o n a l analyses, f a c t o r analyses and multiple regression analyses. At each stage of the process, the data were prepared f o r the subsequent step. 4.1 PREPARATION OF THE DATA The preliminary analysis stage involved i n i t i a l preparation of the data f o r further study. The f i r s t step involved the re-coding of students' responses to background and a t t i t u d e items in t o numerical form. Following t h i s step, responses were aggregated to the c l a s s l e v e l where means and standard deviations were ca l c u l a t e d f o r each v a r i a b l e under i n v e s t i g a t i o n . At the next stage, teachers' responses to items from the Teacher Questionnaire were re-coded to numerical form. These data were then stored i n the form of one record f o r each teacher, containing numerical values assigned to each s i n g l e response item and means ca l c u l a t e d f o r those items which required multiple responses. A t o t a l of 1073 teacher records were produced i n t h i s manner. Class and teacher records were matched at the next stage of an a l y s i s . In some cases no match was found, whereas i n others c l a s s s i z e s were found to be unusually small or large. Since teachers answered t h e i r questionnaires with respect to a si n g l e c l a s s , no more than one match was expected i n cases where they taught Grade 7 mathematics to more than one c l a s s . Unusually small or large classes were deemed to be due to students e i t h e r missing c l a s s code numbers or providing i n c o r r e c t ones. This could have re s u l t e d from non-assignment of c l a s s codes by p r i n c i p a l s or by teachers, or by omission of the assigned code number by one or more students i n each c l a s s . The non-assignment of c l a s s codes or, i n cases where they were assigned, the f a i l u r e of teachers to ensure that a l l students recorded them on t h e i r answer sheets, may have been due to p o l i t i c a l reasons. For example, teachers i n some d i s t r i c t s were i n s t r u c t e d by t h e i r superintendents to omit some of the non-achievement items i n t h i s assessment. This p o s i t i o n was taken, i n part, because of a number of unpopular decisions taken by the p r o v i n c i a l M i n i s t r y of Education. In addition, t h i s was the f i r s t assessment i n B r i t i s h Columbia where teachers were asked to complete a questionnaire which could be l i n k e d to r e s u l t s from t h e i r c l a s s e s . Because of such considerations, c l a s s s i z e s l e s s than 13 and greater than 40 were dropped from the a n a l y s i s . The lower number was selected so that at l e a s t four students i n each c l a s s would have written each of the booklets R, S and T. Since booklet Q was administered randomly to 4 percent of the population, at l e a s t 13 students were required from each c l a s s . An upper bound of 40 was selected since few, i f any, classes greater than t h i s number were known to e x i s t . A d i s t r i b u t i o n of c l a s s s i z e ranges f o r matched records which remained i n the study i s shown i n Table 4. 50 Table 4 Frequency D i s t r i b u t i o n of Class Sizes Class Size Frequency Percent Cumulative Percent 13-18 145 20 20 19-24 255 35 55 25-30 292 40 95 31-40 37 5 100 A t o t a l of 729 classes remained i n the present study with an average c l a s s s i z e of 23. This represented a loss of 32 percent from the t o t a l number of teacher questionnaires which were received. As mentioned e a r l i e r , t h i s l o s s was l i k e l y due to several reasons. I t could have re s u l t e d from missing c l a s s code numbers on Teacher Questionnaires, non-assignment of code numbers to classes, or non-completion of code numbers on answer sheets by students. 4.2 DESCRIPTIVE ANALYSIS OF THE INDEPENDENT VARIABLES In t h i s section d i s t r i b u t i o n s of responses are presented and discussed. Results are presented i n each of the following major categories: student background, teacher background, classroom organization, problem-solving processes, teacher perceptions of mathematics and student perceptions of mathematics. 51 Student Background Variables A t o t a l of f i v e background v a r i a b l e s f o r students were examined: language f i r s t spoken, language c u r r e n t l y spoken at home, time spent on the l a s t mathematics homework assignment, l e v e l of fathers' education, and l e v e l of mothers' education. Students responded to two questions r e l a t e d to t h e i r mother tongue. Results are shown i n Table 5. Table 5 Mother Tongue (Percent) English Non-English Language f i r s t spoken 89 11 Current home language 94 6 Eighty-nine percent of the students i d e n t i f i e d E n g l i s h as t h e i r language f i r s t spoken- The proportion of students for whom English was the language c u r r e n t l y spoken at home was 94 percent. These r e s u l t s show that of those students who f i r s t spoke a language other than English, 45 percent d i d not curr e n t l y speak t h e i r mother tongue at home. For the purpose of aggregating r e s u l t s to the classroom l e v e l , student responses were assigned a "2" i f Engl i s h was selected and a "1" i f a non-English response was chosen. An index number was then determined f o r each c l a s s . Indices were equal to 1 + x, where x was equivalent to the percentage of students f o r whom English was the language spoken. The d i s t r i b u t i o n of index numbers f o r language f i r s t spoken of a l l 52 classes i n the analysis had a mean of 1.89 and a standard deviation of 0.13. For language c u r r e n t l y spoken, the mean was 1.94 and the standard deviation was 0.09. These r e s u l t s r e l a t e to the data shown i n Table 5, where 89 percent spoke Eng l i s h as t h e i r f i r s t language and 94 percent c u r r e n t l y speak English at home. Students were asked which one, of f i v e time i n t e r v a l s , most c l o s e l y approximated the amount of time they spent on t h e i r l a s t mathematics homework assignment. Table 6 reports the r e s u l t s . Table 6 Time Spent on Homework Amount of Time Percent of Respondents None 4 I- 10 minutes 29 I I - 30 minutes 52 31-60 minutes 12 More than 60 minutes 3 The data reported i n Table 6 show that the vast majority of students (85 percent) spent 30 or fewer minutes on t h e i r l a s t mathematics homework assignment. The d i s t r i b u t i o n of responses to t h i s item i s r e l a t i v e l y symmetric, with 52 percent of students spending 11^ -30 minutes on t h e i r l a s t homework assignment and s i m i l a r numbers spending e i t h e r no time or else more than one hour. 53 Weightings assigned to each option ranged from 1 f o r "none" to 5 f o r "more than 60 minutes." The d i s t r i b u t i o n of index numbers f o r classes had a mean of 2.82 and a standard deviation of 0.33. These r e s u l t s show that the average amount of time spent on the l a s t mathematics homework assignment by students i n classes was c l o s e s t to the 11-30 minute time i n t e r v a l . The items on parents' educational l e v e l s received low response rates. These r e s u l t s may i n d i c a t e that students d i d not know the educational l e v e l s of t h e i r parents. For example, 50 percent d i d not s e l e c t an educational l e v e l f o r t h e i r father or male guardian and 46 percent f a i l e d to s e l e c t one f o r t h e i r mother or female guardian. The large omission rate f o r these items could also have resulted, i n part, from the number of s i n g l e parent f a m i l i e s . Table 7 shows r e s u l t s of those students who sel e c t e d one of the educational l e v e l s f o r e i t h e r t h e i r parents or guardians. Table 7 Educational Level of Parents (Adjusted Percent) Level of Education Attended Percent of Respondents Mother Father L i t t l e or None 2 3 Elementary 4 6 Secondary 45 40 Post Secondary 49 51 54 The l a r g e s t proportion of responses by category was "post secondary" f o r both mother and father. For example, 49 percent of respondents indicated that t h e i r mother or female guardian had attended a college, u n i v e r s i t y or some other form of post-secondary t r a i n i n g . This compared to a response rate of 51 percent f o r t h e i r fathers or male guardians. Ninety-four percent of mothers attended school at e i t h e r a secondary school or higher l e v e l compared to 91 percent of fathers. Although these r e s u l t s suggest that parents were well educated, i t should be noted that the question r e f e r r e d to attendance rather than completion at each respective l e v e l . Since schooling i n B r i t i s h Columbia i s compulsory u n t i l age 16, i t i s expected that v i r t u a l l y a l l parents would have attended school at l e a s t to the secondary l e v e l . In the 1987 v a l i d a t i o n study, conducted i n the Surrey School D i s t r i c t , s i m i l a r r e s u l t s were found. Weightings f o r l e v e l s of education ranged from 1 f o r the lowest, to 4 f o r the highest. The d i s t r i b u t i o n of class-index numbers f o r mothers' l e v e l of education had a mean of 3.37 and a standard d e v i a t i o n of 0.36. On the other hand, fathers' l e v e l of education had a mean of 3.39 and a standard deviation of 0.29. Teacher Background Teacher background v a r i a b l e s consisted of seven questions r e l a t e d to experience, preference to teach mathematics, proportion of teaching load, p r o f e s s i o n a l development a c t i v i t i e s and p r o f e s s i o n a l t r a i n i n g . A discussion of r e s u l t s follows. Responses to the question on teaching experience indicated that 55 percent of teachers had taught 11 or more years, i n 5 5 contrast to only 6 percent who were i n e i t h e r t h e i r f i r s t or second year of teaching. The small proportion of teachers with fewer than two years of experience may have r e f l e c t e d cutbacks i n education because of r e s t r a i n t p o l i c i e s of the p r o v i n c i a l government. For example, c l a s s s i z e s increased i n B r i t i s h Columbia during recent years while student enrollments declined. The vast majority, 95 percent, of teachers i n d i c a t e d that, given a choice, they would not avoid teaching mathematics. Only 3 percent reported that they would avoid teaching the subject and 2 percent were undecided. S i x t y three percent of teachers indicated that mathematics took up to twenty percent of t h e i r teaching load. Only 8 percent reported that more than f o r t y percent of t h e i r time was spent teaching mathematics. These data suggest that few teachers s p e c i a l i z e i n teaching mathematics at the Grade 7 l e v e l . This r e s u l t could be due to a l i m i t e d p r a c t i c e of platooning f o r the teaching of mathematics and the small numbers of Grade 7 classes i n many elementary schools. Teachers responded to two questions r e l a t e d to pro f e s s i o n a l development a c t i v i t i e s . The f i r s t asked whether or not the teacher attended a mathematics session at a conference i n the previous three years. The second question dealt with attendance at a workshop (other than at a conference) or an i n - s e r v i c e day i n mathematics during the previous three years. Fifty-one percent had attended a conference and 59 percent had attended a mathematics workshop or i n - s e r v i c e day within that time period. Two questions dealt with professional t r a i n i n g . The f i r s t question asked how many post-secondary courses i n mathematics 56 had been s u c c e s s f u l l y completed, while the second asked f o r the number of s u c c e s s f u l l y completed courses i n mathematics education. Courses were defined as the equivalent of a 1.5 u n i t s e c t i o n at the U n i v e r s i t y of B r i t i s h Columbia. Twenty-two percent of the teachers of Grade 7 mathematics had completed no post-secondary courses i n mathematics and only 12 percent had completed s i x or more. The same proportion of teachers, 22 percent, had not completed any courses i n mathematics education. Since Grade 7 i s part of the elementary school program i n B r i t i s h Columbia, r e s u l t s f o r the question on mathematics courses completed were not s u r p r i s i n g . I t i s common p r a c t i c e at the elementary l e v e l f o r teachers to teach several subjects, and therefore s p e c i a l i s t s with mathematics majors are more l i k e l y to teach at the secondary l e v e l . I t was s u r p r i s i n g and disappointing, however, to f i n d that 22 percent of the teachers had completed no mathematics methods courses. Classroom Organization Teachers were asked a t o t a l of eleven questions which were clustered under the category of Classroom Organization. The teachers were asked to respond with reference to a s i n g l e c l a s s and, with some questions, the most recent one. Questions r e l a t e d to the type of course, frequency of t e s t i n g , time a l l o c a t e d to mathematics, and the proportion of the most recent mathematics period spent on several d i f f e r e n t classroom a c t i v i t i e s . The question which dealt with course type asked teachers to i n d i c a t e whether the program they offered was modified (for slower students), regular, or enriched. Ninety-three percent 57 indicated that t h e i r reference c l a s s was on a regular program. Each of the other two categories received 4 percent of the responses. The high response f o r the regular program suggests that l i t t l e or no streaming i n mathematics at the c l a s s l e v e l occurs at Grade 7 i n elementary schools i n B r i t i s h Columbia. Ten percent of the teachers gave t e s t s or quizzes i n mathematics almost every day. Forty-seven percent gave them once a week and 42 percent once every couple of weeks. Only 1 percent i n d i c a t e d that they gave t e s t s or quizzes e i t h e r only once every reporting period or not at a l l . Seventy-one percent of the teachers o f f e r f i v e mathematics classes each week. Eleven percent o f f e r fewer than f i v e classes per week and 18 percent o f f e r more. Ninety-seven percent of the teachers indicated that the length of periods were between 31 and 60 minutes. In order to determine the amount of time spent i n mathematics each week, a new v a r i a b l e was created by c a l c u l a t i n g products between the numbers and lengths of periods each week. The r e s u l t s indicated that f o r 89 percent of the classes which remained i n the study, the amount of time spent i n mathematics ranged between 190 and 265 minutes per week. The average weekly time spent on mathematics was approximately 223 minutes. One item asked teachers how many students they had c a l l e d on to answer questions during the most recent period. Twenty-one percent indicated that they had c a l l e d upon l e s s than one-quarter of the c l a s s and 45 percent c a l l e d upon more than h a l f . 58 Six questions d e a l t with proportions of the most recent period spent on a number of d i f f e r e n t a c t i v i t i e s . Results are reported i n Table 8. Table 8 Time Spent on Classroom A c t i v i t i e s (Percent) Percent of Class Time A c t i v i t y None 1-25 26-50 51-100 Homework 8 83 7 2 Seatwork 2 30 43 25 Small Groups 59 35 4 2 Work Stations 93 7 - -Computational D r i l l 53 44 2 1 Explaining Topics 10 63 24 3 In 92 percent of the classes at l e a s t some time was spent on homework-related a c t i v i t i e s . This contrasted markedly with the amount of time students spent at work st a t i o n s or a c t i v i t y centers, where i n 93 percent of the classes no time was a l l o c a t e d f o r t h i s . L i t t l e time also was spent on computational d r i l l . For example, a c l o s e r look at the 44 percent who spent between 1 and 25 percent of the period on d r i l l , showed that the majority i n the category (37 percent of a l l classes) spent l e s s than one-tenth of the period on t h i s a c t i v i t y . In 90 percent of the classes teachers spent part of the period explaining new t o p i c s to the e n t i r e c l a s s . This r e s u l t , when combined with 59 others showing l i t t l e or no time a l l o c a t e d f o r work i n small groups or at a c t i v i t y centers, indicates that most teachers appear to organize t h e i r teaching of mathematics i n a l e c t u r e -s t y l e manner. Problem-Solving Processes Teachers responded to eleven questions i n the Problem-Solving Processes category, f i v e of which required multiple responses. The questions dealt with a t t i t u d e s toward problem solving, i n - s e r v i c e a c t i v i t i e s , frequency of teaching problem s o l v i n g and v a r i e t i e s of approaches and resources used by teachers. Teacher a t t i t u d e s toward problem s o l v i n g involved responses to four questions r e l a t e d to t h e i r perceptions of student enjoyment of and achievement i n problem s o l v i n g ; t h e i r s a t i s f a c t i o n i n teaching the t o p i c ; and how easy they found i t to teach. Only 27 percent thought that most of t h e i r students enjoyed problem s o l v i n g and 28 percent expected that most of t h e i r students would perform well on that t o p i c . Only 45 percent of teachers were s a t i s f i e d with t h e i r teaching of problem s o l v i n g and an even lower proportion, 19 percent, found i t easy to teach. Based on r e s u l t s f o r the l a t t e r two questions, there appears to be a need f o r more i n - s e r v i c e on t h i s t o p i c . Responses to the question on i n - s e r v i c e involvement lends support to the suggestion that there i s need f o r more opportunities i n t h i s area. For example, 70 percent of teachers indicated they had not attended any workshops on problem sol v i n g i n the past year and only 9 percent had attended more than one. Thirty-nine percent of teachers answered that they taught problem s o l v i n g every day, as a regular part of the mathematics c l a s s . In contrast, 25 percent indicated that they taught i t only as a u n i t from time to time. Five items from t h i s category involved m u l t i p l e responses. Teachers were asked to s e l e c t from the options l i s t e d , those to which they subscribed. The questions dealt with problem-solving s t r a t e g i e s taught, sources of exercises used, d i f f e r e n t a c t i v i t i e s used to motivate students, problem types assigned and features used i n the classroom. There were f i v e d i f f e r e n t problem-solving s t r a t e g i e s from which to choose. Results are shown i n Table 9. Table 9 Problem-Solving Strategies Taught Strategy Percent of Teachers Look f o r a pattern 84 Guess and check 42 Make a l i s t 70 Make a simpler problem 72 Work backwards 46 The r e s u l t s i n Table 9 show that the most popular problem-so l v i n g strategy taught was "look f o r a pattern." Eighty-four percent of teachers reported they taught t h i s strategy compared to only 42 percent who taught "guess and check." Another question asked teachers to i d e n t i f y the sources they used f o r problem-solving exercises. Results are shown i n Table 10. Table 10 Sources of Problem-Solving Exercises Source Percent of Teachers Textbook 93 Mathematics contests 37 Problem-solving booklets 67 Professional journals 19 Book of puzzles 59 Results i n d i c a t e that the textbook i s by f a r the most popular source f o r problem-solving exercises. Ninety-three percent of the teachers indicated they used i t compared to only 19 percent who used p r o f e s s i o n a l journals as a source f o r exercises. This low response rate to p r o f e s s i o n a l journals i s unfortunate given The Arithmetic Teacher, the elementary journal of the National Council of Teachers of Mathematics (NCTM), has a problem-solving section i n each issue. The r e s u l t , however, r e f l e c t s the low rate of membership i n the NCTM among teachers of Grade 7. Results showing percentages of teachers who used d i f f e r e n t types of a c t i v i t i e s to motivate students f o r problem solving, are shown i n Table 11. 62 Table 11 A c t i v i t i e s Used to Motivate Students Type of A c t i v i t y Percent of Teachers Competitive games 41 Problem of the day 41 Puzzles or brain teasers 81 Li b r a r y f i l e of problems 31 Contests 32 "Puzzles or brain teasers" was the most popular s e l e c t i o n . Eighty-one percent of the teachers indicated that they used t h i s a c t i v i t y . Other s e l e c t i o n s were markedly l e s s popular. For example, the next most popular s e l e c t i o n s were "competitive games" and "problem of the day," each of which were selected by 41 percent of the teachers. The l e a s t popular were "contests" and " l i b r a r y f i l e of problems" selected by only 32 and 31 percent, r e s p e c t i v e l y . Teachers were also asked to in d i c a t e c e r t a i n types of problems they assigned to students. Results are reported i n Table 12. 63 Table 12 Problem Types Assigned to students Type of Problem Percent of Teachers More than one answer 35 Information to be c o l l e c t e d 59 Can be solved i n several ways 89 Can be solved c o l l e c t i v e l y 44 Too much or l i t t l e information 64 Based on the data contained i n Table 12, the most popular type of problem assigned was one which could be solved i n more than one way. Problems which could be solved c o l l e c t i v e l y and those with more than one answer were the l e a s t popular with 44 and 35 percent of teachers, r e s p e c t i v e l y , i n d i c a t i n g that they assigned these types. In order to determine what classroom features were used to promote problem solving, teachers selected from among f i v e d i f f e r e n t options. Results are shown i n Table 13. 64 Table 13 Classroom Features to Promote Problem S o l v i n g Type of Feature Percent of Teachers Problem-solving centre B u l l e t i n board displayed Problem of the week Contests with the c l a s s Students make up problems 14 15 36 52 52 Few teachers indicated that they used these classroom features to promote problem so l v i n g . For example, only 14 and 15 percent of teachers r e s p e c t i v e l y , used e i t h e r a problem-s o l v i n g centre or a b u l l e t i n board d i s p l a y f o r problems. The numbers of d i f f e r e n t approaches, sources of material and classroom a c t i v i t i e s used i n the teaching of problem sol v i n g are reported i n Table 14. I t shows the percentages of teachers who use up to a maximum of f i v e d i f f e r e n t sources or a c t i v i t i e s . Table 14 Number of Problem-Solving A c t i v i t i e s and Sources Used (Percent) Number of Responses Selected A c t i v i t y 1 2 3 4 5 Strategies Taught 16 16 30 16 22 Sources of Exercises 17 23 35 18 7 Vari e t y of A c t i v i t i e s 26 37 25 9 3 Problem Types 14 25 31 18 12 Classroom Features 52 35 10 2 1 65 Table 14 l i s t s the percentages of teachers who selected between one and f i v e of the options f o r each of the m u l t i p l e -response items. For example, i t shows that 68 percent of teachers i n s t r u c t students i n three or more d i f f e r e n t problem-s o l v i n g s t r a t e g i e s . Twenty-two percent indicated they taught a l l f i v e s t r a t e g i e s , compared to 16 percent who taught only one. S i x t y percent of respondents indicated that they used three or more d i f f e r e n t sources f o r problem-solving exercises. Only 7 percent, however, used a l l f i v e sources. The most frequent number of d i f f e r e n t sources used was three, chosen by 35 percent of the respondents. Most teachers, 63 percent, indicated they used e i t h e r one or two a c t i v i t i e s to motivate students. Only 9 percent used four a c t i v i t i e s and an even smaller proportion, 3 percent, used a l l f i v e . Based on the r e s u l t s , three or more d i f f e r e n t problem types were assigned by 61 percent of the teachers. Fourteen percent assigned only one type and 12 percent assigned a l l f i v e . In responding to the question on the number of d i f f e r e n t features used i n the classroom to promote problem so l v i n g , only 13 percent indicated that they had three or more of the features l i s t e d . Most, 52 percent, used only one feature whereas only 1 percent employed a l l f i v e . Teachers 1 Perceptions of Mathematics In Scale R, teachers rated each of ten major t o p i c s i n the curriculum according to t h e i r perceptions of i t s importance, 66 easiness to teach, and enjoyment to teach. Teachers were asked to respond on a f i v e - p o i n t L i k e r t scale, ranging from negative to p o s i t i v e . Results, showing the proportions of teachers who selected the two p o s i t i v e options f o r each item are reported i n Table 15. Table 15 Teachers' Perceptions of Mathematical Topics (Percent*) Topic Importance Easy to Enjoyable Teach to Teach Fractions 81 51 85 Decimals 99 76 94 Percents 97 48 92 Estimation 86 41 67 Basic Facts 90 58 52 Equations 94 34 85 Word Problems 96 15 75 Metric System 93 43 68 Perimeter & Area 85 64 86 Geometry 71 54 83 * Percent of teachers s e l e c t i n g the two p o s i t i v e options. Operations with decimals was rated by teachers as the most important, with 99 percent r a t i n g i t as important or very important. Other t o p i c s rated as important by more than 90 percent of teachers were percents, equations, word problems, and the metric system. The lowest ratings f o r importance were given to f r a c t i o n s and geometry. The low r a t i n g f o r f r a c t i o n s , 67 r e l a t i v e to others, could be p a r t l y because of Canada's adoption of the metric system, i n which greater use i s given to decimal rather than common f r a c t i o n s . Although Geometry received a r e l a t i v e l y low importance r a t i n g , the P r o v i n c i a l Mathematics Revision Committee has given i t greater prominence i n the recent r e v i s i o n of the curriculum. Teachers found decimals the e a s i e s t t o p i c to teach whereas they reported that word problems was the most d i f f i c u l t . For example, only 15 percent rated word problems as easy to teach. This contrasted sharply with i t s importance r a t i n g of 96 percent. A large portion of teachers d i d not report that any of the t o p i c s , with the exceptions of decimals and perimeter and area, were easy to teach. Teachers enjoyed teaching decimals and percents the most. They l e a s t enjoyed teaching the memorization of b a s i c f a c t s , but s t i l l 52 percent rated i t as enjoyable to teach. O v e r a l l , teachers perceived that most t o p i c s were important. They found several t o p i c s d i f f i c u l t to teach but enjoyed teaching almost a l l of them. In order to gain a measure of the r e l i a b i l i t y of each of the three scales, a r e l i a b i l i t y a nalysis was conducted. Results are shown i n Table 16. 68 Table 16 R e l i a b i l i t y Analyses of Teachers' Percept ion Scales Inter-Item Cronbach 1s Scale C o r r e l a t i o n Mean Alpha Importance D i f f i c u l t y to Teach Enjoyment to Teach 0.17 0.31 0.25 0.68 0.82 0.77 Inter-item c o r r e l a t i o n means provided an index of the average degree of ass o c i a t i o n among the items i n each scale. Cronbach's alpha, on the other hand, provided an estimate of each scale's i n t e r n a l consistency based on the item c o r r e l a t i o n s . The D i f f i c u l t y to Teach scale, with an inter-item c o r r e l a t i o n mean of 0.31 and a Cronbach's C o e f f i c i e n t Alpha of 0.82 was the most r e l i a b l e of the three scales. Considering that each scale consisted of only 10 items, t h e i r r e l i a b i l i t y c o e f f i c i e n t s of 0.68, 0.82 and 0.77 were r e l a t i v e l y high. On the basis of these data, i t was decided to sum the scores f o r each of the three scales. As a r e s u l t of the summing process, teachers received a score f o r each of the three scales which were used as index numbers at the next stage of a n a l y s i s . C h a r a c t e r i s t i c s of the d i s t r i b u t i o n s of these indices are reported i n Table 17. 69 Table 17 D i s t r i b u t i o n of Index Numbers f o r Teachers' Perceptions of Mathematics Weighting Standard Var i a b l e Range Mean Deviation Importance 10-50 42.53 4.17 D i f f i c u l t y to Teach 10-50 31.36 6.19 Enjoyment to Teach 10-50 38.79 4.89 These r e s u l t s show that, on average, teachers' ratings of importance were higher than t h e i r other two ra t i n g s . D i f f i c u l t y to teach r a t i n g s , which were the lowest, with a mean of 31.36, were also the most diverse, with a standard d e v i a t i o n of 6.19. Students• Percept ions of Mathematics Students responded to the same to p i c s i n scale R as d i d t h e i r teachers. From t h e i r perspectives they rated each t o p i c i n terms of i t s importance, d i f f i c u l t y to learn, and enjoyment to l e a r n . Table 18 reports the proportions of students who selected the two p o s i t i v e options f o r each item. 70 Table 18 s tudents ' Perceptions of Mathematical Topics (Percent*) Topic Importance Easy to Enjoyable Learn to Learn Fractions 87 64 37 Decimals 66 44 86 Percents 58 83 41 Estimation 88 57 42 Basic Facts 70 41 72 Equations 63 84 51 Word Problems 86 67 46 Metric System 55 59 69 Perimeter & Area 57 60 51 Geometry 53 43 52 * Percent of students s e l e c t i n g the two p o s i t i v e options. Students rated estimation, f r a c t i o n s , and word problems as the most important t o p i c s . Geometry, with 53 percent of the students s e l e c t i n g the two p o s i t i v e options, was ranked as the l e a s t important t o p i c . O v e r a l l , the importance r a t i n g s of students were lower than those of t h e i r teachers. The greatest d i f f e r e n c e i n ratings between students and teachers were f o r decimals, percents, equations and the metric system. Importance ratings of students f o r each of these t o p i c s , were at l e a s t 30 percentage points below those of t h e i r teachers. Equations and percents, with ratings of 84 and 83 res p e c t i v e l y , were reported as eas i e s t to learn by students. Memorizing basic f a c t s , geometry and decimals, on the other hand, were found hardest. Students reported that s i x of the ten to p i c s were easier to learn than teachers had reported they were to teach. For example, dif f e r e n c e s f o r percents, equations, and word problems of 35, 50 and 52 percentage points r e s p e c t i v e l y were found when students' ratings were compared with those of t h e i r teachers. In contrast, only 44 percent of students rated decimals as easy to learn whereas 76 percent of teachers rated i t easy to teach. Operations with decimals was rated by 86 percent of students as a t o p i c they found enjoyable to lear n . This contrasted with a r a t i n g of only 37 percent f o r operations with f r a c t i o n s . Students found eight of the to p i c s l e s s enjoyable to learn than t h e i r teachers found enjoyable to teach. For example, rat i n g s f o r f r a c t i o n s , percents, equations, perimeter and area, and geometry had differences greater than 30 percentage points. There was a p o s i t i v e d i f f e r e n c e , however, i n rati n g s between students and teachers on memorizing bas i c f a c t s . For example, 72 percent of students found t h i s t o p i c enjoyable compared,to only 52 percent of t h e i r teachers. The r e l i a b i l i t y c o e f f i c i e n t s f o r each of the student-perception scales are reported i n Table 19. Both inter-item c o r r e l a t i o n means and Cronbach's alphas are l i s t e d . 72 Table 19 R e l i a b i l i t y Analyses of Student Percept ion Scales Inter-Item Cronbach's Scale C o r r e l a t i o n Mean Alpha Importance 0.22 0.74 D i f f i c u l t y to Learn 0.19 0.70 Enjoyment to Learn 0.25 0.77 The most r e l i a b l e of the scales was "Enjoyment to Learn", with a r e l i a b i l i t y c o e f f i c i e n t of 0.77 and an inter-item c o r r e l a t i o n mean of 0.25. A l l three scales, however, were retained f o r further analysis based on t h e i r r e l i a b i l i t y c o e f f i c i e n t s . A sum was ca l c u l a t e d f o r each of the scales and c l a s s means of the sums were used i n further a n a l y s i s . D i s t r i b u t i o n s of index numbers, comprised of c l a s s means, for the perception scales are reported i n Table 20. The grand mean and standard deviation are reported f o r each of the scales. Table 20 D i s t r i b u t i o n of Index Numbers f o r Students' Perceptions of Mathematics Weighting Standard Variable Range Mean Deviation Importance D i f f i c u l t y to Learn Enjoyment to Learn 10-50 10-50 10-50 37.02 35.59 34.13 2.17 2.47 2.67 Based on these data, students 1 ratings of importance were higher than t h e i r ratings of d i f f i c u l t y or enjoyment. The grand mean f o r t h i s r a t i n g , however, was 5.51 percentage points lower than that of t h e i r teachers. In examining the r e l a t i o n s h i p s between student and teacher perceptions of mathematics, only the importance r a t i n g was examined. Ratings of d i f f i c u l t y and enjoyment were not compared because they were from two p e r s p e c t i v e s — t h e lear n i n g of mathematics and the teaching of mathematics. A c o r r e l a t i o n of 0.15 was found between teacher and student perceptions of the importance of mathematics i n the 1985 study. I t was low, but s i g n i f i c a n t at the 0.05 l e v e l . In an attempt to determine the d i r e c t i o n of t h i s r e l a t i o n s h i p , a cross-lagged panel c o r r e l a t i o n (Campbell and Stanley, 1963) was conducted, using data from the 1987 v a l i d a t i o n study. Time was introduced as a t h i r d v a r i a b l e , i n which c o r r e l a t i o n s between the perceptions of teachers at Time 1 and students at Time 2 were compared with c o r r e l a t i o n s of perceptions of teachers at Time 2 and students at Time 1. The c o r r e l a t i o n between the f i r s t two v a r i a b l e s was 0.16 whereas between the l a t t e r two i t was 0.10. This r e s u l t suggests that teachers' perceptions of the importance of mathematics may have a greater e f f e c t on those of t h e i r students than v i c e versa. 4.3 CORRELATIONAL ANALYSIS The second phase i n the examination of the data involved c o r r e l a t i o n a l analyses to t e s t f o r concomitant v a r i a t i o n between s p e c i f i c v a r i a b l e s . These analyses examined r e l a t i o n s h i p s i d e n t i f i e d i n Chapter 1 and provided input s t a t i s t i c s f o r f a c t o r a n a l y s i s . In order to achieve normality of e r r o r e f f e c t s and to obtain a d d i t i v i t y of e f f e c t s , square-root transforms were applied to the d i s t r i b u t i o n s f o r each v a r i a b l e (Kirk, 1982, p.82). Pearson product-moment c o r r e l a t i o n s were then c a l c u l a t e d among a l l p a i r s of v a r i a b l e s of i n t e r e s t . Results are presented by categories of v a r i a b l e s . Student Background and Achievement Correlations between c l a s s indices f o r student background v a r i a b l e s and achievement on the three c r i t e r i o n domains are shown i n Table 21. S t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s are denoted with an a s t e r i s k . 75 Table 21 Correlations Among Student Background Variables and Achievement Varia b l e SB1 SB2 SB3 SB4 SB5 A l A2 A3 SB1 Lang. 1st Spoken 100 81* -4 14* 20* 2 -1 2 SB2 Lang, at Home 100 -4 12* 15* 5 3 6 SB3 Time on Homework 100 2 -1 5 5 4 SB4 Fathers 1 Education 100 66* 20* 27* 29* SB5 Mothers* Education 100 20* 21* 25* A l Problem Solving 100 72* 61* A2 Test T o t a l 100 83* A3 Applications 100 Note. The c o r r e l a t i o n c o e f f i c i e n t s reported above are based on the computed c o r r e l a t i o n s rounded to two decimal places and m u l t i p l i e d by 100. *p<0.05. F i f t e e n s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s were found among the v a r i a b l e s . Some of the more i n t e r e s t i n g ones are between language f i r s t spoken and language spoken at home; mothers' educational l e v e l and fathers' educational l e v e l ; and parents' l e v e l of education and student achievement on problem solving, t e s t t o t a l and ap p l i c a t i o n s . Among these, r e l a t i o n s h i p s between language spoken f i r s t and currentl y spoken, and between parents' educational l e v e l s were p a r t i c u l a r l y strong with c o r r e l a t i o n s of 0.81 and 0.66 res p e c t i v e l y . Non-significant c o r r e l a t i o n s between language spoken and achievement may be due, i n part, to l i t t l e variance among cla s s i n d i c e s f o r language spoken. For example, 85 percent of the cl a s s indices on language f i r s t spoken and 95 percent on language spoken at home were 1.8 or greater out of a maximum of 2. These r e s u l t s r e f l e c t the high proportion of students with English as a f i r s t language who remained i n the sample. Correlations between subtests were high. They ranged from 0.61, between problem sol v i n g and ap p l i c a t i o n s , to 0.83, between t e s t t o t a l and ap p l i c a t i o n s . A higher c o r r e l a t i o n between the l a t t e r two subtests was expected since a l l of the items contained i n ap p l i c a t i o n s were also contained i n the t e s t t o t a l . However, no t e s t items were common to problem s o l v i n g and ap p l i c a t i o n s . Teacher Background and Student Achievement Table 22 l i s t s the c o r r e l a t i o n s among teacher background v a r i a b l e s and between these v a r i a b l e s and student achievement i n problem solving, t e s t t o t a l and a p p l i c a t i o n s . An a s t e r i s k i s used to denote s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s . 77 Table 22 C o r r e l a t i o n s Among Teacher Background Variables and Student Achievement Variable TBI TB2 TB3 TB4 TB5 TB6 TB7 A l A2 A3 TBI Experience 100 27* 4 4 7* 10* 7* 0 5 1 TB2 Preference 100 5 9* 0 3 -13* -2 0 -5 TB3 Proportion 100 11* 9* 16* 3 -2 -7* -7* TB4 Conferences 100 32* 19* 22* 4 4 8* TB5 Workshops 100 16* 20* 6 12* 13* TB6 Math Courses 100 41* 3 5 7* TB7 Math Ed. Courses 100 2 g* 13* A l Problem Solving 100 72* 61* A2 Test T o t a l 100 83* A3 Applications 100 Note. The c o r r e l a t i o n c o e f f i c i e n t s reported above are based on the computed c o r r e l a t i o n s rounded to two decimal places and m u l t i p l i e d by 100. *p<0.05. F i f t e e n s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s out of a t o t a l of 21 were found between pair-wise r e l a t i o n s h i p s among the independent v a r i a b l e s . C o e f f i c i e n t s of 0.20 or greater were found between experience and preference to teach mathematics, attendance at conferences and workshops, mathematics courses completed and mathematics education courses completed, and attendance at workshops and mathematics education courses completed. A negative and s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n i s shown between preference to teach mathematics and the number of 78 mathematics education courses taken. This r e s u l t may be due, i n part, to responses from teachers who p r e f e r to teach mathematics but who may have attended a post secondary i n s t i t u t i o n where mathematics education courses were not required or e l s e not r e a d i l y a v a i l a b l e . I t may also r e f l e c t r e s u l t s from teachers who were required to take a mathematics methods course but prefer not to teach mathematics. Eight out of a t o t a l of 21 c o r r e l a t i o n s between teacher background v a r i a b l e s and student achievement were found to be s t a t i s t i c a l l y s i g n i f i c a n t . Although the magnitudes of a l l these c o r r e l a t i o n s were low, the strongest ones were between attendance at workshops and student achievement on t e s t t o t a l and a p p l i c a t i o n s , and between the number of mathematics education courses completed and achievement on the same two domains. Classroom Organization and Achievement Relationships among classroom organization v a r i a b l e s and between these v a r i a b l e s and student achievement are shown i n Table 23. A number of s t a t i s t i c a l l y s i g n i f i c a n t , but negative c o r r e l a t i o n s , are included i n the r e s u l t s . 79 Table 23 C o r r e l a t i o n s Among Classroom Organiza t ion V a r i a b l e s and Achievement Var. C l C2 C3 C4 C5 C6 C7 C8 C9 CIO C l l A l A2 A3 Cl 100 -6 4 2 - 7 -15* -11* -3 -2 1 -2 15* 19* 21* C2 100 2 0 -2 9* 5 1 2 16* -4 -1 4 4 C3 100 -40* 5 -7 -6 0 1 1 2 8* 10* 8* C4 100 3 13* 9* 0 -4 4 -3 -4 1 2 C5 100 17* -11* -4 1 4 0 5 2 3 C6 100 -2 2 7* 13* 10* 0 2 -2 C7 100 -1 -8* 3 -26* 1 1 -2 C8 100 22* -6 -11* -2 0 0 C9 100 12* -5 0 2 -1 CIO 100 -6 -3 -3 -4 C l l 100 -3 -6 0 A l Problem Solving 100 72 61 A2 Test T o t a l 100 83 A3 A p p l i c a t i o n 100 Note. 1. The c o r r e l a t i o n c o e f f i c i e n t s reported above are based on the computed c o r r e l a t i o n s rounded to two decimal places and m u l t i p l i e d by 100. 2. Independent v a r i a b l e s were as follows: C l = course type; C2 = frequency of t e s t i n g ; C3 = number of clas s e s ; C4 = length of classe s ; C5 = time on homework a c t i v i t i e s ; C6 = number of students questioned; C7 = time on seatwork; C8 = time i n small groups; C9 = time at a c t i v i t y centers, C10 = time on computational d r i l l ; C l l = time introducing new to p i c s . *p<0.05. 80 Negative and s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s of -0.15 and -0.11 were found between type of course and both number of students questioned and time spent on seatwork. Although these r e l a t i o n s h i p s are not strong, they provide some evidence to suggest that teachers may emphasize these a c t i v i t i e s more with modified classes than with enriched ones. The r e l a t i o n s h i p between time spent on i n d i v i d u a l seatwork and time i n introducing new topics also showed a negative c o r r e l a t i o n (-0.26). This r e s u l t may i n d i c a t e that teachers tend to present new material using a l e c t u r e - s t y l e rather than a discovery approach. P o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s were found between 10 p a i r s of classroom organization v a r i a b l e s . For example, the number of students questioned i n the most recent period c o r r e l a t e d s i g n i f i c a n t l y with the following v a r i a b l e s : frequency of t e s t i n g , length of period, time spent on homework-re l a t e d a c t i v i t i e s , working at a c t i v i t y centers, computational d r i l l , and introducing new t o p i c s . Since each of the preceding a c t i v i t i e s l i k e l y involve question-and-answer exchanges, s i g n i f i c a n t c o r r e l a t i o n s with higher magnitudes were expected. The only classroom organization v a r i a b l e s which showed s t a t i s t i c a l l y s i g n i f i c a n t and p o s i t i v e c o r r e l a t i o n s with student achievement were type of course and number of classes per week. Type of course correlated s i g n i f i c a n t l y with problem solving (0.15), t e s t t o t a l (0.19) and a p p l i c a t i o n s (0.21). The number of classes per week also showed p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t , but lower, c o r r e l a t i o n s with the three achievement domains. 81 Problem-Solving Processes and Student Achievement Table 24 l i s t s c o r r e l a t i o n s among problem-solving process v a r i a b l e s , r e l a t e d to teacher behaviors and classroom a c t i v i t i e s . Relationships between these v a r i a b l e s and student achievement are also reported. 82 Table 24 C o r r e l a t i o n s Among Problem-Solving Process V a r i a b l e s and Achievement Var. PS1 PS2 PS3 PS4 PS5 PS6 PS7 PS8 PS9 PS10 PS11 A l A2 A3 PS1 100 52* 31* 26* 16* 15* 16* 19* 14* 18* 17* 16* 19* 18* PS 2 100 36* 26* 16* 4 16* 19* 14* 22* 14* 22* 30* 26* PS 3 100 24* 9 13* 12* 19* 18* 10* 13* 9* 15* 14* PS4 100 3 12* 2 9* 8* 5 10* 4 2 2 PS 5 100 15* 40* 34* 18* 43* 24* 10* 8* 11* PS 6 100 13* 12* 8* 15* 23* 11* 8* 9* PS 7 100 52* 21* 40* 40* 10* 11* 10* PS8 100 24* 37* 46* 6 10* 11* PS 9 100 16* 22* 6 8* 8* PS10 100 37* 8* 8* 9* PS11 100 7* 5 2 A l Problem Solving 100 72 61 A2 Test T o t a l 100 83 A3 A p p l i c a t i o n 100 Note. 1. The c o r r e l a t i o n c o e f f i c i e n t s reported above are based on the computed c o r r e l a t i o n s rounded to two decimal places and m u l t i p l i e d by 100. 2. Independent var i a b l e s were as follows: PS1 = perception of student enjoyment; PS2 = perception of student achievement; PS3 = s a t i s f a c t i o n with teaching; PS4 = ease of teaching; PS5 = number of s t r a t e g i e s ; PS6 = attendance of i n - s e r v i c e ; PS7 = sources of exercises; PS8 = number of motivational a c t i v i t i e s ; PS9 = frequency of teaching; PS10 = problem types; PS11 = classroom features. *p<0.05. P o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s are shown f o r 50 of the 55 r e l a t i o n s h i p s between p a i r s of problem-83 s o l v i n g process v a r i a b l e s . The two strongest r e l a t i o n s h i p s were between teachers' perceptions of student enjoyment of and t h e i r perceptions of student achievement i n problem s o l v i n g , and between the number of d i f f e r e n t sources of exercises and the number of d i f f e r e n t a c t i v i t i e s used to motivate students. Both p a i r s of v a r i a b l e s showed c o r r e l a t i o n s of 0.52. Other r e l a t i v e l y strong c o r r e l a t i o n s were found between the number of d i f f e r e n t problem-solving s t r a t e g i e s taught and both numbers of sources of exercises (0.40) and d i f f e r e n t problem types used (0.43); the number of d i f f e r e n t sources of exercises and both number of problem types assigned (0.40) and number of classroom features (0.40); and number of motivational a c t i v i t i e s used and number of classroom features (0.46). The r e l a t i o n s h i p s among problem-solving process v a r i a b l e s were stronger than those found among classroom organization v a r i a b l e s . For example, 12 c o r r e l a t i o n s greater than 0.3 0 were found among the former v a r i a b l e s compared to only 1 among the l a t t e r . Twenty-six out of a t o t a l of 33 r e l a t i o n s h i p s between problem-solving process v a r i a b l e s and student achievement were p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t . Most of these were of a low magnitude, with 10 l e s s than 0.10. The strongest r e l a t i o n s h i p s were between teachers' expectations of student achievement and student performance on problem s o l v i n g (0.22), t e s t t o t a l (0.30) and on applic a t i o n s (0.26). Teachers 1 Perceptions of Mathematics and student Achievement Relationships between / teachers' perceptions of the importance of, the d i f f i c u l t y to teach and the enjoyment i n 84 teaching mathematics are reported i n t h i s s e c t i o n . Correlations of i n t e r e s t between these perceptions and student achievement are also discussed. These r e l a t i o n s h i p s are shown i n Table 25. Table 25 C o r r e l a t i o n s Among Teachers' Percept ions and student Achievement V a r i a b l e TP1 TP2 TP3 A l A2 A3 TP1 Importance 100 20* 58* 4 10* 6 TP2 D i f f i c u l t y 100 37* 9* 16* 15* TP3 Enjoyment 100 -1 5 3 A l Problem Solving 100 71* 61* A2 Test T o t a l 100 83* A3 Applications 100 Note: The c o r r e l a t i o n c o e f f i c i e n t s reported above are based on the computed c o r r e l a t i o n s rounded to two decimal places and m u l t i p l i e d by 100. *p<0.05. A l l three of the pair-wise r e l a t i o n s h i p s among teacher rat i n g s of importance, d i f f i c u l t y and enjoyment were p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t . The r e l a t i o n s h i p between importance and enjoyment was the strongest, however, with a c o r r e l a t i o n c o e f f i c i e n t of 0.58. Of the three teacher-perception v a r i a b l e s , the d i f f i c u l t y -to-teach r a t i n g c o r r e l a t e d the highest with student achievement. Correlations between t h i s r a t i n g and student achievement on problem solving, t e s t t o t a l and app l i c a t i o n s were 0.09, 0.16 and 85 0.15 r e s p e c t i v e l y . Although these c o r r e l a t i o n s were not high, they were a l l s i g n i f i c a n t at the 0.05 l e v e l . The only other s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n between teacher-perception v a r i a b l e s and student achievement was between the importance r a t i n g and student performance on t e s t t o t a l . Students' Perceptions of Mathematics and Achievement This s e c t i o n examines r e l a t i o n s h i p s between student ratings of the importance, d i f f i c u l t y to learn and enjoyment i n learning mathematics. I t also looks at r e l a t i o n s h i p s between t h e i r perceptions of mathematics and t h e i r achievement on problem solving, t e s t t o t a l and a p p l i c a t i o n s . A summary of these r e l a t i o n s h i p s i s reported i n Table 26. Table 26 C o r r e l a t i o n s Among Students' Perceptions and T h e i r Achievement V a r i a b l e SP1 SP2 SP3 A l A2 A3 SP1 Importance 100 57* 53* 9* 14* 12* SP2 D i f f i c u l t y 100 59* 7* 15* 14* SP3 Enjoyment 100 17* 27* 19* A l Problem Solving 100 71* 61* A2 Test T o t a l 100 83* A3 Applications 100 Note: The c o r r e l a t i o n c o e f f i c i e n t s reported above are based on the computed c o r r e l a t i o n s rounded to two decimal places and m u l t i p l i e d by 100. *p<0.05. 86 Strong r e l a t i o n s h i p s are shown among student r a t i n g s of the importance, d i f f i c u l t y and enjoyment of mathematics. C o r r e l a t i o n c o e f f i c i e n t s between these p a i r s of v a r i a b l e s ranged between 0.53 and 0.59. Although a l l c o r r e l a t i o n s between student perceptions and t h e i r achievement on the c r i t e r i o n v a r i a b l e s were p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t , the strongest r e l a t i v e r e l a t i o n s h i p s were between the enjoyment r a t i n g and achievement. The c o r r e l a t i o n s between enjoyment and problem s o l v i n g , t e s t t o t a l and a p p l i c a t i o n s were 0.17, 0.27 and 0.19 r e s p e c t i v e l y . I n t e r p r e t a t i o n of C o r r e l a t i o n C o e f f i c i e n t s C o r r e l a t i o n c o e f f i c i e n t s reported i n t h i s chapter show low but p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s between a number of v a r i a b l e s i n each of the input categories and student achievement i n problem solving, t e s t t o t a l and a p p l i c a t i o n s . A r e s t r i c t i o n on the magnitude of these c o r r e l a t i o n s , however, may be due to the e f f e c t s of data aggregation to the c l a s s l e v e l . Burstein (1980) f o r example, points out a number of issues i n m u l t i l e v e l data a n a l y s i s which can impact on r e s u l t s . Within c l a s s variances, i n t h i s case, were not accounted f o r at t h i s l e v e l of a n a l y s i s . Class variances were examined i n each of the achievement domains but no increases i n the magnitude of c o r r e l a t i o n s with input v a r i a b l e s were found. 87 4.4 FACTOR ANALYSIS A f a c t o r analysis of the v a r i a b l e s within each of the input categories was conducted at the next stage of a n a l y s i s . This process was employed to determine common underlying dimensions on which the c r i t e r i o n v a r i a b l e s were located and hence to i d e n t i f y those v a r i a b l e s which were most l i k e l y to be u s e f u l l y included i n a regression equation. The v a r i a b l e s were f a c t o r -analyzed within the categories of student background, teacher background, classroom organization and problem-solving processes i n order to examine the e f f e c t s of inputs i d e n t i f i e d i n the t h e o r e t i c a l model. Since the number of v a r i a b l e s had been reduced to 3 i n each of the student perception and teacher perception categories, no f a c t o r analysis was conducted on these v a r i a b l e s . A c r i t e r i o n commonly used f o r f a c t o r i d e n t i f i c a t i o n i s an eigenvalue of 1.0 or greater (Nie, 1975, p.479). Using t h i s c r i t e r i o n , v a r i a b l e s were grouped in t o f a c t o r s within each input category. The loading of a v a r i a b l e i n t o i t s respective f a c t o r was considered s i g n i f i c a n t i f i t s c o r r e l a t i o n with the f a c t o r was greater than 0.30 (Spencer & Bowers, 1976, p.10). In cases where a v a r i a b l e loaded into more than one f a c t o r at the 0.30 l e v e l , i t was assigned to the f a c t o r i n t o which i t loaded at the highest l e v e l . A discussion of these fac t o r s by major input category follows. Student Background Factors The f i r s t f a c t o r analysis included the f i v e student background v a r i a b l e s : language f i r s t spoken, language spoken 88 cur r e n t l y i n the home, time spent on homework, father's l e v e l of education, and mother's l e v e l of education. Results are shown i n Table 27. Table 27 P r i n c i p a l Components of student Background V a r i a b l e s Eigen- Percent Cumulated Factor value Variance Percent 1 2.0786 41.6 41.6 2 1.4053 28.1 69.7 3 0.9820 19.6 89.3 4 0.3382 6.8 96.1 5 0.1959 3.9 100.0 An an a l y s i s of the data i n Table 27 shows that the f i v e student background v a r i a b l e s would y i e l d two fa c t o r s with a cumulative variance of 69.7 percent. A Varimax r o t a t i o n grouped the v a r i a b l e s (af t e r 3 i t e r a t i o n s ) as shown i n Table 28. Table 28 Rotated Factor Matr ix of Student Background V a r i a b l e s Variable Factor 1 Factor 2 Home Language 0.9318 0.1455 1st Language 0.9258 0.1735 Homework -0.2143 0.0677 Father Education -0.0312 0.9127 Mother Education 0.0900 0.9027 89 Inspection of the loadings into the language and homework fac t o r (Factor 1) show high c o r r e l a t i o n s with the language v a r i a b l e s . The c o r r e l a t i o n of -0.2143 f o r homework, however, i s not s i g n i f i c a n t using +/-0.30 as the c r i t e r i o n (Spencer & Bowers, 1976, p.10). Consequently, i t was dropped from the fac t o r . Both f a t h e r s 1 and mothers' education, on the other hand, loaded s i g n i f i c a n t l y into Factor 2. Teacher Background Factors The second fa c t o r a nalysis involved the seven teacher background v a r i a b l e s : years of experience, preference to teach, proportion of teaching load, attendance at conferences, attendance at workshops, mathematics courses completed and mathematics education courses completed. Results of the p r i n c i p a l components analysis are shown i n Table 29. Table 29 P r i n c i p a l Components of Teacher Background V a r i a b l e s Eigen- Percent Cumulated Factor value Variance Percent 1 1.8828 26.9 26.9 2 1.0976 15.7 42.6 3 0.9853 14.1 56.7 4 0.9564 13.6 70.3 5 0.8738 12.5 82.8 6 0.6667 9.5 92.3 7 0.5374 7.7 100.0 90 The p r i n c i p a l components analysis extracted two fact o r s with eigenvalues greater than 1.0, which accounted f o r a cumulative variance of 42.6 percent. A f t e r three i t e r a t i o n s the rotated f a c t o r matrix grouped the v a r i a b l e s i n t o the two c l u s t e r s shown i n Table 30. Table 30 Rotated Factor Matrix of Teacher Background Variables Va r i a b l e Factor 1 Factor 2 Conferences 0.6584 -0.0034 Workshops 0.6562 -0.0724 Math Courses 0.5917 0.3595 Math Ed. Courses 0.5720 0.4316 Teaching Load 0.4041 -0.1039 Experience 0.0367 0.6892 Preference -0.1083 0.6754 Factor 1 shows s i g n i f i c a n t loadings f o r v a r i a b l e s r e l a t e d to p r o f e s s i o n a l preparation and proportion of teaching load. I t i s notable, however, that the numbers of mathematics and mathematics methods courses completed also loaded s i g n i f i c a n t l y , but not as highly, into Factor 2. Factor 2 co r r e l a t e d s i g n i f i c a n t l y with v a r i a b l e s measuring teaching experience and preference to teach mathematics. This suggests that there i s an underlying a t t r i b u t e common to the v a r i a b l e s i n each of the two fa c t o r s . Classroom Organization Factors A new v a r i a b l e c o n s i s t i n g of the t o t a l time spent on mathematics was created from the product of the number of classes per week and length of period. The new v a r i a b l e replaced these two v a r i a b l e s at t h i s stage of the a n a l y s i s . The r e s u l t i n g ten classroom organization v a r i a b l e s were then f a c t o r -analyzed and the p r i n c i p a l components analysis i s shown i n Table 31. Table 31 P r i n c i p a l components of Classroom Organiza t ion V a r i a b l e s Eigen- Percent Cumulated Factor value Variance Percent 1 1.4730 14.7 14.7 2 1.2625 12.7 27.4 3 1.2041 12.0 39.4 4 1.0685 10.7 50.1 5 1.0025 10.0 60.1 6 0.9615 9.6 69.7 7 0.8629 8.6 78.3 8 0.8105 8.1 86.5 9 0.7047 7.0 93.5 10 0.6499 6.5 100.0 The p r i n c i p a l components analysis extracted f i v e factors with eigenvalues greater than one. These fact o r s accounted for 60 percent of the variance. A f t e r seven i t e r a t i o n s the rotated f a c t o r matrix grouped the v a r i a b l e s i n t o f i v e c l u s t e r s as shown i n Table 32. 92 Table 32 Rotated Factor Matrix of Classroom Organization Variables Variable Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Seatwork -0. 7455 0. 0939 -0. 1494 -0. 1194 -0. 0343 Lecture 0. 6998 -0. 0462 -0. 3075 -0. 2744 0. 0520 Questioning 0. 4998 0. 3432 0. 1120 0. 2324 -0. 1554 Comp. D r i l l -0. 0121 0. 7683 0. 0131 0. 1335 0. 1604 Test Freq. -0. 0263 0. 6633 -0. 0260 -0. 1367 -0. 2440 Small Group -0. 0997 -0. 1497 0. 7605 -0. 0750 -0. 1768 Work Sta t i o n 0. 1666 0. 1898 0. 6957 -0. 0350 0. 2698 Homework 0. 2331 -0. 1081 0. 0047 0. 7852 -0. 1537 To t a l Time -0. 1649 0. 1268 -0. 1164 0. 5313 0. 2053 Course Type -0. 0003 -0. 0731 0. 0220 0. 0167 0. 8760 A l l v a r i a b l e s within each c l u s t e r loaded s i g n i f i c a n t l y into each of t h e i r f i v e respective f a c t o r s . Factor 1 was a composite of time spent on seatwork, time spent on introducing new topics and the number of students questioned i n the most recent period. I t i s notable, however, that time spent on seatwork loaded negatively on the f i r s t f a c t o r and that the number of students questioned also loaded s i g n i f i c a n t l y , but l e s s highly, into Factor 2. The second fa c t o r was comprised of v a r i a b l e s r e l a t e d to time spent on computational d r i l l and frequency of t e s t i n g . Factor 3 combined v a r i a b l e s on time spent i n working i n small groups and time spent at work sta t i o n s or a c t i v i t y centers. Time spent on homework-related a c t i v i t i e s and t o t a l time spent on mathematics clustered into the fourth f a c t o r . Factor 5 consisted of a s i n g l e v a r i a b l e r e l a t e d to the type of course. 93 Problem-Solving Processes The next f a c t o r a nalysis involved eleven classroom process v a r i a b l e s s p e c i f i c a l l y r e l a t e d to the teaching of problem so l v i n g . The p r i n c i p a l components analysis i s reported i n Table 33. P r i n c i p a l Components of Table 33 Problem-Solving Process V a r i a b l e s Factor Eigen-value Percent Variance Cumulated Percent 1 3.0491 27.7 27.7 2 1.6281 14.8 42.5 3 1.0750 9.8 52.3 4 0.9359 8.5 60.8 5 0.8105 7.4 68.2 6 0.7509 6.8 75.0 7 0.6702 6.1 81.1 8 0.6427 5.8 86.9 9 0.5308 4.9 91.8 10 0.4701 4.2 96.0 11 0.4366 4.0 100.0 Based on these r e s u l t s , a three- f a c t o r s o l u t i o n i s suggested. I t accounted f o r 52.3 percent of the variance among va r i a b l e s i n t h i s category. A f t e r f i v e i t e r a t i o n s the rotated f a c t o r matrix, shown i n Table 34, l i s t s the f a c t o r loadings. 94 Table 34 Rotated Factor Matrix of Problem-Solving Process Variables Variable Factor 1 Factor 2 Factor 3 Exercises 0.7697 0.0325 0.0587 A c t i v i t i e s 0.7460 0.1151 0.0495 Problem types 0.6754 0.1080 -0.0366 Strategies 0.6437 0.0385 -0.1151 Class Feature 0.6265 0.0776 0.3735 Test Freq. 0.3588 0.1559 0.1978 Expected Ach. 0.1724 0.8000 -0.2026 Expected Enj. 0.1508 0.7238 0.0126 S a t i s f a c t i o n 0.1062 0.6650 0.1878 Ease to Teach -0.0927 0.5403 0.3872 In-service 0.0843 0.0387 0.8656 The v a r i a b l e s which loaded into Factor 1 involved numbers of d i f f e r e n t approaches and sources of materials used by teachers. They were comprised of the following v a r i a b l e s : d i f f e r e n t sources of exercises and a c t i v i t i e s ; v a r i e t y of st r a t e g i e s and problem types taught; frequency of t e s t i n g problem s o l v i n g ; and number of d i f f e r e n t classroom features used to encourage problem solving, such as a problem-solving center, d i s p l a y board or contests. Variables i n Factor 2 r e l a t e d to teacher a t t i t u d e toward problem s o l v i n g . They included teachers' expectations of students' achievement i n and enjoyment of problem solving, t h e i r s a t i s f a c t i o n with teaching the to p i c and the ease they found i n teaching i t . The s i n g l e v a r i a b l e which loaded in t o Factor 3 r e l a t e d to teachers' i n - s e r v i c e involvement. A l l va r i a b l e s loaded s i g n i f i c a n t l y with t h e i r respective f a c t o r s . 95 Teachers' Percept ions of Mathematical Topics Teachers' perceptions of mathematics were comprised of t h e i r r atings of ten major t o p i c s i n the curriculum. The topics were f r a c t i o n s , decimals, percent, estimation, basic f a c t s , equations, word problems, metric system, perimeter and area, and geometry. Each of these t o p i c s was rated according to teachers' perceptions of t h e i r importance, d i f f i c u l t y to teach and enjoyment to teach. Results f o r each r a t i n g were summed across t o p i c s to obtain a s i n g l e score. In t h i s way a t o t a l score was obtained f o r each of the three r a t i n g s . Since t h i s procedure reduced the number of va r i a b l e s from 30 to 3, no f a c t o r analysis was conducted i n t h i s category. The remaining 3 v a r i a b l e s are re f e r r e d to as facto r s f o r reporting purposes i n the remaining a n a l y s i s . Students' Percept ions of Mathematical Topics Three r a t i n g scores were determined i n a s i m i l a r way f o r students' perceptions of the importance, d i f f i c u l t y to learn, and enjoyment i n learning f o r the same to p i c s as were rated by teachers. Since only 3 v a r i a b l e s resulted, no f a c t o r analysis was conducted. For reporting purposes these v a r i a b l e s w i l l be re f e r r e d to as f a c t o r s . Summary of the Fac tor Analyses The v a r i a b l e s i n each category were reduced i n number as a r e s u l t of the fa c t o r analyses. Table 35 shows the differences i n numbers which resulted. 96 Table 35 Numbers of Variables and Factors Category Number of Variables Number of Factors Student Background 5 2 Teacher Background 7 2 Classroom Organization 10 5 Prob. Solving Processes 11 3 Teachers 1 Perceptions 30 3 Students• Perceptions 30 3 Tota l 93 18 As shown i n Table 35, the number of independent v a r i a b l e s under examination i n the current study was reduced from 93 to 18 fa c t o r s . The number of v a r i a b l e s under Teachers* and Students* Perceptions was reduced from 30 to 3 i n each case by determining t o t a l r a t i n g scores, c a l c u l a t e d by summing raw scores across t o p i c s . Factor scores, on the other hand, which were cal c u l a t e d i n the remaining categories, were determined by f i r s t converting v a r i a b l e scores to z-scores. The o r i g i n a l z-scores were then converted to f a c t o r scores i n standard score form (Kerlinger and Pedhazur, 1973, p. 365). 4.5 MULTIPLE REGRESSION ANALYSIS At t h i s stage of analysis multiple regression techniques were used to analyze the r e l a t i o n s h i p s between each of the c r i t e r i o n v a r i a b l e s and sets of pr e d i c t o r f a c t o r s . The purpose was to determine the best l i n e a r p r e d i c t i o n equation and to cont r o l f o r other confounding factors i n order to evaluate the contributions made by a number of v a r i a b l e s on student achievement i n mathematics. The f i r s t stage involved a s e r i e s of mul t i p l e regressions i n which f a c t o r s from each of the input categories were regressed i n turn on the three achievement domains. In each case a step-wise regression procedure was used to determine which f a c t o r s were the best p r e d i c t o r s of success. In t h i s process, the f a c t o r entered f i r s t was the one which co r r e l a t e d the highest with each c r i t e r i o n v a r i a b l e . Successive factors continued to be entered u n t i l the F - r a t i o was no longer s i g n i f i c a n t . The purpose at t h i s stage of the an a l y s i s was to determine the amount of variance i n student achievement which could be accounted f o r by the fact o r s i n each separate input category. At the second stage of an a l y s i s , f a c t o r s from a l l input categories were entered using the step-wise procedure. The purpose of t h i s process was to develop a general model i n which the t o t a l variance accounted f o r by a l l fa c t o r s could be determined. Using t h i s process any variances which were shared by fac t o r s from d i f f e r e n t categories were taken into account. Student Background The two student background factors were regressed, using the step-wise method, on each of the three c r i t e r i o n v a r i a b l e s . Factor 2, comprised of two v a r i a b l e s r e l a t e d to mothers' and fathers' educational backgrounds, was selected f i r s t i n each case based on higher c o r r e l a t i o n s with the c r i t e r i o n v a r i a b l e s . 98 Factor 2 remained i n the p r e d i c t i o n equation f o r each domain whereas Factor 1 d i d not meet the c r i t e r i a f o r entry. Results are shown i n Table 36. Table 36 Student Background Factors Regressed on C r i t e r i o n V a r i a b l e s Variable Step Factor Mult R R 2 F p Problem Solving 1 2 Test T o t a l 1 2 Applications 1 2 0.2193 0.0481 36.72 <0.001 0.2601 0.0677 52.75 <0.001 0.3005 0.0903 72.14 <0.001 These data show that Factor 2 accounts f o r 4.8, 6.8 and 9.0 percent of the variance i n student achievement on the problem so l v i n g , t e s t t o t a l and a p p l i c a t i o n v a r i a b l e s r e s p e c t i v e l y . The e f f e c t of t h i s f a c t o r i s s t a t i s t i c a l l y s i g n i f i c a n t at the 0.001 l e v e l i n each case. The variances accounted f o r show that while the educational l e v e l of parents accounts f o r a s t a t i s t i c a l l y s i g n i f i c a n t p o r t i o n of the variance i n student achievement on questions which t e s t higher order thinking, i t accounts f o r somewhat more variance on the t e s t t o t a l score and, i n p a r t i c u l a r , on a p p l i c a t i o n questions comprised of routine story problems. Teacher Background Using the step-wise method, the two teacher background facto r s were regressed on each of the c r i t e r i o n v a r i a b l e s . Neither f a c t o r remained i n the pr e d i c t o r equation f o r problem so l v i n g or t e s t t o t a l . However, Factor 1, comprised of 99 v a r i a b l e s i n v o l v i n g p r o f e s s i o n a l preparation, remained i n the equation f o r the a p p l i c a t i o n v a r i a b l e . Teacher background accounted f o r approximately 1 percent of the variance i n student achievement on a p p l i c a t i o n questions. As shown i n p r i o r research, teacher background as cur r e n t l y measured, appears to have no s i g n i f i c a n t e f f e c t on problem so l v i n g or t e s t t o t a l and l i t t l e e f f e c t on a p p l i c a t i o n s . Classroom Organiza t ion The f i v e classroom organization f a c t o r s were regressed on the c r i t e r i o n v a r i a b l e s at the next step of the process. Factor 5, comprised of a s i n g l e v a r i a b l e on the type of course, remained as a p r e d i c t o r f a c t o r f o r a l l of the c r i t e r i o n v a r i a b l e s . Factor 4, which r e l a t e d to the t o t a l time spent on mathematics and the time spent on homework r e l a t e d a c t i v i t i e s , remained i n the regression equation f o r the t e s t t o t a l v a r i a b l e only. A summary of r e s u l t s i s shown i n Table 37. Table 37 Classroom Organizat ion Factors Regressed on C r i t e r i o n V a r i a b l e s Variable Step Factor Mult R R 2 F P Problem Solving 1 5 0.1687 0 .0285 19.54 <0 .001 Test T o t a l 1 5 0.2169 0 .0471 32.93 <0 .001 2 4 0.2338 0 .0547 19.26 <0 .001 Applications 1 5 0.2155 0.0464 32.47 <0.001 100 Classroom Organization Factor 5 accounted f o r 2.8 percent of the variance on problem solving, 4.7 percent on t e s t t o t a l and 4.6 percent on applications.. Factor 2, which remained i n the p r e d i c t i o n equation f o r t e s t t o t a l , accounted f o r an ad d i t i o n a l 0.8 percent of the variance on that v a r i a b l e . A l l factor s which remained i n the equations were s i g n i f i c a n t at the 0.001 l e v e l . Problem-Solving Process A t o t a l of three Problem-Solving Process f a c t o r s were regressed on each of the c r i t e r i o n v a r i a b l e s . Factor 2, which was a measure of teacher a t t i t u d e toward the teaching of problem sol v i n g , remained i n a l l three equations. Factor 1, which r e l a t e d to v a r i e t i e s of approaches used to teach problem sol v i n g also remained i n the regression equation f o r the t e s t t o t a l and a p p l i c a t i o n v a r i a b l e s . The t h i r d f a c t o r , which r e l a t e d to teacher p a r t i c i p a t i o n i n i n - s e r v i c e a c t i v i t i e s , remained i n the pr e d i c t i o n equation f o r the problem-solving c r i t e r i o n v a r i a b l e . Results are summarized i n Table 38. Table 38 Problem-Solving Process Factors Regressed on C r i t e r i o n V a r i a b l e s Variable Step Factor Mult R R^ F p Problem Solving 1 2 0. .1833 0. .0336 24, .89 <0. .001 2 3 0. .2010 0. .0404 15. .35 <0. .001 Test T o t a l 1 2 0. .2377 0. .0565 42. .88 <0. ,001 2 1 0. ,2513 0. .0631 24. .09 <0. .001 Applications 1 2 0. .2212 0. .0489 36, .84 <0. .001 2 1 0. .2418 0. .0584 22. .19 <0. .001 101 Results i n Table 38 show that Factor 2 accounted f o r 3.4 percent and Factor 3 accounted f o r an a d d i t i o n a l 0.7 percent of the variance i n student achievement i n problem s o l v i n g . On the t e s t t o t a l v a r i a b l e , Factor 2 accounted f o r 5.6 percent of the variance and Factor 1 added an a d d i t i o n a l 0.7 percent. The variance i n student achievement on ap p l i c a t i o n s accounted f o r by Factor 2 was 4.9 percent. Factor 1 added an a d d i t i o n a l 1.0 percent. A l l values of the F s t a t i s t i c reported i n the table were s i g n i f i c a n t at the 0.001 l e v e l . Based on these data, teacher a t t i t u d e toward problem sol v i n g , which was comprised of v a r i a b l e s r e l a t e d to teacher expectations of student performance and t h e i r perceptions of effec t i v e n e s s i n teaching problem s o l v i n g accounted f o r some measurable variance i n student achievement on a l l three c r i t e r i o n v a r i a b l e s . Involvement i n i n - s e r v i c e a c t i v i t i e s added s l i g h t l y to the variance accounted f o r i n problem s o l v i n g whereas the v a r i e t y of d i f f e r e n t approaches used i n the teaching of problem s o l v i n g accounted f o r part of the variance i n achievement on the t e s t t o t a l and a p p l i c a t i o n v a r i a b l e s . Teachers* Perceptions of Mathematics At the next stage of analysis the three scores for importance, d i f f i c u l t y i n teaching and enjoyment i n teaching were regressed on each of the problem solving, t e s t t o t a l and a p p l i c a t i o n v a r i a b l e s . Results are reported i n Table 39. 102 Table 39 Teachers' Perceptions Regressed on C r i t e r i o n V a r iables Variable Step Factor Mult R R 2 F p Problem Solving 1 2 0.0944 0.0089 6.64 <0.05 Test T o t a l 1 2 0.1587 0.0252 19.06 <0.001 2 1 0.1810 0.0328 12.48 <0.001 Applications 1 2 0.1468 0.0216 16.26 <0.001 Factor 2, which was comprised of teachers' perceptions of the d i f f i c u l t y to teach mathematics, remained in. the regression equation f o r each of the dependent v a r i a b l e s . I t accounted f o r 0.9, 2.5 and 2.2 percent of the variance i n student achievement on problem so l v i n g , t e s t t o t a l and a p p l i c a t i o n s r e s p e c t i v e l y . Teachers* ra t i n g s f o r importance also remained i n the equation f o r t e s t t o t a l . I t accounted f o r an a d d i t i o n a l 0.8 percent of the variance i n student achievement on that domain. Students' Perceptions of Mathematics Students* ratings of mathematics i n regard to importance, d i f f i c u l t y i n learning and enjoyment i n learning were regressed on the c r i t e r i o n v a r i a b l e s at the next step i n the anal y s i s . Results are reported i n Table 40. 103 Table 40 Students' Perceptions Regressed on C r i t e r i o n V a r iables Variable Step Factor Mult R Problem Solving Test T o t a l Applications 1 1 1 2 3 3 3 2 0.1671 0.0279 20.88 <0.001 0.2714 0.0736 57.79 <0.001 0.1960 0.0384 29.04 <0.001 0.2105 0.0443 16.84 <0.001 Students' ratings f o r enjoyment i n le a r n i n g mathematics (Factor 3) remained i n the p r e d i c t i o n equation f o r each of the c r i t e r i o n v a r i a b l e s . I t accounted f o r 2.8 percent of the variance i n achievement i n problem solving, 7.4 percent on t e s t t o t a l and 3.8 percent on a p p l i c a t i o n s . Factor 2, which was comprised of students* d i f f i c u l t y r a t i n g s , remained i n the equation f o r app l i c a t i o n s and accounted f o r an a d d i t i o n a l 0.6 percent of the variance i n achievement on that domain. 4.6 THE PROVINCIAL MODELS General models f o r each of the c r i t e r i o n v a r i a b l e s were determined through r e s u l t s from the second stage i n the multiple regression a n a l y s i s . At t h i s stage fac t o r s from a l l input categories were regressed on each of the dependent var i a b l e s using the step-wise procedure. By using f a c t o r s from a l l of the input categories, any variance shared among them was accounted f o r before the t o t a l amount of variance was determined. 104 Problem S o lv ing Five out of the t o t a l of 18 fa c t o r s remained i n the f i n a l regression equation f o r problem s o l v i n g . Results of the regression are shown i n Table 41. Table 41 P r o v i n c i a l Regression Model f or Problem S o l v i n g Input Category Factor Number Step Mult R R 2 R 2Ch FCh P S. Back. 2 1 0.1979 0. 0392 0.0392 25.24 <0 .001 Class Org. 5 2 0.2574 0. 0663 0.0271 17.92 <0 .001 S. Percept. 3 3 0.2974 0. 0884 0.0221 15.01 <0 .001 Prob. Solv. 2 4 0.3173 0. 1007 0.0122 8.38 <0 .05 Prob. Solv. 3 5 0.3265 0. 1066 0.0060 4.10 <0 .05 Note. R^Ch = Incremental change i n R^ FCh = F r a t i o f o r the incremental change i n R 2. Based on the data reported i n Table 41, the e f f e c t of a l l fact o r s accounted f o r a t o t a l of 10.7 percent of the variance i n student achievement i n problem so l v i n g . The greatest e f f e c t i s accounted f o r by Factor 2 from student background. This factor, comprised of parents' educational l e v e l s , accounted f o r 3.9 percent of the variance. Factor 5, from classroom organization, accounted f o r the second l a r g e s t amount of variance. I t was comprised of course type and i t explained an a d d i t i o n a l 2.7 percent. The other factors which remained i n the equation and the a d d i t i o n a l amount of variance they explained were as follows: student ratings of enjoyment, 2.2 percent; teachers' 105 a t t i t u d e s toward problem solving, 1.2 percent; and teacher i n -ser v i c e i n problem solving, 0.6 percent. The inputs of schooling d i d not explain much variance i n student performance i n problem s o l v i n g . This r e s u l t may be not only because of l i m i t a t i o n s due to the d e f i n i t i o n of v a r i a b l e s and the aggregation of data but also to many other f a c t o r s which could have e f f e c t . For example, student a b i l i t y and s p a t i a l aptitude are among c h a r a c t e r i s t i c s which are important f o r problem so l v i n g , but are l i k e l y e i t h e r inherent i n students or d i f f i c u l t to teach. Test T o t a l Out of the o r i g i n a l 18 factors 5 remained i n the regression equation explaining variance i n student achievement on t e s t t o t a l . Results are shown i n Table 42. Table 42 P r o v i n c i a l Regression Model f o r Test T o t a l Input Factor Category Number Step Mult R R 2 RzCh Fch P S. Percept. 3 1 0.2808 0.0789 0.0789 52.99 <0.001 S. Back. 2 2 0.3576 0.1279 0.0490 34.75 <0.001 Class Org. 5 3 0.4058 0.1647 0.0368 27.17 <0.001 Prob. Solv. 2 4 0.4300 0.1849 0.0202 15.29 <0.001 T. Percept. 1 5 0.4414 0.1949 0.0100 7.61 <0.05 Note. R^Ch = Incremental change i n R z FCh = F r a t i o f o r the incremental change i n R 2. 106 The e f f e c t s of the facto r s accounted f o r 19.5 percent of the variance i n student achievement on t e s t t o t a l . Students 1 perceptions of t h e i r enjoyment of learning mathematics accounted fo r the greatest amount of variance at 7.9 percent. This was followed by parents' l e v e l s of education f o r an a d d i t i o n a l 4.9 percent, course type f o r 3.7 percent, teachers' a t t i t u d e s toward problem s o l v i n g f o r 2.0 percent and teachers' perceptions of the importance of mathematics f o r 1.0 percent. Students' perceptions of enjoyment i n learn i n g mathematics accounted f o r considerably more variance on t e s t t o t a l than on problem s o l v i n g . I t accounted f o r 7.9 percent of the variance on the former and only 2.2 percent on the l a t t e r . The other two fac t o r s common to both domains also accounted f o r more variance on t e s t t o t a l than on problem so l v i n g . Applications Five f a c t o r s also remained i n the regression equation f o r ap p l i c a t i o n s . Table 43 shows the r e s u l t s . 107 Table 43 P r o v i n c i a l Regression Model f or A p p l i c a t i o n s Input Category Factor Number Step Mult R R 2 R 2Ch Fch P S. Back. 2 1 0.2884 0. 0832 0.0832 56.17 <0. 001 S. Percept. 3 2 0.3498 0. 1223 0.0391 27.56 <0. 001 Class Org. 5 3 0.3974 0. 1579 0.0356 26.10 <0. 001 Prob. Solv. 1 4 0.4184 0. 1751 0.0172 12.81 <0. 001 Prob. Solv. 2 5 0.4282 0. 1833 0.0082 6.21 <0. 001 Note. R^Ch = Incremental change i n R^ FCh = F r a t i o f o r the incremental change i n R 2. The f i v e f a c t o r s which remained i n the regression equation accounted f o r 18.3 percent of the variance i n student achievement on ap p l i c a t i o n s . Parents' l e v e l of education explained the greatest amount of variance at 8.3 percent, followed by student perceptions of enjoyment i n learning mathematics at 3.9 percent, course type at 3.6 percent, v a r i e t y of problem-solving a c t i v i t i e s and materials at 1.7 percent and teachers' a t t i t u d e s toward problem s o l v i n g at 0.8 percent. The regression equation f o r ap p l i c a t i o n s shared 4 out of the 5 facto r s i n common with the equations f o r the other two c r i t e r i o n v a r i a b l e s . However, the l e v e l of parents' education explained considerably more variance i n student performance on t h i s domain than on problem s o l v i n g or t e s t t o t a l . The v a r i e t y of d i f f e r e n t problem-solving a c t i v i t i e s and materials used by teachers explained an a d d i t i o n a l 1.7 percent of the variance i n performance on the a p p l i c a t i o n domain but d i d not enter in t o the 108 f i n a l equation f o r e i t h e r of the other two dependent v a r i a b l e s . This may in d i c a t e that these a c t i v i t i e s have a l i m i t e d , but measurable, e f f e c t on students' a b i l i t y to solve routine story problems but l i t t l e or no e f f e c t on t h e i r performance on items r e q u i r i n g higher cognitive s k i l l s or on o v e r a l l t e s t r e s u l t s . 4.7 THE SURREY MODEL Since pre-test data were not a v a i l a b l e f o r p r o v i n c i a l r e s u l t s , the current study examined r e s u l t s from a large urban d i s t r i c t to determine whether, when pre-te s t scores were c o n t r o l l e d f o r , i t would s i g n i f i c a n t l y a l t e r the f i n a l model. The 1985 P r o v i n c i a l Mathematics Assessment was r e p l i c a t e d through administration of a pre-test i n the F a l l of 1986 and a post-tes t i n the Spring of 1987 to a l l students and teachers of Grade 7 i n that d i s t r i c t . A model was determined f i r s t using only p o s t - t e s t scores, and applying the same procedures as reported f o r the 1985 p r o v i n c i a l data. A second model was then developed i n which pre-te s t scores were used as covariates. Results based on each of the models were then compared to see i f any differences existed. Based on these di f f e r e n c e s , a judgment was made on the appropriateness of using the analysis of survey data, without pre-test information, f o r decision-making at the p r o v i n c i a l l e v e l . 1987 Post-Test Model The same method of analysis used with data from the 1985 Assessment was applied to post-test r e s u l t s from the 1987 109 v a l i d a t i o n study. Since the purpose of t h i s part of the study was to compare findings based on the analyses of c r o s s - s e c t i o n a l data and l o n g i t u d i n a l data, only t e s t t o t a l was used as the c r i t e r i o n v a r i a b l e . As reported e a r l i e r , students from 104 classrooms completed t e s t booklets. A t o t a l of 100 teachers returned questionnaires and matches were found between 97 teachers and classrooms. The f i n a l post-test model was determined by regressing fact o r s from a l l of the input categories, using the step-wise method, on c l a s s means f o r t e s t t o t a l . Four out of a t o t a l of 21 fact o r s remained i n the f i n a l regression equation. Results are reported i n Table 44. Table 44 Post-Test Regression Model Input Category Factor Number Step Mult R R 2 R 2Ch Fch P S. Percept. 3 1 0.4194 0 .1759 0.1759 17.07 <0. 000 S. Back. 1 2 0.5050 0 .2550 0.0791 8.391 <0. 001 Prob. Solv. 3 3 0.5414 0 .2931 0.0381 4.200 <0. 044 T. Back. 4 4 0.5798 0 .3361 0.0431 4.994 <0. 028 Note. R^Ch = Incremental change i n r FCh = F r a t i o f o r the incremental change i n R 2. These data show that a l l factors accounted f o r 33.6 percent of the variance i n student achievement on t e s t t o t a l . Factor 3 from student perception, comprised of students' perceptions of t h e i r enjoyment i n learning mathematics, explained 17.6 percent 110 of the variance i n student achievement on t e s t t o t a l . Factor 1 from student background, which comprised of v a r i a b l e s on language f i r s t spoken and language c u r r e n t l y spoken i n the home, accounted f o r an a d d i t i o n a l 7.9 percent of the variance i n student achievement on the same c r i t e r i o n v a r i a b l e . A d d i t i o n a l variance i n achievement of 3.8 percent was explained by Factor 3 from problem-solving processes. I t was comprised of v a r i a b l e s on teacher s a t i s f a c t i o n i n teaching problem so l v i n g , the number of d i f f e r e n t problem-solving s t r a t e g i e s taught and the number of workshops on problem sol v i n g attended by teachers. Teacher background Factor 4, on the other hand, was a measure of the proportion of teaching load i n mathematics and accounted f o r an a d d i t i o n a l 4.3 percent of the variance i n student achievement. 1987 Longitudinal Model At t h i s stage of analysis, achievement means f o r t e s t t o t a l from the pre-test r e s u l t s were introduced as covariates i n t o the f i n a l regression equation. Using t h i s procedure, student knowledge and behaviors at the beginning of the school year were c o n t r o l l e d . As a r e s u l t , the variances i n achievement which were explained by factors r e l a t e d more c l o s e l y to processes which occurred during the year under examination. Results, showing variances explained, are reported i n Table 45. I l l Table 45 Longitudinal Regression Model f o r Test T o t a l Input Category Factor Number Step Mult R R 2 R 2Ch FCh P Pretest 1 0.6361 0 .4046 0.4046 54.36 <0 .000 S. Percept. 1 2 0.6906 0 .4770 0.0724 10.94 <0 .001 Class Org. 2 3 0.7275 0 .5293 0.0523 8.67 <0 .004 T. Back. 4 4 0.7469 0 .5579 0.0286 4.98 <0 .029 S. Back. 1 5 0.7636 0 .5831 0.0252 4.60 <0 .035 Note. R^Ch = Incremental change i n R^. FCh = F r a t i o f o r the incremental change i n R 2. The f i n a l l o n g i t u d i n a l model f o r t e s t t o t a l retained 5 factor s out of a t o t a l of 22 and explained 58.3 percent of the variance i n student achievement f o r t e s t t o t a l . The pre-test explained 40.5 percent of the variance, with the balance of 17.8 percent accounted f o r by the r e s t of the fa c t o r s which were retained. Included among the other fac t o r s which accounted f o r variance i n student achievement were students' perceptions of the importance of mathematics, classroom organization Factor 2, teacher background Factor 4 and student background Factor 1. An a d d i t i o n a l 7.2 percent of the variance was explained by students' perceptions of the importance of mathematics. Classroom organization Factor 2, which was comprised of va r i a b l e s on the number of students questioned and the proportion of c l a s s time spent working i n small groups explained an a d d i t i o n a l 5.2 percent of the variance i n achievement. Teacher background Factor 4, which was a measure of the 112 proportion of teaching load i n mathematics, accounted f o r an ad d i t i o n a l 2.9 percent of the variance i n student achievement and student background Factor 1, comprised of v a r i a b l e s on language f i r s t spoken and language c u r r e n t l y spoken i n the home, explained an a d d i t i o n a l 2.5 percent. A comparison of r e s u l t s between the two models shows some dif f e r e n c e s . For example, the c r o s s - s e c t i o n a l model explained 33.6 percent of the variance i n student achievement whereas the l o n g i t u d i n a l model, a f t e r the entry l e v e l knowledge and behaviors of students were c o n t r o l l e d , explained only 17.8 percent. The fact o r s and t h e i r composition, which explained the variances i n each model, i n some cases were d i f f e r e n t . Students' perceptions of t h e i r enjoyment of learning mathematics, f o r example, remained i n the post-test model whereas t h e i r perceptions of the importance of mathematics remained i n the regression equation a f t e r entry l e v e l behaviors were c o n t r o l l e d . Mother tongue and proportion of teaching load remained i n both models. However, one f a c t o r measuring the e f f e c t s of classroom organization remained i n the l o n g i t u d i n a l model whereas a fa c t o r on problem-solving processes accounted f o r measurable variance i n student achievement i n the post-test model. 113 CHAPTER 5 SUMMARY AND CONCLUSIONS The purpose of t h i s study was to examine, through the use of survey data, r e l a t i o n s h i p s between inputs of schooling and outcomes, as measured by student achievement i n mathematics. The inputs of schooling were comprised of a number of var i a b l e s grouped under each of the following categories: students' and teachers' backgrounds, students' and teachers' perceptions of mathematics, classroom organization and problem-solving processes. Outcome measures included students' achievement on t e s t t o t a l , problem s o l v i n g and a p p l i c a t i o n s . The t e s t t o t a l v a r i a b l e provided a measure of o v e r a l l performance i n mathematics whereas the problem s o l v i n g and a p p l i c a t i o n v a r i a b l e s were designed to measure student achievement at two d i s t i n c t l e v e l s of cognitive behavior. Problem so l v i n g , f o r example, tested students' achievement on t e s t items intended to measure c r i t i c a l thinking. Test items i n the a p p l i c a t i o n v a r i a b l e , on the other hand, were comprised of routine story problems which were judged by committees involved i n the 1985 P r o v i n c i a l Assessment of Mathematics to be of a lower cognitive l e v e l . A r e l a t e d problem involved exploration of the appropriateness of using c r o s s - s e c t i o n a l survey data from a large-scale assessment to make decisions based on the re l a t i o n s h i p s found among the input and output v a r i a b l e s . To address t h i s question, r e s u l t s from a subsequent l o n g i t u d i n a l 114 study which u t i l i z e d the same instruments were examined f i r s t with p o s t - t e s t data and then, with the i n c l u s i o n of p r e - t e s t data. The intent was to see i f the same general conclusions could be made. 5.1 SIGNIFICANT FINDINGS AND CONCLUSIONS Analyses of p r o v i n c i a l data from the 1985 Assessment were used to address research questions 1 to 6. In order to address question 7, r e s u l t s from the 1987 v a l i d a t i o n study were examined i n two ways. F i r s t , only post-test data were used i n the an a l y s i s to explain the amount of variance i n student achievement i n mathematics accounted f o r by the v a r i a b l e s within each category under study. Second, pre-test data were included as covariates i n the regression equations. The variances subsequently explained were then compared with the preceding r e s u l t s to see what, i f any, d i f f e r e n c e s existed. In the following discussion each of the research questions under examination i s stated f i r s t . Related findings follow d i r e c t l y a f t e r each respective question. 1. What r e l a t i o n s h i p s e x i s t among teacher background c h a r a c t e r i s t i c s and student background c h a r a c t e r i s t i c s ; and between these v a r i a b l e s and students' achievement i n mathematics? Strong r e l a t i o n s h i p s were found between two p a i r s of student background v a r i a b l e s . They were students' language f i r s t spoken and the language they c u r r e n t l y speak at home, and fathers' and mothers' l e v e l s of education. The c o r r e l a t i o n c o e f f i c i e n t between the f i r s t p a i r of v a r i a b l e s was 0.81 and between the second p a i r i t was 0.66. The strong r e l a t i o n s h i p 115 found between language f i r s t and c u r r e n t l y spoken was not unexpected due to the large proportion of students i n the study f o r whom English was t h e i r mother tongue. The high c o r r e l a t i o n found between the educational l e v e l s of both parents on the other hand, i s l i k e l y due to the high proportion of mothers and fathers who attended school at the high school l e v e l or higher. Relationships among teacher background v a r i a b l e s were not as strong as the examples of student background v a r i a b l e s j u s t c i t e d ; however, a number of s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s were found. Examples included c o r r e l a t i o n s between attendance at conferences and workshops (0.32), and mathematics courses completed and mathematics education courses completed (0.41). The r e l a t i o n s h i p found between the number of mathematics workshops and the mathematics sessions at conferences attended during the previous three years may have indicated that many teachers who are a c t i v e i n the area of p r o f e s s i o n a l development l i k e l y attend both types of sessions whereas those who do not attend one form of i n - s e r v i c e probably do not attend the other. The r e l a t i v e l y strong r e l a t i o n s h i p between the number of mathematics courses and the number of mathematics education courses s u c c e s s f u l l y completed may i n d i c a t e that teachers who do not wish to take post-secondary mathematics courses may choose not to take methods courses e i t h e r , provided that option i s a v a i l a b l e i n t h e i r teacher education program. S t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s were found between the educational l e v e l s of parents' and students' achievement on a l l three c r i t e r i o n v a r i a b l e s . The c o r r e l a t i o n s among these v a r i a b l e s ranged from 0.20 between the educational l e v e l s of 116 both parents and achievement on problem s o l v i n g to 0.29 between fathers' educational l e v e l and achievement on a p p l i c a t i o n s . Since parents* l e v e l s of education could be viewed as measures of socio-economic status, the current study confirmed the p o s i t i v e r e l a t i o n s h i p s found between t h i s f a c t o r and student achievement, i n numerous other studies (e.g. Husen, 1967; Murnane, 1981; Horn & Walberg, 1984). The teacher v a r i a b l e s which c o r r e l a t e d most strongly with students* achievement were the number of workshops attended and the number of mathematics education courses completed. Correlations between the number of workshops attended and student achievement on t e s t t o t a l and a p p l i c a t i o n s were 0.12 and 0.13 r e s p e c t i v e l y . S t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s found between the number of mathematics methods courses completed and student achievement on t e s t t o t a l and a p p l i c a t i o n s , on the other hand, were 0.09 and 0.13 r e s p e c t i v e l y . These c o r r e l a t i o n s , however, were too low to draw conclusions. S i m i l a r r e s u l t s were found i n studies by M c D i l l and Rigsby, 1973; Rutter et. a l . , 1979; and Ward, 1979. 2. What r e l a t i o n s h i p s e x i s t among types of classroom organizations and structures; and between these v a r i a b l e s and students' achievement i n mathematics? Relationships between several p a i r s of classroom organization v a r i a b l e s were found to be s t a t i s t i c a l l y s i g n i f i c a n t . For example, the c o r r e l a t i o n between proportion of time spent working i n small groups and at a c t i v i t y centers was 0.22. This r e s u l t may i n d i c a t e that students who are working at a c t i v i t y centers are probably i n small group configurations due to the types of a c t i v i t i e s i n which they would l i k e l y be 117 engaged. Time spent on homework-related a c t i v i t i e s and the numbers of students c a l l e d upon to answer questions had a c o r r e l a t i o n c o e f f i c i e n t of 0.17. This f i n d i n g may r e f l e c t a classroom p r a c t i c e i n which homework-related a c t i v i t i e s include student questioning while discussing the answers. Several s i g n i f i c a n t r e l a t i o n s h i p s between classroom organization v a r i a b l e s c o r r e l a t e d negatively. Included among these were the following: type of course and the number of students questioned (-0.15), and the number of classes per week and length of period (-0.40). The negative r e l a t i o n s h i p between course type and number of students questioned may i n d i c a t e that teachers tend to ask more questions of students i n modified classes than i n enriched ones. A c t i v i t i e s such as d r i l l and p r a c t i c e and discussion of homework, which involve question-and-answer techniques, are l i k e l y emphasized more i n modified classes than i n enriched ones. The negative c o r r e l a t i o n between number of classes per week and length of period was expected. I t i n d i c a t e s that as numbers of classes i n a week increase, t h e i r duration decreases. The only r e l a t i o n s h i p between classroom organization v a r i a b l e s and student achievement of i n t e r e s t which c o r r e l a t e d s i g n i f i c a n t l y was the type of course. Correlations between course type and student achievement i n problem so l v i n g , t e s t t o t a l and a p p l i c a t i o n s were 0.15, 0.19 and 0.21 r e s p e c t i v e l y . These r e s u l t s l i k e l y i n d icate that students i n enriched classes perform better than those i n regular or modified ones. This f i n d i n g i s not s u r p r i s i n g since one would expect that students i n enriched classes are higher achievers on average than those 118 i n regular or modified ones. The magnitude of the c o r r e l a t i o n , however, i s lower than expected. This r e s u l t i s l i k e l y due to the small numbers of classes (7 percent) i n the study which were reported as modified or enriched. 3. What r e l a t i o n s h i p s e x i s t between d i f f e r e n t approaches to the teaching of problem s o l v i n g and students' achievement i n mathematics? A number of r e l a t i v e l y strong r e l a t i o n s h i p s were found among problem-solving process v a r i a b l e s and between these and student achievement. Relationships among the independent v a r i a b l e s with c o r r e l a t i o n c o e f f i c i e n t s greater than 0.20 are shown i n Table 46. 119 Table 46 C o r r e l a t i o n s Among Problem-Solving Process V a r i a b l e s * V a r i a b l e PS1 PS 2 PS 3 PS4 PS1 Enjoyment 100 52 31 26 PS2 Achievement 100 36 26 PS3 S a t i s f a c t i o n 100 24 PS4 Easiness 100 PS 5 PS 6 PS 7 PS8 PS9 PS10 PS11 PS 5 Strategies 100 40 34 43 24 PS 6 In-service 100 23 PS 7 Exercises - 100 52 21 40 40 PS8 A c t i v i t i e s 100 24 37 46 PS 9 Freq./Teach 100 22 PS9 Problem Types 100 37 PS11 Features 100 Note. The c o r r e l a t i o n c o e f f i c i e n t s reported above are based on the computed c o r r e l a t i o n s rounded to two decimal places and m u l t i p l i e d by 100. * Only c o r r e l a t i o n s >0.20 are reported. Results i n d i c a t e that a strong r e l a t i o n s h i p existed between teachers' expectations of students' enjoyment of problem sol v i n g and t h e i r expectations of students' achievement on the same to p i c . In addition, t h e i r s a t i s f a c t i o n with the teaching of problem-solving co r r e l a t e d strongly with t h e i r perceptions of student enjoyment and achievement. Correlations among these v a r i a b l e s ranged from 0.31 to 0.52. These r e s u l t s suggest that 120 a common construct may underlie these v a r i a b l e s . Further evidence of a common r e l a t i o n s h i p among these v a r i a b l e s was gained through the fa c t o r analyses i n which they a l l loaded into the same f a c t o r . These v a r i a b l e s may a l l be a measure of teacher a t t i t u d e toward problem s o l v i n g . Other r e l a t i o n s h i p s of i n t e r e s t were found among the numbers of d i f f e r e n t approaches and sources of d i f f e r e n t materials or resources u t i l i z e d by teachers. For example, the number of d i f f e r e n t sources of exercises and the number of d i f f e r e n t a c t i v i t i e s used to motivate students f o r problem s o l v i n g c o r r e l a t e d at 0.52. Relationships among the other v a r i a b l e s c o r r e l a t e d between 0.24 and 0.46. These r e s u l t s suggest that teachers who use a v a r i e t y of approaches to the teaching of problem sol v i n g l i k e l y also use a number of d i f f e r e n t sources of exercises and motivational a c t i v i t i e s . A common construct may also underlie these v a r i a b l e s since they also loaded i n t o one fact o r at the fa c t o r a n a l y s i s stage. They could be viewed as measures of teacher f l e x i b i l i t y i n the teaching of problem solving. Several s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s were found between problem-solving process v a r i a b l e s and students' achievement i n mathematics. Although they were of a les s e r magnitude than were c o r r e l a t i o n s among the independent v a r i a b l e s , several are worthy of comment. For example, teachers' expectations of students' enjoyment of and achievement i n problem s o l v i n g showed the strongest r e l a t i o n s h i p s with achievement. These v a r i a b l e s c o r r e l a t e d r e s p e c t i v e l y at 0.16 and 0.22 with problem solving, 0.19 and 0.30 with t e s t t o t a l , 121 and 0.18 and 0.26 with a p p l i c a t i o n s . These findings confirm s i m i l a r ones i n studies by Good and Grouws (1977), Evertson et. a l . (1980) and Brophy (1982b). The number of d i f f e r e n t s t r a t e g i e s taught, p a r t i c i p a t i o n i n i n - s e r v i c e a c t i v i t i e s , number of d i f f e r e n t sources of exercises, and number of problem types taught also c o r r e l a t e d s i g n i f i c a n t l y with a l l three of the achievement domains. P o s i t i v e r e l a t i o n s h i p s between d i f f e r e n t approaches used and achievement i n mathematics were also found i n studies by Rosenshine and Furst (1971), Cooney, Davis and Henderson (1975) and Kolb (1977). Based on these r e s u l t s , student achievement was found to be associated with teacher expectations of students, involvement i n i n - s e r v i c e a c t i v i t i e s and with f l e x i b i l i t y of approach to the teaching of problem s o l v i n g . 4. What r e l a t i o n s h i p s e x i s t among teachers• perceptions of mathematics and students' perceptions of mathematics; and between these perceptions and students• achievement i n mathematics? Teachers' perceptions of the importance of, d i f f i c u l t y i n teaching and enjoyment i n teaching mathematics were s i g n i f i c a n t l y r e l a t e d to one another. The c o r r e l a t i o n c o e f f i c i e n t s among these v a r i a b l e s were as follows: importance and d i f f i c u l t y (0.20), enjoyment and d i f f i c u l t y (0.37), and importance and enjoyment (0.58). These r e s u l t s showed that the strongest r e l a t i o n s h i p was between importance and enjoyment ra t i n g s . Relationships between teacher perceptions of mathematics and student achievement were l e s s pronounced. The strongest of these c o r r e l a t i o n s , which were s t a t i s t i c a l l y 122 s i g n i f i c a n t , were between the d i f f i c u l t y to teach r a t i n g and achievement. Students' perceptions of mathematics were more strongly r e l a t e d than those of t h e i r teachers. C o r r e l a t i o n c o e f f i c i e n t s among t h e i r ratings were as follows: importance and enjoyment (0.53), importance and d i f f i c u l t y (0.57), and d i f f i c u l t y and enjoyment (0.59). These data i n d i c a t e that students tend to have common perceptions of the three ratings f o r mathematics. A l l of t h e i r r a t i n g s c o r r e l a t e d s i g n i f i c a n t l y with achievement. For example, student enjoyment ratings c o r r e l a t e d between 0.17 and 0.27 with the three c r i t e r i o n v a r i a b l e s . A number of e a r l i e r studies found s i m i l a r r e l a t i o n s h i p s between student a t t i t u d e and achievement (e.g. Wess, 1970; P h i l l i p s , 1973; Robinson, 1973; Evertson e t . a l . , 1980; and Newman, 1984). These r e s u l t s i ndicate that students' perceptions of mathematics r e l a t e s i g n i f i c a n t l y to t h e i r achievement. On the basis of t h i s f i n d i n g , curriculum developers and teachers need to consider students* perceptions of mathematics as an important f a c t o r i n success i n learning the subject. 5. What diff e r e n c e s , i f any, e x i s t i n the strengths of the r e l a t i o n s h i p s i n questions 1 to 4 when achievement i s measured at d i f f e r e n t cognitive behavior l e v e l s ? Peterson and Fennema (1985) claimed that, f o r the most part, researchers have not investigated whether the classroom processes that f a c i l i t a t e students* learning of low cognitive l e v e l tasks are the same as those i n high l e v e l tasks. A need f o r such i n v e s t i g a t i o n was pointed out by Good (1983) and Rosenshine (1979). An attempt to address t h i s concern was made i n the present study by examining the e f f e c t s of independent 123 v a r i a b l e s upon student performance on routine a p p l i c a t i o n items as well as on higher order problem-solving or c r i t i c a l - t h i n k i n g questions. As reported e a r l i e r , three c r i t e r i o n v a r i a b l e s were examined i n the current study. The v a r i a b l e s were t e s t t o t a l , problem s o l v i n g and a p p l i c a t i o n s . The problem-solving v a r i a b l e was comprised of t e s t items designed to measure student achievement at a high cognitive behavior l e v e l . The a p p l i c a t i o n v a r i a b l e , on the other hand, consisted of items at a lower l e v e l of c o g n i t i v e behavior. For the purposes of discussion at t h i s point, only r e s u l t s i n v o l v i n g these two v a r i a b l e s w i l l be compared. , Relationships between student background v a r i a b l e s and achievement on the two c r i t e r i o n v a r i a b l e s under examination show that although parents* educational l e v e l s c o r r e l a t e s i g n i f i c a n t l y with both, c o r r e l a t i o n s with the a p p l i c a t i o n v a r i a b l e were higher than they were with problem s o l v i n g . For example, c o r r e l a t i o n s found between achievement on a p p l i c a t i o n s and problem s o l v i n g were 0.29 and 0.20 r e s p e c t i v e l y with fathers* l e v e l of education and 0.25 and 0.20 with mothers'. No s i g n i f i c a n t c o r r e l a t i o n s were found between teacher v a r i a b l e s and achievement i n problem so l v i n g . However, 5 of the 7 teacher v a r i a b l e s showed s t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s with the a p p l i c a t i o n v a r i a b l e . A s i m i l a r pattern was found between the r e l a t i o n s h i p s with classroom organization v a r i a b l e s and the two achievement domains. For example, the type of course c o r r e l a t e d at 0.15 with achievement on problem solving, and at 0.21 with .124 achievement on a p p l i c a t i o n s . Most of the problem-solving process v a r i a b l e s c o r r e l a t e d at e i t h e r the same l e v e l or higher with a p p l i c a t i o n s than with problem s o l v i n g . Two notable exceptions were c o r r e l a t i o n s between both i n - s e r v i c e on the teaching of problem s o l v i n g and classroom features used to promote problem so l v i n g . Both of these v a r i a b l e s showed a stronger r e l a t i o n s h i p with achievement on problem s o l v i n g than on the a p p l i c a t i o n domain. Although a l l students 1 perceptions of mathematics were s i g n i f i c a n t l y r e l a t e d to achievement on both c r i t e r i o n v a r i a b l e s , they co r r e l a t e d higher on a p p l i c a t i o n s than on problem s o l v i n g . The only teachers' perception of mathematics which c o r r e l a t e d s i g n i f i c a n t l y with the c r i t e r i o n v a r i a b l e s was d i f f i c u l t y i n teaching. The magnitudes of the c o r r e l a t i o n s , however, were also higher with a p p l i c a t i o n s than problem so l v i n g . Based on t h i s examination of the r e l a t i o n s h i p s of independent v a r i a b l e s with achievement on problem s o l v i n g and a p p l i c a t i o n s , i t i s apparent that they are more strongly r e l a t e d to performance on a p p l i c a t i o n s . A summary of the number of s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s i s shown i n Table 47. 125 Table 47 S i g n i f i c a n t Re la t ionsh ips With Achievement on Problem So lv ing and A p p l i c a t i o n s Input Category Number of Variables Problem-Solving Relationships* A p p l i c a t i o n s Relationships* Stud. Bckgrd. 5 2 2 Teach. Bckgrd. 7 0 5 Class Org. 11 2 2 P. Solv. Process 11 8 9 Teach. Percept. 3 1 1 Stud. Percept. 3 3 3 Tota l 40 16 22 * Note. Number of co r r e l a t i o n s s i g n i f i c a n t at the 0.05 l e v e l . As shown i n Table 47, 16 out of a t o t a l of 40 v a r i a b l e s c o r r e l a t e d s i g n i f i c a n t l y with achievement on problem s o l v i n g . This compared with 22 s i g n i f i c a n t c o r r e l a t i o n s between the independent v a r i a b l e s and achievement on a p p l i c a t i o n s . Of these r e l a t i o n s h i p s , only two cor r e l a t e d more highly with problem s o l v i n g than with a p p l i c a t i o n s . These r e s u l t s may be due to a number of reasons. For example, more time i n school i s spent on application-type items than on c r i t i c a l t h i n k i n g questions. Therefore stronger r e l a t i o n s h i p s could develop between the inputs of schooling and achievement on those types of items. 6. How much variance i n student achievement i n mathematics i s accounted f o r by the e f f e c t s of teacher and student background, classroom organization and processes, and teachers' and students' perceptions of mathematics. 126 As reported e a r l i e r , a f a c t o r a n a l y s i s was conducted with the v a r i a b l e s contained within each input category' Factor scores were determined and the facto r s within each category were then regressed, using the step-wise method, on each of the c r i t e r i o n v a r i a b l e s . One student background fac t o r , parents' l e v e l s of education, remained i n the regression equation f o r each of the c r i t e r i o n v a r i a b l e s . I t explained 5 percent of the variance i n student achievement on problem solving, 7 percent on t e s t t o t a l and 9 percent on a p p l i c a t i o n s . These r e s u l t s i n d i c a t e that socio-economic status, as r e f l e c t e d by parents' l e v e l s of education, has an e f f e c t on achievement i n mathematics. Teacher background fact o r s explained l i t t l e variance i n student achievement. Only one factor, comprised of v a r i a b l e s measuring p r o f e s s i o n a l preparation, remained i n the regression equation f o r a p p l i c a t i o n s . No other teacher background fact o r s explained any appreciable amount of variance. Based on these r e s u l t s i t appears that teacher background, as defined i n the current study, has l i t t l e e f f e c t on student achievement. Classroom organization f a c t o r number 5, comprised of type of program, explained 3, 5 and 5 percent of the variances i n achievement on problem solving, t e s t t o t a l and app l i c a t i o n s r e s p e c t i v e l y . An a d d i t i o n a l 1 percent of the variance i n achievement on t e s t t o t a l was accounted f o r by a second factor, comprised of va r i a b l e s measuring t o t a l time spent on mathematics and time spent on homework-related a c t i v i t i e s . These findings provide some evidence that a d d i t i o n a l time spent i n mathematics 127 and a d d i t i o n a l proportions of c l a s s time spent on homework-re l a t e d a c t i v i t i e s can a f f e c t student achievement. Two problem-solving process fa c t o r s remained i n each of the three regression equations. Factor 2, comprised of va r i a b l e s which measured teacher expectations of student enjoyment and performance on problem solving, and t h e i r s a t i s f a c t i o n of and ease found i n teaching the t o p i c , explained the most variance i n achievement. I t accounted f o r 3, 6 and 5 percent of the variances i n achievement on problem solving, t e s t t o t a l and app l i c a t i o n s r e s p e c t i v e l y . A second f a c t o r , r e l a t e d to p a r t i c i p a t i o n i n i n - s e r v i c e a c t i v i t i e s , explained an a d d i t i o n a l 1 percent of the variance i n achievement i n problem solving, whereas a t h i r d f a c t o r , i n v o l v i n g the uses of d i f f e r e n t approaches and sources of materials accounted f o r the same a d d i t i o n a l amount of variance on achievement i n both t e s t t o t a l and a p p l i c a t i o n s . Results from these regressions show the importance of teacher a t t i t u d e s toward problem s o l v i n g i n explaining variance i n student achievement. They also provide some i n d i c a t i o n that teacher involvement i n i n - s e r v i c e a c t i v i t i e s and t h e i r use of a f l e x i b l e approach i n teaching may a f f e c t student learning of mathematics. Teachers' perceptions of the d i f f i c u l t y i n teaching mathematics explained 1, 3 and 2 percent of the variances i n achievement on problem solving, t e s t t o t a l and app l i c a t i o n s r e s p e c t i v e l y . An a d d i t i o n a l 1 percent of the variance i n achievement on t e s t t o t a l was accounted f o r by teachers' perceptions of the importance of mathematics. Their perceptions of the enjoyment they experienced i n teaching mathematics did 128 not account f o r any su b s t a n t i a l amount of variance i n achievement. Perceptions of the enjoyment students experienced i n learning mathematics accounted f o r 3 percent of the variance i n achievement on problem solving, 7 percent on t e s t t o t a l and 4 percent on a p p l i c a t i o n s . An a d d i t i o n a l 1 percent of the achievement on app l i c a t i o n s was explained by students' perceptions of the d i f f i c u l t y they experienced i n learning mathematics. These r e s u l t s provide some evidence to confirm that the development of p o s i t i v e perceptions of mathematics might be b e n e f i c i a l . Teachers should be cognizant of student a t t i t u d e s and make concerted e f f o r t s to provide students with enjoyable experiences i n mathematics. Factors which remained i n regression equations f o r each input category explained d i f f e r e n t amounts of variances i n achievement on each c r i t e r i o n v a r i a b l e . A summary of the variances accounted f o r i s shown i n Table 48. 129 Table 48 Variances i n Achievement (Percent) Accounted For Input Category Prob. Solving Test T o t a l A pplications Student Bckgrd. 5 7 9 Teacher Bckgrd. 0 0 1 Class Organization 3 5 5 P. Solving Process 4 6 6 Student Percept. 3 7 4 Teacher Percept. 1 3 2 The variances explained f o r achievement i n problem s o l v i n g ranged from none by teacher background f a c t o r s to 5 percent by student background f a c t o r s . This compared with a range by the same fa c t o r s r e s p e c t i v e l y of 0 to 7 percent f o r t e s t t o t a l and 1 to 9 percent f o r a p p l i c a t i o n s . I t i s apparent that f a c t o r s from each category explained considerably more variance i n achievement on t e s t t o t a l and on a p p l i c a t i o n s than on problem so l v i n g . The F i n a l Models One of the goals of t h i s study was to examine the variances i n achievement accounted f o r by fac t o r s within each input category and these r e s u l t s have been discussed i n the preceding section. However, i n developing a f i n a l regression model fo r each of the c r i t e r i o n v a r i a b l e s , i t was important to take into account any common variances which may have been shared among factor s from d i f f e r e n t categories. To address t h i s issue a l l 130 f a c t o r s were regressed, using the step-wise method, on each of the three achievement v a r i a b l e s . Results, summarized with beta weights, from the 1985 P r o v i n c i a l Assessment and the 1987 v a l i d a t i o n study are shown i n Figures 4 and 5. Beta weights are standardized regression c o e f f i c i e n t s , drawn from the f i n a l models discussed e a r l i e r , i n which a l l fact o r s were regressed on the c r i t e r i o n v a r i a b l e s . A comparison of beta weights provides one measure of the r e l a t i v e e f f e c t s that independent v a r i a b l e s associated with each input category have on the dependent achievement v a r i a b l e s (Pedhauzer, 1982). The value of each beta weight indicates the number of standard deviation u n i t s of change that could be predicted i n the dependent v a r i a b l e when the value of the input category changes by one standard deviation u n i t . Since the e f f e c t s a t t r i b u t e d to beta weights are add i t i v e , those weights associated with classroom organization and problem-solving process v a r i a b l e s were summed to produce a beta weight f o r the classroom processes category. The l a t t e r category, o r i g i n a l l y shown i n the conceptual model (Figure 3) introduced i n Chapter 3, i s comprised of those v a r i a b l e s contained i n each of the two sub-categories. The r e l a t i v e e f f e c t s on student achievement i n mathematics predicted by v a r i a b l e s associated with each input category from the 1985 P r o v i n c i a l Assessment are shown i n Figure 4. Results are shown f o r each of the problem so l v i n g , t e s t t o t a l and a p p l i c a t i o n domains. 131 0.50 0.40 Beta Weights 0.30 0.20 0.10 Student Student Classroom Teacher Bckgrd. Percept. Processes Percept. Legend: Achievement i n Problem Solving »• > Achievement i n Test Total - t Achievement i n Applications •> • Figure 4. E f f e c t s of schooling on achievement i n problem so l v i n g , t e s t t o t a l and a p p l i c a t i o n s . Teacher background v a r i a b l e s d i d not show any appreciable e f f e c t on student achievement i n the f i n a l model. L i t t l e e f f e c t i s a l s o a t t r i b u t e d to teachers' perceptions. They predicted a l i m i t e d change i n achievement on t e s t t o t a l , given a corresponding change i n them, and no s u b s t a n t i a l change i n achievement on e i t h e r problem sol v i n g or a p p l i c a t i o n s . Classroom processes, with a beta weight of 0.32, predicted twice the e f f e c t on achievement i n problem s o l v i n g than d i d v a r i a b l e s associated with student background and students• perceptions. The l a t t e r two categories were assigned beta weights of 0.17 and 0.13 r e s p e c t i v e l y . 132 Achievement on a p p l i c a t i o n s could be a f f e c t e d most by change i n v a r i a b l e s associated with classroom processes. The beta weight f o r t h i s category was 0.38, compared with weights of 0.26 and 0.17 f o r student background and students* perceptions r e s p e c t i v e l y . I t was of i n t e r e s t to note that change i n student background might have a considerably greater e f f e c t on achievement i n a p p l i c a t i o n s than on problem s o l v i n g or t e s t t o t a l . Test t o t a l r e s u l t s might be a f f e c t e d l e s s than those on the other two c r i t e r i o n v a r i a b l e s by change i n classroom process v a r i a b l e s . On the other hand, change i n students* perceptions would l i k e l y a f f e c t achievement i n t h i s domain more than i n the other two. The t e s t t o t a l domain was the only one i n which a change i n teachers• perceptions may r e s u l t i n an appreciable change i n student achievement. However, with a beta weight of only 0.10, i t may be considerably l e s s than a s i m i l a r change i n the other input categories. 7. What dif f e r e n c e s occur i n the r e s u l t s found through the analysis of c r o s s - s e c t i o n a l data a f t e r l o n g i t u d i n a l data are included i n the analysis? Results, showing the r e l a t i v e e f f e c t s of input categories from the 1987 v a l i d a t i o n study, are shown i n Figure 5. Cross-s e c t i o n a l r e s u l t s , based on post-test data only are compared with those from the l o n g i t u d i n a l a nalysis i n which pre- t e s t data were included. The only c r i t e r i o n v a r i a b l e reported upon i s t e s t t o t a l . 0.60 0. 50 0.40 Beta Weights 0.30 0.20 0.10 Pre- Student Student Classroom Teacher t e s t Perc. Bckgrnd. Processes Bckgrnd. Legend: Cr o s s - s e c t i o n a l r e s u l t s Longitudinal r e s u l t s Figure 5. E f f e c t s of schooling on mathematics achievement based on c r o s s - s e c t i o n a l and l o n g i t u d i n a l data. Based on data shown i n Figure 5, student background and students' perceptions show considerably stronger e f f e c t s when student e n t r y - l e v e l behaviors are not c o n t r o l l e d than when they are. When these behaviors were not c o n t r o l l e d , the beta weight f o r student background was 0.26 compared to 0.16 when they were. Students' perceptions, on the other hand, were assigned beta weights of 0.27 with the control and 0.42 without. The d i f f e r e n c e s i n student background v a r i a b l e s was expected since they are not l i k e l y to change a great deal between the beginning of a school year and the end. 134 Consequently, r e s u l t s from the pre-test would have accounted f o r a s u b s t a n t i a l p ortion of t h e i r e f f e c t on students' achievement. Likewise, part of the e f f e c t of students' perceptions of mathematics would also be accounted f o r by the pre-test. However, the beta weight of 0.27 f o r students' perceptions, based on l o n g i t u d i n a l r e s u l t s , suggests that student experiences i n a given year can s t i l l have an appreciable e f f e c t on t h e i r perceptions of mathematics. These r e s u l t s i n d i c a t e that some caution should be taken when i n t e r p r e t i n g survey data. Findings, based on student background and perceptions of mathematics, should not be a t t r i b u t e d s o l e l y to current classroom p r a c t i c e s . The r e l a t i v e e f f e c t s of classroom processes on achievement were found to be s i m i l a r i n both cases. The cr o s s - s e c t i o n a l data y i e l d e d a beta weight of 0.23 compared to 0.25 f o r the lo n g i t u d i n a l data i n t h i s category. S i m i l a r r e s u l t s with both sets of data were also found f o r the e f f e c t s of teacher background. Based on these findings, s i m i l a r e f f e c t s r e l a t e d to these two input categories could be expected from survey r e s u l t s . 5.3 IMPLICATIONS FOR DECISION MAKERS Findings from the current study provide some d i r e c t i o n f o r dec i s i o n makers who e i t h e r set p o l i c y or els e are p r a c t i t i o n e r s i n the educational system. Although the findings do not claim to i d e n t i f y causal r e l a t i o n s h i p s , they are based on measures of the strengths of r e l a t i o n s h i p s between v a r i a b l e s . Implications 135 based on the findings are d i r e c t e d at the M i n i s t r y of Education, teacher educators and teachers of mathematics. Among the purposes of p r o v i n c i a l assessments are the c o l l e c t i o n of information to a s s i s t i n d e c i s i o n making f o r the a l l o c a t i o n of resources, the development of curriculum, and d i r e c t i o n f o r future research. (Learning Assessment Branch, 1984). The information c o l l e c t e d , however, i s u s u a l l y l i m i t e d to post- t e s t data from c r o s s - s e c t i o n a l surveys. Based on findings from the 1987 v a l i d a t i o n study, care should be taken when i n t e r p r e t i n g r e s u l t s . For example, although e f f e c t s of classroom processes on student achievement were s i m i l a r a f t e r analysing both c r o s s - s e c t i o n a l and l o n g i t u d i n a l data, there were considerable d i f f e r e n c e s found i n the e f f e c t s of student background and students' perceptions of mathematics on achievement. Relationships found between students' perceptions of mathematics and t h e i r achievement i n the subject have an i m p l i c a t i o n f o r curriculum developers. These r e s u l t s confirm the importance of the a f f e c t i v e domain as an important f a c t o r i n achievement. Based on the findings i n t h i s area, curriculum developers should stres s the importance of t h i s domain i n the design of curriculum and i n the i d e n t i f i c a t i o n of resources to support i t . A number of implications are d i r e c t e d to teacher educators and to teachers. For example, s i g n i f i c a n t r e l a t i o n s h i p s found between student achievement and i n - s e r v i c e involvement, f l e x i b i l i t y of approach and the use of some classroom 136 organizational p r a c t i c e s provide d i r e c t i o n f o r pre-service and i n - s e r v i c e planning. The implications r e f e r r e d to i n t h i s s e c t i o n were di r e c t e d at d e c i s i o n makers at several l e v e l s of the educational system. They were based on findings from the present study. These findings, however, are subject to the assumptions reported i n Chapter 1 and the l i m i t a t i o n s which follow. 5 .4 LIMITATIONS OF THE STUDY Willms (i n press), i n l i s t i n g l i m i t a t i o n s of studies s i m i l a r to the present one, c i t e d problems of aggregated data, s e l e c t i o n bias and q u a l i t y of data. The data contained i n the current study are also subject to some of the r e s t r i c t i o n s he r e f e r r e d to i n the following comment: ...effectiveness studies w i l l continue to be based on data that are l e s s than wholly adequate. National data on schooling outcomes and p u p i l s ' c h a r a c t e r i s t i c s are usually derived from multi-purpose surveys that have a number of competing research goals and p r i o r i t i e s . Researchers u s u a l l y need to make compromises that determine the length and content of t e s t s and questionnaires, the method of data c o l l e c t i o n , and the sample design. Along with t h i s problem, there i s often resistance from p u p i l s , parents, teachers and administration who view the c o l l e c t i o n as an incursion upon t h e i r r i g h t to c o n f i d e n t i a l i t y , or view i t with suspicion, not sharing the goals of those c o l l e c t i n g the data, and fearing the data w i l l be used to hold them accountable (p. 3). Some evidence of resistance from teachers was apparent by the number of classes which had to be dropped from the a n a l y s i s . This was demonstrated i n a number of ways; two of which were 137 non-completion of the Teacher Questionnaire or the absence of a c l a s s code number e i t h e r on student t e s t booklets or on those questionnaires which were returned. E i t h e r of these conditions was s u f f i c i e n t to exclude r e s u l t s of a c l a s s from the analysis since a matching teacher f i l e could not be found. For example, i n the l a r g e s t d i s t r i c t i n the province only 24 matches were found. This was approximately h a l f of the number which were found f o r several other smaller d i s t r i c t s . Since the present study involved a r e - a n a l y s i s of data c o l l e c t e d i n the 1985 P r o v i n c i a l Assessment of Mathematics and a r e p l i c a t i o n of i t , f o r purposes of examining e f f e c t s when pre-t e s t data are introduced i n t o the an a l y s i s , a number of l i m i t a t i o n s were due to pre-determined parameters based on the design and instrumentation used i n the 1985 Assessment. Subsequent methods of analysis used i n the current study added further to these l i m i t a t i o n s . A l i s t of l i m i t a t i o n s follows: Estimation of Class Means Since the 1985 study used a multiple matrix sampling design i n which several t e s t booklets were used, c l a s s achievement and perception means were estimates. Approximately one i n three students wrote a given achievement item and responded to the perception scale contained i n Booklet R. The standard error of estimate was reduced somewhat by the r e l a t i v e l y large number of items i n each achievement domain. The same l i m i t a t i o n applied to c l a s s means f o r achievement i n the v a l i d a t i o n sample. Perception means, however, were based on r e s u l t s from a l l students since a l l booklets contained the perception scale i n that administration. 138 D e f i n i t i o n o f Variables As indicated e a r l i e r , independent v a r i a b l e s i n the 1985 Mathematics Assessment were chosen on the basis of what previous research had to say about t h e i r e f f e c t s on student achievement. However, t h e i r operational d e f i n i t i o n s were based on the multiple choice items used to measure them i n that study. This l i m i t a t i o n i s common to other s i m i l a r studies where independent v a r i a b l e s were defined i n a s i m i l a r way. This p r a c t i c e , however, makes comparisons across studies more d i f f i c u l t since d i f f e r e n t a t t r i b u t e s may have been measured to represent the same v a r i a b l e . Impacts o f Independent Variables on the Achievement of I n d i v i d u a l Students The e f f e c t s independent v a r i a b l e s have on achievement may d i f f e r from one student to another (Luecke & McGinn, 1975). In an attempt to c o n t r o l f o r t h i s some studies have grouped students by performance l e v e l or some other c r i t e r i a and then examined e f f e c t s of the independent v a r i a b l e s on each group. However, student l e v e l data are necessary to u t i l i z e t h i s strategy. Since multiple matrix sampling was employed i n the study i t was not f e a s i b l e to report student l e v e l r e s u l t s . The Class as a Unit o f Analysis The present study used the c l a s s and teacher as un i t s of an a l y s i s . Due to the matrix sample plan employed i n the 1985 P r o v i n c i a l Assessment of Mathematics i t was not f e a s i b l e to use the student rather than the c l a s s f o r t h i s purpose. Burstein 139 (1980) I d e n t i f i e d a number of issues created i n aggregating data to the c l a s s l e v e l . For example, i n t h i s study w i t h i n c l a s s variance was not accounted f o r and the absence of t h i s data l i k e l y had a moderating e f f e c t on the magnitude of c o r r e l a t i o n c o e f f i c i e n t s . These l i m i t a t i o n s shaped the methods employed i n the analyses and the reporting of r e s u l t s . However, they also established parameters l i m i t i n g , the accuracy of conclusions drawn i n the study. 5.5 IMPLICATIONS FOR FURTHER RESEARCH Several findings from the current study have implications f o r f u r t h e r research. These include a need to examine r e l a t i o n s h i p s between classroom process v a r i a b l e s and the achievement of students of d i f f e r e n t l e v e l s of a b i l i t y , and the determination of e f f e c t s of school-related v a r i a b l e s on student achievement. In addition, further research on the nature and e f f e c t s of a number of independent v a r i a b l e s which were found to c o r r e l a t e s i g n i f i c a n t l y with achievement may provide d i r e c t i o n to enhance the effectiveness of the educational environment. The c r i t e r i o n v a r i a b l e s i n the current study were c l a s s means f o r student achievement on three domains. These achievement r e s u l t s were aggregated to the c l a s s l e v e l f o r two reasons. F i r s t , r e l a t i o n s h i p s between teachers' behaviors and how they a f f e c t e d the achievement of t h e i r respective classes were questions of primary i n t e r e s t to t h i s study. To answer these questions the teacher and c l a s s were the l o g i c a l u n i t s of 140 a n a l y s i s . Second, the multiple matrix sampling design used i n the 1985 P r o v i n c i a l Assessment u t i l i z e d more than one t e s t booklet. Consequently student-level r e s u l t s were not meaningful. I t would be of i n t e r e s t , however, to examine the e f f e c t s of teacher behaviors on students of d i f f e r e n t l e v e l s of a b i l i t y . Further research could address t h i s question by u t i l i z i n g a s i n g l e t e s t booklet. Willms and Cuttance (1985) reported that, "...although v a r i a b l e s such as c l a s s s i z e , school s i z e , i n s t r u c t i o n a l s t r a t e g i e s and school expenditure do not appear to have strong d i r e c t e f f e c t s on cognitive achievement, they may have i n d i r e c t e f f e c t s by f a c i l i t a t i n g e f f e c t i v e teaching and c o n t r i b u t i n g to the o v e r a l l function of the school" (p. 290). Further research i n t h i s area which w i l l c o n t r o l f o r school e f f e c t s from outside the classroom may provide a d d i t i o n a l i n s i g h t i n t o the e f f e c t s of classroom processes on student achievement. Included among the v a r i a b l e s which were found to r e l a t e s i g n i f i c a n t l y to student achievement were students' perceptions of mathematics. Further research i n t h i s area could focus on the student behaviors and b e l i e f s which are associated with t h e i r perceptions of mathematics. 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A meta-analysis of the r e l a t i o n s h i p between science achievement and science a t t i t u d e : Kindergarden through college. Journal of Research in Science Teaching, 20, 839-850. 152 Wirvteler, A. (1983) . Causal modelling i n evaluation research. Higher Education Journal, 12, 315-329. Wynne, E.A. (1980). Looking at schools: Good, bad and indifferent. Lexington, Mass.: D.C. Heath. APPENDIX A B r i t i s h Columbia Mathematics Assessment Test Booklets R, S, and T - 1 -SCALE R; MATHEMATICS IN SCHOOL For each of the items i n t h i s scale, three answers are needed. A. T e l l how important you think the topic i s . B. T e l l how how easy i t i s . C. T e l l how much you l i k e the t o p i c . If you are not sure what a topic means, leave i t s three answers blank. 1. Adding, subtracting, m u l t i p l y i n g and d i v i d i n g f r a c t i o n s A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided ea sy very easy d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t Adding, subtracting, m u l t i p l y i n g and d i v i d i n g decimals A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy c. d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t Working with percents A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided ea sy very easy d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t 156 Learning about estimation A. C. not at a l l important very d i f f i c u l t d i s l i k e a l o t not important d i f f i c u l t d i s l i k e undecided undecided undecided important easy l i k e very important very easy l i k e a l o t Memorizing basic f a c t s A. B. C. not at a l l important very d i f f i c u l t d i s l i k e a l o t not important d i f f i c u l t d i s l i k e undecided undecided undecided important easy l i k e very important very easy l i k e a l o t Solving equations A. B. C. not at a l l important very d i f f i c u l t d i s l i k e a l o t not important d i f f i c u l t d i s l i k e undecided undecided undecided important easy l i k e very important very easy l i k e a l o t o Solving word problems B^ £ L not at a l l important very d i f f i c u l t d i s l i k e a l o t not important d i f f i c u l t d i s l i k e undecided undecided undecided important easy l i k e very important very easy l i k e a l o t - 3 -157 8. L e a r n i n g a b o u t t h e m e t r i c s y s t e m n o t a t a l l i m p o r t a n t v e r y d i f f i c u l t d i s l i k e a l o t n o t i m p o r t a n t d i f f i c u l t d i s l i k e u n d e c i d e d u n d e c i d e d u n d e c i d e d i m p o r t a n t e a s y l i k e v e r y i m p o r t a n t v e r y e a s y l i k e a l o t W o r k i n g w i t h p e r i m e t e r a n d a r e a A;_ B ^ C_j_ n o t a t a l l i m p o r t a n t v e r y d i f f i c u l t d i s l i k e a l o t n o t i m p o r t a n t d i f f i c u l t d i s l i k e u n d e c i d e d u n d e c i d e d u n d e c i d e d i m p o r t a n t e a s y l i k e v e r y i m p o r t a n t v e r y e a s y l i k e a l o t 10. D o i n g g e o m e t r y A. B. C. n o t a t a l l i m p o r t a n t v e r y d i f f i c u l t d i s l i k e a l o t n o t i m p o r t a n t d i f f i c u l t d i s l i k e u n d e c i d e d u n d e c i d e d u n d e c i d e d i m p o r t a n t e a s y l i k e v e r y i m p o r t a n t v e r y e a s y l i k e a l o t - 4 -158 STUDENT BACKGROUND INFORMATION 1. W h a t l a n g u a g e d i d y o u f i r s t l e a r n t o s p e a k ? A. E n g l i s h B. F r e n c h C. A n o t h e r l a n g u a g e 2. W h at l a n g u a g e d o y o u s p e a k m o s t o f t e n i n y o u r home now? A. E n g l i s h B. F r e n c h C. A n o t h e r l a n g u a g e 3. W h at p r o g r a m a r e y o u i n ? A. R e g u l a r P r o g r a m i n E n g l i s h B. E a r l y F r e n c h I m m e r s i o n C. L a t e F r e n c h I m m e r s i o n D. " P r o g r a m m e - C a d r e de F r a n ^ a i s " 4. I n t h i s c l a s s M a t h e m a t i c s i s t a u g h t i n A. E n g l i s h B. F r e n c h 5. Do y o u s o m e t i m e s g o t o a L e a r n i n g A s s i s t a n c e C e n t r e i n y o u r s c h o o l f o r h e l p w i t h m a t h e m a t i c s ? A. T h e r e i s no L e a r n i n g A s s i s t a n c e C e n t r e f o r m a t h e m a t i c s i n t h i s s c h o o l . B. Y e s , I d o . C. N o , I d o n ' t . 6. Do y o u s o m e t i m e s g o t o a n E S L c l a s s ( E n g l i s h a s a S e c o n d L a n g u a g e ) i n y o u r s c h o o l ? A. T h e r e i s no E S L c l a s s i n t h i s s c h o o l . B. Y e s , I d o . C. No, I d o n ' t . - 5 - 159 How long d i d i t take you to do your l a s t mathematics homework assignment? A. We never have mathematics homework i n t h i s c l a s s . B. Between 1 and 10 minutes C. Between 11 and 30 minutes D. Between 31 and 60 minutes E. More than an hour 8. About how much time d i d you spend doing homework i n a l l subjects yesterday? A. I didn't have any homework to do yesterday. B. Less than 30 minutes C. Between 30 minutes and 1 hour D. From 1 to 2 hours E. More than 2 hours 9. What was the highest l e v e l of school or co l l e g e attended by your father or male guardian? A. Very l i t t l e or no schooling at a l l B. Elementary school C. Secondary school D. College, u n i v e r s i t y or some other form of post-secondary t r a i n i n g E. I don't know. 10. What was the highest l e v e l of school or co l l e g e attended by your mother or female guardian? A. Very l i t t l e or no schooling at a l l B. Elementary school C. Secondary school D. College, u n i v e r s i t y or some other form of post-secondary t r a i n i n g E. I don't know. 11. Here i s a l i s t of reasons f o r studying mathematics. Which do you believe i s most important? A. To prepare f o r the next year's mathematics course B. To learn how to perform c a l c u l a t i o n s accurately C. To learn how to use mathematics to solve problems i n the r e a l world D. To learn to think l o g i c a l l y E. To learn what mathematics i s 160 - 6 -12. Which of these reasons for studying mathematics do you believe to be l e a s t important? A. To prepare for the next year's mathematics course B. To le a r n how to perform c a l c u l a t i o n s accurately C. To le a r n how to use mathematics to solve problems i n the r e a l world D. To learn to think l o g i c a l l y E . To le a r n what mathematics i s Both answers given for questions 13-16 are c o r r e c t . If you were asked each question, which one of the two answers comes to mind f i r s t ? 13. How much does a b i c y c l e weigh? A. About 15 kilograms B. About 35 pounds 14. What i s the temperature i n t h i s room? A. About 70 degrees B. About 20 degrees 15. How f a r i s i t from Prince George to Prince Rupert? A. About 700 kilometres B. About 450 miles 16. How much gasoline can the gas tank i n a large car hold? A. About 20 gallons B. About 90 l i t r e s 161 1. The statement " t h i r t y i s l e s s than f o r t y - f i v e " i s shown by A. 30 >'45 B. 30 < 45 C. 45 < 30 D. 45 ^ 30 E. I don't know. 2. Which i s the c o r r e c t name for the missing number? 3 x 26 = (3 x [3 + (3 x 6) A. 2 B. 6 C. 20 D. 26 E. I don't know. 3. Joe packs tomatoes 4 to a box. If he has packed 18 tomatoes, which box i s he now packing? A. the four t h B. the f i f t h C. the s i x t h D. the eighteenth E. I don't know. 4. M u l t i p l y : 403 x 59 A. 24 337 B. 5 642 C. 23 777 D. 3 627 E. I don't know. KINDS OF LUNCHES STUDENTS EAT D. 300 E. I don't know. Mike f l i p s 2 dimes. What i s the p r o b a b i l i t y that t h e y " w i l l both land heads? 2 3 D. E. I don't know. Which s t a t i s t i c t e l l s you which event happened the most frequently? A. mode B. mean C. median D. ra ng e E. I don't know. 163 Which pair of l i n e segments shown below have lengths which are i n the r a t i o of 1 to 4? P I 1 1 Q l 1 +• R H S l - H 1 H H I-A. P and Q B. R and S C. P and S D. R and T E. I don't know. 9. The following diagram of a playground i s drawn to a scale of 1 cm = 2 m. What i s the a c t u a l length of the longest side of the playground? 5 cm 10 cm A. 2 m B. 10 m C. 20 m D. 100 m E. I don't know. - 10 -The chart shows the population of the earth at d i f f e r e n t times. Year 1650 1700 1750 1800 1850 1900 1950 Population i n B i l l i o n s 0.60 0.62 0.80 0.95 1.20 1.70 2.55 Which 50 year period showed the l a r g e s t gain i n population? A. 1700-1750 B. 1800-1850 C. 1850-1900 D. 1900-1950 E. I don't know. Which one of the following keys would you push to get back the answer to a c a l c u l a t i o n which you had stored i n the E. I don't know. 165 - 11 -Which one of the following shapes i s a c y l i n d e r ? A. ins i d e the f i g u r e . B. outside the f i g u r e . C. on the boundary of the f i g u r e . D. neither i n s i d e , outside nor on the boundary of the f i g u r e . E. I don't know. 166 - 12 -3 14. Mrs. Smith baked 48 cookies. B i l l y ate — of the cookies and Betty 1 8 ate — of the cookies. In a l l , how many cookies were eaten? A. 16 B. 18 C. 20 D. 24 E. I don't know. 1 5 . Subtract: 51.2 - 4.35 A. 46.95 B. 46.85 C. 17.7 D. 7.7 E. I don't know. 16. Which number i s la r g e s t ? A. f •>• I E. I don't know. 167 - 13 -17. Marbles are arranged i n the shape of a t r i a n g l e on the f l o o r . How many marbles are there i n a t r i a n g l e with 7 marbles i n the base? A. 12 B. 28 C. 42 D. 49 E. I don't know. 18. The stickman below i s Mr. Bi g . He i s 9 paper c l i p s t a l l or 6 buttons t a l l . There i s another stickman, Mr. Short, who i s 6 paper c l i p s t a l l . How many buttons t a l l would he be? C. 4 D. 5 E. I don't know. - 14 -How many p a i r s of p a r a l l e l planes are there i n the following 2 3 4 6 I don't know. A. B. C. D. E. In which t r i a n g l e i s angle X an obtuse angle? 250 g i s how many kilograms? A. 25 B. 250 C. 0.25 D. 2.5 E. I don't know. 169 - 15 -22. You wish to c a l c u l a t e 6% of 85 on the c a l c u l a t o r . Which one of the following sequences of keystrokes w i l l l i k e l y give the c o r r e c t answer? A. BESSEB B. ESSE c EEEEEE D. EEEEE E. I don't know. 0. MC- |MRl ; M - |M 9 ! i ~ E m al uEEB 23. An imaginary computer can draw p i c t u r e s on a t e l e v i s i o n screen. When given the i n s t r u c t i o n MOVE i t w i l l draw a p i c t u r e l i k e t h i s When given the i n s t r u c t i o n MOVE R MOVE i t w i l l draw t h i s p i c t u r e Which one of the following sets of i n s t r u c t i o n s would draw a square? A. R R R R B . MOVE R R R C. MOVE MOVE MOVE MOVE D . MOVE R MOVE R MOVE R MOVE E. I don't know. - 16 -An imaginary computer w i l l input two numbers and p r i n t t h e i r sum these i n s t r u c t i o n s are given to i t : TELL A TELL B WRITE A + P What would the same computer p r i n t i f given these i n s t r u c t i o n s ? TELL A TELL B WRITE A * B A. the sum of the numbers B . the product of the numbers C. the quotient of the numbers D. the d i f f e r e n c e of the numbers E. I don't know. How many d i f f e r e n t routes are there from A to B? You may t r a v e l only up and to the r i g h t . A. 8 B. 10 C. 15 D. 18 E. I A* B Keeping the top up, i n how many d i f f e r e n t ways can the cube on l e f t be p laced i n the square hole i n the f i g u r e on the r i g h t ? O Which one of the fo l lowing diagrams shows the f l i p image of the man shown to the r i g h t ? - 18 - 1 7 2 29. Of t h e f o l l o w i n g e x p r e s s i o n s , w h i c h o n e r e p r e s e n t s a number n i n c r e a s e d b y 5? A. 5 - n B. n + 5 C. 5 < n D. 5 n E. I d o n ' t know. 30. " M i k e p a i d x d o l l a r s f o r y m e t r e s o f r o p e . How much d i d o n e m e t r e c o s t ? " I f x a n d y w e r e g i v e n n u m e r i c a l v a l u e s , w h i c h o n e o f t h e f o l l o w i n g o p e r a t i o n s w o u l d y o u u s e t o f i n d t h e p r i c e o f o n e m e t r e o f r o p e ? A. a d d i t i o n B. s u b t r a c t i o n C. m u l t i p l i c a t i o n D. d i v i s i o n E. I d o n ' t know. - 19 -12m The mat and the f l o o r shown on the r i g h t ate s i m i l a r shapes. How many mats would be needed to cover the f l o o r ? 6m 4f0 A. 4 B. 6 C. 9 D. 10 E. I don * t know. - 20 -A x (V + •) i s equal to A. A x • + V B. A x V + • C. (A x V) + (A x •) D. (A + V) x (A + •) E. I don't know. The s o l u t i o n of 2n + 8 = 20 i s : A. 12 B. 14 C. 6 D. 10 E. I don't know. If n = 5, then In + 4 = A. 14 B. 18 C. 20 D. 11 E. I don't know. Which one of the following expressions represents twice a number l e s s 5? A. 2x + 10 B. 2x - 10 C. 2x - 5 D. 2x + 5 E. I don't know. - 21 -17 5 37. W r i t t e n a s a p e r c e n t , — = A. 5 % B. 0 . 5 % C. 2 0 % D. 5 0 % E. I d o n ' t know. W h i c h o n e o f t h e f o l l o w i n g i s a q u a d r i l a t e r a l ? D. E. I d o n ' t know. 39. W h i c h o n e o f t h e f o l l o w i n g i s a m e a s u r e o f d i s t a n c e a r o u n d a c i r c l e ? A. d i a m e t e r B. r a d i u s C. a r e a D. c i r c u m f e r e n c e E. I d o n ' t know. - 22 -176 40. Divide: 45 ) 1232 A. 25 remainder 7 B. 27 remainder 17 C. 29 remainder 27 D. 207 remainder 17 E. I don't know. 41. 3.008 written i n words i s A. three hundred eight. B. three thousand eight. C. three and eight hundredths. D. three and eight thousandths. E. I don't know. 42. Which one of the following numbers i s largest? A. 0.694 B. 0.07 C. 0.76 D. 0.0816 E. I don't know. 43. M u l t i p l y : 0.01 x 2300 A. 23 B. 230 C. 2 300 D. 23 000 E. I don't know. - 23 -177 44. Which one of the following f i g u r e s i s congruent to the f i g u r e shown to the r i g h t ? B. E. I don't know. 45. Estimate the number of degrees i n angle Y of t h i s t r i a n g l e . Y A. 60° B. 90° C. 30° D. 120° E. I don 't know. 178 46. At what time d i d the highest temperature reading occur? k k Ii :: : A \ Wi l l • m A. 3 am B. 2 pm C. 4 pm D. midnight E. I don't know. 47. Four spinners are shown below. Suppose you LOSE the game i f the pointer lands on 1. Which spinner would you choose? - 25 - 179 48. In the m e t r i c system, what does the p r e f i x " c e n t i " mean? A. j— of the u n i t of measure B. of the u n i t of measure C. 10 times the u n i t of measure D. 100 times the u n i t of measure E. I don't know. 49. 5 metres i s the same l e n g t h as: A. 50 centimetres B. 500 centimetres C. 50 m i l l i m e t r e s D. 500 m i l l i m e t r e s E. I don't know. 180 181 1 -S C A L E S: GENDER AND MATHEMATICS F o r e a c h o f t h e s e i t e m s c h o o s e t h e a n s w e r w h i c h b e s t d e s c r i b e s how y o u f e e l . 1. Men make b e t t e r s c i e n t i s t s a n d e n g i n e e r s t h a n women. A. s t r o n g l y Q > D i g e c < U n c e c i d e d D . A g r e e E . S t r o n g l y D i s a g r e e 3 A g r e e 2. G i r l s h a v e m o r e n a t u r a l a b i l i t y i n m a t h e m a t i c s t h a n b o y s . S t r o n g l y . _ _. „ „ ^ „ '„ S t r o n g l y A. 3 B. D i s a g r e e C. U n d e c i d e d D. A g r e e E. 3 J D i s a g r e e A g r e e 3. B o y s n e e d t o know m o r e m a t h e m a t i c s t h a n g i r l s . S t r o n g l y „ ^. ~ „ j -^ J ^ » ,-. S t r o n g l y A. . J B. D i s a g r e e C. U n d e c i d e d D. A g r e e E. 3 2 D i s a g r e e A g r e e 4. A woman n e e d s a c a r e e r j u s t a s much a s a man d o e s . S t r o n g l y „ ~- ~ o o ^ , r. S t r o n g l y A. . 3 2 B. D i s a g r e e C. U n d e c i d e d D. A g r e e E. 3 2 D i s a g r e e A g r e e 5. My f a t h e r e n j o y s d o i n g m a t h e m a t i c s . S t r o n g l y „ ~- ^ r, J • J J „ » S t r o n g l y A. _. ^ 2 B. D i s a g r e e C. U n d e c i d e d D. A g r e e E. 3 D i s a g r e e A g r e e 6. My m o t h e r e n j o y s d o i n g m a t h e m a t i c s . . S t r o n g l y „ ^. „ „ - , • - , ^ ~ » S t r o n g l y A. . ^ •* B. D i s a g r e e C. U n d e c i d e d D. A g r e e E. _ 3 2 D i s a g r e e A g r e e 7. My f a t h e r i s u s u a l l y a b l e t o h e l p me w i t h my m a t h e m a t i c s homework i f I a s k h i m t o h e l p . . S t r o n g l y „ ~. O M J - J J n * c S t r o n g l y A. 3 1 B. D i s a g r e e C. U n d e c i d e d D. A g r e e E. D i s a g r e e 3 A g r e e - 2 - 182 My mother i s usually able to help me with my mathematics homework i f I ask her to help. Strongly „ . , . Stronqly A. 1 B. Disagree C. Undecided D. Agree E. y * Disagree 3 Agree My mother thinks that learning mathematics i s important for me. Strongly _ _ . ' o „ J ^ ~ « ~ Strongly A. 3 B. Disagree C. Undecided D. Agree E. Disagree 3 Agree 10. My father thinks that learning mathematics i s important for me. Strongly „ „. - „ ^ . ^ „ Strongly A. 1 B. Disagree C. Undecided D. Agree E. 3 J Disagree 3 Agree 11. My father wants me to do well i n mathematics. Strongly „ „. „ „ „ Strongly A. _. 3 - 1 B. Disagree C. Undecided D. Agree E. 3 J Disagree Agree 12. My mother wants me to do well i n mathematics. Strongly „ „ » ^ Strongly A. . 2 B. Disagree C. Undecided D. Agree E. " 1 Disagree Agree 13. G i r l s can do better than boys i n mathematics. . Strongly ' „ „ ^ ^ * P Strongly A. ^. ^ 1 B. Disagree C. Undecided D. Agree E. Disagree 3 Agree 14. My mother i s usually very interested i n helping me with mathematics. Stronqly Strongly A. . y * B. Disagree C. Undecided D. Agree E. . J; Disagree 3 Agree 15. My father i s u s u a l l y very inter e s t e d i n helping me with mathematics. Strongly „ „ „ * ^ ^ * „ Strongly A* r , ^ „ ~ B » Disagree C. Undecided D. Agree E. * J Disagree 3 Agree - 3 - 183 STUDENT BACKGROUND INFORMATION 1. W h a t l a n g u a g e d i d y o u f i r s t l e a r n t o s p e a k ? A. E n g l i s h B. F r e n c h C. A n o t h e r l a n g u a g e 2. W h a t l a n g u a g e d o y o u s p e a k m o s t o f t e n i n y o u r home now? A. E n g l i s h B. F r e n c h C. A n o t h e r l a n g u a g e 3. W h a t p r o g r a m a r e y o u i n ? A. R e g u l a r P r o g r a m i n E n g l i s h B. E a r l y F r e n c h I m m e r s i o n C. L a t e F r e n c h I m m e r s i o n D. " P r o g r a m m e - C a d r e d e F r a n c a i s " 4. I n t h i s c l a s s M a t h e m a t i c s i s t a u g h t i n A. E n g l i s h B. F r e n c h 5. Do y o u s o m e t i m e s g o t o a L e a r n i n g A s s i s t a n c e C e n t r e i n y o u r s c h o o l f o r h e l p w i t h m a t h e m a t i c s ? A. T h e r e i s no L e a r n i n g A s s i s t a n c e C e n t r e f o r m a t h e m a t i c s i n t h i s s c h o o l . B. Y e s , I d o . C. No, I d o n ' t . 6. Do y o u s o m e t i m e s g o t o a n E S L c l a s s ( E n g l i s h a s a S e c o n d L a n g u a g e ) i n y o u r s c h o o l ? A. T h e r e i s no E S L c l a s s i n t h i s s c h o o l . B. Y e s , I d o . C. No, I d o n ' t . - 4 -184 7. How l o n g d i d i t t a k e y o u t o d o y o u r l a s t m a t h e m a t i c s homework a s s i g n m e n t ? A. We n e v e r h a v e m a t h e m a t i c s homework i n t h i s c l a s s . B. B e t w e e n 1 a n d 10 m i n u t e s C. B e t w e e n 1 1 a n d 30 m i n u t e s D. B e t w e e n 3 1 a n d 60 m i n u t e s E. M o r e t h a n a n h o u r 8. A b o u t how much t i m e d i d y o u s p e n d d o i n g h o m e w o r k i n a l l s u b j e c t s y e s t e r d a y ? A. I d i d n ' t h a v e a n y homework t o d o y e s t e r d a y . B. L e s s t h a n 30 m i n u t e s C. B e t w e e n 30 m i n u t e s a n d 1 h o u r D. F r o m 1 t o 2 h o u r s E. M o r e t h a n 2 h o u r s 9. What was t h e h i g h e s t l e v e l o f s c h o o l o r c o l l e g e a t t e n d e d by y o u r f a t h e r o r m a l e g u a r d i a n ? A. V e r y l i t t l e o r no s c h o o l i n g a t a l l B. E l e m e n t a r y s c h o o l C. S e c o n d a r y s c h o o l D. C o l l e g e , u n i v e r s i t y o r some o t h e r f o r m o f p o s t - s e c o n d a r y t r a i n i n g E. I d o n ' t know. 1 0 . What was t h e h i g h e s t l e v e l o f s c h o o l o r c o l l e g e a t t e n d e d by y o u r m o t h e r o r f e m a l e g u a r d i a n ? A. V e r y l i t t l e o r no s c h o o l i n g a t a l l B. E l e m e n t a r y s c h o o l C. S e c o n d a r y s c h o o l D. C o l l e g e , u n i v e r s i t y o r some o t h e r f o r m o f p o s t - s e c o n d a r y t r a i n i n g E . I d o n ' t know. 1 1 . H e r e i s a l i s t o f r e a s o n s f o r s t u d y i n g m a t h e m a t i c s . W h i c h d o y o u b e l i e v e i s m o s t i m p o r t a n t ? A. T o p r e p a r e f o r t h e n e x t y e a r ' s m a t h e m a t i c s c o u r s e B. T o l e a r n how t o p e r f o r m c a l c u l a t i o n s a c c u r a t e l y C. T o l e a r n how t o u s e m a t h e m a t i c s t o s o l v e p r o b l e m s i n t h e r e a l w o r l d D. T o l e a r n t o t h i n k l o g i c a l l y E. T o l e a r n w h a t m a t h e m a t i c s i s 185 5 -12. Which of these reasons for studying mathematics do you bel i e v e to be l e a s t important? A. To prepare f or the next year's mathematics course B. To learn how to perform c a l c u l a t i o n s accurately C. To learn how to use mathematics to solve problems .in the r e a l world D. To learn to think l o g i c a l l y E. To learn what mathematics i s Both answers given for questions 13-16 are c o r r e c t . I f you were asked each question, which one of the two answers comes to mind f i r s t ? 13. How much does a b i c y c l e weigh? A. About 15 kilograms B. About 35 pounds 14. What i s the temperature i n t h i s room? A. About 70 degrees B. About 20 degrees 15. How far i s i t from Prince George to Prince Rupert? A. About 700 kilometres B. About 450 miles 16. How much gasoline can the gas tank i n a large car hold? A. About 20 gal l o n s B. About 90 l i t r e s - 6 - 186 ACHIEVEMENT SURVEY 1. By rounding o f f to the nearest ten, an estimate of 91 x 29 would be A. 270 B. 279 C. 2 700 D. 27 000 E. I don't know. 2. If John had 2300 marbles, how many bags of 10 marbles each could he make? A. 23 B. 230 C. 2 300 D. 23 000 E. I don't know. 3. Sue has 58£. If apples cost 11£ each, what i s the greatest number of whole apples that Sue can buy? A. 4 B. 5 C. 6 D . 47 E. I don't know. 4. As of June 1, 1976, the population of Canada was 22 589 416. Round o f f 22 589 416 to the nearest ten thousand. A. 22 580 000 B. 23 000 000 C. 22 600 000 D . 22 590 000 E. I don't know. - 7 -187 The heavy l i n e shows one edge of the cube. How many edges does the cube have? A. 6 B. 5 C. 9 D. 12 E . I don ' t know. 188 - 8 -7. What i s the numerical value of the computer language expression shown i n the box below? A. B. C. D. E. 2 + 3 - 2 0 2 12 18 I don't know. 8- An imaginary computer has commands c a l l e d PRAX and ADDO PRAX does the fo l l o w i n g : P r i n t out the value of X Do the command c a l l e d ADDO ADDO does the f o l l o w i n g : Add 2 to the current value of X Do the command c a l l e d PRAX What w i l l the following set of i n s t r u c t i o n s do? X = 2 PRAX A. p r i n t out the odd numbers greater than 0 B. p r i n t out a l l integers greater than 2 C. p r i n t out the even numbers greater than 0 D. p r i n t out a l l integers l e s s than 2 E. I don't know. - 9 - 189 L 9. W h i c h o n e o f t h e f o l l o w i n g l i n e s a p p e a r s p e r p e n d i c u l a r t o LM? M B. D. * E. I d o n ' t know. 1 0 . W h i c h o n e o f t h e f o l l o w i n g s t a t e m e n t s a b o u t t h e d i a g r a m s h o w n b e l o w i s INCORRECT? A. P l a n e X i s p a r a l l e l t o p l a n e Z. B. P l a n e X i s o b l i q u e t o p l a n e Y. C. P l a n e W i s p a r a l l e l t o p l a n e Y. D. P l a n e Z i s p e r p e n d i c u l a r t o p l a n e Y. E. I d o n ' t know. - 1 0 -190 11. About how much w i l l t h i s grocery b i l l t o t a l ? •ocono• groceries # $0.43 # $1.67 # $0.17 # $0.93 # $2.89 A. between $3 and $4 B. between $6 and $7 C. between $9 and $10 D. between $12 and $15 E. I don't know. 1 2 . John had 12 baseball cards. He gave of them to Jim. How many does John have l e f t ? A. 4 B. 6 C. 8 D. 9 E. I don't know. 13. Divide: .12 ).036 A. 3 B. 0.003 C. 0.3 D. 0.03 E. I don't know. - 11 -Which one of the following statements i s true? A. 100° C i s the b o i l i n g p o i n t of water. B. 212° C i s the b o i l i n g point of water. C. 32° C i s the fr e e z i n g point of water'. D. 10° C i s the fr e e z i n g point of water. E. I don't know. A ten-year-old boy i s l i k e l y to weigh: A. 35 grams B. 75 grams C. 35 kilograms D. 75 kilograms E. I don't know. The area of each of the 6 SMALL squares shown below i s 4. What i s the perimeter of the LARGE rectangle? A. 10 B. 12 C. 20 D . 24 E. I don't know. 192 12 -17. If on the r o l l of a d i e the p r o b a b i l i t y that a f i v e w i l l appear i s j, then the p r o b a b i l i t y that a f i v e o_r a three w i l l appear i s : A . B. _1_ 36 1 3 D. E. 12 I don't know. 2, 3, 4, 4, 5, 6, 8, 8, 9, 10 18. For a party game each number shown above was painted on a d i f f e r e n t Ping Pong b a l l , and the b a l l s were thoroughly mixed up i n a bowl. If a b a l l i s picked from the bowl by a bl i n d f o l d e d person, what i s the p r o b a b i l i t y that the b a l l w i l l have a 4 on i t ? A . C. 1 2 1 4 1 5 1 10 D. E. I don't know. - 13 -193 19. What i s 24% of $150.00? 100 B. $ 24.00 C. $ 36.00 D. $174.00 E . I don't know. I don't know. 21. Lines that are in the same plane and do not i n t e r s e c t are c a l l e d A. p a r a l l e l l i n e s . B. perpendicular l i n e s . C. skew l i n e s . D. oblique l i n e s . E. I don't know. - 14 - 194 A. B. C. D. E. 30 40 240 19 I don't know. 23. Which one of the following i s the same as "18 more than a number equals. 44"? A. 18n = 44 B. i f - 44 C. n + 18 = 44 0. n = 18 + 44 E. I don't know. 24. "How f a r w i l l Jan walk i f she walks at the rate of 1 kilometre i n 10 minutes?" What a d d i t i o n a l information i s needed to solve t h i s problem? A. where she was going B. how f a s t she was walking C. how long she walked D. how much she was carr y i n g E. I don't know. - 1 5 -195 2 5 . Simplify; A. 0 B. i n f i n i t y C. 6 D. cannot be done E. I don't know. 26. Subtract: 7 - -7 6 C. 7 t - 4 E. I don't know. 27. ESTIMATE the sum: 347.0 + 738.0 + 1.327 A. 1 000 B. 2 000 C. 10 000 D. 100 000 E. I don't know. - 16 -196 28. Which one of the following keys would you push to store the answer to a c a l c u l a t i o n so you could use i t l a t e r on? M + MR C E + [ C j | c E J ! »»/o» ! M C ! |MRI [M- j JM^J E B E B E E E B E B E B B B S S I don't know. 29. The following sequence of keystrokes i s c a r r i e d out on the c a l c u l a t o r : What w i l l l i k e l y be shown i n the display? 6. 69. n u. I C ; i C E . : M C : |MR; i M -jx :; [v~"j jj=j I -E E E E B E E B E E E B fT 1 +/_! ! = 76. 636. E. I don't know. - 17 197 30. To solve for b i n the equation 2b + 3 = 10, the f i r s t step should be to A. d i v i d e both sides by 3. B. mul t i p l y both sides by 2. C. subtract 3 from both sides. D. add 3 to both sides. E. I don't know. 31. Which one of the following statements i s NOT true? A. 2a + a = 3a B. 3a - a = 2a C. a + a = a D. 2a - a = 2 E. I don't know. 32. If y = 15 - x what happens to y as x increases? A. y decreases B. y increases C. y remains the same D. cannot t e l l what happens to y E. I don't know. 53. If x and y are odd numbers, what i s true about x + y? A. I t i s odd. B. I t i s even. C. . I t may be e i t h e r even or odd depending on what x and y are. D. You cannot t e l l at a l l . E. I don't know. - I S -198 34. What i s the minimum number of t i l e s that must be turned so that they are a l l facing the same way? •* • • •i • • • A. 4 B. 5 C. 6 D. 11 E. I don't know. 35. Three tennis players named Pat, Wendy and L e s l i e are walking to the courts. Pat, the best player of the three, always t e l l s the tru t h . Wendy sometimes t e l l s the truth, while L e s l i e never t e l l s the t r u t h . Who i s Pat? A B A. A B. B C. C D. e i t h e r A or B E. . I don't know. 199 36. In which one of the following diagrams i s the second f i g u r e a t r a n s l a t i o n of the f i r s t ? D. i E. I don't know. 37. In which one of the following diagrams i s the second fi g u r e a ro t a t i o n of the f i r s t ? A. B. C. D. E. I don't know. 38. Which unit should be used to measure how much l i q u i d a glass holds? A. k i l o l i t r e B. m i l l i m e t r e C. metre D. m i l l i l i t r e E. I don't know. 200 20 39. P and Q are the centres of the 2 squares shown below. What i s the distance i n centimetres from P to Q? A. B. C. D. E. 2 cm 1 2 2 cm V2 + 2 I.don't know. 2cm 40. The diagram shown below i l l u s t r a t e s a ruler-and-compass method of copying an angle. I f the construction l i n e s are l a b e l l e d as shown i n the f i g u r e , what order could be followed when completing the construction? A. 1, 5, 2, 3 and 4 B. 1, 5, 3, 4 and 2 C. 1, 5, 3, 2 and 4 D. 1, 5, 4, 3 and 2 E. I don't know. - 21 -41. An incomplete f i g u r e i s shown to the r i g h t . Which one of the fo l lowing shows the completed f i g u r e , g iven that m i s a l i n e of symmetry? o m o E. I don' 42. The f igure to the r i g h t shows a cube with one corner cut o f f and shaded. Which one of the fo l lowing drawings shows how the cube would look when viewed d i r e c t l y from above? 202 43. The graph below shows what a boy did during a period of 24 hours. Which i s the BEST estimate for the number of hours he spent watching TV? A. 3 B. 6 C . 9 0. 12 E. I don't know. 44. Pat was t e s t i n g h i s model plane. His f r i e n d s guessed how long i t would stay i n the a i r . The plane stayed up for 17 minutes. Who guessed c l o s e s t to the correc t time? Bob Carol Steven F-. •• •'• [ | 1 yiiii Ijill |Jv:o>:-i:v:v:v.J:->:-:-.v:; < i i ,- ; , it , , ..I m m 2 4 6 8 10 12 14 16 18 20 22 Time (In Minutes) A. Susan B. Bob C. Carol D. Steven E. I don't know. In four months, the v o l l e y b a l l team spent the following amounts t r a v e l l i n g to games: 1st month - $17.9 5 2nd month - $22.40 3rd month - $ 8.25 4th month - $15.80 What was the average amount spent on t r a v e l l i n g each month? A. $10.10 B. $64.40 C. $32.20 D. $16.10 E. I don't know. The median te s t mark was 37 out of 50. B i l l y scored 30 out of 50. How many c h i l d r e n scored higher than B i l l y ? A. more than ha l f B. l e s s than h a l f C. \ exactly h a l f D. none E. I don't know. I am thinking of two numbers. When you add them you get 36. When you subtract them you get 8. To f i n d both numbers the most useful problem-solving technique would be to A. guess and check. B. draw a p i c t u r e or diagram. C. solve a simpler problem. D. work backwards. E. I don't know. 204 24 -4 8 . There are 13 boys and 15 g i r l s i n a group. What f r a c t i o n of the group i s boys? A * 28 B ±1 B - • 15 C. M 13 13 28 I don't know. 4 9 . How many squares must be shaded to show 35% of the s t r i p ? A. 0.035 B. 0.35 C. 3.5 D. 35 E. I don't know. 50 . If 4 v o l l e y b a l l s cost $96.00, how much w i l l 10 v o l l e y b a l l s cost? A. $960.00 B. $240.00 C. $ 24.00 D. $384.00 E. I don't know. 205 - 1 -206 SCALE T; CALCULATORS AND COMPUTERS 1. Do you own a c a l c u l a t o r ? A. Yes B. No 2. How often do you use a c a l c u l a t o r outside school?, A. Never B. Rarely (about once a week) C. Sometimes (a couple of times a week) D. Frequently (almost every day) 3. How of t e n do you use a c a l c u l a t o r i n school? A. Never B. Rarely (about once a week) C. Sometimes (a couple of times a week) D. Frequently (almost every day) 4. Choose the one answer which best describes how your c l a s s used c a l c u l a t o r s on mathematics t e s t s t h i s year. A. Not a t a l l ; we weren't allowed to use them on t e s t s . B. We were allowed to use them on some t e s t s , i f we wanted to. C. We were allowed to use them a l l t e s t s , i f we wanted to. D. We were required to use them on some t e s t s . 5. In what ways do you use a c a l c u l a t o r to do mathematics i n t h i s c l a s s ? (Mark a l l that apply.) A. Not at a l l ; we're not allowed to use c a l c u l a t o r s i n t h i s c l a s s . B. To c a l c u l a t e answer to problems C. To check answers D. For games and fun 207 - 2 -6. Some people say that Grade 7 s tudents should NOT be al lowed to use c a l c u l a t o r s i n s choo l . How do you f e e l about t h i s ? A. S trongly Disagree B. Disagree C . Undecided D. Agree E . S trongly Agree Some people say that i f students are al lowed to use c a l c u l a t o r s , then i t should NOT be necessary for them to l e a r n how to add, s u b t r a c t , m u l t i p l y , or d i v i d e by hand. How do you f e e l about th i s? A. S trong ly Disagree B. Disagree C . Undecided D. Agree E . S trongly Agree 8. Do you have a computer (one that w i l l do more than play games) at home? A. Yes B. No C. I don' t know. 9. What have you used a computer to do? (Mark a l l that a p p l y . ) A. Nothing , I 've never used a computer. B . To p lay games C . To wr i te s t o r i e s or l e t t e r s D. To wr i te my own programs E . To l e a r n about mathematics or other subjec t s 10. Where d i d you get most of your experience with computers? A. I haven't had any experience with computers. B . At home C . In courses taken at school or elsewhere - 3 -208 FOR ITEMS 11-20, CHOOSE THE ANSWER THAT BEST DESCRIBES YOUR OPINION. 11. I would l i k e to l e a r n more about computers. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 12. I f e e l h e l p l e s s around computers. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 13. Every student should be taught, i n school, how to use a computer. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 14. Computers can be used to teach mathematics. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 15. Computers can be used to teach subjects other than mathematics. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 209 - 4 -16. Using computers i s more s u i t a b l e for boys than for g i r l s . A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 17. I f e e l confident about being able to use computers. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 18. I enjoy using computers. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 19. Computers are gaining too much c o n t r o l over people's l i v e s . A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 20. I am able to work with computers as w e l l as most others my age. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree - 5 - 210 STUDENT BACKGROUND INFORMATION 1. W h a t l a n g u a g e d i d y o u f i r s t l e a r n t o s p e a k ? A. E n g l i s h B. F r e n c h C. A n o t h e r l a n g u a g e 2. W h a t l a n g u a g e d o y o u s p e a k m o s t o f t e n i n y o u r home now? A. E n g l i s h B. F r e n c h C. A n o t h e r l a n g u a g e 3. W h a t p r o g r a m a r e - y o u i n ? A. R e g u l a r P r o g r a m i n E n g l i s h B. E a r l y F r e n c h I m m e r s i o n C. L a t e F r e n c h I m m e r s i o n D. " P r o g r a m m e - C a d r e d e F r a n c a i s " 4. I n t h i s c l a s s M a t h e m a t i c s i s t a u g h t i n A. E n g l i s h B. F r e n c h 5. Do y o u s o m e t i m e s g o t o a L e a r n i n g A s s i s t a n c e C e n t r e i n y o u r s c h o o l f o r h e l p w i t h m a t h e m a t i c s ? A. T h e r e i s no L e a r n i n g A s s i s t a n c e C e n t r e f o r m a t h e m a t i c s i n t h i s s c h o o l . B. Y e s , I d o . C. No, I d o n ' t . 6. Do y o u s o m e t i m e s g o t o a n E S L c l a s s ( E n g l i s h a s a S e c o n d L a n g u a g e ) i n y o u r s c h o o l ? A. T h e r e i s no E S L c l a s s i n t h i s s c h o o l . B. Y e s , I d o . C. No, I d o n ' t . - 6 - 211 7. How long d i d i t take you to do your l a s t mathematics homework assignment? A. We never have mathematics homework i n t h i s c l a s s . B. Between 1 and 10 minutes C. Between 11 and 30 minutes D. Between 31 and 60 minutes E. More than an hour 8. About how much time d i d you spend doing homework i n a l l subjects yesterday? A. I didn't have any homework to do yesterday. B. Less than 30 minutes C. Between 30 minutes and 1 hour D. From 1 to 2 hours E. More than 2 hours 9. What was the highest l e v e l of school or co l l e g e attended by your father or male guardian? A. Very l i t t l e or no schooling at a l l B. Elementary school C. Secondary school D. College, u n i v e r s i t y or some other form of post-secondary t r a i n i n g E. I don't know. 10. What was the highest l e v e l of school or c o l l e g e attended by your mother or female guardian? A. Very l i t t l e or no schooling at a l l B. Elementary school C. Secondary school D. College, u n i v e r s i t y or some other form of post-secondary t r a i n i n g E. I don't know. 11. Here i s a l i s t of reasons for studying mathematics. Which do you believe i s most important? A. To prepare f o r the next year's mathematics course B. To lea r n how to perform c a l c u l a t i o n s accurately C. To lea r n how to use mathematics to solve problems i n the r e a l world D. To lea r n to think l o g i c a l l y E. To le a r n what mathematics i s 212 - 7 -12. Which of these reasons for studying mathematics do you bel i e v e to be l e a s t important? A. To prepare for the next year's mathematics course B. To learn how to perform c a l c u l a t i o n s accurately C. To learn how to use mathematics to solve problems i n the r e a l world D. To learn to think l o g i c a l l y E. To l e a r n what mathematics i s Both answers given for questions 13-16 are c o r r e c t . I f you were asked each question, which one of the two answers comes to mind f i r s t ? 13. How much does a b i c y c l e weigh? 15. How f a r i s i t from Prince George to Prince Rupert? 16. How much gasoline can the gas tank i n a large car hold? A. About 15 kilograms B. About 35 pounds 14. What i s the temperature i n t h i s room? A. About 70 degrees B. About 20 degrees A. B. About 700 kilometres About 450 miles A. B. About 20 gallons About 90 l i t r e s ACHIEVEMENT SURVEY The v a l u e o f 572 + 18 0 0 5 + 73 i s A. 18 650 B. 96 410 C. 148 2 0 5 D. 186 410 E. I d o n ' t know. The v a l u e o f 3 + 4 ( 5 + 2 ) i s A. 25 B. 26 C. 3 1 " D. 49 E . I d o n ' t know. S u b t r a c t : 2 0 0 8 - 189 A. 819 B. 1 1 8 1 C. 1 8 1 9 D. 2 1 8 1 E. I d o n ' t know. The g r e a t e s t common f a c t o r o f 24 a n d 30 A. 2 B. 6 C. 1 2 0 D. 60 E. I d o n ' t know. - 9 - 214 5. When t h e i n p u t i s x t h e o u t p u t i s : I n p u t O u t p u t 3 7 4 9 5 1 1 6 13 7 15 8 17 x A. 19 B. 2x - 1 C 2x + 1 D. x E. I d o n ' t k n o w . 6. F o r how many m o n t h s was t h e r a i n f a l l m o r e t h a n 5 cm? 25 E 20 u ^ 15-_J 2 10' £ 5 -\ Jan Feb Mar Apr May Jun Jul Aug S e p Oct Nov Dec M O N T H S A. 3 B. 4 C. 6 D. 9 E. I d o n ' t know. - 10 -T e s t m a r k s : 3, 4, 5, 4, 5, 5, 5, 3, 3, 1, 4, 5, 0, 4, 5 W h i c h o n e o f t h e f o l l o w i n g t a b l e s r e p r e s e n t s t h i s d a t a ? mark f r e q u e n c y mark f r e q u e n c y 0 1 0 1 1 1 1 2 2 0 B. 2 1 3 3 3 3 4 4 4 3 5 6 5 6 mark f r e q u e n c y mark f r e q u e n c y 0 1 0 0 1 1 1 1 2 0 D. 2 2 3 3 3 3 4 4 4 4 5 5 5 5 E. I d o n ' t know. I n t h e d i a g r a m s h o w n b e l o w , a c a k e i s c u t s o t h a t a l l c u t s a r e made t h r o u g h t h e c e n t e r . T o f i n d how many p i e c e s t h e r e w i l l be a f t e r 10 c u t s , t h e m o s t u s e f u l p r o b l e m - s o l v i n g t e c h n i q u e w o u l d be t o A. l o o k f o r a p a t t e r n . B. s o l v e a s i m p l e r p r o b l e m . C. g u e s s a n d c h e c k . D. w o r k b a c k w a r d s . E. I d o n ' t know. - 11 -216 9. Add: \ + \ A. 2 5 1 5 1 6 _5 6 E. I don't know. 10 . If there are 300 c a l o r i e s i n 900 g of a c e r t a i n food, how many c a l o r i e s are there i n a 300 g portion of that same food? A. 27 B. 33 C. 100 D. 270 E. I don't know. 11. Which one of the following shows a discount of 10%? A. 30jz! o f f $3 B. 35jzf o f f $3 C. 40jzf o f f $3 D. 45s* o f f $3 E. I don't know. - 12 - 217 A. AD and DC B. CA and DB C. CB and AD D. AE and EB E. I don't know. 13. Which one of the following i s NOT a parallelogram? A. B. E. I don't know. 14. If two l i n e segments are equal i n length, they are A. h o r i z o n t a l . B. congruent. C. p a r a l l e l . D. perpendicular. E. I don't know. - 13 - 218 15. If 3 cakes are each cut int o t h i r d s , how many pieces are there? A. 1 B. 3 C. 6 D. 9 E. I don't know. 16. The width of t h i s rectangle i s how much l e s s than the length? 18.3 cm  13.6 cm A. 4.3 cm B. 4.7 cm C. 5.3 cm D. 5.7 cm E. I don't know. 17. Written as a decimal, = o A. 0.12 B. 0.8 C. 0.125 D. 0.18 E. I don't know. 219 - 14 -o 18. Which one of the following diagrams shows the s l i d e image of the man shown to the rig h t ? 9 O B. o c. o E. I don't know. 19. In the diagrams shown below, how many pieces the same s i z e as A are needed to cover B? A. 2 B. 3 C. 4 D. 6 E. I don't know. 220 20. 15 -I am a number between 25 and 40. I have a remainder of two when divi d e d by both 6 and by 9. Who am I? A. 26 B. 29 C. 32 D. 38 E. I 21. Think of a c a l c u l a t o r as a person. A person's ears and eyes are l i k e the c a l c u l a t o r ' s A. ch i p . B. keys. C. battery. D. memory. E. I don't know. CE IMC MR M- M+ •as mass QDE10B 22. You wish to c a l c u l a t e the square root of 64 on the c a l c u l a t o r . Which one of the following sequences of keystrokes w i l l l i k e l y give the c o r r e c t answer? A. B. C. D. E. EBB SHE HEEHHE S E E I don't know. C I C E JMC MR M-|[M+] 0 0 0 0 SCHEIE msmB QDESB 221 - 16 23. The table shows the numbers of various coins found i n a box, Coin Number found $1 ( s i l v e r d o l l a r ) 2 50jzf ( f i f t y - c e n t piece) 6 250 (quarter) 1 lOjzf (dime) 3 50 (n icke l ) 8 10 (penny) 3 Which one of the fo l l owing graphs shows t h i s ? OS LL1 co z 1 : 1 i 1 1 a l It Si 10t 25t 50t S1. COIN tr •±i CD Z B. 2 It 5i 10t 25C 50<t 51. COIN UJ m 5 3 Z 2 It 5C 104 25$ 50$ S1. COIN UJ a 5 Z 8T 6 4 D. It 5$ 104 25$ 50$ S1. COIN E . I don ' t know. - 17 -222 2 4 . If the measure of the side of each square i s d u n i t s , how long i s the rectangle? A. 4 + d u n i t s B. 8 + d u n i t s C. 4 x d u n i t s D. 8 x u n i t s E. I don't know. 2 5 . In the formula z~ = R, i f I = 250, P = 1000, and T = 2, then R i s A. B. 1 8 1 2 C. 1 D. 50 E. I don't know. 26. Tom has y marbles and Mary has x marbles. Mary has more marbles than Tom. Which sentence shows t h i s r e l a t i o n ? A. x = y B. x < y C. x > y D. x > 2y E. I don't know. - 18 -223 Which one of the following i s most l i k e a r i g h t t r i a n g l e ? Which one of the following i l l u s t r a t e s the c o r r e c t procedure f o r b i s e c t i n g an angle? 29. What i s the diameter of a c i r c l e with a radius of 4? A . 8 B. 6 C. 4 D. 2 E. I don't know. - 19 - 224 Sparky Spencer spun a spinner 100 times and made a record of h i s r e s u l t s . Outcome A B C Number of times 5 5 30 15 Which spinner d i d he most l i k e l y use? I f the p r o b a b i l i t y that i t w i l l r a i n on a g iven day i s 0.36, then the p r o b a b i l i t y that i t w i l l NOT r a i n i s : A . 0.36 B . 0.64 C. 99.64 D. 99.36 E . I d o n ' t know. The average age of 4 c h i l d r e n i s 6 years . I f the ages of 3 of the c h i l d r e n are 4 y e a r s , 8 years and 3 years , what i s the age of the four th c h i l d ? A . 6 years B. 9 yea r s C . 7 years D. 5 years E . I d o n ' t know. 225 - 20 -A l i s t of i n s t r u c t i o n s for a computer i s c a l l e d a A. program. B. disk. C. terminal. D. memory. E. I don't know. 3 4 . A set of i n s t r u c t i o n s for an imaginary computer i s as follows: 1. Arrange the three names Sandy, Dale, and Pat i n al p h a b e t i c a l order 2. Remove the l a s t name from the l i s t 3. If only one name i s l e f t , stop, otherwise go on to step 4 4. P r i n t out the names i n reverse order 5. Go back to step 2 What w i l l the computer p r i n t ? A. Pat B. Dale, Pat C. Dale, Pat, Sandy D. Pat, Dale E. I don't know. - 21 -Which u n i t would usually be used for the mass of sugar? A. kL B. km C. kg 2 D. km E. I don't know. The temperature on a sunny summer day would most l i k e l y be: A. 5° Cel s i u s B. 25° C e l s i u s C. 55° C e l s i u s D. 85° Ce l s i u s E. I don't know. The thickness of a dime i s about: A. 1 cm B. 1 dm C. 1 m D. 1 mm E. I don't know. Mr. Jones put a fence around h i s rectangular garden. The garden 10 m long and 6 m wide. How many metres of fencing did he use? A. 16 m B. 30 m C. 32 m D. 60 m E. I don't know. - 22 -227 39. A map of B.C. i s to be drawn so that 1 millimetre represents 5 kilometres. If 125 kilometres, be on the map? the a c t u a l distance between Vernon and Penticton i s how many m i l l i m e t r e s apart should these two points A. 125 B. 625 c. 120 D. 25 E. . I don't know. 40. Which one of the following f i g u r e s shows an acute angle? A. D. I don't know. 41. Which one of the following i s a diagram of a l i n e ? A. M N B. M N M -«8—o— N -« M N I don't know. 228 4 2 . Which one of the following patterns can be made into a pyramid? J 4 3 . Along which l i n e can the trapezoid be folded exactly edge to edge and corner to corner? A. q B. r C. s D. t E. I don't know. - 24 -2 2 9 What i s the area of the shaded p o r t i o n of t h i s f i g u r e ? T A. 54 B. 96 C. 120 D. 60 E. I don't know. 45. 3n i s equal to A. "h + 3 B. W. - 3 C. 7k x 3 D. 'D. -T- 3 E. I don't know. 46. Which l i s t contains a l l of the whole numbers which make t h i s a true statement? A. 5 B. 7 C. 0 , 1 , 2 , 3 , 4 , 5 , 6 D. 0 , 1 , 2 , 3 , 4 , 5 E. I don't know. S i m p l i f y : 230 25 -A. 36 B. 64 C. 12 D. 32 £. I d o n ' t know. 48. M u l t i p l y : 12 x 2~ A. 14-r 4 B. 30 C. 33 D. 24-7 4 I d o n ' t know. 49. How many s h a k e s c a n I buy w i t h $ 4 . 2 0 ? 'A A. 2 B. 3 C. 4 D. 5 E. I d o n ' t know. 231 26 50. Joyce has 50^. Which of the following can she buy? 71 8C 3<t 1 0 C A. 3 apples and 3 ice cream cones B. 5 apples and 3 balloons C. 4 ice cream cones and a chocolate bar D. 3 chocolate bars and a p e n c i l E. I don 11 know. APPENDIX B Teacher's Guide Questionnaire GRADE 7 FEACHER'S GUIDE QUESTIONNAIRE 234 - 1 -M A T H E M A T I C S A S S E S S M E N T INSTRUCTIONS FOR THE TEACHER QUESTIONNAIRE You are r e q u e s t e d to use the Answer Sheet a t t a c h e d to t h i s  q u e s t i o n n a i r e i n r e s p o n d i n g to these q u e s t i o n s . T h i s copy o f the Answer Sheet has a s c h o o l f a c i l i t y code bubbled i n , whereas c o p i e s t o be used by s t u d e n t s i n your c l a s s do n o t . T h e i r r e s p o n s e s w i l l be t r a c k e d through use o f the C l a s s Header Sheet i n c l u d e d i n t h i s package. The Answer Sheet has been d e s i g n e d to accommodate both s t u d e n t and t e a c h e r r e s p o n s e s . Four s t e p s to f o l l o w i n c o m p l e t i n g your Answer Sheet f o l l o w : 1. In the s e c t i o n l a b e l l e d Grade, and under the word "Teacher" f i l l i n the bubble f o r Grade 7. 2. I n d i c a t e your Gender. 3. DO NOT complete the s e c t i o n i n d i c a t i n g which form you are answering. 4. Your q u e s t i o n n a i r e c o n s i s t s of f i v e s e c t i o n s . S c a l e R - Mathematics i n S c h o o l S c a l e S - Problem S o l v i n g S c a l e T - C a l c u l a t o r s and Computers Background I n f o r m a t i o n C l a s s s i z e and Textbooks Used P l e a s e complete the e n t i r e q u e s t i o n n a i r e on the d e s i g n a t e d Answer Sheet. Note t h a t bubbles f o r the Background I n f o r m a t i o n and the C l a s s S i z e and Textbooks s e c t i o n s are on the r e v e r s e s i d e of the s h e e t . I f you t e a c h Mathematics to more than one Grade 7 c l a s s , q u e s t i o n s w i t h r e f e r e n c e to one c l a s s o n l y s h o u l d be responded to f o r the f i r s t Grade 7 Math c l a s s which o c c u r s i n the week or i n your t i m e t a b l e c y c l e . A f t e r you have completed your r e s p o n s e s , p l a c e t h i s q u e s t i o n n a i r e and your Answer Sheet i n the same envelope as the green c l a s s Header Sheet and the Student Answer Sheets f o r your c l a s s . A f t e r a d m i n i s t r a t i o n the envelope, t o g e t h e r w i t h a l l s t u d e n t t e s t b o o k l e t s , s h o u l d be r e t u r n e d to your p r i n c i p a l . - z -SCALE R: MATHEMATICS IN SCHOOL 235 Please record your responses ON THE ANSWER SHEET in the section labelled "FORM R". For each of the items i n t h i s s c a l e , three responses are requ i r e d . Consider only the c l a s s designated by your p r i n c i p a l f o r t h i s question-n a i r e . A. T e l l how important you think the t o p i c i s f o r t h i s c l a s s . B. T e l l how easy i t i s to teach the t o p i c to t h i s c l a s s . C. T e l l how much you l i k e teaching the t o p i c to t h i s c l a s s . 1. Adding, s u b t r a c t i n g , m u l t i p l y i n g and d i v i d i n g f r a c t i o n s not a t a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy C. d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t 2. Adding, s u b t r a c t i n g , m u l t i p l y i n g and d i v i d i n g decimals not at a l l important not important undecided important very important very d i f f i c u l t d i f f i c u l t undecided easy very easy d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t Working with percents not a t a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t - 3 - 236 4. Learning about estimation not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t 5. Memorizing basic facts A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t 6. Solving equations A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy C. d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t 7. Solving word problems A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy d i s l i k e a l o t d i s l i k e undecided l i k e l i k e a l o t - 4 -237 8. Learning about the metric system A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy d i s l i k e a l o t d i s l i k e undecided like like a l o t Working with perimeter and area A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy d i s l i k e a lot d i s l i k e undecided li k e l i k e a l o t 10. Doing geometry A. not at a l l important not important undecided important very important B. very d i f f i c u l t d i f f i c u l t undecided easy very easy dis l i k e a l o t dis l i k e undecided like l i k e a l o t - 5 - 238 SCALE S: PROBLEM SOLVING Please record your responses ON THE ANSWER SHEET i n the s e c t i o n l a b e l l e d "FORM S". For Items 1-7, mark the response which best d e s c r i b e s your o p i n i o n about each statement with respect to the c l a s s designated by your p r i n c i p a l f o r t h i s q u e s t i o n n a i r e . 1. Most of my students enjoy problem s o l v i n g . A. S t r o n g l y Disagree B. Disagree C. Undecided D. Agree E. S t r o n g l y Agree 2. Most of ray students perform w e l l i n problem s o l v i n g . A. St r o n g l y Disagree B. Disagree C. Undecided D. Agree E. S t r o n g l y Agree 3. The textbooks I use provide adequate i n s t r u c t i o n f o r developing problem-solving s k i l l s . A. S t r o n g l y Disagree B. Disagree C. Undecided D. Agree E. S t r o n g l y Agree 4. My d i s t r i c t provides adequate a s s i s t a n c e and resources f o r the teaching of problem s o l v i n g . A. St r o n g l y Disagree B. Disagree C. Undecided D. Agree E. S t r o n g l y Agree 5. I am s a t i s f i e d with my teaching of problem s o l v i n g . A. Strongly Disagree B. Disagree C. Undecided D. .Agree E. S t r o n g l y Agree 239 - 6 -6. A l l mathematics teachers should attend a t l e a s t one workshop on problem s o l v i n g each year. A. S t r o n g l y Disagree B. Disagree C. Undecided Agree E. Strongly Agree I t i s easy to teach problem s o l v i n g . A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 8. Which of these problem-solving s t r a t e g i e s do you teach i n your c l a s s e s ? (Mark a l l that apply.) A. Look f o r a pattern B. Guess and check C. Make a systematic l i s t or t a b l e D. Solve a simpler problem E. Work backwards How many workshops on problem s o l v i n g have you attended i n the past year? A. None B. 1 C. 2 D. 3 E. More than 3 10. What sources do you use to provide students with problem-solving e x e r c i s e s ? (Mark a l l that apply.) A. Textbook B. Mathematical contests C. Problem-solving booklets D. P r o f e s s i o n a l j o u r n a l s E. Books of puzzles - 7 -240 11. Which of the following do you use to motivate your students to participate i n problem-solving a c t i v i t i e s ? (Mark a l l that apply.) A. Competitive games B. Problem of the day or week * C. Puzzles or brain teasers 0. Library or f i l e of interesting problems E. Contests 12. Which of the following best characterizes your teaching of problem solving in mathematics? I teach problem solving: A. As a unit from time to time B. Almost every day, as a regular part of the mathematics class C. One period every 2 or 3 weeks D. At the end of a major topic or chapter E. One period a week 13. When you grade students' work in problem solving, for which of the following do you give marks? (Mark a l l that apply.) A. I don't give p a r t i a l credit. It's a l l or nothing. B. For the appropriate diagram or equation C. For the procedures used (computation, etc.). D. For the f i n a l answer E. For checking the answer 14. Which of the following types of problems do you assign to your students? (Mark a l l that apply.) A. Problems with more than one correct answer B. Problems which require students to co l l e c t information C. Problems which can be solved more than one way D. Problems which students work on co l l e c t i v e l y i n groups E. Problems with either too much or too l i t t l e information 15. Which of the following a c t i v i t i e s do you have i n your class? (Mark a l l that apply.) A. A problem-solving center B. A b u l l e t i n board display on problem solving C. Problem of the week D. Problem-solving contests within the class E. Students make up problems for others to solve - a - 241 SCALE T; CALCULATORS AND COMPUTERS Please record your responses ON THE ANSWER SHEET i n the s e c t i o n l a b e l l e d "FORM T". When responding, c o n s i d e r only the designated c l a s s . 1. Do you own a c a l c u l a t o r ? A. Yes B. No 2. How of t e n do you use a c a l c u l a t o r outside school? A. Never B. Rarely (perhaps l e s s than once a week) C. Sometimes (a couple of tiroes a week) D. Frequently (almost every day) 3. In what ways do you have your students use c a l c u l a t o r s i n mathematics? (Mark a l l that apply.) A. Not a t a l l B. For d r i l l and p r a c t i c e to enhance computational s k i l l s C To work on problems D. For other t o p i c s such as estimating, f i n d i n g p atterns, and so on 4 . How are students i n your c l a s s provided with c a l c u l a t o r s ? A. I do not allo w c a l c u l a t o r s i n my c l a s s . B. Each student may br i n g h i s or her own. C. Each student must bring h i s or her own. D. C a l c u l a t o r s are provided f o r the students. 5. In what ways do you use a c a l c u l a t o r f o r n o n - i n s t r u c t i o n a l school work? (Mark a l l that apply.) A. None B. To c a l c u l a t e students' marks, grades, and so on C. To check students' answers on assignments D. To prepare worksheets or t e s t s 242 - 9 -6 . Some people say that Grade 7 students should not be allowed to use calculators i n school. How do you f e e l about this? A. Strongly Disagree * B. Disagree C. Undecided D. Agree E. Strongly Agree 7 . Some people say that i f students are allowed to use calculators, then i t i s not necessary for them to learn how to add, subtract, multiply, or divide by hand. How do you f e e l about this? A. Strongly Disagree B. Disagree C. Undecided D. Agree . E. Strongly Agree 8. Do you have a computer (one that w i l l do more than play games) at home? A. Yes B. No 9. What applications of computers have you had experience with? (Mark a l l that apply.) A. None B. Games C. Word processing D. Computer-assisted instruction E. Programming 10. Where did you get most of your experience with computers? A. I haven't had any experience with computers. B. On my own C. Through special courses 243 - l O -l l . Which of the f o l l o w i n g i s c l o s e s t to the type of computer o r g a n i z a t i o n i n your school? A. There are no computers at a l l i n t h i s s c h o o l . B. There are computers i n some or a l l classrooms. C. Computers are provided i n one or more p l a c e s f o r use by teachers with t h e i r students. D. Computers are provided i n one or more places which are s t a f f e d by s p e c i a l i s t s or resource persons. E. Computers are used f o r a d m i n i s t r a t i v e purposes only. 12. How i s a computer used i n your mathematics c l a s s ? (Mark a l l t h a t apply.) A. Not used B. I use i t as a teaching t o o l to demonstrate concepts. C. Students l e a r n computer programming. D. Students use software packages. E. I use the computer f o r record-keeping. 13. what kinds of computer software do your students use for l e a r n i n g mathematics? (Mark a l l that apply.) A. None B. D r i l l and p r a c t i c e C. E d u c a t i o n a l games to r e i n f o r c e s k i l l s or concepts D. T u t o r i a l (to teach a s k i l l or concept) E. Simulation (to provide a>model of a r e a l s i t u a t i o n ) 14. Which group of students i n your mathematics c l a s s makes the most use of computers i n school? A. None, my students do not use computers i n s c h o o l . B. The low a b i l i t y students C. Students of average a b i l i t y D. High a b i l i t y students E. They a l l make equal use of computers. FOR ITEMS 15-20, CHOOSE THE OPTION WHICH BEST DESCRIBES YOUR OPINION. 15. I f we allow computers to be u:>ed i n s c h o o l , they may take over some of the major f u n c t i o n s of teachers. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 16. The computer software that i s a v a i l a b l e f o r teaching mathematics i s app r o p r i a t e and well-designed. A. S t r o n g l y Disagree B. Disagree C. Undecided D. Agree E. S t r o n g l y Agree 17. Students have an opportunity to be c r e a t i v e when they are taught mathematics by computer. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. S t r o n g l y Agree 18. More boys than g i r l s seem to use computers f o r doing mathematics. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 245 - 12 -19. I t i s e s s e n t i a l that computers become an i n s t r u c t i o n a l t o o l f o r a l l teachers of mathematics. A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree 20. To be s u c c e s s f u l i n modern s o c i e t y , nearly everyone w i l l need computer s k i l l s . A. Strongly Disagree B. Disagree C. Undecided D. Agree E. Strongly Agree - 13 - 246 TEACHER BACKGROUND INFORMATION Please record your responses ON THE ANSWER SHEET i n the s e c t i o n l a b e l l e d "BACKGROUND INFORMATION". 1. For how many years w i l l you have been teaching mathematics as of June, 1985? A. 1-2 years B. 3-5 years C. 6-10 years D. 11-15 years E. More than 15 years 2. I f you had a choice, would you avoid teaching mathematics a l t o g e t h e r ? 3. What percent of your cu r r e n t teaching load i s mathematics? A. 0-20% B. 21-40% C. 41-60% D. 61-80% E. 81-100% 4. To which of the f o l l o w i n g a s s o c i a t i o n s do you belong? (Mark a l l t h a t apply.) A. B.C. A s s o c i a t i o n of Mathematics Teachers B. P r o v i n c i a l Intermediate Teachers A s s o c i a t i o n C. B.C. Primary Teachers A s s o c i a t i o n D. N a t i o n a l C o u n c i l of Teachers of Mathematics E. L o c a l Mathematics PSA F. None of the above 5. Have you attended a mathematics s e s s i o n a t a conference i n the l a s t three years? 6. Have you attended a workshop (other than a t a conference) or an i n s e r v i c e day i n mathematics i n the Last three years? A. B. C. Yes No Undecided A. B. Yes No A. B. Yes No - 14 - 247 7. At what l e v e l should students f i r s t be taught mathematics by someone who s p e c i a l i z e s i n the teaching of mathematics? A. At no l e v e l B. Primary C. Intermediate D. Junior Secondary E. Senior Secondary 8. How many post-secondary courses i n mathematics have you s u c c e s s f u l l y completed? (e.g., For UBC 3 u n i t s = 2 courses) A. 0 B. 1 or 2 C. 3-5 D. 6-9 E. 10 or more 9. How many post-secondary courses i n mathematics education have you s u c c e s s f u l l y completed? (e.g., For UBC 3 u n i t s = 2 courses) A. 0 B. 1 or 2 C. 3-5 D. 6-9 E. 10 or more QUESTIONS 10-20 REFER TO THE SPECIFIC MATHEMATICS CLASS DESIGNATED BY YOUR PRINCIPAL FOR THIS QUESTIONNAIRE. 10. Which of the f o l l o w i n g best d e s c r i b e s the course o f f e r e d to these students? A. F u l l - y e a r course, regular program B. F u l l - y e a r course, modified (slower students) C. F u l l - y e a r course, enriched D. Semester course, r e g u l a r program E. Semester course, modified (slower students) F. Semester course, enriched G. Other 11. On the average, how o f t e n do you give t h i s c l a s s t e s t s or quizzes i n mathematics? A. Almost every day B. Once a week C. Once every couple of weeks D. Once every r e p o r t i n g p e r i o d E. I almost never g i v e t e s t s or quizzes i n mathematics. - 15 - 248 12. How many mathematics p e r i o d s does t h i s c l a s s have each calendar week? A. 3 B. 4 C. 5 D. 6 • E. More than 6 13. How long i s each mathematics per i o d ? A. 30 minutes or l e s s B. 31-45 minutes C. 46-60 minutes D. 61-75 minutes E. More than 75 minutes QUESTIONS 14-20 REFER TO THE LAST MATHEMATICS PERIOD DURING WHICH YOU TAUGHT THIS CLASS. 14. What percent of that mathematics p e r i o d was spent on a c t i v i t i e s r e l a -ted to homework from the previous day (e.g., d i s c u s s i n g , c o r r e c t i n g ) ? A. None B. 1-10% C. 11-25% D. 26-50% E. 51-75% F. 76-100% 15. How many students d i d you c a l l on to answer questions? A. None B. One or two C. Less than one-quarter of the c l a s s D. About h a l f the c l a s s E. Between a h a l f and t h r e e - f o u r t h s of the c l a s s F. Almost every student 16. What percent of that mathematics p e r i o d d i d your students spend working i n d i v i d u a l l y on seatwork? A. None B. 1-10% C. 11-25% D. 26-50% E. 51-75% F. 76-100% 249 - 16 -17. What percent of that mathematics period d i d your students spend working i n small groups? A. None ' B. 1-10% C. 11-25% D. 26-50% E. 51-75% F. 76-100% 18. What percent of that mathematics perio d d i d your students spend working a t s t a t i o n s or a c t i v i t y centres? A. None B. 1-10% C. 11-25% D. 26-50% E. 51-75% _ . F. 76-100% 19. What percent of that mathematics perio d d i d your students spend on computational d r i l l ? A. None B. 1-10% C. 11-25% D. 26-50% E. 51-75% F. 76-100% 20. What percent of that mathematics perio d d i d you spend e x p l a i n i n g new t o p i c s to the e n t i r e c l a s s ? A. None B. 1-10% C. 11-25% D. 26-50% E. 51-75% F. 76-100% 250 - 17 -CLASS SIZE AND TEXTBOOKS DSED P l e a s e r e c o r d your answer t o the next t h r e e q u e s t i o n s on items 1/ 2 and 3 of the s e c t i o n l a b e l l e d Achievement S u r v e y . 1. Which one of the f o l l o w i n g i n d i c a t e s the s i z e o f the c l a s s f o r which you are r e s p o n d i n g to t h i s q u e s t i o n n a i r e ? A. 1 - 1 5 B. 16 - 20 C. 21 - 25 D. 2 6 - 3 0 E. 31 or l a r g e r 2. Which one of the f o l l o w i n g t e x t b o o k s i s used as the b a s i c t e x t i n your classroom? A. E s s e n t i a l s of Mathematics 1 B. Mathematics 1 C. S c h o o l Mathematics 1 D. Contemporary Mathematics E. Other 3. Which of the f o l l o w i n g t extbooks are used as supplementary t e x t s (not the b a s i c one) i n your classroom? Check a l l t h a t a p p l y . A. E s s e n t i a l s of Mathematics 1 . B. Mathematics 1 C. School Mathematics 1 D. Contemporary Mathematics E. Other END OF QUESTIONNAIRE Thank you f o r your c o - o p e r a t i o n APPENDIX C Coding o f Variables 252 Option Codes f o r Independent Variables B. Var i a b l e Item Number Codes by Opt Number A B C D E Student Background SB1 1 2 1 1 SB2 2 2 1 1 SB3 7 1 2 3 4 5 SB4 9 1 2 3 4 -SB5 10 1 2 3 4 — Teacher Background TBI 1 1 2 3 4 5 TB2 2 1 3 2 TB3 3 1 2 3 4 5 TB4 5 2 1 TB5 6 2 1 TB6 8 1 2 3 4 5 TB7 9 1 2 3 4 5 Student Perception SP1 1 1 2 3 4 5 SP2 2 1 2 3 4 5 SP3 3 1 2 3 4 5 SP4 4 1 2 3 4 5 SP5 5 1 2 3 4 5 SP6 6 1 2 3 4 5 SP7 7 1 2 3 4 5 SP8 8 1 2 3 4 5 SP9 9 1 2 3 4 5 SP10 10 1 2 3 4 5 D. Teacher Perception TP1 1 1 2 3 4 5 TP2 2 1 2 3 4 5 TP3 3 1 2 3 4 5 TP4 4 1 2 3 4 5 TP5 5 1 2 3 4 5 TP6 6 1 2 3 4 5 TP7 7 1 2 3 4 5 TP8 8 1 2 3 4 5 TP9 9 1 2 3 4 5 TP10 10 1 2 3 4 5 253 E. Classroom Organization C01 10 2 1 3 - - -C02 11 5 4 3 2 1 C03 12 3 4 5 6 7 C04 13 15 38 53 68 83 C05 14 1 2 3 4 5 6 C06 15 1 2 3 4 5 6 C07 16 1 2 3 4 5 6 C08 17 1 2 3 4 5 6 C09 18 1 2 3 4 5 6 C010 19 1 2 3 4 5 6 COl l 20 1 2 3 4 5 6 G. Problem Solving Processes PS1 1 1 2 3 4 5 PS 2 2 1 2 3 4 5 PS 3 5 1 2 3 4 5 PS 4 7 1 2 3 4 5 *PS5 8 1,0 1,0 1,0 1,0 1,0 PS 6 9 1 2 3 4 5 *PS7 10 1,0 1,0 1,0 1,0 1,0 *PS8 11 1,0 1,0 1,0 1,0 1,0 PS 9 12 1 5 3 2 4 *PS10 14 1,0 1,0 1,0 1,0 1,0 *PS11 15 1,0 1,0 1,0 1,0 1,0 * M u l t i p l e response items 

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