The Illusion of Linearity (eBook)

Contents 6
Contributing Authors 7
Foreword 8
Acknowledgments 12
Chapter 1 A WIDESPREAD PHENOMENON 14
1. INTRODUCTION 14
2. PROPERTIES AND REPRESENTATIONS OF LINEARITY: A CONCEPTUAL ANALYSIS 16
3. MANIFESTATIONS OF OVER-RELIANCE ON LINEARITY 18
3.1 Improper applications of linearity while solving arithmetic word problems 19
3.2 Over-reliance on linearity in graphical environments 23
3.3 Improper application of linearity in probabilistic situations 24
3.4 Number patterns, algebra, and calculus 27
3.5 The overuse of linearity in numerical estimation 29
4. THE CASE OF GEOMETRY 29
4.1 The effect of a linear enlargement (reduction) on area and volume 31
4.2 Introduction to the research program 34
Chapter 2 IN SEARCH OF EMPIRICAL EVIDENCE 36
1. INTRODUCTION 36
2. STUDY 1 37
3. STUDY 44
4. DISCUSSION 50
Chapter 3 SEARCHING FOR EXPLANATIONS: A SERIES OF FOLLOW- UP STUDIES 52
1. INTRODUCTION 52
2. FOLLOW-UP STUDY 1: EFFECTS OF METACOGNITIVE AND VISUAL SCAFFOLDS 53
3. FOLLOW-UP STUDY 2: EFFECTS OF PROBLEM FORMULATION 63
4. FOLLOW-UP STUDY 3: EFFECTS OF AUTHENTIC CONTEXTS AND DRAWING ACTIVITY 68
5. FOLLOW-UP STUDY 4: FROM INDIRECT TO DIRECT MEASURES 85
6. GENERAL SUMMARY AND DISCUSSION 96
Chapter 4 AN IN- DEPTH INVESTIGATION 99
1. INTRODUCTION 99
2. METHOD 100
3. RESULTS 105
3.1 Two sample interviews 106
3.2 Overview of students’ reactions 111
3.3 Phase 1: Solving the word problem 111
3.4 Phase 2: Reactions to the weak form of cognitive conflict 113
3.5 Phase 3: Reactions to the argumentation of a fictitious peer 114
3.6 Phase 4: Reactions to the solution process of a fictitious peer 115
3.7 Phase 5: Reactions to the explicit link with area measurement 117
4. DISCUSSION 117
Chapter 5 A TEACHING EXPERIMENT 121
1. INTRODUCTION 121
2. DESIGNING A LEARNING ENVIRONMENT 122
2.1 Overview of the lesson series 122
2.2 General design principles 129
3. RESEARCH METHOD 131
4. RESEARCH QUESTIONS AND HYPOTHESES 132
5. RESULTS 132
6. CONCLUSIONS AND DISCUSSION 136
Chapter 6 STEPPING OUTSIDE THE CLASSROOM 138
1. INTRODUCTION 138
2. THEORETICAL BACKGROUND 139
3. METHODOLOGY 142
4. RESULTS 146
5. CONCLUSIONS AND DISCUSSION 150
Chapter 7 PSYCHOLOGICAL AND EDUCATIONAL ANALYSIS 153
1. INTRODUCTION 153
2. SEARCHING FOR THE ROOTS OF THE OVERUSE OF LINEARITY 154
2.1 Linearity, intuitiveness and everyday life situations 154
2.2 The effects of schooling 158
2.3 Content-specific effects 160
2.4 Perspectives for further research 161
3. RECOMMENDATIONS FOR THE IMPROVEMENT OF EDUCATIONAL PRACTICE 163
3.1 Concrete suggestions for educational practice 164
3.2 A more comprehensive instructional approach 169
References 175
Index 189