The Illusion of Linearity (eBook)

From Analysis to Improvement

Dirk De Bock, Wim Van Dooren, Dirk Janssens, Lieven Verschaffel (Autoren)

eBook Download: PDF

2007 | 2007
XIV, 184 Seiten
Springer US (Verlag)
978-0-387-71164-5 (ISBN)

Infos

This book presents the reader with a comprehensive overview of the major findings of the recent research on the illusion of linearity.

It discusses: how the illusion of linearity appears in diverse domains of mathematics and science; what are the crucial psychological, mathematical, and educational factors being responsible for the occurrence and persistence of the phenomenon; and how the illusion of linearity can be remedied.

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Linear or proportional relationships are a major topic in mathematics education. However, recent research has shown that secondary school students strongly tend to apply the linear model also in situations wherein it is not applicable. This overgeneralization of linearity is sometimes referred to as the "e;illusion of linearity"e; and has a strong negative impact on students' reasoning and problem solving skills."e;The Illusion of Linearity: From Analysis to Improvement"e; presents the reader with a comprehensive overview of the major findings of the recent research on the illusion of linearity. Although the empirical study of students improper linear reasoning clearly constitutes a new line of research, it owes a great deal to prior work in mathematics education research community (such as the work of the scholars of the Freudenthal Institute on realistic mathematics education). Based on both quantitative and qualitative research, "e;The Illusion of Linearity: From Analysis to Improvement"e; discusses the following issues: (1) how the illusion of linearity appears in diverse domains of mathematics and science and how it is conceptually related to other more general misconceptions identified in the research literature, (2) what are the crucial psychological, mathematical, and educational factors being responsible for the occurrence and persistence of the phenomenon, and (3) how the illusion of linearity can be remedied by appropriate instruction."e;The Illusion of Linearity: From Analysis to Improvement"e; is essential to those working in mathematics education, particularly teacher educators and curriculum research and development.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 30.9.2007
• Reihe/Serie: Mathematics Education Library
• Reihe/Serie: Mathematics Education Library
• Zusatzinfo: XIV, 184 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Geisteswissenschaften ► Sprach- / Literaturwissenschaft
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Sozialwissenschaften ► Pädagogik ► Erwachsenenbildung
• Themenwelt: Sozialwissenschaften ► Pädagogik ► Schulpädagogik / Grundschule
• Schlagworte: Calculus • Curriculum • Development • Education • Educational Research • Experiment • Freud • Mathematics • Problem Solving • Qualitative research • Research • research & development (R&D) • research & development (R&D) • school • Science • Teaching
• ISBN-10: 0-387-71164-3 / 0387711643
• ISBN-13: 978-0-387-71164-5 / 9780387711645