Mathematical Thinking: How To Develop It In The Classroom

by Shigeo Katagiri, Masami Isoda

Overview

Developing mathematical thinking is one of major aims of mathematics education. In mathematics education research, there are a number of researches which describe what it is and how we can observe in experimental research. However, teachers have difficulties to develope it in the classrooms. This book is the result of lesson studies over the past 50 years. It describes three perspectives of mathematical thinking: Mathematical Attitude (Minds set), Mathematical Methods in General and Mathematical Ideas with Content and explains how to develop them in the classroom with illuminating examples.

Product Details

ISBN-13:	9789814350839
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	02/01/2012
Series:	Monographs On Lesson Study For Teaching Mathematics And Sciences , #1
Pages:	320
Product dimensions:	6.10(w) x 9.00(h) x 0.90(d)

Table of Contents

Preface to the Series v

Preface to the Book vii

Acknowledgements xi

Introductory Chapter: Problem Solving Approach to Develop Mathematical Thinking 1

Part I Mathematical Thinking: Theory of Teaching Mathematics to Develop Children Who Learn Mathematics for Themselves 29

Chapter 1 Mathematical Thinking as the Aim of Education 31

1.1 Developing Children Who Learn Mathematics for Themselves 31

1.2 Mathematical Thinking as an Ability to Think and to Make Decisions 32

1.3 The Hierarchy of Ability and Thinking 35

Chapter 2 The Importance of Cultivating Mathematical Thinking 37

2.1 The Importance of Teaching Mathematical Thinking 37

2.2 Example: How Many Squares Are There? 39

Chapter 3 The Mindset and Mathematical Thinking 47

3.1 Mathematical Thinking 47

3.2 Structure of Mathematical Thinking 49

Chapter 4 Mathematical Methods 53

4.1 Inductive Thinking 53

4.2 Analogical Thinking 56

4.3 Deductive Thinking 59

4.4 Integrative Thinking 62

4.5 Developmental Thinking 66

4.6 Abstract Thinking (Abstraction) 70

4.7 Thinking That Simplifies (Simplifying) 74

4.8 Thinking That Generalizes (Generalization) 76

4.9 Thinking That Specializes (Specialization) 78

4.10 Thinking That Symbolizes (Symbolization) 81

4.11 Thinking That Represents by Numbers, Quantities, and Figures (Quantification and Schematization) 83

Chapter 5 Mathematical Ideas 87

5.1 Idea of Sets 87

5.2 Idea of Units 89

5.3 Idea of Representation 91

5.4 Idea of Operation 95

5.5 Idea of Algorithms 98

5.6 Idea of Approximations 100

5.7 Idea of Fundamental Properties 102

5.8 Functional Thinking 104

5.9 Idea of Expressions 108

Chapter 6 Mathematical Attitude 111

6.1 Objectifying 111

6.2 Reasonableness 113

6.3 Clarity 115

6.4 Sophistication 117

Chapter 7 Questioning to Enhance Mathematical Thinking 121

Appendix for the List of Questions for Mathematical Thinking 127

Part II Developing Mathematical Thinking with Number Tables: How to Teach Mathematical Thinking from the Viewpoint of Assessment 129

Example 1 Sugoroku: Go Forward Ten Spaces If You Win, or One If You Lose 137

1 Type of Mathematical Thinking to be Cultivated 137

2 Grade Taught 137

3 Preparation 137

4 Overview of the Lesson Process 137

5 Worksheet 138

Let's Plays this game! 139

6 Lesson Process 140

7 Summarization on the Blackboard 148

8 Evaluation 148

Example 2 Arrangements of Numbers on the Number Table 149

1 Type of Mathematical Thinking to Be Cultivated 149

2 Grade Taught 149

3 Preparation 149

4 Overview of the Lesson Process 149

5 Worksheet 150

6 Lesson Process 151

7 Summarization on the Blackboard 156

8 Evaluation 156

Example 3 Extension of Number Arrangements 159

1 Type of Mathematical Thinking to Be Cultivated 159

2 Grade Taught 159

3 Preparation 159

4 Overview of the Lesson Process 159

5 Worksheet 160

6 Lesson Process 163

7 Summarization on the Blackboard 171

8 Evaluation 171

Example 4 Number Arrangements: Sums of Two Numbers 173

1 Type of Mathematical Thinking to be Cultivated 173

2 Grade Taught 173

3 Preparation 173

4 Overview of the Lesson Process 173

5 Worksheet 174

6 Lesson Process 176

7 Summarization on the Blackboard 183

8 Evaluation 183

Example 5 When You Draw a Square on a Number Table, What Are the Sum of the Numbers at the Vertices, the Sum of the Numbers at the Vertices, the Sum of the Numbers Along the Perimeter, and the Grand Total of All the Numbers? 185

1 Type of Mathematical Thinking to Be Cultivated 185

2 Grade Taught 185

3 Preparation 185

4 Overview of the Lesson Process 185

5 Worksheet 186

6 Lesson Process 189

7 Summarization on the Blackboard 199

8 Evaluation 200

9 Further Development 200

Example 6 Where Do Two Numbers Add up to 99? 203

1 Types of Mathematical Thinking to Be Cultivated 203

2 Grade Taught 203

3 Preparation 203

4 Overview of the Lesson Process 203

5 Worksheet 204

6 Lesson Process 206

7 Summarization on the Blackboard 215

8 Evaluation 215

9 Further Development 216

Example 7 The Arrangement of Multiples 217

1 Types of Mathematical Thinking to Be Cultivated 217

2 Grade Taught 217

3 Preparation 217

4 Overview of the Lesson Process 217

5 Worksheet 218

6 Lesson Process 219

7 Summarization on the Blackboard 230

8 Evaluation 230

Example 8 How to Find Common Multiples 231

1 Type of Mathematical Thinking to Be Cultivated 231

2 Grade Taught 231

3 Preparation 231

4 Overview of the Lesson Process 231

5 Worksheet 232

6 Lesson Process 233

7 Summarization on the Blackboard 242

8 Evaluation 242

9 Further Development 242

Example 9 The Arrangement of Numbers on an Extended Calendar 245

0 Introduction 245

1 Type of Mathematical Thinking to Be Cultivated 245

2 Grade Taught 245

3 Preparation 245

4 Overview of the Lesson Process 246

5 Worksheet 246

6 Lesson Process 248

7 Summarization on the Blackboard 253

8 Evaluation 253

Example 10 Development of the Arrangement of Numbers in the Extended Calendar 255

0 Introduction 255

1 Type of Mathematical Thinking to be Cultivated 255

2 Grade Taught 255

3 Preparation 255

4 Overview of the Lesson Process 256

5 Worksheet 256

6 Lesson Process 259

7 Summarization on the Blackboard 267

8 Evaluation 267

Example 11 Sums of Two Numbers in an Odd Number Table 269

0 Introduction 269

1 Type of Mathematical Thinking to Be Cultivated 269

2 Grade Taught 269

3 Preparation 270

4 Overview of the Lesson Process 270

5 Worksheet 270

6 Lesson Process 272

7 Summarization on the Blackboard 281

8 Evaluation 281

Example 12 When You Draw a Square on an Odd Number Table, What Are the Sum of the Numbers at the Vertices and the Grand Total of All the Numbers? 283

0 Introduction 283

1 Type of Mathematical Thinking to Be Cultivated 283

2 Grade Taught 283

3 Preparation 283

4 Overview of the Lesson Process 284

5 Worksheet 284

6 Lesson Process 286

7 Summarization on the Blackboard 296

8 Evaluation 296

9 Further Development 296