MATHEMATICAL THINKING How to Develop it in the Classroom
Monographs on Lesson Study for Teaching Mathematics and Sciences Series Editors: Kaye Stacey (University of Melbourne, Australia) David Tall (University of Warwick, UK) Masami Isoda (University of Tsukuba, Japan) Maitree Inprasitha (Khon Kaen University, Thailand) Published Vol. 1 Mathematical Thinking: How to Develop it in the Classroom by Masami Isoda (University of Tsukuba, Japan) and Shigeo Katagiri (Society of Elementary Mathematics Education, Japan)
Monographs on Lesson Study for Teaching Mathematics and Sciences Vol. 1 MATHEMATICAL THINKING How to Develop it in the Classroom Masami Isoda University of Tsukuba, Japan Shigeo Katagiri Society of Elementary Mathematics Education, Japan World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Monographs on Lesson Study for Teaching Mathematics and Sciences Vol. 1 MATHEMATICAL THINKING How to Develop it in the Classroom Copyright 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4350-83-9 ISBN-10 981-4350-83-4 ISBN-13 978-981-4350-84-6 (pbk) ISBN-10 981-4350-84-2 (pbk) Printed in Singapore.
Preface to the Series: Monographs on Lesson Study for Teaching Mathematics and Science Lesson study is a system of planning and delivering teaching and learning that is designed to challenge teachers to innovate their teaching approaches, and to recognize the possibilities of intellectual and responsible growth of learners while fostering selfconfidence in all concerned. It operates when teachers develop a sequence of lessons together: to plan (by preparing the lesson in advance, including a prediction of the possible learning), to do (by presenting the class to children observed by other teachers), and to reflect on the learning with the observers (through discussion) so as to improve the lesson for future presentation on a wider scale. It is intended to develop good pedagogical content knowledge that will be useful for the everyday good practice of teachers and the consequent long-term learning of students. The theoretical frameworks in lesson study involve both an overall global theory and local theories that apply in a particular situation for a particular task. These theories which have been developed through a number of lesson studies are intended to support the design of the classroom teaching. On this meaning, lesson study is a re-productive science which produces good practices to develop children in classrooms in various settings. There has already been worldwide growth of research in the first decade of the twenty-first v
vi Mathematical Thinking century that recognizes the role of teachers theories of teaching and learning. Lesson study is a key component that draws together these theories to develop innovative ways of improving teacher practice through sharing observations in the classroom. Evidence of good teaching practice is rarely seen by others, and lesson study provides the opportunity for teachers to share and develop their personal expertise within a wider framework. Lesson study offers well-developed children s activities and teachers actions and interactions in the classroom that can be beneficial for the improvement of teaching and learning in mathematics and science. This monograph series provides teachers, educators, and researchers with illuminating exemplars of the theoretical advances in teaching mathematics and science that are the outcomes of lesson study. It also proposes that teachers, educators, and researchers develop their own teaching approaches and theorize about their own knowledge of teaching to be shared more widely. The series editors welcome anyone to propose his/her theory of teaching mathematics and science in this series and to join the movement of lesson study. Series Editors Kaye Stacey David Tall Masami Isoda Maitree Inprasitha
Preface to the Book For teachers: Are you enjoying mathematics with the children in your classroom? If you develop children who think mathematically, your class will be really enjoyable for both you and the children. This book explains how to develop mathematical thinking in the elementary school classroom. It is especially written for elementary school teachers who are not math majors and wish to teach mathematics in interesting ways. For secondary school mathematics teachers, it will also be useful, because most of the examples are open-ended tasks which will be meaningful to both kids and adults. For researchers: How can you work with teachers to enhance innovation in mathematics education? How can you theorize about it? This book provides you with a theory of mathematics education which has been developed with teachers through lesson study and shared by teachers in their daily teaching practices. This theory supports better reproduction of the mathematics class in order to develop children s mathematical thinking. It already has a wide range of evidence through the lesson studies during the last fifty years. You may recognize that developing the theory of mathematical thinking with schoolteachers in the context of lesson study is also an innovation for mathematics education research, because it provides you with the methodology as in reproductive science. vii
viii Mathematical Thinking Developing mathematical thinking has been a major objective of mathematics education. In today s knowledge-based society, developing process skills such as innovative ways of thinking for problem solving are much desired. Mathematics is also a subject necessary for innovation, as it develops creative and critical thinking in general, and mathematical and statistical thinking in specific situations. In the famous picture Scholars of Athens (ancient Greece), by the Renaissance painter Raphael (1483 1520), there is Euclid showing constructions to his students. At the center of the picture is a student who is explaining his findings to some ladies. This is an image of what ought to be the mathematics classroom: students enjoying mathematical communication among themselves. As well as in ancient Greece and during the Renaissance, mathematics is an enjoyable subject for developing mathematical thinking which is necessary for all academic subjects and useful for the modern world. This is an invariant feature of the subject of mathematics passed on from the age of the ancient Greece school called the Academy.
Preface to the Book ix Parts I and II of this book are written by Shigeo Katagiri, who is the former president of the Society of Mathematics Education for Elementary Schools in Japan, and edited and translated by Masami Isoda, corepresentative of the APEC Lesson Study Project. Katagiri s theory of mathematical thinking is well known in Japan, and also in Korea through Korean editions. If you are a beginner or a schoolteacher who is not a math major, the authors recommend that you try out two or three examples for problem solving in the Introductory Chapter and Part II. If you solve them by yourself, you may begin to imagine how enjoyable this book is. After you have captured some images for enjoying and developing mathematics, you may read from the Introductory Chapter, Part I, and Part II. The Introductory Chapter explains the teaching approach to developing mathematical thinking and provides the views on developing mathematical thinking. Part I explains what mathematical thinking is and how to develop it using questioning. Part II provides illuminating examples using the number table with assessment to show how you can develop mathematical thinking in your classroom. Katagiri s theory is one of the major references for mathematics education research in Japan. It is a pleasure to publish it in English for readers worldwide who are engaged in mathematics education research and mathematics teaching. Masami Isoda, representing the authors
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Acknowledgments The author, Masami Isoda, wishes to acknowledge the technical support given by John Dowsey (University of Melbourne) and Ui Hock Cheah (SEAMEO-RECSAM) for translation. xi
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Contents Preface to the Series Preface to the Book Acknowledgements Introductory Chapter: Problem Solving Approach to 1 Develop Mathematical Thinking Part I Mathematical Thinking: Theory of Teaching 29 Mathematics to Develop Children Who Learn Mathematics for Themselves Chapter 1 Mathematical Thinking as the Aim 31 of Education 1.1 Developing Children Who Learn Mathematics 31 for Themselves 1.2 Mathematical Thinking as an Ability to Think 32 and to Make Decisions 1.3 The Hierarchy of Ability and Thinking 35 v vii xi Chapter 2 The Importance of Cultivating 37 Mathematical Thinking 2.1 The Importance of Teaching Mathematical Thinking 37 2.2 Example: How Many Squares Are There? 39 xiii
xiv Mathematical Thinking 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, 83 Quantities, and Figures (Quantification and Schematization) 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
Contents xv 6.3 Clarity 115 6.4 Sophistication 117 Chapter 7 Questioning to Enhance Mathematical 121 Thinking Appendix for the List of Questions for Mathematical 127 Thinking Part II Developing Mathematical Thinking 129 with Number Tables: How to Teach Mathematical Thinking from the Viewpoint of Assessment Example 1 Sugoroku: Go Forward Ten Spaces 137 If You Win, or One If You Lose (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 play 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
xvi Mathematical Thinking 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 185 Table, What Are the Sum of the Numbers at the Vertices, the Sum of the Numbers Along the Perimeter, and the Grand Total of All the Numbers? (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
Contents xvii 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) Type 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
xviii Mathematical Thinking Example 9 The Arrangement of Numbers on an 245 Extended Calendar (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 255 Numbers in the Extended Calendar (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 269 Odd Number Table (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
Contents xix Example 12 When You Draw a Square on an 283 Odd Number Table, What Are the Sum of the Numbers at the Vertices and the Grand Total of All the Numbers? (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