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6005893632.Pdf Mathematics for Dyslexics and Dyscalculics Mathematics for Dyslexics and Dyscalculics A Teaching Handbook Fourth Edition Steve Chinn and Richard E. Ashcroft This edition first published 2017 © 2017 John Wiley & Sons, Ltd. Edition history: 1e Whurr Publishers Ltd, 1993; 2e Whurr Publishers Ltd. 1998; 3e John Wiley & Sons, Ltd. 2007 Registered Office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Offices 350 Main Street, Malden, MA 02148‐5020, USA 9600 Garsington Road, Oxford, OX4 2DQ, UK The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, for customer services, and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com/wiley‐blackwell. The right of Steve Chinn and Richard E. Ashcroft to be identified as the authors of this work has been asserted in accordance with the UK Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and authors have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging‐in‐Publication data applied for Paperback ISBN: 9781119159964 A catalogue record for this book is available from the British Library. Cover image: Violka08/Gettyimages Set in 10/12.5pt Galliard by SPi Global, Pondicherry, India 10 9 8 7 6 5 4 3 2 1 Dedicated to the memory of our dear friend and colleague, Lyn Rawles Contents Foreword xi 1 Dyscalculia, Dyslexia and Mathematics 1 Introduction 1 Definitions of Dyslexia 3 The Evolution of Definitions of (Developmental) Dyscalculia 5 The Nature of Mathematics and the Ways it is Taught 19 2 Factors that may Contribute to Learning Difficulties in Mathematics 23 Potential Areas of Difficulty in Learning Mathematics 24 General Principles of Intervention 44 Summary 46 3 Cognitive (Thinking) Style in Mathematics 48 Introduction 48 Qualitative and Quantitative Style 49 Cognitive (Thinking) Style in the Classroom 51 Examples 54 Summary 61 4 Testing and Diagnosis 62 A Diagnostic Test Protocol 64 Structure of the Diagnostic Protocol 65 Attitude and Anxiety 73 Thinking Style or Cognitive Style 74 Tests for Dyscalculia 74 Summary of the Test Protocol 74 5 Concept of Number 76 Introduction 76 Early Recognition of Numbers and Their Values 78 The Language of Mathematics (see also Chapter 2) 79 viii Contents Early Number Work 79 Visual Sense of Number 82 Visual Clues to Number Concept 83 Number Bonds/Combinations 84 Place Value 86 Number Bonds/Combinations for Ten 93 Numbers Near Ten, Hundred or Thousand 94 Summary 95 6 Addition and Subtraction: Basic Facts/Number Combinations 96 Introduction 96 Strategies for Learning/Remembering the Addition and Subtraction Facts 99 Overview 107 Subtraction Facts 108 Extension 109 7 The Times Tables Facts/Number Combinations 110 Introduction 110 Rote Learning Strategies 110 Learning by Understanding 113 The Commutative Property 115 Learning the Table Square 117 Summary 135 8 Computational Procedures for Addition and Subtraction 136 Estimation 138 Addition 138 Teaching Subtraction as a Separate Exercise 147 9 Multiplication 152 Introduction 152 Multiplying by Ten and Powers of Ten 153 Multiplication 157 Estimation 163 Extension 163 10 Division: Whole Numbers 165 Introduction 165 Introduction to Division 166 Estimating 172 Division by Powers of Ten 173 Division by Multiples of Powers of Ten 174 Conclusion 174 Contents ix 11 Fractions, Decimals and Percentages: An Introduction 175 Introduction 175 Fractions 176 Decimals 187 Percentages 195 Overview 203 12 Operating with Fractions 204 Introduction 204 Making Segment Sizes the Same 205 Comparing Fractions 207 Combining Fractions 210 Adding Fractions 212 Subtracting Fractions 216 Combined Additions and Subtractions 219 Multiplying by Fractions 221 Multiplying Mixed Fractions 224 Dividing with Fractions 227 13 Decimals 232 Introduction 232 Addition and Subtraction 232 Multiplication and Division by Powers of Ten 235 Division of Decimals 242 Summary 249 14 Percentages 250 Introduction 250 An Image of Percentage 250 Summary 256 15 Time 257 What are the Potential Problems with Time? 258 Reading the Time 260 Time Problems 264 Summary 268 16 Teaching the Full Curriculum 269 Introduction 269 Some General Principles 270 Teaching the Other Parts of the Curriculum 277 Combining the Parts of the Curriculum 282 Summary 286 x Contents 17 Attacking and Checking Questions 287 Practice Examples 287 Preliminary Checks 288 Attacking Questions 290 Methods of Attacking Questions 291 Checking 298 18 Important Elements of a Teaching Programme 303 Introduction 303 Consider the Pupils’ Needs 303 The Structure of the Course 304 Classroom Management: Making the Lessons Suit the Pupils 308 Evolving Expectations and Emphases 309 Internal Assessment 313 Summary 315 Appendices Appendix 1: Books, Journals, Tests and Games 316 Appendix 2: Teaching Materials 321 References 324 Index 337 Foreword Steve Chinn’s newest edition of the 1993 Chinn and Ashcroft book Mathematics for Dyslexics is an important and timely addition to the dyscal- culia or mathematical learning disabilities literature. As Chinn notes in his introductory chapter, basic and applied research on dyscalculia has lagged behind that of dyslexia for decades. The gap has closed somewhat in recent years, but children’s and adults’ struggles with basic mathematics remain relatively neglected. This seemingly benign neglect can have largely unrec- ognised long‐term consequences for students and ultimately the commu- nities within which they reside (National Mathematics Advisory Panel, 2008), independent of their reading skills, cognitive abilities, or ethnic background. For instance, adults who have difficulty solving word prob- lems that require whole number arithmetic, fractions, simple algebra and measurement have compromised employment prospects (Bynner, 1997; Parsons and Bynner, 1997; Rivera‐Batiz, 1992), and difficulties in other areas of life that require quantitative knowledge (e.g. Reyna et al., 2009). These adults would be considered innumerate – either because of an underlying difficulty with mathematical learning (e.g. low working memory capacity), poor instruction, poor motivation, or some combina- tion – and now include 20–25% of adults in the United Kingdom and the United States. Unfortunately, the path to adult innumeracy starts early, probably before the beginning of formal schooling (Duncan et al., 2007; Ritchie and Bates, 2013). One recent study suggested that children who start school with a poor understanding of Arabic numerals, the magni- tudes they represent and the relations among them [e.g. 7 = (6 + 1) = (5 + 2) = (4 + 3)] score poorly on numeracy tests as adolescents (Geary, Hoard, Nugent, and Bailey, 2013). In this study, six‐year‐olds who started school in the bottom 15–20% in this number knowledge were four times more likely to score as innumerate in adolescence on tests that have been shown to predict later employability and wages in young adults. By eight years of xii Foreword age, these at‐risk children learned basic arithmetic as quickly as other chil- dren, but they never narrowed the school‐entry gap. This is not to say that they could not catch up with the right remediation approaches, but it does emphasise the importance of catching and addressing potential difficulties early. The questions then become, what are the early precursors of school‐ entry quantitative knowledge and what are the key mathematical compe- tencies that must be developed to ensure students have functional numeracy skills by the time they leave school? The foundation is a solid understanding of Arabic numerals and the relations among them, and in Chapter 5 Chinn does an admirable job of identifying these core number skills and how they should become elaborated over the course of school- ing (i.e. organized around the base‐ten system). Strong arithmetic skills are just as important as the number skills on which they are built, and Chinn devotes chapters 6–14, inclusive, to various aspects of arithmetic, from learning the multiplication tables (or learning backup strategies for children who have difficulties memorising facts), to rational
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