Learning mathematics in elementary and middle school.: a learner-centred approach PDF
Preview Learning mathematics in elementary and middle school.: a learner-centred approach
EDITION 5 Learning Mathematics in Elementary and Middle Schools A Learner-Centered Approach W. George Cathcart University of Alberta Yvonne M. Pothier Mount Saint Vincent University James H. Vance University of Victoria Nadine S. Bezuk San Diego State University Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Series Editor: Kelly Villella Canton Editorial Assistant: Annalea Manalili Vice President, Director of Marketing: Quinn Perkson Senior Development Editor: Mary Kriener Senior Marketing Manager: Darcy Betts Production Editor: Gregory Erb Editorial Production Service: Nesbitt Graphics, Inc. Manufacturing Buyer: Megan Cochran Electronic Composition: Nesbitt Graphics, Inc. Interior Design: Nesbitt Graphics, Inc. Cover Designer: Linda Knowles Copyright © 2011 Pearson Education, Inc., publishing as Allyn & Bacon, 501 Boylston Street, Boston, MA, 02116. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permis- sion should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or trans- mission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions De- partment, 501 Boylston Street, Boston, MA 02116, or email [email protected]. Chapter opening photos © 2005 LessonLab, a division of Pearson Education, Inc. All rights reserved. Library of Congress Cataloging-in-Publication information was unavailable at press time. 10 9 8 7 6 5 4 3 2 1 WEB 14 13 12 11 10 ISBN-10: 0-13-242099-6 www.pearsonhighered.com ISBN-13: 978-0-13-242099-0 This edition is dedicated to my husband, Steve, and my son, Peter, whose encouragement and support made this work possible. —N.S.B. Brief Contents 1 Teaching Mathematics: Influences and Directions 1 2 Learning and Teaching Mathematics 15 3 Developing Mathematical Thinking and Problem-Solving Ability 39 4 Assessing Mathematics Understanding 60 5 Developing Number Concepts 76 6 Developing Understanding of Numeration 99 7 Developing Whole-Number Operations: Meaning of Operations 126 8 Developing Whole-Number Operations: Mastering the Basic Facts 149 9 Estimation and Computational Procedures for Whole Numbers 170 10 Developing Fraction Concepts 207 11 Developing Fraction Computation 234 12 Developing Decimal Concepts and Computation 259 13 Understanding Ratio, Proportion, and Percent 282 14 Developing Geometric Thinking and Spatial Sense 297 15 Developing Measurement Concepts and Skills 331 16 Collecting, Organizing, and Interpreting Data 363 17 Developing Algebraic Thinking 396 References 435 Index 443 iv Contents CHAPTER CHAPTER Teaching Mathematics: Learning and Teaching 1 2 Influences and Directions 1 Mathematics 15 Influences on Mathematics Education 2 CONNECTING WITH THE STANDARDS 15 Science, Mathematics, and Technology 2 Learning Theories 17 The Diversity of Our World: Learner The Behaviorist Approach 17 Influences 2 The Cognitive/Constructivist Gender considerations 3 Approach 17 Students with special needs 3 The role of the teacher 17 English language learners 4 Learner-centered instruction 18 Curriculum Guided by Standards 4 Cooperative learning 18 TIMSS 4 Jean Piaget 18 NAEP 4 Basic Principles Reviewed 20 Curriculum and Evaluation Standards 5 Begin with Concrete Representation 20 Teaching Standards 5 Assessment Standards 5 Develop Understanding 20 Principles and Standards 5 Modes of representation 20 Integrating the content and process standards 6 Making connections 21 Integrating state and local standards with national Encourage Communication 21 math standards 7 Math journals 22 Curriculum Focal Points 7 Make Connections 23 Teacher Influences 8 Take Time to Motivate Children 23 Directions in Mathematics Education 8 Attitudes 24 Problem Solving 8 Provide Opportunities for Practice 24 Communication 8 Games 25 Puzzles and riddles 25 Reasoning and Proof 9 Surprises 25 Connections 10 The calculator as a practice tool 26 Integration with other school subjects 10 Thinking about Teaching 26 Integration with real-world settings 11 Understanding Individual Needs 26 Representation of Mathematical Ideas 11 Role of communication revisited 26 Equity 11 Language influences 27 Technology 12 Supporting English language learners (ELL) 27 Computation and Estimation 12 Mathematics anxiety 28 Myths about learning mathematics 28 Assessment 12 The Teaching Act 28 Parent Involvement 12 Preteaching Activities 28 Conclusion 13 Identifying children’s learning needs 29 In Practice 14 Using mathematics textbooks 29 Links to the Internet 14 Considering state and local standards 29 Resources for Teachers 14 Planning 29 v vi CONTENTS Sample Lesson: Solving Problems Involving Planning for Instruction on Multidigit Addition 32 Problem Solving 51 Teacher’s postlesson reflections 33 Selecting Appropriate Tasks and Use of the process standards in this lesson 34 Materials 51 The Process of Teaching 34 Problems that are motivating and Models for teaching mathematics 34 culturally relevant 52 Problems with missing, extraneous, or contradictory Postteaching Activities 35 information 52 Evaluation of teaching 35 Problems that encourage the use of calculators, Reflection on teaching 36 computers, and other technology 52 Thinking About the Curriculum 36 Activities that require the use of a variety of The Mathematics 36 problem-solving strategies 52 The Activities 36 Activities that promote communication about mathematical thinking 52 Conclusion 37 Identifying Sources of Problems 52 In Practice 37 Ask children to write problems 52 Links to the Internet 38 Resources for Teachers 38 Clarifying the Teacher’s Role 53 Before children begin to solve the problem 54 While children are solving the problem 54 CHAPTER After children solve the problem 54 Developing Mathematical Examining children’s work 54 3 Thinking and Problem- Organizing and Implementing Solving Ability 39 Instruction 55 Classroom climate 55 CONNECTING WITH THE STANDARDS 39 Grouping children 55 Allocating time 55 Problem Solving: An Integral Part of Assessing children’s understanding 55 Mathematics Instruction 40 Changing the Difficulty of Problems 55 Mathematical Considerations 41 Problem context 55 What Is a Problem? 41 Problem structure 56 Types of Problems 41 Classroom implications 56 Process problems 41 Other Factors Contributing to Children’s Translation problems 42 Difficulties in Problem Solving 57 Application problems 42 Knowledge factors 57 Puzzles 42 Beliefs and affective factors 57 The Problem-Solving Process 42 Control factors 57 Sociocultural factors 57 Understanding the Problem 42 Consider Children’s Preferences 57 Devising a Plan to Solve the Problem 43 Benefits of Using a Problem-Solving Approach Implementing a Solution Plan 43 to Mathematics Instruction 58 Reflecting on the Problem 43 Conclusion 58 Problem-Solving Strategies 45 In Practice 58 Dramatizing or Modeling the Situation and Links to the Internet 59 Solution Process 45 Resources for Teachers 59 Drawing a Picture or Diagram 46 Constructing a Table or Chart 47 CHAPTER Finding a Pattern 47 Assessing Mathematics Solving a Simpler Problem 47 4 Understanding 60 Guessing and Checking 49 Working Backward 49 CONNECTING WITH THE STANDARDS 60 Considering All Possibilities 50 The Assessment Standards 61 Logical Reasoning 50 What Is Assessment? 61 Changing Your Point of View 50 Purposes of Assessment 62 Writing an Open Sentence 51 Phases of Assessment 62 vii Assessment Choices 63 Pictorial and Graphic Representation of Numbers 89 Achievement Tests 63 Standardized tests 63 Symbolic Representation of Numbers 89 Teacher-made tests 63 Numerals 90 Using tests to determine children’s Number Relationships 90 learning needs 63 Order Relations 92 Individualizing Assessment 64 Observation 64 More Than, Fewer Than 92 Using Preassessments to Inform One Greater Than, One Less Than 92 Instruction 64 Part-Part-Whole Relationships 93 Examining Children’s Reasoning 64 Relationship to 5 and 10 93 Conferences 65 Using action language 94 Interviews 65 Bidirectional Relationship of an Equation 94 Performance assessment 66 Estimation 96 Portfolio assessment 70 Conclusion 96 Self-assessment 72 AssessingAttitudes toward Sample Lesson: A Lesson on What Is Mathematics 73 More? 97 Conclusion 74 In Practice 98 Links to the Internet 98 In Practice 74 Resources for Teachers 98 Links to the Internet 75 Resources for Teachers 75 CHAPTER Developing Understanding of CHAPTER 6 Numeration 99 Developing Number 5 Concepts 76 CONNECTING WITH THE STANDARDS 99 ASSESSING MATHEMATICS CONNECTING WITH THE STANDARDS 76 UNDERSTANDING 100 ASSESSING MATHEMATICS Numeration 103 UNDERSTANDING 77 Number Systems 103 The Foundations of Number 79 Numeration Systems 103 Prenumber Activities 79 Classification 79 The Hindu-Arabic Numeration System 103 Examining Children’s Reasoning 80 Base Ten 103 Seriation 81 Positional or Place Value 103 Patterns 82 Multiplicative Principle 104 One-to-one correspondence 82 Additive Principle 104 Conservation of number 83 Number Meanings 83 Zero as a Placeholder 104 Cardinal Use of Numbers 83 Understanding Place Value 104 What Research Says about Place-Value Ordinal Use of Numbers 83 Learning 104 Nominal Use of Numbers 84 Stages in Place-Value Development 105 Counting 84 Grouping 105 Discrete and Continuous Quantities 84 Communicating Mathematics 106 Rote Counting 84 Grouping by Tens 107 Rational Counting 84 Equivalent representations 107 Counting All, Counting On 85 Types of Place-Value Materials 107 Counting Back 86 Examining Children’s Reasoning: Using Models to Skip Counting 86 Represent Place-Value Concepts 108 Representing Numbers 87 Developing Understanding of Two-Digit Concrete Models 87 Numbers 108 viii CONTENTS Introducing Base-Ten Blocks 110 Understanding Addition and Using Place-Value Mats 111 Subtraction 131 Introducing Nonproportional Types of Addition and Subtraction Word Materials 111 Problems 131 Assessing Place-Value Knowledge 114 Examining Children’s Reasoning: Linking to Using manipulatives to represent “Amazing Equations” 131 numbers 114 Examining Children’s Reasoning: Strategies for Understanding tens and ones 114 Solving Story Problems 134 Regrouping with tens and ones 114 Using Models to Solve Addition and Three-Digit Numbers 114 Subtraction Problems 134 Number Meanings: Oral Direct modeling 134 Expressions 115 Modeling Separate problems 135 Developing Number Relationships 116 ModelingPart-Part-WholeandCompare problems 136 Using Hundreds Charts 116 Usingmeasurementmodelstomodel Thinking and Writing about problems 136 Numbers 117 Writing Number Sentences for Examining Children’s Reasoning: Addition and Subtraction 137 Understanding Place Value 118 Understanding Multiplication Understanding Large Numbers 119 and Division 138 Number Names 119 Making the Transition from Adding to Writing Consecutive Numbers 119 Multiplying 138 Magnitude of Numbers 119 Encouragechildrentousethephrase“groupsof” toindicatecreatinganumberofequal Counting to a Thousand and Beyond 120 groups 138 Expanded Notation 120 Help children understand the meaning of each Rounding Numbers 121 quantity 139 Rounding Rules 121 Types of Multiplication Word Problems 139 Equal Groups problems 139 The Computer Rule for Rounding 121 Area and Array problems 139 Estimating 122 Multiplicative Comparison problems 139 Consolidating Number Skills 122 Combination problems 139 Conclusion 123 Introducing Children to Division 139 Division with remainders 139 Sample Lesson: A Lesson on Place Value: Making Groups of Ten 124 Types of Division Word Problems 141 Equal Groups problems 141 In Practice 125 Area and Array problems 142 Links to the Internet 125 Multiplicative Comparison problems 142 Resources for Teachers 125 Combination problems 142 Avoiding misconceptions and dead ends 142 CHAPTER Examining Children’s Reasoning: Strategies for Developing Whole-Number Solving Story Problems 143 7 Operations: Meaning of Using Models to Solve Multiplication Operations 126 and Division Problems 143 Modeling Equal Groups and Multiplicative Comparison problems 143 CONNECTING WITH THE STANDARDS 126 Modeling Area and Array problems 144 ASSESSING MATHEMATICS Modeling Combination problems 144 UNDERSTANDING 127 An instructional sequence for modeling Introduce Operations with Word multiplication and division 145 Problems 129 Another Word about Notation A Model for Beginning with Word and Children’s Language 145 Problems 129 Conclusion 146 Encoding and Decoding Word Sample Lesson: A Lesson on Solving Problems 130 Comparison Problems 147 ix In Practice 147 Fact Families for Multiplication and Links to the Internet 148 Division 165 Resources for Teachers 148 Consolidating Activities for Drill and Practice 165 Planning for Progress Checks Before CHAPTER Developing Whole-Number You Begin 166 8 Operations: Mastering the Setting up Consolidating Basic Facts 149 Activities 166 Using Games 166 CONNECTING WITH THE STANDARDS 149 Puzzles and Riddles 167 ASSESSING MATHEMATICS Using Novel Formats/Novel Ideas 167 UNDERSTANDING 149 Using Computer Software 167 What Are Basic Facts? 152 Conclusion 167 A Three-Step Approach to Fact Sample Lesson: A Lesson on Understanding the Mastery 153 Commutative Property 168 Step 1: Understanding the Meaning of the In Practice 169 Operations 153 Links to the Internet 169 Step 2: Using Thinking Strategies to Retrieve Resources for Teachers 169 Facts 153 Step 3: Using Consolidating Activities for Drill and Practice 153 Addition and Subtraction Facts 154 CHAPTER Estimation and Computational Thinking Strategies for Addition and 9 Procedures for Whole Subtraction 154 Numbers 170 Counting on 154 Counting on in subtraction 155 Counting back 155 CONNECTING WITH THE STANDARDS 170 One more or one less than a known fact 156 ASSESSING MATHEMATICS Compensation 156 UNDERSTANDING 171 Using thinking strategies to organize instruction 157 What Is Computation? 173 Estimation and Mental Mathematical Properties of Addition and Subtraction 157 Computation 174 Commutative property 157 Mental Computation 174 Associative property 158 Estimation 174 Addition property of zero 158 Front-end strategy 175 Fact Families for Addition Rounding strategy 175 and Subtraction 158 Clustering strategy 175 Multiplication and Division Facts 159 Compatible numbers strategy 175 Special numbers strategy 175 Thinking Strategies for Multiplication and Paper-and-Pencil Computation 175 Division 159 Repeated addition 159 An Instructional Philosophy 176 Skip counting 159 Prerequisites 176 Splitting the product into known parts 160 Important Considerations When Teaching Facts of five 160 Computational Procedures 177 Patterns 161 A note about models for computation 177 Mathematical Properties of A note about language 177 Multiplication 162 A note about the role of calculators Commutative property 162 in computation 177 Associative property 163 A note about children’s thinking 178 Distributive property of multiplication Addition 178 over addition 163 Multiplication property of one 164 Posing Story Problems Set in Real-World Multiplication property of zero 164 Contexts 178
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