Table Of ContentEDITION
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
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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
