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Copyright, ii
Introduction, viii
Preface to the Instructor, xi
Acknowledgements, xii
Chapter 1. Problem Solving, 1
1.1. An Introduction to Problem Solving, 1
What is Problem Solving?, 1
Why do Problem Solving?, 2
Learning to Teach Problem Solving, 3
What is a (Mathematical) Problem for Children?, 4
Children’s Beliefs about Solving Problems, 5
How Children Solve Problems, 5
Problem Solving Steps, 6
Problem Solving Strategies – Heuristics, 7
1.2. Patterns, 13
1.3. Mathematical Reasoning, 16
Deductive and Inductive Reasoning, 16
Sense-Making, 17
Key Words, 18
Incorporating Writing and Mathematics, 19
References, 20
Chapter 2. Sets, 22
2.1. Set Theory, 22
A Historical Perspective, 22
What is Number?, 23
How do Children Develop the Concept of Number, 23
Set Theory and Basic Operations, 25
Reasons for Studying Set Theory, 25
2.2. Venn Diagrams, 27
References, 31
Chapter 3. Whole Numbers, 32
3.1. Numeration Systems, 32
The Need for Numbers, 32
Models of Our Number System, 32
Expanded Notation, 35
3.2. Addition and Subtraction, 37
One Child’s Perspective of Addition, 38
Counting Types, 38
Counting Based Strategies, 41
Concept of Ten and Place Value, 42
Thinking Strategies and Learning “Basic Facts”, 43
Subtraction/Take-Away, 44
Two Levels of Difficulty for Subtraction in Context, 45
3.3. Multiplication and Division, 47
Multiplication, 47
From Additive to Multiplicative Thinking, 47
Cartesian Products, 49
Multiplication Thinking Strategies and Basic Facts, 49
How Young Children Divide, 50
Division By and With 0, 52
Order of Operations, 52
3.4. Properties of Whole Number Operations, 55
Commutative Property, 56
Associative Property, 58
Distributive Property, 58
Identity Properties, 59
Closure Property, 59
3.5. Algorithms, 61
Addition and Subtraction Algorithms, 61
Children’s Self-Generated Algorithms, 62
Multiplication Algorithms, 64
Division Algorithms, 65
3.6. Mental Math & Estimation, 75
References, 79
Chapter 4. Number Theory, 81
4.1. Factors and Multiples, 81
4.2. Divisibility Tests, 83
4.3. Prime and Composite Numbers, 85
4.4. Greatest Common Factor & Least Common Multiple, 87
References, 90
Chapter 5. Integers, 91
5.1. Children’s Understanding of Negative Numbers, 91
Number Line Representation, 91
Number as a Quantity Representation, 92
Children’s Use of Negative Numbers in Algorithms, 92
Children’s Understanding of Negative Numbers in a Different Context, 93
5.2. Addition and Subtraction of Integers, 94
Parentheses, 95
Number Line, 95
Black and Red Chip Model, 95
Absolute Value, 96
5.3. Multiplication and Division of Integers, 98
How Children Solve Some Integer Word Problems, 100
References, 102
Chapter 6. Rational Numbers - Fractions, 103
6.1. Fractions, 103
Fraction as a Part-whole Relationship, 104
Fraction as a Quotient, 105
Fraction as a Location on a Number Line or as a Measure, 105
Fraction as a Ratio, 105
Fraction as an Operator, 105
Equivalent Fractions, 106
Improper Fractions, 106
Common Misconceptions with Fractions, 107
6.2. Addition and Subtraction of Fractions, 118
Models for Adding and Subtracting Fractions, 119
Common Misconceptions Adding and Subtracting Fractions, 120
6.3. Multiplication and Division of Fractions, 122
Models for Multiplying and Dividing Fractions, 123
Common Misconceptions Multiplying and Dividing Fractions, 124
6.4. Properties of Rational Numbers, 129
Correspondence of Rational Numbers and Points on the Number Line, 129
References, 130
Chapter 7. Decimals, Percents, and Real Numbers, 132
7.1. Place Value, 132
7.2. Decimals, 135
A Foundation for Decimal Understanding, 135
Models for Decimal Numbers, 137
Rounding Decimal Numbers, 139
Children’s Understanding of Decimals, 139
Scientific Notation, 140
7.3. Decimal Computation, 142
Lining up the Decimal Points, 143
Counting the Number of Decimal Places in the Factors when Multiplying Decimals, 143
Moving the Decimal Point in the Dividend and the Divisor when Dividing, 143
More on Decimal Computation, 144
Partial Sums of Decimals, 145
7.4. Ratio and Proportion, 147
Unit or Unit Rate Method, 147
Scale Factor or Composite Unit Rate, 148
Building-Up, 148
Cross Multiplication, 148
Rate, Ratio, and Scale Factor, 149
Proportional Reasoning, 149
7.5. Percents, 153
Children’s Knowledge of Percents, 153
Mental Computation with Percents, 154
7.6. Rational, Irrational, and Real Numbers, 157
References, 157
Chapter 8. Geometry, 159
8.1. Basic Geometric Concepts, 159
Geometry in Elementary School, 159
Concept Image, 160
van Hiele Levels, 160
Points, Lines, and Planes, 161
Congruence, 161
Parallel and Perpendicular Lines, 162
8.2. Basic Shapes, 166
Shapes, 166
Convex, 166
Shapes, Interior, and Exterior, 166
Triangle, 167
Rectangle, 167
Parallelogram, 167
Invariance of Shape, 167
Computer Tools, 168
8.3. Angles, 175
Angle Measurement, 175
Supplementary and Complementary Angles, 175
Common Problems Children Have Measuring with a Protractor, 176
8.4. Proof/Mathematical Reasoning/Justification/Argumentation, 183
The Sum of the Measures of the Angles of a Triangle, 184
8.5. Three-Dimensional Geometry, 187
What is Important in Three-Dimensional Geometry, 188
Prisms, 188
Euler’s Formula, 188
Mobius Strips, 189
References, 192
Chapter 9. More Geometry, 195
9.1. Transformations or Rigid Motions, 195
More on Congruence, 195
Tessellations, 195
Vertical Angles and Parallel Lines Cut by a Transversal, 196
9.2. Constructions, 200
9.3. Symmetry, 202
9.4. Similarity, 206
What is Trigonometry?, 207
References, 211
Chapter 10. Measurement, 212
10.1. The Concept of Measurement, 213
Measurement as a Developmental Process, 213
The Metric System, 214
10.2. Linear Measurement, 218
Key Concepts of Measurement, 218
The Teaching of Linear Measurement, 221
10.3. Area and Perimeter, 226
Perimeter, 226
The Concept of Area, 227
Children’s Common Misconceptions about Area, 228
Mud Puddle Problem, 229
Key Concepts of Area, 229
The Teaching of Area Measurement, 231
Area Formulas, 232
Area of a Circle, 234
10.4. Volume and Surface Area, 243
Surface Area of a Three-dimensional Figure, 244
10.5. Time, 247
References, 249
Chapter 11. Statistics/Data Analysis, 252
11.1. Data Analysis and Statistical Graphs, 252
Statistical Graphs, 255
11.2. Statistical Deceptions and Examining Statistics Critically, 264
11.3. Mean, Mode, and Median, 268
Mode, 268
Ideal Average, 268
Midrange, 269
Median, 269
Mean, 269
11.4. Variation or Spread, 274
11.5. Statistical Samples, 276
Randomness, 277
Normal Distribution, 278
References, 279
Chapter 12. Probability, 281
12.1. Basic Notions of Probability, 281
The Ratio Concept, 281
Chance, 282
Intuitive Understanding of Probability, 284
Equal Likely Outcomes, 285
Simulations, 285
Randomness, 285
Sample Space, 286
12.2. More Sophisticated Concepts of Probability, 293
Multistage Probability, 293
Independent and Dependent Events, 293
Mutually Exclusive Events, 294
Expected Value, 294
Odds, 295
References, 298
Chapter 13. Algebraic Reasoning, 300
13.1. The Concept of Variable, 301
13.2. Algebraic Reasoning: Generalizing, 306
Generalizing a Specific Problem, 306
Generalizing a General Principle or Property, 308
13.3. Generalizing with Two Variables – Functions, 315
13.4. Graphing: Coordinate Geometry, 319
13.5. The Concept of Equality, 322
References, 324
Answers to Problems and Exercises, 325
Index, 332

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