Chapter 1 Learning Mathematics 1
Societal Needs 2
Calls for Change 3
National and International Achievement 3
The Identification of Standards 5
Principles and Standards for School Mathematics 6
Implementing the Process Standards 8
Describing A Contemporary Mathematics Program 10
Levels of Abstraction 11
Theoretical Foundations for a Contemporary
Mathematics Program 12
Research Snapshot 13
Theory into Practice 17
Closure 18
Teaching Competencies and Self-Assessment Tasks 19
Chapter 2 Teaching Mathematics 21
The NCTM Principles and Standards for School Mathematics 22
The NCTM Professional Teaching Standards: Mathematics Teaching Today 23
An Example of a Teaching Standard 25
Lesson Planning Processes and Purposes 29
Issues in Designing Mathematics Lessons 33
Mathematics Objectives 35
The Role of Motivation and Follow-Up 36
The Assessment of Learning and Teaching 37
Research Snapshot 38
Adjusting Instruction to Meet Individual Needs 38
Classroom Instructional Aids 39
Closure 40
Teaching Competencies and Self-Assessment Tasks 41
Chapter 3 Assessing Mathematics 43
NCTM Assessment Standards for School Mathematics 44
An Example of an Assessment Standard 47
A Classroom Assessment Vignette 48
Purposes for Assessment 49
Assessment Strategies 51
Research Snapshot 53
Closure 58
Teaching Competencies and Self-Assessment Tasks 59
Chapter 4 Technology in K¢"8 Mathematics 61
A Brief History of Technology¢s Influence on Education 62
Calculators and Exploration of Mathematical Concepts 63
Research Snapshot 64
Calculator Features and Functions 66
Examples of Effective Calculator Use 66
Tools for Measuring Motion 68
Computers and Mathematics Instruction 69
Instructional Software in the Mathematics Classroom 71
Application Software in the Mathematics Classroom 72
Virtual Manipulatives 73
Design Issues and Advantages of Virtual Manipulatives 74
Classification Attributes for Virtual Manipulatives 75
Capsule Descriptions of the National Library of Virtual Manipulatives 76
Research Snapshot 80
Multimedia in the Mathematics Classroom 81
The Internet in the Mathematics Classroom 82
Programming Computers and Calculators 83
Closure 84
Teaching Competencies and Self-Assessment Tasks 85
Chapter 5 Problem Solving and Mathematical Language 87
The Nature of Problem Solving 88
Problem-Solving Strategies 91
Research Snapshot 93
Approaching Word Problems Effectively 98
Logic and Reasoning 101
The Language of Logic 101
The Language of Mathematics 103
Closure 107
Practice Exercises for Teachers 108
Teaching Competencies and Self-Assessment Tasks 111
Sample Instructional and Assessment Activities 112
Chapter 6 Number Sense, Numeration, and Place Value 115
A Foundation for Mathematics Development 116
Patterns and Other Relationships in the Primary Curriculum 117
Number Sense 118
Number Relations 123
Extending Numberness Concepts 125
Place-Value Numeration 125
Decimal Numeration 126
Ancient Numeration Systems 131
Research Snapshot 133
Number Bases Other Than Ten 134
Closure 137
Practice Exercises for Teachers 138
Teaching Competencies and Self-Assessment Tasks 141
Sample Instructional and Assessment Activities 142
Chapter 7 Addition and Subtraction of Whole Numbers 145
Creating a Sound Base for Addition and Subtraction 146
Meaning and Models for Addition 147
Developing Basic Addition Facts 148
Memorizing Basic Addition Facts Using Structures 149
Extending Basic Addition Facts Using Place Value 155
Regrouping in the Addition Process 156
Meaning and Models for Subtraction 157
Research Snapshot 158
Developing Basic Subtraction Facts 160
Memorizing Basic Subtraction Facts Using Structures 161
Extending Basic Subtraction Facts Using Place Value 162
Regrouping in the Subtraction Process 163 Closure 164
Practice Exercises for Teachers 166
Teaching Competencies and Self-Assessment Tasks 167
Sample Instructional and Assessment Activities 167
Chapter 8 Multiplication and Division of Whole Numbers 171
Meaning and Models for Multiplication 172
Developing and Memorizing Basic Multiplication Facts 175
Expanding Basic Multiplication Facts Through Place Value 178
Regrouping in Multiplication 181
Meaning and Models for Division 183
Research Snapshot 183
Developing and Memorizing Basic Division Facts 186
Remainders in Division 187
Expanding Basic Division Facts Through Place Value 187
Closure 190
Practice Exercises for Teachers 191
Teaching Competencies and Self-Assessment Tasks 192
Sample Instructional and Assessment Activities 193
Chapter 9 Number Theory and Number Systems 197
The Language of Number Theory 199
The Sieve of Eratosthenes 200
Factor Trees 202
Prime Factorization 203
Divisibility ¢¢Rules¢¢ 205
Research Snapshot 207
Least Common Multiples and Greatest Common Factors 207
Number Systems 209
Closure 218
Practice Exercises for Teachers 218
Teaching Competencies and Self-Assessment Tasks 221
Sample Instructional and Assessment Activities 222
Chapter 10 Algebraic Reasoning: Generalizing
Patterns and Relationships 225
The Content of Algebra 226
Research Snapshot 227
Algebra in Problem Solving 228
Promoting Algebraic Thinking in the Lower Elementary Grades 229
Enhancing Algebraic Thinking in the Upper Elementary and Middle Grades 232
Closure 236
Practice Exercises for Teachers 237
Teaching Competencies and Self-Assessment Tasks 240
Sample Instructional and Assessment Activities 241
Chapter 11 Rational Numbers Expressed as Fractions: Concepts 245
A Fraction of the History of Fractional Numbers 246
Rational Numbers¢"A Definition and Description 247
Research Snapshot 247
The Regions Model for Fractional Numbers 248
The Groups of Objects Model for Fractional Numbers 250
The Number Line Model for Fractional Numbers 251
Knowledge and Understandings Prior to Operations on Rational Numbers Expressed as Fractions 253
Closure 258
Practice Exercises for Teachers 260
Teaching Competencies and Self-Assessment Tasks 262
Sample Instructional and Assessment Activities 263
Chapter 12 Rational Numbers Expressed as Fractions: Operations 267
Research Snapshot 268
Addition of Fractional Numbers 269
Subtraction of Fractional Numbers 274
Multiplication of Fractional Numbers 278
Division of Fractional Numbers 282
Closure 285
Practice Exercises for Teachers 286
Teaching Competencies and Self-Assessment Tasks 288
Sample Instructional and Assessment Activities 288
Chapter 13 Rational Numbers Expressed as Decimals 291
Models That Give Decimals Meaning 292
Place Value¢"A Foundation for Decimals 293
Exponential Notation 295
Addition with Decimal Numbers 296
Subtraction with Decimal Numbers 297
Multiplication with Decimal Numbers 299
Division with Decimal Numbers 300
Relating Decimals and Fractions 301
Scientific Notation 303
Ratio, Proportion, and Percent 304
Research Snapshot 305
Closure 311
Practice Exercises for Teachers 312
Teaching Competencies and Self-Assessment Tasks 313
Sample Instructional and Assessment Activities 314
Chapter 14 Data Analysis: Graphs, Statistics, and Probability 317
A Child¢s View of Statistics 318
Measures of Central Tendency in Data 319
Research Snapshot 320
Organizing and Interpreting Data 321
Taking a ¢¢Chance¢¢ 326
Experimental Probability 326
Theoretical Probability 327
Permutations and Combinations 328
Models for Exploring Probability 330
Closure 331
Practice Exercises for Teachers 332
Teaching Competencies and Self-Assessment Tasks 334
Sample Instructional and Assessment Activities 335
Chapter 15 Measurement 339
The Historical Development of Measurement Systems 340
Contemporary Measurement Systems 341
The Process of Measurement 345
Computing with Denominate Numbers 349
Money and Time as Measures 350
Research Snapshot 352
Closure 354
Practice Exercises for Teachers 354
Teaching Competencies and Self-Assessment Tasks 356
Sample Instructional and Assessment Activities 357
Chapter 16 Geometry: Basic Concepts and Structures 361
Research Snapshot 362
Early Experiences in Geometry 363
The ¢¢Building Blocks¢¢ of Geometry 364
Extending the Basics¢"Curves, Regions, and Rays 366
Angles and Angle Measure 369
Exploring Geometric Constructions 372
Symmetry and Transformational Geometry 374
Other Devices for Exploring Geometric Concepts 376
Closure 377
Practice Exercises for Teachers 378
Teaching Competencies and Self-Assessment Tasks 381
Sample Instructional and Assessment Activities 382
Chapter 17 Geometry: Polygons and Polyhedra 385
Defining Polygons 386
Research Snapshot 386
A Closer Look at Triangles 388
A Closer Look at Quadrilaterals 394
A Closer Look at Circles 395
Area, Perimeter, and Circumference Measurement 397
Defining Polyhedra 401
Closure 409
Practice Exercises for Teachers 410
Teaching Competencies and Self-Assessment Tasks 414
Sample Instructional and Assessment Activities 415
Index 417
