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Copyright, ii
About the Authors, v
Preface, xix
Section I. Teaching Mathematics: Foundations and Perspectives, 1
Chapter 1. Teaching Mathematics in the Era of the NCTM Standards, 1
The National Standards-Based Movement, 1
Principles and Standards for School Mathematics, 2
The Six Principles, 2
The Five Content Standards, 3
The Five Process Standards, 3
Curriculum Focal Points: A Quest for Coherence, 5
The Professional Standards for Teaching Mathematics and Mathematics Teaching Today, 5
Shifts in the Classroom Environment, 5
The Teaching Standards, 5
Influences and Pressures on Mathematics Teaching, 6
National and International Studies, 6
State Standards, 7
Curriculum, 7
A Changing World Economy, 8
An Invitation to Learn and Grow, 9
Becoming a Teacher of Mathematics, 9
Reflections on Chapter 1, 11
Writing to Learn, 11
For Discussion and Exploration, 11
Resources for Chapter 1, 11
Recommended Readings, 11
Standards-Based Curricula, 12
Online Resources, 12
Field Experience Guide Connections, 12
Chapter 2. Exploring What It Means to Know and Do Mathematics, 13
What Does It Mean to Do Mathematics?, 13
Mathematics Is the Science of Pattern and Order, 13
A Classroom Environment for Doing Mathematics, 14
An Invitation to Do Mathematics, 15
Let’s Do Some Mathematics!, 15
Where Are the Answers?, 19
What Does It Mean to Learn Mathematics?, 20
Constructivist Theory, 20
Sociocultural Theory, 21
Implications for Teaching Mathematics, 21
What Does It Mean to Understand Mathematics?, 23
Mathematics Proficiency, 24
Implications for Teaching Mathematics, 25
Benefits of a Relational Understanding, 26
Multiple Representations to Support Relational Understanding, 27
Connecting the Dots, 29
Reflections on Chapter 2, 30
Writing to Learn, 30
For Discussion and Exploration, 30
Resources for Chapter 2, 30
Recommended Readings, 30
Online Resources, 31
Field Experience Guide Connections, 31
Chapter 3. Teaching Through Problem Solving, 32
Teaching Through Problem Solving, 32
Problems and Tasks for Learning Mathematics, 33
A Shift in the Role of Problems, 33
The Value of Teaching Through Problem Solving, 33
Examples of Problem-Based Tasks, 34
Selecting or Designing Problem-Based Tasks and Lessons, 36
Multiple Entry Points, 36
Creating Meaningful and Engaging Contexts, 37
How to Find Quality Tasks and Problem-Based Lessons, 38
Teaching about Problem Solving, 42
Four-Step Problem-Solving Process, 42
Problem-Solving Strategies, 43
Teaching in a Problem-Based Classroom, 43
Let Students Do the Talking, 43
How Much to Tell and Not to Tell, 44
The Importance of Student Writing, 44
Metacognition, 46
Disposition, 47
Attitudinal Goals, 47
A Three-Phase Lesson Format, 47
The Before Phase of a Lesson, 48
Teacher Actions in the Before Phase, 48
The During Phase of a Lesson, 51
Teacher Actions in the During Phase, 51
The After Phase of a Lesson, 52
Teacher Actions in the After Phase, 53
Frequently Asked Questions, 55
Reflections on Chapter 3, 56
Writing to Learn, 56
For Discussion and Exploration, 56
Resources for Chapter 3, 56
Recommended Readings, 56
Online Resources, 57
Field Experience Guide Connections, 57
Chapter 4. Planning in the Problem-Based Classroom, 58
Planning a Problem-Based Lesson, 58
Planning Process for Developing a Lesson, 58
Applying the Planning Process, 62
Variations of the Three-Phase Lesson, 63
Textbooks as Resources, 64
Planning for All Learners, 64
Make Accommodations and Modifications, 65
Differentiating Instruction, 65
Flexible Groupings, 67
Example of Accommodating a Lesson: ELLs, 67
Drill or Practice?, 69
New Definitions of Drill and Practice, 69
What Drill Provides, 69
What Practice Provides, 70
When Is Drill Appropriate?, 70
Students Who Don’t Get It, 71
Homework, 71
Practice as Homework, 71
Drill as Homework, 71
Provide Homework Support, 72
Reflections on Chapter 4, 72
Writing to Learn, 72
For Discussion and Exploration, 72
Resources for Chapter 4, 73
Recommended Readings, 73
Online Resources, 73
Field Experience Guide Connections, 73
Expanded Lesson: Fixed Areas, 74
Chapter 5. Building Assessment into Instruction, 76
Integrating Assessment into Instruction, 76
What Is Assessment?, 76
The Assessment Standards, 76
Why Do We Assess?, 77
What Should Be Assessed?, 78
Performance-Based Assessments, 78
Examples of Performance-Based Tasks, 79
Thoughts about Assessment Tasks, 80
Rubrics and Performance Indicators, 80
Simple Rubrics, 80
Performance Indicators, 81
Student Involvement with Rubrics, 82
Observation Tools, 82
Anecdotal Notes, 83
Observation Rubric, 83
Checklists for Individual Students, 83
Checklists for Full Classes, 84
Writing and Journals, 84
The Value of Writing, 84
Journals, 85
Writing Prompts and Ideas, 85
Journals for Early Learners, 86
Student Self-Assessment, 87
Diagnostic Interviews, 87
Tests, 88
Improving Performance on High-Stakes Tests, 89
Teach Fundamental Concepts and Processes, 89
Test-Taking Strategies, 89
Grading, 90
Grading Issues, 90
Reflections on Chapter 5, 91
Writing to Learn, 91
For Discussion and Exploration, 91
Resources for Chapter 5, 91
Recommended Readings, 91
Online Resources, 92
Field Experience Guide Connections, 92
Chapter 6. Teaching Mathematics Equitably to All Children, 93
Creating Equitable Instruction, 93
Mathematics for All Children, 94
Diversity in Today’s Classroom, 94
Tracking and Flexible Grouping, 94
Instructional Principles for Diverse Learners, 95
Providing for Students with Special Needs, 95
Response to Intervention, 95
Students with Mild Disabilities, 96
Students with Significant Disabilities, 100
Culturally and Linguistically Diverse Students, 102
Windows and Mirrors, 102
Culturally Relevant Mathematics Instruction, 102
Ethnomathematics, 103
English Language Learners (ELLs), 103
Strategies for Teaching Mathematics to ELLs, 104
Working Toward Gender Equity, 106
Possible Causes of Gender Inequity, 106
What Can Be Done?, 106
Reducing Resistance and Building Resilience, 107
Providing for Students Who Are Mathematically Gifted, 107
Strategies to Avoid, 108
Strategies to Incorporate, 108
Final Thoughts, 109
Reflections on Chapter 6, 109
Writing to Learn, 109
For Discussion and Exploration, 109
Resources for Chapter 6, 110
Recommended Readings, 110
Online Resources, 110
Field Experience Guide Connections, 110
Chapter 7. Using Technology to Teach Mathematics, 111
Calculators in Mathematics Instruction, 112
When to Use a Calculator, 112
Benefits of Calculator Use, 112
Graphing Calculators, 113
Data-Collection Devices, 114
Computers in Mathematics Instruction, 115
Tools for Developing Numeration, 115
Tools for Developing Geometry, 116
Tools for Developing Probability and Data Analysis, 117
Tools for Developing Algebraic Thinking, 118
Instructional Software, 118
Concept Instruction, 118
Problem Solving, 118
Drill and Reinforcement, 118
Guidelines for Selecting and Using Software, 119
Guidelines for Using Software, 119
How to Select Software, 119
Resources on the Internet, 120
How to Select Internet Resources, 120
Emerging Technologies, 120
Reflections on Chapter 7, 122
Writing to Learn, 122
For Discussion and Exploration, 123
Resources for Chapter 7, 123
Recommended Readings, 123
Online Resources, 123
Field Experience Guide Connections, 124
Section II. Development of Mathematical Concepts and Procedures, 125
Chapter 8. Developing Early Number Concepts and Number Sense, 125
Promoting Good Beginnings, 125
Number Development in Pre-K and Kindergarten, 126
The Relationships of More, Less, and Same, 126
Early Counting, 127
Numeral Writing and Recognition, 128
Counting On and Counting Back, 128
Early Number Sense, 129
Relationships among Numbers 1 Through 10, 130
Patterned Set Recognition, 130
One and Two More, One and Two Less, 131
Anchoring Numbers to 5 and 10, 132
Part-Part-Whole Relationships, 134
Dot Cards as a Model for Teaching Number Relationships, 137
Relationships for Numbers 10 Through 20, 138
Pre-Place-Value Concepts, 138
Extending More Than and Less Than Relationships, 139
Doubles and Near-Doubles, 139
Number Sense in Their World, 140
Estimation and Measurement, 140
Data Collection and Analysis, 141
Extensions to Early Mental Mathematics, 142
Reflections on Chapter 8, 143
Writing to Learn, 143
For Discussion and Exploration, 143
Resources for Chapter 8, 143
Literature Connections, 143
Recommended Readings, 144
Online Resources, 144
Field Experience Guide Connections, 144
Chapter 9. Developing Meanings for the Operations, 145
Addition and Subtraction Problem Structures, 145
Examples of the Four Problem Structures, 146
Teaching Addition and Subtraction, 148
Contextual Problems, 148
Investigations in Number, Data, and Space: Grade 2, Counting, Coins, and Combinations, 150
Model-Based Problems, 151
Properties of Addition and Subtraction, 153
Multiplication and Division Problem Structures, 154
Examples of the Four Problem Structures, 154
Teaching Multiplication and Division, 157
Contextual Problems, 157
Remainders, 157
Model-Based Problems, 158
Properties of Multiplication and Division, 160
Strategies for Solving Contextual Problems, 161
Analyzing Context Problems, 161
Two-Step Problems, 163
Reflections on Chapter 9, 164
Writing to Learn, 164
For Discussion and Exploration, 164
Resources for Chapter 9, 165
Literature Connections, 165
Recommended Readings, 165
Online Resources, 166
Field Experience Guide Connections, 166
Chapter 10. Helping Children Master the Basic Facts, 167
Developmental Nature of Basic Fact Mastery, 167
Approaches to Fact Mastery, 168
Guiding Strategy Development, 169
Reasoning Strategies for Addition Facts, 170
One More Than and Two More Than, 170
Adding Zero, 171
Using 5 as an Anchor, 172
10 Facts, 172
Up Over 10, 172
Doubles, 173
Near-Doubles, 173
Reinforcing Reasoning Strategies, 174
Reasoning Strategies for Subtraction Facts, 175
Subtraction as Think-Addition, 175
Down Over 10, 176
Take from the 10, 176
Reasoning Strategies for Multiplication Facts, 177
Doubles, 178
Fives, 178
Zeros and Ones, 178
Nifty Nines, 179
Using Known Facts to Derive Other Facts, 180
Division Facts and “Near Facts”, 181
Mastering the Basic Facts, 182
Effective Drill, 182
Fact Remediation, 184
Reflections on Chapter 10, 185
Writing to Learn, 185
For Discussion and Exploration, 185
Resources for Chapter 10, 185
Literature Connections, 185
Recommended Readings, 186
Online Resources, 186
Field Experience Guide Connections, 186
Chapter 11. Developing Whole-Number Place-Value Concepts, 187
Pre-Base-Ten Concepts, 188
Children’s Pre-Base-Ten View of Numbers, 188
Count by Ones, 188
Basic Ideas of Place Value, 188
Integration of Base-Ten Groupings with Count by Ones, 188
Role of Counting, 189
Integration of Groupings with Words, 189
Integration of Groupings with Place-Value Notation, 190
Models for Place Value, 191
Base-Ten Models and the Ten-Makes-One Relationship, 191
Groupable Models, 191
Pregrouped or Trading Models, 192
Nonproportional Models, 192
Developing Base-Ten Concepts, 193
Grouping Activities, 193
The Strangeness of Ones, Tens, and Hundreds, 195
Grouping Tens to Make 100, 195
Equivalent Representations, 195
Oral and Written Names for Numbers, 197
Two-Digit Number Names, 197
Three-Digit Number Names, 198
Written Symbols, 198
Patterns and Relationships with Multidigit Numbers, 200
The Hundreds Chart, 200
Relationships with Landmark Numbers, 202
Number Relationships for Addition and Subtraction, 204
Connections to Real-World Ideas, 207
Numbers Beyond 1000, 207
Extending the Place-Value System, 208
Conceptualizing Large Numbers, 209
Reflections on Chapter 11, 210
Writing to Learn, 210
For Discussion and Exploration, 210
Resources for Chapter 11, 211
Literature Connections, 211
Recommended Readings, 211
Online Resources, 212
Field Experience Guide Connections, 212
Chapter 12. Developing Strategies for Whole-Number Computation, 213
Toward Computational Fluency, 214
Direct Modeling, 214
Student-Invented Strategies, 215
Traditional Algorithms, 217
Development of Student-Invented Strategies, 218
Creating an Environment for Inventing Strategies, 218
Models to Support Invented Strategies, 218
Student-Invented Strategies for Addition and Subtraction, 219
Adding and Subtracting Single-Digit Numbers, 219
Adding Two-Digit Numbers, 220
Subtracting by Counting Up, 220
Take-Away Subtraction, 221
Extensions and Challenges, 222
Traditional Algorithms for Addition and Subtraction, 223
Addition Algorithm, 223
Subtraction Algorithm, 225
Student-Invented Strategies for Multiplication, 226
Useful Representations, 226
Multiplication by a Single-Digit Multiplier, 227
Multiplication of Larger Numbers, 228
Traditional Algorithm for Multiplication, 230
One-Digit Multipliers, 230
Two-Digit Multipliers, 231
Student-Invented Strategies for Division, 232
Missing-Factor Strategies, 232
Cluster Problems, 233
Traditional Algorithm for Division, 234
One-Digit Divisors, 234
Two-Digit Divisors, 235
Reflections on Chapter 12, 237
Writing to Learn, 237
For Discussion and Exploration, 238
Resources for Chapter 12, 238
Literature Connections, 238
Recommended Readings, 239
Online Resources, 239
Field Experience Guide Connections, 239
Chapter 13. Using Computational Estimation with Whole Numbers, 240
Introducing Computational Estimation, 240
Understanding Computational Estimation, 241
Suggestions for Teaching Computational Estimation, 242
Computational Estimation from Invented Strategies, 244
Stop Before the Details, 244
Use Related Problem Sets, 244
Computational Estimation Strategies, 245
Front-End Methods, 245
Rounding Methods, 246
Compatible Numbers, 247
Clustering, 248
Use Tens and Hundreds, 248
Estimation Experiences, 249
Calculator Activities, 249
Using Whole Numbers to Estimate Rational Numbers, 251
Reflections on Chapter 13, 252
Writing to Learn, 252
For Discussion and Exploration, 252
Resources for Chapter 13, 252
Literature Connections, 252
Recommended Readings, 252
Online Resources, 252
Field Experience Guide Connections, 253
Chapter 14. Algebraic Thinking: Generalizations, Patterns, and Functions, 254
Algebraic Thinking, 255
Generalization from Arithmetic and from Patterns, 255
Generalization with Addition, 255
Generalization in the Hundreds Chart, 256
Generalization Through Exploring a Pattern, 257
Meaningful Use of Symbols, 257
The Meaning of the Equal Sign, 258
The Meaning of Variables, 262
Making Structure in the Number System Explicit, 265
Making Conjectures about Properties, 265
Justifying Conjectures, 266
Odd and Even Relationships, 266
Study of Patterns and Functions, 267
Repeating Patterns, 267
Growing Patterns, 269
Linear Functions, 274
Mathematical Modeling, 276
Teaching Considerations, 277
Emphasize Appropriate Algebra Vocabulary, 277
Multiple Representations, 278
Connect Representations, 280
Algebraic Thinking Across the Curriculum, 280
Connected Mathematics: Grade 7, Variables and Patterns, 281
Reflections on Chapter 14, 283
Writing to Learn, 283
For Discussion and Exploration, 283
Resources for Chapter 14, 283
Literature Connections, 283
Recommended Readings, 284
Online Resources, 284
Field Experience Guide Connections, 285
Chapter 15. Developing Fraction Concepts, 286
Meanings of Fractions, 287
Fraction Constructs, 287
Building on Whole-Number Concepts, 287
Models for Fractions, 288
Region or Area Models, 288
Length Models, 289
Set Models, 290
Concept of Fractional Parts, 291
Sharing Tasks, 291
Fraction Language, 293
Equivalent Size of Fraction Pieces, 293
Partitioning, 294
Using Fraction Language and Symbols, 294
Counting Fraction Parts: Iteration, 294
Fraction Notation, 296
Fractions Greater Than 1, 296
Assessing Understanding, 297
Estimating with Fractions, 298
Benchmarks of Zero, One-Half, and One, 299
Using Number Sense to Compare, 299
Equivalent-Fraction Concepts, 301
Conceptual Focus on Equivalence, 301
Equivalent-Fraction Models, 302
Developing an Equivalent-Fraction Algorithm, 304
Teaching Considerations for Fraction Concepts, 306
Reflections on Chapter 15, 306
Writing to Learn, 306
For Discussion and Exploration, 306
Resources for Chapter 15, 307
Literature Connections, 307
Recommended Readings, 307
Online Resources, 307
Field Experience Guide Connections, 308
Chapter 16. Developing Strategies for Fraction Computation, 309
Number Sense and Fraction Algorithms, 310
Conceptual Development Takes Time, 310
A Problem-Based Number Sense Approach, 310
Computational Estimation, 310
Addition and Subtraction, 312
Invented Strategies, 312
Developing an Algorithm, 315
Mixed Numbers and Improper Fractions, 317
Multiplication, 317
Developing the Concept, 317
Developing the Algorithm, 320
Factors Greater Than One, 320
Division, 321
Partitive Interpretation of Division, 321
Measurement Interpretation of Division, 323
Answers That Are Not Whole Numbers, 324
Developing the Algorithms, 324
Reflections on Chapter 16, 326
Writing to Learn, 326
For Discussion and Exploration, 326
Resources for Chapter 16, 326
Literature Connections, 326
Recommended Readings, 327
Online Resources, 327
Field Experience Guide Connections, 327
Chapter 17. Developing Concepts of Decimals and Percents, 328
Connecting Fractions and Decimals, 328
Base-Ten Fractions, 329
Extending the Place-Value System, 330
Fraction-Decimal Connection, 332
Developing Decimal Number Sense, 333
Familiar Fractions Connected to Decimals, 334
Approximation with a Nice Fraction, 335
Ordering Decimal Numbers, 336
Other Fraction-Decimal Equivalents, 337
Introducing Percents, 337
Models and Terminology, 338
Realistic Percent Problems, 339
Estimation, 341
Computation with Decimals, 342
The Role of Estimation, 342
Addition and Subtraction, 342
Multiplication, 343
Division, 344
Reflections on Chapter 17, 345
Writing to Learn, 345
For Discussion and Exploration, 345
Resources for Chapter 17, 345
Literature Connections, 345
Recommended Readings, 346
Online Resources, 346
Field Experience Guide Connections, 347
Chapter 18. Proportional Reasoning, 348
Ratios, 348
Types of Ratios, 349
Proportional Reasoning, 350
Additive Versus Multiplicative Situations, 351
Indentifying Multiplicative Relationships, 352
Equivalent Ratios, 353
Different Ratios, 354
Connected Mathematics: Grade 7, Comparing and Scaling, 356
Ratio Tables, 356
Proportional Reasoning Across the Curriculum, 358
Algebra, 359
Measurement and Geometry, 359
Scale Drawings, 360
Statistics, 361
Number: Fractions and Percent, 362
Proportions, 363
Within and Between Ratios, 363
Reasoning Approaches, 364
Cross-Product Approach, 365
Reflections on Chapter 18, 366
Writing to Learn, 366
For Discussion and Exploration, 366
Resources for Chapter 18, 366
Literature Connections, 366
Recommended Readings, 367
Online Resources, 368
Field Experience Guide Connections, 368
Chapter 19. Developing Measurement Concepts, 369
The Meaning and Process of Measuring, 370
Concepts and Skills, 371
Nonstandard Units and Standard Units: Reasons for Using Each, 372
The Role of Estimation and Approximation, 372
Length, 373
Comparison Activities, 373
Units of Length, 374
Making and Using Rulers, 375
Area, 376
Comparison Activities, 376
Investigations in Number, Data, and Space: Grade 3, Perimeter, Angles, and Area, 377
Units of Area, 378
The Relationship Between Area and Perimeter, 380
Volume and Capacity, 380
Comparison Activities, 381
Units of Volume and Capacity, 381
Weight and Mass, 382
Comparison Activities, 382
Units of Weight or Mass, 383
Time, 383
Duration, 383
Clock Reading, 383
Elapsed Time, 384
Money, 385
Coin Recognition and Values, 385
Counting Sets of Coins, 385
Making Change, 386
Angles, 386
Comparison Activities, 386
Units of Angular Measure, 386
Using Protractors and Angle Rulers, 386
Introducing Standard Units, 387
Instructional Goals, 387
Important Standard Units and Relationships, 389
Estimating Measures, 389
Strategies for Estimating Measurements, 390
Tips for Teaching Estimation, 390
Measurement Estimation Activities, 391
Developing Formulas for Area and Volume, 391
Students’ Misconceptions, 391
Areas of Rectangles, Parallelograms, Triangles, and Trapezoids, 392
Circumference and Area of Circles, 394
Volumes of Common Solid Shapes, 395
Connections among Formulas, 396
Reflections on Chapter 19, 397
Writing to Learn, 397
For Discussion and Exploration, 397
Resources for Chapter 19, 397
Literature Connections, 397
Recommended Readings, 397
Online Resources, 398
Field Experience Guide Connections, 398
Chapter 20. Geometric Thinking and Geometric Concepts, 399
Geometry Goals for Students, 399
Spatial Sense and Geometric Reasoning, 400
Geometric Content, 400
The Development of Geometric Thinking, 400
The van Hiele Levels of Geometric Thought, 400
Implications for Instruction, 404
Learning about Shapes and Properties, 405
Shapes and Properties for Level-0 Thinkers, 405
Shapes and Properties for Level-1 Thinkers, 410
Shapes and Properties for Level-2 Thinkers, 416
Learning about Transformations, 419
Transformations for Level-0 Thinkers, 419
Transformations for Level-1 Thinkers, 422
Transformations for Level-2 Thinkers, 424
Learning about Location, 424
Location for Level-0 Thinkers, 424
Location for Level-1 Thinkers, 426
Location for Level-2 Thinkers, 428
Learning about Visualization, 429
Visualization for Level-0 Thinkers, 429
Visualization for Level-1 Thinkers, 430
Visualization for Level-2 Thinkers, 432
Reflections on Chapter 20, 433
Writing to Learn, 433
For Discussion and Exploration, 433
Resources for Chapter 20, 433
Literature Connections, 433
Recommended Readings, 434
Online Resources, 434
Field Experience Guide Connections, 435
Chapter 21. Developing Concepts of Data Analysis, 436
What Does It Mean to Do Statistics?, 437
Is It Statistics or Is It Mathematics?, 437
Variability, 437
The Shape of Data, 438
Process of Doing Statistics, 439
Formulating Questions, 439
Ideas for Questions, 439
Data Collection, 440
Using Existing Data Sources, 440
Data Analysis: Classification, 441
Attribute Materials, 441
Data Analysis: Graphical Representations, 443
Bar Graphs and Tally Charts, 443
Circle Graphs, 444
Continuous Data Graphs, 445
Scatter Plots, 447
Data Analysis: Measures of Center, 449
Averages, 449
Understanding the Mean: Two Interpretations, 449
Box-and-Whisker Plots, 452
Interpreting Results, 453
Reflections on Chapter 21, 454
Writing to Learn, 454
For Discussion and Exploration, 454
Resources for Chapter 21, 454
Literature Connections, 454
Recommended Readings, 455
Online Resources, 455
Field Experience Guide Connections, 455
Chapter 22. Exploring Concepts of Probability, 456
Introducing Probability, 457
Likely or Not Likely, 457
The Probability Continuum, 459
Theoretical Probability and Experiments, 460
Theoretical Probability, 461
Experiments, 462
Implications for Instruction, 464
Use of Technology in Experiments, 464
Sample Spaces and Probability of Two Events, 465
Independent Events, 465
Two-Event Probabilities with an Area Model, 466
Dependent Events, 467
Simulations, 468
Reflections on Chapter 22, 470
Writing to Learn, 470
For Discussion and Exploration, 470
Resources for Chapter 22, 471
Literature Connections, 471
Recommended Readings, 471
Online Resources, 472
Field Experience Guide Connections, 472
Chapter 23. Developing Concepts of Exponents, Integers, and Real Numbers, 473
Exponents, 473
Exponents in Expressions and Equations, 473
Negative Exponents, 476
Scientific Notation, 477
Integers, 479
Contexts for Exploring Integers, 479
Meaning of Negative Numbers, 481
Two Models for Teaching Integers, 481
Operations with Integers, 482
Addition and Subtraction, 482
Multiplication and Division, 484
Real Numbers, 486
Rational Numbers, 486
Irrational Numbers, 487
Density of the Real Numbers, 489
Reflections on Chapter 23, 489
Writing to Learn, 489
For Discussion and Exploration, 489
Resources for Chapter 23, 490
Literature Connections, 490
Recommended Readings, 490
Online Resources, 490
Field Experience Guide Connections, 490
Appendix A. Principles and Standards for School Mathematics: Content Standards and Grade Level Expectations, A-1
Appendix B. Standards for Teaching Mathematics, B-1
Appendix C. Guide to Blackline Masters, C-1
References, R-1
Index, I-1

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