Helping Children Learn Mathematics , Ninth Edition

Cover Page......Page 1
Title Page......Page 3
Copyright Page......Page 4
About the Authors......Page 5
Preface......Page 7
Acknowledgments......Page 9
Contents......Page 11
INTRODUCTION......Page 17
What Is Mathematics?......Page 18
Needs of the Subject......Page 19
Needs of the Child......Page 21
Needs of Society......Page 23
National Guidelines for School Mathematics......Page 24
Research......Page 25
Textbooks and Other Materials......Page 26
What Is Your Role Now?......Page 27
THINGS TO DO: GOING BEYOND THIS BOOK......Page 28
SNAPSHOT OF A LESSON......Page 30
How Can We Support the Diverse Learners in Our Classrooms?......Page 31
Creating a Positive Learning Environment......Page 32
Avoiding Negative Experiences That Increase Anxiety......Page 33
Treating All Students as Equally Likely to Have Aptitude for Mathematics......Page 34
Helping Students Improve Their Ability to Retain Mathematical Knowledge and Skills......Page 35
How Can We Help Children Acquire Both Procedural Knowledge and Conceptual Knowledge?......Page 36
Building Behavior......Page 37
Constructing Understanding......Page 39
How Can We Help Children Make Sense of Mathematics?......Page 40
Recommendation 2: Actively Involve Students......Page 41
Recommendation 3: Move Learning from Concrete to Abstract......Page 44
Recommendation 4: Use Communication to Encourage Understanding......Page 45
Cultural Connections......Page 46
THINGS TO DO: FROM WHAT YOU’VE READ......Page 47
THINGS TO DO: GOING BEYOND THIS BOOK......Page 48
BOOK NOOK FOR CHILDREN......Page 49
SNAPSHOT OF A LESSON......Page 50
INTRODUCTION......Page 51
What Are the Developmental Characteristics of My Students?......Page 52
What Do My Students Already Know?......Page 53
How Will I Encourage My Students to Talk, What Kinds of Questions Will I Ask, and How Will I Group My Students?......Page 55
What Materials Will My Students and I Use?......Page 58
Planning for Effective Teaching......Page 62
Levels of Planning......Page 63
Planning Different Types of Lessons......Page 64
Meeting the Needs of All Students......Page 71
Cultural Connections......Page 73
THINGS TO DO: GOING BEYOND THIS BOOK......Page 76
BOOK NOOK FOR CHILDREN......Page 77
SNAPSHOT OF A LESSON......Page 78
Assessment of Learning: Summative Assessment......Page 80
Phases of Assessment......Page 82
Making Instructional Decisions......Page 83
Evaluating Student Achievement......Page 84
Ways to Assess Students’ Abilities and Dispositions......Page 85
Observation......Page 86
Interviewing......Page 87
Performance Tasks......Page 89
Self-Assessment and Peer Assessment......Page 90
Work Samples......Page 92
Portfolios......Page 94
Teacher-Designed Written Tests......Page 95
Recording the Information......Page 97
Communicating the Information......Page 98
Cultural Connections......Page 100
THINGS TO DO: GOING BEYOND THIS BOOK......Page 101
BOOK NOOK FOR CHILDREN......Page 104
SNAPSHOT OF A LESSON......Page 105
Introduction......Page 106
Problem Solving......Page 108
Reasoning and Proof......Page 111
Reasoning Is About Making Generalizations......Page 112
Reasoning Leads to Mathematical Memory Built on Relationships......Page 113
Communication......Page 114
Connections......Page 115
Selecting, Applying, and Translating Among Representations......Page 117
Using Representations to Model and Interpret Phenomena......Page 118
A Glance at Where We’ve Been......Page 119
THINGS TO DO: GOING BEYOND THIS BOOK......Page 121
BOOK NOOK FOR CHILDREN......Page 122
SNAPSHOT OF A LESSON......Page 123
INTRODUCTION......Page 124
What Is a Problem and What Is Problem Solving?......Page 125
Teaching Mathematics Through Problem Solving......Page 127
Factors for Success in Problem Solving......Page 128
Choosing Appropriate Problems......Page 129
Finding Problems......Page 132
Having Students Pose Problems......Page 133
Using Calculators and Computers......Page 134
Strategies for Problem Solving......Page 135
Make a Drawing or Diagram......Page 136
Look for a Pattern......Page 137
Construct a Table......Page 138
Guess and Check......Page 139
Solve a Similar but Simpler Problem......Page 140
Looking Back at the Solution Process......Page 142
Managing Classroom Routines......Page 143
Managing Student Needs......Page 144
Cultural Connections......Page 145
THINGS TO DO:FROM WHAT YOU’VE READ......Page 146
THINGS TO DO:GOING BEYOND THIS BOOK......Page 147
BOOK NOOK FOR CHILDREN......Page 148
SNAPSHOT OF A LESSON......Page 149
Number Sense......Page 150
Classification......Page 152
Patterns......Page 155
Conservation......Page 156
Group Recognition......Page 157
Comparisons and One-to-one Correspondence......Page 158
Counting......Page 159
Counting Stages......Page 160
Counting Strategies......Page 162
Developing Number Benchmarks......Page 165
Making Connections......Page 166
Cardinal, Ordinal, and Nominal Numbers......Page 168
Writing Numerals......Page 169
Cultural Connections......Page 170
A Glance at Where We’ve Been......Page 171
THINGS TO DO: GOING BEYOND THIS BOOK......Page 172
BOOK NOOK FOR CHILDREN......Page 174
SNAPSHOT OF A LESSON......Page 175
FOCUS QUESTIONS......Page 176
INTRODUCTION......Page 177
Nature of Place Value......Page 178
Modeling—Proportional and Nonproportional......Page 179
Grouping or Trading......Page 180
A Place to Start......Page 181
Extending Place Value......Page 186
Counting and Patterns......Page 188
Regrouping and Renaming......Page 191
Reading and Writing Numbers......Page 192
Rounding......Page 194
Cultural Connections......Page 195
THINGS TO DO: FROM WHAT YOU’VE READ......Page 197
THINGS TO DO: GOING BEYOND THIS BOOK......Page 198
BOOK NOOK FOR CHILDREN......Page 199
SNAPSHOT OF A LESSON......Page 200
INTRODUCTION......Page 201
Facility with Counting......Page 203
Experience in Talking and Writing About Mathematical Ideas......Page 204
Addition and Subtraction......Page 205
Multiplication and Division......Page 206
Overview of Basic Fact Instruction......Page 209
Build Understanding of the Basic Facts......Page 211
Focus on How to Remember Facts......Page 212
Thinking Strategies for Addition Facts......Page 213
Thinking Strategies for Subtraction Facts......Page 219
Thinking Strategies for Multiplication Facts......Page 220
Thinking Strategies for Division Facts......Page 223
Cultural Connections......Page 225
THINGS TO DO: FROM WHAT YOU’VE READ......Page 226
THINGS TO DO: GOING BEYOND THIS BOOK......Page 227
BOOK NOOK FOR CHILDREN......Page 228
SNAPSHOT OF A LESSON......Page 229
INTRODUCTION......Page 231
Calculators......Page 233
Calculators Are Not Always the Fastest Way of Doing Computations......Page 234
Calculators Are Useful for More Than Just Doing Computations......Page 235
Strategies and Techniques for Mental Computation......Page 236
Encouraging Mental Computation......Page 238
Background for estimating......Page 241
Adjusting......Page 242
Compatible Numbers......Page 243
Flexible Rounding......Page 244
Clustering......Page 245
Choosing Estimation Strategies......Page 246
Cultural Connections......Page 247
THINGS TO DO: FROM WHAT YOU’VE READ......Page 248
THINGS TO DO: GOING BEYOND THIS BOOK......Page 249
BOOK NOOK FOR CHILDREN......Page 250
SNAPSHOT OF A LESSON......Page 251
INTRODUCTION......Page 253
Using Place Value......Page 255
Addition......Page 256
Standard Addition Algorithm......Page 257
Higher-Decade Addition......Page 259
Subtraction......Page 260
Standard Subtraction Algorithm......Page 261
Partial-Difference Subtraction Algorithm......Page 263
Multiplication with One-Digit Multipliers......Page 264
Multiplication with Two-Digit Multipliers......Page 266
Division......Page 267
Division with One-Digit Divisors......Page 268
Division with Two-Digit Divisors......Page 271
Making Sense of Division with Remainders......Page 272
Cultural Connections......Page 273
THINGS TO DO: FROM WHAT YOU’VE READ......Page 275
THINGS TO DO:GOING BEYOND THIS BOOK......Page 277
BOOK NOOK FOR CHILDREN......Page 278
SNAPSHOT OF A LESSON......Page 279
INTRODUCTION......Page 280
Conceptual Development of Fractions......Page 281
Three Meanings of Fractions......Page 282
Models of the Part–Whole Meaning......Page 283
Making Sense of Fractions......Page 285
Ordering Fractions and Equivalent Fractions......Page 289
Mixed Numbers and Improper Fractions......Page 293
Addition and Subtraction......Page 294
Multiplication......Page 295
Division......Page 297
Conceptual Development of Decimals......Page 298
Relationship to Common Fractions......Page 299
Relationship to Place Value......Page 300
Ordering and Rounding Decimals......Page 301
Multiplication and Division......Page 302
Cultural Connections......Page 303
A Glance at Where We’ve Been......Page 304
THINGS TO DO:GOING BEYOND THIS BOOK......Page 305
BOOK NOOK FOR CHILDREN......Page 306
SNAPSHOT OF A LESSON......Page 307
INTRODUCTION......Page 308
Ratios......Page 309
Proportions......Page 312
Percents......Page 316
Understanding Percents......Page 317
Applying Percents......Page 319
THINGS TO DO: FROM WHAT YOU’VE READ......Page 322
THINGS TO DO:GOING BEYOND THIS BOOK......Page 323
BOOK NOOK FOR CHILDREN......Page 324
SNAPSHOT OF A LESSON......Page 325
INTRODUCTION......Page 326
Problems......Page 327
Patterns......Page 328
Relations......Page 330
Language and Symbols of Algebra......Page 332
Variables......Page 333
Routine Problems......Page 334
Patterns......Page 336
Nonroutine Problems......Page 338
Relations: Properties of Numbers......Page 340
Another look at modeling, generalizing, and justifying......Page 342
Cultural Connections......Page 343
THINGS TO DO:GOING BEYOND THIS BOOK......Page 344
BOOK NOOK FOR CHILDREN......Page 345
SNAPSHOT OF A LESSON......Page 346
INTRODUCTION......Page 347
Properties of Two- and Three-Dimensional Shapes......Page 349
Three-Dimensional Shapes......Page 350
Two-Dimensional Shapes......Page 355
Locations and Associated Representations......Page 363
Transformations......Page 365
Visualization and Spatial Reasoning......Page 366
Using Geometric Physical and Pictorial Materials......Page 367
Using Mental Images......Page 368
A Glance at Where We’ve Been......Page 369
THINGS TO DO: GOING BEYOND THIS BOOK......Page 370
BOOK NOOK FOR CHILDREN......Page 371
SNAPSHOT OF A LESSON......Page 372
INTRODUCTION......Page 373
The Measurement Process......Page 374
I. Identifying Attributes by Comparing......Page 376
II. Choosing a Unit......Page 381
III and IV. Comparing an Object to a Unit and Finding the Number of Units......Page 383
Creating objects given the measurement......Page 388
Equivalences......Page 389
Conversions......Page 390
Estimating Measurements......Page 391
Connecting Attributes......Page 392
Perimeter and Area......Page 393
Volume and Surface Area......Page 394
A Glance at Where We’ve Been......Page 395
THINGS TO DO:GOING BEYOND THIS BOOK......Page 396
BOOK NOOK FOR CHILDREN......Page 397
SNAPSHOT OF A LESSON......Page 398
INTRODUCTION......Page 399
Surveys......Page 401
Simulations......Page 402
Analyzing Data: Graphical Organization......Page 403
Quick and Easy Graphing Methods......Page 404
Plots......Page 405
Bar Graphs and Histograms......Page 407
Pie Graphs......Page 408
Graphical Roundup......Page 409
Analyzing Data: Descriptive Statistics......Page 410
Measures of Central Tendency or Averages......Page 411
Measures of Variation......Page 415
Data Sense......Page 416
Misleading Graphs......Page 417
Probability of an Event......Page 418
Randomness......Page 422
Independence of Events......Page 423
Cultural Connections......Page 424
THINGS TO DO:FROM WHAT YOU’VE READ......Page 425
THINGS TO DO: GOING BEYOND THIS BOOK......Page 426
BOOK NOOK FOR CHILDREN......Page 427
FOCUS QUESTIONS......Page 428
INTRODUCTION......Page 429
Factors and multiples......Page 432
Primes and composites......Page 436
Prime factorization......Page 437
Divisibility......Page 438
Polygonal numbers......Page 440
Modular arithmetic......Page 441
Pythagorean Triples......Page 443
Fibonacci Sequence......Page 444
THINGS TO DO:FROM WHAT YOU’VE READ......Page 445
BOOK NOOK FOR CHILDREN......Page 446
References......Page 447
Appendix A......Page 459
Appendix B......Page 467
Appendix C......Page 477
Index......Page 511