Learning mathematics in elementary and middle school.: a learner-centred approach

Cover......Page 1
Contents......Page 6
CHAPTER 1 Teaching Mathematics: Influences and Directions......Page 24
  The Diversity of Our World: Learner Influences......Page 25
  Curriculum Guided by Standards......Page 27
  Communication......Page 31
  Reasoning and Proof......Page 32
  Connections......Page 33
  Equity......Page 34
  Parent Involvement......Page 35
 Conclusion......Page 36
 Resources for Teachers......Page 37
 CONNECTING WITH THE STANDARDS......Page 38
  The Cognitive/Constructivist Approach......Page 40
  Develop Understanding......Page 43
  Encourage Communication......Page 44
  Take Time to Motivate Children......Page 46
  Provide Opportunities for Practice......Page 47
  Understanding Individual Needs......Page 49
  Preteaching Activities......Page 51
 Sample Lesson: Solving Problems Involving Multidigit Addition......Page 55
  The Process of Teaching......Page 57
  Postteaching Activities......Page 58
  The Activities......Page 59
 In Practice......Page 60
 Resources for Teachers......Page 61
 CONNECTING WITH THE STANDARDS......Page 62
 Problem Solving: An Integral Part of Mathematics Instruction......Page 63
  Types of Problems......Page 64
  Understanding the Problem......Page 65
  Reflecting on the Problem......Page 66
  Dramatizing or Modeling the Situation and Solution Process......Page 68
  Drawing a Picture or Diagram......Page 69
  Solving a Simpler Problem......Page 70
  Working Backward......Page 72
  Changing Your Point of View......Page 73
  Selecting Appropriate Tasks and Materials......Page 74
  Identifying Sources of Problems......Page 75
  Clarifying the Teacher’s Role......Page 76
  Changing the Difficulty of Problems......Page 78
  Consider Children’s Preferences......Page 80
 In Practice......Page 81
 Resources for Teachers......Page 82
 CONNECTING WITH THE STANDARDS......Page 83
  What Is Assessment?......Page 84
  Phases of Assessment......Page 85
  Achievement Tests......Page 86
  Examining Children’s Reasoning......Page 87
 Assessing Attitudes toward Mathematics......Page 96
 In Practice......Page 97
 Resources for Teachers......Page 98
 CONNECTING WITH THE STANDARDS......Page 99
 ASSESSING MATHEMATICS UNDERSTANDING......Page 100
  Prenumber Activities......Page 102
  Examining Children’s Reasoning......Page 103
  Ordinal Use of Numbers......Page 106
  Rational Counting......Page 107
  Counting All, Counting On......Page 108
  Skip Counting......Page 109
  Concrete Models......Page 110
  Symbolic Representation of Numbers......Page 112
 Number Relationships......Page 113
  One Greater Than, One Less Than......Page 115
  Part-Part-Whole Relationships......Page 116
  Bidirectional Relationship of an Equation......Page 117
 Conclusion......Page 119
 Sample Lesson: A Lesson on What Is More?......Page 120
 Resources for Teachers......Page 121
 CONNECTING WITH THE STANDARDS......Page 122
 ASSESSING MATHEMATICS UNDERSTANDING......Page 123
  Positional or Place Value......Page 126
  What Research Says about Place-Value Learning......Page 127
  Grouping......Page 128
  Communicating Mathematics......Page 129
  Types of Place-Value Materials......Page 130
 Developing Understanding of Two-Digit Numbers......Page 131
  Introducing Base-Ten Blocks......Page 133
  Introducing Nonproportional Materials......Page 134
 Three-Digit Numbers......Page 137
  Number Meanings: Oral Expressions......Page 138
  Using Hundreds Charts......Page 139
  Thinking and Writing about Numbers......Page 140
  Examining Children’s Reasoning: Understanding Place Value......Page 141
  Magnitude of Numbers......Page 142
 Expanded Notation......Page 143
  The Computer Rule for Rounding......Page 144
 Consolidating Number Skills......Page 145
 Conclusion......Page 146
 Sample Lesson: A Lesson on Place Value: Making Groups of Ten......Page 147
 Resources for Teachers......Page 148
 CONNECTING WITH THE STANDARDS......Page 149
 ASSESSING MATHEMATICS UNDERSTANDING......Page 150
  A Model for Beginning with Word Problems......Page 152
  Encoding and Decoding Word Problems......Page 153
  Examining Children’s Reasoning: Linking to "Amazing Equations"......Page 154
  Using Models to Solve Addition and Subtraction Problems......Page 157
  Writing Number Sentences for Addition and Subtraction......Page 160
  Making the Transition from Adding to Multiplying......Page 161
  Introducing Children to Division......Page 162
  Types of Division Word Problems......Page 164
  Using Models to Solve Multiplication and Division Problems......Page 166
 Another Word about Notation and Children’s Language......Page 168
 Conclusion......Page 169
 In Practice......Page 170
 Resources for Teachers......Page 171
 ASSESSING MATHEMATICS UNDERSTANDING......Page 172
 What Are Basic Facts?......Page 175
  Step 3: Using Consolidating Activities for Drill and Practice......Page 176
  Thinking Strategies for Addition and Subtraction......Page 177
  Mathematical Properties of Addition and Subtraction......Page 180
  Fact Families for Addition and Subtraction......Page 181
  Thinking Strategies for Multiplication and Division......Page 182
  Mathematical Properties of Multiplication......Page 185
 Consolidating Activities for Drill and Practice......Page 188
  Using Games......Page 189
 Conclusion......Page 190
 Sample Lesson: A Lesson on Understanding the Commutative Property......Page 191
 Resources for Teachers......Page 192
 CONNECTING WITH THE STANDARDS......Page 193
 ASSESSING MATHEMATICS UNDERSTANDING......Page 194
 What Is Computation?......Page 196
  Estimation......Page 197
 Paper-and-Pencil Computation......Page 198
  Prerequisites......Page 199
  Important Considerations When Teaching Computational Procedures......Page 200
  Posing Story Problems Set in Real-World Contexts......Page 201
  Using Estimation and Mental Computation......Page 202
  Developing Bridging Algorithms to Connect Problems, Models, Estimation, and Symbols......Page 204
  Examining Student Work......Page 205
  Using Models for Computation......Page 206
  Using Estimation and Mental Computation......Page 207
  Developing the Traditional Algorithm......Page 208
  Examining Children’s Reasoning......Page 209
  Developing Other Algorithms......Page 210
  Examining Student Work......Page 211
  Using Models for Computation......Page 212
  Developing Bridging Algorithms to Connect Problems, Models, Estimation, and Symbols......Page 213
  Developing the Traditional Algorithm......Page 215
  Examining Children’s Reasoning......Page 216
  Determining the Reasonableness of Solutions......Page 217
  Using Models for Computation......Page 218
  Using Estimation and Mental Computation......Page 219
  Developing the Traditional Algorithm......Page 220
  Examining Student Work......Page 223
  Determining the Reasonableness of Solutions......Page 224
  Using Games......Page 225
  Using Computer Software......Page 226
 Conclusion......Page 227
 Sample Lesson: A Lesson on Estimating Quotients......Page 228
 Resources for Teachers......Page 229
 CONNECTING WITH THE STANDARDS......Page 230
 ASSESSING MATHEMATICS UNDERSTANDING......Page 231
  What Do Children Know about Fractions?......Page 233
  Understanding Sharing Situations......Page 235
  Number Sense with Fractions......Page 236
  Different Interpretations of Fractions......Page 238
  Fraction Names......Page 245
  Examining Children’s Reasoning: Meaning of the Unit......Page 246
  Comparing and Ordering Fractions......Page 248
  Relative Size of Fractions......Page 249
  Improper Fractions and Mixed Numbers......Page 250
  Dealing with Equivalent Fractions......Page 251
  Renaming and Simplifying Fractions......Page 252
 Conclusion......Page 254
 Sample Lesson: A Lesson on Comparing Fractions......Page 255
 Resources For Teachers......Page 256
 CONNECTING WITH THE STANDARDS......Page 257
 ASSESSING MATHEMATICS UNDERSTANDING......Page 258
 What Do Children Know about Fractions?......Page 261
  Developmental Activities......Page 262
  Properties......Page 263
  Developing Addition of Fractions......Page 264
  Developing Subtraction of Fractions......Page 268
  General Considerations......Page 269
  Developing Multiplication of Fractions......Page 270
  Developing Division of Fractions......Page 275
  Considering Different Cases of Division of Fractions......Page 276
 Conclusion......Page 279
 Sample Lesson: A Lesson on Estimating Fraction Sums and Differences......Page 280
 Resources for Teachers......Page 281
 CONNECTING WITH THE STANDARDS......Page 282
 ASSESSING MATHEMATICS UNDERSTANDING......Page 283
 Instructional Strategies for Teaching Decimals......Page 285
  Connections to Familiar Concepts......Page 287
  Reading and Writing Decimals......Page 289
  Base-Ten Blocks......Page 290
  Decimal Squares......Page 291
  Number Line......Page 292
 Equivalent Decimals......Page 293
  Ordering and Comparing Decimals......Page 294
  Addition and Subtraction......Page 295
  Multiplication......Page 296
  Division......Page 300
 Writing Fractions as Decimals......Page 301
 Conclusion......Page 302
 Sample Lesson: A Lesson on Comparing Decimals......Page 303
 Resources for Teachers......Page 304
 CONNECTING WITH THE STANDARDS......Page 305
 ASSESSING MATHEMATICS UNDERSTANDING......Page 306
  Ratio and Rational Number......Page 308
  Proportional Reasoning......Page 309
  Equal Ratios......Page 310
  Cross Products......Page 311
  Scale Drawings......Page 312
  Number Sense......Page 313
  Fraction and Decimal Equivalents......Page 314
  Finding the Percent of a Number......Page 315
  Other Procedures for Solving Percent Problems......Page 316
  Everyday-Life Problem Settings......Page 317
 Sample Lesson: A Lesson on Understanding Proportions......Page 318
 Resources for Teachers......Page 319
 CONNECTING WITH THE STANDARDS......Page 320
 ASSESSING MATHEMATICS UNDERSTANDING......Page 321
  Why Teach Geometry?......Page 324
  The van Hiele Levels of Geometric Thought......Page 325
  Comments on the Levels of Thought......Page 327
 Learning about Topology......Page 328
  Mazes and Networks......Page 329
  Three-Dimensional Shapes......Page 330
  Constructing Three-Dimensional Shapes......Page 333
  Polygons......Page 336
  Symmetry......Page 339
  Congruence and Similarity......Page 341
  Rigid Transformations......Page 342
 Learning about Tessellations......Page 344
 Learning about Fractals......Page 345
  Tangram Puzzles......Page 346
  Describing Figures......Page 348
  Dissection Motion Operations......Page 349
 Conclusion......Page 350
 Sample Lesson: A Lesson on Solid Geometry......Page 351
 Resources for Teachers......Page 352
 CONNECTING WITH THE STANDARDS......Page 354
 ASSESSING MATHEMATICS UNDERSTANDING......Page 355
  Children’s Understanding of Measurement......Page 359
  Instructional Sequence......Page 361
  Length......Page 365
  Area......Page 370
  Volume......Page 373
  Capacity......Page 375
  Mass......Page 376
  Time......Page 378
  Temperature......Page 380
  Angle......Page 381
 Sample Lesson: A Lesson on Understanding Perimeter......Page 383
 Links to the Internet......Page 384
 Resources for Teachers......Page 385
 CONNECTING WITH THE STANDARDS......Page 386
 ASSESSING MATHEMATICS UNDERSTANDING......Page 387
  What Do Children Know about Data Analysis and Probability?......Page 391
  Early Experiences......Page 394
  Coordinate Graphing......Page 400
  Histograms, Line Plots, and Stem-and-Leaf Plots......Page 401
  Central Tendency......Page 405
  Variation......Page 408
  Overview......Page 410
  General Teaching Considerations......Page 411
 Conclusion......Page 416
 Sample Lesson: A Lesson on Understanding Probability......Page 417
 Resources for Teachers......Page 418
 ASSESSING MATHEMATICS UNDERSTANDING......Page 419
 What Is Algebra?......Page 423
  Introducing Integers......Page 425
  Adding Integers......Page 426
  Subtracting Integers......Page 427
  Dividing Integers......Page 430
  Repeating Patterns......Page 436
  Growing Patterns......Page 437
  Variables......Page 438
  Misconceptions about Variables......Page 439
  Understanding Expressions......Page 440
  Simplifying Algebraic Expressions......Page 441
  Understanding Algebraic Properties......Page 442
  Understanding the Order of Operations......Page 443
  Early Equations Development......Page 444
  Modeling Algebraic Equations......Page 445
  Real-World Settings for Algebraic Equations......Page 447
  What Is a Function?......Page 448
  Developing Function Concepts......Page 449
  Function Machines and Tables......Page 451
  Graphing Functions......Page 452
  Formulas......Page 453
  Instruction and Assessment Activities......Page 454
 Conclusion......Page 455
 Sample Lesson: A Lesson on Equality......Page 456
 Resources for Teachers......Page 457
References......Page 458
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