Helping Children Learn Mathematics, 7th Edition

by Robert E. Reys (Univ. of Missouri); Mary M. Lindquist (Columbus State Univ.); Diana V. Lambdin (Indiana Univ.); Marilyn N. Suydam (The Ohio State Univ.); Nancy L. Smith (Emporia State Univ.)

Edition: 7th

ISBN13: 9780471151630

ISBN10: 0471151637

Format: Paperback

Pub. Date: 2003-04-01

Publisher(s): Wiley

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Summary

Covers computational alternatives, such as mental computation, estimation, written techniques, and calculators. Emphasizes problem solving, the most important skill in mathematics. Demonstrates effective classroom practices while providing a look into a variety of mathematical lessons at different grade levels. Cites and discusses specific books that can be used to complement and supplement mathematics learning.

Preface	p. iv
School Mathematics	p. 1
Introduction	p. 1
What Is Mathematics?	p. 1
What Determines the Mathematics Being Taught?	p. 2
Needs of the Subject	p. 2
Needs of the Child	p. 3
Needs of Society	p. 4
Where Can You Turn?	p. 5
National Guidelines for School Mathematics	p. 5
State Guidelines	p. 5
Research	p. 5
History	p. 5
Textbooks and Other Materials	p. 6
Electronic Materials	p. 6
Testing	p. 6
Professional Organizations	p. 7
Professional Development	p. 7
Other Teachers	p. 7
What Is Your Role Now?	p. 7
A Glance at Where We've Been	p. 7
Learning Mathematics with Understanding	p. 9
Introduction	p. 9
How Do Children Learn Mathematics?	p. 9
Building Behavior	p. 10
Constructing Understanding	p. 11
What Mathematical Knowledge Should Be Learned?	p. 13
Implications of What We Know about Learning Mathematics	p. 14
Actively Involve Students	p. 14
Learning Is Developmental	p. 14
Build on Previous Learning	p. 15
Communication Contributes Significantly to Understanding	p. 16
Good Questions Facilitate Learning	p. 17
Manipulatives Aid Learning	p. 17
Metacognition Affects Learning	p. 19
Teacher Attitudes Are Vital	p. 19
Experiences Influence Anxiety	p. 20
Gender Aptitudes Are Equal	p. 21
Retention Can Improve	p. 22
A Glance at Where We've Been	p. 23
Teaching: Your Role	p. 27
Snapshot of a Lesson Plan	p. 27
Introduction	p. 28
Recommendations for Teaching from What We Know about Learning Mathematics	p. 28
Expectations	p. 29
Tasks	p. 29
Discourse	p. 29
Tools	p. 29
Learning Environment	p. 29
The Professional Standards for Teaching Mathematics	p. 30
The Importance of Planning	p. 32
Levels of Planning	p. 34
Components of a Lesson Plan	p. 34
Planning Different Types of Lessons	p. 35
Using the Strategic Moment	p. 38
Adapting Textbook-Based Lessons	p. 38
Using Children's Literature to Motivate Mathematics Lessons	p. 39
Points to Consider in Planning and Teaching	p. 39
Maintaining a Focus on the Mathematics	p. 39
Meeting the Needs of All Students	p. 40
Grouping for Instruction	p. 42
The Importance of Questioning	p. 43
The Use of Manipulative Materials and Models	p. 44
The Role of Drill and Practice	p. 44
Incorporating Technology	p. 45
Integrating Assessment and Instruction	p. 46
A Glance at Where We've Been	p. 46
Assessment: Enhanced Learning and Teaching	p. 50
Snapshot of a Teacher's Thoughts on Assessment	p. 50
Introduction	p. 51
Making Instructional Decisions	p. 52
Monitoring Student Progress	p. 52
Evaluating Student Achievement	p. 52
Phases of Assessment	p. 53
Ways to Gather and Make Judgments about Evidence	p. 53
Observation	p. 54
Questioning	p. 56
Interviewing	p. 57
Performance Tasks	p. 58
Self-Assessment and Peer Assessment	p. 60
Work Samples	p. 62
Portfolios	p. 63
Writings	p. 64
Teacher-Designed Written Tests	p. 64
Achievement Tests	p. 65
Keeping Records and Communicating about Assessments	p. 67
Recording the Information	p. 67
Communicating the Information	p. 71
A Glance at Where We've Been	p. 72
Five Mathematical Processes	p. 76
Snapshot of a Classroom Investigation	p. 76
Introduction	p. 77
The Five Mathematical Processes	p. 77
Problem Solving	p. 77
Reasoning and Proof	p. 81
Communication	p. 83
Connections	p. 85
Representations	p. 86
Teaching Mathematics via Problem Solving	p. 87
What Is a Problem and What Is Problem Solving?	p. 88
How Can Problem Solving Be Taught Effectively?	p. 89
Time	p. 91
Planning	p. 91
Resources	p. 92
Technology	p. 93
Class Management	p. 93
Problem Posing	p. 94
What Problem-Solving Strategies Should Be Taught?	p. 95
Act It Out	p. 96
Make a Drawing or Diagram	p. 97
Look for a Pattern	p. 97
Construct a Table	p. 97
Identify All Possibilities	p. 98
Guess and Check	p. 99
Work Backward	p. 99
Write an Open Sentence	p. 100
Solve a Simpler or Similar Problem	p. 100
Change Your Point of View	p. 101
The Importance of Looking Back	p. 101
Generalize	p. 101
Check the Solution	p. 102
Find Another Way to Solve It	p. 102
Find Another Solution	p. 102
Study the Solution Process	p. 102
Using Problem-Solving Opportunities	p. 102
How Can Problem Solving Be Assessed?	p. 102
Observations	p. 103
Interviews	p. 103
Inventories and Checklists	p. 103
Paper-and-Pencil Tests	p. 104
A Glance at Where We've Been	p. 104
Beginning Number Sense: Counting and Early Benchmarks	p. 107
Snapshot of a Lesson	p. 107
Introduction	p. 108
Prenumber Concepts	p. 110
Classification	p. 110
Patterns	p. 113
Comparisons	p. 114
Conservation	p. 115
Group Recognition	p. 116
Counting	p. 117
Counting Principles	p. 118
Counting Stages	p. 118
Counting Strategies	p. 120
Counting Practice	p. 121
Early Number Development	p. 122
Developing Number Benchmarks	p. 123
Making Connections	p. 124
Cardinal, Ordinal, and Nominal Numbers	p. 126
Writing Numerals	p. 128
A Glance at Where We've Been	p. 129
Extending Number Sense: Place Value	p. 132
Snapshot of a Lesson	p. 132
Introduction	p. 133
Our Numeration System	p. 134
Thinking Place Value	p. 134
Grouping or Trading	p. 134
Nature of Place Value	p. 135
Modeling	p. 137
Developing Place Value	p. 138
A Place to Start	p. 138
Counting and Patterns	p. 145
Regrouping and Naming	p. 147
Reading and Writing Numbers	p. 148
Estimation and Rounding	p. 151
Estimation	p. 152
Rounding	p. 153
A Glance at Where We've Been	p. 155
Operations: Meanings and Basic Facts	p. 158
Snapshot of a Lesson	p. 158
Introduction	p. 159
Helping Children Develop Number Sense and Computational Fluency	p. 160
Facility with Counting	p. 160
Experience with a Variety of Concrete Situations	p. 160
Familiarity with Many Problem-Solving Contexts	p. 160
Experience in Using Language to Communicate Mathematical Ideas	p. 161
Developing Meanings for the Operations	p. 161
Addition and Subtraction	p. 162
Multiplication and Division	p. 163
Mathematical Properties	p. 167
Overview of Basic Fact Instruction	p. 167
Get Ready: Starting Where Children Are	p. 169
Get Set: Building Understanding of the Basic Facts	p. 169
Go: Mastering the Basic Facts	p. 170
Thinking Strategies for Basic Facts	p. 172
Thinking Strategies for Addition Facts	p. 172
Thinking Strategies for Subtraction Facts	p. 178
Thinking Strategies for Multiplication Facts	p. 179
Thinking Strategies for Division Facts	p. 183
A Glance at Where We've Been	p. 185
Operation Sense: Calculators, Mental Computation, and Estimation	p. 189
Snapshot of a Lesson	p. 189
Introduction	p. 190
Computational Tools	p. 190
Calculators	p. 192
Mental Computation	p. 194
Computational Estimation	p. 199
A Glance at Where We've Been	p. 208
Operations: Standard and Alternative Algorithms	p. 211
Snapshot of a Lesson	p. 211
Introduction	p. 212
Role of Materials in Learning Algorithms	p. 213
Importance of Place-Value Ideas	p. 213
Addition	p. 213
Column Addition	p. 215
Higher-Decade Addition	p. 216
Subtraction	p. 217
Zeros in the Sum	p. 219
Multiplication	p. 220
Multiplication with One-Digit Multipliers	p. 220
Multiplication with Two-Digit Multipliers	p. 221
Multiplying by 10 and Multiples of 10	p. 223
Multiplying with Zeros	p. 225
Multiplication with Large Numbers	p. 225
Division	p. 225
Division with One-Digit Divisors	p. 226
Division with Two-Digit Divisors	p. 229
Making Sense of Division with Remainders	p. 230
Checking	p. 231
Choosing Appropriate Ways	p. 232
A Glance at Where We've Been	p. 233
Fractions and Decimals: Meanings and Operations	p. 237
Snapshot of a Lesson	p. 237
Introduction	p. 238
Conceptual Development of Fractions	p. 238
Three Meanings of Fractions	p. 239
Models of the Part-Whole Meaning	p. 239
Making Fractions Meaningful	p. 241
Ordering Fractions and Equivalent Fractions	p. 244
Benchmarks	p. 247
Mixed Numbers and Improper Fractions	p. 249
Operations with Fractions	p. 249
Addition and Subtraction	p. 250
Multiplication	p. 251
Division	p. 254
Development of Decimals	p. 255
Relationship to Common Fractions	p. 255
Relationship to Place Value	p. 257
Ordering and Rounding Decimals	p. 258
Decimal Operations	p. 258
Addition and Subtraction	p. 258
Multiplication and Division	p. 259
A Glance at Where We've Been	p. 260
Ratio, Proportion, and Percent: Meanings and Applications	p. 263
Snapshot of a Lesson	p. 263
Introduction	p. 264
Ratios	p. 265
Proportions	p. 267
Percents	p. 270
Understanding Percents	p. 271
Applying Percents	p. 274
A Glance at Where We've Been	p. 276
Patterns, Relationships, and Algebra	p. 279
Snapshot of a Lesson	p. 279
Introduction	p. 280
Patterns	p. 280
Equality	p. 280
Change	p. 280
Generalizations	p. 281
Repeating Patterns	p. 281
Growing Patterns	p. 282
Number Patterns	p. 284
Relationships	p. 284
Number Theory	p. 286
Number Theory	p. 286
Specific Number Theory Topics	p. 287
A Glance at Where We've Been	p. 293
Geometry	p. 295
Snapshot of a Lesson	p. 295
Introduction	p. 296
Geometry in School Mathematics	p. 296
Geometric Characteristics and Properties	p. 296
Location and Spatial Relationships	p. 297
Transformations	p. 297
Visualization and Spatial Reasoning	p. 298
Solid Geometry	p. 298
Describing and Classifying Objects	p. 298
Constructing to Explore and Discover	p. 301
Polyhedron Models	p. 301
Relating Three Dimensions to Two	p. 302
Plane Geometry	p. 305
Properties of a Shape	p. 305
Names of Geometric Shapes	p. 311
Relationships between Shapes	p. 312
Classification Schemes	p. 313
A Glance at Where We've Been	p. 315
Measurement	p. 319
Snapshot of a Lesson	p. 319
Introduction	p. 320
Teaching Measurement	p. 322
Identifying Attributes	p. 323
Units of Measure	p. 326
Instruments for Measuring	p. 329
Formulas for Measuring	p. 331
Comparing Measurements	p. 334
Equivalences	p. 335
Conversions	p. 336
Estimating Measurements	p. 337
Connecting Attributes	p. 339
Area and Shape	p. 339
Volume and Shape	p. 339
Perimeter and Area	p. 339
Volume and Surface Area	p. 339
Perimeter and Dimensions	p. 340
Metric Relations	p. 340
A Glance at Where We've Been	p. 340
Data Analysis, Statistics, and Probability	p. 344
Snapshot of a Lesson	p. 344
Introduction	p. 346
Data Analysis	p. 347
Posing a Question and Collecting Data	p. 347
Surveys	p. 348
Experiments	p. 349
Simulations	p. 349
Displaying Data	p. 349
Quick and Easy Graphing Materials	p. 350
Line Plots and Stem-and-Leaf Plots	p. 350
Picture Graphs	p. 352
Bar Graphs	p. 353
Pie Graphs	p. 353
Line Graphs	p. 354
Box Plots	p. 355
Graphical Roundup	p. 356
Analyzing and Communicating Results	p. 356
Data Sense	p. 356
Analysis	p. 356
Misleading Graphs	p. 358
Communicating Results	p. 359
Descriptive Statistics	p. 359
Measure of Variation	p. 359
Measures of Central Tendency or Averages	p. 360
Choosing Averages	p. 362
Probability	p. 364
Probability of an Event	p. 365
Randomness	p. 367
Independence of Events	p. 368
A Glance at Where We've Been	p. 370
Standards and Expectations, NCTM 2000	p. 374
Blackline Masters	p. 381
Publishers and Distributors	p. 406
Index	p. 407
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