Elementary and Middle School Mathematics: Teaching Developmentally (7th Edition)

“It is fun to figure out the puzzle of how children go about making sense of mathematics and then how to help teachers help kids.” John A. Van de Walle, Late of Virginia Commonwealth University This is the philosophy behind Elementary and Middle School Mathematics: Teaching Developmentally. John A. Van de Walle wrote this book to help students understand mathematics and become confident in their ability to teach the subject to children in kindergarten through eighth grade. Although he could not have foreseen the changes in mathematics teaching over the last three decades, he was at the forefront of the movement towards a constructivist view of teaching, or teaching developmentally. Constructivism says that children construct their own knowledge. They are not blank slates waiting to absorb whatever the teacher tells them. Teachers must understand both mathematics itself and how students learn mathematics in order to teach it effectively. Learning through problem solving is another major theme of this book. Students solve problems not just to apply mathematics, but also to learn new mathematics. Effective problems will take into account where students are, the problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn and not be diluted by non-mathematical activities such as cutting or pasting, and the problem must require justifications and explanations for answers and methods. Learning then becomes an outcome of the problem solving process. The book also addresses in more detail than any other book on the market the effect that the trends of standards-based education, increased pressure to test, and increased teacher accountability have had on teaching mathematics. He addresses the 2000 NCTM Standards in depth, in Chapter 1 on Teaching Mathematics in the Era of the NCTM Standards, through the NCTM icon that appears in the margins throughout the text, and in two appendices in the back of the book. Chapter 5 on Building Assessment into Instruction has also been heavily revised to focus on increased testing pressure, creating more explicit links between objectives and assessment, and including assessments for students with special needs. Additionally, samples of Pearson's reformed-based curricula, Connnected Math Project (5-8) as well as Investigations (K-4), are featured in the text and on the myeducationlab site. Elementary and Middle School Mathematics: Teaching Developmentally is a book for doing math today–for both students who want to become teachers, and the students they will eventually teach. New To This Edition: NEW! Revises Chapter 5 on assessment--Discusses increased testing pressure and accountability, adds more information on equitable assessments, creates more explicit links between objectives and assessment, and includes assessments for students with special needs. NEW! Updates the Literature Connections feature to remove all out of print children’s literature and include more non-fiction, poetry, and other types of readings. NEW! Weaves the Focal Points throughout the chapters as well as links them with the Big Ideas feature–Focal Points have also been added to the Appendix. NEW! Includes expanded coverage of working with diverse learners. NEW! Gives greater emphasis on dealing with math anxiety. Cover......Page 1 Title Page......Page 2 Copyright Page......Page 3 In Memoriam......Page 4 Dedication......Page 5 About the Authors......Page 6 Contents......Page 10 Preface......Page 20 ACKNOWLEDGMENTS......Page 27 SUPPLEMENTS......Page 29 The National Standards-Based Movement......Page 32 Principles and Standards for School Mathematics......Page 33 The Professional Standards for Teaching Mathematics and Mathematics Teaching Today......Page 36 Influences and Pressures on Mathematics Teaching......Page 37 An Invitation to Learn and Grow......Page 40 RESOURCES FOR CHAPTER 1......Page 42 What Does It Mean to Do Mathematics?......Page 44 An Invitation to Do Mathematics......Page 46 What Does It Mean to Learn Mathematics?......Page 51 What Does It Mean to Understand Mathematics?......Page 54 Connecting the Dots......Page 60 RESOURCES FOR CHAPTER 2......Page 61 Teaching Through Problem Solving......Page 63 Selecting or Designing Problem-Based Tasks and Lessons......Page 67 Teaching about Problem Solving......Page 73 Teaching in a Problem-Based Classroom......Page 74 A Three-Phase Lesson Format......Page 78 Frequently Asked Questions......Page 86 RESOURCES FOR CHAPTER 3......Page 87 Planning a Problem-Based Lesson......Page 89 Planning for All Learners......Page 95 Drill or Practice?......Page 100 Homework......Page 102 REFLECTIONS ON CHAPTER 4......Page 103 RESOURCES FOR CHAPTER 4......Page 104 EXPANDED LESSON......Page 105 Integrating Assessment into Instruction......Page 107 Performance-Based Assessments......Page 109 Rubrics and Performance Indicators......Page 111 Observation Tools......Page 113 Writing and Journals......Page 115 Diagnostic Interviews......Page 118 Tests......Page 119 Improving Performance on High-Stakes Tests......Page 120 Grading......Page 121 RESOURCES FOR CHAPTER 5......Page 122 Creating Equitable Instruction......Page 124 Mathematics for All Children......Page 125 Providing for Students with Special Needs......Page 126 Culturally and Linguistically Diverse Students......Page 133 Working Toward Gender Equity......Page 137 Providing for Students Who Are Mathematically Gifted......Page 138 REFLECTIONS ON CHAPTER 6......Page 140 RESOURCES FOR CHAPTER 6......Page 141 CHAPTER 7 Using Technology to Teach Mathematics......Page 142 Calculators in Mathematics Instruction......Page 143 Computers in Mathematics Instruction......Page 146 Instructional Software......Page 149 Guidelines for Selecting and Using Software......Page 150 Resources on the Internet......Page 151 REFLECTIONS ON CHAPTER 7......Page 153 RESOURCES FOR CHAPTER 7......Page 154 Promoting Good Beginnings......Page 156 Number Development in Pre-K and Kindergarten......Page 157 Early Number Sense......Page 160 Relationships among Numbers 1 Through 10......Page 161 Relationships for Numbers 10 Through 20......Page 169 Number Sense in Their World......Page 171 Extensions to Early Mental Mathematics......Page 173 RESOURCES FOR CHAPTER 8......Page 174 Addition and Subtraction Problem Structures......Page 176 Teaching Addition and Subtraction......Page 179 Grade 2, Counting, Coins, and Combinations......Page 181 Multiplication and Division Problem Structures......Page 185 Teaching Multiplication and Division......Page 188 Strategies for Solving Contextual Problems......Page 192 REFLECTIONS ON CHAPTER 9......Page 195 RESOURCES FOR CHAPTER 9......Page 196 Developmental Nature of Basic Fact Mastery......Page 198 Reasoning Strategies for Addition Facts......Page 201 Reasoning Strategies for Subtraction Facts......Page 206 Reasoning Strategies for Multiplication Facts......Page 208 Division Facts and “Near Facts”......Page 212 Mastering the Basic Facts......Page 213 Fact Remediation......Page 215 RESOURCES FOR CHAPTER 10......Page 216 CHAPTER 11 Developing Whole-Number Place-Value Concepts......Page 218 Basic Ideas of Place Value......Page 219 Models for Place Value......Page 222 Developing Base-Ten Concepts......Page 224 Oral and Written Names for Numbers......Page 228 Patterns and Relationships with Multidigit Numbers......Page 231 Numbers Beyond 1000......Page 238 REFLECTIONS ON CHAPTER 11......Page 241 RESOURCES FOR CHAPTER 11......Page 242 CHAPTER 12 Developing Strategies for Whole-Number Computation......Page 244 Toward Computational Fluency......Page 245 Development of Student-Invented Strategies......Page 249 Student-Invented Strategies for Addition and Subtraction......Page 250 Traditional Algorithms for Addition and Subtraction......Page 254 Student-Invented Strategies for Multiplication......Page 257 Traditional Algorithm for Multiplication......Page 261 Student-Invented Strategies for Division......Page 263 Traditional Algorithm for Division......Page 265 REFLECTIONS ON CHAPTER 12......Page 268 RESOURCES FOR CHAPTER 12......Page 269 Introducing Computational Estimation......Page 271 Computational Estimation from Invented Strategies......Page 275 Computational Estimation Strategies......Page 276 Estimation Experiences......Page 280 Using Whole Numbers to Estimate Rational Numbers......Page 282 RESOURCES FOR CHAPTER 13......Page 283 CHAPTER 14 Algebraic Thinking: Generalizations, Patterns, and Functions......Page 285 Generalization from Arithmetic and from Patterns......Page 286 Meaningful Use of Symbols......Page 288 Making Structure in the Number System Explicit......Page 296 Study of Patterns and Functions......Page 298 Mathematical Modeling......Page 307 Teaching Considerations......Page 308 Grade 7, Variables and Patterns......Page 312 RESOURCES FOR CHAPTER 14......Page 314 CHAPTER 15 Developing Fraction Concepts......Page 317 Meanings of Fractions......Page 318 Models for Fractions......Page 319 Concept of Fractional Parts......Page 322 Using Fraction Language and Symbols......Page 325 Estimating with Fractions......Page 329 Equivalent-Fraction Concepts......Page 332 REFLECTIONS ON CHAPTER 15......Page 337 RESOURCES FOR CHAPTER 15......Page 338 CHAPTER 16 Developing Strategies for Fraction Computation......Page 340 Number Sense and Fraction Algorithms......Page 341 Addition and Subtraction......Page 343 Multiplication......Page 348 Division......Page 352 RESOURCES FOR CHAPTER 16......Page 357 Connecting Fractions and Decimals......Page 359 Developing Decimal Number Sense......Page 364 Introducing Percents......Page 368 Computation with Decimals......Page 373 RESOURCES FOR CHAPTER 17......Page 376 Ratios......Page 379 Proportional Reasoning......Page 381 Grade 7, Comparing and Scaling......Page 387 Proportional Reasoning Across the Curriculum......Page 389 Proportions......Page 394 RESOURCES FOR CHAPTER 18......Page 397 CHAPTER 19 Developing Measurement Concepts......Page 400 The Meaning and Process of Measuring......Page 401 Length......Page 404 Area......Page 407 Grade 3, Perimeter, Angles, and Area......Page 408 Volume and Capacity......Page 411 Weight and Mass......Page 413 Time......Page 414 Money......Page 416 Angles......Page 417 Introducing Standard Units......Page 418 Estimating Measures......Page 420 Developing Formulas for Area and Volume......Page 422 RESOURCES FOR CHAPTER 19......Page 428 Geometry Goals for Students......Page 430 The Development of Geometric Thinking......Page 431 Learning about Shapes and Properties......Page 436 Learning about Transformations......Page 450 Learning about Location......Page 455 Learning about Visualization......Page 460 RESOURCES FOR CHAPTER 20......Page 464 CHAPTER 21 Developing Concepts of Data Analysis......Page 467 What Does It Mean to Do Statistics?......Page 468 Formulating Questions......Page 470 Data Collection......Page 471 Data Analysis: Classification......Page 472 Data Analysis: Graphical Representations......Page 474 Data Analysis: Measures of Center......Page 480 Interpreting Results......Page 484 RESOURCES FOR CHAPTER 21......Page 485 CHAPTER 22 Exploring Concepts of Probability......Page 487 Introducing Probability......Page 488 Theoretical Probability and Experiments......Page 491 Sample Spaces and Probability of Two Events......Page 496 Simulations......Page 499 REFLECTIONS ON CHAPTER 22......Page 501 RESOURCES FOR CHAPTER 22......Page 502 Exponents......Page 504 Integers......Page 510 Operations with Integers......Page 513 Real Numbers......Page 517 REFLECTIONS ON CHAPTER 23......Page 520 RESOURCES FOR CHAPTER 23......Page 521 APPENDIX A: Principles and Standards for School Mathematics: Content Standards and Grade Level Expectations......Page 522 APPENDIX B: Standards for Teaching Mathematics......Page 534 APPENDIX C: Guide to Blackline Masters......Page 536 References......Page 548 A......Page 564 C......Page 565 D......Page 566 E......Page 567 F......Page 568 H......Page 569 L......Page 570 M......Page 571 N......Page 572 P......Page 573 R......Page 574 S......Page 575 T......Page 576 Z......Page 577 Elementary And Middle School Mathematics: Teaching Developmentally. Section I : Teaching Mathematics: Foundations And Perspectives -- Chapter 1 ; Teaching Mathematics In The Area Of The Ntcm Standards. -- Chapter 2; Exploring What It Means To Know And Do Mathematics -- Chapter 3; Teaching Through Problem Solving -- Chapter 4; Planning In The Problem-based Classroom -- Chapter 5; Building Assessment Into Instuction -- Chapter 6; Teaching Mathematics Equitably To All Children -- Chapter 7; Using Technology To Teach Mathematics -- Section Ii : Development Of Mathematical Concepts And Procedures -- Chapter 8; Developing Early Number Concepts And Number Sense -- Chapter 9; Developing Meanings For The Operations -- Chapter 10; Helping Children Mast The Basic Facts -- Chapter 11; Developing Whole-number Place-value Concepts -- Chapter 12; Developing Strategies For Whole-number Computation -- Chapter 13; Using Computational Estimation With Whole Numbers -- Chapter 14; Algebraic Thinking: Generalizations, Patterns, And Functions -- Chapter 15; Developing Fraction Concepts -- Chapter 16; Developing Strategies For Fraction Computation -- Chapter 17; Developing Concepts Of Decimals And Percents -- Chapter 18; Proportional Reasoning -- Chapter 19; Developing Measurement Concepts -- Chapter 20; Geometric Thinking And Geometric Concepts -- Chapter 21; Developing Concepts Of Data Analysis -- Chapter 22; Exploring Concepts Of Probability -- Chapter 23 Developing Concepts Of Exponents, Integers, And Real Numbers -- Appendix A; Principles And Standards For School Mathematics: Content Standards And Grade Level Expectations -- Appendix B; Standards For Teaching Mathematics -- Appendix C; Guide To Blackline Masters John A. Van De Walle, Karen S. Karp, Jennifer M. Bay-william. Includes Bibliographical References And Index.