www.downloadslide.com www.downloadslide.com N i nt h E d i t i o n G L O B A L E D I T IO N Elementary and Middle School Mathematics Teaching Developmentally John A Van de Walle Late of Virginia Commonwealth University Karen S Karp University of Louisville Jennifer M Bay-Williams University of Louisville With Contributions by Jonathan Wray Howard County Public Schools Boston Columbus Hoboken Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.downloadslide.com Executive Acquisitions Editor: Meredith Fossel Series Editorial Assistant: Maria Feliberty Executive Development Editor: Linda Bishop Vice President, Director of Marketing: Margaret Waples Senior Marketing Manager: Christopher Barry Program Manager: Maren Beckman Project Manager: Christina Taylor Project Manager, Global Edition: Purnima Narayanan Senior Acquisitions Editor, Global Edition: Sandhya Ghoshal Senior Project Editor, Global Edition: Daniel Luiz Manager, Media Production, Global Edition: M Vikram Kumar Senior Manufacturing Controller, Production, Global Edition: Trudy Kimber Editorial Production Service: MPS North America LLC Manufacturing Buyer: Deidra Skahill Electronic Composition: Jouve Interior Design: Diane Lorenzo Cover Image: © CristinaMuraca/Shutterstock Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on appropriate page within text or on page C-1 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2015 The rights of John A Van de Walle, Karen S Karp, and Jennifer M Bay-Williams to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Elementary and Middle School Mathematics: Teaching Developmentally, 9th edition, ISBN 978-0-13-376893-0, by John A Van de Walle, Karen S Karp, and Jennifer M Bay-Williams, published by Pearson Education © 2016 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, withouteither the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners.The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners ISBN 10: 1-292-09769-8 ISBN 13: 978-1-292-09769-5 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 14 13 12 11 10 Typeset in Janson Text LT Std Roman by MPS North America LLC Printed and bound by R.R Donnelly Kendalville in the United States of America www.downloadslide.com About the Authors John A Van de Walle The late John A Van de Walle was a professor emeritus at Virginia Commonwealth University He was a leader in mathematics education who regularly offered professional development workshops for K–8 teachers in the United States and Canada focused on mathematics instruction that engaged students in mathematical reasoning and problem solving He visited many classrooms and worked with teachers to implement student-centered math lessons He co-authored the Scott Foresman-Addison Wesley Mathematics K–6 series and contributed to the original Pearson School mathematics program enVisionMATH Additionally, John was very active in the National Council of Teachers of Mathematics (NCTM), writing book chapters and journal articles, serving on the board of directors, chairing the educational materials committee, and speaking at national and regional meetings Karen S Karp Karen S Karp is a professor of mathematics education at the University of Louisville (Kentucky) Prior to entering the field of teacher education she was an elementary school teacher in New York Karen is the volume editor of Annual Perspectives in Mathematics Education: Using Research to Improve Instruction and is the co-author of Developing Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K–Grade 2, Discovering Lessons for the Common Core State Standards in Grades K–5, and Putting Essential Understanding of Addition and Subtraction into Practice Pre-K–Grade She is a former member of the board of directors for the National Council of Teachers of Mathematics (NCTM) and a former president of the Association of Mathematics Teacher Educators. She continues to work in classrooms with teachers of students with disabilities Jennifer M Bay-Williams Jennifer M Bay-Williams is a mathematics educator at the University of Louisville (Kentucky) Jennifer taught elementary, middle, and high school in Missouri and in Peru, and continues to work in classrooms at all levels with students and with teachers Jennifer has published many articles on teaching and learning in NCTM journals She has also authored and co-authored numerous books, including Developing Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K–Grade 2, Math and Literature: Grades 6–8, Math and Nonfiction: Grades 6–8, Navigating through Connections in Grades 6–8, and Mathematics Coaching: Resources and Tools for Coaches and Other Leaders She is on the board of directors for the National Council of Teachers of Mathematics (NCTM) and previously served on the Board of Directors for TODOS: Equity for All and as secretary and president for the Association of Mathematics Teacher Educators (AMTE) www.downloadslide.com About the Contributor Jonathan Wray is the technology contributor to Elementary and Middle School Mathematics, Teaching Developmentally (6th–9th editions) He is the instructional facilitator for Secondary Mathematics Curricular Programs in the Howard County Public School System He is the president of the Association of Maryland Mathematics Teacher Educators (AMMTE) and past president of the Maryland Council of Teachers of Mathematics (MCTM) and serves as manager of the Elementary Mathematics Specialists and Teacher Leaders (ems&tl) Project He has been recognized for his expertise in infusing technology in mathematics teaching and was named an Outstanding Technology Leader in Education by the Maryland Society for Educational Technology (MSET) Jon is also actively engaged in the National Council of Teachers of Mathematics (NCTM), serving on the Emerging Issues and Executive Committees He has served as a primary and intermediate grades classroom teacher, gifted/talented resource teacher, elementary mathematics specialist, curriculum and assessment developer, grant project manager, and educational consultant www.downloadslide.com Brief Contents Section I Teaching Mathematics: Foundations and Perspectives Chapter Teaching Mathematics in the 21st Century 25 Chapter Exploring What It Means to Know and Do Mathematics 37 Chapter Teaching through Problem Solving 57 Chapter Planning in the Problem-Based Classroom 81 Chapter Creating Assessments for Learning 108 Chapter Teaching Mathematics Equitably to All Children 128 Chapter Using Technological Tools to Teach Mathematics 151 Section II Development of Mathematical Concepts and Procedures Chapter Developing Early Number Concepts and Number Sense 166 Chapter Developing Meanings for the Operations 191 Chapter 10 Developing Basic Fact Fluency 218 Chapter 11 Developing Whole-Number Place-Value Concepts 246 Chapter 12 Developing Strategies for Addition and Subtraction Computation 271 Chapter 13 Developing Strategies for Multiplication and Division Computation 301 Chapter 14 Algebraic Thinking, Equations, and Functions 323 Chapter 15 Developing Fraction Concepts 363 Chapter 16 Developing Fraction Operations 395 Chapter 17 Developing Concepts of Decimals and Percents 427 Chapter 18 Ratios, Proportions, and Proportional Reasoning 453 Chapter 19 Developing Measurement Concepts 477 Chapter 20 Geometric Thinking and Geometric Concepts 512 Chapter 21 Developing Concepts of Data Analysis 550 Chapter 22 Exploring Concepts of Probability 582 Chapter 23 Developing Concepts of Exponents, Integers, and Real Numbers 606 Appendix A Standards for Mathematical Practice A-1 Appendix B NCTM Mathematics Teaching Practices: from Principles to Actions A-5 Appendix C Guide to Blackline Masters A-7 Appendix D Activities at a Glance A-13 www.downloadslide.com This page intentionally left blank www.downloadslide.com Contents Preface 15 Section I Teaching Mathematics: Foundations and Perspectives The fundamental core of effective teaching of mathematics combines an understanding of how students learn, how to promote that learning by teaching through problem solving, and how to plan for and assess that learning on a daily basis Introductory chapters in this section provide perspectives on trends in mathematics education and the process of doing mathematics These chapters develop the core ideas of learning, teaching, planning, and assessment Additional perspectives on mathematics for students with diverse backgrounds and the role of technological tools are also emphasized Chapter Teaching Mathematics in the 21st Century 25 Becoming an Effective Teacher of Mathematics 25 A Changing World 26 Factors to Consider 27 The Movement toward Shared Standards 28 Principles and Standards for School Mathematics 29 Common Core State Standards 30 Principles to Actions 33 An Invitation to Learn and Grow 34 Mathematical Proficiency 47 How Do Students Learn Mathematics? 50 Constructivism 50 Sociocultural Theory 51 Implications for Teaching Mathematics 51 Connecting the Dots 54 Reflections on Chapter 2 55 Writing to Learn 55 For Discussion and Exploration 55 Resources for Chapter 2 56 Recommended Readings 56 Becoming a Teacher of Mathematics 34 Reflections on Chapter 36 Writing to Learn 36 For Discussion and Exploration 36 Resources for Chapter 36 Recommended Readings 36 Chapter Exploring What It Means to Know and Do Mathematics 37 What Does It Mean to Do Mathematics? 37 Verbs of Doing Mathematics 38 An Invitation to Do Mathematics 39 Searching for Patterns 39 Analyzing a Situation 40 Generalizing Relationships 41 Experimenting and Explaining 42 Where Are the Answers? 44 What Does It Mean to Be Mathematically Proficient? 44 Relational Understanding 45 Chapter Teaching through Problem Solving 57 Problem Solving 57 Teaching for Problem Solving 58 Teaching about Problem Solving 58 Teaching through Problem Solving 61 Features of Worthwhile Tasks 61 High Levels of Cognitive Demand 62 Multiple Entry and Exit Points 62 Relevant Contexts 65 Evaluating and Adapting Tasks 67 Developing Concepts and Procedures through Tasks 68 Concepts 68 Procedures 69 What about Drill and Practice? 71 Orchestrating Classroom Discourse 73 Classroom Discussions 73 Questioning Considerations 75 How Much to Tell and Not to Tell 76 Writing to Learn 77 www.downloadslide.com 8 Contents Problem Solving for All 78 Reflections on Chapter 3 80 Writing to Learn 80 For Discussion and Exploration 80 Resources for Chapter 3 80 Recommended Readings 80 Chapter Planning in the Problem-Based Classroom 81 A Three-Phase Lesson Format 81 The Before Phase of a Lesson 82 The During Phase of a Lesson 85 The After Phase of a Lesson 87 Process for Preparing a Lesson 89 Step 1: Determine the Learning Goals 90 Step 2: Consider Your Students’ Needs 90 Step 3: Select, Design, or Adapt a Worthwhile Task 91 Step 4: Design Lesson Assessments 91 Step 5: Plan the Before Phase of the Lesson 92 Step 6: Plan the During Phase of the Lesson 93 Step 7: Plan the After Phase of the Lesson 93 Step 8: Reflect and Refine 93 More Options for the Three-Phase Lesson 94 Short Tasks 94 Learning Centers 95 Differentiating Instruction 96 Open Questions 96 Tiered Lessons 97 Parallel Tasks 99 Flexible Grouping 99 Planning for Family Engagement 101 Communicating Mathematics Goals 101 Family Math Nights 102 Homework Practices 104 Resources for Families 105 Involving All Families 106 Reflections on Chapter 4 107 Writing to Learn 107 For Discussion and Exploration 107 Resources for Chapter 4 107 Recommended Readings 107 Assessment Methods 111 Observations 111 Interviews 113 Tasks 116 Rubrics and Their Uses 119 Generic Rubrics 120 Task-Specific Rubrics 121 Writing as an Assessment Tool 122 Student Self-Assessment 123 Tests 124 Improving Performance on High-Stakes Tests 125 Communicating Grades and Shaping Instruction 125 Reflections on Chapter 5 126 Writing to Learn 126 For Discussion and Exploration 126 Resources for Chapter 5 127 Recommended Readings 127 Chapter Teaching Mathematics Equitably to All Children 128 Mathematics for ALL Students 128 Providing for Students Who Struggle and Those with Special Needs 130 Prevention Models 130 Implementing Interventions 131 Teaching and Assessing Students with Learning Disabilities 135 Teaching Students with Moderate/Severe Disabilities 137 Culturally and Linguistically Diverse Students 138 Culturally Responsive Instruction 139 Focus on Academic Vocabulary 140 Facilitating Engagement during Instruction 143 Implementing Strategies for English Language Learners 144 Providing for Students Who Are Mathematically Gifted 145 Creating Gender-Friendly Mathematics Classrooms 147 Gender Differences 147 What Can You Try? 148 Reducing Resistance and Building Resilience 149 Reflections on Chapter 6 150 Writing to Learn 150 For Discussion and Exploration 150 Resources for Chapter 6 150 Recommended Readings 150 Chapter Chapter Creating Assessments for Learning 108 Using Technological Tools to Teach Mathematics 151 Integrating Assessment into Instruction 108 What Is Assessment? 109 What Should Be Assessed? 110 Tools and Technology 151 Technology-Supported Learning Activities 152 www.downloadslide.com Contents 9 Calculators in Mathematics Instruction 154 When to Use a Calculator 155 Benefits of Calculator Use 155 Graphing Calculators 156 Portable Data-Collection Devices 158 Appropriate and Strategic Use of Digital Tools 158 Concept Instruction 159 Problem Solving 159 Drill and Reinforcement 159 How to Select Appropriate Digital Content 160 Mathematics Resources on the Internet 162 How to Select Online Resources 162 Emerging Technologies 162 Reflections on Chapter 7 165 Writing to Learn 165 For Discussion and Exploration 165 Resources for Chapter 7 165 Recommended Readings 165 Guidelines for Selecting and Using Digital Resources for Mathematics 160 Guidelines for Using Digital Content 160 Section II Development of Mathematical Concepts and Procedures This section serves as the application of the core ideas of Section I Here you will find chapters on every major content area in the pre-K–8 mathematics curriculum Numerous problem-based activities to engage students are interwoven with a discussion of the mathematical content and how students develop their understanding of that content At the outset of each chapter, you will find a listing of “Big Ideas,” the mathematical umbrella for the chapter Also included are ideas for incorporating children’s literature, integrations with the mathematical practices, and formative assessment notes These chapters are designed to help you develop pedagogical strategies and to serve as a resource for your teaching now and in the future Chapter Developing Early Number Concepts and Number Sense 166 Promoting Good Beginnings 167 The Number Core: Quantity, Counting, and Knowing How Many 168 Quantity and the Ability to Subitize 168 Early Counting 169 Numeral Writing and Recognition 172 Counting On and Counting Back 173 The Relations Core: More Than, Less Than, and Equal To 174 Developing Number Sense by Building Number Relationships 176 Relationships between Numbers through 10 176 Relationships for Numbers 10 through 20 and Beyond 184 Number Sense in Their World 186 Calendar Activities 186 Estimation and Measurement 187 Data Collection and Analysis 188 Reflections on Chapter 8 189 Writing to Learn 189 For Discussion and Exploration 189 Resources for Chapter 8 189 Literature Connections 189 Recommended Readings 189 Chapter Developing Meanings for the Operations 191 Teaching Operations through Contextual Problems 192 Addition and Subtraction Problem Structures 192 Change Problems 193 Part-Part-Whole Problems 194 Compare Problems 194 Problem Difficulty 195 Teaching Addition and Subtraction 196 Contextual Problems 196 Model-Based Problems 198 Properties of Addition and Subtraction 201 Multiplication and Division Problem Structure 203 Equal-Group Problems 203 Comparison Problems 203 Area and Array Problems 205 Combination Problems 205 Teaching Multiplication and Division 205 Contextual Problems 206 Remainders 207 Model-Based Problems 207 Properties of Multiplication and Division 210 Strategies for Solving Contextual Problems 212 Analyzing Context Problems 212 Multistep Problems 214 www.downloadslide.com I-12 Index Knowledge of mathematics prior, 52 teacher, 34 Known facts, 222 Kobett, B., 217, 300 Koch, S J., 631 Koester, B A., 549 Kyriakides, A O., 245 Lamb, L L., 631 Lamon, S., 394 Lampert, M., 56 Langrall, C W., 476 Lannin, J., 217 Lattice multiplication, 309 Law of large numbers, and probability, 592–594 Law of small numbers, probability, 603 Learning assessment standard (NCTM), 110 Learning centers, three-phase lesson format and, 95 Learning disabilities, 95, 130 progress monitoring, 131 Learning mathematics constructivist theory, 50–51 by doing mathematics, 61 productive struggle and, 53 social interaction and, 51 sociocultural theory, 51 theory implications for teaching, 51–54 Learning Principle (NCTM), 29 Learning progressions See Learning trajectories Learning trajectories, 32, 169, 170 Leavitt, S., 426, 452 Leavy, A., 362 Legos, as fraction manipulative, 385 Leinwand, S., 35 Length, 144–145 measurement, 485–491 models for decimals, 433 models for fractions, 367–369, 373–375, 379, 384 Lesson design, 81 Lesson format See Three-phase lesson format Lesson planning See also Three-phase lesson format eight steps in developing, 89–94 probability, 584 Less than sign, 346, 619–620 Let’s Count to 5, Illuminations website, 173 Levels of Cognitive Demand (Smith and Stein), 75 Levi, L., 217, 362 Lewis, M L., 631 Life-long learning, teacher, 35 Life skills, 137 Linear functions, 340–343 parallel/same/perpendicular lines, 342 proportional/nonproportional, 342 rate of change, 341 Linear models, 366, 368, 400–401 Line graphs, 568 Line plots, 561, 565–567 Lines linear functions and, 342 symmetry, 533–534 Listening, by teacher in problem solving, 88 Lists, creating as problem solving strategy, 59 Litton, N., 107 Litwiller, B., 476 Livers, S., 150 Livescribe Smartpen, 164 Lobato, J., 453, 455, 476 Looking at Lines (AIMS), 66 Losq, C., 189 Louis, E., 190 Low-level cognitive demand tasks, 62 Magic of a Million Activity Book, The — Grades 2–5 (Schwartz & Whitin), 270 Making 10, strategy for teaching addition, 227–228 Malm, C., 322 Mamer, J., 581 Manipulatives defined, 46 fractions, 366–370, 372, 383 fraction tasks, 366, 369, 370 misuse of, 47 students with special needs, 133, 144 use with tests, 124 virtual, 158 Man Who Counted: A Collection of Mathematical Adventures, The (Tahan), 394 Man Who Made Parks, The (Wishinsky and Zhang), 426 Martinie, S., 452, 631 Martland, J., 126 Mason, J., 56 Mass, measuring, 504–505 Match and make, problem solving tool, 159 Mathematical thinking, students’, 86 Mathematics assessment standard (NCTM), 110 Math Fact Café website, 241 Math Forum Website, 105, 161 Math Learning Center, 199 Math Play, exponents game, 609, 624 www.downloadslide.com Index I-13 Math Playground, 337, 370 Math TechbookTM Digital Textbook, 164 Math Tools, 161 Math Twitter Blogosphere (MTBoS), 164 McCallum, W., 273 McCoy, L P., 605 McGee, S., 245 McMahon, A., 362 McMillen, S., 605 McNamara, J., 394 Mean, 570, 571–575 balance point interpretation, 572–574 changes in, 574 leveling interpretation, 571–572 Mean absolute deviation, 570–571, 578–579 Measurement, 144–145 angles, 505–507 area, 491–500 capacity, 500–504 comparison activities, 486, 491–492, 500–502, 505, 507 concept development, 477–511 concepts and skills, 478–480 conversion and ratios, 461–464 decimals and, 430–431 developing number sense and, 187–188 fractions and, 364 instruction in, 479 iteration and, 479 length, 485–491 making comparisons, 479 meaning and process of, 478–485 metric system and, 482 misconceptions, 496–497 money, 509–510 nonstandard units, 479, 480 physical models, 479, 492–493, 502–503, 505–506 precision in, 483 role of estimation, 482–485, 493 standard units, 479, 480–482 teaching, 484 tiling, 479 time, 507–509 unit conversion, 488–489 using instruments, 480 volume, 500–504 weight/mass, 504–505 Measurement division problems, 203 Measurement standard, 30 Measures of center, 570–580 choosing, 575–576 Median, 570, 571 Memorization, limitations re basic facts, 220 Metacognition, 77 Metric system, 482 Mewborn, D S., 581 Michigan Algebra Project, 358–359 Mirror symmetry See Lines, symmetry Missing-factor strategies, division, 311 Missing-part activities, 181 Mitchell, A., 394 Mixed numbers, 379, 405, 407–408 Mode, 570, 571 Model-based problems addition, 198–199 division, 207–210 multiplication, 207–210 subtraction, 201 Modeling Middle School Mathematics resource, 338–339 Models decimals, 431–433 for fractions, 366–370, 373–375, 397, 399–402 measurement, 487–488, 502–503 standard algorithm for division, 312–313 standard algorithms for multiplication, 306–308 teaching for equity and, 133 teaching positive/negative numbers, 620–621 visual, 313 Modification, for diversity, 129–130 Mohs, L C., 300 Molina, M., 351, 362 Molisani, J., 126 Monetary units, decimals and, 430–431 Money positive/negative numbers and, 617–618 skills with, 509–510 Monson, D., 452 Moomaw, S., 190 Moreno, J., 581 Motivation, calculator use and student, 156 Multidigit numbers conceptualizing, 267–269 division, 301 estimating activity, 269 multiplication, 301, 304–306 place-value concepts and, 261–265 Multiple entry and exit points, 62–64, 65 www.downloadslide.com I-14 Index Multiplication area problems, 205 associative property of, 210–211 commutative property of, 210 comparison problems, 203–205 decimals, 442–445 distributive property of, 211 equal-group problems, 203, 204 fractions, 408–416 generalizations in, 38, 42 lattice, 309 model-based problems, 207–210 multistep problems, 214–215 with negative numbers, 624–626 partial products, 308–309 problem structures, 203–205 properties of, 210–212, 330–331 reasoning strategies for teaching, 232–238 by single-digit numbers, 303–304 standard algorithms, 306–309 student-invented strategies, 302–306 tasks, 84 teaching, 205–212 written record, 308–309 zero property, 211 Multiplicative comparison problems, 203–205 Multistep problems, 214–215 Munley, J., 605 Murray, H., 80 Museum Shapes (Metropolitan Museum of Art), 548 Music, and fractions, 368 My Favorite No video, 118 My Little Sister Ate One Hare (Grossman), 604 My Mom and Dad Make Me Laugh (Sharratt), 333 National Assessment of Education Progress (NAEP), 27, 335, 361, 363, 378, 381, 389 National Council of Supervisors of Mathematics, 160 National Council of Teachers of Mathematics (NCTM), 28 Calculation Nation, 242 CCSS-M Curriculum Analysis Tool, 160 e-Example applet, 540 Essential Understanding Series, 362 Family Resources Website, 105 Figure This! Math Challenges for Families, 105 Illuminations website, 46, 95, 162, 173, 179, 186, 242, 315, 337, 347, 348, 370, 385, 447, 466, 537, 546, 562, 568, 576, 586, 614, 624 Learning about Number Relationships, 263 position on calculator use, 154 on preschoolers learning mathematics, 167 Principles and Standards for School Mathematics, 108, 110, 176, 271, 512 Principles to Actions, 33, 57, 101, 151 National Geographic Kids Almanac 2015, 299 National Governors Association Center for Best Practices, 30 National Library of Virtual Manipulatives (NLVM), 46, 158, 210, 289, 337, 361, 370, 406, 492, 559, 584, 594, 624 National Research Council (NRC), 31, 47, 191 Committee on Early Childhood Mathematics, 167 Nation’s Report Card, 27 Native language, value use of, 141 Near-doubles, addition teaching strategy, 228–230 Negative numbers, 616–621 Negro League Scrapbook, A (Weatherform), 558 Nifty Nines, multiplication teaching strategy, 234–235 Nonproportional linear functions, 342 Novelty, using with gifted students, 146 Number and operations standard, 30 Number concepts, developing early, 166–190 early counting, 169–172 more than, less than, equal to, 174–175, 185 numeral writing/recognition, 172–173 quantity, 168–169 subitizing, 168–169 Number core, 168–174 Number Devil, The (Enzensberger), 631 Number lines, 211 and computational skills, 284 fraction tasks, 368–369, 373–375, 384, 392, 419 inequalities and, 345–346 model-based problems, 199, 207–208 positive/negative numbers, 620–621 probability and, 593 and variables, 356–357 Number relationships dot cards and, 183–184 extending more-than/less-than, 185 between numbers through 10, 176–184 numbers 10 through 20 and beyond, 184–186 numbers to 100, 185–186 one and two more/one and two less, 176–177 part-part-whole, 179–182 patterned sets and, 183 www.downloadslide.com Index I-15 pre-place value concepts, 184 use of graphs, 188 Numbers absolute value of, 619 imaginary, 627 mixed, 379, 405 patterns in start and jump, 39–40 positive/negative, 616–621 prime/composite, 209 rational/irrational, 627–629 structural array, 329–330 very large and scientific notation, 614 very small and scientific notation, 615 Number sense, 145 calendar activities and, 186 comparing fractions using, 389–391 decimals and, 434–440 defined, 176 developing by building number relationships, 175–186 developing early, 166–190, 219 and fraction operations, 396–398 place-value development and, 246, 247 Number system number combinations and, 324–325 place value relationships, 325–326 properties and, 327–331 Numeracy, family support for, 106 Numerators, 327, 377, 380–382, 390 fraction operations, 406–407 Numerical data, 556–557 Object graphs, 562 Objectives, content/language with students with special needs, 141–142 Observation assessment method, 111–113 and solving equations, 356 Octahedron, 547 Oh, Yikes! History’s Grossest, Wackiest Moments (Masoff and Sirrell), 631 Olivier, A., 80 O’Loughlin, T A., 300 Olympic games, and repeating pattern, 333–334 On Beyond a Million: An Amazing Math Journey (Schwartz), 270 One Hundred Hungry Ants (Pinczes), 216 100th Day of School from the Black Lagoon, The (Thaler), 270 100th Day Worries (Cuyler), 270 One Less Fish (Toft and Sheather), 245 One more than/two more than, addition teaching strategy, 224–225 Openness, assessment standard (NCTM), 110 Open number line, problem solving tool, 159 Open questions, 96–97 Open sentences, 348–349 Operation sense, 191 Operations in mathematics, 191–217 See also Addition; Division; Multiplication; Subtraction contextual problems and, 192–198 exponents, 608–612 formative assessment, 215 model-based problems, 198–201 positive/negative numbers, 621–627 teaching addition, 196–203 teaching subtraction, 196–203 Operator construct, fractions, 364 Oral patterns, 333 Order property See Cumulative property of addition Painted Cube Problem, 607–608 Parallel lines, linear functions and, 342 Parallelograms area of, 497 properties of, 98–99 Parallel tasks, 100 Parents See Families Part-part-whole relationships, 179–182 Partial products, 308–309 Partition division problems See Fair-sharing problems Partitioning, equal-sized portions, 371, 372, 393 Partitioning concept, 69 Partitioning strategies division of fractions, 416–417, 419 multiplication, 303–304 Partition model, division, 312–313 Partnership for Assessment of Readiness for College and Careers (PARCC), 33, 120 Part-part-whole problems, 193, 194, 195, 198 Parts of the whole, fractions, 370–382 Part-whole construct, fractions, 364 Passive learners, 123 Patterned sets, 183 Pattern Fish (Harris), 333 www.downloadslide.com I-16 Index Patterns blocks, 376, 378, 521 in doing mathematics, 38, 39–40 dots, 334–335, 337 geometric, 334 growing, 334–335 looking for as problem solving strategy, 59 oral, 333 perimeter and, 337–338 predicting with, 333–334 recursive, 336 repeating, 332–335 searching for in start and jump numbers, 39–40 T-pattern, 337–338 use in algebraic thinking, 331–335 Peck, R., 581 Pedagogical content knowledge (PCK), 152 Peer-assisted learning, 134 Pencasting, 164 Pentominoes, 538, 544 Pentominoids, 545 Percent, 363, 446–451 circle graphs and, 564–565 contextual problems, 448–450 estimation, 450–451 models, 447–448 problem solving strategies, 472–473 tasks, 86–87 Performance Level Descriptors (PARCC), 120 Perimeter, 82–83 and area, 90, 91–93, 494–496 patterns, 337–338 teaching ELLs, 139 Perpendicular lines, linear functions and, 342 Perry, M., 581 Persistence, teacher, 34 Pete the Cat and His Four Groovy Buttons (Litwin), 189 Phantom Tollbooth, The (Juster), 451–452 Philipp, R A., 631 Pi, 529 Piaget, Jean, 50, 51, 109 Picture graphs, 562 Piece = Part = Portion: Fraction = Decimal = Percent (Gifford & Thaler), 452 Pie charts, 564–565 Place-value development See also Whole-number computation base-ten models, 250–252 concepts, 248–250 connecting addition/subtraction to, 273–282, 292 numbers beyond 1000, 266–269 oral/written names for numbers, 257–261 pre-place-value understandings, 247–248 real world and, 265 written symbols, 259–261 Place-value relationships, 325–326, 428–431 Planning in problem-based classroom, 81–107 in problem solving, 59 Platonic solids, 547 Pólya, George, 58, 59 Polyhedron, 547 Polyominoids, 545 Positive numbers, 616–619, 620–621 Possibility counting, probability, 603 Powers of 10 (Florida State University) website, 608 Practice benefits of, 71–72 defined, 71 drill distinguished, 71 and students with special needs, 137 Precision, in measurement, 483 Prediction probability, 596 as problem solving strategy, 59 with repeating patterns, 333–334 Preschoolers early counting, 169–172 family role in learning counting, 168 learning mathematics, 167 relations core, 174–175 Pressing for details, teaching ELLs, 144 Prime numbers, 209 Principles and Standards for School Mathematics (NCTM), 29–30, 31, 78, 108, 110 Principles to Actions (NCTM), 33, 57 guiding principles, 33 Prior knowledge, in before phase of lesson, 82–83 Probability area representation, 597–599 compound events, 595–600 concepts of, 582–605 a continuum, 587–588 experimental, 589 experiments, 591–595 and fractions, 588 independent events, 595–597 law of large numbers and, 592–594 likely/not likely, 583–587 misconceptions, 602–603 sample spaces, 595–600 simulations and, 600–602 www.downloadslide.com Index I-17 technology resources, 594–595 theoretical, 588–591, 589–591 Problem-based tasks in assessment, 116–118 components, 116–117 examples, 117–118 Problem solving, 57–80 advocating family support for, 103–104 assessment of, 110 calculator use in enhancing, 155–156 children’s literature as context, 65–66 curiosity and, 59, 61 a four-step process, 58–60, 61 fractions and, 44–45 links to other disciplines, 66–67 planning and, 59 strategies for, 37–38, 59–60, 76 teaching about, 58–60 teaching for, 58 teaching through, 61, 69 technological aids, 159 worthwhile tasks and, 61–68 writing to learn in, 77–78 Problem solving standard, 31 Problem structures addition, 192–196 division, 203–205 multiplication, 203–205 subtraction, 192–196 Procedures addressing parental concerns about, 104 problem solving through use of, 68, 69–71 Processes, writing prompts and, 122 Process standards (NCTM), 30, 31 Product-of-measures problems See Area, problems Proficiency in mathematics, 44–49 benefits, 49 components, 48–49, 61 conceptual, 48 five strands of, 47–48, 49 problem solving and, 78–79 procedural, 48–49 strategies for creating, 51–54 Progress monitoring, students with special needs, 131 Project Interactive (Shodor), probability simulation, 601 Prompts questioning, 113 teaching for equity and, 133 writing, 122, 123 Proportional linear functions, 342 Proportional reasoning buildup strategy, 467 cross products, 473–474 defining, 456–457 development through tasks, 69 double number line diagrams, 472 percent problems, 472–473 proportional/nonproportional situations, 457–458 rates/scaling strategies, 467–469 ratio tables, 469–470 strategies for solving situations, 466–474 tape or strip diagrams, 470–472 teaching, 474 unit-rate method of solving, 467 visual models and, 473–474 Protractors, 506–507 Put together problems and take apart See Part-partwhole problems Pyramid, volume of, 504 Pythagorean theorem, 530–531, 543, 630 Quadrilaterals, geometric thinking and, 527–528 Quantity, and ability to subitize, 168–169 Questionnaires, use in self-assessment by students, 123–124 Questions classroom discussions, 73, 75 developing number sense, 187 in diagnostic interviews, 115–116 doing statistics and, 554–556 open-ended on tests, 125 prompts for parents, 105 reflective and learning centers, 95, 141 teaching students with special needs, 141 use in assessment, 112–113 using open, 96–97 Quick images, problem solving tool, 159 Quotient Café, 315 Quotients, ratios as, 454 Random outcomes, probability, 594 Random sampling, 557 Range of a function, 339 measure of variability, 570–571, 578 Rate, ratio as, 454, 473 Rational Number Project (Univ of Minnesota) website, 394 Rational numbers, 627–629 www.downloadslide.com I-18 Index Ratio(s) between type, 461 compared to fractions, 454–455 comparing, 69 as composed unit, 456 covariation, 461–466 defining, 453, 454 equations and, 460 fractions and, 364 and geometric shapes, 462–464 graphing, 465–466 and measurement conversions, 461–464 as multiplicative comparisons, 455 part-to-part, 454 part-whole, 454 as quotients, 454 as rates, 454 scale drawings and, 463–464 tables, 469–470 tasks on, 65, 83 within type, 461 types of, 454 web of associations for, 48 Real numbers, 627–630 Reasonableness, looking for as problem solving strategy, 59 Reasoning, 60 assessment of, 111 and proof standard, 31 strategies for addition, 223–232 teaching strategies re basic facts, 222 Recording devices, 151 Rectangle Division applet, 210 Rectangles area of, 71, 205, 307–309, 340, 497 ratio use with, 462, 464 Recursive patterns, 336 Reeves, C A., 631 Reflection, rigid motion, 534 Reflectional symmetry See Lines, symmetry Reflective thinking, 50, 52–53 Reinforcement, technological tools, 159–160 Reinhart, S C., 80 Relational thinking, 350–352 Relationships correspondence, 336–337 generalizing, 41–42 making explicit, 39, 52 ratios and, 455 Remainder of One, A (Pinczes), 216 Remainders, 207, 425 Renne, C G., 549 Repeated-addition problems See Equal-group problems Repeated subtraction See Fraction operations, whole number divided by fraction Repeated subtraction algorithm, 315 Repeated-subtraction problems See Measurement division problems Repeating patterns, 332–335 Representations assessment of use of, 111 of mathematical ideas, 45–46 Representation standard, 31 Resilience, building, 149 Response to intervention (RtI), 130–131, 145 multitiered models, 130, 131 Responsibility, shared, 100 Revoicing, 144 Richardson, K., 190, 270 Richardson, M G., 605 Rigelman, N R., 80 Rigid motions, geometric thinking, 534–536 Roche, A., 394 Rose, R., 150 Rotation, rigid motion, 534 Roth McDuffie, A., 216 Rounding methods, estimation strategy, 296–297 Rubenstein, R., 362 Rubrics anecdotal notes, 121 criteria/performance indicators, 119, 121 defined, 119 evaluating digital resources, 161 four-point, 120–121 generic, 120–121 task-specific, 121 three-point, 120–121 use in assessment, 110, 119–122 Rulers fraction operations and, 401 making/using, 489–491 measuring with, 480 use in model-based problems, 199 Russell, S J., 300 Sample spaces, probability and, 595–600 Sampling data collection, 557–558 random, 557 Scaffolding, 53, 130 www.downloadslide.com Index I-19 Scale factor, 464, 473, 536–537 Scatter plots, 361, 568–571 Schaeffer, R., 581 Schappelle, M P., 631 Science problem solving context, 66 statistics and, 555 Scientific notation, 99, 613–616 calculator use, 615–616 Scoring, defined, 119 Seeley, C 104 Self-assessment by students, 123–124 students with special needs, 136 use of questionnaires, 123–124 Self-directed learning, 133 Semantic equations, 194–195 Semiotic mediation, 51 Separate problems See Change problems Sequences, 334 Set models decimals, 433 fractions, 369–370, 375, 379–380, 385 Sets, model-based problems, 207–208 Shapes geometric thinking and, 519–532 and inequalities, 347 Shapes, Shapes, Shapes (Hoban), 548 Shares, problem-based assessment task, 117 Shaughnessy, J M., 405 Shaughnessy, M M., 394 Sheffield, L J., 150 Shodor Project Interactive, 337 Shortcut strategy, computational skills, 283, 285 Show Me applet, 164 Similarity, geometric thinking, 536–537 Simplification, as problem solving strategy, 59 Simulation, probability and steps, 601 Simulations, probability and, 600–602 Singapore mathematics, 99 Single-digit numbers, multiplication by, 303–304 Size, fractions and, 371–373, 404 Skemp, R., 45 Slope, 464 linear functions and, 341–342 Slow readers, 84 Smarter Balanced Assessment Consortium (SBAC), 33, 120 Social interaction, in learning mathematics, 51 Social studies, problem solving context, 66–67 So Many Circles, So Many Squares (Hoban), 548 Sophian, C., 190 Sophistication, increasing for gifted students, 146 Sorting, learning rules for, 66 Sphere, volume of, 504 Spinners, 584–587, 593, 595, 598 Split strategy, computational skills, 283 Spreadsheets, 41 Sproule, S., 165 Square roots, 629–630 Stacey, K., 56 Stafford, A., 126 Statistical literacy developing, 550–551 formulating questions and, 554–556 graphical representations, 561–570 interpreting results, 579–580 levels of thinking, 577 mathematics distinguished from statistics, 551–552 shape of data, 552–553 variability, 570–580 Steen, L A., 27 Steinmeyer, M., 56 Stem-and-leaf plots, 565 STEM areas careers in, 147, 149 portable data collection devices and, 158 Stephan, M., 631 Story problems See Contextual problems Strands for Mathematical Proficiency (NRC), 47–48, 49 Struggling students, 129 interventions, 131–135 prevention models, 130–131 providing for, 130–137 Student attitudes, calculator use and, 156 Students with special needs, 51 algebraic thinking, 326, 331, 339, 346, 347, 348, 352, 356 and base-ten concepts, 253–254, 257, 258, 260 building resilience in, 149 and computational skills, 288 counting skills and, 170 cultural/linguistic diversity, 138–147 (see also English language learners (ELLs)) culturally responsive instruction, 139–140 decimals and, 430, 433, 434, 435, 437, 439, 440 disabilities, 135–137 empowering, 140 exponents, 610 www.downloadslide.com I-20 Index Students with special needs (continued) facilitating engagement during instruction, 143–144 and fractions, 378, 385, 389, 407, 410 geometric thinking, 516, 519, 520, 522, 525, 526, 531, 533, 538, 544, 546, 547 interventions, 131–135 measurement, 482, 486, 493, 494, 498 and percent, 448 and personal goal setting, 149 planning strategies, 135–136 positive/negative numbers, 617, 622 prevention models, 130–131 probability, 584, 587 providing for, 130–137 ratios, 456, 460, 462 reducing resistance of, 149 statistical literacy, 559, 560, 572, 575 teaching basic facts, 224, 225, 228, 236, 238 using student stories, 139–140 Stylianou, D A., 426 Subitizing, 168–169 Subtraction, 83, 87, 99 compatible numbers, 275–276 contextual problems in, 192–198 decimals, 441–442 fractions, 398–408 invented strategies for, 285–291 model-based problems, 201 multistep problems, 214–215 and place-value concepts, 273–282 positive and negative numbers, 623–624 problem structures, 192–196 properties of, 201–203 reasoning strategies for teaching, 230–232 single-digit numbers, 285–286 standard algorithm, 293–294 strategies for computation, 271–299 take away, 197, 200, 288–290 teaching, 196–203 as think-addition, 200, 230–231, 288–289 Suh, J., 56, 165 Summative assessment defined, 109 high-stakes tests and, 126 of lesson, 91 Swafford, J., 476 Swamp Angel (Isaacs and Zelinsky), 476 Swanson, P E., 631 Symbolism in addition, 197–198 algebraic, 343–358 in division, 206 equal sign, 197–198 in multiplication, 206 in subtraction, 197–198 Symbols algebraic, 343–358 congruent, 527 decimal point, 429–431 equal sign, 344–352 fractions, 364, 403 greater than sign, 346 inequality sign, 344–352 less than sign, 346, 619–620 parallel, 527 pi, 529 Tables creating as problem solving strategy, 59 statistical variability and, 577 variables and, 360–361 Tablet devices, 151, 158 Take-away subtraction, 288–290 Take from 10, subtraction teaching strategy, 231–232 Talk moves, classroom discussions and, 74 Tangrams, 492, 521 Task Evaluation and Selection Guide, 67 Tasks for assessment, 116–119 collaboration on, 99–100 developing concepts and procedures through, 68–72 evaluating and adapting, 67–68 features of worthwhile, 61–68, 91 lesson planning and, 91 multiple entry and exit points, 62–64, 65 parallel, 99 problem-based, 116–118 relevant contexts, 65–67 short, 94–95 varying size for students with special needs, 136 worthwhile for problem solving, 61–68 Teachers blogs authored by, 164 and gender equity, 148–149 invitation to learn and grow, 34–36 role in problem solving, 59–67, 73–79, 81, 88–89 www.downloadslide.com Index I-21 Teachers of English to Speakers of Other Languages (TESOL), 140 Teaching Principle (NCTM), 29 Team Building Activities for Every Group (Jones), 100 Team Building Games on a Shoestring, 100 TechMatrix, guide to technological tools, 161, 162 Technology See also Technology in mathematics advocate use of by families, 103 calculators (see Calculators) nonmathmatical, 151 skills, 58 writing tools, 78 Technology in mathematics accessibility for all students, 161 addition and subtraction, 624 algebraic thinking and, 330, 337, 341, 343, 348 applets (see Applets) appropriate/strategic use of digital tools, 158–160 concept instruction and, 159 emerging technologies, 162 evaluation criteria, 161 exponents, 608, 609 fraction tasks, 366, 368, 376, 401 geometric thinking, 528, 540, 546 graphing and, 338–339, 561–562 guidelines for using, 160 hundreds chart, 263 learning activities supported by, 152–154 measurement, 496, 499 multiplication with negative numbers, 626 pencasting, 164 portable data-collection devices, 158 probability, 584, 594–595 ratios, 462, 464, 466 scatter plot generators, 361 scientific notation, 614 selecting online resources, 162–163 selection guidelines, 160–161 statistical literacy, 559 and teaching basic facts, 244 teaching tools, 151–165 tools and, 151–152 Technology Principle (NCTM), 30 Temperature, context for positive/negative numbers, 618 Ten-Frames models, 177–179, 259, 278 10 Little Hot Dogs (Himmelman), 189 Tessellations, 537 Tests guidelines in constructing, 124–125 high-stakes, 125 timed (see Timed tests) use in assessment, 124–125 Tetrahedron, 547 Text editing tools, 78 Theoretical probability, 589–591 Twelve Days of Summer (Andrews and Jolliffe), 245 Think-Addition, subtraction teaching strategy, 200, 230–231, 288–289 Think-alouds, 134 Thinking Blocks tool, 199 Third International Mathematics and Science Study (TIMSS), 28 Thompson, D., 511 Thompson, T., 165 Thousands chart, 263 Three-digit numbers, names for, 258–259 Three-phase lesson format after phase, 81, 87–89, 93, 111 before phase, 81, 82–85, 93–94 during phase, 81, 85–86, 93, 111 short tasks, 94–95 students with special needs, 130–131, 141 use in assessment, 111 Tiered lessons, 97–99, 377 Tiling, 479 Tillema, E S., 631 Time, measuring, 507–509 Timed tests, 219, 222, 244 Timelines, context for positive/negative numbers, 619 TI-NspireTM CX graphing calculator/app, 157 Tools defined, 46 fraction models, 366–370, 376 misuse of, 47 for probability instruction, 584 problem solving strategies as, 59 technological, 151–165 writing, 78 Transformations, geometric thinking, 533–538 Translation, rigid motion, 534 Translation tasks in assessment, 118–119 use of templates, 118–119 Trapezoids, area of, 498 Tree diagrams, 43, 595, 597, 599 Trends in International Mathematics and Science Study (TIMSS), 28 www.downloadslide.com I-22 Index Triangles area of, 498 classifying, 67–68 equilateral, 537 geometric thinking and, 519 interior angles of, 531–532 midsegments of, 532 problem-based assessment task, 117 Pythagorean theorem (see Pythagorean theorem) True/false sentences, 348–349 Two color counters, 369, 385, 620 Two-digit numbers as divisors, 315–317 names for, 257–258 Two-event experiments, probability, 595 Two of Everything (Hong), 65, 337 200% of Nothing: An Eye-Opening Tour through the Twists and Turns of Math Abuse and Innumeracy (Dewdney), 581 Two-step problems, 214–215 Uncertainty, teacher comfort with, 73 Understanding for problem solving, 58 technological tools for deepening student, 152 Understanding in mathematics conceptual, 49, 52 defined, 45 relational, 45–47 United States Landfalling Hurricane Web Project, and probability, 591 Universal design for learning (UDL), 137 Unknown facts, 222 Unknown values, variables used as, 352–354 Unmotivated students, 129 Up Over 10 strategy, 177, 227 U.S Census Bureau website, 614 van Hiele, Pierre, 513 van Hiele-Geldorf, Dina, 513 van Hiele levels of geometric thought, 513–519 characteristics, 514 implications for instruction, 518–519 level 0: visualization, 514–515 level 1: analysis, 515–516, 529 level 2: informal deduction, 516–517, 529 level 3: deduction, 517 level 4: rigor, 517 Variability, data analysis, 576–580 Variables, 352–358 independent/dependent, 357–358 number lines and, 356–357 relationships between, 360–361 used as quantities that vary, 356–357 used as unknown values, 352–354 Venn diagrams, 66 Virtual Base Ten Blocks for Addition and Subtraction (NLVM), 289–290 Virtual manipulatives, 158, 337, 559 Visual displays fraction tasks, 366, 419 using with students with special needs, 136 Visualization geometric thinking and, 543–547 as problem solving strategy, 59 Visual/spatial sense, 145 Vocabulary, support for students with special needs, 136, 140–142 Volume of a box with changing dimensions, 359–360 cone, 504 cylinder, 503–504 developing formulas for, 503–504 measurement, 478, 500–504 pyramid, 504 sphere, 504 teaching to students with special needs, 144 Vygotsky, Lev, 51 Walker, E T., 126 Wall, J., 322 Wearne, D., 80 Webb, D., 631 Webcast—Making Math Work for All—A Focus on Equity, 129 Weight, measuring, 504–505 Wentick, M., 300 What Comes Next?, base-ten concept activity, 266–267 What’s Under My Thumb?, teaching basic facts, 239 Wheel of Theodorus, 630 Where the Sidewalk Ends (Silverstein), 476 Whitacre, L., 631 White, J., 80 Whiteboards, interactive, 159, 164, 172 Whitin, D., 511 Whitney, S B., 452 www.downloadslide.com Index I-23 Whole-number computation See also Base-ten concepts place-value development and (see Place-value development) Whole set, problem-based assessment task, 117 Wikis, 78 Wiliam, D., 109 Williams, L., 107 Word walls, students with special needs, 142 World Is Flat, The (Friedman), 26–27 Wray, J., 165 Wright, R., 126 Writing as assessment tool, 122–123 graphic organizers, 77–78 in problem solving, 77–78 prompts, 122, 123 and students with special needs, 136 technology tools in, 78 Wyberg, T., 426, 452 XtraMath drill program, 159 Zbiek, R M., 476 Zero in addition, 225–226 division by, 212 early number concepts/sense, 171–172 Zero property of addition, 202–203 Zeros and Ones, multiplication teaching strategy, 234 Zimba, J., 273 Zone of proximal development (ZPD), 51, 53 www.downloadslide.com This page intentionally left blank www.downloadslide.com Credits Chapter 1: p 3: quote, Steen, Lynn Arthur (1997) Preface: The New Literacy In L A Steen, Why Numbers Count New York: College Entrance Examination Board; p 7: Table 1.1, Adapted with permission from NCTM (National Council of Teachers of Mathematics) (2000) Principles and standards for school mathematics Reston, VA: NCTM Copyright 2000 by the National Council of Teachers of Mathematics All rights reserved; p 7: quote, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 2: p 21: Figure 2.9, John A Van de Walle, Karen S Karp, Lou Ann H Lovin, & Jennifer M Bay-Williams, Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 6-8 Pearson Education Chapter 4: p 66: quote, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 5: p 85: quote: Wiliam, D (2010, September) Practical Techniques for Formative Assessment Presentation given in Boras, Sweden Retrieved from www.slideshare.net/BLoPP/dylan-wiliam-bors-2010 Chapter 6: p 119: list, Based on Whiteford, T (2009/2010) Is mathematics a universal language? Teaching Children Mathematics, 16(5), 276–283 Chapter 7: p 127: quote, NCTM (2014) Principles to actions: Ensuring mathematical success for all Reston, VA: NCTM, p 128: quote, Fey, J T., Hollenbeck, R W., & Wray, J A (2010) Technology and the teaching of mathematics, p 275 In B Reys, R Reys, & R Rubenstein (Eds.) Mathematics curriculum: Issues, trends, and future directions (72nd Yearbook) Reston, VA: NCTM; p 131: National Council of Teachers of Mathematics (NCTM) Using Calculators for Teaching and Learning Mathematics NCTM Research Brief, 2011 http:// www.nctm.org/news/content.aspx?id=31192 Chapter 8: p 146: Table 8.1, Based on Clements, D H., & Sarama, J (2009) Learning and teaching early math: The learning trajectories approach New York: Routledge; p 152, quote, Based on Howden, H (1989) Teaching number sense Arithmetic Teacher, 36(6), 6–11; p 161: quote, © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 9: pp 168, 170, 173, 177, 178, 182, 189: Common Core Standards, © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved p 172: quote, Based on Fosnot, C T., & Dolk, M (2001) Young mathematicians at work: Constructing number sense, addition, and subtraction Portsmouth, NH: Heinemann Chapter 10: p 195: list, Based on Baroody, A J (2006) Why children have difficulties mastering the basic number combinations and how to help them, p 22 Teaching Children Mathematics, 13(1), 22–31 pp 197, 198, 209, 210, 212, 215, 217, 219: Common Core Standards, © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 196: list, Based on Baroody, A J (2006) Why children have difficulties mastering the basic number combinations and how to help them Teaching Children Mathematics, 13(1), 22–31 pp 216–217: Table 10.2, Based on ideas from Forbringer & Fahsl, 2010, and Kamii & Anderson, 2003 Chapter 11: p 224, 225, 231, 234, 239, 242: Common Core Standards, © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 237: list, Based on Ross, S H (1989) Parts, wholes, and place value: A developmental perspective Arithmetic Teacher, 36(6), 47–51; Ross, S R (2002) Place value: Problem solving and written assessment Teaching Children Mathematics, 8(7), 419–423 Chapter 12: p 247: quote, National Research Council (2001) Adding It Up: Helping Children Learn Mathematics National Academy of Sciences, National Academies Press, Washington, DC; p 249: quote, NCTM (2014) Principles to actions: Ensuring mathematical success for all Reston, VA: NCTM; p 255: quote, Based on Carpenter, T P., Franke, M L., Jacobs, V R., Fennema, E., & Empson, S B (1998) A longitudinal study of invention and understanding in children’s multidigit addition and subtraction Journal for Research in Mathematics Education, 29(1), 3–20; p 259: quote: Hiebert, J & Grouws, D A (2007) The effects of classroom mathematics teaching on students’ learning In Frank K Lester, Jr Second Handbook of Research on Mathematics Teaching and Learning (pp 371-404), Charlotte, NC: Information Age Publishing; p 262: quote: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 264: Common Core Standard, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 272: quote, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 13: pp 278, 283, 292: Common Core Standards, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 291, 293: quote: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved C-1 www.downloadslide.com C-2 Credits Chapter 14: p 299: quote: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved pp.301, 303, 304, 312, 319, 320: Common Core Standards, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 303: quote, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 307: Figure 14.1, Van de Walle, J A., Bay-Williams, J., Lovin, L A H., & Karp, K H (2014) Teaching Student-Centered Mathematics Boston, MA: Pearson; p 331: Figure 14.18, Van de Walle, J A., Bay-Williams, J., Lovin, L A H., & Karp, K H (2014) Teaching Student- Centered Mathematics: Developmentally Appropriate Instruction for Grades 6-8 Boston, MA: Pearson; p 334: quote, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 15: pp 342, 357, 360, 362, 366: Common Core Standards, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 343: unnumbered figure, Based on Roddick, C., & and Silvas-Centeno, C (2007) “Developing Understanding of Fractions Through Pattern Blocks and Fair Trade.” Teaching Children Mathematics, 14(3), 140–145; p 344: quote, Siegler, R S., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Wray, J (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE 2010-4039) Retrieved from www whatworks.ed.gov/publications/practiceguides, p 1; p 349: quote, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 16: p 372: list, Based on CCSSO (Council of Chief State School Officers) (2010) Common core state standards Retrieved from http://corestandards.org; pp 373, 375, 384, 395, 396: Common Core Standards, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 17: pp 403, 406, 417: quotes, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; pp 406, 411, 414, 418, 421: Common Core Standards, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 18: pp 431, 433, 436, 437, 443, 444, 446: Common Core Standards, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 432: list, Based on Lamon, S J (2006) Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.) Mahwah, NJ: Lawrence Erlbaum Chapter 19: pp 455, 457, 464, 471, 473, 477, 484: Common Core Standards: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 20: pp 495, 501, 503, 506, 507: Common Core Standards: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 500, 512, 514: quotes, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 21: p 526: quote, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; p 527: quote, Scheaffer, R L (2006) Statistics and mathematics: On making a happy marriage, pp 310–311 In G F Burrill & P C Elliott (Eds.), Thinking and reasoning about data and chance: Sixty-eighth yearbook (pp 309–322) Reston, VA: NCTM; p 530, 533, 537, 540, 544: Common Core Standards: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Chapter 22: pp 560, 562, 564, 566, 570, 573, 576: Common Core Standards: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; Chapter 23: pp 583, 585, 586, 588, 589, 592, 597, 600, 603: Common Core Standards: Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved; pp 592, 593, 604: quotes, Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers All rights reserved Pop-Ups Select Activity Pages, Expanded Lessons, Blackline Masters, and Teacher Resouces from: John A Van de Walle, Karen S Karp, Lou Ann H Lovin, & Jennifer M Bay-Williams, Teaching Student-Centered Mathematics for Grades K-2, vol I Pearson Education; John A Van de Walle, Karen S Karp, Lou Ann H Lovin, & Jennifer M Bay-Williams, Teaching Student-Centered Mathematics for Grades 3-5, vol II Pearson Education; John A Van de Walle, Karen S Karp, Lou Ann H Lovin, & Jennifer M Bay-Williams, Teaching Student-Centered Mathematics for Grades 6-8, vol III Pearson Education; Jennifer M Bay-Williams, Maggie McGatha, With Beth M McCord Kobett, With Jonathan A Wray (2014)Mathematics Coaching: Resources and Tools for Coaches and Leaders, K-12 Pearson; Bay-Williams (2013) Field Experience Guide, Resources for Teachers of Elementary and Middle School Mathematics ... States edition, entitled Elementary and Middle School Mathematics: Teaching Developmentally, 9th edition, ISBN 978-0-13-376893-0, by John A Van de Walle, Karen S Karp, and Jennifer M Bay-Williams,... the technology contributor to Elementary and Middle School Mathematics, Teaching Developmentally (6th? ?9th editions) He is the instructional facilitator for Secondary Mathematics Curricular Programs... taught elementary, middle, and high school in Missouri and in Peru, and continues to work in classrooms at all levels with students and with teachers Jennifer has published many articles on teaching