Learning Mathematics in Elementary and Middle Schools: A Learner-Centered Approach (5th Edition)

This popular text and it's new integrated online resource, MyEducationLab, promotes a learner-centered approach to teaching elementary and middle school mathematics. These resources provide valuable research-based instructional strategies, resources, and activities to help you learn to evaluate how children think mathematically and how to link that knowledge to developmentally appropriate teaching practices . A strong focus on NCTM Standards and developing critical tools to meet the individual needs of students ensures all students and teachers can be successful with math. 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 A......Page 466 B......Page 467 C......Page 468 E......Page 470 G......Page 471 L......Page 472 M......Page 473 N......Page 474 P......Page 475 R......Page 477 S......Page 478 U......Page 480 Z......Page 481

This thoroughly revised edition of Learning Mathematics in Elementary and Middle Schools, Fifth Edition continues to illustrate how to create a leaner-centered classroom with a strengthened pedagogical thread around mathematical thinking, content understanding, and formative assessment.

This new edition helps teachers learn to analyze children’s mathematical thinking in spoken and written work in order to adapt instruction more effectively through:

• opening pre-assessments of mathematics topics in the content area chapters,
• classroom vignettes,
• video that examines children’s mathematical reasoning including clips from the well-known IMAP: Integrating Mathematics and Pedagogy to Illustrate Children’s Reasoning project and additional classroom video
• discussion of children's common mathematical misunderstandings, and
• new sample lesson plans that tie back to pre-assessment topics.

In addition, the book supports high-quality mathematics teaching by emphasizing NCTM Standards and Curriculum Focal Points throughout. With its manageable length, practical strategies and activities to do in the classroom, and NCTM standards focus, this book will prove a valuable reference for readers well beyond their first years of teaching elementary mathematics.

Key Features of this Edition

• NEW! Expanded emphasis on NCTM Standards and Curriculum Focal Points including discussion of the Focal Points in Ch , chapter-opening Connecting With the Standards feature outlining the relevant NCTM Principles and Standards and Curriculum Focal Points, Content Standards boxes and Principles and Standards links throughout the chapters, and Focal Point connections in sample lesson plans underscore content and processes of high quality mathematics teaching.
• NEW! Assessing Mathematics Understanding features at the beginning of each content chapter (Chs. 5-17) offer teacher candidates an opportunity to pre-assess their own understanding of particular content, as well as a tool to pre-assess student understanding with guidance on interpreting the results and a chart identifying follow-up steps.
• NEW! Sample Lesson Plans in every chapter outline how to use pre-assessment results to plan instruction in line with NCTM standards to meet children's needs (chs. 5-17)
• NEW! Special icon highlights Strategies to Reach All Students appearing in select activities that particularly support English Language Learners and students with special needs.
• Revised! Examining Children’s Reasoning features explores children’s problem solving strategies, examines what it says about their math understanding and misconceptions, and draws connections to teaching strategies.
• NEW! Annotations for MyEducationLab A rich online resource of assignments and activities incorporating real videos directly correlated with the text encourages the development of the skills and dispositions necessary for success in the elementary math classroom.