Elementary and middle school mathematics teaching developmentally 9th edition ebook

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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.

Mathematical Proficiency

Chapter 1 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 Common Core State Standards 30 Principles to Actions 33

An Invitation to Learn and Grow 34

29

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

Resources for Chapter 2

56

Recommended Readings

56

55

Becoming a Teacher of Mathematics 34

Reflections on Chapter 1

36

Writing to Learn 36 For Discussion and Exploration 36

Resources for Chapter 1

36

Recommended Readings

36

Teaching for Problem Solving 58 Teaching about Problem Solving 58 Teaching through Problem Solving 61

Exploring What It Means to Know and Do Mathematics 37

What Does It Mean to Do Mathematics? 37 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

Teaching through Problem Solving 57

Problem Solving 57

Chapter 2

Verbs of Doing Mathematics

Chapter 3

45

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

7

8 Contents Problem Solving for All 78 Reflections on Chapter 3 80

Assessment Methods

Writing to Learn 80 For Discussion and Exploration

Resources for Chapter 3

80

Recommended Readings

80

80

Rubrics and Their Uses

Planning in the Problem-Based Classroom 81

81

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

94

96

Open Questions 96 Tiered Lessons 97 Parallel Tasks 99 Flexible Grouping 99

Resources for Chapter 5

127

Recommended Readings

127

Chapter 6 Teaching Mathematics Equitably to All Children 128

Mathematics for ALL Students 128 Providing for Students Who Struggle and Those with Special Needs 130

Providing for Students Who Are Mathematically Gifted 145 Creating Gender-Friendly Mathematics Classrooms 147 Gender Differences 147 What Can You Try? 148

107

Resources for Chapter 4

107

Recommended Readings

107

Reducing Resistance and Building Resilience Reflections on Chapter 6 150

107

Writing to Learn 150 For Discussion and Exploration

Resources for Chapter 6

150

Recommended Readings

150

150

Chapter 5

Chapter 7

Creating Assessments for Learning 108

Using Technological Tools to Teach Mathematics 151

Integrating Assessment into Instruction What Is Assessment? 109 What Should Be Assessed? 110

138

Culturally Responsive Instruction 139 Focus on Academic Vocabulary 140 Facilitating Engagement during Instruction 143 Implementing Strategies for English Language Learners 144

Communicating Mathematics Goals 101 Family Math Nights 102 Homework Practices 104 Resources for Families 105 Involving All Families 106 Writing to Learn 107 For Discussion and Exploration

125

126

Culturally and Linguistically Diverse Students

Planning for Family Engagement 101

Reflections on Chapter 4

125

Prevention Models 130 Implementing Interventions 131 Teaching and Assessing Students with Learning Disabilities 135 Teaching Students with Moderate/Severe Disabilities 137

95

Differentiating Instruction

Improving Performance on High-Stakes Tests

Writing to Learn 126 For Discussion and Exploration

Process for Preparing a Lesson 89

More Options for the Three-Phase Lesson

Writing as an Assessment Tool 122 Student Self-Assessment 123 Tests 124 Communicating Grades and Shaping Instruction Reflections on Chapter 5 126

The Before Phase of a Lesson 82 The During Phase of a Lesson 85 The After Phase of a Lesson 87

Short Tasks 94 Learning Centers

119

Generic Rubrics 120 Task-Specific Rubrics 121

Chapter 4 A Three-Phase Lesson Format

111

Observations 111 Interviews 113 Tasks 116

108

Tools and Technology 151 Technology-Supported Learning Activities

152

149

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

Resources for Chapter 7

165

Recommended Readings

165

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 8 Developing Early Number Concepts and Number Sense 166

Promoting Good Beginnings 167 The Number Core: Quantity, Counting, and Knowing How Many 168

Chapter 9 Developing Meanings for the Operations

Teaching Operations through Contextual Problems 192 Addition and Subtraction Problem Structures 192

Quantity and the Ability to Subitize 168 Early Counting 169 Numeral Writing and Recognition 172 Counting On and Counting Back 173

Change Problems 193 Part-Part-Whole Problems 194 Compare Problems 194 Problem Difficulty 195

The Relations Core: More Than, Less Than, and Equal To 174 Developing Number Sense by Building Number Relationships 176

Teaching Addition and Subtraction

Relationships between Numbers 1 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

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

191

10 Contents Reflections on Chapter 9

216

Writing to Learn 216 For Discussion and Exploration

Resources for Chapter 9

Integrating Base-Ten Groupings with Words 249 Integrating Base-Ten Groupings with Place-Value Notation 249 216

Base-Ten Models for Place Value 250

216

Groupable Models 250 Pregrouped Models 251 Nonproportional Models 252

Literature Connections 216 Recommended Readings 216

Developing Base-Ten Concepts 252 Grouping Activities 252 Grouping Tens to Make 100 255 Equivalent Representations 255

Chapter 10 Developing Basic Fact Fluency 218

Developmental Phases for Learning the Basic Facts 219 Teaching and Assessing the Basic Facts 220 Different Approaches to Teaching the Basic Facts Teaching Basic Facts Effectively 221 Assessing Basic Facts Effectively 222

220

Oral and Written Names for Numbers

257

Two-Digit Number Names 257 Three-Digit Number Names 258 Written Symbols 259

Patterns and Relationships with Multidigit Numbers

261

The Hundreds Chart 261 Relationships with Benchmark Numbers 264 Connections to Real-World Ideas 265

Reasoning Strategies for Addition Facts 223 One More Than and Two More Than 224 Adding Zero 225 Doubles 226 Combinations of 10 227 Making 10 227 Using 5 as an Anchor 228 Near-Doubles 228

Numbers Beyond 1000 266 Extending the Place-Value System 266 Conceptualizing Large Numbers 267

Reflections on Chapter 11

269

Writing to Learn 269 For Discussion and Exploration

Reasoning Strategies for Subtraction Facts 230

Resources for Chapter 11

Think-Addition 230 Down Under 10 231 Take from 10 231

269

270

Literature Connections 270 Recommended Readings 270

Reasoning Strategies for Multiplication and Division Facts 232 Foundational Facts: 2, 5, 0, 1 232 Nifty Nines 234 Derived Multiplication Fact Strategies 235 Division Facts 237

Chapter 12 Developing Strategies for Addition and Subtraction Computation 271

Toward Computational Fluency 272 Connecting Addition and Subtraction to Place Value Three Types of Computational Strategies 278

Reinforcing Basic Fact Mastery 238 Games to Support Basic Fact Fluency 238 About Drill 241 Fact Remediation 242

Direct Modeling 278 Invented Strategies 279 Standard Algorithms 281

Reflections on Chapter 10 245 Writing to Learn 245 For Discussion and Exploration

Resources for Chapter 10

Development of Invented Strategies 282

245

Creating a Supportive Environment 283 Models to Support Invented Strategies 283

245

Literature Connections 245 Recommended Readings 245

Development of Invented Strategies for Addition and Subtraction 285 Single-Digit Numbers 285 Adding Two-Digit Numbers 286 Subtraction as “Think-Addition” 288 Take-Away Subtraction 288 Extensions and Challenges 290

Chapter 11 Developing Whole-Number Place-Value ­Concepts 246

Pre-Place-Value Understandings 247 Developing Whole-Number Place-Value Concepts

273

248

Integrating Base-Ten Groupings with Counting by Ones

248

Standard Algorithms for Addition and Subtraction Standard Algorithm for Addition 291 Standard Algorithm for Subtraction 293

291

Contents 11

Introducing Computational Estimation 294 Understanding Computational Estimation 294 Suggestions for Teaching Computational Estimation

Structure in the Number System: Properties 327 295

Computational Estimation Strategies 296

Making Sense of Properties 327 Applying the Properties of Addition and Multiplication 330

Study of Patterns and Functions

Front-End Methods 296 Rounding Methods 296 Compatible Numbers 297

331

Repeating Patterns 332 Growing Patterns 334 Relationships in Functions 336 Graphs of Functions 337 Describing Functions 339 Linear Functions 340

Reflections on Chapter 12 299 Writing to Learn 299 For Discussion and Exploration 299

Meaningful Use of Symbols

Resources for Chapter 12 299

343

Equal and Inequality Signs 344 The Meaning of Variables 352

Literature Connections 299 Recommended Readings 300

Mathematical Modeling 358 Algebraic Thinking across the Curriculum

Chapter 13 Developing Strategies for Multiplication and Division Computation 301

Student-Invented Strategies for Multiplication 302

359

Geometry, Measurement and Algebra 359

Reflections on Chapter 14

361

Writing to Learn 361 For Discussion and Exploration

361

Resources for Chapter 14 362

Useful Representations 302 Multiplication by a Single-Digit Multiplier 303 Multiplication of Multidigit Numbers 304

Literature Connections 362 Recommended Readings 362

Standard Algorithms for Multiplication 306 Begin with Models 306 Develop the Written Record

308

Chapter 15

Student-Invented Strategies for Division 310 Standard Algorithm for Division 312 Begin with Models 312 Develop the Written Record Two-Digit Divisors 315

Developing Fraction Concepts 363

Meanings of Fractions

313

Models for Fractions

Computational Estimation in Multiplication and Division 317 Suggestions for Teaching Computational Estimation Computational Estimation Strategies 318

317

Reflections on Chapter 13 322 Writing to Learn 322 For Discussion and Exploration 322

Resources for Chapter 13 322 Literature Connections 322 Recommended Readings 322

Chapter 14 Algebraic Thinking, Equations, and Functions 323

Strands of Algebraic Thinking 324 Structure in the Number System: Connecting Number and Algebra 324

366

Area Models 367 Length Models 368 Set Models 369

Fractional Parts

370

Fraction Size Is Relative 371 Partitioning 371 Sharing Tasks 375 Iterating 377 Fraction Notation 380

Equivalent Fractions

Number Combinations 324 Place-Value Relationships 325 Algorithms 336

364

Fraction Constructs 364 Why Fractions Are Difficult 365

382

Conceptual Focus on Equivalence 382 Equivalent Fraction Models 383 Developing an Equivalent-Fraction Algorithm 386

Comparing Fractions

389

Comparing Fractions Using Number Sense 389 Using Equivalent Fractions to Compare 391 Estimating with Fractions 391

Teaching Considerations for Fraction Concepts Reflections on Chapter 15 393 Writing to Learn 393

392

12 Contents For Discussion and Exploration

Resources for Chapter 15

Computation with Decimals

393

Addition and Subtraction Multiplication 442 Division 445

394

Literature Connections 394 Recommended Readings 394

Introducing Percents Chapter 16 Developing Fraction Operations

395

A Problem-Based Number-Sense Approach

396

Addition and Subtraction 398 Contextual Examples and Invented Strategies Models 399 Estimation and Informal Methods 402 Developing the Algorithms 403 Fractions Greater Than One 405 Addressing Misconceptions 406

Multiplication

398

416

Contextual Examples and Models 417 Answers That Are Not Whole Numbers 421 Estimation and Invented Strategies 422 Developing the Algorithms 422 Addressing Misconceptions 424

Reflections on Chapter 16 425 Writing to Learn 425 For Discussion and Exploration

Resources for Chapter 16

425

426

Literature Connections 426 Recommended Readings 426

Resources for Chapter 17

Developing Concepts of Decimals and Percents 427

428

The 10-to-1 Relationship—Now in Two Directions! 428 The Role of the Decimal Point 429

Connecting Fractions and Decimals 431 Say Decimal Fractions Correctly 431 Use Visual Models for Decimal Fractions 431 Multiple Names and Formats 433

Developing Decimal Number Sense

451

451

Literature Connections 451 Recommended Readings 452

Chapter 18 Ratios, Proportions, and Proportional Reasoning 453

Ratios

454

Types of Ratios 454 Ratios Compared to Fractions 454 Two Ways to Think about Ratio 455

Proportional Reasoning

456

Proportional and Nonproportional Situations 457 Additive and Multiplicative Comparisons in Story Problems 459 Covariation 461

Strategies for Solving Proportional Situations 466 Rates and Scaling Strategies 467 Ratio Tables 469 Tape or Strip Diagram 470 Double Number Line Diagrams 472 Percents 472 Equations 473

Teaching Proportional Reasoning 474 Reflections on Chapter 18 475

Chapter 17

Extending the Place-Value System

451

Writing to Learn 451 For Discussion and Exploration

408

Contextual Examples and Models 408 Estimation and Invented Strategies 414 Developing the Algorithms 414 Factors Greater Than One 415 Addressing Misconceptions 415

Division

446

Physical Models and Terminology 447 Percent Problems in Context 448 Estimation 450

Reflections on Chapter 17

Understanding Fraction Operations 396

440 441

434

Familiar Fractions Connected to Decimals 435 Comparing and Ordering Decimal Fractions 438 Density of Decimals 439

Writing to Learn 475 For Discussion and Exploration

Resources for Chapter 18

475

475

Literature Connections 475 Recommended Readings 476

Chapter 19 Developing Measurement Concepts

The Meaning and Process of Measuring

478

Concepts and Skills 478 Introducing Nonstandard Units 480 Introducing Standard Units 480 The Role of Estimation and Approximation

482

477

Contents 13

Length 485

Transformations

Comparison Activities 486 Using Physical Models of Length Units 487 Conversion 488 Making and Using Rulers 489

Area

491

Location

Comparison Activities 491 Using Physical Models of Area Units 492 The Relationship between Area and Perimeter 494 Developing Formulas for Area 496 Areas of Rectangles, Parallelograms, Triangles, and Trapezoids 497 Circumference and Area of Circles 499

Measuring Distance on the Coordinate Plane

Reflections on Chapter 20

Weight and Mass 504

548

Writing to Learn 548 For Discussion and Exploration 503

548

Resources for Chapter 20 548 Literature Connections 548 Recommended Readings 548

505

505

Comparison Activities 505 Using Physical Models of Angular Measure Units 505 Using Protractors 506

Time

543

543

Two-Dimensional Imagery 544 Three-Dimensional Imagery 545 The Platonic Solids 547

Comparison Activities 500 Using Physical Models of Volume and Capacity Units 502 Developing Formulas for Volumes of Common Solid Shapes Comparison Activities 505 Using Physical Models of Weight or Mass Units

537

538

Visualization

Volume and Capacity 500

Angles

533

Line Symmetry 533 Rigid Motions 534 Congruence 536 Similarity 536 Using Transformations and Symmetries

507

Developing Concepts of Data Analysis 550

What Does It Mean to Do Statistics?

551

Is It Statistics or Is It Mathematics? 551 The Shape of Data 552 The Process of Doing Statistics 553

Comparison Activities 507 Reading Clocks 507 Solving Problems with Time 508

Formulating Questions

Money 509 Recognizing Coins and Identifying Their Values

509

Reflections on Chapter 19 511

554

Classroom Questions 554 Beyond One Classroom 554

Data Collection

Writing to Learn 511 For Discussion and Exploration 511

556

Collecting Data 556 Using Existing Data Sources 558

Resources for Chapter 19 511

Data Analysis: Classification

Literature Connections 511 Recommended Readings 511

558

Attribute Materials 559

Data Analysis: Graphical Representations

561

Creating Graphs 561 Analyzing Graphs 562 Bar Graphs 562 Pie Charts/Circle Graphs 564 Continuous Data Graphs 565 Bivariate Graphs 568

Chapter 20 Geometric Thinking and Geometric Concepts 512

Geometry Goals for Students 513 Developing Geometric Thinking 513 The van Hiele Levels of Geometric Thought Implications for Instruction 518

Chapter 21

Data Analysis: Measures of Center and Variability Measures of Center 571 Understanding the Mean: Two Interpretations 571 Choosing a Measure of Center 575 Variability 576

513

Shapes and Properties 519 Sorting and Classifying 520 Composing and Decomposing Shapes 520 Categories of Two- and Three-Dimensional Shapes Investigations, Conjectures, and the Development of Proof 529

523

Interpreting Results 579 Reflections on Chapter 21 580 Writing to Learn 580 For Discussion and Exploration

580

570

14 Contents Resources for Chapter 21

581

Chapter 23

Literature Connections 581 Recommended Readings 581

Developing Concepts of Exponents, Integers, and Real Numbers 606

Exponents

Exponents in Expressions and Equations Order of Operations 608 Integer Exponents 612 Scientific Notation 613

Chapter 22 Exploring Concepts of Probability 582

Introducing Probability

583

Likely or Not Likely 583 The Probability Continuum

Positive and Negative Numbers

Theoretical Probability and Experiments 588

Independent Events 595 Area Representation 597 Dependent Events 599

Writing to Learn 604 For Discussion and Exploration

Resources for Chapter 22

604

Literature Connections 604 Recommended Readings 605

Appendix A Appendix B Appendix C APPENDIX D

595

Addition and Subtraction 621 Multiplication and Division 624

Real Numbers

Simulations 600 Common Misconceptions about Probability 602 Reflections on Chapter 22 604 604

616

Operations with Positive and Negative Numbers 621

594

Sample Spaces and the Probability of Compound Events

627

Rational Numbers 627 Square Roots and Cube Roots

Reflections on Chapter 23

630

Writing to Learn 630 For Discussion and Exploration

Resources for Chapter 23

629

630

631

Literature Connections 631 Recommended Readings 631

Standards for Mathematical Practice A-1 NCTM Mathematics Teaching Practices: from Principles to Actions A-5 Guide to Blackline Masters A-7 Activities at a Glance A-13

References R-1 Index I-1 Credits C-1

607

Contexts for Exploring Positive and Negative Numbers 617 Meaning of Negative Numbers 619 Models for Teaching Positive and Negative Numbers 620

587

Theoretical Probability 589 Experiments 591 Why Use Experiments? 594 Use of Technology in Experiments

607

Preface All students can learn mathematics with understanding! It is through the teacher’s actions that every student can have this experience. We believe that teachers must create a classroom environment in which students are given opportunities to solve problems and work together, using their ideas and strategies, to solve them. Effective mathematics instruction involves posing tasks that engage students in the mathematics they are expected to learn. Then, by allowing students to interact with and productively struggle with their own mathematical ideas and their own strategies, they will learn to see the connections among mathematical topics and the real world.

Students value mathematics and feel empowered to use it.

Creating a classroom in which students design solution pathways, engage in productive struggle, and connect one mathematical idea to another is complex. Questions arise, such as, “How do I get students to wrestle with problems if they just want me to show them how to do it? What kinds of tasks lend themselves to this type of engagement? Where can I learn the mathematics content I need in order to be able to teach in this way?” With these and other questions firmly in mind, we have several objectives in the ninth edition of this textbook: 1. Illustrate what it means to teach mathematics using a problem-based approach.

2. Serve as a go-to reference for all of the mathematics content suggested for grades pre-K–8 as recommended in the Common Core State Standards (CCSSO, 2010) and in standards used in other states, and for the research-based strategies that illustrate how students best learn this content.

3. Present a practical resource of robust, problem-based activities and tasks that can engage students in the use of significant mathematical concepts and skills.

4. Report on technology that makes teaching mathematics in a problem-based approach more visible, including links to classroom videos and ready-to-use activity pages, and references to quality websites.

We hope you will find that this is a valuable resource for teaching and learning mathematics!

New to this Edition We briefly describe new features below, along with the substantive changes that we have made since the eighth edition to reflect the changing landscape of mathematics education. The following are highlights of the most significant changes in the ninth edition.

Blackline Masters, Activity Pages and Teacher Resource Pages More than 130 ready-to-use pages have been created to support the problems and Activities throughout the book. By accessing the companion website, which lists the content by the page number in the text, you can download these to practice teaching an activity or to use with K–8 students in classroom settings. Some popular charts in the text have also been made into printable resources and handouts such as reflection questions to guide culturally relevant instruction.

Activities at a Glance By popular demand, we have prepared a matrix (Appendix D) that lists all Section II activities, the mathematics they develop, which CCSS standards they address, and the page where they can be found. We believe you will find this an invaluable resource for planning instruction.

15

16 Preface

Self-Assessment Opportunities for the Reader As we know, learners benefit from assessing their understanding along the way especially when there is a large amount of content to comprehend. To support teacher learning, each chapter begins with a set of learning outcomes that identify the goals of the chapter and link to SelfCheck quizzes. Self-Checks fall at the end of every major text section. Also, at the end of each chapter the popular Writing to Learn section now has end-of-chapter questions.

Expanded Lessons Every chapter in Section II has at least one Expanded Lesson linked to an Activity. You may recognize some of these from the Field Experience Guide. These lessons focus on concepts central to elementary and middle school mathematics and include (1) NCTM and CCSSO grade-level recommendations, (2) adaptation suggestions for English language learners (ELLs) and students with special needs, and (3) formative assessment suggestions.

Increased Focus on Common Core State Standards for Mathematics and Mathematical Practices What began in the eighth edition is even stronger in the ninth edition. The CCSS are described in Chapter 1 along with other standards documents, and the Standards for Mathematical Practices are integrated into Chapter 2. In Section II, CCSS references are embedded in the text and every Activity lists the CCSS content that can be developed in that Activity. Standards for Mathematical Practice margin notes identify text content that shows what these practices look like in classroom teaching.

Reorganization and Enhancement to Section I If you are a seasoned user of this book, you will immediately note that Chapters 2 through 4 are dramatically different. Chapter 2 has Activity Pages for each of the tasks presented and the chapter has been reorganized to move theory to the end. Chapter 3 now focuses exclusively on worthwhile tasks and classroom discourse, with merged and enhanced discussion of problems and worthwhile tasks; the three-phase lesson plan format (before, during, and after) has been moved to the beginning of Chapter 4. Chapter 4, the planning chapter, also underwent additional, major revisions that include (1) adding in the lesson plan format, (2) offering a refined process for planning a lesson (now eight steps, not ten), and (3) stronger sections on differentiating instruction and involving families. Chapter 4 discussions about ELLs and students with special needs have been moved and integrated into Chapter 6. Chapter 7, on technology, no longer has content-specific topics but rather a stronger focus on emerging technologies. Content chapters now house technology sections as appropriate.

Preface 17

Major Changes to Specific Chapters Basic Facts (Chapter 10) There are three major changes to this chapter. First, there is a much stronger focus on assessing basic facts. This section presents the risks of using timed tests and presents a strong collection of alternative assessment ideas. Second, chapter discussions pose a stronger developmental focus. For example, the need to focus first on foundational facts before moving to derived facts is shared. Third, there is a shift from a focus on mastery to a focus on fluency (as described in CCSS and in the research).

Developing Strategies for Addition and Subtraction (Chapters 11 and 12) In previous editions there was a blurry line between Chapter 11 on place value and Chapter 12, which explored how to teach students to add and subtract. Although these topics overlap in many ways, we wanted to make it easier to find the appropriate content and corresponding activities. So, many components formerly in Chapter 11 (those that were explicitly about strategies for computing) have been shifted to Chapter 12 on addition and subtraction. This resulted in 15 more activities in Chapter 12, seven of which are new.

Fraction Operations (Chapter 16) Using learning trajectories and a developmental approach, the discussion of how to develop meaning for each operation has been expanded. For example, the operation situations presented in Chapter 9 are now connected in Chapter 16 to rational numbers. In particular, multiplication and division have received much more attention, including more examples and activities. These changes are in response to the many requests for more support in this area!

Developing Concepts of Data Analysis (Chapter 21) Look for several important changes in Chapter 21. There are 12 new activities that emphasize topics in CCSS. There also is more discussion on the shape of data, variability, and distribution. And, there is a notable increase in middle grades content including attention to dot plots, sampling, bivariate graphs, and, at the suggestion of reviewers, mean absolute deviation (MAD).

Additional Important Chapter-Specific Changes The following substantive changes (not mentioned above) include Chapter 1: Information about the new NCTM Principles to Actions publication with a focus on the eight guiding principles Chapter 2: A revised and enhanced Doing Mathematics section and Knowing Mathematics section Chapter 3: A new section on Adapting Tasks (to create worthwhile tasks) and new tasks and new authentic student work Chapter 4: Open and parallel tasks added as ways to differentiate Chapter 5: A more explicit development of how to use translation tasks to assess students’ conceptual understanding Chapter 6: Additional emphasis on multi-tiered systems of support including a variety of interventions

18 Preface Chapter 7: Revisions reflect current software, tools, and digital apps as well as resources to support teacher reflection and collaboration Chapter 8: Addition of Wright’s progression of children’s understanding of the number 10 and content from the findings from the new Background Research for the National Governor’s Association Center Project on Early Mathematics Chapter 9: An expanded alignment with the problem types discussed in the CCSS document Chapter 13: Expanded discussion of the written records of computing multiplication and division problems including lattice multiplication, open arrays, and partial quotients Chapter 14: A reorganization to align with the three strands of algebraic thinking; a revamped section on Structure of the Number System with more examples of the connection between arithmetic and algebra; an increased focus on covariation and inequalities and a decreased emphasis on graphs and repeating patterns, consistent with the emphasis in CCSS Chapter 15: Many fun activities added (with manipulatives such as Play-Doh, Legos, and elastic); expanded to increase emphasis on CCSS content, including emphasis on number lines and iteration Chapter 17: Chart on common misconceptions including descriptions and examples Chapter 18: Major changes to the Strategies section, adding tape diagrams and expanding the section on double number lines; increased attention to graphing ratios and proportions Chapter 19: An increased focus on converting units in the same measurement system, perimeter, and misconceptions common to learning about area; added activities that explore volume and capacity Chapter 20: The shift in organizational focus to the four major geometry topics from the precise van Hiele level (grouping by all level 1 components), now centered on moving students from level to level using a variety of experiences within a given geometry topic Chapter 22: Major changes to activities and figures, an expanded focus on common misconceptions, and increased attention to the models emphasized in CCSS-M (dot plots, area representations, tree diagrams) Chapter 23: A new section on developing symbol sense, expanded section on order of operations, and many new activities

What You Will Find in This Book If you look at the table of contents, you will see that the chapters are separated into two distinct sections. The first section consists of seven chapters and covers important ideas that cross the boundaries of specific areas of content. The second section, consisting of 16 chapters, offers teaching suggestions and activities for every major mathematics topic in the pre-K–8 curriculum. Chapters in Section I offer perspectives on the challenging task of helping students learn mathematics. Having a feel for the discipline of mathematics—that is, to know what it means to “do mathematics”—is critical to learning how to teach mathematics well. In addition, understanding constructivist and sociocultural perspectives on learning mathematics and how they are applied to teaching through problem solving provides a foundation and rationale for how to teach and assess pre-K–8 students.

You will be teaching diverse students including students who are English language learners, are gifted, or have disabilities. In this text, you will learn how to apply instructional strategies in ways that support and challenge all learners. Formative assessment strategies, strategies for diverse learners, and effective use of technological tools are addressed in specific chapters in Section I (Chapters 5, 6, and 7, respectively), and throughout Section II chapters.

Preface 19

Each chapter of Section II focuses on one of the major content areas in pre-K–8 mathematics curriculum. It begins with identifying the big ideas for that content, and also provides guidance on how students best learn that content through many problem-based activities to engage them in understanding mathematics. Reflecting on the activities as you read can help you think about the mathematics from the perspective of the student. As often as possible, take out pencil and paper and try the problems so that you actively engage in your learning about students learning mathematics. In so doing, we are hopeful that this book will increase your own understanding of mathematics, the students you teach, and how to teach them well.

Some Special Features of This Text By flipping through the book, you will notice many section headings, a large number of figures, and various special features. All are designed to make the book more useful as a long-term resource. Here are a few things to look for.

15 Chapter

Developing Fraction Concepts Learner OutCOmes

◀ Learning Outcomes [NEW]

After reading this chapter and engaging in the embedded activities and reflections, you should be able to: 15.1 Describe and give examples for fractions constructs.

To help readers know what they should expect to learn, each chapter begins with learning outcomes. Self15.4 Illustrate examples across fraction models for developing the concept of equivalence. checks are numbered to cover and thus align with each 15.5 Compare fractions in a variety of ways and describe ways to teach this topic conceptually. learning outcome.

15.6 Synthesize how to effectively teach fraction concepts.

15.2 Name the types of fractions models and describe activities for each.

15.3 Explain foundational concepts of fractional parts, including iteration and partitioning, and connect these ideas to CCSS-M expectations.

F

ractions are one of the most important topics students need to understand in order to be successful in algebra and beyond, yet it is an area in which U.S. students struggle. NAEP test results have consistently shown that students have a weak understanding of fraction concepts (Sowder & Wearne, 2006; Wearne & Kouba, 2000). This lack of understanding is then translated into difficulties with fraction computation, decimal and percent concepts, and the use of fractions in other content areas, particularly algebra (Bailey, Hoard, Nugent, & Geary, 2012; Brown & Quinn, 2007; National Mathematics Advisory Panel, 2008). Therefore, it is absolutely critical that you teach fractions well, present fractions as interesting and important, and commit to helping students understand the big ideas.

◀ Big Ideas Big IDEAS

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Much of the research and literature espousing a student-centered approach suggests that teachers plan Three categories of models exist for working with fractions—area (e.g., of a garden), their instruction around big ideas rather than isolated length (e.g., of an inch), and set or quantity (e.g., of the class).

Partitioning and iterating are ways for students to understand the meaning of fractions, skills or concepts. At the beginning of each chapter in especially numerators and denominators.

Section II, you will find a list of the key mathematical Equal sharing is a way to build on whole-number knowledge to introduce fractional amounts.

Equivalent fractions are ways of describing the same amount by using different-sized ideas associated with the chapter. Teachers find these fractional parts.

Fractions can be compared by reasoning about the relative size of the fractions. Estimation lists helpful to quickly envision the mathematics they and reasoning are important in teaching understanding of fractions. are to teach.

For students to really understand fractions, they must experience fractions across many constructs, including part of a whole, ratios, and division.

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20 Preface 106

Chapter 6 Teaching Mathematics Equitably to All Children

You may decide instead to break the shape up into two rectangles and ask the student to find the area of each shape and combine. Then have the student attempt the next shape without the modification—you should always lead back to the original task. However, if you decide to begin with rectangular regions and build to compound shapes composed of rectangles, you have scaffolded the lesson in a way to ramp up to the original task. In planning accommodations and modifications, the goal is to enable each student to successfully reach your learning objectives, not to change the objectives. This is how equity is achieved—by reaching equal outcomes, not by equal treatment. Treating students the same when they each learn differently does not make sense.

Complete an accommodation or Modification Needs table to reflect on how you will plan for students in your classroom who have special needs. Record the evidence that you are adapting the learning situation.

Self-Check Prompts [NEW] ▶ To help readers self-assess what they have just read, a self-check prompt is offered at the end of each significant text section. After answering these quiz questions online and submitting their responses, users can review feedback on what the correct response is (and why).

Complete Self-Check 6.1: Mathematics for all Students

Providing for students Who struggle and Those with special needs One of the basic tenets of education is the need for individualizing the content taught and the methods used for students who struggle, particularly those with special needs. Mathematics learning disabilities are best thought of as cognitive differences, not cognitive deficits (Lewis, 2014). Students with disabilities often have mandated individualized education programs (IEPs) that guarantee access to grade-level mathematics content—in a general education classroom, if possible. This legislation also implies that educators consider individual learning needs not only in terms of what mathematics is taught but also how it is taught.

Prevention Models In many areas, a systematic process for achieving higher levels performance for all students includes a multitiered system of support frequently called response to intervention (RtI). This approach commonly emphasizes ways for struggling students to get immediate assistance and support rather than waiting to fail before they receive help. Multitiered models are centered on the three interwoven elements: high-quality curriculum, instructional support (interventions), and formative assessments that capture students’ strengths and weaknesses. These models were designed to determine whether low achievement was due to a lack of high-quality mathematics (i.e., “teacher-disabled students”) (Baroody, 2011; Ysseldyke, 2002) or due to an actual learning disability. They can also help determine more intensive instructional options for students who may need to have advanced mathematical challenges beyond what other students study.

Connecting Fractions and Decimals rtI (https://www.youtube.com/watch?v=nkK1bT8ls0M) 409 response to Intervention. is

a multitiered student support system that is often represented in a triangular format. As you might guess, there are a variety of RtI models developed by school systems as they structure their unique approaches to students’ needs.

CCSS- M: 4.NF.C.6; 5.NBt.a.1; 5.NBt.a.2; 5.NBt.a.3a As you move up the tiers, the number of students involved decreases, the teacher–student ratio decreases, and the level of intervention increases. Each tier in the triangle represents a level of intervention with corresponding monitoring of results and outcomes, as shown in Figure 6.1.

The foundational and largest portion of the triangle (tier 1) represents the core instruction that Give students a collection of paper base-ten pieces created from Base-ten Materials, or base-ten blocks. ask should be used with all students based on a high-quality mathematics curriculum and instructional engLisH them to pull out a particular mix—for example, a student might have three squares, sevenpractices strips, and four (i.e., manipulatives, conceptual emphasis, etc.) and on progress monitoring assessments.

Language Learners “tinies.” tell students that you have the unit behind your back; when you show it to them, they are to figure For example, if usingout a graphic organizer in tier 1 core math instruction, the following high-quality practices be expected in the three phases of the lesson—before, during, and after: how much they have and to record the value. hold up one of the units. Observe what students recordwould as their

Activity 17.2

◀ Adaptations for Students with Disabilities and English Language Learners

Shifting Units

Chapter 6 provides detailed background and strategies for how to support students with disabilities and English language learners (ELLs). But, many adaptations are specific to a particular activity or task. Therefore, Section II chapters offer activities (look for the icon) that can meet the needs of exceptional students including specific instructions with adaptations directly within the Activities.

value. ask students to accurately say their quantity aloud. For eLLs and students with disabilities, it isStates lesson purpose, introduces new vocabulary, clarifies concepts from • Before: needed prior(and knowledge in a visual organizer, and defines tasks of group members particularly important that you write these labels with the visuals in a prominent place in the classroom (if groups are being used) in student notebooks) so that they can refer to the terminology and illustrations as they participate in the activity. repeat several times. Be sure to include examples in which a piece is not represented so that students stuDents with will understand decimal values like 3.07. Continue playing in partners with one student selecting a mix of sPeCiaL neeDs base-ten pieces and the other student deciding which one is the unit and writing and saying the number.

▲ Activities

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Length models. found One of thein bestevery length models for decimal is a meter The numerous activities chapter offractions Section II stick. Each decimeter is one-tenth of the whole stick, each centimeter is one-hundredth, and each milli-

meter is onethousandth.

Any number-line into 100 subparts is likewise a useful have always been rated by readers asmodel onebroken of the most model for hundredths.

Empty number lines like those used in wholenumber computation valuable parts of the book. Some activity ideas are are also useful in helping students compare decimals and think about scale and place value (Martinie, 2014). Given described directly in decimals, the text and in anthe illustrations.

Others two or more students can use empty number line to position the values, revealing what they know about the size of these decimals using zero, one-half, one, other whole numare presented inor the numbered ActivityAboxes.

Every activity bers, other decimal values as benchmarks. large number line stretched across a wall or on the floor can be an excellent tool for exploring decimals. is a problem-based task (as described in Chapter 3) and is set models. Many teachers use money as a model for decimals, and to some extent this is designed to engage students inmoney doing mathematics. helpful. However, for students, is almost exclusively a two-place system and is nonpro-

05/12/14 12:16 AM

portional (e.g., one-tenth, a dime, does not physically compare to a dollar in that proportion.).

Numbers like 3.2 or 12.1389 do not relate to money and can cause confusion (Martinie, 2007).

Students’ initial contact with decimals should be more flexible, and so money is not recommended as an initial model for decimals, although it is certainly an important application of decimal numeration.

multiple names and Formats We acquaint students with the various visual models to help students flexibly think of 196 fracchapter 10 Developing Basic Fact Fluency quantities in terms of tenths and hundredths, and to learn to read and write decimal 65 , and then explore tions in different ways. Have students model a decimal fraction, say 100 the following ideas: Formative assessment Notes. When are students ready to work on reasoning strat• Is this fraction more or less than 12 ? Than 23 ? Than 34 ? Some familiarity with decimal fractions can be developed by comparison with fractions that are easy to think about.

• What are some different ways to say this fraction using tenths and hundredths? (“6 tenths and 5 hundredths,” “65 hundredths”) Include thousandths when appropriate.

65 5 6 or 10 + 100 ).

• Show two ways to write this fraction (100

Formative Assessment Notes ▶

Assessment should be an integral part of instruction. Similarly, it makes sense to think about what to be listening for (assessing) as you read about different areas of content development. Throughout the content chapters, there are formative assessment notes with brief descriptions of ways to assess the topic in that section. Reading these assessment notes as you read the text can also help you understand how best to assist struggling students.

Notice that decimals are usually read as a single value. That is, 0.65 is read “sixty-five hundredths.” But to understand them in terms of place value, the same number must be thought of 13 is usually read the same way as a as 6 tenths and 5 hundredths. A mixed number such as 5 100 decimal: 5.13 is “five and thirteen-hundredths.” Please note that it is accurate to use the word “and,” which represents the decimal point. For purposes of place value, it should also be under3 1 + 100 . Making these expanded forms with base-ten materials will be helpful in stood as 5 + 10 translating fractions to decimals, which is the focus of Activity 17.3.

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egies? When they are able to (1) use counting-on strategies (start with the largest and count up) and (2) see that numbers can be decomposed (e.g., that 6 is 5 + 1). Interview students by posing one-digit addition problems and ask how they solved it. For example, 3 + 8 (Do they count on from the larger?) and 5 + 6 (Do they see 5 + 5 + 1?). For multiplication, 3 * 8 (Do they know this is 3 eights? Do they see it as 2 eights and one more eight?). ■

Complete Self-Check 10.1: Developmental phases for Learning the Basic Facts

teaching and assessing the Basic Facts This section describes the different ways basic fact instruction has been implemented in schools, followed by a section describing effective strategies.

Different approaches to teaching the Basic Facts Over the last century, three main approaches have been used to teach the basic facts. The pros and cons of each approach are briefly discussed in this section.

memorization. This approach moves from presenting concepts of addition and multipli-

cation straight to memorization of facts, not devoting time to developing strategies (Baroody, Bajwa, & Eiland, 2009). This approach requires students to memorize 100 separate addition facts (just for the addition combinations 0–9) and 100 multiplication facts (0–9). Students may even have to memorize subtraction and division separately—bringing the total to over 300 12/5/14 8:29 AM evidence that this method simply does not work. You may be isolated facts! There is strong tempted to respond that you learned your facts in this manner; however, as long ago as 1935 studies concluded that students develop a variety of strategies for learning basic facts in spite of the amount of isolated drill that they experience (Brownell & Chazal, 1935).

A memorization approach does not help students develop strategies that could help them master their facts. Baroody (2006) points out three limitations: • Inefficiency. There are too many facts to memorize.

• Inappropriate applications. Students misapply the facts and don’t check their work.

• Inflexibility. Students don’t learn flexible strategies for finding the sums (or products) and therefore continue to count by ones.

Notice that a memorization approach works against the development of fluency (which includes being able to flexibly, accurately, efficiently, and appropriately solve problems). According to CCSS-M, students should have fluency with addition and subtraction facts (0–9) by the end of second grade and fluency with multiplication and division facts by the end of third grade

In second grade a picture or bar graph can be made with one bar per student. However, is it the best way to showcase the data in order to analyze it? If the data were instead categorized by number of pockets, then a graph showing the number of students with two pockets, three pockets, and so on will illustrate which number of pockets is most common and how the number of pockets varies across the class.

In sixth grade, a dot plot (also called a line plot) could be used to illustrate the spread and shape of the data. Or a histogram can be created to capture how many students fall within a range of songs listened to (e.g., between 0 and 10, 11 and 20, etc.). Or a box plot can be created, boxing in the middle 50 percent to focus attention on the center of the data as well as the range. Each of these displays gives a different snapshot of the data and provides different insights into the question posed.

Preface  21

Creating graphs Students should be involved in deciding how they want to represent their data, but they will need to be introduced to what the options are and when each display can and cannot be used.

The value of having students actually construct their own graphs is not so much that they learn the techniques, but that they are personally invested in the data and that they learn how a graph conveys information. Once a graph is constructed, the most important activity is discussing what it tells the people who see it. Analyzing data that are numerical (number of pockets) versus categorical (color of socks) is an added challenge for students as they struggle to make sense of the graphs (Russell, 2006). If, for example, the graph has seven stickers above the five, students may think that five people have seven pockets or seven people have five pockets.

Creating graphs requires care and precision, including determining appropriate scales and labels. But the reason for the precision is so that an audience is able to see at a glance the summary of the data gathered on a particular question.

Technology Note. Computer programs and graphing calculators can provide a variety of graphical displays. Use the time saved by technology to focus on the discussions about the information that each display provides! Students can make their own selections from among different graphs and justify their choice based on their own intended purposes. The graphing calculator puts data analysis technology in the hands of every student. The TI-73 calculator is designed for middle-grade students. It will produce eight different kinds of plots or graphs, including pie charts, bar graphs, and picture graphs, and will compute and graph lines of best fit.

The Internet also offers opportunities to explore different graphs. Create a Graph (NCES Kids

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◀ Standards for Mathematical Practice Margin Notes [NEW]

standards for mathematical Practice mP6. Attend to precision.

Connections to the eight Standards of Mathematical Practice from the Common Core State Standards are highlighted in the margins. The location of the note indicates an example of the identified practice in the nearby text.

standards for mathematical Practice mP5. Use appropriate tools strategically.

▲ Technology Notes

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Infusing technological tools is important in learning mathematics, as you will learn in Chapter 7. We have infused technology notes throughout Section II. A technology icon is used to identify places within the text or activity where a technology idea or resource is discussed. Descriptions include open-source (free) software, applets, and other Web-based resources, as well as ideas for calculator use.

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chapter 9 Developing Meanings for the Operations

reFLectiOns ON CHAPTER 9 Writing tO Learn Click here to assess your understanding and application of chapter content.

FOr DiscussiOn anD exPLOratiOn ◆◆

The National Mathematics Advisory Panel (2008) deemed number properties as a critical foundation for school mathematics. What is the importance of students learning the underlying principles of the fundamental properties of the operations (commutative, associative, distributive, etc.)? How does this knowledge prepare students for making generalizations and thereby develop their ability to reason algebraically?

1. Make up a compare story problem. Alter the problem to provide examples of all six different possibilities for compare problems.

◀ End of Chapter Resources

2. Explain how missing-part activities prepare students for mastering subtraction facts.

The end of each chapter includes two major subsections: Reflections, which includes “Writing Learn” “For structures Discussion and Exploration,” and Resources, which includes See howto many different and story problem (including unknowns in all positions) you can find in a “Literature Connections” textbook. In the primary grades, look for (found join, separate,in all Section II chapters) and “Recommended Readings.”

3. Make up multiplication story problems to illustrate the difference between equal groups and multiplicative comparison. Then create a story problem involving rates, area or arrays.

4. Why is the use of key words not a good strategy to teach children?

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part-part-whole, and compare problems. For grades 3 and up, look for multiplicative structures. Are the vari-

Writing to structures Learnwith[ENHANCED]. ous problem unknowns in all positions Questions are provided that help you reflect on the well represented? important pedagogical ideas related to the content in the chapter. Actually writing out the answers to these questions in your own words, or talking about them with peers, is one of the best ways for you to develop your understanding of each chapter’s main ideas.

resOurces FOR CHAPTER 9

For Discussion and Exploration. These questions ask you to explore an issue related to The first tells the tale of 100 ants on a applying trip to a picnic. what you have learned. For example, questions may ask that chapter’s content, In an attempt to speed their travel, the ants move from a There are many books with stories or pictures concerning single-file of 100 to two rows of 50, then four rows of 25, collections, the purchase of items, measurements, and so on you to reflect on classroom observations, analyze curriculum materials, or take a position and so forth. This story uses visual representation of arrays that can be used to pose problems or, better, to stimulate chiland students can be given different sizes of ant groups to dren to invent their own problems. Perhaps the most widely on controversial issues.

We hope that these questions will stimulate thought and cause explore factors and products.

The second book describes mentioned book in this context is The Doorbell Rang by Pat the trials and tribulations of a parade formation of 25 bugs.

Hutchins (1986). You can check that one out yourself, as well spirited conversations.

As the queen views the rectangular outline of the parading as the following suggestions.

Literature cOnnectiOns

Bedtime Math Overdeck (2013) This book (and accompanying website) is the author’s attempt to get parents to incorporate math problems into the nighttime (or daytime) routine. There are three levels of difficulty, starting with problems for “wee ones” (pre-K), “little kids” (K–2), and “big kids” (grade 2 and up). Each set of problems revolves around a high-interest topic such as roller coasters, foods, and animals. Teachers can use these problems in class for engaging students in all four operations.

One Hundred Hungry ants Pinczes (1999) View One hundred angry ants (https://www.youtube.com/ watch?v=kmdSUHPwJtc)

bugs, she notices that 1 bug is trailing behind. The group tries to create different numbers of rows and columns (ar-

Literature Connections.

Section rays), but again 1 bug is always a “leftover” (remainder).

Stu- II chapters contain great children’s literature for launchdents can be given different parade groups and can generate formations leave 1, 2, or none out ofconcepts the group. ing into that thewillmathematics in the chapter just read. For each title suggested, Watch a remainder of One (https://www.youtube.com/ watch?v=s4zsaoAlMpM). there is a brief description of how the mathematics concepts in the chapter can be conrecOMMenDeD reaDings nected to the story.

These literature-based mathematics activities will help you engage articles students in interesting contexts for doing mathematics.

Clement, L., & Bernhard, J. (2005). A problem solving alternative to using key words. Mathematics Teaching in the Middle

School, 10(7), 360–365. Readings. In this section, you will find an annotated list of articles and Recommended This article explores the use of sense making in solving word a remainder of One Pinczes (2002) problems replacement for using a keyinformation word strategy. The books toas aaugment the found in the chapter. These recommendations include emphasis is on the meanings of the operations as common student These two books, written by a grandmother for her grandmisconceptions are analyzed. child, help students explore multiplication and division. NCTM articles and books, and other professional resources designed for the classroom teacher. (In addition to the Recommended Readings, there is a References list at the end of the book for all sources cited within the chapters.)

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22 Preface

Supplements for Instructors Qualified college adopters can contact their Pearson sales representatives for information on ordering any of the supplements described below. These instructor supplements are all posted and available for download (click on Educators) from the Pearson Instructor Resource Center at www.pearsonglobaleditions.com/vandewalle. The IRC houses the following: • Instructor’s Resource Manual The Instructor’s Resource Manual for the ninth edition includes a wealth of resources designed to help instructors teach the course, including chapter notes, activity suggestions, and suggested assessment and test questions.

• Electronic Test Bank An electronic test bank (TB) contains hundreds of challenging questions as multiple-choice or short-answer questions. Instructors can choose from these questions and create their own customized exams.

• PowerPoint™ Presentation Ideal for instructors to use for lecture presentations or student handouts, the PowerPoint presentation provides ready-to-use graphics and text images tied to the individual chapters and content development of the text.

Acknowledgments Many talented people have contributed to the success of this book, and we are deeply grateful to all those who have assisted over the years. Without the success of the first edition, there would certainly not have been a second, much less nine editions. The following people worked closely with John on the first edition, and he was sincerely indebted to Warren Crown, John Dossey, Bob Gilbert, and Steven Willoughby, who gave time and great care in offering detailed comments on the original manuscript.

In preparing this ninth edition, we have received thoughtful input from the following mathematics teacher educators who offered comments on the eighth edition or on the manuscript for the ninth. Each reviewer challenged us to think through important issues. Many specific suggestions have found their way into this book, and their feedback helped us focus on important ideas. Thank you to Jessica Cohen, Western Washington University; Shea Mosely Culpepper, University of Houston; Shirley Dissler, High Point University; Cynthia Gautreau, California State University in Fullerton; Kevin LoPresto, Radford University; Ryan Nivens, East Tennessee State University; Adrienne Redmond-Sanogo, Oklahoma State University; and Douglas Roebuck, Ball State University. We are indebted to you for your dedicated and professional insight.

We received constant and valuable support and advice from colleagues at Pearson. We are privileged to work with our development editor, Linda Bishop, whose positive demeanor and upbeat responses on even the tightest of deadlines was most appreciated. Linda consistently offered us sound advice and much encouragement. We are also fortunate to work with Meredith Fossel, who has helped us define the direction of this edition, and helped us with the important decisions that would make the book a better product for pre-service and in-service teachers. We also wish to thank the production and editing team at MPS North America LLC, in particular Katie Watterson, who carefully and conscientiously assisted in preparing this edition for publication. Finally, our sincere thanks goes to Elizabeth Todd Brown, who helped write some of the ancillary materials.

We would each like to thank our families for their many contributions and support. On behalf of John, we thank his wife, Sharon, who was John’s biggest supporter and a sounding board as he wrote the first six editions of this book. We also recognize his daughters, Bridget (a fifth-grade teacher in Chesterfield County, Virginia) and Gretchen (an associate professor of psychology and associate dean for undergraduate education at Rutgers University–Newark).

They were John’s first students, and he tested many ideas that are in this book by their sides.

We can’t forget those who called John “Math Grandpa”: his granddaughters, Maggie, Aidan, and Gracie.

Preface 23

From Karen Karp: I would like to express thanks to my husband, Bob Ronau, who as a mathematics educator graciously helped me think about decisions while offering insights and encouragement. In addition, I thank my children, Matthew, Tammy, Joshua, Misty, Matt, Christine, Jeffrey, and Pamela for their kind support and inspiration. I also am grateful for my wonderful grandchildren, Jessica, Zane, Madeline, Jack and Emma, who have helped deepen my understanding about how children think.

From Jennifer Bay-Williams: I am forever grateful to my supportive and patient husband, Mitch Williams. My children, MacKenna (12 years) and Nicolas (9 years), along with their peers and teachers, continue to help me think more deeply about mathematics teaching and learning. My parents, siblings, nieces, and nephews have all provided support to the writing of this edition.

Most importantly, we thank all the teachers and students who gave of themselves by assessing what worked and what didn’t work in the many iterations of this book. In particular for the ninth edition, we thank teachers who generously tested activities and provided student work for us: Kimberly Clore, Kim George, and Kelly Eaton. We continue to seek suggestions from teachers who use this book so please email us at teachingdevelopmentally@gmail.com with any advice, ideas, or insights you would like to share.

Pearson would like to thank the following people for their work on the Global Edition: Contributor: Somitra Kumar Sanadhya, C.R. Rao Advanced Institute for Mathematical Sciences Reviewers: Santanu Bhowmik, Pathways World School, Aravali Pranab Sarma, Assam Engineering College B.R. Shankar, National Institute of Technology Karnataka, Surathkal

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Chapter

Teaching Mathematics in the 21st Century Learner Outcomes After reading this chapter and engaging in the embedded activities and reflections, you should be able to: 1.1 Summarize the factors that influence the teaching of mathematics.

1.2 Describe the important documents that are a part of the movement toward a set of shared expectations for students.

1.3 Explore the qualities needed to learn and grow as a professional teacher of mathematics.

S

omeday soon you will find yourself in front of a class of students, or perhaps you are already teaching. What general ideas will guide the way you will teach mathematics? This book will help you become comfortable with the mathematics content of the pre-K–8 curriculum. You will also learn about research-based strategies for helping students come to know mathematics and be confident in their ability to do mathematics.These two things—your knowledge of mathematics and how students learn mathematics—are the most important tools you can acquire to be successful.

Becoming an Effective Teacher of Mathematics Before we get started, think back to when you were in pre-K–8 classrooms as a student. What are your remembrances of learning mathematics? Here are some thoughts from in-service and pre-service teachers of whom we asked the same question. Which description do you resonate with?

I was really good at math in lower elementary grades, but because I never understood why math works, it made it very difficult to embrace the concepts as I moved into higher grades. I started believing I wasn’t good at math so I didn’t get too upset when my grades reflected that. Kathryn As a student I always felt lost during mathematics instruction. It was as if everyone around me had a magic key or code that I missed out on getting. Tracy I remember math being very challenging, intimidating, and capable of making me literally sick to my stomach. Math was a bunch of rules and formulas I was expected to memorize, but not to understand. Mary Rebekah

25

26 Chapter 1 Teaching Mathematics in the 21st Century I consider myself to be really good at math and I enjoy mathematics-related activities, but I often wonder if I would have been GREAT at math and had a completely different career if I cared about math as much as I do now. Sometimes I feel robbed. April Math went from engaging, interactive instruction that I excelled at and loved, to lecture-style instruction that I struggled with. I could not seek outside help, even though I tried, because the teacher’s way was so different from the way of the people trying to help me. I went from getting all As to getting low Bs and Cs without knowing how the change happened.

Janelle Math class was full of elimination games where students were pitted against each other to see who could answer a math fact the fastest. Because I have a good memory I did well, but I hated every moment. It was such a nerve-wracking experience and for the longest time that is what I thought math was. Lawrence Math was never a problem because it was logical, everything made sense. Tova As you can see these memories run the gamut with an array of emotions and experiences. The question now becomes, what do you hope your students will say as they think back to your mathematics instruction? The challenge is to get all of your students to learn mathematics with understanding and enthusiasm. Would you relish hearing your students, fifteen years after leaving your classroom, state that you encouraged them to be mathematically minded, curious about solving new problems, self-motivated, able to critically think about both correct and incorrect strategies, and that you nurtured them to be a risk takers willing to try and persevere on challenging tasks? What will your legacy be?

As part of your personal desire to build successful learners of mathematics, you might recognize the challenge that mathematics is sometimes seen as the subject that people love to hate. At social events of all kinds—even at parent–teacher conferences—other adults will respond to the fact that you are a teacher of mathematics with comments such as “I could never do math,” or “I can’t even balance my checking account.” Instead of dismissing these disclosures, consider what positive action you can take. Would people confide that they don’t read and hadn’t read a book in years? That is not likely. Families’ and teachers’ attitudes toward mathematics may enhance or detract from students’ ability to do math. It is important for you and for students’ families to know that mathematics ability is not inherited—anyone can learn mathematics. Moreover, learning mathematics is an essential life skill. You need to find ways of countering these statements, especially if they are stated in the presence of students, pointing out that it is a myth that only some people can be successful in learning mathematics. Only in that way can the chain of passing apprehension from family member to child, or in rare cases teacher to student, be broken. There is much joy to be had in solving mathematical problems, and you need to model this excitement and nurture that passion in your students.

Your students need to ultimately think of themselves as mathematicians in the same way as many of them think of themselves as readers. As students interact with our increasingly mathematical and technological world, they need to construct, modify, communicate or integrate new information in many forms. Solving novel problems and approaching circumstances with a mathematical perspective should come as naturally as reading new materials to comprehend facts, insights, or news. Consider how important this is to interpreting and successfully surviving in our economy and in our environment.

The goal of this book is to help you understand the mathematics methods that will make you an effective teacher. As you dig into the information your vision and confidence will grow.

A Changing World In his book The World Is Flat (2007), Thomas Friedman discusses the need for people to have skills that are lasting and will survive the ever-changing landscape of available jobs. These are specific categories within a larger group that are called “untouchables” as regardless of the shifting landscape of job options—they will be successful in finding jobs. He is the one who defined these broad categories—such as math lover. Friedman points out that in a world that is digitized

A Changing World 27

and surrounded by algorithms, math lovers will always have career opportunities and options.

This is important as science, technology, engineering, and math (STEM) jobs, because of a skills gap, take more than twice as long to fill as other jobs in the marketplace (Rothwell, 2014). This is also aligned with the thinkers who believe students need to not just be college ready but innovation ready (Wagner, 2012).

Now it becomes the job of every teacher of mathematics to prepare students with skills for potential careers and develop a “love of math” in students. Lynn Arthur Steen, a well-known mathematician and educator, stated, “As information becomes ever more quantitative and as society relies increasingly on computers and the data they produce, an innumerate citizen today is as vulnerable as the illiterate peasant of Gutenberg’s time” (1997, p. xv).

The changing world influences what should be taught in pre-K–8 mathematics classrooms.

As we prepare pre-K–8 students for jobs that possibly do not currently exist, we can predict that there are few jobs for people where they just do simple computation. We can also predict that there will be work that requires interpreting complex data, designing algorithms to make predictions, and using the ability to approach new problems in a variety of ways.

As you prepare to help students learn mathematics for the future, it is important to have some perspective on the forces that effect change in the mathematics classroom. This chapter addresses the leadership that you, the teacher, will develop as you shape the mathematics experience for your students. Your beliefs about what it means to know and do mathematics and about how students make sense of mathematics will affect how you approach instruction and the understandings and skills your students take from the classroom.

Factors to Consider For more than two decades, mathematics education has constantly undergone change. There have been significant reforms that reflect the technological and informational needs of our society, research on how students learn mathematics, the importance of providing opportunities to learn for all students, and ideas on how and what to teach from an international perspective.

Just as we would not expect doctors to be using the exact same techniques and medicines that were prevalent when you were a child, teachers’ methods are evolving and transforming via a powerful collection of expert knowledge about how the mind functions and how to design effective instruction (Wiggins, 2013).

There are several significant factors in this transformation. One factor is the public or political pressure for change in mathematics education due largely to information about student performance in national and international studies. These large scale comparisons of student performance continue to make headlines, provoke public opinion, and pressure legislatures to call for tougher standards backed by testing. The pressures of testing policies exerted on schools and ultimately on teachers may have an impact on instruction. These studies are important because international and national assessments provide strong evidence that mathematics teaching must change if our students are to be competitive in the global market and able to understand the complex issues they must confront as responsible citizens.

National Assessment of Education Progress (NAEP). Since the 1960s, at reg-

ular intervals, the United States gathers data on how fourth-, eighth-, and twelfth-grade students are doing in mathematics on the NAEP. These data provide an important tool for policy makers and educators to measure the overall improvement of U.S. students over time in what is called the “Nation’s Report Card.” NAEP uses four achievement levels: below basic, basic, proficient, and advanced, with proficient and advanced representing substantial grade-level achievement. The criterion-referenced test is designed to reflect the current curriculum but keeps a few stable items from 1982 for purposes of comparison (Kloosterman, Rutledge, & Kenney, 2009b). In the most recent assessment in 2013, less than half of all U.S. students in grades 4 and 8 performed at the desirable levels of proficient and advanced (42 percent in fourth grade and 35 percent in eighth grade) (National Center for Education Statistics, 2013).

Despite encouraging gains in the NAEP scores over the last 30 years due to important shifts in instructional practices (particularly at the elementary level) (Kloosterman, Rutledge, & Kenney, 2009b), some U.S. students’ performance still reveals disappointing levels of competency.

28 Chapter 1 Teaching Mathematics in the 21st Century

Trends in International Mathematics and Science Study (TIMSS). In the

mid-1990s, 41 nations participated in the Third International Mathematics and Science Study, the largest study of mathematics and science education ever conducted. Data were gathered in grades 4, 8, and 12 from 500,000 students as well as from teachers. The most widely reported results revealed that U.S. students performed above the international average of the TIMSS countries at the fourth grade, below the average at the eighth grade, and significantly below average at the twelfth grade (National Academy Press, 1999; U.S. Department of Education, 1997).

TIMSS studies were repeated often with the most recent in 2011 in which 57 countries participated. For details, please visit the TIMSS website. The 2011 TIMSS found that U.S. fourth and eighth graders were above the international average but were significantly outperformed at fourth-grade level mathematics by education systems in Singapore, Republic of Korea, Hong Kong, Chinese Taipei, Japan, Northern Ireland, Belgium, Finland, England, and the Russian Federation and outperformed at the eighth-grade level by education systems in Republic of Korea, Singapore, Chinese Taipei, Hong Kong, Japan, Russian Federation, Israel, and Finland.

One of the most interesting components of the study was the videotaping of eighth-grade classrooms in the United States, Australia, and five of the highest-achieving countries. The results indicate that teaching is a cultural activity and, despite similarities, the differences in the ways countries taught mathematics were often striking. In all countries, problems or tasks were frequently used to begin the lesson. However, as a lesson progressed, the way these problems were handled in the United States was in stark contrast to high-achieving countries.

Analysis revealed that, although the world is for all purposes unrecognizable from what it was 100 years ago, the U.S. approach to teaching mathematics during the same time frame was essentially unchanged (Stigler & Hiebert, 2009). Other countries incorporated a variety of methods, but they frequently used a problem-solving approach with an emphasis on conceptual understanding and students engaged in problem solving (Hiebert et al., 2003). Teaching in the high-achieving countries more closely resembles the recommendations of the National Council of Teachers of Mathematics, the major professional organization for mathematics teachers, discussed next.

National Council of Teachers of Mathematics (NCTM). One transformative factor is the professional leadership of the National Council of Teachers of Mathematics (NCTM). The NCTM, with more than 80,000 members, is the world’s largest mathematics education organization. This group holds an influential role in the support of teachers and an emphasis on what is best for learners. Their guidance in the creation and dissemination of standards for curriculum, assessment, and teaching led the way for other disciplines. For an array of resources, including the Illuminations component which consists of a set of exciting instructional experiences for your students, visit the NCTM website.

Complete Self-Check 1.1: A Changing World

The Movement toward Shared Standards The momentum for reform in mathematics education began in earnest in the early 1980s. The main impetus was a response to a need for more problem solving as well as the research of developmental psychologists who identified how students can best learn mathematics. Then in 1989, NCTM published the first set of standards for a subject area in the Curriculum and Evaluation Standards for School Mathematics. Many believe that no other document has ever had such an enormous effect on school mathematics or on any other area of the curriculum.

NCTM followed in 1991 with a set of standards for teaching that articulated a vision of teaching mathematics for all students, not just a few. In 1995, NCTM added to the collection the Assessment Standards for School Mathematics, which focused on the importance of integrating

The Movement toward Shared Standards 29

assessment with instruction and indicated the key role that assessment plays in ­implementing change (see Chapter 5). In 2000, however, NCTM released Principles and Standards for School Mathematics as an update of its original standards document. Combined, these documents prompted a revolutionary reform movement in mathematics education throughout the world.

As these documents influenced teacher practice, ongoing debate continued about the U.S. curriculum. In particular, many argued that instead of hurrying through several topics every year, the curriculum needed to address content more deeply. Guidance was needed in deciding what mathematics content should be taught at each grade level and, in 2006, NCTM released ­Curriculum Focal Points, a little publication with a big message—the mathematics taught at each grade level needs to be focused, provide depth, and explicitly show connections.

In 2010, the Council of Chief State School Officers (CCSSO) presented the Common Core State Standards, which are grade-level specific standards which incorporate ideas from Curriculum Focal Points as well as international curriculum documents. A large majority of U.S. states adopted these as their standards. In less than 25 years, the standards movement transformed the country from having little to no coherent vision on what mathematics should be taught and when, to a more widely shared idea of what students should know and be able to do at each grade level.

In the following sections, we discuss three significant documents critical to your work as a teacher of mathematics.

Principles and Standards for School Mathematics The Principles and Standards for School Mathematics (NCTM, 2000) provides guidance and direction for teachers and other leaders in pre-K–12 mathematics education. This is particularly true in states and regions where they have developed their own standards.

The Six Principles. One of the most important features of Principles and Standards for

School Mathematics is the articulation of six principles fundamental to high-quality mathematics education. These principles must be blended into all programs as building excellence in mathematics education involves much more than simply listing content objectives.

The Equity Principle. The strong message of this principle is there should be high expectations and intentional ways to support all students. All students must have the opportunity and adequate support to learn mathematics regardless of their race, socioeconomic status, gender, culture, language, or disability. This principle is interwoven into all other principles.

The Curriculum Principle. The curriculum should be coherent and built around “big ideas” in the curriculum and in daily classroom instruction. We think of these big ideas as “important” if they help develop other ideas, link one idea to another, or serve to illustrate the discipline of mathematics as a human endeavor. Students must be helped to see that mathematics is an integrated whole that grows and connects across the grades rather than a collection of isolated bits and pieces.

The Teaching Principle. What students learn about mathematics depends almost entirely on the experiences that teachers provide every day in the classroom. To provide high-quality mathematics education, teachers must (1) understand deeply the mathematics content they are teaching; (2) understand how students learn mathematics, including common misconceptions; and (3) select meaningful instructional tasks and generalizable strategies that will enhance learning.

The Learning Principle. This principle is based on two fundamental ideas. First, learning mathematics with understanding is essential. Mathematics today requires not only computational skills but also the ability to think and reason mathematically to solve new problems and learn to respond to novel situations that students will face in the future. Second, students can learn mathematics with understanding. Learning is enhanced in classrooms where students are required to evaluate their own ideas and those of others, make mathematical conjectures and test them, and develop their reasoning and sense-making skills.

The Assessment Principle. Ongoing assessment highlights the most important mathematics concepts for students. Assessment that includes ongoing observation and student interaction

30 Chapter 1 Teaching Mathematics in the 21st Century encourages students to articulate and, thus, clarify their ideas. Feedback from daily assessment helps students establish goals and become more independent learners. By continuously gathering data about students’ understanding of concepts and growth in reasoning, teachers can better make the daily decisions that support student learning. For assessment to be effective, teachers must use a variety of assessment techniques, understand their mathematical goals deeply, and have a research-supported notion of students’ thinking or common misunderstandings.

The Technology Principle. Calculators, computers, and other emerging technologies are essential tools for learning and doing mathematics. Technology permits students to focus on mathematical ideas, to reason, and to solve problems in ways that are often impossible without these tools. Technology enhances the learning of mathematics by allowing for increased exploration, enhanced representation, and communication of ideas.

The Five Content Standards. Principles and Standards includes four grade bands: pre-K–2, 3–5, 6–8, and 9–12. The emphasis on preschool recognizes the need to highlight the critical years before students enter kindergarten. There is a common set of five content standards throughout the grades: • • • • •

Number and Operations Algebra Geometry Measurement Data Analysis and Probability

Each content standard includes a set of goals applicable to all grade bands followed by specific expectations for what students should know at each grade band. Although the same five content standards apply across all grades, you should not infer that each strand has equal weight or emphasis in every grade band. Number and Operations is the most heavily emphasized strand from pre-K through grade 5 and continues to be important in the middle grades, with a lesser emphasis in grades 9–12. This is in contrast to Algebra, which moves from an emphasis related to number and operations in the early grades and builds to a strong focus in the middle and high school grade bands.

The Five Process Standards. The process standards refer to the mathematical processes through which pre-K–12 students acquire and use mathematical knowledge. The process standards should not be regarded as separate content or strands in the mathematics curriculum, rather, they are integral components of all mathematics learning and teaching. The five process standards and ways you can develop these elements in your students can be found in Table 1.1.

Members of NCTM have free online access to the Principles and Standards and nonmembers can sign up for 120 days of free access to the full document on the NCTM website under the tab Standards and Focal Points.

Common Core State Standards As noted earlier, the dialogue on improving mathematics teaching and learning extends beyond mathematics educators. Policymakers and elected officials considered previous NCTM standards documents, international assessments, and research on the best way to prepare students to be “college and career ready.” The National Governors Association Center for Best Practices and the Council of Chief State School Officers (CCSSO) collaborated with other professional groups and entities to develop shared expectations for K–12 students across states, a focused set of mathematics content standards and practices, and efficiency of material and assessment development (Porter, McMaken, Hwang, & Yang, 2011). As a result, they created the Common Core State Standards for Mathematics (CCSS-M) which can be downloaded for free at http://www.corestandards.org/math.

At this time more than 40 states, Washington, D.C., four territories, and Department of Defense Schools have adopted the Common Core State Standards. This represents the largest shift of mathematics content in the United States in more than 100 years. A few states did not opt to participate in the adoption of the standards from the start of their development and at this time others are still deciding their level of participation or reevaluating their own standards against CCSS-M.

The Movement toward Shared Standards 31

Table 1.1

The Five Process Standards From Principles and Standards for School Mathematics

Process Standard

How Can You Develop These Processes in Your Students?

Problem Solving

●● Start instruction with a problem to solve—as problem solving is the vehicle for developing mathematical ideas. ●● Select meaningful mathematical tasks. ●● Set problems in a situation to which students can relate. ●● Use a variety of strategies to solve problems. ●● Have students self-assess their understanding of the problem and their strategy use.

Reasoning and Proof

●● Have students consider evidence of why something is true or not. ●● Create opportunities for students to evaluate conjectures—do they hold true? ●● Encourage students to use logical reasoning to see if something always works or their answers make sense. ●● Demonstrate a variety of ways for students to justify their thinking through finding examples and

counterexamples to use in a logical argument. ●● Invite students to talk about, write about, describe, and explain their mathematical ideas as a way to

Communication

examine their thinking. ●● Give students opportunities to share ideas so that others can understand and actively discuss their

reasoning. ●● Share examples of student work, so students can compare and assess others’ thinking. ●● Present precise mathematical language and notation so that the word usage and definitions can act as a

foundation for students’ future learning. ●● Emphasize how mathematical ideas explicitly connect to students’ prior mathematical knowledge and future

Connections

learning. ●● Assist students in developing the relationships between the mathematics being learned and real world

contexts and in other subject areas. ●● Encourage students to use multiple representations to explore relationships and communicate their thinking.

Representation

●● Create opportunities for students to move from one representation of an mathematical concept or idea to

another to add depth of understanding. ●● Provide problems where students can use mathematical models to clarify or represent a situation.

Source: 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.

The document articulates an overview of critical areas for each grade from K–8 to provide a coherent curriculum built around big ideas. These larger groups of related standards are called domains, and there are eleven that relate to grades K–8 (see Figure 1.1).

The Common Core State Standards go beyond specifying mathematics content expectations to also include Standards for Mathematical Practice. These are “‘processes and proficiencies’ with longstanding importance in mathematics education” (CCSSO, 2010, p. 6) that are based on the underlying frameworks of the NCTM process standards and the components of mathematical proficiency identified by NRC in their important document Adding It Up (National Research Council, 2001). Teachers must develop these mathematical practices in all students (CCSSO, 2010, pp. 7–8) as described briefly in Table 1.2. A more detailed description of the Standards for Mathematical Practice can be found in Appendix B. Kindergarten

Grade 1

Grade 2

Grade 3

Grade 4

Grade 5

Grade 6

Grade 7

Grade 8

Counting and Cardinality Operations and Algebraic Thinking

Expressions and Equations

Number and ­Operations in Base Ten

The Number System

Measurement and Data

Statistics and ­Probability Geometry

Number and ­Operations—Fractions

Figure 1.1 Common Core State Standards domains by grade level.

Rations and Proportional Relationships

Functions

32 Chapter 1 Teaching Mathematics in the 21st Century Table 1.2

The Standards for Mathematical Practice From the Common Core State Standards

Mathematical Practice

K–8 Students Should Be Able to:

Make sense of problems and persevere in solving them.

●● Explain what the problem is asking. ●● Describe possible approaches to a solution. ●● Consider similar problems to gain insights. ●● Use concrete objects or drawings to think about and solve problems. ●● Monitor and evaluate their progress and change strategies if needed. ●● Check their answers using a different method.

Reason abstractly and quantitatively.

●● Explain the relationship between quantities in problem situations. ●● Represent situations using symbols (e.g., writing expressions or equations). ●● Create representations that fit the problem. ●● Use flexibly the different properties of operations and objects.

Construct viable arguments and critique the reasoning of others.

●● Understand and use assumptions, definitions, and previous results to explain or justify solutions. ●● Make conjectures by building a logical set of statements. ●● Analyze situations and use examples and counterexamples. ●● Justify conclusions in a way that is understandable to teachers and peers. ●● Compare two possible arguments for strengths and weaknesses to enhance the final argument.

Model with mathematics.

●● Apply mathematics to solve problems in everyday life. ●● Make assumptions and approximations to simplify a problem. ●● Identify important quantities and use tools to connect their relationships. ●● Reflect on the reasonableness of their answer based on the context of the problem.

Use appropriate tools strategically.

●● Consider a variety of tools and choose the appropriate tool (e.g., manipulative, ruler, technology) to

support their problem solving. ●● Use estimation to detect possible errors and establish a reasonable range of answers. ●● Use technology to help visualize, explore, and compare information.

Attend to precision.

●● Communicate precisely using clear definitions and appropriate mathematical language. ●● State accurately the meanings of symbols. ●● Specify appropriate units of measure and labels of axes. ●● Use a level of precision suitable for the problem context.

Look for and make use of structure.

●● Identify and explain mathematical patterns or structures. ●● Shift viewpoints and see things as single objects or as comprised of multiple objects. ●● Explain why and when properties of operations are true in a context.

Look for and express regularity in repeated reasoning.

●● Notice if patterns in calculations are repeated and use that information to solve problems. ●● Use and justify the use of general methods or shortcuts by identifying generalizations. ●● Self-assess as they work to see whether a strategy makes sense, checking for reasonableness prior to

finalizing their answer.

Source: Based on Council of Chief State School Officers. (2010). Common Core State Standards. Copyright © 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

Watch this video (https://www.youtube.com/watch?v=GlJ44te7jrw) to get a good overview of the CCSS-M from teachers and authors. Additionally, the Illustrative Mathematics Project website provides tools and support for the Common Core State Standards. It includes multiple ways to look at the standards across grades and domains as well as provides task and problems that will illustrate individual standards.

Learning Progressions. The Common Core State Standards were developed with strong consideration given to building coherence through the research on what is known about the development of students’ understanding of mathematics over time (Cobb & Jackson, 2011). The selection of topics at particular grades reflects not only rigorous mathematics but also what is known from current research and practice about learning progressions which are sometimes referred to as learning trajectories (Clements & Sarama, 2014; Confrey, Maloney, & Corley, 2014; Daro, Mosher, & Corcoran, 2011; Maloney, Confrey, Ng & Nickell, 2014). It is these learning progressions that can help teachers know what came before as well as what to expect next as students reach key points along the (Corcoran, Mosher, & Rogat, 2009) road to learning mathematics concepts. These progressions identify the interim goals students should reach on the pathway to desired learning targets (Daro, Mosher, & Corcoran, 2011). Although these paths are not identical for all students, they can inform the order of instructional experiences which will support movement toward understanding and application of mathematics concepts. There is a website for the “Progressions Documents for the

The Movement toward Shared Standards 33

Common Core Math Standards” where progressions for the domains in the Common Core State Standards can be found.

Assessments. The initial idea was to have new summative assessments developed through two major consortia, Partnership for Assessment of Readiness for College and Careers (PARCC) and Smarter Balanced Assessment Consortium, are developing assessments which will align to the Common Core State Standards. These assessments will focus on both the grade-level content standards and the standards for mathematical practice. This process is being put into place to eliminate the need for each state to develop unique assessments for the standards, a problem that has existed since the beginning of the standards era. Yet, there are states developing their own approaches to end-of-year assessment as well.

Principles to Actions NCTM has developed a publication that capitalizes on the timing of the adoption of the Common Core State Standards to explore the specific learning conditions, school structures, and teaching practices which will be important for a high quality education for all students. The book uses detailed classroom stories and student work samples to illustrate the careful, reflective work required of effective teachers of mathematics through 6 guiding principles (see Table 1.3). A series of presentations (webcasts), led by the authors of the publication, explore several of the guiding ­principles and are available on the Principles to Actions portion of the NCTM’s website.

Table 1.3

The six GUIDING Principles from the Principles to Actions

Guiding Principle

Suggestions for Classroom Actions That Align with the Principles

Teaching and Learning

●● Select focused mathematics goals. ●● Use meaningful instructional tasks that develop reasoning, sense making, and problem solving strategies. ●● Present and encourage a variety of mathematical representations that connect the same ideas or concepts. ●● Facilitate student discussions and conversations about important mathematical ideas. ●● Ask essential questions that are planned to be a catalyst for deeper levels of thinking. ●● Use a strong foundation of conceptual understanding as a foundation for procedural fluency. ●● Encourage productive struggle—as it is a way to deepen understanding and move toward student application

of their learning. ●● Generate ways for students to provide evidence of their thinking through discussions, illustrations, and

written responses.

Access and Equity

●● Establish high expectations for all students. ●● Provide supports targeted to student needs (equity not equality). ●● Provide instructional opportunities for students to demonstrate their competence in different ways—creating

tasks with easy entry points for students who struggle and extension options for those who finish quickly. ●● Identify obstacles to students’ success and find ways to bridge or eliminate those barriers. ●● Develop all students’ confidence that they can do mathematics. ●● Enhance the learning of all by celebrating students’ diversity.

Curriculum

●● Build connections across mathematics topics to capitalize on broad themes and big ideas. ●● Look for both horizontal and vertical alignment to build coherence. ●● Avoid thinking of a curriculum as a checklist or disconnected set of daily lessons.

Tools and Technology

●● Include an array of technological tools and manipulatives to support the exploration of mathematical

concepts, structures, and relationships. ●● Think beyond computation when considering the integration of technology. ●● Explore connections to how technology use for problem solving links to career readiness.

Assessment

●● Incorporate a continuous assessment plan to follow how students are performing and how instruction can be

modified and thereby improved. ●● Move beyond test results that just look at overall increases and decreases to pinpoint specific student needs. ●● Consider the use of multiple assessments to capture a variety of student performance. ●● Encourage students to self-assess sometimes by evaluating the work of others to enhance their own

performance. ●● Teach students how to check their work.

Professionalism

●● Develop a long-term plan for building your expertise. ●● Build collaborations that will enhance the work of the group of collaborators as you enhance the performance

of the students in the school. ●● Take advantage of all coaching, mentoring and professional development opportunities and be a life-long learner. ●● Structure in time to reflect on and analyze your instructional practices.

34 Chapter 1 Teaching Mathematics in the 21st Century

Pause & Reflect

Take a moment now to select one or two of the six guiding principles that seem especially significant to you and are areas in which you wish to develop more expertise. Why do you think these are the most important to your teaching? ●

Complete Self-Check 1.2: The Movement toward Shared Standards

An Invitation to Learn and Grow The mathematics education described in this book may not be the same as the mathematics content and the mathematics teaching you experienced in grades K–8. As a practicing or prospective teacher facing the challenge of teaching mathematics from a problem solving approach, this book may require you to confront some of your personal beliefs—beliefs about what it means to do mathematics, how one goes about learning mathematics, how to teach mathematics, and what it means to assess mathematics. Success in mathematics isn’t merely about speed or the notion that there is “one right answer.” Thinking and talking about mathematics as a means to sense making is a strategy that will serve us well in becoming a society where all citizens are confident in their ability to do math (https://www.youtube.com/watch?v=0gW9g8Ofi8A).

Becoming a Teacher of Mathematics This book and this course of study are critical to your professional teaching career. The mathematics education course you are taking now as a pre-service teacher or the professional development you are experiencing as an in-service teacher will be the foundation for much of the mathematics instruction you do in your classroom for the next decade. The authors of this book take that seriously, as we know you do. Therefore, this section lists and describes the characteristics, habits of thought, skills, and dispositions you will need to succeed as a teacher of mathematics.

Knowledge of Mathematics. You will need to have a profound, flexible, and adaptive knowledge of mathematics content (Ma, 1999). This statement is not intended to scare you if you feel that mathematics is not your strong suit, but it is meant to help you prepare for a serious semester of learning about mathematics and how to teach it. The “school effects” for mathematics are great, meaning that unlike other subject areas, where students have frequent interactions with their family or others outside of school on topics such as reading books, exploring nature, or discussing current events, in the area of mathematics what we do in school is often “it” for many students. This adds to the earnestness of your responsibility, because a student’s learning for the year in mathematics will likely come only from you. If you are not sure of a fractional concept or other aspect of mathematics content knowledge, now is the time to make changes in your depth of understanding and flexibility with mathematical ideas to best prepare for your role as an instructional leader. This book and your professor or instructor will help you in that process.

Persistence. You need the ability to stave off frustration and demonstrate persistence.

Dweck (2007) has described the brain as similar to a muscle—one that can be strengthened with a good workout! As you move through this book and work the problems yourself, you will learn methods and strategies that will help you anticipate the barriers to students’ learning and identify strategies to get them past these stumbling blocks. It is likely that what works for you as a learner will work for your students. As you conduct this mental “workout,” if you ponder, struggle, talk about your thinking, and reflect on how these new ideas fit or don’t fit with your prior knowledge, then you will enhance your repertoire as a teacher. Remember as you model these characteristics for your students, they too will begin to value perseverance more than speed. In fact, Einstein did not describe himself as intelligent—instead he suggested he was just someone who continued to work on problems longer than others. Creating opportunities for your students to productively struggle is part of the learning process (Stigler & Hiebert, 2009).

Another random document with no related content on Scribd:

DANCE ON STILTS AT THE GIRLS’ UNYAGO, NIUCHI

Newala, too, suffers from the distance of its water-supply—at least the Newala of to-day does; there was once another Newala in a lovely valley at the foot of the plateau. I visited it and found scarcely a trace of houses, only a Christian cemetery, with the graves of several missionaries and their converts, remaining as a monument of its former glories. But the surroundings are wonderfully beautiful. A thick grove of splendid mango-trees closes in the weather-worn crosses and headstones; behind them, combining the useful and the agreeable, is a whole plantation of lemon-trees covered with ripe fruit; not the small African kind, but a much larger and also juicier imported variety, which drops into the hands of the passing traveller, without calling for any exertion on his part. Old Newala is now under the jurisdiction of the native pastor, Daudi, at Chingulungulu, who, as I am on very friendly terms with him, allows me, as a matter of course, the use of this lemon-grove during my stay at Newala.

FEET MUTILATED BY THE RAVAGES OF THE “JIGGER” (Sarcopsylla penetrans)

The water-supply of New Newala is in the bottom of the valley, some 1,600 feet lower down. The way is not only long and fatiguing, but the water, when we get it, is thoroughly bad. We are suffering not only from this, but from the fact that the arrangements at Newala are nothing short of luxurious. We have a separate kitchen—a hut built against the boma palisade on the right of the baraza, the interior of which is not visible from our usual position. Our two cooks were not long in finding this out, and they consequently do—or rather neglect to do—what they please. In any case they do not seem to be very particular about the boiling of our drinking-water—at least I can attribute to no other cause certain attacks of a dysenteric nature, from which both Knudsen and I have suffered for some time. If a man like Omari has to be left unwatched for a moment, he is capable of anything. Besides this complaint, we are inconvenienced by the state of our nails, which have become as hard as glass, and crack on the slightest provocation, and I have the additional infliction of pimples all over me. As if all this were not enough, we have also, for the last week been waging war against the jigger, who has found his Eldorado in the hot sand of the Makonde plateau. Our men are seen all day long—whenever their chronic colds and the dysentery likewise raging among them permit—occupied in removing this scourge of Africa from their feet and trying to prevent the disastrous consequences of its presence. It is quite common to see natives of this place with one or two toes missing; many have lost all their toes,

or even the whole front part of the foot, so that a well-formed leg ends in a shapeless stump. These ravages are caused by the female of Sarcopsylla penetrans, which bores its way under the skin and there develops an egg-sac the size of a pea. In all books on the subject, it is stated that one’s attention is called to the presence of this parasite by an intolerable itching. This agrees very well with my experience, so far as the softer parts of the sole, the spaces between and under the toes, and the side of the foot are concerned, but if the creature penetrates through the harder parts of the heel or ball of the foot, it may escape even the most careful search till it has reached maturity.

Then there is no time to be lost, if the horrible ulceration, of which we see cases by the dozen every day, is to be prevented. It is much easier, by the way, to discover the insect on the white skin of a European than on that of a native, on which the dark speck scarcely shows. The four or five jiggers which, in spite of the fact that I constantly wore high laced boots, chose my feet to settle in, were taken out for me by the all-accomplished Knudsen, after which I thought it advisable to wash out the cavities with corrosive sublimate. The natives have a different sort of disinfectant—they fill the hole with scraped roots. In a tiny Makua village on the slope of the plateau south of Newala, we saw an old woman who had filled all the spaces under her toe-nails with powdered roots by way of prophylactic treatment. What will be the result, if any, who can say?

The rest of the many trifling ills which trouble our existence are really more comic than serious. In the absence of anything else to smoke, Knudsen and I at last opened a box of cigars procured from the Indian store-keeper at Lindi, and tried them, with the most distressing results. Whether they contain opium or some other narcotic, neither of us can say, but after the tenth puff we were both “off,” three-quarters stupefied and unspeakably wretched. Slowly we recovered—and what happened next? Half-an-hour later we were once more smoking these poisonous concoctions—so insatiable is the craving for tobacco in the tropics.

Even my present attacks of fever scarcely deserve to be taken seriously. I have had no less than three here at Newala, all of which have run their course in an incredibly short time. In the early afternoon, I am busy with my old natives, asking questions and making notes. The strong midday coffee has stimulated my spirits to

an extraordinary degree, the brain is active and vigorous, and work progresses rapidly, while a pleasant warmth pervades the whole body. Suddenly this gives place to a violent chill, forcing me to put on my overcoat, though it is only half-past three and the afternoon sun is at its hottest. Now the brain no longer works with such acuteness and logical precision; more especially does it fail me in trying to establish the syntax of the difficult Makua language on which I have ventured, as if I had not enough to do without it. Under the circumstances it seems advisable to take my temperature, and I do so, to save trouble, without leaving my seat, and while going on with my work. On examination, I find it to be 101·48°. My tutors are abruptly dismissed and my bed set up in the baraza; a few minutes later I am in it and treating myself internally with hot water and lemon-juice.

Three hours later, the thermometer marks nearly 104°, and I make them carry me back into the tent, bed and all, as I am now perspiring heavily, and exposure to the cold wind just beginning to blow might mean a fatal chill. I lie still for a little while, and then find, to my great relief, that the temperature is not rising, but rather falling. This is about 7.30 p.m. At 8 p.m. I find, to my unbounded astonishment, that it has fallen below 98·6°, and I feel perfectly well. I read for an hour or two, and could very well enjoy a smoke, if I had the wherewithal—Indian cigars being out of the question.

Having no medical training, I am at a loss to account for this state of things. It is impossible that these transitory attacks of high fever should be malarial; it seems more probable that they are due to a kind of sunstroke. On consulting my note-book, I become more and more inclined to think this is the case, for these attacks regularly follow extreme fatigue and long exposure to strong sunshine. They at least have the advantage of being only short interruptions to my work, as on the following morning I am always quite fresh and fit.

My treasure of a cook is suffering from an enormous hydrocele which makes it difficult for him to get up, and Moritz is obliged to keep in the dark on account of his inflamed eyes. Knudsen’s cook, a raw boy from somewhere in the bush, knows still less of cooking than Omari; consequently Nils Knudsen himself has been promoted to the vacant post. Finding that we had come to the end of our supplies, he began by sending to Chingulungulu for the four sucking-pigs which we had

bought from Matola and temporarily left in his charge; and when they came up, neatly packed in a large crate, he callously slaughtered the biggest of them. The first joint we were thoughtless enough to entrust for roasting to Knudsen’s mshenzi cook, and it was consequently uneatable; but we made the rest of the animal into a jelly which we ate with great relish after weeks of underfeeding, consuming incredible helpings of it at both midday and evening meals. The only drawback is a certain want of variety in the tinned vegetables. Dr. Jäger, to whom the Geographical Commission entrusted the provisioning of the expeditions—mine as well as his own—because he had more time on his hands than the rest of us, seems to have laid in a huge stock of Teltow turnips,[46] an article of food which is all very well for occasional use, but which quickly palls when set before one every day; and we seem to have no other tins left. There is no help for it—we must put up with the turnips; but I am certain that, once I am home again, I shall not touch them for ten years to come.

Amid all these minor evils, which, after all, go to make up the genuine flavour of Africa, there is at least one cheering touch: Knudsen has, with the dexterity of a skilled mechanic, repaired my 9 × 12 cm. camera, at least so far that I can use it with a little care.

How, in the absence of finger-nails, he was able to accomplish such a ticklish piece of work, having no tool but a clumsy screw-driver for taking to pieces and putting together again the complicated mechanism of the instantaneous shutter, is still a mystery to me; but he did it successfully. The loss of his finger-nails shows him in a light contrasting curiously enough with the intelligence evinced by the above operation; though, after all, it is scarcely surprising after his ten years’ residence in the bush. One day, at Lindi, he had occasion to wash a dog, which must have been in need of very thorough cleansing, for the bottle handed to our friend for the purpose had an extremely strong smell. Having performed his task in the most conscientious manner, he perceived with some surprise that the dog did not appear much the better for it, and was further surprised by finding his own nails ulcerating away in the course of the next few days. “How was I to know that carbolic acid has to be diluted?” he mutters indignantly, from time to time, with a troubled gaze at his mutilated finger-tips.

Since we came to Newala we have been making excursions in all directions through the surrounding country, in accordance with old habit, and also because the akida Sefu did not get together the tribal elders from whom I wanted information so speedily as he had promised. There is, however, no harm done, as, even if seen only from the outside, the country and people are interesting enough.

The Makonde plateau is like a large rectangular table rounded off at the corners. Measured from the Indian Ocean to Newala, it is about seventy-five miles long, and between the Rovuma and the Lukuledi it averages fifty miles in breadth, so that its superficial area is about two-thirds of that of the kingdom of Saxony. The surface, however, is not level, but uniformly inclined from its south-western edge to the ocean. From the upper edge, on which Newala lies, the eye ranges for many miles east and north-east, without encountering any obstacle, over the Makonde bush. It is a green sea, from which here and there thick clouds of smoke rise, to show that it, too, is inhabited by men who carry on their tillage like so many other primitive peoples, by cutting down and burning the bush, and manuring with the ashes. Even in the radiant light of a tropical day such a fire is a grand sight.

Much less effective is the impression produced just now by the great western plain as seen from the edge of the plateau. As often as time permits, I stroll along this edge, sometimes in one direction, sometimes in another, in the hope of finding the air clear enough to let me enjoy the view; but I have always been disappointed.

Wherever one looks, clouds of smoke rise from the burning bush, and the air is full of smoke and vapour. It is a pity, for under more favourable circumstances the panorama of the whole country up to the distant Majeje hills must be truly magnificent. It is of little use taking photographs now, and an outline sketch gives a very poor idea of the scenery. In one of these excursions I went out of my way to make a personal attempt on the Makonde bush. The present edge of the plateau is the result of a far-reaching process of destruction through erosion and denudation. The Makonde strata are everywhere cut into by ravines, which, though short, are hundreds of yards in depth. In consequence of the loose stratification of these beds, not only are the walls of these ravines nearly vertical, but their upper end is closed by an equally steep escarpment, so that the

western edge of the Makonde plateau is hemmed in by a series of deep, basin-like valleys. In order to get from one side of such a ravine to the other, I cut my way through the bush with a dozen of my men.

It was a very open part, with more grass than scrub, but even so the short stretch of less than two hundred yards was very hard work; at the end of it the men’s calicoes were in rags and they themselves bleeding from hundreds of scratches, while even our strong khaki suits had not escaped scatheless.

NATIVE PATH THROUGH THE MAKONDE BUSH, NEAR MAHUTA

I see increasing reason to believe that the view formed some time back as to the origin of the Makonde bush is the correct one. I have no doubt that it is not a natural product, but the result of human occupation. Those parts of the high country where man—as a very slight amount of practice enables the eye to perceive at once—has not yet penetrated with axe and hoe, are still occupied by a splendid timber forest quite able to sustain a comparison with our mixed forests in Germany. But wherever man has once built his hut or tilled his field, this horrible bush springs up. Every phase of this process

may be seen in the course of a couple of hours’ walk along the main road. From the bush to right or left, one hears the sound of the axe— not from one spot only, but from several directions at once. A few steps further on, we can see what is taking place. The brush has been cut down and piled up in heaps to the height of a yard or more, between which the trunks of the large trees stand up like the last pillars of a magnificent ruined building. These, too, present a melancholy spectacle: the destructive Makonde have ringed them— cut a broad strip of bark all round to ensure their dying off—and also piled up pyramids of brush round them. Father and son, mother and son-in-law, are chopping away perseveringly in the background—too busy, almost, to look round at the white stranger, who usually excites so much interest. If you pass by the same place a week later, the piles of brushwood have disappeared and a thick layer of ashes has taken the place of the green forest. The large trees stretch their smouldering trunks and branches in dumb accusation to heaven—if they have not already fallen and been more or less reduced to ashes, perhaps only showing as a white stripe on the dark ground.

This work of destruction is carried out by the Makonde alike on the virgin forest and on the bush which has sprung up on sites already cultivated and deserted. In the second case they are saved the trouble of burning the large trees, these being entirely absent in the secondary bush.

After burning this piece of forest ground and loosening it with the hoe, the native sows his corn and plants his vegetables. All over the country, he goes in for bed-culture, which requires, and, in fact, receives, the most careful attention. Weeds are nowhere tolerated in the south of German East Africa. The crops may fail on the plains, where droughts are frequent, but never on the plateau with its abundant rains and heavy dews. Its fortunate inhabitants even have the satisfaction of seeing the proud Wayao and Wamakua working for them as labourers, driven by hunger to serve where they were accustomed to rule.

But the light, sandy soil is soon exhausted, and would yield no harvest the second year if cultivated twice running. This fact has been familiar to the native for ages; consequently he provides in time, and, while his crop is growing, prepares the next plot with axe and firebrand. Next year he plants this with his various crops and

lets the first piece lie fallow. For a short time it remains waste and desolate; then nature steps in to repair the destruction wrought by man; a thousand new growths spring out of the exhausted soil, and even the old stumps put forth fresh shoots. Next year the new growth is up to one’s knees, and in a few years more it is that terrible, impenetrable bush, which maintains its position till the black occupier of the land has made the round of all the available sites and come back to his starting point.

The Makonde are, body and soul, so to speak, one with this bush.

According to my Yao informants, indeed, their name means nothing else but “bush people.” Their own tradition says that they have been settled up here for a very long time, but to my surprise they laid great stress on an original immigration. Their old homes were in the south-east, near Mikindani and the mouth of the Rovuma, whence their peaceful forefathers were driven by the continual raids of the Sakalavas from Madagascar and the warlike Shirazis[47] of the coast, to take refuge on the almost inaccessible plateau. I have studied African ethnology for twenty years, but the fact that changes of population in this apparently quiet and peaceable corner of the earth could have been occasioned by outside enterprises taking place on the high seas, was completely new to me. It is, no doubt, however, correct.

The charming tribal legend of the Makonde—besides informing us of other interesting matters—explains why they have to live in the thickest of the bush and a long way from the edge of the plateau, instead of making their permanent homes beside the purling brooks and springs of the low country.

“The place where the tribe originated is Mahuta, on the southern side of the plateau towards the Rovuma, where of old time there was nothing but thick bush. Out of this bush came a man who never washed himself or shaved his head, and who ate and drank but little.

He went out and made a human figure from the wood of a tree growing in the open country, which he took home to his abode in the bush and there set it upright. In the night this image came to life and was a woman. The man and woman went down together to the Rovuma to wash themselves. Here the woman gave birth to a stillborn child. They left that place and passed over the high land into the valley of the Mbemkuru, where the woman had another child, which

was also born dead. Then they returned to the high bush country of Mahuta, where the third child was born, which lived and grew up. In course of time, the couple had many more children, and called themselves Wamatanda. These were the ancestral stock of the Makonde, also called Wamakonde,[48] i.e., aborigines. Their forefather, the man from the bush, gave his children the command to bury their dead upright, in memory of the mother of their race who was cut out of wood and awoke to life when standing upright. He also warned them against settling in the valleys and near large streams, for sickness and death dwelt there. They were to make it a rule to have their huts at least an hour’s walk from the nearest wateringplace; then their children would thrive and escape illness.”

The explanation of the name Makonde given by my informants is somewhat different from that contained in the above legend, which I extract from a little book (small, but packed with information), by Pater Adams, entitled Lindi und sein Hinterland. Otherwise, my results agree exactly with the statements of the legend. Washing?

Hapana—there is no such thing. Why should they do so? As it is, the supply of water scarcely suffices for cooking and drinking; other people do not wash, so why should the Makonde distinguish himself by such needless eccentricity? As for shaving the head, the short, woolly crop scarcely needs it,[49] so the second ancestral precept is likewise easy enough to follow. Beyond this, however, there is nothing ridiculous in the ancestor’s advice. I have obtained from various local artists a fairly large number of figures carved in wood, ranging from fifteen to twenty-three inches in height, and representing women belonging to the great group of the Mavia, Makonde, and Matambwe tribes. The carving is remarkably well done and renders the female type with great accuracy, especially the keloid ornamentation, to be described later on. As to the object and meaning of their works the sculptors either could or (more probably) would tell me nothing, and I was forced to content myself with the scanty information vouchsafed by one man, who said that the figures were merely intended to represent the nembo—the artificial deformations of pelele, ear-discs, and keloids. The legend recorded

by Pater Adams places these figures in a new light. They must surely be more than mere dolls; and we may even venture to assume that they are—though the majority of present-day Makonde are probably unaware of the fact—representations of the tribal ancestress.

The references in the legend to the descent from Mahuta to the Rovuma, and to a journey across the highlands into the Mbekuru valley, undoubtedly indicate the previous history of the tribe, the travels of the ancestral pair typifying the migrations of their descendants. The descent to the neighbouring Rovuma valley, with its extraordinary fertility and great abundance of game, is intelligible at a glance—but the crossing of the Lukuledi depression, the ascent to the Rondo Plateau and the descent to the Mbemkuru, also lie within the bounds of probability, for all these districts have exactly the same character as the extreme south. Now, however, comes a point of especial interest for our bacteriological age. The primitive Makonde did not enjoy their lives in the marshy river-valleys.

Disease raged among them, and many died. It was only after they had returned to their original home near Mahuta, that the health conditions of these people improved. We are very apt to think of the African as a stupid person whose ignorance of nature is only equalled by his fear of it, and who looks on all mishaps as caused by evil spirits and malignant natural powers. It is much more correct to assume in this case that the people very early learnt to distinguish districts infested with malaria from those where it is absent.

This knowledge is crystallized in the ancestral warning against settling in the valleys and near the great waters, the dwelling-places of disease and death. At the same time, for security against the hostile Mavia south of the Rovuma, it was enacted that every settlement must be not less than a certain distance from the southern edge of the plateau. Such in fact is their mode of life at the present day. It is not such a bad one, and certainly they are both safer and more comfortable than the Makua, the recent intruders from the south, who have made USUAL METHOD OF good their footing on the western edge of the CLOSING HUT-DOOR plateau, extending over a fairly wide belt of country. Neither Makua nor Makonde show in their dwellings anything of the size and comeliness of the Yao houses in the plain, especially at Masasi, Chingulungulu and Zuza’s. Jumbe Chauro, a Makonde hamlet not far from Newala, on the road to Mahuta, is the

most important settlement of the tribe I have yet seen, and has fairly spacious huts. But how slovenly is their construction compared with the palatial residences of the elephant-hunters living in the plain.

The roofs are still more untidy than in the general run of huts during the dry season, the walls show here and there the scanty beginnings or the lamentable remains of the mud plastering, and the interior is a veritable dog-kennel; dirt, dust and disorder everywhere. A few huts only show any attempt at division into rooms, and this consists merely of very roughly-made bamboo partitions. In one point alone have I noticed any indication of progress—in the method of fastening the door. Houses all over the south are secured in a simple but ingenious manner. The door consists of a set of stout pieces of wood or bamboo, tied with bark-string to two cross-pieces, and moving in two grooves round one of the door-posts, so as to open inwards. If the owner wishes to leave home, he takes two logs as thick as a man’s upper arm and about a yard long. One of these is placed obliquely against the middle of the door from the inside, so as to form an angle of from 60° to 75° with the ground. He then places the second piece horizontally across the first, pressing it downward with all his might.

It is kept in place by two strong posts planted in the ground a few inches inside the door. This fastening is absolutely safe, but of course cannot be applied to both doors at once, otherwise how could the owner leave or enter his house? I have not yet succeeded in finding out how the back door is fastened.

MAKONDE LOCK AND KEY AT JUMBE CHAURO

This is the general way of closing a house. The Makonde at Jumbe Chauro, however, have a much more complicated, solid and original one. Here, too, the door is as already described, except that there is only one post on the inside, standing by itself about six inches from one side of the doorway. Opposite this post is a hole in the wall just large enough to admit a man’s arm. The door is closed inside by a large wooden bolt passing through a hole in this post and pressing with its free end against the door. The other end has three holes into which fit three pegs running in vertical grooves inside the post. The door is opened with a wooden key about a foot long, somewhat curved and sloped off at the butt; the other end has three pegs corresponding to the holes, in the bolt, so that, when it is thrust through the hole in the wall and inserted into the rectangular opening in the post, the pegs can be lifted and the bolt drawn out.[50]

MODE OF INSERTING THE KEY

With no small pride first one householder and then a second showed me on the spot the action of this greatest invention of the Makonde Highlands. To both with an admiring exclamation of “Vizuri sana!” (“Very fine!”). I expressed the wish to take back these marvels with me to Ulaya, to show the Wazungu what clever fellows the Makonde are. Scarcely five minutes after my return to camp at Newala, the two men came up sweating under the weight of two heavy logs which they laid down at my feet, handing over at the same time the keys of the fallen fortress. Arguing, logically enough, that if the key was wanted, the lock would be wanted with it, they had taken their axes and chopped down the posts—as it never occurred to them to dig them out of the ground and so bring them intact. Thus I have