BOBSON WONG is a three-time recipient of the Math for America Master Teacher Fellowship, a New York State Master Teacher, and a member of the Advisory Council of the National Museum of Mathematics. He has served on New York State's Common Core Mathematics Standards Review Committee, the United Federation of Teachers' Common Core Standards Task Force, and as an Educational Specialist for the New York State Education Department.

LARISA BUKALOV is a four-time recipient of the Math for America Master Teacher fellowship and a recipient of Queens College's Excellence in Mathematics Award for promoting mathematics teaching as a profession. She has taught all levels of math, coached the school's math team, and created a math research program for students. As part of her work with Math for America, Larisa has run several professional development sessions for teachers.

LARRY FERLAZZO teaches English, Social Studies, and International Baccalaureate classes to English Language Learners and others at Luther Burbank High School in Sacramento, California. He is the author and co-author of nine books, including The ELL Teacher's Toolbox, and writes a weekly teacher advice column for Education Week Teacher. He is the recipient of the Ford Foundation's Leadership for a Changing World Award and winner of the International Reading Association Award for Technology and Reading.

KATIE HULL SYPNIESKI has taught English language learners and others at the secondary level for over twenty years. She teaches middle school English Language Arts and Social Studies at Fern Bacon Middle School in Sacramento, California, and leads professional development for educators as a consultant with the Area 3 Writing Project at the University of California, Davis. She is co-author of several books including The ELL Teacher's Toolbox.

List of Tables xix

About the Authors xxi

About the Editors xxiii

Acknowledgments xxv

Letter from the Editors xxvii

Introduction 1

Our Beliefs about Teaching Math 2

Structure of This Book 3

Why Good Math Teaching Matters 4

I Basic Strategies 5

1. Motivating Students 7

What is It? 7

Why We Like It 8

Supporting Research 8

Common Core Connections 9

Application 10

Nurturing Student Confidence 10

Motivating Through Math 11

Rewards 14

Motivating Through Popular Culture 15

Motivating English Language Learners and Students with Learning Differences 16

Student Handouts and Examples 18

What Could Go Wrong 18

Using Fear to Motivate 18

Stereotype Threat 19

“Why Do We Need to Know This?” 19

Misreading Students 20

Limitations to Motivation 21

Technology Connections 21

Figures 22

Figure 1.1 Pattern Blocks 22

Figure 1.2 Rotational Symmetry 23

Figure 1.3 Exponential Growth 24

Figure 1.4 Identify a Void 26

2. Culturally Responsive Teaching 27

What is It? 27

Why We Like It 28

Supporting Research 28

Common Core Connections 29

Application 30

Self-Reflection 30

Building a Collaborative Learning Partnership 32

What Could Go Wrong 36

“Color-Blind” Teaching 36

Good Intentions 37

Finding the Right Time or Place 38

Technology Connections 38

3. Teaching Math as a Language 41

What is it? 41

Why We Like It 41

Supporting Research 42

Common Core Connections 42

Application 42

Eliciting the Need for Mathematical Language 42

Introducing Symbols and Terms 43

Translating Between Symbols and Words 45

Making Connections Between Math and English 46

Examples of Confusing Mathematical Language 46

Encouraging Mathematical Precision 48

Vocabulary Charts and Flash Cards 49

Visual and Verbal Aids 51

Word Walls and Anchor Charts 52

Student Handouts and Examples 53

What Could Go Wrong 53

Not Treating Math as a Language 53

Math as a “Bag of Tricks” 54

Technology Connections 55

Figures 57

Figure 3.1 Concept Attainment 57

Figure 3.2 Words and Symbols Chart 58

Figure 3.3 Why the Word “Height” is Confusing 58

Figure 3.4 Draw a Picture 59

Figure 3.5 Functions Anchor Chart 60

Figure 3.6 Polynomials Anchor Chart 61

Figure 3.7 Why the Formula a² + b² = c² is Confusing 61

4. Promoting Mathematical Communication 63

What is It? 63

Why We Like It 63

Supporting Research 64

Common Core Connections 64

Application 64

Open-Ended Questions 64

Guiding Students in Conversation 71

Four-Step Thinking Process 74

Mathematical Writing 79

Differentiating for ELLs and Students with Learning Differences 87

What Could Go Wrong 87

Dealing with Student Mistakes 87

Dealing with Teacher Mistakes 88

Problems in Discourse 88

Finding the Time 89

Student Handouts and Examples 89

Technology Connections 89

Attribution 90

Figures 91

Figure 4.1 Algebra Tiles Activity 91

Figure 4.2 Which One Doesn’t Belong? 92

Figure 4.3 Error Analysis 93

Figure 4.4 Lesson Summary 95

5. Making Mathematical Connections 97

What is It? 97

Why We Like It 97

Supporting Research 98

Common Core Connections 98

Application 98

Equivalence 99

Proportionality 101

Functions 102

Variability 104

Differentiating for ELLs and Students with Learning Differences 107

Student Handouts and Examples 108

What Could Go Wrong 108

Technology Connections 109

Figures 111

Figure 5.1 Addition and Subtraction of Polynomials 111

Figure 5.2 Multiplication with the Area Model 112

Figure 5.3 Division with the Area Model 114

Figure 5.4 Completing the Square 115

Figure 5.5 Determining the Center and Radius of a Circle 115

Figure 5.6 Why (a + b)² ≠ a² + b² 115

Figure 5.7 Ratios and Similarity 116

Figure 5.8 Areas of Similar Polygons 117

Figure 5.9 Volumes of Similar Solids 118

Figure 5.10 Arc Length and Sector 119

Figure 5.11 Proportional Reasoning in Circles 120

Figure 5.12 Four Views of a Function 120

Figure 5.13 Rate of Change 121

Figure 5.14 Characteristics of Polynomial Functions 123

Figure 5.15 Even and Odd Polynomial Functions 124

Figure 5.16 Why f(x) = sin (x) is Odd and g(x) = cos (x) is Even 126

Figure 5.17 Linear Regression 127

Figure 5.18 Long-Run Relative Frequency 129

Figure 5.19 Two-Way Tables 131

Figure 5.20 Conditional Probability 133

II How to Plan 135

6. How to Plan Units 137

What is It? 137

Why We Like It 137

Supporting Research 138

Common Core Connections 138

Application 139

Getting Started 139

Making Connections Between Big Ideas 139

Developing a Logical Sequence 140

Organizing Topics and Problems 141

Summarizing the Unit Plan 141

Being Flexible 141

Developing Students’ Social and Emotional Learning 141

Incorporating Students’ Cultures 142

Differentiating for ELLs and Students with Learning Differences 143

Student Handouts and Examples 143

What Could Go Wrong 143

Technology Connections 145

Figures 145

Figure 6.1 Unit Plan: List of Skills 146

Figure 6.2 Unit Plan: Concept Map 147

Figure 6.3 Unit Plan: Sequence of Lessons 148

Figure 6.4 Sample Unit Plan 149

7. How to Plan Lessons 151

What is It? 151

Why We Like It 151

Supporting Research 152

Common Core Connections 152

Application 152

Defining the Lesson’s Scope 152

Introductory Activity 153

Presenting New Material Through Guided Questions 154

Practice 155

Differentiating for ELLs and Students with Learning Differences 155

Summary Activity 156

Student Handouts and Examples 157

What Could Go Wrong 157

Technology Connections 159

Figures 162

Figure 7.1 Do Now Problem 162

Figure 7.2 Lesson Plan: Standard Deviation 162

Figure 7.3 Lesson Plan: Slope-Intercept Form 166

Figure 7.4 Revised Baseball Field Word Problem 168

8. How to Plan Homework 169

What is It? 169

Why We Like It 169

Supporting Research 169

Common Core Connections 170

Application 170

Sources 171

Homework Format 171

Homework as Practice 172

Homework as Discovery 173

Homework as Transfer 173

Discussing Homework 174

Collecting Homework 175

Grading Homework 176

Differentiating for ELLs and Students with Learning Differences 177

Student Handouts and Examples 178

What Could Go Wrong 178

Students Who Don’t Do Homework 178

Mismanaging Class Time 179

Homework Review Challenges 179

Choosing the Wrong Problems 180

Technology Connections 180

Figures 183

Figure 8.1 Homework as Practice 183

Figure 8.2 Homework as Discovery—Ratios 184

Figure 8.3 Homework as Discovery—Mean Proportional Theorem 185

Figure 8.4 Homework as Discovery—Parabolas 186

Figure 8.5 Homework as Transfer—Similarity 187

Figure 8.6 Homework as Transfer—Bank Accounts 188

9. How to Plan Tests and Quizzes 189

What is It? 189

Why We Like It 189

Supporting Research 190

Common Core Connections 190

Application 190

Types of Questions 190

Test Format 193

Quiz Format 196

Reviewing for Assessments 196

Creating Scoring Guidelines for Assessments 199

Grading Assessments 202

Analyzing Test Results 203

Returning Tests 204

Differentiating for ELLs and Students with Learning Differences 207

Alternate Forms of Assessment 208

Student Handouts and Examples 208

What Could Go Wrong 208

Poor Scheduling and Preparation 209

Assessments as Classroom Management 210

Poorly Chosen Questions 210

Mistakes on Assessments 211

Student Cheating 212

Different Versions of Tests 213

Grading and Returning Assessments 214

Test Retakes and Test Corrections 215

Technology Connections 215

Test Questions, Answers, and Scoring Guidelines 215

Test Review 216

Test Analysis 216

Figures 217

Figure 9.1 Algebra I Test 217

Figure 9.2 Precalculus Test 220

Figure 9.3 Quiz 224

Figure 9.4 Creating Scoring Guidelines 225

Figure 9.5 Blank Test Corrections Sheet 226

Figure 9.6 Completed Test Corrections Sheet 228

Figure 9.7 Test Reflection Form 229

10. How to Develop an Effective Grading Policy 231

What is It? 231

Why We Like It 232

Supporting Research 232

Common Core Connections 232

Application 232

Standards-Based Grading 232

Minimum Grading Policy 234

Point Accumulation System for Grading 236

Differentiating for ELLs and Students with Learning Differences 237

More Than Just a Grade 238

What Could Go Wrong 239

Student Handouts and Examples 240

Technology Connections 240

Figures 241

Figure 10.1 Grade Calculation Sheet 241

Figure 10.2 Completed Grade Calculation Sheet 242

III Building Relationships 243

11. Building a Productive Classroom Environment 245

What is It? 245

Why We Like It 245

Supporting Research 245

Common Core Connections 246

Application 246

Making a Good First Impression 246

Learning Names 248

Getting to Know Students 248

Classroom Organization 249

Classroom Rules and Routines 250

Course Descriptions 252

Soliciting Student Opinion 253

Taking Notes 254

What Could Go Wrong 257

Classroom Tone 257

Mishandling the Teacher–Student Relationship 258

Taking Notes 259

Student Handouts and Examples 259

Technology Connections 259

Classroom Environment 259

Student Surveys 260

Note-Taking 260

Figures 261

Figure 11.1 Student Information Sheet 261

Figure 11.2 Course Description 263

Figure 11.3 Brief Handout 265

Figure 11.4 Full-Page Handout 266

Figure 11.5 Annotated Work 268

Figure 11.6 Double-Entry Journal 269

12. Building Relationships with Parents 271

What is It? 271

Why We Like It 271

Supporting Research 272

Common Core Connections 272

Application 272

Communicating with Parents 272

Addressing Parents’ Math Anxiety 273

Parent–Teacher Conferences 277

Home Visits 277

Working with Parents of Culturally Diverse Students 278

Working with Parents of Students with Learning Differences 279

What Could Go Wrong 280

Student Handouts and Examples 281

Technology Connections 281

Figures 282

Figure 12.1 Parent Communication Script 282

Figure 12.2 Parent Communication Log 283

13. Collaborating with Other Teachers 285

What is It? 285

Why We Like It 285

Supporting Research 286

Common Core Connections 286

Application 286

Discussing Values 287

Planning with Other Math Teachers 288

Interdisciplinary Collaboration 288

Observing Other Teachers 289

Co-Teaching 291

Mentoring 294

Lesson Study 294

Professional Learning Community 295

What Could Go Wrong 297

Lack of Trust 297

Reinforcing Negative Stereotypes 297

Lack of Colleagues 297

Lack of Time 298

Technology Connections 298

IV Enhancing Lessons 301

14. Differentiating Instruction 303

What is It? 303

Why We Like It 303

Supporting Research 304

Common Core Connections 305

Application 305

Differentiation by Content 305

Differentiation by Process 313

Differentiation by Product 315

Differentiation by Affect 320

What Could Go Wrong 320

Student Handouts and Examples 321

Technology Connections 321

Figures 323

Figure 14.1 Tiered Lesson—Literal Equations 323

Figure 14.2 Tiered Lesson—Midpoint 325

Figure 14.3 Curriculum Compacting—Coordinate Geometry 328

Figure 14.4 Tiered Test Questions 331

Figure 14.5 Review Sheet 331

Figure 14.6 Fill-In Review Sheet 332

Figure 14.7 Review Booklet 333

15. Differentiating for Students with Unique Needs 335

What is It? 335

Why We Like It 336

Supporting Research 336

Common Core Connections 337

Application 337

Strengths and Challenges of Students with Unique Needs 337

Techniques to Support Students with Unique Needs 340

What Could Go Wrong 348

Student Handouts and Examples 349

Technology Connections 349

Figures 351

Figure 15.1 Frayer Model (Blank) 351

Figure 15.2 Frayer Model—Perpendicular Bisector 352

Figure 15.3 Concept Map 352

16. Project-Based Learning 353

What is It? 353

Why We Like It 353

Supporting Research 354

Common Core Connections 355

Application 355

Open-Ended Classwork Problems 355

Open-Ended Homework Problems 357

Projects 358

What Could Go Wrong 367

Student Handouts and Examples 368

Technology Connections 368

Figures 369

Figure 16.1 Discovering Pi 369

Figure 16.2 Area of a Circle 370

Figure 16.3 Point Lattice Assignment 371

Figure 16.4 Paint a Room 374

Figure 16.5 Project—Bus Redesign Plan 375

17. Cooperative Learning 379

What is It? 379

Why We Like It 380

Supporting Research 380

Common Core Connections 381

Application 381

General Techniques 381

Differentiating for Students with Unique Needs 384

Examples 387

What Could Go Wrong 398

Student Handouts and Examples 399

Technology Connections 400

Figures 401

Figure 17.1 Jigsaw as Practice 401

Figure 17.2 Jigsaw as Discovery 402

Figure 17.3 Factoring Station 403

Figure 17.4 Peer Editing 404

18. Formative Assessment 405

What is It? 405

Why We Like It 405

Supporting Research 406

Common Core Connections 406

Application 406

Asking the Right Questions 407

Eliciting Student Responses 409

Responding to Student Answers 412

Other Methods of Formative Assessment 412

Differentiating Formative Assessment 413

What Could Go Wrong 414

Technology Connections 415

19. Using Technology 417

What is It? 417

Why We Like It 417

Supporting Research 418

Common Core Connections 418

Application 418

Classroom Organization 418

Mathematical Content 422

Using Technology for Culturally Responsive Teaching 425

Using Technology to Differentiate Instruction 425

What Could Go Wrong 425

Student Handouts and Examples 427

Technology Connections 428

Figures 429

Figure 19.1 Simulation of 1,000 Coin Flips 429

Figure 19.2 Transformations of Functions 429

Figure 19.3 Centroid of a Triangle 431

Figure 19.4 Two Views of a Graph Using Technology 432

20. Ending the School Year 433

What is It? 433

Why We Like It 433

Supporting Research 433

Common Core Connections 434

Application 434

Review 434

Reflection 438

Recognition 439

Maintaining Relationships with Students 440

Differentiating Year-End Activities 440

What Could Go Wrong 441

Year-End Fatigue 441

“What Can I Do to Pass?” 441

Running Out of Time 442

Technology Connections 443

Appendix A: The Math Teacher’s Toolbox Technology Links 445

References 461

Index 515

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