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Math teachers will find the classroom-tested lessons and strategies in this book to be accessible and easily implemented in the classroom
The Teacher's Toolbox series is an innovative, research-based resource providing teachers with instructional strategies for students of all levels and abilities. Each book in the collection focuses on a specific content area. Clear, concise guidance enables teachers to quickly integrate low-prep, high-value lessons and strategies in their middle school and high school classrooms. Every strategy follows a practical, how-to format established by the series editors.
The Math Teacher's Toolbox contains hundreds of student-friendly classroom lessons and teaching strategies. Clear and concise chapters, fully aligned to Common Core math standards, cover the underlying research, required technology, practical classroom use, and modification of each high-value lesson and strategy.
This book employs a hands-on approach to help educators quickly learn and apply proven methods and techniques in their mathematics courses. Topics range from the planning of units, lessons, tests, and homework to conducting formative assessments, differentiating instruction, motivating students, dealing with "math anxiety," and culturally responsive teaching. Easy-to-read content shows how and why math should be taught as a language and how to make connections across mathematical units. Designed to reduce instructor preparation time and increase student engagement and comprehension, this book:
The Math Teacher's Toolbox: Hundreds of Practical ideas to Support Your Students is an invaluable source of real-world lessons, strategies, and techniques for general education teachers and math specialists, as well as resource specialists/special education teachers, elementary and secondary educators, and teacher educators.
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
Motivation-why people do what they do-affects every aspect of schooling. Without motivation, student learning becomes difficult, if not impossible (Artzt, Armour-Thomas, & Curcio, 2008, p. 48). Motivated students tend to have better performance, higher self-esteem, and improved psychological well-being (Fong, Patall, Vasquez, & Stautberg, 2019, p. 123; Gottfried, Marcoulides, Gottfried, & Oliver, 2013, p. 83; Liu & Hou, 2017, p. 49; Reeve, Deci, & Ryan, 2004, p. 22). Conversely, unmotivated students can become disengaged from academics and, in the worst cases, drop out of school (National Research Council, 2004, p. 24).
According to self-determination theory, a theory of motivation developed by researchers Edward L. Deci and Richard M. Ryan, motivation can be intrinsic (doing something because it is inherently satisfying) or extrinsic (doing something because it leads to some other result) (Ryan & Deci, 2000, p. 55). Many times, motivation is difficult to characterize as purely intrinsic or extrinsic. A student may be drawn by an extrinsic reward but may eventually internalize the values and adapt a more intrinsic motivation (Usher & Kober, 2012b, p. 3).
In addition, motivation is not a fixed quantity (Ryan & Deci, 2000, p. 54). Factors like schools, parents, communities, teachers, and life experiences can positively or negatively affect motivation (Usher & Kober, 2012a, p. 7). Students' motivation can vary from class to class-a student who is highly motivated in one class may be completely disengaged in another (National Research Council, 2004, p. 33).
As a result, educators often need to foster both intrinsic and extrinsic motivation. Students who are intrinsically interested in a topic are more likely to seek challenging tasks, think more creatively, and learn at a conceptual level (National Research Council, 2004, p. 38). However, since many academic tasks may not be inherently interesting, teachers also need to learn how to promote different methods of extrinsic motivation (Ryan & Deci, 2000, p. 55).
To sustain motivation, educators often seek ways to encourage students to internalize values. When students do so, they become more persistent and have a more positive sense of themselves (Ryan & Deci, 2000, pp. 60-61).
In our experience, keeping motivational strategies in mind can enhance student engagement, academic achievement, and confidence to do math. Boosting their confidence is particularly important since many of our students experience math anxiety (we discuss it more in the Introduction), which can hinder their academic growth.
Many studies on motivation focus on ways to build inclusive communities that promote learning for all students (Kumar, Zusho, & Bondie, 2018, p. 78). Proponents of self-determination theory argue that people are motivated to complete a task if doing so fulfills basic psychological needs, such as autonomy, relatedness, and competence (Ryan & Deci, 2000, p. 64).
However, some researchers have begun to challenge the idea of a universal theory of motivation, arguing that most of the existing work ignores the experiences and members of historically marginalized groups, such as people of color (Usher, 2018, p. 132). These researchers seek a more culturally responsive framework in which motivation is viewed not just as an individual characteristic but as the product of the social and historical context that shapes students' emotions and beliefs (King & McInerney, 2016, p. 2).
Other studies have focused on the effect of emotions on student motivation (Hannula, 2019, p. 310). Students who feel more anxious about math often have decreased motivation and do more poorly in school (Gunderson, Park, Maloney, Beilock, & Levine, 2017, pp. 34-35; Mo, 2019, p. 2; Passolunghi, Cargnelutti, & Pellizzoni, 2018, p. 282). Discouragement from parents, inappropriate or overly difficult work, and lack of support from teachers can further erode students' self-efficacy-the realistic expectation that making a good effort will lead to success (Usher, 2009, p. 308). In other words, social-emotional learning is tied to motivation.
Research indicates that as students move through the K-12 school system, their attitudes toward math become less positive (Batchelor, Torbeyns, & Verschaffel, 2019, p. 204; Gottfried et al., 2013, p. 70). As a result, keeping middle and high school students motivated in math class can be particularly challenging.
Despite the different approaches and areas of emphasis in the literature, researchers agree on several ways to improve student motivation:
In the Application section of this chapter, we discuss some strategies for improving motivation for all students.
Many of the motivational techniques that we describe in this chapter are related to Common Core standards. For example:
Researchers seeking to merge self-determination theory with culturally responsive teaching (which we describe in more detail in Chapter 2: Culturally Responsive Teaching) have identified five characteristics of effective motivation:
Here are some strategies that apply these characteristics in supporting student motivation.
As we said in the Supporting Research section in this chapter, building students' self-efficacy and autonomy can help alleviate their math anxiety and improve their motivation, which can in turn improve their academic performance.
One way that we nurture students' self-confidence is to use language that supports their choice whenever possible. Saying, "I recommend that you rephrase this definition in your own words," can often be more effective than simply commanding students to write it down. Explaining why completing a task is necessary can help students understand how they can benefit from doing so (Reeve & Halusic, 2009, p....
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