Meet the Authors ix

Introduction xi

1 Overview of Response to Intervention in Mathematics 1

What Is Response to Intervention? 1

      Evidence-Based Instruction 2

      Data-Based Decision Making 2

      Tiered Support 3

Models of Implementation 6

Summary 7

2 Using Assessment to Make Instructional Decisions 8

Step 1: Problem Identification 9

      Core Program Evaluation 13

      Identifying Struggling Learners 13

Step 2: Problem Analysis 15

Step 3: Intervention Plan Development 19

Step 4: Plan Implementation 22

Step 5: Plan Evaluation 24

Summary 24

3 Overview of Evidence-Based Practices for Teaching Mathematics 25

What Is the Core Curriculum? 25

      Description of the Core Curriculum 25

      Evidence-Based Practice for the Core Curriculum 27

      Emphasize Critical Concepts 27

      Teach Critical Foundations to Mastery 27

      Balance Conceptual Understanding, Fluency, and Problem Solving 28

      Use a Combination of Teacher-Centered and Student-Centered Approaches 28

      Follow the CRA Sequence 28

      Select High-Quality Programs 30

Interventions to Support Students Who Struggle in Mathematics 30

      Teach Critical Content During Interventions 31

      Focus on Whole Numbers and Rational Numbers 31

      Basic Facts 32

      Problem Solving 32

      Effective Instructional Methods for Struggling Students 33

      Explicit Instruction 33

      Manipulatives and Visual Representation 33

      Motivation 34

Summary 35

4 Setting the Stage: Increasing Motivation 36

Why Focus on Motivation? 36

Increasing Motivation by Creating Meaningful, Engaging Lessons 37

Using Self-Monitoring and Goal-Setting to Increase Motivation 40

Effective Use of Praise 42

Rewards 45

      I = Instruction with Incentive 46

      N = Not Negative! 46

      C = Criteria 48

      E = Easy 48

      N = Never Leave a Child with No Reason to Try! 49

      T = Time 49

      I = Individualized Incentive 50

      V = Verbal Feedback 52

      E = Evaluate 53

Summary 54

5 Explicit Instruction 55

Instructional Considerations for Struggling Learners 55

The Explicit Instruction Lesson 58

      structional Objectives 60

      Lesson Introduction 62

      Engage 62

      Review of Prerequisite Skills and Concepts 62

      Presentation of New Content 64

      Guided Practice 68

      Independent Practice 70

      Massed Practice 70

      Distributed Practice 74

      Lesson Closure 75

How Explicit Instruction Improves Motivation 76

Summary 76

6 Concrete and Visual Representation 78

Research on the Concrete-Representational-Abstract Sequence 79

Virtual Manipulatives 84

Suggestions for Using the CRA Sequence 87

Summary 89

7 Representing Whole Numbers 91

Developing Number Sense 91

      Counters 92

      Touchmath 93

      Ten-Frames 94

      Number Lines 97

Place Value 101

      Ten-Frames 101

      Base-Ten Blocks 103

      DigiBlocks 106

      Building a Solid Foundation 108

Addition and Subtraction 110

      Developing Conceptual Understanding 110

      Developing Computational Fluency with Basic Facts 116

      Multi-Digit Addition and Subtraction 117

Multiplication and Division 123

      Developing Conceptual Understanding of Multiplication 123

      Understanding Division 132

      Multidigit Multiplication 133

      Multidigit Division 146

Summary 150

8 Developing Computational Fluency with Basic Facts 152

Conceptual Understanding and Computational Fluency 154

Strategies for Addition Facts 158

      Counting On 158

      Plus-One and Plus-Two Facts 159

      The Commutative Property of Addition 160

      Facts with Zero 160

      Doubles 161

      Near-Doubles 163

      Ten-Sums 164

      Near-Tens 164

      Facts Solved by Making a Ten 165

      The Leftovers 166

Selecting a Strategy 166

Strategies for Subtraction Facts 167

      Related Facts 168

      Counting Down: -1 and -2 Facts 169

      Subtracting Zero 170

      Subtracting the Same Number 171

      Decomposition Strategies 171

Strategies for Multiplication Facts 171

      ×2 172

      Fives 172

      Zeros and Ones 173

      Nines 174

      Other Strategies 175

Strategies for Division Facts 178

Mnemonics 179

Using Technology to Practice Basic Facts 180

Differentiating Practice 180

Sumumary 183

9 Representing Rational Numbers 184

Developing Fraction Concepts 184

Using the CRA Sequence with Advanced Fraction Skills 188

Equivalent Fractions 189

Adding and Subtracting Fractions 193

Multiplying Fractions 198

Dividing Fractions 202

Decimals 207

Percent 213

Summary 214

10 Problem Solving 216

Problem Structures for Addition and Subtraction 217

      Representing and Solving Addition and Subtraction Problems 220

      Group Problems 223

      Change Problems 226

      Compare Problems 226

      Extending Addition and Subtraction 229

Problem Structures for Multiplication and Division 231

      Equal Groups 231

      Multiplicative Comparison Problems 234

      Extending Multiplication and Division 235

Scaffolding Support 236

Selecting Appropriate Problem-Solving Materials 238

Summary 239

11 Conclusion: Using RtI to Improve Achievement in Mathematics 242

Locating Materials to Provide High-Quality Instruction 242

Using Data to Inform Instructional Decisions 245

Providing Evidence-Based Interventions 248

Summary 251

Appendix A: Sample Lesson Plans 253

Appendix B: Games to Practice Basic Facts 259

Appendix C: Additional Resources for Teaching Basic Facts 275

References 277

Dr. Wendy Fuchs is an Assistant Professor in the Department of Special Education and Communication Disorders at Southern Illinois University Edwardsville. She teaches the beginning and advanced reading methods courses for pre-service special educators. Dr. Fuchs also serves as the Principal Investigator and Project Coordinator of the Illinois Institutes of Higher Education Partnership, a federally-funded Statewide Professional Development Grant that assists teacher preparation programs integrate MTSS into course content and field experiences. In addition to her work with the IHE Partnership, Dr. Fuchs provides professional development and educational consulting to school districts in Southern Illinois in the areas of Response to Intervention/Multi-Tier System of Supports (MTSS), school improvement, and maximizing student engagement. Dr. Fuchs is a member of the PBIS Statewide Leadership Team, and serves as an executive board member for the Illinois Teacher Education Division of Council for Exceptional Children. Her research interests include effective teaching practices, teacher perceptions of students with disabilities, data-based instructional decision making.

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