Math That Matters

In this insightful math resource for grades 3–8, popular professional developer Marian Small helps teachers understand and facilitate meaningful assessments to advance student understandings. Small shows new and veteran teachers how to do three fundamental things well: identify the most important math to assess; construct meaningful assessments—both formative and summative—to measure student understanding; and provide students with feedback that is clear, timely, and specific. Examples for each grade level are provided, along with details on how to pose questions, analyze errors, and help students understand and learn from their mistakes. The book provides specific guidance for when and how to offer feedback on both correct and incorrect answers in order to advance students’ mathematical thinking. Like other Marian Small bestsellers, Math That Matters combines her special brand of lucid explanation of difficult concepts with fresh and engaging activities.

Book Features:

• Comprehensively addresses different mathematical domains for each grade (3–8).
• Organized around common core curriculum groupings but still applicable to other sets of mathematical standards.
• Includes student examples with suggestions for targeted, constructive feedback.
• Uses a strength-based approach to teacher education and professional learning.

Marian Small is the former dean of education at the University of New Brunswick and a longtime professor of mathematics education. She is widely known for her bestselling books and her in-district work with educators in the United States and Canada. Visit her website onetwoinfinity.ca for in-person and online professional development.

“ Math that Matters is an accessible, practical, and insightful book written for current (and future) math teachers about classroom assessment and feedback…The contribution of this book lies in the elegant way Small has distilled and packaged familiar ideas, and in the rich detail of the examples and illustrations that make the application of these ideas attractive and feasible. The book is a promising resource for teacher educators who want to introduce preservice and inservice teachers to current best practices for assessment and feedback in the mathematics classroom.”

—Teachers College Record

"Our understanding of the power of assessment to improve learning has deepened significantly in the past two decades. . . . Marian Small draws upon the critical research behind this understanding to explain what effective practice looks like. It is essential reading for all elementary educators and has the potential to profoundly affect the quality of mathematics assessment in our schools.”
―From the Foreword by Damian Cooper, president, Plan Teach Assess

“Teachers are often clamoring for concise classroom assessments that can capture students’ conceptual understanding. Clamor no more! Math That Matters is a timely response to that need. Marian Small removes the mystery of how to engage students in learning while collecting assessment data that drive next instructional plans. Using specific examples of discourse-rich pedagogies such as diagnostic tasks, probing questions, thoughtful approaches to delivering feedback, and differentiated performance assessments, she provides models for developing a comprehensive assessment plan.”
—Karen Karp, Johns Hopkins University

"The beauty of this book is that it is simple enough for brand new teachers and complex enough for experienced teachers. New teachers will learn a lot from the examples. Veteran teachers will see themselves in the examples, and the book will help them to shift their practice. The author offers an amazing gift by linking assessment ideas directly to common state standards." ―Felicia Darling, Santa Rosa Junior College

Contents

Foreword, by Damian Cooper ix

Preface xiii

Acknowledgments xvii

1 Linking Assessment and Feedback to Learning Goals 1
How Assessment and Feedback Reflect Our Values 1
What Does This Look Like (More Specifically)? 2
Summary 9

2 Assessment for Learning, as Learning, and of Learning 10
Assessment for Learning 10
Assessment as Learning 15
Assessment of Learning 21
Summary 26

3 Feedback 28
Types of Feedback 29
Reactions to All Work 34
Immediacy of Feedback 35
Anticipating Errors 35
Summary 38

4 Operations and Algebraic Thinking, Grade 3: Assessment and Feedback 40
The Math That Matters 40
Assessment for Learning and Feedback 45
Assessment as Learning 53
Assessment of Learning 53
Summary 58

5 Number and Operations in Base Ten, Grade 4: Assessment and Feedback 59
The Math That Matters 59
Assessment for Learning and Feedback 64
Assessment as Learning 73
Assessment of Learning 74
Summary 78

6 Geometry, Grade 5: Assessment and Feedback 79
The Math That Matters 79
Assessment for Learning and Feedback 82
Assessment as Learning 91
Assessment of Learning 92
Summary 96

7 Ratios and Proportional Relationships, Grade 6: Assessment and Feedback 97
The Math That Matters 97
Assessment for Learning and Feedback 103
Assessment as Learning 112
Assessment of Learning 112
Summary 116

8 Statistics and Probability, Grade 7: Assessment and Feedback 117
The Math That Matters 117
Assessment for Learning and Feedback 126
Assessment as Learning 133
Assessment of Learning 134
Summary 138

9 Expressions and Equations, Grade 8: Assessment and Feedback 139
The Math That Matters 139
Assessment for Learning and Feedback 147
Assessment as Learning 156
Assessment of Learning 157
Summary 160
;
10 Conclusion 162
Some Key Messages 162
Instruction Versus Assessment 163
An Assessment Plan 163

References 165

Index 167

About the Author 171

Contents (tentative)

Foreword

Preface
   What Do We Mean by Assessment in This Resource?
   What Constitutes Feedback?
   What is Special About This Book?
   Organization of the Book 
   Differentiated Assessment
   Acknowledgments

Chapter 1. What Is Foundational in Assessment and Feedback in Mathematics?
   Historical Issues in Mathematics Assessment
   How Beliefs Impact Assessment
   More Specific Examples
      What We Might Value in Grade 3 Measurement
      What We Might Value in Grade 5 Fractions
      What We Might Value in Grade 7 Ratios and Proportion
   Summary of Chapter 1

Chapter 2. Assessment for Learning, as Learning, and of Learning
   Assessment for Learning
      Diagnostic Assessment
      Other Forms of Formative Assessment
      What About the Practice Standards?
   Assessment as Learning
      Focusing on Detail or Not
      Focusing on Answer Versus Focusing on Process
      Success Criteria
      Rubrics
      Using Samples
      Self-Assessment Templates
      Learning to Self-Assess
   Assessment of Learning
      What Information is Collected?
      Should There be Choice?
      How Data are Collected
      When Data are Collected
      How Data are Reported
      Whether Data are Permanent
   Summary of Chapter 2

Chapter 3. Feedback
   Types of Feedback
      Opportunities for Self-Correction
      Communication on Choice of Strategy
      Communication on Perspective Taking
      Communication Related to Problem Interpretation
      Encouragement of Creativity
      Encouragement of Extending
   Reaction to All Work 
   Immediacy of Feedback
   Anticipating Errors
      Overgeneralizations
      Inappropriate Assumptions
      Common Misconceptions
   Summary of Chapter 3

Chapter 4. Assessment and Feedback in Grade 3 Operations and Algebraic Thinking
   The Math That Matters
      Meanings of the Operations
      Properties and Principles
   Assessment for Learning and Feedback
      Diagnostic Task
      Diagnostic Questions
      Tasks Designed for Formative Assessment
   Assessment as Learning
   Assessment of Learning
      Skill Questions
      Concept Questions
      Performance Task and Rubric
      Observation Checklist
      Putting it Together
   Summary of Chapter 4

Chapter 5. Assessment and Feedback in Grade 4 Numbers and Operations
   The Math That Matters
      Place Value
      Adding and Subtracting Multi-Digit Numbers
      Multiplying and Dividing Multi-Digit Numbers
   Assessment for Learning and Feedback
      Diagnostic Task
      Diagnostic Questions
      Tasks Designed for Formative Assessment
   Assessment as Learning
   Assessment of Learning
      Skill Questions
      Concept Questions 
      Performance Task and Rubric
      Observation Checklist
      Putting it Together
   Summary of Chapter 5

Chapter 6. Assessment and Feedback in Grade 5 Geometry
   The Math That Matters
      Using a Coordinate Grid
      Relating and Classifying 2-D Figures by Considering Properties
   Assessment for Learning and Feedback
      Diagnostic Task
      Diagnostic Questions
      Tasks Designed for Formative Assessment
   Assessment as Learning
   Assessment of Learning
      Skill Questions
      Concept Questions
      Performance Task and Rubric
      Observation Checklist
      Putting It Together
   Summary of Chapter 6

Chapter 7. Assessment and Feedback in Grade 6 Ratio and Proportional Relationships
   The Math That Matters
      Multiplicative Comparisons
      Ratio and Rate Language
      Equivalent Ratios and Rates
      Representing Ratios and Rates
      Percents
      Comparing Rates or Ratios 
      Solving Problems
   Assessment for Learning and Feedback
      Diagnostic Task
      Diagnostic Questions
      Tasks Designed for Formative Assessment
   Assessment as Learning
   Assessment of Learning
      Skill Questions
      Concept Questions
      Performance Task and Rubric
      Observation Checklist
      Putting It Together
   Summary of Chapter 7

Chapter 8. Assessment and Feedback in Grade 7 Statistics and Probability
   The Math That Matters
      Statistics
      Probability
   Assessment for Learning and Feedback
      Diagnostic Task
      Diagnostic Questions
      Tasks Designed for Formative Assessment
   Assessment as Learning
   Assessment of Learning
      Skill Questions
      Concept Questions
      Performance Task and Rubric
      Observation Checklist
      Putting It Together
   Summary of Chapter 8

Chapter 9. Assessment and Feedback in Grade 8 Expressions and Equations
   The Math That Matters
      Exponents
      Square Roots and Cube Roots
      Linear Relationships
      Linear Equations
      Systems of Linear Equations
   Assessment for Learning and Feedback
      Diagnostic Task
      Diagnostic Questions
      Tasks Designed for Formative Assessment
   Assessment as Learning
   Assessment of Learning
      Skill Questions
      Concept Questions
      Performance Task and Rubric
      Observation Checklist
      Putting It Together
Summary of Chapter 9

Chapter 10. Conclusion
   Some Key Messages
   Instruction Versus Assessment
   An Assessment Plan

References

Index

About the Author 

• All sales throughout the U.S.A. available via our distributor.
• Co-published with, and available in Canada from, Nelson Publishing Inc.
• All other sales throughout the world available via Eurospan.