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Uncovering Student Thinking in Mathematics, Grades 6-12
30 Formative Assessment Probes for the Secondary Classroom

**Edited by:**

- Cheryl M. Rose - Tobey Education Group, LLC, Education Development Center, Newton, MA
- Carolyn B. Arline - Maine Mathematics and Science Alliance

Foreword by Johnny W. Lott

*"Cheryl Rose and Carolyn Arline provide a valuable resource to teachers who are interested in determining how their students think and answer questions in mathematics classes."*—From the Foreword by Johnny W. Lott

Director, Center for Excellence in Teaching and Learning

University of Mississippi

*Use these targeted probes to assess secondary students' grasp of key mathematics concepts!*

Research findings show that formative assessment is an important strategy for improving student achievement in mathematics. This practical resource by best-selling authors Cheryl M. Rose and Carolyn B. Arline provides tools and examples that allow middle and high school teachers to gauge students' knowledge of core mathematics concepts and strengthen their ability to teach effectively.

The authors provide 30 formative assessment probes—brief, easily administered activities targeting specific mathematics ideas—to reveal common understandings and misunderstandings in student thinking. Field-tested with teachers and students at various grade levels, these powerful diagnostic tools help teachers modify their teaching and identify areas that require more instruction. Written in accessible language, this invaluable book:

- Discusses standards, research results, and practical craft knowledge
- Describes the purpose, structure, and development of mathematics assessment probes
- Helps teachers build on students' current understandings while addressing their identified difficulties
- Offers examples of the faulty thinking students are likely to exhibit and typical obstacles they may encounter

These assessment probes will help teachers can make sound instructional choices and increase the mathematics knowledge of all their students!

Introduction |

What Types of Understandings and Misunderstandings Does a Mathematics Assessment Probe Uncover? |

How Were the Mathematics Assessment Probes Developed? |

What Is the Structure of a Mathematics Assessment Probe? |

What Additional Information Is Provided With Each Mathematics Assessment Probe? |

What Mathematics Assessment Probes Are Included in the Book? |

Differentiating Instruction |

Assessing Points of Entry |

Analyzing Trends in Student Thinking |

Giving Student Interviews |

Promoting Student-to-Student Dialogue |

Developing Vocabulary |

Allowing for Individual Think Time |

Improving Students' Process Skills |

Assessing Effectiveness of Instructional Activities |

Moving Beyond the Individual Classroom |

Summary |

Probe 1: What Is the Value of the Digit? |

Teachers' Notes: What Is the Value of the Digit? |

Probe 1a: What Is the Value of the Digit? Variation |

Probe 2: What Is the Meaning of 2/3? |

Teachers' Notes: What Is the Meaning of 2/3? |

Probe 2a: What Is the Meaning of 2/3? Variation |

Probe 3: Is It Equivalent? |

Teachers' Notes: Is It Equivalent? |

Probe 3a: Is It Equivalent? Variation |

Probe 4: What's Your Estimate? |

Teachers' Notes: What's Your Estimate? |

Probe 5: Is It an Estimate? |

Teachers' Notes: Is It an Estimate? |

Probe 6: Is It Simplified? |

Teachers' Notes: Is It Simplified? |

Probe 6a: Is It Simplified? Algebraic Variation |

Probe 7: Where Is a Million? |

Teachers' Notes: Where Is a Million? |

Probe 8: How Low Can You Go? |

Teachers’ Notes: How Low Can You Go? |

Probe 8a: How Low Can You Go? Variation |

Probe 9: What’s Your Addition Strategy? |

Teachers’ Notes: What’s Your Addition Strategy? |

Probe 9a: What’s Your Addition Strategy? Decimals Variation |

Probe 9b: What’s Your Addition Strategy? Fractions Variation |

Probe 10: What’s Your Subtraction Strategy? |

Teachers’ Notes: What’s Your Subtraction Strategy? |

Probe 10a: What’s Your Subtraction Strategy? Decimals Variation |

Probe 10b: What’s Your Subtraction Strategy? Fractions Variation |

Probe 11: What’s Your Multiplication Strategy? |

Teachers’ Notes: What’s Your Multiplication Strategy? |

Probe 11a: What’s Your Multiplication Strategy? Decimals Variation |

Probe 12: What’s Your Division Strategy? |

Teachers’ Notes: What’s Your Division Strategy? |

Probe 12a: What’s Your Division Strategy? Decimals Variation |

Probe 1: What’s the Measure? |

Teachers’ Notes: What’s the Measure? |

Probe 1a: What’s the Measure? Variation |

Probe 2: Are Area and Perimeter Related? |

Teachers’ Notes: Are Area and Perimeter Related? |

Probe 3: What’s the Area? |

Teachers’ Notes: What’s the Area? |

Probe 4: What’s the Capacity? |

Teachers’ Notes: What’s the Capacity? |

Probe 5: Is It Transformed? |

Teachers’ Notes: Is It Transformed? |

Probe 6: Are They Similar? |

Teachers’ Notes: Are They Similar? |

Probe 7: What Do You Mean? |

Teachers’ Notes: What Do You Mean? |

Probe 7a: What Do You Mean? Variation |

Probe 8: Name of the Graph? |

Teachers’ Notes: Name of the Graph? |

Probe 9: Graph Construction? |

Teachers’ Notes: Graph Construction? |

Probe 1: Equal or Not Equal? |

Teachers’ Notes: Equal or Not Equal? |

Probe 2: Is It the Same as a+b? |

Teachers’ Notes: Is It the Same as a+b? |

Probe 3: M & N’s? |

Teachers’ Notes: M & N’s? |

Probe 4: What’s the Substitute? |

Teachers’ Notes: What’s the Substitute? |

Probe 5: Is It True? |

Teachers’ Notes: Is It True? |

Probe 6: Solving Equations? |

Teachers’ Notes: Solving Equations? |

Probe 7: Correct Representation of the Inequality? |

Teachers’ Notes: Correct Representation of the Inequality? |

Probe 8: Is It a Variable? |

Teachers’ Notes: Is It a Variable? |

Probe 9: Binomial Expansion? |

Teachers’ Notes: Binomial Expansion? |

Probe 10: Is It Quadratic? |

Teachers’ Notes: Is It Quadratic? |

*"Cheryl Rose and Carolyn Arline provide a valuable resource to teachers who are interested in determining how their students think and answer questions in mathematics classes."*

*"The authors contribute to the field by answering the question: How do I help my struggling students who hold a variety of misconceptions or have limited prior knowledge of the content?"*

**University of Minnesota, Minneapolis**

*"The book is timely. The probes will be beneficial to teachers and very useful to districts developing local assessments."*

**Slater Junior High School, North Smithfield, RI**

*"An efficient teaching tool to uncover and correct misconceptions students have across all mathematics strands. This book provides numerous opportunities for increased student achievement that are both powerful and exciting for educators."*

**Springfield Public Schools, MO**

*"A tool for middle school and high school mathematics teachers that combines the effective use of research, formative assessment, and ready-to-go student tasks. This resource belongs int he hands of any mathematics educator who seeks to balance procedural knowledge, conceptual understanding, and skills for each student."*

**New Hampshire Department of Education**

*"Using probes with my students has helped me plan and modify my instruction. The range of student explanations elicit rich mathematical discourse."*

**Kenilworth Middle School, Baton Rouge, LA**

*"One of the most effective tools for teachers to gather valuable formative assessment data. This resource provides an easily accessible, yet powerful way for teachers to elicit and analyze student work that truly reflects student thinking. The probes are targeted to specific concept areas that are easily connected to daily lesson plans and curriculum materials."*