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

Cheryl M. Rose and Carolyn B. Arline
Corwin Press
Publication Date: 
Number of Pages: 
[Reviewed by
Krista B. Hands
, on

This 225 page book is a nice compilation of a variety of ideas. While the title indicates this book is for the purposes of assessment, in that it has assessment probes included, it goes much deeper than the title indicates. To begin with, there is a discussion of types of student misunderstanding and how this gives rise to the need for different types of assessment. This is backed by research and supported by NCTM standards.

There is also discussion of the instructional implications of assessment, as well as the distinction between assessment and evaluation. While we often use these words interchangeably, the authors make it very clear that their probes are not for the purpose of evaluation, ie: grades. Their probes are to unearth student thinking and understanding in a way that will help the teacher make better teaching decisions as they move forward with their class. Slightly over 10% of the book is dedicated to the types of issues described above.

Following the research based discussion above, there are three distinctly different components of the book, all of which are related to the probes alluded to in the book’s title. The three different sections of the book are

  1. Number and Operations
  2. Geometry, Measurement, and Data
  3. Algebra

Within these sections, 30 different probes are presented. These probes are designed to assess student thinking in different areas. Each probe is followed by teacher’s notes regarding the reasoning behind the probe.

Most probes also have variations given to alter the way the probe could be deployed. For example, one may look like a matching exercise, while a secondary option is more of a game format. In addition to the probes, sample responses from students are given, as well as responses from teachers who have used the probe. This would give then reader a chance to see what they might expect as they try the probe for the first time.

I see two distinctly different ways this book could be used. In one frame of mind I can see how a teacher could use this when teaching topics for the first time, trying to get the correct understandings across to her students and assessing her ability to do so. This would enable a teacher to see where she may need to spend more time or address misconceptions.

However, from a personal perspective, working with older students (late high school and early college) I think this would be a great tool to start discussion. I can see it used as an ice breaker or a prior-knowledge kind of assessment. This would allow a teacher to see the misconceptions she may need to address before she can start a related, but more advanced topic.

In my teaching of pre-service teachers, using this book would allow me to address two kinds of issues. First, misconceptions the pre-service teachers themselves may have, and second, the misconceptions their students may have once they start teaching. I am very excited to see this resource and plan to start using it next semester as I work with pre-service teachers at my university.

Krista B. Hands is an Assistant Professor of Mathematics at Ashland University, Ashland, Ohio. Her research is in mathematics pedagogy with a focus in undergraduate mathematics. She teaches mathematics content courses to pre-service teachers as well as other mathematics courses. She resides in Medina, Ohio with her husband, daughter, and son.


Foreword by Johnny W. Lott
About the Authors
1. Mathematics Assessment Probes
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?
2. Instructional Implications
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
3. Number and Operations Assessment Probes
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
4. Geometry, Measurement, and Data Assessment Probes
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?
5. Algebra and Data Assessment Probes
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?
Resource A: Note Template for QUEST Cycle