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Math That Matters: Targeted Assessment and Feedback for Grades 3–8

reviewed by Erik Jacobson & Jinqing Liu - September 24, 2019

coverTitle: Math That Matters: Targeted Assessment and Feedback for Grades 3–8
Author(s): Marian Small
Publisher: Teachers College Press, New York
ISBN: 0807761427, Pages: 192, Year: 2019
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Math that Matters is an accessible, practical, and insightful book written for current (and future) math teachers about classroom assessment and feedback. Although it includes judicious citation of current scholarly writing, it is not so much a scholarly book as a practical book aimed at influencing teachers. The first three chapters describe Small’s approach to assessment and feedback, and these introductory chapters seem to be carefully constructed for a wide audience. Practicing teachers in grades 3 to 8 will not be put off by the generous tone. Furthermore, they will find authentic examples pertaining to the content that they teach. Small uses contrasting examples to argue that teachers’ different beliefs can have profound consequences for assessment and feedback. She goes on to make a subtle and palatable case for the possibility and value of classroom assessment that goes beyond checking for certain skills to provide teachers with evidence of students’ reasoning. On the whole, the book is more about showing than telling. Most of the book’s length is devoted to providing and analyzing detailed sets of examples that illustrate how teachers can elicit informative student responses and reply to them in mathematics classrooms.


In the first chapter, Small begins with the observation that instructional topics do not determine assessment, illustrating how the same curriculum standard can be assessed in very different ways depending on what teachers value, their beliefs about what matters, and their interpretations of the standard. She contrasts two hypothetical teachers’ assessment practices across three examples: grade 3 measurement and data, grade 5 number and operations, and grade 7 ratios and proportional reasoning. The analysis of these examples shows that focusing on skills and procedures versus concepts has strong implications for assessment design and practice.


In the second chapter, Small presents a framework with three categories of assessment: assessment for learning (formative), assessment as learning (self-reflective), and assessment of learning (summative). Each category description is accompanied by examples and discussion. Most novel in this group is the category of assessment as learning, which is focused on students’ self-assessment. Small argues that students will make self-assessments based on the teachers’ values about the relative importance of correct details versus the overall product and the relative importance of the answer versus the process taken to obtain an answer. A key insight from this chapter is that students’ self-assessments will be made based on their perceptions of the teacher’s values, even if these values are left unsaid. Explicitly articulating values as “success criteria” for specific tasks can enable the teacher to better direct and manage student self-assessment.


In the third chapter, Small classifies feedback as communication and describes six different kinds of feedback, each useful for a different purpose. There is also a discussion about ways that teachers can make feedback more useful, for example, by asking questions instead of making comments and cultivating non-evaluative reactions to student work. A particularly helpful section identifies three common opportunities to provide feedback: student’s overgeneralizations, inappropriate assumptions, and common misconceptions.


The remaining chapters all have the same structure. Each chapter targets one big idea for a specific grade and provides a rich set of example tasks, samples of student work for the tasks, and examples of feedback on student work. The big ideas are drawn from operations and algebra in grade 3, number and operations in base ten in grade 4, geometry in grade 5, ratios and proportional relationships in grade 6, statistics and probability in grade 7, and expressions and equations in grade 8. The examples in these chapters can serve as a source for readers to understand the fundamental ideas about assessment and feedback that Small discusses in the first three chapters. She offers a list of diagnostic tasks and questions that allow teachers to learn about students’ prior knowledge. She also provides tasks designed for formative assessment, success criteria for some of the tasks, suggestions about what to focus on and how to ask follow-up questions, student work samples, and examples of specific feedback. Readers will find that these examples are accessible, informatively illustrate the strategies, and are practical in that they can easily be tried and adapted. There is not comprehensive coverage of the grade 3 to 8 curriculum standards, but any grade 3 to 8 teacher should be able to find examples to which they can easily relate.


One particularly welcome feature of this book is that it offers real student samples with detailed feedback for many of the example tasks. These samples are authentic and clearly communicate how feedback can help students gain insight into their own work and thus promote mathematical thinking. For example, on page 86, a student work sample includes the claim that: “all circles have this property… they all have no perimeter.” Small points out that this student likely believed that perimeter could only be used with polygonal shapes. She then provides example feedback to challenge this underlying belief with focused questions such as: “What does perimeter actually mean? If you created a string that just fit around the circle and made a square with that string, would the square have a perimeter?”


Readers may be surprised at the limited discussion of assessment as learning, Small’s term for self-reflective, student-directed assessment. After the promising introduction to these ideas in Chapter Two, little attention is paid to this form of assessment in subsequent chapters. There are examples of success criteria for some of the tasks in Chapters Four through Nine, but rather than providing tailored discussion of this category of assessment as it relates to the big idea or the example tasks, each of these chapters only provides the same, generally worded three sentence discussion of assessment as learning that primarily functions to point readers back to Chapter Two.


Although most of the potential audience for this book will likely be very familiar with assessment of learning (summative assessment), Small introduces some new perspectives on how to conduct holistic and comprehensive assessment in the classroom. She designs questions to collect multiple layers of evidence for teachers to evaluate student learning, such as skill, concept, performance, and observation. She also offers suggestions for how to weight these various pieces of evidence. Asking questions that promote student conceptual understanding is particularly challenging work for many teachers, so readers may find the conceptual questions in this book especially helpful in that regard. In the many examples Small provides, she creatively asks different kinds of questions that push students to understand mathematics ideas deeply. For example, her questions illustrate ways to create cognitive conflict in order to address student learning difficulties, such as by asking, “Why would somebody say that the 5s in 5055 are really not all the same?” (p.73). She also offers ways to ask focused questions that promote deep understanding and connections, such as, “Explain how and why using… 48 and 28 to figure out 68 works” (p. 55).


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.



Cite This Article as: Teachers College Record, Date Published: September 24, 2019
https://www.tcrecord.org ID Number: 23103, Date Accessed: 5/26/2022 2:28:44 PM

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About the Author
  • Erik Jacobson
    Indiana University
    E-mail Author
    ERIK JACOBSON is an associate professor of mathematics education at Indiana University. Recent publications include “Prescribing Structure for Validation Arguments” and “A Replication Framework for Mathematics Education,” and current projects include “Assessing the Structure of Knowledge in Teaching Mathematics” and “Teachers’ Attributions of Mathematical Excellence.” Both projects are focused on developing new assessments to provide actionable feedback to improve mathematics teacher education.
  • Jinqing Liu
    Indiana University
    E-mail Author
    JINQING LIU is a doctoral candidate in Mathematics Education at Indiana University. Areas of interest include pre-service teacher education, professional development of in-service teachers, and fraction teaching and learning. Recent publications include “‘I’m Not Just a Math Teacher’: Understanding the Development of Elementary Teachers’ Mathematics Teacher Identity” and “Integrating Social-Educational and Academic Development in Teachers’ Approaches to Educating Students" (in press). She is currently participating in projects including “Assessing the Structure of Knowledge in Teaching Mathematics” and “Supporting Teachers to Promote Students’ Mathematical Generalization.”
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