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How Students Think When Doing Algebra

reviewed by Amy M. Olson & Rachel Ayieko - August 20, 2019

coverTitle: How Students Think When Doing Algebra
Author(s): Steve Rhine, Rachel Harrington, & Colin Starr
Publisher: Information Age Publishing, Charlotte
ISBN: 1641134119, Pages: 350, Year: 2018
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In How Students Think When Doing Algebra, Steven Rhine, Rachel Harrington, and Colin Starr dive deep into the empirical literature to synthesize over 900 articles with implications for students’ thinking in algebra lessons. The authors state that their purpose is “to accelerate early career teachers’ experience with how students think when doing algebra as well as to supplement veteran teachers’ KCS [Knowledge of Content and Students]” (p. 4). Their focus on student reasoning is designed to help teachers move past whether students’ answers are correct or incorrect to instead focus on helping students build meaningful understandings of algebraic concepts. In this review, we provide a brief summary as well as suggestions for how the book can be used as a resource for teaching.


In the introduction, the authors take an explicitly Piagetian constructivist perspective on student learning. Piaget was fascinated with how children’s thinking differs from adults in ways that result in systematic errors. The authors are similarly fascinated with systematic errors that students make when learning mathematics. The examples they draw from the literature and from their own experience focus teachers’ attention on the difference between how teachers think and how students think about algebra as well as the difference between errors caused by simple mistakes and errors caused by mathematical misconceptions. Examples from student work make clear that when operating from misconceptions, the errors students make are reasonable responses to the tasks they have been assigned. Importantly, the use of student voices in the text gives the reader practice in exploring and reflecting on student misconceptions while they are not simultaneously in the midst of teaching a lesson.


The text is dense and may initially appear intimidating, but the applied focus on helping students to develop meaningful understandings of the core ideas of algebra makes the book worthwhile for both novice and advanced teachers. Each of the five content-focused chapters seeks to answer four questions (p. 22):

1. What is the symbolic representation of the misconception or way of thinking about the algebraic idea? (What does it look like?)

2. How do students think about the algebraic idea or misconception? (What does it sound like?)

3. What are the underlying mathematical issues involved? (Why do they do what they do? When are the misconceptions likely to surface?)

4. How might you address this misconception or way of thinking about the algebraic idea? (What strategies could a teacher use to help students understand the concept?)


This framing helps teachers make explicit connections to how the mathematical misconceptions students tend to have around algebraic concepts appear in the classroom as well as how they are elicited in mathematics education research. Each chapter includes an introduction of the misconceptions students may have in the content area paired with relevant example tasks. These are aligned with specific Common Core State Standards for Mathematics (CCSS-M) in grades six to eight and secondary school. Each section concludes with research-based suggestions for teaching strategies to support overcoming the described misconceptions.


Chapter Two, “Variables and Expressions,” explores the different ways variables are understood by students (e.g., as unknown quantities that must be solved or as abstracts that can be manipulated). The chapter also describes how incomplete and incorrect understandings of variables can result in translation difficulties, similar to what students may experience with word problems. The authors provide strategies for teachers to help students think about the meaning of variables in context and accept that answers can be expressions instead of, or in addition to, numbers. The key concept here is the acceptance of a lack of closure.


Chapter Three, “Algebraic Relations,” builds on the understanding of variables to examine misconceptions about equivalence. The authors deepen the readers’ conceptualization of potential errors by asking teachers to question if student methods are incorrect or simply inefficient. Teaching supports include ideas for working with manipulatives, real-world problems, and algebra apps. Teachers are encouraged to “interview” students about their reasoning (p. 122).


Chapter Four, “Analysis of Change (Graphing),” identifies misconceptions related to ratios and rates that cause systemic errors when students try to represent change or interpret graphs. The inclusion of student-drawn representations from the literature is convincing evidence of the importance of encouraging informal graphing prior to formal graphing in order to open conversations about student conceptions of change. The authors caution teachers about overreliance on procedures and instead stress building students’ conceptual understanding around representational fluency.


Chapter Five, “Patterns and Functions,” focuses on misconceptions that can be created when students are taught to “apply tricks and overuse specific function types” (p. 239). The authors provide suggestions on the learning trajectory of functions. The strength of this chapter lies in pairing common strategies (e.g., the vertical line test, function machines) with scripts of student voices to demonstrate how to identify misconceptions and help students develop correct understandings of functions.


Chapter Six, “Modeling and Word Problems,” identifies story problems as both valued in the standards and valuable for the ways they can support student modeling of familiar arithmetic strategies that teachers can then connect to algebraic thinking. This chapter focus is well-placed at the end of the text because the word problems include all of the algebra concepts discussed. The authors emphasize the importance of promoting understanding of the context of the problem instead of focusing on key words. Given how ubiquitous word problems are, this chapter is probably the most useful for the widest range of grade levels.

The authors wish to support novice teachers to be more effective. However, the density of the text may be challenging for beginning teachers to read, depending on their degree of content expertise. We believe this text is best understood as a reference book and would be helpful for both novice and veteran teachers to reach for when they are planning lessons to ensure they provide opportunities to see, hear, and respond to student misconceptions. To this end, it is appreciated that each chapter makes connections to CCSS-M standards. We further believe that the text would be a useful support to developing preservice teachers’ specialized content knowledge. The framing of the text around central ideas of algebraic thinking can help developing content experts to form coherent understandings of content that may not be well integrated in their own past or current studies or in the existing curricula they are accessing (e.g., function- and equation-based approaches to instruction). The book is also useful for expanding current and future teachers’ knowledge of available technologies (e.g., algebra apps) that can facilitate student learning and teacher professional development.


Overall, How Students Think When Doing Algebra provides a thorough and cohesive understanding of potential student misconceptions when learning key concepts of algebra. We appreciate the ways this book will influence our teaching and believe it is a worthwhile addition to the reference libraries of preservice, novice, and advanced teachers as they strive to develop lessons that provide opportunities for students to reason about difficult algebra-related concepts.



Cite This Article as: Teachers College Record, Date Published: August 20, 2019
https://www.tcrecord.org ID Number: 23058, Date Accessed: 12/3/2021 7:55:24 AM

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About the Author
  • Amy Olson
    Duquesne University
    E-mail Author
    AMY M. OLSON is an Assistant Professor of Education at Duquesne University in Pittsburgh, Pennsylvania. Her research examines the equity implications of teacher beliefs about content, students, and what it means to be successful in the classroom, often in the context of mathematics classrooms. She publishes in a variety of journals, such as School Science and Mathematics, Urban Education, and Teachers College Record.
  • Rachel Ayieko
    Duquesne University
    E-mail Author
    RACHEL A. AYIEKO is an Assistant Professor of Mathematics Education a Duquesne University in Pittsburgh, Pennsylvania. Her research interests focus on opportunities to learn mathematics and how to teach mathematics cross-nationally using large scale data sets and classroom observation tools. She has published articles in journals such as Journal of Mathematical Behavior, Global Education Review, and Teachers College Record.
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