Building Proportional Reasoning Across Grades and Math Strands, K8reviewed by Temple A. Walkowiak  January 12, 2016 Title: Building Proportional Reasoning Across Grades and Math Strands, K8 Author(s): Marian Small Publisher: Teachers College Press, New York ISBN: 0807756601, Pages: 115, Year: 2015 Search for book at Amazon.com Marian Small provides a practical resource for practicing teachers, preservice teachers, and teacher educators in her new book, Building Proportional Reasoning Across Grades and Math Strands, K–8. People often think of proportional reasoning as a skill that students develop in middle school and beyond. However, young children as early as Kindergarten need to engage in activities that build their proportional reasoning and multiplicative thinking (Lamon, 2012). Small explains how proportional reasoning is embedded in the Common Core State Standards for Mathematics (CCSSM) (National Governors Association Center for Best Practice, 2010) from Kindergarten through Grade 8 and provides good questions that can build the proportional reasoning we want all students to develop. The book starts with a brief introduction in which Small defines proportional thinking, how we use it, and where it shows up in the mathematics curriculum. She makes an important point that proportional reasoning is not explicitly mentioned in the CCSSM until sixth grade, but “its roots appear much earlier” (p. 2). With a succinct approach, Small introduces the idea that we use proportional reasoning in our everyday lives. This provides the reader with a shared—and perhaps clearer—understanding before reading further into the book. In the introduction, Small also provides a bulleted list of essential understandings that need to be developed across K–8. She also shows her readers how she has paid attention to the CCSSM Standards for Mathematical Practice by providing examples with page references for each of the eight practices (e.g., model with mathematics, attend to precision, etc.). Unfortunately, while listing examples, Small does not offer explanations why the example sets promote given practices among students. She ends the introduction with very brief attention regarding how the content of this book is aligned with the recent publication of Principles to Actions (2014), from the National Council of Teachers of Mathematics (NCTM), and how the book is structured to directly address the CCSSM content standards related to proportional thinking. After the introduction, there are two parts to the book. The first part has three chapters, one for each grade within K–2, and the second part has six chapters, for the subsequent set of grade levels (3–8). Small titled the first part which focuses on early grades as “opportunities to build a foundation for proportional thinking” whereas the second part is called “focusing on proportional thinking.” Each chapter or grade level section is organized in the same way. Small presents the relevant standard(s) from the CCSSM, explains the underlying mathematical ideas of the standard(s) as they relate to proportional reasoning, and then shares three to seven good questions for students that address their mathematical ideas and build proportional thinking. She explains the underlying mathematical ideas succinctly and helpfully. These explanations can serve teachers well as they increase their mathematical knowledge to support implementation of the CCSSM. Small’s good questions vary in terms of how long it would take a teacher to implement them, but it seems these questions could also be called tasks if you choose to use the language often used in mathematics education (Stein, Smith, Henningsen, & Silver, 2009). For those who have used one of Small’s previously published books, such as Good Questions: Great Ways to Differentiate Mathematics Instruction (2009), there are striking similarities in terms of structure. As in the past, she explains the mathematical content then provides explicit ideas for teachers to use in their classrooms. The simple structure of the book seems to make it easy for readers to use, and past readers of her books will find a similar layout. While Small’s book provides foundational support in a userfriendly format, I would like to offer three critiques that would make the book a more comprehensive resource. First, teachers could benefit from understanding how good questions fit within a lesson or a sequence of lessons. Offering a sequence of good questions that may be asked within a lesson, or across lessons, would help teachers understand how these given questions progress over time within a unit of instruction. For example, Small unpacks a fifthgrade standard in the CCSSM that focuses on “interpreting fraction multiplication in terms of unitizing” (pp. 5859). She provides three good questions after unpacking the mathematical content but, from a teacher’s perspective, it is unclear as to how or when I should be using those questions in my sequence of instruction. Second, and related to the first critique, teachers tend to struggle with anticipating student mathematical thinking, particularly in the elementary grades. If each good question also included anticipated student responses, teachers could likely better plan their lessons with probing questions. Further, providing teachers with anticipated student thinking would likely help them with the aforementioned concern of knowing how to situate good questions within their lessons. Third, I would suggest adding a section to the already helpful introduction of the book. It would be useful for teachers to understand how the standards or mathematical topics progress across grade levels relative to proportional reasoning. A table or other organizational framework would be helpful in giving teachers a vertical understanding of the mathematical ideas being taught before moving to the finely grained details of each grade level. Overall, Building Proportional Reasoning offers a practical resource for educators to build their understanding concerning how proportional reasoning can be developed across K–8 classrooms. The explanations of underlying mathematical ideas for each CCSSM proportional reasoning standard is a particular strength that deserves mention. This book can be used by practicing teachers, preservice teachers, and teacher educators to increase their mathematical knowledge as it relates to the teaching and learning of proportional thinking in K–8 classrooms. Perhaps this book could be paired with another resource in order to provide other types of support that are not presently included (e.g., anticipated student thinking). Small makes a strong case for building proportional thinking throughout the elementary and middle school years instead of delaying this important work until middle school when the topic is explicitly mentioned in the curriculum. References Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). New York, NY: Routledge. National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. National Governors Association Center for Best Practice & Council of Chief State School Officers (2010). Common Core State Standards for Mathematics. Washington, DC: Author. Small, M. (2012). Good questions: Great ways to differentiate mathematics instruction (2nd ed.). New York, NY: Teachers College Press. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standardsbased mathematics instruction: A casebook for professional development (2nd ed.). New York, NY: Teachers College Press.
