Algebra in the Early Grades
reviewed by David Slavit - November 20, 2008
Title: Algebra in the Early Grades
Author(s): James J. Kaput, David W. Carraher, and Maria L. Blanton (Eds.)
Publisher: Lawrence Erlbaum Associates, Inc., Mahwah, NJ
ISBN: 080585472X, Pages: 552, Year: 2007
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This book review is biased. Jim Kaputs place in mathematics education is reverent. And like many others, I have a special place in my heart for his work, but more for who he was and for what he worked. It would be hard for me to write a poor review of this book.
Having said that, I can as objectively as possible say that Algebra in the Early Grades is an important collection of work that addresses a multitude of key issues on the topic. The book is divided into three parts: a) the nature of early algebra (5 chapters) explores the conceptual, representational, and cognitive aspects of the topic from a more theoretical lens; b) students capacity for algebraic thinking (7 chapters), described by the editors as the heart of the book, examines specific instances of childrens reasoning amidst a variety of early algebra topics; and c) issues of implementation deals with instruction, curriculum, and professional development.
While some of the ideas in the beginning section stem from older work (the authors state the book comes from 15 years of thinking), they are collectively a powerful framework for considering the student thinking and implementation issues explored in the later chapters. In Chapter 1, Kaput describes the three strands of early algebra that contain its core aspects: a) structures and systems abstracted from computation and relations (these include generalized arithmetic and quantitative reasoning); b) functions; and c) modeling. Subsequent chapters in this initial section expound and expand on these central ideas, including Thompsons exploration of quantitative reasoning and Smiths discussion of representations of early algebraic ideas.
The many cases of students algebraic thinking presented in the middle of the book are best summarized by Masons approach to the books topic. In Chapter 3, after describing childrens enormous powers for making sense of the worlds they inhabit, Mason states that algebraic thinking is what happens when those powers are used in the context of number and relationships. The chapters on students algebraic thinking seem to take these words to heart. They highlight, through empirical analyses based mostly on student observation, the variety of ways in which students are capable of thinking algebra. All of these chapters are rich in narrative and provide detailed analyses of childrens approaches to and understandings of algebraic ideas and situations. In a wonderful chapter called Early Algebra is not Algebra Early, Carraher, Schliemann, and Schwartz focus on student work in the context of the Candy Box Problem. This detailed account of how Carraher led children to utilize their arithmetic proficiency and pattern-finding skills to generate an understanding of variable is a remarkable story of childrens mathematical potential as well as a specific way in which algebra can be appropriately developed in younger children. Investigating concepts such as slope and extrema, Tierney and Monk also provide a rich account of childrens development of variable and function. The collection of case-based chapters is useful to researchers who wish to formulate frameworks for investigating childrens algebraic thinking, but also to teacher educators who desire specific ways of framing and enacting professional development in this often-overlooked area.
The book then addresses issues of teacher education head-on, exploring, in the latter part, this and a variety of other issues related to implementation, including instruction and curriculum. Dougherty, working with primary grade children in Hawaii, describes the implementation of an early algebra curriculum based on measurement, including volume. She underscores the need for both teachers and students to think about numbers and operations in new and different ways. These include algebraic aspects embedded in units of measurement (a very powerful discussion is provided involving a young girl who states that 3 or 8 could be bigger, depending on the unit), work in base systems other than 10, the introduction of unknowns through the use of fact families, and decomposing numbers using familiar number facts and relationships. Using work from the Cognitively Guided Instruction project, Franke, Carpenter, and Battey further this discussion by detailing important relationships that exist between content knowledge and instruction. By also paying attention to teacher stance, these authors were able to discuss their ability to address their realization that teachers engaged in (early) algebra lacked confidence and worried about being able to productively engage students in algebraic thinking. Drawing from a variety of notions of community, Blanton and Kaput explore ways in which schools and districts can afford teachers with opportunities to develop abilities and stances conducive to early algebra instruction. The notion of stance, an idea currently permeating teacher education research (Wells, 1999; Jaworski, 2006; Cochran-Smith & Lytle, 2001), provides an important perspective to the instructional implementation issues contained in these chapters.
Algebraic learning, and early algebra in particular, was a passion of Jim Kaput. This book is not only a fitting tribute to his work, but a broad account of theory and research into early algebra and algebraic thinking. A multitude of frameworks and findings are provided that are potentially useful to researchers, teacher educators, and practitioners. However, while important as a coherent and wide-ranging work on the topic of early algebra, the book repeatedly draws from older theoretical models and does not seem to advance the field in a new way in this regard. Perhaps this is a by-product of the general lack of recent progress in this field. Nevertheless, a chapter, or expansions within individual chapters, containing a forward-reaching perspective would have served the book well in thinking about ways of building on the solid foundation the book provides. This might have involved a proposal for a future research program with a call to address specific theoretical and empirical areas, a reframing of the conceptual issues at hand, or a practical plan of action for increasing the presence of algebra in the early grades. For example, addressing the ways in which some recent curricular materials emphasize early algebra, potential impacts of state standards and tests, connections between research on early algebra and later algebra, and a greater attention to the broader contexts of professional development, would be welcome additions to this volume, or perhaps a later volume on this topic. Such future work could pay tribute to the important foundational work on early algebra, spearheaded by Kaput and colleagues, found in this volume.
Cochran-Smith, M., & Lytle, S. L. (2001). Beyond certainty: Taking an inquiry stance on practice. In A. Lieberman & L. Miller (Eds.), Teachers caught in the action: Professional development that matters (pp. 45-58). Teachers College Press: New York.
Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9, 187-211.
Wells, G. (1999). Dialogic inquiry: Toward a sociocultural practice and theory of education. Cambridge, UK: University Press.