Algebra in the Early Grades

James J. Kaput, David W. Carraher, Maria L. Blanton, James J. Kaput, David W. Carraher, Maria L. Blanton

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781351577083
• Edition: 1
• Topic: Bildung
• Subtopic: Bildung Allgemein

This volume is the first to offer a comprehensive, research-based, multi-faceted look at issues in early algebra. In recent years, the National Council for Teachers of Mathematics has recommended that algebra become a strand flowing throughout the K-12 curriculum, and the 2003 RAND Mathematics Study Panel has recommended that algebra be "the initial topical choice for focused and coordinated research and development [in K-12 mathematics]."

This book provides a rationale for a stronger and more sustained approach to algebra in school, as well as concrete examples of how algebraic reasoning may be developed in the early grades. It is organized around three themes:

• The Nature of Early Algebra
• Students' Capacity for Algebraic Thinking
• Issues of Implementation: Taking Early Algebra to the Classrooms.

The contributors to this landmark volume have been at the forefront of an effort to integrate algebra into the existing early grades mathematics curriculum. They include scholars who have been developing the conceptual foundations for such changes as well as researchers and developers who have led empirical investigations in school settings. Algebra in the Early Grades aims to bridge the worlds of research, practice, design, and theory for educators, researchers, students, policy makers, and curriculum developers in mathematics education.

I
THE NATURE OF EARLY ALGEBRA

This part leads off the early algebra story with examinations of what algebra and algebraic thinking are, how they relate to more general symbolization processes, and their foundation in naturally occurring human powers that appear at the very earliest stages of human development. Chapter 1 chronicles the history of what we term the algebra problem and defines, from a content point of view, what we mean by algebra and algebraic reasoning. Recognizing that algebra is both an inherited body of knowledge and something that people do, it offers an explicit description of two core aspects and three strands of school algebra that appear in virtually all later chapters of the book.

Chapter 2 describes where these essential aspects come from and how they arise as two related types of more universal symbolization processes. In particular, using examples from elementary mathematics, it discusses how symbols and referents come into being as separate entities in our experience, a deep and subtle constructive process of recording, reflecting, and revising. The authors discuss the common phenomenon of attention switching, where thinking sometimes is guided by what the symbols are taken to stand for, and sometimes by the forms of the symbols themselves (including syntax). They also offer a distinction between non-algebraic and algebraic reasoning based on the purposeful expression of and reasoning with generality, and a distinction between proto-algebraic reasoning and (mature) algebraic reasoning based on the latter’s use of conventional symbol systems.

The two symbolization processes, and especially the first, can be seen throughout the episodes described in the book. However, they come into particular relief in chapter 5, where E. Smith, using the slightly different language of representational thinking and symbolic thinking for the two core aspects, applies them primarily in the context of Strands 2 (Function and Variation) and 3 (Modeling). His classroom episodes are taken from professional development work with elementary teachers and are intended to suggest approaches to mathematics and algebraic thinking that are common across age and grade levels. Of special note is his attention to the important and often neglected role of argumentation and the establishing of certainty, both at the social and the personal levels. Given the importance of generalization and the expression of generality in our characterization of algebraic reasoning, the fact that he regards establishing mathematical certainty as a driver of representational thinking adds a major ingredient to our characterization. In this chapter, E. Smith focuses explicitly on the role of argumentation in establishing generality, but the reader will see argumentation playing this role across many chapters as both a form of student mathematical activity and as a pedagogical goal in teacher practice. Indeed, across virtually all cases, we see the importance of active classroom discussion, where students can test the validity of generalizations, the assumptions behind them, and their appropriate range of applicability.

In chapter 3, Mason argues that children arrive at school in possession of all the powers needed to learn how to think algebraically. He sees these as including imagining and expressing, focusing and de-focusing, specializing and generalizing, conjecturing and convincing, as well as classifying and characterizing objects and processes. They have their roots in the earliest recognition of patterns by infants and in their earliest stages of language development. They continue to develop into the broad cognitive and linguistic competencies that children possess when they arrive at school, especially (but not exclusively) children’s ability to form, argue for, and express generalizations using natural language. He suggests that we would greatly improve our mathematics education enterprise if we were to take these powers seriously as resources for mathematics teaching and learning. A key goal of his chapter is to convince us that these student powers are indeed the same ones, in different contexts and in nascent forms, perhaps, that are used in serious mathematics learning. He closes with a consideration of some of the reasons that these powers are not well tapped in elementary school, including the common assumptions behind the rush to achieve certain topic coverage and procedural skill development and, even more constraining, the low expectations regarding young students’ abilities to engage productively in mathematical activities that we would describe as algebraic.

In chapter 4, J. Smith and Thompson take the point of view that, in order for students to be able to learn and use algebraic statements, these statements need to be experienced as being about something. They suggest that whereas many researchers treat reasoning with and about numerical relationships and operations as a basis for algebraic reasoning, an alternative is reasoning with and about physical quantities and quantitative relationships—quantities such as lengths, weights, times, areas, speeds, and so on, where measurement and units come into play—as opposed to reasoning about numbers, operations, and their properties. By examining a variety of problem situations, they show the depth and richness of extended experience with quantitative reasoning across Grades K–8, how it is of intrinsic value as a way of making sense of many different kinds of situations and phenomena, how it is different from algebraic reasoning, and how algebraic reasoning can be built on a foundation of quantitative reasoning. Relative to the last point, they provide examples illustrating the generality of quantitative reasoning and how it draws the student toward ways of expressing that generality. Hence, it makes the power of algebra immediately apparent once the stage is set. They point out that the traditional elementary mathematics curriculum seldom puts students in this position, where algebra really pays off.

A goal of the chapters in this section is to orient the reader to the whole enterprise of building algebraic reasoning in the contexts of elementary grades mathematics. Although we see much commonality across the chapters, especially in the functional respect given to children’s ways of making sense of their world and the central place given to active student expression of their ideas and processes, we also see that differences in the context, reflected in different strands of algebraic reasoning, yield real differences in how algebraic reasoning emerges.

1
What Is Algebra? What Is Algebraic Reasoning?

James J. Kaput

University of Massachusetts, Dartmouth

This introductory chapter provides a shared road map of algebra in the elementary grades and an historical perspective on why we might need such a road map. Because the landscape is quite varied, it is useful to know where we are in relation to the larger territory and what has brought us to this point. We begin with an account of what we have termed the algebra problem and of the evolution in the research base on learning algebra.

The Algebra Problem

The scholars in this book address what for at least 10 years we have termed the algebra problem. It is especially acute in the United States where, historically, the introduction of school algebra has awaited the completion of a 6- to 8-year computational arithmetic curriculum that has been implemented as independent of algebra. At the turn of the 20th century, with universal elementary schooling already in place, shopkeeper arithmetic was expected for all. Algebra was reserved for the elite, which amounted to a scant 3% to 5% of the population who completed secondary school at that time (National Center for Education Statistics, 1994). Thus, the arithmetic-then-algebra curriculum structure was already well in place as U.S. society evolved toward universal secondary education during the 20th century. The resulting late and abrupt approach to the introduction of algebra was deeply institutionalized and regarded as the natural order of things (Fey, 1984). But, by the dawn of the 21st century, this highly dysfunctional result of the computational approach to school arithmetic and an accompanying isolated and superficial approach to algebra had led to both teacher alienation and high student failure and dropout, especially among economically and socially less advantaged populations. This is the result of the convergence of two unprecedented forces: (a) in response to societal needs for a deeper mathematical literacy, all students are now expected to learn algebra under the “algebra for all” banner; and (b) this expectation is legally codified in high-stakes accountability measures that define academic success in terms of success in algebra.

Political agendas aside, there are some salutary reasons for rethinking and reworking algebra in early grades mathematics. Solving the algebra problem serves four major goals:

1. To add a degree of coherence, depth, and power typically missing in K–8 mathematics.
2. To ameliorate, if not eliminate, the most pernicious and alienating curricular element of today’s school mathematics: late, abrupt, isolated, and superficial high school algebra courses.
3. To democratize access to powerful ideas by transforming algebra from an inadvertent engine of inequity to a deliberate engine of mathematical power.
4. To build conceptual and institutional capacity and open curricular space for new 21st-century mathematics desperately needed at the secondary level, space locked up by the 19th-century high school curriculum now in place.

Changes of this magnitude require deep rethinking of the core algebra enterprise and will not be achieved by minor adjustments such as attempting to fix a first algebra course, starting it a year earlier, or legislating 2 years of algebra for all. As Carraher and colleagues argue, early algebra is decidedly not (traditional) algebra early.

Solving the algebra problem involves deep curriculum restructuring, changes in classroom practice and assessment, and changes in teacher education—each a major task. Further, each must be achieved within the capacity constraints of the teaching population, within the limited time and resources available for in-service and preservice teacher development, and within the constraints of widely used instructional materials. Steps in this direction are underway on several fronts. The Principles and Standards of School Mathematics (National Council of Teachers of Mathematics, 2000) has advocated an increasingly longitudinal view of algebra, that is, a view of algebra not as an isolated course or two, but rather as a strand of thinking and problem solving beginning in elementary school and extending throughout mathematics education. This perspective was reflected in the original Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, 1989) and now is expressed in curriculum and professional development projects and documents that exemplify the principles and standards highlighting the development of algebraic reasoning in the earlier grades (Cuevas & Yeatts, 2001; Greenes, Cavanagh, Dacey, Findell, & Small, 2001). Some of this groundbreaking work is described in the present volume.

The State of Research and its Evolution Over the Past 30 Years

The algebra problem has been brought to the fore in our thinking by research that spans several decades. Through the 1980s, research in algebraic thinking and learning focused on student errors and constraints on their learning, especially developmental constraints. A large body of evidence was accumulated that showed students tended to have fragile understanding of the syntax of algebra (e.g., Matz, 1982; Wenger, 1987) or that they had difficulty in interpreting algebraic symbols (e.g., Clement, 1982; Clement, Lochhead, & Monk, 1981; Kaput & Sims-Knight, 1983) or even coordinate graphs (Clement, 1989). Perhaps we should not be surprised, given the curriculum that students experienced and the fact that for many years most research either measured the given curriculum’s affect on students under traditional classroom circumstances or took the form of brief interventions aimed at teaching symbol manipulation techniques based on the same narrow syntactical view of algebra that defined the dominant curriculum (e.g., Lewis, 1981; Sleeman, 1984, 1985, 1986). However, recent research on the status of student knowledge based in the traditional arithmetic-then-algebra regime has pointed to specific obstacles to algebra learning that computational arithmetic creates for the learning of ...