Developing Algebraic Reasoning through Variation in the U.S.
Historically, algebra in the U.S. has been viewed “as a gatekeeper to a college education and the careers such education affords” (Kilpatrick & Izsák, 2008, p. 11). As such, current curriculum documents emphasize the need to support all students in le
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15. DEVELOPING ALGEBRAIC REASONING THROUGH VARIATION IN THE U.S.
INTRODUCTION
Historically, algebra in the U.S. has been viewed “as a gatekeeper to a college education and the careers such education affords” (Kilpatrick & Izsák, 2008, p. 11). As such, current curriculum documents emphasize the need to support all students in learning algebra (Common Core State Standards Initiative [CCSSI], 2010; National Council of Teachers of Mathematics [NCTM], 1989, 2000). To do so, however, requires a reconceptualization of the preparation students receive for the formal study of algebra (Kilpatrick & Izsák, 2008). In considering this preparation, scholars have indicated that students need opportunities to engage in algebraic reasoning (Blanton & Kaput, 2005; Earnest, 2014; Hunter, 2014; Kaput, 2008; Kilpatrick & Izsák, 2008). Different perspectives exist, though, with regard to the core aspects of algebraic reasoning. Kaput (2008) characterized algebra in two ways. First, he described algebra as an inherited subject or cultural artifact. Second, Kaput portrayed it as a human activity that requires humans for it to exist. In our work, we focus on the latter and explore Kaput’s (2008) view that “the heart of algebraic reasoning is comprised of complex symbolization processes that serve purposeful generalization and reasoning with generalizations” (p. 9). Within this view of algebra, Kaput (2008) described a core aspect of algebraic reasoning as involving “algebra as systematically symbolizing generalizations of regularities and constraints” (p. 11). Although this core aspect appears in some form across all strands of algebra, we are particularly interested in algebraic reasoning as it supports generalizing a pattern through argumentation for the purpose of building towards functions (Kaput, 1999; Warren & Cooper, 2008). This view of algebraic reasoning has permeated recent international curriculum documents (e.g., Ministry of Education, 2007; Ontario Ministry of Education, 2005) as well as U.S. curriculum documents for over two decades. Table 1 provides an overview of the algebraic presence in U.S. curriculum documents, including Curriculum and Evaluation Standards (CES, NCTM, 1989), Principles and Standards for School Mathematics (PSSM, NCTM, 2000), and Common Core State Standards for Mathematics (CCSSM, CCSSI, 2010).
R. Huang & Y. Li (Eds.), Teaching and Learning Mathematics through Variation, 321–339. © 2017 Sense Publishers. All rights reserved.
A. T. Barlow et al.
The inclusion of algebraic reasoning in U.S. standards is informed, in part, by a literature base that supports a need to develop algebraic reasoning in middle school students (Blanton, 2008; Carraher & Schliemann, 2007; Lins & Kaput, 2004; Soares, Blanton, & Kaput, 2005). Note that we define middle school students as those in grades five through eight, approximately 11 through 14 years old. Additionally, algebraic reasoning is described as the process of building general mathematical relationships and expressing those relationships in increasingly sophistic
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