Academic journal article The Mathematics Enthusiast

Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Algebra

Academic journal article The Mathematics Enthusiast

Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Algebra

Article excerpt

Background and Introduction

In recent decades, algebra has become infamous as a gatekeeper of success in school mathematics (Cai et al., 2005; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Stephens, 2008). Moses and Cobb (2001) underscored the importance of algebra by making a comparison between people who lack an education in algebra today to "the people who couldn't read and write in the Industrial Age" (p. 14). Unfortunately, though, it has been documented repeatedly that many students struggle when they reach algebra in middle school or high school (e.g., Kenney & Silver, 1997).

In response to this phenomenon, members of the mathematics education community have called for the inclusion of algebra content in the elementary school curriculum, with the goal of removing the abrupt, often derailing transition from arithmetic to algebra by infusing algebraic ideas into instruction in the elementary and intermediate grades (Kaput, 1998). For example, the National Council of Teachers of Mathematics (NCTM, 2000) suggests incorporating algebra into elementary level curricula:

By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more sophisticated work in algebra in the middle grades and high school, (p. 37)

Research also lends support for the notion of including algebraic ideas in elementary school curricula (e.g., Britt & Irwin, 2008; Schliemann et al., 2003). In her research brief on algebra, Kieran (2007) concludes that the current body of research "emphasizes that arithmetic can be conceptualized in algebraic ways" and "this emphasis can be capitalized on to encourage young students to make algebraic generalizations without necessarily using algebraic notation" (p. 1). Thus, algebra can be infused into arithmetic instruction in a way that is appropriate for elementary-aged children.

Accordingly, algebra topics have been included in recent standards documents as an essential component of the elementary mathematics curriculum. The NCTM's Principles and Standards for School Mathematics (2000) states that students in all grades should develop their understanding of the following algebraic ideas:

* understanding patterns, relations, and functions;

* representing and analyzing mathematical situations and structures using algebraic symbols;

* using mathematical models to represent and understand quantitative relationships; and

* analyzing change in various contexts, (p. 37)

More recently, in the Common Core State Standards (CCSS; National Governors Association Center for Best Practices, Council of Chief State School Officers [NGA & CCSSO], 2010), the Operations and Algebraic Thinking content domain begins in kindergarten and continues through fifth grade, progressing from a focus on understanding properties of, and having flexibility with, the four basic operations, toward a focus on generalizing, describing, and justifying patterns and relationships, and interpreting symbolic expressions.

In light of these standards, it is evident that elementary school teachers are responsible for facilitating their students' development in algebraic concepts, and, therefore, they need to have a deep understanding of the foundations of algebra themselves (Hill, Rowan, & Ball, 2005; Ma, 1999). Moreover, members of the mathematics education community support the notion "that there is a powerful relationship between what a teacher knows, how she knows it, and what she can do in the context of instruction" (Hill, Blunk, et al., 2008, p. 498].

Thus, the mathematical education of PTs in algebra is of critical importance to the quality of the mathematical education of children. This is reflected in the recently updated recommendations of the Mathematical Education of Teachers II (METII; Conference Board of the Mathematical Sciences, 2012] report, which states that kindergarten through Grade 5 teachers need to be able to "[recognize] the foundations of algebra in elementary mathematics" (p. …

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