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

Abstract

Background and IntroductionIn recent decades, algebra has become infamous as a gatekeeper of success in school (Cai et al., 2005; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Stephens, 2008). Moses and Cobb (2001) underscored 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 Industrial Age (p. 14). Unfortunately, though, it has been documented repeatedly many students struggle when they reach algebra in middle school or high school (e.g., Kenney & Silver, 1997).In response to phenomenon, members of education community have called for inclusion of algebra content in elementary school curriculum, with goal of removing abrupt, often derailing transition from arithmetic to algebra by infusing algebraic ideas into in elementary and intermediate grades (Kaput, 1998). For example, National Council of Teachers of Mathematics (NCTM, 2000) suggests incorporating algebra into elementary level curricula:By viewing algebra as a strand in 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 middle grades and high school, (p. 37)Research also lends support for 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 current body of research emphasizes 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 in a way is appropriate for elementary-aged children.Accordingly, algebra topics have been included in recent standards documents as an essential component of elementary curriculum. The NCTM's Principles and Standards for School Mathematics (2000) states students in all grades should develop their understanding of 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 Common Core State Standards (CCSS; National Governors Association Center for Best Practices, Council of Chief State School Officers [NGA & CCSSO], 2010), 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, 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 elementary school teachers are responsible for facilitating their students' development in algebraic concepts, and, therefore, they need to have a deep understanding of foundations of algebra themselves (Hill, Rowan, & Ball, 2005; Ma, 1999). Moreover, members of education community support notion that there is a powerful relationship between what a teacher knows, how she knows it, and what she can do in context of instruction (Hill, Blunk, et al., 2008, p. 498].Thus, mathematical education of PTs in algebra is of critical importance to quality of mathematical education of children. This is reflected in recently updated recommendations of Mathematical Education of Teachers II (METII; Conference Board of Mathematical Sciences, 2012] report, which states kindergarten through Grade 5 teachers need to be able to [recognize] foundations of algebra in elementary mathematics (p. …