The Concept of Slope: Comparing Teachers’ Concept Images and Instructional Content

In the field of mathematics education, understanding teachers’ content knowledge (Grossman, 1995; Hill, Sleep, Lewis, & Ball, 2007; Munby, Russell, & Martin, 2001) and studying the relationship between content knowledge and instructional decisions (Fennema & Franke, 1992; Raymond, 1997) are both crucial. Teachers need a robust understanding of key mathematical topics and connections to make informed choices about which instructional tasks will be assigned and how the content will be represented (Ball & Bass, 2000, Fennema & Franke, 1992). Ma (1999) described this profound understanding of fundamental mathematics as how accomplished teachers conceptualize key ideas in mathematics with a deep and flexible understanding so that they are able to represent those ideas in multiple ways and to recognize how those ideas fit into the preK-16 curriculum. Slope is a fundamental topic in the secondary mathematics curricula. Unit rate and proportional relationships introduced in sixth grade prepare students for interpreting equations such as y = 2x-3 as functions with particular, linear behavior in eighth grade (National Governors Association [NGA] Center for Best Practices & Council of Chief State School Officers [CSSO]), 2010; National Council of Teachers of Mathematics [NCTM], 2006). The focus on relationships with constant rate of change leads to distinctions between linear and non-linear functions (Yerushalmy, 1997) and the idea of average rate of change in high school (NGA Center for Best Practices & CCSSO, 2010). Ultimately, these ideas prepare students for instantaneous rates of change and the concept of a derivative in calculus. The diversity of conceptualizations and representations of slope across the secondary mathematics curriculum presents a challenge for secondary teachers. These teachers must work flexibly and fluently with various representations in many contexts in order for their students to build a coherent, connected conceptualization of slope. Since secondary mathematics teachers need a deep understanding of slope to mediate students’ conceptual development of this key topic, the study reported here investigates both how teachers think about and present slope.