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

Courtney Nagle, Deborah Moore-Russo

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1-18
Number of pages18
JournalInvestigations in Mathematics Learning
Issue number2
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • Education
  • General Mathematics


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