What Does it Mean to “Understand” Concavity and Inflection Points?

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The calculus concepts of concavity and inflection points are often given meaning through the shape or curvature of a graph.  However, there appear to be deeper core ideas for these two concepts, though the research literature has yet to give explicit attention to what there core ideas might be or what it might mean to “understand” them.  In this paper, I propose a framework for the concavity and inflection point concepts, using the construct of covariation, wherein I propose conceptual (as opposed to mathematical) definitions that can be used for both research and instruction.  I demonstrate that the proposed conceptual definitions in this framework contain important implications for the teaching and learning of these concepts, and that they provide more powerful insight into student difficulties than more traditional graphical interpretations.

Location: PME-NA 2016 Conference, Tucson, Arizona

Steven Jones, Brigham Young University