Type of Document	Dissertation
Author	Haciomeroglu, Erhan Selcuk
URN	etd-04262007-145845
Title	Calculus Students' Understanding of Derivative Graphs: Problems of Representations in Calculus
Degree	Doctor of Philosophy
Department	Middle and Secondary Education, Department of
Advisory Committee	Advisor Name Title Leslie Aspinwall Committee Chair Eric Chicken Committee Member Kenneth Shaw Committee Member Maria L. Fernandez Committee Member
Keywords	Integral Imagery Representation Visualization Derivative Calculus
Date of Defense	2007-03-28
Availability	unrestricted
Abstract In this study, I developed cases describing three participants – Bob, Jack, and Amy – and their mental imagery, representations, and methods used to create meaning for calculus derivative graphs. Two research questions were investigated: (1) What is the nature of calculus students’ understanding of derivative graphs; (2) how do calculus students create meaning for derivative graphs? During the clinical interviews, the participants were presented with a derivative graph of a function and asked to draw a possible antiderivative graph as I sought to gain understanding of their mental processes and representations. The participants’ interpretations and representational schemes for derivative graphs were different because of their preferences for mathematical processing. Bob and Jack relied on visual processing and graphic representations (or mental images). For them, the derivative graph represented the slopes of the antiderivative graph, and their images or graphic representations of the slopes of the tangent lines determined the graph of the antiderivative graph. Without the support of analytic thinking, their images hindered their understanding. Amy relied on analytic processing and algebraic representations. For Amy, the derivative graph represented an equation (or a function presented with an equation), and the equation of the derivative graph determined the equation as well as the graph of the antiderivative graph. Without the support of visual thinking, her analytic approach presented different difficulties. This study found that since the participants’ knowledge was strongly associated with one mathematical processing (or representation) and weakly associated with the other mathematical processing (or the other representations), their one-sided thinking or over-reliance on one representation impeded their understanding of derivative graphs. Their difficulties with derivative graphs indicate the importance of reversibility of thinking processes, synthesis of analytic and visual thinking and the use of multiple representations in the complete understanding of differentiation and integration. Derivative and antiderivative graphs with a cusp, a sharp corner, a vertical tangent line, or a discontinuity should be used to encourage students to use formal definitions of left- and right-hand derivatives, differentiability, and continuity as well as help them construct appropriate mental images and representations that will facilitate their learning and understanding of calculus.
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