Type of Document	Dissertation
Author	Stringer III, Eddy W.
URN	etd-07032011-191413
Title	African American Students' Graphic Understanding of the Derivative: Critical Case Studies
Degree	Doctor of Philosophy
Department	Teacher Education, School of
Advisory Committee	Advisor Name Title Leslie N. Aspinwall Committee Chair Kenneth Shaw Committee Co-Chair Elizabeth Jakubowski Committee Member Peter Easton University Representative
Keywords	African American Male Calculus Mathematics Learning Community College Students
Date of Defense	2011-06-08
Availability	unrestricted
Abstract Data suggests that a significant loss of African American students from STEM majors occur between their freshmen and sophomore year. This attrition corresponds to the time period when students encounter the calculus sequence. For this reason, calculus persists as a serious barrier preventing African American students from entering STEM fields. There has been a dearth of research studies on how African American students learn or engage in the learning of calculus. In this study, I developed cases describing two African American participants – Matt and Danny- and their methods used to complete tasks and create meaning for the graphs of functions and their derivatives. Three research questions were investigated: 1. What is the role of graphic representations in African American male community college students’ construction of the derivative in calculus? 2. How do African American male community college students synthesize graphic and analytic meaning of the derivative in calculus? 3. What pedagogical approaches are the most effective in assisting African American students with visual understanding of derivative graphs? During the task-based clinical interviews, the participants were presented with both analytic tasks and graphic tasks and asked to calculate derivatives when presented with analytic tasks (symbols) and to draw derivative when presented with the graphs as I sought to gain understanding of the mathematical processes. The participants’ understanding of the derivative was different because of their preference for mathematical processing. Matt relied on analytic processing and symbolic representation. His understanding of the derivative merely involved the manipulation of formulas, which is dominated by most college mathematics examination. Danny relied on a combination of analytic processing and geometric processing and preferred to primarily operate on graphic representations. His understanding of the derivative was associated with both analytic representations (formulas) and graphic representation. This study found that the participants’ knowledge was strongly associated with their mathematical processing capabilities. Matt’s overreliance on his memory and analytic thinking impeded his understanding of derivative graphs. This one-sided thinking caused Matt’s knowledge (procedural and conceptual) to be disconnected and only understood how to complete tasks when asked questions in the right context. Danny’s harmonic thinking enabled him to complete tasks with much less difficulty than Danny. Danny’s flexibility with his thinking allowed him to understand the changes in the slope of the tangents of graph when he was not presented with a formula to associate with the graphs. Both participants would frequently use verbal –description to aid their understanding of the behavior of derivative graphs when their analytic and visual thinking would fail. The study demonstrates that using graphical representation for functions and their derivative have the potential for producing richer understanding of the concept of the derivative. It also demonstrates that some African American students need further understanding of graphs with a cusp, a sharp corner, a vertical line, vertical asymptotes, or any other discontinuity. It also shows that students are able to connect their procedural knowledge with their conceptual knowledge when students are able to work between both graphic representation and analytic representations.
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