First Advisor

Soto-Johnson, Hortensia

Second Advisor

Karakok, Gulden

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The purpose of this study was to explore the nature of students’ reasoning about the derivative of a complex-valued function, and to study ways in which they developed this reasoning while working with Geometer’s Sketchpad (GSP). The participants in this study were four students from one undergraduate complex analysis class. The development of participants’ reasoning about the derivative of a complex-valued function was captured via video-recording and screen-capture software in a four-day interview sequence consisting of a two-hour-long interview each day. This reasoning was interpreted through the theoretical perspective of embodied cognition. The findings indicated that students manifested embodied reasoning through gesture and speech, through algebraic and geometric inscriptions, and through interaction with the physical environment and the virtual environment provided by GSP. The findings further indicated that students needed to advance their geometric reasoning about the derivative of a complex-valued function in three essential ways in order to reason geometrically about the derivative as a local linear approximation. First, with help from gesture and speech, they recognized that they did not know how to characterize a linear complex-valued function. Second, with help from algebraic and geometric inscriptions, they reasoned that a linear complex-valued function f(z) rotates and dilates every circle by the same amounts Arg(f^' (z)) and |f^' (z)|, respectively. Finally, through embodied reasoning in both the virtual and physical environments, students recognized the need to focus on how a complex-valued function rotates and dilates small circles only. These findings suggest that one approach to improving student learning about the derivative of a complex-valued function is to highlight these three geometric aspects of the derivative, and to offer students opportunities to reason about this geometry in embodied ways listed above.


Mathematics, Geometer's Sketchpad, Student learning


483 pages

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Copyright is held by the author.