The evolution of student understanding of the concept of derivative

Graduation date: 1998This research seeks to answer the question, "What does it mean for a student to\ud understand the concept of derivative?" A structured way to describe an individual student's\ud understanding of derivative is developed and applied to analyzing the evolution of that\ud understanding for each of nine high school seniors during their year-long calculus course.\ud The methodology is a multiple case study. Interviews, including both task-based and\ud open-ended questions, are the primary instruments for collecting data on each student's\ud understanding. Other data collected include tests, written questions, and classroom\ud observations.\ud Several theoretical frameworks contribute to the research: concept image (Tall,\ud Vinner, and Dreyfus), process-object (Sfard; Dubinsky and colleagues), and notions of\ud multiple representations for function, limit, and derivative. I describe the concept of\ud derivative as three layers of process-objects: the ratio or difference quotient, the limit, and\ud the function layers. Each layer may be observed in multiple contexts: graphical (slope),\ud verbal description (rate of change), kinematic (e.g. velocity or acceleration), and symbolic\ud (the symbolic difference quotient definition of derivative). A description of the connections\ud between the various aspects of the concept of derivative comes from the work of Fischbein\ud on paradigmatic, analogic, and diagrammatic models and the work of Lakoff on metaphor\ud and metonymy. The major theoretical result of the dissertation is the development of a structured\ud way of describing the concept of derivative including a diagrammatic methodology for\ud displaying which aspects of the derivative concept a student has demonstrated. This\ud methodology may be applied to other studies and other concepts. The major result of this\ud study of nine students is the realization that the layers and representations of the concept of\ud derivative do not appear to be hierarchical in that none of the nine students learn the aspects\ud in the same order