This research will extend current theories describing how students' acquire an understanding of the function concept, the central concept of the undergraduate mathematics curriculum. The model that emerges will provide a framework for the development of two research instruments to broadly assess students' function knowledge. The Precalculus Concept Assessment Instrument (PCA) and the Calculus Concept Assessment Instrument (CCA) will provide a consistent means of quantifying students' concept development for two large-enrollment undergraduate courses. Investigations of students' problem solving behavior, using interviews, case studies and observations will attempt to identify major beliefs and behaviors that both promote and present obstacles to students' mathematical success and continued mathematical study. The results of these studies will form the basis for the development of a framework and instrument to broadly define and assess undergraduate students' beliefs about knowing and learning mathematics. The emergent instruments will provide objective means for examining patterns in students' function understandings and mathematical beliefs. The resultant investigations will provide insights for guiding the development of precalclulus and calculus level curriculum modules. During the course of this project model workshops will also be offered to disseminate the newly developed instruments and curricular materials.
