This study explores one student's understanding of qualitative methods of analysis for first order differential equations. The student demonstrated a strong graphical understanding of the derivative and used her prior knowledge of population modeling to make sense of various graphical representations. Despite this strong background knowledge, several potential difficulties related to qualitative methods of analysis are uncovered and propositions regarding the sources of these difficulties are offered. A misconception involving the interpretation of direction fields is suggested by the data, which in turn raises questions about the usefulness of phase lines in some recent reform-oriented curriculums.
Paper is available in the following formats: