On the role of student understanding of function and rate of change in learning differential equations

In this research, I utilize the theoretical perspective Knowledge In Pieces to identify the knowledge resources students utilize while in the process of completing various differential equations tasks. In addition I explore how this utilization changes over the course of a semester, and how resources related to the concepts of function and rate of change supported the students in completing the tasks. I do so using data collected from a series of task-based individual interviews with two students enrolled in separate differential equations courses. The results provide a fine-grained description of the knowledge students consider to be productive with regard to completing various differential equations tasks. Further the analysis resulted in the identification of five ways students interpret differential equations tasks and how these interpretations are related to the knowledge resources students utilize while completing the various tasks. Lastly, this research makes a contribution to mathematics education by illuminating the knowledge concerning function and rate of change students utilize and how this knowledge comes together to support students in drawing connections between differential equations and their solutions, structuring those solutions, and reasoning with relationships present in the differential equations.