Knowing solutions to differential equations with rate of change as a function: Waypoints in the journey

摘要

In this paper we illustrate five qualitatively different ways in which students might reason with rate of change as a conceptual tool in order to graphically determine solutions to first order autonomous differential equations. These five different ways of reasoning about rate of change offer an empirically- grounded theoretical account of the waypoints through which student reasoning may progress in increasingly sophisticated ways. The term waypoint comes from the literature on learning progressions, which seeks to identify conceptual landmarks that differentiate ways of reasoning that students are likely to use as they engage in mathematical tasks and solve problems. For each waypoint we give an illustrative example, the associated task and goal, typical inscriptions that students use as a source for reasoning about rate of change, and typical target inscriptions that students use to depict solutions. The paper concludes with implications for research and curriculum development.