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

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 emp...

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Bibliographic Details
Published in:The Journal of mathematical behavior 2019-12, Vol.56, p.100695, Article 100695
Main Authors: Rasmussen, Chris, Keene, Karen
Format: Article
Language:English
Subjects:
Differential equations
Function
Learning progressions
Rate of change
Student reasoning
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Summary: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.
ISSN:0732-3123
1873-8028
DOI:10.1016/j.jmathb.2019.03.002