Using parametric reasoning to understand solutions to systems of differential equations

This paper offers an analysis of how students reasoned with the dynamic parameter time to support their mathematical activity and deepen their understandings of mathematical concepts. This mathematical thinking occurred as they participated in a differential equations class before, during, and instruction on solutions to linear systems of differential equations. Students participated in the following identified mathematical practices related to parametric reasoning during this time period: reasoning simultaneously in a qualitative and quantitative manner, reasoning by moving from discrete to continuous imaging of time, and reasoning by imagining the motion. Examples of this reasoning are provided in this report. Implications of this research include the possibility that instructional activities can build on this reasoning to help students learn about the mathematics of change at the middle school, high school, and the university.