Steven Jones recently published an article titled “Examining Students’ Variational Reasoning in Differential Equations” in the Journal of Mathematical Behavior. Steven has answered a few questions about this article below:
Who were your co-authors on this article?
George E. Kuster, Christopher Newport University
Who would you say is the target audience for this article?
Mathematics education researchers and undergraduate mathematics instructors
What is the big problem you hoped to address with this article?
Reasoning and modelling with differential equations requires significant cognitive effort and is not trivial to learn to do. We wanted to know what role variational reasoning played in students’ understandings of differential equations.
What are some of the key ideas in the article?
1. While students can think of a derivative as a rate of change or as a single numeric value, our results showed that when reasoning about differential equations, students often had to think of the derivative in these two ways simultaneously.
2. Students used extensive covariational reasoning and some amount of multivariational reasoning in order to coordinate the variables at play in a differential equation, such as t, y, and y’.
3. Recognizing “independence” versus “dependence” among the variables was a key reasoning act that helped students make sense of the relationships inside differential equations.
4. We documented a new type of variational reasoning we called “feedback variation.” The idea here is that in a differential equation, the function value automatically determines its own rate of change, creating a feedback loop. Students reasoning about this contained distinct mental actions to coordinate a function’s “feedback” dependence with itself.
Abstract:
In this study, we explored how a sample of eight students used variational reasoning while discussing ordinary differential equations (DEs). Our analysis of variational reasoning draws on the literature with regard to student thinking about derivatives and rate, students’ covariational reasoning, and different multivariational structures that can exist between multiple variables. First, we found that while students can think of “derivative” as a variable in and of itself and also unpack derivative as a rate of change between two variables, the students were often able to think of “derivative” in these two ways simultaneously in the same explanation. Second, we found that students made significant usage of covariational reasoning to imagine relationships between pairs of variables in a DE, and that mental actions pertaining to recognizing dependence/independence were especially important. Third, the students also conceptualized relationships between multiple variables in a DE that matched different multivariational structures. Fourth, importantly, we identified a type of variational reasoning, which we call “feedback variation”, that may be unique to DEs because of the recursive relationship between a function’s value and its own rate of change.