How can emphasizing mathematical modeling principles benefit students in a traditionally taught differential equations course?

- Mixed-methods study compares two instructional emphases for differential equations.
- Lecture observations distinguish analytic techniques and modeling approaches.
- ANCOVA main effect suggests emphasis on modeling principles aids student learning.
- Student exams show connection between a differential equation and its solution.
- Results verify recommendations for teaching and learning of differential equations.

This paper reports a mixed-methods, causal-comparative study of two instructional approaches in a differential equations course for engineering undergraduates. One classroom emphasized decontextualized techniques to solve canonical differential equations. The other classroom emphasized modeling principles to derive and interpret canonical differential equations as models of real world phenomena. Both classrooms were lecture based. This paper presents a brief review of both texts and compares a lesson on the same topic from each classroom in order to illustrate what it means to emphasize modeling principles. An ANCOVA was used to compare student performance on common final exam items while controlling for prior mathematics achievement. The main effect for instructional approach was statistically significant suggesting that a modeling perspective for teaching differential equations aided student learning. This quasi-experimental study corroborates previously reported recommendations for improving teaching and learning of differential equations.