Abstract
In the context of introductory physics, we study student conceptual understanding of differentials, differential products, and integrals and possible pathways to understanding these quantities. We developed a multiple choice conceptual assessment employing a variety of physical contexts probing physical understanding of these three quantities and administered the instrument to over 1000 students in first and second semester introductory physics courses. Using a regression-based mediation analysis with conceptual understanding of integration as the dependent variable, we found evidence consistent with a simple mediation model: the relationship between differentials scores and integral scores may be mediated by the understanding of differential products. The indirect effect (a quantifiable metric of mediation) was estimated as ab = 0.29, 95% CI [0.25, 0.33] for N = 1102 Physics 1 students, and ab = 0.27, 95% CI [0.14, 0.48] for N = 65 Physics 2 students. We also find evidence that the physical context of the questions can be an important factor. These results imply that for introductory physics courses, instructional emphasis first on differentials then on differential products in a variety of contexts may in turn promote better integral understanding.