Abstract
This dissertation presents a series of studies pertaining to introductory physics students' abilities to derive physical meaning from symbolic integrals (e.g., the integral of vdt) and their components, namely differentials and differential products (e.g., dt and vdt, respectively). Our studies focus on physical meaning in the form of interpretations (e.g., "the total displacement of an object") and units (e.g., "meters"). Our first pair of studies independently attempted to identify introductory-level mechanics students' common conceptual difficulties with and unproductive interpretations of physics integrals and their components, as well as to estimate the frequencies of these difficulties. Our results confirmed some previously-observed incorrect interpretations, such as the notion that differentials are physically meaningless; however, we also uncovered two new conceptualizations of differentials, the "rate" (differentials are "rates" or "derivatives") and "instantaneous value" (differentials are values of physical variables "at an instant") interpretations, which were exhibited by more than half of our participants at least once. Our next study used linear regression analysis to estimate the strengths of the inter-connections between the abilities to derive physical meaning from each of differentials, differential products, and integrals in both first- and second-semester, calculus-based introductory physics. As part of this study, we also developed a highly reliable, multiple choice assessment designed to measure students' abilities to connect symbolic differentials, differential products, and integrals with their physical interpretations and units. Findings from this study were consistent with statistical mediation via differential products. In particular, students' abilities to extract physical meaning from differentials were seen to be strongly related to their abilities to derive physical meaning from differential products, and similarly differential products to integrals; there was seen to be almost no direct connection between the abilities to derive physical meaning from differentials and the abilities to derive physical meaning from integrals. Our final pair of studies intended to implement and quantitatively assess the efficacy of specially-designed instructional tutorials in controlled experiments (with several treatment factors that may impact performance, most notably the effect of feedback during training) for the purpose of promoting better connection between symbolic differentials, differential products, and integrals with their corresponding physical meaning. Results from both experiments consistently and conclusively demonstrated that the ability to connect verbal and symbolic representations of integrals and their components is greatly improved by the provision of electronic feedback during training. We believe that these results signify the first instance of a large, controlled experiment involving introductory physics students that has yielded significantly stronger connection of physics integrals and their components to physical meaning, compared to untrained peers.
