A cross-context look at upper-division student difficulties with integration

We investigate upper-division student difficulties with direct integration in multiple contexts involving the calculation of a potential from a continuous distribution (e.g., mass, charge, or current). Integration is a tool that has been historically studied at several different points in the curriculum including introductory and upper-division levels. We build off of these prior studies and contribute additional data around student difficulties with multivariable integration at two new points in the curriculum: middle-division classical mechanics and upper-division magnetostatics. To facilitate comparisons across prior studies as well as the current work, we utilize an analytical framework that focuses on how students activate, construct, execute, and reflect on mathematical tools during physics problem solving (i.e., the ACER framework). Using a mixed-methods approach involving coded exam solutions and student problem-solving interviews, we identify and compare students' difficulties in these two different contexts and relate them to what has been found previously in other levels and contexts. We find that some of the observed student difficulties were persistent across all three contexts (e.g., identifying integration as the appropriate tool, and expressing the difference vector), while other difficulties seemed to fade as students advanced through the curriculum (e.g., expressing differential line, area, and volume elements). We also identified new difficulties that appear in different contexts (e.g., interpreting and expressing the current density). These findings represent one of the first cross-context looks at students' difficulties with integration across multiple courses in the physic curriculum. These results can help instructors and researchers focus instruction and materials development on the aspects of integration that are most challenging in a particular context or persistent across multiple contexts.