Onderzoek naar de rol van calculus in de studie van elektrodynamica

Using mathematical concepts and techniques to describe natural phenomena is a challenging but important task in all branches of physics. Although typically well trained in rote mathematical calculation, students often lack an understanding of the mathematical concepts and fail to correctly apply these in a physics context. Student difficulties related to such issues are particularly apparent in an intermediate electrodynamics course that relies heavily on the use of vector calculus. In electrodynamics, the divergence and curl operators are used to formulate Maxwell's equations in differential form. It is difficult to overestimate the importance of Maxwell's equations, as these laws, together with the Lorentz force law, provide the foundations of classical electromagnetism. Therefore it is essential for students enrolled in an electrodynamics course to profoundly understand the conceptual meaning of these equations and the vector operators that are used to express them mathematically. In this work, we have studied the most prevalent student difficulties with the use of divergence and curl in mathematics and electromagnetism. In addition, we have developed, implemented, and assessed research-based learning materials that are designed to help students with the identified difficulties. In the first phase of our research project, we have studied student difficulties before and after completing the original format of the electrodynamics course at three institutions: KU Leuven, Dublin City University (DCU), and University of St Andrews. To assess students' understanding of vector calculus in a mathematical and electromagnetic context, we adopted a written pretest post-test methodology and analyzed the responses using ideas of phenomenography. We showed that most students were fairly well trained in doing calculations, but lacked a structural understanding of the vector operators, struggled with the interpretation of graphical representations of vector fields in terms of divergence and curl, and often failed to correctly apply Maxwell's equations in differential form in situations involving electromagnetic fields. While the prevalence of specific difficulties varied depending on the institution, the classification of typical errors proved to be appropriate to categorize student responses from all three universities. To improve our insight into students' reasoning processes, and obtain an indication of which elements can cue a correct and productive way of thinking, we conducted individual semi-structured interviews using a think-aloud protocol with eight KU Leuven students who had successfully completed the electrodynamics course. During these interviews, students discussed the divergence and curl of electromagnetic fields using graphical representations, mathematical calculations, and the differential form of Maxwell's equations. We observed that while many students attempted to clarify the problem by making a sketch of the electromagnetic field, they struggled to interpret visualizations of vector fields in terms of divergence and curl. In addition, some students confused the characteristics of field line diagrams and field vector plots, or experienced difficulties with switching between graphical and symbolic representations of vector fields. By interpreting our results within the conceptual blending framework, we showed how a lack of structural understanding of the vector operators and difficulties with graphical representations can account for an improper understanding of Maxwell's equations in differential form. As our earlier studies revealed that students struggle with the use of vector field representations, we have investigated to what extent they can correctly interpret, construct, and switch between field line diagrams, field vector plots, and algebraic expressions of vector fields. We conducted individual interviews, analyzed in-class student activities, and designed a free response assessment tool which was given to second- and third-year physics, mathematics, and engineering students from four different universities. We gained a comprehensive overview of typical errors that students make when switching between vector field representations. In addition, the study allowed us to determine the relative prevalence of the difficulties observed. Although the results varied greatly between institutions, our findings revealed that many students struggled with vector addition, failed to recognize the field line density as an indication of the magnitude of the field, confused field lines and equipotential lines, and did not choose the appropriate coordinate system when writing out mathematical expressions of vector fields. Based on the findings of the studies described above, we have developed, implemented, and assessed research-based learning materials that aim to help students with the identified difficulties. The designed guided-inquiry worksheets aim to provoke prevalent student difficulties and confront students with their incorrect ideas. In addition, they link conceptual approaches to determine the divergence and curl in graphical representations of vector fields, calculations using the vector operators, and Maxwell's equations in differential form. The worksheets were implemented in tutorial sessions at KU Leuven and DCU, and the participating students were encouraged to discuss the questions and their responses in small groups. We established that the KU Leuven students generally performed better on the post-test after the intervention than after the original instruction. The intervention at DCU was less effective, which can be accounted for by a mismatch between the worksheets and the mathematical background of these students. Nevertheless, both the KU Leuven and DCU students indicated they enjoyed the tutorial approach and felt they learned something from the worksheets. We have identified content-specific difficulties that students experience with the use of vector calculus in electrodynamics, and discussed the effectiveness of a research-based intervention in the electrodynamics course. In addition, our findings contribute to the physics education research community's understanding of how students use mathematics in physics and how employing multiple representations can help in this process. We have shown that students who lack a structural understanding of mathematical concepts often exhibit difficulties when these are applied in a physics context. However, we also observed indications that students may struggle with the use of mathematics in physics, even when the mathematical concepts are thoroughly understood. Therefore, a deeper understanding of students' use of the language of mathematics in physics is necessary to further approach physics education from a research perspective.nrpages: 350status: publishe