The ability to switch between various representations is an invaluable problem-solving skill in physics. In addition, research has shown that using multiple representations can greatly enhance a person’s understanding of mathematical and physical concepts. This paper describes a study of student difficulties regarding interpreting, constructing, and switching between representations of vector fields, using both qualitative and quantitative methods. We first identified to what extent students are fluent with the use of field vector plots, field line diagrams, and symbolic expressions of vector fields by conducting individual student interviews and analyzing in-class student activities. Based on those findings, we designed the Vector Field Representations test, a free response assessment tool that has been given to 196 second- and third-year physics, mathematics, and engineering students from four different universities. From the obtained results 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 observed difficulties. Although the results varied greatly between institutions, a general trend revealed that many students struggle with vector addition, fail to recognize the field line density as an indication of the magnitude of the field, confuse characteristics of field lines and equipotential lines, and do not choose the appropriate coordinate system when writing out mathematical expressions of vector fields.

Understanding Maxwell’s equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell’s equations in electrodynamics.

Many students struggle with the use of mathematics in physics courses. Although typically well trained in rote mathematical calculation, they often lack the ability to apply their acquired skills to physical contexts. Such student difficulties are particularly apparent in undergraduate electrodynamics, which relies heavily on the use of vector calculus. To gain insight into student reasoning when solving problems involving divergence and curl, we conducted eight semistructured individual student interviews. 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 attempt to clarify the problem by making a sketch of the electromagnetic field, they struggle to interpret graphical representations of vector fields in terms of divergence and curl. In addition, some students confuse the characteristics of field line diagrams and field vector plots. By interpreting our results within the conceptual blending framework, we show how a lack of conceptual understanding of the vector operators and difficulties with graphical representations can account for an improper understanding of Maxwell’s equations in differential form. Consequently, specific learning materials based on a multiple representation approach are required to clarify Maxwell’s equations.

We have investigated whether and how a categorization of responses to questions on linear distance-time graphs, based on a study of Irish students enrolled in an algebra-based course, could be adopted and adapted to responses from students enrolled in calculus-based physics courses at universities in Flanders, Belgium (KU Leuven) and the Basque Country, Spain (University of the Basque Country). We discuss how we adapted the categorization to accommodate a much more diverse student cohort and explain how the prior knowledge of students may account for many differences in the prevalence of approaches and success rates. Although calculus-based physics students make fewer mistakes than algebra-based physics students, they encounter similar difficulties that are often related to incorrectly dividing two coordinates. We verified that a qualitative understanding of kinematics is an important but not sufficient condition for students to determine a correct value for the speed. When comparing responses to questions on linear distance-time graphs with responses to isomorphic questions on linear water level versus time graphs, we observed that the context of a question influences the approach students use. Neither qualitative understanding nor an ability to find the slope of a context-free graph proved to be a reliable predictor for the approach students use when they determine the instantaneous speed.

The goal of this study is to identify students’ difficulties with learning the concepts of electromotive force (emf) and potential difference in the context of transitory currents and resistive direct-current circuits. To investigate these difficulties, we developed a questionnaire based on an analysis of the theoretical and epistemological framework of physics, which was then administered to first-year engineering and physics students at universities in Spain, Colombia, and Belgium. The results of the study show that student difficulties seem to be strongly linked to the absence of an\ud analysis of the energy balance within the circuit and that most university students do not clearly understand the usefulness of and the difference between the concepts of potential difference and emf

We have developed, implemented, and assessed a teaching-learning sequence that aims to enhance students’ understanding of the divergence and curl of electromagnetic fields. We designed guided-inquiry worksheets based on student difficulties we identified during semiquantitative and qualitative studies and discussions in the literature. A multiple representation approach was adopted to strengthen the link between graphical representations of vector fields, calculations involving divergence and curl, and Maxwell’s equations in differential form. We used the Design, Functions, Tasks framework to optimize learning with the multiple external representations exhibited in our learning materials. These guided-inquiry worksheets used in small-group tutorials comprise short open-ended questions that encourage discussions in small groups of students. They were implemented in three consecutive years in a second-year electrodynamics course at KU Leuven. The intervention was assessed using the same pretest and post-test design adopted to evaluate learning in the original instruction. In addition, we gauged our students’ opinions on the intervention. We observed that our intervention positively affected our students’ structural understanding of the vector operators, their ability to interpret divergence and curl in graphical representations of vector fields, and their conceptual understanding of Maxwell’s equations in differential form. In addition, our students indicated that they enjoyed the teaching approach, felt they learned something from the worksheets, agreed with the difficulty level of the materials, and would like similar tutorials on different subjects.