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.

Most introductory quantum physics instructors would agree that transitioning students from classical to quantum thinking is an important learning goal, but may disagree on whether or how this can be accomplished. Although (and perhaps because) physicists have long debated the physical interpretation of quantum theory, many instructors choose to avoid emphasizing interpretive themes; or they discuss the views of scientists in their classrooms, but do not adequately attend to student interpretations. In this synthesis and extension of prior work, we demonstrate: (1) instructors vary in their approaches to teaching interpretive themes; (2) different instructional approaches have differential impacts on student thinking; and (3) when student interpretations go unattended, they often develop their own (sometimes scientifically undesirable) views. We introduce here a new modern physics curriculum that explicitly attends to student interpretations, and provide evidencebased arguments that doing so helps them to develop more consistent interpretations of quantum phenomena, more sophisticated views of uncertainty, and greater interest in quantum physics.

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.

Just as expert physicists vary in their personal stances on interpretation in quantum mechanics, instructors vary on whether and how to teach interpretations of quantum phenomena in introductory modern physics courses. In this paper, we document variations in instructional approaches with respect to interpretation in two similar modern physics courses recently taught at the University of Colorado, and examine associated impacts on student perspectives regarding quantum physics. We find students are more likely to prefer realist interpretations of quantum-mechanical systems when instructors are less explicit in addressing student ontologies. We also observe contextual variations in student beliefs about quantum systems, indicating that instructors who choose to address questions of ontology in quantum mechanics should do so explicitly across a range of topics

Introductory undergraduate courses in classical physics stress a perspective that can be characterized as realist; from this perspective, all physical properties of a classical system can be simultaneously specified and thus determined at all future times. Such a perspective can be problematic for introductory quantum physics students, who must develop new perspectives in order to properly interpret what it means to have knowledge of quantum systems. We document this evolution in student thinking in part through pre- and post-instruction evaluations using the Colorado Learning Attitudes about Science Survey. We further characterize variations in student epistemic and ontological commitments by examining responses to two essay questions, coupled with responses to supplemental quantum attitude statements. We find that, after instruction in modern physics, many students are still exhibiting a realist perspective in contexts where a quantum-mechanical perspective is needed. We further find that this effect can be significantly influenced by instruction, where we observe variations for courses with differing learning goals. We also note that students generally do not employ either a realist or a quantum perspective in a consistent manner