Students' difficulties with vector calculus in electrodynamics

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.Comment: 14 pages, 11 figure

Student thinking about the divergence and curl in mathematics and physics contexts

Undergraduate physics students are known to have difficulties with understanding mathematical tools, and with applying their knowledge of mathematics to physical contexts. Using survey statements based on student interviews and written responses to open-ended questions, we investigated the prevalence of correct and incorrect conceptions regarding the divergence and curl of vector fields, among both mathematics and physics students. We compare and contrast pre-instruction responses from intermediate-level E&M students at KU Leuven and the University of St Andrews, with post-instruction responses from St Andrews students enrolled in a vector calculus course. The differences between these student populations were primarily in areas having to do with physics-related concepts and graphical representations of vector fields. Our comparison of pre- and post-instruction responses from E&M students shows that their understanding of the divergence and curl improved significantly in most areas, though not as much as would be desired.Comment: Physics Education Research Conference 2015 (submitted

Qualitative investigation into students’ use of divergence and curl in electromagnetism

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.Publisher PDFPeer reviewe

Undergraduate students' difficulties with boundary conditions for the diffusion equation

Combining mathematical and physical understanding in reasoning is difficult, and a growing body of research shows that students experience problems with the combination of physics and mathematics in reasoning beyond the introductory level. We investigated students' reasoning about boundary conditions (BCs) for the diffusion equation by conducting exploratory task-based, think-aloud interviews with twelve undergraduate students majoring in physics or mathematics. We identified several difficulties students experienced while solving the interview task and categorized them using the conceptual blending framework. This framework states that in reasoning, people draw from separate input spaces, in this case the mathematics and the physics input space, to form a blended space, where they make connections between elements from these spaces. To identify difficulties, we used open coding techniques. We observed few difficulties in the physics space. In the mathematics space, we identified several difficulties that we clustered in two main groups: findings about the mathematical meaning of BCs, and findings about reasoning with functions of two variables. Lastly, we identified four ways in which blending failed. Starting from our findings, we formulate recommendations for teaching and future research

Dynamic conceptual blending analysis to model student reasoning processes while integrating mathematics and physics:A case study in the context of the heat equation

In recent years, there has been an increased interest in conceptual blending in physics and mathematics education research as a theoretical framework to study student reasoning. In this paper, we adapt the conceptual blending framework to construct a blending diagram that not only captures the product but also the process of student reasoning when they interpret a mathematical description of a physical system. We describe how to construct a dynamic blending diagram (DBD) and illustrate this using two cases from an interview study. In the interview, we asked pairs of undergraduate physics and mathematics students about the physical meaning of boundary conditions for the heat equation. The selected examples show different aspects of the DBD as an analysis method. We show that by using a DBD, we can judge the degree to which students integrate their understandings of mathematics and physics. The DBD also enables the reader to follow the line of reasoning of the students. Moreover, a DBD can be used to diagnose difficulties in student reasoning

Integrated STEM in secondary education: A case study

Despite many opportunities to study STEM (Science, Technology, Engineering & Mathematics) in Flemish secondary education, only a minority of pupils are actually pursuing STEM fields in higher education and jobs. One reason could be that they do not see the relevance of science and mathematics. In order to draw their pupils’ interest in STEM, a Belgian school started a brand new initiative: the school set up and implemented a first year course that integrates various STEM disciplines, hoping to provide an answer to the question pupils often ask themselves about the need to study math and science. The integrated curriculum was developed by the school’s teachers and a STEM education research group of the University of Leuven. To examine the pupils’ attitude towards STEM and STEM professions and their notion of relevance of STEM at the end of this one-year course, a post-test was administered to the group of pupils who attended the integrated STEM course (the experimental group) and to a group of pupils that took traditional, non-integrated STEM courses (the control group). The results reveal that attending the integrated STEM course is significantly related to pupils’ interest in STEM and notion of relevance of STEM. Another post-test was administered only to the experimental group to investigate pupils’ understanding of math and physics concepts and their relation when taught in an integrated way. The results reveal that the pupils have some conceptual understanding and can, to a certain extent, make a transfer of concepts across different STEM disciplines. However, the test results did point out that some additional introductory training in pure math context is needed

Dificultades de estudiantes universitarios de tres países en el aprendizaje del concepto de fuerza electromotriz en electricidad

El objetivo principal de este estudio es identificar las dificultades de estudiantes universitarios en el aprendizaje de los conceptos de fuerza electromotriz y diferencia de potencial eléctrico en el contexto de corrientes transitorias y de circuitos de corriente directa con resistencias. Para investigar las dificultades de los estudiantes desarrollamos un cuestionario basado en un análisis del marco teórico y epistemológico de la física. Este cuestionario se aplicó a estudiantes de primer año de ingeniería y de física de universidades en el País Vasco, Colombia y Bélgica

Introductory university physics students ’ understanding of some key characteristics of classical theory of the electromagnetic field

In this work, we explore how undergraduate students use classical field theory when describing physical phenomena in the context of introductory electromagnetism. We have extracted five key characteristics of the electric and magnetic field from a historical analysis of the topic. These characteristics informed the creation of a questionnaire comprising six free-response conceptual questions. The questionnaire instrument was administered to undergraduate students in three European countries. Phenomenographical analysis of the student's responses shows that many undergraduates do not have a coherent idea of field theory. We conclude that, rather than focusing on teaching rules with which to calculate, more attention should be paid to the specific characteristics of field theory and the difference between fields and forces, with particular emphasis on the conceptual interpretation of the interaction process