Development, implementation, and assessment of a guided-inquiry teaching-learning sequence on vector calculus in electrodynamics

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.

Figure 1

A model of research-based course transformation. Adapted from Ref. [14].

Figure 2

Excerpt of a worksheet used in the divergence tutorial. In the original format, some blank space was left after every subquestion for the students to formulate their responses.

Figure 3

Excerpt of a worksheet used in the Gauss’s law tutorial in differential form. In the original format, some blank space was left after every subquestion for the students to formulate their responses.

Figure 4

Excerpt of a worksheet used in the Maxwell-Ampère’s law tutorial in differential form.

Figure 5

Questions designed to assess students’ correct and incorrect conceptions of curl and divergence. The correct statements are C4, C6, C8, D1, D6, and D9.

Figure 6

The proportion of KU Leuven students ($N=57$) who indicated that a statement presented in the questions in Fig. 5 is correct on the post-test, compared to the pretest. Only the students who took both the pre- and post-test were taken into account. Correct statements are in bold and underlined.

Figure 7

Question on the pre- and post-test that were given to the students in order to determine to what extent they can decide where the divergence or curl is nonzero in a field vector plot. Fields (c) and (I) are similar and yield the same solution, as is the case for fields (a) and (II).

Figure 8

The proportion of KU Leuven students who determined correctly where the divergence or curl is nonzero in a field vector plot on the pretest [field (c) and (a) in Fig. 7] versus the post-test [field (I) and (II) in Fig. 7], after the original instruction ($N=19$) and the intervention ($N=57$). Only the students who took both the pre- and post-test are taken into account.

Figure 9

Post-test question to determine whether students can decide where divergence and curl are nonzero in field vector plots of electromagnetic fields.

Figure 10

The proportion of students who determined correctly where the divergence and curl are nonzero in a field vector plot of an electromagnetic field on the post-test (Fig. 9), after the original instruction ($N=19$) and after the intervention ($N=57$).

Figure 11

Post-test question that evaluates students’ computational skills and their ability to construct a field line diagram.

Figure 12

Examples of student sketches in response to post-test question 5(c) (Fig. 11).

Figure 13

Post-test question on Maxwell’s equations.

Figure 14

Proportion of students who indicated the divergence or curl is zero in each situation on the post-test (Fig. 13) after the original instruction ($N=19$) and the intervention ($N=59$). Correct answers are in bold and underlined. We omitted the data after the original instruction for situation E4, since this situation was described ambiguously at that time.

Figure 15

Student ($N=29$) responses to eight 5-point Likert scale questions, evaluating their experiences with the tutorials.