Developing a tutorial to address student difficulties in learning curl: a link between qualitative and mathematical reasoning

Many university students have difficulty understanding the concept of curl in our upper-level mechanics courses. This difficulty poses considerable problems when students learn upper-level physics, as the concept of curl is closely related to a wide variety of topics in physics, such as Maxwell's equations, Faraday's law, and conservative fields. However, few studies have considered the reasons that students experience difficulty or ways to help students overcome their difficulty grasping the notion of curl. Therefore, in this study, we try to address student difficulty in learning curl. First, we developed a questionnaire to investigate student difficulties in understanding curl and to obtain comments on a tutorial. The questionnaire involved an explanation and a Likert scale form (degrees 0 to 4). We administered a diagnostic test in which we posed two kinds of curl problems. As a result, students who showed qualitative reasoning to solve two types of curl problems were better at solving a curved-line problem than they were at solving a straight-line problem. In contrast, students who showed mathematical reasoning were better at solving the straight-line problem. Based on these results, we developed a tutorial to aid students in using both qualitative and mathematical reasoning. Use of the tutorial enhanced students' explanations about the curl concept and increased a percentage of correct answers.