We report on semi-quantitative research into students' difficulties with integration in an intermediate-level electromagnetism course with cohorts of about 50 students. We have found that before they enter the course, students view integration primarily as a process of evaluation, even though viewing integration as a summation process would be more fruitful. We confirm and quantify earlier results that recognizing dependency on a variable is a strong cue that prompts students to integrate and that various technical difficulties with integration prevent almost all students from getting a completely correct answer to a typical electromagnetism problem involving integration. We describe a teaching sequence that we have found useful in helping students address the difficulties we identified.

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We deem the two cohorts to be equivalent since in many pretests (not reported in this paper) we obtain very similar responses.

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34.

Of course we are and were aware of the general ineffectiveness of teaching by telling; we were under the mistaken impression that we were merely reminding students of something they had already internalized.

35.

These five tutorial problems entailed: finding the total charge on a thin semicircular disk of radius R with varying surface charge density; calculating the electric flux through a flat sheet due to a point charge located above one of the corners of the sheet; determining the potential due to a uniformly charged rod; calculating by integration the circulation along a rectangular loop of the magnetic field due to a straight current-carrying wire; and calculating the magnetic force on a square current-carrying loop due to a straight wire.

36.

We realize that at first glance, this may look like a “dumbed down” version of Problem 2.10 of Griffiths' textbook.16 We are aware that the answer can be obtained in one line from symmetry considerations, but have found it a useful problem to tackle through integration.

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