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.
Students appear to have interpreted our elaboration on the term “interpret” as intended: “write” was not meant to exclude diagrammatic representations. Students who mentioned area under the curve especially tended to draw diagrams to illustrate their answers.
We use the term “summation” throughout, but note that in the mathematics education literature the term “accumulation” is commonly used.
It cannot be denied that there is an ambiguity in the notation: n(x) could mean “multiply n by x,” although no expert would interpret n(x) that way.
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.
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.
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.
