What does the student know, and what does he misunderstand? How would you help?

Today’s submission comes via Christopher Danielson, who blogs at Overthinking my teaching.

  • This student has memorized that the inverse graph is a reflection of the original function, but a reflection in what way? I’d ask about what inverses are, what inverse functions do, and work with inverse graphs having x and y coordinates reversed compared to the original function. From there I’d hope the student would generalize that the reflection should be in the y=x line, and not any other.

  • Annie of Cleves

    I’d echo what Julia said above.

    To add, though, I’d do the ol’ “College Econ Exam” trick and on the test make them write out their reasoning (even as bulletpoints — like a proof?). That way you and eventually the student (or other students in the class, as an error analysis activity) could pinpoint the exact misconception. I’d expect this student’s response to look something like:

    — the inverse function is a reflection of the first the graph
    — reflect the graph over the line so you have the same thing on the other side

    at which point, one could point out that an issue here is the lack of specificity about over what line you reflect. There is, of course, the possibility that the student knows that you are supposed to reflect over y=x, but doesn’t realize that the y-axis is NOT the line y=x. The written component could identify even more specific misconceptions (AND differentiate between those who have mastered the concept and those who “guessed” correctly).

  • I agree with Annie’s idea, and I think it’d be great if we asked fewer, deeper questions instead of lots of exercises, and asked students to explain their work rather than only giving answers or lists of steps in a procedure.

    I would delete the words “like a proof” from her comment, though: to me a proof ought to be a piece of writing like an essay, not like a bullet-pointed list of statements and reasons. I wish we’d dispense with the whole two-column thing.

  • I really agree on it especially that there are many students who couldn’t easily get it right. It may be their fault why they can’t answer or maybe on their teacher who doesn’t use any techniques in order for him or her to be understood.