Cold Problem Solving

I am teaching real analysis in the fall, and I am beginning to plan it out. Here is one more idea that I would like to record before I forget.

First, some background. I was never very good at real analysis. I like it a lot, but it was over my head as an undergrad (I got B’s in the course, largely because my relative difficulty with the material outpaced my study skills) and I only took one course as a graduate student (similar). Part of the reason why I am teaching it is that I want to learn more about this beautiful subject.

So I am not a great analyst. To use this to my advantage, I am planning on—perhaps weekly—asking my students to give me an analysis problem to solve “cold.” I won’t prepare for it at all; I will solve it on the spot.

I was inspired by a lost blog posting (please comment if it was you—I will happily edit this to credit you) about the difference between the way we discover mathematics and the way we communicate mathematics. Sadly, we normally teach by showing the “communication” rather than the “discovering.” By

  1. Doing problems “cold” in front of the students, and
  2. Not being very good at analysis, but
  3. Having general problem solving skills,

I can hopefully give the students a glimpse into the “discovery” world of mathematics. If I am really on the ball, I will (at least occasionally) take the time to re-write my “discovery” into a “communication” form.

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4 Responses to “Cold Problem Solving”

  1. catatori Says:

    Gutsy, to say the least. You might want to schedule the whole class period for this, just in case. (At least, I would need to do that! 🙂

    • bretbenesh Says:

      I might just do that. Perhaps two class periods.

      Actually, I would be completely happy saying “I’m stumped…I’ll get back to you” if things do not go well.

  2. quantumprogress Says:

    I think this is very interesting. Unfortunately, I teach 9th grade physics, and there are almost no problems I can’t solve cold—though there are occasional questions that cause me to say “I’ll have to get back to you.” I’m wondering how you could replicate something like this in a lower level class, say intro calculus, or even a college algebra course. Do you think it’d be possible?

    • bretbenesh Says:

      Hi John,

      This is a problem in courses where I am very familiar with the material, too (this whole idea relies on the fact that I am not naturally a good analyst). I have two mediocre ideas for you, though:

      1. One the rare occasion when you do say “I’ll have to get back to you” (I do this sometimes, too), spend much more time than normally would. Explain that you are stuck, and that you are going to verbalize your thought process so the students can see how an experienced physicist would think about the problem.

        This, of course, requires you to be flexible with your lesson plans.

      2. Have the students bring in any sort of math, science, or logic problem for you to solve. You would have to somehow define the parameters to avoid questions like: “Mr. Burk: use general relativity to figure out how to time-travel to the past.”

      The former has the advantage that it is genuine and relevant, but has the disadvantage that it is rare that you get stumped in your classes. The latter has the advantage that you could have to genuinely figure out a new problem frequently, but has the disadvantage that it is likely not very relevant to the “content” of the class.

      Can you improve upon my suggestions?
      Bret

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