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Saturday, August 18, 2007

An Uncomfortable Teacher

Part of my difficulty in teaching intro economics classes was that in order to dumb the material down enough for the freshmen business majors to understand, I had to "teach" things that I didn't really believe. I used to think it must be nice to be a math professor, because then everything you taught would be rigorously correct. (You just wouldn't get into, say, Cantor's diagonal argument in a pre-calc class.)

Well now I wonder if that's even true. (I should've asked math professors at the time, but I don't think I ever did.) In any event, my 2-year-old was playing today and showing me different blocks. He held up a cylinder and said confidently, "Circle." So I said "That's right it's a circle." Then he held up a block that was a triangle with depth, and I told him it was a triangle. I.e. I didn't put in a caveat, "Actually Clark, it's just the two-dimensional face of it that's a triangle. And in a few years we'll talk about calculating its volume."

8 comments:

  1. I used to think it must be nice to be a math professor, because then everything you taught would be rigorously correct.

    I don't know, I find myself lying quite a bit when I teach freshman calculus.

    ReplyDelete
  2. Are you serious or ridiculing the premise of my post? Because I think that that might be true, that when you teach calc to freshmen you end up saying things without caveats that you'd give to grad students.

    ReplyDelete
  3. Here's something I handed my calculus teacher in High School:

    Solve: A d2y/dx2 + B dy/dx + Cy = 0

    Multiplying through by dx2:
    A d2y + B dy dx + C y dx2 = 0

    This is a quadratic in dx:
    dx=(-Bdy+-sqrt(B2dy2-4ACyd2y))/2Cy

    Therefore
    dx/dy=(-B+-sqrt(
    B2-4ACyd2y/dy2))/2Cy

    But d2y/dy2 = d/dy (dy/dy) = 0

    Therefore
    dx/dy=(-B+-sqrt(B2))/2Cy
    =(-B +- B)/2Cy
    = 0 OR -2B/2Cy=(-B/C)(1/y)

    Therefore
    EITHER x is constant (doubtful)
    OR x = (-B/C) log y + K
    SO y = exp((-C/B)(x - K))
    = K'exp(-(C/B)x)

    DONE!

    Needless to say, I did not receive a convincing explanation of why this derivation was invalid.

    ReplyDelete
  4. Are you serious or ridiculing the premise of my post?

    I was being quite serious. We lie all the time when explaining to freshmen how things work or why they're true. It makes me cringe every time I get to the section on the Fundamental Theorem of Calculus and completely gloss over why it's true.

    Of course, a bigger lie is the implication that the techniques you learn in the Differential Equations section of Calc. II (or even in a first "real" Diff Eq. class) are actually useful for anything other than solving the problems your teacher assigns you. There's a reason making up your own DEs to solve as practice for the exam never seems to work.

    ReplyDelete
  5. shonk: ain't it the truff? It's much like making up your own integrands to integrate.

    ReplyDelete
  6. It's always fun to con students into trying to solve a non-elementary integral that, superficially, doesn't look any harder than the integrals they know how to do.

    ReplyDelete
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