### 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."

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.

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.

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.

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.

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

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.

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