A common problem in teaching is, I think, telling the [whole] truth vs. actually teaching something. I have two examples of this problem that have been gnawing at my math conscience for a while:

**Separable differential equations**:

This is right out of the first day of an ODE course. We look at a separable differential equation which, I am sure you recall, looks something like , then write it using *Leibniz* notation (which is more precise, at the expense of being somewhat intimidating… a topic for another time!) as

We collect all the terms involving *y* on one side, and all the terms involving *x* on the other (the ability to do so may be taken as the definition of a separable equation):

then integrate

Now, *dy/dx* is not actually a fraction, and on the one hand, you *hope* a student calls you out on this. But it is a useful formalism that, as far as I can tell, everyone uses. The true derivation of the above solution should go:

then we proceed at usual. I am using the variable *u *in the third equation to emphasize the change of variables , so (which again is some formalism that I will duck for now).

In practice, I will not present the second derivation, since the point of the lecture is to learn how to solve separable equations, not to practice changing variables. However, I will encourage students to call out the hand-waving in the first derivation. In a perfect world, I would hope students could justify everything that happens in class, but the last thing I would want to do on the first day of classes is overwhelm students for my mathematical conscience.

**Eigenvalues and the trace/determinant:**

The other situation where I have often felt mathematical-moral-twinges is when eigenvalues make an appearance in a linear algebra course. Two enormously useful identities for checking that you, at least, *might* have gotten the eigenvalues right is that the product of the eigenvalues is equal to the determinant of a matrix, and the sum of the eigenvalues is the trace. The proof is not hard (namely, that a matrix is similar to its Jordan normal form, , where the diagonal of *J* is the [are the?] eigenvalues of *A*, and then recalling that the determinant of a product is a product of determinants, so , and that the trace of similar matrices is the same.) However, Jordan normal form comes, necessarily, later than eigenvalues, and I have never had time to go back to present a complete proof.

I have given students this identity with a “cheat code” caveat in the past. That is to say, I will not test them on it, and they cannot use it as *justification* for calculations, but I point out it would be foolish not to use it to double check calculations: if they get an eigenvalue calculation wrong on the final, I’m not mad, I’m just disappointed.

I am sure there are other examples of topics where this is dealt with, but these have bothered me the most!