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Admittedly a soft question but an important one, I think. The questions I've asked below are questions that can be answered, and not just discussed.

I read this essay yesterday by Gian-Carlo Rota denouncing the structure of most differential equations courses. In the essay, he says that the structure of these courses hasn't changed since the 19th century, almost word for word in some cases, and is filled with redundant lessons.

In particular, he has a problem with the teaching of exact equations, integrating factors, homogeneous differential equations, and existence and uniqueness of solutions, saying that all of these topics and techniques are of no use whatsoever.

He believes the bulk of DE courses should be linear DEs with constant coefficients (not variable coefficients), linear algebra, and Laplace transforms.

Are most people here in agreement with this? If so, how would a better DE course be built from scratch?

EDIT: I just looked at MIT's syllabus for Differential Equations, and it looks like it reflects what Rota said. So there's truth in it, though I haven't checked yet whether or not the course text by Edwards & Penney follows suit.

Korgan Rivera
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  • (a) I agree with a few of the things he wrote (for example, the basic Sturm-Liouville type problems taught in a first ODE class is often not very realistically applicable), but also disagree with a lot (for pedagogical and practical reasons). But a discussion of this type I don't think is the best use of this forum. (b) For the course-design aspect of this question, however, I do feel rather strongly that perhaps the MathEducators Stack Exchange is a much better place to ask this question than here. – user225318 Apr 03 '15 at 19:37
  • @user225318 I was unaware that there was a MathEducators Stack. Thanks! – Korgan Rivera Apr 03 '15 at 19:39

2 Answers2

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I can sort of see what Rota is arguing for but it's kind of a fine line because his arguments should also apply to calculus: "why teach epsilon-delta methods", "why teach volume integration of ridiculous rotated surfaces" etc etc. More to the point, "why teach abstract nonsense to applied-math majors?" I would argue that, aside from giving students an introduction to differential equations, such a class should primarily serve as a reinforcement of recently learned mathematics (see below).

I think Rota's main point is that we shouldn't teach non-linear differential equations in introductory classes. For me, the biggest positive about differential equations as they are taught, is that they give students one of the first direct applications of linear algebra. You really get to see the power, and the abstract usefulness of linear algebra and how it applies to things that students previously didn't think of as vector spaces. It reinforces the power of thinking about kernels, eigenvectors, jordan-block decompositions, etc. I eagerly support differential equations being taught in tandem with linear algebra.

As well, numerical differential equations go a long way to reinforce ideas from calculus like infinitesimals, taylor approximations, etc. So I would emphasize numerical methods more. This is primarily because we cannot solve exactly 99.9999% of differential equations out there explicitly. Moreover, numerical methods would show the importance of understanding basic linear differential equations and how they describe, to first order, the nature of non-linear equations.

On the subject of linearity, teaching first order coupled equations isn't bad either, showing the basic fixed point theory and a very brief overview of nonlinear behavior .

If you're a physicist, you'll learn Sturm-Liouville theory, Bessel functions and all that jazz in a later "math for physics" course. If you're a mathematician, you'll learn the finer details of uniqueness, wronskians and nonlinear dynamics later on.

Here's a caveat though. On the subject of reinforcing knowledge, teaching integrating factor methods and similar techniques goes a long way to improve student's understanding of calculus, specifically integration and differentiation. So I wouldn't completely get rid of that either.

Alex R.
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I would go as far as saying that the problems Rota points out are too localised.

  1. The specialisation of students. We don't try to teach future mathematicians all the fine details of finite element methods (however, we should at least give them the basics at some point), they are needed by engineers. Might as well not teach future engineers the Sturm-Liouville theory in the introductory course, it is unlikely that they will need it. Any discussion can be made only from the point of benefit for a given specialisation of the student. If all possible specialisations have the same course (maths, applied maths, physics, engineers, etc), then something is off with the system. The courses have to be adapted for the needs of students, not to be blanket-covering the DE. Therefore, Rota's critics applies only to his course taught to the specalisation of his students.

  2. The course that is discussed here. If - I don't have the sufficient expertise here - it is the same across the universities in the USA, they I would be highly alerted. If all mathematical schools are the same, then something is off.

  3. Then again, there is the pedagogical benefit easily shrugged off by Rota. I think @Alex R. has expressed this point remarkably well. I especially dislike the passage about P.R. and entertainers in the very end.

I suspect that the subtitle of the essay should read "I want to change my course of DE, but I can't".

As a side note, I don't really see the need to teach the Laplace transform (in introductory course), but this is the subject of a completely different discussion.

TZakrevskiy
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