Guest Post: Bernd Schroeder on teaching non-experts expert material
This is the second in our series of posts by Bernd Schroeder, Wiley author and academic director and program chair of Mathematics and Statistics at Louisiana Tech University.
In this post, he talks about how we can expertly teach non-expert students in a manageable way.
Click here to read his previous post about preparing STEM and non-STEM students for a workplace that demands mathematical skill sets.
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How to Teach Analysis?
As with my other post, the question mark should indicate that this is not a “How To” manual but food for thought instead. I will not claim to be right, but your reaction to what follows can tell you a bit about your own comfort level with change.
At my institution, the sooner students are ready for work in Numerical Partial Differential Equations or Physics, the better. This should be common because by the time most people know all the mathematical details they need for Numerical Partial Differential Equations or Physics, they’re old.
We can make the case that much of the requisite theory builds on pretty deep functional analysis, which builds on measure theory and linear algebra, which is best taken after a first proof class in analysis. So, assuming that these concepts also need time to settle in your mind, a time of 3 years between the first analysis proof and the end of a functional analysis class may not be unrealistic, possibly even too fast.
It is typically the case that a mathematics graduate student who invests 3 years into these fundamentals should have had some of them as an undergraduate. But even if 2 years of graduate school are spent on fundamentals, graduation in a total of 4 years becomes a challenge.
For non-mathematics students, investing 3 years into the mathematical background for the work they do seems unreasonable. Unsurprisingly, many of them do not take “our” (that is mathematics departments’) classes.
In summer 2013, I taught the spectral theorem for unbounded self-adjoint operators on dense subspaces of infinite dimensional Hilbert spaces to a group of 5 students most of whom started their first proof class in analysis in December 2012. (Disclaimer: As I recall it, students asked about the mathematical background for quantum mechanics and I decided to provide it to those who would volunteer for the ride.) The net exposure to analysis for most of my students was two 10-week quarters in which we had a semester’s worth of instructional time, plus the 5 week summer session (another semester’s worth of instructional time). The pace and density of material were quite murderous, as there was nary a result in the early part of the development that was not quoted later.
At the same time, we only left a small number of logical gaps in the presentation: Fubini’s theorem and products of measure spaces, the density of the compactly supported infinitely differentiable functions in Lp, and the Stone-Weierstrass Theorem were discussed, but not proved. We also spent the last day discussing how the powerful functional calculus leads to the Spectral Theorems for Unitary and for Self-Adjoint Operators and did not go through all the technical parts of the proofs, which would have taken two days. Overall, given another 3 weeks, maybe less, we could have done it all without gaps.
Given the short time of exposure, we cannot expect the students to have the same deep connection to the content that an expert has. However, I feel that these students can construct a decent proof in analysis and elementary functional analysis on a regular basis. That is not a bad outcome for being 8 months removed from being first exposed to analysis proofs. Moreover, these students have seen a lot of content that will be useful in their applied classes (spectral theorems, the elements of complex and functional analysis needed to prove them, Lp spaces, convergence of Fourier series in L2 and plenty of in-class remarks attempting to make connections to numerical analysis, physics, etc.). To me, this is preferable to spending a lot of time in training exercises, which leads to students not even seeing the Lebesgue integral in their first year.
Are there gaps? Certainly. Anything that did not directly contribute to progress towards the spectral theorem for self-adjoint operators was omitted. The students have not proved, using ε and N, that the limit of the nth root of n is 1 as n goes to infinity, they have not proved the limit comparison test for series, etc. Is that acceptable? Personally, I find Lp spaces much more important than lots of details on series (just about everything that we needed went back to facility with the geometric series). Similarly, ε-N type training can be provided by analyzing the Dirichlet kernel rather than with training exercises. So, overall, I feel reasonably good about the job we did. Further iterations of this sequence can always be improved by picking the right exercises (and by slowing down a little).
How do you define success? How do you assess it? Here is where judgment calls are needed. It is a virtual certainty that there are problems from a first analysis proof class that would be a lot harder for my students than for students who went the usual route. I also noticed that, although I did not need specifics from the topology classes I took, I was a lot more comfortable with “continuity means inverse images of open sets are open” than my students were: For them, that was one theorem among many with the importance slowly emerging in the course this summer. If that is considered to be a problem, then my experiment (if you will) failed. On the other hand, these students are developing a feel for Hilbert spaces and L2 at a time when other students just learn the definition of an open set.
Is it hard to design such a new approach? Well, let’s say that I was surprised when I thought that I could avoid using the Hahn-Banach Theorem. So I designed the course without proving the Hahn Banach Theorem and the surprise lasted until I ran into a proof (the Cauchy Integral Theorem for Banach space valued functions) that is best done by using a consequence of the Hahn-Banach Theorem. That result could also be proved by simply reworking the proof from complex analysis with the range being a Banach space instead of the complex numbers, but nonetheless …
Along the lines of creating new approaches, the challenge for this sequence of classes is the same as for any change to canonical approaches: First of all, because we are supposed to model logical thought, we have to create something that is logically consistent. After this first step in our student-centered approach, we then have to figure out if it does what we intended to do. For example, the mind can be overchallenged by an approach that is too dense or too fast. You may rightly say that the class was both and, if so, I will not argue against you. However, my experience shows that the mind can stand up to much stricter rigors than we may give it credit for.
Overall, I certainly recommend approaching change with care. The “race to the spectral theorem” above is the product of about 10 years tinkering with the structure of fundamental analysis. Abject failure at any stage would have likely diverted the project from the result described above.
So be careful, and when something does not quite work, learn from it. As long as the bumps in the road can be navigated and you have at least half as much fun as I had with my spectral theory class, you’ll do fine. Just make sure your department head knows and supports what you’re doing. I had a slight advantage there, because I am the department head
I do answer to a dean, though…