# The Unapologetic Mathematician

## Updated teaching philosophy

Just a week or so ago, Trinity College Dublin listed four one-year lectureships for next year. This position would be really great, and I’m still unemployed for next year, so I’m pulling out all the stops. I even sat down and rewrote my research and teaching statements to incorporate the latest developments in both. The latter I think is worth posting here.

The academic mission is twofold: not only are we charged with extending the boundaries of knowledge, but we are equally tasked with communicating it to each other, and to the rest of society. Far too often this responsibility is viewed as at best a necessary evil, but it is essential to the academy as a whole.

Instructing undergraduate students is the largest part of this job. Unlike secondary school teaching, however, at the university level it is important that students take the lead in their own education. A lecturer can present a subject and field questions to clarify the material, but it is not the lecturer’s job to make students learn.

In fact, the goal should be to get students interested in mathematics for its own sake, not just as a tool for other subjects’ use or (worst of all) as a route to a degree and nothing more. I find it helpful to drop references to more advanced areas in the midst of lectures. For instance, a first treatment of power series is enough to give a thumbnail sketch of Fourier series, and Green’s theorem provides enough muscle to prove the Fundamental Theorem of Algebra. Even just a passing mention of generalizations can help show that a given course is not self-contained, but is intimately woven into the rich tapestry of mathematics as a whole.

Of course, not every student will become interested in mathematics, and it’s important not to shortchange those who just need to learn the material. For them I try to keep a collection of good applications of many mathematical concepts to various other fields. A computer science student may be illuminated by examples of running times of algorithms, while an engineering student may be better reached through examples of approximations. Of course, references to the mathematics behind everyday experience can be extremely helpful, if somewhat harder to come by.

In teaching postgraduate students the focus changes. Here the students have already seen the value in mathematics for its own sake. The challenge is to help them make the transition from proving given theorems to stating theorems of their own to prove. It’s also necessary to keep an eye on the current course of mathematics and to help the students learn not only what they need for today’s mathematics, but for tomorrow’s as well. I have not yet had the opportunity to interact with postgraduate students as an instructor or an advisor, but I certainly look forward to the opportunity.

One aspect of the educational side of an academic job that is often overlooked is to communicate one’s subject to the outside world. Mathematicians, particularly, are content to be seen as some sort of elite priesthood, divorced from the “real world”, except as mediated by physicists and engineers. Physicists and astronomers have benefited greatly in the public esteem from popularizers, some excellent (Penrose, Hawking) and some less so (Capra). These authors tap into what I call the “generally interested lay audience”: those who find a basic knowledge of currents in physical sciences to be part of being well-read. However, there is little work on similar popularizations of mathematics.

To this end, I have joined the recent spate of those writing mathematically-themed weblogs. My effort, The Unapologetic Mathematician (with a nod to G.H. Hardy), is primarily concerned with providing a self-contained reference to basic fluency in mathematical concepts. However, I do turn to more advanced material. In the wake of the Atlas Project’s much-publicized calculations of the Kazhdan-Lusztig-Vogan polynomials for the split real form of $E_8$, I set out on an attempt to explain the project’s work to mathematicians working in different fields. David Vogan has praised the effort to spread knowledge of what might otherwise be a niche subject to a much wider mathematical audience, and possibly to some interested nonmathematicians as well.

Not to be left out is the importance of interactions with the local community. I have worked teaching academically talented secondary school students in summer courses on cryptography. I enjoy giving elementary knot theory talks to primary and secondary school mathematics classes. I try to stay current on the state of mathematics education in the local school system, and to do what I can to help improve the programs through which students will pass before ever getting to my own classroom. All of these are part of the educational half of the academic mission.

May 7, 2007 - Posted by | Uncategorized

1. “In fact, the goal should be to get students interested in mathematics for its own sake”

There is no reason at all for this to be the primary goal and when it is the vast majority IS being shortchanged.

If students develop an interest in recreational math by themselves that’s fine.

Comment by Maya Incaand | May 8, 2007 | Reply

2. I have to disagree here. Part of the reason is the straw man you’ve set up of recreational mathematics. I don’t mean to interest students in toy problems, but to show them that the actual practice of mathematics is vitally important, interesting, and worthy of studying on its own terms.

I have a list of students who have taken my courses and added a mathematics major either on its own or as an adjunct to a major in economics, computer science, molecular biology (Yale’s de facto pre-med program), and other fields. These students don’t just learn the mathematics they need in their other field, but about mathematics in general. This training in mathematics qua mathematics functions as training in rational problem-solving skills, and I’m sure that it benefits them more than a few scattered courses as on-paper requirements for another major.

Comment by John Armstrong | May 8, 2007 | Reply

3. Maya, I’ll join with John in a complete disagreement with that statement. First off, there is a fundamental difference between mathematics for its own sake and recreational mathematics. In the latter, you’re basically doing elaborate puzzles, which is fun, but tends to be of non-astronomical depths. In the former, research is being performed, with all that this entails.

The view that mathematics is a “tool science” is one extremely common outside of mathematics, and one I frankly find offensive. The idea that mathematics for its own sake is something weird, belonging to an ivory tower, and something that students should be protected from unless they accidentally discover that there might be more than recipes hiding in mathematics.

I have yet to find any other science which has the same stigma in the popular perception. It would be completely unacceptable to say things like your statement about physics, or literature, or economics, or .. well .. any other science, really. And the reason that it isn’t something people are ashamed of stating when it comes to mathematics is closely connected to what has been described as The Two Cultures. The rift between people who view mathematics as a science, and those who just want the minimum needed to do what they really care about, or even want to avoid any contact with it altogether.

NOT telling students about what might make mathematics worth caring about outside a toolbox seems to me shortchanges everyone. Many don’t know they’re being shortchanged, but nevertheless they are.

Comment by Mikael Johansson | May 8, 2007 | Reply

4. I admit that “recreational” was a bit of a tease or a straw man if you prefer. “Research” is fine.

As it happens I am quite fond of mathematics but I stand by my statement about the way it is taught (at any level except postgrad, too late already).

There must be a diagnosis for the symptoms you describe, “elite priesthood”, “stigma” and so on; there also must be a reason for the continuous decline in interest in the subject (particularly in the West).

If we could agree about that at least then we might progress (or we could become theoretical physicists:-)).

Comment by Maya Incaand | May 8, 2007 | Reply