Examples Are Better than Proofs, and Other Thoughts on TeachingDecember 13, 2011
Careers in the Math Sciences
I was invited by the editor of the careers column to contribute a few thoughts about teaching and also about changing research directions. I have made it short but intense, as it expresses my whole (home-made) philosophy.
Research first, based on a very simple premise: If we can learn one topic (possibly for a thesis), then we can surely learn another. By all means, contribute everything you can in areas you already know. And move on.
That sounds pretty fierce. But the variety of mathematics is wonderful; so many interesting problems are waiting to be discovered and solved. Research is like life---it is renewed and kept alive by change.
As to teaching and lecturing, I wish I had a theory to suggest. Always, the key point is to pay attention to your audience. We get too preoccupied with a proof, or with "covering the material." Examples are usually better than proofs, and uncovering is better than covering. Ask yourself what you hope the audience will take away from the lecture, and say that part at least twice.
When you let the audience help you with the lecture, everybody gains. By the tone of your voice, you are saying "keep up" and "you can see this step." The implication that you are counting on the audience, that they can come through with the help you need, is a terrific message to give.
Gilbert Strang, a professor of mathematics at MIT, is recognized (literally) by students all over the world who have watched his linear algebra lectures at ocw.mit.edu.
Does he follow the research advice he offers in this column? A quick look at some of his book titles suggests that he does. An early interest in finite elements led to a 1973 text with George Fix, An Analysis of the Finite Element Method. In 1996 came Wavelets and Filter Banks with Truong Nguyen, followed in 1997 by Linear Algebra, Geodesy, and GPS with Kai Borre. Always, the textbooks in linear algebra and calculus are revised and updated. Introduction to Applied Mathematics was extended in the 2007 text Computational Science and Engineering, with video lectures on OpenCourseWare. A new book, Essays in Linear Algebra, is now at the printer (and like the others will be available from SIAM).
Sue Minkoff (firstname.lastname@example.org), of the University of Maryland Baltimore County, is the editor of the Careers in the Math Sciences column.