This is going to be a slightly disjointed column. I spent a considerable amount of time vacillating on the question of which of two aspects of the topic at hand to emphasize and in the end I decided on both. Stylistically ill-advised, but both deserve attention.

The topic in question is a course I just had the good fortune to teach on the history of mathematics. The first aspect worthy of the ink is the context for the course. A decade ago one of my colleagues who frequently teaches courses for future high school mathematics teachers became conscious that while we encourage all teachers to be life-long learners, once these folks graduated we were offering nothing to aid that learning – every one of our courses that might be of interest met during regular high school hours. He persuaded the department to set up a series of one-quarter courses, all called Math 497, that would meet once a week for nearly three hours in the late afternoon. This produced a three credit course which it is physically possible for an in-service teacher to attend, and over the years many have done so. The course is also open to regular students and has attracted a fair variety of them, as well as some community college faculty members. My favorite blend is about one third in-service teachers, one third undergraduates majoring in mathematics for high school teachers and one third a motley collection from all around campus, with a community college teacher or two thrown in. It makes for a wealth of perspectives and some really good conversations among the students.

Recently, for a variety of reasons, enrollment of in-service teachers has declined somewhat. Given limited resources, the question of the course's continuation has to be addressed. We've held out, though, partly on the basis of a slightly abstract but highly relevant consideration: as a department we are committed to supporting the K-12 system. We look hard for meaningful ways to accomplish that, and have had some success. Math 497 represents a quiet but solid way of conveying that support.

That covers one aspect of 497 courses – their raison d'être and the benefits we hope accrue from them. Of another I gave a hint above: they're great fun to teach. Part of the reason is the students, who tend to be interesting and interested. Another part is the required course material, or rather the lack thereof. What is required is a topic that has genuine mathematical content, requires thinking but doesn't rely on pieces of mathematics that may have developed rust if a teacher has been in the K-12 classroom for a couple of decades, and is engaging enough to keep people awake for three hours at the end of a full day. In other works, it needs to be something the instructor can really enjoy teaching – what a demand! It can even be something about which one knows a bit but wants to know a lot more. Most of the Graph Theory I know came from an early 497 class, and a colleague became very happily tied up in Knot Theory. Recently (and here's where we switch to the other aspect) I opted to teach a 497 class on History of Mathematics and as a result became a complete convert to teaching both about it and with it.

Given that I chose to teach it, I clearly was not converted from an adverse position. In fact, I have long been interested and wished I knew more. In the past few years this interest was reinforced by the group of mathematicians who have been advocating the uses of history and by one colleague in specific who demonstrated some of them. In the midst of a perennially oversubscribed calendar, however, kind of wanting to know more doesn't cut much ice. Enter Math 497. After thinking about it for a couple of months and weathering a few bouts of cold feet, I submitted a course title of "Where did all this mathematics come from?" and the die was cast.

The decision once made, the question then was how to balance survey and depth, information absorption and relevant activity. In this I was greatly helped by two books. The first was A Concise History of Mathematics, by Dirk Struik. I found it before I committed myself to the teaching, and found it admirable for showing the sweep of events and the spirit of the different ages and cultures. The other was From Five Fingers to Infinity by Frank Swetz. That one turned out to be out of print, but Amazon.com did us proud, and eventually everybody had a copy. It consists of 114 brief chapters, most of them reprints of articles in the NCTM journal, Mathematics Teacher, and each giving a focused insight into some specific topic – the work of a particular person, or the development of some mathematical concept. Each week pairs of students were responsible for reading a small bunch of chapters and choosing one to present to the rest of the class, with the strict requirement that they must give their classmates something to do, not just report out information. The results were generally good and occasionally spectacular. They also maintained (by assignment) an ongoing on-line discussion at a class e-Post, which ranged from the dutiful to the impressively thoughtful. A major project gave them the opportunity to delve more deeply into some one topic. For the take-home final I wanted them to look back through the mathematical developments they had seen all quarter, so I asked them to inspect it all through the lens of E.T.Bell's statement, made in Mathematics, Queen and Servant of Science: "The pure serves the applied, the applied pays for the service with an abundance of new problems that may occupy the pure for generations." Those who were not blown away by the metaphor produced some good insights.

As should be clear, I had a lovely time with all this. I also learned a tremendous amount, including how much more I would like to know. Along with that, I have become ever more convinced that elements of history can enrich and enliven almost any mathematics course. I have even managed to get my toe in the water on that: Euler and Egyptian multiplication were both quite well received, and I have an eagle eye out for my next opportunity.

And to top it all off, I finally know the answer to a question that has been bugging me for years: how did the Romans manage to carry out multiplication and division with such an incredibly cumbersome numerical system? Answer: they didn't. They did it all on an abacus and just wrote down the result!