Once again this newsletter's asynchronicity has reached new extremes.
Sometimes that happens because not enough is going on and sometimes
because too much is going on. In this case, I would say that neither
applies. Lots has been going on, but not too much of it newsworthy,
because it is a natural outgrowth of things I have already talked
about. Math Fairs, for instance, are in the process of becoming an
exciting element of the department's outreach efforts -- but that
process is still a work in progress, so it won't hit the newslines
until it's a little solider. We had a lovely Brown Bag about how to
convince students whom we have accepted into our graduate program to
choose us over other places that may have accepted them, but none of
the numerous ideas that came out of it was earth-shattering -- just
good ways to morph mildly what we already do. There's WaToToM
coming up, so the next silence should be at the short end of the
asynchronicity interval, but for the moment what I have to offer is an
AWM Education column offer. On the other hand, it's about a completely
local event, so at least the void has a reasonable filler. Here it is:
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!