When you go to a talk and the speaker addresses a topic that interests you, it is pleasant, but not too surprising (why else were you at the talk, anyway?) When a second and rather different topic turns up, it is a delightful bonus. But when a third one appears, it is a downright bonanza -- a phenomenon known to soccer players as a hat trick. I just had that happen to me, and I'm still feeling slightly gleeful.

The occasion was a brief presentation by Amanda Hines, a high school (I think) teacher in Edmonton. The first topic was Math Fairs -- not astonishing, since that was the topic of the workshop we were both attending. Upon that subject I have already burbled on these pages (or screens) so rather than re-burble I shall refer you to that newsletter: http://www.math.washington.edu/~warfield/news/news117.html I will merely add that the workshop, which was sponsored by PIMS (the Pacific Institute of Mathematical Sciences) and run at BIRS (the Banff International Research Station -- heavenly spot!!) provided all manner of inspiration, both global and local, to reinforce my intention of continuing doing Math Fairs in our own fair city.

From the Math Fairs, Amanda went on to another undertaking of what sounds like a pretty spectacular mathematics department: Lesson Study. This is an idea that was introduced in Japan over a decade ago, and has been much ballyhooed over here, but only rarely tried. The Bellevue system did try it -- rearranged their schedule to allow the possibility. It may even still be going on, but I haven't heard many recent rumblings. In any case, the general idea is that a set of teachers get together and do an extremely focused study of one particular lesson (or possibly short sequence of lessons). They study it hard from many, many perspectives, then put together what they feel is the best combination of their ideas. After that, one of them teaches while the others watch either directly or by videotape. It is understood that the watchers may be sharply critical -- but of the lesson, not of the colleague who is carrying it out. The results can be extremely impressive, and by the sound of it, Amanda's school's were just that.

While I was still rejoicing in hearing again about something about which I have heard very little for several years, suddenly yet another old friend of a topic arose: it seems that after duly dismantling and reconstructing a couple of items from the standard curriculum Amanda and her colleagues decided to tackle problem-solving -- how, as another speaker put it, to get past the students' initial "I can't!" reflex, and to get them to dive in and tussle with problems. In the course of describing her approach she got so close to the things David Prince had been saying at the Brown Bag a couple of days before that I'm not certain to which of them to attribute some of the remarks.

To make sense of that, I need to divert for a moment to the Brown Bag itself (don't tell my English composition teacher!) That was the one featuring Edwin O'Shea, who has enormously enjoyed teaching sections of calculus heavily populated by students from the MSEP (Minority Science and Engineering Program), and David Prince, who runs the program. There are many reasons why these students are particularly focused and engaged. Among them are some advance preparation the summer before the students' Freshman year and a newly constructed (thanks to a marvelous donor!) study center rather like the departmental one, but considerably more intimate. The one that struck me most, though, was the on-going workshops David runs, in which he challenges the students with extra-tough problems and sees to it that they grapple with them.

So how do you deal with the fact that in any group of students some are going to catch onto a problem while others are still struggling? This is where Amanda and David converged so thoroughly that I'm not sure which of them is responsible for the line I especially enjoyed: "That's a nice solution you've produced. Now go and teach it to at least two people and then you will begin to understand it!"

Zooming out a little, I should place Amanda's talk in context. It was, as I said, part of a conference on Math Fairs. A number of other people talked about their own Math Fairs (including the ones in Sweden and Tai Wan!) Our own contingent included of Lisa Korf from the Math Department and Valarie Reissig from Leschi Elementary School, where we did our Math Fair. We enjoyed bringing in some of our own pictures and descriptions, especially since we had the impression that none of the others schools had quite the same minority make-up.The talks were nearly all informative and useful. They were also a lot of fun, and one of the reasons for that is that they were studded with really neat Math problems. Most of the problems I shall keep up my sleeve, but as a reward for getting all the way through this, you definitely have earned a few, so here are some of my favorites:

It is obvious how to subdivide a square into four squares, and also obvious that you can't divide it into two squares. For what values of N is it possible to subdivide a square into N squares? The squares need not be the same size.

How about an equilateral triangle?

How about a non-equilateral triangle (to be subdivided into triangles similar to the original one)?

Suppose you have a clock with hour, minute and second hands, each hand occupied by a very well trained fly. Every time any hand passes another, the flies occupying the two hands exchange places. Between noon and midnight, how many rotations does each fly make?