First a caveat: this is a report on two Brown Bags and a forum, all of which I was directly involved in. So you have to read it as a report on what was said--or what was intended to be said--rather than what was heard. Though I do hope there was at least a little correlation.

A little background is in order for both pieces. The first week of spring quarter I spent some time at Rutgers. I went without specific agenda, just knowing that their mathematics education bunch has for years been doing things totally attuned to CCML's approach and philosophy and their mathematics department has dealt in some interesting ways with some of the same problems we deal with here. My theory was that I ought to be able to learn a lot just by being there. Not only did the theory prove correct, but I got myself involved in a potential project, about which I will write more if the Annenburg Foundation actually funds it. In the math department I learned about some neat things they have been doing using undergraduates to assist in pre-calculus problem sessions (benefits in multiple directions), and heard a colloquium by Hyman Bass about the results of giving some questions designed for elementary school teachers to his hot-shot freshman calculus students at Columbia. In math ed I learned more about the dizzying array of outreach projects they have going, and in particular about the one which first attracted my attention: they have been carrying out one of the very few longitudinal studies of children's learning. Starting in the mid-eighties they began working intensively with a bunch of kids in one of the New Brunswick inner city elementary schools, doing enrichment sessions a few weeks a year with the whole class and videotaping interviews with a subset of the class which came to be known as the gang of four. They continued over the years. When the kids hit high school they spread all over the city, so the timing switched to after school, but the sessions still continue, and so do the interviews. The result is a terrific record of how mathematical thinking can develop over the years given a situation which encourages its growth.

This last project has produced two artifacts: a CD and a videotape. The tape was the centerpiece for the April 8 Brown Bag. Unfortunately I don't think I did an adequate job of introducing it. Some of the points I find especially fascinating--like the fact that having a tremendously solid grasp on the issue of building towers out of two colors of blocks provided the handle by which ten years later one of the students was able to grasp Pascals triangle--turned out to be decidedly non-obvious in a straight-through first viewing. I think the underlying message about kids' thinking capacities came through pretty clearly, though. There is no missing the glee with which those third graders dived into the discussion of how you could know you really had all of the possible towers, nor the persistence with which they later were still able to tussle with a potentially frustrating problem.

The background for the other two events extends rather further back. For the past several years I have periodically disappeared to France to work on Didactique, or the Theory of Situations. The first several visits centered around translating and editing a book of the articles of Guy Brousseau, who founded the field. When that went to press (Kluwer Academic) I briefly went into a spin, and then launched into what proved to be an even more exciting process: working with Guy and his wife, Nadine, on a wonderful assortment of his ideas, some of them recent, more of them enrichments of ideas he had earlier. What I thought of at first as a project of translating from French to English turned out more a matter of helping translate from the inside of Guy's wonderfully furnished head to the outside world in both languages--much more exciting all round.

For the middle two weeks of April, thanks to support from the Milliman fund, the K-12 Institute and the College of Education, we were able to pursue our projects together here in Seattle. We put the finishing touches on our first article, now on the brink of appearing in the Journal of Mathematical Behavior, and its French twin, dug further into a study on the learning of probability, based on an experiment Guy conceived and Nadine carried out in her classroom a number of years ago, and even dreamed up our next project, which is to re-work the book the two of them wrote in the seventies on the teaching of fractions and decimals.

And in the midst of that, Guy gave a Brown Bag and we both took part in a forum. The Brown Bag centered around the idea of didactical transposition. Basically, the idea is this: mathematics is created in some context, along some personal mental path that generally includes detours, dead-ends and a few disasters. It is then de-personalized, de-contextualized and made as general as possible before being presented to the outside world. This isn't just a matter of cleaning up the mess for company--it is a necessity for communication purposes. And that whole process, in microcosm, is what the student needs to be able to carry out. The first job of the teacher, then, is to set up a context in which the student will be led to create the desired concept, quite possibly with some dead-ends and detours along the way. But then the second job is to help the student de-contextualize the result and recognize that it is a part of the culture of mathematics. Not a simple request, but the alternative of presenting the mathematics in its de-personalized and de-contextualized form produces an undigestible mental lump, and we all know the consequences of that.

The Brown Bag finished with a lively discussion of the fact that the French have two words for our one verb "to know" (savoir and connaitre). By the end of the discussion none of us was sure we knew what we know.

Later in the afternoon, Guy and I joined with Sharyl Burgstahler (Assistant Director of Information Systems, Computing & Communications) in a forum on the New Math, sponsored by Math, Education and the K-12 Institute. Thanks to the breadth of the sponsorship, we had a nice breadth of participants, which made for a great discussion. I led in with a highly subjective account of my own experiences, starting with the excitement of my father and his colleagues as they created the New Math and his distress over some of the reactions, which I analyzed with large slabs of hindsight. Sharyl discussed what it was like to be a math-loving elementary school teacher watching with some incredulity the failure of her colleagues to share her delight in the New Math. And Guy described its progress through France: the Bourbakist frame of mind left the educational establishment wide open to its drier aspects, and in the end "the people didn't just throw out New Math--they threw out Math!" Though we all agreed that the tendency to demonize New Math has led to a general failure to recognize that in fact it left some excellent things in its wake. And that there is a lot that current efforts can learn by looking at its history.

I shall finish this somewhat cumbersome document with one neat news tidbit: two years ago, for two years duration, we had a grant from the Pew Charitable Foundation called Preparing Future Faculty. It enabled us to form some really good ties with Seattle University and Seattle Central Community College, and to send some of our graduate students to each of them to get to know something about life at a two- or four-year college. When the grant ended, the Graduate School funded a continuation of the centerpiece of the grant, so that we were able to postpone the day of having to cease sending students out. And now, just as that extension was about to run out, the AMS and MAA have taken up the project, and, thanks to yeoman efforts of Jack Lee, we have just received one of the new PFF grants. Sound the trumpets!

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