This may read a little differently from the usual, because I actually wrote it to be the education column in the next AWM Newsletter. I realize that for those of you in the AWM this could take all the suspense out of opening the Newsletter, but I figure if you're like me you will have forgotten it by then anyway! So here is how I spent my last week-end--and a jolly good one it was!

Breaking a cycle can be one of life's major challenges. The better established the cycle, the bigger the challenge. One of the best established cycles around is the following: universities complain that they can't fulfill their teaching mission because students are not coming in adequately prepared. Secondary schools whip round and point accusatory fingers at the elementary schools. Elementary schools reply "So who's educating these teachers, anyway?" And around we go again. How to break the cycle? Theories abound, but having long espoused the notion that a student who has gotten as far as middle school has already formed a deeply held view of what mathematics is, what it means to learn mathematics and whether he or she is capable of doing so, I am firmly in the camp of those who hope to attack the situation by helping the people who give elementary students those views, to wit, their teachers. I was therefore delighted when I received an invitation to attend an Elementary Teacher Preparation Content Workshop March 19-21. My optimism grew when I received a large batch of preparatory readings every one of which I really wanted to read. I was not disappointed.

The workshop was at the National Academy of Sciences, sponsored by the National Research Council. The organizers were Gail Burrill, who will shortly progress from being Past President of NCTM to being a past president of it, and Deborah Ball from the University of Michigan and the MSEB. Eighty of us from all over the country, and from every aspect of mathematical teacher preparation spent two and a half intensive days. Friday evening we received our marching orders: until Sunday, we were to work on the first of our two Big Questions: What mathematical knowledge does it take to teach well? More specifically, what mathematics is crucial to the work of elementary school teaching? What can we learn from looking closely at the mathematics that teachers have to teach and analyzing the core tasks and mathematical problems that they have to solve in the course of their work?

Note the absence from this question of "What can we do about it?" That we were commissioned to keep completely out of the conversation until Sunday. Great exercise for those of us who so regularly tell participants in our own workshops "This morning you need to focus on the learning of the mathematics. Don't even think about your classroom until this afternoon!" By way of encouraging this line of thought, we had small group discussions of the readings, a panel discussion on teacher's understanding of fundamental mathematics and a plenary activity on the mathematical knowledge entailed in teaching children to reason mathematically. That last included a videotape of Deborah Ball's third grade class getting a great grounding in the power of generalizations and the beginnings of how to control them. We also had a set of concurrent sessions on the knowledge, skill and sensibilities needed to analyze student thinking, remodel mathematical tasks, analyze student work and manage class discussion, each with an appropriate teacher task available to work on. We each got to two of the sessions and were responsible for filling in our fellow discussers on the others. If any of us had been under the impression that what we are asking of elementary teachers is elementary, we no longer would be. I think the best synopsis of that aspect came from the floor at one of the plenary sessions: we have got to get rid of the tendency to equate basic and simple.

Sunday we were allowed to think about the how. We addressed our second Big Question: How can teachers develop the mathematical knowledge it takes to teach well? Specifically, how might prospective elementary teachers be helped to develop these kinds of mathematical knowledge? Having been duly overwhelmed on the previous days by the magnitude of the task at hand we got to look at some of the alternative and promising approaches being tried. This entailed another set of concurrent sessions, of which I can report on two: Carne Barnett discussed a collection of case studies she and some colleagues have been producing ever since someone commented to her on the oddity of the fact that medical student have medical cases to study and discuss and law students have legal cases to study and discuss, while students of education tend to be hit with straight theory. And that misses a lot. You can dig very deeply into your convictions and blind spots about the teaching of fractions in the course of examining a teacher's report on a lesson culminating in "Which is greater, 4/5 of a dollar or 6/10 of a dollar?" In the other session, a quartet of women from in and around Mount Holyoke reported on Developing Mathematical Ideas, which is a whole series of seminars with the same basic philosophy as the case studies. For me this was particularly exciting, since the NSF project on which I am a co-PI is committed to using those materials widely in Seattle and neighboring school districts very soon.

My small discussion group, which met once each day, had the charge of addressing the distinctly non-small question: "What are some promising ways to help teachers not only develop mathematical understanding but learn to USE mathematical insight and knowledge in the context of practice?" Friday we hurled ourselves at it repeatedly and more or less bounced off, overcome by its magnitude. Saturday we thoroughly enjoyed talking about...all sorts of other issues. Sunday I came in chock full of skepticism, but somehow, as the ideas ricocheted around, it began to feel as if perhaps some shape were emerging, and some ideas being whittled or boiled or squeezed down to a form that might perhaps conceivably have some use. I hope I'm right about that, because in some ways it seems a microcosm of the whole week-end. All such ideas, whittled or otherwise, have now been handed over to Gail and Deborah and the rest of their admirable steering committee. If anybody can whip them into a usable shape, that's the bunch!

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