Normally when I get back from a major mathematical event I come barreling in and hurl myself at the keyboard as swiftly as possible so as to minimize the memory leakage. It's not by chance that this newsletter is an exception. It's been a week since I got back from being a participant in a PMET workshop, and only now is the unruly assortment of thoughts it provoked beginning to have enough coherence for there to be any hope of communicating them. Of course, the fact that I have been on the delivery end of an ECML workshop for three of the intervening days is probably not totally irrelevant, but I think the far more salient fact is that it has taken until now to shake out a loose end solid enough for me to grab and see where following it leads me.

Some background before I wax too philosophical: last January I attended my first meeting of the MAA's Committee on Mathematical Education of Teachers (COMET), expecting to spend it in the neophyte role of observer. I came out very excited, involved to the hilt in two new projects and much intrigued by the results of their previous project: a series of Preparing Mathematicians to Educate Teachers (PMET) workshops. I ascertained that they could use a few old hands along with the folks for whom the workshops were designed, applied with the support of my department and was accepted. As a result I spent the week of the 15th at Humboldt State University taking part in the first of this summer's elementary PMET workshops. I shall refrain from burbling about the gorgeous campus (a student helper said that the alternative expansion of "HSU" is "Hills and Steps University") or scenic surroundings and see what I can do with the substance.

I think even that requires some background. I have been involved in the teaching of teachers for a couple of decades now, and interested for even longer. Along with doing the teaching, I have also gotten progressively more involved with the mathematics education community, have engaged with more issues and become aware of a huge number beyond that. When I began teaching Math 170 (Mathematics for Elementary School Teachers -- the only math course required for admission to the elementary education program), I had a single central theme, which was to convince my students that mathematics was not out to get them. Alternatively phrased, that mathematics made sense and they could do it. I've added a lot since then, but that's still pretty central. Meanwhile, I have spent the years jettisoning one mini-topic after another so as to deepen the focus on problem-solving, place value and fractions. I have learned a lot about the strengths and needs of elementary teachers by working with the wonderful participants in our ECML project. And I have honed my skills at teaching by group-work and become aware of how much more honing they can use.

Outside of the classroom, in the meantime, I have watched the development of a number of themes. One of the most notable is the one which, after the mathematics education community had been studying it for a number of years, burst upon the wider scene when Liping Ma's book took the community of mathematicians interested in K-12 education by storm. Her description of what elementary school teachers need, which has now unambiguously acquired buzz-word status, is "profound understanding of fundamental mathematics". Not the derivative of the sine function, not even the quadratic formula, but why it makes sense to invert and multiply. Several ways of making sense of inverting and multiplying, with an ability to generate good, down-to-earth examples illustrating that sense. Think fast -- how many examples can you generate? This is not trivial mathematics.

This notion has ramifications -- many of them. At the heart of them all is the question: what is the mathematics that teachers need, and how do they use it? I already referred to one discussion of it in my previous newsletter -- a keynote speech by Deborah Ball. One of my fellow PMET senior students just sent out URLs for another of her articles, so I shall repeat mine, add hers, and refrain from further direct discussion of the issue. http://sustainability2003.terc.edu/go.cfm/keynote

Using Content Knowledge in Teaching: What Do Teachers Have to Do, and Therefore Have to Learn? http://www.ed.gov/inits/mathscience/ball.html

Mathematics in the 21st Century: What Mathematical Knowledge is Needed for Teaching Mathematics? Which brings us (finally) to the PMET workshop, and why it threw me thoroughly off balance. I went with my head full of thoughts about what I had learned in and around the classroom, and how I might make that learning be useful to folks just beginning to teach teachers. Patrick Callahan, who ran the workshop, came in with his head full of thoughts about what he had learned from and with Deborah Ball and her associates, and how he might make that learning useful to folks teaching teachers. This resulted in some rapid and not altogether successful gear shifts on my part, and a lot of retroactive re-analysis of the whole scene. The re-analysis began on the last day, when Patrick gave us his background. He had modestly withheld it in the hope of diminishing his centrality, but I found it not only fascinating but very helpful in clarifying many of his choices. Patrick is a professor at the University of Texas, and has long enjoyed and felt successful at teaching (won an award for it, in fact). A few years ago he had a nasty shock. He was teaching an advanced algebra class and found that his students couldn't handle matrices. He ascertained that they had all taken the prerequisite course in which matrices were taught, and decided to determine who had done such a bad job of teaching them. You guessed it -- they had been in his class. Not only that, but it was one that he had felt good about -- and so had they --that was why they had re-enrolled with him. Thus was Patrick introduced to the chasm between teaching and learning, or sometimes even between apparent learning and real learning. He brought the matter up with his landlord, one Uri Treisman (!), and got ever more deeply involved in all the issues surrounding teaching and learning. Eventually (I forget how) he wound up doing some work with Deborah Ball. He was, in fact, one of the people involved in a pilot workshop on which ours was modeled, in which the central figure was Deborah Ball herself. As I understand it (but I should not be trusted too far on this -- I couldn't check details) our workshop had a format very close to the pilot's, with Patrick working on cloning Deborah.

The central element of the workshop was a "laboratory class" of future elementary school teachers. This constituted the third week of a five week course. The students sat around a U of tables while Patrick ran a series of classes on fractions. Meanwhile, all 24 of us participants sat around outside of the U, trying to be invisible and expressionless, but getting up and silently listening in on the discussions of small groups when Patrick gave them group work to do. One thing for sure was that we were all intensely grateful to the students in that class for letting us do so -- none of us could quite imagine doing so, much less doing it with the good grace that they all mustered. And we all learned a lot from what we heard -- neophytes and experienced types alike. Also admirable was Patrick's courage in taking on the teaching under the beady eye of a number of people with a lot of experience and strong opinions to go with it. That's where the going got murkier for me -- I think I kept my face from saying too much, but my notes got pretty emphatic at times. Patrick, who read all of our notes, also took that in good part, which was another admirable feature. After the lab class ended for the day, we would break up into three groups, discussing, respectively, student learning, mathematics in the lesson and teacher moves. Those discussions were among my favorites. Later in the day we had discussions on a number of topics. Most, aside from an afternoon on technology demonstrating some neat software, had an underlying aim of focusing us on the mathematics of teaching. Some succeeded better than others -- that's a very hard topic to get your hands on.

So what, after all that, did I come away with? Lots. For a start, I have been pushed a little farther down the road of contemplating just what mathematics I need to think through, for and with my own students. I have also learned from watching someone else struggle in the perilous and totally essential area of teaching with open-ended questions. I'll attempt to imitate some of the things I thought Patrick handled nicely, and to avoid some of the traps I thought he fell into (much easier to observe from the back of the class than the front). And I have a little more ammunition in my store of ways to try to convince people that teaching teachers is not only crucially important (I think folks are beginning to see that) but mathematically fascinating!

On another plane, the experience has made me conscious of a new parallel. I have long been aware of the way in which the campaign begun two decades (or more) ago to change the way students look at mathematics resulted in a campaign begun a decade (or more) ago to change the way teachers look at mathematics. Now, as a natural consequence, we are in a campaign to change the way teachers of teachers of mathematics look at mathematics. This ties in neatly with another piece of the parallel: my experiences over the years of the CCML and ECML have taught me that if you get together a bunch of teachers who are interested in thinking about the teaching and learning of mathematics you are guaranteed good company and excellent conversations. I now have evidence that that, too, translates: PMET assembled a bunch of people interested in thinking about the teaching of the teaching and learning of mathematics, and the company was so good and the conversations so excellent that I felt that in a week we only scratched the surface. I wanted conversations with more people and more conversations with the ones I did talk with. Frustrating -- but that's one of the better forms of frustration!