The recent gap in newsletters does not indicate a dearth of goings on. On the contrary, it was caused by my really putting my back into my efforts to learn to be in two places simultaneously. Didn't quite work yet, but just you wait!

The bulk of this is an education column for the upcoming AWM newsletter. There are two things I want to insert first, though. The first is a Brown Bag report. On January 13, Monty McGovern ran a well attended and much appreciated session on what he does at (and for and with) his local elementary school. He coaches -- and has done for several years -- a club for teams enrolling in a Math Contest. The contest itself is a one day festivity, and when I expressed my worries about damaged egos, Monty said the kids really do seem to enjoy the day thoroughly and come through in fine shape. Preparation for the contest goes on for many months, with classes before and/or after school, and lots of good problem solving all the time. It showed us a nice, clear example of a mathematician using his professional skills to benefit a bunch of bright kids -- and having a whale of a good time doing so.

A nice spin-off from Monty's Brown Bag was the discovery, in the course of a round of self-introductions, that there were also a number of other folks involved with schools in similar ways. This launched an effort to find out whobody else is in like state -- that's the kind of thing it's really nice to know about.

And one bit of learning that I did at the Joint Meetings I feel compelled to share, even though it was probably the most frivolous thing I picked up all week. I was in a meeting of the Committee on Mathematical Education of Teachers, and we were discussing the variety of educational programs with which the Committee has been working. After one particularly undecipherable reference, the guy beside me moaned "Oh, no! Not another TLA!" In response to my slightly baffled reaction, he explained: "Three Letter Acronym!"

OK -- on to the column:

The Joys of Educating TeachersÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

A couple of decades ago I took on the teaching of the content courses offered to future elementary school teachers by the University of Washington Mathematics Department. My motivation was largely supplied by social conscience: the experience that began my career was directing SeattleÕs Project SEED, which took me into elementary classrooms all over the city. I became aware of how very important elementary school teachers are, and of how much they need and deserve our support. The feeling of obeying my conscience was lovely. Eventually, though, it was joined by another feeling Ð that of needing to know a lot more than I did. This feeling launched me on a whole new phase of my career. In this phase I have gotten to know many folks specializing in mathematics education, first close to home and eventually internationally. I have shared the joys and perils of a massive and wonderful professional development project, working with a team of mathematicians, some of us from the Mathematics Department, some from the College of Education and some from local school systems. I have read some fascinating material on the learning and teaching of mathematics. These experiences have not actually diminished my sense of needing to know a lot more, but the sense of virtue has long since been supplanted by a sense of adventure, not to mention plain pleasure. Last year I was enormously pleased to be asked to join the MAAÕs Committee on the Mathematical Education of Teachers, and even more pleased when I discovered that the committee was sponsoring a series of what are called PMET Workshops Ð Preparing Mathematicians to Educate Teachers (see www.maa.org/pmet). At the recent Joint Meetings in Phoenix, a special session co-sponsored with MER (Mathematicians and Educational Reform) gave a progress report on PMET and its issues. Speakers talked about their experiences with teaching courses for K-12 teachers, or designing them, or writing textbooks for them, and an impressively large collection of folks turned up to listen. The whole afternoon was very exciting, but one moment stood out for me. It happened while Jim Lewis, of the University of Nebraska, was describing his own experiences. It seems that after years of avoidance generated by a memorably hideous experience very early in his career, he recently decided to take himself in hand and make himself undertake the teaching of future elementary school teachers. He joined forces with a colleague from the College of Education, took a deep breath and dived in. He described the course they designed and co-taught, and the huge amount of work he put into it, and a number of surprises and challenges. Then he finished off with what was to him the biggest surprise of all. ÒIt was,Ó he said with a considerable gleam in his eye, Òa lot of fun!Ó

Needless to say, I was delighted with that line. Among other things, IÕll feel much less guilty as I gently twist the arms of my colleagues to persuade them to get involved in teacher education. On the other hand, I find that I still think of the joy of educating future teachers as isolated from the more commonly visible joy of teaching mathematics to future mathematicians.

This week I got a glimpse of a bridge between those joys. The explanation of that glimpse has two components.

This quarter I am teaching for the first time a course the University of Washington instituted a few years ago entitled ÒIntroduction to Mathematical Reasoning and Proof.Ó I find myself grading papers of students who are clearly bright and able, but nonetheless give me three examples when asked for a proof, or negate Òfor all x, x**bÓ, or É I expect anyone reading this could supply many more instances. This has provided me with much food for thought and a fascinating teaching challenge. With these papers fresh in my mind, I attended a talk by Virginia Bastable at this yearÕs conference of the Association of Mathematics Teacher Educators. Bastable is on the faculty at Mount Holyoke College, directs their Summermath program and is one of the co-PIs of the Developing Mathematical Ideas project.**

Bastable was discussing algebraic thinking in elementary school, which she illustrated with a case history from a kindergarten class. The teacher set the class up in pairs to play ÒDouble Compare,Ó in which each child would turn up two cards from a stack of cards with numbers 0 to 6, and whoever had the greater sum would get the cards. As she cruised about the room observing, the teacher noticed that a number of the kids were responding instantly to situations with, for instance, a 6 and a 2 being compared with a 6 and a 4 Ð numbers whose sums a kindergarten child does not havae immediately available. Intrigued, she instituted a discussion of what was going on, and one of the kids articulated nicely: ÒIf two numbers are the same you donÕt have to pay attention to them.Ó Others had good, if less efficient, ways of describing their thinking. She was duly impressed, and in an interview after the class session she speculated on what would have happened if she had tried to get them to articulate one of the other situations that she saw being acted on, where each of one childÕs cards was smaller than one of the partnerÕs cards, but not both (e.g. 1, 4 and 2,6).

Discussion of this case study among the people at the AMTE session was extremely lively. We had a lot of admiration for the teacher, both for her perspicacity in noting the kidsÕ thinking, and for her ability to get them to articulate it. If we can succeed in preparing teachers to do that, we will have made a major accomplishment. It was noted, on the other hand, that we should refrain from feeling critical of those who miss such opportunities. For one thing, the opportunities are much easier to see from the outside or by hindsight than on the spot. On top of that thereÕs a spot of basic human nature: that game was introduced with an eye to getting the kids to practice addition. What the teacher observed was actually a form of avoiding addition by seeing when it wasnÕt necessary, so that following up on her observation involved abandoning her planned topic (one for which there is a lot of pressure, at that!)

Eventually the session continued with a videotape of some first and second graders discussing the relationship among the operations in a set of word problems involving addition, subtraction and the same three numbers. These problems also generated a lot of discussion among the AMTE participants. Towards the end, Bastable described for us the progression she has observed in kidsÕ response to generalization: at first there is none, making each case a new and exciting discovery (not only is 3+5 even, but so is 7+9!). Next comes ÒIt must be true, IÕve tried several examples!Ó Then a little conservatism strikes: ÒI donÕt know for sure, because there are lots of numbers I havenÕt tried yet.Ó That may even lead to ÒYou canÕt prove it, because there are always more numbers to try.Ó And then at last ÒHereÕs the reason why it really has to be true.Ó

This is where I recognized that I was no longer in the land of ÒHow can we educate teachers to help students through this progression?Ó I was in the land of ÒThatÕs where a lot of my students are Ð how can I reach them wherever they are along that progression (some, alas, still not all that far along it!) and coax or entice them forward?Ó ThatÕs an interest I guarantee I share with a lot of folks who have not yet discovered the joys of teaching teachers.

This experience was just a glimpse of a bridge between my preoccupations as a teacher educator and my preoccupations on the general teaching front, as I said. I have a shrewd suspicion that itÕs part of a whole system of bridges. Furthermore, I am beginning to realize I am not the only one jogging around on that system. I look forward to more explorations of the bridges and to finding the company on them!

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