Not long ago a friend whose opinions I value commented that she felt "community" was becoming a buzz word. As a certifed member of the buzz word police, I was put immediately on the alert. And as someone who has invested a lot of herself in the building of community, I reacted with the thin edge of panic. Last week's visit from Gail Burrill served to allay a lot of that panic. There is a very real and live community of people with an interest in mathematics and its teaching. It's pretty loosely woven in spots, and can do with a lot more connecting and tightening, but it is there.
Gail's visit, which lasted roughly from Monday noon to Wednesday midnight, included three lectures to different audiences, a Brown Bag seminar, three formal meetings, several informal meetings and a couple of observations (ridiculous when you string it out that way, but she took it magnificently in stride!) Clearly, a chronological account is not the order of the day. On the other hand, it is also not needed, because one of the stong pieces of evidence for the existence of a genuine mathematical community was the fact that her talk for the Puget Sound Council of Teachers of Mathematics, her colloquium for the mathematics department of UW and of a bunch of colleges, universities and community colleges hereabouts, and her lecture to anyone from the general public who took an interest in the issue had a number of deep themes in common.
One was the "Where are we?" issue. It is occasioned by the barrage of test results and other statistics we are currently sustaining. With no particular difficulty one can become convinced that the end of the world is at hand and American students are no longer learning anything about any mathematics whatever. For this, Gail had a number of counter-statistics. On computational skills, for instance, American students at all levels appear to be holding steady or even improving. Other indicators also belie some of the disaster scenarii. Does this mean the scene is all rosy in this the best of all possible worlds? Well, no. Gail cites, for instance, the respected book on career decision factors whose scale for comparing incomes includes the example: "a physicist with a mid-level income earns $67,000. His or her income Growth Potential is 138%. Adding these two yields a score of $67,138." This in the second edition, mind you, when readers have had a couple of years to induce a correction. Back at the school level, there are some problems from the TIMSS which caused a degree of confusion that no one can account for, particularly given that others of a higher level of technical difficulty caused no such trouble. For instance, a problem about the length of ribbon needed to tie up a highly regular box all of whose dimensions are clearly marked gave much more trouble than a relatively sophisticated word problem involving interpretation of a rate-of-speed graph.
This is not a situation whose solution leaps clear-cut from the page. An accumulation of such examples does, however, reinforce one of the positions of the current math movement pretty firmly: from an exceedingly early age, students need to be given the freedom to tackle problems whose solution is not a routine application of the most recent technique to have been covered in class. They also need to be given the mental space to mess around with the problems, and to follow up on their own ideas, including some dead-end ones, and to get frustrated and deal with it. It's hard to be a really successful problem-solver in the long run if you have been allowed to convince yourself that a math problem that can't be solved in two and a half minutes must be impossible.
This, you will note, requires time. And time is a sticky issue in a system where the curriculum for the year is jammed full of required this'es and that's, and raised to overflowing by the amount of time that must be spent re-teaching material to kids who were not motivated to learn it in the first place because they knew it would be re-taught. Not a good state of affairs. One thing we clearly need to do is re-think the content--not in terms of what can be omitted, but of what really is important in that content.
But there's another issue to deal with that could be even more pertinant on the time and content front. It's not one that can be dealt with once and for all, but one which requires that we adjust ourselves to a constant state of change: to wit, technology. To quote Gail directly: "Students live in a world we do not know, and they have grown up in that world. Technology is not an add on in their environtment, nor is it an end in itself." If you begin to question whether their world really is appreciably different from ours, try informing a young child that he/she sounds like a broken record. No communication will occur. So from a pedagogical point of view, are all these calculators a Good Thing or a Bad Thing? Answer: no. They are whatever we make of them. If we try to teach an unrevised and unadjusted curriculum, aimed at producing mathematical clones of the splendid specimens we were as we departed the hallowed halls of high school we will emphatically fail. Furthermore, if we did succeed, that would be a failure of another sort, because the skills now needed are not the same as those needed a couple of decades ago. Nor, if we define the skills too closely, are they the ones that will be needed in another couple of decades. What we need to be providing is mental flexibility, and problem solving ability and communication skills. But one needs to exercise care in describing that, because it can be taken (and frequently is) as advocating ditching the traditional "basic skills". Not so. Students should, for instance, definitely learn how to multiply multi-digit numbers. On the other hand, they should learn it in a way that has some meaning. If you don't know why you are writing that little 3 up there and that big 6 down over there, are you really that much better off than if you were hitting the keys of a calculator?
But along with providing a pretty sticky bunch of decisions about how to handle sundry areas of mathematics teaching, calculators do provide access to some kinds of problems that require casual crunching of hefty numbers. Gail had a lovely example from a Manhattan parking lot involving reasoning from context and then trying out a batch of different multiplications of whole columns of numbers--a nightmare by hand, but a snap on a TI-83, and a problem her students have really grooved on.
One other theme common to the three talks and sundry conversations was the importance of multiple solution paths leading to the same answer. She illustrated this at one talk with a nifty problem involving chickens standing on each other's backs on a baggage scale (sorry--you hadda been there!) for which, in about three minutes, she got more different ways of arriving at 12.6 kilos than I could keep track of. She says she has yet to get fewer than six solutions. But think what we are asking of teachers when we require them to deal with problems like that. It's a lot easier to spend your time working on getting students to use The Process as instructed to get The Answer to This Kind of Problem than to open yourself up to accepting a bunch of solutions of which you know some will be correct and some flawed, often with pretty subtle differences. Only snag is that the former denies thestudents both the autonomy and the flexibility that are so essential in the current scene.
Gail also had one set of challenges I'd like to reproduce directly: "Think about: How students learn is as important as what they learn, and what students learn is more important than what you think they learned." and, directed specifically to mathematicians, "Help teachers identify the important ideas students should know about mathematical topics; and lend support to teachers as they attempt to define important mathematics."
Reverting to my original theme, I take this last admonition as
evidence that Gail shares my conviction that we are all in this
together. But the image that most reinforces that for me is not
one stemming from any of her talks. It is rather a description
that she gave over dinner (Copper River salmon--we didn't totally
maltreat her!) of the process by which she put together the AP
statistics course at her high school. She felt insecure of her
statistical knowledge, so she got a statistician friend from the
University of Wisconsin in on the act. He would explain a concept,
and she would set up a lesson about it, and he would say "Wait
a minute--that bit over there is going to lead them to believe
something untrue" and she would re-write it and run the new
version past him. Huge amounts of joint effort and a great class
emerging from it. Now that, to me, is a stellar example of community
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