This is a newsletter on a single topic, to wit, last week's visit by Alan Schoenfeld, and I've a horrible feeling that even a whole newsletter can't do the topic justice. In fact, there's internal evidence that the case is hopeless: Wednesday evening as he arrived, Alan remarked with justifiable pride that for the four occasions on which he was to speak here he had produced four orthogonal talks. Since I plan to cover all of them, that puts this newsletter firmly in four dimensions. Beware!

Before I begin this risky venture, it might be well to give a small amount of background information. One of my great pleasures in recent years has been bringing to the department from time to time speakers who seemed to me to have things to say on the educational front that would be of interest to folks whose major focus is on the mathematical front. Starting with Leon Henkin, whose visit ante-dates the first of these newsletters, we have had quite a number of such speakers, generally both well-received and well-enjoyed. There's been a gap, though: early on it became clear to me that one quintessential example of such a speaker was Alan Schoenfeld. Characteristically, though, one of the reasons he is so interesting is that he does so many interesting things, and the implications of that on scheduling are clear. Last week's visit is the outcome of a level of persistence that would have been downright rude, except that he gave me permission to persist. I'm mighty glad he did.

On to the talks. I can't see any reason not to report them in chronological order, though I wouldn't put it past me to smuggle ideas from one into another, consciously or un-. Lest I appear to be laying claim to a better memory than I really have, though, I should admit from the start that I am working from the overhead transparencies that Alan kindly let me photocopy.

We got things off to a running start with a Brown Bag. For that I had hit Alan with the challenge of giving just part of a talk, with some nice provocative lines in it for stirring up conversation. So he decided to address "Purposes and Methods of Research in Mathematics Education" -- a good choice, since he neatly distilled a number of contrasts in points of view in it. One such contrast was between the purposes of research in mathematical education from his point of view and from the point of view of many mathematicians. The former has a pure side : to understand the nature of mathematical thinking, teaching and learning, and an applied side: to use such understanding to improve mathematics instruction. The latter tends to boil down to "Tell me what works in the classroom." In fact, this latter point of view leads to some questions I'd say most of us have heard, and probably have a lurking expectation of an answer to: "Are large classes as good as small classes?" "Can we prove that calculus reform worked?" To which, he points out, the only possible reply is "What do you mean by 'as good'? 'worked'? Any answer you give has to involve values. Is your measure the scores on similar exams given at the end of the course? The amount remembered in subsequent courses? The number of students who finish the course liking mathematics enough to take another math course? The number who ultimately wind up as mathematics majors?" No one should claim to have answered the first questions unless they have specified the answers to the later ones. Alan reported that he had succeeded in convincing the NSF of this when they were wanting (naturally enough) a report on the impact of calculus reform. I call that an accomplishment!

Meanwhile, looking past the questions Mathematics Education research does not address, we get to the matter of what it can provide, which Alan characterized as theoretical perspectives for understanding thinking, learning and teaching; descriptions of aspects of cognition; existence proofs for what students can learn and accomplish; and descriptions of (positive and negative) consequences of instruction. Even within this provision, though, there are contrasts between the fields, because, as Henry Pollak put it "There are no theorems in Mathematics Education", and correspondingly no absolute proofs. There is evidence, some of it pretty darned compelling, and a fair range in the reliability of results. Just don't look for a Q.E.D. Or alternatively, if you find one, be suspicious.

That bunch of statements did, indeed, provoke some good discussion, of which my memory, alas, has floated off into the fourth dimension. All I remember is my pleasure in the fact that while I was sitting there thinking "Gosh! Cool!", Ron Irving was coming up with a couple of remarks that caused Alan to respond with "Right! Good comment! It's this way..."

Talk #2 occurred rather later on Thursday, in a very different locale. This one was in some sense the origin of the entire series, in that it is the one that motivated my most recent (and finally successful) invitation. One of the mandates of our Preparing Future Faculty grant is to build community among the mathematics departments of UW and other local colleges and universities. With that in mind, the PFF steering committee decided that this would be a jolly good occasion for enticing graduate students over to the campus of Seattle University. Wynne Guy took on all of the arranging, and produced an absolutely spiffy dinner in Casey Commons -- a fifth floor dining room with balconies overlooking the campus's bright green central plaza, complete with fountain and just enough wandering students keep it realistic. Good food, too. Her efforts at arranging and our joint efforts at advertising really paid off: we had upwards of forty people, including a delightfully hefty representation of graduate students, and a number of faculty members not only from our partner institutions of Seattle University and Seattle Central Community College, but from a number of others as well. Many lively conversations happened over dinner, some of them kicked off by Alan's pre-dinner talk entitled "Can we understand how and why teachers do what they do "on line" as they teach? And if so, why should anyone care?" This was more of a nuts-and-bolts talk, looking relatively closely at the kind of effort that is the heart of Alan's current research. He and his students and colleagues take a video-taped (or in one case audio-taped) class and study it and analyze it and analyze it and study it and... until they feel they really understand the sources of the decisions the teacher is making every step of the way. Much of this involves the teacher's "KGB" -- Knowledge, Goals and Beliefs -- sometimes inquired into in advance and sometimes deduced from the interactions. Alan showed us a videotape of Deborah Ball leading some third graders in a discussion about even and odd numbers and about their own learning process, and then diagrammed the discussion as a recurrent cycle of the teacher eliciting and responding to student ideas. His demonstration was a tour de transparency force which it would be ludicrous to attempt to reproduce, so I shall cut to the chase: what's the point? This, as I recall, is where Alan built most heavily on one of his recurrent metaphors: Cognitive Science (the study of learning) is running a course parallel to that of Medical Science, but with a multi-decade delay built in. Prior to the early 20th century, medicine was a collection of chunks of information and results, but no kind of a science, because it was in total intellectual disarray. In the course of the century a lot got codified, articulated and generally organized, with standards of proof and modes of communication and other aspects all developing as it went. Cognitive Science got started roughly in the eighties, and has a long way to go, but it has already produced some major advances. Its underlying motivations are the belief that the more you understand something the better you can make it work, that when you understand how something skillful is done you can help others do it and that it is possible to develop tools for describing developmental trajectories. Alan also pointed out one additional motivator for him: it's fun!

That was Thursday. Friday we had to share Alan with other people, because a couple of other folks responded to the news that he was coming as to a battle trumpet: "Action!!" This had the merit that with very little effort we got to be in on two more talks (though before I get too flip about it, I should add that the second talk was partially supported by the Milliman fund, to which much thanks.) The morning talk was sponsored by the Center for Multicultural Education, run by Jim Banks in the Education Department. It was entitled "Mathematical Literacy and Civil Rights: Issues of Equity, Standards and Testing". Alan led in with some high-impact quotations from a book I feel a definite need to find and read, Robert Moses' "Radical Equations: Math Literacy and Civil Rights". Moses' contention (on which his Algebra Project is acting with great energy) is that "the absence of math literacy in urban and rural communities throughout this country is an issue as urgent as the lack of Black voters in Mississippi was in 1961." In particular, technological literacy is becoming more and more of a gatekeeper to the kinds of jobs that have the capacity to get someone out of the poverty cycle. Data from the NRC and from TIMSS and from Jonathan Kozol's "Savage Inequalities" back up the statement that the attrition rate from the mathematical pipeline is huge for all the population, but vastly more so for females, African Americans and Latinos. I suspect that the data came as a shock to the largely non-mathematical audience, but then again, so did the NCTM "Principles and Standards" with which Alan followed up the grim statistics. And after giving a thorough introduction to that document and describing the development of curricula designed to be aligned with the original "Standards", Alan produced a set of data that wowed even those of us that were up on everything that came before: it seems that someone has finally set up a big enough, well enough studied implementation of two "standards-based curricula" (Everyday Math in elementary schools and the Connected Math Project in middle schools) to produce some data that have real meaning. In Pittsburgh, in fact. And the data are wonderfully encouraging. Even on basic tests of basic skills the percentage of students scoring at or above standard has skyrocketed while the percentage of those way below standard has declined (you have to watch that one -- if the top students are going up but the bottom ones are going down you're not in good shape.) On concepts and problem solving there was nowhere to go but up, but that they did mightily. In short, a highly heartening batch of evidence. But note one thing it illustrates: the process which led to these data began in the mid-eighties with the early research into mathematics teaching and learning. The data were released a few weeks ago. That, by a lightening calculation, is 25 years -- in a country where you're lucky to get people to allow 25 weeks before demanding visible, testable, unquestionable outcomes for an experiment. A problem, that.

From there, Alan went on to a more general discussion of issues and obstacles in improving education. Specifically, he listed four: 1) high quality curriculum, 2) a stable, knowledgeable, professional teaching community, 3) high quality assessment that is aligned with curricular goals, and 4) stability and mechanisms for evolution. The first he had already covered. The second, in terms of how unprofessionally teachers tend to be treated, is a hot button for me, but his basic position was that all we can do from where we are is be a support -- movement on that front must come from the teachers themselves. In the third he introduced a term which was new to me: WYTIWYG. What You Test Is What You Get. I love it! The fourth is a place where he feels that other countries tend to be a lot better off than we are -- we do pendulum swings (Basics; New Math; Back to Basics; Standards-Based Mathematics; Back to Back to Basics,...) This needs to stop! **** On to the last, and here (fourth dimension or no) I should be on really familiar ground, because its ideas were much discussed in a teaching/learning seminar a bunch of our graduate students ran a few years back. It topic was Developing Problem Solving Skills: Learning to Think Mathematically (or like an engineer, or,...) With that he packed the hall, 4:00 Friday or no. And kept everybody awake, to, which was even more of an accomplishment. He had a great fund of examples, because a lot of what he said stemmed from his efforts in teaching a course specifically geared to teaching problem-solving (which in turn stemmed from his having discovered Polya's "How to Solve It" and been appalled that all the tactics and strategies in there were ones he had had to discover on his own). Reducing a highly engaging hour to a rather dry nutshell, I shall give just the barest of outlines: the five basic aspects of thinking mathematically are

1) Subject matter knowledge: facts, procedures, etc. (and here he spent some time on the constructivist perspective -- the idea that what we 'see' as reality is necessarily a construction and interpretation of reality, and that that construction is made using interpretive frameworks which determine, and may well distort, what we see. Part of teaching is trying to be aware of our students' frameworks.)

2) Problem-solving strategies, as distinct (very distinct!) from the way a mathematician in lecturing mode would tend to present a problem's solution. Tactics include looking for patterns,exploiting an easier related problem, and the like. Alan reported with glee that after a semester of working on those tactics, students often beat him on the draw on an unfamiliar problem.

3) "Control": It's not just what you know, but how and when (and whether) you use it. Here he had some examples contrasting students and mathematical experts at work on the same problem. Characteristically, students latch onto some idea and work on it for dear life, losing all touch with how (or whether) the idea actually contributes to the desired solution.

4) Beliefs: your sense of mathematics as a discipline. This one included a rather chilling set of beliefs inculcated in children by their schooling. One is that answers don't really need to make sense, since a word problem is just a cover story for an arithmetic problem anyway. Another is that no math problem should take more than five minutes to solve, and if a problem takes more than 10 - 15 minutes it must be impossible.

5) Practices: Acting as a member of the mathematical community. For this he advocates that we need to create an instructional environment in which the students are -- at a level appropriate for them -- doing mathematics. And by way of motivation, he described some neat things that happened when he simply turned a college class loose on the Pythagorean Theorem. And with that I must stop. It's frustrating not to be able to do justice to a really great set of talks, but at least this gives you an idea. And I plan to have the transparency copies in my office (in a notebook, yet even) for anybody who's in a reasonable range to be able to look at. --

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