In my last Brown Bag announcement I mentioned that I had seen a few examples of worksheets designed to promote the use of calculators as a tool for thinking rather than a substitute for it, and expressed my conviction that those worksheets represented the tip of a really nice iceberg. I then invited people to validate my conviction by bringing their worksheets to the Brown Bag. Well, I am not as yet in a position to speak of the size of the iceberg--those of you awaiting a bound tome should cease waiting--but of the quality, yes indeed. We had a small number of worksheets with a huge spread in content and with enough variation in approach and effect so that just looking at three of them generated a full hour's discussion.

The first worksheet we looked at was exploratory in nature and dealt with parametric equations. Linda Martin and Julie Harris had put it together for that deadly day before Thanksgiving when only half the class is there, but were pleased enough with the results so that they plan to use it at the introductory phase next time. The general context was that students who have developed a pretty good intuitive grasp of graphs that look like y=f(x) can have a feeling of having stepped off the edge of a cliff when they find themselves graphing (x(t),y(t)). Which I guess makes the worksheet in question an attempt at a parachute. Basically, it consists of a sequence of graphs for the students to reproduce on their own screens, with a monotone increase in level of complication and a monotone decrease in number of hints. For extra credit they had the option of either reproducing a Lissajou figure or creating something on their own. The former I didn't see, but the latter included something entitled "Grumpy face" (I gather one eye turns up circular and one shut) and a remarkably realistic EKG read-out. Clearly some at least of the students had quite a good time, and equally clearly a parametric graph just became a lot less mysterious.

The next worksheet up was a multi-faceted item. At the previous calculator discussion, Michael Keynes, who is Newton's Method afficionado, had mentioned that he had put together a worksheet on the subject. Judith Arms picked up on the comment, borrowed the worksheet and added her own twist at the end, then gave it to her TA's to use. Eric Scott, who is one of said TA's, was able to report on how it went. Corresponding to the multitude of facets, the discussion had a multitude of directions, some of which I missed out on because of being in the which-button-was-that mode (I have had a graphing calculator for all of eight days, now!) The gist of Eric's I can report because it was so painfully familiar: they were getting a lot out of it, but needed about three times as long. And to me the meatiest of the rest of the conversation centered around the goal of the worksheet. Michael produced it because he feels that really being able to use Newton's method puts students much more closely in touch with the functions themselves, but that without the aid of a calculator the computations are so heavy they get in the way of that connection. I think everyone agreed to the underlying idea. "But", said John Roth, "I contend that when you say 'Hit the enter button several times' you lose that connection." Point taken, and we then discussed how to put the connection back in (not difficult, I gather.) Michael is back at the drawing board on that bit.

The third was by Jan Ray, who buzzed over from Seattle Central to join the discussion (and actually found a parking space--the day's miracle.) It is aimed at consolidation of existing ideas. Specifically, she wanted students to solidify their understanding of the global behavior of polynomial, exponential and logarithmic functions. The instructions have a deceptively innocent air: "Find the appropriate scales for the x- and y-axes so that the graph in the resulting window shows how the difference between these functions behaves as x approaches infinity."-- but woe betide the student who tries to do it by trial and error. One could drain several batteries in the attempt. Jan produced it in the hope that students would do a bit of discussing, but the expectation that a lot would toss it off a shade briskly. As it turned out, most spent an appreciable amount of time in discussion, and came up with excellent comments and observations--and an unexpectedly solid ability to deal with the concepts on their next test.

Needless to say, with a certified expert on hand the conversation did not restrict itself to the specifics of that particular worksheet. We finished the hour with a more generic discussion. A lot of good points and interesting comments went by, but only one sticks quotably in my mind: someone asked Jan whether she found that there were students who tended to grab their calculators too swiftly and make inappropriate use of them. "Certainly", she replied. "Likewise their pencils."

The other Brown Bag which has not been newsified is an earlier one on the Port Ludlow expeditions. It wound up more like a synopsis that a de-briefing, which makes this a synopsis^2. I shall report directly on Port Ludlow. The occasions in question were a pair of orientation sessions designed to introduce the participants in our NSF project to ideas and goals of the project--not to mention to us and each other. The participants in question are all of the middle school math teachers in the Seattle, Bellevue, Mercer Island, Lake Washington, Northshore and Shoreline school districts--somewhere between 250 and 300 of them, which is why two sessions were required.

One of our goals was to deepen their understanding of some mathematical concept in a way which would embody the kind of teaching we want to help them to develop. We opted for symmetry and used as our vehicle Marion Walter's nifty Mirror Puzzle book. We built a series of activities starting in effect at a look-for-the-line-of-symmetry level and working up to articulating strictly verbal directions for describing particular visual situations and to creating puzzles of their own based on the ideas they had arrived at while working on Marion's. This they tore into with considerable vim and occasionally spectacular results.

Another goal was to raise their consciousness of the new standards of mathematics teaching that the state of Washington is working its way into. This was a more delicate issue to address, because those standards are one of the major motivators for our participants, but can be really overwhelming gulped whole. We scheduled a full afternoon session for it, and discussion was lively, if highly varied, and seemed to have the right flavor.

A third, and perhaps the most essential, goal is the one in our title: "Building a Community of Mathematics Learners". That means connecting them with each other--and getting them together in a pleasant ambience with a certain amount of straight talk-time certainly promoted that. It also means building a connection between all of them and all of us, and in that we were greatly aided by the fact that on both of the week-ends there were UW mathematicians, both faculty and graduate students, who were willing to commit their Saturday to coming over to Port Ludlow and meeting people and joining conversations and just generally being present. They were a great boon.

So much for Brown Bags. I shall finish off with a tid-bit which is totally second-hand--not my norm, but this one is too good to pass up. Neal Koblitz just got back from MSRI, where he was invited to attend a symposium on the role of education in research institutions. He reports that he enjoyed it thoroughly. I gather there were a number of thought-provoking short talks (including one by Steve Monk, which is hardly surprising!) and small group discussions (Neal joined the one on the responsibility of universities toward pre- and in-service teachers--hurrah!) and a lot of good conversations among interesting and interested people. As far as I am concerned, simply the fact of the existence of the symposium is great news, and the fact that it was so well done makes it downright stupendous!