Human nature being what it is, our lives tend to be dotted with gatherings motivated by a desire to figure out what went wrong. It was therefore refreshing to be asked last month to run a gathering designed to figure out what went right. Needless to say, I agreed at once.
To set the context a little: the University of Washington Mathematics Department has a long-standing, highly casual series of Brown Bag Seminars on Teaching and Learning - occasions to gather and discuss whatever educational issues come to hand. I'm its organizer, and frequently the process by which such an issue comes to hand consists of a colleague pointing out something novel or perturbing or otherwise worthy of pursuing. I was therefore not surprised to receive an e-mail telling me that one of our teaching assistants had collected some wonderful projects from his class, and suggesting that a celebratory Brown Bag might be in order. I duly trotted 'round to have a look and found an absolutely stunning display of models of topological surfaces, all produced in a sophomore course on vector calculus. Pete, the instructor in question, responded to my request that we do a Brown Bag on them with bemused pleasure: "Oh, yes! I'd love to discuss it. I can't figure out why it worked so well!"
What follows is a combination of ideas that arose in that discussion and products of my subsequent ruminations. We did not, alas, discover a recipe or even an algorithm, but some of the points are worth noting and/or pondering.
For starters, we asked Pete how he came to assign this project. His reply was highly illuminating. It seems that he was one of the many who did not go directly from college to graduate school. In his case the problem was that despite having majored in mathematics he was unconvinced that the field held any real attraction for him. Then he ran into a book on topological surfaces, with not only gorgeous illustrations but also nice mathematical discussions. The power of their fascination swept him right back to school. He wanted his students to have a glimpse of that fascination - to realize that mathematics contains some really neat things. The glow in his eyes as he said that went a long way towards explaining his success.
The next question was how he set it up. There, I felt, was the area most susceptible to profitable imitation. To my eye (an eye well sharpened by hindsight) he managed to give exactly the right amount of instruction. He gave them references to three or four books and a couple of good, well-linked web sites (with a thumbnail sketch of each one), and told them to choose a surface to model and go for it. No holds barred on choice of material or style, no restrictions on type of surface. When they finished, they were to write a page or two about what they learned. And that was all: no suggestions about how he himself would do it, no scoring rubric. In fact (and this is a wonderful antidote to some of the jaded comments we tend to make about today's students) very few points attached to it - less than 5% of the course grade, as I recall. So grade-grubbing definitely does not account for the student who produced a cloth Möbius band with a zipper all the way around its center and a wooden rack to put it on. Or the one who cut out dozens of pieces of foam-core and produced the most elegant skeletal Klein bottle I have seen. Or the one who produced a six page report with downloaded color illustrations. These were students who had taken ownership of the project and were engaged and excited by it. There were other spectacular projects, too, as well as some where enthusiasm and a bright idea had suffered a head-on crash with real world materials, but could still be perceived in the slightly wobbly results.
On the other hand - and this is a downside which I maintain is inextricably entwined with the success - a certain number of class members had duly noted the lack of specific requirements and taken the line of least resistance. One, for instance, turned in a slightly dilapidated paper Möbius band stuck to a scant page report that was clearly a synopsis of two references at the most. A more detailed assignment might have prevented that, but would also have tied down the students who flew so high. Sitting in the midst of the whole collection, we arrived at an easy consensus that that would have been too steep a price to pay.
Looking for conclusions from all this: what did make this project tick? Well, for a start, there was Pete's incandescent enthusiasm and absolute conviction that this was something his students could do and really enjoy. Those two elements can neither be imitated nor faked, though in searching for something to assign they make an excellent goal. Then there was the nature of the instructions, with a comforting assortment of ways to get started, and highly non-restrictive instructions beyond that point. And there was the element of novelty, which adds a little sparkle in most contexts. That seemed to sum it up.
Thinking it over, though, I became conscious of one more element which may well be the most fundamental of all of them. Somehow, by some combination of intelligence, good instincts and - let's face it - sheer, blind luck, Pete managed to come up with a project which was exactly tough enough to stretch and challenge his students, but not to defeat them or scare them off. Now if somebody could just produce a formula for thatŠ --