The past week and a half have provided us with a wonderful trio of highly diverse talks, any one of which deserves a newsletter unto itself. I seem, however, to be facing a choice between doing them the gross injustice of condensing them into a third their deserved space and doing them the even grosser injustice of leaving them totally unmentioned, so I shall opt for the less gross of the two.
The first of the series, entitled Probability by Surprise: the Pleasure of Paradoxes, was given by Susan Holmes on Tuesday the 17th. I suspected I was going to like it when she lead in with a quotation from Laplace: "The mind, like the sense of sight, has its illusions, and just as touch corrects those of the latter, so thought and calculations correct the former... One of the great advantages of probability calculus is that it teaches us to distrust our first impressions." My suspicions were confirmed when she went on to describe the origins of the probability course in question. She developed it first to meet the needs of a class of math-phobic psychologists in France, where math phobia is not the socially acceptable phenomenon it is here. To entice them out of their shells she presented them with probability paradoxes, pointing out that they were dealing with problems that confuse absolutely everyone, but that they could figure out, and that while doing so they could use themselves as a sample space of one to study the process of getting the mind around the concept. She used, for instance, the famous three card problem, where from a set of cards one of which is red on both sides, one blue on both and one red on one and blue on the other one card is drawn and shown to have a red side. The question is then the probability that the other side is also red. Any intuition worth its salt will gallantly supply an answer of 1/2, and persuading it otherwise can be a really lively occupation. They proved quite a success with the class of psychologists, so Susan had the complete set up her sleeve when she got to Stanford and was handed a class of exceedingly bright, thoroughly cocky undergraduates each convinced that he or she knew all the answers. You can guess the next phase. All in all a neat course.
In fact, the course was so much up my alley, and so charmed me, that she completely snuck past my guard on another front. I tend to be highly techno-skeptical, even (blush!) in some areas where demonstrably excellent results are to be had. But what she has done is to set up a web site with simulations of a batch of these situations, and to make the playing of the games, or carrying out of the experiments, mandatory in preparation for class. She reports gleefully that students may be totally resistant to the idea of cracking a book before class, but they will sit by the hour pointing and clicking at, for instance, a set of doors and come in with much more feeling for the Monty Hall problem than she ever used to be able to count on. Very convincing. Though I must admit the skepticism kind of kicked back in when several of the javas (which can be tracked down from her home page of http://www-stat.stanford.edu/~susan/) proceeded to freeze my computer.
Susan's lecture caused a largish classroom in Anderson Hall to overflow. The next evening's lecture, by her husband Persi Diaconis, caused Kane 210 to overflow, which is even more of an accomplishment. The lecture was on Coincidences, and to appreciate his slant on them you it really helps to know a bit about his background: he ran away from home at age fourteen to apprentice himself to a world-class magician, learned the trade and became a pro himself. Ten years later in a bookstore a friend handed him Feller's Introduction to Probability Theory and he discovered A) that it looked interesting and B) that he couldn't read it. So he remedied the latter situation. Thoroughly. If I remember the chronology correctly it was less than seven years later that he received his Ph.D. in Statistics from Harvard. He has since pursued a career which has simultaneously wowed both statisticians and pure mathematicians, thereby putting himself into a sample space which he shares with very few others.
Now picture a roomful of psychologists observing a young man with a deck of cards demonstrating his claims to ESP. Once you've got the picture, add to the observers Persi Diaconis. I don't think I need say much more... His lecture was full of entrancing stories and examples. Between them came some nice illustrations of ways in which quite elementary mathematics could be brought to bear to produce a neat computation of the probability of the occurrence of some event or combination of events, most of which turned out to be rather less unlikely than one would originally tend to rate them.
Diaconis' talk was very much geared to a general audience. The third of the trio of talks described herein was the exact reverse: at last Thursday's Brown Bag, Bob Palais produced with much energy and conviction a message whose gist was "Look, fellow mathematicians! Just see how we keep throwing cognitive hurdles in the path of our students!" Some of these hurdles he just decried, for instance pi -- consider how much more reasonable it would feel to the average person in the street if the measure of a quarter circle were pi/4 rather than pi/2, and think how many formulas with pi in them also have a 2 that you have to keep dealing with. He's not going to attempt a change, though, I am happy to say (talk about tilting at windmills!) The changes he would like to make are less flashy, but have a good deal solider core. Far too much of mathematics from algebra on up, he feels, comes at students as a collection of unconnected and correspondingly unmotivated individual topics. He would like to see choices made at the elementary level on the basis of more advanced areas where the earlier ideas can return and resonate. He has worked out a number such longitudinal notions. One is a nice geometrical construction which provides a basis for a proof of the law of cosines, for Fourier Transforms and for a number of things in between. Another is a more general theme that could be used from elementary school up through calculus, that of pairs of things that undo each other. In the midst of describing those, he came up with a vocabulary issue that I found really striking because it had so completely escaped my notice: the use of the work "limit" out there in the real world really isn't our use. "All right, WIllie, that is the absolute limit!!" does not imply that Willie's behavior has been oscillating around some value toward which it is gradually settling. In fact, there is a clear mathematical term for the item that the everyday world intuition produces when handed the word "limit". That term is "bound." Oy!
I certainly didn't take in all of the ideas that Bob zinged past us in the Brown Bag, and if I had, I don't guarantee that I would subscribe to all of them. What I do subscribe to very firmly is the point of view that he so enthusiastically embodied: if you want to teach somebody mathematics, you must constantly try to see what that mathematics looks like from the other person's point of view. Now there's a slant I will support any day! --