For this year's joint meeting I decided to mix it up a bit and attend a wider variety of talks. I think I had a better time because of it. On this first day, I attended some MAA talks on teaching, some history of math talks, and my friend James' contributed talk.
Talk 1: "Instant Insanity: Using Colored Blocks to Teach Graph Theory" by Stephen Adams
Instant insanity is a game using four colored blocks. The object is to stack the blocks into a column so that each color appears at most once per column. It's easy to set three up, but pretty difficult to get four. Stephen had built his own blocks, and since he had a small class, he let them attempt the game on their own, and keep the blocks during the subsequent lecture. Graph theory can be used to label the blocks in such a way that a relatively simple algorithm can then be applied to achieve a successful stacking. I think this is an interesting way to introduce graphs. I like how it uses visual, tactile, and formal reasoning skills on the part of the student.
Talk 2: "A Magic Trick That is Full of Induction" by Robert W. Vallin
The students have shuffled the cards. They are behind your back. You have told them that you can tell red cards from black cards by touch, and, of course, they are skeptical. Miraculously, you consistently alternate pulling and red and black cards, without looking. Now they want to know the trick. It's actually quite simple, you prep the deck to alternate red and black ahead of time. So even though you let the students cut the deck repeatedly, those don't change this alternating feature. But what about the riffle shuffle you performed before putting the cards behind your back. How did that not mess up the order? Well it did, but in a very specific way, so that if you pull cards in the right way, you will still alternate red and black. That's the read magic. What's the proof that it works? It goes by induction.
Robert brought this up as an interesting way of introducing students to induction proofs, more interesting than the usual summation formula at least. If you extend the concept to shuffled permutations, it sometimes works. There is a nice characterization of exactly when it works (Gilbreath permutations), and related to this characterization is a problem which needs to be proved with strong induction. So a magic trick could be the motivating example for all of the theoretical material involved in teaching induction in a discrete class.
Talk 3: "From L'Hopital to Lagrange: Analysis Textbooks in the 18th Century" by Robert Bradley
At the beginning of the 18th century, calculus (although developed by Newton and Leibniz) was still only understood by very few people. Over the course of the 18th century it spread across the intellectual world, in part because of key textbooks. L'Hopital wrote one called "Analyse des infiniment petits" in 1696, with a second edition in 1715. From Leibniz to the Bernoullis to L'Hopital the axioms for infinitesimals had been reduced to two. L'Hopital didn't deal with functions, but curves, and so although his book contains derivative rules for rational functions, it does not have the chain rule. It also does not have rules for transcendental functions.
Euler's textbook was called "Calculus Differentials", and was published in 1750, although there is an unpublished version from 1727. In the 1727 versions, he does talk about functions, begins with finite differentials, then talks about log and exponent, and uses similar axioms to L'Hopital. He does not talk about trig functions. In 1750 he has changed his viewpoint on infinitesimals. Now there algebraically 0, but not geometrically. Euler also wrote a pre-calculus book, "Introductio." In this book he defines functions, distinguishes between variables and constants, and uses series and power series representations of functions. He also defined the trig functions from the unit circle.
The last book Robert discussed was the "Theorie des Function Analytiques" by Lagrange. Lagrange extends Eulers view, and begins the book with power series representations. He does to avoid using infinitesimals. From power series, derivatives are algebraically definable. He is the first of the three to talk about derivatives as functions, and although his reasoning is fundamentally circular, he talks much about rigor and purity in mathematics. Apparently this approach became the standard one in the 19th century, surviving the actual rigorization of calculus. It wasn't until the 20th century that the rigorous sequence of limits -> derivatives -> power series was incorporated into calculus textbooks.
I'm definitely interested in looking at these books myself at some point. It was interesting to see how some aspects of the pre-cal/calc curicculum have been around for a long time, and others (like late transcendentals) are relatively recent. Also this trickle of pure math results dripping down into lower math curriculum is really interesting.
Talk 4: "Jean le Rond D'Alembert: l'enfant terrible of the French Enlightenment" by Lawrence D'Antonio
D'Alembert was a very prominent Frech mathematician who was a contemporary of Euler. He was very influential in both the French and German Academies. I wasn't aware that he was the inventor of the ratio test for series, which is kind of cool. This talk was really entertaining: D'Alembert's life is very dramatic and Lawrence presented it well. D'Alembert was eventually enemies with just about everybody, including Euler (who basically scooped him on a result about equinoxes). He was also co-editor, along with Diderot of the Encyclopedie, a mega collection of writings of enlightenment thinkers. Its translated and on-line here http://encyclopedie.uchicago.edu/, which is awesome for us now. I didn't write any of them down, unfortunately, but there were some really hilarious quotes from D'Alembert.
Talk 5: "Packing Measure of Super Separated Iterated Function Systems" By James Reid
James discussed one of his recent research projects, which was to find an algorithm for computing the packing measure of a certain family of fractals. Two things about this are very cool. The first is that these dimension and measure computations are usually not even remotely computable. The dual notion, Hausdorff dimension and measure, certainly are not computable in the broad context that James was working in. The second is that, although the packing measure had been given a nice formula for one dimensional examples, people had yet to push this to higher dimensions. Although James' formula only applies to objects whose packing dimensions is bounded by 1, they are in the plane, R^3, and so on. It will be interesting to see if having packing dimension greater than 2 is really an obstacle, or if the nice behavior will somehow continue. Oh yeah, the talk has some really fun pictures of fractals generated by polygons as well.
No comments:
Post a Comment