The new semester is almost upon us here; our classes start up next Wednesday. I am ashamed to report that I am delinquent with a referee’s report; I’ll work some weekends to catch up.
Of course, we come in with “new ideas” which include evaluating things like this:
“Most people like to talk about how in college we need to develop critical thinking skills”, said Mike Starbird near the beginning of this talk yesterday, “but really, who wants to hear “Oh, yeah, Soandso, he’s really critical”?”. This, Starbird says, is what led him and coauthor Ed Burger to coin the phrase “effective thinking”. Because that is something one would like to be called.
The talk was affected by some technical difficulties, which meant that the slides Starbird had prepared with mathematical examples were unavailable to us. But, following his own advice, Starbird rose to the challenge and gave a talk, without slides, and using the overhead projector for the examples he needed to draw. As usual, his delivery and demeanor were both charming and informative (I am lucky enough to have both taken a class from him and taught a class with him), and the message on what strategies to follow for effective thinking, and to get our own students to be involved in effective thinking, was received loud and clear.
The 5 elements of effective thinking, as Starbird and Burger describe in their eponymous book, are the following: understand simple things deeply, fail to succeed, raise questions, follow the flow of ideas, and everything changes. The first couple he described by using examples of mathematics in which each strategy led to deep insights about a problem. For “understanding simple things deeply”, Starbird showed us a new, purely geometric, way of proving that the derivative of sin(x) is cos(x).
Note: Professor Starbird was one of my professors at the University of Texas. I took a summer class from him which involved the class going over his technical paper called A diagram oriented proof of Dehn’s Lemma
(Roughly speaking: Dehn’s Lemma says that if a polygonal closed curve bounds an immersed polygonal disk whose self intersections lie in the interior of the disk, then that given curve also bounds an embedded polygonal disk (e. g. one without self intersections). Dehn’s Lemma is especially interesting because the first widely accepted “proof” proved to be false; it wasn’t rigorously proved true into years later.)
Ed Burger was a Ph. D. classmate of mine; I consider him a friend. He has won all sorts of awards and is now President of Southwestern University.
I have to chuckle at the goals; at my institution we mostly teach calculus, which is mostly for engineers and scientists. The engineering faculty would blow a gasket if we spent the necessary time for finding deeper proofs that the derivative of sine is cosine.
And yes, we are terribly busy with this or that: on the plate, right off of the bat, is a meeting on “reforming” (read: watering down) our general education program, a visit day, among other things (such as search).
It has gotten to the point to where things like a “department lunch” went from being something fun to do to being “yet another frigging obligation”.
I’ll have to find a way to keep my creative energy up.
So, what I’d like to “think about”:
1. I have a couple of papers out about limits of functions of two variables. Roughly speaking: I gave new proofs of the following:
1. A real valued function of two variables can be continuous when evaluated over all real analytic curves going through the origin and yet still fail to be continuous. (see here)
2. If a real valued function of two variables is continuous when evaluated over all convex functions running through a point, then that function is continuous at that point. This result does NOT extend to .
So, what is so special about ? Is this really a theorem about curves through a planar set of points with a limit point? Or is more going on….can this result extend to results about differentiablity?
Then there is something that sparked my interest.
There is this very interesting result about Bezier curves and their control polygons in 3-space: it is known that a Bezzier simple closed curve can be unknotted but have a knotted control polygon. What else is there to explore here? Can only certain differences appear (say, in terms of crossing number or other invariants?) Here is another reference.
I’d like to sink my teeth into this. It doesn’t hurt that I am teaching a numerical methods course. 🙂