College Math Teaching

January 17, 2014

The New Semester: Spring 2014

Filed under: academia, advanced mathematics, algebraic curves, analysis, knot theory, research — Tags: — collegemathteaching @ 11:34 pm

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 C^1 functions running through a point, then that function is continuous at that point. This result does NOT extend to C^2 .
(see here)

So, what is so special about C^1 ? 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. 🙂

April 27, 2013

Unsolicited advise to young professors at heavy teaching load universities: Go to Research Conferences anyway!

This is coming to you from Ames, Iowa at the Spring American Mathematical Society Meeting. I am here to attend the sessions on the Topology of 3-dimensional manifolds.

Note: I try to go to conferences regularly; I have averaged about 1 conference a year. Sometimes, the conference is a MAA Mathfest conference. These ARE fun and refreshing. But sometimes (this year), I go to a research oriented conference.

I’ll speak for myself only.

Sometimes, these can be intimidating. Though many of the attendees are nice, cordial and polite, the fact is that many (ok, almost all of them) are either the best graduate students or among the finest researchers in the world. The big names who have proved the big theorems are here. They earn their living by doing cutting edge research and by guiding graduate students through their research; they are not spending hours and hours convincing students that \sqrt{x^2 + y^2} \ne x + y .

So, the talks can be tough. Sure, they do a good job, but remember that most of the audience is immersed in this stuff; they don’t have to review things like “normal surface theory” or “Haken manifold”.

Therefore, it is VERY easy to start lamenting (internally) “oh no, I am by far the dumbest one here”. That, in my case, IS true, but it is unimportant.
What I found is that, if I pay attention to what I can absorb, I can pick up a technique here and there, which I can then later use in my own research. In fact, just today, I picked up something that might help me with a problem that I am pondering.

Also, the atmosphere can be invigorating!

I happen to enjoy the conferences that are held on university campuses. There is nothing that gets my intellectual mood pumped up more than to hang around the campus of a division I research university. For me, there is nothing like it.

This conference
A few general remarks:
1. I didn’t realize how pretty Iowa State University is. I’d rank it along with the University of Tennessee as among the prettiest campuses that I’ve ever seen.

2. As far as the talks: one “big picture” technique that I’ve seen used again and again is the technique of: take an abstract set of objects (say, the Seifert Surfaces of a knot; say of minimal genus. Then to each, say, ambient isotopy class of Seifert Surface, assign a vertex of a graph or simplicial complex. Then group the vertices together either by a segment (in some settings) or a simplex (if, in one setting, the Seifert Surfaces admit disjoint representatives). Then one studies the complex or the graph.

In one of the talks (talking about essential closed surfaces in the complement of a knot), one assigned such things to the vertex of a graph (dendron actually) and set up an algorithm to search along such a graph; it turns out that is one starts near the top of this dendron, one gains the opportunity to prune lower branches of the group by doing the calculation near the top.

The weather couldn’t be better; I found time over lunch to do a 5.7 mile run near my hotel. The run was almost all on bike paths (albeit a “harder” surface than I’d like).

April 1, 2013

Fun for my Facebook Friends

Filed under: advanced mathematics, knot theory, topology — Tags: , — collegemathteaching @ 9:59 pm

Fun question one: can anyone see the relation between the following three figures? Note: I made a (sort of subtle) mistake in one of them….the one where the graph lines are showing)




Fun question two (a bit harder):

What is the relation between these figures?



And for the win: what is going on here? (this is ambiguous)


Ok, I’ll help you with the last one: imagine this process (one solid torus (think: doughnut or bagel) inside a larger one, and repeat this process (think: those Russian dolls that are nested). If you then take the infinite intersection, you get a simple closed curve (not obvious) that is so badly embedded, it fails to pierce a disk at any of its points (and certainly fails to have a tangent vector anywhere).

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