Midweek melange

It's Wednesday. It's been raining all day, and the rain has hardly ceased, even now that the sun is down.

We're one student away from filling our last slot in this upcoming summer's REU, having received a seventh acceptance today. I spent an hour or so this afternoon hammering out a list of learning goals for the program, and a schedule of activities through which we will work towards realizing those goals. Like the goals I typically set for my classes, the list includes content-oriented targets like mastery of graph theory, group theory, etc., but also less traditional goals such as gaining confidence in communicating mathematics to others, and building the authority to challenge unproven results.

How well will we fare?

Well, how'd we do last year? We can't possibly do worse, can we? (Famous last words...) Here's a brief report card on 2007, filled out from the perspective offered by a year of hindsight:

Recruitment: A. We got great kids, and what's so marvelous is that we did so so late in the game, having not secured funding until a month or so after most REUs had already made their hires. That delay allowed us to catch the best of the best of the younger crowd, many of whom had missed the first round of REU applications. We lucked out. I feel honored that I had the chance to work with such a talented group of students, many of whom are surely destined for great things.

Logistics: A. We handled housing well, we covered all the human resource aspects admirably. As far as I'm aware (aside from one snafu with one of the subsistence checks when something didn't get signed in time), all of the paperwork came off without a hitch. Yay, we're good pencil-pushers!

Seminar: B+. For the most part, we hit the nail on the head. I think the structure of our opening week-and-a-half was sound, and it did a good job of preparing the students for what would come in the next weeks. I don't think we adequately anticipated the stress it would induce in some (all?) of the participants...but we adjusted for that, and pulled up in time. This time around we'll know what to expect, we'll be able to ease up when needed.

Structure: B+. Again, things moved along smoothly, for the most part. The students did a great job of establishing semi-regular meeting times with their respective faculty mentors, the students did a great job in keeping their noses to the grindstones, the weekly meetings were generally productive. Those meetings, though, were awkwardly scheduled, and I'm not sure that the students played as strong a leading role as they could have: in the future, we might be able to challenge the students to take authority in these sessions. Moreover, we didn't have a chance to include any "guest" research talks by faculty from UNCA or elsewhere, as we'd hoped we'd be able to do. (This I've already remedied this time around: I've sent out three invitations to colleagues from other institutions, and have already received one positive reply.)

Social organization: A. We couldn't have done it without the students, who got along admirably well. Not only did they not kill each other, they became fast friends. By the summer's close, the care and concern they showed for one another was evident. (I hope this year's crowd will come to the conclusion that they need to have a talent show, too...although nothing's going to top the 2007 crew's rendition of "A Whole New World.")

Research outcome: Incomplete. It's hard to say at this point how "much" the students will have generated when all the dust has settled. Wilhelmina and Francoise have got a nearly submission-ready manuscript they've been sitting on for a while now, Ned's work with me will make a nice section in the paper whose prequel has been tentatively picked up by a nice graph theory journal, and Kendrick's name appears on an as yet unsubmitted manuscript my next-door colleague here has put together. Let's hold off on this one.

Long-term outcome: Another incomplete. I'm not trying to cop out here; we're still too close to this past summer to measure long-term outcomes, but I like to think that the program made a primarily positive impact on the budding careers of a handful of talented young mathematicians. As far as I'm concerned, if five years from now I run into one of the participants at a conference just after she's presented on her dissertation research, and she's able to say that her experience was a worthwhile one and helped her decide what she wanted to do with her career, then we've dealt ourselves a royal flush.

Changes this year? Reflecting changes in my own pedagogical style over the past year or two, the program already exhibits more conscious design and attention to explicitly stated learning goals. Writing plays a more central role, with an introduction to LaTeX coming in at the program's beginning (towards the end of the first week) rather than at its end (towards the end of the sixth week). Indeed, written progress reports will be expected of this year's students, in addition to the weekly meetings. We're also going to make meeting times explicit, and as I mentioned above we've already begun scheduling guest speakers. Finally, and perhaps most importantly, we'll be encouraging the students to seek out their own problems more actively: though we'll still have ready stockpiles of personal problems from which the students will be able to draw, the participants will be encouraged to seek out problems that entice them, hopefully from within the fields in which the participating faculty specialize. "All right, y'all, that's everything you need to know about chromaticity of Cayley graphs. Here's a survey paper. Dig in!"

I'm excited. Now that we're in the thick of it instead on the fringe, this year's selection process has been more of a roller coaster than last year, and with as many noes as yeses the thrill of the chase has gotten my blood pumping. We'll have our team set up soon, and I'll probably take one more shot at getting folks to blog about themselves before they get here. (Last year's awful attempt failed pitifully...I'm pretty sure that I mercifully deleted the pathetic little webpage that limped along painfully for a few weeks...yup! Just checked: all gone.)

What else is up?

For a few weeks now I've meant to say a bit more about my Graph Theory class, let me take this time to do so.

In-class presentations are for the most part much improved, especially in the past few weeks. The students who take the time to craft solid proofs ahead of time are executing marvelous performances and are sometimes uncovering techniques I would not have considered. Though their methods are not always the most efficient, they're authentic, through and through. Today in particular saw a handful of nearly immaculate proofs: one problem asked the students to prove that the path metric induced by a subgraph could only exceed the path metric of the original supergraph, another asked for an explanation for what breaks down when one tries to define the path metric on a disconnected graph, yet others asked properties of eccentricity. All solutions were skilfully executed.

Where some of the students are having trouble is in the written submissions. How so? Well, c'mon, people, even if all the problem says is "find the chromatic polynomial of the complete graph on n vertices," I can't jolly well in good conscience give the same grade to some who just hands me the formula, ex nihilo, as to someone who includes a half-page proof of that same formula. Trust me, from now on I'm going to explicitly include wording like "give a formula for... and prove that your formula is valid." I'll say that, if you'd like me to, but I claim that at this point I shouldn't have to say this, it ought to be assumed that at this level we prove our claims.

But we all know that when we assume, we make an ass out of "u" and me.

For the most part, Graph Theory's a blast. I'm still having fun, I think most of the students are finding it a worthwhile experience and are learning a lot. For Friday, I've asked them each to write a few paragraphs about what they feel is working well, and what could stand to be changed for the closing third or so of the semester. I'm eager to see what they've got to say. I've already talked to two of them about modifying the "review" problems at the end of each problem sheet, to allow these problems to be more group-centered and in-class. We'll see we can make that work, if others are up for trying it out.

It's all good.

I'm getting tired, I'm going to slink off in a moment, but just a quick word about my Calc II kiddies: two days into sequences, they're doing great. "These are fun!" one of my students said. They've already asked great questions and have exhibited profound intuition and insight. I think they'll be able to wrap their minds around this stuff comfortably. I'm also happy to report that Taylor series are playing a crucial role in my own ongoing research right now, so I should be able to bring that in as a "real-world" example of series methods before the semester's through. Huzzah! This crap is useful! Who'd'a thunk it?

Well, more anon, likely. For now, I'm off like a prom dress, as my college buddy Jennifer was fond of saying. Ta for now.