Week One progress report

How we doin'?

If I had to say off the bat, I'd have to come back with a "not too shabby."

The students are fast, sharp, and diligent. They excel not only at the formal mathematics but at the auxiliary activities: LaTeX, Mathematica, you name it. They're kickin' butt and takin' reservations. I've already begun to think of them as able colleagues, and I and my fellow faculty have encouraged them to adopt the same viewpoint. Hell, no doubt in less than a month they'll be slowing down to let us catch up to them.

I don't know how much of their apparent progress this past week can be chalked up to the design of our program (a design which I feel is much tighter, clearer, more well organized...well, just so darned much more effective than last year's) and how much to the strengths of the students themselves. I'll be eager to know their take on the matter.

Meanwhile, I'd like to take a moment to look back over the first week and hold it up against the learning goals I'd put forth for the program's participants...how well do they match up?

1. Have a grasp on the fundamental concepts of graph theory, group theory, metric geometry, and dynamical systems; and understand cutting-edge open problems in these fields. We're there. I figure that in the first week we've given them the equivalent of one half to two thirds of a semester each of graph theory, abstract algebra, and dynamical systems. (Not to mention a year's worth of seminars, as Tip has said.) They've soaked it up, and they've been far more than passive note-takers. We've given them some pretty challenging exercises, all of which they've navigated with aplomb. Their mastery shows best in the brilliant questions some of them have asked (Norton's got a thing for topology, judging by his constant queries about that area) and in the frequent and astute observations they've made (late yesterday afternoon Thalia noted an omission in a project Tip and I have been working on for over a year; it was easily fixed, but her insight led us to find and fill the gap in our method...summer research has begun in earnest!). I think they've got the "fundamental concepts" down and are ready to hammer away at the open problems. A couple have already started to apply the concepts we've covered to original ideas. Uwe's application of graph theory to analyze the efficiency of saxophone fingerings is absolutely delightful!

2. Be able to make effective use of research databases, including MathSciNet and arXiv. We spent an hour or so on Wednesday talking about these databases. Our conversation, directed largely by Camilla's questions about how one becomes a referee, how one becomes an editor, how much publication is expected of faculty, and so forth, took us, on this third day of the program, deeper into the issues surrounding professional development than we ventured all summer last year. I don't think the conversation's tangents were distracting; the students did admirably on their first research assignment. On Friday each submitted the results of at least two literature searches, each including at least three academically rigorous references. The annotated search results proved the students' comfort in working with the relevant databases. I think they're ready to ply their skills to authentic searches.

3. Feel comfortable in generating new questions and topics for research, either by modifying and generalizing existing statements, or by branching off into uncharted territory. The verdict's still out on this hard-to-assess issue, but judging from the number of questions several of the students have put forth during the past week, I don't think we're going to have to worry about this one. A few of the students are more reticent than the others, and it's hard to say whether this is out of shyness or lack of skepticism. Time only will tell.

4. Appreciate the qualities that make for a friendly and effective research community. Progress towards this cognitive goal will be exceedingly hard to measure without hearing from the students themselves. We've got our first meeting (by conference call) with Ophelia (our evaluator) on Monday, and by then I hope she's had a chance to glean some feedback from the students.

5. Understand how to make high-level use of a computer algebra system such as Mathematica. On Thursday morning we spent about two hours plowing through several Mathematica notebooks put together by my currently-honeymooning colleague Nostradamus. About half of the students have had previous experience with Mathematica, and a couple others had worked with its main rival among computer algebra systems, Maple. As far as I could tell, there were no major snags. The most heartening indicators of student success are the applications the students have already found for the software. Uwe's made extensive use of its high-level functioning as he's undertaken his analysis of saxophone key structures. During the last few days he's developed incredible proficiency in using Mathematica's Combinatorica package. Meanwhile a few of the others put the software to work in crafting a T-shirt design featuring a complete graph on 11 vertices, each of which holds the name of a program participant. (Ain't they sweet?)

6. Have facility in working within the LaTeX typesetting environment. They made mincemeat of yesterday morning's typesetting exercise, in which I asked them to mimic as closely as they could a one-page sample document I gave them. Though I could have made it substantially harder, that one page featured a number of minor obstacles (double quotation marks, positive horizontal spaces, etc.) as well as some tricky mathematical typesetting conundra (arrays, commutative diagrams, variably-sized delimiters). Three or four of the students have used LaTeX before, and they made short work of the exercise (Norton finished in about 50 minutes, over a half hour faster than the final finisher). Even the least experienced of the bunch managed to put together a beautiful document, and in a totally reasonable amount of time. Just a week in, and they've all got text editors and compilers installed, ready to be pressed into service. I'm eager to see how their first "papers" turn out; they're due on Monday.

And speaking of Monday's assignment...

7. Be familiar with the structure of a mathematical research paper, and be able to construct such a paper. Monday's assignment calls for them to produce a brief report mimicking the structure of a formal mathematical research paper. At this point the content is unimportant; I merely asked them to write on a topic we've talked about at some time during the first week. What's important is that they begin to understand how a mathematics paper is structured. Though I'm by no means expecting perfection from the students on Monday, I've already come to expect excellence.

8. Have confidence when communicating mathematics orally, to an audience of peers or to an audience consisting of research professionals. At this point we've had no structured oral presentations, but all three of us faculty facilitating the program have asked the students to perform impromptu exercises at the board. To a one they've performed with skill and grace. As I mentioned in Tuesday's post, on that day the students ran the show for roughly four and a half hours, during which time only infrequently were I and my colleagues at the board. Surely their first oral progress reports, to be given next Friday morning, will be strong.

9. Understand the dynamics of, and feel comfortable working in, a collegial research group. As somewhat of an outsider, it's difficult for me to say how the students are functioning as a unit outside of the seminar setting. However, it seems to me that the students have already formed a tight-knit social group. They're supportive, they stick up for one another, they're good at reading and meeting one anothers' needs. Camilla's fantastic at sensing the mood of her friends and conveying it to the faculty if there's any doubt we don't grok what's going on. With her confidence, intelligence, and outspoken character, she's a natural leader. Meanwhile, as I'm sure you can judge from the comments to the above learning goals, the students' interactions with faculty bespeak a high level of comfort in working with us. I think we've already made good progress towards meeting this particular goal.

10. Possess a healthy skepticism and authority to challenge as yet unproven results. This is a tough one to measure as yet. Once the students have begun to put together coherent research agendas, I'll be able to say more about the strength of their skepticism. Nevertheless, if confident questioning is any indicator of the students' sense of authority (and I believe it is), these students are well on their way to developing the ownership of mathematical knowledge it will take to make them exceptional scholars.

11. Be ready to present their findings to an appropriate audience at a regional or national conference. This one too is difficult to assess, and might remain difficult for some time yet. The students have hardly begun to present to one another, so it's understandable that they might be nowhere near ready to present to a broader and perhaps (though as likely perhaps not!) more sophisticated audience. I expect great strides will be made in this direction before August 1st comes around.

12. Have a sense as to the structure of the mathematical community at large, and understand their own place in it. This goal too is a bit nebulous, and evidence of its achievement will really only become evident by the end of the program, at which point it might be witnessed by maturity, poise, and overall mastery of the tasks demanded of a professional research mathematician. However, as I said above, we've already had a number of serious conversations about what it means to be an academic mathematicians, and I delightfully anticipate many more before the summer's through.

Keep it up, my friends! You're doing wonderfully, and you've made the first week a joy. I look forward to the next seven weeks, and beyond, in which time many of you will no doubt become my colleagues, coauthors, and coworkers in the mathematical community. What a wonderful time it will be!

Well, by Wednesday we'll be done with our intensive "seminars," at which point the students will be cut loose to seek out research problems and begin their work in earnest.

Before I close this entry out, I'd like to mention that I've now heard back from one of the Calc I students whom I approached regarding her poetry. I'm happy to say that her responses to my "interview" questions showed honesty and depth of thought, and they showed that she really did give a good deal of consideration to the exercise last Fall semester. I also heard from Magdalena, another of those students I contacted: she indicated that she's "still working on" her responses, implying that she's taking time to compose robust and meaningful replies. I can't wait to read what she's written.

I'll check in again in a few days. Until then, your homework: think back on the most meaningful academic experience you've ever had, and explain in 250 words or less why it left its mark on you. Spelling counts.