Monday, March 26, 2007

Pi Day pix

As promised, here are a few pictures of the Pi Day Festivities at UNCA, held on March 14th on the Main Quad of the UNCA campus. (The event drew over 50 people!)

This first is from the very start of the pie-eating contest. We had one poor soul bow out after the first few seconds when it became clear to him that he wasn't going to be in the running. This allowed Nadia (our only female entrant), standing there in the back, to have a seat.

Here Telemachus and Bocephus go head-to-head a good way through the contest. Bocephus ended up winning the title in the First Annual UNCA Math Department Pie-Eating Contest. Below he displays proudly the leavings of his impromptu meal:

His award? An official UNCA Math Club T-shirt. Hey, I can't neglect mention of the many π fans who showed up at the event. Below Twyla gets into the spirit with a makeshift placard:

The pie-eating contest was followed by the π reciting contest. Ulrich won this handily, belting out 64 places after the decimal in under a minute. Rock! Below he holds his trophy, a mind-boggling wooden math game:

We just gotta do this again next year!

Notes to self

I've given a bit more thought into making a few minor changes in my Calc I course design. Regarding Calc I next semester:

1. Let's think about laying down the law on the first day: what irks you? What irks me? Let's each agree not to do those things, shall we?

2. Let's give that student-weighted grading thingamajig a shot, shall we?

3. Lots and lots o' test corrections!

4. I'm going to mitigate the HW lottery in the following fashion: out of every week's assigned problems, I'll still grade roughly 3-5 of them thoroughly, offering robust feedback and commentary. But...for every problem I don't grade carefully, I'll offer up with a check or an "X." If the checks outnumber the "X"s on the student's assignment for the week, in pop a few more points. This way the students get a few extra points for covering all their homework bases, and they also get a minimum of feedback ("right" or "wrong," essentially) on every problem. Besides, it's something I can feasibly do given the time that I have to grade. It's win-win!

Coming soon: a few pix from the recent Pi Day Extravaganza!

Saturday, March 24, 2007

Running on empty

I have this to say regarding my Number Theory class this semester: it's the first class I've ever taught that I feel is running itself.

I've had low-maintenance classes in the past, and I've had semesters in which I've taught a course I'd taught just the semester before (last Fall's Calc II sections, for instance, saw a repackaging of a large amount of the material I used with last Spring's Calc II folks), but never before have I had a course that just sort of...does it for itself.

That's not to say that I'm not putting any effort into the class (I am), and that's not to say that problems haven't arisen (they have), but the problems have been little ones (like yesterday's goof when I used an inappropriate power for the RSA cryptosystem on the handout I'd written up...oopsies!) and the effort I've expended has paid off to an extent I've never before experienced. I feel that after an hour or so of preparation for the class I can walk in, plop the worksheets in front of the students, and let them take it away. I feel comfortable in that class. I don't worry about it at all, it's very stress-free.

This is largely because of the students in that class. They're strong, they're independent, for the most part they're comfortable working together. Their talent (not to mention Deidre's lightning-fast calculator skills) makes my job an easy one. I'm looking forward to their presentations, beginning in a few weeks. I think Karl is still planning on undertaking a project dealing with the structure of arithmetic functions, and I know Bocephus and Simon are looking into the Riemann Hypothesis and how it related to the distribution of primes. (Simon came to me the other day with a copy of Selberg's paper on the elementary proof of the Prime Number Theorem.) I don't know what some of the others have up their respective sleeves, but I'm betting it'll be good. I'm going to get them to nail down their ideas during the next week.

My Calc I students got my cheese a little bit this weekend, I have to admit. I spent an hour or so this morning grading their latest projects (which were by and large good) and the latest homework assignments. Hmmm...I'm concerned, primarily for those that are clearly not putting effort into the homework. Not surprisingly, those that are turning in the homework regularly (even if they don't complete it so beautifully as they'd like) are the ones at the head of the class. For the most part these are the same folks I see in the Math Lab all of the time (Tiffani, your efforts are paying off!).

Maybe I'm not motivating them properly, not making it clear enough that doing the homework matters? Am I being too nicey-nice, too much of a big softie? The thing is, see, they're just not getting it done. I've never had this much trouble getting a class to just do the homework. I'm not asking for perfection, just completion. I understand that homework is a testing ground, it's where one learns by making a few mistakes here and there. Appropriately, it's low-stakes: any one homework problem counts for so little of the final grade (roughly 0.25% per problem, as opposed to maybe 1.5% or 2% for an exam problem of comparable difficulty, if that's the kind of thing you're worried about) that one shouldn't be concerned about messing up now and then.

Maybe that's the problem, that I'm making it too low-stakes?

Or maybe the "homework lottery" that's worked marvelously for four sections of Calc II during the past two semesters just isn't the thing for Calc I students, or for this particular set of Calc I students?

It's something to think about.

Wednesday, March 21, 2007


Monday in class an excellent question came up: someone (I think it was Tomassino) asked if permutations behave like combinations in the following fashion: "is it true that P(n,k) is the same as P(n,n-k)?"

"I don't know," said. "Let's find out. A minute or so later, we'd completed the computations. Of course, it was little more than three or four lines of simple arithmetic, but the lesson learned (I hope!) was more than simply how to manipulate a few factorials. Rather, "I want you all to know that the authority to do mathematics, to ask questions and to solve them, to prove things, to come up with new theorems and new theories, does not inhere in me. It doesn't lie in your textbook, it doesn't lie in the 'experts,' whoever they are. The authority lies in the mathematics itself, and therefore in anyone who takes the time to learn the mathematics. It lies in the logically sound arguments and valid computations of which mathematics is built. Anyone who can learn the rules of logic and algebra and adhere to them correctly and consistently has authority to do mathematics, and so to ask questions, to answer them, to create new mathematical ideas. Anyone. The authority is in you, if you take the time."

As much as I despise the term (primarily for its blatant capitalist and patriarchalist overtones), "ownership of" the material, or better yet, "partnership with," the material, is an end towards which I hope I help my students strive.

The math ain't mine. It ain't the domain of the experts, the pointy-heads, the mathematical gurus that rest on high in chaired positions in Harvard and Berkeley. Hell, it ain't even theirs.

It's everyone's.

Thursday, March 15, 2007

Happy day after Pi Day!

Hey, All!

Yesterday was the 301st Anniversary of the naming of π (celebrated), and the Math Club event I helped to put together went off splendidly. Over 50 people (mostly my Number Theory class combined with Quidnunc's Linear Algebra class, and a few assorted hangers-on) gathered to watch 6 folks compete in the pie-eating contest and 2 in the π-reciting event. Bocephus finished off about 90%-by-volume of his pie in 3 minutes and 14 seconds, giving him the victory in the first activity, and Ulrich recited 64 places after the decimal to garner the win in the second.

Many photos to come soon.

Meanwhile, my classes are chugging along nicely (I don't think anyone was too distraught over classes being cancelled on Friday). In Calculus we're almost done with shortcut rules, in 280 we're set to talk about relations and functions, and in Number Theory we're headed back to the text to talk about more on congruence arithmetic for a little while before tackling a couple of primality testing algorithms. The first of my Senior Seminar students' presentations comes next week, too, as Beulah will speak about hyperbolic geometry and how it inspired M.C. Escher. She's shown me her slides, and she did a great job in putting them together. If she can work out the timing, I think it'll be a fantastic talk.

Now, I've gotta hit the road to Georgia, hoping to make it to Statesboro in time for this afternoon's Project NExT-Southeast events. Tomorrow morning brings our panel on PBL/IBL. I'm looking forward to that, and I hope we get more than the 9 pre-registered participants.

Sunday, March 11, 2007

(Re)start your engines...

All righty, then.

Tomorrow we recommence, revving up for the straightaway dash to the end of the semester.

This is as good a time as any to take stock of where we are in the semester, content-wise. Accordingly, I'm going to ask folks in each of my three classes to spend around half of their respective class periods tomorrow in reviewing what we've done so far: what have we learned? What techniques have we developed? How does it all fit together?

I've been doing a good deal of reading on pedagogy over the break, from the text for this semester's Learning Circle, Maryellen Weimer's Learner-centered teaching: five key changes to practice (Jossey-Bass, San Francisco, 2002), and Alife Kohn's No contest: the case against competition (Houghton-Mifflin Company, Boston, 1986). The latter does not deal strictly with pedagogical theory, but I came to it through Weimer's text, and I've found its insights useful in designing new classroom concepts.

A digest of ideas:

1. "Our classrooms are now rule-bound economies that set the parameters and conditions for virtually everything that happens there" (Weimer, p. 96; emphasis mine). A page later: "our classrooms are now token economies where nobody does anything if there are not some points proffered" (p. 97, again my emphasis). This economic image is an oft-used and apt metaphor for the give-and-take between the student and the professor, and I've come across it in one text after another. Surely some such variety of exchange is inherent in whatever classroom structure one could imagine, but my question is: must the classroom economy always be a capitalist one?

Given the research that Kohn lays out (suggesting that competition in the classroom and elsewhere is generally detrimental to both group and individual achievement), doesn't it make more sense that the classroom economy be one in which cooperative values serve as the "gold standard" for the course's currency? To carry the metaphor one step further, what if we redesign the economy so that it takes on a more "communist" hue?

For instance, I can envision, in a sufficiently small course (no more than, say 7 or 8 students), an untimed, class exam. Either in lieu of or in addition to a stand-alone individual exam, the entire class would be asked to complete a few problems as a unit, the professor sitting by as an observer and as a "clarifier," roles she or he typically already plays in proctoring an ordinary final exam. All students participate in generating solutions, offering ideas, helping to synthesize ideas already put forth. At the outset of the exercise, a single student could be chosen as a scribe in order to create a single solution to the problems presented, and perhaps no solution could be submitted which had not been "ratified" by every person present.

Yes, yes: there are problems with this idea. For instance, there would almost inevitably be "slackers," those who would get the same grade as everyone else without having participated at all, whether out of lack of knowledge or out of shyness. The more outgoing students would also have a tendency to monopolize the discussion.

A compromise between this innovation and the "traditional" exam format might look something like Weimer's study group exams, presented on pages 89-90 of her text. I think Weimer may have turned me off of this idea with her heavy-handed treatment of the "best" students who chose not to participate in the group exam (p. 90).

2. An idea transversing both Chapters 2 and 5 of Weimer ("The balance of power" and "The responsibility for learning") is the following: grant the students the opportunity at the semester's outset to, within reason, decide the distribution of point values for various types of assignments. This student-led distribution could occur on the first day of class, students breaking into small groups to meet one another and discuss the pros and cons of weighting this sort of assignment that much, and so forth. After giving each small group the change to come up with some rough guidelines, the class could be reconvened as a whole, and ideas shared. A consensus can then be approached: how much will this be worth? Once point values are arrived at, we'd record the result and all stick to the deal.

Obviously there should be some initial parameters outside of which the students would not be allowed to deviate. For instance, in Calc I class, I would ask that each of homework, quizzes, projects, and exams count for some percentage of the class's points, and I would likely set some minimum values (HW must be worth at least 10%, quizzes at least 10%, and so forth). But from there, the students would be on their own. I'd even let them throw in extra requirements, like attendance, if they saw fit to include them.

This arrangement has the benefit of providing students a chance to take control of the grading system to some extent, and thus while it gives them greater power (and less excuse for complaining should they not keep up!), it also invests them with commensurate responsibility.

3. Through Kohn's text I've found some interesting tidbits on pedagogical competition, from other sources: Morton Deutsch, in Education and distributive justice: a social-psychological perspective, Yale University Press, New Haven, 1985, writes: "If educational measurement is not mainly in the form of a contest, why are students often asked to reveal their knowledge and skills in carefully regulated test situations designed to be as uniform as possible in time, atmosphere and conditions for all students?" (p. 394, from Note 48, Chapter 2 of Kohn). Good question. As a fairly non-competitive soul myself, I hate in-class exams and see little purpose to them in the long run. It was this line, in part, that made me think up the class exam scheme in (2) above.

Also, Kohn says on one of the works of the brothers David and Roger Johnson ("The socialization and achievement crisis: are cooperative learning experiences the solution?," Applied Social Psychology Annual 4, L. Bickman ed., Sage, Beverly Hills, 1983): "In fact, even the widely held assumption that 'students learn more or better in homogeneous simply not true.' A review of hundreds of studies fails to support this assumption even with respect to higher-level students" (Note 28, Chapter 3 of Kohn). There's some ammo for the folks who take flak for "making the smart students work with the dumber ones."

All in all, I'm enjoying both books. Weimer, though I'm not always agreeing with her and I find her tone a bit condescending at times, has given me a good deal of practical ideas, while Kohn's work has been a great fount of references to other authors who purport to prove claims I've heard bandied about before but have never been able to track to the source.

Monday, March 05, 2007

Spring Break!

Well. Yeah.

Here we are.

Here in the South (where they wouldn't know real winter weather if it smacked 'em upside the head with a two-foot blizzard and subzero wind chill) it snowed yesterday, and though there are two weeks left before spring actually begins, we're on Spring Break.

For me this means it's a good chance to get ahead in class prep, since March and April are going to be busy travel months for me (two conferences, two colloquia on the schedule so far), and I'm not going to want to fall behind while gallivanting about the eastern United States. It's also a good chance for me to post here for the first time in a looooooong time...

The funny thing is, I'm less busy this semester than I was last semester, despite the fact that I'm doing three preps this time around (and for the first time ever). Most of the slack is due to the fact that 365 is not one of the classes I'm teaching...I put so much of myself into that course, and I let so much of myself (including sense of self-worth, I fear I must say) get wrapped up in how well I pulled it off. Even when I was done getting ready for 365, I was never done worrying about it.

I have to say that I'm enjoying this semester a lot more than I did the last, perhaps to a large extent because I've managed to distance myself personally from my classes. That's not to say I'm not being myself in class, as I am, and it's not to say I don't care about the students, their learning, or their welfare. I mean only that I recognize that the success or failure of the class, however that might be measured from day to day and week to week, reflects in no way on me as a person.

And the funny thing is, I think I'm doing a much better job with both of my upper division classes this semester than I did with 365 in the fall. Things are running a bit more smoothly. I've found in both 280 and 368 a good balance between me standing at the front and yammering like a talking head and the students working the entirety of the class period with minimal direction from me.

368 is purring along particularly nicely. The text is fantastic, and eminently suitable for the manner in which I'm teaching the course. I've found that by distilling each chapter into a short worksheet I can ask the students to do most of the computations and the bulk of the simpler proofs, leaving me to stand off to the side to lend a hand on the trickier arguments as they arise. The presentations are rotating nicely in that class, and I've had no trouble convincing people to volunteer to present. Right now we're off-text, working our way through a couple of weeks of analytic number theory. We spent the last week on divisor sums and Dirichlet convolution, and the coming week brings an estimate of the average number of divisors for large numbers (a value that tends to ln(x) in the limit). After that we'll return to the text to do a little cryptography before heading off towards elliptic curves and more about Fermat's Last Theorem.

280's easin' on down the road, too. Freeing myself from a text was the right thing to do for this class, I believe. Though it's meant that I don't have a ready reference immediately at hand, it's also meant that I can run the course using the in-class worksheets without having to defer to a text that covers topics in just such an order, that provides at-best weak explanations or overly difficult exercises. I'm happy with the day-to-day goings-on, though people aren't nearly as eager to present HW solutions as the 368 students are. (I'm chalking that up to the relative mathematical inexperience of the 280 crowd; it's nothing unexpected, nor is it to be sorely lamented.) I'm very pleased with the improvement I'm seeing in a lot of the students' work. I'm thinking particularly of folks like Neville, Sylvester, and Una, a trio whose first homeworks were lackluster, but in whom there was definitely potential. In the past few assignments from them, I've seen much stronger structure, clearer arguments, cleaner logic, better use of notation. All three of them have put in long hours in the Math Lab, and it shows. Kudos! Of course, I'm getting stellar performance from people like Fiona and Elmer, folks I knew I could count on to do well. From everyone, there are struggles to be won, but I think overall we're at a good place to be by midsemester. Right now the order of business is combinatorics: combinations and permutation rule the day, and by the end of next week we should begin talking about functions and relations.

Oh, yeah, and the University's Writing Intensive Committee has ruled: 280 is now officially a WI course, from here on in!

Finally, I'd be remiss if I left unmentioned my Calc I class. They're a laid-back bunch, and after 280 and 368, which are often hectic and fast-paced, it's often nice to wind down the day with the 191 folks. We're moving a bit more slowly than I typically do, but I truly think it's the right thing to do. There are a number of people in this class who haven't had math for quite a while, who aren't so confident of their math skills as they might could be, and the slower pace is letting them absorb the material more meaningfully than if we were simply blazing through it. We're just now working our way through the Product and Quotient Rules, and by next week's end should be ready to consider some interesting applications.

So that's the score.

Big-picture-wise, I'm still waiting to find out whether or not I'll be getting this REU picked up for the coming summer...the fact that I've not yet heard could be a good thing. Ever the optimist am I. If we don't land that big fish, I'm going to try to rustle up a couple of research assistants for the summer so's I can have a crack at a couple of problems from geometric group theory I've had on the back burner for several months.