Overdue notices

I'm overdue to post (so saith a few faithful readers, including my mother-in-law, my wife, and two of my most diligent students). Enough, already! So here I am, I'm writing.

I've been here, I've been busy. The last few weeks have been exciting ones, and next week promises its own supply of hecticness (hecticity?).

What's new, pedagogically speaking?

Well, yesterday was the deadline for applicants to this year's REU. This afternoon I counted up the number of distinct applicants from whom I've received at least one document (application, statement of purpose, transcript, or letter of recommendation): 64. Not bad. That's about what we had last year, though I believe the ratio of men to women is higher this year than last. I've yet to break it down geographically, but I think this year's pool is more widely distributed throughout the country than last year's. I'll probably begin looking at the applications in earnest over Spring Break. No sooner: next week's going to be a bear. Tuesday brings a colleague from Davidson College to campus (hello, Twyla! thanks in advance for making it out! and sorry again for the confusion in scheduling!) for the Research Seminar. Then comes the Math Literacy Summit we've been planning for several months, highlighted by the public lecture and keynote address given by Dr. Robert P. Moses. That's Wednesday night and Thursday day. Saturday sees the start-up of Super Saturday once more (I'll see if I can enlist some student helpers...the downside is the onset of Spring Break, taking many students away from campus). By Saturday afternoon I'll be beat.

What else is new?

This past Friday my Graph Theory gurus had their first exam, an in-class ditty that represented my first attempt in almost two years at writing an in-class exam for an upper-division course. It's known far and wide that I'm not a big fan of such a format for seminar courses, and it was hard for me to write an exam that I felt made fair demands of the students' knowledge and was doable within the alloted time. The end result, I feel, was a hair (and no more!) on the long side, and a tad too easy. (I'd be interested in knowing how the students feel about both of those appraisals.) As it is, the students went down to the wire time-wise, several staying for an extra ten minutes to finish up, while the average was high, around 86%. Maybe I didn't ask for enough proofs? Maybe I was a little light in grading? I don't know. The only question that seemed regularly to ensnare the unsuspecting was the problem asking the students to compute the number of homomorphisms from the star with 3 vertices to the star with 4 vertices: there was a broad array of answers to that question.

How's the day-to-day activity in that course been? The students' presentations of solutions are getting tighter, more succinct, more precise. For the most part their diagrams are more descriptive and intuitive than they'd been during the first few weeks, and their proofs are more straightforward and understandable. Most interesting are the differences in presentation styles between the various members of the class. Some are silent until all has been written on the board; their presentations then consist of little more than "voilà! Pas de lacune a remplir!" Some are so verbose they can barely put chalk to board to draw a single tittle without prefacing it with a megillah of mathematical exposition. I wonder to what extent they find others' presentations are affecting the style of their own? (This sounds like a perfect question for a mid-term evaluation!)

We've gotten to the point where the students have a grasp of the basics, I can probably branch off in whatever direction I'd like to in terms of the ground material. The most recent problem sheet (the seventh, available here), deals primarily with components, paths, and chromatic polynomials. I believe I'll make trees the focus of the next sheet, unless someone has a better plan. Lorelei mentioned the other day to me that she'd like to see more applications, so I'll do what I can to work those in (colorability has many applications, and they'll soon be ready to take off in that direction). Markus came to me on Thursday, indicating that his relatively light schedule is granting him plenty of time to take on some independent study in graph theory, so I gave him some reading on graceful labelings, maybe he can join Trixie and Sieglinde in their pursuit for new graceful trees.

Speaking of which, we'll soon have our strongest showing at an MAA Southeast Sectional meeting since my arrival at UNCA: at least five faculty members and four students have indicated interest in going to the meeting, and I'm trying to get three of these students (including Trixie) to present in the student poster session.

And speaking of Trixie, how's Calc II? They too completed their first exam this past week, and overall the grade distribution was pretty fair, with a course average of about 76% after corrections were made. Oddly enough, though, there was a profound difference between the two sections of the course: the first section's post-correction average was about 68%, the second's around 86%. I kid you not. After corrections 12 out of 16 students in the second section got either an A or a B (8 As, 4 Bs), while only 10 of 30 students in the first section earned that bragging right.

What, as they say, is up?

Could it be class size? Admittedly I find it much easier to engage the second section as a whole and as individuals, owing largely to the fact that it's got about half as many students. Moreover, the students are less intimidated by speaking up in front of one another, and by presenting on the board. They're also much less reluctant to get into groups and work on problems together. Whether any of this has anything to do with the size of the class, I don't know, but I can't help but think that class size plays at least a small role. (Incidentally, I'm happy to report that I'll be teaching two sections of Abstract Algebra I next fall, each considerably smaller than the traditional single sections that have been run in the past. Our program has been so successful in courting majors that we're having to run two sections of the upper-division courses. America gonef! [You'll have to excuse the Yiddishisms, I've been making my way through Leo Rosten's delightful Hooray for Yiddish: a book about English. I'm easily influenced].)

Could it be the time of day? But one might think that the sluggish 9:00 a.m. class would be more ideally situated in that regard than the post-lunch but usually-punchy 12:45 p.m. class. Or maybe I just think it that way because I'm a morning person. Perhaps there's something to it: the morning section's usually slow-to-rile and torpid, the afternoon section's much more get-up-and-go.

Could it be...the luck of the draw? I've got great students in both sections, but they're just more highly concentrated in that smaller second section. Maybe it's just coincidence that the second section's so much more lively.

Whatever the cause, the difference between the two sections is as that between night and day. I love both of them, but I find myself often wishing wishing wishing that the first section would wake up and stop dragging its heels!

What else?

Faculty talks have ended in the Senior Seminar, I capped them off with a presentation this past Wednesday, on open problems in graph theory. I'm proud of the fact that we had three speakers from off campus, and that we'll soon have at least two more visitors coming to speak in the Research Seminar. I truly believe that our department should attempt to cultivate a more active research environment, and I think we're well on our way towards achieving that goal.

Student talks begin in the second week after Spring Break, two-by-two they'll fill up the last six weeks of class. I'm looking forward to those talks, the topics look to interesting.

What else?

The Writing Intensive committee (well, technically it's a subcommittee, but who's keeping track?) has sprung back into life, continuing our analysis of WI applications and beginning our conversation on the assessment of the success of already-WIed courses. This is a sticky wicket of a tricky schtick: How are we to judge whether a WI course has met the goals it laid out for its students? What materials must we demand of the course instructors in order to perform a proper assessment? How many of a course's learning goals relating to writing must be met in order to call the course a success? And if a course is less than entirely successful, what consequences do we as a committee mete out? It's unrealistic to aim for the ideal right out of the gate (assuming the ideal can be articulated from the outset anyway): all but the perfect instructor is going to stumble here and there, and no course is flawless in design and execution. Therefore it's pointless to pull someone's WI away should perfection not be attained. We don't want to smite those who fail in providing this or that element of their class's purported learning experience. Instead, we wish to encourage the instructor to look carefully at her course's goals, to look at the students' products in attempting to meet those goals, and say, "this was done well. But this, when asked of the students, proved unrealistic. Better I ask that they reach for the moon with their hands at their sides!"

How many of us are so reflective? I'd like to say that I am, but who am I to say?

In the first of what will be several meetings of the writing assessment project this past week (didn't I say I've been busy?) I told my colleagues that last semester's 280 course taught me to be truly conscious of the role played by writing in my own particular discipline...indeed, I think I learned more in that class than my students did. My approach to writing as a tool for learning has changed because of that class.

Writing is playing less of a role in my Calc II class this semester than it did in either of last semester's classes, and while I've not shone a spotlight on writing in Graph Theory, it's ever present. (The work I've done in 280 over the past year is most evident in the structure of the students' proofs on the blackboard: I'm thrilled whenever I see clear statements of hypotheses, an explicit indication of proof technique, summarizing sentences that indicate when and why a proof has been concluded, and so forth. In all only two or three of that class's students didn't have 280 with me, and almost daily I see elements of my own idiosyncratic style that have rubbed off on them.) I'm going to take a few minutes during this coming week to refocus the students' attention on writing and encourage them to keep an eye on the criteria for solid mathematical writing as they put together their solutions to the problems selected for written submission.

What else?

Um...hmm...giving a recruitment spiel on the upcoming REU and speaking ongoing graph theory research to a wonderfully receptive and warmly inviting crowd at Morehead State University in Kentucky (y'all were great, thank you so much for having me!), writing about a dozen or so REU rec letters for current and old students, joining a couple of colleagues in a presentation to the university's Foundation Board, agreeing to help organize this coming May's Writing Intensive workshop, and shaking off a nasty cold that took me out of commission for a few days...see? There's a reason I've not been around!

If you'll now excuse me, it's Maggie's birthday (which one, I will not say, though I doubt she'd mind if I did, she's not embarrassed by her birthdays), and we've got to go celebrate it in the manner of her choosing.

As usual, all comments, questions, queries, suggestions, insinuations, epiphanies, innuendi, graffiti, scritti politti, revelations, retorts, ripostes, and recriminations are welcome on the comments page.

Until next time, live well, and try to learn something new today.

Oh, hey!

Hey, sorry I've not checked in for a bit!

I think about writing, I really do.

And then something else gets my attention. Some small fire pops up and needs putting out, someone comes by with a ten-minute diversion, or I just say to myself, "gee, I'd like to finish reading that Singer story I started this morning before the sun came up."

Some of my favorite of his stories involve the framing device wherein a motley crew of wayfarers, scholars, beggars, etc., find themselves holed up in a snowbound Hasidic study house somewhere in semirural fin-de-siècle Poland. There's a coziness to those tales, an intimacy, that makes them more believable, more real than they already are. You get the sense from that device that Singer himself told that tale by the flickering light of tallow candles, or at least overheard the story as it fell from the mouth of some unnamed wanderer who spoke of the spirit who haunted his second wife and caused her to suffer horribly and cavort wildly and brought her (and him with her) to shame in the eyes of his town's most devout Jews.

But I digress.

I've meant to say that we've done away with the soccer ball (mercifully!), and the last few classes of 473 have recovered much more of that sense of excitement with which the semester began. People have been better in not speaking out of turn, though, and it's led to more polite exchanges with less cross-talk and more consideration for others' rights to have a say. All in all, it's been an improvement.

One our class's quietest students led us off with the very first presentation after the soccer ball's eternal banishment, and it made for fifteen minutes of silence as he very meticulously wrote most of his proof (of the fact that the subgraph relation is an order relation) on the board before explaining it. (I can't help but think that a week before, there would have been a half-dozen interruptions during this time, by onlookers eager to offer their 34.5 cents on the problem's solution, but all of us did a remarkable job of sitting on our hands and biting our tongues.) The explanation was solid, and though not quite complete it was almost entirely correct. One or two others interjected helpful suggestions to move the proof to the finish line. It took about half the class, and it made for some tense moments, but it was well executed.

On Wednesday Joachim "solved" the first of the "review and discussion" problems I've begun adding to the problem sheets, at the suggestion of one of the students. These problems ask the solver to recap the definitions, theorems, and examples considered in the given problem sheet, providing the class with a "where are we now?" moment. I think these'll be useful in focusing the class's attention on the highlights and reminding them of key definitions and results.

I'd like to see the students improve their ability to interpret definitions; there was a bit of confusion over the definition of "bounded degree" on Problem Sheet 4. Or has it been that I've not made the definitions as clear as I might have? It's likely a combination of the two, we could all stand to do a little better. I have to remind myself that (a) I'm not writing to my research peers when I write these definitions, and (b) I'm not going to take extraordinary pains to describe these definitions to the students in person; it's up to them to interpret, draw examples for themselves, understand. I'm happy to help them over the hump if they come to me with questions, but I expect them to make the effort alone to understand a definition and apply it properly. After all, one of the learning objectives of this course asks the students to develop an ability to read a mathematical article and interpret and understand it, alone. I'd like for them to be able to read a fairly low-level math paper unassisted by the end of the semester, and that'll more often than not entail wading through a few new definitions on their own.

Nevertheless, I've got to insist on absolute clarity on my own part. I'm going to pay special attention to my definitions from now on, to make sure they're clear as crystal. Students, if they're not, please call me on it!

Meanwhile, Calc II has been chooglin' along. My morning section is a soporific one, but the early afternoon section, a bit smaller, is more lively, more engaged. I've only lost one student from that second section from the start of the semester, and two from the morning section. We're in the middle of methods of integration right now, about two weeks away from the first exam of the semester. So far the students have been really good about getting homework in, with only a few stragglers. Aside from a couple of folks whom I've carried over from last semester who look like they're crusin' for a losin', most everyone's eager to do well, a phenomenon that's a welcome change from Calc I, in which there are always a handful of folks who don't really give a rat's ass and are just drifting along until the end of the semester.

News flash, by the way: I found out that I'll be teaching Precalc (!), of all things, this coming Fall, along with two sections of Abstract Algebra. Woo hoo! This'll be the first time I'll have taught Precalc ever, and the first time I'll have taught Algebra since coming here. I'm excited on both counts.

The time has come for me to say adieu, as I must away to dinner in Greenville with our grad school buddy who now teaches at Furman U.

Farewell, and have a wonderful weekend, what remains of it!

Vox populi

The students have spoken!

Some of them, anyway. I thought I'd post the feedback I've received so far on this last Friday's class. The core issue is class structure: soccer ball or no? A few students brought up residual issues, but this single one remained at the center.

Saith one:

I find the best form of government is a benevolent dictatorship. Think about that premise and I think you'll be able to maintain an iron fist over the class without compromising the spirited environment.

In response to this point of view said I in an e-mail: "Point taken, kindly. I'll put my iron fist in a velvet glove and see how things go."

This student's take was echoed by a colleague:

I definitely felt like my thoughts were being forced to remain in my head, during Friday's class. I do agree that maybe some of those thoughts might be better off there, but what's most disappointing about that feeling is that it felt like much of the passion and the fun that existed in the first two class periods was sucked out of the experience.

I think that the soccer ball should be eradicated from the world of Graph Theory 473. What should be put in it's place is some self awareness, and some consideration on the part of those who are speaking, a few rules giving the 'presenters' more authority while they are in front of the class, and possibly a comment from the prof. when things start to get a little out of hand.

Man, I just went back and read your suggestions and ideas for next class. [See the excerpted e-mail from my previous post.] I didn't realize that you had already written the same ones that I did. Oops. Oh well, that's how I feel.

One student waxed a bit more philosophical:

Even though it was painful, I am glad we had the class we did on Friday. For me, two big ideas came out of what occurred during class. The first is, that in setting up time to ensure that we build a solid foundation for what we are learning, I believe that we are avoiding some long term pitfalls that might only have come up in the last weeks of the class. I feel like what really came out of the horse that we beat to death from problem sheet #2 was that carefulness leads to the deeper meaning that we are trying to glean. I know that there are Algebra and Calculus ideas that I learned only well enough for testing purposes and now wish that I understood more principally. I like the idea the idea of the review problem and the carefulness in answering problems.

Second, I think some good things came out of the 'structure discussion.' While I do not like the soccer ball, I do like the metaphorical one. I don't think that things were out of hand before Friday, in fact I love this class. I do think that we might not have had this discussion until things were out of hand though, and that would have been more painful. I like the 'sitting in a circle' idea. I believe that half of what we were concerned about will be fixed by that one addition. Also, because our Friday talk was not based in failure but in improvement, I think it gave us an early opportunity to be a little more conscious about how we do want to shape the class. I would guess that most of us gave that a little more thought after the discussion. So I guess the second 'big idea' for me was that awareness that came out of Friday. How could that hurt us!

Ultimately, some kind of order is necessary, as acknowledged by the folks above, and by our last commenter:

My thought is that there definitely needs to be some type of system, because the first few days it did get kind of crazy and was hard to really understand people's thoughts. But I also think we are all college students and should be able to respect one another enough to listen when they talk and then put in our thoughts (without interrupting). And people should be able to ask questions!

The plan from here: let's abandon that awful soccer ball, let's grant the presenter the authority to open or close the discussion, let's allow for open conversation while discussion is "on," with the understanding that that requires respect for one others' points of view and rights to have a say as well, and let's let me have another go at better moderating the discussion should the need for moderation arise (see my "velvet-coated iron fist").

Friday was painful for me, too. As the second commentator above pointed out, it felt a lot less fun, and when it comes to research (let's face it, folks, we're doin' research here) fun shouldn't be undervalued. I want this class to be fun and engaging, and I think we can manage that without the help of the soccer ball.

I'm glad we've had this conversation, I think it's helped us all to understand better what it takes to make up a healthy learning environment, and I'm glad it's happened in Week 2 instead of Week 11, allowing us twelve more weeks of organized, respectful, blissful interaction!

I'll check in again tomorrow and let you know how things go down.

Chaos

I've got mixed feelings about today.

Calc II felt all right: the first section was a little sleepy, but the second was more engaged and seemed to enjoy the sugar fix provided by the shortcakes used to illustrate the method of cylindrical shells.

Graph Theory?

Hmmm...

After Wednesday's class got a bit rowdy, with lots of cross-talk, overdubbing, interruptions, and just plain ol' mayhem, we decided that maybe we ought to try out a means of directing the discussion. I suggested the possibility of getting a small plush object to toss around: she/he with the ball was the one who got to speak. It seemed a bit puerile, but I didn't want discussion to get out of hand, lest people start zoning out, not understanding what's going on, what's being said, what's being proven.

So Theodoric brought in a plushy soccer ball, and we tried it out.

The atmosphere was...

...well, to me it seemed a bit dead. I'm not sure the deadness was completely unwelcome, but I don't want to kill off the natural enthusiasm that folks are having for the course.

I think some people had difficulty getting the attention of whoever it was who had the ball at any given time, others felt like they weren't going to stoop to "playing the game" of getting the ball before speaking and so said nothing...for the most part we stuck to the plan and didn't speak until given the ball...but it felt stilted, juvenile.

Having to choose between lively and occasionally cacophonous debate on important mathematical topics and stultifying silence, I'll take the debate, even if it means a little chaos every now and then. As I put it to the students in an e-mail exhorting them to write to me and let me know how they felt (their comments will be posted here as they trickle in):

My own two bits, for what it's worth: we've got a room full of 16ish smart, eager people, and I know that there are time when we've all got something to say. I want to keep the class lively and the discussion excited, but I also don't want it to descend into utter chaos. I'm not doing my job well if I let the class devolve into a kindergarten class. That said, if people are overwhelmingly for it, I'll be open to trying to use the soccer ball again (thanks for bringing it, Theodoric, by the way), but my feeling is that (a) you're all mature enough to not interrupt one another and to not crack jokes when other people are trying to explain something, and (b) I can do a better job at moderating discussion should it need moderation. I'd like to come in on Monday and try to go without the soccer ball, we'll let the person at the board lead the discussion (opening it up once he/she is through presenting), and if things begin to get rowdy, I'll exercise my authority and rein it in.

We'll see what they have to say.

Mathematically, we finished off a single problem today, proving that a subgraph of a simple graph is also simple. Our proof, built up in bits and pieces, was ultimately a careful one. One person starting things off with an intuitive explanation, a second tag-teamed with a more solid justification, and a third stepped in to nail it down with some clear notation. The result was a pretty clean proof, and I'm glad we took the time to make it rigorous. Remember, folks: I'd like you to be able to understand these theorems, but you should also be able to prove them.

That's all for now. I'll post student comments on the Great Soccer Ball Fiacso of 2008 as those comments come in.

'S no doubt we're snowed out...

...thus I had no meetings today.

Bummer.

I was looking forward to seeing how my UGs were progressing on their graph theory endeavors.

However, I did get out of an all campus-meeting.

I found out that the University of South Carolina's Combinatorics Seminar will be running on Thursdays at 12:30, which will allow me to attend at least semiregularly, hopefully with students in tow.

Meh. It's been a snowy gray day. I'm going to get back to my research...

Graph Theory: Day 2

As the snow storm descends on the Asheville area, I'll take a moment to briefly chronicle this afternoon's mathematical goings-on.

I felt a bit out-of-step in my first section of Calc II today. I never really got into my stride, somehow, and I felt awkward. The awkwardness carried over into the second section, with whom I felt more at ease, but still stretched thin. I'm looking forward to Friday in both of those sections, I'll be leaving much of the work up to them. Then Monday will bring the first of several food-based exercises, always favorites with the students.

These two classes were more than made up for by Graph Theory.

Right away the atmosphere was a positive one: before class, as people were still trickling into the classroom in dribs and drabs, everyone was chatty, jovial, open. The students joked, compared solutions. Everyone seemed relaxed, ready. I put some colored chalk on the front table and went to the side board, where I wrote "Correctness / Completeness / Clarity / Composition," urging the students to intone these words as a mantra as they prepared their presentations.

Then we began.

Things went well from the start: when called, each student took to the board to the sound of applause from her or his colleagues. Everyone was quiet and respectful during presentations, and each success was met by another round of applause and cheers.

The first few presentations went smoothly; it was Problem 4 that caused a bit of hullabaloo.

"Problem 4. Draw as many fundamentally different graphs as you can, each having order 4 and size 3, also writing each as a triple."

Its the fourth and fifth words here that brought down the house: there was (understandably! I'd somewhat hoped that this problem would provoke a discussion) a great deal of disagreement regarding what was meant by "fundamentally different"; it'll be another week, at least, before we define graph isomorphism. (Brigitte actually said a few words about "bijections" that were very close to the mark, but her quiet voice didn't carry so well amidst the hubbub.) The chimerical nature of this phrase, coupled with the immense number of graphs having the properties desired, led to uproar. Poor Joachim, attempting to answer the problem as fully as he could, was interrupted by a chorus of overly helpful classmates: everyone wanted a piece of the problem, and the next ten minutes were spent in taking unruly turns at trying to pin down the meaning of those elusive words, "fundamentally different."

Ultimately it became clear that we all had more or less the same idea as to what those words meant.

The discussion was lively, even heated, but ever respectful and supportive: no one attacked anyone else, corrections were friendly ones, and even when there was disagreement, the disagreement was civilly made.

The next three problems were relatively humdrum; Problem 8 caused a bit more furor, though without the controversy attending Problem 4. Quincy was called on the complete Problem 8 (asking for an enumeration of the maximal number of edges in an order-n graph without multiple edges), and he offered a nearly-complete proof of his (correct) formula, the sum 1 + 2 + 3 + ... + n.

"Did anyone have a different proof?" I asked. Sylvester offered that he did, and he went to the board to provide an inductive proof of his (equally valid) formula, C_n,2 + C_n,1. Throughout both presentations, everyone was quiet, attentive. Sylvester's proof brought us to the end of the period, midway through the first problem sheet.

Afterward Quincy characterized the mood of the class as "fun, but serious." "We all mean business, we're taking it very seriously," he said. "But we're having a good time with it." He had a blast, as did his friend Norbert, and as did Nadia, who spent some time after class trying vainly to convince Olivia to join our class.

I am positively delighted with the way class came off today: the students took control. They constructed their own mathematical meaning while engaging in lively, sincere debate about deep mathematical issues. If we can replicate today's success over and over again for the next several dozen class periods, I'm going to end this semester as the happiest man on Earth (not that I don't already hold claim to that title).

I'm already looking forward to Friday.

I'm also looking forward to tomorrow: barring too-hellish weather, I'll be trudging into campus to fulfill a number of bureaucratic commitments, and to meet with Sieglinde and Trixie, my budding freshperson graph theory research team. Trixie's progress on the problems I pitched her over break has been nothing short of astounding: I met with her yesterday and she showed me the pictorial essence of the results she's come up with, and they look solid. Sieglinde's indicated progress too, and I can't wait to see what she's got in store. They're both sharp are tacks and a kick to work with.

On that note, it is wearily but happily that I bid you a good night, I'm off to do some relaxing reading before calling it a day. Adieu!

One down, a whole bunch to go

Day One.

A difficult beginning?

Not really.

The day was tiring, but pleasant.

I felt uncharacteristically (for a first-day-of-semester) comfortable in my first section of Calc II, and hardly more perturbed in my second section. There was a bit more nervousness in Graph Theory, but overall my ordinary first-day jitters subsided quickly.

As I suspected would be the case, I had a hard time getting to sleep last night, and once asleep I couldn't stay asleep. More than once I awoke to find the room still black in deep night. The early morning hours dragged, and I swore that the six hours or so hours I'd allotted myself were among the longest I've ever lived, wide-eyed and ceiling-staring.

My alarm went off at 5:30, and I was strangely refreshed. I showered quickly, had breakfast, and headed out, puzzling over various schemes for constructing expander graphs in my head as I walked into campus.

It was just growing light as I arrived, and it was near enough to the start of the first class period (not mine, this semester) for the earliest of the students to be poking their overly-punctual heads into their 8:00 classrooms as I entered Rhoades Hall.

The next hour or so was spent putting together a few odds and ends I'd need for my first Calc II course of the day; organizing my notes, syllabi, handouts; putting a couple of finishing touches on the website; placing the Skittle-filled candy machine in the Math Lab; responding to a few early-morning e-mails.

Then came class. By the afternoon's end there'd be 32 people in the class, only 9 of whom I've not had the pleasure of working with before. (Belladonna, chagrined, pointed out that there are only 6 women in the section. I mentioned that there's often a precipitous drop-off from Calc I to Calc II, that women more heavily populate the biological sciences than the mathematical ones.) Class seemed to go rather smoothly, despite the massive amount of crap I wanted to get through by the first day's end. After an exercise in which I asked the students to compile a list of dos and don'ts when constructing a safe and effective learning environment, and after a brief review of the essentials from Calc I, I left them with the assignment sheet for Confectionary Conundrum.

Trixie and I had a chance to catch up after class and spend a few minutes talking about the graph theory she'd been working on over break. One of her friends, a fresh face to me, lingered too, and she took a few minutes to explain to him, very well, the problems she's been considering. She's made great progress, and if she manages to push it much further, I don't think a presentation at MIGHTY would be out of her reach.

The next few hours saw me doing all manner of busywork, and by 12:45 it was time for Round Two, duked out with a smaller section consisting of a mere 18 students (half of whom are women!). The smaller section is sure to make for a more intimate environment, and already I can tell that people are more comfortable speaking up in front of each other than are the folks in Section 1. I have high hopes!

Graph Theory, having shrunk by a student, now has 16 students. After obligatory welcomes and niceties, I explained to the students the structure of the course: they'll be in charge, hands on the wheel and the feet on the gas, directing the flow and the pace of the course. I'll be there with a road map if they need it, but I'll try to keep it tucked away in the back of the glove compartment, beneath a pile of oil change receipts and a pack of 10-year-old once-minty chewing gum. Their presentations of problem solutions will dominate class time, and through their work with one another I hope that they will learn to become colleagues in discovery. (For a complete description of the "Moore method" means I'll be utilizing, please consult the syllabus.)

I apologize for the highly simplified, blow-by-blow account of the day's proceedings. Honestly, I don't have much left in me to make the day sound any more poetical than a pedestrian succession of events. It was a good day (superlative, as first-days-of-semesters go), I'm glad I've lived it, and I look forward to many more like it this semester.

I'm just beat.

Well, Calc II continues tomorrow, I'd best be off to get some R 'n' R before beddie-bye.

Giassou!

All beginnings are hard

"All beginnings are hard."

These are the first words of Chaim Potok's In the beginning, a book I read long ago and decided just this afternoon to read again.

They're true, no matter to what "beginnings" refers.

They're also fitting words to have in mind as I begin tomorrow, the first day of my 31st (if I'm counting correctly) term of teaching at the college level.

I'll tell ya one thing, folks: after a bit it might get easier, and it goes more smoothly, but the nerves never go away. I'll probably be up half the night tonight wondering how it's going to go down.

Saith Potok: "I say it to myself today when I stand before a new class at the beginning of a school year or am about to start a new book or research paper: All beginnings are hard."

In one form or another I've taught Calc II six times before, and I've assisted in three other Calc II classes (again, if I'm counting correctly). It's my favorite class to teach, hands down: there are so many beautiful concepts, wonderful and broad-based applications, and computations that require not only mathematical dexterity but also almost poetical creativity...how can one not love this class? I feel like I've finally gotten Calc II where I want it, but I'm sure my students this semester (roughly 35 of whom are coming back from previous semesters with me) will be able to teach me something new.

Graph Theory will be presenting new challenges: for one thing, I've never taught the course before. Moreover, I don't think I've ever had such a high concentration of proven talent: I strongly encouraged a lot of our ace students to take this course, and my advertising efforts paid off, giving me 17 students representing the cream of our crop, all but a few of whom I've worked with in previous courses. I'm looking forward to seeing what I can get out of them, and to seeing what kinds of new ideas we can uncover. (By the way, a special shout-out goes to my colleague Fosdick on the West Coast, who's also teaching Graph Theory for the first time this semester!)

How will it go tomorrow?

I'll keep Potok's words with me as I start the day off.

Beginnings are hard.

If you're reading this and like me are girding your loins to go into tomorrow's fray, please keep in mind that I'm sure to be as nervous as you are, as jittery, as excited. I know the beginning'll be hard, but I also know that if we keep at it, we'll be capable of wonderful things together, and that the semester will bear that out.

I'll be there, in Rhoades 105, by 9:00 a.m., bright and early. I hope you'll be there with me.

Until then, take care, and have a pleasant tomorrow.

Write or wrong

Hey, hey, hey! It's a brand new year, folks!

So far this year I've done little related to my teaching, I've spent most of my time reading (gasp!) for pleasure. I've capped off seven books in the last two weeks...golly, it's nice to have free time. This morning, for instance, I finished Wangari Maathai's Unbowed: a memoir, a recounting of her life in Kenya and the founding of the Green Belt Movement, the organization primarily responsible for her receipt of the 2004 Nobel Peace Prize. Before that it was a collection of short stories by Guy de Maupassant, Georges Bernanos's The diary of a country priest, John Griffin's classic Black like me, and a pair of books by Jonathan Kozol and Kurt Vonnegut, reviewer elsewhere in this blog. It's been great to be free to read again, something I'm sadly unable to do much of during the school year.

I've also been mulling over what I'd like to accomplish through my teaching during the coming year.

Last year could be characterized by consciousness: I believe that more than anything else I learned to become fully conscious of my pedagogical efforts, and cognizant of the effect my deliberate actions would have on my students. I made conscious efforts to structure my assignments developmentally, to engage students in meaningful, conscious discovery. I believe that my conscious focus paid off, I feel as though my 280 class benefitted enormously, for instance, and the effort that went into Newton v. Leibniz was repaid tenfold by the students' growth through the project. (By the way, I heard back from Prof. Bornstein, she was delighted to hear from me, and wrote me a wonderful letter on her own ideas on teaching. She looks forward, as do I, to continued correspondence. I need to write back to her...)

So what is it that will characterize my teaching in the coming year?

Discovery, perhaps? That will certainly be a central theme of my upcoming graph theory course, in which I'll be challenging students to rebuild the discipline from scratch.

Or maybe authority? Might I focus my energy on encouraging my students to take the reins in their own studies, to ask the questions that need to be asked, to take responsibility for their own futures?

The line between these broad territories is an unclear one. I look forward to seeing how my classes take shape in the coming weeks.

At present I'm ready for Day One (now a week and a half away, on Monday, January 14th), freshly printed syllabi, worksheets, problem sets, and project outlines covering my desk. All I've got to do now is get some Skittles for the candy machine, in order to be ready for the third installment of Calc II's Confectionary Conundrum, an exercise whose execution I've now got down to an artform.

Good news came yesterday in the form of an e-mail from Texas: both my individual presentation and the panel presentation I'm putting together with a couple of my UNCA colleagues (one from the Writing Center and a second from Sociology) were accepted by the organizers for May's 9th International Writing Across the Curriculum Conference at UT-Austin! This is exciting. It'll be the first time I'll have had a chance to speak at a non-math-related conference, about a subject that's quickly becoming my second specialty (writing in the mathematics curriculum). In my individual presentation I'll be talking about the use of the "homework committees" and other structured peer-review exercises to encourage student self- and peer-assessment and self-authorship. Our panel will discuss the ways in which discipline-specific writing is taught, nurtured, and evaluated in the liberal arts setting. My portion of that program will invite non-mathematicians into the world of mathematical writing, indicating the similarities between math writing and writing in other disciplines. By highlighting the grammatical structures, syntactical rules, stylistic conventions, and assessment criteria that characterize mathematical writing and by comparing these aspects with corresponding aspects of writing elsewhere, I hope to dispel the notion that math writing must be an alien enterprise to non-mathematicians.

This is going to be an exciting conference.

Meanwhile I'm only a couple of days away from departing for San Diego, site of my fifth Joint Mathematical Meetings. A lot going on there (judging an UG poster session, presenting in the expander graphs and Ramanujan graphs special session, glad-handing every mother-lovin' person I can find to drum up support for my REU), but I'm already looking beyond it to May, bringing not only the Writing conference but also my next graph theory conference, to which I hope to drag a few students. (One of my freshpeople is making great progress on graceful labelings over this break! I told her to expect me to try to get her to go to this conference in May. More on that as events warrant...)

Problematic

The last couple of days (as the coming couple of weeks will also) have seen preparation for my courses this coming semester. While Calc II is something I could do with my eyes closed, I'm spending a bit more time in getting ready for Graph Theory.

I spent a few hours yesterday afternoon putting together the first problem sheet for that class.. It asks for a nice mix of examples and proofs, all interspersed throughout a series of definitions that flesh out the basics of graphs. I'm quite pleased with it. It's got a little more than one question per person currently registered for the course, so I hope that everyone will have a chance to get involved through this first set of exercises. I don't think it should take us more than the first week or so to get through it. I'm going to get started on the second sheet soon, but it won't be made public immediately; I have a hunch I'll want to make adjustments once I see how the first sheet goes over. I may, after all, find that I'm aiming too high, or too low, or expecting too much or too little...the students may want to take the class in a totally different direction. We'll see how it goes.

After tinkering with various ideas for that class's exam structure, I've decided to just play it by ear and ask the students to help me design the exams when the time comes: they'll help in the construction of the tests, and in their organization.

Tidbits: I'll also be asking each of the students to read and present on at least one mathematics research paper, and I'm toying with the idea of giving a small amount of extra credit for learning and using LaTeX in the preparation of written homework problems.

Besides this, nothing much to report, teaching-wise.

Further bulletins as events warrant.

Monday review of books

I've finished two books in the last twenty-four hours, including one that I capped off in a single three-hour sitting this afternoon: Jonathan Kozol's The night is dark and I am from from home (Simon and Schuster Inc., 1990), and Kurt Vonnegut's Mother Night (Dell 1999). Although neither directly relates to teaching at the university level, the former deals with Education with a capital 'E,' and in reading the latter on the heels of the first, I couldn't help but be impressed by dramatic differences between the two, as well as some deep parallels.

Both books I read just now by accident: Kozol's book I came across at an ongoing book sale in the basement of the Swannanoa Branch of the Buncombe County Library system this past Tuesday, and just yesterday afternoon when I accompanied Maggie in to keep her company during her short holiday weekend work Vonnegut's novel was on the top of the bin we pulled from the Pack Library's return kiosk. The title was fresh in mind, recently recommended by one of my students; it was one of the small few of Vonnegut's novels I'd not read before.

Odd that I should have read them so close in time to one another, so different the views they express, on the face of it.

Kozol's book paints a bold picture in absolutist strokes, uncompromising, often laced with ugliness and spite. The world he portrays is one in which unassailable good does battle daily with the purest of evil. The battle takes place throughout our society, our author assures us, but not surprisingly his focus rests on the nation's public schools. His thesis: "the first goal and primary function of the U.S. public school is not to educate good people, but good citizens. It is the function which we call -- in enemy nations -- 'state indoctrination' " (p. 1). To support this thesis he erects argument after argument, exposing for the reader the "straightforward lies" told by the educational establishment, laying bare the artfully hidden connections between the adversity felt by the nation's poor and the plenty enjoyed by its wealthy, and indicating the means by which teachers and administrators encourage the children in their charge to substitute passive concern for meaningful action. Certain themes recur, indicators of our nation's evil: My Lai (the book was written during the period from 1969 to 1975), Kent State, socioeconomic discrepancies in infant mortality rates, income opportunities, and access to medical care. Every aspect of our educational system, we're convinced, goes to supporting the perpetrators of these crimes.

Kozol's criticism is harsh, his tone cold and merciless. In fact, many of his accusations are simply unjust, and many of his comparisons inapt, insulting, and hurtful. Fortunately an older, wiser, Jonathan Kozol recognizes this fact, and for the 1990 edition of the book I just finished reading has written a set of critical annotations offering a rebuttal to the words of his younger, more hot-headed, self. "It was a time [1969-1975] when people who grew up to love their nation felt a sense of shock and shame. If, like myself, they had also staked their early years to try to bring about some serious social change, they were also likely to feel bitterness and rage. This much background may explain, though it does not excuse, the passages within this book in which I seem to speak about America not as a good but flawed society but as a land of unabated cruelty and evil" (pp. 18-19).

In his critical notes Kozol often decries his own words as "exaggeration," "abhorrent," and "reckless overstatements" (pp. 240-243).

I have to say that I agree.

While I agree also with the general thrust of many of his arguments, his tone is so continually demeaning that by the end of the book I felt a disgust which surely must have lost him many allies. Ultimately this is the problem with the book, the way it's written: though I too believe that the American educational system is deeply, deeply flawed (for many of the reasons Kozol highlights in this book, and for others reasons that didn't obtain when it was first written), I believe that through the sort of hectoring he does with The night is dark, he is likely to lose the support of those most ready and able to answer his call for change.

The chapter that struck closest to my mark, of course, was that titled "Colleges and Universities." Here Kozol sounds much like an acquaintance of mine in Illinois, who insisted on referring to the University of Illinois as "that little university down the street" on his spite-spewing radio program, all the while both he and his wife drew their paychecks from that same university. Here Kozol bars no holds: "the university is built on blood and nourished by injustice," and functions "both to sedate the lives and to protect the conscience of the university population, to insulate the college common and the paneled dining quarters of the college faculty and deans from either knowledge, memory or recognition of the pain, the description and the devastation of those tens of thousands who live just beyong their reach and recognition" (pp. 214-215). The claim that university faculty have difficult jobs to perform he calls the "Fiction of Hard Work" (p.215); the seersucker-wearing, lime-sherbet-slurping ivory-tower intellectuals with whom Kozol populates his college campuses (he has a thing for lime sherbet, mentioning it at least three times) are said to lead nearly effortless work lives. This would come as a surprise to me and to my colleagues (I can name several) who work 70-plus-hour weeks on a regular basis. I'm not claiming that we dig ditches for a living, I'm simply claiming that it takes a great deal of time and, yes, intellectual effort, to create meaningful coursework, serviceable assessment measures, significant learning experiences, and insightful evaluations and subsequent revision of our own work; to work with one another in planning and implementing truly impactful activities that involve not just the university but also the surrounding community we are chartered to serve; to craft original ideas and to incorporate those of others into our own, and to spread these ideas far and wide; and yes, to challenge our students to do more than say, but to do as well; to do more than care, but to act as well.

I found that chapter insulting, as did Kozol himself, in retrospect: "too many words like 'evil,' 'brutal,' 'fraud' demean these pages" (p. 251).

At the end of the day, I feel that Kozol's downfall is his insistence that in order to wash away the brand of "hypocrite" we must not simply put in a few hours each week at the homeless shelter, tutor math in our neighborhood elementary school, or work weekends for Habitat for Humanity; we must instead disavow all connection with the bloodthirsty machine that is the U.S. government and its corporate allies, we must pull our names from the university class rolls or quit our university posts, we must throw away all but the clothes on our backs and the staffs in our hands, shaking the sand from our sandals at any house where we are made unwelcome, making of ourselves martyrs for the cause of the have-nots, when often such martyrdom removes us from the positions in which we are most able to affect meaningful change. While I agree that too many people lead relatively comfortable lives blissfully unaware of the sometimes-causal connection between their own comfort and others' agony, I find ludicrous, for instance, Kozol's claim that he is hypocritical who, with access to quality medical care for his family, does not eschew that care available to him but instead brings his family to one of the understaffed and undersupplied clinics that service the nation's poor (pp. 204-205). Kozol's insistence on absolutist actions like this out-and-out boycott is the stuff of relatively immature extremism.

In final assessment, again, I find myself agreeing with Kozol in much that he says, but his message is lost in the brutal static of his medium. He loses much by demonizing not only his "enemies" (whoever they are; in sketching them his job is poorly done. As near as I can tell, his attitude is a Bush-like "yer either fer us or ag'in us" in which a nebulous Marxian elite plays the role of bogeyman), but also his allies. The world he paints is black and white, right and wrong, good and evil, and he makes it very clear on which side of each respective terminator he stands.

Perhaps not surprisingly, Vonnegut's world is a different one. The novel's anti-hero, Howard W. Campbell, Jr., served the Americans as a spy by adopting the role of a high-ranking Nazi official charged with promulgating anti-American propaganda. Throughout the novel he tells us of the pragmatism that kept him alive during the war and after it: to himself he ascribes neither guilt, nor a sense of loss, nor a loathing of death, nor heartbroken rage, nor a unlovability, nor a sense of the cruelty of God (pp. 231-232). He steals, lies, deceives, schemes, and somehow even when caught manages to purchase freedom by one means or another. His world is not black and white, but gray, a full-blown cloud of obscuring mist. In order to perform the greatest acts of espionage, he was forced to author and deliver the most hateful of anti-Semitic screed. Which outweighs the other, the sinner or the saint? Impelled by whatever situation he finds himself in, our hero eventually loses all sense of black or white and opts for suicide over freedom when freedom might lend him another opportunity to pick sides.

Still there are parallels between Kozol's work and Vonnegut's. Most notably, both admit that in order for an ordinary human to live with herself, she must adopt a certain degree of hypocrisy. Early on, Kozol quotes Tolstoy's The Kingdom of God is within you: "the men of the ruling classes -- the honest, good, clever men among them -- cannot help but suffer from these internal contradictions...We cannot pretend that we do not see the policeman who walks in front of the windows with a loaded revolver, defending us, while we eat our savoury dinner or view a new performance...We certainly know that if we shall finish eating our dinner, or seeing the latest drama, or having our fun at a ball, at the Christmas tree, at the skating, at the races, or at the chase, we do so only thanks to the bullet in the policeman's revolver and in the soldier's gun" (quoted on p. 62). Kozol limns a similar modern American myth: "the lives of children of the white and well-to-do within a land like ours exist upon a plateau of relaxed and innocent intent: one which turns at times, if we so wish, to passages of benefaction, at other times to academic labors, string quartets or summer garden parties, yet one which is at all times disaffiliated from the exploitation that it rests on, uncontaminated by the blood that nourishes the soil" (p. 63).

Compare this with a passage from p. 223 ff. of Vonnegut:

I have never seen a more sublime demonstration of the totalitarian mind, a mind which may be likened unto a system of gears whose teeth have been filed off at random...The dismaying thing about the classic totalitarian mind is that any given gear, though mutilated, will have at its circumference unbroken sequences of teeth that are immaculately maintained, that are exquisitely machined...The missing teeth, of course, are simple, obvious truths, truths available and comprehensible even to ten-year-olds, in most cases. The willful filing off of gear teeth, the willful doing without certain obvious pieces of information -- That was how a household as contradictory as one composed of Jones, Father Keeley, Vice-Bundesfuehrer Krapptauer, and the Black Fuehrer could exist in relative harmony --That was how my father-in-law could contain in one mind an indifference toward slave women and love for a blue vase --That was how Rudolf Hoess, Commandant of Auschwitz, could alternate over the loudspeakers of Auschwitz great music and calls for corpse-carriers --That was how Nazi Germany could sense no important differences between civilization and hydrophobia...

For Kozol, school is the file with which the teeth are removed.

Vonnegut also seconds Kozol's observation that in totalitarian societies there is safety in imitation and danger in originality (that, in fact, true originality can be punishable by death): "'What harm is there in writing what's already been written? Real originality is a capital crime, often calling for cruel and unusual punishment in advance of the coup de grace' " (Vonnegut, pp. 206-207). Vonnegut stops short of labeling our own society a totalitarian one, as Kozol does throughout his work.

I suppose we all, those of us well enough off to be writing or reading these words right now, must pull a bit of ethical legerdemain in order to sleep well in a world as rife with inequality as is our own, and I suppose we all must mask our true intentions from time to time in order to preserve the integrity of the society in which we live and work.

How much is too much?

And how, if the balance is upset, do we reset it?

I'm going to have to think some more about these books, and how they relate to what I do, with what tasks they charge me.

Until I next check in, I welcome my readers' insights: have any of you read these books? What do you think?

Promises, promises

At last!

Many apologies for the delay in getting around to posting on this topic, after all of the promises I’ve made. It’s taken a while to gather all the necessary permissions from my students (who shall remain anonymous in all of the following comments), the semester’s end brought the usual concomitant tumult, meeting after meeting with administration about this matter and that added extra delay, and the last couple of weeks have been devoted both to finishing up a number of loose ends and to enjoying a relatively unstructured, somewhat work-free languor.

I did want to share my thoughts (and my students’) on the Newton v. Leibniz project, since at least one of the two reenactments proved to be a very memorable learning experience for a number of the class’s students. Though there was certainly room for improvement, I feel that my management of the project was quite effective, and many of the students put a great deal of effort into making the activity a successful one.

Below I’ll highlight some of the successes of the project: what did the students take from it? What did they learn about math, about math history, about themselves and the way in which math plays a role in their lives? What did they take from the classroom reenactment itself? On the other hand, I’ll point out some of the project’s flaws, and ways various students have suggested the project be modified the next time it’s assigned.

First, I might say a few words about the character of the reenactments.

The first section’s trial reenactment was satisfactory, but comparatively lifeless, characterized by a lack of preparedness on the part of some of the trial’s principal parties (the teams comprising Newton and Leibniz themselves, along with their counsel, and the teams comprising the scientists’ friends and colleagues). Two of the, shall we agree to say, less modest students in the section had taken on the roles of Newton and Leibniz, and it was clear that they did not always take these roles seriously. (This levity was a source of distraction for at least two of these students' colleagues.)

The second section’s reenactment was positively splendid: it was lively, heated, authentic, and
characterized not only by preparedness and seriousness, but also sincere concern for the discovery of "truth." Nearly every member of each of the principal teams prepared meticulously, and this level of preparation was clearly on exhibit in the resulting trial, in which students engaged in excited debate. Those cast as friends of one party or another fit seamlessly into their parts, attorneys on either side deliberated feverishly in reaction to the other side’s statements and questions, objections tore through the air like shrapnel. The result was an electrifying activity, and a pleasurable one: several students expressed a wish that the trial could be extended to the following class period. Said one: "the execution of the trial was a blast and I would have been willing to stay an hour after class just to continue it."

What was it that held the first section back?

"I have to admit, I was disappointed by the trial," said one student. "I understand that it’s important to have fun with projects, but there’s a different between having fun and not taking it seriously. I feel like, in general, the groups of colleagues and the historical experts came to class prepared to present information, they collaborated, and took the project seriously. I was dismayed to find that the primary groups…were relatively unprepared, and spent a great deal of class time joking with one another."

One of this students’ colleagues in the class agreed: "The arguments and the overall project could have been given a heightened sense of importance…Coming from my personal thoughts and experiences working on this project, I thought it had the potential to develop into a really intense trial, maybe not to the caliber of the To Kill a Mockingbird trial scene, but pretty close. I thought that everyone loved to argue which would ignite a heated trial."

I later asked this student for advice on making the experience more worthwhile. He gave examples of a mock trial in which he’d taken part in high school, indicating techniques his high school teacher had used to make the most of her project. He explained how in studying Beowulf his English class had put Grendel on trial, and the teacher herself played an active part in the trial by assuming the role of Grendel. "She knew her stuff about it, being an English teach," the student indicated in an e-mail to me later, "this makes it hard on the team who is unprepared." I agreed that in this way the teacher can "call out" those who are underprepared. The disadvantage of this set-up is that the pivotal role of Grendel (comparable to that of either Leibniz or Newton in our class’s rendition) is then taken away from a student, who might gain much from being cast in that part. The relative merits of either side could be debated. I very much appreciate this idea, and will definitely consider it when running the activity again.

Others were discomfited by the trial experience itself, whether or not they felt it was a success: "overall, I like the idea of a mock trial, but I found this experience to be really stressful." After listing a litany of inconveniences and discomforts experienced during the lead-up to the trial, the student admits that "despite all of this, I kind of enjoyed it." One of his fellow students agrees, clearly a lot more comfortable with the activity itself: "the actual trial was the part I found most enjoyable, because I got to get outside of the person that I normally am and really attack the other team. Being an attorney ended up being more fun than I thought it would be."

What did they take from the experience, this first section?

A number of students indicated that they learned more about Leibniz than they did about Newton, given the relative obscurity of the former mathematician. Several students mentioned learning a good deal of mathematics and the history to go along with it. A few insights were more purely epistemological, dealing with the nature of mathematical knowledge and its acquisition than with the math itself: where does math come from? How does it arise? Who can lay claim to its discovery?

One student became more aware of the "social" aspect of mathematical discovery: "from this project I have learned that in math everything is a collaborative effort with [one’s] colleagues or predecessors, nothing is every truly invented on [one’s] own." Another student agreed with this assessment and reflected on the "human" provenance of mathematics: "if anything else, from this project I have taken away the important idea that a mysterious Calculus book wasn’t laid in someone’s lap. It took many years to discover and perfect." And no angels were they, the folks who cobbled calculus together. In the words of yet another student, "these [Newton’s] colleagues were not so innocent in their daily lives, and their moral drawbacks were made public, so as their future representatives we had to make Newton’s colleagues look trustworthy and knowledgeable. In all honesty I never realized such dramatic lives were led by the founders of mathematics and the sciences."

Others took from the trial more pragmatic insights, ones that might be incorporated usefully into other settings: "it was interesting to learn and expand my knowledge of Newton and Leibniz but what stuck out even more was the small details," says one student. "The seeming less significant aspect of the project was the most interesting to me. The most beneficial was the fact that this assignment involved, writing, mathematics, the overall use of language, communication, cooperation with group members. This was interesting because it was universal and very helpful in other subject areas…Thanks for incorporating more than just math into this project."

How did the second section experience the same assignment?

Overall, excitedly, and positively. I don’t believe I received a single negative comment from the second section. Not a one. The nearest thing to a negative comment came from a rather reticent student who knew herself well enough to anticipate her strengths and weaknesses: "this is not the kind of project that I like to express myself with since I’m more of a creative writing person. The poem that’s coming up will be better for me to express myself." (Indeed, she would do that; hers is one of the poems appearing in the second set of poems, in the previous post.)

On the other hand, most were thrilled by the trial experience: "with the three-second attention span of a goldfish, it really amazed me how focused I became on this project," said one. She continues: "I enjoyed working on this with my teammates, I had fun digging up facts during research, and, more than anything else, I loved picking apart the other [team’s] arguments. So many things about this project interested me, but one aspect has really stayed with me. I was inspired and charmed by the level of dedication that came out of the students, including myself, during this project…what amazed me was the dedication of the students during this project. Weeks of planning and research culminated into a wondrous thing."

Her colleagues agreed: "I think the whole approach was creative and stimulating," said one. Several indicated disbelief at first, but later conversion. Many made revelations, about math and its history, and about themselves, much like those uncovered by the first section’s students. Here’s a representative testimonial:

"When the Newton v. Leibniz project was assigned I was, in all honesty, unmotived and in no way looking forward to the trial date. The assignment somehow seemed to be kind of irrelevant and distracting to what we were actually learning in class. After the actual trial on Monday though, I cannot stress how wrong I had been…Math had seemed like one of those things that has just always been around. I never thought twice about the history behind Math because I had never had a need to do so. By doing the [trial] however, one seemingly obvious thing was made evident: someone somewhere had to actually derive these concepts and be the first to think of them. Before, I had just thought of math as math. I never took into account that there is more to it than that. Math isn’t just math; math is a conglomeration of ideas and concepts over time. A major part of math is in its history and to truly understand math, one cannot ignore its past…Math is a collection of evolved ideas over time, whose history is almost as important as its functions."

And a second:

"When I enrolled in calculus in order to fulfill my biology requirement, I expected that the semester would be full of long, grueling problems which would bring back painful memories of math classes long past. And so it has (although with a lot less stress than I had anticipated). But if you had told me at the beginning of the semester that I would be doing a project later in the semester that I would actually enjoy, I’d figure you were crazy. But I did enjoy Newton vs. Leibniz, probably for two reasons: one, there were no math computations involved; and two, I felt I could relate it to what I am interested [in] more than anything else we’ve done this semester."

And a third:

"Upon the receiving the details of the assignment, I would be lying if I say that I didn’t let out a few groans. First of all, it was a group project. Group projects have seemed to be the epitome of evil to me for the past several years of my life. I kept flashing back to previous group projects in high school where it seemed that I pulled most of the weight of the project, whether or not it was intentional…Second of all, the assignment seemed a bit silly. I had only briefly heard about Leibniz and I felt as if there was no point to argue the Leibniz side of the dispute. Obviously, I was a bit wrong…From this project, I have changed. I have changed my outlook on the idea of group projects. Everyone in my group was amazing, and they all did their part. I realized that I needed to stop being so paranoid and pessimistic [about] others in my groups. I actually like group projects a little bit, now, and it is nice to work and talk with others who I would have otherwise possibly never talked to."

Though this last student had a change of heart about working on a project as one member of a team, others had different experiences: "this was one of the most interesting projects I have ever participated in and not in a bad way. At the beginning of the project I was very [skeptical] and did not know what to [expect]. Even though I had a rough start, in the end I thoroughly enjoyed myself. I learned a lot about myself and what role I play in groups. This may be cliche, however, I truly did learn about myself and realized that I do not like working in a group at all, unless I can pick the members in my group and that does not always turn out great. I realize this is a big problem because for the most of my remaining life I will be expected to work in groups."

As in the first section, some students had much to say about how the project had lent them insight into the process through which mathematical knowledge is constructed: "it’s [mathematical discovery is] a constant building of ideas on top of each other, almost like a pyramid with each layer built being built upon a previous layer. Newton of Leibniz could not have done their work without the work of mathematicians such as Isaac Barrow, Barrow couldn’t have done his work without the influence of scientists before him." As this student indicates, this "influence" causes difficulties when it comes time to ascribe credit: "The idea of creating new concepts also made my group and myself question what it means to actually invent something…Whoever created it first gets the credit, right? Well, then I started thinking, the more complex something is the more parts there are, the more parts there are the more other people are involved in the process."

Assigning credit is made more difficult by the fact that real people were involved, a fact brought to the fore by the students’ research during the project. "We often think of scientists as being ‘morally superior’ to politicians in that they are interested in obtaining the facts," says one student, who continues, "while I am sure this is true to an extent, scientists want fame and success just as much as politicians want power…In researching Newton vs. Leibniz, I was reminded that scientists are only human." This student was able to relate his experience in his section’s trial to his leisure reading: "It’s reinvigorated my interest in reading the last two books of the Baroque Cycle [a series of historical fiction novels], the last of which specifically deals with the calculus controversy. I can’t wait to see how Neal Stephenson portrays Newton: embittered, jealous, or sincere in that an injustice was done? In either case, I know it will be an entertaining read."

Others were able to make similar connections to their interests outside of class. Our friend with the three-second "goldfish" attention span hopes to recycle the courtroom concept for her own classes once she begins her career as a teacher: "as impressed as I was with the level of participation during the trial, I began to formulate a way I could use the idea of a trial in my own classroom. After reading a little of Change of Basis, I realized that this was not a strict model and could be applied across different disciplines. It was reassuring to know that while the process and student involvement stayed with me, it also stayed with my professor before me."

Perhaps the most ecstatic comments came from a student who had an epiphany concerning his own approach to mastery of mathematics: "for me it is much easier to understand mathematics when I say what I am writing down, in verbal form in my head," he explains. "I have always done this and before I had no idea why. Now that I have reasoning and have explored the subject, I have a better understanding of how to teach myself and be a good student of Calculus and the broader field of Mathematics. As a person that does not make breakthroughs like that on a regular basis, I was floored to learn more of how I learn. All due to this project!…If I were you I would keep assigning this project to Calc I classes."

Never fear, I believe I shall.

While the first section’s experience of the reenactment of Newton v. Leibniz offered mixed results, I was overwhelmed by the energy and alacrity with which the second section embraced the project, and gratified by their positive feedback. In all honesty the project was a bitch of a row to hoe, but it’s clear to me that the harvest was a rich enough one to warrant sowing these seeds again. Made wiser by the suggestions my students have offered me, and simply by the experience itself, I’m sure the next installment will be more successful than the last. I think I can safely say I’ve put together a significant learning experience with this trial.

And with this post, I’ve fulfilled a promise I’ve now made for several weeks.

I hope to post again soon with comments on my upcoming courses (choose-your-own-adventure testing in Graph Theory), on my teaching-related Winter Break leisure reading (Jonathan Kozol’s vitriolic The night is dark and I am far from home), and so forth.

Now, however, the night is indeed dark, and I should soon away to bed.

Round two

I've got one more meeting in an hour and a half, and commencement tomorrow...but then everything...classes, projects, meetings, grading...EVERYTHING...is over for the semester.

Give me a week, and I'll be bored out of my ever-loving skull, itchin' to get back into the classroom.

The last couple of days have been a whirlwind of excitement (grant-related activity, proposal-writing, this meeting and that, toying with various research ideas), and I'm looking forward to starting a few weeks of mercifully highly unstructured busyness.

For the moment I've got nothing to say, really, but I thought I'd post the second of what will be three rounds of student-authored math-themed poetry.

I should say up front that I've received permission from all of the students from whom permission was sought to post their writings; the only two poems I'm not including in this post are rich concrete constructions in which the pagination plays a very meaningful role. I'll have to see if I can render the effect in Blogger, I might have to settle for obtaining the original MS Word files from the students themselves and posting links to those.

For now, I give you

Mathematical Poetry, volume 2

Let's begin with a simple haiku, courtesy of Mark. Mark indicates that he felt justified in forgoing the tradition in American haiku of making mention of a season, since his poem deals intimately with the theme of Nature:

Nature is Chaos, by Mark

Nature is Chaos,
Math brings order to Chaos,
Math Quantifies life

***

Cory's poem deals with the sweet taste of victory, written after Leibniz had been exculpated in his section's reenactment of Newton v. Leibniz; Cory was one of several of his classmates who described the experience as an immensely positive one:

Newton v. Leibniz, by Cory

It started with a proposition
Which led my team to the path less traveled
Whitney, Cory, Ben, and Justin...Leibniz and his colleagues
We were doomed from the start
Newton this, Newton that, all in favor say Newton
With the invention of calculus at hand
We had to bring it home for the one and only Leibniz
As we ravaged the library and surfed the web
Little by little it pieced together
Closer and closer victory was ours
As underdogs we had no choice but to be prepared like no other has ever prepared
Days grew long
Nights became dawn
And before we knew it the hourglass was empty
The epic trial of Newton v. Leibniz was here
In only minutes we could see that we were better prepared
Minutes passed and minutes left before the deciding factors
As the trial came to an end Leibniz and his colleagues left the room with their heads held high
And a sense of pride lingered in their step
BAM
It was here
The verdict
Dead quiet
Leibniz found not guilty
Therefore he published first
And was crowned the king of calculus
It came with no surprise to Leibniz and his colleagues
WE WERE PREPARED FOR GLORY
AND NOTHING LESS!

***

The following is the only e-based poem I received, strangely enough. I love the way the author has made use of the fact that e's first several digits alternate between 1s/2s and 8s/9s: this alternation leads to a series of quick exclamations followed by more prolix elaboration.

e, by Anonymous

Great Calculus.
.
Can put to use in everyday situations.
Crazy.
Challenging yet the only subject that is constant.
Mind boggling.
Sometimes you want to pull your hair out.
Satisfying.
Confusing/makes perfect sense at the same time.
Intricate simplicity.
If used can solve many problems we face.

***

Ben's work, below, paints an idealized picture of the way the world might look through a mathematician's eyes. I'm not sure exactly what's meant by the cryptic closing lines.

mathematician, by Ben

late nights and classroom lights
chalk board dust and old book must
soda redbull coffee and tea
oh! the world the mathematician sees

to take heed - a desire planted so deep

to justly discover and describe
the fine natural truths held inside
deep blue seas, big old trees. its
size, shape, speed, that he sees

joy and fulfillment abound
beautiful! the answer is found
and my dear! the truth will set you free
oh! i long to see the world that he sees

but as his time ends, there is one regret -

his curiosity reigns supreme
and learning all is but a dream

some regret a dollar lost
or an unforeseen cost

but as it comes fro'
his time to go

things unknown
leave him
'lone

the world the mathematician sees
is the world the mathematician must leave

this is the pain the mathematician knows

***

I felt the next poem was one of the strongest submitted, I've mentioned it in a previous post as its author was working it out in its preliminary form. Its theme is one of the most personal I encountered as well. Stylistically, the author expresses an affinity for the chiastic form the lines took on the page, and, not surprisingly, professes to be a fan of alliteration!

Frustrated, by Anonymous

It used to come so easy.
Never being challenged or troubled
Always loving the beauty and complexity of it,
Now getting bogged down in the cumbersome intricacies,
Confused not knowing how to help myself,
Frustrated with the fucking functions,
Wanting to get back to the beauty,
Seeking guidance.

***

Elise's piece was a particular clever one. Wanting to work with the Fibonacci sequence, she soon came up with the idea of laying out a flower garden, in which the nth flower named not only occurs Fn times, but also possesses Fn petals in its floral structure!

I think this poem would make a lovely children's book, if we could find the right illustrator.

My Fibonacci Flower Garden, by Elise

In my garden I have many flowers
The bloom elegantly after May day showers.

Here, I must bring you on a tour
Unless you've seen it all before.

It first starts off with a single white calla lily
That a friend gave me named Billy

Then next is a single pitcher plant
This is home to some ants.

After that is two euphorbia flowers
Couldn't you just look at them for hours?

Then there are three trilliums
Don't tell, I picked them from a mausoleum!

Then come five columbines
I swear, they bloom all the time.

And eight bloodroots
Aren't they just so cute?

Next to those are thirteen black-eyed Susan
Susan shouldn't have been cruisin' for a bruisin'.

And twenty-one asters
They seem to grow faster.

Then last but not least, thirty-four field daisies
I hope I don't seem crazy.

Now that you have seen my garden
Please beg your pardon.

It is now time for you to go
It looks like I have to put on another show.

***

Nathan's pi-based poem used the number's decimal places to count off the syllables occurring in each line, rather than the words. The effect is to render each line tighter, more succinct. What results is a geometric tour involving everyone's favorite constant:

Wonderful Pi, by Nathan

Beautiful.
Ace.
Vast bottomless
Pi.
Without existence
Not a Circle would ever compute.
Golden.
True value never known
Yet still you exist.
Essential.
Impossible, no
Inside an orbicular world
Your escort, radius, assists
finding area and volume
Infinite. Great. Infinitely great.
You, my friend,
Are pi.

***

I love this next poem, an ode on an abbreviation I use frequently when grading: "w.b.c" stands for "wrong, but consistent," and arises whenever a student makes use of an incorrect calculation but does so in the correct manner, resulting in a solution which is wrong, but consistent with an incorrect intermediate value. "W.b.c" is good, because generally it means the student's got the right idea, but just made a minor boo-boo in her computations. Partial credit-wise, "w.b.c." might mean only a point or two off.

Clearly our anonymous author has been haunted by this abbreviation:

Untitled #1, by Anonymous

Progress.
I stare and stare at the door
Feeling the approach of a familiar stranger
(Like pointless anticipation
Of an unnamed and unknown event)

Lingering.
On the very brink of my thoughts
The solution totters
Awaiting enlightenment (as a form of escape)
From the chamber of my mind

Disappointment.
Wrong but consistent.
Partial credit.

***

Sarah's poem, much like Sam's (see the previous post!), poses a mathematical puzzle embedded in its lines. In addition to performing this feat of numerical legerdemain, she based the structure of her poem on the sequence of prime numbers, the only student to do so.

Magical number, by Sarah

Magical number
Add twenty-four
Then multiply me by two
Half of my last digit is one
Eventually there will be a number that is right for me
Multiply my very first digit by six then subtract it all from me
Average my two digits and you will find a golf term
Triple me to see consecutive numbers
I am only one digit
Count backward to
See me!

***

In unpoetical news, I've just come back from the final faculty meeting of the calendar year, at which our fearless Learning Circle leader delivered directly into my hands the text for the LC in which I will be participating: Learning partnerships: theory and models of practice to educate for self-authorship, by Marcia Baxter Magolda and Patricia M. King. It should prove an interesting follow-up to Meszaros's book from this past semester.

This is the book we should have read first, perhaps?

Allow me now to end this post and set to work reading the paper I've been asked to referee for the Journal of Combinatorial Mathematics and Combinatorial Computing. (Say that one three times really quickly...)

I sing the verses eclectic

I've just been granted leave to share a number of my students' poems with my readership! Much thanks to the students who have gotten back to me and let me put their work on display, both those of you who'd rather remain anonymous, and those who'd like their names plastered proudly on their work!

In no particular order, then, I give you...

Mathematical Poetry, volume 1

The first I pulled from the stack at random, a little ditty by Sam, features a hidden message (read every seventh letter starting in the right position, and you'll find it...I cheated and typed the text into Mathematica to get it to paste together every seven letters):

Though It's Early, I'll Be There Surely!, by Sam

My, the amazed small one
At hardish attention will surmount
A test ahead
By reading a bunch, in easy bed stead

At this you can be delighted
As we, farsighted
Show our faces
In our favorite old "Man, is it eight?" places

A math class adored
I named thy award
It's found in my 4th
My paper explains

***

This next one is a stroke of parodical genius...I hope Lisette didn't spend too much time working on this send-up. I have a hunch her background includes a measure or two of Shakespeare (and her nights a good deal of euphoria-inducing caffeine):

Mathbeth, by Lisette

Act lim_(x -> 1+) 2x + 2, Scene lim_(x -> infinity) x^(1/x)

Student: By the scratching of my pencil
The answer slowly this way comes.
Open, mind,
the catacombs
Of transcendental sums.
But hark! Look you how the utensil moves!
(Enter Pencil Stage Right)
Pencil: How now, caffeine-riddled youth
What is it you do?
Student: A quest without an end.
Pencil: Horrors! Not another, fiend!
Student: Yet by your yellow wood I swear
'Twas not by my own choice!
Pencil: However the task was first derived,
To a veteran of your caravans,
Make good this oath with eye and voice:
Look to the ink of stalwart pens
In your aimless waste if parchment.
For never will your proofs amend
Those errors in your quotient.
Day upon day you dulled my lead
As ere you chased the numbers 'round.
So many times I thought, perhaps
You'd finally reached your limit,
Then watched my world shake upside-down
To briskly hide your mishaps.
I grew quite bald from misadventures
With wild domains and vicious powers
No more! I say again, No more!
My lead is soft, my wood is fragile.
Find some youth with a liquid core
And a shiny plastic shell.
We of wooden constitution
Have failed our last equation.
(Exeunt Pencil)
Student: So I, alone
But for my mug of coffee black,
Must weary waste the night away
With a knavish rogue called BIC.
(Flourish, Exeunt)

***

Farrah's oeuvre was one of the finest of the pi-based poems I received. The imagery (as much olfactory as visual) that it conjures up is magnificent, and the ironical twist at the end is superb. In her words, "I chose the topic of this poem because I was extremely hungry...then my thoughts on food and frustration led me to think about Pi which is delicious food that make my apartment smell like pastry and fruit and it is a pretty neat number...My favorite thing about the poem is the shape the poem made when centered on the page."

I hope the Blogger's pagination will render a similar effect, Farrah!:

Motivation for a Sweet Tooth, by Farrah

Broccoli carrots kale
.
Chopped
Garlic onion ginger tamari
Tofu
Cooked in a hot wok
Delicious food fast from my two burner hot plate
Aroma fills
My tiny apartment for many days
One room living makes for
A constant smell
Garlic permeating the whole place
Maybe I should have made some delicious Pi

***

This next one is one of a handful of personal professions of mathematical affinity that I received. (I'm happy that some of my students are willing to own up to a love for mathematics!) Carrie's not the only one to speak of a certain "balance" or beauty in math, as you'll see below.

For Me, by Carrie

Some see it as confusion
Something messy, some ugly
Something they are forced to do.
I am not these
I see patterns everywhere
A balance here that is seldom elsewhere
There are answers, solutions
And it's not hard
For me

But I talk to different people
They avoid it
Keep it at a distance,
Far far away
Higher levels
Fewer people
People think that you are crazy
It's still not that hard
For me

Everyone has things they are good at
Some have writing
Some have art and music
Some will go on to speak to the masses
These are the things that I find hard
These are what I avoid
Just give me a math problem
Because it's not hard
For me

***

Our next piece is another deeply personal one, a reflection on the author's feelings about not only mathematics but the way in which it is encountered. After it was written, I talked with the author about the fact that every line begins with an 'I': this pattern was accidental at first, but once noticed became an intentional goal that was harder and harder to achieve with each line. I find that pattern interesting, given the self-focus of the poem:

Imperfect, by Anonymous

I am ready to be challenged
Ironically this is the first thing which has been problematic
It is difficult for me to be patient and understanding with others
Irritation seems to be my main state of emotion
Intricate concepts are something I crave
Implicit differentiation as surprisingly stimulating
I fear my arrogance will be my downfall
I am ready to be challenged with others who feel the same way

***

I've selected a few of my favorite haikus by this next author, who turned in no fewer than ten of them! The slightly humorous, yet still insightful, tone I found in these poems was delightful. The author has asked to remain anonymous.

Confusion

A tall ladder falls
At twenty feet per second
Why would it do this?

The Number "Pi"

Three point one four one
five nine two six five three five
eight nine seven nine...

Math in Daily Life

Patterns on my bunks,
They resemble the graphs of
cosine and sine curves.

The Inventors

Did Leibniz invent
calculus or did he steal
the work of Newton?

My thoughts about math

Sometimes I like math
more than art; veggies and fruits
are boring to draw.

***

Here's another admission of fondness for mathematics; I have to admit that I feel much as the anonymous author does.

Math, by Anonymous

Math
To put into poetic verse
Seems quite contradictory
Free form verse
Clashes the firm ways of
Math
To put into words
Seems rather absurd
Verbs and nouns
Make no substitution
For the numbers and functions of
Math
Does not need words to explain
It stands on its own
Its equations speak their own verses of poetry
Numbers create their own images
In forms of graphs and shapes
Ideas are not conveyed through rhymes,
Or rhythms,
But through logic
And reason
Poetry is not needed to show the beauty of math
Its charm is found beneath its theories, proofs and functions.

***

Mark's pi-based piece, below, involves not only mathematics but biology as well, giving a brief natural history, and perhaps a glimpse of the future. His work was heavily informed by his view of the universality of mathematics:

The Universe to 44 decimal places, by Mark

Hydrogen and oxygen,
Bonding,
Create habitat for life.
Abruptly:
Orderless masses become perfect spheres
These fragments yield amino acids within fermenting, fruitful seas,
Beneath a
Young light and even younger sky.
Molecules converge as RNA grows,
Followed by DNA.
Double helixes entwined with data,
Data that unleashes its complexity upon every duplication.
Cells are born, and form intricate relationships of survival,
Synergies of molecules endure harsh primordial seas.
Mitochondria and bacteria converge, becoming what both couldn't alone.
Fins dominate waves,
Eyes guiding,
Gills supporting all,
Including gaping jaws, merciless to those who're obsolete.
But the sun touches
Not only the sea, but the land
As well.
The sun reaches leaves, creating new opportunities,
As a toughened foot blemishes
The smooth sand.
Tethered to water,
A bind that only the amniotic egg cuts.
A fiery rock flies,
Bringing chaos.
An instant unravels an eternity of adaptation.
Hairy hands grasp wooden clubs, utilizing new-found digits.
Musket drawn, soldiers hunt shadows.

The moon:
A pearl-white foot disturbs dust, an eternal footprint.
Generations later, another foot climbs a sanguine summit,
Its owner re-imagining humanity.
Europa:
Ice-cold, alien, yet sharing same miraculous elements:
Hydrogen and oxygen, yet yielding entirely different organisms.
Sol
Keeps all of this life alive.
Precious, yet inconsequential, a speck in a milky-white galaxy,
Itself very insignificant,
One galaxy amongst countless, but all under one universe.

***

My final poem for this post (I've got a bunch more for which I'm waiting for permission to post) was one of the strongest of those submitted verses that worked off of the Fibonacci sequence. I love the way John makes use of the sequence's exponential growth to allow each line to elaborate further on the previous ones, until the final line is reached, where it seems a thematic turnaround is made. (I apologize for the decreasing font size in the final line; I've done this to try to accommodate the line on a single line of the screen...I think it's gonna fail anyway!)

Math, by John

Math
Math?
Math is...
Math is everything;
The nature of everything is math.
The dynamics of today still follow original patterns.
And all the patterns in life and nature are based upon a formula.
But I wonder: do the laws of nature dictate these math formulas, or does nature follow universal laws created by math?

***

That's all for now, folks! I'm glad that I've been able to fulfill one of my promises. Later this week I'll post some more poems, and I'll get around to dissecting that Newton v. Leibniz thingamajobber.

For now, please have a pleasant eve.