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...)