Bounce

In a recent post I fretted a bit about one of my sections of Calc III, which section seemed to me a bit underprepared for class this past Monday. I worried that their apparent lackadaisicalness (if that is indeed a word) regarding Problem Set 4, on which we were working in class on Monday would lead them to be unready for today's class, in which they would be presenting their solutions.

I stand corrected. That section bounced back, showing themselves up to the challenge. Every single student called on to present did so, and did so with aplomb. I was particularly impressed by Dionne's willingness to work all of the way through the dreaded #61, which asked for a proof that two non-parallel vectors in the plane span the entire plane. Dionne, one of our promising young majors, has some exposure to linear algebra and is currently enrolled in Foundations, so she's no stranger to the proof genre. With a little help from a couple of her colleagues, she beasted that problem.

Yes, they bounced back, but not before I exhorted them to keep up with their work outside of class. Don't just come ready for the problem you think you'll be presenting (padded with the one or two preceding problems for insurance); come ready to present any one of them...and ready yourself as soon as you can so that when you're offered time in class to hash out the details, you can do so without delay.

Good work, everyone! I have to admit to a bit of nervousness at running my first Moore-method class in four or five years, but so far you're all making the most of it. Thank you for that, and for all that you do.

Feedback, as ever, is appreciated.

Fishbowl

Today I attended the first of several meetings of the faculty Learning Circle in which I'm taking part this semester, on Parker J. Palmer's and Arthur Zajonc's The heart of higher education: a call to renewal (transforming the academy through collegial conversation), and hot on the heels of this we held the first meeting of the Curricular Sustainability Subgroup of CRTF. The former meeting prepared me well for the latter: big talk and deep thoughts about integrative pedagogy helped inform the reflection my colleagues and I did on the general education programs at several of our peer institutions.

Most such programs differed from ours in one or two fundamental ways: either they comprised significantly fewer required courses (no more than a handful of fundamental classes scattered across the curriculum, generally a few to each major division, and maybe including a writing course or a math requirement) or they placed more stringent demands on the student but offered her a multiplicity of ways of meeting those demands (with copious menus of courses to meet requirements for history, humanities, or "world cultures" requirements).

What could go wrong? Both designs assume students (and faculty!) can pack a great deal of interdisciplinary engagement into a broad array of courses. This assumption seems like a risky one at schools like Wesleyan College (Connecticut), where students must complete only 9 general education courses, 3 from each division, all with different departmental prefixes, and hundreds of courses can count toward the gen ed requirement...wouldn't an optimal interdisciplinary experience require the instructors in all of relevant courses to work in concert to help students realize this experience? Honestly, I'm not sanguine about the ability of an entire faculty to coordinate such a widespread effort.

I wonder, though: if we were to balk at implementing a curriculum like this, would it be out of fear of handing the reins over to our students? Maybe as we contemplate changes to our own ILS program, we should consider that our students may be less risk-averse, more daring, and more willing to try new things than we are: they might not need as much help as we think they do as they muck about in their meaning-making.

In the Learning Circle several of us noted how our students can thrive on the chaos and disorder that ensues when we dare to step away from center stage. Though the exercises we plan for our students might not lead them where we expected them to, they cannot be labeled "failures," so long as they tell us something about ourselves.

I was reminded of a particular meeting of the MATH 280 course I taught a few years back (in Fall 2007, I think?). It was a small section (15 or so students), and full of bold and fearless thinkers. One day toward the end of the semester our conversation meandered from the preplanned course when one student, Quincy, asked a pointed question about modular arithmetic. Rather than finish up the exercises I'd planned for us, we wandered off (all of us, as a class, one student, and then another, and then another, suggesting steps to take) in search of patterns that would help devise solutions to very general equations. The end of class came on quickly (time flies when you're having fun), but not before every one of us in the room (including me) had a far better understanding of modular arithmetic than we'd had half an hour before.

"The Math Department should run a 'fishbowl' course," Quincy said as we walked back to Robinson Hall from our classroom in the basement of Karpen. "At the beginning of the semester, we'd fill a few dozen scraps of paper with brief notes about various topics in mathematics, and at the end of every class someone would pull a scrap at random. Whatever topic was on that scrap would be the basis for the next class discussion, a free-form exploration of that topic."

Though this sort of course would be messy, hard-to-manage, and perhaps occasionally fruitless, to this day I still think it sounds like a hell of a lot of fun.

More to come on the Learning Circle and CRTF...and not so long now before this year's CWPA meets up in the wilds of Wildacres, just yards from the Blue Ridge Parkway. Stay tuned!

MATH 280 exams, revisited

A couple of posts back, I reprinted the final question on my MATH 280 students' final exam. In asking them to reflect on the course and find something worth their whiles in all that we've done together this semester, I had hoped that the students would fulfill several beneficial purposes:

1. They would notice connections and correspondences between disparate ideas we've studied, which they didn't notice before.

2. They would take the time to generate some questions of their own about the course ideas more intriguing to them.

3. They would get a bit more experience in communicating mathematical ideas in relatively informal (and non-technical) terminology. (After all, I wasn't asking them for proofs, just conversation.)

4. They would gain a sense of accomplishment in recognizing just how much we've studied this semester.

5. They would give me guidance for designing future iterations of this course by indicating to me which topics they found most interesting, most challenging, and most relevant.

The students did all of those things, and in composing a page or two in response to each of their responses to this question, I've been able to extend our conversation about ideas mathematical well beyond the time frame of the course. The semester's now over for this class, but I have a hunch that a good number of the students will spend some time over the summer following up on some of the ideas they found most interesting about the course.

I am definitely going to include this kind of exercise on every final exam from now on.

We'll see in a couple of days how my Calc II students' final portfolios come together: will it see a similar success, or a more meager one? Stay tuned...

...incidentally, my next post on this blog will be my 500th. After nearly five years of blogging on my teaching, it's come to this. If you're a regular reader (or even just a casual one), please let me know in the comments: what would you like to see me blog about in #500? And what would you like me to make of this space in the next 500 posts? Do tell!

One more vote for confidence

The second-to-last question on my MATH 280 students' final exam, designed, as was the last question, to get the students to perform a bit of reflective self-analysis, reads:

Go back over all of the proofs you've written in our class this semester, and select the single proof which you believe to be the clearest, the most complete, and most correct...that is to say, the BEST...of all of the proofs you've written this term. You may consider proofs in homework assignments, exams, or textbook chapters. Write a couple of paragraphs explaining why you selected the proof you have. Why is it that this proof represents the best you've got?

I'm currently grading this question, and I'm gratified to find that a very large number of the students are selecting the proofs that they are because those particular proofs were the ones which helped instill confidence in their abilities to do math, and to complete proofs. To me this fact underscores the importance of confidence and other affective aspects of learning. Kudos to my colleagues who work as hard at developing students' affective engagement as they do at ensuring cognitive mastery!

Lovely LaTeX

Though it'll be a few days before my MATH 280 students submit their final take-home exams to me, I'm already excited about their responses to the last problem on the exam, a very open-ended one which reads as follows:

Go back over all of the topics we've talked about in our class this semester, and select one about which you'd like to know more. (Please try to make your topic a relatively narrow one..."binomial coefficients" would be a good choice, for instance, whereas "combinatorics" would be a bit broad, and "proofs" would be faaaaaaaar too general.) Would you like to see more examples of a particular topic, or further applications of it? Would you like to know more about where it's used in mathematics, or how it connects with other topics we've considered? Do you have a specific question about it, or are you simply interested in learning more about it in general?

Having chosen a topic which you find sufficiently intriguing, write a few paragraphs about that topic. (As always, please use complete, grammatically correct sentences!) Your discussion should indicate clearly why you find your topic intriguing and what about it you'd like to know more about. If you have specific questions, please feel free to raise them. If you have answers to those questions, even conjectural ones, please feel free to share those as well! My goal here is not to get you to answer questions so much as to ask them.

One of my students caught me in the hall a half hour ago and asked if "LaTeX" would be a valid topic for this question. I only hesitated a moment before responding with alacrity: "that would be awesome!" I'd really like to learn more about how students make use of LaTeX, and if they feel it helps them (a) express themselves mathematically with greater effectiveness, (b) present themselves as authentic creators of mathematics, and (c) engage mathematical ideas at a higher cognitive (or metacognitive) level.

To this last point: I hypothesize that in using LaTeX a student is forced to include an additional reflective/analytical stage in her writing process, at which stage the student more carefully than before or otherwise scrutinizes word choice, notation, and other aspects of her writing. I'd love to study this more carefully...maybe some of my Charleston posse would like to take this on with me as a side task. Any other takers?

Wrapping up

The end is near! (I realize I've been saying this a lot lately, but it's more and more true each day.) We've got just under a week of classes left, with only one more meeting of MATH 480 and three each of Calc II and 280.

My last "regular" meeting of MATH 179 was this morning. We dedicated the day to a free-wheeling discussion of the past semester, focusing on the three central purposes of the class: (1) learn how math is experienced from place to place and across time, (2) learn a bit about a liberal education in general and UNCA in particular, and (3) get some practice in writing academically in a specific discipline, and for specific purposes.

Students pointed out some of the course's strengths: it helped fine-tune their writing skills, gave them a broader view of mathematics, and offered a forum where they could talk about random school-related matters with students in similar situations without fear of looking like fools. They pointed out some rough spots, too: the first text we used (by Ascher) was awful, and the 9:25 time slot wasn't the most conducive to student excitement. A couple of students indicated that I should teach the course again, and I suspect that the next time around it'll go much more smoothly. The students' take-home final exam, due in a week or so, asks them to reflect more intentionally on the three points I mentioned above; I'm sure I'll get from it an even greater sense of the course's strengths and weaknesses.

The MATH 280 students get their last exam tomorrow, too, and like the last it'll be a collaborative one. I was tremendously gratified by the students' reception of the last exam: the learning environment the students created and cultivated as they worked on the exam was a rich and fertile one. I have no doubt that many of the students, particularly those who may struggle more mightily than their quicker peers, learned more from working with one another than they would have from any other form of assessment. (Incidentally, I've been keeping track of collaboration networks on the collaborative exams I've given over the past couple of semesters, and I hope to begin tracking the data more carefully to try to analyze correlations between indices of collaboration like number of collaborators and measures of success like initial exam scores.)

The last 280 exam asks students to provide a few more "traditional" proofs, but, like its counterpart in MATH 179, it also asks the students to reflect on the course and identify specific topics they found interesting and write at length about those. My hope is that the exam will help me to discern which areas students are most interested in, and about which they feel most "iffy." This'll help me plan the course more appropriately in the future.

Speaking of course-planning...I've already got some ideas in mind (involving multiplication tables) for first-day activities for MATH 461 (Abstract Algebra I) this coming fall. I'm excited! I know it's several months off yet, but I'm looking forward to it already.

Before I go, I have to give a shout-out, to my colleague Kelli and to our first-year junior Kamryn, for spearheading the successful founding of a UNC Asheville chapter of the Association for Women in Mathematics (AWM): wonderful first meeting! I look forward to seeing what activities the students can put together in the coming months.

P.S. -- I sent the first draft of my manuscript to the publisher today. Much happiness.

In other words

Every now and then a student says something that makes me beam with pride. In my last post I asked my current students (particular those in MATH 280 and Calc II) to write and let me know what they needed to help finish out the semester strong. A large number of them responded, offering not only ideas for end-of-semester activities, but pep talks for one another and advice on helping each other across the finish line.

I wanted to pull a rather lengthy excerpt from one student's response that I thought did a better job of explaining what learning...and, in fact, doing...math is all about than anything I've ever said.

In this student's words,

Before college-level math, there was no wrong-to-be-right for a lot of students (myself included). A concept was explained and then you applied it and moved on. Right is right and wrong is wrong.

Especially before this class I would have found it very hard to believe that I could spend 6 hours on a problem, and that 5 of them would be spent barking up the wrong tree. I then would have found it near-impossible to believe that those five hours were vital to the process.

As far as the group work goes, it helped a lot when every time I found myself up a wrong tree, there seemed to be another student or two up there with me, searching for some sort of elusive coconuts. We then climbed down together and gave each other a boost up the next (for better or worse!).

The upshot: don't be afraid to be wrong; you'll almost certainly get it wrong before you get it right, and there's no consequence for going down a dead-end street. I could not have said it better myself. Literally.

I get to hang out with these people all day, every day? I'm a lucky guy!

Home stretch

We've got two weeks of classes to go, including 2 meetings of MATH 480, 4 meetings of Ethnomathematics, 6 of MATH 280, and 8 of Calc II. I can't believe we're here already! This semester's gone faster than any I can recall, and it's left us all stressed and strained.

This past week's conference gave me a much-needed break, and I honestly feel much more on top of things now than I did a week ago. I felt much more relaxed going into both of my classes today, especially given that I'm feeling pretty good about where my Calc II and 280 classes are right now.

As this semester's been so dramatically abbreviated (it's almost two weeks shorter than spring semester of last year, for instance, after snow days are factored in), I've spent no small amount of time this term stressing out over how much I'd be able to "get through" in my classes. I fear MATH 280's suffered a little bit because of this, and I'm going to have to trim back on the "special topics" I like to include at the end, just to get the bare minimum in. Ditto Calc II: bye-bye differential equations, hello Taylor series; the latter I deem indispensable, the former not so much.

Despite the frustration I've felt at gettin' it done, so to speak, I'm feeling better about where we are. I think at this point I've just got an (pardon my language) "ah-fuck-it" attitude about the term. By that I don't mean that I'm cutting the students loose and letting them drift in the wind; nothing could be further from the truth. In fact, I'm relaxing my expectations for the courses and focusing on making sure every last student understands fully every last thing we work on together in these closing weeks. If that means we have to cut a quiz here or modify an assignment there, so be it. I want us all on the same page as we cross the finish line.

Students: if you're in one of my classes and you're reading this, you can help me out immensely by responding in the comments to this post. Please take five minutes to write back to me there, indicating one thing you'd like to see happen in your course before the semester's up that'll help you make the most out of our remaining time together. Tell me which class you're in (anonymously, if you'd prefer), and let me know what one thing we can do to help each other across the finish line.

Shall we dance?

Shall we dance?
(for my Spring 2011 MATH 280 class)

We stumble, falling from step to step, counting each one
out loud like teenage boys working out their first waltz.
We are thankful that the lights are low, and that no one but the chaperone
can see us trod on our partners’ toes.

Yet not long ago we struggled just to stand. Now we’re hovering
at the gym floor’s edge, nervous fingers tugging slack into our neckties
and fretting at the seams of strapless gowns. “Omigod he’s coming over!”
Don’t just stand there! Shall we dance?

Whelmed

The end is near. I can almost see the end of the semester from where I stand. And, unlike many of my colleagues, friends, and students, the rest of my semester should be relatively unbusy (compared to the past month or so) after this coming week. I no longer feel overwhelmed; I'm simply...whelmed. However, Webster's Free On-Line Dictionary lists "whelmed" as a synonym for "overwhelmed," so maybe that's not accurate.

The past few weeks I've felt the urge to post here, but have been at a loss for what to post about. Anything that I felt was worth saying was too trivial to mention or to comprehensive to put into a one- or two-page post.

I thought about writing on some of the Neat Teaching Ideas my colleagues offered up in the Project NExT-SE session at the MAA conference in Tuscaloosa this past weekend. My UNCA colleague Kelli talked about the "peer mentoring" program she's been using in her Calc I class here this semester, and my Project NExT colleague Kade talked about using "math moments" to expose lower-level math students to nifty ideas from higher mathematics, like the Four Color Theorem and Russell's Paradox. I've done this sort of thing in the past, but not recently, and I've never tried putting a peer mentoring program into place. I'm going to try both out in Precalc next fall.

I thought about writing on the feeling I had driving back from Alabama, a feeling of calm, serenity, and oneness, as, just for a moment, I felt like I saw with perfect clarity my role as a teacher and learner. I felt for a moment as though I understood precisely how what I do affects what my colleagues do and reciprocally, and precisely how I help my students to learn as they help me to do the same. It was a pleasant moment.

I thought earlier today about writing on a common category error my MATH 280 students tend to make...one which I didn't mention in class this morning as I was debriefing them on their latest homework sets. Namely: students frequently confuse conjunction ("and") of mathematical statements with intersection of sets, and disjunction ("or") of mathematical statements with union of sets. There's little to do but practice in order to overcome this confusion, training oneself through repetition to recognize the different between a set or a class on one hand, and a statement or a mathematical claim on the other.

Snippets, random snippets. If you've got something to say about any of them, feel free to chime in. In fact, feel free to chime in even if all you have to say is utterly non sequitur; I always love hearing from my readers, and I want to know where you are right now: puzzled and perplexed? Curious and questioning? Or simply stressed, and tired, oh so tired?

Hang in there, my friends. The end is near. Have a seat beside me and tell me a simple tale; I'd love to hear it.

The end is near

I sure hope so, anyway. An already-short semester made nearly a week shorter by three snow days to start it off has meant that nearly every faculty member on campus has been piling on the work, making up for lost time. Everyone, faculty and students alike, is overworked this semester. The stress level is extraordinary.

This is one reason I felt bad effectively asking many of my 280 students to redo their last homework set even as they bear into the next one...and get ready for the next take home exam, to be handed out tomorrow. They took it well, though, and judging from the revisions I've seen since yesterday morning, they've made good on the opportunity: the revisions look great! I think it was the right move.

I hope this coming weekend's conference will be a welcome relief to the handful of faculty and ten or so students who are going. (Danielle, who was in my Linear class last term, jumped at the chance to go and get away from her dreaded CSCI 273 course for a little while.) This year's MAA Southeastern Sectional meeting, in Tuscaloosa, will be the second for several of the students and the first for a few others. My student Tonio is presenting on the work he did with me a year ago, we're fielding a Jeopardy! team for the second year running, and our department's faculty are presenting on everything from low-stakes writing activities (guess who?) to peer mentoring programs for Calc I courses. I'm just looking forward to the drive and a night or two to sort out some of the last revisions on the book before submitting it.

I've got one more conference coming up, at the end of next week. The annual Conference on College Composition and Communication ("Cs," as the comp/rhet people call it) takes place next week in Atlanta. It's my first. I'm looking forward to hammering out the next step on the research on REU writing Bella, Damian, and Nicola and I will be taking again this summer, and to meeting a few more folks in the composition community. (Moreover, from what I've heard, the publishers' receptions are pretty much the bomb...I've already been told about Bedford/St. Martin's shindig at Turner Field.)

Almost there...almost there...hang in there, my friends; we're almost there.

Change of plans

Around four or five this afternoon I got back from a trip to Belmont University this past Friday (great trip, fun talk, good friends, etc.), and went right up to my office to pick up the MATH 280 homework that was to have been shoved under my door. 23 out of 36 students had submitted their work "on time," and two more slyly slid their work under my door while I sat there grading Calc II quizzes.

I held off on grading the 280 homework (a set dealing with set equality and properties of set operations, like commutativity and associativity of union and intersection, and so forth) until I got home. Dinner made, I then settled into my chair to start.

The first problem was rough (the proof I'd requested has a more subtle structure than do the others in the set), and I'd expected that. The second, though, was rougher, and this I'd not expected. Several people did fantastically. Several others had the right idea but their exposition was ragged and raw. Several more still, though, had only a very loose grip on the problem.

I stopped grading there, knowing that were I to keep going, I'd see more of the same. After all, these kinds of proofs (of set equality) are tricky at first, but they're also very formulaic, and once you've got one of its kind, you've got 'em all. I'm almost certain that whatever mistakes a student made on Problem 2, she'd repeat on Problem 3, and again on Problem 4, and so forth. I doubt it would make much difference were I to keep commenting.

Rather than finish off this problem set, I'm going to return the ungraded homework sets tomorrow, giving students the option of revising their work after we've had a chance to go through a variety of approaches to Problem 2 together at the outset of class.

We'll see how it goes.

Travelers' tales

It's been a much better day today.

A highlight: another meeting of the minds in my office, involving three 280 students who'd stopped by after class for a debriefing on their recently-returned homework assignments. I'd mentioned in class just an hour before how beneficial others had found such debriefings, and these three had come hoping to reap the same benefits.

The meeting was wonderful. Sibyl, Nigel, and Quark joined me in a rough circle in front of my desk, and we went through the homework together, one problem at a time. Sometimes it took no more than a minute to iron out the wrinkles they'd worked themselves into, and other times we spent ten or twelve minutes puzzling through a problem's subtleties. They fed my wisdom and of their own, which was often richer. It's good for the students to see that others often have the same struggles they do (more than once they made the same kind of mistakes on the same problems), and it's good for them to hear their peers' explanations (often more lucid than my own). We ended the meeting with my giving them a few impromptu inductive exercises for practice with the technique. (Prove: n nonparallel lines in the plane determine n(n+1)/2 + 1 regions, bounded or unbounded.)

I left our meeting invigorated. I love this aspect of my job.

A second highlight: I realized this afternoon, while rereading Stanislas Dehaene in preparation for MATH 179 tomorrow, why it is I've had such a hard time getting into the mathematical aspects of my Ethnomathematics course this semester: the book, by Marcia Ascher's Mathematics elsewhere: An exploration of ideas across cultures, is awful, at least for my course. It's a bad fit. It's boring, condescending, pedantic, preachy, and is aimed at entirely the wrong level. (It's too dry to intrigue most math majors but too mathematically sophisticated to appeal to non-mathematicians.)

By comparison, Dehaene's book, though not strictly speaking about ethnomathematics (it deals more with the psychology and neuroscience of math and math learning), is intriguing, engaging, and written in a manner that's neither condescending nor confusing. I think the students will find it much more interesting than Ascher's text. We'll start drawing from it tomorrow and next week.

It's been a good day; I'm looking forward to another one tomorrow.

A closing note: my thanks to all of you who offered me support after my day of frustration yesterday. I'm sure regular readers will recognize that blogging is a cathartic exercise for me (need evidence? Check out the "anxiety" and "bitching" tags on the right!), and I often feel tremendously better after writing.

Calm

Late this morning I stopped for about a minute on the landing midway between the ground and the second floor in the stairwell at the far east end of Robinson Hall. I leaned against the wall and let the cold yellow brick rub into my back. I breathed deeply, hung my head, and closed my eyes. For a few minutes then I felt like crying. Then it passed.

I've often felt overwhelmed lately, what with four preps (i.e., teaching four different courses this term), a book draft due in dwindlingly few weeks, and an ever-growing glut of REU applications piling up in my in-box...the budget situation doesn't help, and I'm increasingly annoyed by the website migration, committee governance responsibilities, and the almost farcically bureaucratic assessment-related demands on my time brought about by the university's upcoming reaccreditation site visit. (Do I really need to write another impact statement? How long before we've simply assessed ourselves to death?)

I think perhaps what's most stressing and distressing to me lately (and may even be the root cause of the passionlessness I've lately expressed for my discipline?) is the distance I seem to have drifted from the aspect of my job I enjoy the most: hands-on, one-on-one work with my students. I don't feel like I get as much of that work as I used to, but when I do it's effect is as if godsent.

I had a few refreshing moments today. Kip, Kamryn, and Miriam all took me up on the offer I made my 280 students yesterday: I encouraged them each to come by and conference with me about the comments I made on their most recent homework assignments. Such conferences give me a chance to contextualize the issues I found in their writing, and to offer them more substantial guidance as they develop as writers and provers. The conferences also give them a chance to ask questions, to probe more deeply the problems I posed to them, and to get help on putting together new proofs that build on the old ones they've put in place before.

Every one of the conferences I had today was a helpful one, at least for me. Every one helped me normalize my expectations with those of the student I met; every one helped me better understand each student's weaknesses and strengths. (For instance, one fears being a "fraud" or an "impostor" because her strongest proof mimicked one we'd done together as a class.) I now feel as though I know each student who met with me just a little bit better than I know any of their classmates.

Moreover, as I hinted above, these conferences gave me a much-needed connection to the most meaningful aspect of my work I've too often felt missing lately. Meeting with these 280 folks, and strolling through the Math Lab and helping out with the odd trig-sub integral, really helped me get over my morning's mini-breakdown. I hope I was able to help a few other folks to wrestle down a few of their own personal demons.

We're all stressed, I can tell. But we're none of us alone. Let's all lean on each other. I'll slay your dragons if you slay mine. Together we'll make it through.

Editing is underway

As I write this the first MATH 280 "editing party" of the semester is going on in the Math Lab. Six of the class's 36 students showed up to look over one anothers' submissions for Chapter 1 of the textbook.

I'm eager to see how well the students receive this assignment this term, especially in such a large class!

UPDATE: ten people have now shown up, and the party is hoppin':

Fun with L-tiles

Today was that funnest day of the semester in MATH 280, when everyone gets to take a trip back to the third grade as we play with L-tiles in order to understand the idea of induction. This day is always a fun one, but I don't think any 280 class has ever had as much fun as these folks did. They loved it!

Play

I've gotten to the point where I don't think I can think unless I'm writing my thoughts out.

"Write, write, write," I exhort my students. "Write out your thoughts, in whatever form they take." "Play!" is a variant of this exhortation.

It's taken me over a decade and a half to realize that what I call "play," and what my wonderful mentor Jim Hagler no doubt called by this same word when he encouraged me with it during my own years as an undergraduate, is simply a mathematical version of low-stakes writing. "Play" can comprise words and numbers and notation, but it can also comprise visual elements such as pictures, charts, and graphs.

What purpose does "play" serve? The same purpose served by any writing-to-learn activity. I reminded my 280 students this morning that the very act of rendering concrete the abstract contents of our convoluted brains can help us make sense of those contents. The recursive processes involved in "play" (drawing, drafting, doodling, then reflecting, recollecting, and redrafting) help us organize our thoughts in ways we never could if we kept those thoughts confined.

Case in point? This afternoon I was working with one of the three math students whom I'm guiding in undergraduate research this term. Lulu's a terrifically intelligent student with whom I had the pleasure of working two years ago this term when she enrolled in my 280 course. She was easily one of the top two or three students in that large class, and since then she's distinguished herself by continued success in several upper-level math courses. Right now she's doing original work with me on the Möbius function of various subgroup and subgraph lattices, and she's making great progress so far. (Damn it...I've written so damned many rec letters this season that my blog posts are starting to sound like one...)

Today I was introducing her to the Deletion-Contraction algorithm for the recursive computation of chromatic polynomials. To illustrate the algorithm, I thought to use it to compute the chromatic polynomial of the cycle on 6 vertices, C₆. We began our work playfully, tinkering with the graphs which resulted from applying the algorithm to the initial graph. It soon became evident (through the simple stick charts which we drew) that we'd only find our answer if we knew something about the chromatic polynomial of P₆, the path on 6 vertices.

We let play push us forward. P₆, itself not the initial object of our investigation, soon led us further astray, as P₆ gave way to P₅, and this to P₄, and so on. A few computations (and a hand-wavy inductive argument) later, and we had a formula for the chromatic polynomial of P_n.

Play pushed us back, back to C₆. P₆ subdued, we turned to C₅, also intimately involved. But analogy (again recognized by the stick charts we'd drawn) allowed us to develop a pattern for this graph, too, and we soon realized a sort of inclusion/exclusion formula for the chromatic polynomial we sought.

Whether or not you follow the technical details above, I hope you can recognize this: the tinkering and tweaking we did on the blackboard in my office, including both playful formulaic experimentation and purposeful doodling, is precisely what helped us to develop the formula we sought. There is simply no way we could have arrived at that formula had we simply stared into space and "thought about" the graph C₆. Though the writing I was doing as Lulu and I worked this problem together helped me to communicate some new mathematical ideas to my young mentee, the writing also served the more critical role of making real my once-cluttered/now-clear thoughts.

Goethe famously said "I call architecture frozen music." Might writing merely be frozen thought?

I'll play with that.

From Greenville

My first visit to East Carolina University (ECU) has been a successful one so far, at least from my point of view. (You can ask my interlocutors in the Math Department and the university’s current WAC [Writing Across the Curriculum] Academy if they feel otherwise.) I’ve been here for a day and a half now, and it’s been pretty much nonstop action.

I got into town on Wednesday night around 7:30, having left Asheville just minutes after wrapping up my early-afternoon Calc II class. There was still enough time left on the evening to catch a late dinner at Copper & Vine with my wonderful colleagues and hosts Libby and Xavier (holla!), faculty in the ECU Writing Center.

Yesterday morning gave me some free time to finish off a couple more rec letters before the real work began. Over coffee and tea at Starbucks Libby and I went over the feedback she’d written on Chapters 3 and 6 of my book. She seemed to be apologetic about offering her comments with an eye toward reader response, but I assured her that this is exactly what I need. Besides pointing out potential points of confusion, her comments offered up several sources with which I wasn’t familiar before, and easily a dozen new ideas for low-stakes writing activities, methods of providing feedback on writing, and means of making the writing process smoother and more meaningful for student writers.

We bustled off to lunch at the Starlight Café (if nothing else when this trip is done I’ll able to tell you where to eat in Greenville) with a few more folks from the Writing Center, and shortly thereafter it was time for my first performance, a research talk in the Mathematics Department.

I was well-received and my talk was somewhat well-attended…but somehow the energy wasn’t there for me. It’s no secret to people close to me that I’ve recently found myself a little disillusioned by mathematics: there’s little sense of urgency to it, there’s little authenticity to what I do. This isn’t to say that math is pointless and purposeless; it’s simply that lately I fear I find little purpose in it for myself beyond play-like manipulation or action-under-constraint, a sort of glass-bead game or extended Oulipian exercise. I fully recognize that the mathematics I do has little, if any, immediate impact on the world and I feel that on the other hand most other aspects of my career and personal pursuits and passions (teaching, writing, and poetry) have relatively profound impact.

Couple this with the fact that mathematicians are, let’s face it, not the most outgoing or socially “ept” of individuals by and large, and I feel as though I’m alienating myself from my own discipline.

Meanwhile I’m moving closer and closer to another: right now teaching writing to me seems a much more personal, passionate, fulfilling, rich, and worthwhile activity. While one in ten thousand people might need to know what the independence polynomial of a path-like graph tells us about that graph’s connectivity, it’s impossible to know too well how best to express oneself. It doesn’t hurt that nearly every one of my colleagues in composition and rhetoric is warm and welcoming. I’m sure the discipline’s got trolls of its own; it may just be that they do a better job of hiding them.

Straight from my math talk I walked across the quad with my Quimby (another of the ECU Writing Center folks…both he and Libby attended my talk in the Math Department, brave souls!) to yesterday evening’s meeting of the WAC Academy. This six-week faculty development series challenges its participants to find more meaningful ways to involve writing in their disciplinary courses. Perhaps unsurprisingly (this is always the case in such workshops) there is a preponderance of younger faculty involved in this program, but the various disciplines are broadly represented in this semester’s group, spanning from psychology and biology at one end to drama and dance at the other.

The focus of the evening was The Writing Process, reverently rendered in capitals. Libby (who just now stopped by where I’m sitting as I write this sentence), the WAC Academy’s fearless leader, started things off by leading a “writing into the day” activity. She asked participants (and me) to illustrate the process they performed when writing their most recent substantial writing project. I portrayed my book-writing project as an Iron Chef-like jumble of too many skillets on too many stoves which gradually became more like an ever-revolving Ferris wheel, one chapter on each of several gondolas that got written on each time it spun past the spot where I stood on the ground. It was an intentionally playful exercise that definitely helped me to pinpoint where and when it is I do my best writing, and what I need to perform that writing.

Unsurprisingly, several others’ writing portraits belied decidedly different processes. One writer combined passion and intellectualization in a picture of a brain with giant red lips: a “brainkiss.” (This is the same man whose list of New Year’s resolutions included the perfection of the “cloud hug.”) A scholar in child development portrayed her writing of a grant proposal as a regular succession of polygons, signifying a step-by-step process which began with an idea and ultimately ends by branching into new ideas (represented by the same circle which symbolize the initial idea). Cyclic and circular, linear and direct, all manner of processes were represented. Surely this variety in the final product is expected, and the “discovery” of such manifold means of writing is one of the take-home points of the exercise.

My portion of the evening’s activities gave the participants a taste of the textbook writing exercise I’ve used in MATH 280 and, with less success, in Topology. (I find it interesting that as I type this sentence my 280 students are hard at work, I hope, in outlining the first chapter! I just texted Iris to see if all is well there.) I asked the participants to brainstorm ideas for the first “chapter” of a text on the ideas they’ve encountered in the first week and a half of the Academy, select five topics on which to base “sections” for the text, and then in pairs write two or three sentences on each of those topics. The emphasis through and through was on the underlying process. Though we didn’t have time to review and revise the writing they’d done, I believe the exercise was well-received and helpful.

As always, the informal and after-hours activities provided as meaningful a learning experience as the formal ones. Dinner at Wasabi 88 gave me a chance to swap traveler’s tales of disciplinary dragons, budgetary woes, and ideas for potential future collaboration. Several of my writing colleagues came, as did three of the brand new ECU faculty. Boudica, Irving, and Monica talked about the frustrations and felicitations they’ve felt during their first year at a new school. They dream big, talking about cross-listed courses they’d like to teach and plans they have for using collaborative learning and boundary-blurring pedagogies in their classes. (Think “theater of the oppressed” and “literature of the wilderness.”) The future is in good hands.

After dinner I spent a couple more hours putting together materials for my final performance at ECU, a presentation I’ll be giving in about two hours to the ECU Writing Intensive faculty. I’ll be hitting the highlights of writing-to-learn, indicating several useful types of WTL activities and letting participants play with a few of them. (I’m toying with the idea of assigning a lipogram…)

It’s been a busy, but bountiful, trip. I’ll be glad to be home, and I’ll have returned a richer man.

So far

So far, though it took a little while to build up steam, this semester's been great.

After three straight days of canceled classes and (and a couple more late starts thrown in the next couple of weeks), we got enough ground to get some traction, and I'm starting to feel like I'm in the groove.

Both my Calc II and 280 classes are large...I'd even call the 280 class huge: 33 in Calc II and 36 in 280. Yes: three, six. I'm always anxious about classes this large (I seem to attract them), but the students in both have set my mind at ease: they're all very engaged, outgoing, and willing to both ask and answer questions, in both classes. I've got fantastic students in both, and they're making the classrooms' atmosphere lively, spirited, safe, and fun.

I think I've done a good job in encouraging a healthy learning environment so far this term, downplaying grades, up-playing collaboration, throwing in a good number of writing-to-learn opportunities...the students are receiving this well. I sense a greater-than average willingness to learn on these students' parts. It's going to work out well.

I was particularly excited about today's 280 class. This morning we had our "LaTeX seminar," in preparation for which I asked everyone to install a compiler and a text editor on their computers and bring them, if possible, to class. Roughly three quarters of the students had laptops with them, and most of these (after a few fits and starts and glitches involving flavors of Texmaker and odd configuration settings) were able to get LaTeX up and running within five or six minutes. And they liked it. Comments like "This is so cool!" were fairly frequent. It's the best reception LaTeX has ever gotten in a 280 course.

Meanwhile Ethnomathematics is steaming along, picking out a course through the Marshall Islands. Literally, actually. We've spend the last week talking about the mathematical aspects of the mattang, the stick charts used by traditional Marshall Islander navigators in order to plot their path from one atoll to another in the sprawling and sparse archipelago. The students have even been working on building their own mattang out of various materials, including everything from pipe cleaners to pretzel sticks.

Late last week I tried to get the students to cast aside their "Western" assumptions about the make-up of maps by asking them to create maps from our classroom to various important offices and organizations on campus, maps which could not make reference to human-made objects, could not use any sort of reference to fixed units of distance (feet, miles, paces, etc.), and would be followable by someone who had not had a hand in creating the map in the first place. I realized after the fact that I should have included additional stipulations: no text, and mandatory use of "nontraditional" materials: the vast majority of the texts included copious textual commentary and were drawn on paper. (I admit that I'd hoped to see more "tactile" map-like objects like the mattang.)

Nevertheless, the students are doing well in what I think is a highly nontraditional course. I'm eager to see what they come up with for the brochures I'm asking them to make, one for each of several important campus services (like the Math Lab and the Student Health Center).

That's all I'll say for now. I'm sorry this is a bit of a banal post, but I've had nothing heinous happen so far this term. I could say something about the book (coming along, in bits and pieces) or the state's budget situation (in a word, bleak...think "how in the hell can we keep providing the quality of education we pretend to provide?"), or even about my upcoming visits to various writing programs around the state, but I'll leave those for other posts to come soon.

Trust me

Elon's over.

At last.

And after several months of planning, it came off all right. And the planning was worth it, as annoying as it often was: the students (particularly the younger ones!) got a lot out of it.

The drive out was a pleasant one (with only a light spray of rain), and the three-hour trip to the conference site gave the students (several of whom didn't yet know each other well) open up a bit and build bridges. As I told them then, for the first of several times during the next 48 hours or so, "you don't really know someone until you've got to a conference with them."

Obligatory alcohol-induced first-night revelry out of the way, the conference began on a high note. After an hour at the Project NExT session, I headed over to the first round of Math Jeopardy! in which our school has ever fielded a team. The Asheville team faced off against three others, and though our folks stumbled a bit at first (with a score at one point dipping into negative numbers), they quickly rebounded. Georgia Southern's team ran through the differential equations category, but with those questions depleted the UNCA team blazed through the category on "trans" words, pulling ahead, the ten or so of us in the audience exultantly pumping our fists with each right answer. A brief slump put them in second place going into Final Jeopardy, and it was only because of conservative wagering that our team came in second in the round...and ultimately ninth out of thirty-four teams. I was floored, and beaming with pride.

The conference itself began with a plenary talk on female mathematicians in the time of Euler. One of the central figures in the talk was the Italian mathematician Maria Gaetana Agnesi (of "Witch" fame), a topic I knew would be of interest to Ino, a student I've come to know as something of an Italophile. Sitting behind her, I tried to judge her level of enthusiasm during the talk. My suspicions were confirmed when I talked to her after the lecture was over, noting the excitement in her eyes. I've since suggested she might look into the possibility of translating the work of some other classical Italian mathematician as a research project straddling math and her beloved language.

The REU panel my colleague Lancelot and I put together was well-received, with nearly 100 people attending the panels and talks. I most enjoyed the student panel, on which three UNCA students, each of whom had taken part in a very different summer research experience, and one of the participants in my 2009 REU led a discussion regarding the aspects of a successful REU program. The discussion was open and frank, touching on negative aspects of summer programs as well as positive ones. As Lancelot said, our panels (both the faculty-led one and the student-led one) each could have been an hour long, and the conversations would have remained as rich at the end as at the beginning.

My new department-mate Kelli and I had a chance to catch up for a bit after the REU session ended, and after she and I and Lancelot hung out for a bit shooting the breeze I headed back to the hotel to lead our entire UNCA contingent to a Thai restaurant I happened to spy on the hotel's dining list: most of us trekked halfway across town to enjoy a surprisingly lovely dinner together in the company of twenty or so of our closest friends.

I spent most of last night putting the finishing touches on my talk for this morning, a fifteen-minute piece titled "The role of trust in teaching and learning." Here I had a few words to say about the ways in which the affective effects our learning environments as strongly as does the cognitive. More engaging to me than my talk were the conversations with my students that led up to it, in which I asked them to offer me their own views on trust in teaching. In every one of these conversations (ones similar to which I would recommend my fellow teachers to have with their own students) the students were lively and animated. I suspect that students may be so animated because they're not often asked questions like "how do you feel about the work you're doing in class?" or "how might your professors most easily earn your respect and trust?"

I think my talk was somewhat well-received, judging from comments people made to me afterward. I was a little worried that its rather unorthodox topic and methodology (highly qualitative, empirical, and anecdotal) would rub some of the more traditionalists the wrong way, but I'm not overly concerned. I feel strongly about what it is I had to say, and I can live with whatever discomfiture it may have caused a few of the gray-beards.

During the next couple of hours I had a chance to catch up with previous years' REU students (Dione from 2008 and Daria from 2009, who'd already spoken and served on a panel in the REU session the day before) and take in a couple of talks given by a few of my Asheville students: Siegfried's discussion of logarithmic concavity was breezy and easygoing, and Uriah and Ulrich's overview of last semester's textbook assignment in 280 was fantastic. Not only did they offer a full and accurate picture of the assignment's structure; they also highlighted the ways in which the assignment served as an extremal example of a writing-to-learn project and the ways in which bridges to other aspects of mathematics could be built from the abutments it offered. (Ulrich mentioned his discovery of the software package Geogebra, inspired by his need to create attractive graphics for the textbook. Reminder: the latest version of the textbook can be found by following this this link.)

The conference over, we hit the road again. I wish I'd had the van miked on the way home: Uriah, Iris, and Ino and I spent much of the drive back to Asheville talking about various things, including AP exams, life plans, Southern accents, and, yet again, trust in teaching. By the time we'd returned to the parking lot we'd left a little over two days before, several of the students were zombified from over-stimulation and lack of sleep, but everyone seemed happy and all professed to have been stimulated by the past days' goings-on. The younger students in particular, including those who may still have been on the fence regarding the pursuit of a math degree, were sold. Several are already ready for their next conference.

I'm tired. But happy. And proud. They're wonderful people.