Leaps and bounds

Two days in, and going strong.

We've now wrapped up two days of intensive seminar-style introductions to graph theory and group theory (to say nothing of mountains of paperwork, campus tours, and bureaucratic snafus of various orders of magnitude), and the students seem to be taking it just fine.

Example: after three and a half hours in which we plowed through four weeks' worth of graph theory (we worked off of modified versions of my Moore-method notes for last semester's 473 course...the out-of-class homework problems adapted well to their new niche of in-class exercises), their "homework" was to strike out on their own and track down definitions, examples, and applications pertaining to 30 different graph theoretical concepts. The kids came through in spades, spending the first four and a half hours of our working time together in taking turns presenting the material they'd come up with. Although I suspect there was a good deal of division of labor (certain students "claimed" certain problems and broke up the workload along clearly demarcated lines) it was just as clear that many of the topics had been multiply-researched, and thoroughly, at that.

They kept each other honest, too: they weren't shy about comparing divergent definitions, asking questions to reconcile apparent contradictions, demanding clarification on points that weren't so obvious at first. One of the students is particularly bold about asking for elaboration, it's marvelous to have her there since her boldness and the elaboration it requires are no doubt leading to greater understanding on everyone's part, including my own. My thanks to you, I hope you know who you are!

Indeed, few of the students are shy about taking part. Two or three are ever eager to strut their stuff on the board, and a few more are just as comfortable in directing the action from the cozy quarters of their desks. A couple are quieter than the others and are therefore harder to read, but I have a hunch they're all following along pretty well. (Two of the students have yet to take an abstract algebra course and today's dessert course included and introduction to geometric group theory, so those two folks might have felt a bit pinched at the end. I hope they roll with the punches and persevere, I know they're both highly intelligent and will weather this first week's storm if they stay the course.)

I've got very high hopes for this group of young mathematicians. They're already top-notch scholars, and I hope we can effectively take them to the next level.

In other news, I've just received the copy of Oulipo: a primer of potential literature (Warren Motte, trans. and ed., Dalkey Archive Press: Champaign, IL & London, 2007) I ordered. Oulipo is a French acronym, standing for Ouvroir de Littérature Potentielle, a consortium of artists (most French) who in 1960 began a movement dedicated to the production of rule-based works of literature. The generative rules that govern the poems they create are highly mathematical in nature, and any serious study of math and poetry demands that Oulipo's work be considered.

Meanwhile I just this morning sent "interview" questions to seven of my students from last Fall's Calc I classes, asking them to reflect not only on the poems they created for that course, but also on the process that led to the poems, and on the thoughts and feelings that governed that process.

I have to admit I'm not entirely sure what I'm going to do with the data I glean from these interviews. I think I'll only be able to find that out once I've got the student responses in front of me.

For now, I'll put down my pen and away myself to bed. It's late, and I've got another long day that begins early tomorrow (at 8:45 in the OneCard office, where I must meet with our Excellent Eight to sort out a minor matter involving their inability to check out library materials on their cards).

Feel free to let me know if you're out there!

All in

We're set.

In ten and a half hours we'll begin officially the 2008 installment of our REU.

I'm excited.

And nervous.

These are some smart people I'll be working with; it's evident in their conversations, their interests, in the way they carry themselves and express themselves. They're sharp.

What'll they be like as colleagues, tomorrow and beyond? As I was telling Thalia last night, I hope that this summer we can succeed in forming a community of learners in which no one is deemed as The Expert™, in which we all feel free to question and learn from one another as equals. The only difference between me and her is that I've been doing it for a few more years than has she; time will erase that advantage too, and if she keeps at it ten years down the line we'll be sitting in a side room at some conference, yammering away and remembering our conversation ten years back.

Last year's group: where are they now?

I've been in fairly close contact with five of last year's eight.

A couple of days ago I invited Kieran to come down at some point in the next eight weeks and give a talk about his current research program. He may be able to make it, he's not sure. He's got a lot to get ready for in the fall, when he begins graduate work at the University of Kentucky. He's got his name on a paper with me and my next-door neighbor in the department, on one that's been submitted to Complex Systems.

As I mentioned a few days back, I've been spending some time during the last couple of days retooling a draft of a paper I'll be submitting with my name alongside Wilhelmina's and Mirabel's. I may shoot for Communications in Algebra again; we'll see. Mirabel will be headed southward in the fall; she begins grad work at Wake Forest in August. I hope I'll get a chance to do some more work with her.

I'll be incorporating work by Nestor into the sequel to my paper that will soon appear in Discrete Mathematics; I don't think this sequel will sit for long before getting snapped up by a pretty decent journal.

And twice this past half-year I've seen Opal at conferences. I don't know if she intends to keep on keepin' on once she finishes her undergraduate degree, but I'm sure she's bound to do good things.

Where will this year's eight find themselves in the coming years?

Six of the eight are rising seniors (three last year), one is a rising junior (five last year), and one a rising sophomore (none last year), so this year's crowd is slightly older on average. As a whole they seem a little more sure about themselves and their mathematical futures than last year's group did. Grad school's a likely feature in all of these kid's futures.

I'm sure I'm setting myself up for a heavy round of rec-letter-writing in the fall.

That thought in mind, I'm off to bed; tomorrow promises an early start and a long, long day, crowned by (havah nagilah!) bowling night. I'll so be ready for that by the time 9:00 p.m. rolls around.

Who's who

Six in, two to go, both of whom will arrive in the next seven hours or so.

Loads of meetin' 'n' greetin' and gettin' to know one another goin' on. Everyone seems to be getting along so far (some more well than others...I'm going to have to separate certain people during the introductory seminars), I don't foresee any obstacles to this crowd's forming a hard-core tight-knit learning community within the first few days!

Further bulletins as events warrant; for now, everything appears to be copacetic.

Twenty-five minutes to go...

...At least that's what it feels like.

Desdemona, our REU student who's scheduled to arrive first, should be here within the next hour and a half.

Am I ready?

I've spent the last couple of weeks putting together material for this year's program, and now I've got a list of learning goals, a rough timeline, a list of about 45 possible research questions, five handouts involving set theory, graph theory, and metric geometry (and several more on the way), five handouts on Mathematica, four handouts on LaTeX and the structure of mathematical research papers, an exercise asking students to uses research databases to track down primary source material, and a map of North and Central Asheville, complete with markers indicating locations of yummy eateries and other useful establishments. (All of these are available on the REU website, here.) Not to mention a couple dozen research and survey articles on various topics of interest to me.

Still I don't feel ready.

I'm just anxious, giddy as a schoolboy on the first day of class.

I met with Lulabelle and Casanova yesterday to talk about the direction our new year-long writing assessment study is going to take. Suffice it to say we've got a mountain of data (both quantitative and qualitative), making the selection of research questions a leviathan task. For now we're focusing on collecting the final round of data from the previous year's participants, including syllabi and course assignments, as well as statements regarding these folks' experience in constructing and applying the writing rubric we all worked so hard to put together.

Lulabelle's visibly more relaxed since we've finished up with Austin. I hope she's able to get some good nights' sleep in these days.

After our meeting yesterday I stayed on to help her out with reading math papers (a scary proposition for a non-math-type person!), and then we talked about Kevin Rathunde's work on family environments that are most conducive to optimal experience in children. (I came across a reference to Rathunde's work in reading Csikszentmihalyi's Flow for our upcoming summer learning circle, and tracked down the original work, appearing as a chapter in Csikszentmihalyi and Csikszentmihalyi (ed.) Optimal experience: psychological studies of flow in consciousness (Cambridge, NY: Cambridge University Press, 1988).

Central to his study are five traits (all starting with 'C'; a fitting companion piece to my "Four Cs" rubric!): clarity (of rules and purpose, providing a stable set of experience parameters), choice (giving children the opportunity to direct their own experience within clearly defined parameters), centering (of activity in the present, making it an "autotelic" activity worthy of pursuit for its own sake), commitment (or trust, giving children a feeling of security that will enable them to boldly strike out on their activities), and challenge (ensuring that children are given tasks that are adequately interesting).

A question to ponder: in what ways are these traits best developed and nurtured in the classroom, and to what extent do they there serve the same roles as they do in a "well-flowing family" context? (I can already see how these traits mesh well with what I've always thought of as "good teaching practice.")

That's a question for later, since in the middle of my typing the above paragraph, Desdemona arrived with her family in tow. She's now safely moved into her dorm room, alone for the night but not for much longer.

I'm so excited!

And I just can't hide it!

I'll attempt to keep control.

But I'll still like it.

Until next time, ponder the following: just how long did it take Dr. Csikszentmihalyi to learn to spell his name? And how long does it take him to type it?

Channel assignment

Channel assignment

Were you a step away from me,
and I a step from you,
then you could hear
were I to speak.
Yet another step between us
would make two,
and to speak would mean to scream.
Beyond two steps
no sound could penetrate:
we’d sit alone
on silent isles
until some friends between us stood
to mediate,
one a step from you, one a step from me,
with a single step between them, too.

Sleepless in SoTL

One of the questions that arose during yesterday's midmorning meeting with my colleagues Ophelia (official evaluator for my REU) and Fidelis (world-renowned expert in science education assessment who just happens to have retired to the Asheville area) is: "How does student technical writing improve over the course of a summer REU, given that this learning goal is given attention?" Fidelis assures us that next to nothing is known about process when it comes to REUs, especially regarding the attainment of specific learning goals surrounding professional development: writing proficiency, oral presentation, acculturation to the career of a professional academician. Any one of these would make an excellent target towards which to aim the arrows of study.

I wonder why it is that I had writing on my mind?

This summer's REU will feature much more intentional instruction in writing, the use of research databases, and the use of Mathematica. Fortunately I've got access to a good store of material relating to the second two activities: I can recycle a lot of the handouts on research I used in Fall 2006 for the now-infamous MATH 365 course, and I can borrow liberally from (or just simply reuse) the Mathematica files my colleague Nostradamus made up for the REU last summer.

As far as writing is concerned, it's always a pleasure to make up some new material. I've already got a few handouts that I gave to the students in Week 6 last summer, introducing them to the rudiments of LaTeX. Instead of holding off for six weeks, they'll be given those handouts right away, and they'll get the ball rolling in Week 1:

Week 1, Wednesday: students will be asked to download freeware LaTeX compilers and text editors (OSTeX, MiKTeX, TexnicCenter, WinEdT, etc.).

Week 1, Friday: by the end of the day, students will be asked to submit their first assignment in LaTeX, a rudimentary paperlet discussing one concept with which we've dealt during the first week's seminars. The paperlet will have the form of a research paper in miniature, structured roughly as follows:

1. Introduction:what exactly is the concept you've chosen to write about? (Give a simple definition.) What is its history? What is known about it?
2. Elaboration and examples: say a bit more about the concept by elaborating on your definition from above. In what context does your concept live? What concepts are related to it, and how? Give a few examples of your concept.
3. Conclusion and discussion of future work: Bring your work together, summarizing the findings, and indicate open questions surrounding the concept you've chosen to talk about.

I hope that by asking them to so structure their work right away they'll grow accustomed to this genre.

Week 2, Friday: students will be asked to submit a research prospectus indicating at least three topics into which they would like to look during the coming weeks, including an abbreviated (and unannotated) literature review for at least one of them.

Weeks 3 through 7, Friday: students will be asked to submit "progress reports" taking the form of mini-research papers. These reports will be structured much as the paperlets described above and will continually update the reader on the students' active research. A student may choose to report on only one week at a time, or to maintain an ever-up-to-date comprehensive report summarizing all work performed until the present.

Week 8, Friday: the "final version" of a paper will be submitted. If all goes well, this paper should be a fairly close approximation to something publishworthy.

Last year only Wilhelmina's and Mirabel's paper approached publication quality by the end of the summer (speaking of which, I've got Wilhelmina's most up-to-date version sitting in my in-box, I hope to get to that today...), though with the intentional iteration highlighted above, I have hopes that many more of the students will leave with a strong piece of work in hand.

In other writing news, I'll be meeting with Lulabelle and our colleague Casanova on Thursday to discuss the directions our new writing study might take. We've got a ton of data to sift through, and any number of ways we could conceivably slice it. Between faculty journals and conversations, student pre- and post-tests, student assignments, the faculty-developed rubric and the results of its application to the students' work, there's a lot to work off of.

As I indicated towards the end of this previous assessment project, I have a hunch that this study is going to say more about faculty writing practice than it is student writing practice; this is why I'm interested in tracking the elusive "instructional intentionality": how is it that faculty's attention to writing instruction manifests itself in the classroom, and to what extent does that attention pay off in (at least perceived) significant student proficiency gains?

Beh. Mumbo jumbo mumbo jumbo...

...finally, I should say that I've already heard back from most of the former Calc I students I'd written regarding their poetry from last Fall semester. I need to make up an "interview" form to send them so that I can help them reflect on their writing process.

Well, I'm going to get going and take a look at my graph theory notes from this past semester's seminar course; I have a hunch I'll be able to modify the problem sheets to serve as a basis for next week's graph theory instruction. We may end up playing a lot of it by ear and making it up as we go along.

Avanti!

Back

Damn.

That was productive.

It's going to take me a while to unpack all that I've carried back from Austin.

It's been a long time since I've had a trip that fruitful; I've come away with so many ideas, insights, opportunities for collaboration, connections to new colleagues...what a wonderful crowd these WAC folks make!

It's safe to say that there's a great deal of interest in writing in the math curriculum.

Lucky me.

People were enamored with the work Lulabelle's put together, and with my piece of that project. If anything, though, the writing folks were more interested in hearing about my math poetry project from the Fall 2007 semester. I've already contacted several of the students from that Calc I class and asked if they'd like to work with me by providing me with reflections on their creative process. I'd like to know: what did they get out of the exercise? Did it help them to engage mathematics, and if so, how? I've spent the last hour or so looking over Arthur Young's Poetry Across the Curriculum and other related projects. This is such wonderful work!

As I said, it's going to be a long time before I'm done unpacking the mental baggage I've brought home from that conference. Already, though, I've begun thinking about how I can assess this summer's REU students' writing as they proceed from novice article authors to more seasoned technical mathematical writers, and I've begun puzzling through the process of assessing just how it is that a writing instructor's intentionality leads to students' gains in writing proficiency. And I'm coming up with ways I can use blogs in Abstract Algebra and wikis in Precalc this coming fall. And Lulabelle and I are meeting with Casanova on Thursday to discuss the direction(s) we'll be taking in our follow-on to the last assessment project.

And...

...and...

...and...

...too...much...to...think...about...

...and with only a week to go before the REU begins.

Until I return with a longer and more meaningful post, here's a question for all of you out there who happen upon this blog: would you rather be bored shitless or busy as hell? Discuss. You will be graded on ramblingness, incoherence, and lack of a thesis sentence.

Think 'pon it carefully. Surely your response says more about your personality than a thousand Cosmo quizzes.

Jump

Tomorrow has me jetting off to Austin (from what I'm to understand just about the only livable place in Texas, and just about the only place in that state I've never been) to take part in the Ninth Biennial International Writing Across the Curriculum Conference.

That's a mouthful.

It's also mildly frightening, since it'll be the first time I've ever presented at a decidedly non-math conference before. Frightening but exhilarating at once. The feeling I've got going into this shindig is not unlike the one I had when I made my first research presentation at a non-group theory conference a little more than two years ago now. (As a group theorist by formal training, I felt a bit of a dilettante in dabbling with graph theory, so I felt it a pivotal moment when I started going to/presenting at graph theory conferences.) It's a jump.

It's fitting that I've started reading Mihaly Csikszentmihalyi's Flow: the psychology of optimal experience (Harper: New York, 1991), the centerpiece for the summer Learning Circle in which I'm participating.

A relevant passage, read just last night (p. 61):

What people enjoy is not the sense of being in control, but the sense of exercising control in difficult situations. It is not possible to experience a feeling of control unless one is willing to give up the safety of protective routines. Only when a doubtful outcome is at stake, and one is able to influence that outcome, can a person really know whether she is in control.

This conference should be fun.

Just your run-of-the-mill first-day-of-class anxiety dream

It took most of the dream I had the other night (i.e., about 5 milliseconds in real time, most likely) to put my finger on why it was I felt underprepared for the dream-class in front of which I was standing: I'd not yet pulled up my students' photos on OnePort, so I was completely unable to put names with faces. While I usually get 80% to 90% of the names right on the first day, in this dream I was shooting blanks: everyone was a stranger.

Why is it that so many of my anxiety dreams involve classroom underpreparedness?

Yes, I'm always fully clothed in these dreams...usually I'm anxious I haven't planned the activity for that particular day of class.

Ah, but class is done for the year here, commencement's come and gone, and the late-spring conference season is winding down (one to go, in Austin at the end of next week...I'm nearly prepared, and I'm very much looking forward to it!). I've got a brief lull of about two and a half weeks right now, time in which I've got to get my REU ready to go for the summer. I've got a list of about 40 problems made up for the REU. I hope that by the end of the program's first week I can throw the list to the kiddies and let them spend the weekend looking up primary source material they can then mine for their own interesting problems.

Speaking of commencement (as I did, briefly, above), I'd like to send my heartiest congratulations out to all of my wonderful students who have now passed from these hallowed halls of academia and on to bigger things. You've done wonderful work here, and I'll miss you all.

Just a few days back Fiona sent me a picture her stepmother took of Fiona, Niobe, and me, all clad in regalia (ah, pomp and circumstance). Maggie's made it the desktop image on her computer. (She's entranced by the novelty of the photo, as I'm not often seen decked out like that.)

Since returning from Nashville the other day, my time's been focused on applying the writing rubric our research team's developed to the discipline-specific writing assignments gathered by me and my colleagues. I've had the honor of reading management research papers and reflections, and response essays and creative writing assignments from a course in French composition. These works include everything from taxonomies of corporate social responsibility through original poetry pitting Baudelaire against a red balloon.

I've found it very hard to apply our rubric to the French papers, in particular to the creative writing assignments whose brevity and informality make it difficult to assess a student's mastery of items such as "be able to contextualize a thesis within the body of existing knowledge." For both assignments from the French course, and for the brief reflection papers from the management course, I've noticed certain of the items on our eleven-item rubric blending together. I wonder if my experience is similar to that of my colleagues (some of whom have the misfortune of having to slog through my 280 students' homework papers!); if so, it may be that the rubric is more difficult than we had suspected to apply universally, or it could just need some subtle tweaking. I'll be interested to hear how our conversations go on Tuesday when we all meet to compare notes before Lulabelle and I take off for Austin with the assembled data.

And speaking of assembled data, this morning Lulabelle sent me a nice thick spreadsheet full of very nice numbers showing significant gains on a number of items from the pre/post-test administered in my 280 class last Fall. I wonder how my students' responses compare with those of my colleagues' students.

Exciting!

Yes, I'm a nerd.

Anyway, I'll leave it at that for now. Maggie's likely to be home from work in a half-hour or so, and I'd like to free up my evening schedule to spend some time with her.

Arrivederci!

Déjà vu

It's raining outside, Maggie's asleep on one of our new couches, it's the perfect day for updating the blog.

I've just returned from the second of my three May conference-season journeys, this one to Nashville, and I'm spending the day recuperating before a new busy week-and-a-half that ends with me taking off to Austin, Texas for the 9th biennial International Conference on Writing Across the Curriculum. Most of the next week will be spent preparing for my presentations there, and in applying our team's newly-minted rubric to piles of papers from Management and French Literature courses taught by my colleagues participating in the study.

I could have it worse: my friends Cordelius and Lulabelle have to wade through a stack of my 280 students' papers as they try to apply the rubric to nearly a hundred pages of intermediate mathematics. I spent a few hours making up a "primer of mathematical writing" to help them begin to read math. Therein I indicated that most mathematical notation is little more than shorthand, that mathematical terminology, though daunting, has the advantage of being precise and unambiguous, and that the most unfamiliar genres of mathematical writing (namely, definition, proposition, and proof) are fairly straightforward and exhibit clear idiosyncratic qualities that make them quite accessible to someone patient enough to examine their stylistic elements.

I hope it helps.

Wednesday and Thursday of this past week were spent in faculty workshops on writing, led by me, Lulabelle, and a handful of our colleagues. Wednesday's focus was on writing for freshmen, particularly those in our "liberal studies introductory colloquium" courses in which new students are exposed to the liberal arts tradition through seminars on a variety of subjects spanning the school's academic spectrum. I felt Wednesday's session was a bit lackluster, I sensed only tepid commitment from a number of the participants, I myself was still getting over a cold that had knocked me flat the week before, and I was having a hard time granting myself the authority to speak as a "writing expert." It was the first workshop outside my discipline I've ever had to run, and I felt like something of a pretender or a charlatan, standing in front of a roomful of my colleagues and trying to convince them that I knew what in the hell I was talking about.

By Thursday I felt better, both physically and psychologically. What's more, the audience seemed a bit more receptive: I sensed much firmer conviction on the part of that day's participants than on that of their colleagues from the day before. Thursday's crew was there to talk about discipline-specific writing, the sort that (presumably) goes on in higher-level coursework. I felt good about Thursday's workshop.

Immediately after the workshop ended on Thursday I hit the road to Nashville, arriving at my hotel just before 9:00 Central Time. I called up Titania, my friend from grad school, and we met for coffee at Fido (greatest coffee shop on Earth), my home away from home much of the four years I'd lived in Nashville.

We talked for a few hours, sounding like we were a hundred years older than we were five years ago. We talked about students and committees and conferences and enrichment programs and research and course design and assessment...we talked like the real university faculty that we are.

She, like two or three others of my colleagues who graduated from Vanderbilt within a year or so of me, has just received tenure.

I'd be lying if I said I didn't wish I were in the same boat. I'm starting to feel anxious. It's not that I'm worried about dismissal; I passed this year's reappointment process with flying colors. I'm just looking forward to the symbolic stability that tenure confers. I'm looking forward to the sense of permanence that comes with it. Yet I lost three years on the tenure clock in moving from a research postdoc to a tenure-track gig at a liberal arts institution, so I've got at least another year to go (likely two). Don't get me wrong, I can understand the powers that be not wanting to credit me with three full years of time served, but I'm a little disappointed that I wasn't able to get at least a year or a year-and-a-half of credit for the teaching I did in grad school and afterward in Illinois. It would be one thing if I'd gained nothing through that experience, but in all immodesty my talents speak for themselves.

I'm sorry, I hope you'll grant me this brief bout of pridefulness.

As Fido was prepared to close, Titania and I took to the street and headed over to Vanderbilt's campus. We continued to talk as we walked, reminiscing on our time in grad school (which time both of us enjoyed immensely), updating one another on mutual friends' goings-on. After a while we found ourselves at the math building, Stevenson Center's Building #1, and happily the doors were unlocked.

Our first stop was the first subfloor, where still stand the math grad students' offices. I know three of the current grad students (one's an ex-mentee and coauthor, a second's research path has crossed my own, and I was the reviewer for a paper by the third), the rest were just a smiling sea of young mathematicians who happened currently to be squatting in the offices we'd once called our own. We wandered through the dimly-lit office suite, trying to remember who was in which office in our own time. We were able to place most of our colleagues, but one or two offices were hard to fill.

We inspected the classrooms farther down the hall. "I had semigroups in here, with Gould," I told Titania.

"I had topology," she replied.

"I defended in this room," I said in another.

"Why'd you defend down here? I was in one of the rooms upstairs."

"I don't know."

She told me about how she once broke the sliding blackboard in the larger lecture room at the end of the hall. "I grabbed it from one side only and pulled it off the track," she explained.

Not much had changed.

After a bit we went back down the hall and took the stairs to the third floor, the first one above the ground. We stopped for a moment to peruse the list of faculty, most of whom we recognized. I mused that the Fields Medalist had no office number assigned to him.

At the end of the hall on the left was the room in which I first taught a course of my own, about which I blogged back in September 2007. It's changed very little since the night on which I went to campus to prepare for class the following morning: the blackboards are the same, the desks are the same, there still superfluously hangs from the ceiling above the side board an ancient set of maps of the world. "I never figured out why those are there," Titania said.

Half of the countries depicted thereon no longer exist.

That classroom remains, though.

I don't believe I ever led a class in that room after Fall 1999, so it's been over eight years since I've taught there.

Odd.

I shudder as I think back to the first writing assignment I made up for my Calc I kiddies back in 1999. Its prompt was little more than "Write a 5-10 page paper on a mathematical topic of your choosing. This is due on November eleventy-first." Granted, I gave my students hints on how to go about picking a topic, how to get started on their research, how to vet and cite their sources, which is more than many faculty would give. (Hell, most math faculty don't even think about assigning writing projects.) But I did next to nothing to structure or stage the assignment, to allow for rough drafts, to incorporate feedback, to provide writing instruction. The completed papers I returned to the students were line-edited, as much attention paid to misplaced commas as to thematic vacuums and ill-used evidence. Small wonder the products of my students' efforts were so hit-and-miss.

Ouch.

I've come a long way as a teacher in the past ten years. I wonder what experience the next decade will bring?

For now, I'm off to fold some clothes and rearrange our bookshelves. I'll be back soon, I hope.

In the meantime, if you've got something to say, please say it.

End of the line

Today was the last day of class. As I said to my second section, such endings are bittersweet: I'm happy to see them succeed, go forth, move on...but I'll miss them. They're smart, fun, for the most part hardworking. I'll miss them, every one of them special to me in some way.

This week's been a surreal one: I've been riding a roller coaster of research ups and downs over the past 48 hours, and that's eaten up most of my time.

For what it's worth, the end of this semester couldn't have come too soon, I'm going to enjoy the few months of busy but unstructured time I've now got coming to me.

Run down

As I was rounding the south-bending curve of I-240 West on the way home this evening I caught a quick glimpse of a dead possum. He wasn't the first of his kind I've seen dead at the side of the road, and surely not the most brutally beaten. I sped by at fifty miles an hour and never came closer than fifteen feet from where he lay, so my view was only brief. Somehow, though, the sight of his body keenly and clearly connoted death: final, absolute, without ambiguity.

This past Monday I laughingly offered to shoot myself after bowling a disappointing 113 in league play. "Just shoot me," I said to one of my teammates, making a gun of my hand and aiming it at my right temple. A split-second later I thought of Elmer, and I felt like an ass.

Elmer was a student in my 365 course, the one for which I began writing this blog nearly two years ago. He was a fantastic student in all three classes I shared with him (Calc II, Calc III, and Linear Algebra), and I'd grant that he had stronger mathematical aptitude than any other non-math major I've yet taught at UNCA. He had a promising career as a research chemist, and was a favorite with all of his faculty here. He was on his way towards stardom at UC Berkeley.

This Monday afternoon I found out that Elmer had hung himself last week.

Death stalks, watching from the wings. It's never far away.

Reminders of mortality have been thick of late. Two of my best students have lost loved ones in the past two weeks: one a brother's father; the other a childhood friend. One to sickness, death coming unsurprisingly on steady-stepping feet; to the other death came with blinding rapidity, swift and startling.

We do what we can to cope, to go on living, to make do, and each of us makes do in our own way.

I realize after re-reading the last few (scattered and infrequent) posts I've made to this blog that I've said little lately that's purely pedagogical. All semester I've been preoccupied with matters one could only describe as "personal," even as they effect my teaching: "I feel" this, "I sense" that...

A funny thought occurs to me: yesterday I was charged with writing up an evaluation form to be used in our capstone course (MATH 480), which I'm coordinating this semester. (Context: we currently don't solicit from the students course evaluations for this class, though I've often wondered why this is. We're hoping to offer the students a chance to give us more feedback on the course, thus the eval form I wrote.) I resisted a colleague's suggestion to do away with the Likert-scaled items "How confident would you now feel in preparing a presentation on a mathematical topic?" and "How confident would you now feel in researching an unfamiliar mathematical topic?", indicating the need to assess not only cognitive learning goals, but also affective ones. In the items stayed.

Exercise #1: Complete the sentence: "I feel..."

...as though this semester is best characterized by improved awareness of one's self in one's teaching. Last semester the lesson I best learned was one dealing with intentionality: if one hopes to improve students' ability to write mathematically, one must intentionally design one's course to address students' writing abilities. If one hopes to encourage students to work well as members of an academic team, one must intentionally design activities that bring students together as a team.

This semester I've learned (as a reading of my recent posts should suggest) that the "me" that I am outside the classroom is never very far from the "me" that I am in it.

My frustrations follow me when I walk through the classroom door. So do my annoyances, my pet peeves, my likes and dislikes, my (sometimes overly) generous and giving nature, my distractibility that leads to entertaining tangents. What you see is what you get: I prefer harmony to conflict, and the carrot to the stick. I'm bad at playing The Heavy, and when I play that part I do it clumsily, erringly.

Exercise #2: Complete the sentence: "I am confident that..."

...I've learned a lot from my students this semester, perhaps more than they've learned from me.

More than anything else, I've learned about perspective.

A Brief History of Perspective

Once the world was flat,
and folks were formless blobs.
Life was dull, to say the least.
We knew little of each other,
and cared even less.
We wandered around clumsily back then,
always bumping into one another.
"Excuse me," you might say.
"Harumph," I might reply: "Watch your step! I was here first."
"I didn't know..."
We'd argue, but it wouldn't matter anyway, for
there was no telling who stood where in relation to
this
or that
or another.
Then angles rose up, ministers of space,
connoting volume,
motion,
action.
All at once I could be
behind you, or in front of you, or at your side,
and in any case it would be clear where I meant myself to be.
"She was barely seventeen," you tell me.
"Fresh as a lavender bud,
clean as April air."
Who was she?
She wasn't numbers derivatives integrals sequences series...
...She was, however, day lilies dalliances mischievous winks curfew-breaking cruises down
the highway at unimaginable speeds.
Angles did her in.
They gave her room to move about, they let her up from the page...

What do we mean to each other, in the end?

"Let's do it," I told Cassio at the end of this semester's Parson's lecture. Prof. Mary Lou Zeeman had just finished her talk on mathematical modeling in biology, and Cassio had asked her afterward about how modifications of her models might look in the context of global economics and politics.

"Let's look into it." He was excited.

Exercise #3: Complete the sentence: "I feel that..."

...I've done my best work this semester when the lines that mark the borders between my teaching self, my research self, and my self self have been most indistinct.

I feel good about the undergraduate research projects I've helped to get underway this semester. Several students have begun meaningful, original, authentic research projects as a consequence of being in the right place at the right time. They've been excited by the potential of mathematics to answer questions they themselves have asked, and I've been lucky enough to be there, in the right place at the right time, to steer them on a course towards their respective goals.

I feel good about the work I've done this semester with the Super Saturday program, particularly that done with the help of my student assistants. I'm proud of them, I'm proud of their dedication, their sense of purpose.

I feel good about the work I've done with the Writing Intensive Subcommittee, and with my colleagues on the Writing Assessment Pilot Project. I'm proud about our findings, and I enjoy our meetings immensely. (Am I sick for looking forward to them?)

I feel good knowing that more than once this semester I've touched students' lives in a positive way. I've excited them about mathematics, I've helped them to access hidden talents, and just by being myself I've managed to make a difference.

Is it ever enough?

Exercise #4: Complete the sentence: "I want..."

...to work with my students and colleagues to create the perfect learning environment, an edenic haven in which the woes of the outside world can be forgotten long enough for us all to come together and put together some beautiful math.

Towards this end, I want to be a hero: I want to be able, singlehandedly, to fend off stalkers and suicides and seasonal affective disorder. I want to be a human restraining order. I want to walk tall and stand firm.

I want to be strong.

Exercise #5: Find meaning in your teaching.

How did I come to teach at a liberal arts college?

I'm not sure my decision to come here was a fully conscious one.

From my adviser, six years ago, after I'd just accepted my postdoc at Illinois: "it's not hard to get your first postdoc. Getting the second one is the trick."

From my wife: "I'm just not convinced you wouldn't be happier somewhere where you had graduate students!"

From my mother-in-law: "I think you should get a job at Princeton. Their campus is beautiful."

Why me, why here?

The past three years have taught me a boatload about who I am and what I'm doing and why I'm doing it.

I'm here because I'm a people person. It's here only that I have a chance to know the students as well as I do, and to work with them so closely. I do what I do here because...

...well, because there's nothing in the world I'd rather do.

I. Fucking. Love. My. Job.

I am still amazed that I get to do what I do well and willingly, and that I get paid to do it.

I am still amazed at the satisfaction I get out of my work: the thrill of a new theorem, the wondrousness of a student's sudden epiphany, the warm fuzzies that come from a productive committee meeting.

I'm sick, I'm telling you.

I'm also making less and less sense as I near the end of this post.

And I am nearing the end.

Indeed.

And I've as yet said nothing about the talk I'm giving at Wake Forest tomorrow (on research of which I'm very proud) or the conference I'll be attending in West Virginia this weekend.

There. I said it.

Oh well.

Like other recent posts, this post has been not so much about my teaching as it has been about me. I hope you've made it this far without gouging out your eyes with a butter knife.

I'd like to end with a brief (assuredly incomplete and mostly anonymous) list of my heroes this semester. Maybe you'll recognize yourself.

Before I go, let me make an intensely personal exhortation: if you're reading this, please write to me. Write anonymously, write with your name in big, bold letters. Write comments, dialogues, diatribes. Write poems, stories, confessions. Write apologia, hagiographia. Write pompously, funnily, sarcastically, banally. Write in streams of consciousness, write in haiku. Write however you'd like to, just write. Write, write, write. Tell me I'm right on, tell me I'm full of crap. I'd like to know that you're out there, I'd like to know what you think. It's comforting, it gives me perspective: if I know where you are, I can better know where I am, and I hope that once we all know where we stand in relation to one another we can work together to make our ways meet up in the middle.

Heroes, Spring 2008 Edition

• The Calc II student who, since he cleaned up his act about a third of the way through the semester, has fought his way from an almost certain failing grade to within striking distance of a B. His homework's often messy, but his exam grades have skyrocketed. I'm immensely proud of him.
• The several first-year students who make up my Unofficial Freshman Graph Theory Research Group. Their passion for mathematics is evident and unassailable. They've already made several new discoveries, and I have no doubt that before they leave this school they'll make many more.
• My wife, for putting up with my workaholism, for often being satisfied with seeing me only one hour here, another hour there. Without her help and support I would have gone insane years ago. Without her love I wouldn't be able to make it through the day.
• My Super Saturday assistants, who labored week after week at the thankless task of corralling hyperactive ten-year-olds, convincing those kids that math is a worthwhile endeavor.
• The student who was brave enough to acknowledge that this semester she's been a sub-par scholar, that she's had neither the time nor energy she'd like to devote to our class. I respect her and admire her perseverance, and I hope I'll get a chance to work with her in a future class, when she'll have more time to commit our common cause of learning.
• My best friend, for inspiring me with new teaching ideas, entertaining me with stories of her own classroom, and letting me kvetch to her about whatever it is that's bugging me. Sometimes I think I work as hard as I do just to keep up with her.
• My Graph Theory class, for standing by me through my first fully Moore-method course. My thanks go to them for all of their hard work, and for sticking with it and trusting me enough to put together a halfway decent discovery-centered course.
• The dead possum I saw on the roadside on the way home this evening. He was more than a dozen pounds of bruised flesh and broken bones: he was fragility, humility, mortality. He was, in a sense, me. He brought me here, and here I am. I am now who I'll be tomorrow when I'm talking about my research in Winston-Salem, when I'm working with my students on Friday morning.
  

I'll be there. Will you be with me?

Pile-up on page 565

It rained all day in Asheville today, and I spent most of the afternoon lying on the couch reading Meyer Levin's The old bunch, a novel that started slow but has definitely grown on me now that I'm about 60% of the way through it. Levin's sense of character is rich and deep, and by the time I reached page 565 I realized just how well-developed the characters have become, how well I feel I know each one, how well I know their motivations, how I understand what makes them do what they do.

At the top of that page my mind drifted for a bit, back to the presentation on my REU that I gave yesterday afternoon to an odd assortment of colleagues, administrators, and community hangers-on. On one slide I summarized my colleague Ocarina's analysis of the REU students' survey data, an analysis she based upon William Glasser's choice theory. Once the students' comments had been coded as corresponding to various topics ("having fun," "faculty support," "making progress," and so forth), Ocarina was able to trace the students' development through the milestones of Glasser's social trajectory: forming, storming, norming, performing, and adjourning.

For a few minutes I thought about the stages through which the various characters in Levin's novel were going, according to my naive (and likely highly inapt) analysis: though Sam's storm was over and he'd reached a period of normalcy, Joe was yet storming, trying to find a place in his world. Harry, meanwhile, was performing, as Sol had been since the end of the novel's first book. Mitch? Norming? Performing? Straddling a gap between the two?

"Back then, where was I?" I thought, thinking back to when I was as old as the kids in the book on page 565. Was I norming? Performing, yet? I was through storming, I think: by the time I was twenty-five, I think I'd nearly sorted myself out. I knew what I wanted to do with my life, at least, and that's more than many could say at that age. I had a pretty good idea about what it is I believe, about many things, though it would be inaccurate and presumptuous of me to say that my life was static, resolved, and fully meaningful. Nevertheless, the tempestuous growth spurts that made my twenty-year-old self alien to my eighteen-year-old self, and my twenty-two-year-old self as alien to me at twenty, had subsided, and one would have to thumb through the first 565 pages of my history in order to learn of them at all.

565 pages.

Who could say what makes me do what I do, without at least leafing through those pages?

This evening at a friend's potluck I met several people I'd never met before, and spent a dizzying two and a half hours trying to keep up with their newness, trying to take it in and learn about who these people were. Musicians, mostly, most of whom knew each other already, had written several chapters together. Between them all, they'd spent a century or two on Earth, learning this, doing that, dabbling, babbling, building friendships, making noise...We're all of us walking around with hundreds of pages of history, most of them forgotten, unreadable, unknowable.

It's amazing that people get along as well as they do, given that we're all walking about with unindexable, unabstractable users' manuals buried in our brains.

My students are still forming, still storming. Some (I have one particular student, from last semester, in mind as I write this) are so storm-tossed they can't make it to the classroom more than once in a while. Some are quick at finding their places and soon settle in for an expert performance. (Blackwell claims to have completed formulas for the remaining two cases of his graceful labeling I'd asked him to describe technically.)

These kids are at an awkward age: they're about half as old as they think they are, twice as old as they act. Yet they're often wiser than we give them credit, and though they often find critical thought difficult, their minds are quick to absorb new ideas and new information. They're malleable and manipulable, but can be taught skepticism. They are arch, and smartly shrewd. They're clever and conniving, and are capable of utterly unfair and solipsistic acts, yet among them are some of the most selfless people I know.

I love these people, I love my students. I feel honored to do what I do. Halfway through my second section of Calc II the other day I turned from the board and said, "if I won the lottery tonight, I'd still be back here tomorrow."

This morning's Super Saturday class was the last of the semester, and we finished up with "Bending Space and Time," the activity for which I always get my Calc students making poster board polygons. It went over as well as it typically does, and the kids were actually more or less on-task (thanks in a large part to Sieglinde's and Tallulah's ability to rein them in with stentorian teacherly commands). All but a couple of them dutifully pieced together the polyhedra they were asked to build: cubes, dodecahedra, tetrahedra, octahedra, and finally a bit of free-form spherical building before we finished with a small chunk of the hyperbolic plane.

This particular activity is one of four "solid" Super Saturday classes I've got now. Having run this thing for four semesters, I've finally found four activities (the others: "Games on Graphs," "Build Your Own Fractal," and "Math Treasure Hunt") that go over consistently well, leaving me with two slots yet to fill with dependably interesting and educative classes. "Codebreaking and Codemaking" is an exciting activity, but needs some tweaking: the current incarnation, involving binary arithmetic, is often above the heads of all but the brightest students in the class.

I'll do some brainstorming over the summer to think of a couple new activities to try. (Ideas? Send 'em in, I'd be delighted to know if you have thoughts on the matter.)

I don't know if I've mentioned yet that we've finalized our list for the coming summer's REU: it took about a week to fill the first seven of our eight slots (with only three or four "noes"), and another week past that to fill the eighth position (with another four or five "noes"). Now, we're set: four are men, four are women (the same as last year) seven students will come from liberal arts institutions, and one from a Ph.D.-granting school (again, the same); and three come from schools in the Southeast, as opposed to last year's four. (The Midwest gained one slot this year, three instead of two, while the Northeast remained stable with two students.)

I'm looking forward to this year's program, to be conducted more "intentionally" in many ways, and more freely in others. While I already have a topic or two in mind for specific students, we'll be letting them loose to find their own ways once we've given them a week or so of a head start.

Ah, for now bed beckons. I hope to return soon, when events warrant further commentary. Feel free to chime in if you've got something to say.

Prejudice and Pride

A classic 1968 study by Robert Rosenthal and Lenore Jacobsen (summarized in Pygmalion in the classroom) posited the theory that students' intellectual development is a function of the expectations placed upon them by their instructors. According to Rosenthal and Jacobsen, if teachers expect a lot from their students, and if they make these expectations clear, students will tend to rise to the occasion; if teachers' expectations are low, students will work only to leap over the low-placed bar.

Since that time various studies have called these results into question, but the power of the study still pervades a good deal of thought on pedagogy at all levels.

I must admit I've had it in mind as I compare the achievements of my two Calc II sections this semester.

Last week saw their second exam, on which the results of the first were more or less replicated: of the 19 students who received As and Bs before revisions were allowed, 10 of them were in the second section, which has only 16 students; the 29-person first section was home to under half of the overachievers. There were 10 Fs in the first section (several of them very low Fs), a section whose average score was about 68% (contrasted with the second section's 82%).

The in-class attitudes of the two sections are dramatically different: the first is torpid, laconic, nearly silent. They respond only when it's practically demanded of them, they work together in groups only reluctantly, they mutter answers almost incomprehensibly when an answer is called for. If they don't understand something, they let it slide by without question. The second section is vibrant, lively, jocose, responsive. They offer answers willingly, they ask questions unashamedly. They have fun, that much is clear.

The out-of-class attitudes are similar. The first class is lax, the second thoroughgoingly diligent. While the second section takes pride in its work, the first seems to do its work grudgingly. The second section's denizens log more hours in the Math Lab and coat their work with elbow grease; many in the first section are content to transcribe the solutions manual and call it good, assuming they bother with the homework at all. For example, with 10 weeks of homework behind us, the second section has submitted 156 out of a potential maximum of 16 x 10 = 160 assignments, a 97.5% submission rate, a mark simply unmatched by any non-upper-division course I've ever taught. I've not counted the submission rate for the first section, though I would guess that it's around 75%.

Please keep in mind that here I'm making generalizations: there are wonderful students in both classes, students who are active, proactive, interactive, attentive, and as dedicated to their studies as the finest of scholars. Keep in mind too that I have nothing personal against anyone in either section. To a one I like my students, I respect them, cheer for them, I want the best for them.

Which is why it's so damned difficult, puzzling over what it is that makes the one section so different from the other.

This brings us back to Pygmalion.

Is it something I do, or something I don't do? Is it something I can control, something I can adjust, tweak, in some way modify, so as to help the class run more smoothly, more effectively? Is it that my first section's students' development is being hampered by some set of expectations on my part? Am I, in the very act of writing this, undermining a search for a solution? By wondering aloud about the differences between my classes, am I admitting that I hold them to different standards, that I place one above the other, that I am prejudiced before I even set foot in Rhoades 105, and that that prejudice is somehow affecting, for good or for ill, the achievements of the students I meet with for an hour a day, four days each week?

It's absolutely incredible how much more tiring it is to work with a sluggish class than it is an active one. My energy is not limitless (much as I try to pretend otherwise), and more than once this calendar year I fear I've shown hints of exhaustion as my efforts to lead that first class onward peter out. Tuesday's class was a particularly rough one. "We can stop here, if y'all want," I said peevishly at one point ten minutes from the period's end, having waited for nearly half a minute for some kind of response, any kind of response, to my request for a pretty straightforward sum. "1+1/4" was all I needed to hear, yet silence was all I got. I felt like a schlemiel.

I was pissed as hell after that, not at my students, but at myself, for letting their unresponsiveness get to me as it had.

You see, I feel helpless when all that I do or try to do fails to excite, fails to entice, to allure, even to amuse (there are days when I'd be satisfied with that). I hint, I prod, I show, I cajole, I even bribe...I sit back and look on, I wheedle, needle, hint, and direct. I nudge, nurture, insinuate, and elaborate. I illustrate and animate, I offer up worlds of wonder full of mathematical mystery...what more can I do?

Or am I doing enough? Is it working? Am I getting through? Are they learning?

They must be learning, at some level. I must be getting through. There are signs, after all.

After all, as I said above, there are beautiful minds in both classes, and those minds are making progress: Section 001 is home to the author of all of my courses' most beautiful homework, a weekly technicolor fantasie of positively gorgeous (and nearly flawless) solutions. It's home to two freshmen (freshmen!) who are eagerly undertaking graph theory research under (and beyond) my direction. It's home to a quiet and unassuming young woman who made away with a perfect score on this last exam. It's home to one of my most regular Super Saturday volunteers, a brilliant young woman whose talents are remarkable, and whose career I'm sure will take her far. It's home to a couple of my brightest engineering students, one of whom willed himself most of the way from a C to an A last semester, borne on the back of his tireless efforts.

These are smart, smart people, and I'm annoyed with myself for being annoyed with their unresponsiveness.

I'm going to ask more of them in the semester's closing weeks: I'm going to crank out more worksheets, more Mathematica exercises, more interactive games. I'm going to get them up and bouncing about. I'm going to challenge their inertia and pry them from their seats. By gum, I'm going to get them moving.

Then there's Graph Theory.

Over dinner on the first night in Charleston (about which, more later) I had a delightful conversation with Sylvester and Nadia regarding the way our class has shaped up. These two, strong students both, had been too busy to submit their homework from the previous week: a paragraph or two describing their experience in our class, indicating both effective and helpful aspects of the class and what might be modified to make for a better learning environment during the waning weeks. "So, what do you two think?" I asked at the Starfish Grille. (Note to self: avoid this establishment in the future. The food is bland, the service dour, and the prices, though "Charleston cheap," still ain't "student cheap.") Egbert (auditing the class) and Trixie (nowhere near it) looked on.

Always outspoken, Nadia was happy to lend her opinion: it's all right, but she feels that fifty percent (her estimate) of the folks in the class aren't working as hard as they should be, aren't taking it seriously enough. Although she recognized that she's gotten better at presenting and communicating mathematics as the semester's gone by, I get the feeling that she felt certain people were holding the class back, and that I'd do more of the teaching. Sylvester seemed to concur.

I reminded these two that though by now they're old hands at advanced mathematics (having worked their ways through nearly two semesters of real analysis and other assorted beastliness), about a third of the class is fresh out of 280, and another third are one semester removed from 280 but have taken very little beyond that course. This course, for some, is the first course in which one encounters proofs for more than simply the sake of proofs. Thus there's a bit of trepidation on these peoples' parts: it's harder for them to take a stand on a nontrivial proof, it's harder for them to make themselves clear. Though the intuition may be there, the explanation is harder to come by.

The feedback from the rest of the class? Most of the others had primarily positive things to say. A couple regretted that the class seemed to move a bit more slowly than they'd like it to, and this comment was understandable, coming from the people who made it. Most have thoroughly enjoyed the structure of the class and have gotten a lot out of it. It seems we've come a long way from the awkward first weeks (including the awful soccer ball affair). The most concrete request was for a more real-time, group-oriented approach to the "review problems" at the end of each problem sheet. On Monday we'll try this out, picking apart the definitions, theorems, and problems in small groups and discussing the results as a class.

I'll let you know how it goes.

So, yeah, how was that MAA meeting in Charleston? (This brings us to the "Pride" in "Prejudice and Pride"...)

Every educator worth her weight in textbooks knows the feeling of pride that comes from being on site to witness her students' successes. "Them're mine!" you feel like shouting. "They're goin' home with me!" You feel a spark inside when your student boldly asks a good question at the end of a talk, you feel a glow when she defends the results of her own research.

For Charleston, Sylvester, Nadia, and Trixie all put together posters showcasing the research they've done over the past few months. (Trixie felt underprepared, yet she was the only one to finish her poster before leaving town; the other two threw theirs together at the last minute, literally. Indeed, it was five minutes into the judging period when they picked up their posters and launched themselves into the display area. The hour and a half leading up to that moment was seen through a frenzied haze of spray adhesive and hastily-scissored poster board. Trixie had watched nonchalantly from the sidelines, alternating between watching the action and fiddling with her Gameboy.)

It wasn't all work, of course. After a pleasant drive down, we had a brief break before finding dinner and taking a twilit walk on the Folly Island beach. The next morning let the kids stroll around downtown while I took part in some faculty development whatnot, and then the conference came.

Conference highlights:

• several hours' of research and relaxation with my good friend and colleague, Griselda
• warm fuzzies on hearing Sylvester and Nadia ask fantastic questions at the end of one of the conference's talks
• the elation of making a breakthrough in one's research (followed by the realization later that day, during the long and drizzly drive home, that the breakthrough was an erroneous one)
• hearing Trixie tell of an exchange between her and one of the poster judges, who had been rather critical of her design: "So, are you a junior or a senior?" "Actually, I'm a freshman..."
  
• the tired contentedness of driving a vanload of sleeping students back from their first academic conference, at which they'd made a hell of a splash

It was good.

I needn't have gone as far as the South Carolina beach to find students to be proud of this past week: Trixie's friend Blackwell has jumped on board the labeling lorry and has managed to find his own graceful labeling of a class of spiders similar to those Trixie claimed. We worked together for over an hour yesterday afternoon, hammering out a technical description of his labeling. (I even managed to sign him on as a math major! My hope is that peer pressure will finally cause Trixie to cave...) Throw his work in with Trixie's and Sieglinde's, and with the impromptu caterpillar labeling enumeration project begun the other night with Umberto and Nadia, and we've got a heckuva graph theory group coming together up here in the mountains.

And while we were away in Charleston, Tallulah led the Math Discoveries Super Saturday class. She and a few of her friends skilfully guided our troupe of elementary schoolers through the mathematical treasure hunt I'd planned for them. To hear it from Tallulah, though it poured a bit the night before and all was a bit rain-soaked, the kids had a blast. My warmest thanks go out to Belladonna, Tallulah, Sieglinde, and any others who helped them this past weekend; without wonderful students like you, these efforts wouldn't be nearly so meaningful for the kids. I really do think that we're in the business of changing lives for the better, and you're playing a big part in that venture.

I'm fixin' to wrap up this here post, but I'd like to end it on a note as high as my opening note was low. Let me be more frank than I've been since my cathartic post from December 8th, 2007.

I'm tired right now. Though overall this semester's not been as busy as last fall's, the past week or so has been a rough one on me, and I'm aware that I've not been as patient and peppy as usual. I've been short, curt, and I hope not quite rude. I've let my frustration show, and I'm frustrated by this fact.

Be patient. Remember that I'm human too, and can falter and fail as well as soar and sail, and that I need your help to make sure our classes succeed.

If you're reading this, please tell me what I can do to help you out. Let me know if you've got any hints, tips, clues, or suggestions. Post anonymously, if you'd like to, or send me an e-mail. One way or another, lemme have it. I'd love to end this semester on a high note, but I can't do that alone.

Do the Charleston!

I returned just a few hours ago from the MAA Spring Southeast Sectional Meeting held this year at The Citadel in Charleston, SC. (Two conferences in Charleston, in one year!) I spent much of my time with four of our stalwart students, three of whom presented posters in this morning's undergraduate poster session...

...I'll have much more to say about the conference, likely tomorrow...but for now, I'm fair and squarely exhausted and so must bid adieu...

Funny

For the past few months the neon over the entrance to a neighborhood tanning salon has been broken, yielding the following hilarious (and soon appropriate, to my Calc II crew) mathematical
result:


We'll see if the students find it as funny as have I.

Midweek melange

It's Wednesday. It's been raining all day, and the rain has hardly ceased, even now that the sun is down.

We're one student away from filling our last slot in this upcoming summer's REU, having received a seventh acceptance today. I spent an hour or so this afternoon hammering out a list of learning goals for the program, and a schedule of activities through which we will work towards realizing those goals. Like the goals I typically set for my classes, the list includes content-oriented targets like mastery of graph theory, group theory, etc., but also less traditional goals such as gaining confidence in communicating mathematics to others, and building the authority to challenge unproven results.

How well will we fare?

Well, how'd we do last year? We can't possibly do worse, can we? (Famous last words...) Here's a brief report card on 2007, filled out from the perspective offered by a year of hindsight:

Recruitment: A. We got great kids, and what's so marvelous is that we did so so late in the game, having not secured funding until a month or so after most REUs had already made their hires. That delay allowed us to catch the best of the best of the younger crowd, many of whom had missed the first round of REU applications. We lucked out. I feel honored that I had the chance to work with such a talented group of students, many of whom are surely destined for great things.

Logistics: A. We handled housing well, we covered all the human resource aspects admirably. As far as I'm aware (aside from one snafu with one of the subsistence checks when something didn't get signed in time), all of the paperwork came off without a hitch. Yay, we're good pencil-pushers!

Seminar: B+. For the most part, we hit the nail on the head. I think the structure of our opening week-and-a-half was sound, and it did a good job of preparing the students for what would come in the next weeks. I don't think we adequately anticipated the stress it would induce in some (all?) of the participants...but we adjusted for that, and pulled up in time. This time around we'll know what to expect, we'll be able to ease up when needed.

Structure: B+. Again, things moved along smoothly, for the most part. The students did a great job of establishing semi-regular meeting times with their respective faculty mentors, the students did a great job in keeping their noses to the grindstones, the weekly meetings were generally productive. Those meetings, though, were awkwardly scheduled, and I'm not sure that the students played as strong a leading role as they could have: in the future, we might be able to challenge the students to take authority in these sessions. Moreover, we didn't have a chance to include any "guest" research talks by faculty from UNCA or elsewhere, as we'd hoped we'd be able to do. (This I've already remedied this time around: I've sent out three invitations to colleagues from other institutions, and have already received one positive reply.)

Social organization: A. We couldn't have done it without the students, who got along admirably well. Not only did they not kill each other, they became fast friends. By the summer's close, the care and concern they showed for one another was evident. (I hope this year's crowd will come to the conclusion that they need to have a talent show, too...although nothing's going to top the 2007 crew's rendition of "A Whole New World.")

Research outcome: Incomplete. It's hard to say at this point how "much" the students will have generated when all the dust has settled. Wilhelmina and Francoise have got a nearly submission-ready manuscript they've been sitting on for a while now, Ned's work with me will make a nice section in the paper whose prequel has been tentatively picked up by a nice graph theory journal, and Kendrick's name appears on an as yet unsubmitted manuscript my next-door colleague here has put together. Let's hold off on this one.

Long-term outcome: Another incomplete. I'm not trying to cop out here; we're still too close to this past summer to measure long-term outcomes, but I like to think that the program made a primarily positive impact on the budding careers of a handful of talented young mathematicians. As far as I'm concerned, if five years from now I run into one of the participants at a conference just after she's presented on her dissertation research, and she's able to say that her experience was a worthwhile one and helped her decide what she wanted to do with her career, then we've dealt ourselves a royal flush.

Changes this year? Reflecting changes in my own pedagogical style over the past year or two, the program already exhibits more conscious design and attention to explicitly stated learning goals. Writing plays a more central role, with an introduction to LaTeX coming in at the program's beginning (towards the end of the first week) rather than at its end (towards the end of the sixth week). Indeed, written progress reports will be expected of this year's students, in addition to the weekly meetings. We're also going to make meeting times explicit, and as I mentioned above we've already begun scheduling guest speakers. Finally, and perhaps most importantly, we'll be encouraging the students to seek out their own problems more actively: though we'll still have ready stockpiles of personal problems from which the students will be able to draw, the participants will be encouraged to seek out problems that entice them, hopefully from within the fields in which the participating faculty specialize. "All right, y'all, that's everything you need to know about chromaticity of Cayley graphs. Here's a survey paper. Dig in!"

I'm excited. Now that we're in the thick of it instead on the fringe, this year's selection process has been more of a roller coaster than last year, and with as many noes as yeses the thrill of the chase has gotten my blood pumping. We'll have our team set up soon, and I'll probably take one more shot at getting folks to blog about themselves before they get here. (Last year's awful attempt failed pitifully...I'm pretty sure that I mercifully deleted the pathetic little webpage that limped along painfully for a few weeks...yup! Just checked: all gone.)

What else is up?

For a few weeks now I've meant to say a bit more about my Graph Theory class, let me take this time to do so.

In-class presentations are for the most part much improved, especially in the past few weeks. The students who take the time to craft solid proofs ahead of time are executing marvelous performances and are sometimes uncovering techniques I would not have considered. Though their methods are not always the most efficient, they're authentic, through and through. Today in particular saw a handful of nearly immaculate proofs: one problem asked the students to prove that the path metric induced by a subgraph could only exceed the path metric of the original supergraph, another asked for an explanation for what breaks down when one tries to define the path metric on a disconnected graph, yet others asked properties of eccentricity. All solutions were skilfully executed.

Where some of the students are having trouble is in the written submissions. How so? Well, c'mon, people, even if all the problem says is "find the chromatic polynomial of the complete graph on n vertices," I can't jolly well in good conscience give the same grade to some who just hands me the formula, ex nihilo, as to someone who includes a half-page proof of that same formula. Trust me, from now on I'm going to explicitly include wording like "give a formula for... and prove that your formula is valid." I'll say that, if you'd like me to, but I claim that at this point I shouldn't have to say this, it ought to be assumed that at this level we prove our claims.

But we all know that when we assume, we make an ass out of "u" and me.

For the most part, Graph Theory's a blast. I'm still having fun, I think most of the students are finding it a worthwhile experience and are learning a lot. For Friday, I've asked them each to write a few paragraphs about what they feel is working well, and what could stand to be changed for the closing third or so of the semester. I'm eager to see what they've got to say. I've already talked to two of them about modifying the "review" problems at the end of each problem sheet, to allow these problems to be more group-centered and in-class. We'll see we can make that work, if others are up for trying it out.

It's all good.

I'm getting tired, I'm going to slink off in a moment, but just a quick word about my Calc II kiddies: two days into sequences, they're doing great. "These are fun!" one of my students said. They've already asked great questions and have exhibited profound intuition and insight. I think they'll be able to wrap their minds around this stuff comfortably. I'm also happy to report that Taylor series are playing a crucial role in my own ongoing research right now, so I should be able to bring that in as a "real-world" example of series methods before the semester's through. Huzzah! This crap is useful! Who'd'a thunk it?

Well, more anon, likely. For now, I'm off like a prom dress, as my college buddy Jennifer was fond of saying. Ta for now.

Pi Day festivities and other assorted goings-on

I've now had two days to recover from the hedonistic revelry that accompanied this year's observance of that most hallowed of days, Pi Day, March 14th, and I've a few minutes of time in which to sit down and chronicle the occasion.

This is the second straight year we've put a bit of effort (more this year than last) into celebrating this immovable mathematical feast, and that effort paid off, with roughly fifty folks, mostly Math Department students and faculty and their close acquaintances, in attendance at the 1:59 ceremonies.

What went on?

For some weeks now Stanley (our Math Club president) and I have been mulling over various means of approximating π probabilistically that would lend themselves to audience participation. The classic Buffon's needle experiment (implemented here on George Reese's homepage at the University of Illinois, Urbana-Champaign) could be replicated by allowing passersby to chuck hot dogs into an enclosure with a ruled surface, enabling a running tally of hot dogs that strike a line. The cost of implementing this procedure would be rather high, unless we wanted to reuse the same hot dogs over and over and over(an icky proposition)...plus there's the need for constant supervision of the enclosure, and we've called upon our students quite a bit lately, what with the recent Math Literacy Summit. To bring the cost down, we thought then about replacing the hot dogs with pixie sticks, which would be more inexpensive and likely more accurate (there would be less error incurred by the thinner width of the pixie sticks), but we'd still have to ensure the event was continually monitored, in order to tally up the results of the experiment.

Then I hit upon the idea of just doing a simple Monte Carlo area estimate: build a small square enclosure, and let people chuck spare change into it throughout the day. At the day's end, collect all of the coins that lay within a circle centered at the enclosure's middle, and divide by the total number of coins present. This ratio should be roughly π/4, the ratio of the circle's area to that of the square. Assuming a fair degree of faith in human nature, there'd be no need to oversee the experiment, since the coins would only minimally interfere with one another. All we'd have to do is put the booth up in the morning and take it down at night after carefully cataloging the location of the coins.

This we did. I spent a few minutes on the evening of the 13th drilling holes in the plywood and posts, and then Maggie and I schlepped the assemblage up to campus, along with the roughly 18 pounds of pie we'd bought for the pie-eating to take place on the following afternoon.

First thing on Friday morning, I went downstairs and slapped the enclosure together. The edges bowed outward slightly, but it was very roughly square and would serve well. I tacked an explanatory note to each side of the enclosure, along with an encouragement for people to chuck their change into the square:


Classwise, it was a humdrum day. For whatever reason (I attributed it to hangovers resulting from an overly exuberant demarcation of Pi Eve the night before) attendance at my morning Calc II class was exceptionally bad, and I felt no qualms in devoting the class period to working on the current class project, asking students to compute the centroids of various pieces of poster board. (Funny story about that project: as I handed out the project this past Tuesday, Louella asked me, "so, were you a creative writing minor in college?" when she read the project description, with the following text: "...the Math Lab will be home to eight small shapes cut from festively-colored poster board. (They are bundled together with a binder clip, hanging from a tack over by the coffee pots.) There are three colors represented: four are blood red, three are Day-Glo orange, and a final shape is a nice soothing green, as fresh as a newborn magnolia leaf.") The class was a relaxing one, the students were laid back, and had fun working together to get a good head start on the project. I hope Monday's class will be similar, when I'll circulate various worksheets asking the students to consider various applications of integrals not considered by the textbook.

The second section of Calc II was as fun as the first, and we wrapped up just in time for me to bolt upstairs to gather what we'd need to set up for the 1:59 celebration: pies and plates, camera and stopwatch, prizes, and a print-out of π to a thousand places. Even as I came back down from the first of two trips to my office, students and faculty were beginning to gather. By the time the ceremony got underway with a dramatic reading of π (pictured below) there were about thirty or forty people assembled, and more still would come to watch the pie-eating contest in a few more minutes.

After a few words about the occasion, I began the reading of π with a bold recitation of the first 25 places, handing the script to a student who would continue where I'd left off. Emotions bubbled close to the surface as student after student took turns reading digits.

Here's a shot showing the thrilling denouement of the dramatic reading:


Next came the pie-eating. Five stalwart students came forth to vie voraciously, and each was seated with a pound of pie in front of him or her (four hims, one her). At the appointed moment, they set to, chomping away for 3 minutes and 14 seconds.

At the end of the carnage, little was left of most of the pies but the skeletons of empty crusts. Norbert, an engineering student in my first Calc II class, was declared by the several faculty judges to be the winner, with Nicodema, the contest's sole female entrant, coming in a close second. The fearsome five gathered for a group photo at the contest's end:


Next came the π-memorizing contest (more appropriately, perhaps, the π-reciting-from-memory contest). Just that day I'd announced to both of my Calc II classes that if they sat down and committed twenty or thirty places to memory they'd probably stand a good chance of winning the competition, since I expected a fairly weak field. Little did I know that last year's winner, Ulrich, had returned to defend his championship. He intended to best his previous record of 64 places with a public recital of the first 150 places of π.

Loath to let Ulrich get away without a challenge, Trixie came forward and belted out 48 places unerringly, offering an incredible extemporaneous memorization. Here she is, the midst of her performance:


Her recitation was followed by a flawless 45 places, and it was then up to Ulrich to hold his own.

This he did, rattling off 150 places with only the slightest pause now and then. Here he is below, in the midst of his recital:


After this, there was plenty of opportunity for hangers-on to mingle and partake of a leisurely piece of pie themselves. My department chair then gathered everyone present and took a photo of the whole throng. I count 44 people in the picture, and I know there were at least four present who were not captured "on film" (what does one say these day? "On flash" doesn't have quite the same ring to it...):


So it is that with heavy hearts we say farewell to another Pi Day, only to wait another year before again marking this felicitous occasion. (By the way, by the day's end, our enclosure had gathered over 300 coins, yielding an estimate for π that was around 2.79. I've got the data in my office, I'll post them later when they're in front of me.)

After a brief bit of frenzied clean-up, I was off to Graph Theory. There we finally managed to finish off the now notorious Problem Sheet 7, dealing with chromatic polynomials, components, and the basics of trees. Proof-heavy and definition-intensive, this sheet was a definite departure from the previous ones, and it challenged even the strongest students in the class. "Now that we've got the fundamental of graph theory under our belts, we're able to consider some of the deeper concepts and techniques, and that's what this sheet has been asking us to do," I told them. We're now set to begin the next sheet, in which is introduced and investigated the path metric on a given graph. That's where we'll find ourselves on Monday.

I regret that I've not had the time lately to update this blog as much as I'd like to...and when I've had the time, I've hardly had the strength, as busy as I've been. Now that the Numeracy Summit has passed, and now that the bulk of work on our NSF grant is completed, and now that the REU applications have been read and evaluated (we're about halfway through the selection process as I write this), I will likely have a lot more time on my hands, and I'll be less tired when I have it.

I've got a good deal of travel coming up, about which I'm very excited. For instance, in a couple of weeks it'll be down to Charleston for the Southeastern Sectional MAA Meeting, several students in tow and several colleagues by my side. Trixie will be presenting a poster there on her work in graceful graphs, I'm very proud. Whether or not she chooses to pursue a math degree, this experience will be a fantastic one for her.

Aside from travel, there's the REU to get ready for (we're opting for a "less directed" approach this year, offering students a bit more room to explore), and the Parsons Lecture (featuring Mary Lou Zeeman) is just a few weeks away. Much to do, much to do!

I'll be sure to check in whenever I can.

Pre-Pi Day poetry

Integrals

My attempt to solve
an integral tells me more
about me than it.

Happy Pi Day, everyone!

Breakin'

It's Spring Break.

Which doesn't, sadly, mean I have nothing to do. It only means that what I've got to do (and there is a great deal of it) needn't be done on a rigid schedule.

I've got several job-related tasks to take care of in the next week, ranging from the quotidian (prepping for class once school resumes a week from tomorrow) to the leviathan (going through a stack of roughly 75 applications for this coming summer's REU). I've got a couple of meetings tomorrow, one with a student (my independent study in order theory), another with a colleague (Writing Intensive stuff). After that, I'm looking at a nearly completely unstructured week.

I'm in need of some unstructured time, after the busyness of this past week. Dr. Robert P. Moses, noted civil rights leader and founder of the Algebra Project, came for his visit this past Wednesday, and between the public lecture on Wednesday evening (a talk about the degree to which the Constitution ensures a quality education, at which I was delighted to see several of my students!) and the ensuing Math Literacy Summit held on Thursday, there was no shortage of excitement and things to do in our department.

The session I chaired at the summit (a talk on numeracy as it relates to health issues, given by a psychologist at the Duke University Medical Center) led me to the book I'm now reading, Stanislas Dehaene's The number sense: how the mind creates mathematics (Oxford University Press, 1997). This is proving a truly fascinating read!

Dehaene is a psychologist specializing in the neurobiology of mathematical acquisition, his book is a record of many of the facts that have been discovered concerning the way in which people learn mathematics, they way they organize its ideas in our minds, the way math is retrieved from memory. At its most basic level, our sense of mathematics is very little advanced beyond that of many animals, who share with us a precise sense only of the numbers 1, 2, and 3; beyond this is a roughly-reckoned haze of numeric quantities. Dehaene compares our mental conception of number as an "accumulator" with approximate graduations allowing us to give rough estimates of large quantities, but which fails to give precise values for these same quantities.

A few snippets:

• Even as soon as a few days after birth, babies are able to discern between the numbers 2 and 3. (See p. 50.)
  
• We (adults included!) are susceptible to "the magnitude effect": it's harder for us to discern the difference between 90 objects and 100 than it is the difference between 10 objects and 20. Various factors (symmetry, density, etc.) militate and mitigate this effect. (See pp. 71 ff.)
  
• Studies show that when asked to compare numbers, such as 5 and 7, and state which is the larger, instead of behaving reflexively and answering based upon our knowledge that the symbol "7" represents a larger quantity than the symbol "5," we instead convert each of these abstract digits into collections of the requisite number of objects before performing the comparison on these collections. (See pp. 75 ff.)
• We have a tendency to "compress" numbers as they grow, storing them in our minds as though on a logarithmic scale. One corollary of this behavior is that when asked to provide a random sample of numbers in a certain range, people will tend to elect an overrepresentation of smaller values, as though these were more widely spaced than their larger compatriots. (See pp. 77 ff.)
• Since adults compute sums and products (for example) by retrieving the resultant quantity from a memorized table, those whose native languages have exceedingly short names for the ten numerals (like Chinese and Japanese) are able to more efficiently memorize the desired sums and products, and so perform much more quickly and with fewer errors than their counterparts speaking other tongues. (See pp. 130 ff.)
  

These are just a few of the fascinating facts I'm learning about the development and refinement of mathematical thought processes in and by the human mind. Ultimately, one of Dehaene's primary points is summed up nicely on pp. 118-119: "Although our knowledge of this issue is still far from complete, one thing is certain: Mental arithmetic poses serious problems for the human brain. Nothing ever prepared it for the task of memorizing dozens of intermingled multiplication facts, or of flawlessly executing the ten or fifteen steps of a two-digit subtraction. An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack proper resources."

To be continued, I'm sure.

For now, I'm off to enjoy some more of this wonderfully unstructured time, probably by knocking off a few more pages of Dehaene's book. Highly recommended!