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!

Overdue notices

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

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

What's new, pedagogically speaking?

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

What else is new?

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

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

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

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

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

What, as they say, is up?

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

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

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

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

What else?

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

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

What else?

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

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

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

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

What else?

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

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

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

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

Oh, hey!

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

I think about writing, I really do.

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

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

But I digress.

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

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

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

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

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

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

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

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

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

Vox populi

The students have spoken!

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

Saith one:

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

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

This student's take was echoed by a colleague:

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

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

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

One student waxed a bit more philosophical:

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

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

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

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

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

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

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

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

Chaos

I've got mixed feelings about today.

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

Graph Theory?

Hmmm...

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

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

The atmosphere was...

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

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

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

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

We'll see what they have to say.

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

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

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

...thus I had no meetings today.

Bummer.

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

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

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

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

Graph Theory: Day 2

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

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

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

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

Then we began.

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

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

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

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

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

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

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

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

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

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

I'm already looking forward to Friday.

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

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