Dialectical driver's ed

How's it going?

Three days into Calc I, and it 's difficult to say which way the class is going to break this term: the students are still strongly engaged in class and contributing well to the problem-centered classes, but I think a few are hesitant about the rather non-traditional format of the class. As yet we've not done a great deal of computation, haven't derived or delved into any formulas, haven't stated any theorems. A number of the students are used to doing all of these things within the first couple of days of a math class and so they've got a sort of deer-in-headlights, "WTF?" sort of look in their eyes.

The formulas will come soon, I'm certain, and I hope that with them a sense of familiarity and comfort: we're about to enter into a discussion of tangent lines and secant lines and their slopes, a discussion will include a great deal of computation and review (and practice) with algebra.

I'm not sure quite how quickly we'll get there: it really all depends on the pace at which the students end up driving the course. As I mentioned to my first section this morning, they're sitting in the driver's seat, working the pedals, holding the wheel. Every now and then, as instructor, I'll reach over and grab the wheel with one hand to steady it or add a few degrees to a particularly too-tight turn.

So far the structure of the course is "dialectical," an exercise in turn-taking: I've proposed an initial course heading, and now we've walked for a while. Now, at the end of the first day of hardcore hiking, having surveyed the site at which we've made camp, I've worked out an itinerary for our next day's travel together. We'll see where that takes us. We'll make camp again, make some next-day plans, and set out again in the morning. I know the stops we need to make, and we'll make them all, but it's up to the students to figure out which we hit when, and in what order.

We'll figure it out.

Meanwhile, I am loving Linear. The class is huge (the largest I've yet taught at UNCA), but has great cohesion. The students are eager to work and are making great progress, and I feel that I'm doing a much better job of structuring the discovery process than I did the last time I taught the class. I've put a lot more thought into precisely which skills are needed to solve which problems, and we're making a (long) beeline toward our first major goal: analyzing the long-term behavior of a simple Markov process (like the game we played on the first day).

Today we figured out how row operations mirror the operations needed to solve a system of linear equations. Next, after posing and solving a couple of realistic problems requiring the solution of a linear system, we'll motivate and move toward matrix multiplication. From there, it's on to matrix inverses, invertibility, determinants, eigenvalues, and, at last, an analysis of Markov processes like those we began with. At every stop we'll solve another real-world problem or two, get some practice with computation, and write a bit about what each concept really means.

I figure it should take us something like 5-7 weeks to get where we're going. After that, we'll talk about linear transformations, vector spaces, changes of basis, etc.

Exciting!

Meanwhile, I'm reading some fascinating works on writing and rhetoric in preparation for various parts of my book. Most recently I've begun Deirdre McCloskey's The rhetoric of economics (2nd edition, 1998, Madison: The University of Wisconsin Press), a delightful rhetorical analysis of economic discourse in which she dissects the ways in which economists convince one another and others of the truth of their assertions. According to McCloskey, much of what many economists consider unassailable scientific truth is really rhetorical "smoke and mirrors."

Not that there's anything wrong with "rhetoric," a word which McCloskey takes pains to save from those who would use it only in a pejorative sense: rhetoric is ubiquitous and unavoidable, and McCloskey merely asks that we be intentional with the rhetoric we use. We all use metaphors and other rhetorical devices; it's simply important that we be aware of the devices we employ. As one might put it, extending the tried-and-true "lens" metaphor, it's no crime to have poor eyesight, as long as you can admit that you need to put your glasses on in order to see properly.

Her opening chapter, which begins with a close reading of passages from canonical articles in economics, has made me think about the way in which I begin some of my own mathematical papers: what sense of authority am I conveying though my choice of words? What world am I building? What agent is acting? What am I asserting about the truth? I'd like to look into my own writing to do a careful reflective analysis.

But with all that I've got on my plate right now, such an analysis has got to wait for a little while. I hope to finish off the first draft of the introduction to More Than Numbers tomorrow, and set to work on the slightly slanted history of WAC, WID, and WTL which will make up much of Chapter 1.

Much to do, much to do, and so little time...but so much fun!

Loose ends

In order to tie together the ideas contained in a couple of recent posts (this one and this one), I offer the following:
This diagram, like that contained in the last post, represents what mathematicians call a finite state automaton. Such a creature is little more than a very simple computer, capable of generating and recognizing strings that fall into a class called a regular language.

To be more precise, imagine that you're plopped into the diagram above in one of the ten states (the little yellow boxes with labels on them). You're then asked to move about the diagram by following the red edges in the direction indicated by the arrows, and every time you take an arrow from state s to state s' you write down the letter s', and if s' is one of the consonant states, you can write down any consonant you'd like. The only catch is that you can only follow an arrow heading out of the state in which you find yourself at the current point in time.

For instance, if we're plopped down on the state labeled "consonant₅," we can either begin by writing down a consonant (following the red arrow back to "consonant₅") or by writing down an a and heading over to the state labeled "a." From there we must either write a consonant (and thereafter any number of consonants for a while) and head to "consonant₁," or go directly to "e" by writing an e.

Any piece of writing we can compose in this fashion without causing our walk from state to state to "block" and come to a screeching halt is said to be accepted by this finite state automaton. I'll leave it as an exercise to the reader (I love that phrase!) to check that this automaton accepts precisely the poems generated by the permutation method I described in this post, in which your permutation is (a,e,i,o,u) and your sequence is {1,1,1,...}: you have to use the vowels a, e, i, o, and u in strict rotation, but consonants can be thrown in willy-nilly. (And yes, "willy-nilly" is a technical term [just kidding].)

How does this relate to Hamlet's soliloquy, the subject of my last post? The diagram in that post shows another finite state automaton, and this one is the minimal automaton of the same sort that accepts Hamlet's famous "To be or not to be" speech. By "minimal" I mean that this automaton has only those transitions from state to state that one needs in order to generate that soliloquy: any more edges would give you more flexibility than you'd need, and moreover you'd be able to generate more than you'd like. For instance, as one of Erdrick's students astutely noticed, in Hamlet's automaton you can only reach "z" by coming from "u," and from "z" you can only travel back to "z" or to "l," suggesting that the only z-word you're allowed to build is "puzzle." This is indeed the case, and this limitation puts a strong constraint on the sorts of written work you can produce using this automaton, and as I'm fond of pointing out, poetic constraint is a heckuva lot of fun.

Indeed, there are dozens of poetic possibilities from this point, and I've already started writing a paper that describes a number of them. I'll mention a few interesting constraints, and I hope my readers will help me to explore these and share with me the results of their explorations:

1. Given the minimal automaton for one piece of writing (maybe one that already exists, like Hamlet's soliloquy), generate poems that are "entailed" by that piece of writing, in that they too are accepted by the first piece's automaton.

2. Write the shortest possible piece of semantically and syntactically meaningful prose or poetry whose minimal automaton is the "complete" automaton consisting of all 26 potential states and all 26² transition arrows. I envision lines like "'To Iraq!' Zelda cried."

3. Find the longest possible semantically and syntactically meaningful piece of writing that is completely determined by its minimal automaton, in that the only meaningful (in whatever way you choose to define it) pieces accepted by that automaton are "subpieces" of the original one. For example, even the simple sentence "Me too!" gives rise to an automaton (below) which accepts not only "me" and "too," but also "met" and "to":


4. As my colleague Erdrick has pointed out in private correspondence, maybe you want to introduce whitespace states? Such a modification might be either enabling or disabling, depending on how it's handled...

In any case, there'll be much more to come about this in future posts as I develop the technique, write the paper I referred to above, and actually write some damned poems that fit some of the above constraints. If you'd like to play around a bit, I'd love to see what you come up with. Just remember that you saw it here first!

More face time

"I don't know if you knew this," said one of my Abstract II students to me just moments ago, "but I'm planning on being a teacher."

"Excellent!"

"Yeah, so...the more face time as we can get in front of class, the better."

"Excellent!"

"I know I griped a lot about the way graph theory was taught, but that was right around the time I was beginning to think about teaching."

I hope more students appreciate Moore's method!

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!

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.

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.

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.

Graph Theory: Day 2

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

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

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

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

Then we began.

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

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

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

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

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

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

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

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

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

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

I'm already looking forward to Friday.

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

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

One down, a whole bunch to go

Day One.

A difficult beginning?

Not really.

The day was tiring, but pleasant.

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

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

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

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

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

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

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

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

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

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

I'm just beat.

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

Giassou!

All beginnings are hard

"All beginnings are hard."

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

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

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

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

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

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

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

How will it go tomorrow?

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

Beginnings are hard.

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

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

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

Problematic

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

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

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

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

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

Further bulletins as events warrant.

A few days late: what's up?

I promised another entry last Saturday, now that Saturday's come and gone, and beyond it another week, come and gone. Promises, promises...

Last Saturday brought my first Super Saturday of the season, in which I and three of my stalwart students spent an hour and a half (thanks Deidre, Belladonna, Sieglinde!) with seven little 9ish-year-old munchkins, teaching them the ins and outs of binary arithmetic, and how it can be used to make hard-to-break codes.

This lesson I used last semester with the Spring installment of Super Saturday. The kids then were on average a year or two older and a bit quicker out of the chute: I remember that it took only a few minutes to run through binary arithmetic, almost all of them had seen it before and knew all about it already. In this new group only one of the seven had seen it before, and he was willing enough to let the others catch up. We spent about a half hour learning to count by twos, and then another forty minutes or so running through an example of the code. Time ran out while they were working through their own examples, but I think most of them were getting the hang of it in the end.

Later that day...I finished grading for the weekend sometime in the middle of Saturday afternoon, and left teaching alone until Sunday night, when Griselda and I talked on the phone for about four hours, the bulk of our conversation, as usual, about our respective classes, colleagues, and institutions. She gave me an encouraging word regarding my plans for a Moore-method graph theory course next semester (verbatim: "Do it! Do it!"), as I'd asked her to do several times by e-mail.

Then came Monday, ushering in a hellish week of work, work, work. I was able to get ahead in my class prep on Monday morning, whipping together several note sets, worksheets, and handouts for both of my courses. I got everything set through to this coming Monday in 280 and through Wednesday in Calc I, freeing me to work on other things, like grading 280 homework sets and papers (oh, by the way, here's a link to a compilation I made of what I felt were the most insightful comments in their Professional Proof Analysis papers), helping out with Who Wants to be a Math Major?, putting on Part II of my Research Seminar talk, meeting with several advisees, putting some finishing touches on the grant proposal I'd be bringing with me to Fayetteville for the NSF Day held there on Friday, getting exams ready for my Calc kids' Thursday thrill, touching base with all three of my independent study students, and hauling ass eastward for a less-than-24-hour whirlwind tour of the drizzly and depressingly drab city of Fayetteville. Tons of ideas took shape there, and on the way my highly experienced colleague Quimby giving me plenty of food for thought. On the road there and back, he was able to help me sort out and solidify several promising ideas for future initiatives in both teaching and research. I was also able to meet up with folks at nearby colleges whom I'd not met before. Here's a big fat CoB shout out to Winslow and Queshia, and to Bonnie!

Last night I staggered, overdressed, into Robinson Hall at the ungodly hour of 10:00 p.m., having spent little more than 24 hours away and not having been home since the morning before. Strangely enough, the third floor hall lights were on, the Math Lab door was open. My 280 students Quincy and Olivia were there, working away on their homework from my class. (It's no wonder they're doing so well...) After calling Maggie to bring the chariot round and pick me up, I unlocked the installation blocker on one of the Math Lab computers and downloaded a workable LaTeX editor for Olivia, something I'd promised to do for her long ago. She was, to understate the matter, happy.

Super Saturday returned this morning, six kids, four student helpers, all from the second section of my Calc class. For an hour and a half we worked away on fractals. After an easy definition and a few simple examples, I brought out the by-now-beloved L-tiles. The elementary-school kids were ecstatic when they beat the college kids in creating the L₂ from four L₁s, and it took little more time before both teams worked together to build first several separate L₄s, and from these an L₈. Next came a music video built upon the Mandelbrot set and a round of free-form fractal building, finished off with a level-2 Sierpinski pyramid. I think we all had fun this morning, but I was most heartened by the turnaround shown by the class's youngest child: Boudica began the morning too terrified to even sit next to the class's other young girl, and she spent much of the period working by herself. While working on her own fractal, she sat next to my Calc I student Henrietta, a warm and friendly young woman (whom I've been lobbying heavily regarding a math major...she's clearly passionate about the subject) who was able to put Boudica at ease. By the time her parents came to pick her up, she was hell-bent on putting together a few more components for our communal pyramid, and Boudica didn't want to leave. "I want to come back next week," she said. "This is awesome!"

I spent the next hour doing a bit of grading, and after lunch with Maggie at Noi's Thai Kitchen (mmmmm...spring rolls...), I got a couple more hours in. And (you'll be amazed by this!) after dinner came...more grading! I'm finally done grading all of the Calc kids' exams, and about a third of the way through the homework.

Whew.

Damn.

I'm tired.

The last couple weeks have both been 90-hour weeks.

And you know what?

I'm pissed.

Weeeeeeell...not pissed. Just disappointed.

Why's that? Let me answer with a question.

What's up, Calcsters? What happened?

Let's just look at the numbers: the course average (both sections) on Thursday's exam was just a hair under 68%. While 9 out of 54 people taking the exam got As (including a 99 and, yes, a single 100) and another 8 got Bs, there were also nearly as many Fs. There weren't many low Fs, but a few too many Fs overall to let me sleep easily tonight. The performance was perceptibly bad: on Thursday afternoon, Neville, easily one of my first section's best students, offered me a humble apology as he handed in his homework. I nearly melted.

"I wanted to apologize for my performance on this morning's test."

"Neville, it's okay. We all have our off days."

"I just know I did really bad. I've never done that bad on a math test before."

"Did you feel that it was too hard of an exam?"

"No! That's the thing! I knew how to do all of the problems, I just looked at it and went blank."

"Even so, there's no need to apologize."

"I just didn't want you to think any less of me."

"If there's one thing a good teacher can't afford to do, it's let his students' performance affect the way he feels about them personally."

So I ask again, what happened? I agree with Neville: I don't think the exam was too hard. It was hard, to be sure, but not too hard. Moreover, it was very similar to the practice exam, and some of the exam problems people did worst on are ones dealing with topics through which we spent a great deal of time working (like related rates).

As I started grading the homework just a few hours ago, I got a clue as to what might have gone wrong: it's clear that nearly none of the students did any of this most recent homework set before taking the exam. Though I assigned the related rates problem set early last week, early enough so that I was able to offer those who wanted it the opportunity to turn in the homework a week early and get feedback on it for study purposes, most of the students didn't get started on it and the two accompanying homework sets (exponential and logarithmic models, and linear density) until the wee hours in the lead-up to the exam, or later still, after the exam was over.

I really hate to use this word this term, y'all, but there's one word for that sort of academic behavior, and that word is stupid.

Settle down, settle down: I'm not calling you stupid, I'm just saying you made a stupid move. There's just no other way to put it.

You simply cannot expect to well on a difficult exam if you've not done the homework for the sections with which the exam deals by the time you've taken the exam. (One or two of the brighter students might be able to get away with this kind of crap, but they're living on the edge. Don't follow their reckless example.)

I'm feeling disappointed right now. I'm feeling disappointed, helpless, and, to be frank, a little hurt.

Disappointed: you're smart kids. You wouldn't be here if you weren't. UNCA's a tough school to get in to. You're the cream of the crop, in many ways. You're smart cookies. Every last one of you has the potential to do well in my class, and I'm disappointed that many of you are clearly not doing what's needed to wrestle down and master some of the concepts we're dealing with in this class. You can do it, that's what frustrates me: you can do it, but you're choosing not to.

Helpless: what else can I do? I sometimes feel like I've done all that I can. I work my rear end off each week, forcing me to ask myself whether I need to invoke the tired old aphorism and work smarter, not harder...but how so? I pride myself in my ability to blend traditional teaching methods with more innovative ones; what better way to make most effective use of the time given to me? I troll the Math Lab, helping out any students I come across in my travels. I offer up nearly endless office hours, my open-door policy allows students to drop in and chat just about whenever they want to. I put together practice exams and solutions, similar, as I always say, to the actual exams in length, content, and format. I give review sessions, in some of which are solved problems that will appear verbatim on the next day's test. I do a lot, and moreover, a bit immodestly, I must say that I think I do it well. Some might say very well.

Mother of Claude, what more can I do?

Thus, helpless.

Hurt: why? As I told Neville, it's unwise for a teacher to let his students' performance affect the way he feels about them personally...moreover, I shouldn't let my students' performance affect the way I feel about myself. But I do, if only a little bit. I feel hurt because I fear a disheartening answer to the question, "do they just not care?" And I want so much for you to care. I want you to care, as much or more than I want you to succeed. I don't feel that I've succeeded in guiding you through my course unless you not only understand, but you also share at least some of the excitement that I feel for what I do. See, I very truly believe that mathematics is a beautiful thing, I care deeply about it, and I bring my enthusiasm about it with me to the classroom. My excitement isn't an act, it's genuine, and I hope that excitement rubs off on you, if only a little.

Where's that leave me? Where's that leave us?

Look, y'all: I know the score. As a rule, you're young, you're full of life. You're free, many of you for the first time in your lives. Most of you are being pulled in a zillion and a half different directions, and you're trying to figure your lives out. I understand that it's hard to manage your time, your space, your resources, in some fashion that allows you to finish everything you need to do in your day in time for you to plop your head restfully down on your pillow before the next morning comes.

Yeah, you've got a long list of things to do.

And guess what? My class, and the work it comes with, are on that list.

You signed up for it, you've fought through the semester this far, we've got about five weeks left. Let's stick with it, all right?

So where do we go from here? Here's a short to-do list:

1. Do the exam revisions. Seriously. Do them. Even for those of you who scored in the 40s, it shouldn't take you more than a couple of hours. Sit down, get out your book, find yourself a quiet hour or two in the library or your dorm room or your apartment, and do the revisions. Do them neatly, fully, correctly. If you all can manage half credit back, you can bring the class average on the exam back up to a more-than-respectable-in-fact-pretty-damned-good 84%. As you do the revisions, try to understand how you messed up when you did. Keep in mind that I allow revisions not because I want you to get a high score on the exam (that's a happy bonus), but because I feel that your understanding of the ideas we've talked about comes before everything else. That being the case, I want you to take this chance to grapple with the concepts that may have eluded you before.

2. I've said it before, I'll now say it again, and I hope that maybe with those pretty sad-lookin' exam grades standing right behind me, you'll all take me seriously this time: do the homework. Seriously. Do it. Get started on it early. "Early" does not mean "on the evening before the homework is due." Rather, "early" means "right when the homework is assigned." That night. Set aside a half hour or an hour to get started. Don't feel the need to do it all at once, unless you find yourself having fun and cruising right through it. Do a few problems, enough to get the hang of what you're doing. If you get stuck, get as far as you can, write down whatever ideas you've got for solving the problem, and move on. Make a note of your difficulties so you can catch me after class with them, or bring them to me or to the Math Lab folks later on. Work on the homework a little at a time. Don't get frustrated. If it's just no fun anymore, put it down for a few hours and come back to it later. I don't assign all that much of it; almost without exception the work I give you in any week should be doable within three or four hours all told, and if you spread that out over the week you shouldn't have more than a half hour or an hour a day.

Hey, if you want to be where the action is, go to the Math Lab, hang out there, do your work there. You'll find a lot of your friends in the Math Lab: a lot of you already come and spend a good deal of time there, so you won't be alone. There's absolutely no stigma attached to hanging out there. It's a free service, they've got cheap coffee, cheap tea, and often a ready supply of free food. Smart people roam the Math Lab. They're paid to help you. It's a good place to be.

All that I ask is that if you make use of some of the Math Lab's resources, like the solutions manuals, don't abuse them. That is, don't allow yourself to become fully reliant on them, don't use them as crutches. A few of you, I can tell, "complete" your homework by copying the solutions from this book. You might not mean to, you may have every intention of doing the work yourself. But you come on in, you camp out in front of the manual, and you start to work. Here's how it might go:

"Okay, let's get started! Problem number 17...all right, here we go. Okay, I know how to get this started...there's f(x)=sin(x)...okay, now let's see...hmmm...what's next?...[15 seconds later]...I'm stuck...let me just peek real quick here...oh, yeah! Duh! Okay...now...yeah. Hmmm...I know this...[10 seconds later]...just another peek. Zomg! I knew that! I'm almost done now, I can do this in my head...write it down...and...there! Okay, let me just check my answer...[peeks]...wait, how'd they get that?...oh, yeah, I see...[erases]...that's what I meant. Duh. Okay, next one, number 22..."

How can you effectively make use of the manual? Use it only to check your final answer. If you didn't get the right answer, walk away from the manual and see if you can find your mistake yourself: look back over your work. Does the problem present itself? It might. Many errors are easy to catch yourself. If you can't find anything wrong, flag down the Math Lab assistant, elbow your friend working next to you, or come across the hall to my office. Ask someone else to take a look; generally another set of eyes, even if it belongs to someone less mathematically adept than you are, will do a good job of spotting something amiss. If you find a mistake, revise. Give it another go. Only go back to the solutions manual once you've got a revised answer.

By the way, I can tell if you're relying too heavily on the solutions manual when you complete your homework. (I'm not a fool, however well I play one in the classroom.) If you're addicted to the manual, you're only hurting yourself, since it'll nearly always kick you in the pants come exam-time. Here's a tip: a high percentage (over half) of the people who are nailing the exams to the wall ("nailing" being defined as "getting As"), including the one person who's received the highest scores on both of the exams so far, are doing sub-optimally on the homework. They're not doing horribly, but they're not doing perfectly either.

You wanna know why that is? Because they're actually doing the work. They're actually (horrors!) making mistakes. Because even though they're really smart, they're also really human. They're making honest mistakes, and they're giving themselves the chance to learn from their mistakes. Their homework isn't perfect because they're legitimately trying, and not just copying the answers from a book, however innocently that act of copying might be.

And by the way, if I sound cliché in offering up all of this tired old advice, I apologize. I'm only shoveling all of this shit because it happens to be true. Believe me, I don't wanna sound like an old-fart fogey, any more than you want to hear me sitting here lecturing you. I wouldn't say any of this if it weren't true, and if I didn't care.

3. Come and see me if you're having difficulties. I'm not a scary guy, I'm probably one of the most approachable professors on campus. (I'm a bit of a nerd, but I can't really help that. At least I recognize that fact, and I revel in my nerdiness.) Don't worry, I don't hate you because you're doing poorly. I don't hate you at all. As I'm fond of saying, I've had two students out of the thousand or so I've taught over the past decade whom I really just couldn't stand. And you are neither of them.

I don't hate you, I'm not mad at you. I might feel bad for you, that you've gotten a rough start, that you've slipped so far behind. Whatever position you're in, whether you're one of the best students in the class who's having a momentary lapse of reason or one of the weakest students in the class who's struggling mightily to just keep up, I'm ready, willing, and I hope able to help you. It's my job. It's what I get paid to do. More than that, it's what I'm most passionate about in life. Funny, isn't it? It's funny that you've got someone sitting there nearly 60 hours a week who gets all fired up about the possibility that you'll traipse into his office and ask him for help? Funny.

4. In class, take notes (please don't think I don't notice you when you're not doing so, and don't think I don't see the correlation between not taking notes and doing poorly on the exams...as I said before, I'm not a fool, and I'm probably one of the most observant people you've ever met), ask questions, and if I ask you to take part in a group activity, please take part in that activity. Some of the group activities seem corny, I know. (Just wait for the Mean Value Theorem exercises coming up this week!) I'm a bit of a cornball, it comes with the nerdiness. But cornballery can be fun if you just let yourself be taken away by the cornballitude. What's more: I've spent thousands of hours designing my classroom activities, and I don't do anything in the classroom unless I think it serves a useful purpose, so nothing's ever corny for the sake of corniness. Please keep in mind that my primary goal is to help you understand what we're talking about on any given day, and I'll do anything I damned well can to meet that goal. I'd really appreciate your cooperation in this endeavor.

You know what? I'm feeling less disappointed than I was, less hurt, and less helpless. I feel like I've actually done something, I've managed to unburden myself. Writing this entry, though it's cost me another three hours, has really helped me work through this issue.

I'll finish grading your homework tomorrow, folks, and I promise that I'll be in a better mood. I'm there already, in fact, as the end of this leviathan post draws into sight.

We've got four or five weeks left in the class, and a fair chunk of hard work ahead of us. I won't promise that it'll be easy, but if you stay with me and you keep on top of the work, I promise you'll make it through alive.

So how 'bout it, huh? Are you ready?

I'm ready.

Let's do it.