Not your grandmother's calculus class

I'm writing this from my boyhood home (right down to the house) in Helena. Thanksgiving's still a couple of days off, I'm enjoying a little "leisure time" in the snowy wilds of Western Montana, and I thought I'd take an hour or so out to pen a new post, mostly by finishing off one I'd begun a while back but hadn't posted.

What is it that I love about teaching?

It's the little things, y'know? Little successes, minor (or not so minor) victories.

The Calc I students spent the past few days putting together their final writing assignment for the Newton v. Leibniz project, a reflection paper (the prompt for which can be found here) describing whatever it is they feel they got out of the project. I offered to take a gander at any rough drafts they'd like to slide across my desk, though only two of my 55 students submitted such a draft.

One of these drafts was a remarkable one, and its finalized version still stands as one of the strongest I've yet read.

After class early this past week, Uta quietly handed me a rough draft, and my spirits soared as I read it.

An excerpt (thanks, Uta, for letting me quote this!):

"When the [Newton v. Leibniz] project was first brought up in class, I assumed that it was going to be easy to research the historical dates of important events and the people involved. I was completely wrong in my assumption. While researching, I noticed that several different sources had different dates around the same year of when a document was published or a letter written. It was extremely frustrating, considering I wanted my research to be as historically accurate as possible. During the trial, I noticed some of the historical experts had slightly different dates than what I had found in my research, which was completely understandable. If the individuals in the 1700's had a difficult time proving dates and events, pinpointing specifics, then it is completely normal for people now to deal with the same conflicts."

Of course, I'm assuming Uta's not just saying what she thinks I want to hear, but she's never struck me as the sort of student who would do that.

I gave no lecture on the complexity of historical data and the inferences which can be drawn from them, no prompt asking students to describe the quality of the resources they were able to find. This excerpt was wholly unsolicited, purely personal, reflective.

I'm a happy camper.

I've read all of the reflections from the first section, and the second section beckons, sure to be read tomorrow. I've gleaned some more quotables from these reflections, many of which I hope to quote once I get clearance from the respective authors.

A few general comments of my own: I should begin by noting that the reflection papers that were most meaningful and helpful to me were those written by students who dared to say what they weren't so sure I wanted to hear. Those who were willing to be critical, to challenge me, to provided open and honest feedback, were among the more interesting and useful papers to read.

For instance, a couple of students mentioned being disappointed in the trial, feeling as though their own preparedness had been wasted by classmates who hadn't taken time to put together a decent case. Both of these students had advice (very little of it highly specific, sadly) for me on how to improve the project the next time around. I might solicit further information from them.

The students made a number of interesting observations about the nature of discovery and about the particular case we'd considered. Several made a distinction between "invention" and "discovery," several made reference to the collaborative nature of scientific study, and a few spoke of the social construction of knowledge, though not in such specific terms.

I'm looking forward to reading over the reflections from the second section. That section's trial was more animated, truer, more well-organized. The principal parties in that section's trial clearly had bought into the trial's premise and went at each other like two tomcats in a backyard brawl. The result was a more authentic re-enactment, with more excitement and engagement.

Since then the Calcsters have finished off their third mid-semester exam, this one take-home. I spent most of the plane trip back to Montana grading them, finishing during the trip's final leg. They did quite well, considering the difficulty of the exam. There were several As, several more Bs, and very few people failed (far fewer than on the in-class exams). I'll be offering the chance to do revisions once more, since besides the final writing assignment (constructing a math-themed poem...see the prompt here), the kids won't have a huge amount of work before the final exam.

For now, I'm going to take a little more time off. I'll get to the reflections tomorrow, and check back in then. I've got more to say about my 280 folks, too, as our time together draws to a close and I look ahead to administering our Writing Assessment project's post-survey.

Adieu!

Numerical nightmares and Newtonian nattering

Could it be that I became a mathematician in order to exorcise personal demons?

At dinner tonight, I was telling Maggie about an unpleasant dream I had last night.

As in many of my similarly unpleasant dreams, I had lost myself in the heart of a giant hotel, one replete with cascading staircases, ornate chandeliers, and atria with glass ceilings towering to dizzying heights. I'd lost myself, I couldn't find the room I was seeking, didn't know if it even existed. Somehow I sensed I was late for something. Hours, days, even weeks late.

"Were you at a conference?" Maggie asked. I suspect so, and I said this.

"It's weird how many 'conference anxiety' dreams I have," I commented.

What could it mean? Is this indicative of some fundamental insecurity in the quality of my research? Or does it belie a more general sense of confusion or befuddlement, some sense of unease at my place in life?

Tonight's conversation reminded me that I'd mentioned a childhood anxiety dream to Quimby's wife over dinner last night. She'd mentioned that she'd had a dream in which she was asked to count out some exceedingly large sum of money, and just as she was finishing, she lost count. When I was a child, I then explained, I had a recurring dream in which I'd been asked to count to astronomically huge figures, and the simple act of counting so high terrified me. "I woke up in a sweat," I swore.

Not falling, for me. No death by drowning, hanging, shooting, or fire. No ordinary nightmare would do.

For me it had to be counting.

The most common theme truly was an astronomical one: I'd be hauled out into the midst of a far-off asteroid belt, where I'd be left, presumably with some sort of life support system, and having found myself a cozy place to sit on one of the larger planetoids, I'd set about counting.

Millions gave way to billions, and billions to trillions and so on...fast forward, and before I knew it my dream-me was sitting there, enumerating the asteroids with fictitious numbers I'd invented for the sole purpose of completing my never-ending task.

Jung would have something to say about some sort of Sisyphean archetype.

I would often awake from these dreams too scared to lie back down for another hour. I'd often have to walk it off, pacing the kitchen floor in the sharp blue light of the microwave's LCDed digits.

I don't remember how young I was when I first had these dreams...twelve? Ten? Surely no younger than that.

Number.

Maybe Number has always had something to say to me, and I've spent the last few decades learning how to say something back.

Next week I'll be asking my Calc I students to call upon metaphor, synecdoche, all kinds of poetic imagery, to explore their feelings about mathematics. I hope that they'll take care to have fun in writing a poem about math, but I hope also they'll take the exercise seriously enough to discover something truly meaningful about their own feelings towards mathematics.

What does Number say to them? What can they say back?

Meanwhile, I really ought to say something about today's trials.

They were great.

I gotta admit, what they pulled off today was a tough task: the active participants were asked to think on their feet, to come up nearly instantly with viable explanations for actions that happened thousands of miles away hundreds of years before they were born. They had to continually adapt to newly-discovered evidence, they had to keep tabs not only on their own lines of attack but also on their opponents'. They had to shift chimerically from one train of thought to another, without skipping a logical beat. And they had to do it all in front of a couple dozen peers and their instructor.

Both trials saw their fair share of ers and uhs and uncomfortable pauses (as I put it in a congratulatory e-mail I sent to both classes), but given the leviathan load with which they were tasked, a fair amount of hesitant deliberation was understandable.

They did well.

The exchange got downright vitriolic at times. In my second section, especially, the litigants threw themselves into their roles and engaged in vigorous and rigorous debate. Once or twice during that section I was sure it would come to fisticuffs. Throughout the affair, hastily scribbled notes were passed from colleagues to legal teams, and there was ceaseless whispering between the debaters on either side. There were numerous and strenuous objections. I tried to be as fair as possible in allowing both sides some, but not too much, leeway; I hope this is how my actions were perceived. I attempted to quash speculation and focus on fact, but given the fineness of the line between the two, it was a difficult rope to walk.

Beatrice, who along with Farrah had spent a good deal of time during the trial passing notes to Leibniz's legal team, mentioned after class that in hindsight both of them wished they'd proposed to take a more active role than that of Leibniz's colleagues. "We didn't realize until today that we'd've had fun in that role," she said. Cassio, as Leibniz, did an admirable job in defending himself, and he was ably helped by Xavierina, his lead counsel. On the other side, Tallulah was chief counsel for Newton, played, astonishingly enough, by Newton, who was rather reticent and let Tallulah and Ambrose do most of the talking.

I talked with both Tallulah and Cassio a bit after class. Both thoroughly enjoyed the experience. "At first I thought, 'how are we going to stretch this out to fifty minutes?' " said Cassio. "Now I wish we'd had more time." Others agreed. "Why couldn't we continue on Wednesday?" a few folks asked.

So little time, alas!, so little time.

I thank you, one and all, for putting your time and effort into this endeavor, to make it the success that it was today. What will you have to say about the experience in your reflective essays? I can only guess at this time.

Another exploratory moment

Okay, so I'm not giving an update (yet) on how Newton v. Leibniz went down...I've just remembered that I'd meant to spend a few moments this weekend chronicling last Friday's 280 meeting, in which we took a detour from my planned path and did a little mathematical bushwhacking.

Given numbers n and k, let Z[_k] be the set of integers modulo k, and define f_n from Z[_k] to Z[_k] by f_n(x) = xⁿ mod k.

The questions I'd planned to ask: is does f₂ = f₃ hold when k = 2? This question was quickly dispatched, as was the corresponding question when k = 3.

From here, it was impromptu math.

Quincy and Uri and a handful of others were curious enough to press the matter further: suppose k = 2 and we want to know when f_m = f_n; what then?

This question too was a simple one, and was easily answered.

What about when k = 3?

Not a problem: noting that 3 = -1 mod 3, we were able to tackle this question as well...

...the course had become an expedition, we were making up mathematics as we went along.

I felt a bit bad for several people in the class, since it seemed as though four or five individuals in the class were deeply involved in the discussion while the others were relatively disinterested onlookers.

There were no objections, though.

It was exciting!

Now I've got to go again...I've now got most of the final drafts of the N. v. L. arguments, and I'll be taking those home tonight to look over them.

It's been a lot of work on all of our parts. I hope this project has proven meaningful to the students. I look forward to reading their reflections, due next Monday.

Ciao!

Super Saturday pics

For those who'd like a window on the world of Super Saturday, I thought I'd post a few pictures from the past few weeks of the class. I apologize for the delay in getting these up! Many thanks go to Tallulah, Beatrice, and Belldonna for serving as photographers over the past few classes.

First, on "Build Your Own Fractal" day, the little kids show off their L₄, assembled from L1s:



Now, the college kids get their turn:



A bit later on, everyone's busily working at putting together tetrahedra for the Sierpinski pyramids we worked on as a team (we finished two level-3 pyramids, but were too short on time to finish the two more we'd need to make a level-4):


A week later, amidst a tense game of Toss and Sort: can everyone find their way back to their home vertex?


This past week, while "Bending Time and Space," yours truly looks on as Betty Sue throws herself into the unenviable task of sorting out the hundreds of polygons we'd cut up, making it easier for the kids to find what they'd need:


Trixie helps a few of the younger folks put together those polyhedra. As mentioned before, her hand-painted polygons were a huge hit:


The boys had a chance to show off their finished products:

...As did the girls:


Next post: how'd Newton v. Leibniz go down?

Never too many cooks

Howdy, folks!

Yesterday's installment of Super Saturday set a "Math Discoveries" record for most student volunteers, with six stalwart students showing up, breaking the mark of five, met earlier this semester and originally set in Fall 2006 when five students came out to help organize and oversee a series of mathematical games. Many thanks go to nearly-omnipresent Beatrice and Belladonna, the irrepressible Tallulah and her roommate Betty Sue, who isn't even in my class but thought it might be fun to come along, Sieglinde, admirably representing my morning Calc I class, and first-time shower Trixie, whose hand-painted polygons were a hit with everyone (I likened them to stained glass, and one of the little kids thought they bore a favorable resemblance to wood chips). Thanks also go to the eleven Calcsters who, though unable to attend, each contributed a hundred or more poster board polygons to the effort, they were very much appreciated.

As with any successful Super Saturday, I'm not sure who had more fun, the college kids or the young 'uns. Both big and little fingers had a hard time at first in managing the assembly of cubes, dodecahedra, and icosahedra. Once we got the hang of it, though, it was smooth sailing, and it was hard to stop. Several of the little kids left for home with their own polyhedra and handfuls of unwed polygons, some just took a few to use as templates to trace their own. Tallulah and Betty Sue vowed to spend some time this weekend making more polyhedra with which to adorn their room (maybe they'll start a craze?). Several of us old-timers hung out well after class was over, finishing off particularly large polyhedra and chatting about next semester's schedules.

A moment of crowing: I've won two of these six over as Math majors, and I've still got hopes for a third!

Another member of my morning section approached me last weekend regarding a Math major, and I was heartened to hear while speaking with him after class on Friday that he was convinced in part because of this blog. Orville, you have no idea how happy that makes me! Welcome aboard! Any questions you've got about the major, please ask. I like to think that my approachability and others' is part of what makes our department and our major such a popular one, and such a strong choice. I really do believe that ours is not only one of the best programs on the campus, but also the friendliest.

Going backward in time...on Friday afternoon I spent an hour with a couple of my colleagues in talking over Chapters 3 and 4 of Bob Moses's Radical equations. Both remain somewhat cynical, one regarding the entire venture, another regarding the relevance of the Civil Rights Movement in all of this math talk. For instance, one doubted the strength of the "ball bouncing" analogy invoked by Moses: if you want to get their (the kids') attention, says Moses, go to the corner and start bouncing a ball. At first they might not take notice, but gradually they'll come, and they'll ask questions to find out what it's all about, and before long you'll have a game going. "I can't really see myself bouncing the ball," my colleague admitted.

"I don't think Moses's point is that every person reading this book has to be a ball-bouncer," my other colleague pointed out. "Everyone has a part to play in this, and many people will be acting behind the scenes in some organizational capacity, you don't have to stand on the streetcorner with a ball."

"I don't think Moses anticipates that every person reading this book is going to rise up and become a part of the movement," I added. "If for every ten people who read the book only one hops aboard, then that's fine."

Something is better than nothing, someone better than than no one.

I feel that our discussion was a good one, and it's helping me to understand the weaknesses of Moses's approach, as well as its strengths.

This past week I approached the director of the Teaching Fellows program at UNCA, asking her if she thinks she'd be able to interweave a reading of Moses with her program, much as she did with Jonathan Kozol's The shame of the nation last year. I've yet to have a real-time conversation with her on the matter, but from our one e-mail exchange I think she might be up to the collaboration. I'd love to get some of my students on-board with the reading circles. How 'bout it, readers, are you up for it? (Don't try to hide: too many of you have outted yourselves as regular readers for me to think you're not out there!)

I've had some more thoughts about what a more learner-centered Calc I class might look like...having been reminded this past semester just how loathsome "word problems" are to math-minded freshpeople, perhaps it would be best if they spend a semester never seeing a problem that isn't a "word problem." That is, from Day One through Day Sixty (or however many days there are), every example considered would be embedded in the context of some application, no matter how simple or straightforward. No computation would be without at least some interpretation requiring a modicum of extractive analysis. Sure, it would be a bear at first, but the students would grow stronger and stronger as the semester wore on, and by the time they'd get to topics in which "word problems" are traditionally replete (related rates?), it would be nothing new to them, and they'd breeze on through.

Something to think about.

The difficulty, clearly, would come in fitting computations classically done for their own sakes out with realistic applications. I ain't despairing. It can be done, it'll just be difficult.

By the way, for those keeping score at home: I don't think I'm arguing for a return to a "reform" curriculum for calculus. I'm not, for instance, suggesting that the formal definition of a derivative be put off for weeks beyond its natural point of introduction. I'm suggesting that a more "traditional" curriculum be retooled to accommodate meaningful and motivating examples.

Before I close this post, I should mention that tomorrow we try Newton v. Leibniz. I'm happy in that I think both of the primary parties in the trial, in both classes, have done a good job in preparing. The worst-case scenario would have involved Newton and his team damning the torpedoes and cruising ahead with full sail while Leibniz et al. drifted around in front of them on a chunk of creosote-soaked flotsam.

We'll see how it turns out. For the time being, I've got some Mathematica code to write to help me out with my research, and I promised a few Calc I kiddies that I'd try to slap together a practice version of the third mid-semester exam, to be handed out this coming Thursday.

What gets you hot?

What is it that makes one love math?

And can one teach another to find whatever that is inside of oneself?

My ex-student and good friend Mariposa, now a middle-school teacher in Virginia, today told me a story about one of her sixth graders. During class she'd given the students a brief history of science, in which she'd lingered on Newton, indicating that he'd found a way to compute the area of objects that were squirrely and squiggly, and not at all so well-behaved as prim and proper objects like squares and triangles, and that the way he took led him to calculus.

The student in question clearly thought long and hard about how one might go about finding such areas, and in class the next day he submitted to Mariposa an unsolicited brief that told how to find the area of a curvilinear figure: one needed to trace a graph around it and then divide it up into little boxes whose side lengths you knew ("they can either be centimeters or inches"), counting the boxes so enclosed to compute the area.

Not far off. Not far, at all.

Remember, we're talking about an eleven-year-old here.

Right now I've got a handful of students in whom I see that sort of passion for math, that willingness to think. They're spread pretty evenly between my 280 class (a class more-than-usually-heavily-weighted in the direction of the Atmospheric Science department) and my Calc I sections...feeding that passion proves hard, though.

Most of the time the Calc students simply don't seem to have the time to spend on "extra" math problems, so even if they seem naturally predisposed to think about math and I pitch them an odd problem from graph theory that's well within their reach, they don't have a chance to follow up on it.

And by the time they get to 280, many students (busier now, but generally with a more well-tuned work ethic and time management skills to match) look on math as a job, and not something to be taken lightly, for its own sake and for nothing more.

I should be happy that the Problem Group meetings are still drawing a solid core of five or six students, even though we're scrubbed from the Putnam list this year and the Virginia Tech Exam has come and gone. And that we're still getting a small several students coming to the research seminars on Tuesday afternoons.

Still and all...

...How does one best make math intriguing, interesting, exciting, sexy?

Through relevance? Is it enough to point out how it's useful, where it naturally arises? This seems to satisfy some.

Or aesthetics? Expander graphs, fractal drumheads, Julia sets, tournament diagrams...is mathematical beauty in the eye of the beholder?

What about mystery? I suppose that as many as are turned off by the challenge of the unplumbed depths of mathematics are spurred onward by it.

To this end, the counterintuitive is a big draw: that chaos can be quantified, that infinity can be counted. There's something of the numinous in the statement that there is no such thing as the set of all sets, or that every set can be well-ordered, or that with a period of three, every period is possible.

Even in calculus, there is profundity: I fear many mathematicians today fail to appreciate the beauty of something as simple as the Fundamental Theorem, so familiar is it, and so little changed over the past three hundred years. Something so old, so well known, and so well accepted can't be interesting, can it? Does it take its rederivation by an eleven-year-old to make us remember how fucking incredible it really is?

Discovery is something we should demand from every one of our students. Maybe that's what I'm getting at here: without the chance to discover anything at all, it may be damn near impossible to discover passion.

I'm tired of walls between research and teaching.

The next person to invoke the triumvirate of Teaching, Research (or its less formidable little sister, Scholarship), and Service gets a smack upside the head with a wet herring.

There's that famous shot of the Berlin Wall, hundreds of elated German citizens crawling over its face like ants, pounding away at it with sledges, hammers, fists, watching it crumble below them in front of them, no more to split their city in two.

No more.

So tell me, readers (I know I've got a few of you, judging from conversations with colleagues, students, and assorted others in my life): what gets you hot? Why math, or why not?

I'd really like to know.

If/Then

If A, then B.

If I teach, then they will learn.

If I say it, then they ought to understand it.

If I indicate the relevance, then surely they will see it.

If I dub them "experts," then experts they just might become.

If then two years pass, what happens to memory?

If five?

If ten?

If more?

Then what?

If, day in, day out, condition heaped upon condition, if, if, if:

"If you do this, then..."

"If you do that, then..."

If after if, ifs with fuzzy consequents:

If, if, if...

...then what!?

If uncertainty is eternal...

If the way is shown, the way might then be taken.

If a change is made, a change might then remain.

If I show I am excited, then they might be excited, too.

If I show I care, then maybe so might they.

...then?

If A, might B?

Super Saturday IV

I'm not sure why I got up at 6:30 this morning...at least I got a head start on the grading. I'm halfway through the Calc HW at this point, but 280'll call this afternoon.

At this moment I'm sitting in Zageir 227, 45 minutes remaining before our fourth Super Saturday session begins. I've got nothing to do for a bit, so I thought I'd take a moment to check in here.

How'd the week wind up?

In 280 we talked about injectivity and surjectivity, concepts that are substantially more puzzling in the abstract than they are in the familiar context of real-valued functions. We had a bear of a time working through a proof of the equivalence of the two most frequent characterizations of injectivity. There was something about the subtlety of the contradiction that was tripping up even the best and the brightest in the class. The confusion likely had to do with the convoluted structure of the statement we were trying to prove. Roughly speaking, the statement looks like (A ⇒ B) ⇔ (C ⇒ D), for various statements A, B, C, and D, so at one point we're dealing with a hypothesis that looks like (A ⇒ B) ∧ C. Say huh?

In Calc I we rapped about the way a function's graph is determined by its derivatives. On Monday I'm going to have them do a few exercises as a class to drive home the meaning of properties like concavity and inflection, and how they relate to increasing and decreasing.

Oh, by the way: they made substantial recovery on their exam revisions, raising the class average to roughly 77%, up from the woeful 68% last week. Several students offered touching apologies or narratives on their submitted revisions. Magdalena, for instance, mentioned how she saw herself in the inner monologue I included in the post following last week's grading frenzy. "Oh my god, he's inside my head!" she said. She vows never to become so reliant on the solutions manual again.

They're good kids.

Newton v. Leibniz takes place a week from Monday; this coming Monday I'll get the rough drafts of their "arguments," many of which I will share with opposing teams so that each legal team may adequately prepare a defense having had access to the other's evidence and data. So far most of the teams seem to be functioning pretty well together, with a couple of exceptions. There's been a little tension stemming merely from personality differences present in one or two of the groups. In a perfect world, perhaps...

...you know, when I was a student, I didn't so much like working in groups. I didn't have to do it all that often, but when I did, I got it done with as soon as possible and moved on. I can't say now how I must have come across to my teammates. Bossy, maybe? Headstrong? I hope not. I know that I'm the sort of person who wants to get it done "right," even if that means that I have to take the entire project onto my own shoulders and do every last bit of the work myself. Better that than that the project turn out shoddily because one of my teammates had the audacity to not be as smart as I was.

I suppose I might have been bossy, without meaning to be.

And I wasn't always my sweet current self, either. Though sarcasm is my native language, over the years I've learned to tone it down with others whom I don't know so well. Sarcasm, spite, vitriol. My mother loves telling the story of how I frustratedly called one of my kindergarten colleagues "stupid" because she couldn't grasp the nuances of the game I was trying to play with her. I was insufferable as a young 'un, but I've gotten more mellow with age.

Why can't we all just get along?

More later...I'll let you know how my first session of the departmental Moses Learning Circle went yesterday afternoon (short version: I'm not so sure my colleagues are sold on the claim that mathematical literacy is a civil rights issue, or the claim that improved numeracy is a societal panacea), but for now I've got to get ready for the SS kiddies who'll be here in a few minutes (Tallulah and Deidre are already here, ready to lend a hand).

Avanti!

Continued!

Lunch went well!

I need to stop hanging around with Quimby, he keeps making more work for me.

Short version: I'm now in dialogue with some folks over in psychology regarding work they're doing to put together a grant to get support for a lab they'd like to put together. There'd be a substantial human element to their program, involving, for instance, student researchers. Quimby thought that my experience with the REU would come in handy to them, and that their need for researchers might dovetail nicely into the ongoing STEM-related NSF grant we're all busily working away at over here...it's all good.

After lunch I got waylaid in the Math Lab by student after student...Calc I folks cleaning up their exams (good job on those so far!), 280 students making good progress on the homework and the take-home exam...a good group.

Topic for a future post: how in the hell do you manage blips and burps in group dynamics when students are working together? This is a cause for much consternation. When personalities clash, well...

...to be continued, again...

Harder than it looks

Teaching is hard.

What makes it this way?

After 280 wrapped up yesterday, I was walking back to Robinson Hall with Quincy, and he was rehashing the experience he'd had less than an hour earlier with two others from the class, their committee report on the latest problem set. He was worried that what I had seen as a successful endeavor had gone horribly awry; it hadn't gone as he'd expected it to. It wasn't as smoothly executed, perhaps, and he hadn't been able to clearly get across (pardon my split infinitive) the "subtle nuance" he was trying to point out in the proof they were critiquing.

"It's really hard to lead someone to say what you'd really like to hear her say, without telling her to say it," I assured him. "That's called teaching." From a pedagogical standpoint, I felt that their committee report had been a solid one.

Most heartening to me is the fact that the students have begun to break away from the "here's the right answer" style of committee report with which they began the semester. "We want this to be a discussion, so if you have anything to say, just come out and say it," Nicolette exhorted her peers yesterday. (Like Quincy, I think she feared that their team just wasn't saying whatever it was they'd have to say to get the others out of their seats.) They were aiming at a different model for their presentation: rather than simply hand out the "correct" proof of the indicated proposition, they intended instead to guide their peers to a proper understanding of the weaknesses of the proofs they saw, and of the necessary elements of a valid proof. The second team had the same goal, and they strove towards it with different steps. In lieu of doling out a proper proof, they instead gave an outline of the elements such a proof would need, indicating where they felt people might have tripped up most commonly. Neither presentation was fully explicit, both focused on the process instead of the product, both adopted a more mature attitude regarding the course content than most earlier discussions had.

In case you can't tell, I'm pleased!

I cornered one of the other students in the Math Lab after class and asked her how she would compare this semester's installment of the course with last semester's. (She'd been enrolled in the course in the spring until health issues forced her out about 2/3 of the way through.) She's enjoying it much more this time around, and she feels like she's learning more effectively. The format, she says, is much stronger (what about it? This was unclear. Is it the idea of the homework committees? The structure of the worksheets? The revisions? I'm not sure...I'll have to probe further), this particular group of students is more open to the idea of learning in this way.

Perhaps what's working well for her is the more conscious focus on writing instruction, in particular writing as a discipline-specific endeavor. She indicated that partly as a consequence of our class she's seen her writing improve in all of her classes. (Her partner, who proofreads all of her written work for her [what a kind soul!], has noticed this as well " 'I only had to add a couple of commas for your last paper,' " my student reported her partner's words.) With the clearer writing has come a clearer understanding: her scores this time around are noticeably better than her scores in the spring. As I told her yesterday, I'm so happy I could hug her.

I am dying to find out what kind of responses we get on the exit surveys for the writing assessment project. This is an exciting study!

Yesterday too I had the first of two interviews with one of the Writing Center's new student consultants. I prepared myself by going over the interview questions I'd been sent in advance, and when Beulah came by, I was all ready...perhaps too ready...with my responses. I actually printed out the abstracted answers to her interview questions that I'd typed up for myself, and asked if she'd like a copy. She accepted them happily. "That's fewer notes I have to take!" she said.

"I hope it's okay that I give you those," I said. She assured me that would be fine.

Our subsequent interview was a brief one, but she asked some good questions. "In what math classes do you feel that writing is important?" I warned her that I was answering for myself and not for my colleagues up and down the hall, and I told her I felt that writing was of pivotal importance in any class, including any math class. It wasn't until I said it out loud that I realized how strongly I feel that way, and how rare that feeling might be among my colleagues. (By "colleagues" I mean my colleagues in the profession, not necessarily the other folks here on the Third Floor.) I cannot imagine teaching a class in which writing didn't figure into the curriculum in some way, whether it's a conscious focus of the class, as it is in 280, or whether it takes the form of a few simple papers on vaguely mathematical topics, as in Calc I.

What else do I need to say right now? I thought to make a dent in the list of topics about which I wanted to say a little, but it seems like I'm running to stand still. (This is a good thing, to be sure, but a frustrating one...I need to invent a way to increase the length of the day by 20% or so.)

With the help of my students, I'm busily amassing good ideas for activities to take place in existing classes, and for classes we could offer as a part of our curriculum:

1. Last week the idea of math-themed poetry arose from two completely unrelated sources. We batted the idea around a bit in both of my Calc I sections, and it came to the fore as a topic of discussion on listserv of the MAA Special Interest Group on Math and Art, of which I am a member. I've decided I'm going to make math and poetry the focus of the last of my Calc I projects for the semester, a short one that'll cool the students down after the leviathan efforts they'll have expended on the Newton v Leibniz project. I'm going to incorporate a bit more guidance and instruction into this project than I did the last time I asked students to construct mathematical poetry, a sad little project I put into action during my grad school days at Vanderbilt. (Totally unrelated note: Vanderbilt, at 5-3, has the same record right now as Florida. Go 'Dores!)
2. I'm liking more and more the idea Quincy pitched a couple of weeks back regarding a "Random Seminar," in which participants, students and faculty alike, did research into and subsequently constructed classroom exercises around mathematical topics pulled from a goldfish bowl placed at the center of the room. It could work. It would take some fine-tuning, but it could work.
3. Quincy pitched another good idea to me by e-mail. A bit less ambitious, this one involves a component exercise for the 280 course: each student has a turn in which she presents a particular nasty proof she's been struggling with to the rest of the class, receiving feedback, suggested revisions, and so forth. Sounds kinda Moore-ish to me. Maybe we'll get a chance to do this a little bit before this semester's over. (Quincy, you're gonna love next semester's graph theory course!)
4. Another course idea that'll take some work to clean up is an "Unmath Seminar": students are asked on the first day of class to make an alteration to some fundamental axiom of mathematics, somewhat akin to denial of Euclid's Fifth Postulate. From that point on, students are asked to construct an internally consistent mathematical system that obeys all laws that are consequences of the assumptions made at the outset. If inconsistencies arise or inordinate difficulties ensue, seminar participants would be allowed to return to the starting point or some other point intermediate in the construction in order to modify their assumptions to make the resulting system more amenable to analysis. This whole project would be hard, and would require a good deal of advance planning to make the exercise worthwhile. Moreover, the students taking the course would have to buy into the project completely to make it work.
5. On a less ambitious tack, at some point I'd like to offer special topics courses on lattice theory and set theory.

So much to do!

I've got lunch this afternoon with Quimby and a couple of our cognition folks over in the Psych Department. I have no idea what they're going to spring on me, but I've no doubt it'll prove to be interesting. I've yet to send an e-mail to my colleague in mass comm whom Quimby recommended to me as an interested partner, regarding my idea for a "communicating mathematics" course.

For now, I'm off to class...to be continued!

Bile and bluster

I've now had nearly 72 hours to come down from my last post.

In retrospect, perhaps I was a bit harsh.

I mean, I'm pretty sure that my students aren't out to depress me with their eight-o'clock apathy and their poor performance on the exam. ("Did we really make you cry?" asked one of the students in the second section. I think the answer I gave was sufficiently vague so as to leave the matter unresolved.)

At the end of the day, they're good people, they had a rough week last week (didn't we all?), and enough of them have expressed enough remorse over their collective blanking, biffing, and blooping on the exam to make me regret my vituperative post in the wee hours of Saturday night.

Meanwhile, my 280 class has been doing wonderful things I fear I've not yet lauded enough. I'm very happy, for instance, with their Professional Proof Analysis papers, particularly insightful comments from which I've selected and compiled to share with the class as a whole. A sample:

"I think [the structure of the Hass and Stewart proofs] reflects a contemporary view that introductory Calculus is more about using Calculus than deeply building an understanding of why it works....[T]o the exploring student, it suggests the text is offering statements like this is true and here is why, instead of by using these ideas we arrive at this helpful result to introduce the material."

And:

"Along with the actual analysis of the proofs, I believe it is worthy to acknowledge that the Weir and Thomas proof was composed by three authors, as opposed to the one author each of the other had. This is important because it allows for immediate revision of each author’s ideas to produce a tighter, more inclusive proof (I now see why work in groups during class)."

We've been working on order relations for the past couple of classes, and a couple of the students seem to have a particular knack for this stuff. Timofei has even expressed interest in signing up for a reading course on lattice theory with me next semester. I think that would be wicked awesome. I pulled Davey and Priestley (Introduction to lattices and order) off the shelf to browse through it with him after class yesterday, it would make an excellent text for a reading course, and he could get it cheap cheap cheap. I don't think it'll take much convincing to make him take the plunge. I've got a soft spot in my heart for order theory, I don't think I'll ever leave it alone for long.

So what's going on in the pedagogical scene, besides the day-in-day-out of my classes?

Yesterday we had our first post-season meeting of the self-authorship Learning Circle. I and three others assembled to share our "homework": each of us was to produce a short narrative describing what self-authorship meant to us (perhaps from the point of view of a practitioner within our respective discipline), illuminating it for the benefit of one of our peers who may never have heard of it.

My narrative on self-authorship in mathematics was informed heavily by my ongoing experience in this semester's 280 course. My description came off sounding a bit clinical in comparison with Thibault's downright hortatory "manifesto" (his word, not mine) on self-authorship in theater. Echoing Goethe, he indicated the fundamental need for a student of theater to be true to one's self. Meanwhile Nola's narrative struck me as a bit more catholic and all-encompassing, with an emphasis on the mutuality of the relationships arising in Baxter Magolda's Learning Partnerships Model. I liked aspects of all three narratives. In discussing them, I found particularly insightful Nola's observation (as I described to her my experience of watching my older students interact with my younger ones) that Vygotsky's "zone of proximal development" is arrived at in the Math Lab.

I see the same dynamic in the conversations between the students taking part in our Math Problems Group (which convened a couple of hours ago this evening): the difference in ability between the weakest regular attendees and the strongest is noticeable but not overwhelming, and all of them possess the background needed to interact proficiently (if not fluently) with one another, yet a few have perceptibly stronger skills than their peers and often serve as coaches for the others.

Tonight I served up a particularly nasty Putnam problem from last year's exam (one that only a handful of the top solvers in the country scored well on), and I let 'em at it. It took 75 minutes to arrive at the rudiments of an argument, though by well before then they'd convinced themselves that they had the right answer. It was fascinating watching them at work, watching them scratch away on their paper, listening to them communicate with one another and share their ideas. Three of the attendees independently arrived at the same conjectural solution, allowing for minor variations on a theme. While Nadia continued to work on her own, Nikolas met up with Beulah and Simon and formed a consensus on a particular formula for the number we sought. Soon Nadia gave up on her computations and joined the others, and they spent another fifteen minutes clarifying their thoughts, and then another fifteen minutes spinning their wheels before we decided to table that problem and start another easier one, a simple problem on graceful graph labellings. As I suspected, this last problem only took Nikolas about five minutes to solve, and we ended the night on a high note.

Now?

I.

Am.

Tired.

Well...

...we'll see how the calc kiddoes are doing tomorrow. I've talked with a few of them about test corrections, including a couple more who agreed that it wasn't that hard an exam, they just had the mother of all brain farts last Thursday.

I'm not sure what it was about last week (something in the air?), but I don't think there was a single person worldwide who was in top form.

Except maybe Matt Ryan.

But that's a different story.

Anyway, before this becomes a football blog, I'd better call it a night.

I'll try to get another post in tomorrow to talk about seven gajillion and one ideas I got from Quimby in Fayetteville. I'd also like to talk about my interview tomorrow with one of the new Writing Center student consultants, the status of the Robert Moses Learning Circle I've put together for our department, the NSF grant I'm nearly done writing, and my final decision to run my upcoming graph theory course using the Moore method.

Stay tuned!

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.

Long, long, long

As I'd suspected would be the case, the past week and a half or so has been absotively, posilutely insane, and I've hardly had a chance to keep on top of each day's work as it's come due. Between travel and teaching, City Council meetings, grading, grading, and a little bit more grading, Learning Circles, research, and reappointment shit, it's been a rough one. Ergo, no posts for a bit now. Many apologies, etc. Today's the first day I've not felt torn in several directions at once; once or twice this morning I actually had a chance to sit at my desk and ponder my next task before instinctively setting to it.

So what's up?

The first draft of the 280 students' Professional Proof Analyses, in which I asked them to apply our course rubric for superlative mathematical writing to three proofs of the same theorem, each written by three different authors, were fantastic. The students raised excellent points, made perspicacious observations, dug deeply into the "Four Cs" of the rubric (Correctness, Completeness, Clarity, and Composition) and applied these principles consistently and clearly. By one means or another most of them accounted for variable audiences, the evolution of exposition through time, the difficulties entailed in the analysis of a single proof extracted from within the context of the entire textbook, the subtle epistemological differences between putting a proof before a proposition's statement and the converse configuration, and so forth. And these were just the drafts! I was able to honestly say as I handed them back that those papers were among the strongest mathematical writing I've yet seen in any of the classes I've ever taught. How much of this is due to the students' inherent skill in constructing well-thought-out essays, how much to the clarity with which the assignment was designed and implemented, and how much to the fact that I feel I'm a much better teacher of mathematical writing than I've ever been before, is hard to say. I like to think it's a combination of all of the above.

I'll be polling the class more formally on Monday after the final drafts are handed in (this will be one of the assignments collected for the purposes of the Writing Assessment study, incidentally), but preliminary estimates show that the proof of the second part of the Fundamental Theorem of Calculus appearing in Hass, Weir, and Thomas's latest edition came out on top, beating Stewart's 2nd edition out in terms of completeness and composition, both of them beating out Abraham Schwartz's 1967 treatise that made use of outdated and relatively unfamiliar notation and awkward terminology. There was some dissent on this point, though: a few of the class's strongest students argued that in terms of completeness and composition, Schwartz's proof had the others beat. I believe the students' sense of completeness might come from Schwartz's explicit construction of Riemann sums; the other authors hide most of the messy details inside references to other theorems and corollaries contained elsewhere in the text, and so might feel a bit more scanty than Schwartz.

Class itself has had its ups and downs during the past couple of weeks. Last Friday I was in no mood to talk about permutations, so after a few minutes going over suggested corrections on the most recently-graded homework, I gave an impromptu lecture on Russell's paradox, proper classes, the infinity of infinities, and the cardinality of the reals. The topics are engaging, the students asked fantastic, insightful questions, and we all had a good time: that's how I wish every class could be. Quincy suggested that perhaps I should try to get my Chair to allow me to teach a "Random Seminar," in which topics are drawn from a fishbowl at the room's center, and teacher and students together spend a few weeks digging into the topic so chosen, convening as needed to fill each other in on the details. Sounds like fun, but it would be require an incredible amount of work on the parts of both the teacher and the students, and it would take some tweaking before it would fly.

The past week saw us slog through the remainder of our work on combinatorics, leaving us ready to tackle relations. On Wednesday equivalence relations proved a bit dodgy for the students, so I took some time yesterday to make up an additional handout that dealt more concretely with equivalence relations, asking the students to construct explicit examples of relations with certain properties, on small sets. I took several of the students aside after class, one at a time, and asked them if they felt the worksheet helped ground their understanding, and the consensus was that yes, it did. I'm glad.

Calculus, meanwhile, has been a hoot, but with the conference I attended all of last weekend, I've felt out-of-whack with regard to that class. I wasn't able to grade last Friday's homework over the weekend, as I nearly always do, so I didn't get it back to them until Wednesday this week, and that's made me feel as though I'm a bit behind. (Likely, the students couldn't give a rat's patoot.) I do feel more on top of things now that I've had a chance to catch up...just in time for this weekend's homework. Huzzah!

We've been talking about related rates, an ever-vexing topic that never fails to confound student understanding at first. The last couple of days we've been talking about exponential and logarithmic models, including the semilogarithmic model for network growth that my colleague and I came up with this summer. (I hold out hopes that by infusing my teaching with my research and vice versa I might catch a few students early in the game and entice them into considering a Math major...it might be working: Tallulah seems open to the idea of undertaking a little undergraduate research soon.)

The most exciting events concerning Calc I have to do with the Newton v. Leibniz project. Role proposals were due on Wednesday, and every one of them made it across my desk before zero hour. The proposals were...entertaining. Some were quite formal, serious pieces of persuasive writing, offering solid arguments for why I should make one appointment over another. Others were simply silly. I had fun reading them. In assigning roles I attempted to balance the strength of the individual proposals with the cumulative "happiness" of the class as measured by the number of teams receiving their respective top choice. In Section 1 I was able to grant five of the eight teams their top choice for roles, while only three of the eight teams got their number one choice in Section 3. There were a few long faces in that section, I know a couple of the teams had high hopes for their first bid and had only half-heartedly lobbied for their second. On the other hand, there was genuine excitement on some people's parts in both classes. I think enough students are going to get something truly meaningful out of this project that it'll be worth the trouble it's taking to organize it.

What else? The conference in Charleston was a conference. I learned a bit, met some new people, managed to insinuate myself a bit more deeply into the graph theory and combinatorics community. (Today I was offered a chance to referee some of the papers for the proceedings, to appear next year in the Journal of Combinatorial Mathematics and Combinatorial Computing. Exciting stuff!) I came back with several interesting problems to think about, including one that I pitched to one of our brightest junior majors almost as soon as I got home. He picked up on the general idea almost immediately and is already working on the problem I gave him.

Ummm...what else? Hmmm...yesterday I finished my reappointment binder and got it in to my Chair a week before it's due. I'm happy with it. I tried to cut my "candidate's statement" down a bit, but I really wasn't sure how to. I'm certain I included more documentation than was necessary: at roughly 75 pages, it's probably about three times as long as it needs to be, but I don't do anything halfway (or a third of the way, for that matter).

What else? Hmmmm...

...I'm sorry, I'm really tired right now, and should probably get to bed.

I'll try to post again tomorrow, as I really do have many thoughts I'd like to commit to paper (well...to...whatever passes for paper this millennium) before they escape me for all time: Fabian's astute observation that a many-authored proof might more quickly than a solo effort reach a sound and stable equilibrium, my conversation with Tallulah about a student's authority to assess the rightness and wrongness of a mathematical computation, and so forth. But sleep would do me well tonight, if I'm to survive the onslaught of the Super Saturday kiddies tomorrow morning. I'll likely have several stalwart students by my side to help me out, but no such Saturday goes by without my renewed appreciation for the role played by our nation's middle school teachers.

Until tomorrow, then!

The show must go on

So it is: the learning experiences one remembers best from years past are those in which one played the most active role as a student.

The Latin American Civilization and Culture course I took during my last quarter as an undergraduate, some...holy crap...um...eleven years ago, was taught by one of the finest teachers I've ever had, Dr. Miriam Bornstein-Gomez. (Truly the entire faculty of the University of Denver's Spanish Department is fantastic, and I was made happy this evening to find, upon looking up the department's website, that all but one of the professors I had during my years there are still teaching.) She herself was strong and strict (I feared her for the first few weeks of class, honestly) but supportive and approachable, and her class was exciting and fun. She challenged her students to seek out every ounce of their talent, every day. An advocate of active learning, Profesora Bornstein felt there was no better way for us to learn about the atrocities committed during Pizarro's conquest of the Inca empire than to let her class put Pizarro on trial for the crimes he'd committed.

As the sole male member of the class of a dozen or so (and one of the most proficient speakers), I suppose I was the natural choice to play Pizarro, and so I was cast. I was assigned a defense attorney, played admirably by a young woman named Shailini, whose name I still remember because, truth be told, I had a bit of a crush on her and had vainly asked her out earlier that semester. Opposing us in the classroom court would be a pair of prosecuting attorneys who would have the right to call on any one of a handful of witnesses, played by other students in the class. The remaining students fulfilled the role of the jury, while Profesora Bornstein presided as judge.

I remember some impressions of the proceedings, but few details. I seem to recall that I adopted a cocksure attitude and tried to shift the blame onto the conquered Incas. In my good but not completely fluent Spanish, my attitude probably came off more comic than cocky, and I recall Profesora Bornstein laughing several times during the trial.

Strangely enough, though I can recall with punctilious detail the layout and orientation of the classroom (it was a small room, tucked into a tight corner on the third floor of the General Classroom Building, one wall of windows looking out over the quad to the building's east), I can't remember what verdict was rendered.

Nevertheless, the fact that I remember anything at all from that course, in a field that was not my major, more than a decade ago, is a testament to good teaching.

Today I distributed the first handout dealing with the next team project in Calculus, Newton v. Leibniz. Over the next five weeks, each team in each class will be asked to (1) "audition" for one of the roles in the trial (Newton and counsel? Leibniz and counsel? colleagues of either? historical or mathematical experts? jurors?), (2) write a brief or letter of support or document describing initial findings, as appropriate, before the trial, (3) fulfill an active role in the trial's proceedings, up to and including rendering a verdict, and (4) write a brief paper reflecting on the experience of the trial once everything else is done.

A few of the teams in the first section seemed to show some interest in the project already, exchanging knowing glances and nodding when I spoke of the "casting call" that'll start the project off. The second section's response was a bit more subdued, but I'm chalking this up, at least in part, to the fact that their teams have been shuffled around, whereas the first section's teams have remained the same. (The team evaluations for the first section were overwhelmingly positive ones, with a number of people saying explicitly that they'd like to stay in the same teams; there were no negative comments. Meanwhile there were a few teams in the second section that didn't function quite so smoothly for various reasons, so I felt a rearrangement was definitely in order there.)

I encouraged them to throw themselves into this project, to be creative, to have fun with it. As is true every semester, I've got a good number of talented students in both sections, and I'm looking forward to seeing what they can pull out of their hats.

Today I also distributed the handout describing the 280 folks' next written assignment, Professional Proof Analysis, in which the students are asked to apply our "Four Cs" rubric to three proofs of the same (hopefully somewhat familiar) mathematical result, the second part of the Fundamental Theorem of Calculus, written by three different sets of university textbook authors (Stewart, Schwartz, and Hass-Weir-Thomas). I'm interested to get their take on these proofs, if some will stand out from others in one category or another, if they really all seem the same. Whatever they come up with, I'm sure the results will be interesting. This is definitely one of the assignments I'll collect for the Writing Assessment Pilot.

Yeah, I'm excited.

I only hope the students are half as excited about these projects as I am.

Philosophy 201

FYI: as promised the other day, an updated version of my teaching philosophy is now available at this site. Enjoy!

Thirteen hours

A rare conjunction of coursework came this weekend, and I found myself with four substantial assignments to grade over the last 48 hours. On Friday Calc I students submitted their sixth set of homework problems and their first team projects; 280 students turned in their fourth homework sets and their first take-home exams.

It took thirteen hours to grade it all: a couple of hours on Friday night, roughly eight hours yester(Satur)day, and another three this morning.

Highlights?

My first section of Calc I clocked in with their highest homework percentage yet.

The team project write-ups were uniformly good, with two or three exceptions, including one project that was so stellarly composed I felt it warranted a few extra points on top of the full measure. (This team was the only one of sixteen that had me look over a rough draft of their project before revising it to create the final version. I have two words for this team's leadership: "you," and "rock.")

The 280 exams were hit 'n' miss, the hits coming more frequently on the more difficult questions people spent the most time on ("Prove that every year has a Friday the 13th" and "obtain and prove carefully by induction a formula for the nth derivative of sin(x)"), the misses coming through careless errors in translating English propositions into quantifier notation. I'm hoping that with the opportunity to revise, students will be able to put a high shine on these rough drafts.

The 280 homework on induction was quite good, I think people are understanding the mechanics of an inductive proof pretty well. I've made a mental note, however, to spend a little bit of time tomorrow reminding them of the direction arrows must point when "reducing" the desired proposition to an "obvious" one. This tricky point was the number one cause of errors in the HW set.

The 280 folks are writing well. I'm much more conscious of the writing going on in the course this semester than I was last Spring, and I sense that the students are already stronger (or at least more self-conscious and self-aware) mathematical writers now than last year's students were by the semester's end.

Tomorrow?

Tomorrow, two of my departmental colleagues will be sitting in on my Foundations class in order to gather material to write letters on my behalf for the my reappointment file and for the teaching award for which I've been nominated. Woo hoo. I must say I'm sorry that it's nothing tremendously exciting we'll be discussing tomorrow, just some basic combinatorics (the Pigeonhole and Addition Principles, most likely). Too bad the committee reports won't come until Wednesday.

Now?

Now, I'm decompressing.

Meh.

I think I've forgotten how to relax.

Random thoughts, volume 2

I had a few interesting conversations today, with colleagues and with students. I also found myself unfairly piqued during my second section of Calc I, and I feel an apology is in order to my students.

Let's start there: Quiz 4 came today, asking for a brief rundown on the two fundamental interpretations of the derivative. As I would soon forcefully point out to my students (post-quiz), it's not as important to me that they memorize the formula for the derivative, nor that they master every one of the rules for differentiation we will soon study, as it is that they understand what derivatives mean, and how it is that we can see them in nature, and put them together to help us understand natural phenomena. As I put it to them, Mathematica can do all the derivatives for us, and much more quickly than we ever could. What Mathematica can't do is study a natural process, recognize that there is an interrelationship between two or more dynamic quantities, chart those interactions over a long enough course of time to posit a model that describes the way they depend on one another, and use that model to put together the derivative that gets fed into Mathematica at the end of the line. Mathematica, in this sense, represents the mathematics of the past, when it wasn't yet the case that there was a handy formula that one could apply to find the derivative of a given function. That was then. This is now, and the future is yet to come. The mathematician of the future needs to know more than a mechanical rules for finding derivatives (as important as they are to be able to apply well); she needs to know how to use derivatives, how to interpret them.

Of course, when I said all this to them, it came out ne'er so fluently as it did just now, above. Figures, huh?

Performance on the quiz was...meh. It wasn't horrific (I've given harder quizzes), but it was by no means stellar. A couple of my best students cornered me after the second section and asked if they were going to be okay from this point on. Tallulah: "because I didn't do well on that quiz." "I doubt anyone did," I told her. "The folks in the first section pretty much biffed it. It was a hard quiz, largely because you're not used to being asked questions about concepts rather than computations." She's a fantastic student, she'll recover splendidly.

I realized even as it was happening that I was (unfairly) letting my frustration with my students' conceptual misunderstandings get the best of me for a few minutes during that second section's class. I threw markers and punched the board like I always do, but I did so with more vigor than is typical, partly to dissipate my frustration. "How dare they not get this? Damn it, what, they think they can coast in this class if they spit up a formula or two?!"

My righteous indignation subsided as the class went on, and I realized by the end that if they'd not focused on the concepts over the calculations, it was as much my fault, and my colleagues' faults, as it was the students', for not asking them to refocus their attention elsewhere in the first place. I've got to try harder at that myself. For some reason, I have to admit, Calc I has proven the most resistant of all courses to redesign along the lines of discovery and application-based learning. Only now am I beginning to understand what a truly problem-based Calc I class might look like, and I admit that this semester I'm falling far short of that mark.

I promise a less angry, less frustrated tone tomorrow, folks: you really are a great bunch of students, and I enjoy working with you very much. Let's make tomorrow's class a good one, huh? I'll bring some donuts tomorrow morning, and we'll start off with a couple of conceptual exercises to get our creative juices flowing. Sound like a plan?

Good.

From the second section of Calc I, it was off across the quad, to the second of the semester's Writing Intensive meetings (the first was this past Monday) for me. As I anticipated, I'm enjoying working on this committee, conferencing with a group of peers who feel as strongly and as passionately as I do about writing-to-learn and writing-across-the-curriculum and writing in general. I'm starting to get a good sense for the way writing is integrated academically, campus-wide, rather than simply in my own courses and in those of my math colleagues. The bar is quite high; the quality of writing instruction university-wide is solid. Nevertheless, there is room for improvement, and I found myself in a heated exchange of hallelujahs with Lexington, the WI committee's acting chair, as we walked back to our shared building after the committee meeting.

We agreed that the university has made tremendous strides forward in terms of embracing writing-across-the-curriculum, undergraduate research initiatives, outcome-based curricula, discovery learning opportunities, and so forth...but that there's also a lot of work to be done before perfection is reached. "If we're going to advertise that we're using discovery learning," Lexington said, "we've got to start doing just that, and to do that we're going to have to get serious about giving people the resources they need to do that." We agreed that we need to try to drive class sizes down (I mentioned my conversation with my own Chair last week regarding getting my Calc I classes capped at a lower level), we need to offer kids the opportunity to engage in alternative classrooms early and often, we need to make a focussed, directed, campus-wide effort to provide these opportunities to students from the get-go.

After a brief stop at my office and a moment in the Math Lab to unstick the stuckness of a few of the Calc I kids in computing the derivative required of them in the team project, I was off across the quad again to 280. Davina caught me before class with a few concerns about her service on one of this week's homework committees. For one, she wasn't sure about what to write on a person's submission if he said something like, "I'm stupid, I can't figure this out," or something along those lines. "That's a hard one," I agreed. More substantially, she wasn't sure she was giving the right kind of feedback, and she felt like she was being hypercritical, telling people to reword this, change that, and so on. I suggested that she might try to balance positive and negative feedback, and to offer comments like, "I'm having trouble understanding this, could you make this more clear?" or "This is a really good insight, it really helped me to see this point more clearly!" I later reiterated some of these ideas to the whole class, and wrote on the board: "Recall that the purpose of the committee work is not to homogenize, but rather to help people to clarify their own individual ideas."

The subsequent committee reports were good ones. I was particularly impressed with Davina's discussion during the presentation she and DeWayne gave on the homework problem they'd been assigned. I admired the way she was willing and able to come out and say that her serving on the committee definitely helped her to better understand the concepts involved in the problem they'd reviewed: "seeing how other people did it made me see how I could make my own writing more clear and more concise." I'm glad she came out and said that, and I hope her sentiment is shared by the others. I'm certainly going to ask the students about their committee experiences explicitly when I pass out midsemester evals in a week or so.

Came then (after another half hour of set theory) the trek back to Robinson Hall, where I'd spend a few more hours before heading home. I finished grading the second section's Quiz 4s, on which they did marginally better than the first but still not wonderfully. I also got a chance to work with a number of the teams as they struggled through their projects (they're all doing quite well, from what I can tell), and I met up, one-on-one, with several of the 280 students, helping them to polish various drafts of homework problems. They're definitely developing an appreciation for more and more subtle nuances, meanings of stereotypical mathematical phrases ("thus...," "for every...," and so forth), and clarity, clarity, clarity in writing. (A funny, and very heartening note, if I may: at the close of today's committee reports, I reminded the students to keep an eye on the rubric I'd handed out last week as they worked on their math writing, and I asked them if they could remember the "four Cs." In nearly complete unison, they intoned: "correctness, completeness, clarity, and composition." I didn't have to say a goldarned thing. I was a happy man.)

While I was finishing off those Quiz 4s, Cuthbert, Chemistry colleague of Lexington and a big, big man in undergraduate research, came by to ask me if I wouldn't mind providing him with the titles of the projects the REU kiddies worked on this past summer: he was putting together material to take to a regional conference on UG research, indicators of the sorts of things that can be made to happen in a liberal arts science curriculum. I was happy to oblige, and not an hour later I'd send him a list of the topics. We then got to talking about the inclusion of research components in UG courses, and he asked if I'd done much of that. "Depending on how you defined research," I told him, "I do that in nearly every class I teach." I told him a bit about the course that spawned this blog (MATH 365, Linear Algebra I, taught last Fall semester), and he asked if he could have information about the projects those students worked on. I obliged him there, too, sending him the prompts for a few of the research projects (Monopoly, Traffic Patterns, and Come on, Feel the Noise).

We hit a few more points in our short conversation, enough to give me a sense of déjà vu, feeling as though I was reliving the conversation I'd had earlier with Lexington. We talked about the increasingly interdisciplinary nature of scientific study; the need for broader, deeper, more meaningful implementation of discovery learning in our courses; the need for more robust interdisciplinary course offerings than an occasional and disingenuous cross-listing: we need team-taught classes, classes that provide a true interface between one field and another, like one Cuthbert mentioned that took place at his previous institution, in which beginning chemistry students learned their ways around a lab while generating real, unprettified data that could be fed to statistics students who would put it through realistic rigorous analysis, the results of which analysis could be funneled back to the students who ran the lab experiments in the first place in order to help refine their techniques. We agreed that a course that combined the concepts of physical chemistry with those of linear algebra would be a tremendous boon to both the Chem and Math departments; I have no doubt that we'll be talking about this again soon.

Thence, from my office, to dinner at Sorrento's (offering the best Italian in town in a cozy out-of-the-way den halfway to Oteen that sadly never seems very busy), and then home. Once home, I finally had a chance to call Bedelia, my colleague late of Harvard, now of Lesley University, and catch up with her. As it was bound to be, our conversation turned to...teaching! Imagine that! We talked for a bit about a program she's like to get off the ground, a sort of summer prep course to get local (in her case, Bostonian) college-bound kids up to speed on the math skills they might need to succeed in a challenging entry-level math course, while providing them with some useful advice on making the transition to college more smoothly. I compared the program she was describing with UNCA's own SOAR program, and she seemed to agree with the comparison, only she emphasized the local nature of the plan she envisioned. Then we talked shop for a bit, I about my own six courses (if one counts the senior seminar and my three independent study students), she about her Quantitative Reasoning and Pre-Calc classes.

We're both having a good time.

Hey, I've rambled on long enough, it's time to stop. I'm going to make a smoothie, maybe watch an episode of Mr Bean or some other such nonsense, and call it a day. Thanks for reading, I hope you'll free to comment, I always appreciate your feedback!