Professor Alexander Ol'Shanski'i worked wonders with his magical constants.
He'd begin a proof by laying out several numbers that were related to one another through precisely premeditated proportions: "we let epsilon [always pronounced "ep-SIGH-lon"] be given, and we choose lambda less than 53 epsilon, and we let c be chosen so that c squared is greater than 7 lambda minus 3 epsilon over 2..." These numbers were always given from memory, as though he were performing a familiar liturgical paean in some gnostic ritual.
As the proof progressed, the pieces of the proof, including the magical constants, fell into place like tumblers in a lock. If you waited patiently and bothered to put the pieces together you'd see why it was each particular choice was made, you'd see why every constant had been chosen to have exactly the value that it had, no more and no less. The tapestry was woven so finely as to deflect the sharpest logical blade. The resulting proof was a thing of unrivalled beauty.
Professor Ol'Shanski'i was the architect of several other, relatively constant-free, of my favorite proofs from my graduate school days. I recall a day on which he led our Representation Theory class through the verification of some theorem or another whose content now escapes me. At the time the proof left me with a warm feeling of empowerment, of being in on a deep secret. That's how such a proof made me feel: with its careful construction, its meticulous and methodical progression, its ultimate culmination in an elegant and often surprising result from the most arcane abysses of mathematical thought, it could hardly fail to leave an avid acolyte like me spellbound, basking in the warm glow of its ethereal nimbus.
Thus, it seems like sacrilege to me to say that I worry about the message the non-mathematically inclined might get from all of this mathematical masturbation.
Now, when I exhort my mostly mid-level math students in 280 to make their proofs clear, readable, and intuitive, I'm asking them to construct the antithesis of a "magical constant" proof. A good proof should let its reader in, not shut her out. It should beckon to the reader and invite understanding through a gentle mental climb. It should provide toeholds and grips, enough for the reader to reach its peak. Don't say "let epsilon be at least 7 delta...," say instead "notice that our goal is to find a value of epsilon so that if this condition is met...because of this, we must choose an epsilon that is at least 7 delta..."
Mathematicians on the whole do a whole lot of mutual ego-stroking and chest-thumping. "We're so damned smart that no one really understands how smart we are."
Mathematicians on the whole are lousy ambassadors for their field.
What are we doing to let non-math-type people in on the tricks of our trade? It's clear that such in-letting needs to be done: have you ever known another subject in which people are so proud to boast loathsome incompetence?
"What do you do?"
"I'm a mathematician."
"Oh, I hate math. I'm so bad at it."
I'd kill to hear someone say in earnest, "You're a sex therapist? Oh, I hate sex. I'm so bad at it."
Many view math as an inaccessible sanctuary, a place where only geniuses may tread, and there with only the lightest and carefullest steps. Instilling confidence in my students, convincing them that yes they too can do mathematics, encouraging them to embrace their inner mathematicians (yes, we all have one)...these are the hardest parts of my job. Anyone can add, subtract, multiply, divide, differentiate, integrate...given the will to do so and the dedication to give it an honest effort.
I really believe that.
Maybe I'm just naive. Maybe I'm just an idealist, seeing the world through rose-tinted glasses.
Can you blame me? I'm having a really good week. Every teacher will recognize those "breakthrough" moments, at which a crack appears and a shaft of sunlight blazes through. I've had a dozen or so in the past three days.
Today's most momentous breakthrough came when one of my students dropped by my office with a rough draft of her poem, titled simply "Frustration."
"This is really cool," I said after finishing the first read. "This is really cool," I said again. And then again. "I realize I just said the same thing three times," I said, "but...this is really cool!" Then we had a somewhat more lucid and fruitful conversation about the poem.
It spoke sincerely of her feelings on mathematics, how sometimes its "cumbersome intricacies" frustrate her to no end. "Cumbersome intricacies": I love that choice of words! I wonder how conscious she was of that choice. As I noted to her, the words "cumbersome intricacies," cumbersomely intricate themselves, phonetically capture the sense of building tension and frustration she's attempting to convey; in those words the form and the content of the poem combine and give her writing the energy it needs to reach the climax at the piece's middle.
The only suggestion I could offer her was to think about anchoring it more firmly to mathematics by providing a metaphorical object for her frustration: is there one thing she could point to, specifically, that frustrates her?
We both hit upon the same concept, simultaneously: "those functions!" we declared, in unison. We'd worked on those together.
"Don't be afraid to say something like 'why can't those fucking functions behave!' " I told her, knowing from a prior conversation that she'd not object to the salty language. (One of my colleagues was walking by my office at the time and made a comment about how violent math had become.) "Like I said in the prompt for this assignment, you should feel free to use any words you want, obscenity, profanity, whatever you want to say, as long as it means something to you." She loved the idea, and she set about revising her work right away.
I was positively tickled by her work. As I see it, she's gotten out of this project exactly what I'd hoped my students will get. I hope they'll find out something about themselves, as students, as students of math, as human beings. I hope they'll explore the way they think about math, and how it makes them feel: is it frustrating? Is it warming? Is it exciting, sexy, fun? I hope they'll learn how it is that they learn, I hope they'll learn that writing about math can be useful, it can help them better access and organize their own ideas, it can help them make sense of their own thoughts, and to fit them together with others' thoughts as they work together to construct new knowledge.
She's done that.
This is really cool.
The first two of several student presentations in 280 came today. Although Dewey could stand to work on eye contact a little bit, he showed great mastery of the content he'd chosen to work on (a couple of countability proofs involving unions of countable sets), and his organization and boardwork were solid. With only a minor slip here and there, I thought his presentation was splendid.
Twyla and Calliope, my pair of graduating seniors, followed this with a flashy (methinks at times distractingly flashy?) PowerPoint presentation on fuzzy logic. I appreciated the central example they chose: how might two people of different ages construct different fuzzy membership functions for the same concept, namely, age itself.
"Twyla, who is 21," began one slide, "views anyone under 25 as definitely young." After that, your youth slinks quickly away.
"Calliope, who is 31," began the next slide, "believes you're definitely young if you're under 40." For her, youth slid downward, but not nearly so precipitously as in Twyla's measure.
By Twyla's measure I had partial membership in geezerhood, but I was still a spring chicken in Calliope's eyes, which are only a year younger than my own.
After their presentation, they told me how glad they were that I'd suggested the topic to them, that they'd both learned a lot and had had a lot of fun in preparing their presentation. I told them that their presentation had made me want to get back into fuzzy logic, with the aim of teaching a special topics course on the subject at some point soon.
I'm going to miss them. I've had the pleasure of having Calliope in three of my classes now, and I'm sure she'll do well in grad school.
It is now after 11:00 p.m.
I've had no more than five hours of sleep on any given night this week.
I am going to bed.
Fare thee well, fair reader, fare thee well.
Friday, November 30, 2007
Professor Alexander Ol'Shanski'i worked wonders with his magical constants.
Wednesday, November 28, 2007
It's been a heartening day since I last checked in.
I'm happy to say that today's installment of 280 proved a useful one, as far as I can tell.
A few weeks ago I spent a bit of time designing a suitable peer review component for the third and final exam for the course. Since time permitted neither exam revisions (as I'd allowed for the first exams) nor a by-now-typical committee-based peer review of one of the exam problems, I decided to allow those who completed a draft of a particular problem (namely, the first on the exam) to take part in an in-class peer-review activity in which participants were divided into groups of three at random, allowed to discuss their approach to the problem within these small groups, and finally given the chance to share their groups' discussion with the reconvened class at large.
Without fail, everyone completed a draft of the indicated problem, and group discussion was lively and apropos. After ten minutes, we met again as a class, and several of the most eager folks in the class took turns presenting their solutions to bits and pieces of the problem.
At one point Quincy scrawled on the board both a certain proposition and a "proof" of this assertion. Although the proof was a flawless justification of the proposition we really sought, the proposition as stated was incorrect. One of his peers pointed out the error, and with a slight modification, the theorem read correctly.
"Same proof, different theorem," I said. "I need a t-shirt that says that."
I'm impressed with how willing these folks have become to get up to the board and perform math in front of their peers: even Dewey, a relatively reticent soul, spoke up once or twice today when he believed his friends to be in error. And Fiorello didn't skip a beat before taking the board to slam down a nearly perfectly composed proof of one equivalence relation's transitivity.
I'm going to miss this class, it really has been one of my favorite so far at UNCA. I've learned as much from them as they've learned from me.
Today I've had several other things to be happy about pedagogically, professionally, philosophically.
This afternoon I had a brief tête-a-tête with my 280 student Keiko, who over the past couple of weeks has made tremendous strides in coping with equivalence relations. It's clear to me that she's truly understanding them, not just going through the motions. Though she's still making little errors here and there, the mistakes are typographical and not logical. I'll take an armload of typos over a single conceptual slip-up any day.
In an e-mail from Barrymore, one of my first-section Calcsters, I got some of the most useful teaching ideas I've ever received from a student. A veteran of several mock trials in high school, he offered me some advice on how to make the trial experience a more useful one, a more intense one, a more authentic one. His advice centers on introducing the instructor as an actor in the drama, perhaps as the defendant, or perhaps the plaintiff. As students are called on to challenge not their peers (who may be more or less knowledgeable about the subject at hand, depending on their level of preparedness) but rather their assumed-proficient professor, the care with which they must construct their arguments is concomitantly heightened, and the stakes are upped.
Barrymore therefore suggested having faculty play the roles of Newton and Leibniz, while students are asked to play the lawyers, witnesses, and colleagues.
I'm not sure how I feel about this advice. It's solid, to be sure...I'm only wondering if the benefits described above would be outweighed by the loss of the students' opportunity to play the leading roles.
Barrymore also suggested that the various experts should be asked to meet with the respective legal teams in order to perform "depositions" of sorts, to make sure both sides agree on a consistent set of evidence. I like this plan.
The specificity with which Barrymore was able to offer advice showed that he has really thought about this project. I respect his judgment and will certainly consider his input when I put this project together again.
Just half an hour ago I got an e-mail from Bethesda, eight pages into her final paper on the issue of computer proofs she's writing for our independent study on the history of math technology. She's frazzled. She feels uncomfortable making claims about the proof of the Four-Color Theorem, the proof of which she can barely understand (especially its migraine-inducing implicitness). She's questioning the validity of computer proofs, questioning what it truly means to be able to prove something in the first place. In one long-running paragraph she spat out a dozen or so insightful observations whose perspicacity made me want to weep with joy.
I'm going to have to think for a bit before offering a robust reply.
It's days like this that make me glad that I do what I do.
I'm off to bed now. It's nearly eleven, I've been up since four this morning, and I spent nearly thirteen hours on campus getting "caught up" after a "break" during which I worked for nearly a day altogether.
What's wrong with me that I work so hard and yet love that work so much?
One parting note: I've received the go-ahead from several Calc I students to quote from their reflections. Excerpts to come soon!
I'll be off to my second section of Calc I in a little while here, but I just wanted to check in and let y'all know that I wasn't eaten by grizzly bears in Montana. I've made it back home safe 'n' sound, and ready to tackle the last (less than a) week of classes.
What's up with teaching?
I spent a cumulative 20 hours or so over break in grading Calc I exams and 280 homeworks, and in reading my Calcsters' reflections on Newton v. Leibniz. I've extracted 19 excerpts, ranging in length from short sentences to entire paragraphs, from these papers, in order to quote them here. I'm in the process of gathering my students' permission to do so, but I hope to publish those excerpts soon, along with my own interpretation of the events that elicited the comments.
Many wrote on discovery versus invention: what does it mean to "discover" something? How can one claim credit, or is it even worthwhile asking who gets credit for what?
A number pointed out the humanizing effect that the project had on their understanding of mathematics, and of mathematicians: suddenly these titanic personages from history seem more lifelike, in all of their humility, failing, and pettiness. Through their faults are magnified their successes, and all of math becomes a more "human" enterprise.
Several reported periods of intense focus and excitement about the project. It came as no surprise to me that the most passionate reflections came from the second section, whose reenactment was decidedly more intense and authentic. Not a single person from the second section reported any disappointment in the trial, other than that it should have been allowed to take up two class periods instead of one. (I'll be sure to budget time accordingly next time around...several people explicitly said that this project should be retained in future incarnations of the course.)
In these reflections at least two of the Calc students expressed excitement about penning a math poem, and I've received a request to read over one students' rough draft.
Who knows what I'll learn from these poems?
What does math mean to them?
Is it wild, or warming, or simply terrifying?
And how can writing help them access their feelings about math, and then express them?
How is it that they make mathematical sense of the world, how do they fill the math-shaped holes that pop up around them?
This morning I picked up the post-surveys to be used in math 280 class to wrap up the writing assessment study we're currently working on. I can't tell you how eager I am to see what we can find out from the pre- and post-surveys and the differences between them. I typed up an additional question to which I'd like to know an answer: "Has this class affected your perceptions about writing in disciplines outside of mathematics? Please explain."
My hope is that my students, in reflecting consciously on writing in mathematics, will actually have learned something about writing in other disciplines.
The way I see it, it's kind of like how I had no earthly idea what in the hell the future indicative tense was in English until I learned what it meant in Spanish. Ditto the genitive case: German helped me out there. In such a way I hope that in asking students to focus on making their math writing correct, complete, clear, and well-composed, I've actually asked them to do the same with their writing in general.
Random observation for future elaboration: this semester, more than in all previous semesters of teaching combined, I've become more consciously aware of the appropriate pacing and spacing of assignments, from a developmental point of view.
Random note to self: I must write to Profesora Bornstein and let her know how the trial went...
...now, off to class: avanti!
Thursday, November 15, 2007
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.
Monday, November 12, 2007
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.
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.
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 fn from Z[k] to Z[k] by fn(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 fm = fn; 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.
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?
Sunday, November 11, 2007
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.
Sunday, November 04, 2007
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.
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 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, 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...
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.
If A, might B?
Saturday, November 03, 2007
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).
Thursday, November 01, 2007
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...
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:
- 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!)
- 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.
- 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!)
- 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.
- On a less ambitious tack, at some point I'd like to offer special topics courses on lattice theory and set theory.
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!