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."
"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.
...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.
Tuesday, October 30, 2007
I've now had nearly 72 hours to come down from my last post.
Monday, October 22, 2007
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.
The last couple weeks have both been 90-hour weeks.
And you know what?
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?
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?
Let's do it.
Friday, October 19, 2007
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!
Friday, October 05, 2007
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.