Arrrrrr...avast!

The "pirate" section of Topology is on. Beginning on Tuesday, February 2nd, I'm going to try holding an intermittent "unofficial" (or "pirate," named for the manner in which we'll have to hijack a vacant classroom somewhere on campus) section of Topology, splitting the class in two in order to attempt to provide a more cozy and intimate learning environment to the students in the class. It looks like I'm going to have roughly 8 or 9 in the unofficial section and the remaining 15 or 16 in the section meeting at the official time.

We'll see how it goes.

In other Topology-related news, we've had two installments of homework presentations (operating in lieu of homework committees this semester), and the second flowed much more smoothly than the first. Students took turns explicating four of the five homework problems which were due today, and they all did a fine job. I was particularly impressed by the way in which the proofs they offered were for the most part merely sketches. It's best that the student presenters leave lacunae to be filled by their fellow scholars in their individual write-ups: to provide a complete and error-free proof would be counterproductive.

What I'd like to know at this point, from the students in Topology with whom I've worked in earlier classes (280, Abstract Algebra, etc.), is: which system do you prefer so far, homework committees or homework presentations? Obviously they both serve different purposes (one offers substantial opportunity to peer-review while the other is an instance collaborative learning in the extreme), but it might very well be asked which serves its purpose more effectively.

Which one is the most useful to you? Which one helps you learn better?

We'll see.

I'll leave by noting my frustration at Mother Nature for dumping several inches of snow all over what should have been the writing workshop I've been co-organizing, scheduled to take place in Black Mountain this afternoon. Snowpocalypse II caused my writing colleague Betty Lou and me to push the workshop back to next Friday.

A highly personal post

I'm demanding of my students. I push them hard. I ask them to challenge themselves on highly nontraditional projects, and on computationally and conceptually difficult assignments. I get them to work together and to present their ideas in front of one another, no matter how shy they are. I don't let up: from Day One through the Final Exam, they're hard at work, writing textbooks, critiquing each others' work, and pushing their intellectual limits.

I push my students in the same ways I push myself.

It's no secret that often try to be superhuman, and sometimes I succeed. I'm proud of the progress I've made in my teaching, of the advances my research (and that of my colleagues and students) has made, and of the work I've done in service to my department, my university, and my profession. I feel that I've achieved a lot, especially since I began this blog over three years ago.

But I've recently come to realize that I can't do it all. Even superhuman strength has its limits.

I've done such a good job of making myself ready to be "the strong one" for so many people in so many different ways that I've forgotten what it's like to lean on others when I need their strength.

This has hurt me.

I don't want to hurt.

I've decided that I'm going to relearn a lot of things I've likely forgotten.

I'm going to relearn how to let myself rest from time to time.

I'm going to relearn what it means to rely on others.

I'm going to relearn what it's like to be human, and not superhuman.

To those who may think this post may seem inappropriate for my blog, whose focus is pedagogical: anyone who's worked with me as a student or as a colleague knows that my teaching style is a very personal one. I believe it's crucial that teacher and student understand one another as persons, as human beings, before real learning can take place. I believe that for effective learning to take place, teacher and student must trust one another, and respect one another, and that teaching and learning are highly personal endeavors.

I don't know that I intend any specific exhortation by this post, except to say that if you know me and if you see me, in class or in the hall, on the quad or in my home, please know what it is that I'm trying to do: I'm trying to be a better person, and in order to be a better person I might just need your help, your support, and your understanding.

I have nothing but the deepest admiration and respect and warmth for my students, my colleagues, and my friends. Thank you for all that you've done for me, and for all that you do, and for all that you'll do in the coming days and weeks and years we spend together.

I feel more human now than I ever have before.

Two weeks in the can

We just wrapped up Week Two here at UNC Asheville, and because of the MLK holiday I've yet to teach a full week of classes this term. In fact, I'll be skipping out on my Friday Calc II class this coming week, so it won't be until Week Four that I manage to teach a full week.

Things are...well, I think they're settling in.

I still feel unsettled, though. Missing the second half of the first week because of the Joint Meetings really threw me off. Nevertheless, the classes are starting to come together, the enrollments are stabilizing, and we're getting work done.

My Calc II students did quite well on their first miniproject. (It was the perennial favorite, "Confectionary Conundrum," in which they are asked to estimate the number of gumballs it would take to full the candy machine I keep on my office desk. They're meant to use Riemann sums, but there are always a few folks who come up with other clever ways of getting an estimate.) And I've got great attendance so far, which is somewhat troubling, since there are 36 people enrolled in one section and 37 in the other, while the room is designed to seat about 32.

Meanwhile my Topology students have weathered their first round of homework presentations. The presentations (of which there've been two so far, delivered by four students) were a bit different, as is to be expected, from the homework committee presentations in past classes. The presentations were less coy and more direct; this is unsurprising, given that in the homework presentations in this class the students are asked to sketch draft solutions for their peers, whereas the purpose of the committee presentations in past classes was not to solve the problem, but rather to highlight common errors and pitfalls, and indicate any clever methods one or more students may have discovered.

I'm hoping that as the homework sets become more difficult, the students will put more thought into their presentations, planning ways to offer their friends helpful hints without giving away the full story. Being able to sketch a proof is as important a skill as being able to write it out in detail.

I'm also hoping that students will start to take advantage of the unlimited revision and resubmission policy I've got for the course.

I think one of the things that's kept me from settling into a groove is that much of the work I've done so far this semester is administrative rather than pedagogical. We've got a lot going on in our search for a new faculty member, and I've been scrambling around putting out fires for the Writing Intensive Subcommittee (dealing with student petitions, setting a new meeting schedule, uploading course proposals, planning next week's workshop, puzzling out assessment) and the ILS Oversight Committee (another new meeting schedule, dealing with the Academic Policies Committee, and putting out a bajillion little fires regarding Clusters). And I'm trying to organize the growing list of students who are planning on going to the MAA's Southeast Sectional meeting at Elon University in March. (Twenty students have committed to going now, and I've got my sights set on nine or ten more.)

Ugh.

Okay, I just realized that I've said nothing interesting or insightful in this post, so I'll end it here and wait until I've got something more meaningful to share before I write again.

Clusterf**k

I know I've not posted lately. I know it, I know it, I know it...things have been rather busy lately, what with a hectic Joint Mathematics Meetings in San Francisco (and thus a severely truncated first week of class), a second week of class spent trying to play catch-up, and a serious need to stick every finger I've got in the quickly-collapsing dike that is the Integrative Liberal Studies Clusters program.

I promise a more meaningful post soon.

Day Two

Topology met this snowy, icy morning, although we were a few people short (due, no doubt, to the hazardous road conditions brought about by the nasty weather). Those who came were excited to be there, though, and no one seemed to have much trouble jumping right into the thick of it.

Even if we can't manage to split the class into two smaller sections (one meeting "informally," commandeering a free classroom here or there), I think the semester will go smoothly for this course: most everyone knows each other quite well, having weathered several previous courses together, and no one seems particularly shy about speaking up in front of the others or about writing on the board. I think it'll be fine.

The feedback I've gotten so far has been very positive. Generally the students respond very well to the problem-based, handout-driven class activities, and almost all of the groups that gathered this morning to work on the handouts together functioned smoothly and efficiently, with animated (and apropos!) conversation.

I should mention there was glee on the part of some regarding the disappearance of the homework committees. We'll see next week how their replacement for this course, the "homework presentations," go over.

On to Day Two of Calc II...

Topology

Sometimes our closeness cannot be
a simple reckoning of distance,
a tally of the steps from you to me;
for in a purer sense
it's in the neighborhoods we share,
and in the friends to which we're bound,
and in the circles where
we find our common ground.

Day One, part 2

One class down, one to go. The first section of Calc II went pretty well. As usual, I felt rushed, since there's just so much to be done on the first day. I left a few things unsaid, but nothing that can't be un-unsaid (would that be "said"?) tomorrow. It'll be all right.

Only fifteen minutes before I head off to the first meeting of the second section.

Day One

It's Day One, all right, but it sure doesn't feel like it yet. There are a few quirks in my schedule this term that are going to take some getting used to:

1. I teach on every single day. Generally I teach four out of five days, and those days are typically very heavy ones, but I've usually got either Tuesday or Thursday to recuperate. Speaking of heavy days...

2. Tuesdays, the days I most frequently have off, are my heaviest days this semester: all three of my courses meet on that day.

3. I'm used to having my first class no later than 10:00 a.m. (frequently 8:00 or 9:00), but on Monday, Wednesday, and Friday of this term I don't teach until 1:45 p.m. This pushes me back to 5:25 p.m. on Mondays, the latest I've ever had class. That's going to be weird, especially for the first month or so, as it will be dark by the time class gets out. On the up-side, I've only got one class on Fridays.

4. Finally, and perhaps strangest of all, I teach two sections of Calc II, one which meets three days a week (on Mondays, Tuesdays, and Thursdays), and one which meets four days a week (on every day but Thursdays). This is going to cause my two Calc II sections to go out of sync with one another in a New York minute, making it damn near impossible for students to miss one section and show up at the other if convenient to do so, as they occasionally did last semester without any damage done.

Weird.

In other news, I spent a little while this morning walking around and scoping out potential classes for our "pirate" section of Topology. I found a couple of candidates during the 10:00 to 11:00 time slot. This could actually work.

Okay, I'm off to try to get some more bureaucratic BS done before my first class meets, at 1:45. Avanti!

First Day Jitters™

Yup.

Here they are again.

A day early, in fact. Those First Day Jitters™ have returned for another encore performance.

I've no reason for jitters, of course, but that doesn't stop 'em.

I'm prepared through next Tuesday in both of my courses (through next Thursday, likely, in Topology); I already know roughly 2/3 of my 90-odd students, and the faces of those students I don't know are now committed to memory; I'm coming off of a really good semester and looking forward to another good one.

Nope, I still don't feel quite there.

It's the newness.

It's the adventure.

It's the embarkation.

I've been doing this for almost 15 years now, but it never gets any less exciting.

God, I love my job.

Please know, my friends, that when class begins tomorrow I'll be just as nervous (probably more so) than you are.

So be gentle.

Let's make this one a semester for the ages, shall we? Let's teach one another, let's learn from one another. Let's complete compelling projects. Let's prove powerful theorems. Let's plan eye-popping conference talks. Let's give it our all. Let's get out there and make each other proud.

Let's do it, huh?

Two calculi

A derivative is a recipe,
an integral is a poem;
the last has dreams of majesty,
the first is enchained by form.

A textbook for 280

At last.

Working for about four hours this morning, La Donna and I put the finishing touches on the (pen)ultimate draft of the MATH 280 textbook, which can be accessed here.

Why "(pen)"? It would be nice to throw in a few more diagrams (there are spaces and other placeholders left for such diagrams here and there), and one of the folks in the class volunteered to come up with a cover design...but aside from relatively minor matters like that, the thing's done.

Is it perfect? No.

Is it in places sketchy and awkwardly-worded? Yes.

Is it an authentic effort, the culmination of 15 students' legitimate attempt to make sense of a number of difficult concepts drawn from high-level abstract mathematics by putting together a "textbook" which addresses those concepts? You bet yer sweet bippie.

I'm happy. I think it's gone well, and the feedback I've gotten on the project is for the most part good. With better organization on my part, the same project, assigned to my Topology students, should run even more smoothly.

Excellent work, y'all! You've all done a wonderful job both as writers and as editors.

The fun begins anew in a week.

2010

Well, it's a new year. And I've made a good start of it, I think.

This morning, after my first run of 2010, I spent a couple of hours getting a couple more days ahead in Topology (the Joint Meetings are exing out the second half of the first week of class, so I'm doing all that I can to get some traction before the semester actually gets started, since things will be getting crazy quickly once we're underway).

I've now posted the first two handouts and the first two homework sets, which should take us through the end of the second week of class.

I'm happy that this course is a "terminal" course: I'm simply not sure how "far" we'll get given the typical student-centered design of the class. I am, however, taking steps to "streamline" the flow of the course so as to maximize the material we'll be able to explore.

For instance, I'm making efforts to bring "later" concepts in early on so that students are introduced to them sooner rather than later. For instance, the second homework set already includes the definition of the Hausdorff separation condition, which Munkres doesn't address until over a hundred pages in, and I'm already weaving together treatment of continuity and closed sets, both of which come a bit later in Munkres as well.

Also, I'm taking a "just-in-time" approach to review of ideas from set theory and relations (i.e., material from MATH 280), putting off such review until it's needed. Most of that review will probably take place in the homework sets.

I'm happy about the way the course is coming together so far, and I'm eager to get underway!

Spring syllabi

At last!

The syllabi for my Spring 2010 courses have been posted on the now-up-and-openly-accessible course websites for Calc II and Topology. Supplementing these are the omnipresent links to my now-standard writing stylesheet, and for the budding topologists I've included a more-robust-than-ever introduction to LaTeX, now including a guide on inserting graphics files into LaTeX documents.

What about those syllabi? I've tried to take all of the feedback my students offered to me by e-mail (about 20 students eventually got back to me with their suggestions!) as I put them together, and I've tried to stay somewhat true to my goal of de-emphasizing grades (and thereby at least implicitly emphasizing collaboration), all while minimizing the "bureaucratic overhead" that was probably the number one complaint my 280 students had about their course this past semester, as measured by the course evaluation forms I got back a couple of weeks ago.

A few additional comments:

1. Rinse and repeat. I've backed off from explicit portfolio grading, but I've included the portfolio-esque unlimited revision-and-resubmission option in both courses: exams and projects in Calc II and everything in Topology can be revised and resubmitted as many times as the student would like to in order to craft an increasingly correct, complete, clear, and well-composed piece of mathematical writing. These systems of unlimited R 'n' R should help to cut down on the complexity of grading: if you're not happy with a grade, revise and resubmit. Again, if you'd like. And again. And again and again and again. I like this idea in theory and I'm excited to see how well it's going to work in practice.

2. Homework presentations. For the first time in two years I'll be teaching an upper-division course without homework committees, replacing the committees in Topology with "homework presentations" instead. This move too is designed to reduce "bureaucracy" by eliminating the deadlines associated with committee problem submissions and committee presentations, and it introduces a good deal of flexibility: as you can see on the syllabus, any number of students (including a single student!) may opt to join in on a particular homework problem's presentation, and there's only minimal bureaucracy involved in assessing participation in presentations. Speaking of flexibility, I believe the new syllabi offer tremendous...

3. Flexibility in assessment. Though there's no explicit mention of portfolios, in both courses I've left the exact manner of grading up to the students: in the first full week of class one of our tasks will be to decide, as a class, how each sort of assignment will be assessed, and with what weight it will affect students' grades.

4. Homework in Calc II. In Calc II, I will still be requiring homework, but at least half (probably more) of the homework will be handcrafted by yours truly, and I'll include frequent "short essay" questions demanding interpretation and explanation in addition to more traditional problems requiring no more than computation. Although I've still not made up my mind on the matter, I believe I'll grade the homework by giving each problem a number from 0 to 3:

0: no attempt, or a clearly half-assed attempt, made at solving the problem.

1: an honest attempt has been made, but the method of solution is not correct, and/or the student's work is very unclear and logically jumbled.

2: the method of solution is correct, but perhaps there are one or two minor computational errors; the writing is clear and well-composed, but insufficient explanation is given when explanation is called for.

3: solid; there are no errors (save perhaps a transcription error or a minor, minor error in computation), the student's logic is clear and straightforward, and all necessary explanations are sufficiently complete and are given in complete sentences.

It'll probably be a relatively easy matter for most students to get a 2 on a problem, but I'll warn them that getting a 3 is typically going to require careful attention to mathematical exposition as well as mathematical computation.

The syllabus for Calc II doesn't allow for unlimited revision and resubmission of homework problems, but if there's enough call for it, maybe I'll consider it.

5. The "textbook" assignment. The Topology crew is going to follow in the footsteps of this past term's 280 folks in writing a textbook for the course. As I did this past fall, I'll randomly assign topics from each "chapter" to students in the class. As in the fall, we'll meet up at "editing parties," but I'll be making these parties a bit more formal (going so far as to bring homemade goodies to them!) in order to encourage attendance. Finally, I'm taking La Donna's suggestion and writing a template the students will be asked to use in order to craft their textbook submissions; this will encourage at least basic typographical consistencies between the sections of the book.

Incidentally, La Donna and I will be meeting next week to hammer out some more edits on this past semester's textbook. I'm excited to see what we can make of it.

Before I go, I wanted to provide a link to the pedagogical Live Journal authored by my collaborator in crime, Cogswell. He and I will be sharing ideas with one another (often openly on our respective blogs) as we move forward with our courses this semester. I believe he's thinking about using the textbook assignment in his abstract algebra course, but is understandably skittish about the amount of work it might entail (he's the proud father of a brand-spankin'-new baby and has therefore clearly got other major commitments!).

More power to ya, Cogswell!

Course coordination

Cogswell (as he'll be known here), a colleague whom I've met in various places (while a grad student at UW-Madison he spoke in the Group Theory Seminar at UIUC, and later I ran into him while he was doing a stint as a preceptor in Harvard's Math Department), has suggested that we put our heads together as we design our courses for the upcoming semester, and as we put them into action.

A lovely idea! We've got many of the same ideas regarding teaching, and we're eager to try out some of the same methods.

I think our collaboration will be a fruitful one, and I'll look forward to trading traveler's tales with him.

Collaboration II: Electric Boogaloo

Today's collaborative extra credit session for Calc I is slightly better attended than Monday's was, with 26 people plugging away at problems while they partake of tooth-rotting holiday-themed treats, 7 more than the 19 who showed on Monday.

I'm not sure if this should be surprising: final exams end this evening, so in a way it's shocking to see so many people still engaged enough to make it to this session; on the other hand, perhaps enough people are desperate enough to do anything to add a few points to their grades that attendance is thereby boosted.

I don't sense desperation on most people's parts, though. Of course, everyone wants to get a good grade, but as a whole the students in these two sections of Calc I have done a good job in focusing their efforts on understanding and not on realizing largely artifical benchmarks of excellence. "I think our class already de-emphasizes grades," one of my students told me just a couple of hours ago as we were talking about my plans to further de-emphasize them next semester in Calc II. "I've felt all along that as long as I'm working on the homework and keeping up then I'm going to get a B."

"For the most part, that's true," I told her. "If you're doing what you need to to stay involved and engaged in class, and you're finishing the homework and doing decently on the exams, you'll get a C or a B, and most people in my classes get Cs and Bs. If you go above and beyond the basic expectations, you'll get an A, but you have to work pretty hard to get a D or an F."

I talked with her a bit about what a portfolio-based course would look like, and I admitted that I still haven't worked out all of the details for myself. "You have to turn in a grade at the end of the semester anyway, right?" she asked. "How would you do that?"

"It would be determined by looking at the products of the work you'd done throughout the semester and making sure that it demonstrates your achievement of various learning goals that we'd agreed upon in advance. Maybe we'd have said 'You need to be able to compute integrals of these types,' or maybe 'You need to show that you know some basic problem-solving techniques,' and I'd look to see that your portfolio contains assignments that show you can compute those integrals, and assignments that show you can solve some complicated problems."

I think we both ended the conversation with a better understanding of what our class would look like if I switched to portfolio-based grading, but I indicated that I'm still not sure that I'll implement that system in Calc II next semester. "I may try it out in my upper-division class," I told her, "and if it works out well there I'll contemplate using it the next time I teach a calc class of some kind."

But is this fair? I think now: one of the aspects of my own teaching I'm most critical of is the relative eagerness with which I apply techniques like inquiry-based learning and discovery learning and whatnot in my upper-level courses and eschew those same techniques in lower-level courses. To some extent this is understandable, since my lower-level courses are generally considerably larger than my upper-level ones, and such student-centered methods are much more easily implemented in smaller classes. Would portfolios present the same difficulties?

I don't think so. So why not go for it? Maybe I'm just clutching uncharacteristically conservatively at tradition, afraid to take that long, long leap all at once, preferring a few baby steps in its place.

I'll sort it out.

For now I'm going to sit back, close my eyes, and enjoy the pleasant hum of my students' voices as they puzzle through their extra credit problems.

Words to watch out for

Here are two tremendously off-putting words one might encounter in reading others' teaching philosophies: lecture (particularly when used without exception as a synonym for "class") and cover (or any variation thereof).

Random Walk, Take One

I've got a few photos to share from yesterday's inaugural random walk, sponsored by Algebra al Fresco. In all we had 16 different people take part (never more than 13 at a time), including three members of the general community who found out about the event through local advertisements. Good fun!

First, a demonstration of some of the equipment we used to help us plot our course:


Four-sided dice led the way at each four-way intersection; traditional six-sided dice directed us (with two values for each route) at three-ways. Only once or twice did we have to flip a coin for "either/or" choices.

It being a rather chilly morning (about 30 degrees Fahrenheit at the outset of the walk at 10:15), an early consensus decision was made to stop at Izzy's Coffee Den on Lexington Avenue, where walkers warmed up and partook of toasty caffeinated beverages:


Out on the road again, we stopped at every corner to conference on our next move. One person would roll, and another one or two, before the roll, would call the directions, pointing: "1, 2: that way...3, 6: that way...4, 5: that way..." It was a genuine group effort.

I don't know why La Donna's laughing in the picture below, but she's always laughing at me about something. ("You look goofier than me," she tells me. She's probably right.)

Pauses, though by necessity frequent, were never long, and we were soon on our way again. Below, we stride confidently down College Street, Ino (one of three of my Calc I students to show up...way to represent, y'all!) leading the way.

By my reckoning we took 38 steps, visiting 22 distinct intersections. Of these intersections, 11 were visited once, 6 twice, and 5 three times (none more frequently than this). As anticipated we never strayed off of the map of downtown Asheville I'd printed out for folks to follow as we walked. The furthest-flung intersection we ever reached from our starting point at the southeast corner of Pritchard Park was eight blocks to the east, the roundabout at the corner of College and Oak.

I'd be happy to provide readers with further statistics upon request.

All in all, it was a great social event, a good way to get some exercise on a brisk and beautiful late autumn morning, and a fair bit of nerdy fun. For sure there'll be another in the spring!

Collaboration takes coordination

We're about 50 minutes into the collaborative extra credit problem session (and potluck) I'd planned in order to give my Calc I students some sort of group-learning activity for the final exam.

So far, they're working together wonderfully.

18 students showed up by the session's beginning, and after a few minutes to organize the foodstuffs and return graded assignments, I pitched to them three problems that are considerably harder than the problems appearing on the exam itself, in the hope that their collaborative effort would make the problem-solving process a relatively easy one. I've asked them to submit their own individual solutions to these problems by the end of the session in order to receive extra credit, but they're allowed to work together in whatever way they'd like to in order to prepare those solo solutions.

Immediately the student broke into three groups, of 3, 4, and 11 students, respectively. (One student has now left, bringing the 11 to 10, and another just arrived, bringing the 3 up to another 4.) At first there was silence as the students started feeling the problems out on their own. Within about five or ten minutes, though, the chatter began, and in the murmurs they made I detected clear signs of honest collaboration: some students understood one part of a problem, and others another. They began comparing their solutions and sharing their methods. One student went to the board to log a tricky computation he'd just performed. "Just so y'all know," said another student, "he's writing up the step in Number 2 where we got stuck."

Another student has now gone to the board to indicate an approach for the third problem.

This is working well!

If only I can get them to eat some more of the lovely guacamole and pumpkin muffins that a couple of the students brought.

The calm at the center of the storm

It's early, early, early on Sunday morning on the weekend between the last week of class and finals week. I've had a good stack of grading to get through, but I'm about two thirds of the way through that, and tomorrow morning brings the most recent Algebra al Fresco event, a random walk through downtown Asheville.

The semester's ending well. I've been worried about how my philosophical frustrations from a few weeks back might have been adversely affecting the students in my classes (especially Calc I), as I've feared they may have been buffeted by ever-shifting winds. But signs point to students' weather the storm rather well. One Calc I student spoke of being "inspired," and one 280 student said that, despite the difficulty and relative disorderliness of the textbook project, he learned "50% of his understanding of the concepts from the project." That comment, coupled with the envy my Spring 2009 students have shown for my current ones' getting a crack at this assignment, has convinced me that the same assignment, modified to fix the weaknesses it's shown this semester, should be a part of the curriculum for next term's Topology course.

I've still not put much thought into the precise structural details of that course, but it's starting to come together. It's not going to be as tightly structured, and it's going to be highly collaborative, and all assignments will allow unlimited revision and resubmission. Beyond that, who knows?

For now, it's late, and I'm off to bed. I hope to post again tomorrow with pictures of the random walk!

There's a first time for everything

I'm a pretty bad "sympathy crier": if I see someone close to me crying, I'm almost certain to join in.

I've had a good number of students cry in my office before (usually out of stress, sometimes out of disappointment), and there've been a few times when I've gotten in on the action.

But I'd never teared up in class before, like I did today when I was telling my 280 students what a wonderful job they've done on so many different things this semester, and how proud I am of them and how much I appreciate all of the work they've put into our course.

To those students: this class is one of the most challenging in our curriculum. I've asked you to prove deep theorems, and to perform complex computations. I've asked you to write proofs, explanations, dialogues, a textbook. I've asked you to review each others' work almost weekly, offering advice to your peers in both in-class presentations and written comments on each others' papers. I've asked you to master a technical typesetting environment, which is in many ways asking you to master a new tongue. You've done all of this willingly, eagerly even, without complaint. At all of these tasks, varied and difficult, you have succeeded beyond my expectations. As individuals and as a class, you've shone. You've soared.

Let me say this again, just so we're clear: I am proud to be your professor. It's working with students like you all that makes me love my job so much and makes me realize that I would never be happier doing anything else.

Thank you. I will miss our class tremendously.