Report card

So how did I do?

As the semester draws to a close, I'm reflecting back on how well my classes went this term.

Useless, I know: I'm so self-conscious now about asking loaded and essentially pointless questions like "how did I do?", knowing as I do that the best measure of the success of a class is not whether the professor's plans were executed smoothly, not whether her or his lectures were flawless, but rather how much the students were able to learn from the class.

I know this, I know this, I know this. I know that to ask how I "did" as a teacher without knowing precisely how much my students have learned from the class assumes that effective learning is a result of effective professorial procedure, so assessment of that procedure yields an accurate measure of quality of learning.

Yet, as a product of the product-oriented academic establishment that continues to pile everything up into one heap to stamp it with a single letter grade (even while talking from the side of its mouth about portfolios, peer assessment, and learner-centered methodologies), I can't help asking: how well has it "gone"?

I'll soon find out from my end-of-term-evaluations how my students feel, but for the time being I include below some self-evaluation.

Calc I. Overall grade: A-

I feel this class has "gone" quite well. There's some room for improvement, but all in all I can't complain.

I've not taught Calc I for a year and a half now, and after having led four sections of Calc II, one of Calc III, and one of Advanced Calc, all since last teaching Calc I, I wasn't very fresh. I found myself stretching to come up with new ideas for mini-projects (that part went okay, I think), and awkwardly incorporating the team quizzes born in last semester's Linear Algebra class.

What's "gone" well? Class meetings in general have been smooth ones, and I feel that my presentation of the underlying concepts (for what they're worth) has been strong. The smooth flow of class truly has been the work of the students, who've shown nearly no inhibitions when it comes to working together, both in and outside of class. I feel I've developed a good rapport with most of the students, a sort of trust that makes our interactions more fruitful.

What's "gone" not so well? I feel that perhaps I've lingered too long on some topics. I'm not sure I've made as strong an effort as I should have to engage the students outside of the classroom, or to encourage them to do the homework. The "random grading" that I've done for the last four sections of Calc II hasn't gone over so well with the Calc I class, if one is to take any message from the relative infrequency of homework completion. I may rethink this form of assessment before Fall comes. But what will work best to get the students to do the work? Homework quizzes? Grading all of the homework?

Of course, as I've said at length above, it doesn't matter how well I "taught"; ultimately, the question that must be asked is "how well did they learn?" Only the students can answer this question. (Students: thoughts?)

Foundations. Overall grade: B

I started out the semester with exceedingly high hopes, and I'm not altogether certain I've met the goals I'd set out for myself (thus the relatively lackluster grade). I think what's hurt me most is my underestimation of the difficulty of the concepts we've covered: I've forgotten just how difficult it is to master the idea of induction, or how it's far from clear at first what is the structure of a proof by contraposition. I've forgotten that the idea of "relation" isn't immediately intuited, but rather takes time to understand. I've forgotten what it's like to be a budding mathematician, that in the beginning more than at any other time the art takes a great deal of patience and hard work to master, that that mastery comes more easily to some than to others, and that most students will struggle with it. As a consequence, I think I might have assumed that my students are at a higher stage of development as learners than they truly are, opening a chasm between me and my expectations on one side, and them and their abilities on the other. Might this be what's led to the recently-ballyhooed decline in attendance towards the semester's end? (It's hard to stay engaged if your efforts end only in frustration.)

Granted, it was my first time teaching this course at UNCA (and my second time teaching such a course anywhere), so I suppose I'm allowed to err in my assessment of their level of development as learners. I'll know this more well going into the same course next Fall.

Nevertheless, I can think of several students who have excelled as independent learners, who, I feel, have gained immeasurably from the class. I hope they recognize themselves.

To repeat a useless question, for what it's worth: what's "gone" well? The general dynamic of class, with its highly participatory nature, has been a healthy one, by and large. I've had compliments on this dynamic from a number of students who have found it effective, who have said that it makes them feel less anxious about the difficulties in approaching the math, having others to share those difficulties with. Overall, this class, which I've managed in more or less the same way I led Linear Algebra last semester, went far more smoothly than the latter class. (I have a feeling the few students...four, I think...that I've carried over from that class to this one would agree with me.) Students were more engaged in this current class, more eager to contribute, and less trepidatious, and I myself felt more at ease. I felt comfortable with the amount of "lecturing" that I did. I felt this balanced well with the student-centered portions of the class.

What's "gone" not so well? I'm not sure I did as well as I might have in managing the student homework presentations, particularly as regards students' peer evaluation of these presentations. For instance, I didn't challenge the students to challenge each other's solutions, I didn't force them to take the responsibility for the mathematics presented. I think I spent too much time worrying about how long the presentations were taking, and not enough time worrying about how well the students were guiding themselves and each other towards stronger, clearer proofs.

I'm also not sure that I did as well as I should have in demanding boilerplate proofs of everyone. But should I have asked for this?

It's like this: if I've got a student whose grasp of the most basic logical conventions, even at this end of the semester, is, to put it nicely, minimal (and there are a few such students in the class), the last thing I'm going to be worried about in their proofs is whether they ended their sentences with periods. Although good grammar leads automatically to stronger proofs, if the student's handling of the concept "if/then" is nothing more an awkward pawing, no amount of textual emendation short of out-and-out rewriting is going to make a clear proof out of a mish-mash of barely coherent semi-mathematical ramblings.

I'm not sure I'm making any sense.

Again, students: what do you feel?

Number Theory. Overall grade: A

Easy A. With knobs on. I've loved this class, I can't point to a single thing that I feel has gone poorly. The smallness of the class (not to mention the inherent motivation of the students) has led to nearly seamless in-class activities. The students' homework presentations have been strong ones, almost without exception. The worksheets I've constructed based on the text (the first text I've used in years that I feel strongly positively about) did a good job of distilling the essential information into class-long activity guides. The students have cooperated well with each other, have shown genuine interest in the subject, and have unswervingly followed my lead into some pretty dense and detailed detours (like the theory of arithmetic functions and basic analytic number theory). I've already heard from several of the students that they agree with this assessment: they've learned a lot, and they've had fun. This has been one of my favorite classes yet at UNCA.

There you have it. I hope my students will read and feel free to post their own comments (even if anonymously).

Cough cough cough...

Oh my.

I just need a few days off, is all, but when am I going to get them?

As the semester really starts to heat up (don't you love the fast pace of these last few weeks?), I up and decide to come down with the mother of all colds.

To my students: I apologize for the raspiness (and sometimes absence) of my voice, and for any perceived shortness of breath and of temper. If I seem frustrated, it's not with you, it's with my damned lungs. Too, I thank you in advance and retroactively for your patience and understanding, and for your help in ensuring that I only talk when needed, and then only as much as I need to to make my point.

So...what have we got to do to nail things down?

In calculus we'll be finishing up with a little bit more on curve sketching, a treatment of L'Hôpital's Rule, maybe a little Newton's Method. Y'know. Fun stuff. I'll be fitting in one more miniproject, a couple more quizzes, and one more exam, on Chapter 4, before the final wraps things up.

Only a few more days remain in Number Theory and 280; in each, I've got two more days to say my piece before I turn things over to the students. I'm excited about the presentations folks are putting together. In 368 we'll devote whole class periods to the Riemann zeta function and the Riemann hypothesis, to groups of arithmetic functions, to extended properties of Gaussian arithmetic, and to algebraic cryptography. In 280, folks are putting together 15-minute presentations on Ramsey theory, the Fibonacci sequence, Euler's identity, the cardinality of sets of subsets of the naturals, on Pythagorean triples, and so forth. I'm looking forward to it.

Meanwhile, on Sunday I've gotta drive a few hours to the north to James Madison U. to give another talk on detecting hyperbolicity using asymptotic connectivity, assuming I'm well enough.

Oh, the pizza man just drove up outside, I'd best be off for now.

Remind me to post again later on the idea that struck me while lying awake in a codeine-induced stupor last night: The Multimedia Menger Sponge Project. And about another (a 9th! horrors!) student pet peeve I thought of this afternoon. Not to be missed!

Running on empty

I have this to say regarding my Number Theory class this semester: it's the first class I've ever taught that I feel is running itself.

I've had low-maintenance classes in the past, and I've had semesters in which I've taught a course I'd taught just the semester before (last Fall's Calc II sections, for instance, saw a repackaging of a large amount of the material I used with last Spring's Calc II folks), but never before have I had a course that just sort of...does it for itself.

That's not to say that I'm not putting any effort into the class (I am), and that's not to say that problems haven't arisen (they have), but the problems have been little ones (like yesterday's goof when I used an inappropriate power for the RSA cryptosystem on the handout I'd written up...oopsies!) and the effort I've expended has paid off to an extent I've never before experienced. I feel that after an hour or so of preparation for the class I can walk in, plop the worksheets in front of the students, and let them take it away. I feel comfortable in that class. I don't worry about it at all, it's very stress-free.

This is largely because of the students in that class. They're strong, they're independent, for the most part they're comfortable working together. Their talent (not to mention Deidre's lightning-fast calculator skills) makes my job an easy one. I'm looking forward to their presentations, beginning in a few weeks. I think Karl is still planning on undertaking a project dealing with the structure of arithmetic functions, and I know Bocephus and Simon are looking into the Riemann Hypothesis and how it related to the distribution of primes. (Simon came to me the other day with a copy of Selberg's paper on the elementary proof of the Prime Number Theorem.) I don't know what some of the others have up their respective sleeves, but I'm betting it'll be good. I'm going to get them to nail down their ideas during the next week.

My Calc I students got my cheese a little bit this weekend, I have to admit. I spent an hour or so this morning grading their latest projects (which were by and large good) and the latest homework assignments. Hmmm...I'm concerned, primarily for those that are clearly not putting effort into the homework. Not surprisingly, those that are turning in the homework regularly (even if they don't complete it so beautifully as they'd like) are the ones at the head of the class. For the most part these are the same folks I see in the Math Lab all of the time (Tiffani, your efforts are paying off!).

Maybe I'm not motivating them properly, not making it clear enough that doing the homework matters? Am I being too nicey-nice, too much of a big softie? The thing is, see, they're just not getting it done. I've never had this much trouble getting a class to just do the homework. I'm not asking for perfection, just completion. I understand that homework is a testing ground, it's where one learns by making a few mistakes here and there. Appropriately, it's low-stakes: any one homework problem counts for so little of the final grade (roughly 0.25% per problem, as opposed to maybe 1.5% or 2% for an exam problem of comparable difficulty, if that's the kind of thing you're worried about) that one shouldn't be concerned about messing up now and then.

Maybe that's the problem, that I'm making it too low-stakes?

Or maybe the "homework lottery" that's worked marvelously for four sections of Calc II during the past two semesters just isn't the thing for Calc I students, or for this particular set of Calc I students?

It's something to think about.

Happy day after Pi Day!

Hey, All!

Yesterday was the 301st Anniversary of the naming of π (celebrated), and the Math Club event I helped to put together went off splendidly. Over 50 people (mostly my Number Theory class combined with Quidnunc's Linear Algebra class, and a few assorted hangers-on) gathered to watch 6 folks compete in the pie-eating contest and 2 in the π-reciting event. Bocephus finished off about 90%-by-volume of his pie in 3 minutes and 14 seconds, giving him the victory in the first activity, and Ulrich recited 64 places after the decimal to garner the win in the second.

Many photos to come soon.

Meanwhile, my classes are chugging along nicely (I don't think anyone was too distraught over classes being cancelled on Friday). In Calculus we're almost done with shortcut rules, in 280 we're set to talk about relations and functions, and in Number Theory we're headed back to the text to talk about more on congruence arithmetic for a little while before tackling a couple of primality testing algorithms. The first of my Senior Seminar students' presentations comes next week, too, as Beulah will speak about hyperbolic geometry and how it inspired M.C. Escher. She's shown me her slides, and she did a great job in putting them together. If she can work out the timing, I think it'll be a fantastic talk.

Now, I've gotta hit the road to Georgia, hoping to make it to Statesboro in time for this afternoon's Project NExT-Southeast events. Tomorrow morning brings our panel on PBL/IBL. I'm looking forward to that, and I hope we get more than the 9 pre-registered participants.

Spring Break!

Well. Yeah.

Here we are.

Here in the South (where they wouldn't know real winter weather if it smacked 'em upside the head with a two-foot blizzard and subzero wind chill) it snowed yesterday, and though there are two weeks left before spring actually begins, we're on Spring Break.

For me this means it's a good chance to get ahead in class prep, since March and April are going to be busy travel months for me (two conferences, two colloquia on the schedule so far), and I'm not going to want to fall behind while gallivanting about the eastern United States. It's also a good chance for me to post here for the first time in a looooooong time...

The funny thing is, I'm less busy this semester than I was last semester, despite the fact that I'm doing three preps this time around (and for the first time ever). Most of the slack is due to the fact that 365 is not one of the classes I'm teaching...I put so much of myself into that course, and I let so much of myself (including sense of self-worth, I fear I must say) get wrapped up in how well I pulled it off. Even when I was done getting ready for 365, I was never done worrying about it.

I have to say that I'm enjoying this semester a lot more than I did the last, perhaps to a large extent because I've managed to distance myself personally from my classes. That's not to say I'm not being myself in class, as I am, and it's not to say I don't care about the students, their learning, or their welfare. I mean only that I recognize that the success or failure of the class, however that might be measured from day to day and week to week, reflects in no way on me as a person.

And the funny thing is, I think I'm doing a much better job with both of my upper division classes this semester than I did with 365 in the fall. Things are running a bit more smoothly. I've found in both 280 and 368 a good balance between me standing at the front and yammering like a talking head and the students working the entirety of the class period with minimal direction from me.

368 is purring along particularly nicely. The text is fantastic, and eminently suitable for the manner in which I'm teaching the course. I've found that by distilling each chapter into a short worksheet I can ask the students to do most of the computations and the bulk of the simpler proofs, leaving me to stand off to the side to lend a hand on the trickier arguments as they arise. The presentations are rotating nicely in that class, and I've had no trouble convincing people to volunteer to present. Right now we're off-text, working our way through a couple of weeks of analytic number theory. We spent the last week on divisor sums and Dirichlet convolution, and the coming week brings an estimate of the average number of divisors for large numbers (a value that tends to ln(x) in the limit). After that we'll return to the text to do a little cryptography before heading off towards elliptic curves and more about Fermat's Last Theorem.

280's easin' on down the road, too. Freeing myself from a text was the right thing to do for this class, I believe. Though it's meant that I don't have a ready reference immediately at hand, it's also meant that I can run the course using the in-class worksheets without having to defer to a text that covers topics in just such an order, that provides at-best weak explanations or overly difficult exercises. I'm happy with the day-to-day goings-on, though people aren't nearly as eager to present HW solutions as the 368 students are. (I'm chalking that up to the relative mathematical inexperience of the 280 crowd; it's nothing unexpected, nor is it to be sorely lamented.) I'm very pleased with the improvement I'm seeing in a lot of the students' work. I'm thinking particularly of folks like Neville, Sylvester, and Una, a trio whose first homeworks were lackluster, but in whom there was definitely potential. In the past few assignments from them, I've seen much stronger structure, clearer arguments, cleaner logic, better use of notation. All three of them have put in long hours in the Math Lab, and it shows. Kudos! Of course, I'm getting stellar performance from people like Fiona and Elmer, folks I knew I could count on to do well. From everyone, there are struggles to be won, but I think overall we're at a good place to be by midsemester. Right now the order of business is combinatorics: combinations and permutation rule the day, and by the end of next week we should begin talking about functions and relations.

Oh, yeah, and the University's Writing Intensive Committee has ruled: 280 is now officially a WI course, from here on in!

Finally, I'd be remiss if I left unmentioned my Calc I class. They're a laid-back bunch, and after 280 and 368, which are often hectic and fast-paced, it's often nice to wind down the day with the 191 folks. We're moving a bit more slowly than I typically do, but I truly think it's the right thing to do. There are a number of people in this class who haven't had math for quite a while, who aren't so confident of their math skills as they might could be, and the slower pace is letting them absorb the material more meaningfully than if we were simply blazing through it. We're just now working our way through the Product and Quotient Rules, and by next week's end should be ready to consider some interesting applications.

So that's the score.

Big-picture-wise, I'm still waiting to find out whether or not I'll be getting this REU picked up for the coming summer...the fact that I've not yet heard could be a good thing. Ever the optimist am I. If we don't land that big fish, I'm going to try to rustle up a couple of research assistants for the summer so's I can have a crack at a couple of problems from geometric group theory I've had on the back burner for several months.

It's been a long time...

...since I rapped at y'all. What's up?

We've been chugging away in all three classes, and things are running well all around. The folks who spent last semester with me in Linear who stuck things out to find themselves in 280 agreed with me when I mentioned it yesterday before class: 280's going a lot more smoothly than did 365.

This is large due, I think, to the fact that I've discovered the means of class preparation that works best for me: engage in thorough medium-range planning, punctilious day-to-day planning, and don't sweat the long term. I think mapping out 365 in advance was ultimately a mistake, as it created the illusion of an artificial schedule, a framework which didn't really exist. It existed on paper because it was "supposed" to exist there, but I never really brought it to the classroom.

This semester, both 280 and 365 are being constructed more or less one week at a time, and hours of careful planning goes into each worksheet and homework set, but I'm not worrying about where we'll be in six, seven, eight weeks. I suspect we'll finish the first 20 or so chapters of Silverman in 368, before veering off into primality testing and Fermat's Last Theorem in the context of Gaussian integers...I suspect that after proof methods and propositional logic I'll lead the 280 folks into the wonderful world of axiomatic set theory, and then do some theory of relations...I suspect these things, but I'm hardly going to commit them to paper until we get there. Trust, though, that once we get there, the treatment will be careful, meticulously crafted, and exact.

So what are we doing just now?

This week's Induction Week in 280, and yesterday's struggles reminded me how hard it is for even the most intelligent student to grasp induction when it's first introduced. (To say nothing about the difference between limits and indices in sigma notation...) We'll do a few more examples tomorrow, introduce Strong Induction, and use it to prove the well-orderability of the naturals.

In 368 we're doing congruence computations left and right. Fermat's Little Theorem is our main course tomorrow, when we'll learn how to find the last digit in 4²⁰⁰⁰¹⁸ without a calculator.

In Calc I? We're getting nitty-gritty tomorrow with the epsilon/delta definition of a limit. Since they've already seen and drawn a good number of "banded graphs," I'm hoping it's not going to be too much of a stretch.

But it always is.

Anyhow, that's where we are right now. Humdrum, ho dee hum.

Back in the saddle again

Yesterday was my first day teaching in almost a week, what with last week's abridged schedule precipitated by my travel to give a talk in Denver, and this week's late start brought about by a few flakes of Sunday evening snow.

So Tuesday it was, a roomful of semi-sleepy Calculus kiddoes. We're almost done with Chapter 1, and we'll soon blister our way through Chapter 2, limits 'n' continuity.

Ah, limits, how I've missed ye!

Today looks to be a full day: we begin proofs proper in 280, and we'll solve some linear equations in Number Theory. More o' the same for Calc, with a refresher on inverse trig functions. Joy.

I'm finding it hard to get motivated today, who knows why?

A bit tired, I suppose.

Too bad I've got no cheese to go with this whine.

No time to write, but...

I just wanted to let y'all out there in the world at large know that...

1. 280 went well, with two solid demonstrations to inaugurate our student presentations. We followed that up with a discussion of compound statements, in which everyone played an active role. Although I'd take last Friday over today, I'd take today over Monday.

2. 368 was a good ol' time today. Our first student presentations went splendidly there. Oswald and Bertrand showed us a neat fact involving primitive Pythagorean triples in which the "odd shorter leg" is divisible by 5. Leonardo and Egbert had a quicker proof to give, and managed it handily. We continued by talking over Euclid's Algorithm for GCDs.

3. We had our first group quiz in 191 today, and people worked together on it WONDERUFLLY. I'm very happy with y'all! And...

4. ...I heard just hours ago that Fall 2006's MATH 365 is now officially Information Literacy Intensive! You may commence rejoicing. And yes, Fiona was the first person I told.

Monday, Monday

Meh.

Yesterday felt a little...off. Not that any class went particularly poorly, but I don't feel that any of them "zinged" either.

In 280 we put a stopper in negation, spending the hour negating all sorts of statements with quantifiers. I think order of quantifiers and scope of variables is still confusing to enough people that I spent an hour or so last night and wrote up a handout with 17 more examples of quantifier statements, organized by "Statement/Negation/Reordered quantifiers." I hope that'll help folks to get a better handle on such statements as we go into tomorrow's class on compound statements. We start things off with presentations of the solutions to two of the first set of homework problems.

In 368 yesterday we pounded away at Pythagorean triples. It took us about 30 minutes to prove the characterization of all primitive integral triples. There was call to use our characterization to come up with a massive primitive triple, so we took s = 625 and t = 729 and generated

(st, (s²-t²)/2, (s²+t²)/2),

just to look at it. They're working on a different Diophantine equation for next week's homework. Tomorrow will start off with presentations on the second homework set, and we'll go from there to work on basic divisibility.

In Calc I we finished up with exponential functions and various perturbations thereof that can be used as more realistic population models. After working with Mathematica on the logistic equation, several of the students expressed interest in obtaining the free version of the software they've got coming to them under our site license. Score.

Today's a bit laid back, I've got a few student meetings and some prep work to do, but nothing crushing...there is an all-campus faculty/staff meeting in a few hours, but nothing else to keep me away from the office until class. Calc I today: shiftin' and scalin' and other nonsense with functions.

Oh yeah...

Hello, all!

Second day of class for a couple of my classes, I've finished 280 and Number Theory today. Lemme tell ya, 280 went awesomely. I had a good ol' time. I mean, how hard is it to have fun when you're making up crazily false mathematical statements?

In all earnestness, I did have a lot of fun in class today, and I think we were right where we need to be, all over mathematical statements and their truth, quantifiers singly and in conjunction with one another. There was willing, active participation, there was effective group communication, there were apt, authentic, and clever examples galore (my, do I have some smart students!). There were laughs, and no tears. There was excitement. I'd be a happy man if every class I lead were to come off as well as this one did.

368 went well, too. 45 minutes of the class were spent by the students giving their presentations on the first chapter of Silverman. We had some very robust discussions on twin primes, Dirichlet's theorem and its consequences, primes of specific forms, proofs of the infinitude of primes, factorizations...we were deep enough into the discussion that we only just had enough time to spit out a few small facts about Pythagorean triples (Silverman, Chapter 2) before calling it a day.

Hijinx and hilarity outside of class, too: Karl and I spent a half hour before class looking at connections between triangle-square numbers and Pell equations, examining the limiting behavior of a curve whose infinitely many integer-valued loci give rise to triangle-square numbers.

I love these students! I'm feeling like the luckiest teacher in the world today.

Wednesday next is the first day of presentations in both classes, and I'll be fishin' for volunteers come Monday. I'm pretty sure I'll find no shortage of willing confederates: both classes are stocked with solid students chocked full of mathematical derring-do.

Now, off to Calc I in the few minutes. Dare I ask for a third wonderful class?

...And the hilarity continues!

368 went off pretty well. It turns out we'll have a 13th member of the class who hasn't registered yet, but who will be signing on soon. The more, the merrier, but unfortunately 13 is has a much smaller number of divisors than does 12 (in fact, here's a note to 368 students: 12 is actually abundant. What does this mean?), a fact which makes 12 a class size much more amenable to fair and equitable group work. Nevertheless, we'll prevail!

We spent most of the class period today just jawing about various sorts of numbers, focusing on prime numbers and their importance in the realm of cryptography. The difficulty of factorization came up in our conversation, as did the AKS algorithm for primality testing, theoretical considerations involved in counting primes without explicit testing, and subtleties dealing with the Prime Number Theorem. Karl (continuing in the use of pseudonyms) asked a fantastic question regarding the estimate provided by the function x/ln(x), as to whether one could tell if it were an underestimate or an overestimate. We wrapped up by checking the accuracy of this approximation for 10¹⁶ and 10²¹, and neither gave substantial error (Deidre, I fear you might have made a calculator error in class...). Cool beans. We'll start off class on Friday with brief presentations on different sorts of natural numbers, and questions that can be asked about them.

Meanwhile, in my one section of Calc I, I had a great time today. The first 15 minutes or so were ho-hum as I was busy yammering away about functions and how they're defined, but once I turned the class over to them, I had a lot more fun. We spent 15 minutes or so coming up with and graphing some real-valued functions, breaking out Mathematica for the first of what will be many times this semester. Afterward, I let them take a crack at designing plausible functions to describe several phenomena occurring in geoscience, chemistry, physics, and the humanities. (As always, you can check out the prompts for this exercise on my website.) Boy, who'da thought that you could get folks in a Calc I course to come to fisticuffs over a hypothetical reader's interest in Northanger Abbey?

Ah, and soon Friday comes...

First Day, take 2

Yesterday didn't count, did it?

I mean, I only had one class.

I'm not sure why we bothered to start on a Tuesday. Weird.

Ah, c'est la vie.

We'll call today the first day of class, though, okay?

I've just wrapped up our first meeting of 280, our intro to proofs and other foundational aspects of mathematics. It went pretty well. It's at the perfect time of day, after the early-morning sleepiness has passed and before the lunchtime restlessness sets in. As a consequence, everyone (your humble narrator included) was bright-eyed and bushy-tailed, awake and up to the challenge. We had a good ol' time building a theorem from scratch, and proving it. Based only on data suggesting that ODD + ODD = EVEN and so forth, we came up with a clean statement of the appropriate theorem, and we managed to prove one of its three prongs before the metaphorical bell rang. On deck for Friday: fun with quantifiers! Oh boy!

In half an hour my number theory folks meet for the first time. I'm looking forward to that one with eager anticipation. I had two more students enrol just this morning; it's a good thing I checked my rolls before heading over, I was able to print off a few more copies of everything.

Before I leave, here's a shout out to Nikola, up in Maine! Nope, no 0-degree weather here, sadly. I miss the cold. I did, however, and much to the surprise of some of my students, wear shoes to walk across the quad this morning.

Aaaaaaaaand...

I spent a few hours in the office this morning putting the finishing touches on the first-day activities for my three classes.

Three! This'll be my first semester with three preps. Wow. Zowie. Havah nagilah!

Fortunately, I've taught two of the classes before. One of them often enough that should I be called upon to do so, I could teach it with my eyes closed. (Fear not, gentle Calculus I folk, I shall not attempt this feat, nor shall I give you any less than a full measure of my preparations and attention!)

At this point, everything's printed up (save the handout for 368; the copier ran out of toner at the last minute) and posted on-line. I invite you to check it all out on the respective websites.

One nice thing about this semester is the relatively small number of students I'll be working with. After working with over 100 students last semester in one way or another, I'll be dealing with about 60 this time around, with 18 registered right now for my Calc II class, 24 in 280, 10 in Number Theory, and 3 for MATH 480, with a single undergraduate researcher dolloped on top. Even if I see a mad rush of calcfolk charging into my little afternoon calc hidey-hole in the next couple of days, I shouldn't have more than 60-65 people to deal with. And I know almost half of them already. Of my 24 MATH 280 students, f'rinstance, I've met 15 of them, and 13 of these people I've had in previous classes. (Some of them in as many as FOUR classes...you know who you are!)

As regular readers will note, this semester's starting up with a lot less fanfare and foofaraw on my part, and that's probably a good thing. I don't plan to undertake any major renovations in my courses this semester, though there'll definitely be some tweaking here and there. I'll let those tweaks and squeaks speak for themselves, though, rather than advertise them in bold 'n' brassy "hello world!" internet announcements.

It's getting rather dark outside, I hope this doesn't presage some sort of ominous goings-on tomorrow.

Better check the weather forecasts...

Primathlons and proofs

Work continues apace! I've managed to get all of the syllabi for this semester posted (280's can be found here, and 368's here.) In addition, I've got the first day's activities for 280 written up and ready to run, I'll get those posted soon.

I've got in touch with Deadrick, Nikola's colleague up in Maine who's just started teaching number theory for the first time up there in the Moose-covered hinterlands. Strangely enough, he's using the same book that I am, too! We're going to take this chance to keep in touch and compare notes as we proceed. I hope he'll take the opportunity to read along and post his thoughts during the semester.

The funnest aspect of 368 might be the Primathlon, a semester-long prime-finding event I'm sponsoring for the 368 folks in order to give them ample opportunities to earn extra credit. The deal: throughout the semester, they'll be allowed to submit proofs of primality to me (which proofs must consist of personal verification by algorithmic or theoretical means that the purported primes truly are prime). For such proofs they will earn extra credit, depending on how big are the prime numbers they find: for finding a prime with at least 10^k digits, they will earn k percentage points of extra credit in the class. Judging from the length of time it took my fairly sophisticated (relative to an introductory number theory course) strong pseudoprime primality test (with a handmade modular power function), run on Mathematica, to verify the primality of a 3000ish-digit primorial prime, I have a feeling I'm not going to be handing out extra credit like candy.

We'll see, though. I know most of the folks in that class already, and there are some pretty hardcore math majors among them!

Things get underway on Tuesday with Day One of Calculus I. I've got a dynamic exercise I'm going to use to make that first loooooooooooong class pass reasonably quickly. I've got a little bit of prep to do in all three classes before then, but I'm mostly there.

More later!

Priming the pump

Here we are again!

It's the last day of the old year, and I'm really starting to get things in order for the start of classes, coming up in a little over two weeks.

I've got (and have had for a couple of weeks now) the syllabus for Calc I put together and posted on-line. I've taught that class often enough that I'm sure I could do it with my eyes closed and both arms held behind my back...which is exactly why I need to challenge myself to do it differently, better, this time around. Not that I've taught it poorly in the past, but I believe that now I'm capable of running this course so much better still that it'll make my previous efforts look like those of a first-year grad student. (I ain't knockin' on first-year grad students, some of them are hella good teachers; what they lack is experience.)

What'll be different about this coming semester? I plan on teaching this course in much the same way I've taught the last four sections of Calc II I've had: lots of application-oriented projects (which are, for the first time ever, built into the syllabus), structured team activities, including the ever-popular team quizzes, carrying over from last semester's MATH 365 course.

Then there's 280, our "Foundations" (read: "Proofs") course. To be honest, I haven't given it much thought, though that'll change in the next couple of weeks.

For 368, the course with the hifalutin' name "Theory of Numbers" (it's "number theory," people! "Number theory"!), I'm envisioning something much more akin to a seminar than a lecture. I may just have to take a page from Maryellen Weimer's playbook and let the students come up with their own course, selecting the assignments they'd like to complete from among a smorgasbord I place before them.

There is one goal I want to lay before them and make a sort of lodestone for the semester: what's the largest number you can prove is prime? I might make it a contest between the members of the class, to see who can come up with the biggest provably prime number before the semester is out. This'll spur them into reading about all sorts of primality tests, involving everything from basic modular arithmetic and Fermat's Little Theorem, through quadratic reciprocity and Dirichlet characters, all the way up to Dirichlet's theorem on prime congruences, and the Riemann Hypothesis itself!

Obviously this is a bit to bite off, let alone chew. But I have a feeling we'll get farther if I let them lead the race than if I serve as a pace car.

I'm off for now...I hope to get a working syllabus up for the other two courses before the week is out and I head down to New Orleans.

The proverbial Morning After

Well, it's all over and done with.

I think.

There are a few minor details to wrap up with one or two folks, but nothing major. The third exams, the last journal entries, the presentations and concomitant posters are all graded and ready to hand back to those eager enough to come and retrieve them. (C'mon, y'all! Come and get 'em!)

Moreover, I've finished grading.

Ouch.

In the end, I'm really disappointed that I've got to hand out grades: how difficult is it to boil down all of the interactions, inquiries, examples, applications, projects, presentations, portfolios, and other assorted whatnots we've collaborated on in the past few months, and end up with a residuum summed up by a single letter?

I tell you what: put simply, it's a bitch.

But it's done. And ultimately many people in the class did very well. There's a pretty large number of As and A-minuses, a fair smattering of Bs and Cs of various sorts, and only a puny handful of anything lower.

Beyond the grades, there are the lessons learned. I hope that in the case of our class we've been able to transcend cliche and put some truth into that truism. I can't speak for the students in the class (I'd love it if they'd take the time to speak for themselves in the comments to this post!), but I know I've learned a lot.

1. I am never, ever going to do this with a class this size again. Ever. I'm figurin' the upper limit for this method is something in the ballpark of 15 students. At that point I could have an eminently manageable 5 teams of 3 folks each. With that small a number of teams, I could make the rounds in the classroom during group exercises and be sure of hitting everyone at least once. I could schedule team meetings more regularly to ensure frequent updates, and the teams would be small enough to allow for easier scheduling of research meetings. We started out with thirty-three students and ended with thirty, and as hair-pullingly frustrating as the size of the course sometimes made the daily proceedings, I'm quite frankly awed that more people didn't drop midway. I have nothing but admiration for the patience and dedication of those that stuck with it.

2. This method of learning is not for everyone. Those that fared best were those who were more used to courses run along these lines, and those whose learning styles are at odds with those assumed by a more "traditional" classroom. For instance, those who identified as "visual" learners were likelier to find our class useful. Others, more used to the run-of-the-mill lecture format, felt a bit out-of-place and longed for those infrequent days when I'd stand at the front of the room and yammer. As the semester wore on, I developed a balance between the applications-based guided discovery exercises I'd envisioned for the course and a more lecture-led semitraditional format, all based upon the worksheets I turned out, one or two per week.

3. This method of teaching is not for everyone. As folks who've had me for other courses can attest, my teaching style is probably best characterized by the word "enthusiastic." A number of other words have been used to describe my teaching (few of which, fortunately, are unprintable), but this is the word which predominates in my teaching evaluations at the end of every semester (runners-up include "approachable" and "accessible"). And honestly, without the charisma and energy that I put into my classes, I have NO IDEA how I would have made it through this semester. WARNING: if you plan on teaching in this manner, make sure that you've got lots of free time, and boundless energy. Even with all of the preparation I did in advance, I was still blown away by just how much I had to do to keep up with the work. A lot of this labor was on account of the size of the course, but much can be attributed to the method alone.

4. Some innovation is appreciated. Team quizzes, for example, went over enormously well. I'm keepin' those: you can be sure that every course I teach from this point forward will include some variant of that activity. This blog's been a popular feature, too. For a while there, before everyone was occupied with exams and presentations and other geegaws, I was getting at least one or two comments per post (and as many as 14 at one point), which ain't bad considering all of the other faaaaaaaaar more interesting blogs there are out there that my students could be reading. (By the way, a shout out to The Comics Curmudgeon, one of the baddest blogs on the internet.) Change of Basis, too, will live on, in modified form, as I move into the planning stages for next semester's classes. Look forward to my continued chronicling of my teaching adventures. There are several of you from Linear who are continuing on with me, either in 280 or in Number Theory...keep reading, folks, and keep posting!

I've learned more lessons than this, but those are the biggies.

I've gotta go for now, but hey, 365ers! I really would like to hear what you feel you've gotten from this class, so please feel free to leave a comment or two on this post: let me know what you've learned, what lessons you'll take with you.

This'll likely be the last post I make on 365 for quite a while, but I'll be back soon with updates regarding next semester's courses. And I'm slated to give a talk on the writing component of our class in January at the big annual Joint Meetings of the American Mathematical Society and Mathematics Association of America. I'll be sure to let you know how that goes. (And yes, Fiona, I'll let you know as soon as I hear about the Information Literacy Intensive status for the class...but I'll be seeing you in 280 in the Spring, so I know you won't be going anywhere!)

Au revoir, then. To everyone in my class: thank you. Thank you for your hard work, your time, your cooperation, your willingness to try something new, your everything. Take care, and have a wonderful Winter Break!

Atomic radii

Today was probably a breath of fresh air for a lot of folks as we considered determinants of 2x2 and 3x3 matrices, a topic familiar to most from Calc III and other classes. We did some pretty straightforward geometric and algebraic computations, and after the team quiz Konrad took over for a little while and presented work he'd put together over the weekend. By computing the volume of the unit cell in iron crystals and applying a little mathematical legerdemain involving the mass of such a cell and of a single iron atom, we were able to determine both the general crystal structure (body-centered) of iron, as well as the radius of the iron atom. Konrad did a great job in putting his material together, and he explained it well, too. He was a bit short on time, though, and I have a feeling not everyone picked up on all of the nuances of his material, so I'll soon be posting solutions to his exercises on the course website tomorrow.

I spent the weekend getting this coming Friday's classwork together. While I'm down at Clayton State University speaking on the large-scale geometry of infinite graphs, my 365 folks will be going over each other's preliminary reports and offering each other feedback on those reports. Now to choose a "facilitator"...

I decided this weekend that I'm going to continue this blog after the semester's over, at which point it will become a forum for discussing all of my classes. Next semester sees me teaching a section of Calculus I, one of Number Theory, and one on the Foundations of Mathematics. As much as I love teaching calculus, these last two should prove a laugh and a half. A good deal of fun! I'm already looking forward to continuing to work with several of the MATH 365 folks (not to mention one or two of those in Calc II right now) as they work their ways into my Number Theory and Foundations courses.

Well, until tomorrow's Problem Session, adieu!