Playin' the fool

Consarn it...

Lorelei came to 280 a bit before class began and found herself a seat in the back where she could sit and observe, herself unobserved. Towards the end of last week she'd checked with me to make sure she could sit in and look in on class for one of her psych classes. "Sure, I said."

Today, I began by returning homework from this past week. I'm so used to her presence in my classes that my initial thought was "Oh, Lorelei's not handed anything in...that's not like her." I went back to her seat and said, "Um...I don't think I got yours. Did you hand it in under my door?"

"Are you serious?"

"Yeah, I don't have it here."

"No really, are you kidding?"

"No."

"Umm...I'm not in your class."

The clouds parted. I bashed Lorelei repeatedly with my class notes.

Much hilarity...must go to Number Theory.

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.

...We're off!

First day of class!

Sadly, I got very little sleep last night, and so at this point I am pretty much dead on my feet.

A light day to open things up this semester, only one class to teach. (I have been here since 7:30 this morning, though, putting together activities for the remainder of the week.) My first installment of Calc I went well enough. Our "average-velocity-becomes-instantaneous-velocity" exercise came off okay, but didn't spur huge involvement. This class seems a quiet bunch for the most part, unless everyone was simply as tired as I am. I fear I might have to wake some folks up a bit with more interactive exercises. Don't think I won't do just that!

I apologize for incoherence...I'm just about ready to fall asleep on my djfffir46o-3gfklb...

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!

Nawlins

How-dee, y'all!

I'm just back from the city where creole and cajun cookery reign in the kitchen, where Zydeco music blares from nearly every storefront, where Bourbon Street flows with cheap liquor and smells like things unmentionable.

And where nearly 6,000 mathematicians assembled for the past few days to meet and greet and hash over the latest advances in mathematics and math pedagogy.

I had a chance to meet up with a HUGE number of friends, faculty, and colleagues from graduate school, and from my postdoc at UIUC (hello to all the Vandy/Illinois folks, especially Nikola up in Maine, one of my most regular readers!), and to meet a passel of new folks who share my interest in developing and implementing alternative pedagogical methods.

The session on writing and discussing mathematics in which I spoke about the writing component of our recently-completed MATH 365 was fantastic. I got a number of great ideas for group writing exercises and projects, some of which I hope to soon incorporate into my courses.

My mind is still awash, and not yet fully unpacked, so it'll be a little while before I assemble all my thoughts on these matters.

More to come soon, though, I promise...

280, Day One

I've hit upon a good (I hesitate to call it perfect) first-day activity for my 280 folks, the budding mathematicians being introduced to rigorous proofs for the first time.

We'll cogitate, construct, and verify the correctness of a theorem, a process which will involve sifting through a bit of empirical evidence, assembling from this data a hypothesis, shaping it into some meaningful mass with the help of a from-scratch definition or two, and demonstrating the validity of the resulting theorem with some careful logic and mathematically precise language. Emphasis will be placed on the "process" involved, including coming up with a hypothesis, expressing it with clear definitions, symbols, and terminology, and verifying its truth with a persuasive argument.

I don't want to give the game away completely, so I'll keep the topic under wraps for now, suffice to say it's a rather straightforward problem I'm sure everyone'll be able to handle.

In other news, I've finished my talk for the New Orleans meeting on the writing component of this past semester's 365 class. It was bloody hard to boil all that we did writing-wise in that course down into a ten-minute talk, and I hope what I've come up with does the trick.