Sunday, October 29, 2006

Somewhat tangential update: beards at two months

Hey! Just thought I'd check in with the latest photograph of our ongoing facial hair fracas. Roughly two months in, here's how it stands:

Howzaboutit? (Note my especially spiffy Elvis wife hates it when I wear that in public...)

Friday, October 27, 2006

Out of town

Hello, faithful CoB readers! I'm comin' to ya this morning from the raininess of Central Georgia, where I'm giving a colloquium talk at Clayton State University, invitation courtesy of my colleague LuAnn. Props, LuAnn! They've been exceptionally kind hosts so far, I've met her Chair, and even her Dean. I almost get the feeling I'm on an interview, I've met so many people. Very hospitable.

Right now I've got an hour or so to kill before my talk, so I thought I'd write a long-postponed update.

I spent several hours yesterday morning poring over the students' first preliminary reports on their research projects. By and large they're solid, and though some can stand a little improvement, all teams appear to be well on the way towards potentially strong final projects.

Today Beatrix is leading class (thanks again, Beatrix!), facilitating as timekeeper while the teams exchange their preliminary reports in order to get a few more sets of eyes to look them over. Once that's done and they've had a chance to absorb the comments I and their peers have left for them, they'll have the opportunity to rewrite and resubmit on Monday, if they so desire. I hope they so desire...not only because they're able to gain a better grade, but because they'll gain the experience stemming from going over their own work once more, with a slightly more critical gaze.

Monday brings Cramer's Rule, and Wednesday brings eigensystems. I've been looking forward to using that 'E' word since the beginning of the semester. I'm excited! It should prove a boon to all of the research teams: the traffic mappers and the Monopoly players can predict long-term behavior in their Markov chains, the rainwater harvesters can decouple any complicated systems of equations that pop up in the course of their analysis, the crystal gazers can determine the directions in which their units cells are perturbed from the norm, and so on. I hope I can help each team incorporate these new ideas into their respective programs.

Under the heading "upcoming events" falls the next exam, on deck for Monday, November 6th. The majority of people seem to favor another take-home exam, but I will give more time for it than was given for the last one, handing this one out on Monday and collecting it on Friday. I plan to have two problems, one of which involves a single overarching idea which will be divided into several stand-alone pieces, and failure to complete any one of them will not affect one's ability to complete the others. I'll let you know more about that once plans are solidified.

Looking back on this past week, I think we've had a strong one. (The week before last, maybe not so much.) On Monday came determinants in the form of cross products and parallelepipeds, and with them cane atomic radii. Wednesday was spent working through the determinant scorecard, a laundry list of determinants which demonstrated explicitly what happens when one modifies matrices in certain ways. The students seemed to like that one. A few said so, and told me that it made more sense to see it in concrete examples than to read through a bunch of notationally dense math that purports to say that row exchange results in a flip of the sign in the determinant. Well said, Studenten! Well met, well played, well well!

I talked for a bit with LuAnn this morning about her guided-discovery combinatorics class down here at CSU, and it sounds like she might be having some of the same ups and downs with her folks as I've had with my own. For her too is the uneasy feeling that comes from using a nontraditional method, the sense you're on a tightrope without a net as you toe-heel-toe-heel-toe-heel your way to the other side. Experiencing class the way we're doing it is something like taking turns driving while out on a family roadtrip across the country. Every now and then I need to take a nap and let the students take the wheel and the navigator's seat. I might doze, but the map's right there in the glovebox (does anyone still keep gloves in those?), if they need to use it. Unfortunately half of the time it looks like it's written in Hmong or Swahili. Nevertheless, the more one drives, the better one drives, and the more one's sense of direction improves. It's good for the mind, good for the soul.

I envy LuAnn's class here; she has only 7 students to my 31. That's not to say that I'd want to lose a single one of my students! But with that small of a class, I'd bet our linear course would be running much more smoothly. As it is, twists and turns aside, we're doing all right. I feel it's going very well for most of us, we're chugging along through the semester like a 300-pound linebacker who's just recovered the other team's fumble and is now on his way to the goal line. But I'm humble enough to admit that I'll do this faaaaaar better the next time I choose to use this style, and I sure as heck won't try it again with a class this size! To my students, yet again, my warmest and sincerest thanks for your hard work and patience as we all learn together. I look forward to working with many of you again next semester in MATH 280 and Number Theory! (And no, I won't be running those in the same way...look forward to my traditional nontraditionalness...those of you who've had me for other classes might know what to expect...)

Please treat Beatrix well today, and have a productive peer review session. Please do let me know how things go by commenting on this post. I'll be back on Monday, when I'll take the wheel again for a little while. Until then, drive safely!

Monday, October 23, 2006

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!

Saturday, October 21, 2006

Fun for all ages

I just got done with the second of six installments of this semester's Super Saturday Math Discoveries program for 4th through 8th graders. (I think most of our group is closer to the 4th grade end of that range.) I had a great time today, and I think the kids did, too (to get kids can't-sit-still excited about math is not an easy task, and we've pulled it off for two consecutive weeks) say nothing of my volunteers, including the ever-indefatigable Fiona, who did a fantastic job in directing the young 'uns as they learned and played the game Toss 'n' Sort, the same graph theory game we played way, way back in the early days of our MATH 365 course. Yay, Fiona! (If any others from MATH 365 are reading this and would like to help out with Super Saturday, by all means let me know, we can certainly use the help!)

Yes, today was fun, and I feel like I learned as much as the kiddies did. Fun for a day, but I'm quite sure that I don't have the patience to be an elementary school teacher. Those of my students in Elementary Ed, I salute you!

Wednesday, October 18, 2006

"Live 'n' learn," or, "D'oh!"

One of the drawbacks of running a class in this format is that it's often hard to tell just how much we can "push the envelope."

We've now spent a bit of time talking about linear transformations, and I daresay most of the folks in the class are pretty adept at (1) testing algebraically whether or not a given function is a linear transformation using the defining characteristics of linear transformations, (2) determining the action of a linear transformation algebraically, given its action on a set of basis vectors, and (3) setting up a matrix which performs the given linear transformation. I know some of us have even begun to tackle the problems from Section 3.4, wherein we consider linear transformations in arbitrary "exotic" vector spaces.

But there's a word which appears frequently in the previous paragraph, and that word is "algebraically": indeed, so far as a class (some of the MATH 365 folks, like the crystal-gazers and the computer graphics programmers, whose research topics are quite geometric in nature, are excepted here) we have not considered linear transformations from a geometric point of view.

While I've already received one electronic request to remedy this oversight on Friday when we have a linear transformation free-for-all, it will do no good as far as today's quiz was concerned, in which I asked a rather bold question requiring the student to translate a geometric action into a linear transformation, and thence into a matrix.

While it was an ambitious question to ask, I'm glad to see that (a) a fair percentage of people in the class answered the first portion of the question nearly (or entirely!) correctly, and (b) an even more sizable chunk of the class mastered the second portion of the question splendidly, viz., constructing a matrix to mimic the transformation once its action on a basis was known.

In retrospect, I feel that the quiz was unfairly difficult (my bad), but I hope that all will soon come to see that the question is not an unreasonable one.

Onward! let us march, into a maelstrom of matrices and a hail of linear fire, as Friday brings us to consider linear transformations as they relate to every other aspect of a vector space's structure. Onward, onward, ONWARD!

Tuesday, October 17, 2006

Project progress

From third grade on I took part in one of those "accelerated learning" programs with which many of the 365 folks (and other assorted readers) are likely familiar. Ours was called "Project Promise."

Nerd that I am (at least I can admit that), many of my most pleasant memories from elementary school come from activities we took part in during that program. We played "Balderdash," that fun game where the object is to B.S. each other by creating false definitions. We constructed our own archaeological dig sites by burying "sociologically significant artifacts" in a tub of dirt...and then we excavated each other's tubs, trying to figure out what we could learn from the objects we found. (That was cool!) We did a lot of the other standard smart-kid stuff: dropping eggs off of rooftops, building rubber band-powered locomotives, and so on...

I've now met with four of the eight teams (one of the crystal-gazing teams, one of the traffic modelers, the waste-water people, and the Monopoly players), and the other four teams have all scheduled meetings with me before Friday's end. Good, good, good! There's a good deal of work going on, and my impression so far is that people are more on task than they thought they were. I'm beginning to look forward to seeing the fruits of these folks' labor.

I also spent an hour or so last night in the math majors study room with Deidre, dinking around with Mathematica, working on getting it to do some basic image manipulation. I played with it some more this morning while I was proctoring a Calc II exam, and I managed to figure out how to turn a color image into a gray-scale image, which allows us to do some funky linear algebra-type stuff to it. Wicked.

Fiona reminded me that I'd looked into getting Information Literacy Intensive status for this course...sounds like another weekend project...

Here's a questions I've been asking myself: would I teach a course this way again if the class were so large as this one is? I've enjoyed everything I've gotten out of it so far, and I think most of the students appreciate it, too...but it's a heckuva lot of effort, and things would likely run much more smoothly (for all involved parties!) were the class to be smaller.

Any thoughts on this?

Saturday, October 14, 2006

Come together, right now

With the latest journal entries (and a few informal meetings with some of the teams), I'm beginning to get a sense as to how far into their respective research projects the course's teams are. While there's a little discomfort here and there, it appears that most teams have been able to find at least enough resources to get a good start on the actual research.

The team of chemists is working on narrowing their focus within the topic of crystallography, having found a wealth of information on that subject.

The two teams working on traffic patterns seem to be taking their projects in very different directions, which is what I'd hoped would happen. Both are now considering models for traffic flow, beginning to understand how they work and how they might be used. One team is even looking towards finding real data to test and refine the models they come up with.

The team working on wastewater treatment has also looked into models involving differential equations, and into data to refine those models.

The Monopoly team has found a bundle of sources analyzing the game from a mathematical standpoint. One of these mentions those two magic words that drive a linear algebraist wild: "Markov process." I talked for a bit yesterday with a couple members of that team, and we came up with a plan of action for their research in the coming weeks. Dare they consider a multiplayer model for the game?

All in all, the research projects are coming along nicely.

There's no shortage of "extracurricular activities," either. Yesterday I showed the class how a 35 x 36 matrix naturally arose in my research during analysis of a problem from graph theory: one equation short of a fully determined system! And yesterday I received a paper from one of my atmospheric science folks, an article in a very recent atmos journal dealing with Markov processes as they arise in weather forecasting. I'm going to take that home with me today and flip through it...perhaps the student who brought this in might be interested in leading a class on this topic in a few weeks, once we've got eigenvalues under our belts?...

...Speaking of which: in the next couple of weeks we're going to be blazing through determinants, with the primary goal of understanding them well enough to approach eigenvalues/vectors, since these puppies are the COOLEST things since sliced bread, and will prove eminently useful in just about every research project.

Stay tuned!

Wednesday, October 11, 2006

In or out?


Just got back from a whirlwind tour of Champaign-Urbana, and boy are my arms...wait, no, that's not right...

I spent a bit of time this past weekend (Fall Break here at UNCAland) wandering around my old hometown of Urbana, Illinois, remembering what it was like to teach at a school much larger than UNCA.


There's not as strong an emphasis there on one-on-one interaction. There's not as much time available for face-to-face meetings, for individualized attention. It's a wholly different dynamic.

Meanwhile, back at the ranch, things got back into gear in MATH 365 with some work with another "exotic" vector space, the collection of all polynomials in a single variable x. We reviewed the idea of bases and linear independence in the context of an arbitrary vector space, and we made some tentative moves towards coordinatization, to be continued on Friday.

So here's the question I'm grappling with regarding the next exam: in-class or take-home? I've had a few folks say that they'd rather continue with the take-home format, which offers a good deal of time to work the problems out, pick them apart, develop a robust understanding of them, learn from them. Others have said that the take-home exam was very stressful, that they'd feel more comfortable taking a more "contained" in-class exam: it might be painful, but after an hour, the pain is gone.

Let me put the question out there for all of you MATH 365 folks: in-class or take-home? Which would you prefer, and why? I'd really appreciate hearing from you on this issue, anonymously, if you prefer!

Thursday, October 05, 2006

After math aftermath

Well, boy howdy!

I've just finished grading the first exam. There were a few folks who didn't do so well, but there was also a large number of Bs and As. Lots of people did splendidly. Huzzah!

In the end the class average was roughly 77%, heavily weighted towards both sides in a sort of bimodal distribution.

I was particularly happy on how well people did on the first (and most difficult) problem on the exam, which asked the students to adjust the flowrates in a system of pipes in order to balance the new inflow of two different solutes occurring in varying concentrations in each pipe. (For those who are intensely, perhaps obsessively, interested in the exam, you can find a copy, with solutions worked out, here.) The most fun question was the third, in which folks were asked to construct a geometric representation of the solution set to a given linear system. Kaytlynne's opus magnum in plywood and Day-Glo paint stands two feet tall and is shown below gracing my office desk:

How 'bout that?

Yesterday saw most teams hit the brick wall in the latest worksheet, a question which asks them to find a basis for the space of polynomials of degree at most n. We'll start from there tomorrow as we begin to build a bridge between polynomial spaces and real Euclidean space.

Wednesday, October 04, 2006


...And I'd hoped that I'd kicked this blasted cold when it first snuck up on me this past weekend.

I'm hoping that class will lead itself pretty well today as we continue to work through that vector space of polynomials, I don't much feel like talking, and I certainly don't feel like offering up my typical hucksterish stentorian tone...right now I can barely croak a whisper.

Tuesday, October 03, 2006

You're gonna make me cry...

What a day!

This exam has been a bruiser. While the computations themselves are not all that difficult, the concepts are the slippery sort that are hard to grasp at first, and a number of folks in the class have had quite a struggle in wrestling them to the ground.

They're pretty hardcore "word problems," after all, and two of them involve concepts which, though analogous to application we've covered in class, are not derived entirely from examples we have seen before. These are the sort of problems one has to grow in to, the kind from which one can learn as the problems are solved.

Progress is being made. I spent much of the day in the company of a number of the 365 students...looking at the list, I count 20 of the 32 students in the class with whom I've spent some face-to-face time today. Now that's dedication! These folks are hard workers.

Even for me, it's not all wine and roses, it's not all stress-free. I've been on pins 'n' needles all day, put on edge by my students' discomfort. I feel their stress, their frustration. Every time I give a take-home exam, I'm made to remember how painful it was for me to take one of these blasted things (insert memories of pounding out page after page of Frame Theory for Prof. Wayland-Walters while listening to way-too-loud Nine Inch Nails, thereby annoying the living crud out of my downstairs neighbors...).

On the other hand, more than once today I've had one of those magical moments which makes it all worthwhile. Fiona's "Aha!" moment in tonight's Problem Session (now capitalized!) was a grand one. And this afternoon, minutes after quitting the second of my Calc II sections, I was joined by one of the 365 folks in my office so she could have me look over what she'd done so far. She was worried she was TOTALLY off the mark, but almost everything she'd done was bent headlong in the right direction.

It was enough to make me weep with joy, and I did.

"Your eyes are watery. Are you crying?"

"I'm happy," I said. "This is one of those teaching moments."

"Oh god, now I'm going to cry."

Later on, several 365 folks would admit to having shed a tear or two over this exam.

It's a toughie, but we'll soon be through it.

Monday, October 02, 2006

Splendor in the grass

It being a beautiful early October day today, we took our show on the road and laid ourselves out on a patch of the quad outside of Karpen Hall:

The day's activities consisted of laying the foundation for a study of general vector spaces. Above, the hard-working academicos and -cas are puzzling their way through spaces of polynomials.

I also handed out the first take-home exam today, due Wednesday. We'll see how that goes. I'm looking forward to being able to take a seat on Wednesday when Fiona and Niobe take charge!

Sunday, October 01, 2006

Maybe I oughta start a band...

It's Sunday, I've finished grading for the weekend (most of it came from my Calc II sections), and I think I've managed to dodge the cold that threatened, unseasonably, to lay me low.

I talked with Griselda for a while last night. Now that we've finally got a phone that doesn't die after about an hour, she and I can talk for somewhere near as long as we typically do at conferences without the aid of technology.

Her IBL proofs course is going smoothly, it seems, and it sounds like she's had no shortage of "teaching moments." Her students are responding well, including one she mentioned whom she'd had for calculus in an earlier semester, and who had come out of the shell she'd hidden herself in in that previous course. Griselda related a story of this student's performance in a recent class, in which the student, by no means the strongest mathematically in the class, had held her own (rightfully!) against most of the rest of the class in an impromptu on-the-board proof. Huzzah!

Once, current and future teachers who are reading this may recognize such events as the moments that make it worthwhile to be a teacher. I mean, let's face it: there are far more "prestigious," and certainly more lucrative, careers Griselda and I could've gone into. Griselda's actually been out there, in the private sector.

Hmmm...not for me. I feel honored (some would say blessed) that I get to spend my life doing something that I love doing, and that I'm pretty darned good at...and I get paid for it. Imagine!

I've had a number of "motivating moments" this semester: the first day of class got me all fired up, and the problem sessions (where everything just seems to click) never fail to put a smile on my face. Then there are those many "aha!" moments that come as I'm drifting from team to team in the midst of class discussion. This past Friday made me smile: at the end of the day, when we completed the "Choose Your Own Matrix" exercise and we realized that everything we've talked about so far this semester is related to everything else (rank becomes column space dimension becomes number of pivots becomes a criterion for invertibility, which is equivalent to uniqueness of solution, which means that the nullspace has dimension zero, which...), well, that was one of those moments for me. Was it good for you?

I found out that a friend of mine on the West Coast is soon going to go to see the Decemberists perform. Not a bad band, though not exactly my cup of tea. They came through here several months ago, and I almost convinced myself to go catch their act.

The reason? I went to high school with one of the members of the band. Funny, huh?

It's no great secret that I've not kept in touch with the folks I knew in high school; I wasn't particularly close to more than a handful of them, and even those I knew well and cared about somewhat have drifted away over the last dozen years or so. The few folks whose whereabouts I know include a member of a pretty well-known alternative folk band, an economist for the State of Montana, an engineer working for a defense contractor on armor technology, a technician on staff at the Smithsonian, and a cell biologist in Sicily.

Then there's me.

Maybe I oughta start a band.

I've got better things to do, though.

I'm fully aware that "better" is in the mind of the believer.

I wouldn't trade my life for anyone else's in the world.

Tomorrow: Taylor series, revisited. Let's dig into vector spaces!