Sunday, December 31, 2006

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.

Sunday, December 10, 2006

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.


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!

Thursday, December 07, 2006

Math aftermath

Well, we did it! We survived. We finished.

Well, they've finished, anyway. I've got a stack of grading as high as the Petronas towers to clean off. I'm about halfway through the third exams, on which folks are doing uniformly well, and then it's onto the papers and the presentations.

The presentations? They went well, for the most part. There were a couple that could have stood to be a little more robust, a bit more fully fleshed-out, but by and large they were informative, and fun. I felt that there were two that stood out from the others in terms of clarity, content, and comprehension, not to mention preparedness and smoothness of delivery. It was exciting to see the aftermath of everyone's hard work. I'm sure the papers will reflect the same sort of diligence.

For now, I'm getting back to the grading, but I'll check in later with more insights as they come.

Wednesday, December 06, 2006

Today's the day!...

...We get things underway in about an hour now. I've been putting together my score sheets, getting the food ready, setting up tables, and whatnot for about an hour now, and I think we're just about ready. I'm hoping that the next hour will see folks showing up to put their files on the desktop so we can effect smooth transitions from talk to talk.

I'm confident things are going to go well.

Tuesday, December 05, 2006

One more day...

Well, we're finally here. The end of the semester is upon us. Yesterday brought us the last day of class, tonight's will be the final problem session, and tomorrow will see the Symposium on Linear Algebra and Its Applications.

During the past few days I've read through five or six drafts of final papers; written a ton of Mathematica code to simulate moves in Monopoly, to translate between various representations of color, and to analyze traffic flow; and helped several teams plow through some fairly dense source material. All in all, the projects have come along nicely, and I know I'm not the only one looking forward to tomorrow's presentations. Several folks have told me how excited they are, how much fun they've had, and how proud they are of the work they've done.

I don't recall if I've yet mentioned that about a week ago I submitted an Information Literacy Intensive proposal for the course. If we get that picked up, that's one more checkbox all of the people under the ILS system can put a mark in. Woo hoo!

Right now, I've got to go and do a little tweaking with some of the color simulation Mathematica code I worked up the other day. Toodles!

Tuesday, November 28, 2006

Hello, hello!


I've not been as regular a poster as I c/should be lately! Yes, it's here, folks: that dreaded End o' the Semester Crunch. It's biting down hard on everybody, including yours truly. I didn't managed to finish quite as many of the tasks I'd hoped to cap off over the oh-so-short Thanksgiving Break we had last week, so this week's found me in the midst of a dizzying whirlwind of activity.

Strangely enough, though, things are starting to come together.

Linear-wise, we're a little over a week out from our Symposium. I've just made up! That reminds me, I should post them on the website!

That's right, at 3:00 next Wednesday afternoon, we'll be gathering in Room 105 of Rhoades Hall to regale each other with traveler's tales of linear algebra applications to everything from wastewater management to computer graphics. I'm looking forward to it. Today alone I helped one group to use wavelets to generate computer music; another to come up with a reasonable model for their traffic transition matrix, given the observed data they'd gleaned from an hour's worth of counting cars in downtown Asheville (photos forthcoming!); and third to sort out a pretty complicated model for the flow of water through a conservation-conscious household. Busy day in Linearland.

Regarding the upcoming third exam, I floated the idea of an in-class exam, figuring that might help unburden these horrifically busy students a wee bit by freeing up some out-of-class time...but ALL but one person (and I've heard from well over half of the class) is looking for another take-home exam. I guess I'll get to work on that one!

What else? stalwart and studious person showed up to last night's Problem Session, to match the one who showed up to the previous one, just prior to Thanksgiving Break. Not once have I been totally stood up! And tonight's session promises to be well-attended, I've already had several folks RSVP.

I'd best be off now, after this relatively short missive; there's a bit to be done before today's final Senior Seminar presentations, coming up in about half an hour!

Ta-ta for now...

Sunday, November 19, 2006

...Sunday night!

Well, it's over and done with.

The Harvard talk seemed to go over very well. It was an interested audience to which I spoke, a small group of preceptors and graduate students in the Harvard Department of Mathematics. They asked good questions, and they most definitely kept me on my toes.

After we got the ball rolling with the Markov Dance, I described the basic philosophy of the course, and then got into the nitty-gritty details of the way the course is put together. Much of the time we had a hearty dialogue going, in which we engaged in a discussion of the course and its design. They were really interested in finding out more about the team quizzes, the nature of the worksheets we work through, the source of our applications, the dynamics of the group work we've encountered, and how the size of the class has affected the way it's been run. I gave honest answers, often aided by the 11 pages of comments (from which I quoted heavily) you all gave me in your last journal entries. (Thank you all, thank you, thank you, thank you!)

I had a much-needed rest on Saturday, hanging out with Bedelia and her honey, Eugenia, and their beautiful daughter, Isadora. We hung out in their Somerville apartment, ate crepes, and joined them in a walk to the Cambridge Public Library. Good times!

I'm tired. Very tired.

I hope that all went well with all of you this week, and that you've had a chance to look over each others' preliminary reports before revamping them along lines penciled in lightly by your colleagues. We'll all be back together again tomorrow afternoon, when we'll consider Fibonacci-like applications that arose in my own research this past spring, and are arising again as a consequence of the conversations I had with my colleagues in Tennessee this past week.

To be continued!...

Friday, November 17, 2006

Live, from Harvard Square, it's...

Hey, folks!

It's been a busy few days, and I really am a little bit homesick and missing the green, green grass of UNCA as I pass the midpoint of my whirlwind tour of...well, two schools. I don't know how rock stars do it. (Anyone familiar with the song "Math Prof Rock Star," by Jim's Big Ego?)

I had a pleasant (however brief) stay in Murfreesboro, TN, where, on Wednesday last, I gave a research talk in graph theory to a crowd of folks from Middle Tennessee State University. I followed this talk with a very fruitful two-hour discussion with the couple of colleagues who'd invited me out to speak (props, Xavier! Thanks much, Caspar!). I look forward to working with them closely in the next few months.

Meanwhile, I'm sure you're all interested in knowing how the deal at Harvard is going. I haven't given my talk yet, but I had lunch with the principal pointy-heads, and they seem open to learning all about what we've been doing for the past several weeks. I'm going to open things up by asking them to do the Markov Dance; we'll see how well they perform those steps! From there we'll discuss the structure of the course, its planning, its successes and failures, its pitfalls and its triumphs. I'll be sure to post the PowerPoint slides on the course website when I get back (please remind me to do so if it slips my mind!).

I will also be sure to share your comments with them as well; I've created a digest of your feedback through the latest journal entry (the one asking, "whaddaya wanna say to this Harvard folks?") to share at relevant points in our discussion this afternoon.

Oh, and to those who had asked: yes, I have taken and will continue to take some pictures. I hate to disappoint you, Farina, but the folks up here in Cambridge don't look all that different from the faculty and students at UNCA.

Only, we're prettier!

On that note, I'll end for now. I'll try to post again this evening and let you know how things went.

Take care, and...oh!...have fun in class in a few minutes (those of you who are taking part in the optional peer review session)!

Thursday, November 09, 2006

It's that time again

Judging by my relative reticence lately, you've probably come to the correct conclusion that things have been


around here the past week or so. The next time I agree to give three talks in three weeks, please shoot me. Although I'm excited about my travels, I'm not looking forward to the busy-ness next week will bring (how do my soccer students do it?): Wednesday sees me heading out to Murfreesboro, Tennessee to give a research talk in the colloquium series at Middle Tennessee State University, and after driving home on Thursday morning, we find ourselves (me 'n' the missus) flying up to Boston on Thursday night so's I can hit the folks at Harvard with the 411 on how our class is run. Many thanks go to...oh, what name have I given you in past posts?...let's just say many thanks go to Bedelia, one of my bestest friends and my host while up in Cambridge.

I asked the 365 folks to use their most recent journal entries to let me know anything they'd like me to pass on to the Harvard audience, beginning with the prompt, "As you all know, I will be traveling to Boston in a week and a half to speak on the style of study we have undertaken in this class. Please include in your journals anything about this class (good, bad, ugly, or beautiful) you would like the folks up at Harvard to hear about." The responses have been, as far as I can tell, honest and heartfelt. It warms me to know that these hard-working people feel comfortable expressing not just the good but also the bad. I hope to put together a packet of all of their comments (unabridged, unexpurgated) to share with interested parties in Cambridge.

I've also decided how I'm going to begin my presentation at Harvard...but to make sure Bedelia doesn't spill the beans (she reads this blog semiregularly, I believe), I'm not going to say anything about it...

Meanwhile, the course moves onward apace. We've spent several classes now on eigenvalues and eigenvectors, since these are by far my favorite concepts in linear algebra, and, I believe, among the most useful. We've looked at eigenstuff from a number of points of view, including computational, geometric, and algebraic. We've looked at applications to win/loss records, traffic flow, crystal structure, and, most recently, heat flow. Tomorrow we'll work at diagonalizing a discrete time model for heat transfer in a one-dimensional rod. (Doncha wish you were in this class?)

On Monday I handed out the second take-home exam. I underwent a protracted internal deliberation regarding the format of this second exam, and though at one time I considered making this second exam in-class, when I found that the overwhelming portion of the class preferred to see another out-of-class exam, provided they were given a bit more time, I went with another take-home test. (I feel such tests are substantially more appropriate for our course, anyway.) I feel this exam is a good deal easier than the last in a number of ways: although it's probably longer than the first, it involves more straightforward computation, and is less "theoretical." Already I've noticed fewer tears (on the students' parts as well as my own) and less stress. I think we'll make it through this one okay.

Alas, I must now away and come up with some cool big numbers with which to entertain the Super Saturday kiddies this weekend as we learn to count to infinity...

Thursday, November 02, 2006


It's been a little while since I've posted, this week's been slamming.

I do have a lot to say, a lot I've been thinking about as my Hahvahd trip nears and I have to say something about IBL in the context of our 365 course. I want to write a bit more later in reflection on my goals for the course, and how well we're meeting them right now (in particular as regards the learning goals I'd set out in the syllabus).

I've still got some fun course materials to type up, though, so I'll be brief at present.

For the time being, I hope that the inventors won't mind me sharing with my readership the following linear algebra drinking game, made up this past weekend:

Equipment: TI-81-or-later calculators, one per person. Drink (non-alcoholic, of course!).

Object of game: players compete by constructing 6x6 matrices on their calculators, and then computing the determinants. The first to obtain a matrix with determinant lying between 10 and 20 takes a swig. Repeat as desired.

Change of Basis reminds you to drink responsibly.

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!

Thursday, September 28, 2006

When you assess... you make an ass out of "e" and "ss"?

I sounded good at the time.

I've just finished making up the first exam for 365.

It's a take-home exam with four questions on it (three of which are divided further into subquestions). There are some computations involved, but nothing too heinous, and nothing that can't be facilitated by the use of a calculator or Mathematica.

There are a number of applications considered, and though a couple of them are novel, they're clearly laid out, and quite simple. The other applications are similar to others considered in class. I think the exam will test interpretation of computations better than it will test anything else. So you know what the rank of a matrix is? Great! Now how do you use it?

I've also written up the plans for tomorrow's class, where we'll consider one more application, this one related to my research, before we play a game of "Choose Your Own Matrix" in order to flex our rank-finding muscles.

On deck for Monday: vector spaces! Fun! And on Wednesday, we might have some guest teachers...

Wednesday, September 27, 2006

Oh, and...

I have no idea how many of you are reading this blog, but I have a sense that it's a goodly percentage of the class. I'm curious: what's it like to be able to peek behind the curtain?

I hope my often senseless ramblings serve some useful purpose!

Where do we go?

I gotta be honest: I'm beginning to think I'm manic-depressive when it comes to this course.

At one moment, I'm on top of the world, and at the next, well...maybe I'm just blowing the little things out of proportion.

What do you think?

At the end of the problem session this last Monday night, I was practically euphoric. We had a good turnout, we had fun, as usual, we (and I'm including myself in that "we") learned a lot, we made a good deal of progress.

And in class today, when one of the applications "clicked" with someone who'd been struggling with it for a long while, well...those "clicking" moments are the moments that teachers live for.

And of the five people I've now approached about leading discussions and activities in the class, all five are excited about it and willing to rise to the challenge. You have no idea how much this encourages me.

But I know that some folks are concerned that we're not moving fast enough...

...and some folks are concerned that we're moving too fast.

Some folks are concerned that they're not getting the "formulaic" aspect of the course, laid out in theorems and proofs and computations galore by the textbook...

...and some folks are concerned that they just can't see the pictures that we use to describe the applications that come up in class.


I know the answer.

The answer is: I can't.

And that...that weakness...that inherent and unavoidable failure...gets to me.

I just can't make sure that everyone's completely satisfied. No one can do it. It just can't be done.

I guess the best I might be able to manage is to "maximize" satisfaction.

But how on Earth can I do that?

I realized something the other day about the way I've designed this course. On the one hand, it's a good thing that I've taken the trouble to create such robust methods for communication with the class: I've had no trouble developing a rapport with the students which I think is often underrated, I've been able to quickly and (I hope) accurately assess when people are having difficulties, and I've therefore been able to remediate problems before they get too nasty. I think that I've been able to do this far more effectively than I would have had I been teaching the course in a "traditional" way. And I'm not saying that's a bad thing. But...

...I also feel like ignorance could be bliss. Bliss for me, at any rate. If I cut all these lines of communication, I would no longer know right away if something were amiss...for instance, the way things are set up now, I can tell almost immediately if someone's having difficulty with a particular concept or example, and I can take steps to fix that as soon as possible.

The "old" way, someone could fail an exam before I'd even know what was up. And in all that time, during which that poor student was struggling, maybe faking his or her way through the homework and in-class examples, managing a mediocre but not pathetic grade on the quizzes, I'd be totally unaware of that struggle, and I'd assume all was well.

Maybe that's why I might be going more slowly through this material than I should be: I've got my fingers on the pulses of every body in that class, and if even one of those pulses starts racing, I hit the emergency brakes, and we slow down.

I know I shouldn't be doing that, I know I can't expect everyone to do equally well, but how can I do otherwise? How can I let myself fail in my role as a teacher, if I know someone's not understanding?


I don't know.

So here's my plan for the next couple of days: we'll get that quiz in on Friday (at last!), and then we'll do one more example before we call it quits for Section 2.1. Section 2.2 will take us on a short tour of notation and terminology, and then, after I've handed out the take-home exam, we'll start some really cool stuff on Monday, when we take on arbitrary vector spaces. The applications are very quickly going to become much more interesting, more realistic, more varied in nature.

What do you think? How are we doing, as a class? How are you learning, as students? And how am I helping in facilitating that learning, as the, for lack of a better term, "teacher"? I really want to know.

Tuesday, September 26, 2006


So just guess which branch of mathematics played a role in the pure mathematical research I was working on this afternoon?

Hmm? Any guesses? I'll give you three, and the first two don't count.

Here's the problem. I've got a set of distinct integers ai, 2 or larger, arbitrarily many of 'em, maybe something like: {2,4,5,8,13}. Now you get to pick any positive number a (doesn't have to be an integer) lying strictly between the smallest element of the set and the biggest. In our example above, I could take 5.4, maybe.

The question is: how many different ways can I pick "weights" x1, x2, x3, and so forth, one for each element of my original set, so that each weight lies on [0,1], the sum of the weights is 1, and the sum a1x1 + a2x2 + ... +anxn is the number a? Above, for instance, we'd get to pick 5 weights, and we'd have to get the sum from the previous line to add up to our chosen value of 5.4.

I'll leave it as an exercise for the reader (a common math ploy!) to provide the details, but I'll cut to the chase: you can turn this problem into a linear system, and from there into a matrix, with 2 rows and as many columns as you have numbers ai in your set. Since you can have arbitrarily many elements in that set, you can end up with boatloads of columns (yes, "boatloads" is a technical term). And as soon as you've got more than 2 columns...what happens? I'll let you work it out.

Cool. This has some pretty heavy-duty implications in the land of graph theory.

I was so excited about figuring this out this afternoon that I went across the hall to the Math Lab, found Kaytlynne and Theophila working hard on homework from their other courses, and bugged them about the cool way linear algebra had worked itself into my work.

Neat, huh?

Tomorrow we'll get a few more folks from the class up at the board. Nadia and Imogene did a great job in getting us started on mapping out crystal structures. We'll pick it up from there.

Monday, September 25, 2006

Problem Session III

Hey, hey, hey!

I felt that class came off pretty well today. We began with an example of linear algebra applied to chess, in which we analyzed the moves of the knight in terms of the basis {[1,2],[2,1]}. Fun galore! From there we spent some more time on the applications begun last week, to crystal structure and roving bases that track the structure of a tropical storm.

And at tonight's problem session, we got our attendance back up to 11, with some new faces (and the sorely-felt absence of a few familiar folks), and with the SAME six teams represented as in the past two weeks. (Come on, you folks in the other two teams...what's keepin' ya?) Having now received permission from those depicted to post the pictures from tonight and last week, I've included (below) candid shots of the evening activities.

Also, I've approached four of the students from the class with the idea of letting them have the chance to lead the classroom activities, and all four were up for it! (I'm doing a little dance of jubilation as I write this. Don't believe me? Ha! Try and prove it!) All seemed excited by the idea, though one of the stalwart students expressed a little fright, which would be mitigated by being allowed to lead class with the assistance of a good friend. I have confidence in all four of these students' abilities, and will soon be prepping them for this responsibility. I'm excited!

Meanwhile, here are the promised pix from tonight and last Tuesday:

Okay, y'all who haven't come yet: see how much food you're missing out on? A veritable smorgasbord of goodness! Two sorts of apple bread (store-bought and homemade), homemade brownies (thanks, Fiona!), grapes, and your choice of refreshing carbonated beverage! Mmmmmm...MMMM!

And check this out: can you get any more excitement than the Row Reduction Races?!!? I mean, really! Look at these tough competitors from Week 2, pounding out those row operations on their calculators, each one hoping to be crowned the monarch of matrix manipulation!

And here it is! The exultant victor! Just look at that reduced row-echelon form! Yowza! It's this sustained, unceasing excitement that led to tonight's increased crowd, shown below:

Food, friends, linear algebra...a natural combination. I ask you, does it get any better than this?

Friday, September 22, 2006


I'm tired.

It's been a long week.

It's been a long semester.

And we're only one third of the way through.

This week's been rough on a lot of folks. I know of several tough classes in which there were exams this past week. I've seen a number of glazed-over looks and catatonic stares, and attendance has been low in all of my classes, especially today.

Friday. Wonderful, wonderful Friday.

It's been a rough week on me, too, I have to admit. Two fifteen-hour campus days started things off, followed by three only-slightly-shorter 11 hour romps. Mathematical action from dawn until dusk. I'm glad I've got an insuperable supply of energy, I've had to draw on it this week.

I knew I'd be putting a lot of effort into this class. I'd tried to front-load it as much as possible by doing a good deal of the planning before the semester began, and to a large extent that planning's paid off: I've had less to do on the fly than I would have had I not put together beforehand a goodly portion of the projects and class activities, much of the documentation, and a boatload of resources. Still, though, there are some midcourse adjustments to be made, a few wrongs to be righted.

What's gone wrong so far?

Nothing big, really. Minor missteps, here and there.

For instance?

For instance, I've been overestimating how much we can "cover" in a single class period. Given the more open, bidirectional format of the course, the exchange of information is certainly not as "efficient" as it would be were I standing at the board lecturing the whole time. I'm just now getting a feel for how many in-class exercises we can cap off before Doctor Bob's unofficial class clock strikes 3:35ish.

I also underestimated the difficulty some folks would have in reading the textbook. I know it's a dense read, my friends. Most upper-division math texts are, especially the first time around. I'm thinking back to the first math text I had to really read (Richard Strichartz's The way of analysis, the basis for my first two semesters of real analysis): though I now find it a better-than-average text for its topic, quite funny in places and very lucidly written, I know at the time that I was first reading it it might as well have been written in Sanskrit.

Nevertheless, the text is a decently good one, as linear algebra texts go. It does a fair job of clearly developing the necessary theoretical points while saving time for more mundane computations. And I hope the "Key Points" slides I've been providing in class have softened the textbook's blow, and that our time in class has helped to clarify any obscurities encountered in the reading.

If it's any consolation to my students, reading math doesn't necessarily get any easier: you can't imagine how many times I have to read every page of some papers in order to understand them as well as I'd like to.

Still, as hard as the reading can be, it's crucially important that every one of us does her or his best to keep on top of it. Quite frankly, the in-class exercises (which are meant to be challenging, but not heinously so) will be much more enjoyable for those who've done the reading and have prepared themselves for class. Should one expect to do at all well in a literature seminar centered upon a certain novel if one has consistently gone to class without first reading the book? I hope the answer to that question is "no."

All in all, I think that I'm doing just about all that I can to make things go smoothly.

What more can I do? So much of the class is up to the students, I can only go so far.


Well, well.

Well, I'm off to read a few of the journals submitted on-line.

Tonight, and then tomorrow, and then...and then tomorrow again. Then we'll start anew.

Yes, I'm tired.

More to come.

Meanwhile, let me throw out a question to my students: how many of you out there in linear algebra land would like a shot at leading the class discussion for ten or fifteen minutes? What would you do with that time if I gave it to you to use? I really would like to know, feel free to write!

Tuesday, September 19, 2006

The few, the proud

Tonight's showing was a little sparser, largely, I would guess, on account of the heavy exam schedule burdening a lot of folks this week. I know there was a crippling Organic exam yesterday, and there's a Complex exam tomorrow in our own department. Lots goin' on, and people are having to divide their time.

For the few who came tonight, there was much food and much fun. There are pictures to post, but as usual I want to get permission before posting them.

In all, eight folks showed up, representing 6 teams. Good spread! I feel like we got A LOT done tonight. Y'all are going to be experts by the time this semester is over. Good work! And keep it up...

Hear ye, hear ye!

Harken, all ye! Good news is in the offing!

I was informed just minutes ago by the illustrious chair of our great university's Writing Intensive Committee that...the recently-applied-for WI status for MATH 365 has been approved!

You may commence rejoicing.

I'll see you all this evening, or, barring that, tomorrow.

Sunday, September 17, 2006

Show 'n' tell

Hey, howdy!

Conversations with a few of you folks over the last couple of days, both on this blog, and over e-mail, have made me think a lot about pulling off a good research project.

For those of you who haven't been reading all of the comments lately, someone responded to my post from yesterday, anonymously encouraging me to light a fire under y'all's behinds and get you going on those research projects. In reply to this, I wrote another comment giving some tips on how to get a good start on these projects, and how to sustain them well as the semester progresses.

And then, while writing back and forth with another member of our class about undergraduate research, I got to thinking: hey, since when am I the sole font of knowledge in this classroom? Many of you have already done (and perhaps are doing) undergraduate research, and I'm sure you have lots of good ideas about how to start off a good research project, how to go about keeping track of your sources, how to record your ideas and later write about them...and so on.

I'm not claiming that my hints won't help, I'm sure they will. But I'm sure that many of you have lots of other hints you could share with your peers in this class, to make the ride a bit smoother.

So, I'm kindly asking all of you who've had some experience in research before: what's helped you out? Do you have any secrets to impart? Ideas you'd like to share, particularly, at this time, regarding how to get started? I'd greatly appreciate if you'd chime in by responding to this post, even if you do so anonymously.

Thanks, all! And I'll see you in class tomorrow.

Saturday, September 16, 2006

Shameless self-promotion

Howdy, all Change of Basis fans!

It's Saturday, and I'm just now fixin' to put together a solution sheet for the latest round of representative problems, as I've taken to calling the recommended textbook exercises for 365.

I've spent the afternoon up until now working on a statement of my teaching philosophy, an exercise which meant several hours of agonizing over the right choice of words. Teaching is something I could easily prattle on and on and on and on and on about for chapter after chapter (surprise, surprise), so a good deal of the difficulty lay in cutting what I had to say down to the requested three pages. I was forced to think about the elements of teaching and learning which mean the most to me.


I'll think some more about that matter. What's on your mathematical minds right now?

In the meantime, I wanted to put out a reminder for my students in 365: please don't forget that Tuesday night at 6:00 we'll have our second problem session, location TBA. I'll be sure to bring a loaf of my homemade apple bread (mmmmmm! apple bread...), and I've already heard murmurred rumors of brownies and soft drinks from some of the students. Please feel free to bring snacks, music, beach balls...whatever will make the session more helpful, more comfortable, more fun. No matter what else you bring, bring yourselves, and let's see if we can break last week's attendance mark of 11!

Wednesday, September 13, 2006

Story time

Today saw us going over matrix inverses again, more fully than we'd done on Friday. I felt that Friday we'd not yet had a chance to really get a good grip on the geometric meaning of an inverse, so we retraced our steps into that territory before learning how to run a Markov process in reverse.

Although I think most of the folks in the class got the hang of it by the end of the class, there were a few who remained confused. (One student came by after class, frustrated as hell at not picking up on a lot of what we'd done. We spent about fifteen minutes working through the applications we'd considered, one-on-one. By the end I think it was clear that the student understood more of what was going on than the student had thought at first.)

Not knowing what in the wide world of wackiness is going on is truly frustrating.

And there's no doubt about it: frustration sucks.

Frustration works its ways differently on different people. Some cry, some scream, some just lose all capacity for rational thought.

The most frustrating incident of my academic career came in my third year of graduate school, at the end of the Fall 2000 semester at Vanderbilt.

That'd been a tough semester. If memory serves, I had a somewhat traditional algebraic topology course, a couple of seminar classes, including one on combinatorial group theory and another on small cancellation theory, and I was auditing a class on advanced group theory. Meanwhile I was teaching a pretty cool Calc I class, helping to put together our department's undergraduate seminar in mathematics and compose our in-house precalc review text, and conducting research that would later become my first published article.

By early December, I was pretty much gone.

The seminar on small cancellation theory was an exciting one. The instructor, one Professor Peanut, was (and is) one of the world's foremost experts on combinatorial group theory, universally respected in his area, and well-known outside of mathematics for his theories on the use of mathematics in deciphering biblical prophecy. Big, big man in the mathematical community. A very kind gentleman, he terrified me nonetheless with his stature and renown.

He was a man of deep conviction and numerous idiosyncrasies. An orthodox Jew, he davened constantly while reading his Torah, something he did during just about every break available to him. He played with his beard absentmindedly, stroking its great gray length with thick stubby fingers. He drank Diet Coke as though he owned stock in it. If he were not the one speaking but only participating as an audience member, he would sit in the front row (an open can of Diet Coke in front of him), his attention fixed on the speaker for all of about ten minutes before sleep overtook him and he dozed off, his chin sinking through his beard to rest upon his chest. He would sleep for five or six minutes at a time before waking and looking around as though to make sure no one had caught him in his nap. Five minutes later the cycle began anew.

Strange travel restrictions forced him to spend with us only half of the semester he'd promised our department, so it was not until mid-October that he came to the United States to begin the seminar course we'd scheduled for the fall. Once underway, we (the five of us registered for the course and a few faculty hangers-on who wanted to sit in) decided the best way to make up for lost time would be to meet once a week for twice as long as we'd originally meant to meet. So began our three-hour Thursdays.

The class was fascinating. Peanut would lecture to us extemporaneously on cutting-edge research, much of which he'd developed only weeks or even days before. Unlike his long-time colleague and co-author, Professor Cashew, who for the past year had been a full-time faculty member at Vanderbilt, Peanut was not a slave to detail and would almost invariably omit precise computations and rigorous proofs, opting instead to give outlines and general descriptions of his intricate arguments. Though he painted in broad strokes, the pictures he produced seemed clear to us, and only occasionally would we have need to stop him in his exposition. Much of the beauty of math lies in the creativity that goes into the construction of new techniques, and Peanut's were among the most delicate and delicious I've ever seen. Though his exposition was too sparse to allow me to get at the innermost workings of his mathematical thought, I felt I had a good understanding of the subject.

I realize now how difficult it is to learn to paint without actually getting the chance to hold the brush yourself.

We spent nine or ten evenings in these seminars, and in that time I filled perhaps four dozen pages with notes on Peanut's techniques. Supplementing this were about thirty pages of poorly mimeographed notes Peanut himself had written (by hand, in very nice and readable Russian script) in preparation for writing one of his papers. We had a good written record of the course, but given its lack of precision (like his speech, my notes are filled with phrases like "quite big," "big enough," "very small," and so forth), to do much math with what we had would be like to building a warp drive from pencils and chewing gum after watching an episode of Deep Space Nine.

I must take a moment to describe one of the course's faculty fans. Professor Pistachio was (and is) a brilliant man with an acerbic wit and a certain impishness that came out most noticeably when torturing undergraduates. On his website he keeps a page which admits that his students often complain he is too sparing with positive feedback. To counter this, he includes on the page a MIDI file of soothing music, after several seconds of which he intones carefully and lovingly, "good job. Good job. You like the word 'great' better? Great job..."

"Sarcastic" doesn't cover it. (In his defense, I must say that I consider Pistachio one of my favorite professors in graduate school, an excellent expositor of mathematics, and one of the best writers of mathematics I've ever known: it's hard to write clear technical mathematics articles, and he does this exceedingly well.)

At the end of the last class, Peanut dismissed us. "That is all," he said, and with a kindly smile he set down the chalk. If only it were so! From his seat to my left, Professor Pistachio said, in Russian, "what about their exam?"

We sank. We'd expected to not be tested; it was an unwritten understanding. After all, this was a seminar course, and those who participated did so because they wanted to learn the material. We weren't undergrads anymore, we didn't have to be prodded and cajoled by some penny-ante system of praise and punishment. We didn't need an exam!!!

More to the point, we weren't ready for an exam. What would he, could he ask of us? How could we be tested on details we'd never learned?

"Ah yes," Peanut replied softly. "There is the matter of this exam. I think perhaps it should be...oral examinations. You will each schedule a time to meet me individually, yes?"

We left the room dejectedly. I trudged back to our basement offices with my colleagues Damon, Maurice and Nathaniel. We discussed our plans. "We've got a week," we agreed, "before he leaves the country. We can study. We have notes."

But we were all busy. We had finals to write and proctor and grade, we had research, we had other classes. Our time was precious, and we could spare little of it to digest the abstruse methods we'd been given in big, unchewable chunks over the preceding weeks.

"I'm going to do it tomorrow," I told my friends the next day.

"Daring," they agreed.

"I just want to get it over with," I said. I sent Peanut an e-mail and made an appointment at 4:00 the following afternoon.

I spent the next 24 hours cramming like never before. I read over my notes again, forwards, backwards, sideways, diagonally. Like Peanut with his biblical gematriyot, I sought meaning in every word.

4:00 the next day came way too quickly. I knocked timidly on the open door of the office Peanut shared with a few other visiting faculty members. There was no answer. I knocked again, and again there was no response. I padded into the office softly and peered around the cubicle divider which separated Peanut's desk from the one nearest to the door. At his desk he sat, squinting at the screen of his laptop. "Professor Peanut?" I asked. He looked up and smiled and gestured for me to sit in the chair across from him. I sat, and the test began.

"First, can you tell me how to describe the automorphisms of the...icosahedron?"

What in the HELL?!!?!!? This question had NOTHING to do with ANYTHING we'd talked about in the last several weeks. NOTHING. Nada. Zippo. My mind spun.

"You have had a basic course in group theory?" Peanut asked, his tone belying at least a hint of annoyance.

"Yes," I insisted. "But..." He scribbled a sketch and a hint or two on a piece of paper and thrust it across the desk towards me. I stared at it blankly. "Um...let me think..." And he did. He folded his fingers together and sat back in silence as I struggled to make connections between what I knew of icosahedra and what I'd learned in Peanut's course. There wasn't much to go on, but I managed to retrieve a few tidbits concerning fixed point sets and fundamental domains, and with these I pieced together a good bit of BS.

I looked up. Peanut, true to form, was asleep.

"Professor Peanut?" I asked gently. "Professor Peanut?" He slept on. I leaned back and sighed, and the hot light of the setting sun cut into the room through the slats of the office's Venetian blinds. I squinted and looked outside at the walls of the physics building next door. I wished myself far away.

I tried again to rouse Peanut, and at last succeeded. I showed him my notes, and he nodded and made a few ambiguous and uninterpretable noises. Was I right on or full of crap? I had no idea. As he let me get back to work, I looked wonderingly at the notes I'd produced. I'd been in Peanut's office for almost an hour and had done almost nothing. Quite frankly, I felt like shit.

Peanut let me struggle for another quarter-hour, and at the end he let me off the hook. "Okay," he said. "That is enough." I'd made a half-assed attempt at solving one easy question, and in producing that one solution I'd demonstrated minimal knowledge of a few minor techniques we'd covered in the class. The worst was yet to come.

"Now I must assign a grade," said Peanut. "I'm afraid I do not understand...what does this mean, an 'A'? What does a 'B' mean?" I gave him a briefing on the American grading system, perhaps curving each category a bit too generously. Peanut seemed satisfied. His next question came sock in the gut.

"What grade do you deserve?" he asked.

After the worst academic performance of my life, I'd been asked to assess its merit.

The lowest grade I got in grad school, I gave myself.

I left Peanut's office numbly and dumbly. I went to my office and let the others know how it had gone, and then I left. I spent the rest of the evening in a stupor, eating flavorless food and having mirthless conversations.

Yeah, frustration sucks.

I hope that if you're ever frustrated with this course, like our friend who spent some time with me this afternoon, you'll be willing to share that frustration with me. It feels better when it's spread around, and once it's out there, we can do something about it.

Okay, that's enough for tonight. I'm sure you've all enjoyed this bedtime story. Feel free to post away and share your own moments of frustration! Meanwhile, let's hit that reading for the weekend, and get ready for some hardcore Tinkertoy action on Monday!

Monday, September 11, 2006

Beard update

Oh yeah, and here's the 10-day beard update, yours truly on the left fact, we're standing in the same order as before:

How ' bout that?

And yes, I did trim off the cheeks.

After hours

We had the first of our after-hours problem sessions tonight, and I feel it went very well.

The session was a fantastic opportunity for all of us: while the students in attendance (11 of the class's 33, representing 6 of the 8 teams) seemed in complete agreement that the session helped to solidify their understanding of the underlying computations, I felt that I gained tremendous insight into the students' thoughts on the class overall. We had some refreshingly open and honest conversations about how people are feeling about the course right now, and I'm heartened. I love these people!

We'll have another such session next Tuesday night, but be forewarned: I fear I'm going to have to start asking for a couple of bucks a pop (or "a soda") to defray the cost of refreshments!

My mind is aflutter with thoughts, but sadly I'm too tired to track any one or two of them down to pin them to the page, so I'll leave it at that.

I will end with another exhortation to those who are reading this blog (and after tonight, I know there are quite a few of you!) to please feel free to post responses to my own posts.

Tomorrow is another day!

Saturday, September 09, 2006

For the best

365's been on my mind a lot since class got out yesterday afternoon.


Maggie tells me it's too much on my mind. She's probably right, as she often is when it comes to questions about me. I hate to sound cliche, but truly she knows me better than I know myself.

I logged on hoping to have something to say, but I'm having trouble finding words to wrap around my thoughts.

Am I overthinking this course?

I'm enjoying myself immensely, but I still find myself feeling unsettled. I think much of this has to do with the wacky schedule UNCA's served up for us this semester: a long first week took the steam out of everyone's engines before the first break rolled around; Labor Day stumbled in awkwardly after two full weeks had passed, granting us a shortened week this past week; and next week we've got an artificially truncated schedule with the cancellations in honor of the installation of our new chancellor. We've not yet had a chance to find our groove.

With a handful of nonmathematical classes, 365's borne the brunt of this clumsy scheduling. We've already spent class time on the syllabus and technical writing, and this coming Monday will bring us to the library.

I certainly don't regret setting time aside for these topics, and when I next teach this course you can bet they'll still play a major role, yet it'll be nice to have a purely mathematical straightaway in front of us by the end of this coming week.

This coming Wednesday we'll take another look at matrix inverses. Given the importance the geometric interpretation plays in a number of the applications we'll consider later on in the course (and that these folks'll consider throughout their careers!), I want to make sure everyone's got them down. Giving another day to inverses has the added bonus of allowing those who are struggling with the reading to get caught up.

In each of his first two journal entries, one of my atmospheric science students threw down the gauntlet, demanding to know why I'd yet to bring out any atmos applications, given that 3/11 of the class are ATMS majors. In answer to his challenge, I insisted that most of the anywhere-close-to-realistic applications of linear algebra to the atmospheric sciences are too complex to use as in-class examples. I felt this answer was pretty lame, though, so I hit the library yesterday afternoon between classes and checked out three texts on numerical weather prediction and linear climatic models, and spent a few hours early this afternoon slogging through a chapter or two of one of them, learning about matrix methods in airmass modeling. Although the physical equations governing the models are indeed complicated ones, if they're treated as black boxes, there are reasonably simple applications involving as few as four equations which might be workable in class. I'll have to have a crack at it. (To my challenger, if you're reading this: howzabout them apples?)

Okay, I've had more to say than I'd thought I would. Maggie'll be calling soon, to get a ride home from work. I'd best be going.

She's probably right: I'm overthinking this course. But I'd rather overthink than underthink.

I think.

Friday, September 08, 2006

Dear Diary

Last night I finished reading the first of my students' journal entries. Wonderful insights! These folks have some fantastic ideas, a couple of which I'm already beginning to integrate into the structure of the class.

For instance, beginning this coming Monday I'll be holding informal (but informative!) problem sessions on Monday evenings, for those who'd like the chance to work on the HW with each other in a setting conducive to study. We'll start out in the Math Study Room on the second floor of Rhoades, but if that place doesn't "feel right" for some after-hours linear festivity, we may pop over to some other campus location in the following weeks.

By and large people seem to be settling in and having fun. The two main concerns on most people's minds are (a) the readings, and (b) those darned articles! I think I might have introduced some sample bias on that second point, since the second journal entry solicitation specifically asked the students to write on the issue of the project: "How do you feel about...?" Regarding the first point, while a number of folks are taking the reading in stride and handling it nicely, some others are having trouble digesting the pulp and getting to the real meat of the matter. I sent a pretty detailed e-mail yesterday afternoon giving some more hints as to how to read the text more effectively. I hope that e-mail helps folks direct their studies and make better use of their precious time.

For the most part, though, people are enjoying the class, liking their teams, and getting comfortable with the format. Almost without exception people are having fun with the team activities and the classroom atmosphere, and people seem confident that their teams can handle the semester's rigors.

Today's agenda's got matrix inverses written all over it. After looking at the geometric interpretation of inverses, we'll find out how to make the Markov Dance run backwards.

Wednesday, September 06, 2006

Monday, Monday...


Class went pretty well today, I thought. (As always, I'd be interested in hearing the student take on this assessment...) The team quizzes were great, the individual quizzes were better, and there was some serious row-reduction of matrices goin' on in our hands-on approach to chemistry. Due to the size of some of the matrices which arose from those chemical reactions (6-by-7 in some cases!), I don't think many people managed to get their reactions to balance through linear algebraic methods. (One notable exception was Deidre: while at first her coefficients didn't come out right, she soon caught her small error in arithmetic a few row operations back, and hey presto! the balanced reaction popped out. Props, Deidre, if you're reading this!)

Anyhow, things went well, and I'm happy to say that it looks like folks are really getting the hang of using matrices to solve linear systems.

A couple of folks have mentioned that it's hard to focus one's attention on the highlights of a day's reading without knowing exactly what those highlights are. To try to address this weakness in the course design, I've begun (with today's class) to provide a short "focus" session at the beginning of class, in which we spend about five minutes hitting the major points of interest from the reading for that day's class. Jeremiah recommended this particular idea to me this morning, and I think it worked out really well today. (Props, Jeremiah!) I'll try to make that a regular part of the class, and students should feel free to comment on how it went today, and how we can make it work better in the future.

On deck for Friday: matrix inverses. I plan on bringing in some geometric applications that might be of especial use to any of the folks whose projects involve geometry (Crystal Gazers and Screen Flickerers take note).

Sunday, September 03, 2006


(My partner in conversation below is a fictionalized amalgam of real people and of inner discourse. He is a cipher, into whom I've placed words as freely as he's spit them out.)

Ethelred: Why?

Me: Why not?

Ethelred: Why not? Because you can't possibly cover as much. I mean look: how many of your students are going to be able to state the Cauchy-Schwarz Inequality, let alone prove it? Now it's come and gone, and you've never said a word about it in class.

Me: How many of my students in a traditional Linear class would be able to state the Cauchy-Schwarz Inequality a week after it's "covered"? And how many of them will ever use it?

Ethelred: Okay, bad example. But what about the Triangle Inequality? How are they supposed to understand more general inner product spaces later on if you're not even making sure they get the tools they need to understand the most important one?

Me: They've understood that most important one all their lives, just have them try to draw a triangle where one side's longer than the other two put together. So what if they don't know "why" that must be?

Ethelred: Now you're being pedantic.

Me: Probably.

Ethelred: You have to admit that Friday's class didn't get as far as you'd hoped it would.

Me: Maybe. Maybe at first. Maybe at first I thought that we'd fallen a little bit behind.

Ethelred: Ah. Ha.

Me: At first.

Ethelred: And then?

Me: And then I thought, "you know, if we'd finished up that chemistry exercise, we'd have worked our way through the solution of a linear system by row-reduction of matrices. We'd have finished up Section 1.4. We'd have been ahead..."

Ethelred: So you're still thinking in terms of "coverage"?

Me: "Coverage" is a four-letter word.

Ethelred: But you don't deny that it's on your mind?

Me: I don't deny it. A decade of teaching more or less one way, even if that one way's got some bells and whistles on it every now and then, doesn't disappear overnight. But you interrupted me.

Ethelred: Sorry.

Me: "...and they would have pushed their way ahead on their own. By themselves. They would have worked out the application for themselves. I wouldn't have done a single example for them."

Ethelred: So what?

Me: So how would you have had me teach Sections 1.1 and 1.2? "Definition, theorem, proof, theorem, proof, incomprehensibly inapplicable theorem, unnecessarily dense proof, useless abstract example, definition, long and boring (however impressive-looking) list of properties...half of which everyone'll forget a week after the exam, all of which half of them will forget a week after the exam..."

Ethelred: You're being pedantic again. There's a lot to say for the traditional Linear Algebra classroom. Are you really doing your math majors a favor? They, at least, deserve to see worked-out proofs of the major theorems from linear algebra. They, at least, deserve to gain access to the more technical aspects of the material.

Me: They'll get that in the reading. Besides, what would you have me do, sacrifice the rest of the class for the sake of the math majors?

Ethelred: There are a good many of them in there.

Me: There are just as many, if not more, atmospheric scientists, and you can bet most of them could give a monkey's red rear end for whether or not the Cauchy-Schwarz Inequality holds in any given inner product space.

Ethelred: Whether they care about it or not, exposure to that fact is good for them. Damn it, Patrick, you've said it yourself all of these years: above all else, the best thing that a student can get out of a math class is a little exercise in critical thought.

Me: And who's to say I've changed my mind? Besides, the person who said that was a younger version of my self-as-teacher, an alpha model. An inchoate form, Patrick-of-five-years-ago. I've rested safely and soundly in my old teaching method for years now, and though it's worked for me well all those years, there was something missing.

Ethelred: What was missing? Patrick, you've always been a good teacher. Why change?

Me: I've always been good, but why not be better? And I'm not sure I've been as good a teacher as I should have been. "What's missing?" you ask. Fair question.

Ethelred: And?

Me: Substance. Structure. A coherent overall frame. A big picture.

Ethelred: Overrated.

Me: Really? You say I want to teach critical thought. Fair enough. What does that mean?

Ethelred: And?

Me: No, really, what does that mean, to teach critical thought? It's something I've taken for granted, something I've assumed myself capable of, something I've assumed I could recognize, that others could recognize, when they'd achieved it. It's something I've always thought inhered in mathematics, so fundamentally so that you couldn't get out of a math class without having exercised it at least somewhat. But that's not so: you can make it through Calc II without even so much as rubbing your elbows up against critical thought, though you both sat side-by-side in a crowded classroom the whole semester. The problem is that the students don't recognize critical thought. And I've grown so accustomed to thinking that I recognize it that half of the time I don't, either.

Ethelred: What, so you have to tell them that they've had a "spiritual experience" in a math class in order for them to actually have one?

Me: Not necessarily, but you may at least have to set them up for one. I'm not sure I was doing that before, however much group work I was doing, however many cupcakes I got them to cut up, however much fun and interactive my classes might have been.

Ethelred: And what's so grand about this Linear class?

Me: The other day Livonia was in my office, hangin' fire, chewing the fat. We were talking about the class, and all of a sudden she said something like, "I have a feeling this class is really going to help me with my upper level math classes. It's hard to do, but I'm going to need to know how to learn math on my own."

Ethelred: So?

Me: I pulled up the syllabus on-line and scrolled down to the learning goals: "be aware of your own learning styles, and of how to make effective use of those styles."

Ethelred: Hmm.

Me: Just a few minutes later, she was talking about her team's research proposal, and she mentioned how important understanding of the issues involved in their top choice would be later in her career: "that's the kind of thing I'm going to have to do, take a problem and take it apart to understand it, and then put it together again." I pointed back to the syllabus: "dissect a real-world problem in order either to analyze the way in which linear algebra can be applied in order to solve the problem, or to explain why linear algebra might not be applicable."

Ethelred: She's a bright kid.

Me: Oh, yeah. You can bet dollars to donuts she is. That's a roomful of smart people I've got in there. Don't think I'm not afraid of letting them down, of not helping them to find the best damned education they can get.

Ethelred: And your list of learning goals the way to do that?

Me: It's not just a list. It's not just a touchy-feely component of a lifeless document that becomes irrelevant as soon as it's printed. It's the shadow of something far more important, of an underlying structure, of an edifice that I built before the class even began. That list is a continual reminder to myself of everything I've got to do in the classroom before I head over there at 2:35 on Mondays, Wednesdays, and Fridays.

Ethelred: And what do your students have to do? Do you think that they're thinking about these things, too? Haven't they got enough to think about, with all the reading they've gotta do, all of the writing, all of the crap you've got them doing in the class itself?

Me: I don't know. I hope I've set it up in such a way that they don't have to think about the list, that every one of those fifteen goals is met without them knowing it. That's my intention, anyway, and whether or not I'm succeeding...well...we'll see how it turns out. Edward was asking me the other day about teaching one of the freshman colloquia: did I think I'd want to do that at some time soon, did I have any plans, any ideas for such courses? I said that yes, I had an idea that I'd been kicking around for a while that would make a perfect LSIC course, but that I wasn't ready to teach it yet. "It's probably a pretty good idea to get settled into the department before branching out like that," he said. Or something to that effect. That got me thinking: is that what I'm doing? I don't think that it is.

Ethelred: You're not getting settled?

Me: No, that's not it. I think I've done a fantastic job of getting settled into the department. I don't think that I'll ever be done doing that, of course. In fact, you're probably in trouble if you ever start to think that you're done getting settled. I mean that I'm not just sitting around in the math department, waiting for someone to give me permission to take a step outside Rhoades-Robinson: 365 isn't just another math class, and I have to say I took a little offense at the implication that that's all it is.

Ethelred: Is that what he meant to say?

Me: Maybe, maybe not. But that's the way I heard it. This course is so much more than that. You want a writing intensive course? You got it. You want an information intensive course? You got it. You want a course that provides a legitimate research experience? You got it. You want interdisciplinarianism? You got it. It's a more integrative course than just about any LSIC course I could ever dream up. This course is the perfect example of everything UNCA claims to be about.

Ethelred: Hmm. What are you thinking now?

Me: Now? Not much. It's late.

Ethelred: Get to bed. Tomorrow's here already. You need some sleep.

Me: I know.

Ethelred: Your students'll be there tomorrow. They're good kids. You've got 'em hooked now. They'll come back.

Me: I hope so.

Ethelred: I know so.

Me: We'll see. Thanks for talking this out. It's helped.

Ethelred: No problem. Good night.

Me: Good night.