Priming the pump

Here we are again!

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

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

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

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

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

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

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

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

The proverbial Morning After

Well, it's all over and done with.

I think.

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

Moreover, I've finished grading.

Ouch.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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!

Hello, hello!

Argh!

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 flyers...ooo! 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? Hmmm...one 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 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!...

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)!

It's that time again

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

ABSOFRICKINLUTELY INSANE

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...

Quickie

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.

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 shirt...my wife hates it when I wear that in public...)

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!

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!

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)...to 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!

"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!

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?

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!

In or out?

Wow!

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.

Weird.

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!

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.

Oy...

...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.