I've not laughed so hard in a long, long time as I did this afternoon at some of the passages the 365 folks came up with for incoherent instructions for the multiplication of two 2x2 matrices. I'm going to have to try to get permission from some of them to anonymously post their work. It's simply classic.
After a brief discussion of the criteria by which written work will be evaluated in the course, we went through with the "Write a Bad Paper" exercise, to which people took with gusto. It was quite clear from the get-go that clear and consistent terminology and notation play a major role in understandable mathematical writing, as their absence signaled without exception dense and opaque technical prose. Irene was happy enough to share her subteam's creation with the rest of the class: a nice long string of "multiply the first first element by the other first first element and add this to the product of the first second element by the second second element..." earned a round of applause. Niobe struggled to read through the work Hugo and Bethesda had created, in which, among other faults, the 'n' in "and" bore a tilde.
The students also had the opportunity to provide feedback to one another on their work, taking care to remain respect, helpful, and relevant.
I had a peek at a few of the rough drafts for project proposals, too: it seems there's a fairly high level of excitement about these. I'm very much looking forward to reading both the proposals and the students' journal entries, which I'll get my hands on on Monday.
During the last few days, I've also had a chance to talk to a number of the students individually. I've met some of those I didn't know before this course, I've learned more about those I did know, and I've found that all of the groups (with a few minor bumps and gurgles) appear to be working admirably well together. There were a few kinks to work out in one group in which communication had not been so grand, but things seemed to be running much more smoothly today than they had on Monday.
Another issue: a couple of students have expressed concern that every other member of their respective teams seems a bit quicker than they are. One of these students suggested having a team that convened only for the purpose of in-class exercises, one that would be made up of those more reflective souls who might not pick things up so quickly and would prefer the setting of a slower pace. As I indicated to this student, my only problem with this proposition would be that such a team would likely slide even further behind the others on in-class activities. I also assured this person that all individuals' contributions would become evident and invaluable as research gets underway: the more reflective, prone-to-pondering people are often liable to catch things the blitzing brainstormers overlook. I recall my first year of graduate school at Vanderbilt: there were four of us in that brutal Moore-method topology course, and though two of us came with considerably stronger backgrounds, our surety more than once turned into cocksureness and we glossed over important yet subtle details which one of the two "slower" students would catch, having to think each step through carefully and critically.
It'll all come out in the wash, you'll see!
Wednesday, August 30, 2006
I've not laughed so hard in a long, long time as I did this afternoon at some of the passages the 365 folks came up with for incoherent instructions for the multiplication of two 2x2 matrices. I'm going to have to try to get permission from some of them to anonymously post their work. It's simply classic.
Monday, August 28, 2006
I'm just about to wrap up another day, and I thought I'd check in on the ol' blog, not having written anything for a couple of days.
Things have been going pretty well in MATH 365, as far as I can tell. I spent a couple more hours on Sunday morning putting together some Mathematica programs to help people deal with various issues (visualizing vector addition and getting practice with matrix multiplication), and then writing a brief somethingorother for Wednesday's class on mathematical writing: I should be all set for that class's "Write a Bad Paper" exercise. I'm looking forward to it!
I was pleased to get a chance this morning to work one-on-one with Irene, one of the students I'd not met before this class. She's putting a lot of effort into the class, and I admire her courage, because I know she was a little hesitant about her preparedness for the course. I look forward to learning more about the other students in the class with whom I've not yet become acquainted.
I experienced a minor setback in getting the course to qualify as a Writing Intensive course this afternoon, when I was informed by our school's diligent WI committee chair that I'd inadvertently filled out the wrong form in constructing the WI application. No worries: some relatively minor modifications and my current document will serve, with a few frills added for trim. I hope to finish that up tomorrow, depending on the busyness of my schedule.
Class today (Monday) went well, as far as I could tell. I had a better feeling about it than I did about last Friday's class: Friday seemed to go by so quickly, and I got the sense that on that day we didn't get to the heart of the topics we discussed. Today, on the other hand, there was just a single focal topic (the use of matrix multiplication and matrix powers in understanding a simple stochastic economic model), so our attention was more singularly drawn. Although I had originally planned on using the PDF outline I'd posted on the course website as a guide through the trenches, I decided after all to just throw them into the pool and see if they sank or swim.
And they swam, every last one of 'em. Not a single team failed to come up with and elaborate upon the idea of matrix multiplication. I wish I'd had more chance to work with each team individually, but from the floating and flitting about the room I was able to accomplish, I noted a good deal of creative work on the part of many students, and for the most part there was robust communication within each team. (After class, a few folks noted a kink or two here and there, owing to relative reticence of certain team members, or to unfamiliarity with one another, but these issues should not be hard to resolve. On no one's part was there any malice or animosity.)
I spent about half an hour earlier this evening in sending e-mails to a few of the folks in the class, polling them to see how they felt their respective teams managed today. Every response I've received so far has been positive, so I'm heartened. I've been informed by one of its members that even the one team I was a little concerned about fared quite well today. Rejoice!
And so, happily, here comes sleep. More tomorrow, to be sure!
Saturday, August 26, 2006
Well, after a few hours in the office on this glorious Saturday morning, I'm ready for Monday's class, and nearly so for Wednesday. The project proposal guidelines are posted, as is the matrix multiplication activity on deck for Monday. I think that should go well.
I'm satisfied with how class went on Friday (yesterday), although I'd be eager to know how the students felt about it. (Post away, people!)
As planned, we spent the first few minutes involved in the team portion of the first two-part quiz. (We had a late start due to some people who straggled in after class began.) That out of the way, we did some brainstorming on properties of vectors and affiliated formulas, and then we looked at vectors in their natural habitats (PowerPoint slides available here). The RGB vectors went over pretty well, and I think folks got the hang of it rather easily. We didn't have as much time as I would have liked to tackle chemical equations, and I'll be passing out a brief handout on that topic taken from David C. Lay's text Linear algebra and its applications on Monday.
All in all, vectors being review for most of the people in the class, I don't think it went poorly. Monday's activity (renewed analysis of the opening day's Markov Dance) is a bit more structured, and should get the gears turning on matrix multiplication.
Thursday, August 24, 2006
The team assignments have been made, following a different prioritization of criteria than I had originally intended. I don't think any loss of function will result, though.
I'd originally meant to put teams together based first upon schedule compatibility, second upon location of residence (both of which facilitate meeting arrangements), third upon like academic interests, and finally upon compatibility and complementarity of comfort levels in speaking in front of others. I'd gone so far as to put together a rough draft of teams made by following these criteria.
While walking into campus this morning, however, I had second thoughts, and seriously considered the criteria Maggie'd suggested to me, letting common academic interest be the primary criterion. (Thanks, honey!) It really does make sense: since this course is a terrifyingly unfamiliar one for most of the participants (including me!), any measure of "safety" or "security" I can provide the students as they prepare for the semester will surely encourage them further. If this means grouping students with folks from their own departments, then that's what it means.
As a result, the criteria I ultimately used to craft the groups are ranked as follows:
- commonality of academic major or interest
- compatibility of schedules
- location of residence
- complementarity of comfort levels in speaking in front of others and in doing mathematics
I spent about an hour this morning putting together notes for tomorrow's discussion of Vectors in Their Natural Habitats. The materials to be used (a PowerPoint presentation and a Mathematica file) are available on the course website for those who are truly interested.
Wednesday, August 23, 2006
I've obtained permission from Ursula and Imogene (yes, pseudonyms) to post the following shot of them tossing the icosahedron about in a dramatization of our in-class activity yesterday. (In which Imogene actually took part!) Note the icosahedron itself, just leaving Ursula's hand, dark against the white wall behind. (I shoulda been an SI photographer...)
...and a whole lot more.
I think we had a good class today (although everyone involved should feel free to disagree!). We spent the better part of class discussing just what it is that makes teamwork work, and then we plowed on through that syllabus. I hope I did a decent job assuaging people's fears of the course (in particular, of that terrifying "final article" and the scary "symposium presentation").
Of course, we started things off with a rousing game of Toss 'n' Sort. With the foresight for which I am well and widely known, I neglected to take pictures of the action in progress, but I managed to catch a couple of the folks in the class in the hall afterward, and I got them to do a re-enactment (I'll post a picture if they let me know it's okay!). Unfortunately, the icosahedron we were using as a ball suffered horribly:
To be continued...
As I expected, a few students (including some I know will do well!) have expressed a little bit of hesitation regarding the course...the "newness," the perception that they will be engaged in an "experiment."
As I expected, I may to have to do a little work to counter the perception that this is the "wrong" way to teach math, simply because it's not the way math's often taught. I'll have to take care to explain that this class is by no means an "experiment," however new this method of learning is to the students, to me, and to the department: the methodology I'll be using is a very well-tested and well-validated one that has proven to be a supremely effective mode of education.
Sure, there'll be some bumps along the way. I'm certain of that. But I'm just as certain that if we put forth the effort, we'll all ride over those bumps smoothly enough.
Tuesday, August 22, 2006
Now that the dust had cleared...I checked the rolls this morning to find that two folks have dropped from 365 (did I scare them away?), and with the two folks I let in this morning, we're sitting on a healthy 30 students. There are three more who need to get registered for whom I've already either filled out exception forms or indicated that I would do so, so we might end up at 33. Fine by me.
I wanted to indicate one minor meta matter to blog regulars: I've re-enabled commenting for non-Blogger registrants, so you CAN post anonymously if you would like to, but I've enabled word verification for comment posting, which shouldn't be a hindrance unless you have profound visual impairment of some sort.
Thus...feel free to comment away. I'd prefer if you provide your identity in some form when commenting (particularly if you're a student of mine!), but I certainly can't require it.
Today's preparation for 365 involves setting up the graph for tomorrow's Toss 'n' Sort escapade, and getting things ready for tomorrow's discussion of team work.
Monday, August 21, 2006
This day's finally drawing to a close.
I've had a little more opportunity to reflect on how 365 went today. I'm still very satisfied with our start.
The Markov Dance went off without a hitch. Many thanks go out to our wonderful team leaders, Bethesda, Kent, and Lorelei (pseudonyms all), for their solid organizational skills and the professional way in which they quickly systematized the process. We were able to do about 15 iterations of the "Dance" in the first 20 minutes or so of class, and as I mentioned in the previous post, the numbers came out beautifully. Well done, all! (If you're a member of the class who's reading this, please pat yourself on the back.)
After we'd done dancing about, it only took a fairly simple hint to get people thinking about how to provide a mathematical structure for the Markov process we witnessed. We spent the next 20 minutes or so understanding this underlying linear model, introducing along the way the two most fundamental objects in linear algebra: the matrix and the vector. (Doooon't worry: much to come on both of these topics in the ensuing weeks!)
I'm especially proud of the ease with which everyone in the class jumped right into working in teams. Again, well done!
We'll spend most of class on Wednesday discussing further aspects of team work, particularly as it regards the research projects on which everyone will be working: how does one work effectively as a member of a team? What expectations will I have for teamwork in this course? What projects will need to be completed in teams?
Of course, we'll start off Wednesday with a fun class exercise. I'll probably go with Toss 'n' Sort, since that game can simultaneously involve as much as a third of the class in a manageable fashion. (I've already made sure none of my colleagues who precede me in Karpen 033 mind me setting up the game board in advance!)
Good day, good day.
But even good days have their ends...
At least from my point of view, MATH 365 went well.
We started things off with the Markov Dance, and the numbers cooperated, giving us the same distribution after several steps, whether we started with a uniform distribution, or all clustered at one point. The same: I couldn't have planned it any better.
People were more than happy to tackle the analysis of the underlying equations, and with the help of a few stalwart volunteers, we managed to pick apart our first linear algebra application.
Well done to everyone in the class! (And for those of you in the class reading this, please feel free to log on and post your own perspective.)
More to come later, after I've had a chance to digest.
The first section of Calc II went splendidly this morning! We got through everything we needed to get through, we had a few laughs, we got to know each other a bit, and we even did some very successful group work!
If the rest of that day goes this well, tonight I'll put my head to my pillow a happy man.
I've got the faces down for Section 2, and now it's on to Linear...
I've been sitting in my office poring over my notes, my syllabi, my students' pictures (trying to commit the latter to memory), and I find that I can't concentrate: I've done so much to prepare for these classes (MATH 365 in particular) over the past several weeks that I feel as though there's nothing much to be done this morning.
My first section of Calc II begins in just over an hour.
More to come, I'm sure...
Friday, August 18, 2006
I'm supposedly taking the day off today, giving myself a three-day weekend before classes start up on Monday.
I woke up this morning at about 2:00 a.m., unable to sleep. It was hot, the dogs were shoving me out of bed, and I had The Polecats' "Make a circuit with me" stuck in my mead. None of these factors are conducive to sleep.
As I lay awake, I got to thinking about MATH 365 (what else?): what could go wrong?
I've already identified resistance to what could be called tactfully a "highly non-traditional mathematics classroom environment" as the number one challenge that I must overcome in putting this course into action. But there are other pitfalls, too.
I'm a little worried about the team work: every time I've run with team work in the past, there's been at least nominal resistance on the part of some students. This could be because a number of students fail to appreciate the benefits that come from learning to work in teams...heck, I was one of those students: as I admitted in an earlier post, I hated team work as a student, and I often fought against it by endeavoring to do all of the team's work myself. What bugged a few folks in my Calc II classes last semester (the first course I'd taught in a long time that had had a somewhat substantial team work component) was the impermanence of the teams: one of my best students (for whose opinion I had a great deal of respect) questioned the efficacy of "changing out" the teams at various points in the semester, indicating that he would have gotten more out of the experience had the teams persisted throughout the semester.
That student also questioned my commitment to team work, and though at the time I was a little bit offended by this implication, I realize now, after all of the thought and effort I've put into designing team exercises and activities for this coming semester's Linear course, that he had reason to question my commitment: my plan for team work last Spring was, at least compared to this Fall's class, inchoate and incomplete, at best. I wasn't as committed as I could have been, as I should have been, I must admit. I wasn't committed enough to choose the proper team assessment techniques, to provide framework to facilitate team work, or to schedule class time for students to spend in team-(and confidence-)building exercises.
I've done my homework this time. I feel ready now, but will the students?
It's also going to be hard for me to remove myself entirely from the role of "lecturer." As my students (and some of my colleagues) know, I'm less of a "lecturer" than many, if not most, math teachers...but my role in the upcoming course will demand that I remove myself even further from that bully pulpit: I'm anticipating having to consciously (at least at first) distance myself from standing at the front and giving out all of the answers. I'm going to have to grow even more accustomed to long, awkward silences.
This post falls under L. Dee Fink's 11th Step in the planning process: answering the question, "all right, what could go wrong?"
Answer: a lot. But we'll see it through.
Wednesday, August 16, 2006
Four full days (and the spare bit of today) lie between me and the first day of the fall semester (Griselda began her IBL proofs course today...I wonder how that's gone?), and I'm finally feeling almost ready for it.
Today's accomplishments included getting the syllabus finalized and posted (yay! eight glorious pages!), updating the graphics and content on the course websites (both for 365 and for the two sections of 192 that I've paid woefully little attention to in the past few weeks), "formally" choreographing the Markov Dance (the description can be found here), and writing the last two of the research projects...those too are posted (on the course homepage, here), just to save me a little time when the li'l munchkins inevitably come looking for spare copies because they lost their originals on the way from the classroom to the dining hall.
I also just wrote up a proposal to put together (with Griselda) an IBL paper session at the 2007 MAA Sectional Meetings down at Georgia Southern. Fun, fun, fun.
I think I'm pretty much set for Day 1. That's scary.
Every now and then the thought crosses my mind: "What in the $?%&?!@! are you doing?!!!?"
And then it passes.
I'll be fine.
Tomorrow's the first big campus-wise faculty meeting of the year. I figure I'll come a little bit early for that, coordinate the last few pre-class details that need coordinating, and then call it good.
Monday, August 14, 2006
It's coming together, folks, it's coming together.
I've spent the last six hours working away on 365. The fruits of that labor are quite sweet, but not yet entirely ripe. (I'm enjoying this metaphor...can I take one step further?...) I hope my students will be able to eat of them before they get all rotten and wormy.
Oh! No! Bad, bad metaphor!
In any case, I'm too loopy right now to get much more done, so I'm going to head home. In the meantime, though, here's my progress report: Today I've...
- put together and posted a website for the course schedule, including scheduled times for all quizzes and exams,
- put together a homework website,
- laid out a grading system: it's strictly points-based, wherein each student's grade is reckoned out of 500 points (70% will be individual, 30% will be team), and
- written roughly 70% or 80% of the syllabus, which is already about 5 pages long.
I'll soon have the syllabus posted on the course website, and I'll include a link to it from here.
For now, I'm beat. I'm going home.
Saturday, August 12, 2006
A note to those who would like to post their comments on my course design (including colleagues and students and other assorted well-wishers): I am eager to hear your ideas! Please let me know what you're thinking about my course as I plan and progress in its implementation! I want this blog to be a vehicle for discussion, not just unidirectional transmission by yours truly here.
That said, I must apologize for resetting the comment-posting options in order to rule out anonymous posts. This blog, like every other on Blogger, most likely, has been the target of blogbots that go 'round and post automatically generated comments, and I'd rather not have those clutter up the comments pages. To help cut down on this practice in the future, I've made it so that you have to be a registered Blogger user in order to post a comment. To register is free and non-intrusive (you can do so here), so I hope that people won't see it as a tremendous obstacle to letting me know how they feel. Nevertheless, I regret having to place this hurdle in the path of would-be conversation partners.
Please, post on!
That's all for now.
Moving on: one week to go. A new beginning is just 'round the corner.
Knoxville's 2006 MathFest brought me a few almost sleepless nights and a boatload of new ideas for 365. Many of these came courtesy of my best friend and partner in pedagogical crime to whom, in keeping with the rule of anonymity promised before, I shall refer as Griselda. (At least until she reads this and tells me to cut it the hell out.)
Griselda's teaching an inquiry-based learning (IBL) version of her school's proofs class, her confidence and skills bolstered by what sounded like a marvelous workshop out in CA a few weeks back. She's got some great ideas, and I had a chance to look over the first 42 problems she's designed for her students to work through in semi-Moore-method fashion. (Someday when I'll less tired maybe I'll explain to the reading public exactly what the Moore method is, but meanwhile those who are truly intrigued can follow this link to learn Moore about it.)
One change I'm thinking of making to my plans is to adjust the grading of the "two-part" quizzes, in order to make at least some of the credit given for such a quiz "gratuitous," perhaps letting the team portion of the quiz be extra credit to be added to the individual score for each team member.
We'll see. My goal for the coming couple of days is to hammer out the dings in the project descriptions, to get the website up and running, and to write the syllabus.
Oh yeah: Griselda and I plan to put together an IBL session at the MAA Sectional meeting at Georgia Southern next spring. Stay tuned!
Tuesday, August 08, 2006
It hit me while lying in bed this morning, the topic for the final research project: linear codes! What's not to like about linear codes? While not incredibly interesting, even rudimentary codes like repetition codes and basic parity checks involve vector addition and the notion of subspace. And how quickly these give way to Hamming's work, where all of a sudden matrix multiplication comes into play!
Yes, that's what I'll do...
...of course, I'm leaving town for a few days, but I'll hammer out this project when I get home.
Also, come hell or high water, I'm going to start that syllabus as soon as I get back, too.
Until then, faithful readers...
Saturday, August 05, 2006
I've got a few more weeks' worth of classes planned now, taking me most of the way to the midpoint of the semester. Feast your eyes on the following:
Day 10 (Wednesday, September 13th): things get rolling with Quiz 5, on spaces, subspaces, and spans. From there we'll split into teams and work in Round-robin fashion on solving homogeneous equations. [Note to self: I need to come up with a good application exercise here!] HW for Day 11: Read pages 91 through 99, finishing off Section 1.6.
Day 11 (Friday, September 15th): No quiz, but teams should sign up to come by sometime in the following week to send a couple of representatives to my office to provide a progress report on the team's project. The goal for the day is to work through one of those awful "equivalence" theorems which are all the rage in linear algebra: we'll spend most of the day putting together our own proof of Fraleigh-Beauregard's Theorems 1.16 and 1.17 on various equivalent forms of uniqueness of solution. HW for Day 12: Read pages 125 through 134 (Section 2.1), and make some noise in your journals about what's going on in your math world.
Day 12 (Monday, September 18th): Quiz 6, on linear independence, starts us off, and from there we go into some examples of finding spans and verifying linear independence. HW for Day 13: Read pages 136 through 140 (Section 2.2) of the text, and take a gander at the worksheet for the Pipe Problem which will be the focus of class on Day 13!
Day 13 (Wednesday, September 20th): Quiz 7, on matrix ranks, starts things off. But the pace quickens with quickness, and we spend the rest of the day on the Pipe Problem, in which we try our darnedest to get the right levels of salt solutions in a tank in a constant state of flux. Oh, the salinity! More than linear combinations, we're starting to talk about convex combinations! HW for Day 14: Read pages 179 through 186 (Section 3.1), paying careful attention to the first four examples from that section, and read through the worksheet on A Variety of Vector Spaces. You'll wanna know these forwards, backwards, and put together in any linear combination!
Day 14 (Friday, September 22nd): Vector spaces, vector spaces, vector spaces! You want 'em, we got 'em! We'll spend the first part of the class in a think-share-write-SHARE exercise meant to brainstorm the properties we want these "vector spaces" to have. From there we'll launch into an examination of each of the eight examples from the homework (four from the book, four from the worksheet): one per team, we'll have some mini-presentations on what makes each of these guys a vector space. HW for Day 15: Read pages 190 through 201 (Section 3.2), and let your journals know about vector spaces. How do these relate to your research project?
Day 15 (Monday, September 25th): After Quiz 8 (on vector spaces), we'll revisit the examples from Friday in order to examine their dimensions, bases, subspaces, and so forth. Trying to anticipate what might go on on Day 16, what do you notice about all but two of the examples from Day 14? HW for Day 16: Read pages 204 through 211 (Section 3.3) of the text. Pay special attention to how you might apply these ideas to the examples from the last few classes.
Day 16 (Wednesday, September 27th): We'll spend yet another day on the examples at which we've been pounding away since Friday. In particular, note how all but two of those examples look a heckuva lot like real Euclidean space! And what, indeed, about those last two? Can we "fix" them somehow, by chucking the set of real numbers out of the window? Hmm...! HW for Day 17: read over the worksheet Flyin' High (projective geometry and computer graphics).
Day 17 (Friday, September 29th): We start off with Quiz 9, on the worksheet from the previous class. We'll then spend the next half hour or so talking more about this worksheet: what's really going on here? Lines became other lines, or points, even! Things got squashed from three dimensions down to two, just like when 3-D objects are rendered in 2-D by a computer's graphical display. What's at work here are linear transformations, which will be dealt with formally in the HW for Day 18: read pages 142 through 152 (Section 2.3) of your text, taking care to note how linear transformations relate to matrices: is there really any difference?
It's comin' together! Now I'm off to pester my colleagues...
Friday, August 04, 2006
...one to go. As I see it, if I've got teams of three or four students working on each project, I need eight of 'em. I wrote up a seventh this afternoon, outlining the linear algebraic basics of a research program involving random trees on which I was working feverishly this past fall. Those lucky ducks who find themselves working on this project will have the added incentive of the possibility for publication if they come up with anything significant! On the other hand, they'll have the debilitating handicap that they'll be ducks.
Other than that one project, I've not done much for 365 today...most of the morning was spent pounding away on the REU proposal which is (say it with me now) al...most...DONE! (Thanks goes out to my colleagues for getting me the information I needed from them...)
More to come soon, I'm sure. Don't miss a single exciting installment of...
Thursday, August 03, 2006
The astute observer may have noticed that I've yet to provide any pretty pictures of all this whatnot that's been going on planning for 365.
It's time to change all that! It's time for action!
Well, of a sort. (Don't count on prettiness yet.)
I'm sure that once we're underway a number of wonderful photo ops'll present themselves. For now, you're gonna have to be satisfied with a couple of self-aggrandizing shots of the "meta" materials I've assembled to help me track the course design. The first photo below gives you a view of the inchoate course plan (left), alongside an exceptionally-well-organized (for me, anyway) tabulation of the learning goals for the course:
Note the presence of other mental clutter in the background, as well as a shameless plug for MAA's wonderful program, Project NExT. For those who might want a closer look at that spiffy course plan, here you go:
Cute, huh? Nine days down, roughly thirty-six to go...
Tuesday, August 01, 2006
Dog will hunt, y'all!
I've got the first nine days or so of the semester laid out.
This is hard.
Here's the condensed version (transcribed here with the hope that somewhere I will have a legible copy to which I myself can refer back to):
Day 1 (Monday, August 21st): Welcome! With no warning, let's get together and do the Markov Dance! For twenty or thirty minutes, we'll mimic the motion of people from Rich to Middle-Class to Poor, exploring population dynamics through simple linear systems. What does this mean? By the end of the day, we'll be able to see how linear equations, and maybe even vectors and matrices, fit into all of this! HW for Day 2: Read the syllabus, and fill out the questionnaire, both provided at the end of Day 1.
Day 2 (Wednesday, August 23rd): We'll spend our time working in groups, playing a little Toss 'n' Sort, and talking about what sort of teamwork I'll be expecting throughout the semester. We'll talk about expectations for research journals and research meeting notes, and how I would like folks to make use of the course blog and message board. HW for Day 3: Read Sections 1.1 and 1.2, and begin looking over the project descriptions, provided on-line.
Day 3 (Friday, August 25th): By this class, I'll have decided upon teams for the research projects. The first of many fun Two-part Quizzes will be given on this day, to give everyone a start in working with the teams they'll be in for the next few months. After that, we'll do some work with vectors. After briefly brainstorming properties of vectors, we'll look at some specific incarnations: RGB vectors, chemical equations, and realizations of abstract vectors in two- and three-dimensional Euclidean space. HW for Day 4: Read Section 1.3 of the text, re-read the project descriptions, this time as a group. Communicate with your group regarding these projects!
Day 4 (Monday, August 28th): After beginning Quiz 2 on vectors, we revisit the Markov Dance, in order to gain a deeper understanding of the steps we took. Specifically, we'll focus now on the matrices that arise in the Dance. The first of the "What If...?" exercises comes in at this point: what if we allow our matrices to vary over time? After using this exercise to examine the effects of matrix multiplication, we do some brainstorming on the various properties of matrices and their algebra. (The format for this brainstorming?: think-share-write-SHARE.) HW for Day 5: write in your journals about linear algebra so far, and get together with your teams to write rough drafts of your project proposals. You should have a rough draft in hand before class on Wednesday.
Day 5 (Wednesday, August 30th): Meta work! We spend the day investigating good and bad mathematical writing. After a little think-share-write-SHARE on what makes a good paper good, I'll pass out the rubric which will be used for the course's writing assignments. The fun then gets underway in earnest with the much-ballyhooed "Write a Bad Math Paper" exercise on something simple, like the definition of a vector. Each person will work with one other to write a horrifically bad paper, which will then be exchanged with that produced by another pair, who will then have the task of providing useful feedback. These are all skills that the teams'll wanna keep in mind when they do the...HW for Day 6: polish up those proposals for the various research projects. Also, read pages 51 through 56 and come ready for action involving systems of linear equations.
Day 6 (Friday, September 1st): Now that we know what systems of linear equations are, we'll revisit a couple that we've seen already, namely the Markov Dance and the Chemical Equations Balancing Act. How do we set up these linear systems? How are they translated into matrices? How do we solve them? We'll run a few examples, and we'll cap the week off by handing in those proposals on which you've worked oh so hard! HW for Day 7: Write in your journals regarding your proposals: do you feel good about them? Which project do you think you'll land? Also, finish reading Section 1.4, pages 57 through 73.
Day 7 (Wednesday, September 6th): Two-part Quiz 3 starts things off, covering the solutions to systems of linear equations. From there, we'll consider a new case study: the structure of a simple economy, with various inputs and outputs (resource: Lay, pp. 57-59). How can we understand this economy through what we've learned so far? HW for Day 8: Read pages 73 through 77, from Section 1.5.
Day 8 (Friday, September 8th): Two-part Quiz 4 gets us movin', where we're talkin' 'bout inverting elementary matrices. Once that's out of the way, we'll do some more complicated inversions with a step-by-step team exercise (Round Robin? Flock Around?), and then we'll go back to Applicationland by revisiting the simple economy constructed on Wednesday. HW for Day 9: Read pages 78 through 87, write in your journals about your team's work so far.
Day 9 (Monday, September 11th): Field trip! We'll meet with one of the university's reference librarians to learn more about the various research tools available for use on the projects. We'll also kick off the Reference Scavenger Hunt by breaking off into teams and tracking down a reference or two using MathSciNet and the Ramsey Library catalogue. HW for Day 10: Read pages 88 through 99 (Section 1.6), and complete the Reference Scavenger Hunt, handed out at the end of the day and posted on-line.
Dagnabbit, I need a break...more to come soon!
I took the time yesterday to look up some resources on problem-based learning. While much of the literature still concerns itself with medicine (the field in which PBL first arose and became widespread), there were a few sources that gave information on PBL in nonmedical settings, and there were a number of medical sources which provided useful hints on making PBL work in a more generic setting. (The Problem-Based Learning Initiative at Southern Illinois University has proven helpful for some of their suggestions.)
First and foremost among their hints (surprise, surprise) is that in the PBL setting, students must be held responsible for their own learning. This ratchets up the stakes: there are definitely greater risks involved when you're asked to take charge of your own learning. The stakes are high for me, too, obviously: I've got to make sure that the students have access to the skills they need to face the challenges I'll be presenting to them as the semester progresses, if we can ever hope to achieve that elusive "Flow" state.
With stakes this high, things are bound to be interesting this coming semester. I'm going to start off the syllabus for the course with a line or two from David Mamet regarding the vibrancy of theatrical drama produced by actors who've raised their stakes as high as they can go.
I finished the day yesterday by swinging by the room where I'll be teaching 365 next semester. It's pretty spacious, and the desks'll be easy to move. There's not a good deal of blackboard space, but I'll manage, as I hope to be less dependent on boardspace than I've been in previous courses. There's plenty of light, too, and that should be nice.
Having finished up with the design of learning activities appropriate to the stated course goals, I'm going to spend some time this afternoon putting together the "overall scheme of learning activities" based upon the seven primary foundational concepts I want the students to master (vectors, matrices, vector spaces, linear independence, linear transformations, determinants, and eigenstructures). Castle-top diagrams? Nah...that's just not my style. Maybe more of a flowchart. Who knows? I figure I'll end up with some more "wall art" of some form, though.