I had a good time with my Calc I kiddies today, mostly because they did the lion's share of the work. (We spent the day working through several archetypal examples of tangent slope computations, observing patterns along the way.)
So far (six class periods in) several students have expressed mild apprehension regarding the structure of the course. It's noteworthy, though, that they're all enjoying the class and are clearly willing to ride it out and let themselves grow more comfortable as the semester goes on.
Most unsettling to them, I think, is the decentering of formulas that's gone on. There's no call to memorize formulas...in fact, there have been almost no formulas needing memorization. Central, instead, is concept: rather than hand them ready-made formulas for computing slopes of tangent lines to the graphs of various families of functions, I've asked them to derive those formulas from scratch from a single formula, emphasizing constantly why that formula works and where it comes from.
Most of the students are used to being told to memorize; they're used to "theorem proof theorem proof formula formula formula definition example example theorem proof" math courses.
Most of the students are not used to being told to understand; they're not used to "why is that what are we looking for how can we find it how does that formula work how can we use it to our gain" math courses.
I give them tremendous credit: they're patient, they're hard-working, and they're doing great.
They're also fun to talk to, in class and out of class. We jawed a bit today about the following pretty well-documented pedagogical fact: though it's painful (for both the professor and the pupils) to sit in a silent classroom in which everyone's staring at everyone else, when a greater lag time is allowed for between a teacher's question and a student's response, the response that comes is generally richer, more complete, and more meaningful than the one that will come from continual once-every-few-seconds prodding on the part of the professor.
As a part of that same conversation, I mused out loud about why it is that students seem more eager to write a solution on the board when I'm out of the room than when I'm sitting right there in front of them. "Is it because you feel safer making mistakes in front of each other than you do making them in front of me?" I asked.
"No," someone said. "I think it's just because we feel bad if you come back in and we haven't written anything."
There could be something to that.
In any case, I'm looking forward to following up tomorrow (limits!), and to continuing our discussion of matrix algebra in Linear.
For now, I've got to finish a few more paragraphs of my book (tonight: the unique challenges faced by writers in quantitative disciplines) before calling it a night.
Tuesday, August 31, 2010
I had a good time with my Calc I kiddies today, mostly because they did the lion's share of the work. (We spent the day working through several archetypal examples of tangent slope computations, observing patterns along the way.)
This one's for my current calculus students (others may answer in the comments hypothetically):
Given: we've now computed the slope of a tangent line to a variety of low-degree polynomials, at an arbitrary point. We've also seen a couple of applications of such slopes (minimization and velocity). One thing we've not yet done (intentionally) is made our method of computation precise.
Would you rather spend the next few days...
1. ...computing more slope formulas (for arbitrary powers and sums/differences/constant multiples of those powers...and maybe even some trig and exponential functions),
2. making our method of computing those formulas more precise, or
3. look into some more applications of slope formulas?
Keep in mind that we will do all of the above at some point soon...I just want to know where you want to go first.
What'll it be? The polls are open!
Sunday, August 29, 2010
(Caveat: I've got writing process on the brain, as I just started work on Chapter 3 this afternoon, in which I hope to discuss writing as process, and how to effectively stage that process, in the quantitative disciplines.)
For future reference: when asked to write a dialogue in which you're to help a hapless friend who's hopelessly lost regarding one or another mathematical concept, it helps to involve two people in the conversation.
The second interlocutor plays a crucial role: with this person in play you, playing the role of The Expert, need not anticipate every difficulty that might arise to trouble your poor partner in conversation. This partner is the one who need only act naturally in order to bring to light the most subtle and sophisticated aspects of the problem you've been posed. That is, this partner is your foil, the dunce whom you can task with asking every question that might conceivably come up when learning a tricky technique.
With two people talking it through, what might otherwise be a boring monologue or lecture (in which posing trivial questions and then immediately dispatching them would seem stilted at worst or pedantic at best) is now a dynamic dialogue, where nearly real-world interaction is possible.
Talk with your dialogue partner as though you would talk with a real-life friend; help him as you would help a real-life friend, let his ignorance and your expertise take turns driving the conversation forward as you uncover the truth together.
Could there be a better low-stakes writing-to-learn activity?
Friday, August 27, 2010
In Linear Algebra today I used rampant tidal locking as an explanation for the fact that this past week has easily been the longest one ever.
It's been a long week, but a good one, and I've gotten a lot done. More importantly, I feel I've set the appropriate mood in all of my classes. The Calc I kids are slowly starting to come to life, and Linear's just getting better and better.
I've now had "get to know ya" meet 'n' greets with about 20 or 25 of my new students, and these have been uniformly fun. As always my students very in every conceivable way: age, prior education, academic background, major, motivations for study. I've got young, old, quiet, bold, math-minded, math-shy, self-directed, and diffuse. A couple share my love of poetry, one my fascination with early human evolution. Some are sarcastic, like Dwight, the student in my second Calc I section who showed both bravado and cojones by merely modifying the work I'd done on the board to solve an earlier problem, instead of transcribing his own solution from scratch. Others are retiring, like Bertie, whose description of greatest common factors was practically unintelligible owing to the quiet of his voice.
So far, aside from a little bit of hesitation on the part of a couple of the more outspoken students, all of the feedback I've gotten on the structure of the courses (both highly student-centered, Calc I to an extent I've not taught it in the past) has been positive. A couple of the Calc I students have even commented appreciatively on the use of writing. One thanked me for asking them to write in order to more deeply understand the difference between two oft-confused rules for exponents (when do you add 'em, when do you multiply 'em?): "I didn't do so well the last time around in calculus, but I feel like I'm going to do a lot better this time."
One of my nontraditional students (I'm always happy to have them in my classes: they're generally so much less shy about speaking up) expressed some concern in my meeting with her on Tuesday morning. "Am I going to be prepared for Calc II by the end of this semester?" I assured her that we'll do all of the computations we'd do in a theorem-driven calculus course, and we'll learn all of the concepts we'd learn in that setting...we'll just approach them all from a much more useful angle.
"Don't worry," I said. "We'll get there."
In fact, freed from having to put together a punctilious list of soon-forgotten limit laws, we're making good time: just a week in and we've already computed the slopes of several tangent lines, having figured out that that's what we need to do in order to solve the problem posed to us on Wednesday morning.
Yup, classes are going well.
In other news, I've finished the first draft of an introduction to my book. I'm quite satisfied with it. It charts my personal history with writing in the discipline of mathematics, highlighting the ways in which I've grown as a writing instructor as I've progressed in my career and indicating the empty spaces in the writing literature which need to be filled. Tomorrow I hope to start work on either the first or the fifth chapter (these are the two whose structure I can most clearly envision right now).
Now, bed calls. Students: please let me know what you thought of this first week. Where do we go from here?
Oh, and: 400th post!
Wednesday, August 25, 2010
How's it going?
Three days into Calc I, and it 's difficult to say which way the class is going to break this term: the students are still strongly engaged in class and contributing well to the problem-centered classes, but I think a few are hesitant about the rather non-traditional format of the class. As yet we've not done a great deal of computation, haven't derived or delved into any formulas, haven't stated any theorems. A number of the students are used to doing all of these things within the first couple of days of a math class and so they've got a sort of deer-in-headlights, "WTF?" sort of look in their eyes.
The formulas will come soon, I'm certain, and I hope that with them a sense of familiarity and comfort: we're about to enter into a discussion of tangent lines and secant lines and their slopes, a discussion will include a great deal of computation and review (and practice) with algebra.
I'm not sure quite how quickly we'll get there: it really all depends on the pace at which the students end up driving the course. As I mentioned to my first section this morning, they're sitting in the driver's seat, working the pedals, holding the wheel. Every now and then, as instructor, I'll reach over and grab the wheel with one hand to steady it or add a few degrees to a particularly too-tight turn.
So far the structure of the course is "dialectical," an exercise in turn-taking: I've proposed an initial course heading, and now we've walked for a while. Now, at the end of the first day of hardcore hiking, having surveyed the site at which we've made camp, I've worked out an itinerary for our next day's travel together. We'll see where that takes us. We'll make camp again, make some next-day plans, and set out again in the morning. I know the stops we need to make, and we'll make them all, but it's up to the students to figure out which we hit when, and in what order.
We'll figure it out.
Meanwhile, I am loving Linear. The class is huge (the largest I've yet taught at UNCA), but has great cohesion. The students are eager to work and are making great progress, and I feel that I'm doing a much better job of structuring the discovery process than I did the last time I taught the class. I've put a lot more thought into precisely which skills are needed to solve which problems, and we're making a (long) beeline toward our first major goal: analyzing the long-term behavior of a simple Markov process (like the game we played on the first day).
Today we figured out how row operations mirror the operations needed to solve a system of linear equations. Next, after posing and solving a couple of realistic problems requiring the solution of a linear system, we'll motivate and move toward matrix multiplication. From there, it's on to matrix inverses, invertibility, determinants, eigenvalues, and, at last, an analysis of Markov processes like those we began with. At every stop we'll solve another real-world problem or two, get some practice with computation, and write a bit about what each concept really means.
I figure it should take us something like 5-7 weeks to get where we're going. After that, we'll talk about linear transformations, vector spaces, changes of basis, etc.
Meanwhile, I'm reading some fascinating works on writing and rhetoric in preparation for various parts of my book. Most recently I've begun Deirdre McCloskey's The rhetoric of economics (2nd edition, 1998, Madison: The University of Wisconsin Press), a delightful rhetorical analysis of economic discourse in which she dissects the ways in which economists convince one another and others of the truth of their assertions. According to McCloskey, much of what many economists consider unassailable scientific truth is really rhetorical "smoke and mirrors."
Not that there's anything wrong with "rhetoric," a word which McCloskey takes pains to save from those who would use it only in a pejorative sense: rhetoric is ubiquitous and unavoidable, and McCloskey merely asks that we be intentional with the rhetoric we use. We all use metaphors and other rhetorical devices; it's simply important that we be aware of the devices we employ. As one might put it, extending the tried-and-true "lens" metaphor, it's no crime to have poor eyesight, as long as you can admit that you need to put your glasses on in order to see properly.
Her opening chapter, which begins with a close reading of passages from canonical articles in economics, has made me think about the way in which I begin some of my own mathematical papers: what sense of authority am I conveying though my choice of words? What world am I building? What agent is acting? What am I asserting about the truth? I'd like to look into my own writing to do a careful reflective analysis.
But with all that I've got on my plate right now, such an analysis has got to wait for a little while. I hope to finish off the first draft of the introduction to More Than Numbers tomorrow, and set to work on the slightly slanted history of WAC, WID, and WTL which will make up much of Chapter 1.
Much to do, much to do, and so little time...but so much fun!
Monday, August 23, 2010
Well, Linear went wonderfully, too.
I am tremendously happy about the way this first day went, and in all three classes I think I owe the smoothness and success to several different sources:
1. The students (duh). They were almost without exception engaged, focused, and excited to be where they were. They clearly came ready to learn, and I hope I didn't disappoint.
2. I came ready to teach, not to recite a bunch of bureaucratic nonsense from the syllabus. I truly feel that the act of delaying the handout of the syllabus by 40 minutes (at the end of the first period instead of at the beginning) does more than any other single act to place the class's focus on the proper point (learning) and not on the improper ones (grades, lateness and attendance policies, etc.).
3. Student-centered, student-driven first day activities. Both courses began with activities whose pace and ultimate outcome were determined by the students. I provided an overarching structure, but the students led in the act of discovery. Of course, this is what happens all semester long in my classes, but it's important to do this on Day One. Do NOT put it off. The more you're able to get the students working together, helping each other, and sharing ideas on the board on the very first day of class, the more comfortable they'll be doing these things all semester long.
4. Knowing their names. It cannot be overstated: no single act more quickly earns the students' respect and admiration (by showing them respect and admiration) than learning their names as soon as you can, as many as possible on the first day.
It's going to be a great semester. Dare I say...best ever? We shall see.
Students: thank you for making today a wonderful one. (And a welcome to all of you who are joining one of my courses for the first time! Post a comment, if you'd like!)
Two down, one to go. My second section of Calc I was more wide awake, but a bit more unfocused. They did a great job in the "memory dump" of review topics from algebra, trig, and precalc, however.
It's going to be a good semester, that's for sure.
Now I've got to prepare for my 35-student (!) Linear Algebra I section.
One down, two to go. Every person on my roll for my first section of Calc I was in attendance, and no one not on my roll was there. This has got to be a first.
They were lively, fun, and relaxed. They took the focused freewrite with which I started things off in stride. They got into groups with no hesitation, and they showed no trepidation about coming up to the board. They'll be great.
It's going to be a good semester!
Saturday, August 21, 2010
It's been a while, huh?
As you might suspect, I've had a lot going on, keeping me from posting here for the past, oh...two months? Ouch.
What's going on?
The REU's over, but the work has really just begun: the students this year were fantastic. They were smart, fun, and funny, and they worked tremendously hard, producing a ton of interesting mathematics. (We should get at least five papers out of it in the next couple of months, and maybe more beyond that in the follow-up.) And this year it was "buy eight weeks, get one free": six of the eight students ended up traveling to MathFest in Pittsburgh, PA to present on their summer research.
The REU took up much of my time, but I've had a number of other things going on, as well, among them: (1) working on student learning outcomes for both the Mathematics Department and the Writing Intensive program, (2) putting together the ILS Oversight Committee's report to the Academic Policy Committee (still not submitted, but finally almost finished), (3) planning my contribution to the short course I was helping run at MathFest, (4) working with my College of Charleston peeps on our paper on the rhetoric of mathematical writing, (5) looking ahead to my round table discussion of program assessment for this year's CWPA (coming up in a few weeks), (6) laying the groundwork for the book I'm now slated to write for Jossey-Bass (w00t), and (7) trying to get ready for my Fall classes, which start on Monday.
(1) Student learning outcomes: although I see the purpose, ultimately, of programmatic assessment, too highly institutionalized assessment tends to become bureaucratic and corporate. I don't want to say more about my role in this university-wide process this past summer other than that it was frustrating at times, had its minor joys, and though it was ultimately fulfilling in that I feel like some good will come of it, I worry about the intentions some people in administration may have for codifying the university's student learning outcomes as rigidly as is being done. 'Nuff said.
(2) The ILSOC report to the APC: this is more of the same. I see the purpose, and the writing of this document is a worthwhile activity, but one which runs the risk of being overly parliamentary and pro se. 'Nuff said.
(3) MathFest short course on Sage (an open-source computer algebra system): I actually wrote to my colleague Oscar (one of the co-organizers of the short course) in mid-July, a few weeks before the course was to take place, expressing my insecurities about putting together an hour or two of meaningful material.
"Nonsense," was the substance of his reply; "you'll do fine. Write about what you know." So I threw together several worksheets full of Sage code purporting to implement a number of higher-level graph theoretical algorithms and called it good. The worksheets actually went over really well in the short course and filled a rather comfy niche in the program. I felt good about what I'd done.
The moral of the story: just do it.
(MathFest, by the way, was a blast. Pittsburgh's a nice city with lots of pretty runs and nice architecture, the REU students were fun to hang out with, as always, and the MathFest program was replete with fun stuff to see and do. A good time had by all.)
(4) Math rhetoric: as I mentioned in my last post, oh so long ago, Bella and Damian came up from the coast to spend a couple of days with me, hammering out a plan to finish the paper we'd begun (with Nicola, who couldn't make it) on the rhetoric of the writing my REU students had produced in 2008 and 2009. They had a lovely time while here, visiting with and interviewing the REU students from this past year, working on our draft at the time, and just hanging out. I'm happy to say that as of last night we have a draft that's nearly ready to submit to the comp/rhet journal Across the Disciplines. It's the first of what we hope will be...well...at least more than one article on the rhetoric of mathematics and its learning, and the role writing plays in inducting students into the scholarly mathematical community.
(5) CWPA: I'm already looking forward to the Wildacres retreat! It's going to a be a full house this year, several dozen of us crowding into the nooks and niches of the Wildacres North Lodge to join in round table discussions of assessment with various forms and functions. I plan on presenting a brief "natural history" of the assessment that's gone on in our WI program during the past three years, exposing to public view the layers of rock comprising the "pilot" assessment Lulabelle headed up in AY 2007-8, the refinement she and I performed in the following year, the plans for assessment she and I laid out in the lead-up to the university's reaffirmation of accreditation (notice I haven't mentioned SACS yet in this post...), and the modifications those plans underwent as I limned the WI program's assessment of student learning outcomes this past summer.
It's sure to be riveting.
Actually, I'm more looking forward to hanging out with Damian and Bella, and with them planning the next phase of our study of the REU students' writing.
(6) Book: Oh yeah...I'm now under contract to write a text (working title: More Than Numbers) on the role of writing (specifically, writing to learn and writing in the disciplines) in mathematics and the mathematical sciences. Jossey-Bass received well the proposal I pitched to them several months back (in March, I think?), and after some fine-tuning of the project, I was brought on board. I'm excited, but at the same time terrified. I'm confident it'll all work out well, though. I've already done a ton of enlightening reading (if you get a chance, check out Scott L. Montgomery's The scientific voice...it's a fascinating read that, along with several other sources, has helped me put together a theory of the schizophrenic nature of undergraduate mathematical writing), and I have some great ideas. Furthermore, my consulting editor for the project is someone a few of whose books I've read and whom I respect very much; I'm sure she'll be very helpful to me.
I will say this, however: if between now and next March I start randomly babbling to you about reader response theory or the role of writing in the mathematical finance classroom, just smile and nod your head.
(7) Fall 2010: yup, we get underway in two days. Yikes. I'm actually really excited: I've got the first week mapped out in all three of my courses, and I've plotted out what I think are clear and navigable problem-centered paths for both of the lower-level courses (Calc I and Linear Algebra I) to follow.
Calc I is going to begin with a brief review, and then the students will work toward developing the skills needed to solve an optimization problem from economics. They'll need to "invent" the method of secants and tangents, the definition of a limit, and the definition of the derivative; and they'll need to learn how to compute some simple derivatives in order to solve this problem. Once all of that groundwork is laid, we'll spend some time working with the textbook and picking up some useful formulas for limits and derivatives, but everything we do is going to be application-driven and learner-centered. I hope to not have to lecture for more than 10 minutes at a time for the whole damned semester.
Linear's going to be the same: Day One features a variation on the "Markov Dance" that started the semester off the last time I taught this course, in the semi-mythical beginnings of this blog. The limiting behavior of this system is going to be the motivating problem for everything we do in the first half of the course, as we work our way through everything we'll need to do to get to the fun and useful stuff (eigenvalues, diagonalization, and Markov processes): solving linear systems, inversion of matrices, solvability criteria, and finally eigenvalues. I'm going to downplay theory and upplay (what a cool word!) functionality. I hope to get to the fun stuff by the sixth or seventh week, at which time we can spend almost all of our time on applications. It's going to be fun.
I should mention that I've modified my grading schemata somewhat, but not substantially, from last semester: students in both of the above courses will still have an opportunity to perform potentially unlimited revision on most things they hand in, but I'm tempering the grading scale so that it's no longer possible to earn full points back with each revision. I think the system I've set up is a generous and reasonable one, though, and I believe it'll still serve to motivate students to learn rather than to grub for a good grade.
Meanwhile, the Senior Seminar's speaker schedule is chocked full, with four visitors (all versed in speaking to undergraduate audiences) coming to share their knowledge and spend a little time with the students. It's going to be a good semester in that course.
Okay, that's all for now...I'm going to try to post more regularly during the coming semester, although I can't promise it'll be more than anxiety-tinged stream-of-consciousness rambling about this or that idea for the book.
Fare thee well!