Wednesday, September 26, 2007

Random thoughts, volume 2

I had a few interesting conversations today, with colleagues and with students. I also found myself unfairly piqued during my second section of Calc I, and I feel an apology is in order to my students.

Let's start there: Quiz 4 came today, asking for a brief rundown on the two fundamental interpretations of the derivative. As I would soon forcefully point out to my students (post-quiz), it's not as important to me that they memorize the formula for the derivative, nor that they master every one of the rules for differentiation we will soon study, as it is that they understand what derivatives mean, and how it is that we can see them in nature, and put them together to help us understand natural phenomena. As I put it to them, Mathematica can do all the derivatives for us, and much more quickly than we ever could. What Mathematica can't do is study a natural process, recognize that there is an interrelationship between two or more dynamic quantities, chart those interactions over a long enough course of time to posit a model that describes the way they depend on one another, and use that model to put together the derivative that gets fed into Mathematica at the end of the line. Mathematica, in this sense, represents the mathematics of the past, when it wasn't yet the case that there was a handy formula that one could apply to find the derivative of a given function. That was then. This is now, and the future is yet to come. The mathematician of the future needs to know more than a mechanical rules for finding derivatives (as important as they are to be able to apply well); she needs to know how to use derivatives, how to interpret them.

Of course, when I said all this to them, it came out ne'er so fluently as it did just now, above. Figures, huh?

Performance on the quiz was...meh. It wasn't horrific (I've given harder quizzes), but it was by no means stellar. A couple of my best students cornered me after the second section and asked if they were going to be okay from this point on. Tallulah: "because I didn't do well on that quiz." "I doubt anyone did," I told her. "The folks in the first section pretty much biffed it. It was a hard quiz, largely because you're not used to being asked questions about concepts rather than computations." She's a fantastic student, she'll recover splendidly.

I realized even as it was happening that I was (unfairly) letting my frustration with my students' conceptual misunderstandings get the best of me for a few minutes during that second section's class. I threw markers and punched the board like I always do, but I did so with more vigor than is typical, partly to dissipate my frustration. "How dare they not get this? Damn it, what, they think they can coast in this class if they spit up a formula or two?!"

My righteous indignation subsided as the class went on, and I realized by the end that if they'd not focused on the concepts over the calculations, it was as much my fault, and my colleagues' faults, as it was the students', for not asking them to refocus their attention elsewhere in the first place. I've got to try harder at that myself. For some reason, I have to admit, Calc I has proven the most resistant of all courses to redesign along the lines of discovery and application-based learning. Only now am I beginning to understand what a truly problem-based Calc I class might look like, and I admit that this semester I'm falling far short of that mark.

I promise a less angry, less frustrated tone tomorrow, folks: you really are a great bunch of students, and I enjoy working with you very much. Let's make tomorrow's class a good one, huh? I'll bring some donuts tomorrow morning, and we'll start off with a couple of conceptual exercises to get our creative juices flowing. Sound like a plan?


From the second section of Calc I, it was off across the quad, to the second of the semester's Writing Intensive meetings (the first was this past Monday) for me. As I anticipated, I'm enjoying working on this committee, conferencing with a group of peers who feel as strongly and as passionately as I do about writing-to-learn and writing-across-the-curriculum and writing in general. I'm starting to get a good sense for the way writing is integrated academically, campus-wide, rather than simply in my own courses and in those of my math colleagues. The bar is quite high; the quality of writing instruction university-wide is solid. Nevertheless, there is room for improvement, and I found myself in a heated exchange of hallelujahs with Lexington, the WI committee's acting chair, as we walked back to our shared building after the committee meeting.

We agreed that the university has made tremendous strides forward in terms of embracing writing-across-the-curriculum, undergraduate research initiatives, outcome-based curricula, discovery learning opportunities, and so forth...but that there's also a lot of work to be done before perfection is reached. "If we're going to advertise that we're using discovery learning," Lexington said, "we've got to start doing just that, and to do that we're going to have to get serious about giving people the resources they need to do that." We agreed that we need to try to drive class sizes down (I mentioned my conversation with my own Chair last week regarding getting my Calc I classes capped at a lower level), we need to offer kids the opportunity to engage in alternative classrooms early and often, we need to make a focussed, directed, campus-wide effort to provide these opportunities to students from the get-go.

After a brief stop at my office and a moment in the Math Lab to unstick the stuckness of a few of the Calc I kids in computing the derivative required of them in the team project, I was off across the quad again to 280. Davina caught me before class with a few concerns about her service on one of this week's homework committees. For one, she wasn't sure about what to write on a person's submission if he said something like, "I'm stupid, I can't figure this out," or something along those lines. "That's a hard one," I agreed. More substantially, she wasn't sure she was giving the right kind of feedback, and she felt like she was being hypercritical, telling people to reword this, change that, and so on. I suggested that she might try to balance positive and negative feedback, and to offer comments like, "I'm having trouble understanding this, could you make this more clear?" or "This is a really good insight, it really helped me to see this point more clearly!" I later reiterated some of these ideas to the whole class, and wrote on the board: "Recall that the purpose of the committee work is not to homogenize, but rather to help people to clarify their own individual ideas."

The subsequent committee reports were good ones. I was particularly impressed with Davina's discussion during the presentation she and DeWayne gave on the homework problem they'd been assigned. I admired the way she was willing and able to come out and say that her serving on the committee definitely helped her to better understand the concepts involved in the problem they'd reviewed: "seeing how other people did it made me see how I could make my own writing more clear and more concise." I'm glad she came out and said that, and I hope her sentiment is shared by the others. I'm certainly going to ask the students about their committee experiences explicitly when I pass out midsemester evals in a week or so.

Came then (after another half hour of set theory) the trek back to Robinson Hall, where I'd spend a few more hours before heading home. I finished grading the second section's Quiz 4s, on which they did marginally better than the first but still not wonderfully. I also got a chance to work with a number of the teams as they struggled through their projects (they're all doing quite well, from what I can tell), and I met up, one-on-one, with several of the 280 students, helping them to polish various drafts of homework problems. They're definitely developing an appreciation for more and more subtle nuances, meanings of stereotypical mathematical phrases ("thus...," "for every...," and so forth), and clarity, clarity, clarity in writing. (A funny, and very heartening note, if I may: at the close of today's committee reports, I reminded the students to keep an eye on the rubric I'd handed out last week as they worked on their math writing, and I asked them if they could remember the "four Cs." In nearly complete unison, they intoned: "correctness, completeness, clarity, and composition." I didn't have to say a goldarned thing. I was a happy man.)

While I was finishing off those Quiz 4s, Cuthbert, Chemistry colleague of Lexington and a big, big man in undergraduate research, came by to ask me if I wouldn't mind providing him with the titles of the projects the REU kiddies worked on this past summer: he was putting together material to take to a regional conference on UG research, indicators of the sorts of things that can be made to happen in a liberal arts science curriculum. I was happy to oblige, and not an hour later I'd send him a list of the topics. We then got to talking about the inclusion of research components in UG courses, and he asked if I'd done much of that. "Depending on how you defined research," I told him, "I do that in nearly every class I teach." I told him a bit about the course that spawned this blog (MATH 365, Linear Algebra I, taught last Fall semester), and he asked if he could have information about the projects those students worked on. I obliged him there, too, sending him the prompts for a few of the research projects (Monopoly, Traffic Patterns, and Come on, Feel the Noise).

We hit a few more points in our short conversation, enough to give me a sense of déjà vu, feeling as though I was reliving the conversation I'd had earlier with Lexington. We talked about the increasingly interdisciplinary nature of scientific study; the need for broader, deeper, more meaningful implementation of discovery learning in our courses; the need for more robust interdisciplinary course offerings than an occasional and disingenuous cross-listing: we need team-taught classes, classes that provide a true interface between one field and another, like one Cuthbert mentioned that took place at his previous institution, in which beginning chemistry students learned their ways around a lab while generating real, unprettified data that could be fed to statistics students who would put it through realistic rigorous analysis, the results of which analysis could be funneled back to the students who ran the lab experiments in the first place in order to help refine their techniques. We agreed that a course that combined the concepts of physical chemistry with those of linear algebra would be a tremendous boon to both the Chem and Math departments; I have no doubt that we'll be talking about this again soon.

Thence, from my office, to dinner at Sorrento's (offering the best Italian in town in a cozy out-of-the-way den halfway to Oteen that sadly never seems very busy), and then home. Once home, I finally had a chance to call Bedelia, my colleague late of Harvard, now of Lesley University, and catch up with her. As it was bound to be, our conversation turned to...teaching! Imagine that! We talked for a bit about a program she's like to get off the ground, a sort of summer prep course to get local (in her case, Bostonian) college-bound kids up to speed on the math skills they might need to succeed in a challenging entry-level math course, while providing them with some useful advice on making the transition to college more smoothly. I compared the program she was describing with UNCA's own SOAR program, and she seemed to agree with the comparison, only she emphasized the local nature of the plan she envisioned. Then we talked shop for a bit, I about my own six courses (if one counts the senior seminar and my three independent study students), she about her Quantitative Reasoning and Pre-Calc classes.

We're both having a good time.

Hey, I've rambled on long enough, it's time to stop. I'm going to make a smoothie, maybe watch an episode of Mr Bean or some other such nonsense, and call it a day. Thanks for reading, I hope you'll free to comment, I always appreciate your feedback!


Anonymous said...


I must say, your dedication to your students is amazing. Seeing how much effort you put into teaching, makes me want to be a better student.(and that's saying alot!)

So, I just wanted to thank you for that, and let you know I enjoy the blog. It makes for good reading right before bed.

-a calc 1 student

Anonymous said...

Patrick, even though I was once a teacher, I have to admire your dedication. I am not mathematical, but I feel I'd be willing to learn if I had you for a teacher. You inspire me. Beth

Chris said...

As I said before, your dedication shines though your teaching. Best math class/teacher I have ever had.