Saturday, September 27, 2008

Chapter 1. From either/or to both/and; from trespass to transcendence

"You're really brave to come here," was a comment I got from a few different people on a few different occasions during this past week's Carolina Writing Program Administrators conference at the Wildacres conference center just off the Blue Ridge Parkway. As a mathematician at a writing conference, I was a bit of a black sheep.

However, the camaraderie and conversations I shared with my Carolina colleagues on Monday, Tuesday, and Wednesday showed that without doubt we all face the same challenges in helping our students meet their academic goals. The further I go in my study of writing and writing pedagogy, the more similar teaching writing and teaching math begin to seem. As frequent readers of this blog can attest, I'm often struck by the parallels between the disciplines themselves.

Last week's conference was a particularly enriching one for me. I've returned more than typically energized by all that I've learned from my new-found colleagues. I'd like to share some of my insights on this conference in this forum, my blog, my personal reflective space.

This is the first of twelve posts I've planned in which I hope to lay out some of the ideas I've brought back with me from Wildacres. I make absolutely no claim about the originality of these ideas, and since I aim merely to trace outlines of some very complex notions I have no doubt that my exposition will often be skeletal and wan. I'll leave it to my readers to fill in details as they feel fit, and as always I welcome input from my colleagues and my students, my fellow-travelers in academia: please feel free to post comments!

The theme of this first post is false dichotomy and overcoming it: how does one achieve transcendence through the act of trespassing from one academic territory into another? How does one craft a "both/and" from an "either/or"? How does one efface the line between research and teaching, between control and collaboration, between "coverage" and meaningful student-centered learning? I'll address each of these apparent dichotomies in turn, and a final false dichotomy (between mathematics and writing) will lead us to the next chapter, in which I'll take a look at the parallels between first-year composition courses and entry-level mathematics courses like UNC Asheville's Nature of Mathematics or Precalculus.

Let me begin with a dichotomy that doubles itself by laying bare (one of my writing colleagues recently claimed) distinctions between academic disciplines whose modi operandi bear witness to profoundly different means of interrogating ideas.

(Incidentally, the sentence I've just written conveys a subtle such difference: just this past week I had a mildly heated discussion with Lulabelle [a sociologist, recall] over whether or not an inanimate object is capable of "witnessing" anything. In mathematics it's common to say that a computation or an instantiation of some variable "witnesses" the truth of some statement or another. Lulabelle adamantly resisted my attempt to put such language into a table she was creating for my handouts at Wildacres.)

"Research" and "teaching" generally constitute two of academe's three pillars of support (the third is "service"). One, two, or all three of these vessels into which is poured copious amounts of faculty time may be construed broadly or narrowly, depending on the whims of a particular institution's leadership. Even when granted leave to grow with utmost manifest destiny these lands, though they frequently border, don't often overlap. This is made most evident when it comes time to account for a faculty member's expenditures of effort. Even at UNC Asheville (where holism undoes parochialism on a regular basis) the faculty record, to be completed annually by every faculty member, asks faculty to categorize their achievements under the headings "Teaching," Scholarship, Creative, and Professional Activity," and "Service."

I spurred a fervent academic discussion on the way back from last week's conference when I pooh-poohed hard-and-fast distinctions between research and teaching, suggesting a dialectic between the two that synergizes my activity in both.

My teaching informs my research: in my students I find not only collaborators and future colleagues but also sources of novel insights and new perspectives. In mathematics (as in other fields) the solution to a puzzling problem may be reaped not from years of toilsome groundwork in a heretofore fallow field but rather from a few days' reworking, in some novel fashion, of a field already furrowed, sown, and freshly watered. With fresh ideas students are frequently able to do the work that's needed.

Moreover, as I've discovered again and again, my research informs my teaching: in countless classes I've been able to enliven the exposition of various topics from theoretical mathematics by bringing a particular course's techniques to bear on the research problems with which I'm grappling at the time. The Fundamental Theorem of Calculus played a crucial role in much of my recent work on random graphs, so I trot it into my Calc classes. My analysis of asymptotic connectivity required mastery of Markov processes, and I put my Linear Algebra students to work on those. Right now I'm merely waiting for my Abstract students to learn about group homomorphisms so I can challenge them to write their own (G,φ)-grams.

To me, research and teaching are inseparable, and I credit my successes in both of these areas to my willingness to let them intermingle.

I told this to a colleague who teaches here in Literature and Languages, and she responded by indicating that the situation is different in her field. There, she claims, it's more difficult to involve students in undergraduate research efforts since it's no easy matter to carve out bits of one's own research program to make fodder for undergraduate inquiry. Whereas many of my projects can be dissected into chewable chunks, the ways in which knowledge is interrogated in her corner of the humanities frequently make overseeing undergraduate research difficult for her colleagues.

I must admit my ignorance regarding undergraduate research in literature, or in composition theory, or in rhetoric. Is it so difficult? Why can it not be, as in mathematics, seamlessly grafted onto typical classroom experiences? What forms does research take in these fields? Are these forms of inquiry inaccessible to most undergraduate students? Does meaningful inquiry require the student to long immerse herself in a body of literature so deep as to preclude its plumbing by an undergraduate with a typical courseload? Are there no ways to effectively circumvent such arduous preparation?

I don't ask these questions to be obtuse or snarky, I ask because I don't know the answers.

So I drone Ben Steinianly into the blogosphere: "Anyone? Anyone?"

I cannot imagine teaching without challenging my students to engage themselves in their own original inquiry. There's no more authentic learning experience, no better means of encouraging ownership of a discipline's core concepts, than to involve students in real research. It's the ultimate form of student-centered learning.

Speaking of which brings us to another dichotomy that was addressed only occasionally at the CWPA last week: "coverage" versus student-centered classroom methods. It's understandable that this topic didn't get so much attention at this conference, as entry-level mathematics courses like Precalc are likely much more content-driven than their correspondents in the writing curriculum. In introductory writing courses, I imagine little is lost in taking time out of class to ask students to write in groups, to perform peer review, to engage in ungraded freewriting exercises: these projects all serve the course's learning goals in direct and obvious fashions.

Meanwhile, traditionalist math teachers kvetch: "I can't use group work in my class, we've got so much stuff to get through." "It'll take the students three hours to uncover this concept that I'd explain more clearly in three minutes." "If they get to Calc I without having seen the Law of Cosines, they'll be in a world of hurt, I can't slow down."

I can sympathize. I've been there. I know how it goes. Even now I catch myself about to mutter or moan a comment like one of the above, as I'm struggling to keep up with where I "should be" in my Precalculus class. On the verge of sacrificing a group project or some other active-learning exercise (one that at the very least gets them taking turns coming up to the board to jot down a number or two, a graph, or even just a point in the plane), I stop myself and say, "whooooooa, now. The hell, Patrick? What are you thinking?"

"What are millennials [students born after 1982] like?" asks Nora, one of my colleagues at CWPA, in a handout she passed around during her presentation. In partial answer Nora offers that they "gravitate toward group activity...are fascinated by new technologies." Millennials' learning styles include "teamwork, experiential activities, structure, and use of technology...collaborative style." What's to stop us from giving them every opportunity to engage in these effective practices?

The false dichotomy rears its hairy head, bellowing an eight-letter four-letter word: "coverage!"

To which I bellow back: "bullshit." Indeed, the astute reader of this blog will note that there are a few words I never use, because it pains me to do so; "cover" is one of them.

Quick quiz: I've got fifteen minutes left of class, and we're working on cleaning up some trigonometric minutiae. Are my students better served by watching me babble about the Law of Cosines or creating and solving their own problem that makes use of more fundamental computations involving sines and cosines themselves?

The former concept they'll use maybe once in each of Calc I and Calc II, and a few times Physics, assuming they go that route. And if they need the law at those times, it'll be presented to them anew and re-explained anyway. (Such "just-in-time" explanations are often the most effective, especially if the topic is one that's unintuitive or difficult to motivate: by deferring the concept's introduction until the time at which it's needed, students aren't asked to remember pointless facts and figures any longer than they have to.) Meanwhile, the latter concepts are foundational and come up continually in later coursework, and through direct engagement with them the students better understand their workings at a practical, as well as theoretical, level.

No sacrifice is made here. While I might remove a few of the ornamental baubles and bijoux that typically adorn the face of an introductory course, all that needs to be "covered" is "covered," and the face is left intact. Students then engaged the rudiments of my discipline more directly, more meaningfully, than they would have otherwise. All we need to do, as practitioners of content-rich academic fields laden with centuries and sometimes millennia of knowledge, is divorce ourselves from the rather arrogant notion that our students need to know every last fact about our fields of study.

Guess what, folks? They don't. They need to master a few core concepts, and they need to become adept problem-solvers and critical thinkers. Everything else? Icing on the cake. Accepting this fact is difficult for some of us. After several years of graduate training in highly specialized and technical fields, it's hard to remember that our students don't need to know everything there is to know about CAT(0) spaces, or about Sterne's use of the subjunctive in Tristram Shandy. We need to let go.

And we need to let go of the reins and let the students take the helm every now and then. They must be allowed to be our collaborators, not our convicts, and although guiding them in collaboration requires that our students be given a bit more control over their studies, it by no means implies a loss of control on our parts. The most meaningful learning experiences I've ever shared with my students have come at moments when everyone in the room agrees (often tacitly) on a common course of action and each person present assumes control over her or his own role in the action taking place. I can think of three such relatively recent moments off the top of my head:

1. The day, roughly two-thirds of the way through the Fall 2007 semester, on which my 280 class and I agreed to go off-script and just blundered our way about through the wilds of set theory for over a half-hour. That was the day on which Quincy suggested something along the lines of a "fishbowl seminar" in which all participants spend each day's hour discussing a topic drawn at random from a fishbowl placed at the room's center. (I'm still not sure that wouldn't work...) I blogged about that class in this post.

2. My second section of Calc I's rendition of the classic Newton v. Leibniz debate last fall. (Blog posts related to this class include this one, in which the students shared their reflections on the experience, and this one, in which I provide my own initial reflections.) It was clear that everyone had prepared carefully, and after a few minutes most people seemed to have forgotten their real identities as they let the experience take hold of them.

3. The second day of "class" during this past summer's REU, on which the students took turns presenting their findings on the roughly 40 terms and topics from graph theory I'd given them the day before. With one exception (Dione was a bit shy at first) they strode boldly to the board, one after another, and by lunchtime we'd ticked off nearly every item on the list.

In each of these experiences, every individual involved took control and directed her own actions to create a meaningful part in a collaborative project. None of these projects required abrogation of authority. None required the teacher to play the part of dictator, nor the students the part of the peasantry. Each was more than a monologue, more than a dialogue: each was a polylogue. The result in each case was what L. Dee Fink would surely call a "significant learning experience."

Control versus collaboration, coverage versus deep conceptual mastery, research versus teaching: each of these tugs-of-war is played out every day in classes across the college curriculum. With several steps forward, one combatant pulls the other across the line dividing one territory from the other...but if we're wiling to admit that the line between the two is arbitrary, then trespass becomes transcendence.

Of my new-found friends in the writing community who may be reading this, I ask: how do these false dichotomies, these plays for power between imaginary opposing forces, manifest themselves in your classrooms?

I'm curious to know the extent to which our disciplines run parallel to one another. I'm eager to learn more about the form your instruction takes, the ways in which you engage your students in your courses and programs. I'm hoping you will take the time to share your thoughts.

In my next CWPA-inspired post I'll continue by discussing parallels I've noticed between the pedagogy of writing and the pedagogy of mathematics, parallels which no doubt witness (there I go again!) more profound similarities between the disciplines themselves.

Until then, have a pleasant evening, I wish you all well.


Phillip said...

hey patrick,
I still think the fish bowl thing could work. here are 2 more constructed ways:
1) this could be done in a class like math 381. IT would accomplish the same task as 381 does now, but the discussions would lead to the questions for that week. Thus getting the students to help shape the way we connect the concepts.
You already know what topic need to be hit over the semester, so you could limit the fish bowl to those areas of math.
2)A special topics class could be done on a given set. Say integers, and then the discussions on sets, graphs, modeling, algebra, ... whatever was in the bowl, could all be addressed using that set. This would really draw the connection between types of math because each would relate back to a principle idea.
In both these cases, I think that the students would be empowered by having driven the experience. Further, they would feel more free to "poke around" in mathematical questions of which they might be interested, but have heretofore dismissed because they have not been encouraged to explore topics in such a manner. After all, most of the math we study started with an unabashed sense of curiosity and a willingness to be wrong. The ladder of course is more painfully discouraged in a university setting, but that is a different subject.
see you tomorrow

Anonymous said...

Patrick, I am amazed and intimidated by your writings. Why do I always feel so moronic? I admire your intelligence and your abilities. Your students are the lucky ones! Beth