Mathematical Formulations

I was unfortunate in my allotment of high school math teachers. In sophomore year, I was assigned to the class of a man I'll call Mr. Unferth. He was the largest man I had ever seen. He started at the floor at six inches wide and rose six feet seven inches into the air in roughly a V shape. He had enormous thighs, a trim waist, and the largest arms I have ever seen on a human being: his biceps were the size of cantaloupes. He had been hired by my school, a small private school with a proud football tradition, to be the offensive line coach, and apparently the only place they could find for him as a teacher turned out to be the Geometry class in which we were stuck with each other.

Unferth was, I came to realize some years later, a man victimized by his successes. Physically powerful through a genetic endowment which had been honed by thousands of hours in the weight room and on the turf, he had little sense of nuance, not much patience (especially with dweeby little upstarts like me), and no flexibility whatsoever. I doubt that he had ever encountered an obstacle he was unable to simply push out of the way, and he was not given to compromise or self-reflection.

Unferth's teaching methodology was strict and unvarying. He would ask us to take out our homework from the night before. He would read, with a not inconsiderable effort, the answers to the homework out loud to us from the teacher's manual. We were to check our own work. He would then ask one of the students to read the section of the text which explained the next set of concepts and showed example exercises. That completed, he would tell us to do get started on the homework, which was always whatever sets of questions appeared in the text. We'd settle in to work, he'd settle in studying the football playbook, and we'd start with the review of the homework the next day. Occasionally, with evident reluctance, he would venture to ask if there were any questions, but the set of his chin and the intensity of his gaze as he stared about the room basically daring any of us to call him on it, communicated very clearly that he was merely being rhetorical. In point of fact, questions made him nervous, and followup questions just basically pissed him off.

The said thing is that I actually liked geometry, as far as the subject matter went. I generally got the concepts and enjoyed doing the proofs, the problem being that I had the really annoying (to Unferth) habit of skipping over some of the steps that seemed self-evident to me, and so when he sat down to grade my quizzes he was flummoxed if the sequence of steps I had on my answer sheet did not match exactly the sequence of steps laid out in the teacher's manual. And God forbid I should attempt to either explain myself or ask him for clarification. It was a very long semester. I ended up with a D.

Senior year, in another school, I had for my Trig teacher a Jesuit priest I'll call Father Brown, known to his students as "Pa Brown" or just "Pa." Pa was a shortish, balding, husky man with glasses and an enormous pot belly which hung over the sash he wore around his waist as part of his habit, which was inevitably wrinkled and covered with food and chalk stains. My memories of Pa are much clearer than my memories of whatever it was that he was trying to teach us about math. I remember that the book was a slender hardback volume with a red cover and that it was packed densely on every page with text charts and formulas and nothing else, not so much as line drawing. Pa's daily ritual was not unlike Unferth's, although Pa was a much smarter man, and his methods of intimidation more oblique. Once we were settled in class, he would call on a student ("Mr. Dlugos, come to the board.") and dictate a problem to be solved. Then we would all watch the student intently work through problem. Once he was done, Pa would begin the process of interrogation, checking to see if Mr. Dlugos was sure of his answer or wobbling (most often the case), and then giving the other students in the class the chance to show up Dlugos or, more often, compound his ineptitude with their own, keeping Pa amused and feeding into his talent for sarcastic invective. While I somehow managed to stay under Pa's radar, and wound up with a grade which was neither remarkably good nor remarkably bad, I can honestly say that I don't remember a single thing about that class except the word "extrapolation" and the mental picture of Pa himself, in all his dishevelment. The content of the course was all just routines and words and abstract concepts that melted into one another and vaporized, and which seemed to bear no relation to anything in the physical world at all.

Which is perhaps why I was so blown away when I opened the New York Times magazine in December of 2004 and saw the article which eventually gave rise to this slide show, showing the work of Japanese photographer Hiroshi Sugimoto. As the intro to the slide show indicates, Sugimoto had apparently run across a set of plaster models that were manufactured in the early 20th century to help students understand complex mathematical formulas. I was struck, then as I am now, with how beautiful the pictures are as pictures. But I was just blown away by the realization that somehow, in some way that neither Pa Brown nor any of my other math teachers had ever managed to convey to me, those inscrutable sequences of numbers corresponded to and could actually be made to generate three-dimensional forms. My experience with high school math made me into a math avoider. I used to count it as one of my successes that I managed to graduate from college without taking a single math course. Now I'm not so sure it was a success, and I wish I had come to my studies of math with a different set of understandings and a different set of mentors. I know enough about education and creativity and critical thinking at this point to understand that there is no reason why study of mathematics can not be inspiring and generative and aesthetically satisfying. It's not a question of whether, it's a question of how.

Hand Me That Tool, Will Ya?

To tell you the truth, I don't remember a whole lot about elementary school. I remember the faces of a few of my teachers. I have pretty clear memories of Sister Mary Vincent, my fifth grade teacher, mostly because she was about 110 years old and a variety of creative ways of expressing her displeasure, including one memorable instance in which she picked one of my classmates up by the hair, dragged him to the blackboard, and slammed his head into the board five or six times to put the fear of God into him.

One thing I do remember is that I was absent for a week or so during third or fourth grade, I think it was, when the class covered the process for deriving a square root. I remember coming back to school and having the teacher tell me I'd have to make it up on my own following the instructions in the text. I gave it a shot, and never did figure it out. It remained a conspicuous hole in my mathematics background up until the first hand-held calculators showed up in the early 70's, at which point deriving a square root became something any moron with fingers could do. Not that I can ever recall ever having the need; the derivation of square roots has turned out to be about as useless a math skill for the average person to possess as one can well imagine.

Reason I bring this up, yesterday I got to sit in on a joint meeting of the middle school and high school math departments, one of the first such meetings that has occurred at my school in many a moon. Todd and Will, the respective department chairs, had asked the teachers to read in advance a packet of articles relating to math pedagogy, and had arranged to have the 30 or so teachers divide up into groups, one group for each of the topics that the articles had covered. My group turned out to be the group discussing calculators, and it wound up being a very interesting discussion. The original focus of the discussion had to do with the reality that while middle school students are at this point in time allowed, even encouraged, to use calculators for routine arithmetic operations, students in at least some of the grade nine classes are forbidden to do so.

I haven't spent a lot of time (any time, actually) researching the arguments of the pro- and anti-calculator factions in education or tracking the history of the debate, which has of course been around ever since calculators became ubiquitous and cheap. But I get the gist of the conflict, and got a quick review of the core positions as the discussion unfolded. The anticalcs argue that use of the calculator gives kids an excuse never to learn the basic computational skills without which one will never truly understand, much less master, mathematical thinking. The procalcs argue that there are many very bright kids who are good mathematical thinkers who for organic or developmental reasons struggle with arithmetical operations, so that use of the calculator actually frees them up to go further and deeper at the conceptual level than they ever would be able to if they were forced to do the (likely to be faulty) arithmetic on their own. Out of the instance of the freshman program, larger questions emerged. What is the logic of allowing, or forbidding, calculators in the first place? Should there perhaps be a consistent schoolwide (K-12) policy or position statement that says when and why we are going to ask, or forbid, the students to use calculators? What are the mathematical non-negotiables? Are there some operational procedures — the derivation of square roots, for example —that the invention of the calculator has made irrelevant or unnecessary for students to learn? Is long division one of those operations? Multiplication? Addition? Where do you draw the line?

From there we bumped up to the next larger frame of reference: laptops. We are well on our way to the implementation of a One-to-One laptop program in grades 4-12 (as of this moment, only grades 11 and 12 remain technologically unenhanced.) Laptops have not only multifunction calcuators of various kinds wired in, they have very sophisticated mathematical modelling programs as well, like the one that one of our physics teachers used on the opening chapel ceremony, which was built around a theme of "hearts beating as one," to give a scientific demonstration of a drum head doing a complex series of simultaneous vibrations. The concept came across very clearly, despite the fact that there was perhaps no one in the audience, with the possible exception of the teacher himself, who could have done the math necessary to produce the demonstration onscreen. So when kids walk into the math room, do we tell them to check their laptops at the door? Do we define certain types of activities or lessons or assessments as being laptop-friendly or laptop-hostile? Bottom line, is telling math students they can't use calculators any more or less reasonable than telling English students they can't use word processors? Spellcheckers? Dictionaries? When is a tool a tool, and when is a tool a crutch? Answers may vary.

Finding My Way Home

It's been a month since I've posted anything. For the middle two weeks of that month I was on vacation, visiting my family in Florida and North Carolina. Now I'm in an extended transition into my new office across the quad from my old office. Most of the boxes of materials I had packed up have not been brought over yet, so I'm in a sort of in-between space, getting to know my new location, establishing some new routines, and re-establishing some old ones. This, for example. I'd like cultivate the habit of writing something every day. Sometimes I get into that pattern, and I like it when I can make it happen; but sometimes it just gets to be too much. One of the things that I noticed when I was traveling and not writing was that I was consistently having vivid and robust dreams. I don't know if that's a function of being on the road and out of my element, whether it's the stimulation of new sensory data that's putting my brain on overload, or whether it's some sort of spillover effect from not having the outlet of writing. But my sleep for the last three weeks has been fitful and punctuated by interconnected dreams. I wake up with a dream in my head, turn over, and fall back into the same dream transformed, bent, re-channeled. In the morning I wake up, and unless I make some effort to capture the dream and make some notes, within a few minutes all the images and all sense of narrative structure has fled.

While I was in North Carolina I ran across an article in the July 28 New Yorker by Jonah Lehrer called "The Eureka Hunt," (abstract here) in which he reflects on the sources of good ideas. At one point he talks about the importance of relaxation — even drowsiness — in generating insight:

The insight process…is a delicate mental balancing act. At first, the brain lavishes the scarce resource of attention on a single problem. But, once the brain is sufficiently focussed, the cortex needs to relax in order to seek out the more remote association in the right hemisphere, which will provide the insight. “The relaxation phase is crucial,” Jung-Beeman said. “That’s why so many insights happen during warm showers.” Another ideal moment for insights, according to the scientists, is the early morning, right after we wake up. The drowsy brain is unwound and disorganized, open to all sorts of unconventional ideas. The right hemisphere is also unusually active. Jung-Beeman said, “The problem with the morning, though, is that we’re always so rushed. We’ve got to get the kids ready for school, so we leap out of bed and never give ourselves a chance to think.” He recommends that if we’re stuck on a difficult problem, it’s better to set the alarm clock a few minutes early so that we have time to lie in bed and ruminate. We do some of our best thinking while we’re still half asleep. (43)

I know that it's true that a lot of the ideas I have gotten for poems have come in that half-waking moment. But I don't often have the self-discipline to make the effort to capture them when I'm just waking up.

But what I'm doing now is not entirely different. It's the end of the day. I could be, and soon will be, on my way home, but right now I have a little time to relax, the time and the inclination to sit here and type. And it's, well, pleasant, to be sitting her in my new office at the end of the day following the lines of thought that my fingers are spinning out for me.

The walls and the shelves of my new office are, for the time being, mostly bare. I've spent a lot of time the last few days sorting through materials, throwing out some of them as I was packing, and throwing out others as I have been unpacking. I've got books and files that I brought with me from Massachusetts when I moved here ten years ago, some of which I haven't looked at since then. Doesn't seem likely I'm suddenly going to need any of them soon. So once the rest of my stuff arrives, I'm going to be weighing each item in mind's eye and tossing what I don't really need. Travel lighter.

I'll end with a poem by Eamon Grennan from which his book Matter of Fact, which I read on the plane to Florida and re-read on the plane back, derives its title. Grennan is thinking about Cezanne, and about how the methodology of the the artist and the writer ("excavating form from facts") might be said to overlap. The poem consists of two sentences, a statement and a question, which taken together articulate and exemplify a process of writing that makes perfect sense to me. Yes, this is what we might aspire to be able to do. That Grennan succeeds so often at it is one of the reasons I enjoy reading him so much.

Cezanne and Family

When he was excavating form from facts—
finding the geometry of trees and Mont Sainte Victoire—
he was doing what I'd like to find
a byway to, translating ravages of daily dross

into an illuminated shape or two, simple as light
but holding all the prickly specific unspeakable
matter of fact, a grasping-at (think the thousand
cuts of colour), paint laid and layered, angling

into a new veracity), that offers a centre
but no easy symmetry, coming to a point, yes,
but letting the disorderly goings-on of nature
go on, undisciplined as they are

and no containing them. Could it be like families,
I wonder, the way they don't ever or rarely ever
make clear and formal sense, and yet the facts
add up and we stand there, astonished by them?