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.