I am inspired, by several previous blog entries, to write about my own mathematical awakening, and what I’ve learned from reflecting on it.
I went to New York City Public Schools, in the Bronx. I always enjoyed arithmetic and mastered it easily. I remember not knowing what ‘fractions’ were, but don’t remember learning about them, any more than I remember learning to read. The understanding came to me naturally, and I hardly noticed the process. Even first year algebra didn’t seem like a learning process, more like a set of exercises. So I had mastered a lot of mathematics (well, a lot of algorithms) before I really understood what it was I was learning.
A revelation came in ninth grade, when I was 13. Ms. Blanche Funke, a good math teacher in JHS 135, took some of us during lunch and organized us as a math team, to compete against other local Junior High Schools. Now this is work I have since spent decades doing, and I know now what could have been done. But Ms. Funke didn’t quite. Her idea was to give us advanced training in textbook algebra—not to find ways to make us think differently about that same algebra.
So she gave us the definition of an arithmetic progression, and the standard formulas. And a problem something like: “Insert 3 arithmetic means between 8 and 20.” I loved this work. Plug into one formula, get the common difference, then plug into another formula and get the three required numbers. I could see what I needed to do and took joy in starting the work.
But next to me was my friend David Dolinko, and he was busy drawing something in his notebook—some diagram of a molecule in chemistry. (Professor Dolinko has lately retired from the UCLA School of Law). I poked David. “C’mon. Let’s do this problem. It’s fun!”
David looked at me, as if annoyed at the interruption: “Oh, I did that already. Eight, eleven, fourteen, seventeen, twenty.” And went back to his drawing.
That moment changed my world. Suddenly I realized that these formulas had meaning, were trying to express something. They were expressing that the numbers were ‘equally spaced’. So David could just pick them out—the numbers were small—and didn’t have to bother with the algebra. Algebra has meaning. And if you know its meaning you can use it more effectively. Suddenly, instead of black and white, I saw the world of algebra in color.
I thought about this a long time. The colors attracted me more and more. I wasn’t just good at mathematics. I enjoyed it, and enjoyed being good at it.
Well, the next year I was still sitting next to my friend David, in the last seat, last row of a classroom in the Bronx High School of Science. We were taking geometry, the classic neo-Euclidean syllabus, taught by one Dr. Louis Cohen. He was a somewhat impersonal teacher, or so we thought, but a master of his discipline. And of teaching it. So one day he had covered (I don’t remember how) the theorem that the angles of a triangle sum to 180 degrees. The lesson had gone quickly, so he filled the time with some ‘honors’ problems: the sum of the angles of a quadrilateral, some problems with exterior angles, and so on. And to cap it off, he drew a five-pointed star on the board:
Not a regular figure, but just any one that came to hand, using the usual technique of following the diagonals of an imaginary pentagon. He then asked for the sum of the angles at the points of the star.
My hand shot up, seemingly of its own accord. “180 degrees,” I said, without quite knowing why. And to my horror, Dr. Cohen strode calmly down the aisle to my desk, with a piece of chalk in his hand, handed me the chalk, and asked me to explain to the class how I had figured this out. But I didn’t know how I had figured it out. I just saw it, with intuitive clarity. What was I going to do?
I was lucky that we sat in the back of the room. As I saw him coming towards me, I began to analyze my own thoughts. And as I walked to the front, I figured out what to say. To this day I remember my hand trembling and my voice shaking as I pointed out certain triangles, certain exterior angles, and got the angle measures all to ‘live’ in the same triangle. Dr. Cohen praised me, then gave a slicker version of the proof that must have clarified it for the other students. Of course, there are better ways even than his to prove this statement. If the reader can’t think of a nice proof offhand, take a look (for example) at https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-polygons/v/sum-of-the-exterior-angles-of-convex-polygon (accessed 6/2018). The argument can be adjusted to cover non-convex polygons.
Why is this important? Well, it is important for us to understand that the language of mathematics is a language of thought. And that thought is synonymous with intuitive thought. We sometimes get caught up in the expression of our intuitions, and fail to go back and make clear, even to ourselves, what we are talking about. This phenomenon has deep implications for teaching. How we do this, how we know it has happened, how we integrate it into the teaching of mathematics as a forma language, are all questions we must struggle with. But they are not questions that we can beg. We must somehow be sure that students can eventually understand our results on an intuitive level, whether or not we communicate with them on this level directly. Without that, we are teaching algorithms—even algorithms of proof—and not mathematics.
I invite readers to contribute their ideas to this blog about how to make mathematics accessible on an intuitive level.