How I Discovered Mathematics

I didn’t discover mathematics once but several times over. Each time felt like returning home.

Meaning

“If one must choose between rigour and meaning, I shall unhesitatingly choose the latter.”

- Rene Thom

I do most things slowly, always have, from the time I was a small child. Of course, dawdling along never made me popular. My older brother, borrowing (as he often did) a well-worn joke, used to tell me I had two speeds. The faster of the two put me in the relaxed company of tortoises and snails, though they inevitably grew impatient always waiting around for me to catch up. And the slower of the speeds wasn’t so much a speed as a state, usually a nighttime state in a cozy bedroom under warm blankets. So there you have it: my two speeds: slow and off.

So it is no surprise that in the first grade attending school in an upper midwestern state I was among those the stern teacher sent out into the hallway. We were a small group, and the larger bulk of kids moved on while we tramped out, following orders and carrying booklets containing a motley, tedious collection of addition and subtraction exercises: 3 + 8 = __ or 9 – 4 = __. Each exercise had no natural connection to the one before it, nor to our surroundings, the headache-inducing, dull school hallway. We were not, in working these exercises, doing math, despite that common expression. We were doing calculations, and disenchanting ones at that. We plodded along, lying around on the floor, sharing yawns. And continuing to go slowly. Why would a dull hallway speed us up, or, for that matter, teach us anything?

And why was I doing arithmetic exercises at the pace of a ground sloth? Note here something important: I didn’t ask why I was learning arithmetic slowly, but doing exercises slowly.

In second grade, I moved on to a different school in an entirely different midwestern state, but not much changed until my third grade year. For the first month of that year, the teacher, Mrs. Dare, did not smile. With her brown hair worn up, she was tall and intimidating. But there was a method behind her initially cold exterior. Behind that authoritative facade was a wise, warm heart. I don’t recall any bullying or other disruption in her classes. Once she established a respectful order so we could learn in quiet and peace, she lightened up, smiled regularly, and we loved her. We had fun.

Early that year, perhaps in the first week or so, she placed a transparency on a projector showing a 3 x 3 array of black dots illustrating the elementary mathematical truth that 3 x 3 = 9. I was smitten. With that visual display, multiplication clicked for me, and I loved it like I loved my favorite jigsaw puzzle (the one with a cartoonish picture of a decrepit, haunted house and monsters popping out everywhere). I approached mathematics like a jigsaw puzzle: you take in the big picture on the box first, absorb its shades and shapes, and then work to fit all the pieces together. I never did memorize multiplication tables: I vaguely recall refusing to do so, or maybe I just didn’t have to. Asked to multiply 9 and 7, I drew an array - 7 rows of 9 dots or 9 rows of 7 dots - so I had a picture of the entire problem, and then I counted all the dots. I started with what the operation of multiplication means, at least in much of everyday life. I always wanted to take each and every problem back to first principles and visualize at once how these principles combine in the final result. I didn’t get any faster at working through exercises, in fact I probably got slower, but my understanding skyrocketed – and would often come quickly.

When you think carefully and fully about what you are learning, sifting through many nuances for thorough understanding and unexpected connections, it takes longer, even when your mind is quick. The fastest hikers slog through thicket-filled, rocky canyon floors, especially if they check out any of the cave openings along the sides, or any of the intersecting branches from neighboring ravines. And when you want to start with the big picture before filling in the details, taking in the basic structure of the network of chasms as a whole first, well that takes time and resources.

My approach to calculation is visual. When I mentally compute 3 x 7, I imagine the numbers three and seven as pieces of a jigsaw puzzle, and in my mind’s eye I lay seven threes down end-to-end, or three sevens down end-to-end, and get a (linear) picture of the number twenty-one. Numbers are movable, overlapping pieces of an indefinitely long puzzle. They aren’t colored in any way, for me at least. They aren’t even black and white: color simply doesn’t apply. But there is shape, and it’s long and linear, and flows off into the potential infinite.