A paper about the experience of mathematical beauty and its neural correlates was just published in Frontiers in Human Neuroscience. It reminded me of how I fell in love with mathematics just before my senior year of high school.
Before that time, I found math a little frustrating. I was decent at it and got good grades, but something critically important about math escaped me. You could say that I had no trouble with math exercises, but I wasn’t skilled at solving math problems. I didn’t know it at the time, but I lacked a sufficiently deep understanding of math and problem-solving in general. All of this changed when I unexpectedly rekindled my childhood interest in logic puzzles.
Near the end of junior year of high school, I saw a curious little book on my English teacher’s bookshelf: Fantastic Book of Logic Puzzles, by Muriel Mandell and Elise Chanowitz. I pulled it off the shelf, started reading the back cover, and was immediately taken by such statements as the following:
“The puzzles you’ll meet inside this book are the world’s greatest and most baffling.” (This was not even nearly the case, of course.)
“To solve them, you need a plan of attack and logical reasoning…You’ll go on to solve tougher puzzles than you ever thought possible.” (True!)
It piqued my interest because it reminded me of how I used to enjoy solving logic grid puzzles as a seven- or eight-year-old. My English teacher made a copy of it for me and I started working through it immediately.
Unlike the exercises I had encountered in math classes, which reinforced facts or blind techniques, the logic puzzles in this book placed a heavy emphasis on deeply understanding the situations at hand. You almost certainly wouldn’t solve most of them if you weren’t thinking carefully.
I finished the book, started studying for the SAT, and realized something very important: I never learned geometry. I mean, I took geometry in ninth grade (I was now about to start twelfth grade!), and I did “well”, but now that I knew how deep understanding felt, I realized I didn’t actually understand geometry on any significant level.
Nor did I know algebra. Or trigonometry. Or precalculus.
So, I sat down and gradually worked through all of these subjects from scratch while studying for the SAT. I had learned, while solving logic puzzles, to constantly step back and question my assumptions. I learned to hold only valid assumptions, and as few of them as possible. I applied these habits of mind as I boldly attempted to prove theorems and other mathematical relationships by myself. It was hard work, but enjoyable and addictive.
I became a mathematical explorer. Math was no longer frustrating but fun. I got into the habit of attempting to solve any test of ingenuity that came my way. Paul Zeitz, in The Art and Craft of Problem Solving, asserts that this exploratory/investigative mindset is key to gaining skill at solving significant problems. The problem-solver lays siege to the problem at hand, circling it until an opening is found–until the crux point of the problem is resolved–and then the problem surrenders its secrets to him, often in a trivial way, and is solved.
Gradually, I gained confidence and became skillful at math, doing very well in senior year calculus and in any other class (chemistry, physics) requiring accurate, logical thinking, as well as on standardized exams. No longer intimidated by math, I switched from biochemistry to computer science after my freshman year of college and deliberately took the hardest math and science classes I could (much harder than entry-level “pre-med” classes), enjoying them greatly and doing very well in them, and, eventually, getting into one of the best computer science doctoral programs in the world.
They say the bumblebee doesn’t know it shouldn’t be able to fly. I didn’t know I couldn’t become good at math, so I kept at it, and kept surprising myself, until I became pretty good. I never did become great, but that’s okay.
(Who knows? Maybe I would have become great if I had not switched careers and had continued to apply the principles of deliberate practice. Maybe you could become great at some skill that currently escapes you, if you just drop the invisible script of “I don’t have the necessary talent” and get down to business!)