How I Fell in Love With Mathematics…and Only Then Became Good at It

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!)


To Be or Not to Be Creative: a Meditation

In one of her books about writing poetry, Mary Oliver tells the would-be poet that danger is always lurking somewhere.  The would-be poet’s dread about sitting down and writing, when he could be bread-winning to avoid some imagined impending disaster, isn’t a good excuse to skip out on practice sessions.  She’s right, of course, to a reasonable extent.

I bring this up because creative activities are often seen, by would-be creatives, as too risky:  their return on investment is not worth the cost in time, energy, and money.  However, as a great friend pointed out, if you stop watching TV, and if you don’t currently have or take care of kids, then, unless your life is otherwise extremely demanding, you have enough free time, energy, and money for a few serious creative activities.  (If you have kids, you probably still have time to be creative; you can even involve them in your creative activities, which could give them an early start to something amazing later on!)  The choice between being “practical” or creative now reduces to how important and stimulating those creative activities are for you:  if they give you a sense of fulfillment, then you might as well dive in.

Additionally, the successful creative habit (e.g., regular drawing, painting, music-making, etc.) is self-reinforcing.  Any activity that does not somehow reinforce its continued practice will soon extinguish itself.  So, for example, if you are a busy physician constantly managing other people’s problems, and drawing provides a meditative/soothing counterpoint to your day, then drawing may be self-reinforcing for you.

Creativity doesn’t have much to do with external validation.  (See Vincent van Gogh.)  The person who creates for external validation is the person who, if he somehow even manages to establish a creative habit, will stop as soon as that validation (such as sufficient “return on investment”) diminishes or disappears.

Your life is like a sailboat adrift on the ocean: you direct your sailboat, as well as you can, toward your goals, but have little control over the storms and threatening waves that come your way.  Sooner or later, you will die.  You, like everyone else, are biodegradable.  So, while you’re still around, you might as well create something we can remember you by.

How to Draw Anything You See, Part 2: Negative Space

When you draw what’s in front of you, you’re drawing that particular thing you’re looking at, right?  Well, what if, instead of drawing the thing you want to draw, you draw the shapes around it?  What happens?

Surprisingly, your drawing turns out more pleasing and often more accurate than if you directly drew the object of interest!

The empty shapes/spaces bordering your subject constitute what is referred to as negative space.  Particularly when drawing objects with simple shapes, paying attention to the negative space can result in a drawing or painting that is much more pleasing and harmonious than otherwise.

In these images of a vase, the negative space is colored black and resembles two faces along the sides of the vase:


To see how much of a difference it makes to pay attention to negative space, try drawing the chair, below, two times:  the first time, draw the chair directly.  The second time, draw the empty spaces/shapes surrounding the chair.  You may notice that your second drawing is more pleasing than your first.

wooden chair

Negative space is something to always keep in mind as you draw.  Respecting it will make your drawings more accurate, pleasing, and harmonious.