Speaking of failure and its uses, I had a conversation today with a brilliant artist who never “got” math in school, and had thereafter stayed away from all things mathematical. At the moment we were talking, we happened to be on a bus outside of Torino, Italy, passing by an orchard. The trees were planted in a square grid pattern. Periodically the bus would pass a row of trees, whereupon the trunks of whatever row we were in front of would visually merge into a beautifully perfect line.

I told my friend that if the orchard were infinite, and each tree trunk infinitesimally thin, you could think of each tree as a rational number (a ratio of two integers). I asked him to imagine standing at the corner of the orchard, and to think of each row is a numerator and each row as a denominator. If you look out at the horizon, the bits of sky you see between the trees are all the irrational numbers.

He told me that made perfect sense, and that if they’d only presented math that way in school he’d have had no problems. Because I’ve been thinking about the positive benefits of failure, I told him that I thought that maybe the reason so many people have trouble learning math and science in school is that they are taught without good failure modes.

If you write an essay in your high school English class, and the teacher wants to give you suggestions about how to do better, they can always say “You have some interesting ideas. Try using smaller words and less description, and focus on the main concept.” Or something like that.

No matter what sort of improvement you might need in your essay, it wasn’t a failure – just a path to success. And this is true for many subjects in school, such as art, history, drama. All of the failure modes work – they produce something interesting, something for which you can get positive validation from your teacher, and to get progressively better you can take things from there.

But math and science are often not taught that way. Rather, they are taught in terms of “this is the way things work, and if you don’t get it, you’ll get the wrong answer”. You are told that *this* is Avogadro’s Number, *that* is Newton’s law, to find the roots of a quadratic polynomial you need to apply *this* correct formula. Everything is rules, rules, rules. And if you don’t follow those rules, then the message is that maybe you’re just not cut out for this stuff.

But of course in reality – as opposed to our schools – math and science are far closer to English and history and drama. They are wide open spaces, opportunities for play, fields that stretch out in all directions ripe for endless experimentation and innovation.

For much of my K-12 education, I wasn’t learning math mainly in a classroom. Instead, I would spend joyful hours in the library or the school courtyard trying things out, asking myself crazy questions, posing puzzles for myself, and then trying to work out the answers. The fact that sines and cosines make circles, and that you can express both sine and cosine with a series of equations in a simple pattern – that was just the starting point for exploration. Suddenly the derivatives of those equations had meaning, they were things I could play with and make something out of, like the trees in that orchard of rational numbers.

Of course I would often fail – set out in a direction only to encounter a brick wall. But that was ok. I always learned something from the failure, and it never bothered me when I got stuck. I would just strike out in another direction, try something else, because I was always having fun.

In other words, the failure modes were not punitive – they were part of the exploration and play, like in a well designed computer game. And that’s still the way math and science are to me today, only now I can also write a computer program to try something out, to see for myself whether my crazy theory is correct. My theories often turn out to be wrong, and I still hit brick walls quite often. But even in those cases, I feel like I have had fun with my little chemistry set, and it never feels like I’ve been wasting my time.

One of my hopes for the use of computer games in the classroom for teaching math and science is that they might help those of us who had a hell of a lot of fun exploring those worlds (and still do) to share our sense of exhilaration with a new generation of kids. The key to fun learning is to try things, explore, break something just so you can put it together a different way to see what happens.

Like I said yesterday, failure is indeed an option. In fact, if you’re doing it right, failure can be your friend.