Another mathematician's lament

Via Reddit this morning I came across a 25 page essay by a research mathematician turned maths teacher named Paul Lockhart. The essay is called "A Mathematician's Lament". Here's the discussion on reddit, the introductory article and here is a .pdf of the essay itself. Since things on the web have an annoying habit of disappearing after a few years, I'm locally hosting a copy of the essay here. It's a fantastic piece of writing which is essentially a critique of the way that mathematics is currently taught to students at high school. I'm really encouraged to see the overwhelmingly positive response to this on Reddit and I've been motivated to write a bit on the subject matter myself.

Although it's not something I really talk about lot (because I usually doubt the ability of non-mathematicians to appreciate it), I have been, in my mind, critical of modern mathematics education for a long time. There's no way I could not be, having experienced both the dizzying intellectual highs of Galois theory and the soul destroying drudgery of being told "here's how to solve a particular kind of problem. Now solve these 50 instances". Here is a brief overview of mathematics in my life.

I never dreamed of being a mathematician. I was not good at it during high school. I wasn't /terrible/ at it, mind you, and I wouldn't say I was afraid of it, but I usually achieved scores somewhere in the 70-80% bracket on my assessments. I was average at best. It was rare for me to enjoy mathematics, too. I didn't usually hate it, but sometimes I did. I remember quite clearly walking to the mathematics classes toward my final year of high school with feelings of intense dread. This was during the death march of preparation for the final exam at the end of the year. There was no actual teaching involved, just revision. We would be given gigantic problem sets and told to work through them, for the entire lesson (sometimes for two consecutive lessons!). Our teacher would sit silently at her desk and wait for students to approach her for individual help with problems they couldn't solve. And that was it. An near hour or near two hours of tedious silence and solitary drudgery through hordes and hordes of unmotivated questions.

I did enjoy physics, though. Once again, not at first. At some stage I read Stephen Hawking's "A Brief History of Time" and I was absolutely fascinated. That book drew me into physics with absolute force. I followed it quickly with Brian Greene's "The Elegant Universe" (a fantastic book from which was eventually - perhaps inevitably - made a rather mediocre television series). These books got me fascinated by physics and I was kept that way by a physics teacher I had later, one who had only relatively recently left university, who had real theoretical physics research experience and could still radiate enthusiasm for man's quest to understand the universe. By the time I finished high school, I had a qualitative, lay person's understanding of the broad concepts involved in relativity and quantum mechanics and modern cosmology. I could not wait to go to university to study theoretical physics and learn all the gritty details. As anybody who has taken a first year physics course can probably appreciate, I was disappointed to say the least. First year physics scratches the very surface of relativity theory, and I'm not sure even mention was made of quantum mechanics. It was largely a rehash of what I had already learned in high school, except now we were allowed to do it with calculus (which of course could not be done at high school, because very many people in the physics classes were not also taking a maths course that included calculus). To be fair, doing this sort of stuff with calculus is the way to do it and we did need to be taught that. We also had to spend many hours doing experiments that I thought were stupid; measuring the density of brass using archaic equipment, or verifying that momentum really is conserved. I won't say that I didn't enjoy first year physics, because on the whole I probably did. But it was not what I had been longing for. Not by half.

At the same time as physics had been generally disappointing me, something quite unexpected was happening on the other side of campus. By necessity, I was taking a lot of maths classes at the same time. But this was not the dreary drudge work I had done in high school. This was a fantastic, brave new world! For the first time in my life, mathematics was not a disparate collection of unrelated problem classes which I was told how to solve without explanation and then made to grind through a multitude of. Mathematics was a whole, a flowing poetry of ideas which built on each other successively. It was a cohesive fabric woven of rigorous thought, a crystalline iceberg of internal consistency. I hadn't seen anything like it before.

Half way through my second year of university, I took a deep breath and followed my gut instinct by dropping out of my physics degree and taking up place in mathematics. It was not an easy choice to make - I deliberated over it for literally months. Furthermore, there was more motivation involved than I've written about here (I got quite swept up in the rage against reductionism in physics that the authors of several books on chaos theory and emergence exposed me too, but that's something for another entry), but in very large part my decision to switch from one discipline to the other was a direct result of the rapturous joy I felt at my first true exposure to real mathematics. I am in the utmost agreement with anybody who expounds Lockhart's sentiment that by presenting mathematics to the young people of the world in a way which robs it of its creative aspects, which shatters it into disparate, boring parts without even hinting at the interconnected beauty of the whole, we are predisposing them to hate it, to fear it, to fail at it. We are also denying them the ability to enjoy one of mankind's greatest achievements. It should be done differently.

Lockhart's ideas about teaching mathematics in a way which emphasises its simple naturalness and creativeness - its underlying core of just thinking about things, developing a feel for problems, experimenting with potential solutions, refining ideas - deserve a lot of attention. Unfortunately, and somewhat pessimistically, I don't hold out a lot of hope for progress in this direction. The reason for this is that almost everybody who is in a position to change this situation have themselves already had their ability to appreciate maths destroyed by the drudge and grind style of education. Nevertheless, it's encouraging to see these ideas being well articulated. If you know anybody involved in mathematics education, I encourage you to point them in the direction of Lockhart's writing.