Bad Mathematics? (part 2)

Yesterday, I meant to write a short introduction to a wonderful piece of writing often known as "Lockhart's Lament", but which I prefer to call by its original title, "A Mathematician's Lament".

Paul Lockhart, explains Devlin's Angle, is a self-taught mathematician who now teaches grades K-12 "subversively" - i.e. "the real thing". Apparently Devlin's Angle is the first place the "lament", written in 2002, was actually published; though it has apparently been quietly circulating through the mathematical and teaching communities and appearing on other blogs for some time.

Let's say that yesterday's post was what maths seems to be, and Lockhart's Lament is what maths actually is.

I struggled a little at first. I could only conclude that I had never had the slightest idea what mathematics was. Then pieces began sliding together, like those of a jigsaw puzzle. And the picture they made was a great blast of light, as overwhelming as a Shakespeare tragedy. Yes, I was lazy at school, and stubborn. I could have got further than I did. But I had fallen into the same trap millions of other bored students had - of thinking of mathematics as a set of rules, as I had been told it was, not as a set of patterns to discover. I had thought that to think up anything for yourself in mathematics must require years of dedicated postdoctoral study - not that it is as simple as working out things for yourself about characters in a book or flavours in a recipe.

It's so simple. The method maths teachers use tells you everything before you can think it out for yourself. But if you work something out for yourself, you care, and always remember.

In place of discovery and exploration, we have rules and regulations. We never hear a student saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no mention of the aesthetics behind this choice, or even that it is a choice.

Maths is like everything else - "rules" are what we invent, to organise what we discover. They're a conclusion, not a driving force. Same with definitions. It drove me mad in History to have to learn the definitions of "Benthamism" and "Utilitarianism" without being allowed to see any examples. Not to mention learning the pluperfect subjunctive for Spanish without the least idea how to have a proper conversation (and once I was in Granada doing that, all that grammar was suddenly much easier to learn as part of the package). Definitions are not a starting point but a tool we develop as we go along.

Why on earth do we teach maths like that? Because, evidently, few teachers - and certainly not the government (Lockhart's Lament presents a similar enough story to my own experiences that I make no apology for generalising) - have the faintest idea what mathematics education has the potential to be. As Lockhart puts it,

Everyone knows that something is wrong. The politicians say, “we need higher standards.” The schools say, “we need more money and equipment.” Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “math class is stupid and boring,” and they are right.

Fear of maths is blamed, which we try to address with fancy textbooks and desperate cuteseyness. Popular culture is blamed, which in my opinion just makes everybody feel helpless and alone. Popular culture has as much power as we choose to give it. Nobody's going to be able to come and say, "Maths is cool" and have the whole population believe them, because the real culprits here are cowardice and a lack of imagination.

If I hadn't seen the Bertrand Russell bit, I'd blame standardised exams, which expect all pupils to have reached point X by age Y, and allow for neither particular weaknesses nor strengths nor interests. But obviously the problem has been going on for longer than that. It's basically a lack of freedom to learn things for their own sake. Learning is so important that it's been hijacked by commercial, utilitarian forces, who miss the real point of knowledge:

". . . [I]t is far easier to test someone’s knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning."

(Now if that's not also true of curriculum-worshipful science teaching, I don't know what is.)

Well - people tell me it's never too late to learn. I do have some maths textbooks. But I don't know what anything in them means, because they present the rules first, and then the exercises, and never any explanations or challenges to think. I suppose I could figure some of them out, back to front. In the meantime, I am taking up a challenge Lockhart left in his paper (I do hope he won't mind if I post this small scanned snippet - I've been prowling around trying to discover how to contact him to ask, and I am happy to delete this if necessary):

That was a mean trick, Dr Lockhart! Now, imagine that the longest side of the triangle is the base - just as in the original triangle. (If you haven't downloaded the document, get on with it, it's well worth it - the original triangle is on page 4.) That makes it easy to draw a line from the tip to the base. Notice that that gives us three small triangles, which - if we fold the piece of paper - we'll see are all equal! The three of them take up exactly half the box; two of them are part of the triangle, and the third is not. That means that triangle takes up exactly a third of the box.

But what if the triangle was just a bit more or a bit less slanted? Aha. That was where Dr Lockhart was clever. He made one of the sides of the triangle (I'm afraid I can't remember what the sides of the triangle are called, apart from the hypotenuse, which Tom Lehrer has ensured I will never forget) exactly a third of the way across the rectangle. What if it isn't? What if its tip isn't in the corner?

Well in that case, I'm not sure the chopping will work at all. I reckon you'd just have to give the triangle a box of its own, and measure its height and width and compare it to the height and width of the original box. But I may be wrong.

I also tried - feebly - to solve the pyramid in a cube challenge, but apparently I got that wrong too.

So my interest is kindled, my appetite awoken. I now know what I've missed. Is it better or worse? It hurts a lot more now. But it's still good, I'd rather know. And I've got something to aim for now, if only I knew where to start looking. Paul Lockhart's suggested method of teaching mathematics is the following, although he sadly consigns it to impossibility because "it's too much work!":

So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.

But I have no such teacher. Nor do most people. Where can we find one?

I have a plan. But I can't do it myself, because I have neither the maths nor the programming skills (something else to aim at perhaps). I suggest a website like the Galaxy Zoo Forum dedicated to mathematics. Not to listing rules, but to exercises available of the kind Lockhart suggests, at all different sorts of levels. On this website, we'd be able to draw and measure shapes and do calculations easily; graph paper would turn on and off at a command; playing would be the process, rather like galaxy classifying. The freedom to waste time and make mistakes and go off on tangents is essential, because it's not a test, it's a journey in as many different directions as you wish to travel at once.

As a lot of the explanations would be in words, not proofs or symbols, it would need people, not machines, to give feedback. That's where things would get harder. That would be where the forum came in. I wonder if such technology will ever be available to all? Well, I certainly didn't imagine Galaxy Zoo would ever be available before it was.

There are a couple of threads on the forum you might be interested to read: a Maths Problems one and one I started about the Mathematician's Lament. I said in the latter that Lockhart's suggested teaching methods are very similar to how we teach each other at the zoo - I hope you'll agree!

There is so much more I could say about this excellent document - but I think reading it for yourself is the best thing. I've lost count of the number of times I have. It took me several months to stop sniggering nastily about the fictional art teacher at the beginning:

". . . I’ve got a degree in Painting myself, but I’ve never really worked much with blank canvasses. I just use the Paint-by-Numbers kits supplied by the school board.”