A Mathematician's Lament
A long time ago I read something that really summed up the dire state of Maths education. Happily, I came across the exact same thing quite recently. Please, read this:
It really explains so well what has gone wrong. I still, although I'm quite used to it by now, get shocked by how few people think they can "do" maths, and how "difficult" maths is perceived to me. If I had to sum up the process of "doing maths" I would say it is
1. Pattern spotting
3. Proving/disproving the conjecture.
Humans are naturally great at 1, just watch an episode of "Deal or no deal?" to see the ridiculous and elaborate patterns people have spotted and are basing their decisions on.
2 is also very easy. All you have to do is make up a rule. Doesn't have to be true and you don't have to give any evidence whatsoever towards it being true (that's part 3). 3 gets nasty, yes. It requires a great deal of creativity and perseverance. For research mathematicians, they don't even know if their conjecture is true or false, so may spend ages attempting a proof that is doomed to fail simply because their conjecture is false. And you can't tell from the conjecture how easy the proof will be either. If you're given something to prove that is at the right level for you, then proving stuff is great. You have to be so creative, much of the time you end up bringing in areas of maths that seem totally unrelated. It gets really frustrating ("character building!") when you can't do it though, which is why I think part 3 is something to be thankful that someone has done for you.
This is why maths shouldn't be seen as "difficult". Maths is reliable, it always works. If some conjecture has been proven then you can use it, build all sorts of things on it, and it won't fail you. In science, gradually new theories replaces an old ones (the earth is flat, anyone?) so what you're learning isn't concretely true like maths is. Plus if you're unlucky, something will be changed while you're still around (Pluto isn'ta planet!?). Maths doesn't get outdated, like media or technology or ICT might be. Much of the maths we learn is very very old. It gets built on and added to, rather than be replaced. In short, the fundamentals of what makes maths maths are awesome and everyone should have a natural affinity towards it.
But what we learn, and teach, in school rarely fits into one of those 3 catergories. It's what could be put as
4. Applying the theorem.
(Conjectures get an upgrade to theorem when they are proven to be true.) Applying a theorem is what much of school maths seems to be. Here is Pythagoras' theorem. Now apply it to all these very boring little triangles. If I'm really trying I'll make you find a variety of sides, vary the notation, and perhaps even draw some triangles at funny angles. I'm going to try really hard with my teaching to do more of parts 1-3 (especially 2, who can'tmake up any old wild claim?).