Why is maths different?

Humans struggle at a lot of intellectual endeavours. But for some reason, of all of them, maths seems to have an unreasonable level of arcane-ness (arcanity?). People who haven't spent much time thinking about maths are not only unable to appreciate mathematical arguments, but seem to be unwilling to. The same isn't true of other disciplines.

Why is this?

Here's a metaphor: Imagine that the only person who was able to hear a piece of music is the person playing it. So someone is playing a piece of music on the piano. You're watching, and you can only watch their hands move over the keys. If you're observant you may start to get an idea of which keys they press at the same time, and which ones tend to follow each other.

But then they give you the score, and you sit down at the piano and start to play through it yourself. Now you can hear it, and you begin to understand how it fits together and what makes it beautiful.

That's what maths seems to be like. You can't understand or appreciate the complexity or beauty of a piece of maths unless you're creating it yourself. And you can't create it yourself without having learnt a bit about it. And you probably won't want to learn anything about it until you've had a small experience of hearing it.

Algebra

Reading the comments to this article. There's more than one comment with the sentiment "I cannot recall the last time I used algebra!"

If you've ever been in a situation where you had to try to calculate something without having all the information, you've used algebra. Shopping without knowing the prices. Deciding what to spend your tax return on before you get it. Figuring out how big a home loan you can afford.

Let's say you're organising an event, and you want to get an idea of how much profit you're going to make. You know what your expenses are, and what the ticket price is, but you don't know how many people are going to turn up. What do you do? I'm a mathematician, and I'll tell you what I do: I work out a pessimistic estimate, and an optimistic estimate, and keep those two numbers around. I know that the real ticket sales will be somewhere in between, and now I have a rough idea of what's possible, and what's likely. I don't work out the precise linear relationship between ticket sales and profit.

You don't need to write any of this stuff on paper in order for you do be doing things mathematically. Of course if you do, you can get much more accurate results. But most of the time we don't need more accurate results, we just guesstimate. But it's still mathematical thinking.

Confusion

I don't think it's always necessary to avoid confusion. At least when teaching someone. If just trying to communicate then yes, confusion is bad, but teaching is different. At some point you need to learn how to deal with confusion. Not being exposed to it means you're being shielded.

Maths activities/problems

Buckets of fish! - http://jdh.hamkins.org/buckets-of-fish/

Fractivities - http://fractalfoundation.org/resources/fractivities/