Explaining what real math is to a high school student I think, after reading through some of the questions here and their answers, that there are many people here who share my opinion on high school mathematics that it's quite different from "real mathematics" as taught at university. I'm not only talking about the difficulty, but much more about the way it is taught and what the focus is on. High school mathematics focuses on procedures and very stereotypical problems, while real mathematics is, in my opinion, much more about insight and creativity. 
As Keith Devlin puts it here: "in high school, you learn to drive the car, at university, you learn to take the car apart and, if you pursue the subject far enough, you learn how to build your own car."
I'm looking for ways to explain what real mathematics is to high school students using problems, analogies, exercises... they will understand. Good books with this purpose in mind are also welcome. While anything that can spark the interest of a high school student is welcome, what would seem to be particularly interesting is one specific topic in which a comparison is drawn between what you do with it a high school and what you (can) do with it at university.
One of the reasons I ask this is because, while I have always been fairly good at mathematics in high school, I thought it was boring (and I still think high school math, or at least in the way it is taught, is boring :)). It was only later, at university, that I discovered what a wonderful subject mathematics is. I would like to share this insight with high school students who feel the same way as I did when I was in high school. 
 A: Paul Lockhart (a mathematician and former professor at Brown University who left academia to become a high school math teacher) has recently published a book, Measurement, designed to reveal the beauty and wonder of mathematics to those who may have only experienced the drudgery and tedium of a normal high school mathematics curriculum.
He's also published an essay that laments that drudgery and tedium, appropriately titled "A Mathematician's Lament." 
In fact, Keith Devlin actually devoted a column to it, and wrote the introduction to the published edition of the essay.
It's a passionate rant that anyone who's been through high school mathematics, student or professor, will love to read.
A: I have not read all of these, but I have been told they are very nice books:
Mathematics: A Very Short Introduction by Tim Gowers;
What is Mathematics? by Richard Courant and Herbert Robbins;
The Princeton Companion to Mathematics by Gowers et al.
How to Solve It by George Polya
The Companion is a personal favorite of mine, and offers something for everyone: an introduction to the idea of mathematics and mathematical research, a basic overview of some fundamental structures, short expository articles on almost every topic in mathematics, essays and opinions by mathematicians on a variety of topics, et cetera, et certa. I hope to have read the entire thing (not just selected pieces of it) by the end of my lifespan.
It is of course unreasonable to expect a high school student to read any of these, but perhaps you will find something in them that inspires you.
A: As pointed out before, there are many books written about mathematics that are intended for people attending high school, many good ones even.
I always have the feeling that certain types of puzzles are particularly well suited for explaining what mathematics actually is: those where there's a group of prisoners that is allowed to leave prison, once a certain condition is satisfied, but punished heavily when this is not the case. For example:
"In some totalitarian country, there is a prison. On a day, the dictator of the country decides that it would be funny to give a group of 20 prisoners a challenge. The prisoners would be given seperate cells and no means of communication, except for one. There was one particular cell with a lightbulb and switch. Every single hour one of the prisoners would randomly be taken out of his cell and be transported to this particular cell. He might then switch the light on or off. As soon as one of the prisoners would declare that every one of them had been in this special room at least once and if this were correct, all prisoners would be released. However, if it was wrongly proclaimed, something ominous would happen. Before the start of this 'experiment', the prisoners had the chance to discuss their strategy. What should they do?"
The reason to choose this puzzle that is somewhat lengthy to write down, is that I think it contains quite a lot that one could find in "real mathematics". Moreover, lots of people can both understand the puzzle and at least one of its solutions.
What makes me think it's a good way to explain a bit what mathematics is? First of all, people need to understand the puzzle instead of just knowing it. Lots of problems one encounters in maths, one just hears without understanding them fully. After that, there is no designated path that leads to a solution - most people will arrive at the same solution, but in rather different ways. It is not, as it is with most problems in high school, clear cut what needs be done. In solving this puzzle, one needs creative thinking or else one won't be able to solve it. Apart from creative thinking, one of course also needs a strong dose of logic to come to a solution, which of course fails to be the case in most high school problems.
A second aspect that makes it suitable, is that after solving it, one can immediately start asking other similar questions that are not quite the same: what would happen if there was more than one room with a lightbulb? How about more bulbs in one room? Is it still possible if the prisoners wouldn't know the exact amount of prisoners? Etc. Something similar is often the case in maths. One knows a problem, works hard on it and after some time solves it. One then often starts to think of similar problems, possible generalisations of the work.
And then there's another thing: one could, after solving, ask the question if the found method is the most efficient in the sense of time it will on average take for the prisoners to leave the prison. This is somewhat comparable to the concept of elegance in proofs. It often happens that one arrives at some conclusion in a very ugly way, while knowing that there should be a more beautiful or elegant solutions. And then one still keeps thinking about a problem, even when it's solved already.
I'm very interested what others around here think about this - if they agree that this kind of puzzles might help in giving high schoolers an idea of what maths is.
A: I would get them started on learning actual math. Things like basic real analysis, number theory, and combinatorics; number theory, when correctly presented, is intuitive and interesting--very much unlike the drudgery of calculus. The standard high school curriculum is very computation-heavy, so the emphasis should be making students think deeply of the ideas of mathematics that they may have taken for granted when studying only computational techniques.
Also, encourage them to apply for programs where they will experience real math. For example, a great place for young mathematicians to get their start in the US is the Ross Mathematics Program.
A: High-school mathematics is mostly technical prerequisites to calculus. The reason it's not "real math" is NOT that real math is more advanced; the reason it's not real math is that the way the curriculum is organized requires sticking to technical prerequisites. There are books in any good high-school library, some published by the MAA, that present real math to high-school students. There is C. Stanley Ogilvy's Excursions in Geometry (in print with Dover Publications, I think) and others of that sort. 
