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.
Best Answer
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.