> Finally, so what if they never need it. Show it to them anyway - they > can go home and confuse their parents with their new knowledge. Those > who know me know one of my favorite phrases: "Math for math's sake!" > Do it just for the fun of it.
This is reminiscent of Conway's comment--something like "teach them not to ask how it applies in the real world"--and bears some examination, I think. Some people mentioned that complex numbers are behind the beauty of the Mandelbrot and Julia sets. Of course, you don't have to know that to appreciate their beauty or explore them, but I find that knowing a little more about things like that tends to make their beauty (and my appreciation of it) more intense. In much the same way, simply knowing a few constellations adds a lot to the pleasure of looking at the night sky, and every incremetal bit of knowledge (that some particular star has a radius the same size as our orbit around the sun, or where scientists think a black hole might be located, or which direction the center of the Milky Way is) adds that much more.
To me, opening people's worlds to this kind of enjoyment and wonder is the biggest (or at least one of the biggest) motivation for teaching math to everyone. My experience has been that people that hate math (or at least hate the math they have seen) and don't do it well still make it all right. But they miss out on a lot of the wonder in the world by not knowing math's beauty. I often think that we would do well to show kids more of things like graph theory, abstract algebra, etc. while they are still curious and eager to learn rather than killing all of that with addition drills and long division. (Naturally, that statement should be qualified, but I do think there is a lot of truth in it).
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