Brian Greene makes the point about science, and it holds just as true for math: “We rob science education of life when we focus solely on results and seek to train students to solve problems and recite facts without a commensurate emphasis on transporting them out beyond the stars.”
Interested in communicating with a faraway friend without allowing anyone to eavesdrop on you? Of course you are; this problem affects a middle-schooler’s daily life, and yet it is also the basis for modern commerce: communicating your credit card number over the Net without allowing a thief to eavesdrop is a non-negotiable requirement for our economy. How is this problem solved? First, you try for a while. Then let’s talk about prime numbers and see what they can do for us.
Interested in taking a rocket to the moon? Well, if you want to drive a spaceship, you probably ought to understand how gravity works differently from your intuition when you’re far from Earth. Let’s talk about ellipses, and while we’re at it, let’s predict the next approach of Halley’s comet. (And if you’re a high-school sophomore or so, we can talk about inverse-square laws while we’re at it.)
What if we picked up a signal for an alien civilization on our radio telescopes tomorrow? Do you think they’d speak English? Not a chance. Let’s think about how we would communicate without a shared language, a shared culture, even a shared planet? Well, there’s one thing we have in common with any race advanced enough to send us a signal like that: they know math. Let’s think about how that might work…
One of the biggest rarely-asked questions in math is simply “why should I care?”
I love answering that one.