From an email to friends 2 years or so ago after Chris Morse's talk on math skills being correlated with chemistry performance, and a subsequent animated discussion wherein some of my (engineering student) buddies expressed frustration and dislike for the study of math. I love math. You could say that it was puppy love (I was around 10-11 when I first really got into abstract math), one of my first serious relationships* with an academic discipline, with an amiable breakup in mid-high-school when I realized this wasn't what I was meant to spend the rest of my life with. I'll never be a professional mathematician; my talents and my passions lie in other (but closely related) realms, but math and I will always be good friends.
*Most teenagers date other teenagers. I dated math, physics, and creative writing, flirted with computer science, and occasionally got set up on blind dates with theater tech and improv.
Uh... moving on.
Making sandwiches is a limiting reagent problem. Splitting worms is an acid-base problem. Another way to say "applications to real life" is "isomorphism to ," and that to me is what math is all about; finding patterns, finding connections, finding what things that are really weirdly different are, on the other side of the equals sign, actually the same. If you learn how to see that a donut is actually a coffee cup (topology), then maybe it will help you discover that... a coffee cup is really an insulator, or cardboard is a construction material, or this tiny circuit is equivalent to this big ugly one.
So it's concrete and it's not concrete. It's immediately applicable and it's just kind of fun. Math is a formal statement of what human beings do naturally; find patterns and equivalences. Learning how to read math, speak math, think math, and play in a mature way with the material is important for our growth as expressive, creative human beings. Why does Rob Martello deconstruct and then reconstruct our essays? We can write, we can talk, we do this communication thing already. But understanding it, studying it, and getting to that depth in it helps us do it so much more effectively.
J. Random Blogger "who rites about lol dosnt this sux" is not as effective a communicator as Noam Chomsky criticizing the same thing fluently. And it's not that Chomsky is merely expressing the same thoughts in a better way than J. Random. Because he's studied writing (among many other things), he has a better mental toolset for thinking and developing those thoughts in the first place. Part of this toolset is drawing, part is linguistics, part psychology, part history... and part math.
This doesn't say anything about who we should teach what math to how (and quite honestly, we don't have the right to determine that for anyone but ourselves and perhaps our children), but I do math because it brings me moments of small joy. I do it for the tiny, lovely little connections where things click together; I'm sure you've experienced this in other disciplines (a line makes a drawing fall into place, one word in a business plan suddenly opens up possibilities for a whole new enterprise, a subtle guitar passage sends a shiver through your spine).
Last night I wrote about wavelets. I looked at the formula which was full of Greek letters that would probably cause most people to shut it out immediately. They see indecipherable scribbly lines. I see tiny dancing waves, tiny waves bobbing up and down on the surface of a sound or an ocean, massing together and combining their sameness to make something different. Ben talks about modular arithmetic; I see transparent colored blocks raining down in circles, clicking into place and falling down together in chunks like Tetris every time you match n of them together. It's pretty.
It is easy to look at a bunch of lines and think it's weird, it's hard, it has no relevance. But if you know Japanese, you can see that oh, it is a poem about the moon rippling through the pond. Haiku is always better in the original Japanese. I can explain wavelets to you without formulas, but it will never be the same.
I won't use most of the math I learned here when I graduate. Heck, I've forgotten all the formulas already. What I will use is the way of thinking about problems that I've learned from the math I learned here. As others have pointed out, college (and to some degree life) is about learning how to learn. Math is one way of thinking that is important to learn. I can honestly say that my comfort with math has opened many doors to me; it was because of math that I got into computer science and by extension programming and being comfy with code and chips, which in turn led me to electrical engineering.
It was math (graph and information theory) that gave me a handle on signals and systems, which in turn led me to a deeper intuitive understanding of how feedback works between organizations and people at Olin. It's math that is behind my artistic ability, even - I can render three-dimensional organic shapes because I can render three-dimensional nonorganic ones because I worked on a 3d graphics engine in high school because I was learning linear algebra. I'm sad to say that math has given me a leg up and helps me stand out, because I honestly wish that more people could see this.
If you don't know what you will come across in the real world (and for the most part, we don't; we are still very young and very, very green) then math is one of the best investments you can make in terms of learning. I'm not talking about plug n' chug memorization; that's crap. But a deep understanding of how to find patterns and connections, and how to formally and unambiguously express, manipulate, and model those patterns and connections so that others can understand them - well, that's universal. And that is the math I know and love.
I need to write about math the same way Alan Lightman (Einstein's Dreams) writes about physics. In fact, you'll all probably appreciate Lightman's work; go to the library and get either Einstein's Dreams or Good Bonito, and read one of the short stories. He shows as much of the beauty of formulas in words as I think is humanly possible without seeing the formulae themselves.