> "The thing I was missing is that math is just a bunch of symbols that you manipulate to get new symbols out."

This is the Formalist philosophy of mathematics, which isn't the only one (as you might imagine given that it has a name):

http://plato.stanford.edu/entries/formalism-mathematics/

Formalism is at odds with Platonism, to pick an obvious antithesis, as Platonism holds that mathematical entities do indeed have a reality beyond us finite, mortal creatures and our symbols:

http://plato.stanford.edu/entries/platonism-mathematics/

There are, as you might imagine, a lot more philosophies of mathematics:

http://en.wikipedia.org/wiki/Philosophy_of_mathematics

That said, at the K-12 level, formalism is what students need most; make mathematics a game and show them how well the game applies to reality. That unreasonable effectiveness is quite astounding the first million times, and it never really quits being astounding:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_...