Great article! I've always thought that having math education focusing on formulas is having poetry and literature focus on grammar and spelling -- the painting and music analogies are dead-on.

But that's exactly what happens: Formulas and equations (grammar) are one way to express a beautiful idea. Unfortunately we end up focusing on the syntax and not the semantics, the underlying meaning.

Shameless plug, but I try to do my part to make math intuitive at betterexplained.com.

What a great essay.