Perhaps my place to start is a quote I placed on my faculty webpage that seems to have caught Kristin's eye: The quote is from A Mathematician's Lament:
Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it. Remove this from mathematics and you can have all the conferences you like; it won’t matter. Operate all you want, doctors: your patient is already dead.This sentiment speaks to me, and is part of a long and very strong essay penned by Paul Lockhart, a mathematician currently teaching at Saint Ann's School in Brooklyn, New York. It was written in 2002 and only published by the Mathematical Association of America in 2008. The whole essay is a complete analysis of why the mathematics we, as mathematicians, "see" is definitely not what is presented in school. It also talks about what a huge failing this is, and what we may be able do about it.
It is a long 25 pages. But the first page is a rather eye-catching analogy between how we approach the teaching of math at early ages, and how we approach the teaching of music.