I read Paul Lockhart's essay, A Mathematician's Lament, almost 15 years ago as I begun my exploration into mathematics. It opened my eyes to what mathematics really was. Recently, I remembered the essay and found the book, a lightly edited version with another part added at the end that shows how mathematics could be taught.
The best way to think about his argument is in the beginning: what if music was taught to kid without letting them actually listen or play music until grad school? What if we taught notation, circle of fifths, counterpoint, and harmony as just abstract ideas, to prepare a few students to go on to actually enjoy what music actually is? Or, what if art was taught to children as color theory, perspective, and historical facts, without letting kids actually draw? What if art classes were tested whether they know the mechanics of "good art", and teacher and curriculum success was judged by the students scores?
The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.
This is the state of mathematics today. Growing up, I was okay at "math", but was always bored. We learned to quickly do speed multiplication tests, learned rules of long division, quadratic formulas, chain rules, and on and on. All by rote, all without any sense of wonder or exploration. We needed to learn algebra because "it's needed for pre-calculus", which was "needed for calculus", which was needed for...? Most people leave math eventually, largely hating it—because they aren't "math people"—without actually "needing" what was taught. (Confession: I have used calculus quite a bit, but these days I'd use Wolfram Alpha for anything non-trivial anyways.)
Math is an art that explores pattern and logic, and is fundamentally about exploring problems. These problems are motivated by curiosity and wonder. Paul Lockhart shows exactly how mathematics pedagogy went wrong, and shines the light upon a better path that opens the door for more people to truly appreciate the wonder of math. This essay (and book, who's additions are indeed useful!) is for people who have never seen math as I'm talking about it, for those who aren't "math people". I can't recommend it more for anyone curious about the world, or who can help point children towards this special place of pattern and wonder.