What is Algebra?

I'm biased, but I love discussions of the nature of mathematics, especially when they're targeted at mathematical novices. Algebra was one of the areas of math that I spent quite a lot of time on in college, and it holds my interest like no other. Keith Devlin has a very interesting analysis of what algebra actually is:

The important thing to realize is that doing algebra is a way of thinking and that it is a way of thinking that is different from arithmetical thinking. Those formulas and equations, involving all those x’s and y’s, are merely a way to represent that thinking on paper. They no more are algebra than a page of musical notation is music. It is possible to do algebra without symbols, just as you can play and instrument without being ably to read music.

It's this kind of insight that is typically not present in most mathematical courses, and I feel it's a huge part of why so many people have significant math anxiety. They don't really know what they're doing, and more importantly, they don't know why they're doing it. Removing the big picture means that the entire context of math is gone, and so each new chapter in the mathematics text book becomes disconnected from every other.