No ideas but in things

Jordan Ellenberg is perhaps the only mathematician to quote William Carlos Williams in order to make a (debatable) point about mathematics: that it is all about assertions that refer to the world.

I agree with him completely about the value of doing your homework:

Getting these rules in muscle memory is what you need if you want to get fluent in algebraic computations.

There’s just one thing missing, but it’s a big thing: the fact that algebra is made of sentences, that it means something, that it refers to something outside itself. An algebraic statement isn’t just a string of symbols with an x stuck in there somewhere. It’s an assertion about a relationship between quantities (or, when you get to more advanced algebra, between functions, or operations, or even other assertions.) Without that animating idea, algebra is a dead and empty exercise....

Computation is important! We lose just as badly if we generate students who have some wispy sense of mathematical meaning but who can’t work examples swiftly and correctly. A math teacher’s least favorite thing to hear from a student is “I get the concept, but I couldn’t do the problems.” Though the student doesn’t know it, this is shorthand for “I don’t get the concept.”