Gowers on examples

Tim Gowers wrote My favorite pedagogical principle: Examples first a few days ago. (The comments are worth reading too.) As an example, he gives an axiomatic definition of a field, and then compares this with defining a field by saying "look, you know about the rational numbers, the real numbers, and the complex numbers; fields are `like these', and now here's what we formally mean by that.) People have said most of what I'd want to say about this topic, but there's an interesting question here. It's been pointed out that it's often a good idea to read mathematics "out of order". But then why doesn't the writer write things out of order, since most people's natural impulse is to read things in the order they're written? Presumably the writer understands the material better than the reader, and therefore has a better idea of what order things should be presented in.