[api]

The problem is really very simple.

First you teach the basics of the language. Then you teach how to express concepts in that language and what those concepts mean. Finally, you teach how to manipulate those concepts to build new higher-order forms.

Mathematics is taught like this:

First, students are shown how to manipulate symbols they do not understand. During this process, sometimes (if you're lucky) these symbols are explained in a piecemeal and oblique way. Sometimes conceptual meaning is discussed at the end to wrap things up (oh by the way this is what you'd use this for, now let's move on), but this is rare. Mostly you just get elaborate dances of symbols thrown at you with no explanation to tie what you're doing to any problem, reality, or conceptual meaning. In the end most students end up memorizing these meaningless opaque incantations and never understand why anyone would be interested in math.

[cma]

I don't think just doing everything from first principles is reasonable. Building up the natural numbers from set theory etc. would just be frustrating for kids.

Like trying to teach them their native language by grammar diagrams etc. instead of immersion

http://en.wikipedia.org/wiki/Language_immersion

Math is not exactly the same as natural language, but there are tradeoffs to doing it one way or the other and there needs to be balance.

[ska]

This may have been how you were taught, and that's unfortunate.

It is not, however, "how mathematics is taught".

[j2kun]

Ah, well I wasn't speaking about mathematics education (in, say, high school). Because there are a whole lot of problems with that beside the notation.