One reason math is hard to teach is that you have someone who found something easy to understand trying to each it to someone who finds it hard to understand. This post at Language Log is an interesting example of someone in a different field running into the same problem.
Linguists use the idea that language constructs are recursive. The example that Mark at Language Log gives is that of “stone traffic barrier”. Here, “stone” and “traffic” are both adjectives. Adjectives can modify not only single-world nouns, but more complex phrases that stand in for nouns. Here, “stone” modifies “traffic barrier”. If you parenthesize it like it’s a piece of mathematics, you would write “(stone (traffic barrier)”. Mark comments that he’s regularly surprised how difficult people find the concept.
It also makes me wonder if mathematics is hard to teach at a basic syntactic level in a way that we would never appreciate. Syntactically, mathematics is defined entirely in terms of recursive syntactic constructions, as the Wikipedia page for well-formed formula illustrates. Getting across the idea of order of operations of arithmetic would be pretty hard if you’re talking to somebody who hasn’t learned the idea of recursive syntax in the first place. It makes me wonder if learning how to diagram English sentences would make it easier to learn algebra.