**Title**: Reading, Writing, and Learning from Mathematical Language

**Abstract:**

Years ago a small fraction of Barbie dolls included “Math class is tough!” in their four-sentence linguistic repertoires. Protests followed, and the offending utterance was expunged. Let’s reject right now any suggestion that math class is uniquely, or even especially, tough for Barbie’s target audience. Still, Barbie had a point. Math classes *are* tough, and not just for elongated plastic effigies. Learning anything deep and difficult will always be challenging. Perhaps less obvious is that “math class” may be tough for reasons connected to the nature of mathematics itself, and especially for students first seriously encountering theory and proofs. The villain, and the hero, is the special language of mathematics.

That we mathematicians use language in unusually precise and complex ways is hardly surprising: communicating technical ideas and fine distinctions naturally requires extra linguistic effort. I suspect, though, that we under-appreciate how formidable students often find it to parse correctly the genuinely complicated mathematical sentences they meet in, say, a first course in real analysis, which may involve multi-layered universal and existential quantification.

I’ll pursue these ideas with examples drawn mainly from elementary real analysis, and suggest ways to help students engage not just the difficulty and subtlety of mathematical language, but also its power and economy as a tool for mathematical thinking.