Explaining pure mathematics to non-mathies

[noam_rion]

I'm currently studying in the undergraduate Pure Mathematics program at the University of Waterloo. Every so often I meet someone who finds it really interesting that I study math, and wants some idea of what it's like. I struggle over how to explain abstract math to someone having no college/university-level math experience. They're often surprised I like working with numbers, and I have trouble explaining that what I do involves very little calculation, at least of the sort they're familiar with from high school mathematics.

Any suggestions for how to explain what it's like? Are there any good examples that can be shown to someone who doesn't have much formal mathematical training?