When I tell people that I study mathematics they tend to have one of two reactions:
1. They make some impressed-sounding noise, or mention that they were terrible at maths at school, and then quickly make it clear that they wish to change the subject.
2. They are genuinely interested, and want me to tell them exactly what it is that I study.
The second reaction is the one that I fear most! And at this point it is usually me who tries to change the subject. I have an ongoing competition with myself to increase the length of time which I can spend explaining my research to someone before their eyes glaze over and their body language starts to say “I want to be somewhere else now”. I am currently up to about 15 seconds. And the subject I am currently working on (somewhere between graph theory and galois theory) is fairly accessible compared to some of the more exotic branches of mathematics!
The main problem in explaining pure mathematics to a non-mathematician is the level of abstraction involved in the subject. Most people’s view of mathematics is that it deals with numbers, and it is hard for people to imagine what exactly it is that mathematicians do…add and subtract really big numbers? Many seem to find it difficult to imagine how it could be that all the mathematics that could be done hasn’t been done already.* People rarely encounter abstract mathematics before university; and for good reason, as the transition from dealing with concrete quantities in a familiar setting, to treating those quantities and that setting as merely one very special case in a vast world of abstraction, can be rather bewildering.