Mathematics: The Most Misunderstood Subject

Agreed. I'm currently in your same position, only a few years ahead (in my second year of grad school, currently). When I try to explain math research to people, I too am left with strange looks.

I think most people honestly think that mathematicians sit around and multiply larger and larger numbers together.

Part of the problem of explaining the true nature of mathematical research is one of language. For example, I study Lie algebras and representations. People ask me "What are Lie algebras?" and pretty much the only definition I can give that is understandable is "Lie Algebras are a kind of algebraic structure that is useful in many fields of science, including quantum physics." This is an answer many people understand (on the surface), but it really doesn't say anything.

What's worse, most of the time, when we justify or explain our research, it is by connecting it to fields that the person may be more used to (astronomy, biology, physics, economics, etc.). But for those of us in pure math, we really do not think about these fields in our day-to-day work. That is, I study Lie algebras because of their beautiful structure and the interesting combinatorics behind them, not because they are useful in some other field.

So a better answer to the question "What are Lie algebras?" would be something like "Lie algebras are vector spaces with additional algebraic structure that gives rise to beautiful and deep combinatorics. They occur naturally as certain sets of square matrices, and are a kind of generalization of the ideas of symmetry." However, this is mostly unintelligible to most people who haven't taken some mathematics beyond calculus, and I find that it sounds condescending to a lot of people, which turns them off from listening to any more explanation.

What I've taken to saying lately in response to a question along the lines of "What do you do in math research?" is something like "Mathematicians create new knowledge from existing knowledge. They take things that the human race already knows, and using only logic, they deduce new things. This allows them to find fascinating relationships all throughout the world."

I find this response to be pretty good. It's mostly accurate, it's mildly interesting, and best of all, it's short.

It is hard to explain the value of higher math, but it's worth it.

Perhaps a better answer to "What are Lie algebras?" is to respond in terms that mean something to your audience. Avoid words like "vector" and "combinatorics".

Instead use metaphors. Like Rubix cubes. Tell them Lie algebra is a way to solve Rubix cubes faster. And also other similar puzzles that are way harder than Rubix cubes. It's true enough for casual conversation and probably more interesting than a vaguer answer.

When I'm asked about math there are a couple of things.

Firstly, I ask them about Pythagoras. Most people know of it, and I phrase it in terms of cardboard cutout squares. Take three squares cut from a heavy material, and make them so that A and B together weigh the same as C. Arrange them so their sides lie on a triangle. Not only is it always possible, but the triangle you get always has a right-angle.

Why? How do we know? As it happens, the reasoning as to why it's true is wonderfully elegant, and totally accessible.

Secondly, I ask - do you think mathematicians know about numbers? Here's something. Take any positive number. If it's even, halve it. Otherwise, triple and add one. Keep doing this, and what happens. So far every number anyone has every tried ends up in a ...->1->4->2->1->... cycle. Does it always happen?

No one knows.

Possibly it's useless, but there's a bunch of stuff people thought would be useless, and they've given us micro-processors, SatNav, cryptography, error-correcting codes, and a million other things.

Who knows what will be useful? After all, if we knew what we were doing, it wouldn't be called "Research".

Oh, Collatz problem. I usually use Goldbach or twin prime conjectures to explain the difficulty of seemingly simple problems, but come to think of it, Collatz is even better, because it does not involve any complicated concept at all -- Goldbach and twin prime conjectures are about primes, and sometimes people do not even know what primes are.