Yeah, I know, math is hard. You know what else is hard? Any subject if you study it thoroughly enough. People seem to have a much different relationship with math than other subjects, however, so that whenever I admit I am, indeed, pursuing a graduate education in mathematics, their first impulse is to tell me about how hard math was for them. Do you often tell your doctor about how you failed organic chemistry? Are people mentioning all the time to reporters how they were never any good with sentence structure? This is why we have a specialized society.
I am a graduate student in mathematics. We are not a group known for our prowess in social situations. But people, you aren't doing us any favours when the first thing you tell is that our chosen life is so difficult, or boring, or something you could never do. Most of us can only come back with a weak "haha yeah I dunno". We aren't all super geniuses (some are, I am not). We spend years learning math. It all seemed like crazy gibberish to me, too, before I had spent 5 years (and counting) learning it.
I think people are afraid that we are constantly on the precipice of launching into a proof of the Sobolev Embedding Theorem or something, and feel the need to convince us that that would be a bad idea. Don't worry, we are not. If you are not studying math, anything I talk to you about that is vaguely math related will probably be related to baseball as well. I spend a good amount of my time trying to explain calculus to people who think they are being punished for something, I don't have an urge to continue trying teach when I am at a party. Now, if you want my take on GMRES or the conjugate gradient method, I'll be happy to give it. But don't worry people, I am not scouting bars for potential students to talk at about my studies. I'll leave that to the political scientists.
People also like to ask what I do all day, as if they think mathematicians spend their days stroking long white beards (hopefully attached to the same person stroking them) and staring at clouds, occasionally stopping to scrawl out a proof on a chalkboard. Perhaps you think we spend our time calculating integrals, that our research looks a lot like your calculus homework (I wish we could still credit for "discovering" the chain rule, I'd be published already). My day is not that uncommon. I go to class, work on homework or research, read papers, teach, and dick around on the internet just like everybody else.
I also tend to get (mostly from middle aged men in bars for some reason) the question "what do you do with a PhD in math?" I guess it's hard for someone in pharmaceutical sales or business management to figure out why anyone would pay a mathematician. They see us, think about their old calculus homework, think about profit margins or something like that, and think to them selves "you can't sell a theorem to anybody". The easy way out of this is to lie and say I like teaching and want to be a professor. People accept this as perfectly reasonable and noble, and exactly what they would have guessed. This is because most people think a professor is more or less the same thing as a high school teacher. Don't get me wrong, I would never knock teaching as a profession. I have had (and am related to) some great teachers, and have great respect for that profession. But I (and most professors and TAs at major research universities) do not have a passion for teaching. I don't mind it, and I know it is important so I work hard to be the best TA I can be. But I am not here to teach. I want to "do research". I want to find out things no one yet knows about our world, our universe, and (and here is the domain only math can reach) fundamental truth. I don't care where I am doing it.
But how do I explain that to someone with no idea what a mathematician can discover? How do I convince the guy in the bar that, no, I do not want to be an actuary, and yes, I am aware that that profession pays well?
So here's My Point. Don't be afraid of math. Everyone loves watching planet earth on TV, and reading things about sustainable energy in popular science, but nobody wants to hear about the article in the SIAM news. People think they won't understand any article about what is going on in math. Well, you don't really understand that article you just read on the boy who was "cured" of HIV. Raise your hand if read about that and now can explain why and how it happened. Anyone? Ok, fine, anyone who is not a biologist? But you found it interesting and you took something from it anyway. So don't be afraid of those SIAM articles. Google SIAM and click on "siam news". If you are feeling brave (or maybe you are already not so afraid of math) go to ams.org and click around. Read the featured column. You don't have to understand the math, just take a look at what it is accomplishing. You might be amazed. You might even be interested. And it might make our next conversation a lot less awkward.