Okay here's the thing...
In general, everybody knows arithmetic. This is an inevitable consequence of going to school. After only the first grade you know the beginning of adding and subtracting, and by the time you hit grade six, you're a master of multiplication, division, and operator precedence. High school teaches you trigonometry and algebra, and if you're lucky a bit of calculus... but then it stops.
A vast number of people have not looked beyond algebra into the heart of mathematics. For these people math is something that people do with numbers. This is wrong. Ask any mathematician. To them, the numbers that we use in day to day arithmetic are boring and just a little bit dirty - they have no place in a mathematical paper. The few, (well, many) important numbers, once deduced, are wrapped in greek letters as not to hurt the eyes. Nobody cares about 3.14159... when π will do just fine. So if there's no numbers, what's there to talk about?
The most important question in mathematics is "why?". Everybody knows Pythagorus' theorem, but many of them don't actually know why a2+b2=c2. Guess what, there are over 300 reasons why; over 300 proofs of pythagorus' theorem, and each one a thing of beauty. (Though I concede beauty is in the eye of the beholder.)
You see, mathematics is all about making tools for the masses. Pythagorus' theorem is very useful in day to day applications, but mathematicians don't sit around applying it to things and then writing about it. "Dear Diary: today I measured a very large triangle." No. Mathematicians make the tools - the theorems - and then use them to make more powerful tools. At the bottom of all mathematics are the axioms - simple statements like 1+1=2 which are assumed to be true without proof (and are unproveable). All of mathematics is built, theorem upon theorem, from these axioms. We go from 1+1=2 to calculus, and we know that all the steps in between are logical and true because mathematicians have walked the path to reach them.
These thoughts were brought on by a discrete math class I had to take in university. Discrete math is math about very simple things - counting, adding, grouping in sets - and building complex ideas from them. It was amazing to sit down knowing all the axioms I had available, and the moves I could make to build my own theorems and proofs. But the problem is that not everybody gets this joy. Up until the end of high school, math teachers tell you things without really explaining why. True, the proofs are much more difficult than the results, but without proof mathematics is just a whole bunch of things you do with numbers. No wonder people get sick of it, and no wonder they go around thinking math is boring.
Now, I propose that the first sort of math taught in high school, before trig or polynomials, be discrete math. That way people could get a glimpse of what mathematics truly is, maybe they wouldn't hate it so much, and maybe they wouldn't think of mathematicians as boring nerds because really they are studying the universe and answering the eternal question "why?" one miniscule piece at a time.