I think part of it is that most math is taught as equations. Those are useful, but only after a person can do word problems. I also think we start on math too soon, before a person's reasoning capacity is developed, and then move too slowly once their reasoning capacity is developed.

If we started a few years later, other than counting and simple math (what you can do with a pile of pebbles), and then focused on solving real world problems (like Nemo's example, or figuring out change among multiple people), we would get a better educated (mathematically) population.

It would probably be a good idea to learn math in the order we as a species learned it, as well, since that would ensure that no prerequisite knowledge could be left out. Finally, I think that unit analysis is a critical skill virtually untaught in public schools. (Ex: we know, even if we've never learned that time = distance over rate, that hours is equal to miles over miles per hour, assuming we know basic rules about how you can alter equation forms without altering the answer.)

While you are fixing public schools... or at least trying seriously to do so... may I suggest an introductory course in economics and finance.

In my opinion, there is no area of study in which today's students are more woefully ignorant.

When I was in school (mid 80's-mid 90's), economics were not taught at all - in any way, shape or form. No lessons in supply and demand, the way the stock market works, how taxation affects the economy, budgeting, debt & deficits, etc. - nothing at all. Everything I've ever learned about the way the economy works, about our system of free market capitalism, I learned at the feet of men like Reagan, through my own common sense and logical reasoning, and living in the real world.