Relevant (an article I wrote): http://j2kun.svbtle.com/you-never-did-math-in-high-school
As an anecdote for you, since you're a mathematician: during school I had zero patience for math. I would often study just enough to get a passing grade. On the other hand, I enjoyed physics classes a lot, it involved more problem solving and holistic vision than memorization (since you can just memorize a couple formulas and derive everything else you need - e.g. the second and third equations of motion from the first). Maybe I would have enjoyed math classes more if it involved more intuition and less rote? I'm sure you'll come across kids like I've been during your voluntary lectures.
PS: I'll be following your blog.
EDIT: I just read your article, I think our math program is like you describe it should be.
I agree there is some baseline level of rote manipulation people can't get away with, but it's gone too far.
http://nces.ed.gov/fastfacts/display.asp?id=97
-- says that in 2009, just 16% of U.S. high school students took calculus. And a common complaint by college professors is that high school calculus courses teach rote procedures without explaining calculus in any depth -- the students know how to solve calculus problems, but they don't actually understand the subject.
When the AP test rolled around I was really struck. The problems were about the flow rate of water descending into a tank and what happens when you put a drain here with this rate of flow, etc. Completely different from our earlier problems which could be solved by rote memorization. And from my perspective - one of a student who knew a bunch of calculus "rules" but lacked understanding of how calculus really worked - this was incredibly difficult.
I got a 3, which is passing, but opted to take beginner-level engineer's calculus when I got to college anyway. Even if you "learn" calculus in high school, it does not teach the level of mathematical maturity required to understand the higher level manipulations. This kind of thought is the foundation for any good engineer, and I have no regrets about retaking the course.
I jumped to AP Calculus without taking the pre-requisite AP class everyone else had taken, and I thought I was so cool and smart. By the weeks end I no longer felt so cool and smart - I was completely lost.
As a side note, I can't stress enough how bad the education system is in Brazil (in the sense of lack of structure, motivation, etc.), even at college level. Almost everything I know is self-taught (including English). I sometimes hear complaints about public school in the USA, and there's a meme that americans are "dumb", but it seems miles ahead to what I had access to.
- Introduction to mathematical thinking [1]
- Introduction to Mathematical Philosophy [2]
- Machine Learning (actually a CS course, but involves linear algebra and some calculus) [3]
- Calculus: Single Variable [4]
[1] https://www.coursera.org/course/maththink
[2] https://www.coursera.org/course/mathphil
College-level calculus burned me so much, I didn't had the algebra background necessary to grok it, much of it is memorizing heuristics and "tricks", and college professors (specially in my public uni) didn't had much patience either. I even did well in other math-related classes, but calculus was a chimera. Unfortunately, I fell so much behind the curriculum that I had to drop out of college to start working full-time.
Lately I've been picking up other math-related online classes, and I finally realized that I had never had real exposure to mathematics, and that high-school curriculum is something else. So, there's that too... we are mostly scammed into thinking we've learned something during high-school, and then life happens and you actually figure out you don't know shit. Hopefully I'm a curious person, so I'm always learning something by myself, mostly to broaden my knowledge in other topics, but to not let the gears grind too.