The question of why is there so much crappy education is interesting. I think part of the answer is that society is well-served by the failings of the educational system in that widespread educational crappiness helps support the class system. I think it would be fairly straightforward to turn 90% of the population into well educated upper middle class types, using techniques like btilly describes...
... But if we did that, who would drink Coors and drive forklift for Walmart and not complain about it?
So I think that a huge function of the educational system is to educate a large part of the population badly. I think that the teacher training system serves this is as well, by selecting for mediocre teachers and then making them more mediocre via training. Additionally, in the schools themselves a lot of effort is made to classify young people into dumb and smart categories, usually unfairly (http://en.wikipedia.org/wiki/Pygmalion_effect), and this classification stamps them for the rest of their lives and creates a population of hopeless lower class workers.
If there are SOME good teachers in the system like btilly, then SOME kids go on to get out of their class, which is perfect in that (1) we need to claim that it is possible to pull yourself up by your bootstraps, and (2) we need to recruit SOME kids to grow up to fill management positions, but not too many that they all can't find careers. Historically, underemployed educated people are the people who become union leaders.
The beauty of it is that the people who fail think it is their fault!
(Sorry to rant about the educational / class system in general, rather than the topic of how to teach advanced math, but I couldn't stop myself.)
"But if we did that, who would drink Coors and drive forklift for Walmart and not complain about it?"
Robots? If the education system elevated more of the population, there would be a shortage of blue-collar workers, more pressure for system automation, and more well-educated minds to develop the robots. Of course, there would be no shortage of immigrants to fill the blue-collar positions, but perhaps in the long term, a "better" equilibrium would be reached.
... But if we did that, who would drink Coors and drive forklift for Walmart and not complain about it?
So I think that a huge function of the educational system is to educate a large part of the population badly. I think that the teacher training system serves this is as well, by selecting for mediocre teachers and then making them more mediocre via training. Additionally, in the schools themselves a lot of effort is made to classify young people into dumb and smart categories, usually unfairly (http://en.wikipedia.org/wiki/Pygmalion_effect), and this classification stamps them for the rest of their lives and creates a population of hopeless lower class workers.
If there are SOME good teachers in the system like btilly, then SOME kids go on to get out of their class, which is perfect in that (1) we need to claim that it is possible to pull yourself up by your bootstraps, and (2) we need to recruit SOME kids to grow up to fill management positions, but not too many that they all can't find careers. Historically, underemployed educated people are the people who become union leaders.
The beauty of it is that the people who fail think it is their fault!
(Sorry to rant about the educational / class system in general, rather than the topic of how to teach advanced math, but I couldn't stop myself.)
Call me paranoid, but there you go.