There are numerous essays and newspaper blurbs lamenting the poor state of mathematical education in the US. Here is a typical example: Presidential panel bemoans state of math education.
What I see as the problem is that advanced mathematics is introduced in language that is unfit to inspire any but the few that were genetically destined to be mathematicians (or physicists).
Ask a recent college grad what an Eigen value or Eigen vector is. I give you 100:1 odds you'll get a blank stare. Okay now ask them to read this explanation from a popular Math web site. I bet their face will be even blanker. Now ask them to read this wonderful little explanation. Chances are the lights came on.
This is not to say that the later explanation will allow a person to do the math. But this is certainly where Math education, even at the highest levels, should begin. Illustrate why the problem is important, give a sensory picture to go along with the abstractions. Some might believe that this is how most Mathematicians teach but that is simply not the case. Mathematics is a very macho profession and many mathematicians believe its beneath them to offer intuition prior to rigor. The sad truth is many of them could not come up with compelling intuitive explanations even if they wanted to. It was not the way they were taught either.
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Just found this: http://www.ted.com/index.php/talks/view/id/228
Alan Kay gets it.
My experience with mathematics teaching in the US is that it is not really done by people that know mathematics themselves. Before getting to intuitive explanations, one has to understand the science, which is where I think the problem lies. When you know little, there's little you can transmit, no matter the amount of good will and skill you have for communicating. The problem mentioned in your post may become more important at higher levels of education (college and up), although again, my experience with US textbooks is that they are way clearer than the ones I learned from, which were more abstract and formal than anything I've seen in the US.
Here's a hypothesis - let me know if it makes sense to you: I believe the problem is economical - if you're smart enough to teach mathematics well, you're smart enough to find a job that pays better than as a high-school teacher.
Perhaps industry steals some talent but I don't think that is the main problem. I think that to teach something well you need the kind of empathy that comes from struggling to learn yourself. Many mathematics professors are so talented at math that they fail to appreciate how difficult it can be to some people. That makes them naturally favor the few students who would pick up mathematics with no help at all because of there sheer talent. Hence the cycle continues.
I agree that math could be taught in a better way; in fact most subjects could be taught in better ways.
But if this would be the main problem with teaching math in the US, why aren't other countries encountering the same difficulties when they use more formal teaching methods, because I don't think there is a country that has textbooks that are more explicit than the US ones. The textbooks I learned math from were inspired from Russian textbooks and were even more concise (meaning they included less examples) than the originals - way, way more abstract than any textbook I've found here, with the only exception of Papadimitriou's "Computational Complexity", which is a graduate textbook.
Empathy helps a lot, but what is more important is to regard math as some important piece of your education and not as some weird, useless class that is a waste of time - it's a matter of attitude towards math, a matter of culture. Here in US, I don't feel people respect math as something worth spending your time with.
Let's look at Hollywood movies - whenever a mathematical genius is depicted, he's never shown working hard to prove some result - his achievements are always the result of some 30 min brainstorming or of some sudden inspiration - it gives you the impression that you're either good at it or it's not worth trying - and that's bad, because no one is an innate mathematician - some people may be more gifted than others, but everyone has to do some hard work to achieve success. Even comics sort of show you the same thing - heroes gain powers accidentally and they never have to work hard to get things done - at least not until Frank Miller revitalized Batman with "Dark Knight Returns" ;)
Math won't be easy even with intuitive explanations - it may be easy initially, but there will come a time when you'll need to leave the crutch of practical examples and be able to reason using abstract models only. And to be able to do that, you'll need someone showing you the way who has more than empathy on his side. And you'll also need to do some hard work,
I think the problem with math education is complex and a single explanation doesn't do justice to it.
Best regards!
Perhaps we agree more than we disagree. You are correct that U.S. students don't value math. Most likely there are aspects of our culture that does not leave much room for appreciation of anything unless immediate application toward economic gain is seen. If this is true then there are two choices: change the culture (unlikely) or leverage the culture by providing examples up front of how math applies to problems with economic reward. It’s still sad because, for me, the intellectual discovery is its own reward.
The easiest solution is to get Charlie Epps to teach every math class.
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