I'm really not convinced by this. Sure, math education is far from ideal. But the solution cannot be to make it more "touchy-feely" and make everybody feel good.
Courant, in his book "What is mathematics?", says the field historically seems to go through alternating phases, from wild, creative, adventurous phases in which new things are discovered -- often inspired by "real-world" problems in physics or engineering -- to more sober, rigorous phases in which another generation takes over and basically cleans up the findings of their predecessors (making proper definitions, constructing and refining the proofs etc.).
But this is far from what math _education_ can be about.
The biggest obstacle to math education is the meme that mathematics is "only for intelligent people". Mathematics at the basic, cleaned up level has almost nothing to do with "intelligence" and a lot to do with memory. I know this first-hand because I used to be really bad at maths and then one day decided to learn it anyway (and did). Most often, when someone doesn't get some result or can't figure out the derivation, it's because he doesn't remember or has never learned the relevant facts. [EDIT: Also, if you think you need to be "intelligent" (however defined) to do math, you will give up sooner when you don't understand sth because you think "uh, I'm too dumb" instead of asking "what am I missing".]
Mathematics is a kind of language, and as in any language, most people will actually need to sit down and memorize vocabulary. It's mundane, it's not fancy-creative, it's a pain in your behind, but once you succeed in automating certain things, the process takes on a momentum of its own.
This "dragon mom" stuff I occasionally read about in the NYT is silly IMHO but it's just as silly for a 22 year-old, who by her own admission doesn't even know the standard curriculum of a plain vanilla mathematics education, to want to be "the ambassador of mathematics" (http://www.nytimes.com/2011/01/18/science/18prof.html). First you do the hard work, then you can fly.
But the solution cannot be to make it more "touchy-feely" and make everybody feel good.
Why? That certainly isn't the type of response that would cut it in mathematics, since it is just an assertion without even a hand-wave to support it.
Also, I'm not sure you really understand her argument. Her point isn't that we should to make math classes easier so that people feel good about math; her point is that students might be more interested in math class if it contained some mathematics.
The primary aim of math class up through calculus is to teach kids to perform the calculations needed to do physics, which is needed to do engineering, which is generally perceived (correctly I think) to be greatly beneficial to society. This is fine, but there is a huge chasm between this and the types of things that actual mathematicians study. (My brother once commented that if people had any idea what was actually going on in the math department, there is no way in hell they'd fund it).
The best counter to Vi's point, I think, is "why would we want to make kids appreciate pure math?" It's really fun and really interesting, and personally I would love for more people to get to experience math, but I'm sure the same could be said of classical music, or poetry, or knitting or pretty much anything else that people find stimulating. There may be some benefit to it, but I don't think that benefit has been quantified.
btw... sitting down and memorizing vocabulary is an awful way to go about learning a language. Communicating in a langauge every day, reading a ton and trying to think in that language are much more effective.
I like/appreciate your reply a lot (except the first line ;) and would comment more in response if I didn't have to run just now.
But let me at least thank you for this: "The primary aim of math class up through calculus is to teach kids to perform the calculations needed to do physics, which is needed to do engineering, which is generally perceived (correctly I think) to be greatly beneficial to society."
This is the best summary I have yet seen about why math is taught the way it is (in high-school, not in university) -- and by extension why many people, incl. myself, feel or felt uncomfortable with math up to high-school.
Specially because it should be economics and budgets and financial stuff the focus of math in those formative years. And statistics too!
Not everyone will be an engineer, but almost everyone will have to deal with money in their life. And the world would be 1000 times better if only people could understand a little more statistics.
Math education should at the very least include a description of what math is and why it's interesting. Current math curricula don't let students know that math can be engaging, and so they're not engaged. The ambassador thing is to tell "normal" people what math is about. That's fundamental, not "touchy-feely".
> Math education should at the very least include a description of what math is and why it's interesting.
But let's be realistic about this. "Interesting" is an opinion, and the number of students who don't already find it interesting who would if we took the time to explain it would be fairly small.
I disagree. If you look at the comments left on Vi Hart's YouTube videos, it's clear that there are many, many people who only realize that math can be interesting after watching her videos. http://www.youtube.com/user/Vihart
It has to do with more than memory, right? At least if you want to come up with new things, whether directly in mathematics itself, or to use it creatively in some other field. The idea is that intelligence is partly indicated by how you can visualize things that cannot be seen, and often when someone cannot figure out a derivation of get at a result, it's more likely because they are not seeing the big picture.
I don't mean to say that people in general cannot be taught to see mathematics, it's just that they need good teachers and textbooks.
Edit: of course, all this comes only after having grasped the basic terminology at your fingertips.
Courant, in his book "What is mathematics?", says the field historically seems to go through alternating phases, from wild, creative, adventurous phases in which new things are discovered -- often inspired by "real-world" problems in physics or engineering -- to more sober, rigorous phases in which another generation takes over and basically cleans up the findings of their predecessors (making proper definitions, constructing and refining the proofs etc.).
But this is far from what math _education_ can be about.
The biggest obstacle to math education is the meme that mathematics is "only for intelligent people". Mathematics at the basic, cleaned up level has almost nothing to do with "intelligence" and a lot to do with memory. I know this first-hand because I used to be really bad at maths and then one day decided to learn it anyway (and did). Most often, when someone doesn't get some result or can't figure out the derivation, it's because he doesn't remember or has never learned the relevant facts. [EDIT: Also, if you think you need to be "intelligent" (however defined) to do math, you will give up sooner when you don't understand sth because you think "uh, I'm too dumb" instead of asking "what am I missing".]
Mathematics is a kind of language, and as in any language, most people will actually need to sit down and memorize vocabulary. It's mundane, it's not fancy-creative, it's a pain in your behind, but once you succeed in automating certain things, the process takes on a momentum of its own.
This "dragon mom" stuff I occasionally read about in the NYT is silly IMHO but it's just as silly for a 22 year-old, who by her own admission doesn't even know the standard curriculum of a plain vanilla mathematics education, to want to be "the ambassador of mathematics" (http://www.nytimes.com/2011/01/18/science/18prof.html). First you do the hard work, then you can fly.