Sometimes personal, sometimes a bit about science or math or politics or whatever.
There are a lot of things here I agree with (e.g. the overemphasis of rote memorization and under-utilization of exploratory approaches) but that's no fun to talk about. I have too many things I could discuss to be thorough, so instead I will bring up three points:(1)Let's say we are happy to accept that mathematics is an art and that mathematics education should therefore resemble art education. I'm not sure how much the author knows about art education. (To be fair, I also don't know a lot about it.)We don't do much in the way of art education in the K-12 universe. I'm familiar with three basic types of courses - the futz around and draw/paint/etc. stuff with no expectation from anyone that the teacher is going to particularly help you improve (and art is produced that generally only a parent could love); the art/music "appreciation" course in which the student is fairly passive and perhaps learns a bit about art history; and the courses (like band) that run more like traditional math classes in that they emphasize technique.There is not a widespread demand for significant time given to art in school because (1) it isn't (perceived as) "useful" either in applications to fixing society's problems or in improving a student's future job prospects, (2) kids start drawing, singing, dancing, etc., before they ever start school and parents will pay for their kids to continue classes outside of school, and (3) people do not perceive creativity as something that can be taught.So basically, it seems to me that virtually all artists develop their skill through a combination of self-study (e.g. through looking at art, playing with it) and classes outside school.It is my impression (others may be able to speak to this with more information/authority) that the more serious the art school is about training real artists, the MORE emphasis placed on developing technique and the LESS emphasis placed on creative exploration. For example, a student is accepted to the ultra-elite American School of Ballet in NYC. Are they asked by their instructor: What do you think a snowflake would dance like? How about a swan? I think not. Indeed, such a student has many years of grueling, repetitive work ahead to master technique, develop stamina and facility, and to embody the aesthetics and values of the faculty. How many great dancers did not undergo this sort of process early in their training? Art teachers are also well-known for their bluntness in telling a would-be artist "You have no talent; give it up." I don't think this is what he's talking about.The visual and performing arts are, generally speaking, quite accessible to the public at large. One needn't be trained in the arts to enjoy Mozart, Matisse, and Swan Lake (and Michael Jackson). Some works are less accessible than others, of course, but the point is, people with no exposure to art curriculum could still enjoy a lot of things through a perhaps predominantly emotional reaction. I am not aware of any human culture that did not have some kind of art (anyone?). But math - even the kind of math he's talking about? For all his talk about various proofs being elegant, charming, etc., I'm not sure this kind of "intellectual art" really falls into the same category.to be continued...
(2) He doesn't think that it's possible to teach teaching. Where are all the new math teachers going to come from? How do we find the people to teach the very first wave of "math as art" if current math teachers can't do it? I mean, great, he's personally a math PhD who has dedicated himself to K-12 education. Who else is there? Kids who do well-enough in today's "confused heap of destructive disinformation" to get quant-related college degrees can walk into jobs that pay good money. Kids who do well in his fantasy math curriculum are going to be willing to walk away from the jobs they can get to become K-12 teachers? God, esp. K-5!(3)This brings up a point Robert made that cracked me up: A mathematician who had an Erdos number of 1 as a college dropout, and whose entire K-12 teaching experience is in a super-elite private school entirely dedicated to creative learning, is a questionable source for talking about what is wrong with and what is viable in K-12 education. Personally, I think it's great that he can present a nice proof developed by one of his 7th grade students...at his ridiculously non-representative, mega-elite school. I would be a lot more impressed with this if he were working with anything approaching a normal student population. If he had experience in public classrooms, he might not summarize the students as saying "math class is stupid and boring" but "math class is stupid and boring and hard."Oh, and one thing I was curious about:(4)What's this guy's take on the math curriculum in places like China and India? Don't they have even more strict, conscribed programs than we do? Have they been unsuccessful at producing mathematicians? How about Europe? Russia?
I think my overall (mostly uneducated) feeling on math education is that there needs to be a great emphasis on thinking your way through problems. Kids need to actually have the experience of trying to solve problems without having been told how sometime in their math lives. It is kind of the math equivalent of reading from primary sources in history or coming up with ideas for essays in English.But I don't see how to structure a whole curriculum around his ideas.
There is a large gap between "great emphasis on working your way through problems" and what he is suggesting, IMO, with a lot of people in the mainstream of math ed working on approaches that incorporate/emphasize exploration. Also, I know his article is several years old, but I believe that some of the specifics of his rants are no longer true, such as the use of that particular kind of formal proof in h.s. geometry courses.I can't decide, on balance, whether I think this sort of extremist rant makes the more moderate reformers seem sensible by comparison or if it creates the impression that people who support more exploration in math teaching are clearly bonkers.
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