I didn't see anybody give their experience with the "new math" experiment that was done a while back. If you think about this from a little distance you can see that there are two camps in mathematics education, the "Creatives" and the "Pedantics", the "new math" was an attempt by the "Creatives" (who Lockhart is clearly a member of) to inject "thinking" and "creative thought" into the mathematics curriculum. It was a total bust, primarily because the teachers teaching it really didn't understand the intentions behind this new curriculum and they reduced it to rote. Those in favor of "back to basics" would be in the "Pedantics" camp and have been making a comeback recently.

So here in a nutshell is the two opposing camps arguments.

Creatives argument against the Pedantics -- The Pedantic curriculum is a soul destroying exercise in rote and memorization leaving no room for a child to feel any inspiration or creativity.

Pedantics argument against the Creatives -- The Creatives assume the world to be filled with inspired teachers that won't reduce any curriculum to a pedantic exercise. If the quality of teachers is such that they can only teach pedantic material, you might as well have the children learn something useful and constructive even if it is boring and soul destroying.

I am an ex-mathematician and I am firmly in the "Pedantics" camp. I hate to see children that cannot add two digit number to two digit numbers without a calculator. That is the world that well meaning "Creatives" create.

Also, is there really that strong a correlation between the percentage of students that pass standardized tests on calculus and the overall success of the community? Russia has a very strong educational system, see what that got them. The general population of the U.S. would be considered be woefully uneducated by the standards of many other countries. But if you were to take any country with as large an immigrant population, I suspect you would see similar numbers. Over time the immigrants are absorbed into the main stream and their children do better. But could it be possible that these immigrants are also the source of the vitality of the U.S. economy and their education (or lack of it) is not the primary reason for why they make this nation so successful?

I always wonder what the parents are thinking when they push their kid through the entire K-12-BA school system at such a fast rate. Do they really believe that the kid is better set up for life at that point then if they were to take their time and send the kid to a better college where the kid would get an early exposure to professors who are leaders in research in their respective fields? Better yet, have the kid skip out on the normal curriculum and find some more challenging instructional texts that the kid can do with remote supervision by a professor at a local college.

There is a quote that I remember from one of my professors in college. When talking about a particularly bright kid, he said "that kid is sufficiently bright that our classes and curriculum are more of a hindrance than an aid". The implication is that accomplishing a BA degree is many times more about grinding through material than something that shows a real difference between those who are truly educated than those who are not. I can still remember Jay Leno's "Jay Walks" where he would ask college graduates questions that many (admittedly unusually smart) 12 year old kids (still not in high school) without a BA could answer easily.

I have read enough biographies of those who end up with great achievements in their fields (Einstein and Newton being good examples). Almost none of them had abnormally accelerated the normal learning track. They almost always just side-stepped it. For example, in his teen age years Einstein was thinking thoughts that only specialized masters of those fields would be able to fully comprehend. Did his track record in school reflect this? No. (And if you follow Newton as a model, then you should force your child to be a farmer for a while so that they really will do anything they can to avoid that fate).

This common fallacy comes up again. Poker is not just an estimation of probabilities. Poker is a raw and pure example of a very complicated multi-way "game theory" problem. Such problems are far from NP-complete (not computable in polynomial time relative to number of inputs on a computer) and much more interesting than they might first appear. Nobel prizes in economics have been given for insight into such problems. There is a book called "Mathematics of Poker" (those interested can do a Google search) which uses Poker to introduce game theory. Please read that before trivializing the theory that underlies poker.

A good poker player models his or her opponents and then creates optimized exploiting strategies -- strategies which are far more sophisticated than you might first imagine (once you have started giving up chances to raise up pots with strong hands in order to catch bluffs you have started down the first step of this long road). This optimization takes into account that your opponents are trying to exploit you and how well you believe you have misled them. The skills to master this are not too different from the skills required by a chess grandmaster (the analogy is not exact -- poker is more like a large look up into a vast reservoir of experience than a deep computational think, but an expert can beat up an non expert very quickly -- even a non expert who knows all the odds).

The one thing I found strange about String theory is that it made Physicists study Algebraic Geometry (with Sheaves and such). Algebraic Geometry got started as a field when mathematicians tried to link up the algebraic properties of polynomial equations (what "algebraic solutions" does it have is one of the questions you might ask) with the differential/topological (how much curvature does it have, how many "multi-dimensional" pseudo-holes does it have -- think about the questions of curvature and "holes" you might ask about multi-holed doughnut in many dimensions -- this is a gross simplification, but I am grasping for intuitive analogies).

What I remember about Algebraic Geometry is that it was one of the harder fields of study in all of mathematics and only a few mathematicians in the world could wield the theory with any real authority and skill (Faltings is a famous such Mathematician). At the time it made me worry that maybe humans would never be clever enough to truly figure out the rules of the universe. Because if we are already have to understand some Algebraic Geometry to get a handle on the current most respected "theory of everything", what would happen if the "theory of everything" required one level of abstraction complication beyond that? There has been a constant progression of theories in Physics from the less abstracted to the crazier highly abstracted (quantum mechanics and general relativity already can only truly be understood by at most a few hundred people in the world). Maybe this time we are going beyond the ability of us poor human mortals to understand.

In defense of String theory, though it may give no predictions, it does give those who study it a feeling of "enlightenment", as if they are getting a potential intuitive understanding of how the universe is put together. Studying mathematics in general can create such a feeling (I think in general that is why mathematicians love their field of study), but it is way cooler to think that the theory and the real world might have some linkage. Also, from what I understand, competing theories all have the feeling of artificial glitchy repairs to existing theories without granting much enlightenment. If you give me a bunch of data, I can create an equation which will spit out the data. But if the equation does not offer insight into the nature of the data (for example, you cannot see that it is actually a "repeating wave pattern of visual distortions"), then though it may be useful, it really does not offer much in the way of "enlightenment".

Some may view the usage of the word "enlightenment" as an allusion to some type of religious feeling. That may be, but it is NOT connected to any type of statement that could be read as "this vision that I see is true and those who disagree with me are morally inferior beings and will be viewed as a lesser person by the higher powers that rule the universe". In fact I suspect that those who disagree with me about the worthiness of understanding mathematics may have spent more time worrying about their morality (as opposed to their "faith") and may actually be superior human beings (and may be viewed as such by the higher powers in the universe).