> The transition into geometry is particularly difficult for some. In the United States, the typical high school curriculum suspends algebra after one year to teach geometry, then resume algebra again, calling it algebra 2. Thus geometry starts and ends abruptly. In my opinion, this is not the best arrangement, but that’s what the higher educational power of the United States decided, and I only work here.<p>I took algebra 1 in 7th grade and geometry in 8th grade, I don't really remember anything from either class, not even the teachers really. Though admittedly I don't remember much from my other classes at the time either, the two most influential I think were a typing class that forced me (by punishing non-two-handers) to go from 30-40wpm with 'finger pecking' to 80-100wpm using both hands, and a nicely rigorous 8th grade English class.<p>Regarding the problem of arithmetic inability, I don't think it's that much a problem. The real problem is teachers thinking students should be just as proficient in the same areas as they were, learning the same things as they did, and that's where public math education fails the most. "I agree that the problem lies with the other people more than with the students. The most profound engine of civilization is the inability of a larger and larger fraction of the population to do the basic things needed to survive. Many people fail to realize this." <a href="http://www.theodoregray.com/BrainRot/" rel="nofollow">http://www.theodoregray.com/BrainRot/</a><p>I'm also disgusted with the way probability is presented, both here and in most classes.<p>I thought his last point about abstraction was made weakly, especially when he writes things like this: "Calculators become useless at this stage." Clearly someone's never used a TI-89. (But you don't even need that amount of power most of the time.)<p>Then stuff like: "This is the point at which studying by memorization fails." When in fact most calculus classes go about memorizing a bunch of derivative and antiderivative identities, formulas for Taylor polynomials, etc. There is very little abstraction in high school/freshman Calculus, and from my own experience and the experience of several teachers I've had people are usually only weak in Calculus if they're already weak in Algebra. (Which feeds nicely into his point about Foundations.)<p>The point that he hints at with the issue of infinities is better expressed with the issue of implication, both logical and probabilistic. Even if you can't actually add up all those fractions, following the math rules deductively leads to the implication that they do without having to go out and check. I think this problem can more easily be solved in the sciences, but that's another can of worms.
Classroom setting is really not the best setting for studying math. Studying new materials in math needs complete concentration, a 2-3 minutes mind-wondering can cost your the entire period, since new concepts/abstractions in math build on top of each other.<p>It's very difficult to concentrate for a 40 min period, especially considering students just came from other classes that could be just as mentally challenging.<p>I think Khan academy does a good job at this. A 15-minute long video is a much more consumable chunk of knowledge.