Thanks for the further replies. Here's another FAQ section:
3) LINKS ABOUT LEARNING MATHEMATICS AND OTHER SUBJECTS
I was first introduced to a mathematician writing about how to teach elementary mathematics when a parent told me back in the twentieth century about the article "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education,"
http://www.aft.org/pdfs/americaneducator/fall1999/wu.pdf by Professor Hung-hsi Wu. His writings have been very influential on my thinking about mathematics education. In June 2010, I had the privilege of meeting Professor Wu in person at a teacher training workshop in St. Paul, Minnesota.
EXECUTIVE SUMMARY: The simple things in mathematics are the hard things. Any learner of mathematics, and more generally any learner of any subject, has to know the foundational principles of the subject thoroughly, and there is always more to learn about how the "basics" fit together to make ideas.
A link that furthered my process of pondering how to teach mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma.
http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf Another review of that excellent book by mathematician Roger Howe
http://www.ams.org/notices/199908/rev-howe.pdf is also food for thought. In some countries, elementary mathematics is not considered "easy" mathematics, but rather fundamental mathematics, which must be understood in full context to build a foundation for later mathematical study.
Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
http://www.amazon.com/gp/product/0387982892/ Richard Rusczyk, a champion mathematics competitor in high school and now a publisher of mathematics textbooks, among other ventures, has written an interesting article "The Calculus Trap":
http://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap& I particularly like this article's statement,
"If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. Students of like interest and ability feed off of each other. They learn from each other; they challenge and inspire each other."
which is one reason to encourage able mathematics learners to learn together. I had the privilege of meeting Richard Rusczyk twice in the summer of 2010, once at a Summit of Davidson Young Scholars program participants, and then again at the Minnesota State High School Mathematics League coaches conference. Rusczyk thinks it is crucial for bright students to avoid the "tyranny of 100 percent," in which they only get school homework assignments that are easy enough to do perfectly. He thinks it is very important for the development of young learners to face problems that are hard enough to challenge a learner, so the learner learns how to persist in problem-solving and not give up too soon.
Another good article about a broader rather than narrower mathematics education is "Mathematics Education." Notices of the American Mathematical Society 37:7 (September, 1990) 844-850.
http://arxiv.org/PS_cache/math/pdf/0503/0503081v1.pdf by William Thurston, a Fields medalist.
"Another problem is that precocious students get the idea that the reward is in being 'ahead' of others in the same age group, rather than in the quality of learning and thinking. With a lifetime to learn, this is a shortsighted attitude. By the time they are 25 or 30, they are judged not by precociousness but on the quality of work."
Thurston explains why a broad mathematics education is useful in helping mathematics research advance in his article "On Proof and Progress in Mathematics." Bulletin of the American Mathematical Society, 30 (1994) 161-177.
http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf Timothy Gowers, a mathematician who is both an International Mathematical Olympiad gold medalist and a Fields Medal winner, wrote "The Two Cultures of Mathematics"
http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
to point out that both solving problems and building and understanding theories are important aspects of mathematics. The development of mathematics is limited when mathematicians only do one or the other.
Terence Tao, answering the question "Does one have to be a genius to do maths?"
http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ says, "The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities."
