The book mentioned in this blog post is a pretty good book. I like another book

http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773

for slightly more advanced math learners, including those I teach in the more experienced section of the math classes I teach in my town. Below is a set of quotations and links about math education I send out to parents of newly enrolled students in my classes:

LINKS ABOUT LEARNING MATH AND OTHER SUBJECTS

I was first introduced to a mathematician writing about how to teach math when a parent told me a decade ago about the article "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education,"

http://www.aft.org/pubs-reports/american_educator/fall99/wu....

by Professor Hung-hsi Wu. His writings have been very influential on my thinking about math education.

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/pubs-reports/american_educator/fall99/ame...

Another review of that excellent book

http://www.ams.org/notices/199908/rev-howe.pdf

is also food for thought.

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 math competitor in high school and now a publisher of math textbooks, among other ventures, has written an interesting article "The Calculus Trap":

http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calcul...

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 math learners to learn together.

Another good article about a broader rather than narrower mathematics education is

http://www.math.sunysb.edu/~mustopa/thurston_edu.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."

I recently read an issue of the MAA Focus newsletter of the Mathematical Association of America, and in the newsletter I saw a link to an article by a math professor

http://www.ams.org/notices/200502/fea-kenschaft.pdf

who is very concerned about the quality of elementary mathematics education in the United States. She provides many interesting examples of ways elementary teachers can think more mathematically about elementary mathematics and thus teach better.

A very interesting article about overcoming learning challenges and thriving from them

http://www.stanfordalumni.org/news/magazine/2007/marapr/feat...

reports the work of psychologist Carol Dweck, who shows how "growth mindset" can make learners smarter.