Thanks for the many interesting comments. I'll reply jointly here to a few of the comments, and invite further discussion on some points made in the submitted article that I'm curious about.

From a top-level comment:

http://news.ycombinator.com/item?id=4084559

As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)

From a second-level comment, which has already received some helpful replies:

http://news.ycombinator.com/item?id=4084426

After Vector Calc, I wanted to go back to the fundamentals, to understand instead of remembering.

There is a FAQ page on the Epsilon Camp site

http://www.epsiloncamp.org/FAQ.php

that includes some Frequently Asked Questions articles about learning mathematics for deeper understanding. The FAQ article "Problems versus Exercises"

http://www.epsiloncamp.org/faq/faq_1.php

relates to what kind of work to set for yourself to build deeper understanding, and the FAQ article "Learning Mathematics"

http://www.epsiloncamp.org/faq/faq_3.php

points to writings by various mathematicians, including the book Numbers and Geometry by John Stillwell, about how to appreciate mathematics as a deep, connected subject.

The submitted article mentioned "Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000)." The most striking example of this that I remember from a news report was a Wall Street Journal series in the 1990s that followed two young men in an inner city ghetto, one who was a good high school student and the other who was a street criminal. The street criminal usually skipped high school, but happened to show up the day students could take one of the major standardized tests (probably the PSAT, if I remember correctly). The street criminal, who sold illegal drugs among other activities, scored just as well on the test as the more regularly attending student who had learned most of his mathematics from school lessons. That's a rather stark illustration of what's missing in school lessons for children who don't have an outside-of-school environment for learning mathematics.

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

The article also says that many students say, "You have to be willing to accept that sometimes things don't look like - they don't see that you should do them. Like they have a point. But you have to accept them." I wonder how that relates to the quotation attributed to John von Neumann,

http://en.wikiquote.org/wiki/John_von_Neumann

"Young man, in mathematics you don't understand things. You just get used to them."

And from a third-level comment:

http://news.ycombinator.com/item?id=4084865

I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me.

I ask, because when I studied mathematics in school, I had a drive to understand the general principles first before I launched into working on my homework, while some of my classmates were successful--at least in the context of school--by working on the homework and DEVELOPING some level of understanding as they tried to figure out answers for the homework. (I was in a "tracked" mathematics class, taking algebra in eighth grade in an era when most Americans took algebra in tenth grade, if at all, and most of my classmates had parents who were engineers or medical doctors and could ask their parents for help at home if the school lessons were confusing, as they often were.) I also have a very strongly visual approach to grappling with mathematical problems. So when I first learned algebra, which was presented to me as a bunch of "Do this to the equation, and then do this" with little rationale, I found that very dissatisfying. Later in the school year, we learned about coordinate graphing of systems of equations in the Cartesian plane, and I remember thinking, "Why didn't you tell me this in the first place?" For historical reasons, and perhaps for reasons of what most learners consider most easy, usually purely procedural algebra for solving systems of two equations in two unknowns has been taught in school before graphing systems of equations in the coordinate plane. But for some learners, it would be easier and more accessible to reverse that order. What do you think about the issue of students working first according to instructions, to DEVELOP understanding a la the von Neumann quotation, versus getting the "big picture," perhaps explicitly visually, before working on problems.

I'll comment also that the approach taken to learning mathematics in school in most of the newly industrialized countries of east Asia and southeast Asia is plainly superior to the United States approach for at least two reasons:

1) the school textbooks in those countries explicitly encourage students to THINK about why a procedure will or will not work, and about how many different ways there might be to solve a problem, and

2) the school textbooks show multiple representations of most mathematical concepts, building from "concrete to pictorial to abstract" as in the Singapore Primary Mathematics series

http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....

and the follow-up New Elementary Mathematics series

http://www.singaporemath.com/New_Elementary_Math_s/47.htm

which interleave arithmetic, number theory, geometry, and algebra in increasing depth and interconnection throughout all grade levels.