The more important difference between mathematics education in Japan and mathematics education in (say) the United States is how hard the problems are and the encouragement to pupils in Japan to try to figure things out for themselves.
I put instructional methodologies to the test by teaching supplemental mathematics courses to elementary-age pupils willing to take on a prealgebra-level course at that age. My pupils' families come from multiple countries in Asia, Europe, Africa, and the Caribbean Islands. (Oh, families from all over the United States also enroll in my classes. See my user profile for more specifics.) Simply by benefit of a better-designed set of instructional materials (formerly English translations of Russian textbooks, with reference to the Singapore textbooks, and now the Prealgebra textbook from the Art of Problem Solving),
the pupils in my classes can make big jumps in mathematics level (as verified by various standardized tests they take in their schools of regular enrollment, and by their participation in the AMC mathematics tests) and gains in confidence and delight in solving unfamiliar problems. More schools in the United States could do this, if only they would. The experience of Singapore shows that a rethinking of the entire national education system is desirable for best results,
but an immediate implementation of the best English-language textbooks, rarely used in United States schools, would be one helpful way to start improving mathematics instruction in the United States. There are accessible descriptions for teachers in the United States of the system in Singapore, for example, "Beyond Singapore's Mathematics Textbooks: Focused and Flexible Supports for Teaching and Learning" American Educator, Winter 2009-2010 pages 28-38.
begins with the statement "In American schools, mathematics is taught as a dark art. Learn these sacred methods and you will become master of the ancient symbols. You must memorize the techniques to our satisfaction or your performance on the state standardized exams will be so poor that they will be forced to lower the passing grades." This implicitly mentions a key difference between United States schools and schools in countries with better performance: American teachers show a method and then expect students to repeat applying the method to very similar exercises, while teachers in high-performing countries show an open-ended problem first, and have the students grapple with how to solve it and what method would be useful in related but not identical problems. From The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999):
"Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x
"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77
A great video on the differences in teaching approaches can be found at "What if Khan Academy was made in Japan?"
Tokenadult, I always enjoy your well-informed comments on topics such as this. I think you may have misinterpreted Stigler's 1999 "The Teaching Gap", though, as many of us did. In that book, he reports on a study of math teaching in the US, Japan, and Germany, and finds Japan's results to be far superior to the others and their teaching methods very different, and different in exactly the way you describe.
But he did a followup study involving more countries to see if most or all high-performing countries used the Japanese approach. It turned out that they did not. Some were more like the US than the US.
Here (http://timssvideo.com/sites/default/files/Closing%20the%20Te...) is one place where Stigler reports his updated findings and attempts to debunk your (and my) initial conclusion---a conclusion that seemed strongly justified by his 1999 book---that, as you posted above, "The more important difference between mathematics education in Japan...and the US is...the encouragement to pupils in Japan to try to figure things out for themselves."
He points out that another high-performing country in his second study, Hong Kong, was more US-like and less Japan-like on this spectrum than the US itself. On the dimension you're calling "more important", the low-performing US is between the high-performing Hong Kong and the high-performing Japan.
His conclusion was that the main factor was not having kids figure things out for themselves but having teachers carefully teach kids the relationships among things. It didn't matter if the US kids spent time practicing procedures. The Chinese kids spent MORE time practicing procedures and did better, but then the Chinese teachers spent time directly pointing out important relationships, which the US teachers didn't do much of. The Japanese kids had to spend a lot of time figuring things out for themselves, but then the teachers would gather them together and carefully lead them to see relationships that they hadn't seen when working by themselves. The US teachers would tell kids to figure things out for themselves and basically leave their learning to whatever they managed to figure out.
Given equal IQ, time on task, etc., it's the effectiveness with which mathematical relationships are made clear to the students (part of which requires significant procedural drill, which Japanese kids do after school) that matters most. A lot of time is wasted in the US doing procedural drill with no conceptual understanding, with even more wasted on constructivist "discovery" methods whereby kids are supposed to somehow teach themselves and each other the mathematical relationships, and all of this led by teachers who aren't required by their union to even know anything about mathematical relationships much less teach them.
Thank you, SiVal. I didn't see contact information in your user profile (and indeed the contact information in my user profile is rather subtle until I do a personal website update). So here I will say thanks for your comment. I'll be revising some FAQs based on what you wrote. Feel free to contact me off-site if you'd like to discuss these issues more. (Much of today I am updating my personal website on its seventeenth birthday, and then I'll have to finish a revised FAQ promised to another participant here a few days ago, a response to a link that shows up too often in discussions on the topic of international educational comparisons.)
I know that there was a previous link on HN about the math education approach and materials used by Phillips Exeter which very much followed this approach. Where the students are allowed to discover the relevant mathematics in a structured way so that by the time they learn the "standard" way of expressing a line they understand intuitively why it must be so.
It was very interesting, and actually quite similar to the way I was taught physics in university tutorials - I think I've commented on HN before about the way we "discovered" an approximation similar to Born-Oppenheimer as part of a cleverly designed series of small group tutorials.
edit: I just noticed that the previous link was in fact posted by you.
Regarding the "dark arts" quote: not my experience. I'm not present at my kids lessons but the homework indicates a wide variety of approaches, methods and applications are used. The math textbooks are much more interesting and varied than the ones I had as a kid.
Teachers in the U.S. who expect students to struggle with math problems instead of just giving them answers after the student puts in the most minimal effort can get in big trouble for doing so. (Source: My math teaching career.)