The traditional method of mathematics education is not the same thing as the subject of mathematics. It can be taught with computers in a constructionist manner, it can be taught straight from Euclid (the old "traditional method"), it can be taught (poorly) with "New Math", or to use an example that should be familiar: it can be taught with bingo cards.
These are not all equivalent, obviously, but none has a monopoly on learning.
[Edit: I'd also point out that Japan and South Korea also have an extremely competitive and intensive educational environment (with correspondingly high rates of school refusal and suicide among students) and a huge amount of cultural buy-in that can't simply be transplanted. I believe these are much bigger factors in the relative performance of those countries than the differences in the content of their education.]