Ideally, how would math be taught?

Rather than "memorize formula, solve problems, tommorow add more to formula, solve more problems, repeat," how should math ideally by taught?  What kind of teacher would you need?  What kind of student interaction would there be?  And are there any close examples from other countries or societies, past or present?

Well, countries that tend to have high scores in mathematics and science typically teach the practical applications of math rather than concentrating on theorems and whatnot, or, even worse, dumbing down mathematics by completley ignoring material that students may find difficult, such as "word problems" or irrational numbers.  In fact, I occasionally voulunteer at my local community center as a math tutor, tutoring 7-10th grade math, and I consistently find that very important subject matter is often skipped, or at the very least simply glanced over.  Word problems are very rare until 10th grade algebra, in the 8th grade remedial math class, 471/23 is 20, remainder 11, not 20.47826086956522..., (and remains so until the end of the year) and in practically every algebra class, students cover material such as polynomials, rational expressions, etc, without being explained even the first thing about what they are, or what they are for.  Analytic Geometry is only discussed by students taking the honors algebra course (about 20 kids out of 500 a semester), so while these kids might be able to factor (x+2)(x+3) into a second-degree polynomial, they wouldn't be able to tell you the characteristics of its graph.  When I tutor these kids, I try to cover this material for them as best as I can, and I often see marked improvements in the few kids who are willing to listen to me talk about it, but for the majority of them, they're simply victims of the typical American school district.