For the sake of example, how was Soviet Science high school different from Stuyvesant/Brooklyn Tech/Bronx school of Science? (Just to name the ones I know of).
I am not very familiar with these schools, I am in CA. I think the biggest factor for me during my high school years was several hours per week of 1-1 discussions with currently active mathematicians and grad students. While formal math is really hard to read, a lot of ideas are very elegant and actually simple. It’s hard to find these ideas in a formal math text, but you can explain a lot of things in 1-1 setting while still allow student to “discover” core concepts on their own. Mathematics is not about solving 100 problems of one type ;)
This is what infuriates me about college level mathematics from about 12 years ago at least. Why can't they also explain it like they do in those 1-1 sessions instead of making calculus textbooks that assume you know set theory before set theory is taught in some of the worst most obtuse writing there is.
There are good reasons why formal texts are important but I can’t agree more that this is not a good way to teach math if there is nothing else available. But you also need to learn how to read these texts at the end because this is the way mathematicians communicate to each other ;)
Re 1-1: the biggest advantage is that students can rediscover things. It’s one experience to read or listen to a proof, and it is completely different experience when you discover it yourself with small hints from the teacher.
I don't think formal texts should go away, just things equivalent to those well done, plain spoken, 1-1 explanations need to be right beside them in the textbooks. They often are not.
Also math tends to have a documentation problem that could learn a lot from software engineering. Single letter variables are not acceptable in most coding for a simple example.
And another point, high school and college level math is coming from the 18th and early 19th centuries. All of it can be explained without formalization from 1920-30 and students for non-mathematical degrees will understand it much better. I can explain sets to elementary school kids in 10 minutes ;)
The very point of 1-1 sessions is to start with an awareness of your existing knowledge background (set theory or not?) and to aim for real mastery in the most effective way. Unfortunately 1-1 sessions don't scale, while at least khan academy does.
I know lots of UC math/cs profs who meet 1-1 with high school students every week. Currently, the process is pretty informal, where local high school students just knock on the doors of professors until one agrees to regular meetings. UC Riverside is in the process of opening up a high school directly on the campus grounds to make the process more formal.
I'm not familiar with Stuyvesant, TJHSST and the like. The best Russian schools follow the Konstantinov [1] tradition of "listki" (handouts), where each student is essentially guided to re-invent the math curriculum by solving problems. There're no textbooks or lectures. You solve problems from the handouts at your own pace and discuss them with a TA when ready (typically volunteers from among former graduates of the same school).
And another factor, also coming from 1-1 tutoring, is that nobody waits for others. You can move ahead as fast as you can and actually you get more attention if you are ahead of class. As the result, while everyone knows the same fundamental theorems, some students have a chance to learn A LOT more.
