Here are interesting reviews of russian math circles. Friends who experienced such math circles (in the USSR and copies in other countries) told that it works great, even if one later decides for not becoming a mathematician.
Gromov expressed in his interview here some very interesting thoughts: "Now there are no more Abels... It means that they have been destroyed. The education destroys these potential geniuses – we do not have them! ... There are some experiments on animals which indicate that the way you teach an animal is not the way you think it happens. The
learning mechanism of the brain is very different from how we think it works: like in physics, there are hidden mechanisms. We superimpose our view from everyday experience, which may be completely distorted. Because of that, we can distort the potentially exceptional abilities of some children. ... There are very interesting experiments performed with Chimpanzee and Bonobo apes and under which conditions they learn, or even how you teach a parrot to talk. How do you do that? The major factor is that it should not see the teacher. You put a mirror between you and the parrot and then you speak behind the mirror. The parrot then sees a bird – it talks to a bird. But if it sees you, it will learn very badly. That is not an obvious thing. The very presence of a teacher, an authority, moves students in a particular direction and not at all the direction the teacher wants them to move. With all this accumulated evidence, you cannot make any simple decision."
Edit: Alexandre Borovik links in his blog to this very fascianating essay by Gromov: "We introduce a concept of an ergosystem which functions by building its ”internal structure“ out of the ”raw structures“ in the incoming flows of signals. ... We shall argue in this section that the essential mental processes in humans andhigher animals can be represented by an abstract universal scheme and can bestudied on the basis of few (relatively) simple principles."
In general, I guess one should have the time frame of brain development in mind: One 'developmental window' is ca. from birth to age 4, the other around puberty. In both, many new nerves/nerve-connections build up, but untrained ones are then deleted. Reports of recent studies suggest that pre-birth risks fro the brain like stress, lead exposure, malnutrition are important too and widely underestimated.
Comes the dawn of 'Algernon mathematicians'
Edit: Here is the link to an article about a project to identify "good teaching": "The most stunning finding to come out of education research in the past decade: more than any other variable in education—more than schools or curriculum—teachers matter. But we have never identified excellent teachers in any reliable, objective way."