Everybody who is in graduate mathematics had a moment where they realized that mathematics was "their thing", and they decided to dedicate their academic career to it. I don't know of many people who decided this before their fourth semester or so, but I do know a few who have had a life dream of being a mathematician since high school (they went to some charter school or something). For me personally, I entered undergrad as a physics major, and when I took my first abstract algebra course (as a just-for-fun elective) in my fourth semester, I loved it, I felt this feeling of "How have I not seen this [abstract mathematics] before?" I had had 13 years of mathematics education - from first grade up elementary school through three semesters of calculus and differential equations in my first two years of college - and never once had I even heard of these abstract mathematical structures - groups (-> universal algebra, free term algebras, commutative algebra), topological spaces (-> algebraic topology -> category theory), analysis (-> hardcore set theory), combinatorics (-> finite set theory)... etc. etc. etc.
Why? Why had I never seen these before? Was it because I wouldn't have understood them before? I certainly don't think this is true. I think as early as middle school, but definitely by 9th grade, if I had been introduced (in a suitable "fun" way, perhaps) to basic set theory, groups (set with an operation), topology (set with some sub-structure), combinatorics (playing with finite sets)... I would have understood these, I think. The standard two-semester undergraduate series in these topics, maybe with a slight slow-down to accommodate the lack of "mathematical maturity", could be understood by an (advanced) high schooler.
I have heard of examples where a student's prodigous interest in math in middle school will get recognized by a teacher and they will get put on some magic track to math success - they will design a cross-curriculum with the local university which allows them to spend the mornings doing their rote high school graduation requirements - biology, history, etc. - but in the afternoons they will go to a class or two at college - usually number theory or something basic like that. In the extreme (e.g. Gatton Academy) the students are effectively attending college for the last two years of their high school career.
But this system is obviously completely ad-hoc cannot reach everyone. Back to me personally, I never met a single "real mathematician" until I came to college. I was a "smart kid" - I got a 5/5 on the AP Calc test, 35/36 on my math ACT, 800/800 math SAT, etc. I did all the things "smart kids" are supposed to do - in particular I was a member of the math club at my high school. But, what happens in a math club? Who runs the math club? Who selects the topics? I'll spare you the sob story: it was essentially an engineering club. All of the fun activities were fundamentally engineering - launching rockets, designing bridges, etc. etc. I did not learn any math in the math club.
I entered college and decided to be a physics major. Here for my first two years I took calculus 1-3, diff eq, matrix algebra. Even in these classes, it felt very rote and "computational" (possibly because this university's engineering department is so big that it dictates the curriculum of the calculus courses since 75% of the students taking them are in the college of engineering...? just a crackpot theory). The professors would drop subversive hints sometimes - "They don't really want me to teach you guys this, but I'm going to spend today talking about the epsilon-delta definition continuity. This will not be on the exam." - but even then, I only got the feeling that there was some secret world behind calculus in particular - still no inkling that there existed a grand structure of abstract mathematics underlying everything.
Anyway, all this is a vague rant to suggest that somehow there should be opportunity for smart, interested students to access "deeper depths" of mathematics - to know that mathematics is a field where people still do research today, and that, bluntly, this shit goes mad deep. Ideally at a very early level - the earlier you start, the further head-start you have on your finite lifespan. I am already 21 years old! (Galois already made substantial advancements in mathematics by my age, and even had enough free time remaining outside of mathematics to die!) If I had learned about this stuff in high school, I could have had a 2-4 year head start on where I am now! It makes me feel stupid by comparison to "what I could have been". I feel like a large part of my life was wasted (not the point of depression, of course, but it is a nagging thought I've been having recently).
[Of course, it could be argued that such a structure would be impossible to implement without forcing the "uninterested" students through so much math that "they will never need in everyday life". After all, only a very small percentage of the world's children grow up to be abstract mathematicians.]
[Hypothetically the argument could be made, "Seventh graders just aren't ready for abstract math - they still have to learn about adding and multiplying and fractions and slope of a line in the plane." I don't believe this for a second. All it takes is a sufficiently strong teacher-student connection (which can often be obtained by simply a good enough or passionate enough teacher).]
What is the education-theoretic position on this phenomenon? Are there any avenues in place to combat this sad occurrence?
