As a teenager who dislikes the current schooling system, what I am most curious about is actually why something like the "mathematical circles" do not exist in the west.
To some extent, a similar social group is formed by robotics teams in my experience. A dedicated teacher/coach and a bunch of people who like electronics can really get amazing things done. Why is this the case?
I am under the impression that such things do/can exist, but ad hoc. For example, if you were to take extracurricular lessons at a RSM or similar private tutoring business, you could/would probably find a number of similarly inclined teens who might be interested in creating a circle.
Similarly, my experience attending a science & math magnet school in the 90s was that -- basically mirroring my later experience in college -- a subset of the kids taking advanced math classes in high school naturally tended toward hanging out & studying/practicing/researching together.
More formally, there are tons[1] of local, state, regional and national math competitions that target elementary, middle and high school students, and -- just like robotics -- it's up to volunteers (teachers, students, parents) at the school level to decide whether to invest time & resources to create a local team/club.
When I was in school, we had MathCounts teams at younger ages, and Math Olympiad (and Science Bowl & Science Olympiad -- my team made it to the national event in Science Bowl, actually) in high school. I'm under the impression that this is pretty common, at least in urban/suburban areas.
Maybe you already know or could try to find other peers who like math, and see if you can get something started? I think there's a good chance that if you approached a supportive teacher with "my four friends and I want to start a math club, can you help us?", it would get the ball rolling. Perhaps you could put something up on the bulletin board or equivalent you kids have these days, data kiosks or hovergrams or whatever.
Oh boy did I try and try. Unfortunately, my school suffers from two problems: it is very small - so less potential candidates for both teachers and students, and it has a culture of anti-intellectualism; for some reason, everybody hates maths, thinks it is hard and so on. I mean I tried to do an engineering/robotics thing. Even that was unsuccessful. People were _afraid_? of doing things.
Probably cultural - if a society celebrates physical prowess (sports etc. especially for males) and physical appearance (fashion etc. especially for females) but doesn't celebrate intellectual prowess for either gender - often the opposite - then children, teenagers, and young adults won't view working hard to improve their mental abilities as something worthwhile that they will be rewarded for. It's a symptom of a fairly unhealthy society.
There are a great many historical reasons why this situation arose - aristocrats feared the rise of well-educated groups within the serf class that might challenge their power, and plantation owners famously forbid teaching their slaves how to read and write, and some of that thinking persists to this day.
Regardless, people who spend roughly equal time on developing both their mental and physical capabilities via deliberate (and time and energy consuming) practice are the ones that tend to turn out healthiest and happiest.
To some extent, a similar social group is formed by robotics teams in my experience. A dedicated teacher/coach and a bunch of people who like electronics can really get amazing things done. Why is this the case?