Uh, look, guys who didn't like my post: If you read carefully and think a little, just a little, not much at all, just a very little, you will see that my post is about the only item, ray of light, for the concerns of the OP.
Look, guys: How the heck to get the students with native ability, but otherwise disadvantaged, doing well? Well, as in the OP, that's tough.
But, but, but, but, but, but, if we think a little, just a little, and actually read my post, just a little, guys, not much, we will see that I offered a solution.
And the solution is, and may I have the envelope please? Here it is: Go for math and learn to work the exercises. By that, let me clarify -- work the exercises and learn to do that. Now to be more clear, in the relevant math books, there are exercises, lots of them, and the good news for students is to learn to work them, not the students, the exercises, the exercises in the books. Did mention the exercises? Look, guys, it's the exercises. Let me spell it out for you -- e.x.e.r.c.i.s.e.s. Clear enough?
No tutors, charter schools, special schools, YouTube, courses, credits, grades, teachers.
Q. How to do that?
A. First, get a book. Second read the book. Third, and may I have the envelope, please? Here it is: Amazing. Work the exercises.
Q. How is a disadvantaged student to know this?
A. Uh, I posted it here, and maybe they read it here.
Q. But, but, but, how is a disadvantaged student to be able to work the exercises?
A. Uh, did I mention (1) get a book and (2) read the book? Sorry, I intended to mention those two.
Q. But, but, ..., how can they do anything in a poor school with a disadvantaged background?
A. Uh, did I mention work the exercises? I thought I mentioned the exercises.
Q. But, how can the disadvantaged students do that in a poor school with poor other students, with poor or no good teachers in math, and poor or no good math courses?
A. Gee, I thought I covered that here: (1) Get a book. (2) Read the book. (3) Work the exercises.
I guess I've been just too brief; I needed to write a 1 million byte PDF with 90 chapters with the first 30 on "Get a book", with the next 30 on "Read the book", and the last 30 on "Work the exercises".
Q. But, but, but, how, how, how, again, again, again, without a teacher?
A. Uh, I thought I covered that: (1) Book. (2) Read. (3) Exercises.
Q. Impossible.
A. BS. Instead, that's not impossible. Indeed, it is a good way. But no sense in my insulting a good idea -- it's the best way. I'm still insulting a good idea -- for my second step, research, it's the only way. Or, for this last, how many full profs do you see in classes trying to learn? None. They teach themselves, alone, no courses, credits, grades, teachers, books, or exercises. So, in fact, learning math is possible even without a good book and exercises.
Look, guys, that's how I learned math. Yup, there were some courses. But in grades 9-12, I mostly ignored the teachers. Instead I did (1)-(3) -- we remember those. Indeed in grade 10, plane geometry, the teacher was really ugly and offensive. So, in class I ignored her and had my head down sleeping. She thought I was a bad student. BS. On the objective state exam, I was 1-2 in the class. At my high school, of 1-2-3 on the Math SATs, I was 2.
In plane geometry, why? I worked the exercises, not the trivial ones she assigned, not the easy ones in the main part of the book, but the more difficult supplementary ones in the back of the book, 100% of them, about 10 a day. All of them. One took me from Friday afternoon to late Sunday. In class on Monday the teacher couldn't work that one; neither could anyone else.
Calculus? I went to a college I could walk too. They wouldn't let me take calculus. So, I (1) got the book, (2) read the book, (3) worked the exercises. Alone. No teacher involved.
The next summer I was taking a course in German at a good college with a good math department I was to transfer to. There was one exercise in the calculus book where I did want to ask a prof. So, I walked to the math department and asked the first math prof I'd seen. He was willing to help. Fine, one exercise. I explained I wanted to start as a sophomore that fall with their sophomore calculus, and he gave me a fast oral exam and concluded I was well prepared. Then he said that he couldn't give me credit for the freshman course without my taking it. I told him I wasn't asking for credit and just wanted to start on their sophomore calculus. I did. Made As. So, actually, I never took freshman calculus. Taught it, learned advanced calculus, applied it (once saved FedEx with it), published peer-review original research in it, but never took it.
It went on that way: Sometimes the courses helped a little, but the secret was (1)-(3). As a college senior I got a copy of the challenging Kelley, General Topology, got a reading course, used (1)-(3), gave a lecture a week to a prof, one week the material, the next the exercises, through the book. The prof helped me with NOTHING. I did fine.
Later I did much more using (1)-(3). Then I went for a Ph.D. at one of the world's best research universities. On the qualifying exams, I did the best on three of the exams, the three where I'd taught myself.
Look, guys, if a student in high school uses (1)-(3), does well with that material, wanders into the math department of a college or university, picks the first math prof they see with some gray hair, asks a few questions, shows that they can work the exercises I listed together with others in relevant books, then, dollars to donuts, they, presto, bingo, just got a scholarship as a math major at that college. No damn joke.
Look, guys, that was all 99% fully clear from just my first post.
Look, guys, some students with disadvantaged backgrounds, with no books or teachers, do really, really well at basketball. Somehow they do.
If such a student wanders into the gym at a university, stands at the three point line, and tosses in, nothing but net, 9 out of 10, can dribble, can run, can run the basic plays, then guess how they get through college? Sure, presto, bingo, they just got an athletic scholarship.
Are we learning yet?
The OP was struggling with such issues, and I just explained that for math there is a solution, (1) get a book, (2) read the book, (3) work the exercises.
What books? Okay. In high school, five books: First year algebra. Plane geometry. Second year algebra. Trigonometry. Solid geometry. There are plenty of good books, old, used, cheap.
Still in high school, plus one more:
Calculus. College calculus. IMHO AP Calculus is junk, written by people who didn't understand calculus. So, do college calculus. The book should have some analytic geometry in the beginning -- that's fun material so learn that, too. There are plenty of highly polished calculus books, old, used, dirt cheap. Learn calculus from such a text. To do still better, also work about half the exercises from two other such books. Do that while still in high school, as soon as get through trig and solid.
Then take the SAT that tests knowledge in math. Get at least 700.
Walk to a college, walk into the math department, with the SAT paper showing that SAT math knowledge score, with the rest I mentioned, and get a scholarship as a math major.
How to pay for the SAT tests? Write up very neatly the solutions to the exercises I posted in this thread, mail them to James Simons at Renaissance Technologies, East Setauket, New York, and ask him to pay the Math SAT fee for you. Dollars to donuts, you will get the fee. For some of those exercises, even Simons may not have seen them or even know where to find them -- no joke, trust me!
Then in college, and no law against doing these in high school, books abstract algebra, linear algebra, advanced calculus, measure theory and functional analysis, high level probability based on measure theory, intermediate statistics and then high level statistics. Will then know the statistics in the OP much better than the author of the OP. Maybe also do optimization, linear programming, network linear programming, non-linear programming and the Kuhn-Tucker conditions. For more, read anything by D. Bertsekas at MIT and D. Luenberger at Stanford. For linear algebra, start with, say, Hoffman and Kunze (free as a PDF on the Internet) and, then, by all means, Halmos, Finite Dimensional Vector Spaces (written at the knee of von Neumann). Take the Math GRE, get 800 (I did), and apply to math grad school. Expect to go for free. I did.
For some of the exercises I posted, will have to cover much of this college material first.
Do some research, publish it, see if you can call it your Ph.D. dissertation. I did that, independently, but used something else I'd done, alone, independently, before grad school for my dissertation.