My writing was fine! It was effective! My first post on that topic was too short. My "rambling" -- actually it was well enough organized -- worked because yo

My writing was fine! It was effective! My first post on that topic was too short. My "rambling" -- actually it was well enough organized -- worked because you understood it just fine. As I mentioned, to solve some of the exercises I gave, will need the college material. That one about a convex polytope is an example. It turns out, that is not an easy exercise. In particular, surprisingly, it does not yield to the usual ideas in, say, W. Rudin, Principles of Mathematical Analysis. My first post was partly an exaggeration: Anyone who can work all those exercises might just be admitted to, say, second year in a Ph.D. program. The exaggeration was so extreme that, sure, anyone who can work those definitely has done a lot of independent reading and is in nearly all respects quite far above college level material. The questions you asked about the books do need to be answered, but to save length mostly I omitted answers. Actually, the omitting is okay: Finding good books is part of the work. So, how to do that? Sure, for the high school books: (1) do an Internet search and see what the most popular/recommended books are/have been. (2) Go to the local Board of Education, talk to their subject matter specialists, see what books they recommend. (3) Go to high schools recommended as being really good and see that books they are using. (4) Anything else can think of. For your concerns that a student could get stuck on the exercise about convex polytopes, they likely won't encounter such a problem because, as I warned, it needs some college material. But maybe the challenge of the problem would motivate a determined student to do what they could to get a solution, and that might be good. For that question about the high school material, that material is so simple and the best books should be so well written that there should be no problems -- I never had such a problem in first year algebra, ..., solid geometry. For a college calculus text, again, the best of those books have been so highly polished for so long that, again, no teacher should be needed. In particular, I did make clear to avoid the AP Calculus materials I regard as junk. Then in what I wrote, the next book would be abstract algebra. I didn't get stuck there, either, but I didn't try to work all the hardest exercises. I might have included, "If in the whole book, omit, say, 10 exercises, fine. Why? Because some of the exercises might actually have errors in their statements, are placed in the book before the material needed for a solution, or for a solution actually need material outside the book and much more advanced." With that cautionary, wise, flexible, non-rigid, and non-absolute statement, your concerns about students getting stuck and needing a good teacher shrink. Sure, as I stated, my suggestion to solve the problem of the OP, that is, for disadvantaged students, was only for math and only for talented students. That for such students it appears, from my background, yours, and much more, that my partial solution to the terrible floundering around struggles of the OP are quite good. It's no joke: A lot of talented, disadvantaged students should be able to do this. Gee, guys, a lot of talented, disadvantaged students do really well at basketball which in some ways is more difficult, e.g., as I mentioned, no teacher, text, or exercises. And no answers in the back of the book. Heck, I wasn't disadvantaged, but, still I got a Ph.D. in applied math from one of the world's best research universities, and 85+% of everything I learned for that degree and 90+% of all the pure/applied math I have learned was from independent study much like I described, e.g., how I did really well with high school plane geometry, college calculus, and more. I omitted some of the details of what I did, a LOT, with linear algebra, ordinary differential equations, statistics, numerical analysis, and more. Again, my point is: For the struggles in the OP, there is a partial solution: For a disadvantaged but talented student, go for math with a lot of independent study. Why? For one, the approach of just get a book, read the book, work the exercises can work really well there, all the way through qualifying exams in the math department at Princeton (on their Web site at least at one time they stated that courses are introductions to research by experts in the fields, no courses are given for preparation for the qualifying exams, and students are expected to prepare themselves for the exams on their own). For the grim stuff in the OP, my suggestion is a very good to know lesson. In some good ways, it's much better than anything I knew until well into my Ph.D. Had I and/or my parents understood this lesson, then starting in about the sixth grade I could have just raced ahead and been ready for math graduate school by high school graduation. You seem to understand some of what I wrote. If you want, then add to what I wrote, e.g., with some books, say, MathOverflow, and maybe more. And maybe there are some good massive open on-line courses (MOOCs). But I'd say to be careful: The time I looked at Khan Academy on calculus, I concluded that they didn't understand calculus well and had bad material. And I've commented on the AP Calculus material. Uh, sure, James Simons has been running Math for America or some such. Well, there maybe some efforts there to help talented students who want to rush ahead in math and use that as a way to get free college and grad school educations. Sure, Simons was math chair at Stony Brook so knows a lot about math education. And he may be able to recommend some really good books! Generally, a student who has done really well on their own in math deserves and maybe often can get a lot of praise and scholarship offers. Or, IIRC, good graduate departments in math have a lot more tuition scholarships than they have applications from good students.