Advanced education? Had a lot to do with that, did really well on Math SATs, ..., got a Ph.D. in pure/applied math from a world class research university, taug

Advanced education? Had a lot to do with that, did really well on Math SATs, ..., got a Ph.D. in pure/applied math from a world class research university, taught in three well known universities, have published in applied math, statistics, computer science, and artificial intelligence. For this thread, my views are different, and I suggest an approach that is simpler, easier, faster, cheaper, safer, etc. First, K-12 and anything advanced are miles apart. Basically the staff in K-12 is unable to help, teach, or guide students to anything both advanced and worthy. Partly the staffs set up advanced work as ways to challenge students to test them and see who survives -- it is like they hate the students. A lot of students get seriously hurt, and nearly no students get benefit. Net, for advanced work, f'get about K-12 -- the staffs are not qualified and will do more harm than good. Second, some good paths to advanced work are available in some of the top universities. E.g., I knew a student who as an undergraduate got a lot of tutoring from unique world class mathematician A. Gleason at Harvard. But getting into Harvard as a ugrad is super difficult. For the usual colleges or state universities, their faculties are poorly qualified. My suggestion: For the advanced work in academics (doesn't apply to the arts) just get a good path through a broad Master's in pure/applied math and pursue it via independent study of the best texts with some occasional contact with a carefully selected good mathematician. Pursue the path starting ASAP while in K-12 and then college, and don't expect any of the teachers to be able to help -- don't even mention your efforts to them; you don't want them involved. So, right learn calculus, set theory, abstract algebra, linear algebra, advanced calculus, ordinary differential equations, measure theory, Fourier theory, functional analysis, probability theory, number theory, and some applications to parts of physics, engineering, statistics, optimization. Go to some talks in good math departments and there look for research problems. Might also attend talks in physics, engineering, and computer science departments. Might also try other departments, e.g., maybe some having to do with medicine. Commonly they have a lot of important problems to solve and a lot of data but are not very good at math, e.g., statistics. So, might find a problem you can solve to get some results important in their applied work using some math, maybe partly new; then publish that. Try to publish at least one research paper. Write the paper; pick a journal; submit your paper; see what happens. Take the SAT exam in knowledge of ugrad math -- with the study of the topics above, should be able to blow that away like Michael Jordan in his prime trying out for a junior high basketball team. Pick some universities, the top in the world (it does make a difference), and at those some of the best profs (it does make a difference), make personal contact with the profs, pick a prof (make sure he is a nice guy and won't be mean to you), and apply for a Master's and to their Ph.D. program. May get to skip the Master's and go directly into research for a Ph.D. dissertation. By the way, never study any high school AP (advanced placement) materials. Instead, e.g., to learn calculus, just get the best freshman college calculus books (several) you can find (see what Harvard, Stanford, etc. are using; ask people; ignore Khan Academy and similar Internet sources). Now have your coveted advanced education. Likely, with what I have outlined, it can be easier to get a Ph.D. at Harvard than to be accepted as a ugrad at Harvard. Eventually you will discover that in high end academics "advanced" study, learning, and knowledge are not much respected; the high end understands clearly that no one can carry the library around between their ears. Instead the respect goes to research, finding new results -- sometimes that requires a lot of prerequisites, and sometimes not. Independent study? Need to learn to do that anyway. All researchers and all teachers, profs, and professionals who want to keep up have to do that. Independent study in math is relatively easy -- just be able to work the exercises. If also can prove the theorems, then there's no doubt you have done well enough. But, do get a good college math prof to read some of your first writing as you prove theorems -- need to learn how to write proofs and at the beginning have someone who knows how check your work. For a research paper with no exercises, learn, say, to prove the theorems. Why pick math? Because the best work in all fields of science and technology mathematizes the field. So, for whatever field you want, math, physics, chemistry, biology, engineering, business, ecology, ..., the best background you can have is math.