I had to teach a doctor’s daughter from Alexandria who showed up to my physics recitation not knowing what a function was, despite having taken AP calculus AB. And how did this happen in the public schools in Alexandria? Because the gym teacher taught it and everyone got 1s. Furthermore, the school board gets bonuses for kids taking AP tests and teaching gifted classes and then hands the teaching jobs out to their sycophant favorite teachers
Starting with first algebra through my applied math Ph.D., nearly everything important that I learned I got heavily from independent study. (1) Loved plane geometry. Slept in class then worked ALL the more difficult supplementary exercises. (2) For my first year of college, went to a cheap state school, partly because I could walk to it. They put me in a math class beneath what I'd had in high school and would not let me take first calculus. For their class, a girl I knew also in the class told me when the tests were, and I showed up for them. For calculus, I got a copy of the book they were using, not a bad book, and started in and did well covering the first year. For my sophomore year, transfered to a fancy college, took an oral exam on first calculus, then got into their second year, did well, and was caught up. (3) Linear algebra? Sure, went through Halmos carefully word for word. About a fine point, wrote a letter to Halmos and got a nice answer. Also worked through Nering's book -- Nering was a student of Artin at Princeton. Later did a lot in linear algebra applications, e.g., in statistics, numerical issues, etc. (4) In grad school, got pushed into their course in 'advanced linear algebra'. When the course got to the polar decomposition, I blurted out in class "That's my favorite theorem!". Blew away everyone else in the class. Partly intimidated the prof. In grad school took an advanced applied math course then in the summer went over the class notes word by word. Wrote the prof a letter improving on one of his theorems. Back in class, took a 'reading course', and from the study in the summer saw a problem and solved it with some surprising math, two weeks. Later published it -- so, technically it was a dissertation.
Point: Self study can work well. Obviously: Once a prof reading research papers, nearly always have to use self study, and the papers are generally much less polished than good textbooks.
So, I recommended to students short on money just to do some self teaching and show up, demonstrate what learned, and ask for a scholarship.
