[cgriswald]

Most high schools and high school teachers don’t teach in a way that particularly enables students to take this advice. “These are the rules. This is how the math works. Do the math. Also these are the classes you have to take to be an $X.” A relatively good teacher will engage with a student who is enthusiastic about the material in this way but many won’t: “You’ll learn about that in $future_course” without even an attempt to talk about it in the abstract.

Without some guidance on which book to choose, (1) could condemn a reader to mathematical purgatory. In any case, I’d also say that taking multiple math courses in high school helped me to understand that math is most often about relationships and that there were multiple ways to describe the same relationships. This was in spite of no one bothering to really teach that idea. Previously I was mostly focused on axioms and rules (probably because of how I was taught), which, while useful, missed some important stuff. I also prefer to read through multiple texts because it gives me a break, let’s me see new connections, and if I come back to the text after working on something else and still know what’s up I know I really grok the material. (It’s insanely easy for me to memorize and apply rules; that’s not indicative of understanding.)

(3) is good reading practice in general, but also, it’s a skill most people haven’t developed. It’s very easy to say, “Pfft, I know all this,” and just skip something without actually really grokking the information; it’s too easy to over-estimate understanding. For most people I’d say at least work through any proofs and the most difficult questions in each section. This is actually something I think we should be learning in middle school at the latest.