Lots of great comments. I'll get back to the rest, but I was curious about this.
One other thing that I haven't heard people say about AOPS but I feel is true is that AOPS is much more theoretical and proof oriented which is great, but it lacks the practical application problems that most of the other math courses have. Purists can argue that applications isn't really necessary and I won't argue, but our public schools have a lot of data analysis and statistics and approach to practical problems. I saw only a small amount of that in Algebra 1 of AOPS, and I want my dd to have the "alignment with state curriculum" as well as problem solving. So doing both works well for us.
This makes me like AoPS even more. But it makes me wonder, how exactly are their classes structured. Do they lecture about some theory, with adequate proofs/explanations? Or do they lead you to "discover" things via a sequence of problems/exercises/questions? Or a combination of these? Or what?
For comparison, the K12Inc Prealgebra course simply stated formulas (after adequately defining n!)
P(n,r)=n!/(n-r)!
C(n,r)=n!/[(n-r)!r!]
without any explanation of why LHS=RHS. (Usually they give some kind of explanation for things, but not this time.) Then they have some "worked examples" which are easily skipped past, then a routine quiz.
How does AoPS take a student who hadn't seen n! before, and lead them to know these formulas and understand why they are true? And what other activities surround the learning of that particular piece of maths? (Or substitute any other piece of maths for the purposes of this discussion.)
How does AoPS work?
