I sort of agree, but your advice is targeted at an entirely different group of students than the article's. If someone is confused about how to interpret sin^2(

I sort of agree, but your advice is targeted at an entirely different group of students than the article's. If someone is confused about how to interpret sin^2(x) vs sin^{-1}(x), there's no amount of "understanding" that's going to help them (it really is just idiosyncratic, inconsistent notation), but a checklist like the article might clue them in. I think students often underestimate what "understand" means in a math class. The progression you're describing ("the first couple of semesters are foundational...") is definitely not universal in US undergrad programs, though. There's just not that much coordination between professors.