Individual school districts and occasionally schools have enormous autonomy compared to the norm in Europe. So two schools in the same state can each have a class called Algebra II, with literally zero overlap in the material covered. Yesterday I read about a high school math teacher who decided to teach partial differential equations, normally, I believe what Americans call Calculus II in university, as an elective. There are good schools, but the system is very far from uniform. Partly this is because it wasn’t designed from the ground up to teach nationalism with education fit in around that goal. Puritan New England was the first society with mass literacy. Schools were locally funded, run and organised and that organisation of local rather than state administration persists, possibly in every state, certainly in most. Education came before nationalism so there was never a state system designed from the top down to turn everyone into Americans, nationwide, though many reformers gave it a good try.
No, partial differential equations would be the fourth semester.
Calculus I is differentiation. Calculus II is integration. Calculus III is vector calculus, with stuff like curvature in 3 dimensions. Differential Equations would be the next class.
The AP test covers differentiation and, optionally, integration. It's the first semester or two. This is what a good high school student will do unless the school itself is really bad or really small.
I remember Calc I (1st semester) being limits, differentiation and integration. And Clac II (2nd semester) being partial differential equations. This was in the engineering school though... it may have been different for other schools in the university.
This is the original poster's point. There is no consistency. Even in university. Some schools use quarters, some semesters, some trimesters. Some have letter grades, some percentages. Some are pass/fail freshman year, but are graded in subsequent years (e.g. MIT). What makes up "Calc I" at university varies tremendously.
