Otherwise you couldn't teach calculus without teaching number theory and set theory and probably some algebraic structures and mathematical logic too.
We actually did learn number theory, set theory, basic logic and algrebraic structures such as rings, groups and vector spaces.
In Germany every student has to select two subjects called "Leistungskurse" in which he gets more classes. In my case I selected math and physics which meant we had 5 hours worth of lessons in those subjects per week.
When I went to high school in Israel we had a similar system, but extra math wasn't an option (at least not at my school).
A big part of an undergrad math (or CS) degree is spent on these subjects. I don't believe the study everything, prove everything you do level is attainable with 5 hours per week for 3 years at the high-school level, even with a very good self-selected student group.
Another month has passed and here is a new rationality quotes thread. The usual rules are:
And one new rule: