A more rigourous approach was tried after WW2, when Americans feared the Soviets were edging ahead mathematically/scientifically. It was called "New Math" [0]. For an example of the type of textbook high school students were taught from, check out Dolciani's Modern Introductory Analysis (the 1960s and 1970s editions only; the later editions were dumbed down, especially when Dolciani died) [1], which starts with set theory, logic, field axioms, and proof writing techniques.