One of my precalculus students (or is it the hyphenated pre-calculus?) thought that he was studying calculus. He figured that precalculus was a kind of calculus, just as differential calculus is a kind of calculus.
What does that prefix “pre-” mean, anyway? To him it meant something like “in advance”: prepaying is paying in advance; to preplan is to plan in advance (in contrast to postplanning, which is much easier); if you preheat an oven, you heat it in advance of cooking; and a pre-approved credit card has been approved in advance of applying. So why shouldn’t precalculus just be calculus in advance?
I, on the other hand, thought of the prefix “pre-” as meaning “before” or “preparatory”: precalculus comes before calculus, or it prepares you for calculus. There are plenty of examples of this meaning as well: prehistory comes before history; a preadolescent will be an adolescent, but s/he isn’t a type of adolescent; and a precancerous condition comes before cancer.
How do we tell the meaning of this prefix when we meet an unfamiliar word?
It’s amazing that anyone actually learns English.