What does it mean to “learn calculus”

In a recent post, Calculus on the Road, I pointed out that studnts who didn’t know the definition of a derivative were making good grades in their calculus class.  So what does “learning calculus” mean?  At some “elite” schools it doesn’t mean knowing any formal definition of a limit (part of the definition of a derivative).  For example, this is true at the Univ. of Mich.  (and even for what has to be learned at Mich., the grading scale there seems quite lenient) At some places “learning calculus” does mean learning the definition of a limit.  For example, this is true at Caltech and Reed.  In some places anything that almost any place labeled college calls calculus is considered calculus.  Here is an excerpt from one of the top fifteen ranked school’s policy “…The material in [our calculus sequence] is more-or-less the standard material found in the Calculus I-II-III courses at other colleges and universities that operate on a semester system.  Therefore, coordinating calculus courses taken at such schools with those offered [here] is usually not much of a problem.  However, there will be variations from school to school on items such  as how technology or computer software is used, or the order in which topics are presented in certain textbooks….”

Is it true, as that last school implies, that Calculs is Calculus is Calculus?  Or is it true that, though it may be good for some students to learn a little about Calculus and its applications, we need more than that for people who will eventually be working on airplane design, or bridge design, or financial derivitives, or etc…?

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