In addition to just practicing a lot, I strongly recommend learning discrete stuff: combinatorics, basic number theory, basic discrete math, graph theory, linear algebra, algorithms, theory of computation, probability (not the theory, I mean get a book of problems), even group theory. If you really are philosophically minded, set theory and logic are good ideas too.

Some of these might sound scary, but learning the basics is not harder than calculus - get a few introductory books, and choose one you like. And pretty much everything I listed will be more useful to you in computer science / programming. I'm not saying you don't need to know calculus, it's just that studying the above will more directly teach you mathematical reasoning, as well as being of direct use in CS.