With Analysis, I like to walk the student through a fairly simple problem with a lot of detail to emphasize why the details of the definitions are important and provide counterexamples as to why the statement would not be true if it were not for this one property (proof based). In calculus, I like to talk about what I would do if I were looking at a problem and had no idea what to do; what to look for and things to try (in addition to emphasizing understanding the concepts in class)