I will be working as a TA in Calculus 1 this semester. In Norway, this means leading a study group with about 20 people (most of the days, I guess around five or six will show up), writing up the exercise-numbers on the blackboard, and answer any questions they have about them. More than likely, I will get students without an interest in mathematics (biology students, chemistry, geology etc.)
If the students have no interest in learning the rigorous and (subjectively) interesting parts of the subject, should I still force these concepts upon them? Many of them will be forced into taking more advanced subjects later (Linear Algebra, Calc 2). However, the nature of Calc 1 allows you to get a C, or maybe even a B, by simply cramming methods. I am having a hard time making up my mind about this, because it is very likely that my students wishes (and maybe what's best for them in terms of grades) will be in strict conflict with my philosophy as someone who hopes to become a good expositor of mathematics.
(The best example may be the formal definition of the limit. This can be taught well, as in, "this is why it works", or by simply reducing it to a humanized algorithm, which will often give you a correct value of delta.)