Mathematics at highschools is quite different from that in universities. Instead of calculating numbers and finding solutions to specific problems, freshmen end up proving theorems and figuring out what happens in general. (By in general I mean, for example, that we are not very interested in the integral function of $xe^{\sin x}\log\cosh x$, but in the fact that every continuous function can be integrated and the integral satisfies the fundamental theorem of calculus.) I have heard statements like "I like math but I always hated when we had to prove stuff", which means that the student will hate mathematics the way it is presented and studied at universities, and studying mathematics would probably be a bad idea for the student. The way of thinking about mathematics as a whole and the collection of mathematical tools makes a big transition when one enters a university (at least in Finland), and it is easy for a teacher to make fallacious assumptions about students' proficiency in basic tools in doing and communicating mathematics.
How can I, as an instructor of some kind in a freshman course, help students adapt to the new situation? Is there something that I should be very careful about? In your experience, what is important when trying to get abstract ideas (and the need for them) across to freshmen? I understand that it is not uncommon that a freshman class is taught by someone who has long ago forgotten how a freshman thinks, and although (because?) the teacher has made his best to make the material mathematically elegant, the students get little grasp of what is going on. I would like to prevent this from happening in my class if I ever get assigned a freshman course. It would be great to hear experiences about supporting students in this transition.
Note: Educational systems are different in different countries, so "first year university mathematics" is not globally well defined. The transition I wanted to support the students in is the one from calculation-oriented mathematics to proof-oriented mathematics.