I want to help each student improve their playing and musical understanding. This means working with them on technical weaknesses they have; filling in gaps in their repertoire; exposing them to ideas they haven't encountered and increasing their level of confidence. The details are different with each student. This is, by the way, why you can't really learn an instrument from a book.
Sometimes I tell a student that I can't really teach them how to play; they have to teach themselves. The reason for this is so they know that their own efforts are what is crucial. A teacher just guides, encourages and introduces. The actual work in the trenches is all up to the student. You can inspire a bit and point in certain directions, but all the wood-shedding is the job of the student. How do you get to Carnegie Hall? Practice, practice, practice!
I wonder how much of this is true of disciplines other than music?
Pretty good playing for her age!!