My teacher must be very good..
In his book about teaching guitar 'not pulling strings' Joseph O'Connor makes an interesting observation:
'The less you play for your students, the better they think you are'.
So, some students observe: 'My teacher NEVER plays for me.
She must be very good.'
I say she, because male piano teachers are in a frightful minority.
Moreover I play for my students all the time, which means there are no secrets there for sure. When after the lesson they ask me to play something, I never refuse.
Another little student, Peter, after one year's piano lessons asked with some worry: 'When will I be able to play the piano?'
That is indeed an interesting observation. He obviously had expected after some time to be able to play the piano and not having to practise anymore. Also he probably would have expected that it would gradually become easier instead of harder.
That is an obstacle the teacher has to avoid. Which means, that the student must gradually become convinced that his skills are indeed increasing and many things he can now do easily.
The student will want to learn 'to play' the piano and 'playing' must therefore be an import factor of the piano lesson.