Teaching 'How-to' not 'What-to'
Playing the piano involves a great deal of being able to play without any music at hand. Which is in fact improvising. For this reason from the first lesson there is always some time reserved for playing without music. Which is in fact putting the scales and chord to practice.
'Practice' NOT 'Practise!
Playing a four bar phrase in one of the keys which the student has mastered is something each of my piano students can do easily. The small beginners of 7 years old can do it after their third lesson or so on any white key using five notes.
So often we are so overwhelmed by the details of a new piece of music, that we forget about the music behind it. We forget rhythm, the prase structure and the mood and character of the music, all in order to have all the notes right.
Sometimes you hear students argue:
"Yes, but I'll put that all in later". How wrong and in-efficient. At every stage we have to realise we are practising music. We are performing, even if only we ourselves are the listener. I only we were listening.
I still remember when I was a student and went to a piano lesson my teacher used to scratch all sorts of remarks over the music and it annoyed me beyond end, because I had arrived at the level, where after a year or so I would come back to the piece I had worked on and had difficulty seeing the threes through the wood. Soon I developed a habit of borrowing my music teacher's book for the pieces I was working on. She would then write all over her own pages and when finished I still had my own clean copy!
Some people find the written marks very important and I was once sitting in a music centre after working on a Scriabin Sonata. A highly respect musician came in and wanted to know what I was working on and I give him my book.
He looked at the pages of a Dover Edition, which I like because they do not offer fingering, and arrived at the conclusion that I obviously was not studying seriously, because I had written nothing over the pages.
Hence with me as a teacher my students keep their pages clean, because only very rarely will I write over their pages. It should go in their heads and not in their books. And with most of my students that seems to work out Ok.