I want to go back to what Louispodesta wrote, because there are some important things behind what he wrote.
We have to look at what happens in teaching itself. Now, Louis gives a specific portrait, which is how his lessons went. A lot of lessons do go that way - I think this is the "traditional" pattern that is still quite prevalent, and personally I am not that enamoured by that model. There are also other ways that music is taught - other routines and systems. My impression is that really good teaching is rare. You also have the complicated question of students working with the teacher, the parents' (parent's) role, and how one thing can affect the other. For example, a student "not practising" may hamper the teacher's effectiveness - but what the teacher does may affect the student's practising, or whether an erstwhile compliant young student will ever play the piano again once he is set free.
Where there is smoke there is fire.
I was forced to look into this myself some years ago, after having had 5 years of music lessons as an adult on a different instrument, with a child having entered college for a music degree and what happened there. I'd been thrown into the music lesson scene unprepared, having never experienced lessons and came out some years later reeling. I had had a simple assumption: Playing a musical instrument is a skill involving physical actions, training of the ear, and certain intellectual and aesthetic understanding of music. The teacher's aim is to develop those skills, and there is probably some sequential way of getting there. Follow the teacher's lead, and you'll obtain all of that. That's what I thought. In many cases that is not what it is!
There are lots of variants:
- In the "traditional, old fashioned" model, the student gets a series of things that the teacher's teacher, and teacher's teacher's teacher did. You "do" Czerny, and Hanon, and Inventions, and Sonatas, and scales and chords and arpeggios. By "doing" them you will get the skills somehow. The teacher may say this is wrong (that F# should be F), "bring out the voice" without teaching how to do it (doesn't know how), browbeat the student who fearfully tries to make it sound right at home, and may get some kind of a semblance of "better" through sheer willpower and fear. I've known people who went through that scenario. I don't know if this was Louis'.
- You have the scenario of outside events, like DC described in the thread of why she quit teaching. The teacher prepares the student to compete in competitions (an outside event), to ace exams, or shine at recitals. Yes, skills are part of this to some extent. But you can choreograph the student, have him imitate you, and have him play the same piece for a long time, polishing beyond polishing. Does the student learn to understand what is within the music, to interpret it? Can he read and examine the music in order to interpret and work on music on his own eventually? Can he solve problems? Or is he just left with a few polished pieces to make his parents proud?
- You have "going through the method books", maybe shallowly, and maybe as fast as possible so that Suzie reaches grade 8 faster than neighbour Mary. A variant for adults is to give shortcuts, and/or favourite pieces, so that not much time needs to be taken in practising - both in terms of minutes per day, or number of years of lessons.
- You have people who can sort of play the piano, and hang out their shingle as piano teachers. Or people who play piano extremely well, but have no idea how to teach it. Especially at the very important beginner level. What use is it to guide in advanced interpretation, bringing out this or that note, when the student hasn't figured out how to produce forte vs. pianissimo? That kind of thing.
These are some of the goings on. I am not convinced that there is any single formula, and I definitely would not want to impose one. There are some excellent teachers out there, but I couldn't tell anyone how to find one in their own location. The excellent teachers are probably often the least visible, because they don't play the game, and they may not select excellent well-trained students whose playing will help cement their own reputations. In addition to which, the hardest and most important job, often not acknowledged, belongs to the teacher of the beginner who sets everything up.
Phew. Too long.