Wait - when you get a new teacher they simply send you home with a piece? Why must it be pieces at all? Is that what they "teach" - pieces? Something doesn't sit right with this story.
Yes, that's what the teachers I've had have done. I don't know why it must be?
The last teacher when she asked my goals I said - bearing in mind I'd gone over pretty much everything you can read in this thread "to play anything, even 2 notes, so it feels comfortable without pain, and I have control over how I play them. Whether that's staccato or legato, P or F or whatever" and I told her I would play anything at all...exercises, hanon, scales, pieces - whatever genre. Whatever she thought it would take so I could make some progress.
She gave me 2 pieces to do that week. Field nocturne No. 5 and another that I can't remember - but similarly beyond my ability [I've never opened the book at that page again, I didn't get past the first few bars of the nocturne. I made no more progress with these new pieces than I did with any of the, probably several hundred by now, that I've attempted before]
But IME, yes, today piano teachers teach pieces. Well, you can see Bernhard's threads on this [using pieces as the basis for teaching - whether it's music or technique], so it isn't that surprising.
What is surprising is that these teachers don't actually teach anything. This is perhaps why they diverge from what Bernhard claims to do with pieces. Or perhaps to be more fair, I suspect they do teach stuff to kids. They see an adult - one that can read music and someone who, albeit is no authority on the subject compared to some here, but nevertheless knows a heap of composers and pieces that their average student doesn't.
Perhaps they dismiss the tale of woe as though, I'm acting like many people who can play but say they can't? I've had this in the past where someone will ask me about my playing and I'll start to tell them the truth, and my SO will say "Oh he says he can't play" and at that point they think you're being modest somehow.
After all, you see Marik talking in here as though he can't play - so I'm basically hitting that wall built by people who can play but are either insecure or modest about their playing [or having a bad hair day or whatever] I was in a piano shop once shopping for an adjustable stool to see whether this was the problem with my arm, and said the problems I was having and the lady in the shop said she couldn't play to my SO when she asked. I joked and said "I don't think anyone can play the piano IRL, except people like Horowitz we see on the TV....I don't know anyone who can play" - and she sat down and started playing really well.
So, how to say you're crap when people that play say the same thing?
As I said, what basically happens is you start to play that new piece, perhaps just the right hand or left hand, and stumble through it a bit at the lesson. They might write some notes on the music during this phase [like if you kept hitting B instead of Bb they might write it in, that kind of thing...or if you get the timing wrong, they might emphasise it in some way and so on]
But bear in mind I can read the music, I can find the notes on the piano. Perhaps with a kid they are doing more. Then they send me home with the piece to "practise" it.
As you perhaps are thinking, there's nothing at this stage that I can't do by myself. I can pick a piece, read the music and learn the notes, and then play them roughly in the right order. It sounds completely crap, but I can do it.
I've never had a teacher [or anyone else for that matter] who has physically seen me playing and said something like "well, look, this is what you're doing wrong" - in the way Marik hints that he would if he could see someone.
The next lesson you perform whatever it was you practised. Presumably, if you can do whatever that was, you move on. I wouldn't know. What happens is they will stop me at some point where I make a mistake during that performance. That could be anything or everything, pretty much at random.
I might play a wrong note and then they'll go over that bar...or they might decide I should play the whole piece at half tempo because my timing is out. Or use a metronome. So that's another week. At the next lesson playing it at half tempo they might decide that you're playing the left hand too loud and the next week is playing the left hand softly.
How to do any of that stuff - i.e any information at all remotely resembling, for example, the info that is in the youtube videos and website I linked to earlier, has never been addressed by any teacher I've met in person. In terms of musical interpretation that is never addressed either.
All it basically is, is someone listening to my playing, stopping on some kind of technical mistake and pointing that mistake out in terms of the music, not in terms of what I might be doing wrong to cause that musical effect nor what I might do to correct it. Nothing that would require me to be there. I could post a recording and you could say "the left hand is too loud" - how to play softly? Well, afaict, a teacher will just say "press the notes with less force" - which is fairly obvious.
The really annoying part is that it is random...one week to the next jumping from one technical thing to the next, never solving it and never moving forward because my playing is completely random and inconsistent. Ultimately, I cannot play the piece and no matter how much I practise, whether it's the whole piece or just 1 bar of it, it never gets any better - beyond that early stage before you know the notes. If I try a harder piece, say a chopin waltz, my reading isn't so fluent, so it might take a while before I get to that stage, but I can butcher the first 8 bars of a Chopin Waltz, just as well as I can butcher pieces that are so simple they aren't even grade 1.
What I can't do is take any piece to a stage where I can play it.
That's pretty much my experience with piano teachers.