I am taking piano lessons as an adult (at roughly an early intermediate level), an hour each week. For the most part I am having tremendous fun, and my teacher is extremely kind, but a little bit of frustration is starting to set in. Her advice aligns with other things I have read on the internet, and I'm pondering how strictly to stick to her advice.
I would like to try playing a lot of different music, some of it above my level, without worrying too much about perfecting it. For example this summer (shortly after I'd started) there was a month break in the lessons and I spent a lot of time hacking through pieces like Bach Inventions without worrying too much about my mistakes. It was a lot of fun, and I felt like I was definitely improving -- even if I knew it would be a very long time before I could play these pieces at performance level.
My teacher discourages this approach (and discourages me attempting Bach's Inventions at all). She is picky about phrasing, dynamics, and articulation -- for me, the musical equivalent to being asked to eat vegetables instead of pizza. (Yes, vegetables are good for you!) She asks me to keep practicing the same pieces week after week, even after I have gotten pretty good at them. Even though l have room to improve, I confess to getting a little bored of the music. I don't feel like I'm improving as fast (even though I suspect I actually am).
I am not naive -- I understand that all of this is very likely to my long-term benefit, at least if I aim to develop to an advanced level. Indeed, as a professional math teacher my own approach has a huge amount in common with hers!
I understand that I should spend a substantial proportion of my practice time taking my teacher's advice fully to heart. Nonetheless -- would making additional time for some "sloppy" and "fun" practice also contribute to my long-term improvement?