Some piano students are impressed when their teacher seems to know after hearing only a few notes played at the beginning of the lesson, how much practice they have done. There is a famous quote of Jascha Heifetz that goes:" If I don't practice one day, I know it; two days, the critics know it; three days, the public knows it." Growing up, my musician mother would say to me: If you don't practice one day, you'll feel it. Two days without practice, and the neighbors will notice. Three days of no practice and the whole world will know! I am not sure how attentive the neighbors and the rest of the world are, but I certainly notice how every day of non practice affects my playing. Or, positively put, how daily practice boosts the playing.
The heart and soul of acquiring a new skill lies in the practice. That is, in the daily repetition of new content until it turns into a habit and makes room for more new content to be processed.
This is familiar to anyone who has learned a new language, sport or dance, since you can only learn something new when whatever you learned during the former stage has been internalized.
Learning piano is the same. You go to your lesson and get new pieces or new ideas about how to play whatever you are working on. Over the week you go over this new input and by repeating and molding it you gradually turn it into something that comes easily, intuitively, and frees you for receiving new content in your next lesson.
Imagine receiving a new plant in the lesson. If you forget to water it over the week, it will dry up and shrivel away. When you do water it though, your plant will grow and expand and become part of the flourishing garden you are creating and nurturing.
In computer terms, it is like adding a new document to a file, that will then get interlinked with other files, so it has become an inseparable part of the system.
Here is a link to an article about what goes on in the brain when you learn something new.
Obviously, the quality of the practice has enormous importance. The ideal would be daily brilliant and efficient practice sessions, full of motivation, interest and inspiration. If that isn't always possible, having 7 less imaginative (but correct!) drill sessions will be more effective than one fantastic practice session and nothing else. Like hammering a nail into a wooden board, one big blow won't be sufficient- it will take many smaller taps to get the nail into the board.
By continuously repeating the new content, you are acquiring certain skills that will be hard to change or uproot later on. The hammer taps mentioned above could get the nail into the board alright, but in the wrong angle! This is one of the reasons it is so important to be certain you are practicing correctly, and why weekly lessons are considered the standard. Until you have reached the stage where you are absolutely sure you can independently stay on the right track, you need the external control of a teacher who will constantly correct you and make sure the practicing you are doing is actually taking you forward!
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