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If it happens that you have more time and long break between piano lessons (school holidays - no piano lessons and plenty of time), how reasonable is the idea to take and learn something on the own initiative? For instance, some other song that does not look more difficult that the one I learn with the teacher?

From the one side, it is definitely possible to learn something wrongly. When I started with the teacher, I already knew a couple of songs - the teacher said I play these horribly, and probably she was true (now I play these songs less horribly).

From the other side, it will be more lessons in a few weeks, so the teacher probably will correct any really habits. Is such a "mixed learning" fully ok or it may be better to wait?

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I think most teachers would love it if their students wanted to do extra-curricular learning. As long as the break is not too long, they can always help you correct any bad habits that are picked up as a result. Also, the process of unlearning a bad habit is, itself, a good thing to experience and learn. –  Jeffrey Kemp Apr 5 at 8:01
    
I should also point out as a teacher myself, that if a student comes in with something that they have been working on on their own it gives me new insight into what they want to learn. –  BobRodes Apr 21 at 21:31

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up vote 6 down vote accepted

Pro active supplemental learning should be encouraged and only tempered by proper instruction when it applies to your technique or anything related to the way you control your body, arms, wrists, and fingers.

Managing the precious surplus time you have and aiming it at practicing scales, arpeggios, and chords, learn new ones, improvise with what you have already learned, and so on can only help to master your craft.

Also consider performing with others in an ensemble as a way to practice listening, communicating, and collaborating, as it will facilitate a higher level of mastering your instrument.

While singing in a choir may not help you make better octave reaches with your fingers, it will improve your overall ability to read, listen, and make music. Remember music is a highly collaborative artform. Even a soloist will tell you that they are collaborating with the composer to interpret the music to the audience.

However, when if comes to such areas that need instruction e.g. the correct fingering for a passage in a new piece or the proper way to execute an accelerando in the context of the literature, it may be best to wait for such instructions as you may have to 're-learn' what you taught yourself. Although, youtube may have some pretty good examples of the 'correct' way to play so and so's sonata, fugue, nocturne, etude, or fantasy, this may be the exception. Best to ask your teacher for any such links.

For now, I suggest you work with your piano teacher to set reasonable boundaries on what would benefit you most with self learning vs what you should wait for with regard to instruction.

At the professional level many pianists will hire an expert teacher best suited for the repertoire they are preparing. Although you can learn to master the piano, the learning will never end, there is always something new to learn, something inspiring to aim for.

If you get to the point where you think you know it all, and can play it all, it may be a good time to start to think about being a painter.

UPDATE: I just want to be clear that I have nothing against painters, actually I very fond of their centuries of great works. It may be more clear if I had written, if you think you know it all... try something new like painting as it is wide open, will humble you, and put you at the beginning again...a fresh look as in the Zen of a beginner, may actually help you move to a higher level on the piano as well.

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