Many years ago I was asked this very sad question by three different students in the same week. "I went back and played some of my old songs this week. Is that okay?" My students, or least some of them, were under the impression that they were only allowed to play their assignments and nothing else. I'm not sure why they felt that way, but I have to take full responsibility.
Since then I make it a point to encourage them to play old songs, make up new ones, or take a crack at some new music on their own. Often it's a part of their homework. I actually assign "old favorites". I constantly remind them that it is an important part of learning. Just cranking out pretty notes isn't the complete job. Having a deep and intimate understanding of the song is what turns it into true music.
Of course I don't want to fail my students like I feel I did those many years ago, so I tend to prattle on and on. The teenagers are the best at letting me know when I may have gone on a bit too much. Only a teenager can/will roll their eyes out loud. But anything worth doing is worth over doing, so I won't let up.
The payoff comes when the student begins to have an easier time learning the mechanics of the song. They get the notes and timing right - mostly. We make a few adjustments and then we're set to do the fun part. We discuss interpretation.
I love talking about and demonstrating music interpretation with my students. Learning to explore the music within the confines of the song is a true exercise in subtlety. The composer tells us what notes to play, when to play them, and how long each note should last. Play louder here, softer there, now fast, now a little slower. But these terms are vague and the musician needs to give them specific meaning.
These discussions with teens are somewhat difficult because many of them aren't used to teachers who tell them that their opinions have worth. They want direction, which I'm happy to provide, but they are reluctant to take the lead. But eventually, with some degree of prodding by me, they start to find their voice. And as they gain confidence, the quality of their playing rises. Then comes the breakthrough moment and they play a song with true musicality. As the last note is still ringing they look at me for approval and see me smiling like an idiot.
Explaining the taste of chocolate is futile. If you've ever tasted it no explanation is necessary, but if you've never tasted it no explanation is sufficient. And so it is with playing music with your own interpretation. But personal interpretation is an important and time honored artistic tradition. Three painters in a studio will paint the same scene in three different styles. Shakespearian actors are able to find their own voice within the play. Even people reading my words now will read them in their own voice.
So I push, nag, and cajole students into playing confidently with their own style. Officially I'm doing this for their own good. But truthfully I'm doing it for the most selfish reason. I get to spend my work day listening to nice people playing beautiful music beautifully.