I'm sure that psychologists have studied the age at which children become meta-cognitively aware of the differences between long term and short term memory. This is something I need to try to find out.
I came to the realization, with one of my teenaged students, that he wasn't practicing with his long term memory in mind. In The Perfect Wrong Note, William Westney writes some about ingraining the correct version of a piece in our mind. He laments the kind of practicing where a mistake is encountered, corrected and then never addressed again. What he's getting at, I think, is that there's the conscious, working, short term memory. Then there's the unconscious, storage, long term memory. Usually we practice so that our short term memory understands the right or the wrong way to play something, but we don't realize that there's no guarantee that our long term memory knows the difference.
What gets stored in our long term memory will be the version that we've practiced most, the one that can be chunked and related to some piece of information already in our long term memory.
Some day I'm going to try an experiment with a more advanced, teenaged or early twenties student:
I'll assign them the task to memorize a few paragraphs of some text, maybe something musical to distract them from the point. I'll instruct them to only memorize the text for 10 minutes a day, each day for one week. Each day they'll take notes about what they did, how they approached memorization. I assume the way they'll practice memorizing text will be more conducive to long term memory storage, than the usual way we practice piano.
I'm very passionate about practicing in efficient and effective ways. I've written an e-book about it, which you can get for free here. No matter how hard I try, I often feel like my ideas and tactics aren't appreciated, or even taken up at all, by my students. But I think we naturally will approach memorization of text differently.
Ultimately, the problem is that we don't think about learning a piece of music as a matter of memorization. We're not memorizing the notes from the get-go. We're memorizing the gestures, the choreography of the piece, which are necessary to play it with fluidity, without making major mistakes. This is called implicit memory. I hope that the idea of memorizing text will help students connect to a more efficient method of practicing, where long-term memory is the ultimate goal.
"Modern performers seem to regard their performances as texts rather than acts, and to prepare for them with the same goal as present-day textual editors: to clear away accretions. Not that this is not a laudable and necessary step; but what is an ultimate step for an editor should be only a first step for a performer, as the very temporal relationship between the functions of editing and performing already suggests." -Richard Taruskin, Text and Act