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That ugly beast--peer pressure --may well be rearing it's head here, also. 11 is a tough age where kids are moving from little kids to adolescents and it is pretty common for kids to go along with the much-less-talented crowd and goof off; if they show achievement, it means they care about something, and that is definitely not something to let on. Do as bloke says and you might salvage the student; sometimes you just have to wait the kid out. Unfortunately, you also lose some of the best, but it is really almost beyond your control. No magic here!

Fly the kid out to Amarillo... Schlep would love to sit by this one!

Seriously, if you could find a performance opportunity where he could sit by you in one of your ensembles, and see that indeed, he can play, this might help alleviate feelings of fraud. In graduate school, my teachers (trombone and euphonium) would find/make opportunities for me to play alongside them. I cannot tell you how much this helped to solidify what we had built in lessons. My high school and undergrad teachers probably would have done the same... but I could not play back then.

Reminding him how far ahead he is of most people could help a bit. Nothing like an ego boost to improve one's demeanor... I was definitely flaunting a repertoire of tunes comprised of F, Eb, D, C and Bb at his age, along with almost every other tuba player I know. He should definitely be proud of himself.