This is really similar to one of several school-behavior complaints we've received about my own kid -- constant noisemaking, sometimes seemingly unconscious, sometimes apparently a bid for attention. I'm trawling through this forum looking for anything that sounds familiar, so was relieved to see this question. School is bemused, deeply irritated, and all out of behavior management ideas.
OP or others in this thread: if you're still hanging out here, how does this look for your kid, three years on? Has anyone had any success in teaching a kid like this to zip it in school?
