We definitely have bratty restless kids when they are under-stimulated. We see it first at home, and then it occurs in school.
We even see it over vacations. DD7 usually lasts about 1 week with nothing planned before she needs some sort of intellectual stimulation. Last week over Christmas break - 2 weeks long here, we had her do some time on her school's math website because I could see the brat creeping out and since we were staying with my parents we didn't want brat child to show up. After that she was great and good to go for the rest of our time there. My parents thought I was nuts having her do math over vacation until they saw the change in her attitude and how much she liked it.
I like the idea of sending in a notebook of other things to do when school work gets done - I might try that.