I'd say this is very much a gifted issue.
If you don't provide him with something to do after he's finished his class work, what is he going to do to learn to entertain himself? Maybe he can get a book from the teacher's box of books available to the class, but they're probably beneath him. If he's allowed to get books from the school library, he probably can't get enough to last until the next trip, if he's even allowed to get books at his level. You'll probably need to supply books for him on a daily basis.
Kids rarely spontaneously do math, you need to provide a framework to build on. 1st grade math probably isn't keeping him busy very long. Sudoku, logic problems, and other workbooks are great to keep them entertained. GS11's teachers welcomed these additions.
Many kids resort to undesirable behavior when allowed to entertain themselves. It's not strictly a gifted issue, but when the gifted kid has more time on his hands it can be a bigger issue for him.
