I am almost sure that my six-year-old son is gifted. He gets in trouble in school for talking, interrupting, and goofing around, but this improves with structure. His teacher is sweet and positive but does not provide extra or advanced work, even though he asks me for it and she knows he can do it and is very bright. She thinks it is more important at his age to fit in with his peers and not be singled out in any way. Would testing him at this point serve a purpose? A school psychologist said he didn't think it was necessary if my child wasn't going to receive special services as a result. He also said if I know he is gifted, why test?
The reason we have our gifted children tested is so that we will fully understand their capabilities and so that the school will hopefully provide an appropriate education for them. It's not unusual that the teacher is not offering him more, because that is not in her curriculum plan at this point. The problem here is that if giftedness is ignored and unchallenged in the primary grades, the student may have learned to show only what is expected and that to be the same as everyone else is what is desired by the teacher. By the time the gifted program is available, the student is still gifted, but he hides it well.
Many teachers take the time to provide something a little extra to a student whom they know is gifted, even if no program is being officially provided. And many parents provide outside activities and special learning fun at home to augment their child's regular education. If you have read some of the checklists on gifted behavior, and you believe your son fits the bill, I would go for it and have him tested -- though the school may not agree to do it.
Even an identification of giftedness does not allow for your son's "pay attention to me" behavior that you describe. A structured behavior plan in school where he earns points for not interrupting -- where points can be redeemed for fun things like computer time -- can still work at this age. Good luck.