Thank you for the suggestions. So far we have mostly let him be. He is in an alternative school where they don't get homework or even grades so there was no pressure for him to perform. We take him to extracurricular classes when he shows interest but often he says he prefers to learn on his own. I really didn't want to get carried away because of the gifted label and end up putting too many expectations on him.
However, next year he will be attending a "normal" school where he will be required to take tests, turn in assignments and so on, so my old worries are returning.
I guess I have always been somehow in disbelief about his scores because it seems with such high scores his intelligence should be more obvious, but my references might not be accurate, since I suspect many of my relatives might have been gifted too.
