Definitely broaden your perspective to more than just math. Anything academic, including the arts should be looked at. My son got his college degree in math (plus comp sci) and his youth included all sorts of varied experiences, not just math/comp sci. btw- music and math are related. Again- let him be a child as well. Nothing wrong with a kid his age have a lot of unstructured time in the summer. The gifted camps run by various organizations are wonderful if your kid qualifies (usually based on an ACT or SAT test taken through one of the programs such as CTY- different regions have different sponsors).