As one who has gone from morbidly obese to now just mildly overweight by distance running, I can tell you that a sedentary lifestyle leads to a drowsiness and difficulty concentrating. My brain function has improved dramatically along with my fitness. Habits formed by the discipline of a consistent fitness program reinforce habits required for academic disciplines.
Yup, and little kids have more energy and are more hyper than older kids. When you run around your brain gets more OXYGEN, and it LOVES oxygen. That is why getting someone who has had a brain injury or a stroke on oxygen right away is important. And there are new oxygen therapies for even older brain injuries using high pressure and oxygen.