At what age should a child be brushing his teeth independently?

Around 8-9. Usually around 8 or 9 years old. I do recommend that a parent oversee the brushing every once in a while to ensure that an adequate job is done. Also try to incorporate flossing into the routine to help prevent cavities between teeth.
Age 10. Start brushing your child's teeth/gums around 6 months of age. They can begin to assist you around 15 months. However, they cannot do the rapid alternating motions required to brush well until about age 10. Until then, your child needs you to help them brush effectively.