If self defense is your primary goal for you child, than look at programs like Kids Krav Maga. Krav Maga is based on the Israeli Special Forces self defense program that makes defending yourself as simple and as effective as possible. Other martial arts can accomplish this goal but usually focus on other aspects of martial arts. If your child wants to compete in martial arts look at more sport based styles such as Brazilian Jiu Jitsu, folkstyle or catch wrestling programs, karate, or kickboxing.