Yes what is necessary is the child bully is expelled from the school, but it may not stop the problem as the bully may seek out the victim in other places, cyber bullying has become popular extending the duration and reach. We have to get real and make bullying criminal behavior just as assault, abuse, molestation, etc is criminal behaviour and have a mandatory reporting policy for teachers. This means you have a detention and reeduction program for bullies and it should include parential attendence and counselling. You also have to have proper supervision in recesses.
It must be recognised that the problem stems from the family dynamic and it might be that it can only be dealt with in absolute terms; exclusion from the school population
