Both, depending on facts. First, you should know that contact sports have huge protections, and if this was during a game or practice, you would likely not have a case. Since it was during recess, it could fly, but be aware that suing a school as a govt entity is not easy and there are many hoops to jump through. This is how bullying can be dealt with under the law:
Get the school's written policy on bullying (complaining is not enough, you need to get the written policy). The school is obligated to follow it and investigate. However, if their investigation does not turn up anything actionable in their opinion, they are not obligated to act further. An Educational Rights attorney can be very effective in dealing with the schools on bullying issues, so you can consult one if you are not satisfied with the school's investigation and results. Also, consult a personal injury attorney for the injury to your child.
Have an attorney send a legal letter to the parents of the child because they are responsible for his actions as long as he is a minor. They can have liability as long as they knew, or should have known, he had this propensity (the letter will be proof they were put on notice). You can sue them for anything their kid does.
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The laws concerning public school immunity from injury claims are complex in the extreme and need to be addressed by a North Carolina education attorney. Locate one using the Find a Lawyer tab on this page. Entire law school courses are based on this one aspect of education law.
As for a claim against the student and/or parents, the law concerning these bullying claims vary widely from state to state and are changing quickly with new legislation. Again, contact aan education attorney in NC.