According to New York's courts, that assumption is unwarranted; to the contrary, it is rather difficult to prove that the school was in fact negligent.
Consider the 2005 case of Botti v. Seaford Harbor Elementary School District 6. In that case, a little girl was injured when, during a school recess, she fell from the rings that were suspended from the monkey bars in the defendants' playground, and suffered serious personal injuries. Even though a school aide saw this little girl fall from the monkey bars on two prior occasions, and the evidence indicated that the aide still may have encouraged the plaintiff to continue using the apparatus, the Court nevertheless held that the case must be dismissed because the plaintiff failed to raise a triable question regarding whether the supervision afforded by the school was adequate or not. The courts have been particularly harsh on cases where the plaintiff was injured in a sudden, unforeseeable incident.
On the other hand, there are instances where the courts have carved out an exception to this rule, and allowed the parents' negligent supervision claims to survive, particularly where the school acknowledged that its own equipment was unsafe for games of tag, yet they allowed the children to engage in this game for a prolonged period of time (see, e.g., Oliverio v. Lawrence Public Schools), or where the school's equipment was itself unsafe and defective.