Accidents happen anywhere, anytime on any clock. How old is your child? Why was he outside walking to a class? In elementary school, the outside time is recess time. Only in Senior high school (grades 11-12) are there open campuses where students walk from one building to another.
We do have insurance but every August at the beginning of the school year, there is a cheapo accident policy sold and I buy it for my daughter because it is extra coverage for us. For someone with no insurance it is good accident coverage, manyn times lower than $100 for a school year total.
What do you expect from the school? Are you trying to sue them for medical plus anything that pops up in the future? How are you communicating with the school? If you are communicating with a harsh tone, etc you won't get as far.
Take emotions out of your decisions. Gather all info and proceed with facts. When people make decisions based on emotions, things can easily backfire.
For instance, I have a friend who is a well known attorney. Her client was in a minor car accident but took care of all costs involved with the other person. The other person wanted more, more, more and refused all communication until a jury heard the case. Well, the jury heard the case. Within 5 minutes of deliberation, they came back and awarded the guy suing $1. Why $1 you ask? Because that prevents him from being able to appeal, he was given his choice of a trial by jury.
I am not saying in anyway you are going after money from the school district. Some people would. Just be realistic and leave emotions out of your decisions.