The injuries are not sufficient to warrant the time and costs that go into a lawsuit. If you want to pursue it you need to serve a Notice of Claim if this is a public school.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 577-9797 or via email at Eric@RothsteinLawNY.com. I was named to the Super Lawyers list as one of the top attorneys in New York for 2012. No more than 5 percent of the lawyers in the state are selected by Super Lawyers. The above answer is for informational purposes only and not meant as legal advice.
I assume you child is at a public school. It is a specific violation of the Rules of the Board of Regents to use corporal punishment against a student, i.e. any physical force upon a student as a means to punish him. The only exception is if alternatives cannot be reasonably employed. Go ahead and file a notice of claim within 90 days of the incident against the school. That way your foot is in the door to sue if it comes to that. It will also let the school know you mean business. If the facts bear out that what you said is true, then your child may be vindicated.
If you found this "helpful" or "best answer," please click it with my appreciation. My response is for educational purposes and does not constitute legal advice nor creates an attorney client relationship which requires all the details and a personal conference.
Be aware that infant compromise orders can flummox your recovery completely.
The city will enter into an agreement quickly to settle your case for a sum, but if you do not have an attorney retainer filed with the office of court administration and in possession of a retainer number, you'll never get to file your infant compromise order. You'll have to create a "pro se" retainer just to file and get a retainer number. The clerk of your local court will not accept your proposed infant compromise order without it.
There are many pitfalls in representing children either as a lawyer or a "next of friend". Don't try this yourself. Seek competent and experienced counsel to go through the maze of tort actions involving minors.