I would not do this. Even if it is not against the law in NY, it is probably against a school policy and you're just buying yourself more grief, especially because you'll be recording minors.
Get a written copy of the school/school district's policy on bullying. The school is obligated to follow it and investigate. However, if their investigation does not turn up anything actionable in their opinion, then they are not obligated to act further.
If this kid threatens your kid, the cops should be called every time. Also, have a local attorney send a legal letter to this kid's parents. As long as he is a minor, they are legally responsible for what he does as long as they know, or should know, of this propensity--the letter will put them on notice. Hopefully, the legal letter will get their attention. If not, then, you can sue them in small claims court, or if you want more than your small claims court can give, hire an attorney to go to superior court -you sue the parents for whatever cause of action your child would have against the other kid.
If you can show the court a course and pattern of harmful conduct and that your child will suffer immediate and irreparable harm if this child is not kept away from yours, you could also try getting a restraining order. Schools tend to be afraid of this because then they have to monitor and keep them apart.
You should also get a consultation with a local educational rights/administrative law attorney--they are specialists at dealing with the schools and often are experienced with bullying issues as well. If your child has special needs, you need to hire the educational rights attorney right away.
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New York is a one party consent state so your child can secretly record anyone he or she is taking to but the school could have rules against this.
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Unless your child knows about the recorder and consents to you sending it you cannot do it. In addition, the school may have rules against recording which could lead to your child's suspension from school.