What kind of notification is required to terminate a contract?

How termination can happen may be spelled out in yor contract, or it may be silent on the procedure required, so maybe there's nothing in the contract itself. The entire contract would have to be reviewed to see what it says about giving notice in general, and giving notice of termination specifically, and whether there's any other provision in it that affects this issue.

Assuming the contract was breached because of the board director's failure to give you sooner or proper notice, your damages would be the month of salary, if you actually performed services, but your inquiry implies there were no "duty days" in July and you felt you had completed all requested services by June but still expected to be paid for July because of the length of your contract.

Assuming that the school accepted your services during the month of July, then they'd have to pay for them, and could not claim you had been terminated a month earlier. But if you didn't do any of your "duty days" in July, I don't see how you can make the argument that you should be paid for July if you didn't actually perform any services.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.