SPRINGFIELD — The Illinois Supreme Court on Wednesday appeared inclined to rule that public school districts may prevent teachers from splitting their paid sick leave across extended periods of nonwork days.The justices, except for Justice P. Scott Neville Jr., who abstained from oral arguments, seemed to be searching for a limiting principle, one that would allow teachers to use at least one month of paid leave but cut short the time in the school year when those days could be redeemed.One of four civil appeals …