With nightmare visions of a gunman stalking school halls, districts often rush to hire police officers to patrol their campuses after news of a school shooting. Critics of that impulsive response, which has been in high gear nationwide since the Sandy Hook Elementary School shootings in December, acknowledge the concern for student and staff safety that drives the addition of school resource officers, as such police are often known. But they say the rarity of deadly school incidents must be weighed against the likelihood that an influx of officers will raise the stakes on school discipline and funnel students into the juvenile-justice system for matters administrators should handle in-house.