In Boston, the public schools have asked the police to step up visits to elementary schools throughout the day on Monday.

In Denver, psychologists and social workers were prepared to visit students.

Schools in New York City were encouraged to review safety measures, which include posting security officers in lobbies and requiring identification from all visitors.

And the Chicago school district urged principals to conduct lockdown drills, with reminders to stay low, keep quiet and turn off classroom lights.

Parents, teachers and school administrators in Newtown, Conn., confront the most immediate and raw tasks of helping children respond to the horrifying killings of 20 children at Sandy Hook Elementary School.

But across the nation, as schools prepared to resume on Monday, officials and parents spent the weekend worrying about not only how to talk to students about what happened, but also how best to discourage its happening again.

Tom Boasberg, the superintendent of schools in Denver, said he had not yet determined whether to ramp up drills. “When you read the story of what happened at Sandy Hook, you realize, ‘Holy cow, they did a lot of things right,’ ” he said.