“A lot of people are saying that by calling it bullying, we are not taking it as seriously as we could, and not treating it as damaging and harmful as it really is,” said Amanda Nickerson, director of the Alberti Center for Bullying Abuse Prevention at the University at Buffalo. “Some are arguing we have to elevate this to other levels.”

Some municipalities have enacted local laws that can lead to parents being fined for their children’s actions. In North Tonawanda, N.Y., near Buffalo, parents may be fined $250 and jailed up to 15 days if their children engage in bullying behavior. Gouverneur does not have such a law.

Ms. French, the schools superintendent, said that she agreed with the police’s decision to treat the episode as a crime, as did Mayor Ronald P. McDougall of Gouverneur.

But some anti-bullying advocates said they believe the drive to turn young children into criminals over such behavior has gone too far.

“Parents are out for blood,” said Ross Ellis, the founder and chief executive of Stomp Out Bullying.org, a national organization. “I had a mother call me who wanted a 3-year-old on the playground arrested. I get that you don’t want your child beaten up, but it’s got to stop on both ends.”

Ms. Ellis said that the girls accused of bullying should receive counseling and that school officials and the girls’ parents should meet to explore the roots of the behavior. “It’s a terrible thing that happened, but make it a teachable moment,” she said.

Gouverneur, which has 4,000 residents and is about 30 miles from the Canadian border, is 95 percent white and 1 percent black, according to 2010 census data.