CLEVELAND -- At Martin Luther King High School one morning, two students forgot momentarily for whom the school was named. One girl told a friend that another girl wanted to fight her. When the two saw each other in the hall between classes, hard looks led to harder words. "You're a stupid bitch," one said to the other in what was to be the gentlest thought expressed.

As it happened, a vice principal overheard the tough talk. She interceded -- with a difference. Instead of just breaking up the fray, which is the customary work of vice principals, she told them to head for the mediation room. That afternoon, the two girls, plus the third who turned out to be lying about the proposed fight -- came together with five student-conflict mediators.

Ninety minutes were needed to cover the basics of mediation: Get the story from all sides, examine the problem, explore and choose solutions and leave with a signed agreement.

I wasn't at this particular session but the mediators -- Tiffany Jones, Laurella Mayes, Shakir Ball, Michael Lowery and Demarko Williams -- ran through the details of that case and many more. The names of the students are worth knowing because they are teaching the adults of Cleveland -- and the rest of us if we'll listen -- that if you work at peace, peace works. King High School has one of the lowest rates in the city of suspensions for fighting.

A local verifier of that is Carole Close, a public school social studies teacher for 23 years and King's faculty adviser to the mediation program called WAVE -- Winning Against Violent Environments. Close works 400 hours a year beyond her regular teaching assignments -- weekends, after school, in the summer -- to spread the program to other schools in Cleveland. When society asks what should be done about stopping violence in schools, Close has an answer: Educate the young about the methods of nonviolence. "It should start early," she says in the mediation room after her five students left for the day. "Elementary school -- even kindergarten -- is the time. High school can be too late."

At Martin Luther King High School, which is in the Hough area of Cleveland (where some of the nation's harshest racial clashes occurred in the late 1960s), almost one out of 10 students is involved in the mediation program. King is the school that local politicians praise when they want to say there is hope for inner-city public education.

Nationally, mediation and conflict resolution programs are what school boards, faculties and parents increasingly turn to when overwhelmed with violence in schools. While some refer to Gandhi -- "To reach real peace in this world we shall have to begin with the children" -- and others agree with Maria Montessori -- "Establishing lasting peace is the work of education" -- all are realizing that resolving conflicts nonviolently is as necessary a subject in schools as math, science and English.

The National Association for Mediation in Education, the five-year-old Amherst, Mass., membership organization, reports that it had 200 programs listed in its 1988 directory. This year's directory will see at least a 50 percent increase.

Similar growth is reported by the Grace Contrino Abrams Peace Education Foundation in Miami. Its founder and director, Fran Schmidt, who has been innovating in the field for more than 20 years, has been listening as well as teaching: "The message from the young is clear. Our children want a world without violence, without drugs and without the kind of anger that destroys people. Our challenge is to change their perception of peace from being a golden fantasy into an achievable reality by teaching them the real-life skills that can be used to make every day a little better. Peace isn't winning the lottery. It's building relationships, it's compromising, it's trust and risk."

Perhaps more than any other state, Ohio has the philosophical conditions for mediation and conflict resolution programs to flourish. The source of energy and vision is the governor, Richard Celeste. He is the nation's first governor to spend tax dollars on a peace commission. The $1 million enterprise is carrying out Celeste's 1987 State of the State Address in which he named peace education one of the four major goals for his second term.

Reactionaries in the state are calling the Ohio Commission on Dispute Resolution and Conflict Management the "Peace, Love and Dove Commission," which is good for a laugh but not much more. Celeste, the Peace Corps director in the Carter administration, has exactly the right response: "Peace isn't some abstract, mushy notion. Everyone has a responsibility to create a more peaceful environment."

If anyone doubts the wisdom of that, ask the student mediators at King High School. The two girls who were in the hallway fight have become friends. Ex-enemies are everywhere here.