Mr. Ebewo, through his lawyer, declined to comment for this article. His case took years to reach a hearing because of a state law that requires the city to show evidence that it has given the teacher a chance to improve and instruction on how to do so. Any missing file could jeopardize a case, lawyers for the department said.

And the hearing will probably go on for months, because of a rule the city agreed to four years ago. In an effort to impose more order on the process, the city and the union agreed to set up a panel of arbitrators to hear such cases regularly. There are only so many arbitrators, however, and lawyers can handle only so many cases at once, so the arbitrators hear a case only five days a month.

Howard C. Edelman, who has been an arbitrator on teacher cases for 20 years, said it was hard to judge the success of the city’s effort “because these are the hardest cases.” He said he preferred hearing cases of misconduct — for example, a teacher accused of corporal punishment — to hearing cases of incompetence.

“What you have 99 percent of the time is an administrator testifying about something they observed one and a half to three years ago,” he said. “They try to refresh their memory before the hearing, but you get into these arcane issues of were you asking a pivotal question of the students, and what are pivotal questions?”

One arbitrator recently stepped down from a case after the department said he had fallen asleep during a hearing. (The arbitrator said he might have “dropped off once or twice.”) The urge is not uncommon. Alan R. Viani, another arbitrator, said he had done “one or two cases, and it made me want to put a bullet through my head.”

In general, the department considers bringing incompetence cases when teachers have received consecutive unsatisfactory ratings from their principal. There are currently 190 teachers who have received at least two consecutive unsatisfactory ratings in the past three school years, and 586 more who received such a rating last year.

Fifty-five teachers are currently facing incompetence charges, eight others are in arbitration and one is awaiting a decision from a recently completed arbitration.