The Education Department originally said that teachers from the reserve might be placed in schools that are part of Mayor Bill de Blasio’s Renewal program, in which the city is spending more than $500 million to turn around failing schools, but this week said that would not happen.

As of Oct. 4, schools listed approximately 800 teacher vacancies, according to an Education Department spokesman, though that might not reflect the exact number of jobs available; he said he could not give a breakdown of where the vacancies were.

In a system in which only 1 percent of teachers earn the lowest possible ratings, of ineffective or unsatisfactory, 12 percent of the teachers who were in the Absent Teacher Reserve at the end of the last school year had received one of those ratings in 2015-16.

Nearly 40 percent worked in schools that were closed for poor performance. Another 30 percent were “excessed” from their positions because of budget cuts or because the number of students enrolled at their schools fell. Principals are sometimes forced to let go of teachers they would like to keep for those reasons, but they can also use them as a way to get rid of low performers. The rest landed in the pool because of legal or disciplinary charges.

Troubling Histories

As of last spring, roughly a quarter of the teachers in the reserve were in it five years earlier. These teachers are the most likely to be problematic, since their history suggests that no principal has been willing to hire them, or that they have actively avoided getting a permanent assignment. Among this group, principals say, are teachers who are incompetent or mentally unstable.

“So many of them, in my opinion, weren’t capable of leading a classroom again, or ever were,” said Matt Williams, the founding principal of a small high school, Bronx Design and Construction Academy, who left the department in 2014.

To identify teachers from the reserve with troubling histories, The New York Times cross-referenced two sets of records: the Education Department’s Excessed Staff Selection System, which lists available staff and openings in the system, and arbitrators’ decisions in disciplinary cases, which are available from the New York State Education Department.