Many principals said they felt they were getting more support now than they got under the Bloomberg administration. In interviews, principals cited the emphasis on collaboration among schools, the new professional development programs being offered and the fact that schools are no longer assessed primarily on their students’ test scores as examples of a more collegial, less punitive atmosphere.

“The feeling is less threatening,” said Kenneth Zapata, the principal of I.S. 75 Frank D. Paulo on Staten Island. Although the students at I.S. 75 score better on state tests than the citywide average, Mr. Zapata said the Bloomberg administration’s school report cards put principals in competition with one another because schools received points based on how much their students improved on the tests compared with students at similar schools.

“Not pitting us against each other, but that we’re all in it together, sharing best practices — to me, that’s the one thing that’s changed the most,” Mr. Zapata said.

But some principals have expressed concern that the teachers’ union — now that it has allies at City Hall and at Tweed Courthouse, the Education Department’s headquarters — has become too empowered, and that department officials will no longer necessarily support them when they have disagreements with union representatives. Two high school principals in Manhattan and Brooklyn said they chafed at receiving an email signed by Ms. Fariña and the union president, Michael Mulgrew, saying principals could not routinely collect lesson plans from teachers or dictate what they put in their lesson plans. A principal of an elementary school in Queens said his union chapter leader sent an email to his staff announcing who his new superintendent was before he had been informed. All three principals declined to be identified, fearing reprisals by the union and the Education Department.

Ms. Fariña was known as a principal who convinced ineffective teachers to leave, sometimes even finding them other jobs. During her visit to M.S. 334, she took the principal by surprise by asking him, “How many teachers do you feel that you need to encourage to find other ways to spend their time?” Mr. Samerson laughed awkwardly and asked, “Is that a trick question, Chancellor?” before she assured him that she was serious. He replied that he had three teachers who received low ratings last year, but that one of them was attending a professional development program and was eager to improve.

Statistics so far, though, do not reflect that concern for getting subpar teachers out of the classroom. The number of so-called 3020-a cases — proceedings to remove tenured teachers, for either ineffectiveness or wrongdoing — initiated by the city in 2014 was down almost one-fourth from 2013, to 306.

Devora Kaye, the department’s spokeswoman, did not offer a reason for the decline, but said that the number of cases would return to 2013 levels this year.