Last year, Zipporiah Mills emailed schools Chancellor Carmen Fariña to inform her that she had recently been diagnosed with breast cancer and would soon be stepping down as principal of P.S. 261 in Boerum Hill, Brooklyn.

Within minutes, the head of the country’s largest school district was on the phone with the school’s secretary, asking to speak with Principal Mills. Weeks later, after Mills underwent surgery, Fariña appeared at her school with a bouquet of flowers.

“Those humane things you do to build culture in a school building, to build trust,” said Mills, who is now retired from the school system, “she lived by that.”

The story — one of thousands that parents and educators across the city could tell about their personal interactions with a chancellor whom many simply call “Carmen” — is not just an illustration of the character of the schools chief, who announced Thursday that she will retire in the coming months after four years on the job. It is also a guide to her school-by-school, person-by-person approach to leading a school system with as many students as Rhode Island has people.

Fariña’s many fans see her hands-on style — coupled with her insistence on building on the system’s strengths rather than blowing it up and starting fresh — as her greatest gift, and its own type of innovation. (It is also the polar opposite philosophy of her predecessors under former Mayor Michael Bloomberg, who privileged systemic disruption over targeted tweaks.) When Fariña, 74, who began her half-century career in the city schools as a teacher, was tapped as schools chief by newly elected Mayor Bill de Blasio in 2014, one principal called it a “dream come true.”

But Fariña’s detractors see her faith in the perfectibility of the school system she knows so well as a critical flaw. To them, her encyclopedic knowledge of the city’s 1,700-odd schools and many of their leaders led her to miss the forest for the trees — which, they say, is a system that serves high-achieving and privileged students well, but leaves many disadvantaged students of color behind.

“Like all of us, I think her strength is also her weakness,” said Clara Hemphill, director of education policy and Insideschools at the New School’s Center for New York City Affairs. “She knew an awful lot of principals and visited an awful lot of schools. But sometimes you need to step back and look at the whole picture.”

Fariña’s school visits are legendary. In her first year as schools chief, she toured roughly 200 schools, sometimes conducting five 90-minute visits in a single week. Drawing on her years as the equally feared and revered principal of P.S. 6 on the Upper East Side, she would walk briskly from classroom to classroom inspecting students’ work, sizing up the hallway decorations, and issuing a steady stream of commands to principals, teachers, and anyone else within earshot.

On one visit, with reporters in tow, she advised a principal to observe a weak teacher multiple times a day. “Not everything has to be knocking people on the head,” she told the gathered press, explaining her method for nudging out ineffective educators. “But if that’s what it takes, we’re happy to do that as well.”

Fariña also attends monthly town-hall meetings, where she engages parents on topics as granular as the amount of homework teachers should assign and as grand as how schools should prepare students to live in the world. Early on, she was known to phone individual parents later on to provide answers that she didn’t have on the spot.

The visits, the Q-and-As, and the hundreds of parent and teacher emails she reads are rich data sources for Fariña, who favors the qualitative over the quantitative. “This is about stories,” she said at a press conference Thursday. “It’s not about the statistics.”

Armed with that anecdotal evidence, Fariña acts as the system’s chief troubleshooter. She tells individual teachers to incorporate more writing into their lessons, school guards to smile at parents, and principals to bring her their complaints so she can “take it one issue at a time.”

One of her first moves as chancellor was to grant principals’ more email storage space. The small-bore initiative, like many that followed it, spoke to Fariña’s philosophy that school-system transformation must be driven by educators — and that they can function at full capacity only when feeling motivated, celebrated, and supported. “You can’t have people working well if their morale is low,” she explained last year, adding that she gauges her own success not just by how well students score on exams, but also by the number of teachers and principals who remain in the system.

Fariña’s emphasis on joy and cooperation over intensity and competition as the fuel of school improvement marked a sharp contrast with Bloomberg’s worldview, which considered radical change a moral imperative in a system where the majority of students still cannot meet grade-level benchmarks. From the start, she made it her mission to reverse many Bloomberg-era policies — to “undo while you’re doing,” as she memorably put it. That meant diminishing the office of new-school creation, eliminating school letter grades, and halting the closures of all but a handful of chronically underperforming or underenrolled schools.

She oversaw de Blasio’s signature education initiatives — the vast expansion of free prekindergarten and a half-billion-dollar effort to “renew” floundering schools — but her own pet projects were more narrowly focused on the craft of teaching and running schools. She created a school-to-school mentoring program (modeled on a system she employed as a superintendent), promoted new social-studies materials (she wrote a popular social-studies curriculum when she was a teacher), and pushed for 80 minutes of weekly teacher-training time to be written into the teachers contract she helped negotiate. “Training is everything,” she has said.

To enact her agenda, Fariña undertook what may the most sweeping change to come out of her office, rather than City Hall: an overhaul of the education-department hierarchy that re-empowered superintendents, who had been demoted under Bloomberg in an effort to put more power in principals’ hands. The move, which returned the system to a traditional power structure, shifted more authority back to the central office, with superintendents once again serving as the chancellor’s “eyes and ears.”

The reorganization was meant to standardize practices and establish clear chains of command across the vast school system. But it has also left some school leaders feeling micromanaged and beholden to superintendents whose small staffs allowed them to offer directives but not hands-on help, which is outsourced to new support centers.

Under Fariña, “principals have less autonomy and less authority over their schools,” said a principal who requested anonymity because she said she feared retribution for speaking out — “despite the continual suggestion that there’s trust and collaboration throughout the system.”

Even if some educators have chafed under Fariña’s watch, test scores and graduation rates have steadily improved across the city — though not at the troubled schools Fariña said this week she intends to close, despite the mayor’s promise of rapid turnarounds. “You cannot deny that our school system has moved a great deal under Carmen Fariña,” said United Federation of Teachers President Michael Mulgrew, a reliable de Blasio administration ally.

However, Fariña herself is unlikely to judge her success by such real-time indicators. Instead, she will be watching what happens to her agenda after she leaves.

“If your successor comes in and the first thing they’re going to do is get rid of some of the policy,” she said shortly after becoming chancellor, “then it probably wasn’t good policy in the first place.”