The populist poetry of Mayor-elect Bill de Blasio’s campaign has officially entered prose mode with his appointment of Carmen Fariña as chancellor.

Fariña is a longtime educator who was a teacher, principal, superintendent, and instructional chief during her four-decade career in the city schools. After a seven-year retirement, she returns as chancellor to confront the pressing policy issues that face the nation’s largest school system.

Her appointment came with a pledge of a “progressive agenda” but few details about her positions on specific policy issues. Yet in the coming months, Fariña will have to bring her extensive experience to bear on thorny terrain that includes union contracts, charter schools, universal pre-kindergarten, struggling schools, curriculum, and much more.

Below, we run through a few of the education conundrums de Blasio and Fariña must confront — and some answers they might consider — as they begin the messy work of governing.

How should the city provide universal pre-K?

De Blasio’s central campaign pledge — to fund full-day pre-kindergarten for all New York City children by taxing the city’s highest earners — is both tricky and ambitious.

On the funding front, de Blasio must either convince Republican legislators and Gov. Andrew Cuomo to sign off on his proposal to raise income taxes for the city’s top earners, or figure out another way to generate $530 million over five years to pay for the pre-K expansion.

Fariña will likely leave the negotiations to de Blasio’s political staff. But if the money comes through, she’ll play a major role in figuring out how to provide full-day preschool to about 50,000 additional four-year-olds. Where will they go? Who will teach them? And what will they learn?

Fariña was already proposing space solutions months ago, suggesting at a public forum that real estate developers be required to build early childhood education centers that would also serve as community centers for middle school students. Underused city school buildings, out-of-use Catholic schools, public-housing community centers, and local nonprofits could also all potentially host the new pre-K seats.

Fariña’s résumé suggests that she would pay close attention to the quality of the instruction offered by the new programs. But while there are successful early childhood programs that could be mined for curriculum, finding and training people to staff the programs could be a steep challenge, in part because pre-K teacher pay can lag nearly $19,000 behind that of a starting kindergarten teacher.

Setting up the sites and hiring the teachers so that the expansion starts by next September, as de Blasio has promised, will not be easy.

“Realistically, getting that done in a year strikes me as extremely optimistic,” said Teachers College professor Aaron Pallas.

How should the city work with the teachers union?

Sources say Fariña was not the top choice for some in the United Federation of Teachers, which did not endorse de Blasio in the primary election. But Fariña’s experience as an educator earned her quick support from the city’s teachers and principals unions, and her remarks today indicated that she is serious about showing respect for the work that educators do.

Whether that tonal shift will translate into material gains for city teachers remains to be seen. After years of bitter clashing with the Bloomberg administration, the UFT hopes to win retroactive pay raises and other desired contract terms in talks with de Blasio and Fariña. Raises for the UFT and the principals union, which have gone for years without contracts, could amount to more than $3 billion.

Again, the financial picture is likely to remain mostly in City Hall’s purview. And de Blasio has said full retroactive pay for all municipal unions won’t happen, that any retroactive pay must be offset by cost savings, and that the UFT’s snub of him in the primary means he is not beholden to the union.

But Farina, who as a young teacher joined her colleagues in striking over the loss of planning periods and other affronts, will influence any changes to work rules for educators. She has said she wants to see a reduced role for test scores, suggesting that she might be receptive to requests to change the city’s teacher evaluation rules, and she is also likely to support calls to give teachers more time for training and collaboration.

Whatever deals de Blasio and Fariña hammer out with the UFT will send a strong message to the city’s 75,000 teachers, said David Steiner, the state’s former education chief.

“The shape of a contract is inseparable from the shape of the work in the schools,” he said.

How should the city handle struggling schools and teachers?

One of the biggest challenges that de Blasio and Fariña face, like their predecessors before them, is how to turn struggling schools around. The Bloomberg administration focused on closing low-performing schools and opening new ones in their place, an approach that followed a system of intensive support for weak schools that did not result in many closures.

Neither de Blasio nor Fariña has said that school closures should be taken completely off the table, but both have said the step should be taken only after aggressive efforts have been made to help schools improve. De Blasio has said he’ll establish an “early warning system” and deploy experienced principals and other staff to struggling schools.

And Fariña has proposed pairing pair principals from schools with similar student populations but widely varying student achievement to exchange ideas about what works and what doesn’t — something that she did as superintendent of District 15.

“Principal-to-principal, teachers-to-teachers, are the best vehicle to professional development that I know,” she said this fall.

Whether that will be enough to revamp chronically low-performing schools or bring about widespread change remains to be seen. According to a 1999 New York Times profile, Fariña’s strategies to overhaul P.S. 6 on the Upper East Side included introducing selective admissions criteria — a move that would hardly solve citywide issues.

Fariña will also have to decide how to adapt the Bloomberg administration’s school and teacher rating systems. De Blasio has promised to do away with the A-to-F letter grades that schools have gotten annually since 2007. But he has not said whether his administration would tinker with the existing system, for example by improving the school “peer group” system or adding more information for families, or overhaul the ratings system more substantially. Given that elementary and middle school grades have been based almost entirely on test scores, it seems likely that Fariña would push for change.

She’ll have somewhat less latitude in adjusting a new teacher evaluation system that UFT President Michael Mulgrew called an “unmitigated disaster.” There, Fariña will have to figure out how, if at all, to change the rules while still adhering to state law.

How should the city deliver support to schools?

Of all of the education policy questions facing the new administration, the one where Fariña has the most experience — how to deliver support to schools — is also where she has the most latitude to make change.

After several shakeups, the Bloomberg administration landed on an organizational structure that lets each school join a multi-borough support network of their choosing, while still being evaluated by local district superintendents. Critics of the arrangement say it can be confusing and result in too little guidance for struggling schools, while advocates say it allows schools to get exactly the help they need.

Whether to preserve the network structure was a major debate during the mayoral campaign, and some 120 pro-network principals recently wrote de Blasio urging him to create a “hybrid system” where principals who like their networks could keep them, while others could lean more on their districts.

Fariña hasn’t publicly spelled out a future for the network structure, but her long track record in the system will give her ample insight to draw on when devising a system of school support. As a former district and regional superintendent, she will have as good answers as anyone about how to balance superintendents’ authority with school autonomy and how to incorporate local stakeholders without cutting off cross-district ties.

Fariña’s avowedly progressive pedagogical preferences suggest that she might want to preserve the way that principals affiliate based on philosophy under the network model. But Elizabeth Phillips, the principal of P.S. 321, said that in her experience, Fariña successfully fostered collaboration among principals from schools with different cultures and student demographics within the district structure.

“One of the things I was so impressed with was how she brought together principals from very diverse districts,” Phillips said.

Even though Fariña doesn’t have to get anyone’s permission to craft a school support system, her vision could be constrained by politics. Even as critics of the current structure, like Mulgrew, say they’re open to compromise, it’s clear they’re pushing for big changes. “There’s no tweaking here,” the union leader said. “It’s just not working.”

How should the city treat charter schools?

During the campaign, de Blasio did not mince words about charter schools, saying they diverted attention and resources from the city’s traditional public schools and that the Bloomberg administration added “insult to injury” by offering them free rent in public-school buildings. He promised to halt such space-sharing arrangements and charge rent to some charter schools.

The question now is whether he can actually rein in charter schools without undermining the high-performing ones or pushing their powerful backers into attack mode.

One possibility would be to phase in the rent plan over several years, giving the schools time to adjust their budgets or look for cheaper facilities, said Pallas, the Teachers College professor. Another way to appease both sides could be to create a system for studying and sharing the innovations of the most successful charter schools, in an outgrowth of the collaboration that Fariña has said should be fostered.

Pedro Noguera, a New York University education professor, cautioned the new mayor not to devote too much time to making decisions about charter schools, which serve only about 6 percent of the city’s public school students.

“The more you’re fighting about charter schools, the less time you have to solve other problems,” he said.

That’s an attitude that Fariña seemed to share when she appeared with Diane Ravitch, the education historian who has fiercely criticized charter schools, at a Brooklyn school earlier this month. While she said she did not support the rigid approach of some charter schools, she also said the charter-district divide can mask areas where district schools should improve.

“Let’s worry more about what we need to do and how we need to do it positively than worry about them,” Fariña said. “Because, you know what, they also have our kids in their buildings, and to me, kids are kids.”

How should the city establish and advance broad ideals?

De Blasio and Fariña will have to decide how to deliver on other pieces of campaign rhetoric, including how to give parents a real voice in some decisions. They are starting that right away, they said today, by bringing the Department of Education’s Division of Family and Community Engagement directly under Fariña’s supervision and by scheduling a series of meetings with parent leaders in each district.

Fariña will also have to take stock of what is and isn’t working in terms of curriculum at a time when teaching and learning are receiving renewed emphasis. She has praised the new Common Core standards, which are meant to propel students toward college readiness, but is certain to scrutinize the curriculum choices that the Bloomberg administration made to help schools transition to the standards.

That scrutiny will come with the eye of someone steeped in literacy instruction and with a reputation for imposing instructional approaches on schools. When Fariña retired in 2006, then-UFT chief Randi Weingarten said that while the union respected her, it had also seen her dictates as “micromanaging of classroom instruction.”

Now, Fariña’s curriculum choices carry even higher stakes: Some educators and parents are already unhappy about the transition to the Common Core, and missteps early on could cost the new chancellor dearly.

The new administration will have the chance to reexamine broader ideals, too. School choice — letting families pick among diferent kinds of schools — was a hallmark of the Bloomberg era, but also a lightning rod for critics who say the system favors families able to navigate the sea of choices. Those critics will want De Blasio to revamp the school choice and enrollment system so that it benefits more families and a more diverse group of students end up at the most selective schools.

As a district chief, Fariña opposed tracking, or achievement segregation, within schools. But her record from P.S. 6 — and today, when her appointment was made at a middle school that screens students in part by their test scores — also suggests a tolerance for selective schools within the broader system.

How she and de Blasio choose to balance the needs of individual families, single schools, and the broader system in order to give all city students a fair shot at a great education cuts to the core of the de Blasio message, said Noguera, the education professor.

“If he recognizes that inequality is the big question facing New York, how is that going to show up in his education agenda?” Noguera asked. “And it’s got to be much more than just preschool.”