From left, Christine Quinn, Kelly Shannon, Scott Stringer and Vicki Sando, founder of the Greenroon Environmental Literacy Laboratory, and state Senator Tom Duane.

Manhattan Borough President Scott Stringer and City Council Speaker Christine Quinn are likely in for a year of confrontation as they prepare for their prospective mayoral bids.

But on Friday morning at the opening of P.S. 41′s new 7,000 square foot rooftop garden, they were happy to agree on one thing.

“We agree that you’d be on any mayor’s short list for chancellor,” Stringer said to P.S. 41′s principal Kelly Shannon during a speech in the Greenwich Village elementary school’s gymnasium.

The Democratic primary is still a year away, making serious contenders unlikely to make any declarative statements on education or anything else. But who a mayor considers — and eventually selects — to be his or her chancellor is one of the most telling hints for how a candidate plans to guide education policy, which is shaping up to be a defining issue in the race.

“There are two major decisions the next mayor’s going to make in this town. The first is, who’s gonna be the police commissioner? And then who’s gonna be the schools chancellor?” Stringer said.

Stringer later clarified that his comments were meant to be more of a reflection of Shannon’s ability to run a school and coordinate a large scale project that took six years and cost more than $1.5 million to complete than an indication of who he’d pick to run the school system if he were elected mayor.

But he still praised her abilities as principal. “The skill set that she demonstrates is really a skill set that we should look at for whoever the next chancellor may be,” said Stringer, who combined with Quinn contributed $1.3 million to the project.

The garden party eventually moved upstairs to the converted roof, which is now home to more than two dozen species of plants, solar paneling to fuel rooftop electricity and a water fountain that runs only when the sun is out. The roof, called the Greenroof Environmental Literacy Laboratory, opened last week for students, who teachers said would use it to enhance learning units on solar energy, plant biology and the scientific method.

When asked to discuss what they considered important qualities for a schools chancellor to have, Stringer and Quinn agreed again. Both said an extensive education background was important, but neither would say it was absolutely necessary.

“I think there are a lot of different kinds of folks who can bring a lot to our education system,” Quinn said.

There isn’t much precedent in New York City for schools chancellor appointment since control of the system was moved under the mayor’s office. Bloomberg’s shortlist in 2002 consisted of just one person with an extensive background in education, then-Cleveland schools chief, Barbara Byrd-Bennett, according to the Daily News at the time. Byrd-Bennett, now second-in-charge below Jean-Claude Brizard in Chicago, was then-Deputy Mayor Dennis Walcott’s pick, but the job ultimately went to Joel Klein, a lawyer with business experience.

Bloomberg’s two succeeding chancellors, Cathie Black and Dennis Walcott, had barely any experience working in schools.

James Kemple, executive director of the Research Alliance for New York City Schools at New York University, said he believed that both classroom and management experience were important for the job. But “the most important factor for success will be the capacity to attract a leadership team that brings all of these skills to the task,” he said.

For her part, Shannon didn’t shy from the flattery and said she was honored to be mentioned. But she said as a veteran teacher and principal who has worked for 18 years in the school system, “experience is important.”

“You have to remember to bring it back to the people who are living on the front every single day,” Shannon added. “Make sure their voices are heard.”