On Tuesday, about 40 teachers listened, with notebooks open and pens ready, as Sharon Burns lead a discussion on classroom learning.

Hired by Bank Street College of Education to prepare teachers for the first year of the mayor’s expanded universal pre-K, Burns asked the teachers how they’d help students learn to problem-solve.

“Zippers,” one teacher offered. The answer made sense. “Some kids keep trying and trying. Then a lightbulb will go off,” she explained. “That will be the kid who wants to teach the others.”

Weeks away from the first day of school, the city has put a lot of faith in Bank Street to train a whopping 4,000 teachers to ready them to teach the city’s four-year-olds. The program cost $2.2 million and will include four separate three-day sessions, the first day of which was Tuesday.

The training offers a glance into the city’s goals for universal pre-K, which includes familiarizing educators with the pre-K Common Core standards and with ways to help students develop social and emotional skills. And, consistent with Schools Chancellor Carmen Fariña’s past initiatives, the program is designed to help teachers share their own learning and connect with colleagues who can help them in the future.

In its effort to reach as many pre-K teachers as possible, the city offered the program free to anyone teaching universal pre-K this fall. That means teachers from public schools, charters, and community-based organizations. They range in experience from brand-new teachers, to new to pre-K, to veteran early education teachers. Some were very familiar with the terminology and philosophies in the handbook, others were brand new to it.

“New York City is very uneven,” said Shael Polakow-Suransky, the former deputy chancellor who is now the president of Bank Street College of Education. “It’s an opportunity to shift the practice.”

Some teachers said one obstacle is overcoming parents’ — and some teachers’ — view of pre-K as glorified child care.

“This is not babysitting,” said Bank Street’s dean, Virginia Roach.

“The biggest challenge is getting people to understand that children are learning through play,” said Donna Fuentes, a pre-K teacher at the Ralph Hirschkorn Child Care Center in Queens.

Participants take home a hefty handbook designed to help teachers turn fidgety little-kid behavior into learning opportunities, for example. What does that look like in the classroom? If you have a student who bumps into other kids, the handbook suggests giving that child extra space on his rug square to help him define his personal space.

Polakow-Suransky offered a personal example. During a pre-K class at Bank Street, he noticed that one student had a large glob of playdough and another student had none. Resisting the impulse to pointedly ask that student to share, Polakow-Suransky instead asked him how he thought the other child felt without any playdough.

The student parted with a pea-sized ball of dough, then a little more. “It was enough to make it work,” Polakow-Suransky said.

Many of the teachers in attendance were already familiar with these ideas, but others had a ways to go.

“It’s a little more than I expected,” said James Cervantes, who is set to start as an assistant teacher at Alpha Academy, a new school still enrolling kids. “I honestly didn’t know it’s that detailed for little kids.”