The number of New York City children registered to start pre-kindergarten on Thursday — now over 51,000 — represents more kids than in the entire school systems of cities like Seattle or Cincinnati. While Mayor Bill de Blasio touted the enrollment as an early sign of success for his pre-k expansion, the work of assessing and refining what happens in the classroom begins this week.

City officials announced on Wednesday that they will spend about $2 million on an independent study of pre-k programs to assess what works. About 200 sites will participate in the assessment.

And officials will separately assess children’s outcomes over time.

The mayor recently told an audience of pre-k teachers that his expectations are high. “I get up every morning and say, ‘We’ve got to make this work, we’ve got to make it fantastic. Not just good. It has to be great.’”

Deputy Mayor Richard Buery, who is managing the expansion, said the number of kids walking into pre-k classrooms today represents the intensive labor of thousands of workers — teachers, principals, administrators — but also fire officials, building and health inspectors, investigations officers, contract lawyers, and parent outreach workers.

“We’ve basically moved heaven and earth to get rid of every bureaucratic obstacle that could stand in the way,” said Buery.

Hundreds of new teachers were trained. Contracts were processed quickly, and classroom furniture companies ratcheted up production.

But critics said the rollout is happening too quickly.

Only days before the start of school, the city revoked contracts for nine pre-k programs because of health and safety problems, ranging from rodents to financial misconduct.

Of the 3,000 pre-k workers who gave fingerprints, city officials said Wednesday that 245 of them would be unable to work in pre-k centers because of a criminal record.

And in neighborhoods with high demand, parents complained there are not enough seats, and the application process was confusing.

“Maybe this is the hiccup part of his initiative,” said Reginald Swiney, who said he called 12 different programs near Sunset Park, Brooklyn, but could not find a seat for his son.

The mayor defended these setbacks, and said the commitment to safety is critical to the program's success.

“I hold a simple standard: Would I send my own child to one of these programs? The answer is yes,” he said.