Chancellor Dennis Walcott read to a group of 4-year-olds at the Bank Street Head Start center in November 2011.

American Federation of Teachers President Randi Weingarten fueled mayoral candidate Bill Thompson’s attacks on Public Advocate Bill de Blasio’s plan to fund universal pre-kindergarten, calling Thompson a “doer” and de Blasio an idealist.

“We need a mayor in the city of New York who will take this idea and actually get it done and not base it on a tax that may never materialize,” Weingarten said during a call with reporters that the Thompson campaign arranged.

Since last week, Thompson and his allies have been criticizing de Blasio’s plan, which would raise taxes on New Yorkers earning over $500,000 a year to fund universal pre-K. They say de Blasio’s plan relies too much on approval from Albany and does not consider that the state doesn’t even use all of the state pre-K funding that it gets.

Their first point is a fair one. De Blasio’s plan would require legislative approval, a step he says would come readily but which could be a heavy lift. The New York Times cited this shortcoming to explain why it did not endorse de Blasio.

But on the second point, about the unused state funding, Thompson’s campaign’s math does not add up. Calculating the true cost of expanding pre-K to all city 4-year-olds is a challenging task, pre-K advocates say, but no matter how the numbers are crunched, they suggest that the city would need more funding.

Last week, State Sen. Diane Savino said the real issue is getting the state to change laws so that the city can use funding earmarked for half-day programs to pay instead for full-day slots, which families in the city prefer.

Savino was accurate in saying that the city returned $31 million in pre-K funding to the state in 2011. But in the 2011-2012 school year, the city received $225 million from the state for general education pre-kindergarten classroom instruction, and only about $2.1 million went unspent, according to the the Independent Budget Office. Last year, the city again received about $225 million and left about $4.9 million on the table.

How many pre-K seats would last year’s leftover funds have paid for? Fewer than 800, using the city’s average spending of about $6,300 per student that the IBO has calculated.

And considering that many more children in public pre-K in the city attend half-day, the real number of new full-day pre-K seats that the leftover money could create is almost certainly far fewer.

“Clearly the funds we are returning are far from sufficient to expand full day pre-k by 10,000 slots and convert thousands of part-day slots to full day,” said Doug Turetsky from the IBO.

Today, Weingarten dialed back Savino’s comments by saying that while some money already exists for universal pre-K, additional funding might be needed — and hard to come by.

“You need to have a plan first about how to implement it before you ask people for a funding increase,” Weingarten said. “When you put out a plan that says we’ll raise money from the rich so we can fund pre-K, when Albany has already called it dead on arrival, sounds non-serious.”

Weingarten said she wasn’t “smart enough to know” how much money is needed to fund universal pre-K in New York City. (Thompson has proposed expanding pre-K but has not quantified the degree of expansion he hopes to achieve.)

Betty Holcomb, director of public policy for the Center for Children’s Initiatives, said nailing down that number is indeed a complex endeavor that researchers in the state are actively tackling. There are many different factors to take into account, she said, such as if other funding sources will be mixed in, the exact length of the day, and how many days a year it is available.

But she said the bottom line is clear.

“There’s no magic bullet and we’re open to all proposals. We’re not supporting one plan over another,” Holcomb said. “But we definitely believe that there has to be further investment to get there.”

Listen to GothamSchools’ Geoff Decker discuss the pre-K issue on the Brian Lehrer Show: