The city has been fudging its school-vacancy numbers to prevent charters from expanding, critics charge — citing a report quietly released by Mayor de Blasio’s own education honchos.

Kids come as they are. Good schools find their curiosity and build their hope.

The topline job of schools is to renew a student’s sense of possibility and give her what she needs to make it real.

I think of schools that do this as the real renewal schools — to borrow a phrase.

Hundreds of city children are at risk of being “educationally homeless” next fall — victims of Mayor de Blasio’s refusal to give charter schools room to grow.

Research backs up what common sense dictates: Students in good schools learn more in a year than those in poor schools.

For instance, Stanford University has found that students attending New York City charters learn the equivalent of five more months of math in a year than their peers in traditional schools. Comparing good traditional schools against low performers also reveals large gaps in learning.

Instead of heading home after the bell rang Wednesday, thousands of New York charter school teachers donned gold shirts bearing the words “Teachers Demand Great Schools” and headed to Foley Square to eat hot dogs, wave signs and call on Mayor Bill de Blasio to expand the number of students in charter classrooms.

WASHINGTON — U.S. Education Secretary John King on Wednesday weighed in on a swirling schools controversy, criticizing what he called “arbitrary caps” on the growth of high-quality charter schools, publicly funded but, in many cases, privately operated K-12 schools in 42 states and the District of Columbia.

The NAACP, arguably the nation’s most influential civil rights organization, has approved a resolution that calls for a ban on new charter schools.

It is unfathomable that the NAACP is trying to halt a movement working to reverse the achievement gaps for black and brown children in public school education nationwide.

The city Department of Education just beat a temporary retreat from its clumsy plan to desegregate schools on the Upper West Side — but the whole sorry tale exposes DOE’s more fundamental failure to offer good schools in minority neighborhoods.

Thousands of parents and teachers descended on Foley Square Wednesday to demand that Mayor de Blasio get behind charter school proponents’ plans to increase the number of charter seats across the city to 200.000 by 2020.