New York City is paying private contractors more than $1 billion this year to operate a little-known special education program for 3- and 4-year-olds, nearly double the amount it paid six years ago.

The program serves 25,000 children with physical, learning, developmental and other disabilities. While the number of children in the program has risen slowly in recent years, annual costs have soared to about $40,000 per child, according to an analysis of city education spending by The New York Times.

The city pays private contractors to provide classes, as well as individual instruction at homes, day care centers and nursery schools. Children may also be prescribed speech, physical and occupational therapy in half-hour sessions, several times a week.

The prekindergarten program is far more expensive in New York than it is elsewhere, and oversight by the city and state has often been lax, according to interviews with officials, regulators and contractors.