The eight children, ages 5 to 11, who attend the Brooklyn Autism Center Academy need intensive individual instruction to cope with a neurological disorder that can make achieving academic progress slow and grueling.

During the course of the day, one teacher is paired with each child. After successfully completing a task, students are rewarded with a spoonful of vanilla pudding, time on a piano or a few minutes in a bouncy castle. The system repeats itself, interspersing work with small breaks.

“Every child with autism can learn,” said Jaime Nicklas, 32, the school’s educational director. “If they are not learning, it is our responsibility to change our teaching procedure, so they can make the progress they are capable of.”

But this type of focused instruction comes with a high price: The academy’s annual tuition is $85,000. The parents of one of the students, Ruby Kassimir, 5, the only girl in the school, took out a home equity line of credit on their home in Queens to help pay the tuition. “There just aren’t that many options available,” explained Ruby’s mother, Sue Laizik, a project coordinator at Columbia University.

As the number of autism diagnoses has risen, the extraordinary cost of educating the children has become a growing point of contention. In 2001, the city’s Department of Education listed 3,278 students with autism; by 2008, that figure had more than doubled to 6,877.

The public school system is required by law to provide an appropriate education for such children, even if it means paying for private school tuition if there is no public school option (although, as Ruby’s parents found, getting the school system to pay is not always easy).

“The crux of the matter is that we need to have a public debate about how much are we willing to invest in making individuals who are disabled, and sometimes profoundly disabled, have a meaningful level of membership in society,” said Gil Eyal, a sociologist at Columbia University who has done research on autism.

Of the more than 6,800 children with autism recorded by the city’s public schools, 4,200 are enrolled in special education classes with a small student-to-teacher to ratio, 285 students are part of a program where children with autism are taught alongside regular education students and 28 are in a charter school with a one-to-one ratio between teachers and students. That school, the New York Center for Autism Charter School, is the only public school in New York City offering intensive one-on-one instruction.

Other autistic students attend private schools from a list of those approved by the state, and their tuition, which ranges from $30,800 to $48,100, is paid by the city’s Education Department. Finally, if parents are dissatisfied with any of the options offered by the public schools, they can choose another private school, one not on the list, at their own expense and seek to have the cost reimbursed by the city.

For all special education students, the department paid $88.9 million for private school tuition last year, compared with $57.6 million in 2007. “Private school tuition claims are a growing burden for us,” said Michael Best, the Education Department’s general counsel.

Ms. Laizik, Ruby’s mother, entered her daughter in the lottery for the New York Center for Autism, and said she broke into tears when she learned that Ruby had not gotten one of the spots on the waiting list. “That’s when it really hit me, how hard it’s going to be,” Ms. Laizik said.

Three separate evaluations of Ruby, between the ages of 2 and 5, emphasized the need for one-to-one instruction.

So when she was not able to enroll Ruby in the public charter school, Ms. Laizik sent her to the private Brooklyn Autism Center Academy and filed a claim with the Education Department seeking tuition reimbursement.

After a hearing, a departmental judge ruled in March that Ruby’s parents were entitled to a 30 percent tuition reimbursement because the city had failed to offer Ruby appropriate placement. They are now appealing to the New York State Education Department’s Office of State Review for the remainder of the tuition.

For the parents of autistic and other special-needs children, springtime is usually when they hear back from the city’s Education Department about their claims for private school tuition reimbursement.

During the 2007-8 school year, there were 4,375 reimbursement hearing requests for special education students, 462 of them for children with autism.

“We are concerned that some parents see this as a way for us to pay for private school,” Mr. Best said. “It’s not supposed to be a vehicle to get private school tuition if there’s something appropriate available in the public schools.”

But parents of autistic children and their advocates argue that any hope for progress requires the kind of concentrated intervention that the public schools cannot always provide.

“The giant elephant in the room, if one in 150 children are being diagnosed with autism, is that they have the same life expectancy,” said Gary S. Mayerson, a lawyer who has represented more than 1,000 families, including Ruby’s, making claims for tuition reimbursement. “Either invest the money now for effective programming or find that your efforts are inadequate.

“At some point you may be staring at the prospect of an even more expensive residential placement — and the state will be footing the bill.”

Despite the onerous financial burden they are confronting, Ruby’s parents are pleased with their decision, having seen that their daughter has made significant progress since she started at the school in September.

Ultimately, the goal is for Ruby to be able to learn in a mainstream classroom.

“The thing that stays with me the most is what kind of life she will be able to have,” said Ruby’s father, Ron Kassimir, 51, an associate provost at the New School.

His wife, Ms. Laizik, added, “When you have a child like Ruby, you realize how much of a role you have to play in that outcome, how involved you have to be to affect that outcome — especially early on, when the stakes are so very high.”