But that surge, coupled with the madrasas’ poor record in nonreligious subjects, high dropout rates and graduation of young people with few marketable job skills, worried the government. It responded by making primary education at public schools compulsory in 2003, allowing exceptions like the madrasas, provided they met basic standards by 2010. If they fail, they will have to stop educating primary school children.

Image A kindergarten class at Al-Irsyad Satya Islamic School, in Kota Baru Parahyangan, Indonesia. This school, which used the Singapore school as its model, opened two years ago. Credit... Norimitsu Onishi/The New York Times

“That forced the madrasas to shift their curriculum away from being purely religious schools,” said Mukhlis Abu Bakar, an expert on madrasas at the National Institute of Education, a teachers college.

Last year, the first time all six madrasas were required to sit for national exams at the primary level, two failed to meet the minimal standards, though they still had two more years to pass.

Al Irsyad, which was the first to alter its curriculum, outperformed the other madrasas. But neither it nor the others made any of the lists of best performing schools or students compiled by the Education Ministry in Singapore.

Mr. Mukhlis, who also was a member of Al Irsyad’s management committee in the 1990s, said the madrasas still had a long way to go to catch up with mainstream schools. While Singapore’s teachers are among the most highly paid civil servants, the madrasas have had trouble attracting qualified teachers because they rely only on tuition and donations to operate, he said.

“I think Al Irsyad has not achieved a level where I would say it is a model for Islamic education,” he said, “but somehow the system it has in place could become one.”

Still, it began drawing students who would not have attended a madrasa otherwise. Noridah Mahad, 44, said she had wanted to send her two older children to madrasas but worried about the quality of education. With Al Irsyad’s adoption of the national curriculum, she felt no qualms in sending her third child. “Here they teach many things other than Islam,” she said. “So Muslim students will have two understandings: the Muslim and the outside world.”