Professor Fryer said that under his program, fourth graders and seventh graders who take the new round of mandatory standardized tests that the city is introducing in the fall would be rewarded with at least $5. They would get more money for high scores, with a cap of $25 for fourth graders and $50 for seventh graders. In addition, each participating school would receive $5,000.

Money for the payments will come from private backers, Professor Fryer said, because there would be no public money available for them.

The prospect of cash introduced into the classroom has made some local educators uneasy.

“It makes me really nervous,” said Maggie Siena, the principal of Public School 150 in TriBeCa. “I suspect paying kids for achievement in any way tends not to work.”

Ernest Logan, the president of the principals’ union, also expressed concerns. “We are troubled by additional pressure being placed on children to achieve perfection,” he said. “What really matters in education is continued student progress, not perfect test scores.”

Professor Fryer and other educational scholars have argued that some children, especially those from impoverished backgrounds, lack the foresight and role models to be self-motivated.

“The fundamental problem with education and motivating kids to learn what they need to learn is that the payoffs are so distant,” said Tom Loveless, a senior fellow at the Brookings Institution, a left-leaning research organization. “So it’s very hard to motivate students to do well. Good students get that motivation from somewhere — from peers, their parents, how they’re raised — but the kids who are unmotivated have a very hard time understanding that what they do today pays off decades from now.”

Eric Nadelstern, the chief executive of the school system’s empowerment initiative — which gives principals more autonomy to run their schools — lauded Professor Fryer in a recent e-mail message to principals. “He has my enthusiastic support,” Mr. Nadelstern said. “I encourage you to give the program serious consideration.”