ATLANTA (AP) — For parents and politicians hungry for better schools, the idea of paying teachers more if their students perform better can seem as basic as adding two and two or spelling ‘‘cat.’’

Yet just a handful of schools and districts around the country use such strategies. In some states, the idea is effectively illegal.

That could all be changing as the federal government wields billions of dollars in grants to lure states and school districts to try the idea. The money is persuading lawmakers around the country, while highlighting the complex problems surrounding pay-for-performance systems.

Some teachers, like Trenise Duvernay, who teaches math at Alice M. Harte Charter School in New Orleans, want to be rewarded for helping students succeed. Duvernay is eligible for $2,000 a year or more in merit bonuses based on how well her students perform in classroom observations and on achievement tests.

‘‘It’s a reward for doing what we all have a passion to do anyway — making sure our kids master the skills they need in order to be successful,’’ Duvernay said.

Other teachers, like Debra Gunter, a middle school math teacher in Cobb County, Ga., say teachers can’t control which kids walk into their classrooms.

‘‘Your mother and father just got a divorce, your grandfather died, your boyfriend broke up with you: those kinds of life-altering events have an effect on how you do in class that day, through no fault of the teacher whatsoever,’’ said Gunter, echoing the position espoused by major teacher unions.

Some researchers have found student achievement improves when teachers get performance bonuses. Others have found no correlation.

Matthew Springer, director of Vanderbilt University’s National Center on Performance Initiatives, said the problem is that there are only a handful of valid studies, most from other countries.

‘‘I think the jury is still out,’’ he said.

The push for performance pay programs dates to 1950, but has mostly failed because districts and states didn’t get buy-in from teachers and couldn’t come up with objective ways to measure performance.

School districts in most states calculate pay based on seniority and level of education. For example, teachers who get master’s degrees generally get a pay bump.

In a massive survey of the nation’s teachers released in March, most said they value non-monetary rewards, such as time to collaborate with other teachers and a supportive school leadership, over higher salaries. Only 28 percent felt performance pay would have a strong impact and 30 percent felt performance pay would have no impact at all. The survey was conducted by Harris Interactive and paid for by the Bill & Melinda Gates Foundation and Scholastic Inc.

Still, lawmakers and education officials in many states are pushing the idea.

Washington, D.C., schools just reached a tentative agreement with their teachers’ union that would allow teachers to earn annual bonuses for student progress on standardized tests, among other benchmarks.

In Georgia, Gov. Sonny Perdue is pushing for a law requiring teacher salaries to be based on student test scores and other academic factors rather than years of experience and education.

Oklahoma lawmakers are considering a similar bill that would create a pilot program for teacher bonuses. In Louisiana, Colorado, Florida and Minnesota, where a few local districts have been offering merit pay to teachers for years, lawmakers and governors are aiming to create statewide programs.

The states and D.C. hope to win some of the $4.35 billion in highly competitive federal ‘‘Race the Top’’ money available this year to states that embrace education reforms like merit pay and charter schools. Tennessee and Delaware were initial winners of the money, garnering $600 million in part because of their teacher merit pay programs and their use of student achievement data in teacher evaluations.

Other states — particularly those like Georgia, Florida and Colorado that were among the 16 finalists for the grant competition — are hoping to get performance pay laws passed in time to reapply for the money in June.

‘‘We want to reward our educators who are truly making gains with our students,’’ said Perdue, who has used Georgia’s position as a finalist for the federal money to urge lawmakers to pass his performance pay bill. ‘‘To some, it’s become more of a job than a calling or a passion.’’

Powerful teachers’ unions in many states are fighting performance pay proposals, arguing that they lack thoughtful planning for how performance will be measured, and advocating that teachers should instead be paid more overall.

‘‘If you don’t engage teachers in the process of what the incentives are — they put them out there, and teachers don’t understand them and don’t believe they will work or be workable — they are not going to be incentives that mean anything. They’ll actually do the opposite. They will demoralize people,’’ said Randi Weingarten, president of the American Federation of Teachers.

Meanwhile, trouble brews in some states merit pay programs that already exist. A judge ruled recently that an Arizona performance pay program is unconstitutional because it’s open to only 28 out of more than 230 school districts

In Florida, just eight of 67 districts participate, although a bill before lawmakers would create a $900 million pot to woo more districts to the program.