The presidential candidates are trading blows over education, with each senator declaring the other unequipped for the task of improving America’s schools.

Senator Obama threw the first gauntlet yesterday with a major speech; the proposal of a $1 billion policy package that includes unconventional choices such as charter schools and performance-based pay for teachers, and a television ad declaring that Senator McCain “doesn’t understand” how to make American schools better.

The McCain campaign shot back a fast response, charging that Mr. Obama has a “failing” record on education and releasing a TV ad arguing that the Illinois senator’s greatest accomplishment in the field was to push for comprehensive sex education for kindergartners.

Behind the rhetorical barbs are stark differences in how each candidate would approach the issue as president — as well as some open questions on important specifics that neither candidate has clarified.

The package Mr. Obama unveiled yesterday would significantly expand the role of the federal government in improving schools; the dollar figure by which he would expand the U.S. Education Department’s budget is $19 billion, an increase of nearly one-third from current discretionary spending, $59 billion.

Mr. McCain is proposing no increases in domestic discretionary spending, including in the field of education.

Mr. Obama has also been more critical of using the kind of “accountability” measures that were President Bush’s trademark with the No Child Left Behind law — efforts to define schools as failing or not based on their students’ test scores, and to punish those schools where students were not scoring well.

While Mr. Obama has criticized the law as pushing art and music out of curriculums and focusing students too much on test-taking and not real learning, Mr. McCain has embraced a national effort, led by the New York City schools chancellor, Joel Klein, to expand No Child Left Behind’s philosophy of using test scores to zero in on and eliminate failing schools, principals, and teachers.

Mr. McCain has taken advantage of this difference to jab Mr. Obama, saying he is too beholden to teachers unions to make a serious attempt to change schools.

“Senator Obama wants our schools to answer to unions and entrenched bureaucrats,” Mr. McCain said in his speech accepting the Republican Party’s nomination. “I want schools to answer to parents and students.”

Mr. Obama’s education proposals yesterday appeared to have been designed to answer this criticism, which has been voiced not just publicly by Mr. McCain but also privately by some Democrats who worry that Mr. Obama is not challenging teachers unions strongly enough.

“Democrats, I’m speaking to Democrats now, Democrats have to realize that fixing No Child Left Behind by itself is not enough to prepare our children for a global economy,” Mr. Obama said in Dayton, Ohio, yesterday. “Being against No Child Left Behind itself is not an education policy.”

The list of initiatives Mr. Obama went on to propose would maintain a strong federal presence in education, but in a different spirit from Mr. Bush’s efforts. Mr. Obama proposed at least three innovation funds that would transform the U.S. Education Department into a kind of venture philanthropy organization, encouraging projects deemed successful to replicate — and shutting down those that do not seem to be working.

Some of Mr. Obama’s proposals put him at odds with many teachers unions and parent groups around the country.

Mr. Obama proposed doubling spending on charter schools, to $400 million from the current $200 million; he said he wants to expand successful charter schools — and to encourage states to shut down charter schools that are failing.

Charter schools are privately managed public schools whose faculty are often not represented by teachers unions, major forces in Democratic Party politics.

They are a special target in Ohio, where Mr. Obama made his address.

The Ohio teachers union recently asked the Internal Revenue Service to examine the finances of local charter schools, and the Democratic governor, Ted Strickland, has moved to put a moratorium on new charter schools.

Mr. Obama also resurrected his promise to encourage teachers to be paid based on their performance, not just the number of years they have been teaching.

He said the overall spirit of his program was bipartisanship.

“It’s been Democrat versus Republican, vouchers versus the status quo, more money versus more reform,” Mr. Obama said. “There’s partisanship and there’s bickering, but there’s no understanding that both sides have good ideas that we’ll need to implement if we hope to make the changes our children need.”

The executive director of the lobbying group Democrats for Education Reform, Joseph Williams, who has pushed for Democratic candidates to take positions against teachers unions, said he was pleased with Mr. Obama’s speech.

“As Democrats we’re not supposed to be the ones who admit that there are any bad teachers,” Mr. Wiliams said. “So for him to go into an important state and do that, it’s a very different tone for a Democrat running for president.”

A former Republican education official, Chester Finn, the president of the Thomas B. Fordham Foundation, said Mr. Obama’s proposals did not sound new, but instead familiar and overly ambitious.

“They’re all in their way nice ideas for things,” Mr. Finn said. “But they sorely overreach what the federal government, or the president, can actually make happen.”