Spending has certainly jumped. But a huge part of the increase — about half, according to economist Richard Rothstein — has been dedicated to serving students with disabilities who were not guaranteed (and often did not receive) a free public education until the 1970s. Schools are also serving far more immigrant students who come in speaking a dizzying array of languages.

As for the academic flat line: The percentage of kids scoring “below basic” on the National Assessment of Educational Progress — widely considered the most reliable measure — has plummeted in both reading and math in both fourth and eighth grade for every racial group except Native Americans. Average reading and math scores for each subgroup in the fourth and eighth grades have also climbed steadily over the past 20 years.

But demographic changes in U.S. schools mean that a greater percentage of test takers now come from groups that traditionally score lower on the NAEP tests, such as Hispanic students. So when test scores are aggregated nationwide, it doesn’t look like there’s been much progress — even though taken individually, each group of students has dramatically improved.

The NAEP tests do show one clear trouble spot: high-school students. NAEP reading and math scores for 17-year-olds haven’t budged much since the 1970s. Ravitch suggests that’s because it’s hard to get older teens to take a no-stakes test seriously; she points to other gains, such as improvement in the high-school graduation rate, to show progress.

Peterson is more pessimistic. What good are the gains in fourth and eighth grade, he asks, if they’re not sustained through high school?

Another red flag comes from ACT scores. The organization that gives those tests recently calculated that just 66 percent of high school graduates are prepared for college-level English, 45 percent for math and a woeful 30 percent for science.

The Common Core, which 45 states and D.C. have adopted to guide instruction in math and language arts, aims to improve those numbers by setting uniformly high standards instead of the state-by-state patchwork that now prevails. Ravitch opposes the standards but they’ve been embraced by a broad coalition from businesses to unions, President Barack Obama and former Florida Gov. Jeb Bush.

4. It’s not an education problem, it’s an equity problem

On nearly every measure of academic performance, poor kids fare poorly.

To Ravitch and her supporters, the solution is obvious — schools in poor communities need more money and more resources to support families struggling with hunger, unemployment and unmet medical needs.

Reformers counter that it makes no sense to pour more money into schools with a long history of low test scores or dismal graduation rates. Instead, they push to close the worst performers and open the system to competition from alternatives like charter schools, which are publicly funded but privately run, sometimes by for-profit firms. They also push to weed out bad teachers by evaluating K-12 educators in significant part based on how much they raise their students’ scores on standardized tests.

The evidence here is muddy. Some charters have done exceedingly well at raising the test scores of low-income students. Others have not. Some of the most successful don’t serve anywhere near as many of the hardest-to-reach kids — those who are disabled, destitute or still learning English — as the struggling neighborhood schools all around them.

As for the new teacher evaluation systems, they have identified more low performers. But they’re far from precise.

When New York City calculated teacher “value-added” ratings last year, city officials acknowledged that a teacher rated at the 50th percentile of her peers might actually have been as low as the 23rd — or as high as the 77th, a huge margin of error that persisted even when the city used three years of student test data to smooth out bumps, city officials said.

So what’s the bottom line here?

Ravitch argues that “corporate reformers” and “privatizers” have a vested interest in making it sound like teachers and schools are failing so they’ll be invited to run their own schools or sell educational technology at a profit. Reformers say that’s ridiculous and accuse their critics of prioritizing adult concerns like teacher union jobs over children’s needs.

The debate has grown so contentious — even nasty — that the two sides often talk past one another, except to hurl insults. That frustrates Duncan.

He has been blunt in his critiques of public schools, arguing that too many have unacceptably low standards for their students.

Yet in an interview with POLITICO, Duncan said he has little patience for those who argue that public education is a failed enterprise. Acceptance of the status quo bothers him just as much, he said.

“Yes, the trends are very encouraging, but yes, relative to our international counterparts we have a long way to go, so I feel a fundamental sense of urgency,” Duncan said. “We have to continue to get better — faster.”

CORRECTION: An earlier version of this article incorrectly stated the name of the organization that gives ACT tests. The tests are given by the ACT.