Students in the District’s public schools made significant gains in math and reading achievement during the past two years, including progress among black, white, Hispanic and low-income students, according to national exam results released this week.

But even with the District’s overall improvement on the 2013 National Assessment of Educational Progress, there continue to be wide disparities in performance according to race, income, disability status and parents’ educational attainment.

Over time, achievement gaps between the city’s most- and least-advantaged children, historically among the largest in the country, have narrowed in some cases but have grown — sometimes significantly — in others.

That poses a challenge for officials who have bet that the changes of recent years — such as universal preschool, new academic standards and teacher evaluations, the rise of charter schools and mayoral control of traditional schools — are creating a public education system that can change the trajectory for poor children.

“The achievement gap is still clearly a pressing issue for us,” said Abigail Smith, deputy mayor of education. Smith said that although the gaps are too large, they would be even more worrisome if only white or high-income students were improving. “In fact, everyone is on an upward trajectory. We just need to keep accelerating the pace.”

NAEP measures the math and reading performance of fourth- and eighth-graders and is administered by the federal government every other year. Scored on a scale of 0 to 500, it is widely regarded as the most consistent measure of K-12 academic progress.

But different children take the test each year, making it difficult to draw conclusions about what causes scores to rise or fall, especially amid demographic change. The proportion of white fourth-graders in the District — who score higher than fourth-graders in any state in the nation — has doubled since 2003, from 5 to 10 percent. The proportion of black fourth-graders — whose scores are among the lowest in the nation — fell from 85 to 73 percent.

The District’s black-white achievement gap has consistently been the largest in the country when compared with other states and urban school districts, and it was about twice the national average in 2011. In fourth-grade math, the District’s gap was 60 points in 2003. It shrank to 53 points in 2007, but since then — the same year then-Mayor Adrian M. Fenty (D) took control of the schools — it widened again, to 55 points.

The black-white gap has narrowed more steadily in fourth-grade reading but still stands at 62 points, unchanged from 2011.

The gap between Hispanic and white students is smaller, but it is still among the country’s largest.

“My takeaway is that we are leaving low-income children behind. We are leaving the neediest kids behind,” said Mary Levy, an education finance lawyer and watchdog of D.C. schools.

Schools Chancellor Kaya Henderson said black and Hispanic students are improving at a faster rate than the national average, crediting the school’s turnaround efforts in recent years.

“I don’t know that it’s necessarily gap-closing, but all of my boats are rising,” Henderson said. “When you concentrate on teacher quality, you get results. When you radically increase the level of academic rigor, you get results.”

There are some bright spots for the city’s achievement gap. In eighth-grade math, for example, the black-white gap shrank 20 points between 2005 and 2013.

But by other measures, the gap is growing, including between children of the least- and most-educated parents. On the 2007 reading exam, nine points separated eighth-graders whose parents dropped out of high school from those whose parents graduated from college. By 2013, that gap had more than doubled, to 23 points.

It is nearly impossible to track the performance of poor children because the method for identifying low-income students in the District has changed since 2011.

A child’s poverty status is measured by their eligibility for a free or reduced-price lunch. Until last year, children became eligible for free meals by turning in forms showing household income. Now, if 40 percent of children in a D.C. school are in foster care, homeless or receive welfare benefits, every child in the school is deemed eligible for free meals.

The change in the District is a test of a new federal policy meant to ensure that more hungry kids have access to free meals. It means that some children who are not actually poor but who attend high-poverty schools are now included in the low-income category, said Jack Buckley, commissioner of the National Center for Education Statistics.

That change is “masking whatever is actually happening,” said Buckley, who said his office is concerned about and working to address the inability to track the progress of poor children. He cautioned against drawing conclusions about the progress of the District’s poor children based on the 2013 test results.

Those results show that the proportion of children eligible for free meals rose significantly and that they made some gains. But the gap between them and their more-affluent peers also swelled.