WASHINGTON — Teens from Asian nations dominated a global exam given to 15-year-olds, while U.S. students showed little improvement and failed to reach the top 20 in math, science or reading, according to test results released today.

American students scored below the international average in math and about average in science and reading.

The top average scores in each subject came from Shanghai, China’s largest city with more than 20 million people. Singapore, South Korea, Japan, and Hong Kong were among the participants with students scoring at the top on average in each subject. Vietnam, which had its students participate for the first time, had a higher average score in math and science than the United States.

Education Secretary Arne Duncan called the results a “picture of educational stagnation.”

“We must invest in early education, raise academic standards, make college affordable, and do more to recruit and retain top-notch educators,” Duncan said.

About half a million students in 65 nations and educational systems took part in the 2012 Program for International Student Assessment, or PISA, which is coordinated by the Paris-based Organization for Economic Cooperation and Development, or OECD.

Most results come from a sampling of scores from countries as a whole, but in China it was given in select regions. The Education Department’s National Center for Education Statistics released the results.

The test, which is given every three years to 15-year-olds, is designed to assess how students use what they’ve learned inside and outside of school to solve problems.

U.S. scores on the PISA haven’t changed much since testing started in 2000, even as students in countries such as Ireland and Poland have shown improvement and have surpassed U.S. students.

“It’s hard to get excited about standing still while others around you are improving, so I don’t want to be too positive,” said Jack Buckley, commissioner of the National Center for Education Statistics. But he added that the country has a respectable foundation on which to build.

American students historically have not had high marks on international tests. Factors often cited include high rates of child poverty and population diversity.

In contrast to the PISA results, American fourth- and eighth-graders over time have made some progress in reading and math on an assessment referred to as the Nation’s Report Card, even though recent results found the vast majority of the students still are not demonstrating solid academic performance in either subject.

In the education community, Finland has drawn notice for its past test performance, but this year its average PISA score dropped in all three subjects, most pronounced in math. Finland’s students did better on average than those from the U.S.

Overall, NEA President Dennis Van Roekel said among the Asian nations dominating the test, “The one thing they all have in common is that they make a real commitment to education for all kids and nothing deters them from that vision.”