No state or any of the 21 urban school districts that participated in the tests raised scores in both subjects and grade levels. But the District of Columbia repeated some of the strength it showed in 2013 by raising fourth-grade reading and math scores and holding eighth-grade reading and math scores steady. Kaya Henderson, the public schools chancellor, attributed the progress to strong teacher recruitment and training, preschool for 90 percent of the district’s 4-year-olds and a new mandatory curriculum.

In New York State and New York City schools, math scores on the national tests fell in fourth grade but held steady in eighth grade, while reading scores were flat in both grades throughout the state and in the city.

As states have adopted the Common Core — guidelines for what students should know and be able to do between kindergarten and high school — many teachers have adjusted their curriculum and instructional methods, particularly in math. Students are asked to use math to solve real-life problems and find different ways to come at the same answer rather than simply repeating formulas.

Some educators suggested that some of the changes have sowed confusion among teachers and students that could be reflected in the national test scores. “Right now, what’s going on in many states is a wholesale change in math instruction,” said Daniel Koretz, a professor of education at Harvard. “We don’t know what’s happening with that in classrooms.”

A study released Monday showed that some items included in the national assessments are not covered by the Common Core before the grades in which they are tested.

“Knowing other kinds of math isn’t going to help you unless you’ve been taught it,” said Fran Stancavage, an author of the study. “The Common Core moves the sequence around, so there are lots of things that used to be taught before fourth grade that are now showing up in higher grades.”

Randi Weingarten, the president of the American Federation of Teachers, linked the drop in test scores to recent educational policies as well as the economic downturn and its aftermath. “Of course we are disappointed” with the scores, she said. “But they should give pause to anyone who still wishes to double down on austerity and make competition, scapegoating teachers, closing rather than fixing schools, driving fear, and testing and sanctioning the dominant education strategies.”