Virginia students saw modest gains on state-mandated math tests while average scores on reading and writing exams stagnated, according to state results released Tuesday. But nearly half of the state’s districts saw drops in reading scores as many school districts brace for wavering accreditation status this fall.

The state’s annual Standards of Learning tests gauge student academic performance compared with their peers across all grade levels in the commonwealth. The scores help education officials track student achievement in Virginia, which was one of a few states that has not adopted the new national Common Core State Standards touted by the Obama administration.

Of the state’s 132 school districts, 110 divisions improved their average scores on math tests and 13 remained the same last school year compared with the previous year. On state reading tests, students in 64 school divisions saw overall reading scores drop and 76 districts recorded losses on writing exams.

State education officials warned this month that as many as one-third of Virginia’s 1,800 schools could lose full accreditation this fall because of dropping reading and science exam scores as state tests increased in difficulty.

In 2013, the state gave full accreditation to 1,414 of Virginia’s 1,827 schools. Virginia Department of Education spokesman Charles Pyle said that this year as many as 250 schools could join the 395 that last year were “accredited with warning,” a dramatic increase from 2009, when 15 schools earned the lower status.

The improvements in math scores are significant because it marks the third time students took a more rigorous exam that the state introduced in 2011. The revised tests push students to solve challenging multi-step problems and are part of a larger effort among state officials to help Virginia students prepare for college and careers.

A total of 74 percent of Virginia’s students passed their grade-level math tests, compared with 71 percent a year earlier.

“The statewide focus on teaching students to be problem solvers and to apply what they have learned in mathematics in real-life situations is producing results,” Steven R. Staples, state superintendent of public instruction, said in a statement.

But pass rates statewide for students in grades three, seven and eight were below 70 percent. In reading, average pass rates for students in grades three through eight did not exceed 76 percent.

Fairfax County showed almost no gains in reading scores. Student pass rates in grades three, five, six and eight dropped or remained the same, with third-grade pass rates falling five percentage points. Math scores overall mostly improved in the county, where fourth-grade pass rates gained five percentage points.

More than 50 of the county’s 139 elementary schools achieved perfect pass rates on certain math exams. Some schools, such as Colvin Run Elementary, saw a 99 percent pass rate on the sixth-grade math exam and a 100 percent pass rate on the seventh-grade math exam. Schools spokesman John Torre said that most of the students taking the seventh-grade exam were placed in smaller advanced math classes; many of the elementaries that saw perfect math pass rates were schools that offer advanced academic classes.

In Arlington, students across all grade levels saw improvements on reading and math. On average, 75 percent of eighth-graders passed the math exams, compared with 67 percent a year earlier. In all nine math tests, the school district exceeded the state average pass rates by nine percentage points, according to a county analysis.

Loudoun and Prince William counties saw slight overall improvements in pass rates in both reading and math tests, and Alexandria students showed slight improvement in reading pass rates while math scores were inconsistent.

David Lehr, an economics professor at Longwood University in Farmville, Va., recently published a study on state SOL scores that showed the greatest predictor of academic achievement was student socio­economic background.

In the study, Lehr examined the 2008-2009 SOL data from more than 1,100 Virginia elementary schools and found that ethnicity and family income showed the most accurate predictor of academic performance. The more students eligible for free or reduced-price meals — a federal measure of poverty — the lower the school scored on the SOL tests. Lehr’s study also showed that higher percentages of minority students also affected a school’s scores. Demographic trends in the Northern Virginia area show that a majority of the youngest students now entering the school systems are immigrants who come from poor families.

“If you think that raising salaries or lowering class size or hiring teachers who are more educated will be the key to substantially higher pass rates, this paper should give you pause,” Lehr said in a statement. He said the data shows that “there is no magic bullet for SOL pass rates.”