Fairfax County math teacher Vern Williams has never used class time to review for state tests. The multiple-choice exams have always been easy, he said, for his algebra and geometry students at Longfellow Middle School.

Now that’s changed. As state testing begins this week across much of Northern Virginia, students are facing a new and more difficult generation of math exams. They’re designed to require critical thinking and multistep problem-solving, not just memorization and arithmetic. Pass rates are expected to drop, perhaps steeply.

Williams said that for the first time he respects the tests enough to spend time preparing his young mathematicians to take them.

“I’m telling my students we really need to take this seriously,” he said.

The new state tests — given to all Virginia students in grades three through eight, as well as those finishing courses in Algebra I, Algebra II and Geometry — herald the arrival of an era of harder standardized tests not just in the commonwealth but also across the country.

Sean Pratt, 14, listens to a presentation in Stephanie Nichols’ math class in 2009 in Arlington. A new, more difficult statewide test is being put into use this year. (Evy Mages/For The Washington Post)

The tougher exams grow out of a broad effort to ratchet up academic expectations and make sure graduates have the skills they need to compete in their post-high school lives.

The District and most states — including Maryland but not Virginia — have agreed to a shared set of national standards that are widely considered more demanding than what states had previously expected of students. Tests based on those “common core” standards will be administered for the first time during the 2014-15 school year.

“We expect them to be more difficult,” said William Reinhard, a spokesman for the Maryland State Department of Education. “The world demands it.”

Virginia, meanwhile, has stuck with its own state Standards of Learning, revising them in recent years to demand a deeper level of understanding from students.

Last year, the social studies tests became more difficult. Next year, English and science exams will get tougher.

This year, it’s math. And parents should not be alarmed if their children get lower scores on the new tests than they did on previous versions, said Charles Pyle, a spokesman for the Virginia Department of Education.

“It’s an indication that the state is expecting more, not that the student is necessarily learning less,” he said.

Some multiple-choice questions have more than one correct answer; others require students to go beyond plugging numbers into formulas and instead manipulate the formulas to reach an answer. Rather than being asked to calculate the volume of a box, for example, students might be asked to figure out how the volume of a box with a given height — six inches, say — would change if that dimension shrank by half.

In sixth grade and up this year — and in all grades beginning next year — the tests will be delivered online. Students might be asked not only to interpret graphs but also to create them. Sometimes they’ll have to supply their own response instead of plucking an answer from among a set of choices.

The new tests were first given in the fall, when a relatively small fraction of the state’s middle- and high-school students — less than 10 percent — are ready for end-of-course exams.

Pass rates on Algebra I fell from 84 percent in the fall of 2010 to 49 percent in the fall of 2011. Algebra II and Geometry students saw similarly substantial drops.

Those low scores prompted several local school superintendents to lodge protests with the state board in March. They argued that the untimed exams — which took some students longer than three hours to complete — were tests of physical endurance instead of academic mastery.

The new tests could put a school’s reputation at risk, the superintendents said, not to mention its state accreditation status. And high school seniors who fail required tests could lose the chance to graduate on time.

“What are we doing to our students and teachers by jumping so high so quickly?” asked James Merrill, chief of Virginia Beach schools.

In an April letter, the Virginia Association of School Superintendents requested a grace period during which math scores would not count and teachers could adjust to the new expectations.

That request was rejected this month by David Foster, president of the Virginia Board of Education, who said higher standards couldn’t be put on hold.

In Fairfax, the state’s largest school system, with 177,000 students, where the testing window opened this week and lasts until early June, Superintendent Jack D. Dale said he wished the state had provided more sample test problems and other resources to help teachers prepare their students for the new exams.

“I don’t mind increasing the rigor, but we should do this in a thoughtful way,” he said in an interview. “We have to change some of our teaching practices, and that requires time.”

Some parents and educators said they worry that the longer, harder tests will prove stressful for students.

But several Fairfax parents, teachers and principals — including at schools that have struggled to meet test-score targets in recent years — said they hope the exams will encourage and reward the kind of teaching that pushes past rote memorization.

“I applaud anything that makes the SOLs less about regurgitating facts and more about stimulating the critical thinking skills in our children,” Christine Adams, PTA president at Bailey’s Elementary School for the Arts and Sciences, wrote in an e-mail.

Sonya Swansbrough — principal of Poe Middle School, where many students are living in poverty or learning English as a second language — agreed.

The old tests were “just a minimum standard — this is the basic that all students should be able to achieve,” she said. “We don’t want to stop at that. Nobody wants their children to have minimum standards.”