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BOICEVILLE, N.Y. — It started with a speech in the fall, to parents who had gathered in the auditorium to learn what to expect during the nascent school year.

“I spoke at the open house and said, ‘We hope you’ll opt out of the tests,’” said Heather Roberts, vice president of the Bennett Intermediate School parent teacher association. Last year, 92 percent of eligible students in this district, the Onteora, took their standardized English tests. “Jaws dropped.”

Soon there were forums, T-shirts with snappy slogans and fliers translated into Spanish. During pickups and play dates, in classrooms and at lunch, parents and students would ask one another: “Are you opting out?”

By the first day of testing in April, two of every three students in the district who were expected to take the exams were refusing to lift their pencils.

Across New York State, a small if vocal movement urging a rejection of standardized exams took off this year, maturing from scattered displays of disobedience into a widespread rebuke of state testing policies.

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Heather Roberts, vice president of the parent teacher association for Bennett Intermediate School in the Onteora district, with her sons Axel, left, and Nash. “I spoke at the open house and said, ‘We hope you’ll opt out of the tests,’” she said. “Jaws dropped.”CreditEmma Tannenbaum for The New York Times

At least 165,000 children, or one of every six eligible students, sat out at least one of the two standardized tests this year, more than double and possibly triple the number who did so in 2014, according to an analysis by The New York Times.

As the vanguard of an anti-testing fervor that has spread across the country, New York’s opt-out movement already has become a political force. Just two months ago, lawmakers from both parties, at the behest of Gov. Andrew M. Cuomo, a Democrat, increased the role of test scores in teacher evaluations and tenure decisions.

Those same legislators are now tripping over one another to introduce bills that guarantee the right to refuse to take tests. The high numbers will also push state and federal officials to make an uncomfortable decision: whether to use their power to financially punish districts with low participation rates.

“We’ve written letters to legislators for years, until we were blue in the face, and they didn’t listen,” said Eric Mihelbergel, a founding member of New York State Allies for Public Education, a test-refusal group. “But they’re listening now, now that we’re opting our kids out.”

At the same time, some education officials and advocacy groups fear, the opt-out movement will reverse a long-term effort to identify teachers and schools — and students — who are not up to par, at least as far as their test performance goes. Of particular concern is that without reliable, consistent data, children in minority communities may be left to drift through schools that fail them, without consequences.

This month, a dozen civil rights groups, including the N.A.A.C.P. and the Leadership Conference on Civil and Human Rights released a statement saying they were opposed to “anti-testing efforts” because tests provide data that is crucial for catching and combating inequities in public schools.

“When parents ‘opt out’ of tests — even when out of protest for legitimate concerns — they’re not only making a choice for their own child, they’re inadvertently making a choice to undermine efforts to improve schools for every child,” the statement said.

In New York State, the movement swelled to its current numbers from virtual nonexistence in only two years, fueled by a potent cocktail of union and parent activism. Karen E. Magee, president of the New York State United Teachers union, expressly urged parents this year to skip the tests in order to subvert the new teacher evaluations. A surge of activity on social media, especially Facebook, helped show parents a way to protest if they believed their school system overemphasized testing.

“If a certain amount of people opt out, then they’ll have to change it,” said Nanci Leiching, a parent at Bennett. “That’s the goal.”

Though school officials are required to administer the tests, given to children in grades three through eight, many made it clear to parents that it was acceptable to skip it. In Suffolk County, Joseph V. Rella, the superintendent at Brookhaven-Comsewogue, was a vocal critic of the tests, and in his district, 80 percent of students sat out at least one exam. “I’m not encouraging parents to refuse the test, but just to think and be informed,” he said in an interview. But, he added, “If I had kids that age, they wouldn’t go near that test.”

In Onteora, a district of the Hudson Valley framed by lilac-colored mountains, the school board convened a “Task Force for Testing Reduction,” and released a policy statement that said no laws or regulations “expressly prohibit parents and students from refusing” the assessments. Tony Fletcher, the board president, said that he had decided to opt out for his own fourth grade son. Last year, his son took the tests.

“People were saying, ‘Just opt out!’” said Rajahn Dash, a sixth grader at Bennett, whose mother, Lisa Dash, said she had decided to opt him out after watching a video on Facebook.

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By the first day of testing in April, two of every three students in the Onteora district, in the Hudson Valley, who were expected to take the exams had opted out. CreditEmma Tannenbaum for The New York Times

As the number of people who planned to refuse to let their children take the test grew, according to Ms. Roberts, the PTA vice president, dynamics began to shift and the pace of refusals quickened. In a way, it became easier to just go with the flow, she said. Those who bucked the trend, it seems, felt some pressure to keep that decision to themselves.

“The people having their kids tested tended to be more quiet about it,” said Wendy Wolfenson. She joined the test reduction task force but ultimately decided to have her daughter, Grace Molmed, a third grader, sit for the exam this spring because she thought the results would be informative. When she found other parents planning to have their children participate, they felt camaraderie, and maybe a little relief, she said.

Grace told her mother that seven or eight children, out of about 20 students in her class, had taken the tests.

In 440 of the state’s 721 districts, The Times found that at least 165,000 students opted out of at least one test, based on information from districts and local news reports. That figure is four times the number of pupils who had refused the tests in those districts last year. In at least 60 districts, refusers outnumbered the test takers.

In the other 250-plus districts, including New York City’s, by far the state’s largest, data was either unavailable or officials did not respond to requests for information.

Just two years ago, 95 percent student participation in tests, the minimum standard called for by federal requirements, was nearly universal; this year, among districts with available information, only a few reported meeting that standard. In Fairport, near Rochester, for example, participation dropped from 96 percent to 33 percent in a single year.

Official numbers will not be released until the summer, but the total is quite likely to be at least triple the number who opted out in 2014, when 49,000 students without an officially valid excuse, like an illness, did not take the English test, and 67,000 did not sit for the math test, according to the state education department.

A spokeswoman for the city’s Education Department declined to provide an estimate of how many students sat out the assessments, but said the department did not anticipate having less than 95 percent participation. Last year, less than one-half of 1 percent of city students skipped the test, according to the city.

The refusal movement this year was strongest in middle-class districts, although the numbers rose in virtually every district analyzed. The districts where refusal remained low tended to be either very poor or very rich.

“Parents of children in poverty are working, trying to put food on the table,” said Edward Kliszus, the superintendent at Port Chester-Rye Union Free School District, where 61 percent of students are poor enough to qualify for lunch subsidies. “They’re not paying attention to this. It’s not a priority.”

The refusal movement sprouted after states instituted tougher tests in recent years aligned to the Common Core standards, which, in many districts, caused scores to plummet.

The Onteora district, which is largely white and middle class, performed above the state average on the tests last year, but not by much. Just 39 percent of district students tested as proficient on the English test, compared to 31 percent statewide.

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Clockwise from top left, Anna Millenson, Ann McGillicuddy, Michele Garner, Christian Caloro, Teddy Byron and Josh Luborsky are all parents of students who opted out of state testing. Ms. McGillicuddy sits on the Onteora Board of Education. CreditEmma Tannenbaum for The New York Times

Many local parents, however, said they had their children skip the tests not because they were afraid of the results, but because they felt they put too much stress on students, for example, or because they wanted to make a statement on behalf of teachers.

In March, Governor Cuomo, dismayed at the large percentage of teachers getting high ratings, succeeded in tying teacher evaluations and tenure decisions more closely to the tests. If fewer than 16 tests are available to apply to a teacher’s score, however, which appears likely in many cases this year, districts will have to produce an alternative rating method, such as using the scores of other students in the school.

Some refusal advocates point to Mr. Cuomo’s effort as a moment that galvanized parents to opt out. But Jim Malatras, the governor’s director of state operations, said the rise in test refusals did not signal a political miscalculation on the part of the governor. This kind of reaction, he said, was to be expected from any substantial shift in policy.

State education departments are responsible for ensuring that districts attain 95 percent participation, using enforcement tools like increased monitoring or withholding certain funds. If the federal education department decides the state is too lenient, the state can be penalized with the same measures, forcing its hand.

However, the department has never withheld money from a state because of low participation — in the past, states have responded in a way the department deemed sufficient, a spokeswoman said — and doing so may be both unpalatable and politically untenable, because many districts that receive substantial federal aid are in poor areas.

Robert Pondiscio, a senior fellow and vice president for external affairs at the Thomas B. Fordham Institute, a conservative education policy organization, said that rather than enforcing the rules, government officials may very well retreat.

“You could write a really good history of education writ large about our tendency in this country to go from one extreme to the other, and this has all the hallmarks of that,” Mr. Pondiscio said. “This is not a prediction, but it would not surprise me to see New York, or someplace else, go from testing every kid within an inch of their life to testing nobody, ever.”

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