Given that caveat, how accurate or useful is polling data? Education polls often ask unprepared people to make “finely nuanced distinctions” without the requisite background, said Andrew Rotherham, the co-founder and a partner at Bellwether Education Partners, a national nonprofit in Washington, in an interview last year. “You get a result, but you also get a lot of noise.”

The Common Core’s popularity has waned as more Americans have become familiar with the standards, which are opposed by an animated contingent of liberal and conservative voters, though for different reasons. Liberal critics charge the standards were written without sufficient input from parents and teachers, while conservatives see the standards as a federal intrusion on states’ rights. (The U.S. Department of Education created incentives for states to adopt the standards, but did not require their adoption.)

Two years ago, 65 percent of Americans in an Education Next poll supported the standards. That year a PDK/Gallup poll noted that nearly two-thirds of U.S. adults have never heard of the Common Core. Fast-forward to this month and nearly every adult is in some way familiar with the standards, PDK/Gallup says, and support for the standards is considerably lower.

For the Common Core, the “proof will be in the pudding,” said Paul Peterson, a Harvard professor who oversaw the Education Next poll. “If the standards do end up being fully implemented and students start learning more, then the public opinion might change.”

When the PDK/Gallup questions on standards are put next to the Education Next findings on the Common Core, the responses are not out of alignment, Peterson said: People are generally in favor of setting higher expectations for students across states but they also want local teachers to have leeway in how those goals are met. (In an essay analyzing the two polls, he and his colleague Martin R. West wrote that the “two surveys are complementary, because they ask about different topics.”)

Indeed, in a call with reporters, Joshua Starr, the former superintendent of Montgomery County Public Schools in suburban Maryland and the newly named chief executive officer of PDK, made points similar to Peterson’s:

The divide between the percentage of parents who believe those standards should be higher and then the percentage of parents who don’t support the Common Core, that’s very interesting to me in signaling the potential for a lack of understanding of what the Common Core is.

The furor over allowing parents to pull their students from taking state tests is a newer phenomenon, but can affect the legitimacy and even funding of high-stakes tests. With too many students foregoing taking the assessments, the results may be limited in what they can tell the public about how much students know in math and reading. And according to the federal No Child Left Behind law, states technically risk losing some federal funding if less than 95 percent of students take the high-stakes exams.