For the fourth time within a month, state education officials have tossed out a question on the standardized tests after finding that errors by Pearson, the test maker, made the problem virtually impossible for students to solve.
The question, which appeared on the fifth-grade math exam, was administered in a trial-run last year and vetted by teachers. Yet it was not until after it was included on a real test last month that teachers found and reported a major flaw: the problem asked students to find the perimeter of a trapezoid that could not exist within the bounds of mathematics.
Officials said the item was meant to test students' knowledge of perimeter and geometric ratios. But because of the way the problem was worded, to arrive at the mathematically correct answer, fifth graders needed to know the Pythagorean theorem and how to find an imperfect square root, concepts they typically do not encounter until middle or high school.
The "right" answer, supplied by Pearson, was actually wrong.
Dennis Tompkins, a spokesman for the State Education Department, said on Monday that the question would not be counted in students' scores.
"The item was developed by Pearson to assess fifth graders' conceptual understanding of perimeter, not the application of the Pythagorean theorem," Mr. Tompkins said.
This year’s standardized math and reading exams were already causing anxiety in classrooms around the state. They were longer than in years past, and a new evaluation system made the results more critical to teachers’ jobs. But a series of gaffes, including a confusing reading passage about a pineapple, has given new fodder to opponents of standardized testing and even led the chancellor of the Board of Regents to complain.
The English language arts and math exams, which were administered on six days between April 17 and April 27, were the first produced by the Pearson testing company under a five-year, $32 million contract.
State education officials specifically asked the company to make the exams, given to third through eighth graders, less tricky, with fewer questions designed to trip up students. Answer choices like “none of the above” or “all of the above” were banned.
But some teachers and parents have argued that this year’s exams were unusually confusing. First, eighth graders reacted to a parable about a talking hare and a pineapple by mocking the passage on Facebook and other social media sites.
Later, state education officials warned principals that the fourth-grade math test had a question for which there were accidentally two correct answers.
And an eighth-grade math question had no right answer because of a typo.
In a telephone conversation this week, Merryl H. Tisch, the chancellor of the Board of Regents, which oversees the State Education Department, said that Pearson would have to do better.
"My answer to Pearson is, 'No, no, no,'" she said.
"I take full responsibility for all of these errors, I do, I do, do,” she added. “And I would hope that Pearson as a producer of these exams would join me in this responsibility."
Pearson rarely responds publicly to questions about its tests, and did not do so this week.
At a Regents meeting on Monday, John B. King Jr., the commissioner of the State Education Department, suggested that the public outcry had less to do with the content of the exam and more with students’ access to social media and teachers’ concern about the new evaluation system, in which at least 20 percent of their rating will be based on their students’ test performance.
"In every administration of a test we've had, there's always the occasion of a typo,” he said. “There is the occasional question that we have to adjust."
He announced last week that the confusing questions about the pineapple and the hare would not count toward students' scores. The math question with no right answer also was tossed.
Opponents of standardized testing have seized on the gaffes as the latest example of its failures.
"There is much more scrutiny now because of the state's having ratcheted up the stakes for kids, schools and teachers, as well as increasing their length and cost to schools, as well as the fact that the state is determined to keep them secret," Leonie Haimson, the executive director of Class Size Matters, wrote in an e-mail.
Ms. Tisch said that despite the errors, the tests were still a valid measurement of students’ academic abilities.
"Does it invalidate the test? Definitely not," she said. "And anyone who says that is really just trying to push back, not only against the test, but against testing in general and against teacher evaluations in specific."
Compounding the confusion over the faulty fourth-grade math question were the state's instructions on how schools should deal with it. They were told to inform any stumped fourth grader who asked about the question that there were two correct answers, but students who did not inquire might be left scratching their heads.
Many teachers brushed aside the state’s advice and warned students.
"She made it really big on a big piece of paper so everyone could see it," Martine Arranz, 9, a fourth grader at Public School 261 Philip Livingston in Boerum Hill, Brooklyn, said of her teacher. "She said it would be two answers."
This is not the first year the state has had to send out last-minute notices warning about errors on the exams. In fact, the state’s Web site lists examples of Spanish translations gone wrong and mislabeled answer choices. In 2004, the Braille edition of the fourth-grade science test listed the answers for one question in the wrong order. The list is incomplete.
Pineapplegate also has a predecessor: Browniegate. In 2006, parents called on the state to throw out the results of the fourth-grade reading exam, claiming that a listening passage had been so confusing that no child should be penalized for getting it wrong.
The story was about an arrogant rooster who was tricked by Brownie, a cow, into getting up early each morning and crowing. The test asked children to write an essay about how and why the cow’s behavior changed over the course of the story.
Parents objected on the grounds that 9-year-olds were being asked to perform psychoanalysis. The state disagreed, saying, "The question was not confusing to children."
Editor's Note: Our astute readers pointed out a typo in our graphic. It has been corrected. Thank you, readers.