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Saying its college admission exams do not focus enough on the important academic skills, the College Board announced on Wednesday a fundamental rethinking of the SAT, ending the longstanding penalty for guessing wrong, cutting obscure vocabulary words and making the essay optional.

The president of the College Board, David Coleman, criticized his own test, the SAT, and its main rival, the ACT, saying that both had “become disconnected from the work of our high schools.”

In addition, Mr. Coleman announced programs to help low-income students, who will now be given fee waivers allowing them to apply to four colleges at no charge. And even before the new exam is introduced, in the spring of 2016, the College Board, in partnership with Khan Academy, will offer free online practice problems and instructional videos showing how to solve them.

The changes are extensive: The SAT’s rarefied vocabulary challenges will be replaced by words that are common in college courses, like “empirical” and “synthesis.” The math questions, now scattered across many topics, will focus more narrowly on linear equations, functions and proportional thinking. The use of a calculator will no longer be allowed on some of the math sections.

The new exam will be available on paper and computer, and the scoring will revert to the old 1,600-point scale — from 2,400 — with top scores of 800 on math and 800 on what will now be called “evidence-based reading and writing.” The optional essay, which strong writers may choose to do, will have a separate score.

Once the pre-eminent college admissions exam, the SAT has lost ground to the ACT, which is based more directly on high school curriculums and is now taken by a slightly higher number of students. Last year, 1.8 million students took the ACT and 1.7 million the SAT.

The new SAT will not quell all criticism of standardized tests. Critics have long pointed out — and Mr. Coleman admits — that high school grades are a better predictor of college success than standardized test scores. More colleges have in recent years become “test optional,” allowing students to forgo the exams and submit their grades, transcripts and perhaps a graded paper.

For many students, Mr. Coleman said, the tests are mysterious and “filled with unproductive anxiety.” And, he acknowledged, they inspire little respect from classroom teachers: only 20 percent, he said, see the college-admission tests as a fair measure of the work their students have done.

Mr. Coleman came to the College Board in 2012, from a job as an architect of the Common Core curriculum standards, which set out the content that students must master at each level and are now making their way into school.

He announced plans to revise the SAT a year ago and almost from the start expressed dissatisfaction with the essay that was added in 2005. He said he also wanted to make the test reflect more closely what students did in high school and, perhaps most important, rein in the intense coaching and tutoring on how to take the test that often gave affluent students an advantage.

“It is time for the College Board to say in a clearer voice that the culture and practice of costly test preparation that has arisen around admissions exams drives the perception of inequality and injustice in our country,” Mr. Coleman said Wednesday. “It may not be our fault, but it is our problem.”

While test-preparation companies said the SAT was moving in the right direction, with more openness and more free online test preparation, the changes were unlikely to diminish the demand for their services. “People will always want an edge,” said Seppy Basili, a vice president of Kaplan Test Prep. “And test changes always spur demand.”

The suggested changes were well received among many educators, but Mr. Coleman’s comments about the ACT drew harsh words from an executive of that company.

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