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When the “math wars” began in the 1990s, on one side were those who argued for a new focus on concepts and reasoning rather than drilling students on their times-tables. On the other were the traditionalists, who said the progressive approach allowed students to become unmoored from the building blocks of the subject, leaving them unprepared for more advanced mathematics.

The writers of the Common Core math standards have sought a middle ground.

“There are explicit expectations for knowing the times-table from memory, and that’s going to take dedicated work toward that end. So this isn’t fuzzy math,” said Jason Zimba, a professor of physics and math at Bennington College in Vermont and lead writer of the math standards. “On the other hand, some of the curricula we have are weak on applications, so kids don’t ever get to see what it’s good for, or what it’s used for.”

Students will spend time memorizing and practicing formulas in Common Core classrooms. And they’ll spend an equal amount of time modeling to understand concepts they’re learning about—using the seasons or a business cycle to understand trigonometry, for example.

“I think we’ve had curricula that swing too far to one side or the other on these things,” Zimba said. “The notion of rigor in Common Core involves equal intensity about conceptual understanding, procedural skill, and fluency and application.”

The creation of the math standards was in large part an editing process. Experts have mostly agreed that previously, American math classes tried to cover too much ground, leaving students without the deeper grasp of central concepts that would serve them best in more advanced mathematics. So the Common Core math standards tackle fewer topics, and also move students more slowly through arithmetic, subtraction, multiplication and the other operations that build up to more complex math, particularly algebra.

“These standards are focused in a way that we didn’t have before in the sense that they really try to say in each grade-level, this is what you need to learn so you can move on,” said William McCallum, math department chair at the University of Arizona and a member of the work team for the Common Core math standards. “A lot of curricula tend to keep teaching the same thing over and over again, and never doing it in a particularly deep way.”

Even critics have praised the focus, and also the way that the Common Core math standards address some of these basic areas, especially fractions.

“It’s based on pretty solid research on what is done in high-achieving countries,” said Milgram. “Mathematically, it’s summed up in one little phrase: Fractions are numbers. And it’s made emphatically clear in the Common Core standards.”

“They are not pieces of pizza and they are not little blocks, and they are not a certain number of dots in a bigger set of dots,” he added.

Using pizza to teach fractions isn’t banned, Zimba said. But the idea that fractions are actual numbers that fall on the number line—rather than pieces of something larger—is emphasized.

Other aspects of the Common Core math standards—mostly at the secondary level—have raised concerns among a handful of mathematicians, however.

For one, experts have worried that the standards are encouraging a way of teaching geometry that may not only be above the heads of students, but also hard to grasp for teachers. The standards start with transformational geometry, a way of visualizing congruence by, for example, transposing figures over one another or flipping them into mirror shapes. The authors of the standards say it’s a way to help students grasp fundamental concepts in geometry. Mathematicians, though, worry that what may seem like a simple way of teaching students is actually a highly complex approach more appropriate for college math majors that could reduce the emphasis on the rules and formulas of geometry.

“It’s true that the transformations are the beginning of geometry,” said McCallum. But, he added, “They’re exaggerating what’s in the standards.”

The main critique of the math standards, however, is that they don’t include a full course of Algebra I until high school.

William Schmidt, the Michigan State researcher, has found that “internationally, the focus of eighth grade for all students in virtually all of the TIMSS countries—except the United States—is algebra and geometry.” A National Center for Education Statistics report in 1999 found that 40 percent of U.S. eighth-grade mathematics lessons included arithmetic topics such as whole number operations, fractions and decimals. These topics were much less common in Germany and Japan, where eighth-grade lessons were more likely to cover algebra and geometry.

Algebra in eighth grade prepares students to take more advanced classes in high school, which in turn better prepares them for college and a possible career in science, technology, engineering or math (what are known as the STEM fields).

Research has found that black, Hispanic and economically disadvantaged students are much less likely than their peers to take algebra in eighth grade. Those groups are also less likely to enroll in advanced math classes later in high school. The disparities have turned access to algebra into a civil-rights issue. In the last decade, more states have pushed eighth-graders to take algebra in order to close the gaps and also to meet demands that they better prepare students for STEM careers.

“If you do algebra in grade 8, then you have four years—and if you need to repeat, you can repeat, or you can reach calculus by grade 12. It’s not mandatory for being accepted to colleges, but selective colleges expect it,” said Ze’ev Wurman, a former U.S. Department of Education official under George W. Bush who participated in the creation of California’s highly regarded math standards. (In adopting the Common Core math standards, California rescinded its previous requirement that students take Algebra I by eighth grade.)

“If you don’t prepare everyone, then essentially you only have the privileged kids who are prepared to take [advanced math],” he added.

Research suggests teaching algebra to all students by eighth grade may be ineffective, however. Many students fail because they are unprepared, and even fall further behind. And Zimba says the standards include “an awful lot of algebra before eighth grade,” even if they don’t technically include an Algebra I course. “By the time you’re in eighth grade, you’re solving two equations and two unknowns. It’s highly rigorous,” he said.

McCallum said the eighth-grade standards, though not called Algebra I, cover “what happens in normal Algebra I in high school.”

But Zimba also acknowledges that ending with the Common Core standards in math could preclude students from attending elite colleges or pursuing STEM careers.

“If you’re a young person who wants to become an engineer, or who wants admission … to an elite university, you would be advised to take mathematics beyond the college- and career-level,” he said. “If you want to take calculus your freshman year in college, you will need to take more mathematics than is in the Common Core.”

He argued that it isn’t the role of the standards to close racial and socioeconomic gaps between those who go down that path and those who don’t. “You can simply graduate from high school, you can graduate college- and career-ready [via the Common Core], or you can graduate STEM-ready,” Zimba said. “It would be great if policymakers would make sure underprivileged communities were aware of these distinctions.”

McCallum said the standards make it easier to help students who want to push ahead, however. The Common Core includes directions for alternative pathways that are more advanced than the regular pathway, and which allow a student to complete courses in calculus or something equally rigorous, like statistics, by the end of high school. “It’s always been the case that you need to take more math if you want to be ready for a STEM career,” he said. “There’s always going to be differentiation in high school. So this is not a new thing.”

The main challenge with the new standards, McCallum said, will be ensuring teachers are ready to handle a tougher set of requirements for their students. “A lot of teachers who are used to teaching math as a sort of ‘do-the-math’ subject, they’re going to be called on to have a deeper understanding of what the math is all about,” he said. “For many states, these are simply higher standards than they had before. That in itself is a hard thing.”

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