- Quanta Magazine

# Meet the New Math, Unlike the Old Math

IF WE COULD snap our fingers and change the way math and science are taught in US schools, most of us would. The shortcomings of the current approach are clear. Subjects that are vibrant in the minds of experts become lifeless by the time they’re handed down to students. It’s not uncommon to hear kids in Algebra 2 ask, “When are we ever going to use this?” and for the teacher to reply, “Math teaches you how to think,” which is true—if only it were taught that way.

To say that this is now changing is to invite an eye roll. For a number of entrenched reasons, from the way teachers are trained to the difficulty of agreeing on what counts in each discipline, instruction in science and math is remarkably resistant to change.

That said, we’re riding the next big wave in K-12 science and math education in the United States. The main events are a pair of highly visible but often misunderstood documents—the Common Core math standards and the Next Generation Science Standards (NGSS)—that, if implemented successfully, will boldly remake the way math and science are taught. Both efforts seek to recast instruction in the fundamental ideas and perspectives that animate the two fields.

“What we did in reorganizing the content of school mathematics was long overdue,” said Phil Daro, one of three lead authors of the Common Core math standards.

The changes go beyond the contentious new methods of teaching arithmetic that have grabbed headlines and threatened to blunt the momentum of Common Core math. Both documents developed out of decades of academic research on how children learn, and they reflect similar priorities. They exhibit an elegant rethinking of the basic structure of knowledge, along with new assertions of what’s important for students to be able to do by the time they finish high school.

“Overall, there’s a movement towards more complex cognitive mathematics, there’s a movement towards the student being invited to act like a mathematician instead of passively taking in math and science,” said David Baker, a professor of sociology and education at Pennsylvania State University. “These are big trends and they’re quite revolutionary.”

Pedagogical revolutions are chancy endeavors, however. The Common Core math standards were released in 2010 and NGSS in 2013. Now, years on, even enthusiastic early adopters of the Common Core like the state of New York are retreating from the standards. While the ultimate impact of both the Common Core and NGSS is still uncertain, it’s clear these standards go beyond simply swapping one set of textbooks for another — to really take hold, they’ll require a fundamental rethinking of everything from assessments to classroom materials to the basic relationship between teachers and students.

The Old New Math NGSS and the Common Core are a significant departure from the way science and math have been taught, but they didn’t come out of nowhere. In fact, they’re consistent with a trend that’s been slow-boiling for a half-century.

In a 2010 paper, Baker and colleagues analyzed 141 elementary school math textbooks published between 1900 and 2000. They found that what kids were learning changed considerably during that period. Until the 1960s, basic arithmetic accounted for 85 percent of math instruction. By the end of the century that proportion had dropped to 64 percent, with the balance of instruction devoted to more complex topics like advanced arithmetic and geometry.

“When you step back historically and sociologically, it’s clear education has really ratcheted up along these cognitive dimensions,” Baker said. “The idea that education is like men’s ties and just goes through this cycle of wide and thin is not true.”

Pedagogy has shifted as well. During the same period in which students began to learn more complex mathematics, leaders in science and math education launched complementary pushes to teach students to think more like real scientists and mathematicians. These efforts included the “New Math” of the 1960s and similar plans that decade to teach science as an “enquiry into enquiry,” as one leading expert of the time put it. Later manifestations of the impulse away from rote instruction include curricular standards created by the National Council of Teachers of Mathematics in the 1980s and the enthusiasm for “inquiry-based” science in the 1990s.

All of these initiatives had the right idea, but their implementation was off, say developers of NGSS and Common Core math. “Inquiry” is a habit of mind among scientists, but in the 1990s it was taught as its own curricular topic: Last week we learned about DNA, this week we’re going to learn about inquiry.

“Inquiry became almost an empty word, where it didn’t really matter what the inquiry was about,” said Heidi Schweingruber, director of the Board on Science Education at the National Academies of Sciences, Engineering, and Medicine, which provided guidance for the development of NGSS.

The same problem happened in math. For the last 50 years, reformers have wanted to teach kids to reason mathematically, to think nimbly about topics like quadratic equations that otherwise come off flat. Instead, in programs that employed the New Math, students often ended up playing logic games.

“The push toward conceptual understanding and understanding rich mathematical ideas sometimes ended in practice with students just engaged in activities and messing around,” said Robert Floden, dean of the College of Education at Michigan State University.

It’s not surprising that ambitious changes like these would be hard to implement. After all, teaching kids to adopt a scientific mindset is a subtler and more complex task than having them memorize the parts of a cell. For one thing, it requires teachers who inhabit that mindset themselves, and they’re harder to find. For another, it takes a more patient perspective than the prevailing one in public education, which expects teachers to post a learning objective on the board before each class and end every unit with a multiple-choice test.

Less Is More How does one adjust the course of a curriculum that’s been gathering inertia for decades? The developers of NGSS and Common Core math started by reducing the mass of content that had accumulated over the years, often in haphazard fashion.

“Mainly, the US mathematics curriculum prior to the Common Core was a geological accretion of additions, mostly, and [some] compressions over 50 years,” Daro said. “There was a lot of mathematical junk food and traveling down rabbit holes and up cul-de-sacs.”

Schweingruber made a similar point. “The US has a mile-wide, inch-deep curriculum with tons and tons of things and ideas for kids to learn, but not an opportunity to go in depth,” she said.

As the authors got down to work on Common Core in 2009 and on NGSS a year later, some of their first discussions were about what to leave in and what to take out. “It required some argument on the part of folks in the framework about what that baseline really would look like,” Schweingruber said.

The final documents omitted a number of familiar topics. The NGSS writers eliminated instruction in the rote formula for stoichiometry calculations (the process for quantifying elements at different stages of a chemical reaction) from the high school chemistry curriculum. Daro and his collaborators on Common Core math, William McCallum of the University of Arizona and Jason Zimba of Student Achievement Partners, decided the technique of “simplifying” answers didn’t add much to mathematical understanding, so they took it out.

By removing content, the creators of Common Core math and NGSS hoped to expose core disciplinary ideas. A good example of this is how the Common Core teaches proportionality. Before, proportionality occupied about 10 percent of math instruction in grades six and seven. The main outcome of all that instructional time was that given two equivalent fractions, students could cross-multiply in order to find a missing term.

“What they’re learning is: The way you find the fourth number is by setting up this gadget called a proportion,” Daro said. “That’s not really learning anything about proportionality, that’s learning how to get answers to problems in this chapter.”

Common Core math doesn’t mention cross-multiplying, and it cuts out the special case of finding a missing fourth term. Instead, it focuses on the idea of a ratio, which begins modestly in sixth grade and develops all the way through calculus. Students begin by looking at a table of equivalent ratios—also presented as a double number line—and progress to the understanding that the slope of a line is a ratio.

“[The Common Core writers] said, look, let’s figure out what’s important about fractions and choose a path through them, which leads to ratio and proportion, which leads to linear functions, which leads to aspects of algebra,” said Alan Schoenfeld, a professor of education and mathematics at the University of California, Berkeley.

The understanding of slope as a ratio feeds into an even more fundamental emphasis in Common Core math: the analysis of functions. By thinking about the slope of a line as a ratio, students get in the habit of analyzing the parts of a linear function so they can see how changes in elements of the function affect the relationship between inputs and outputs.

Daro sees this shift from solving equations to analyzing functions as one of the biggest conceptual changes in the Common Core.

“The important line of progress is the line that begins with the theory of equations, a 19th-century central focus, to calculus and analysis, which is 20th-century [mathematics],” he said. “It’s a move from spending almost all your time solving equations towards analyzing functions.”