Do the Math!

The need to reform U.S. mathematics instruction dates back to the early 19th century, when cutting-edge educators railed against rote instruction and called for new teaching methods so students could better understand meaning in the complex system of quantitative reasoning.

Associate Professor Jon Star, Ed.M.'93, says those calls for reform have surfaced each generation since, with the latest iteration in the Common Core State Standards — one that focuses on more in-depth study of fewer topics in a coherent national framework. Student understanding these days comes through knowledge of mathematical operations as well as the development of problem-solving strategies that get discussed in class and explained on tests.

"Each successive generation makes that claim, and everyone is after meaning," says Star. "We're getting better at articulating what we mean by meaning."

Today, math educators have a full plate. Adoption of the Common Core Standards last fall has shifted the curricula across the nation to hone aptitudes in critical thinking and problem solving. Stiff high school graduation requirements have expanded math offerings and pressured educators to find ways to teach increasingly complex math concepts to a broader range of students. New teacher evaluation systems, meanwhile, judge educators on their students' performance on statewide standardized tests, or the pre- and post-assessments they've devised to determine how much their students have learned that year in their classrooms.

"There's so much going on, from the outside and on the local level," said David Fleishman, Ed.M.'93, superintendent of schools in Newton, Mass. "We have new, rich problems developed for the Common Core and the new evaluation system, and the pressure on math teachers is significant, even if they are all good."

Jennifer Walsh, Ed.M.'03, who chairs the math department at Sleepy Hollow High School in New York City's northern suburbs, had these issues on her mind one morning in early September during her first-period Fundamentals of Algebra class. It was designed for ninth-graders who had struggled through their eighth-grade math classes and were deemed at risk of failing the statewide math Regents exam, which is among the tests that must be passed to earn a high school diploma. Walsh's Fundamentals course stretches over two years, instead of one, to give her students enough time to master concepts that, she reminded her students, date back to 1800 B.C.

That day, Walsh taught them about the order of operations, the rules that govern the order in which different calculations are made in a complex problem, such as 28/4 – 2(8–7) + 5(3). It was a fast-moving class, with students proceeding at different paces.

Walsh needed the students to learn the nitty-gritty rules that govern how you solve such problems. But she also wanted them to talk freely about their mathematical reasoning, engaging them in lively discussions, and encouraging them to speak with each other to figure out how best to proceed.

To learn the right progression, Walsh suggested they use the acronym PEDMAS, which lays out the proper order: parenthesis first, then exponents, division, multiplication, addition, and, finally, subtraction. She warned, however, that the acronym wasn't failsafe because when it came to multiplication and division, you had to carry out the function that came first in the problem.

"Math teaches you how to think, and that number sense is really crucial in daily life," says Walsh, who that day also administered the course's pre-assessment, the results of which will create a baseline for students, so the district can measure what she's taught by year's end. "It teaches you how to solve puzzles and figure things out. You might struggle, but when you start with a problem and figure out how to solve it, you feel good."

The accountability movement and the standards movement have both placed math instruction near the top of the 21st century reform agenda, with math concepts now taught as early as preschool. No Child Left Behind has students tested statewide in math in grades three through eight. In many states, they must pass four years of high school math and an exit exam to graduate.

Recently, President Obama joined the math bandwagon. In July 2012, Obama proposed creating an elite corps of master teachers in a $1 billion program to reward teachers in the STEM fields — science, technology, engineering, and math. Teachers selected for the program, who commit to one of these fields for several years, would be eligible for a $20,000 annual stipend. The administration kicked off the program by making $100 million available immediately.

The development of online learning has added to the mix. Khan Academy in California has created thousands of 10-minute videos on math topics, tied to grade-level expectations, which have proved useful for both teachers and educators around the world.

"The flipped classroom is the way to make it happen," says current Ed.L.D. student Karl Wendt, referring to instruction done online and homework done in the classroom. Wendt is working at Khan's campus in Mountain View, Calif., for his one-year residency. "It lets students at multiple levels get access to the content."

Math and English stand at the center of the Common Core Standards, the set of policies developed by the National Governors Association, the Council of Chief State School Officers, and the business group Achieve (run by Professor Robert Schwartz, C.A.S.'68, for five years), which has been adopted by 46 states and is expected to be reflected in the curricula of all states by 2015. It's part of a drive to make all students "career and college-ready."

Civil rights leader Bob Moses has called math literacy the key to 21st-century citizenship in our technology-based society. Math literacy is seen as the foundation for many 21st-century careers in the STEM fields. The Bureau of Labor Statistics projects that employment in STEM occupations will grow by 10 percent to 3.5 million in 2020.

"I truly believe that math is the path to power," says Leba Heigham, Ed.M.'95, assistant principal of the Linden S.T.E.A.M. Academy, a K–8 school in Malden, Mass., and a former math coach. "Just look at the opportunities for students with a strong math background — computer software, financial applications, there are lots of burgeoning fields. The skills embedded in math instruction will be the 21st-century skills: They teach habits of mind, so you can use logic to build arguments."

Mary Maxwell West, Ed.M.'74, Ed.D.'81, a senior research associate at the Program Evaluation and Research Group at Lesley University, says studies have established a connection between math mastery and college success. A 2006 study by the U.S. Department of Education, for example, found that success in high school algebra, and algebra II in particular, was highly correlated with college attendance and graduation. The National Science Board in 2004 found that completion of rigorous math courses in high school was a predictor of college success, across race, ethnic, and socio-economic lines.

However, she says a 2004 study by Robert Balfanz and Nettie Legters at Johns Hopkins University also found that many ninth-graders who fail algebra and are held back from entering tenth grade end up dropping out.

Andrew Hacker, a professor of sociology at Queens College, in July 2012 published a provocative essay in The New York Times Sunday review section, questioning why all students need to attain such high levels of mathematics mastery to graduate from high school.

Hacker makes his claim in New York, where the state Board of Regents did its part to raise the bar in math by mandating a score of 65 to pass on the integrated algebra Regents exam required of students. But in a nod to just how hard it is to master algebra, the state put a curve on the test results so that students in 2012 had to answer just 30 percent of the answers correct to earn their 65, according to the state Education Department spokesman Jonathan Berman.

"Yes, young people should learn to read and write and do long division, whether they want to or not," Hacker wrote. "But there is no reason to force them to grasp vectorial angles and discontinuous functions. Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions?"

But Star says cutting out rigorous math could consign many students to dead-end jobs in our increasingly quantitative world.

"It's clear that not every student will need to know algebra in his or her future jobs," he says. "But if you want kids to have an equal shot at getting those jobs, you need the algebra foundation. And people are solving for X more than they think."

As states raise the bar on math, students are learning more complex subjects, earlier on in their educational lives. This trend has presented particular problems for some elementary school teachers, whose pedagogy has typically centered around literacy.

S.T.E.A.M. Academy's Heigham says some of these elementary-school generalists are stymied by all the changes. Her school is in the first year of a state innovation grant to focus instruction on five areas: science, technology, engineering, arts, and math.

"It used to be that instruction focused on addition, subtraction, multiplication, and division," says Heigham, who has also served as a math coach and school principal. "Now there's more geometry, patterning, and pre-algebraic skills" — in elementary school.

There's more math, more often. In the early 1980s, 25 percent of high schoolers didn't take algebra, and students could earn a high school diploma after completing just two years of math. Today, 34 percent of eighthgraders take algebra classes, up from 11 percent in 1986, according to a recent report from the National Assessment of Educational Progress.

"If you have a penchant for math, you should be able to keep advancing, just like in music classes," says Linnea O'Brien, Ed.M.'92, who teaches math at Kamiak High School in Mukilteo, Wash. "You just keep making it happen for those students."

However, she acknowledges that managing a class of students of varying abilities presents huge challenges, especially when she faces classes of 32 to 34 students each day. She has solved that problem by encouraging students to help one another by developing what she calls a "culture of math chat," in which students help each other to solve that day's problems, with O'Brien moving around the classroom to help those with special issues.

"I give a new concept every day, teach them a new layer, give them a little warm-up, and they start buzzing," she says. "They are engaging, thinking, talking, and putting their pencils to paper. I'm a catalyst as I walk around the room, but they are also catalysts to each other."

Math didn't reach the pinnacle of the U.S. education hierarchy until the late 1950s, following the flurry of action in response to the Soviet Union's launch of the Sputnik satellite. Star says that in colonial America, math wasn't taught until students matriculated at college. Math was used by tradesmen, but not by the wealthy elite. By the early 1830s, he says, math reformers were making one of their first stands, arguing that educators needed to move away from rote learning, so they could better understand the meaning of math.

Such calls to move from rote learning to meaning have cropped up regularly ever since. That tension between drilling to understand math concepts and reasoning to understand what math means continues a century later.

The New Math movement from the mid-1950s to the early 1970s, says Star, created new ways to think about teaching and learning math. Educators looked at big ideas and provided young students with exposure to important math concepts at an early age, devising experimental curricula that had kindergartners learning about sets, an abstract concept now taught much later.

"The idea was that even young children would have access to important mathematical concepts and jump right in," he says. "But New Math met its demise in the late 1960s because the elementary school teachers had trouble teaching it. It wasn't the way they had learned math. There was a perception that it was hard to teach, and then-President Richard Nixon pulled the plug on federal funding."

The backlash to New Math brought a return to back-to-basics, which held sway until 1989, with the publication of the National Council of Teachers of Mathematics, which identified process standards that addressed not just what students should learn but also showed what good teaching looked like when math was taught properly. These standards, which stressed reasoning and communication, were in ascendency until the early 2000s, when states developed standards that were assessed through No Child Left Behind.

The Common Core Standards, which are now being phased in across the nation, aim to sharpen the focus of math instruction and bring more coherence to the practice. They come in response to criticism of U.S. math instruction as being far too broad without the kind of indepth instruction practiced in national school systems in Europe and Asia, where students often outperform their U.S. counterparts on tests like the Trends in International Mathematics and Science Study. The study has been carried out every four years since 1995 by a sampling of more than 400,000 students from 59 nations in grades four and eight. In 2007, the United States placed ninth in grade-eight math, behind students in Asian and European nations. The 2011 test results were released in December. For eighth-grade math, there was no significant difference in the score between 2007 (508) and 2011 (509).

Star says the Common Core will bring a greater coherence to the topics across states, on a national scale.

"Everyone will be doing the same thing," he says. "Critics had called our math programs fuzzy and illdefined. Now it's scaled back, and the idea is that we will do a smaller set of things, really well." The statewide standards, and now the Common Core, with its national scope and rigor, will, some say, bring more uniformity to the mixture of instruction that for so long characterized U.S. math pedagogy.

Of course, not everyone thinks this is all positive. Alice Capson, M.A.T.'72, of Overland Park, Kan., who retired from teaching in 2010 after 28 years in the classroom, recalls her first years, with teachers like herself having independence in her classroom, in which she covered a course's topics at her pace and in the order she saw fit.

Almost three decades later, during her last years in the classroom, math instruction had become a team effort, with her school's math instructors meeting often to discuss topics, sharing lesson plans, and developing common assessments to make sure everyone was on the same page.

"Everybody had to do the same thing, every class, from August through March," she recalls. "We all had the same special quizzes. It was nice working together, but it kind of stifled creativity."

Looming at year's end was the state of Kansas' annual assessment, which her students needed to pass. So every day, between 15 and 50 percent of her class time was spent having students do problems from old statewide tests and then going over the individual questions.

"We had to pay more attention to the slower kids; we couldn't write them off," she said. "I worked with them to show them that math was something they could do and get satisfaction from it. They'd come in prepared to work and with a positive attitude. And they'd realize they could do it."

— David McKay Wilson's last piece in Ed. focused on the use of data in education.