Equity versus excellence, conceptual understanding versus procedural expertise, and classroom experience versus ivory-tower pedigree. When it comes to math instruction, opposing camps of experts dispute the correct set of priorities to emphasize among the above when directing how math is taught in K-12 classrooms.

Recently, the debate over math instruction has been mistakenly rolled into the debate over the Common Core and other state math standards. These standards are a set of academic targets. Their stated purpose is to create a roadmap for what students need to know in which grades to become college and career ready. Backers of the standards say they were intentionally written to be flexible—so states, districts, and educators wedded to different teaching styles could continue to teach math in the manner their leaders thought best.

Opponents of the standards have argued that, in actuality, the way the standards were implemented constituted a top-down imposition of how reading and math were to be taught in classrooms. Nonetheless, many of the social media posts objecting to “Common Core math” are actually objections to *practices* used to teach the math, rather than the underlying *content* of what is being taught.

Herein lies another frequent point of confusion. In recent years, political backlash to the Common Core led many states to “repeal” the standards. While these repeal efforts have satisfied the grassroots who objected to Washington interference in state curriculum policy, in most states, Common Core repeal meant changing the name used to refer to the academic targets, while keeping the substance of the standards in place. While many states no longer use the “Common Core” in a strict sense, in actuality, the majority of states’ standards are still aligned to the principles undergirding the Common Core targets.

Despite the conflation between the packaging of state standards, the actual substance of the academic targets, and teaching procedure, the real debate over the practice of how to best teach math is a longstanding one with serious implications that predates the development of the Common Core.

Alan Schoenfeld is a professor at the Berkeley School of Education with a high-level background in mathematics. In a 2004 paper entitled “Math Wars,” he traced back the politics of the teaching practices debate to the early 1900’s.

In essence, the paper argued that, historically, the political left has backed the teaching of a more accessible mathematics that prioritizes understanding concepts over rote repetition. The political right, meanwhile, has backed a focus on traditional mathematics that stresses rigor, memorization, and procedural skill.

In an email exchange with InsideSources, Schoenfeld said that the Common Core had become a “Rorschach test” that people use to project their feelings about education. So while the debate over math practices is real, and the Common Core aligned math standards actually are more amenable to progressive teaching styles, the math standards themselves do not mandate how the subject should be taught.

“I’m willing to bet that the majority of the people who blog about the Common Core and blame it for this or that pedagogy haven’t read the Common Core and don’t know that whatever pedagogy they’re reacting to was the product of a local decision,” wrote Schoenfeld.

So while it is true that some left-wing progressive groups have allied with the broader coalition of right-wing opposition to the Common Core aligned standards movement, the teaching practices commonly associated with the math standards are actually rooted in the political left’s historical preference for how math should be taught.

A physics professor fired the latest salvo in the ongoing debate last month by penning for Forbes: “Why Writing About Math Is the Best Part of Common Core.” The author’s work was a response to an article in the Atlantic from 2015 “Explaining Your Math: Unnecessary at Best, Encumbering at Worst.”

The left-leaning “reformers,” cited by Schoenfeld, argue that learning math should be made easier through visual models and encouraging students to write to prove understanding. The right-leaning “traditionalists” argue that the models quickly become crutches, and writing about math is a distraction from hours spent working with the numbers.

**The Case for ‘Reformed’ Math Instruction**

Steve Leinwand is a lead research analyst at the nonpolitical, nonprofit American Institutes for Research (AIR). His strong views in favor of more accessible math instruction are rooted in his previous work on the board of the National Council of Teachers of Mathematics. His arguments do not reflect the stance taken by AIR, which stays out of these debates as an institution.

“One of the positive and powerful qualities of the Common Core is that it supports learning mathematics in ways that gives kids opportunities to communicate” along with the expectation that those communications reflect “a manifestation of their reasoning,” said Leinwand.

Leinwand was speaking in support of the item that appeared in Forbes, in which a physics professor with school-aged children wrote: “explanation is an essential part of math and science. In fact, open communication of results among scientists is the crucial step that separates modern science from alchemy, and necessarily entails *explaining how you got your answer*” (emphasis in the original).

Leinwand’s argument is that educators should do everything they can to make mathematics accessible to all students by using all available teaching aids, including graphs, charts, pictures, words, and equations rather than solely teaching how to crunch numbers.

“The idea of practicing and practicing and regurgitating a procedure flies in the face of everything we know about how to take this body of knowledge called mathematics and have it work for everyone,” rather than just those at the top of the class, said Leinwand.

He argues that despite the noise, the Common Core, and the similar math standards states have held onto, are working for parents, students, and teachers. He rejects the arguments made by the traditionalists, whom he said are only animated about protecting their supply of high achieving math majors, rather than ensuring that each student has a solid foundation in the subject.

“Why the hell, after 100 years of that kind of math teaching” said Leinwand of the traditionalists, “was the U.S. doing so poorly, until recently?”

**The Case for ‘Traditional’ Math Instruction**

Jim Milgram is one of the traditionalists. A former mathematics professor at Stanford, Milgram was a lead author of a set of rewritten standards in California in the 1990’s that marked a backlash against the teaching practices promoted by Leinwand and the National Council of Teachers of Mathematics.

More recently, Milgram was a top math representative for California during the Common Core standards writing process. When the process was over, Milgram was the highest profile participant to refuse to endorse the math standards. He was featured prominently in a documentary “Building the Machine.” That movie was produced by the homeschooling lobby, which has been outspoken in its opposition to the Common Core.

While “Building the Machine” focused primarily on the hardball methods used to pound out the standards, which Milgram agreed were objectionable, his real opposition isn’t to the idea of having national standards, but instead to the quality of the Common Core. He argues that the final product is not rigorous enough.

“From the beginning I thought national standards were a very good idea,” but in the final calculus, he said, “the standards are nowhere near what they would have to be to prepare students for both college and the workplace.”

While he agreed that the Common Core standards were better than those used in the majority of states, he feared that a small group of large states with high standards dating to before the Common Core would be backsliding by adopting them. He argued that these states, such as California, were doing a good job of preparing students for college math and a competitive high-tech workplace. By designing a low bar intended for access for every student in the country, Milgram thought the Common Core’s backers were making a mistake that would hurt high achievers.

Furthermore, Milgram disputed the idea that forcing students to use the imprecision of language, or graphical aids, would help them master concepts in the long term.

The foundation of mathematics “is just as rigid as anything you are ever going to see—it doesn’t change—it hasn’t changed for thousands of years,” he said. “There is no place for imprecision in describing what you are doing in mathematics and why it works.”

“What happens with kids who are taught this way” he said of the math instruction associated with the Common Core, “is they do alright, until about 5th or 6th grade, and then they fall apart.”

According to Milgram, this is because rather than anchoring the students in hard math, they become grounded in imprecise models. These instructional devices—which Leinwand backs and Milgram equates to crutches—become far less helpful as the math gets more complicated, typically when ratios, rates, and more complicated fractions are introduced, he argued.

**It’s Personal**

In addition to their disagreement over what the research says is the best way to teach math, Leinwand and Milgram—who have met in person and have known of each other for years—both question the fitness of the other to be making decisions about how math is taught to the millions of students in K-12 schools.

Milgram, the Stanford math professor whose father was also an accomplished mathematician, argued that it takes someone of his pedigree to fix what ails the American school system.

“We see the whole subject,” he said of the community of top-flight math professors he said he belongs to. According to Milgram, it takes someone of his background and an understanding of the top of the field to craft a solid K-12 math curriculum.

After dismissing Leinwand as “an ass,” Milgram said: “call him up and ask him how many courses he’s taken at the college level […] the answer is none.”

Leinwand, meanwhile, described Milgram as “a despicable human being” and “a bully.”

“Sure he knows higher order math—how does that prepare you to tell elementary teachers what they should do?” asked Leinwand. “[Milgram] couldn’t walk into a 4th grade class and teach 4th graders fractions if his life depended on it.”

Part of the personal animosity has to do with the weak outcomes for students that each accuses the other of producing. Leinwand believes that Milgram is only concerned about the top math students—the future mathematicians. Milgram believes that Leinwand targets only the lowest common denominator and is gutting middle and high school math courses in service to a warped view of equity.

The disproportionate focus on equity means, “once the inequalities in the class start to appear, there is enormous pressure—really enormous pressure—to not only suppress the top kids, but equally suppress the average kids and bring down the whole class to the lowest kid,” according to Milgram.

Leinwand disagreed. He said Milgram and some top mathematicians have “hoodwinked” educators into serving the needs of the top two percent of students who go on to become math majors. The best students, aren’t hurt by Common Core math instruction, they simply get put on an accelerated path, he argued.

“The problem is not the top two percent, the problem is the bottom 60 percent,” he said. “Milgram doesn’t give a damn about that, he just wants his pipeline of math majors”

**Finding a Middle Ground**

Despite the personal recriminations, there are some, like Schoenfeld, the Berkley affiliated author of “Math Wars,” who have spent years working to forge a peace.

Schoenfeld has said that he has a foot in both camps. In addition to his own work on K-12 education, he also has a doctorate in pure mathematics.

“Leinwand and Milgram are like matter and anti-matter; put them together and you get an explosion,” he wrote. So while those two are unlikely to ever come to a consensus, Schoenfeld believes there is a happy medium between the two positions they represent.

While Shoenfeld identified with Milgram’s emphasis on precision—“Of course getting the ‘math’ right counts”—he also believes there is room for expanding the focus of the discipline to include questions of comprehension (a big part of the argument championed by Leinwand).

“If you teach for skills you get skills, but if you teach for skills, concepts, and problem solving in the right ways, you get all three, with no loss in skills,” he wrote.

In general, because the Common Core focused on emphasizing content over teaching practices, (and because substantively similar standards have become solidly entrenched), Schoenfeld said that the “math wars” over teaching procedures have actually died down since their height in the 1990’s, despite some “notable exceptions.”

Leinwand said this is because he has “beaten” Milgram. And while Milgram expressed dismay at the state of mathematics instruction in the U.S. today, there is still enough attention given to the energy in the grassroots opposition to Leinwand’s preferred teaching practices that the policy pendulum could swing back in Milgram’s direction.

Recent reports revealed that a push to give New Mexico’s accomplished education secretary, Hanna Skandera, a top spot at the federal education department was scuttled over her support for the Common Core. Even current U.S. Secretary of Education Betsy DeVos disavowed her earlier support of the Common Core to become a palatable nominee.

Like many of these debates, looking at test scores or research studies in the immediate future is unlikely to definitively end the argument over Common Core-aligned math instruction. Though many states first adopted the standards in 2010, implementation has had its hiccups, and backers say the effects of the more rigorous standards are only just now being felt.

While Milgram and Leinwand exchanged barbs over test scores in states where the other has previously led the charge in changing teaching practices, it is often difficult to draw a line of cause-and-effect between a standards reform and student achievement in the immediately subsequent years. As Schoenfeld states, it takes roughly a decade before curriculum, textbooks, and professional development aligned to a new set of standards are fully put in place and the standards become fully alive.

“Anyone who claims success or failure within the first couple of years of new standards is whistling Dixie.”

## 5 responses to Understanding the Debate Over Math Instruction

I wonder if they still do rote verbal multiplication tables together in the third grade? Worked for me, long before we had pocket calculators. And for everyone else in the class who could listen and speak.

We have been educating kids since the beginning of this country when my ancestors arrived in Salisbury in 1638. All those people who got us to the moon in 1969 probably learned their multiplication tables in the third grade as well. Same for the guys who won WW I and II. That B 17 was a product of the 1930’s Engineering. And then along came: Long Division… in the fourth grade!

I wonder how Bill Gates learned his multiplication tables? Might be worth asking him …. Or Perhaps The Donald…one of the most successful business guys in his generation.

Back to basics… worked for me and millions of others.

Unfortunately, many of your classmates don’t know why 12*3=36. I hold a master’s degree in math education. I am all for drilling students on their multiplication tables, but not until after they have a firm grasp on addition and understand that multiplying is repeated addition. (ie, that 12*3 = 12+12+12). Then, and only then, are they ready to drill those multiplication facts. As you can see, I am not a part of either faction referenced in the article. I advocate for a balanced approach that wants conceptual understanding accompanied by memorization of related math facts.

Don’t they do addition in Kinder Garden, or at least by first grade? Don’t they have to know about making change for lunch costs? 10 cents + 10 cents +10 cents = 30 cents.

When addition is introduced depends on the school system. My daughter was introduced to the concept of addition in kindergarten and has learned addition facts and algorithms, including carrying, in first grade. I am not saying that multiplication tables shouldn’t be taught in third grade. I agree with you that our students should have their addition, subtraction, multiplication, and division facts memorized. I am saying that they shouldn’t be taught until a child first understands what multiplication means. Regarding lunch costs, few kids carry lunch money any more, at least not at my daughter’s school. It is charged to the student’s account and the parents pay it off at the end of the month. (Or the parents pre-pay, depending on preference.) With the prevalence of debit and credit cards, making change is becoming a lost skill.

Tough to be a dinosaur.. no wonder the kids can’t make change anymore. “I’ll give you a nickle and you can give me a dollar back.” What??

If they can’t put in on the register or console the amount tendered, they can’t figure out the change unless the machine tells them.

And I read also where huge numbers of kids using the school lunch programs are paid for by the govt for “low income” families. But no one ever checks to see if that is really the fact.

But I see a lot of pictures of these kids now playing with boxes of beads or something similar in their “math classes”. How about using instead fake pennies, nickels, dimes and quarters ? Might that not help them become familiar with pocket change?

But I think they want to go to the “cashless society” in the future, where it will all be “funny money.” I still carry cash, and love to ask, “Do you still accept cash?”

Ask any small business owner. I recently negotiated a discount for cash on 2 used laptops I bought. Owner said, “I have to wait 2 days to get the credit card money in my account, and I have payroll to make.” So I went to the bank and came back and got 5% off when I paid with cash.

Always worth a try!