View: The ideology of math education

WASHINGTON, March 6 (UPI) -- There's more to the struggle over math "reform" than meets the eye.

On the surface it appears to be simply a question of emphasis. Parents' groups, along with allied mathematicians and scientists, stress the importance of learning traditional computational skills. The education establishment that runs primary, middle and secondary schools -- led by the National Council of Teachers of Mathematics -- places great importance on understanding math "concepts."

Each side pays homage to the concerns of the other.

But deep ideological and political differences underlie the arguments. Ultimately, they involve the control of credentials in a closed-shop industry, jobs, resources, race, and power.

These issues roiled beneath the surface Monday at an American Enterprise Institute forum titled "Does Two Plus Two Equal Four? What Should Our Children Know About Math?"

Moderator Lynne V. Cheney is a senior fellow at AEI, where she focuses on education standards and policy. She served as chairman of the National Endowment for the Humanities from 1986 to 1993. And yes, she is the wife of Vice President Dick Cheney.

Cheney disclosed that she has been critical of "reform math," which she preferred to call "NCTM math" because, she said, both sides agree that the National Council of Teachers of Mathematics endorses it. She said the approach generally de-emphasize drill and memorization, encourage the use of calculators from kindergarten on, and recommends that students discover methods of addition and subtraction, multiplication and division for themselves.

"Parents groups from California to New York have organized to ensure that more traditional mathematical instruction is at least an option," she said.

Cheney used an example from what she called an NCTM-inspired program called Mathland. Students were asked to solve this problem: "I just checked out a library book that is 1,344 pages long. The book is due in three weeks. How many pages will I need to read a day to finish the book in time?"

The traditional method would be to divide the number of days (21) into the number of pages, getting 64. But, Cheney said, students today are often not taught long division. She held up a huge poster board covered with numbers, displaying the work of the student that Mathland featured as exemplary. "This particular student added up 21s until reaching 1,344," she said.

Later in the program, NCTM President Lee Stiff said that he would never recommend such a method with numbers that large. But Stiff, a professor of mathematics education at North Carolina State University in Raleigh, said the technique is useful in teaching math concepts with much smaller numbers.

"It is not the position of the council that computation is harmful," Stiff said.

Former NCTM president Gail Burill expressed similar sentiments.

But a concerned parent wasn't buying it. Elizabeth Carson is co-founder of NYC HOLD, a New York-based coalition of parents, educators, mathematicians and scientists. "That's what they all say," she told United Press International in an interview following the forum. "'Of course, we all want our children to be able to compute fluently.' But that's not what goes on behind the scenes and in the programs. Because they firmly believe that learning computation restricts the flow of ideas."

Carson said long division "was completely excised from the programs for quite a while until the backlash got so large that they at least had to make nods toward teaching it.

"In 5th grade they're still drawing pictures to solve problems in multiplication and division."

Her son, who attends the public schools in New York City, was drawing clumps of sticks to solve multiplication problems in the 6th grade, she said.

In the 7th grade, he and his classmates were asked to find the area of a circle. Four weeks were devoted to the task. Traditionally, children were given the formula, but apparently these junior Archimedes were supposed to rediscover the uses of pi.

"Where's algebra?" Carson asked. "This is inquiry-based learning taken to an absurd end. But in schools of education, inquiry-based learning is IT!"

Michael McKeown, who identified himself as "a parent activist," is also a skeptic. He is a professor of medical science at Brown University and cofounder of Mathematically Correct, which he described as "a bipartisan citizens' group working to improve mathematics education."

In 1989 NCTM introduced its first standards document, influencing curricula around the country. McKewon said the standard was "deliberately revolutionary." This standard was generally low in content and high in inventive method. It de-emphasized drill, memorization, and "algorithms," which are established procedures for solving problems.

"They emphasize ... generic problem solving, methods such as guess-and-check, stressing conceptual understanding, statistics and data analysis. In addition, they abjure direct teaching and practice in favor or discovery learning and creation of ad hoc methods."

The Californian Mathematics Framework of 1992 "was a particularly radical interpretation of the standards," McKeown said. The California standards of 1997 "rebounded" from the 1992 framework. The NCTM introduced its new standards in 2000.

McKeown said that some educators associated with NCTM standards have written that algorithms are not only unhelpful for learning arithmetic but actually hinder development of numerical reasoning.

But McKeown pointed out that the algorithms of arithmetic, such as long division, help children understand the decimal system and prepare them for algebra. Therefore, he said, the teaching priorities inherent in the NCTM standards are badly flawed.

Those standards have been favored, McKeown said, because of the "foolish romanticism" that children are hard wired to learn math in the same way they are hard wired to learn language.

Venturing below surface disagreements about emphasis, McKeown said misconceptions about math education based on generalizations about race and sex are "less flattering." He quoted former NCTM president Jack Price as saying that traditional methods of teaching math are adequate for "Anglo males" of relatively high socioeconomic status but that women and minority groups do not learn the same way. McKeown represented the NCTM's philosophy in this way: High-status white males (and Asians of both sexes) learn math best deductively in competitive environments, while minorities and non-Asian women learn math best inductively in collaborative environments.

"Various people who support the NCTM style of math have suggested not only that we need math for reasons of ethnic or gender differences, but if you oppose the NCTM standards - their content or their methods - that perhaps you are trying to keep one social group up while keeping other groups down," McKeown said.

He endorsed the belief of Lee Stiff, the current NCTM president, that research data should determine the merits of NCTM standards and said that none of the high-performing California schools with poor, black students uses NCTM-style math as its main program. In 1995 he called a principal of one of these schools to ask the secret of its success. "She answered, 'No new books.'"

McKeown disputed claims, ostensibly based on "brain research," that "discovery learning and inventive methods" work best in math education. In fact, he said, the best research shows that repetition and appropriate repeated stimulation are critical for long-term memory and mastery, which free short-term memory for the problems at hand.

Attempts to teach generalized high-level problem-solving skills are largely ineffective, he said, because such problem solving is based on mastery of domain-specific knowledge.

It is not enough to give lip service to the teaching of computational skills in a program that holds such skills to be unimportant, McKeown said.

Gail Burrill, a past president of NCTM, is in the science education division of Michigan State University. She said as someone who taught high school math for more than 25 years, she believes students should master skills and that computation is important. But she said classroom teaching has often focused too narrowly on the memorization of information, "giving short shrift to critical thinking, conceptual understanding and in-depth knowledge of subject matter."

"Research about teaching suggests that learning may be hindered by presenting too many topics too quickly, presenting isolated sets of facts that are not organized or connected, and not helping students to know where when and why to use their knowledge," Burrill told the AEI forum.

Many mathematics curricula emphasize not so much thinking as procedural objectives that substitute for thinking, she said, adding that when the new standards are implemented intelligently, it produces students with a deeper understanding of math.

David Klein, a professor of mathematics at California State University at Northridge, was highly critical of the NCTM approach.

Klein said no single U.S. institution has caused more damage to the mathematical education of children than the National Science Foundation's Education and Human Resources Division, which funds education projects from Kindergarten to grade 12. It has promoted the worst fads of the past decade, he said.

In both his prepared remarks and in discussion, Klein disparaged colleges of education, which produce most U.S. math teachers. In countries where students score high on standardized tests, teachers have degrees in mathematics, not math education, he said.

Klein said both parents and academic mathematicians are rebelling against the NSF and NCTM programs, which radically de-emphasize basic skill, encourage "rampant" calculator use beginning in Kindergarten, and falsely claim to teach conceptual understanding. "Instead they squander valuable class time on aimless projects with little or no intellectual content," he said.

These programs, 10 of which the U.S. Department of Education listed in 1999 as "exemplary" or "promising," are particularly harmful to children with limited resources, Klein said. "Upper-middle class parents can afford tutoring to compensate for what the National Science Foundation has done to their schools," he said, noting that the tutoring industry has "skyrocketed" across the country as a result.

Like McKeown, Klein denounced as "misguided in the extreme" the idea that minority groups and women do not learn math the same way as white males.

Klein urged President Bush to find new leadership for the NSF's Education and Human Resources Division.

NSF Director Rita Rossi Colwell was in the audience, and Cheney asked her to respond. Colwell said the NSF is in the process of addressing the math and science partnership programs Bush proposed, that she wants to hear all sides, and that Klein's concerns will be looked into. She also said she believes in drill.

Stiff, speaking for the NCTM, said council members want students to be able to do all the traditional operations but also to understand the structure of arithmetic and algebra.

"It's really about change. ... Meeting the needs of students changes over time," he said. "It doesn't stand still. We don't look back and say these were the good old days and things should be this way forever, because times have changed. We need students who are flexible and resourceful problem solvers."

But how much has math changed? Can Stiff make the case that high school students with high math SAT scores in 1960 would be unprepared for today's challenges?

McKeown called "pernicious" the notion that the world is changing so fast that "mere facts" are soon outdated. "Newton's laws still hold for almost all situations we'll encounter," he observed, and algebra's still algebra."

Carson told UPI that the NCTM constantly casts anyone opposed to its reforms as being afraid of change.

"Lee Stiff has said many, many times that parents are simply afraid of a classroom that is unfamiliar to them. It's not true. ... It's not how they're teaching; it's what they're not teaching. In fact, Lee Stiff has been quoted as saying that his organization is trying to change the definition of what the basic skills are."

Revisionists feel free to experiment in prosperous school districts, Carson said, because parents there will spend hundreds of dollars a week on tutors to ensure that their children get the content that is slighted in the classroom. "And then they sell the programs on the basis of this privileged demographic."

Stiff told the forum that data show that NCTM standards are "beginning to make some headway."

But Tom Loveless, director of the Brown Center on Education Policy at the Brookings Institution, questioned that conclusion through a series of graphs representing math scores on Novell Education Academic Partner tests.

Loveless distributed a graph which, he said, showed that with the introduction of NCTM math reform in 1990, "suddenly the breaks were slammed on" to math competency that had been improving since 1982. Tests of students in Iowa showed a rapid decline in the computational skills of eighth-graders since 1990.

Loveless conceded improvements in geometry, problem-solving skills and in the ability to estimate. But the former elementary school teacher said the tradeoff is not a good one. Arithmetic is a non-negotiable skill that children must learn, he said.

Carson told UPI that proponents of "constructive" approaches falsely claim that traditional methods turned children off to math and either hint or say that skeptics are "elitist, racist, or afraid of change."

What motivates the reformers?

It's a business, and it's a secular creed, Carson said. "The ideology that drives the NCTM reform is the same one that dominates the schools of education. There is a belief system - someone described it as a religion - that children learn best in these collaborative environments where they construct their own knowledge, where they're not lectured to, and where they really work well with others with no drill and few objective assessments."

U.S. Department of Education panels that give these programs the stamp of approval "are full of ideologues," Carson said.

It's also a craft monopoly that drives the best teachers out of the profession, she said. Classroom teachers in New York City who feel that they are letting down their children are punished for speaking out against these programs, she said.

Even tenured teachers can get bad evaluations and be punished in more insidious ways. "They don't get their supplies on time, they get the most difficult students, they don't get that special project, maybe they don't advance," Carson said. Teachers, students and parents are "pawns" in the game.

Carson lamented the fact that not only does the federal government fund constructivist math education schemes, but such private philanthropists as Bill Gates and George Soros also underwrite the programs. "They are sold on the lie that this is way to help poor children. It's incredibly racist to presume that these kids cannot be challenged in the same way others have been challenged."

Carson said NCTM-style math education is closing the "performance gap" in test scores by lowering the achievement of above-average students.

"Everyone says parent involvement is the answer," Carson said. "It's ludicrous. Parents don't have time to do what's required to hold this system accountable. It will never work."

Parents resist believing that educators don't have the best interests of their children at heart, she told UPI.

Carson sees bad faith and anti-intellectualism on the part of the educational establishment that goes beyond preferred styles of teaching math. The basics of every subject have been de-emphasized, she said.

The worried parent sees a form of political radicalism at work. She characterized the thinking of educational reformers as she understands it: "We don't see that this elite group that became our mathematicians and physicists has done so well for our country. Look at all these people who are suffering. We have so many things wrong with our culture and with the world."

Carson continued: "There's a revolution going on. Don't ever think there's not. I've seen it from the beginning when my son was taught that Christopher Columbus was a bad man. The emphasis was on children making up their own reality in a sense -- urging children to think that what came before and what experts in any area say is suspect.

"It gets mixed up with race issues, equity issues, where there's a mistrust of experts because of the assumption that those experts are trying to keep you down."