Debate Over How to Teach Math Takes Cultural Turn

As a kid growing up in Nicaragua, Guillermo Mendieta was taught by his father to work out the square roots of license plate numbers in his head. Learn to analyze numbers, the elder Mendieta would preach, and “you can basically write your own ticket in any field.”

Mendieta, now 38, listened. He became a much-honored Los Angeles teacher who is passionate about the power of math to open doors in the new economy--especially for young Latinos and African Americans.

But those doors, barely ajar for many blacks and Latinos, will slam shut, Mendieta argues, if California schools turn back the clock and return to a traditional method of instruction that emphasizes repetitive drills and memorization of multiplication tables.

The Los Angeles Unified School District, now in the midst of a bitter debate over how to teach math, has emerged as the critical battleground in the state’s long-running math wars. The fight pits advocates of a return to traditional methods in all schools against those who favor a stress on hands-on activities to translate abstract concepts into concrete problems.

Traditional methods have divided the world into those who “get it” and the vast majority who don’t. Mendieta and his allies argue that teachers must have a broader repertoire of techniques if more students are going to acquire the math skills demanded by the workplace.

Insisting that old-fashioned methods have a disproportionate, negative impact on the poor and minorities, they have cast the debate as a struggle for economic and social justice.

Mendieta, who recently resigned in protest from a 35-member committee that will make a recommendation to the Los Angeles Board of Education next month on what approach to use, contends that traditional instruction dooms most students to failure, regardless of race or ethnicity.

But he argues that blacks and Latinos have the most to gain from methods that make math accessible to larger numbers of students because they have lagged furthest behind in terms of achievement. For support, he and other reformers point to dramatic increases in the number of minority students taking college-preparatory math classes in high school.

The importance of Los Angeles in the debate is a result not only of its size, but also of its previous commitment to the “reform” methods. The struggle has reverberated nationally.

“I’ve received hundreds of phone calls and letters of support and e-mails,” said a buoyant Mendieta, who works with less-experienced teachers through the Achievement Council, a Los Angeles nonprofit organization that addresses educational inequities. “I see this as a beginning . . . in a long, hard but wonderfully rewarding and just struggle.” To draw attention to the issue, Mendieta announced Thursday that he will go on a hunger strike.

Adversaries Reject Racial Argument

Mendieta’s adversaries agree that the stakes are high, but they reject the idea that traditional math instruction has favored the interests of white males and Asian Americans, two groups who have traditionally excelled in the subject.

“If anyone is racist or sexist, it is those who claim that women and minorities are unable to deal with traditional mathematics,” said Barry Simon, a Caltech professor who holds an endowed chair in math and theoretical physics.

California’s new academic standards suggest that all high school students--not just those bound for college--should have the equivalent of three years of rigorous math instruction. The California high school exit exam, which today’s eighth graders eventually will have to pass to graduate, will include a significant amount of algebra. In Los Angeles, those students also will have to pass geometry.

The drive for standards, which is taking hold nationwide, has been propelled by growing international economic competition and a new information-based economy that puts a premium on mathematical skills.

Unfortunately, there is no conclusive data with which to sort out the claims and counterclaims. In fact, as the fight between traditional and reform approaches to math has played out over the last several months within the Los Angeles Unified School District, the most compelling numbers show widespread failure regardless of the teaching method.

But even without data to make their case, the traditionalists have been winning all of the key battles at the state level over standards, textbooks and tests. Now local districts throughout the state are having to change course to remain in sync with the new standards dictated in Sacramento.

Los Angeles must decide which textbooks to use, whether to return to a traditional sequence of algebra, geometry and advanced algebra in high school and how best to help students failing math to meet the new standards.

Some districts are sticking resolutely with reform. Hundreds have hedged their bets, buying workbooks to boost basic skills. Others, such as the Sacramento Unified School District, have returned to a more traditional approach.

Traditionalists believe that understanding math comes not from activities, but from intensive work with formulas, equations, fractions and the like until those concepts become second nature. In particular, they fault reformers for allowing students to rely on calculators instead of learning basic math facts.

“Mathematics is just building upon and building upon, and if at any point the foundation is air, the whole building is going to collapse,” said Simon, one of a growing number of top university mathematicians who have become activists on the traditional side.

Mendieta complains that lessons that focus almost exclusively on number crunching “castrate the curiosity of the vast majority of children before they leave middle school.” Teachers should focus more on comprehension--for example, using tangible objects such as beans or blocks to provide physical representations of such abstract ideas as borrowing and carrying in addition.

For all the back-and-forth over philosophy, both sides recognize that American public schools have not succeeded in teaching math as well as those in many European and Asian countries.

“In reality, everybody has failed,” said Robert Collins, an L.A. Unified administrator trying to reach a compromise among the factions on the math committee. “There’s no evidence that we were ever successful teaching large numbers of urban children mathematics, anywhere.”

More than half of the ninth-graders in L.A. Unified got grades of D or F in algebra or its equivalent last year. The failure rate in the remedial math course taken by ninth-graders, which starts with basic arithmetic, was even higher.

The failure rate varies from school to school but is equally high for both traditional and reform courses. “We may have changed the curriculum but we haven’t changed the failure rate,” Collins said.

National Record Shows Few Successes

Nationally, performance of students in their first high school math classes has been dismal for many years, said John A. Thorpe, the executive director of the National Council of Teachers of Mathematics. Teachers and researchers offer a variety of explanations.

High school teachers say students come to them weak in basic arithmetic. Sixty percent of the eighth-graders in L.A. Unified, it is estimated, do not yet know their multiplication tables.

Many teachers say that students are unmotivated, that they skip class and don’t do homework. Robert Drake, a math teacher at Roosevelt High School, said many students are simply convinced they lack a talent for math.

“They say, ‘It’s OK if I fail. My mom knows I’m bad at math,’ ” Drake said.

Mendieta blames traditional math for fostering that defeatist attitude. “By nature, kids are born with the desire to learn,” he said. “But the overwhelming majority of the students under the traditional curriculum end up hating mathematics, not understanding mathematics or thinking very poorly of their own mathematical competence.”

The reform method was launched a decade ago by the National Council of Teachers of Mathematics. A key element was the creation of “integrated” courses, which combine the teaching of the various branches of mathematics--algebra, geometry, probability, statistics and so on. The point is to promote comprehension by showing how various branches of math are interconnected. The classes also tend to cover fewer topics but in greater depth.

Since the advent of those courses in Los Angeles, the number of African American students completing three years of high school math with a grade of C or better has risen by more than 40%, according to district analysis; the rise is nearly as high for Latino students.

The numbers of students eligible for the University of California and the numbers of students taking Advanced Placement courses in math also were up dramatically.

Statistics like that have convinced some officials, including L.A. school board President Genethia Hayes, that the nontraditional math classes need to be kept. “I will advocate as hard as I can with my colleagues to make sure this particular door never gets shut for children of color,” Hayes said. “I really do see this as an issue of social justice.”

But critics say college-prep courses that fly the reform banner are weak and leave students unprepared for college. Moreover, they say, although more students are taking AP courses, they’re not necessarily passing the tests showing they have learned the advanced material.

Committee member Paul Clopton, a San Diego statistician and co-founder of Mathematically Correct, compiled data showing that students taking integrated courses score lower across the board on state standardized tests than do those in traditional courses.

The number of students taking the SAT exam at L.A. Unified schools using the reform approach is down substantially and so is the average score on the math portion.

“It seems as if the attitude is that one way to get more people to pass algebra is to redefine what the subject is,” Simon said. “You’ve got to teach people certain basic skills and facts.”

James W. Hiebert, a professor at the University of Delaware, recently analyzed the existing research. He concluded that all programs in use today--including traditional ones--are experimental.

“We’ve never had a systematic trial of these things over time and then chosen the one we think is best,” he said.

Even so, he said, “presuming that traditional approaches have proven successful is ignoring the largest database of information we have.”

Well-implemented reform programs can teach all students concepts without sacrificing their ability to compute, he said. “My opinion is that good materials and good teaching are good for everybody.”

Reformers concede that their method is vulnerable to criticism because poorly trained teachers have, in some cases, substituted meaningless activities for instruction. People on both sides of the debate agree that far too many teachers lack a fundamental knowledge of math.

Karen Fuson, a math education researcher at Northwestern University, has spent years writing and testing a new math program for elementary school students that combines traditional and reform methods and works well with disadvantaged children.

Bolstering students’ conceptual understanding while giving the proper weight to basic skills is difficult “and nobody has worked out all the pieces and how they fit together,” she said.

But she believes that data she has gathered so far “clearly indicate that it is possible for urban kids to learn much more than they do learn now.”

The key, Fuson and others say, is good teaching. Teachers not only have to ask students to compute, they also have to make sure students understand why the computations work and what the answers represent.

“Of course it’s quite difficult to do,” she said.

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Old Math, New Math

Since World War II, math instruction in the United States, and especially California, has changed course time and again--with little improvement in test scores.

Early 1950s: Students, grouped by ability, learn math through drill and memorization.

Mid-1950s: “New math” begins to spread; students are taught underlying rules and theorems of math, as well as how to calculate.

1957: Sputnik launched; worries about lapses in math instruction lead to federal teacher-training programs.

1962: Opposition to “new math” bubbles up in academic journals; lack of training cited.

1970s: Back-to-basics begins to edge out “new math.”

1985: Reform gains ground; California issues framework for math instruction that urges schools to emphasize problem-solving and understanding.

1989: National Council of Teachers of Mathematics publishes standards for reform methods.

1992: California embraces methods that further de-emphasize basic math skills.

1994: Adoption of textbooks based on reform ideas sets off backlash; term “fuzzy math” is coined to characterize reform.

1995: State panel is appointed to restore balance to math program.

1995: Los Angeles Unified School District receives $15-million federal grant to train teachers in reform methods.

1996: Study finds that American eighth- and 12th-graders lag behind peers in most of the developed world in math skills.

1997: California requires students to memorize basic skills, but also stresses problem-solving and in-depth understanding.

1999-2000: L.A. Unified begins to overhaul its math program to reflect the more traditional state standards.