March 2002
Does Two Plus Two Still Equal Four?

Despite efforts to improve mathematics education in the United States, a majority of children still cannot perform at a basic level and an achievement gap between white and minority students continues to persist in that subject. In the ongoing debate over mathematics instruction, two divergent views have emerged. One emphasizes the systematic mastery of basic skills and standard methods. The other de-emphasizes the direct teaching of basic skills, encourages the use of calculators from kindergarten on, and recommends that students discover methods of addition, subtraction, multiplication, and division for themselves. The latter view is often referred to as “reform mathematics.”

A seminar held at AEI on March 4 examined the evidence regarding the success or failure of this second approach. AEI senior fellow Lynne V. Cheney moderated the discussion. Michael McKeown, a professor at Brown University, delivered the main presentation. McKeown cofounded Mathematically Correct, a group formed to strengthen mathematics education in California. His talk was followed by comments from Gail Burrill, past president of the National Council of Teachers of Mathematics; mathematics professor and researcher David Klein; Lee V. Stiff, the current president of the NCTM; and Tom Loveless, a scholar at the Brookings Institution and the director of the Brown Center on Education Policy.

Lynne V. Cheney
AEI

The purpose of this seminar is to discuss a method of teaching mathematics called “reform math” by supporters and “fuzzy math” by those who oppose it. In the name of neutrality, I will call this method of teaching “NCTM Math,” since both sides agree that the National Council of Teachers of Mathematics encourages this approach.

Instructional programs based on NCTM recommendations generally de-emphasize drill and memorization, encourage the use of calculators beginning in kindergarten, and recommend that students discover computational methods for themselves. Proponents of NCTM math argue that such an approach encourages conceptual understanding, but many parents have become concerned that students who learn mathematics in this way will never perform mathematical operations efficiently and automatically. Throughout the United States, these parents have organized groups, such as Michael McKeown’s Mathematically Correct, in an attempt to ensure that traditional mathematical instruction remains an option.

Michael McKeown
Brown University

In 1989, the NCTM introduced the first version of what it believed to be comprehensive K–12 mathematics standards. Deliberately revolutionary, the standards emphasized the way mathematics was taught as much as, if not more than, they emphasized the mathematical content that should be taught. The standards downplayed aspects of that content such as mastery of skills, standard algorithms, and analytical methods. They instead emphasized inventive methods, generic problem-solving, and conceptual understanding. They abjured direct teaching and practice in favor of discovery learning and creation of ad hoc methods.

Many states revised their state mathematics standards to reflect those proposed by the NCTM, and math programs reflecting the NCTM standards were adopted nationwide. But the standards were not without opposition. In 1992, for example, California adopted a particularly radical interpretation of the NCTM standards. Following a backlash by many parents and mathematicians, California changed direction in 1997 and adopted standards emphasizing mastery of standard algorithms and skills. Since then, curricula and tests in California have followed those standards. In 2000, partly in a response to criticism of the 1989 standards, the NCTM introduced new standards, called Principles and Standards of School Mathematics.

Why are standards so important? The standards movement is based on an appealing three-part idea. First, develop standards that indicate what students should know, when they should know it, and to what extent they should know it. Second, develop tests that reflect the content of the standards. Third, develop curriculum linked to the standards. While these elements are appealing, the problem arises in implementation. Poor standards can lead to ineffective testing and bad curriculum.

There are several fallacies behind the NCTM standards:

·        We are hard-wired to learn mathematics. This is false. We are hard-wired to learn basic concepts about numbers, but not about simple arithmetic or more complicated techniques, such as algebra.

·        Brain research shows that discovery learning and inventive methods are the most effective methods for teaching math concepts. The best research shows that repetition is critical for long-term memory and mastery.

·        Skills and knowledge are no longer as important as problem-solving abilities in today’s fast-changing world. This simply is not true, as algebra and other traditional methods remain relevant. In addition, problem-solving skills are based on the mastery of a specific set of knowledge. 

I have asserted that many aspects of the NCTM standards are badly flawed and, in some cases, leave children unprepared. If so, why have schools adopted and favored these standards? One reason could be romanticism—the notion that if we place children in a math-rich environment to explore and discover for themselves, they will learn the basics of math. Another possibility is the belief that while traditional teaching methods are effective in teaching white males, women and minorities learn differently and thus need different methods.

To test the latter assertion, I analyzed data from high-achieving California schools where minorities make up at least 90 percent of the student population and at least 70 percent of students are from low-income families. I identified the top four schools and found that none of them used mathematics programs based on the NCTM standards.

In conclusion, some basic terms need to be defined. 

·        Standards. Everyone likes the idea of standards, but quality matters.

·        Balance. Though we all want balance, the actual weight given to different factors matters.

·        Skills. Anyone can claim to favor skills, but if standards do not stress the proper skills, those skills will be de-emphasized.

Gail Burrill
Michigan State University

As a math teacher, I believe that students should master skills and that computation is important. The focus, however, should be on helping students become problem-solvers.

What does it mean to say that we want students to understand mathematics? Students who understand mathematics are able to solve problems and make connections between what they know and what they are learning. They can make decisions using mathematical reasoning, represent and analyze situations, and communicate their thinking to others.

How do we accomplish this? Research provides some pointers. Research about learning suggests that students learn when they are actively involved in choosing and evaluating strategies, considering assumptions, and providing feedback. They learn by building or transferring knowledge from previous experiences. Struggling with concepts on their own enables students to benefit from a lecture that brings ideas together. Research on teaching suggests that presenting too many topics too quickly, presenting isolated sets of facts, or neglecting to provide a context for how students should use mathematical knowledge may hinder learning. In light of this research, traditional methods of teaching too often focus narrowly on memorization without focusing on critical thinking, conceptual understanding, and in-depth knowledge of subject matter. The development of intellectual competence requires more than the accumulation of discrete pieces of information.

The NCTM’s standards support the kind of teaching advocated by this research. Traditional curricula often fail to help students understand the structures within a discipline and instead emphasize mastering procedure. New and promising curricula based on NCTM standards support the kinds of learning and teaching that research suggests are effective. Evidence is mounting that when properly implemented, the new curricula produce students with mathematical understanding.

There are still challenges to face. Quality and adequate teacher preparation and development are needed, computers and calculators must be used effectively, and teachers must be given necessary resources.

David Klein
University of California at Northridge

No institution in the United States has caused more damage to the mathematical education of children than the education and human resources division of the National Science Foundation, the division that funds K–12 education projects. The NCTM and colleges of education created fuzzy mathematics, and the NSF (along with private foundations) supports it.

In 1999, for example, the Department of Education released a list of ten “exemplary and promising” mathematics programs that it recommended for the nation’s schools. More than half of these programs were developed using NSF grants, while others are promoted using NSF money. Why are parents and hundreds of mathematicians concerned about these “exemplary and promising” programs? Because these mathematics programs are among the worst in the country. They radically de-emphasize basic skills in arithmetic and algebra and introduce calculators in kindergarten. They claim to teach conceptual understanding, but do not. Instead, they waste valuable class time on aimless projects with little or no intellectual content.

In November 1999, I faxed an open letter to then–education secretary Richard Riley, signed by more than 200 mathematicians and scholars (including seven Nobel laureates), asking the Department of Education to publicly withdraw the “exemplary and promising” list. The NCTM denounced the letter and affirmed its support for fuzzy math programs. These programs continue to be promoted by the NSF.

NSF and NCTM fuzzy math programs harm all schoolchildren, but especially lower-income children. While middle- and upper-class parents can afford tutoring to compensate for weak math programs, this option is not available to children from low-income families.

The NCTM has adopted the view that female and minority students have learning styles that are different from those of white males and Asians of both genders. This view is consistent with NSF-funded math books, which stress inductive reasoning instead of deductive reasoning, the core of mathematics. This approach is misguided. Teachers such as Jaime Escalante, renowned for helping minority children excel in mathematics, physics, and computers, prove that children of all races can clearly excel in classical, content-rich educational environments.

The NSF needs new leadership for its education and human resources divisions, leaders who can stop the damage the NSF is causing to mathematics education in America.

Lee V. Stiff
National Council of Teachers of Mathematics

Fuzzy math existed in traditional mathematics. You may have understood how to do basic computation, but did you understand place value or grouping and regrouping? If you did not understand it, it was unclear to you. And I maintain that if something is unclear to you, it is fuzzy to you. Fuzzy math existed long before the creation of the 1989 NCTM standards.

We are not in much disagreement. We all want students to be able to add, subtract, multiply, and divide, but the NCTM wants them to do more. It wants them to understand the mathematics they are learning, so that when they learn algebra, they understand the structure of arithmetic and can then understand algebra.

The NCTM wants every child to have a high-quality mathematics teacher. That means we need to support our teachers. Middle-grade teachers must be capable mathematics learners so that they can impart the vision of the Principles and Standards of School Mathematics (PSSM). We must also improve the preparation of elementary school teachers. We support requiring college students who are preparing to become teachers to take more mathematics before they enter the classroom. In other countries, teachers are mathematicians and math educators by training. We want that for our children. Why do we not require that?

Standards show the way. Forty-nine of fifty states have endorsed the ideas found in PSSM. It is all about change. Today, we need students who are flexible and resourceful problem-solvers. Knowing basic facts does not make you a great problem-solver. If you want young people to use knowledge, they must be given sufficient opportunities. The new standards give students those opportunities.

We have evidence that what we are doing with standards is working—in Pittsburgh, in Philadelphia, in Los Angeles, in Puerto Rico, in North Carolina, and in Massachusetts. Data from standardized tests, such as the National Assessment of Educational Progress, tell us we are making some headway.

We understand what we need to do. We need to get better teachers in the classroom and support them. We need to go to urban settings and solve the problem of poorly qualified teachers. Less than 50 percent of students in urban settings have a chance of having a teacher with any credentials for teaching mathematics. It is our responsibility to provide young people the opportunity to learn mathematics.

I want to direct my comments to the evidence. I will focus largely on arithmetic because I consider arithmetic essential to mathematics. Arithmetic is a non-negotiable thing that children must learn.   

The National Assessment of Educational Progress consists of two tests: the Main NAEP and the Trend NAEP. The Main NAEP is based on the NCTM standards, but the questions on the Trend NAEP have remained the same since the 1970s. A graph of fourth-grade scores for both tests shows that, over time, math scores for the Main NAEP test increased. Geometry, data analysis, and problem solving are far more important on the Main NAEP test than on the Trend NAEP, and that is why students are scoring better. If you look at the Trend NAEP, you will see a basically flat line. Why have these scores remained flat?

Since the federal government does not break out and report trends in arithmetic on the Trend NAEP, we used publicly released test items to determine students’ performance in arithmetic over time. In the 1980s, all age groups showed improvement. Beginning in 1990 (the year NCTM math was introduced into schools), growth in arithmetic scores stops and, in some cases, even begins to decline. Further, an analysis of test items related to fractions shows that scores for seventeen-year-olds plummeted after 1990. This analysis is not conclusive, but it does need to be taken seriously.

It is important to determine whether students can add, subtract, divide, and multiply by the end of fourth grade. The NAEP test needs to be revised to report data specifically on arithmetic proficiency.

At the state level, Iowa provides another source for trend data since it has been giving the same test since 1978. The Iowa test reports two categories of data: “Computation,” which measures student’s ability to perform basic arithmetic, and “Total Math,” which encompasses everything outside of computation. Trend analysis of Computation scores reveals a rapid decline in the skills of eighth graders in Iowa since 1990. Total Math scores have remained high because students are learning more geometry and problem-solving skills. While the increase in Total Math scores is good, it should not come at the expense of basic proficiency in arithmetic.

The NCTM often implies that before its standards were introduced, students were uninterested in mathematics. The data show quite a different picture. When students were asked if they agreed with the statement, “I like mathematics,” the responses of fourth graders remained unchanged between 1990 and 2000, while eighth graders increasingly say they do not like mathematics. Student interest in mathematics has not increased. When the responses are controlled for race and ethnicity, the data shows a sharp decrease in the number of African-American seventeen-year-olds who like mathematics.

The evidence remains inconclusive, but it does indicate that we have paid a price by using NCTM math. That price is too high. If there is evidence supporting any gain from NCTM math, it has not yet been produced.

AEI research assistant Stephanie Lundberg prepared this summary.