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The two city districts that made the greatest strides in math on the latest national assessment relied on similar strategies: building students鈥� conceptual math skills and investing in professional development in that subject for elementary and middle school teachers.

While administrators in Boston and San Diego say that many factors were at work in their gains on the recent National Assessment of Educational Progress, they believe their approach is giving students greater ability to solve a broad variety of math problems and preparing them for more complex mathematics later in school.

鈥淚t鈥檚 not enough to memorize algorithms,鈥� said J. Chris Coxon, the deputy superintendent for teaching and learning in the 57,000-student Boston district. 鈥淚f [schools] are going to be successful in teaching math, they have to be open to students鈥� using different ways of solving math problems.鈥�

Conceptual math generally seeks to cultivate students鈥� overall understanding of different math concepts, and lessen their reliance on memorizing set formulas and procedures. San Diego and Boston officials, as well as many math and curriculum experts, believe that approach is helping students learn to solve problems in a variety of ways, as well as preparing them for higher-level math.

Both districts鈥� math efforts have received grant money in recent years through the National Science Foundation. The independent federal agency has been a strong supporter of conceptual math.

NAEP provided results last month as part of what is known as the Trial Urban District Assessment. The test showed average math scores for 11 urban districts rising at both the 4th and 8th grade levels. Such gains were not echoed in reading, where performance remained mostly stagnant.

Overall, urban schools remained well below national averages in both subjects, with greater percentages of students showing only 鈥渂elow basic鈥� skills. Boston and San Diego, by contrast, each had combined improvements at the 4th and 8th grade levels in math that exceeded those nationwide gains.

Still, some math researchers are reluctant to attribute the districts鈥� improvements in scores to a commitment to conceptual math, or any single educational strategy.

鈥淚t鈥檚 not wise to read too much into one wiggle in NAEP data,鈥� said Tom Loveless, a senior fellow at the Washington-based Brookings Institution, who has studied state and local math scores in depth.

Many Approaches

Researchers and school officials have debated for years the most effective way to teach math. The so-called 鈥渕ath wars,鈥� which gained intensity in the late 1990s, featured supporters of a more conceptual approach to teaching math arguing from one perspective. Backers of conceptual math see evidence of its effectiveness in foreign nations, particularly in Asia, whose students routinely outperform their U.S. peers on tests of math skill.

Critics of that approach, however, said American students鈥� shortcomings were most likely the result of a lack of fundamental math skills.

In recent years, both camps have emphasized that a balance between conceptual math and basic skills is needed.

San Diego and Boston officials say they had no interest in revisiting those math conflicts.

Kris Acquarelli, the mathematics director for the 137,000-student Southern California district, said the school system has made changes in its approach to math over the past five or six years, some of them with the help of philanthropic and corporate backers, such as the San Diego-based technology giant Qualcomm Inc.

A central focus of San Diego鈥檚 plan has been to make math more digestible for teachers and students alike. While the district uses state-approved math textbooks at all grade levels, many elementary teachers鈥攑articularly inexperienced ones鈥攈ave difficulty choosing lessons and determining which areas of math should receive the most emphasis.

To help them, San Diego officials produced a number of additional materials, including 鈥渃urriculum maps,鈥� which point to important concepts in textbook chapters; pacing guides, which help teachers balance the amount of time spent on various topics; and modules, which give teachers ideas for individual lessons.

鈥淭hey help teachers navigate the curriculum,鈥� Ms. Acquarelli said of those tools.

An overriding goal is to broaden students鈥� overall problem-solving skills, rather than just encouraging them to memorize formulas. 鈥淲e want to make sure they learn the meaning behind the mathematics,鈥� she said, 鈥渞ather than just rules and procedures.鈥�

San Diego is also aggressively promoting professional development, according to Ms. Acquarelli. The district arranges workshops throughout the year for K-12 teachers, some mandatory, some optional. It also has placed math resource teachers, instructors who are especially strong in the subject, in its lowest-performing elementary schools.

More Math for Teachers

The National Council of Teachers of Mathematics, in Reston, Va., has encouraged states and teacher-training programs to require more math of aspirants. Teachers need to have taken enough math courses to understand how basic concepts lead to more complex topics, NCTM President Cathy L. Seeley said.

鈥淵ou need to know more than just elementary math to teach elementary math,鈥� said Ms. Seeley, a research fellow at the University of Texas at Austin.

Michelle Collins, a 4th grade teacher at Ibarra Elementary School in San Diego, considers herself fortunate. Her master鈥檚-degree training included an emphasis in math. But when she joined the district five years ago, many elementary instructors struggled to find the best way to use textbooks or interpret the goals outlined in California鈥檚 state standards, she recalled. The pacing guides and workshops have helped, she said.

鈥淲e talk openly about our comfort level in math,鈥� Ms. Collins said.

How Much Progress?

Boston administrators have heard elementary teachers voice doubts about their math-teaching abilities, compared with their comfort with literacy lessons, Mr. Coxon said. The district has established math-leadership teams and math coaches at its schools, to try to provide 鈥渃onstant support鈥� in the subject, he said. It has also arranged professional-development sessions focused on math at night and on weekends.

Urban Standings

Boston and San Diego showed the most improvement in math performance among selected urban districts on the latest adminstration of the National Assessment of Educational Progress.

*Click image to enlarge.

SOURCE: National Center for Education Statistics

鈥淭eacher training is pretty much the cornerstone of whether we鈥檒l be successful,鈥� Mr. Coxon said.

Especially, he said, because Boston is teaching math much differently than it had a number of years ago. The district has phased in the use of two sets of curriculum programs that emphasize conceptual math. In elementary schools, it uses Investigations in Number,Data, and Space, crafted by the Cambridge, Mass.-based TERC, formerly the Technical Education Research Center, a nonprofit that specializes in math, science, and technology education. In its middle schools, the district uses Connected Math. Both programs were developed with NSF funding.

Other urban districts face challenges in trying to ensure that math is taught consistently. Chicago officials recently counted 86 different texts and editions of math textbooks in use in their K-8 schools, said Martin Gartzman, the chief math and science officer for the 427,000-student district.

Three years ago, the district launched a math and science initiative, aimed in part at fostering a more coherent curriculum. The district is now encouraging schools to adopt conceptually driven math curricula in the K-8 grades. Chicago鈥檚 scores on the urban NAEP improved, but not by a statistically significant margin.

鈥淲e鈥檝e got a long way to go,鈥� said Mr. Gartzman.

Those who are wary of the conceptual approach to math instruction say other factors may be behind the districts鈥� improved NAEP scores. Martha Schwartz, a co-founder of Mathematically Correct, which advocates the strong teaching of basic math skills, attributes San Diego鈥檚 progress in part to a greater emphasis on basic content in California鈥檚 math standards in the late 1990s.

鈥淲e have strong standards that people have pushed for,鈥� said Ms. Schwartz, who lives in the Los Angeles area.

Mr. Loveless noted that San Diego is just one of many California districts that made headway on state math assessments in the past five years.

While she is encouraged by San Diego鈥檚 NAEP scores, Ms. Acquarelli does not believe that an allegiance to conceptual math, or any single strategy, is enough to help a district.

鈥淭he language gets in our way,鈥� she said, referring to debates about math instruction. 鈥淲e have similar goals. 鈥� Are we where we want to be? Absolutely not. We want to continue to make steady progress.鈥�