Math instruction in the United States is “an inch deep and a mile wide,” leaving students with little knowledge because teachers try to cover too much territory, according to a massive international comparison of student achievement and teaching practices. Compared to teachers in countries with higher math achievement, American teachers are tackling an ever wider range of math topics each year, the Third International Math and Science Study (TIMSS) found.

“In other countries, they might spend a month on a topic while we spend days on a topic,” says William Schmidt, the U.S. research coordinator for TIMSS. The “inch-deep” coverage makes it harder for students to remember what they learned. “Then next year, since they’ve forgotten it all, we have to review it.” As a result, extensive time is spent each year on the same basic skills.

What’s working against the U.S., researchers say, is a complex set of cultural ideas about school organization, curriculum design, local autonomy, the role of educational policy and even teaching itself.

The most troubled spot for American mathematics education is in the middle-school years, researchers say, drawing on data collected in more than 40 countries in 1994 and 1995. Where American students scored just above the international average in 4th-grade math, they had slipped below the average by 8th grade; by 12th grade, the U.S. was among the lowest-scoring countries.

Researchers blame this pattern on the heavy repetition of basic skills that begins in 5th grade and persists through grade 8. Students fall so far behind in those years, Schmidt explains, that they never have a chance to catch up. “The middle school math and science curriculum is an intellectual wasteland,” he says. “We seriously deprive our kids of intellectual work during those years.”

Part of the problem, he believes, is that the U.S., unlike other countries, tends to think of middle school as the end of elementary school rather than the start of secondary school. One manifestation is that American 8th-graders are still working on arithmetic while 8th-graders in other countries have moved on to algebra and geometry.

TIMMS researchers also blame repetition on a prevailing philosophy about math curriculum design: that math topics should be revisited each year but at a higher level, a practice known as spiraling. The theory is that concepts are best learned when they are presented in gradually increasing complexity. The reality, Schmidt finds, is that topics are repeated at the same low level. “From the evidence we have, any attempts to produce a spiral curriculum have just had kids go in a circle.” Countries that study a topic in depth and move on get better results, he says, because kids retain what they learn.

Textbooks also aggravate the inch-deep, mile-wide problem. TIMSS found that in the United States, only 25 percent of topics taught in the typical 8th-grade math textbook were new since the 4th grade. For most countries in the study, 75 percent of the topics were new since the 4th grade. American math textbooks are longer as well— averaging 530 pages at the 4th-grade level, compared to the international average of 170.

Textbook publishers aren’t wholly to blame, researchers say. To sell textbooks, they aim to please a broad range of consumers: States with different academic standards. Districts with their own guidelines or philosophical approaches. Back-to-basics parent groups. Professional organizations that issue standards. “Any group that wants to get together can say ‘Here’s what kids should know in math,’ and if enough people say it, then the textbooks will put it in, and you end up with a mess,” Schmidt says.

Countries that outperformed the United States in TIMSS tended to have more uniform standards. Schmidt advocates a set of national standards for academics at each grade level. “There are no magic bullets here, but a coherent set of standards would go a long way towards providing a foundation for serious reform.”

Other TIMSS research suggests that reforming math education in the U.S. also will take a change in deeply ingrained teaching practices.

As part of TIMSS, researchers videotaped a random sample of American, German and Japanese math lessons and interviewed the teachers about their classroom practices. In the United States, nearly all the teachers said they were familiar with standards documents, such as those issued in 1989 by the National Council of Teachers of Mathematics (NCTM), and had implemented them in their classrooms. Very few, it turned out, actually had.

The NCTM standards, for example, call for rote computation to be replaced with activities that convey a deeper conceptual understanding of mathematics. To reach that goal, the standards suggest the use of “math manipulatives,” technology, group work and other strategies. The TIMSS videotape study revealed that teachers had in fact adopted the teaching strategies but ignored the mathematics.

Instead of computing without understanding concepts, students were engaged in hands-on activities without understanding concepts, says NCTM President Glenda Lappan. Teachers failed to communicate clearly the underlying concepts in the math activities and make sure students understood them, she explains. Rather, teachers seemed to look on the instructional strategies as an end in themselves. “You can put kids into groups and say you’re meeting the standards because [they] say you should get students actively involved. But that’s not the point of the standards.”

**Reform pendulum**

“It’s very difficult to change something that is as deeply ingrained as teaching methods,” says video study author James Stigler of the University of California at Los Angeles. “Teaching methods are highly culture specific. You learn them through years and years of being a student.”

Consequently, educational reforms swing like a pendulum between “back-to-basics” and “progressive” teaching methods—with only superficial impact on the classroom, Stigler says. When test scores don’t rise as a result of one reform, “everyone assumes that the [reform] must be wrong, so we go out and try to do the opposite. Nobody ever goes to check what effect policies have on classrooms. They have very little.”

Teaching practices are surprisingly uniform across the United States, Stigler says, despite the value Americans place on local control and teacher autonomy. Whether from New York City or the deep south, California or Montana, teachers in his video study displayed a remarkably similar teaching style. German and Japanese teachers also had a style particular to their culture.

American math lessons followed this pattern: The teacher would present the procedure for solving a particular kind of problem and then guide the class through solving example problems on the board or overhead projector. Then students would work problems on their own while the teacher helped individual students.

Japanese math lessons also followed a distinct pattern: The teacher would present a complex and unfamiliar problem. Students would struggle to solve it and then present different solutions, which the class would discuss. One or more problems would be worked through in this way. At the end of the lesson, the teacher would highlight the central concepts in the lesson.

What characterized American lessons was “a real emphasis on procedures and skills over understanding.” Stigler explains. As part of the study, a panel of mathematicians reviewed transcripts of the lessons in which countries were not identified. The math content of each lesson was rated high, medium, or low. None of the American lessons received a high rating; 89 percent were rated low. German lessons landed fairly evenly in each category. Nearly 90 percent of Japanese lessons earned a medium or high rating.

Stigler’s study also identified cultural differences in the way teachers work together. Japanese teachers often visited each others’ classrooms and critiqued the lessons they observed. American teachers avoided criticism and instead engaged in a “vague sharing of ideas or discussions about individual kids,” he says.

Stigler suspects that the real reason American teachers don’t collaborate more effectively is that they have so little time to do it. “There’s no tradition of what to do with collaborative time. They have no idea of how to structure it.”

Where American elementary teachers might spend nearly all of a six-hour school day teaching, Japanese teachers spend only four hours of an eight- or nine-hour day in the classroom. In Japan, teachers have desks in a shared office space where they meet and refine lessons together.

**Cause for hope**

Some experts say that it’s a cultural assumption that the kind of collaboration expected in the business world isn’t necessary in teaching.

“Imagine that you’re a business person and you have to make a crucial presentation that’s critical to the success of your client and the future of your organization,” says Terry Dozier, the U.S. Secretary of Education’s Special Advisor on Teaching. “But you have no time to confer with your colleagues, no time to prepare. You have to do it at home, unpaid, alone, on your own time. And you have to do it day after day, 180 days a year. What we’re asking of American teachers is unbelievable.”

Despite the obstacles to improving math instruction in the United States, experts see cause for hope. School reform advocates have made professional development time for teachers a priority issue, and some districts are taking note, according to Dozier. “Achieve,” a group of governors and CEOs, is helping states work together on creating more uniform academic standards, Schmidt reports. The National Science Foundation has funded math programs that Lappan says are more closely aligned with NCTM standards than are traditional textbooks. The math programs are just now becoming commercially available.

Stigler thinks instruction can improve if Americans stop relying on public policies to change teaching. Instead, he says, they should take a lesson from the Japanese, who expect gradual improvements through the collective efforts of teachers and have no concept of education reform. “They don’t write reports and hand them out to teachers and assume it will change the way they teach. That’s a very American thing.”