A comprehensive examination of videotaped math lessons unveiled here last week provides rare insights into how classroom teachers approach the subject in the United States and six other countries.
The study, “Teaching Mathematics in Seven Countries”, is available from the National Center for Educational Statistics. (Requires Adobe’s Acrobat Reader.)
The videotapes of 638 lessons in 8th grade math are a “virtual microscope for the mathematics classroom,” through which teachers and researchers alike can “see and understand the variety of ways good mathematics teaching can take place,” said William J. Frascella of the National Science Foundation. He is the director of elementary, secondary, and informal education for the NSF, which financed the United States’ share of the project along with the Department of Education.
Researchers also examined teaching strategies in Australia, the Czech Republic, Hong Kong, the Netherlands, and Switzerland, and reanalyzed videos of Japanese classrooms taped in 1995.
The study found that American middle school teachers use teaching approaches similar to those of their counterparts in higher-achieving countries. But the U.S. teachers, the report says, omit one critical ingredient: the underlying mathematical ideas that help students understand how the skills they’re learning are part of a logical and coherent intellectual discipline.
“Higher-achieving countries focus on developing conceptual underpinnings of the problems,” said James Hiebert, a professor of education at the University of Delaware, in Newark, and one of the researchers.
Students in each of the other six countries studied outperformed their U.S. peers in the 1995 Third International Mathematics and Science Study, known as TIMSS. Every country but the Czech Republic outscored the United States in a 1999 follow-up, called TIMSSRepeat.
When American teachers discuss problems that require multiple steps and an understanding of mathematical concepts, Mr. Hiebert said, their focus shifts to the skills needed to solve the problem, not the concepts behind it.
The findings suggest that the long-standing debate over teaching basic skills or conceptual understanding should shift to how to teach both, Mr. Hiebert said at a news conference held here in Washington to release the report.
Moreover, the study suggests that the debate must extend beyond the kind of curriculum teachers should use and to their own professional training, according to James W. Stigler, a professor of psychology at the University of California, Los Angeles, and the director of the video study.
Even if a curriculum requires students to engage in conceptual thinking, most American teachers appear to be unprepared to talk about the process they use to arrive at their answers, he said.
“If they don’t know how to implement it at a high level, it turns into a routine procedural level,” Mr. Stigler said.
The video project is the final piece of TIMSS—Repeat. On the math and science tests in that follow-up study, U.S. 8th graders scored in the middle of the 38-country pack on both the math and science sections. A separate analysis compared student achievement in 13 states, nine districts, and five regional consortia with that of children in other countries. (“A World-Class Education Eludes Many in the U.S.,” April 11, 2001.)
In a 1995 video study, researchers compared practices in the United States, Japan, and Germany. They discovered that Japanese teachers employed an approach dramatically different from that of teachers in America and Germany.
Japanese teachers challenged their students to solve problems that prove mathematical concepts, and they rarely gave assignments that required the repetitive use of skills. Teachers in the United States, in contrast, focused on applying the skills they had shown students how to use. (“New Images of Teaching,” April 9, 1997.)
In the newly released study, researchers discovered that Japan has a distinctive approach to teaching math. Other countries with high achievement use other strategies, though with similar success.
For example, the Japanese teachers who were videotaped for the study spent just 17 percent of their time on what were deemed easy math problems, while the other six countries devoted 60 percent or more of their class periods to such problems.
Likewise, Japanese teachers spent much of their time helping students solve problems, while teachers in Hong Kong lectured almost exclusively and aided students’ understanding by prompting them with difficult questions.
Despite the different teaching styles, students in Hong Kong and Japan both excelled on TIMSS tests, according to Mr. Stigler, who is also the president of LessonLab, a Santa Monica, Calif., company that offers professional- development programs that enable teachers to review videos of excellent teaching practices.
While American teachers employed many of the same teaching techniques as those used by their colleagues in other countries, they failed to make the connection between the mathematical concepts and procedures that required solving complex problems, the TIMSS-Repeat study found.
Such connections might force students to conjecture why a solution might be true, and then generalize about other conditions under which it might be true.
In Japan, more than half the problems teachers asked students to solve entailed conceptual thinking. No other country offered such problems more than 25 percent of the time. U.S. teachers presented such problems 17 percent of the time, about the same as their counterparts abroad.
But, the researchers say, U.S. teachers fell short in leading class discussions about such complex problems. When discussing the solutions, none of the American teachers prompted his or her students to connect the procedures they had used to solve the problem with the mathematical ideas behind the solution. A third of the time, the teachers simply gave the answer.
American teachers “are more focused on getting the answer and less focused on getting students to engage in serious mathematical concepts,” Mr. Stigler said in an interview.
By contrast, almost half the time, teachers in Japan and Hong Kong coached students into conjecturing and generalizing why an answer was true, the analysis of the videotapes found.
A lack of deep mathematical knowledge may be the reason why U.S. teachers spent more time reviewing previously covered material than teachers elsewhere, Mr. Hiebert suggested.
American teachers spent more than half their time reviewing content previously taught, compared with 24 percent of the time devoted to that purpose in Hong Kong and Japan. Only the Czech Republic’s teachers dedicated more time to review—58 percent—than did U.S. teachers.
If students don’t understand the concepts, Mr. Hiebert said, “then it’s likely that they’re going to forget, and the teachers are going to have to go back and review and review.”
The videotape study’s findings reinforce the need for better professional development and teacher preparation, according to one leading math educator.
“We need to be talking more with prospective teachers about how to make the connections,” said Johnny Lott, the president of the National Council of Teachers of Mathematics and a professor of mathematics education at the University of Montana-Missoula.
Meanwhile, current teachers need time to “think about what they’re doing,” he added.
Videos of model lessons from four of the countries are available from Mr. Stigler’s company. Teachers can view them for a fee and see ways to teach that they might never have considered trying.
“Teaching has been shown to make a difference,” Mr. Hiebert said. “Teaching is something that teachers can do something about. A lot of other factors are out of the control of teachers.”
The mathematics report is the first of three studies emanating from this latest video project. Next year, researchers will release a similar report comparing science-teaching practices across the same seven countries. The final report will analyze changes in teaching practices in the United States from 1995 to 1999.
Just as researchers are finishing the final pieces of TIMSSRepeat, 50 countries are in the middle of collecting student-achievement data for the next version of the project. Students in the Southern Hemisphere took the tests late last year, and students in the United States and the rest of the Northern Hemisphere are scheduled to take the tests this spring.
The program is now known as the Trends in International Mathematics and Science Study, but has the same acronym.