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William Jackson begins class at 12:20 p.m. by posting a cartoon drawing of two rabbits on the chalkboard.

Before he tells his 8th graders why that picture will help illustrate their current lesson, Jackson reminds them of the problem they solved the day before. If someone is paid one penny per day for a job and that wage doubles every day, then 20 days later, the person will be making more than $5,000 a day.

That鈥檚 the power of exponential growth, he says. Today, he tells his students that they will learn a different method of growth, known as the Fibonacci sequence. And the rate at which rabbits reproduce will play a significant role in understanding it.

In Jackson鈥檚 math class, students are encouraged to work in groups. They help each other understand the day鈥檚 assignment and discuss the best way to solve the problems. 鈥擝enjamin Tice Smith

The way Jackson is teaching mathematics now is different from the way he taught four years ago, different from the approach of most of his colleagues in this urban district about 15 miles from New York City, and different from the methods of most American math teachers.

Since 1997, Jackson and his colleagues at Paterson Public School No. 2鈥攌nown here simply as School 2鈥攈ave been investigating the findings of an international study of teaching methods in various countries and how well students learn by those methods. Most schools, experts in the subject say, haven鈥檛 heeded the lessons from the Third International Mathematics and Science Study鈥攌nown as TIMSS鈥攁nd are teaching exactly the way they were before the U.S. Department of Education and its counterparts around the world released that ambitious analysis of teaching methods, curriculum, and student achievement.

The 1996 Study found that elementary students from the United States scored above average in international comparisons, but then their standing dropped in middle school and high school. The researchers found the U.S. curriculum a 鈥渕ile wide and an inch deep,鈥� often covering topics in many grades, but never encouraging students to acquire a deep understanding of the material.

For Jackson and his colleagues, the most compelling findings from TIMSS were its results on teaching methods. U.S. teachers show students how to perform mathematical functions, but don鈥檛 challenge them to learn the underlying concepts of mathematics, according to the study.

While the research has been the subject of widespread coverage in newspaper articles, books, and academic journals, experts say few educators are going through the research and making changes in the way they teach.

鈥淭here are people who are deeply involved in it, but that鈥檚 a relatively small number,鈥� says Karen Holoweg, the senior program officer for the Center for Science, Mathematics, and Engineering Education at the National Academy of Sciences, which has published a guide to help schools apply the lessons from TIMSS. 鈥淭hen at the other extreme, there are people who have read about [TIMSS] in the newspaper and that鈥檚 about all.鈥�

In the middle are many school administrators who are exploring what the TIMSS findings might mean for their classrooms, Holoweg says鈥攂ut few are as far along as Paterson鈥檚 School 2.

Jackson, who has taught at School 2 for 16 years, has discovered a new way to impart knowledge. He and his fellow teachers have written mathematics curricula for the 7th and 8th grades that is a mix of New Jersey鈥檚 academic standards and the Japanese curriculum. More important, they鈥檝e changed the way they teach in an effort to help their mostly high-poverty, immigrant students understand the concepts of mathematics, not just how to operate formulas.

鈥淭IMSS definitely was the catalyst for everything we have done,鈥� Jackson says.

The rabbits on Jackson鈥檚 chalkboard are emblematic of the problem he is going to have his students solve. It鈥檚 a tactic that Japanese math teachers commonly use. For Jackson, the cartoon is only the first of several tactics he has ready for today鈥檚 lesson.

If this pair of rabbits鈥攁 male and a female鈥攁re born today, they will have to mature for one month before they can mate, he tells the class. Once they mate, they will wait a month before they reproduce a new pair. He shows how the warren of rabbits would look after the second month: It would include the original pair and a pair of newborns.

Assuming this pattern continues, he asks, how many rabbits will be in the warren at the end of one year?

At 12:30, he tells the students to work on their own, stopping to explain the instructions in Spanish to those who didn鈥檛 follow him in English and encouraging Bengali students to translate for others who didn鈥檛 understand.

Five minutes later, he gives the whole class a hint by posting the results as of the third month. At that point, the original pair would have mated a second time and had a new set of offspring, the poster shows, but the 1-month-old pair wouldn鈥檛 be old enough to mate. That means there would be three pairs of rabbits at the end of the third month.

When he鈥檚 done, he encourages students to work in groups.

Three minutes later, as the buzz of students working on their own starts to heighten, two boys approach a visitor and ask what answer he calculated. 鈥�2,048,鈥� he says, showing that he predicted the warren would double in size every month.

鈥淚 saw a different pattern,鈥� one of the students, named Michael, tells the visitor, who completed four courses of college-level math more than a decade ago.

Since every pair at the beginning of the month will not be parents by the end of the month, the size won鈥檛 double, Michael says.

Instead, it would go up in smaller increments. At his desk, he had drawn pictures of what the first six months looked like and then noticed a pattern. If he added the total from the previous two months, he would get the number for the next month.

After the first month, there was one pair. After the second month, there were two. After the third, three (or 1+2). After the fourth, five (2+3). After the fifth, eight (3+5). After the sixth, 13 (5+8). If you take the pattern out to 12 months, 233 pairs of rabbits will be in the pen.

A few moments later, Michael approaches Jackson and explains his answer.

鈥淒o you think that鈥檚 right?鈥� the teacher asks. Michael nods, and the teacher says, 鈥淒id you see if anyone else gets that answer?鈥�

鈥擝enjamin Tice Smith

How Jackson鈥檚 students learned about the Fibonacci numbers pattern is common in Japan and almost unheard of in the United States.

As part of TIMSS, researchers at the University of California, Los Angeles, hired videographers to record 231 8th grade mathematics lessons in Japan, Germany, and the United States.

The researchers expected to find that Japanese and German teachers used the same methods across their countries because those countries have centralized curricula. But they were surprised to find that even though the curriculum varies in states and districts across the United States, American teachers tend to employ the same teaching methods as their colleagues around the country.

The typical U.S. lesson, according to James W. Stigler and James Hiebert, the principal researchers on the project, consists of the following script: The teacher asks a quick question that requires a short answer. Then the teacher checks homework, distributes a worksheet, and monitors how students handle the questions. If students get stuck on a problem, the teacher explains how to solve it. At the end of the period, the teacher offers a quick lesson review.

鈥淲hat we鈥檙e looking at is an American way of teaching,鈥� says Hiebert, who is a professor of education at the University of Delaware in Newark. 鈥淚t wouldn鈥檛 matter where you go, you鈥檇 see pretty much the same method.鈥�

The videotapes reveal a dramatically different approach in Japan. The teacher begins by reviewing what students learned the day before, and then the teacher assigns a problem for students to solve using that method. Ten minutes into the class, the heart of the lesson starts. The teacher assigns the problem for the day and tells students to begin solving it. Unlike problem-solving in the U.S. classes, where students are expected to employ a technique the teacher has explained, the Japanese problems are designed to prove a mathematical principle. Often, the problems will challenge the students to prove an underlying concept.

Soon, the teacher invites students to work with groups. While students work, the teacher observes and responds when asked for help. Instead of showing the class how to solve the problem, the Japanese teacher coaches students by asking them questions that may help lead to a solution.

By the end of the period, students have put their work on the blackboard and are invited to explain their findings to their classmates, according to The Teaching Gap: Best Ideas From the World鈥檚 Teachers for Improving Education in the Classroom, the book that Stigler and Hiebert wrote explaining their research.

By 12:49鈥攋ust 19 minutes after Jackson assigned the problem about rabbits鈥擬ichael is ready to present his findings to the class. He has written his work on the blackboard, just as he explained it to the visitor and Jackson a few minutes earlier.

Once Michael tells how he arrived at his answer, one of his classmates is offering a solution. He found the same pattern in a slightly different way.

鈥淎t what point did you stop drawing rabbits and see the pattern?鈥� Jackson asks.

鈥淎t about six months,鈥� he says.

After the boy is done, Jackson asks the class: 鈥淗ow many people got 233 pairs of rabbits?鈥�

Approximately half the students raise their hands.

School 2 is an unlikely place to find innovation. Eight years ago, a team of New Jersey officials who took over the management of the 24,600-student Paterson district labeled the K-8 school one of the four worst in the city. The new administration fired the principal and brought in Lynn A. Liptak, the principal of one of the city鈥檚 highest-performing schools, to turn School 2 around.

Liptak began to explore ways to overhaul the school, which draws many of its 720 students from the immigrant Hispanic community and homeless shelters. Almost all of the students receive free or reduced-priced lunches.

At a 1997 seminar, both Liptak and Jackson heard of a radically different way to teach mathematics. Frank L. Smith, a professor of educational administration at Teachers College, Columbia University, showed a group of Paterson teachers excerpts from the videos collected by Stigler and Hiebert.

While the possibility of teaching differently excited Jackson and Liptak, other teachers rebelled.

鈥淭hey saw it as a dig at the United States,鈥� Jackson remembers. 鈥淢ost of them were just outright hostile.鈥�

鈥淭hey saw it as another mandate that they should be hostile toward,鈥� says Smith, who is a consultant to the school district as part of the state鈥檚 takeover team. 鈥淚nstead of being eager to learn, they became quick to be hostile.鈥�

The lack of receptivity is a standard reaction, according to Hiebert and others who are investigating ways of learning from the TIMSS results. Educators get defensive because they see the research as yet another criticism of American schools. Hiebert and others say they try to emphasize positively that the Japanese method may help students better understand mathematics, and therefore American teachers might want to learn from it.

鈥淥ur teachers are very open to understanding that,鈥� says John R. Breckett, the superintendent of the Lake Shore district in suburban Detroit. 鈥淚f you say, 鈥榃e have to do this because our achievement levels are pathetic,鈥� then you鈥檝e lost them.鈥�

While most Paterson teachers rejected the TIMSS message, Jackson and Liptak decided it could help them teach math differently and perhaps more effectively.

鈥淲hat really sold it was that videotape study,鈥� Liptak says. 鈥淲hen we saw what math instruction could look like, that was powerful.鈥�

In the summer of 1997, Jackson set out to compare the state鈥檚 mathematics standards with any other materials he could find, including the voluntary standards set by the National Council of Teachers of Mathematics, the Japanese centralized curriculum, and textbooks from Korea, Singapore, and Hungary.

Over the next two years, he and a team of School 2鈥檚 mathematics teachers crafted a unique curriculum for the 7th and 8th grades and wrote lesson plans.

The hardest part, the teachers say, was writing each individual lesson. Some mornings, the teachers would meet for just an hour, Jackson says, and they鈥檇 get so frustrated that they鈥檇 go home. Other days, they would charge ahead and make a lot of progress.

Often, the biggest roadblock was the teachers鈥� knowledge. They knew they had learned the algorithms when they were in 8th grade, but realized they hadn鈥檛 mastered the concepts that supported them.

It鈥檚 a common complaint among those who are trying to learn lessons from TIMSS.

鈥淔or the majority of the teachers involved here, they鈥檙e finding that they鈥檙e learning the concepts along with the kids,鈥� Breckett says. The 3,200-student Lake Shore system has molded its 1st grade math curriculum and teaching methods using the findings from TIMSS.

鈥淥ur teachers understand math and can teach it, but a lot of them don鈥檛 have the in-depth knowledge to know how math works,鈥� Breckett adds.

Once the teacher has learned the math concepts and prepared the lesson, School 2鈥檚 teachers say, the students are more likely to grasp the lesson.

鈥淲e had to make sure we were clear on the concept of how the algorithm works,鈥� says Nicholas B. Timpone, one of the School 2 teachers who spent last summer writing 7th and 8th grade math lessons with Jackson. 鈥淚鈥檓 so familiar with the math in them that it鈥檚 so much better for the kids.鈥�

By 1:05, Jackson is explaining the Fibonacci sequence. Just as Michael and his classmate found, the next number is the sum of the two previous ones. While the pattern doesn鈥檛 yield growth as quickly as the one in which a worker鈥檚 pay doubles every day, it does start to expand. The pattern eventually gets to 233, jumps to 377, and then to 610, and so on.

鈥淵ou might be thinking, so what?鈥� Jackson says to his class. 鈥淭hese Fibonacci numbers are very important numbers. You see them in nature.鈥�

The example of breeding rabbits is true, and it鈥檚 one of the examples that the 13th-century Italian mathematician known as Fibonacci cited in explaining his discovery. Other places the pattern appears, Jackson tells the class, include the mating patterns of honeybees, and in the branches of some trees and plants, such as the Achillea ptarmica, otherwise known as the sneezewort.

American teachers are unlikely to incorporate honeybees or sneezewort into their lessons, say some advocates of remaking U.S. math instruction, because they don鈥檛 get the support they need.

Jackson has been paid to develop the curriculum and write lesson plans over the past three summers because Liptak, his principal, has tapped a variety of local, state, and federal professional-development funds to pay him and others.

But that鈥檚 a luxury for most schools.

鈥淭he limitation in the United States is the lack of support for teachers to grapple together with the teaching of mathematics so it makes sense to the students,鈥� says Patricia Wang-Iverson, a senior associate for Research for Better Schools. The Philadelphia nonprofit organization runs the regional Eisenhower Consortium for Mathematics and Science Education, one of the sources for Jackson鈥檚 work.

鈥淭hey don鈥檛 have the sustained support to learn how to do it differently,鈥� adds Smith, the Teachers College professor.

In addition to paying faculty members for summer work, School 2 is starting other efforts to engage teachers in Japanese-style pedagogy.

In his role as math facilitator at the K-8 school, Jackson runs a 鈥渓esson study鈥� period once a week. Teachers from all grade levels gather to share their ideas of how they will write lessons that encourage in-depth thinking skills and use materials to illustrate them.

In Japan, teachers engage in such sessions regularly as a way to hone what they are doing in the classroom. They even make it part of their regular workday to observe and learn from other teachers. In the United States, teachers usually spend their free periods on their own, not consulting with their peers, and often use the free time to perform work unrelated to classroom instruction.

鈥淚鈥檝e learned how to teach math by not just giving an algorithm, but teaching kids how to construct knowledge,鈥� says Sandy Joseph, a 2nd grade teacher at School 2. 鈥淚 never had the idea that kids could construct their own knowledge.鈥�

School 2 also has formed a partnership with a school run by the Japanese government for expatriates in the United States. The American teachers get help in running their lesson study and receive advice from Japanese educators who practice the methods that their U.S. colleagues are trying to learn.

For Liptak, the combination of experiences is better than any seminar the teachers attend outside the school.

鈥淭hey鈥檙e really talking about teaching and learning,鈥� the principal says. 鈥淭hey鈥檙e talking about what happens in the classroom. That is professional development right where it belongs鈥攊n the classroom driven by teachers.鈥�

When Jackson鈥檚 class isdismissed at 1:35 p.m., the blackboard has four more cartoons posted on it, along with several proofs by students written in white chalk, and the teacher鈥檚 notes in colored chalk. Students have come up with an algebraic formula to explain the Fibonacci sequence.

But Jackson isn鈥檛 so sure that the lesson has been a success. Students had difficulty understanding exactly how the rabbits鈥� breeding patterns progressed. Some assumed that all rabbits were ready to breed; others thought only two pairs were added every month.

At the end of the class period, he鈥檚 already brainstorming ways to revise the lesson.

The experience shows just how slowly the new teaching methods are taking hold here. Jackson and his colleagues dedicated hours to learning the material and designing a creative way to present it, but he鈥檚 still not satisfied that what they鈥檝e done will help the students grasp the concepts.

That means progress is proceeding at a painstaking rate. After three years of work, School 2 has lesson plans and curricula for only two grade levels. By next year, the school may be ready to unveil a new 6th grade curriculum, Jackson says.

Other teachers in situations similar to School 2鈥檚 also talk about working at a slow pace.

Michigan鈥檚 Lake Shore district is in its second year of work, but has established only a 1st grade curriculum and is piloting one for 2nd grade.

鈥淲e鈥檙e just now getting to the point where teachers are getting beyond the barriers they put up, and are now into the point where they are more open,鈥� says Breckett, the Lake Shore superintendent. While the pace seems slow, advocates of the change say they鈥檒l take what they can get.

鈥淚f you want to fundamentally change the culture of schools, its going to be a slow process,鈥� says Hiebert, the University of Delaware researcher. 鈥淚f we could figure out a way to achieve slow but steady progress, I鈥檇 be really happy. We don鈥檛 even have that in most places.鈥�

鈥淭he whole speed of change is slow,鈥� agrees Holoweg of the National Academy of Sciences. 鈥淚t鈥檚 the changes that are implemented slowly that last.鈥� Changes that are suddenly foisted upon schools and teachers 鈥渢end to be relatively short-lived,鈥� she says.

At School 2, educators are seeing enough progress to keep going. Last year, the 8th graders at the school passed the state math exam at a higher rate than the Paterson average. When the school鈥檚 students move into high school, they are more likely than students from other schools to enroll in Honors Algebra and Algebra 1. Last year, all of the School 2 graduates who took the honors course passed, as did three-quarters of its graduates in Algebra 1.

Says Smith, the Teachers College professor: 鈥淭his came from a school that was declared one of the four worst in the system, and is now one of the most creative and productive.鈥�