Recalling her 8th-grade math class, 15-year-old Samantha, now a sophomore at a North Side public high school, shakes her head with exasperation. “The first half of the year was nothing but fractions. We started out with ‘what a fraction is.'”

Fractions were old hat to Samantha—she’d learned them in 6th grade and relearned them in 7th. But her classmates were slow to catch on, she says. “These kids, you could sit there and tell them, ‘It’s a fraction!’ and the next day they would be like, ‘What is that?'” That’s why the teacher kept reviewing. “She didn’t like anybody to fail. She was a good teacher. Finally, she gave up and said, ‘Okay, if you fail, you fail.'”

But with so little time for 8th-grade math, Samantha struggled her freshman year in high school. “I wasn’t prepared for algebra,” she says.

Samantha’s story is not unusual. The typical 8th-grade math teacher in the Chicago public schools is emphasizing 5th-grade math content, according to a study to be released later this month by the independent Consortium on Chicago School Research.

“That means that [students] are being held accountable on tests for material they were never given an opportunity to learn,” says study co-author Julia Smith. “And the accountability can be severe,” she notes.

Students who fall below certain math and reading scores on the Iowa Tests of Basic Skills (ITBS) might have to repeat a grade. Adding insult to injury, retained students might repeat the same below-level material as well. “We’re talking about a cycle of failure here,” Smith says.

Instruction lags furthest behind at schools that are large, poor or have high mobility rates, which reflect the number of students who transfer in and out during the year, the study found. The good news: Even at the most disadvantaged schools, faculties that worked as a team could quicken the pace of instruction.

According to the study, Chicago elementary students get off to a good start, with most 2nd-graders adding and subtracting two-digit numbers, among other skills. The slowdown begins in 3rd grade, where many 2nd-grade skills are repeated.

By 8th grade, most math classes lag three levels behind—reviewing long division and decimals rather than tackling quadratic equations. Only 10 percent of 6th- to 8th-grade math classes in Chicago public schools are keeping pace with grade-level content, the study found.

Not surprisingly, 33 percent of last year’s 9th-graders failed algebra, a required course for freshman.

The Consortium findings are based on a 1994 survey of 2,293 teachers at 266 Chicago elementary schools. Researchers compared the math topics teachers reported covering to those assessed on the ITBS, which assess skills commonly taught at each grade level nationwide.

In the comparison, schools in the city’s poorest neighborhoods fared worst. Mixed-income schools kept pace with grade-level content, while schools with more than 90 percent low-income students tended to focus on 3rd-grade math in the 8th grade.

“It’s not unheard of” for 8th-grade teachers to teach how to tell time, says Smith. More typically, 8th-grade teachers at the slowest-paced schools review multiplication, long division and fractions.

While Chicago students have made solid gains on the math section of the ITBS since the 1994 survey, “The basic story is the same as far as what kinds of schools are making progress and what kinds are not,” says Consortium Director Anthony Bryk.

The Consortium is analyzing teacher surveys collected in 1997 and, based on continuing field observations, expects the picture will look much as it did in 1994, Bryk says. From 1994 to 1996, the Consortium’s research team observed over 800 math and language arts lessons in elementary and high schools; it was startled by the endless repetition of basic skills.

For instance, researchers saw similar introductory lessons on the parallelogram in 2nd-, 5th-, 8th- and 10th-grade classrooms. In language arts, students at all levels rarely advanced beyond basic story elements—”Who is the narrator? What is the setting?”—to more advanced topics, such as foreshadowing, flashback and irony.

Classroom observations suggested a number of reasons for slow pacing: low expectations for the amount of work students could complete, time spent reviewing for standardized tests and difficulty coping with reluctant students. Students would insist they hadn’t learned something before when, in fact, they had. Although some students might only need a brief refresher, teachers often chose to reteach a topic step by step, just to be sure, researchers found.

Schools that received many midyear transfer students tended to review material, as did large schools where teachers had less contact with each other. “The more unfamiliar you are with the students in front of you, the more unfamiliar you are with what teachers have taught before, the easier it is to keep repeating the same material,” Smith says.

Poor schools in Chicago tend to have large and highly mobile populations. In addition, Smith says, parental support for instruction at low-income schools tends to be lower, and students come to school with distracting personal problems that teachers must address. “There’s a lot of pressure to slow down.”

To counter that pressure, teachers need to pull together, Smith explains. “It takes a lot more energy to try to get students to do things that they don’t know how to do. And that energy has to come from somewhere. There has to be support for it.”

Schools that valued teamwork— where teachers said they met regularly to discuss instruction, visited each others’ classrooms and felt a collective responsibility for student learning— outpaced schools where teachers worked alone, the 1994 survey revealed.

The difference was most striking for schools with high poverty levels. On average, at low-income schools that rated high for teamwork, 8th-graders focused on 6th-grade math; those with low ratings spent most of their time on the 3rd-grade level.

Communication simply may not be part of a school’s culture, notes BetsAnn Smith, a study co-author. Recently she heard some 3rd-grade teachers complain how poorly prepared their students were. She asked them what they thought the problem was. “They said, ‘We have no idea what the 2nd-grade teachers do.'” When I asked if they had ever spoken to the 2nd-grade teachers about their students’ shortcomings, they told me, ‘That’s not our job, and we wouldn’t be comfortable doing that.'”

“I don’t think there is anything unusual about that situation,” she adds.

Yet another obstacle to improving math instruction is poor teacher preparation. To earn certification in Illinois, elementary school teachers need only six semester hours of math; they often choose watered-down courses aimed at education majors. As a result, elementary teachers tend to think of math as rote computation, experts say.

Higher standards

Many states, including Illinois, have adopted learning standards for students that call for a deeper conceptual understanding of mathematics. Under the new standards, a student may be expected not simply to compute an answer, but also to justify it using words, concrete objects or diagrams—a task teachers themselves find difficult, according to Northwestern University Education Dean Penelope Peterson.

For example, she says, “Why, when you divide by a fraction, do you have to invert and multiply? You ask teachers to explain concretely, to draw diagrams. They can’t do it. They’re used to teaching it as a rule.”

When math students are taught only to memorize rules, they often don’t retain what they learn, she adds. “If you have the basic conceptual understanding, you can always reconstruct the rule. If you only learn the rule and you forget it, you can’t reconstruct the meaning.”

Spurred by standards adopted in 1989 by the National Council of Teachers of Mathematics, many states now require algebraic and geometric concepts to be taught in elementary school. The ITBS has reduced the number of basic computation problems in favor of those that require more analysis, such as equations.

Teachers are ill-prepared to teach the more demanding content, experts say. “I think most elementary school teachers are afraid of algebra,” says Zalman Usiskin, director of the University of Chicago School Mathematics Project. “They don’t want to touch it.”

Nationally, only 18 percent of math teachers in grades 5 through 8 have any kind of math background—a major or minor in math or math education, either as an undergraduate or graduate student.

Teachers in Illinois can earn state “endorsements” in the subjects they teach by pursuing advanced coursework. A middle-school math endorsement requires three college credits in methods for teaching math and 15 in college-level mathematics. In Chicago, only 448 elementary teachers have an endorsement in math, fewer than one per school.

The lack of expertise has an impact on student achievement, many principals believe.

At Tilton Elementary on the West Side, for example, only 14 percent of students scored at grade level on the ITBS in 1998. The school does not have a single teacher endorsed in mathematics. “I have trouble finding staff that is comfortable with math instruction,” says Principal Faye Terrell-Perkins.

Low comfort level

With help from DePaul University’s School Achievement Structure, Tilton has carved out time for teachers to meet and discuss teaching strategies. The school also wrote up a curriculum aligned with state and district standards and the ITBS. Each teacher has a schedule of skills to teach. Administrators monitor classrooms.

Despite this organization, “If your teachers don’t feel comfortable [with math], they’re reluctant to do the kind of pacing that needs to occur,” Perkins notes. “Math and science are two fields that, unless you have the skills and the staff development you feel you need to do a great job, teachers tend to do the minimum.”

In most urban districts, 7th- and 8th-grade teachers specialize in one or sometimes two subject areas, according to Usiskin of the University of Chicago. In Chicago, however, many 7th- and 8th-graders spend a full day with the same teacher. “When we tell people from other cities that there are still so many self-contained classrooms at 8th grade, they’re astonished,” he says.

Without a specialty, teachers don’t have time to learn a subject in enough depth to teach effectively at the upper elementary level, he adds. “If you’re teaching so many subjects, you’re torn.”

Over half of Chicago’s elementary schools have their 7th- and 8th-grade teachers specialize, estimates the board’s manager of mathematics support, Telkia Rutherford. Officially, the state and the board require that teachers who specialize be endorsed in their subject areas. Practically, the board doesn’t keep track of which schools are organized into departments; therefore, it can’t enforce the rule.

The board does encourage schools to “departmentalize” in the 7th and 8th grades. When schools don’t, Rutherford says, it’s usually because they have only one or two teachers at a grade level. Districts that organize into middle schools or junior high schools avoid that problem; Chicago is unusual in having elementary schools that extend through 8th grade.

Textbooks can also slow the pace of instruction—research has found that most are highly repetitive. In a famous 1987 study, a University of Chicago doctoral student reviewed the country’s three leading elementary math textbook series. If any material on a page—even a single problem—had not been covered previously, he counted the page as new. In 8th-grade books, only 30 percent of pages had any new material. More recent research on textbooks conducted by the Third International Math and Science Study (TIMSS) had similar findings.

Despite the shortcomings of most textbooks, teachers often rely on them heavily, according to board Curriculum and Instruction Director Andrea Kerr. “They start at page one and go through the book. What they’re covering is what the publishers say they should be teaching—it may or may not have to do with what we say is important in Chicago.”

Central office is encouraging schools to use textbooks selectively and to organize a curriculum based on state and district standards, as well as the ITBS. To guide schools, the board distributed “programs of study” last fall for grades 9 and 10 and in June for grades 6 to 8. The programs outline a more detailed curriculum based on district standards.

Taking curriculum definition still further, the board has prepared 180 day-by-day lesson plans in the four core subjects for kindergarten through the 12th grade. Chief Executive Officer Paul Vallas stresses that the lesson plans are optional and intended to serve as a model and support. One hundred Chicago teachers authored them, working part time from January to July. Thirteen elementary schools and seven high schools have volunteered to try them out this fall.

The board is particularly interested in addressing two problems: teachers who lack content knowledge and schools that lack expertise in curriculum design.

“A teacher will tend to teach to the area they feel most comfortable in and neglect others,” Kerr notes, adding that teachers with a math or science background are in short supply.

She also points out that teachers need only one class in curriculum and instruction for certification. “That does not prepare you to build curriculum,” she says. Some schools may not have anyone, even a principal, with the necessary background.

Plans have limits

Vallas has high hopes for the lesson plans, saying they “will probably have a greater impact on the quality of instruction than anything we’ve done in the area of accountability or any impact that the after-school or year-round programs have had.”

Some experts are less optimistic. Even lesson plans of high quality, they say, are at best a good first step.

For one, lesson plans won’t solve the problem of poor school organization, says Bryk of the research consortium. “I’m doubtful that you can just write [lessons] down and have everybody go back to their classrooms and still coordinate their work.” Students probably will not master the material at the same rate, so teachers will still have to decide whether to reteach a lesson or move on. “Unless teachers communicate with each other, you’re [still] not likely to get coordination across classrooms and grade levels,” he says.

What Bryk recommends: More time during the school day and school year for teachers to meet and collaborate, and more funds in the central office budget earmarked for professional development.

Lesson plans will do little to shore up teachers with a weak math background, says Paul Sally, a University of Chicago mathematics professor. “If teachers don’t have the conceptual foundations and the applications, then it makes no difference what you give to them, they’re just going to follow [a lesson plan] by rote. They don’t even have the capability of interpreting it.

“There’s no curriculum that can anticipate all the necessary strategies for solving problems or that can react to comments or questions by students,” he adds.

For the least competent teachers, the lesson plans might “give them an oar in the water,” he says, “just one oar.”

What Sally recommends: a coherent, five-year plan developed by central office to bolster teachers weak in the content areas. “The focus is always the students, but it’s the teaching corps that’s going to be here five years from now,” he notes.

Central office might consider building a corps of teachers who could do staff development in their own buildings, he suggests. So far, schools have relied mainly on universities and other outside consultants to provide staff training. But with so many teachers and so much content to cover, “no university program can do it all,” he says. “There’s no chance.”

Improving math instruction should be thought of as a long-term project, contends Prof. James Stigler of the University of California at Los Angeles. Stigler is a TIMSS researcher who studied teaching styles in Japan, Germany and the United States. “If you think that just handing out lesson plans is going to work, you’re wrong.”

For example, Japanese teachers work from model lesson plans developed by colleagues. But these lessons undergo continuous and intensive revision. Virtually every teacher belongs to a “lesson study group” that will work on refining a lesson on a single topic for the entire school year. Along the way, teachers test the lesson out in the classroom, pick it apart and revise it again. They do this with an attention to detail that is astonishing to Americans, according to Stigler.

The long haul

During one lesson study meeting recorded by TIMSS researchers, teachers spent an hour discussing whether students should be asked to solve the problem 13 minus 6 or 13 minus 7, and how they would react depending on the example used, he says.

“They spend a lot of time role-playing, thinking how students will interpret this question. What problems might they have?” In the end, teachers write up the lesson and publish it.

Where the Japanese see improving instruction as a long-term, gradual process, he says, Americans tend to opt for quick fixes that are soon dropped in favor of another short-lived approach.

The real question for the Chicago Public Schools, he says, is: “What is the process by which these lesson plans are going to be improved over time?”

What Stigler recommends: A 10-year plan for continuous critique and revision of the lessons, “because that’s realistic. Then I think teaching might improve.”