Picking Up the Pace
Stanley Kowalski stalks around his darkened classroom, his clogs clattering on the floorboards and his ponytail swaying, as he punches keys on a calculator.
As he programs symbols into the machine, the letters and numbers appear simultaneously on a special overhead-projection screen connected to the hand-held device.
Alternately peering at the screen and pecking away at their own calculators, his students follow along.
The finished equation, when loaded into each student's graphing calculator, reads: Y = X¢ + BX + C.
"If B is 2,'' Kowalski asks, "then this is a what-degree equation?''
Several students, without hesitating, answer: "A second-degree equation.''
Even ignoring the technological trappings of the lesson, a quick glance around the room at Suitland High School in suburban Washington reveals that this is not a typical high school mathematics class.
Students sit in chairs arranged in groups of four, facing one another. The cooperative grouping, it quickly becomes apparent, encourages the 11th and 12th graders to discuss strategies for solving problems among themselves as Kowalski observes them at work.
Occasionally, when a student asks for help, Kowalski will squat near a desk to coax out an answer. But more often, if a little self-consciously because visitors are arrayed on one side of the room, he'll badger students into thinking through a problem for themselves.
"What are you asking me for?'' he admonishes. "Work with your teammates.''
Collectively, these nontraditional touches stand as signs that distinguish this course from more conventional classroom settings.
Its unusual nature becomes even more apparent when you contrast the students' readiness to answer and apparent grasp of difficult math concepts with their prior academic records.
"These are not students who would necessarily have chosen to go on to college,'' confides Elfreda Saunders, one of Kowalski's colleagues. "They are C and D students.''
Kowalski's students, in the Prince George's County, Md., school system--a district of some 109,000 students, the majority of them African-American--are among roughly 900 students nationwide who are piloting a new approach to teaching and assessment designed to draw more students into studying advanced math.
The program, known as "Pacesetter: Precalculus Through Modeling,'' is a reform initiative of the College Board, the New York City-based, nonprofit organization that sponsors the Scholastic Assessment Test.
Pacesetter--which is based on highly regarded national standards for what students should know and be able to do in math--is envisioned as a "capstone'' course that students would take late in their high school careers after completing algebra and geometry.
It is designed to help students understand how to apply concepts in algebra, geometry, and trigonometry to solve "real world'' problems.
One goal of the program is to increase the number of students who study four years of math--only 9 percent of students currently do so--while improving their preparation to enter the workforce.
But the Pacesetter approach, which officials say is designed to build on the successes of the College Board's Advanced Placement program, also is designed to force educators to look critically at the types of math they offer to non-college-bound students.
By developing Pacesetter as a challenging capstone course, the College Board hopes to enhance the quality of the mathematics American schools offer to the majority of their students.
"If you set high standards at the upper years, you force improvements down'' to make prerequisite and introductory-level courses more challenging, argues Janice Weinman, the College Board's executive vice president for programs.
In combination with another College Board initiative called Equity 2000, which encourages all students to enroll in algebra and geometry, the College Board expects Pacesetter to create what it calls a "push-pull effect.'' Eventually, the board hopes the combined programs will encourage all students to enroll in a challenging high school math curriculum.
Hard evidence of the program's efficacy is, after just a single year of the pilot effort, difficult to come by. But teachers at Suitland High and at other pilot sites believe that Pacesetter, while not without some snags, seems not only to engage students much more deeply in learning than traditional curricula do, but also to reinvigorate their own teaching.
The College Board, meanwhile, is poised to pilot a Pacesetter English course next year. And, in years to come, it will launch Pacesetter science, history, and foreign-language courses, too.
Building on national subject-matter standards, new forms of assessment, and a strong emphasis on teacher development, Pacesetter provides a template for effective curriculum reform to which other programs should aspire, officials at the College Board contend.
"Inherent in the word 'Pacesetter' is the concept of a series of courses that will set the pace for the future of what teaching and learning should look like,'' says Lola H. Greene, the board's project manager for Pacesetter.
For now, College Board officials are guardedly optimistic about the lessons learned in the math pilot.
"Is it premature to draw conclusions about the success of the program?'' Weinman asks. "I think it is. But I think one can make some observations.''
Teachers and students do seem to be "struggling a little bit'' with the course content, she admits. "But they're struggling because it really involves a very active learning process as opposed to the passive learning that so often characterizes math courses.''
So far, the number of schools that have asked to offer Pacesetter math and to pilot Pacesetter English next year seems to indicate widespread interest in the program, Greene says. When letters went out to 100 College Board member districts asking them to apply to teach Pacesetter math in the coming school year, 37 responded affirmatively. "We had not expected such an overwhelming response,'' she says.
When Pacesetter math expands next year, some 4,000 students nationwide will be enrolled in the course, quadrupling the number in the pilot sites this year.
Observers also note that the Pacesetter program will be a proving ground of sorts to test ways to incorporate national curriculum standards into actual teaching.
To insure that the course materials are in line with national standards, the College Board continues to develop its individual Pacesetter courses with the cooperation of national professional societies in the various subject areas.
Pacesetter math, for example, draws heavily on standards for math teaching and content developed in the late 1980's by the National Council of Teachers of Mathematics. The N.C.T.M. standards--which frequently are cited as national models for standards-setting across the curriculum--de-emphasize rote computation in favor of real-world applications of math and teaching students how to use math as a means of communication.
Incorporating these standards into a working curriculum has been part of the challenge of the Pacesetter program's first year. "The course standards are very high because they reflect the N.C.T.M. standards,'' Weinman says. And, she adds, "not every teacher has had experience in teaching those standards.''
But, she points out, in the 15 districts piloting Equity 2000, many of which also are offering Pacesetter, students are learning the kinds of math that the standards emphasize. "In these districts, we have forced a 'detracking' of the system,'' she argues. "By the time these students reach Pacesetter, all of them will be able to reach these high standards.''
Still, critics say, it is an open question whether a critical mass of districts will be able to afford the costs associated with offering Pacesetter. Although the College Board has footed the bill for the pilot program, districts that implement the math program next year must pay $1,950 annually to cover the costs of the extensive teacher training required, as well as an additional fee of $30 per student.
Many districts plan to use money from the U.S. Education Department's Eisenhower Science and Mathematics Program to cover the course costs, Greene notes. Still others hope funding recently made available as part of the National Science Foundation's Urban Systemic Initiative will help support Pacesetter.
Despite its early shortcomings and difficulties, however, officials argue that Pacesetter is a keystone of effective reform.
"This, for the College Board, is a high priority and a very important initiative. It represents our commitment to equity and to excellence,'' Weinman says. "It really represents the future of this organization.''
But Pacesetter is more than a testing ground for implementing standards-based reforms. It also challenges the type of teaching required by those standards. The program's revamping of assessment calls for teachers to dump their traditional multiple-choice tests in favor of open-ended and performance-based examinations.
Pacesetter students, for example, keep journals of their progress in class as a way to emphasize the concepts of math as a means of communication. The journals allow students to express their frustrations--and share their successes--as they progress through the course, teachers say.
Kowalski, who was at first skeptical of the journal-keeping, now enthusiastically supports the individual assessments. In fact, he's even incorporated the idea into his other courses at Suitland High. But, he admits, "I never would have done that without Pacesetter.''
In accord with the N.C.T.M. standards, Pacesetter math also emphasizes complex problems to which there may be one, two, or several different answers--a radical departure from traditional math classes.
What's more, assessments focus on problems derived from areas not traditionally addressed in math. "Real-world problem-solving is a major focus of this course,'' Greene says. "Just as there are no simple solutions in the real world, there are no simple solutions in this course.''
Students in the pilot phase of the Pacesetter program, meanwhile, recently completed the final examination for the course, called the "culminating assessment,'' which the Educational Testing Service of Princeton, N.J., developed in conjunction with a team of teacher advisers.
The three-part examination, notes Ernest W. Kimmel, the Pacesetter project director at the testing firm, tested a variety of skills stressed in the N.C.T.M. standards.
The final assessment centered on a hypothetical brush fire, such as the ones that recently devastated portions of California, and how students could apply math learned in the course to predict how the fire could spread and what natural factors might inhibit the blaze.
As part of the three-hour exam, students received materials about brush fires and their behavior, which they were allowed to read as background and discuss with other students.
In the first phase of the actual exam, which they took individually, students then were given a map of a hypothetically fire-devastated area and were asked to calculate the rate of change in the intensity of the fire and the spread of the blaze to decide whether a particular town on the map was in danger. Of course, they were required to use a mathematical model to support their answers.
In the second part of the exam, students wrote short essays describing three factors that might lead them to change their calculations, including such variables as rainfall and the direction of the slope around the town. In the final section, students designed a scale that would allow firefighters to gauge at a glance how much fire hose remained on a reel.
As Kimmel remarks, the questions are quite unlike those found on traditional math tests. "It reports a multidimensional performance. They take a problem that occurs in the real world, choose from their repertoire of math skills, and propose a solution to that problem,'' he explains. "It demonstrates that they can do more than just crank through an equation.''
Pacesetter teachers respond that the test was clearly written but that it was too exhaustive for the time allotted.
"The assessment was a very well-written test, but it was difficult,'' says William Fisher, who teaches the pilot math program at Central High School in Battle Creek, Mich. "The students had the skills to solve the problems. They just didn't have the time.''
Although the results of the culminating assessments are yet to be tabulated, Kimmel concedes that when they are, teachers' hunches about the complexity of the exam will likely be proved correct. "I think the teachers are dead on,'' he says. "I think this is probably too difficult for the students. For most of these kids, this has been the only year of their math in which they've had to think in new ways about math.''
Kimmel adds that the teacher-advisers to the assessment project will meet soon to discuss how the assessment might be modified to make it less time-consuming.
Despite their misgivings about the culminating assessments, teachers generally speak favorably of the Pacesetter approach and what it has done for them and their students.
"It has definitely shown the students a different way of looking at math, and of solving mathematical problems, than they have had in the past,'' says John S. Pluta, the chairman of the math department at Mill River Union High School in North Clarendon, Vt., another Pacesetter pilot site.
But Kowalski points out that there have been inconsistencies in the way students have been selected for the program at the 10 pilot sites, which were specifically chosen to insure racial and socioeconomic diversity.
Pluta, for example, says students in his district's Pacesetter course "would probably have taken a fourth year of math anyway.''
The students at Suitland High School would not have done so, Kowalski says, which will greatly complicate any comparative assessment of student achievement in the pilot phase.
Teachers, however, seem to be cautiously optimistic that their students have benefited from the course.
"I don't think the fact that they're in this Pacesetter course has turned on 100 percent of the students to mathematics,'' Fisher says. But, he adds, students at Battle Creek's Central High do seem far more likely to remain in the Pacesetter program than in traditional pre-calculus courses.
Some 35 percent of his students in traditional pre-calculus, he estimates, would drop out before the end of the year. The Pacesetter dropout rate at Central High, he adds, has been roughly 6 percent.
Aside from student achievement, however, teachers in the program seem to agree that it has challenged them as educators and added to their enjoyment of teaching.
"I think I have learned a lot of math,'' Fisher says, "and I know my colleague feels exactly the same.''
Pluta also notes that, as he expected, there are still many bugs to be worked out of the course. "We seem to spend longer periods of time on a given task than maybe we anticipated,'' he says. "But you have to understand that because it is a pilot program we [teachers] are learning a tremendous amount.''
Several teachers also say that the national scope of the program and its emphasis on teacher development and professional networking have brought a refreshing change from their daily routine.
"There is consistent, constant teacher assistance and professional development,'' says Greene. Teachers with questions about the program, for example, can call a telephone hot line--staffed by curriculum specialists--for answers. And an electronic-mail system will soon allow teachers in the program to communicate with each other.
Pluta says he's collaborated far more with his colleagues than he did before, especially because several high schools in nearby communities are also Pacesetter pilot sites. National Pacesetter meetings, where teachers gather to share experiences and offer advice, have also helped break down the isolation of the classroom.
"Seeing that everybody has the same problems,'' Pluta says, "has been extremely valuable to us.''
Teachers in the program also say using graphing calculators in class has, in many cases, dramatically increased student interest.
"That particular tool has excited a lot of the kids,'' Pluta says. "They feel less of a burden in terms of doing a lot of the tedious number-crunching that they have in the past.''
Graphing calculators, which are now manufactured by several different companies, are hand-held devices with built-in screens that have roughly the same computing power as early Apple II microcomputers.
The devices allow students to plot complex functions at the touch of a button, changing variables at will and doing away with much of the tedium of paper-and-pencil graphing.
"Students were almost ashamed to be in Pacesetter,'' Kowalski says. "Now, they carry around this calculator almost like a badge of honor. It's almost like a school letter.''
One otherwise lackluster student at Suitland High, adds Kowalski's colleague Saunders, learned how to program the machine with amazing skill and became far more involved in the Pacesetter course after demonstrating his handiwork to students and teachers.
"I saw him do it, and I couldn't believe it,'' says Sterling Marshall, Suitland High's principal. "That kind of brilliance is lost in a traditional math setting.''