PUMPing students though the math pipeline

Mathematics teacher agree on at least on thing: there's something wrong with high school mathematics. The effects show up in falling SAT scores, in failure and dropout rates, in disclaimers by adults who are "no good at math" and in the frustration that many feel with simple tasks like balancing checkbooks and making change. Dating back to Sputnik in 1957, Americans have readily acknowledged that there's a problem; yet the erosion of math skill has continued. According to John A. Anderson, a professor of psychology and computer science, a case can be made that curriculum reform is part of the problem. "American mathematics education has been in a constant state of reform ever since the new math movement," Anderson points out. "American teachers face the need to change what they teach more often then their counterparts in other countries, [and they] complain bitterly about that fact."

Curriculum reform by itself, he writes, will not raise student achievement levels. "Without other changes, American students will be as bad at the new curriculum as they were at the old."

To break the cycle, Anderson and a team of Carnegie Mellon researchers and Pittsburgh public school teachers have spent more than 10 years developing and refining a computer-based instructional tool called the cognitive tutor. Built on an understanding of the way the mind solves problems, the cognitive tutor "models" the paths towards a correct answer, explaining a point if necessary, or nudging a student away from a potential dead-end.

Their cognitive computer project, known as the PUMP Algebra Course and Computer Tutor, is part of an innovative mathematics curriculum at Langley High School in the Sheraden section of Pittsburgh. Before the introduction of the tutor in 1993-94, about a third of Langley freshman enrolled in elementary algebra. Now all students are required to take the course. Math test scores are up, and failure and dropout rates are down. One benchmark test shows algebra students reaching the same level of competence in a third of the time previously required. By all accounts the project is poised for a breakthrough. "I haven't been involved with a single school interaction that hasn't been positive," says Victor Fishmen, the Carnegie Mellon licensing officer who is marketing the tutor. "No one says, 'This just isn't for us. Our students don't need this.'"

The core idea, summed up last year in a journal article, is that "Instruction should be designed with reference to a cognitive model of the competence that the student is being asked to learn." Rather than just furnishing an answer key, the system "models" or mimics the thought processes involved in solving the problem.

The tutor is not reserved for the brightest students. Educators are turning to it as a tool for teaching algebra to students who previously might never have taken the course.

Langley dropped its general math courses when it began PUMP algebra, and Fox Chapel High School in the affluent suburb north of Pittsburgh is preparing to do the same. Fox Chapel offered four PUMP algebra courses this past year, populated by sophomores, juniors and seniors who earlier took a pre-algebra course. This fall there will be no more pre-algebra, and PUMP courses will double to eight, says math department chair Robert Janosko.

No cute humanoid name

The tutor does not have a cute humanoid name. It is not a toy that has students work a problem, press a button and match answers with the machine. It has no bells or whistles, and there is no CD-ROM version in a snazzy box. The developers talk about making the project "ugly by design" so users don't get the wrong idea. "The tutor is not meant to replace the teacher, " explains Albert T. Corbett, a cognitive scientist at Carnegie Mellon who has worked on the project for about a decade.

Corbett explains the "tutor" is actually something of a misnomer. Not unlike a high-end calculator, the tutors "appear to work better if they present themselves to students as non-human tools to assist learning rather than emulation's of human tutors, " Anderson, Corbett and team members Kenneth R. Koedinger and Raymond G. Pelletier explained last year in a journal article.

Indeed, the typical weekly schedule calls for three classroom sessions and two computer lab sessions. Anderson and the others followed a teacher-is-always-right philosophy in working with master teacher William Hadley in developing the cognitive tutor. In-service training and technical support provided by Hadley and other teachers are key components in the package now being offered to other school districts.

The project has attracted more than $8 million in university, federal and private funding to date, generating a basic algebra course that is already up and running in six Pittsburgh area high schools- Langley, Brashear and Carrick in the city, Hampton and Fox Chapel in the suburbs, and Beaver Middle School farther north. Public schools in Milwaukee and Pensacola, Fla., expect to use the algebra course in the fall. Jacyln Snyder, the math department head at Langley, is teaching the pilot geometry course there, and the curriculum and programs for Algebra II are already under development.

"We lose about a third of the kids every year, " Hadley says. Before PUMP, when only a third of the students signed up for algebra, about half failed. The next year about half of the survivors typically failed geometry. One goal of the PUMP program has been to remedy the situation and keep more students in the mathematics pipeline. Corbett explains, "We wanted to develop a classroom that would allow all kids to enter academic math with a reasonable chance of success."

In the words of Fishman, the licensing officer, "If you can't do Algebra I, you aren't going to college, and if you can't go geometry, you aren't going to do as well."

Lab days bring students to class

In the computer lab at Langley on a Wednesday morning, Vincent Hawkins is the first to take his seat at the start of class. He turns on the Macintosh and up pops a screen divided into four quadrants-an algebra word problem in the upper left, a "grapher" in the upper right, a worksheet in the lower left, and a space for messages and equations sharing the lower right quadrant, with a "skillometer" showing at a glance the student's command of skills and concepts.

Hawkins works steadily throughout the period, mostly ignoring two students at his left who are eventually separated after they switched off each other's computers, wiping out the day's work.

Teacher Fred Brinton is circulating around the room, helping students set up the charts that appear in the lower right quadrant. There on a miniature two column spreadsheet, they chart the independent variables from story problems. "Hours shoveled" and "money made" in the case of a snow shoveling problem, or time and distance for a problem about a trip to Philadelphia via turnpike at a fixed speed. "Is it less than an hour or more than an hour?" Brinton asks a student struggling with a problem about how long it will tale to travel five miles. The student doesn't know. So Brinton tries something else: "What fraction of an hour is six minutes?" The student lights up at this suggestion, saying it takes 6/60 of an hour. Brinton suggests reducing that fraction, and the student uses the calculator to do it.

If a student is following one of the many paths to a solution, the tutor lurks in tacit approval. But if the student makes an error or starts down a dead-end path, the tutor can produce two types of instructional messages based on an approach called model-tracing. A bug message explains why the operation was an error, and a help message points towards a correct solution.

By the end of the hour, Hawkins has quietly solved three problems replete with charts and graphs. "You learn faster on this," he says, beaming.

The passing rate in Langley algebra courses is about the same as it was before PUMP, even though all the students are now taking the course, says Snyder. Judging from all available measures including absentee rates, the students like the tutorials. On lab days, the average class attendance is higher. Snyder says the program has stabilized the algebra course. "With the computer and the curriculum we're able to level off the weaker points. One is the teacher's ability to control the class..... It's not often that you'll hear 'OK, quiet down, everyone in your seat.'" Working on their own problems at their own rate, students tend not to talk with each other.

Janosko, the Fox Chapel High School math chairman, says, "If you as a teacher think you're going to sit back and watch it happen, you're wrong. you work from the second you get into the room." The difference, though, is that all students working with the tutor can remain on task, so a teacher offering individual help to one student is not encouraging the rest of the class to look out the window.

"We know that they like it, that they have fun, that they're motivated, that their time on task is high, " Fishman says. He remembers a student at Fox Chapel High School saying the tutor "has a way of putting it in your head."

In the '70s and '80s at Fifth Avenue High School and later at Brashear, Hadley was frustrated by teaching a general math course where the curriculum was mostly devoted to rote calculation.

"That just doesn't prepare them for anything. It prepares them to be a $2.98 calculator."

As math department chairman at Brashear and later as part of the Pittsburgh Urban Math Project, Hadley had been trying to direct students- especially minority students- away from general math courses. "One of my personal goals was more kids in algebra," Hadley says. "I started putting together an algebra curriculum to make the kids more successful. Of course I didn't have any money."

Along came Anderson, who turned to the school district in 1990 in search of a master teacher to work on a project that was already well underway. As Hadley describes it, "I had a curriculum, and he [Anderson] had money to develop tutors, so it was a marriage made in heaven."

Students like it. Teachers like it. Test scores go up. Corbett has figures documenting that PUMP Algebra I students are more than twice as likely as non-PUMP students to pass Geometry and enroll in Algebra II. And this is only the beginning.

"Anything you can do to keep kids in school is going to make a difference," Hadley says.