Math software allows students to subtract fear, multiply confidence

Psychologist builds an artificially intelligent math tutor.

By Rebecca A. Clay

Remember the humiliations of high school math class? If you ever had to stand at a blackboard struggling over a difficult problem in front of your peers, you probably hoped that one day someone would come up with a better way to teach math.

For thousands of students making their way through the algebra-geometry sequence, that dream has become a reality. Forty-five schools are now using artificially intelligent math tutors developed by John R. Anderson, PhD, and other psychologists in the Pittsburgh Advanced Cognitive Tutor (PACT) Center at Carnegie Mellon University.

Called ?cognitive tutors,? the software programs are designed to think along with the students who use them. Students typically spend two days a week in computer labs, working on individual problem-solving skills as a supplement to the more traditional classroom instruction they receive the other three days of the week. The result is not only individualized instruction but an opportunity to learn without fear.

?By far and away the most common comment I hear from students is, ?Nobody knows I?ve made a mistake when I?m using the tutor,? says Anderson, the center?s co-director and the university?s Walter VanDyke Bingham professor of cognitive psychology and computer science. ?There?s a sense of not having to be embarrassed by your errors, of being free to learn.?

Adaptive control of thought

As a cognitive psychologist, Anderson has always been interested in how the mind works. In fact, his lifelong goal was to develop a cognitive theory so well specified that it could be simulated on a computer.

In the 1970s, he developed a cognitive model called the Adaptive Control of Thought (ACT) theory. ACT posits that cognition arises from the interaction of declarative knowledge?factual information such as the multiplication tables?and procedural knowledge?rules about how to use declarative knowledge to solve problems.

To test the theory, Anderson developed software that used the ACT model to solve cognitive tasks, such as geometry proofs. His model passed the computer simulation test and became the basis for his artificially intelligent tutors.

To understand how the tutors work, think of them as brains wired according to the ACT theory?s dictates. Anderson and his computer science colleagues at the PACT Center program each tutor with the procedural knowledge necessary to master a given subject, plus some so-called buggy rules that represent students? most common errors. As a student uses the program, the tutor keeps track of which procedural rules he or she has mastered. The constantly updated student profile that results allows the tutor to follow the student?s thinking and anticipate his or her next move.

This ability to think along with users comes in especially handy when a student gets stuck or makes an error. Because the tutor understands how the student got into trouble, it can offer appropriate feedback ranging from brief messages to remedial instruction. If the tutor decides the student hasn?t fully mastered a particular procedural rule, it pulls out a problem involving that rule to give the student extra practice.

?One frequent error that is low-level but can lead to disaster is that student enters a figure in the wrong column,? says Anderson. ?Based on their past behavior, the tutor interprets what their intentions really are and presents a message informing them, ?This column is for that purpose. Don?t you really want to put this quantity into that other column???

The fact that each student is working on a different problem often alarms teachers at first, says Anderson. Because tutors can answer most questions themselves, however, teachers are free to concentrate on students whose difficulties go beyond the tutors? capabilities.

From the lab to class

The tutors debuted in 1992, when a National Science Foundation grant allowed the PACT Center to partner with a group called the Pittsburgh Urban Math Project in developing an algebra tutor and curriculum. The researchers tested the resulting course in a Pittsburgh high school and compared the success of children using the tutors with students in traditional algebra classrooms. Students using the tutors turned out to be twice as successful as students using traditional learning methods at the program?s year-end assessments and performed 10 percent to 15 percent higher on standardized test items. Today the program claims a one letter-grade improvement.

Those kinds of results make schools willing to pony up $25,000 for software, teacher training and technical assistance. Since word of the program?s success has gotten out, calls have been pouring into the PACT Center from high schools and even colleges offering remedial math classes.

Tutors and accompanying curricula are currently available for geometry and algebra I and II. Anderson and PACT Center co-directors Albert T. Corbett, PhD, and Ken Koedinger, PhD, hope to expand into other subjects.

Anderson?s real pride, however, comes in putting psychological science into practical use. ?That?s probably been our singular accomplishment?the degree to which our systems have been able to survive in real classrooms.?

Rebecca A. Clay is a writer in Washington, D.C.