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Math Tutors Wanted

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Algebra class at Brashear High School is proceeding normally, students working with individual tutors as they try to devise the appropriate formula to solve a word problem and then plot their solutions on a graph.

Brashear High, perched atop one of the rolling hills on the outskirts of Pittsburgh, is a typical urban high school with all of the problems and disadvantages that the term implies.

In one corner of the classroom, a student sits slumped over his desk, apparently sound asleep, or at least oblivious to the activity around him. Later, school security guards deliver to class a recalcitrant student, who minutes later gets sent to the central office after a brief and unpleasant exchange with the algebra teacher, Bill Hadley.

But the disruptions seem to highlight the fact that the majority of students here appear to be engrossed in working with their tutors to learn algebra. They don't seem to notice that the tutors, while arguably "intelligent,'' are unable to speak or understand spoken language. Nor do they appear to care that their "teaching assistants'' can never leave the confines of the plastic casings and glass screens in which they "reside.''

Their tutors, part of the Pittsburgh Urban Mathematics Program, are the latest members of a new generation of computer software. And researchers at nearby Carnegie Mellon University and a handful of area high school teachers are looking to the state-of-the-art "intelligent tutors'' to help the district fulfill its commitment to teach every student the essentials of algebra.

"Our view is that the ideal situation is when the teacher is free to go around with specific students and provide a lot of individualized instruction,'' says John R. Anderson, a Carnegie Mellon professor of cognitive psychology who developed the software. And Anderson thinks the intelligent tutors at work at Brashear High and one other area high school will help free up teachers to do just that.

Imitating Human Learning

Based on cognitive research in artificial intelligence, these Macintosh personal computers have essentially "learned'' the lessons they now are "teaching'' students. Anderson's theory of cognition holds that humans solve problems by applying a set of "production rules.'' Versions of those rules, he says, can be programmed into a computer that can then emulate human learning to a remarkable degree.

But observers of advances in artificial intelligence are quick to note that scientists and the news media both have frequently fostered the misconception that machines capable of behaving in ways indistinguishable from humans "are just over the horizon.''

In one particularly egregious example, notes researcher Daniel Crevier, a national news magazine in the 1970's hailed a robot that had "learned'' to move about and to pick up and arrange a series of blocks as "the first electronic person.'' In fact, Crevier writes in his recently published book, A.I.: The Tumultuous History of the Search for Artificial Intelligence, "the robot could barely negotiate straight corners.''

Anderson's intelligent tutors, which he designed with funding from the National Science Foundation, set out to master mathematics, not movement. And unlike its more ambitious predecessors, Anderson says, his software works. No, it's probably not up to the task of teaching computers the nuances of a multitude of skills that humans take for granted to, let's say, move around. But Anderson's software does appear to work quite well for such routinized tasks as solving mathematical problems.

The software design allows the computer to compare student responses to a particular problem with a theoretically optimum solution, interpret errors, and offer the appropriate hints to guide learners who have strayed into unproductive channels of learning back toward an effective problem-solving strategy.

"Sitting inside the computer is a cognitive model that is busy solving the problem alongside the students,'' Anderson explains. As the students interact with the tutor, he continues, "we can follow each particular step as they're solving the problem and respond [with on-line help] to where they are in that particular problem.''

"This software individualizes instruction,'' adds Ken Koedinger, a research computer scientist at the university who helped create the program. "And the way it individualizes is by understanding cognition.''

The basic level, for example, called the problem-solver, begins by presenting students with an equation and asking them to solve for a variable. Students who have daily classroom exposure to the basics of algebra seem capable of progressing rapidly through the increasingly difficult problems the computer presents, solving each equation, step by step. But for less-competent students, or for those whose high school algebra courses are long behind them, the computer offers many levels of on-line help at the touch of a button.

"Because we know what the student knows and doesn't know, we can select problems that are appropriate for that student, and we can decide when to promote the student through the curriculum,'' Anderson adds.

The advantages of this flexibility became obvious in a recent class at Langley High School, the second Pittsburgh school to receive a set of the intelligent tutors. While some students moved rapidly through their problem sets, others were able to concentrate on practicing skills they had not yet mastered.

Bridging the Gap

Despite the tutors' adaptability, Anderson has found that a frustrating political and philosophical schism separates the potential of the software to individualize the curriculum and its practical application in the classroom. Committed to a curriculum that values equity in learning and seeks to teach algebra to all students, district officials are careful about how they want to see the tutors used.

They are particularly leery of any device that appears to give some students an academic edge over others, Anderson says. While "it's certainly possible for students to go at very different rates,'' he explains, that is not a goal of the program.

In keeping with the district's curriculum, which is based on standards developed by the National Council of Teachers of Mathematics, the program's ultimate goal is to help every student learn algebra, a course often described as a "gatekeeper'' for future academic advancement.

To attain its goal, the district has added another course to its traditional algebra curriculum to emphasize practical problem-solving over such skills as factoring polynomials, which the average student is far less likely to encounter after high school.

"The traditional algebra curriculum is [structured] the way it is to prepare kids to take calculus,'' explains Hadley, the algebra teacher, who has worked closely with the Carnegie Mellon research team to bring the intelligent tutors into Pittsburgh classrooms. With the addition of the more practical course, students now have another option.

Academically able students can still enroll in algebra in the 8th grade and proceed to calculus. But other students can now fulfill their 9th-grade algebra requirement by taking the less theoretically oriented course.

Anderson would prefer that the district use the intelligent tutors to teach a more traditional curriculum. And, in fact, when he first introduced an earlier version of the alegbra tutor almost a decade ago, it was designed to support what most educators would consider a "standard'' algebra course like Brashear High's traditional version.

Similarly, Anderson's team developed a tutor to help students learn Euclidean geometry. But when the district decided to de-emphasize the formal proof that is the hallmark of the Euclidean method, Anderson's efforts to deploy the geometry tutor proved unsuccessful.

After consulting with district officials, however, Anderson grudgingly conceded the need to modify the algebra software. Based on his previous experiences, Anderson says, "I try to back away from any arguments about what the mathematics curriculum should teach.''

The new version of the software, which allows teachers to write problems of their own into the system, is more compatible with the district's view of how algebra should be taught.

The updated program is also more successful than its predecessor, thanks to advances in computer technology that have eliminated the need for extremely sophisticated, and expensive, machines to run the software. "The computational powers of the machines are just now becoming available in a form that a school district can afford,'' Anderson notes.

If officials do expand the tutoring system into other schools across the district, Hadley argues, one of its major functions should be to free teachers to work individually with students not using the computers. In an ideal world, he says, students would spend perhaps one-third of the school year working with the electronic tutors, with the rest of the time spent working cooperatively in groups to solve "real world'' algebraic equations.

On a recent day, for example, students were working on word problems Hadley had programmed into the computer. A typical problem asked them to devise a method to calculate the likely number of points a basketball player would score in a given number of games, based on the player's record.

"A problem like this, if you do it in class, could take you two days,'' he says. Using the tutor, the most able students can set up tables listing the various variables and equations needed to solve the problem, solve the equations, plot their solutions on a graph, and receive on-line help on a single screen--all in a matter of minutes.

Raising Comfort Levels

Athough Anderson admits that much research needs to be done to prove that his intelligent tutors increase academic achievement, anecdotal evidence does indicate that the students who use them are more engaged in their learning.

"Teachers are substantially unaware of the achievement gains, but they are aware of the gains in classroom management,'' Anderson says. "The big thing in the classrooms are the reports that the students are much more motivated.''

Anderson also points to a similar tutoring system that has helped Carnegie Mellon computer-programming students learn computer languages in one-third the time. But, he concedes, using the tutors to help highly motivated college students and putting them to work in a public school classroom are clearly different propositions. In fact, since embarking on the project a decade ago, Anderson and his team of researchers have learned a great deal about the give-and-take between theoretical research and workable classroom practice.

And, according to Hadley, they've learned their lessons well. "The teachers feel like collaborators on the project,'' he says. "When the teachers say, 'We don't like this,' [the researchers] change it.''

Still, Hadley says, when it comes to making sure teachers are comfortable with the machines, there's a good deal of work yet to be done.

Anderson agrees. "We've gotten very different results with the system depending on who the teacher is. And it seems to be a function of the ease the teacher has with the software.''

When compared with their students, teachers often experience a longer learning curve. Indeed, it seems, students find working with their intelligent tutors much less daunting.

"I did that, you stupid machine,'' one student yells at the screen before again pecking away at the keyboard. Her tutor, meanwhile, waits silently until the lesson resumes.

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