Technology-Based Math Curriculums, Custom Built for Today’s Classroom

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As a high school principal, I have long been aware of the debate surrounding technology in the classroom. “Don’t throw money at computers unless you’ve got a good plan for how to use them,” I’ve heard. “Don’t let the computer replace the teacher,” others preach.

I’ve always held a certain degree of skepticism about the ability of a computer to teach a child. This is a skepticism I continue to embrace today. The fact is, computers alone can’t teach. However, since I’ve been principal at Langley High School, I’ve realized that the right computer-based curriculum can help teachers reach tremendous results with students.


Langley’s Relationship with a Nearby University

When I came to Langley High School in 1992 as the vice principal, one of the first things to impress me was the relationship that our math teachers had developed with researchers at nearby Carnegie Mellon University, a leader in computer science. Nearly everyone in Pittsburgh is interested in benefiting from the power of this world-class university, so I knew the Langley teachers must have brought a great deal of knowledge and passion to the table.

One of our leading math teachers, Bill Hadley, had been working nights and weekends in his basement, trying to design a better curriculum for teaching algebra. Word about his project was spreading, and soon he was meeting with some of the top cognitive psychologists in the world, including Carnegie Mellon’s John Anderson, Ken K'edinger and Albert Corbett. They were working on a computer model to solve the same teaching problems, but they needed advice from the field.

One of the first things Hadley told them was that students were tired of contrived problems. They were much more interested in calculating the life span of a threatened rain forest, or how many trips a Medevac helicopter can make on a set budget, than in solving meaningless equations with no context.

The researchers at Carnegie Mellon were working on a hunch of their own. They believed that students were more successful in solving problems in which they had solid numbers for their starting point, but did not know the ending point, instead of starting from an unknown point to reach a known goal. Word problems were easier, and students were more comfortable knowing their starting point.

Using this and other research, the Carnegie Mellon team developed a Cognitive Tutor that was programmed to build a profile of the learning patterns of its users. They asked whether Langley would be interested in becoming the university’s first pilot school to use the Cognitive Tutor. The formal relationship began in 1991.





Fundamental Differences from Traditional Curriculums

Each time I visit math teacher Jackie Snyder’s classroom here at Langley, my amazement is renewed. The classroom is very quiet, but an energy buzzes throughout, as students peer at their screens and attempt fresh approaches to their problems. It is clear that the students are engrossed by the problems and are truly motivated by the “learn by doing” process they experience with the Cognitive Tutors.

The Cognitive Tutor tracks a student’s learning style and pinpoints flaws in reasoning. As mistakes are made, the computer gives the student clues for rethinking the problem so that he or she can get back on track. A noticeable difference between this model and past teaching models is the elimination of the “sage on the stage” effect, in which a teacher explains the problems to passive students, who must then try to apply anything they retained to the problems in their textbooks.

Cognitive Tutor learning happens while the students work, keeping them involved all along the way. If a problem becomes insurmountable, the computer won’t give away the answer. The student will know that he or she can ask for help from the teacher.

Cognitive Tutor presents problems to students in three ways that differ from traditional curriculums:


•     It uses real world problems that students will encounter in the work world, involving real business issues.


•     The Cognitive Tutor puts the goals set for students in context. Instead of telling them that they will need to solve 3x + 6 = 5x - 4, we tell them they will have to solve the following problem: “One company charges $3 per item with a shipping charge of $6. Another company charges $5 per item but gives a $4 rebate. How many do you need to order before your company is cheaper?” The on-screen “skill-o-meter” lets kids know how close they come to meeting these goals.


•     Problems are presented on computers about 40% of the time, and the other 60% of student time is spent in a classroom without computers, using the textbook. Students come to understand that algebra is an abstract tool that can be useful in many different contexts.


By experiencing goals in the context of real world problems, Langley students thrive and learn math. Teachers report increased attendance on days when the Cognitive Tutor is used. We have heard kids say some great things about the program:

“This stuff actually makes sense.”

“I used to fall asleep in math class. Now I like to come to class.”

“It knows what I have to work harder on.”


Advantages Over Traditional Approaches

In independent observations of a classroom using the Cognitive Tutor, it was found that teachers:


•     Spend more time speaking to students one-on-one.


•     Spend extra time with students who need more assistance.


•     Need to spend only half as much time reviewing materials the subsequent semester.


•     Report that students are actually excited about learning math.


Not only are the problems based on real world contexts, but the classroom format also mimics real world work life, as the graph below illustrates.


Achievement Results

The Langley evaluations showed that Algebra students who used this approach significantly outperformed similar students in traditional courses, including achievement gains of up to 25% in skill and up to 100% in problem-solving. Over time the research also showed that students using the Cognitive Tutor were twice as likely to complete Geometry and enroll in Algebra II.

Additionally, studies have shown that U.S. students learning through the Cognitive Tutor curriculum achieve scores up to 30% higher than those reported in the TIMSS study. Despite Langley’s drop in total enrollment over the years from about 1,300 to 800, the number of students in Algebra II and Elementary Functions (a fourth year math class) remains the same.


Collaboration Is the Lesson

To any principal who intends to improve teaching practices, I would recommend developing strong relationships with local universities. The results can be very rewarding and long-lasting. I was fortunate to arrive at a school that already had developed strong bonds with Carnegie Mellon — and I will continue to work toward fostering new projects with the university.

We have come to further understand the value of collaboration by watching the students at work with each other, giving each other pointers, talking in an analytical way about the mistakes they make and articulating the rationale behind the preferred solution. This collaboration is an integral part of classroom activities.

A total of 15 years of research went into the Cognitive Tutor project. Ten years of field-testing brought together teachers, administrators and researchers. The input of real students helped in streamlining the curriculum. At Langley alone, 2,500 students have used the Cognitive Tutor.

Today the Cognitive Tutor program is having success in a variety of classroom settings: urban, suburban, rural, vocational, private, public and alternative. Students and teachers across the country have been interested in Cognitive Tutor, leading to the formation of Carnegie Learning, Inc. in 1998. The company is working to develop new Cognitive Tutor curriculums and to bring the product to the Internet.

Recently the U.S. Department of Education deemed the Cognitive Tutor Algebra I curriculum as one of five “exemplary” programs. Chosen from more than 60 submitted programs, the Cognitive Tutor received exemplary status based on the evidence of its effectiveness in improving student achievement in multiple sites with multiple populations.





Start-unknown Problems



When Ted got home from his waiter job, he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he earned $81.90. How much per hour d'es Ted make?



Starting with some number, if I multiply it by 6 and then add 66, I get 81.9 What number did I start with?



6x + 66 = 81.90




Result-unknown Problems


When Ted got home from his waiter job, he took the $81.90 he earned that day and subtracted the $66 he received in tips. Then he divided the hourly wage. How much per hour d'es Ted make?



Starting with 81.9, if I subtract 66 and then divide

by 6, I get a number. What is it?



(81.90 - 66) / 6 = x


Researchers at Carnegie Mellon found that when problems were put into the context of a story, students were more successful in solving them. They were also more successful in solving problems in which the result was unknown, rather than starting from an unknown point. Langley High School was the first pilot site for the Cognitive Tutor.




Technology at Langley Has a Bright Future

Not only d'es Langley use the Algebra I, Algebra II and Geometry Cognitive Tutors, but Langley is also piloting the English Tutors developed through Carnegie Mellon. The Tutors have inspired us to continue improving the technological capabilities of Langley. In fact, we are partnering with AT&T to become the first school and community in Pittsburgh to adopt Fast Ethernet, a connection that is 2,000 times faster than a 56K home modem. Fast Ethernet allows online homework posting, connectivity between schools and community centers, student access to personal accounts and e-mail, and much more. This new level of connectivity will be implemented in the fall of 2000. For more information about Carnegie Learning and the Cognitive Tutor program, call (412) 683-MATH or visit the Web site at


Lawrence Hubbard is principal of Langley High School, a public school located in the Sharaden section of Pittsburgh.


E-mail: [email protected]

This article originally appeared in the 10/01/2000 issue of THE Journal.