A Different Kind of Math Tutor

Research on how people think results in cognitive tutor programs that help kids do better in math.


Algebra class may be less difficult and a bit more fun these days, thanks to research on how human cognition works. Developed over two decades by psychologist John Anderson, the Adaptive Character of Thought (ACT-R) theory is a framework for understanding how we think about and attack problems, including math equations. The theory reflects our understanding o human cognition based on numerous facts derived from psychological experiments.

ACT-R suggests that complex cognition arises from an interaction of procedural and declarative knowledge. Declarative knowledge is a fairly direct encoding of facts (such as Washington, DC is the capital of the United States, 5 + 3 =8); procedural knowledge is a fairly direct encoding of how we do things (such how to drive or how to perform addition). According to the ACT-R theory, the power of human cognition depends on how people combine these two types of knowledge.


The ACT-R theory provides insights into how students learn new skills and concepts, and, in doing so allows teachers to see where students may need extra practice to master the new work.

Practical Application

Dr. Anderson and colleagues at Carnegie Mellon University have used this research to develop cognitive tutors, computer-tutoring programs that incorporate the ACT-R theory in the teaching of algebra, geometry and integrated math. The tutors are based on cognitive models that take the form of computer simulations that are capable of solving the types of problems that students are asked to solve. The tutors incorporate the declarative and procedural knowledge imbedded in the instruction and monitor students' problem solving to determine what the students know and don't know. This allows instruction to be directed at what still needs to be mastered and helps insure that students' learning time is spent in a more efficient manner. Students work on a concept until it is fully understood. Students who are having conceptual problems will be drilled on in that area, while those who have mastered the concept move on to other areas.

The most widely used cognitive tutor program - now known as Carnegie Learning's Cognitive Tutor - combines software-based, individualized computer lessons with collaborative, real-world problem-solving activities. The program now serves more than 150,000 students in most of the nation's largest school districts. Field studies have shown dramatic student achievement gains where the program is in use.

In 2003, the U.S. Department of Defense Schools awarded a contract that will use Cognitive Tutor mathematics curricula in its 224 public schools in 21 districts located in 14 foreign countries, seven states, Guam and Puerto Rico. These schools have approximately 8,800 teachers serving 106,000 students.

Cited Research

Anderson, J. R. (1983). The architecture of cognition. Cambridge: MA: Harvard University Press.

Anderson, J. R. (1993). Rules of the Mind. Hillsdale, NJ: Erlbaum.

Anderson, J. R., Corbett, A. T., Koedinger, K., & Pelletier, R. (1995). Cognitive tutors: Lessons learned. The Journal of Learning Sciences, Vol. 4, pp. 167-207.

Anderson, J. R. (1996). ACT: A simple theory of complex cognition. American Psychologist, Vol. 51, pp. 355-365.

Anderson, J. R. & Lebiere, C. (1998). The atomic components of thought. Mahwah, NJ: Erlbaum.

American Psychological Association, October 23, 2003