Mathematics is a tool humans invented to help deal with commerce, navigation, agriculture, and government, and other real-world applications. They often require that we extend our mathematical knowledge beyond the exact procedures that we have been taught. Many people have mastered the mathematical knowledge they have been taught in school, but they still display serious difficulties when challenged to extend that knowledge to new situations and often produce nonsensical answers. This research will study adolescents and college students who have mastered middle-school mathematics including fractions and negative numbers, and challenge them to extend this knowledge to deal with a novel mathematical concept.
This research will focus on a little-known class of mathematical problems, called pyramid problems or trapezoidal numbers, that have a natural geometrical interpretation to help guide their understanding. Their novelty to the general public makes them ideal for study and they require no calculations that go beyond middle school mathematics. Participants will be taught to solve pyramid problems that involve small positive integers and then they will be challenged to extend the relationships to deal with large numbers, fractional numbers, and negative numbers. To understand developmental trends, adolescents will be compared with college students. The data to be collected will involve a combination of performance measures, verbal protocols, and fMRI brain imaging patterns. Separate studies will investigate the basis of individual differences in successful knowledge extension, the effect of metacognitive engagement on success, and the role of the geometric interpretation in guiding inferences about the mathematical relationships. This research will take place within the theoretical framework of the ACT-R theory, a computational model of mathematical problem solving. The critical test of this theory will be its ability to predict the rich pattern of data collected. Having such a theoretical framework will be important for generalizing the results from pyramid problems to helping students extend their mathematical knowledge more generally.
The critical contribution of this research is that it goes beyond the question of how to teach a specific mathematical competence and addresses the question of how to prepare students to extend their mathematical knowledge and discover new mathematical relationships. Placed in the context of a formal computational model of cognition, it would make a major contribution to cognitive science and neuroscience by moving theory beyond the learning of well-defined procedures to the mechanisms responsible for the generation of new knowledge. This research will take place within the context of the Cognitive Tutors, which are currently deployed in many American classrooms, reaching over 500,000 students. The computational model developed in the project can be transitioned to these tutors and would enable a major enhancement in the kinds of competences that these tutors teach.
