This project, to be conducted by researchers at the University of California, Berkeley, and Carleton College, will develop an online math tutor to help high school and college students learn algebra. The tutor will diagnose students' misconceptions about algebra by asking them to solve a series of math problems. The website will be made available to students anywhere, making it possible to collect large amounts of data on algebra problem solving that will be used to refine the technological approach, develop computational models of student learning, optimize the design of tests, and identify effective strategies for online learning and teaching. This project will advance the work of the REAL (Research on Education and Learning) program in studying the cognitive basis of STEM (science, technology, engineering, and mathematics) learning, as well as the Cyberlearning program in discovering how to design and effectively use learning technologies of the future.
The approach used for knowledge diagnosis in the online tutor will be Bayesian inverse reinforcement learning. This approach combines ideas from machine learning and cognitive science. Students' responses will be modeled using Markov decision processes, a decision-theoretic formalism that indicates how a rational agent should take a series of actions to achieve a goal. This will make it possible to calculate the probability that a student would take a particular series of actions, given his or her knowledge of the domain. Bayesian inference will be used to invert this model, providing a probability distribution over the knowledge state of the student given his or her actions. Applying this approach to algebra problem solving will make it possible to identify students' misconceptions from freeform solutions to algebra problems. The proposed research will require addressing challenges such as working with the infinite, structured state spaces that are needed to describe algebraic equations, accommodating composite actions (such as skipping a step in a derivation), modeling learning, and optimizing the design of assessments.
