Diagnosing misconceptions about algebra using Bayesian inverse reinforcement learning

这个项目将由加州大学伯克利分校和卡尔顿学院的研究人员进行，他们将开发一个在线数学导师来帮助高中生和大学生学习代数。导师会通过让学生解决一系列数学问题来诊断他们对代数的误解。该网站将向任何地方的学生开放，使其能够收集大量关于代数问题解决的数据，这些数据将用于改进技术方法，开发学生学习的计算模型，优化测试设计，并确定在线学习和教学的有效策略。该项目将推进REAL（教育与学习研究）项目在研究STEM（科学、技术、工程和数学）学习的认知基础方面的工作，以及网络学习项目在发现如何设计和有效使用未来学习技术方面的工作。在线辅导中用于知识诊断的方法将是贝叶斯逆强化学习。这种方法结合了机器学习和认知科学的思想。学生的反应将使用马尔可夫决策过程进行建模，马尔可夫决策过程是一种决策理论形式，表明一个理性的主体应该如何采取一系列行动来实现目标。这将使计算一个学生在给定他或她的领域知识的情况下采取一系列特定行动的概率成为可能。贝叶斯推理将被用来反转这个模型，提供一个概率分布在学生的知识状态给定他或她的行为。将这种方法应用于代数问题的解决将有可能识别学生对代数问题的自由形式解决方案的误解。拟议的研究将需要解决诸如处理描述代数方程所需的无限结构化状态空间，适应复合动作（例如跳过推导中的一步），建模学习以及优化评估设计等挑战。

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.