Online educational technologies offer opportunities for providing individualized feedback and detailed profiles of students' skills. Yet many technologies for mathematics education assess students based only on the correctness of either their final answers or responses to individual steps. In contrast, examining the choices students make for how to solve the equation and the ways in which they might answer incorrectly offers the opportunity to obtain a more nuanced perspective of their algebra skills. To automatically make sense of step‐by‐step solutions, we propose a Bayesian inverse planning model for equation solving that computes an assessment of a learner's skills based on her pattern of errors in individual steps and her choices about what sequence of problem‐solving steps to take. Bayesian inverse planning builds on existing machine learning tools to create a generative model relating (mis)‐understandings to equation solving choices. Two behavioral experiments demonstrate that the model can interpret people's equation solving and that its assessments are consistent with those of experienced teachers. A third experiment uses this model to tailor guidance for learners based on individual differences in misunderstandings, closing the loop between assessing understanding, and using that assessment within an educational technology. Finally, because the bottleneck in applying inverse planning to a new domain is in creating the model of possible student misunderstandings, we show how to combine inverse planning with an existing production rule model to make inferences about student misunderstandings of fraction arithmetic.

中文翻译：

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通过贝叶斯逆向规划评估数学误解

在线教育技术提供了提供个性化反馈和学生技能详细资料的机会。然而，许多数学教育技术仅根据学生的最终答案或对个别步骤的反应的正确性来评估学生。相比之下，检查学生对如何解方程的选择以及他们可能错误回答的方式提供了机会，让他们有机会更细致地了解他们的代数技能。为了自动理解逐步解决方案，我们提出了一个用于方程求解的贝叶斯逆向规划模型，该模型根据学习者在各个步骤中的错误模式以及她对问题解决步骤顺序的选择来计算对学习者技能的评估采取。贝叶斯逆向规划建立在现有机器学习工具的基础上，以创建将（错误）理解与方程求解选择相关联的生成模型。两个行为实验表明，该模型可以解释人们的方程求解，并且其评估与有经验的教师的评估一致。第三个实验使用该模型根据误解中的个体差异为学习者量身定制指导，关闭评估理解和在教育技术中使用该评估之间的循环。最后，由于将逆向规划应用于新领域的瓶颈在于创建学生可能误解的模型，因此我们展示了如何将逆向规划与现有的生产规则模型相结合，以推断学生对分数算术的误解。