Abstract

We developed and tested strategies for using spatial representations to help students understand core probability concepts, including the multiplication rule for computing a joint probability from a marginal and conditional probability, interpreting an odds value as the ratio of two probabilities, and Bayesian inference. The general goal of these strategies is to promote active learning by introducing concepts in an intuitive spatial format and then encouraging students to try to discover the explicit equations associated with the spatial representations. We assessed the viability of the proposed active-learning approach with two exercises that tested undergraduates’ ability to specify mathematical equations after learning to use the spatial solution method. A majority of students succeeded in independently discovering fundamental mathematical concepts underlying probabilistic reasoning. For example, in the second exercise, 76% of students correctly multiplied marginal and conditional probabilities to find joint probabilities, 86% correctly divided joint probabilities to get an odds value, and 69% did both to achieve full Bayesian inference. Thus, we conclude that the spatial method is an effective way to promote active learning of probability equations.

摘要

我们开发并测试了使用空间表示来帮助学生理解核心概率概念的策略，包括根据边际概率和条件概率计算联合概率的乘法规则，将优势值解释为两个概率的比率，以及贝叶斯推理。这些策略的总体目标是通过以直观的空间格式引入概念来促进主动学习，然后鼓励学生尝试发现与空间表示相关的显式方程。我们通过两个练习评估了所提出的主动学习方法的可行性，这些练习测试了本科生在学习使用空间求解方法后指定数学方程的能力。大多数学生成功地独立发现了概率推理背后的基本数学概念。例如，在第二个练习中，76% 的学生正确地将边际概率和条件概率相乘以找到联合概率，86% 的学生正确地将联合概率相除以获得赔率值，69% 的学生同时做了这两项以实现完整的贝叶斯推理。因此，我们得出结论，空间方法是促进概率方程主动学习的有效方法。