After being taught how to perform a new mathematical operation, students are often given several practice problems in a single set, such as a homework assignment or quiz (i.e., massed practice). An alternative approach is to distribute problems across multiple homeworks or quizzes, increasing the temporal interval between practice (i.e., spaced practice). Spaced practice has been shown to increase the long-term retention of various types of mathematics knowledge. Less clear is whether spacing decreases performance during practice, with some studies indicating that it does and others indicating it does not. To increase clarity, we tested whether spacing produces long-term retention gains, but short-term practice costs, in a calculus course. On practice quizzes, students worked problems on various learning objectives in either massed fashion (3 problems on a single quiz) or spaced fashion (3 problems across 3 quizzes). Spacing increased retention of learning objectives on an end-of-semester test but reduced performance on the practice quizzes. The reduction in practice performance was nuanced: Spacing reduced performance only on the first two quiz questions, leaving performance on the third question unaffected. We interpret these findings as evidence that spacing led to more protracted, but ultimately more robust, learning. We, therefore, conclude that spacing imposes a desirable form of difficulty in calculus learning.

在教给学生如何执行新的数学运算后，通常会在一组练习中给学生提供几个练习题，例如家庭作业或测验（即集体练习）。另一种方法是将问题分布在多个家庭作业或测验中，增加练习之间的时间间隔（即间隔练习）。间隔练习已被证明可以增加各种数学知识的长期保留。不太清楚的是间距是否减小实践中的表现，一些研究表明它确实如此，而另一些研究表明它没有。为了提高清晰度，我们在微积分课程中测试了间隔是否会产生长期的保留收益，但会产生短期的练习成本。在练习测验中，学生以集中方式（单个测验 3 个问题）或间隔方式（3 个测验中的 3 个问题）解决各种学习目标的问题。间隔增加了期末考试对学习目标的保留，但降低了练习测验的表现。练习成绩的降低是细微的：间隔只降低了前两个测验问题的表现，而第三个问题的表现不受影响。我们将这些发现解释为间隔导致更持久但最终更稳健的学习的证据。因此，我们，