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1. Introducing desirable difficulty in engineering mathematics with spaced retrieval practice

   Bego, Campbell R.; Ralston, Patricia A.; Lyle, Keith B.; Immekus, Jason C. (July 2021, Proceedings of the 128th ASEE Annual Conference and Exposition)

   null (Ed.)

   This study examined the difficulty introduced by spaced retrieval practice in Calculus I for undergraduate engineering students. Spaced retrieval practice is an instructional technique in which students engage in multiple recall exercises on the same topic with intermittent temporal delays in between. Spacing out retrieval practice increases the difficulty of the exercises, reducing student performance on them. However, empirical research indicates that spaced retrieval practice is associated with improvements in students’ long-term memory for the retrieved information. The short-term costs and long-term benefits of spaced retrieval practice is an example of desirable difficulty, when more difficult exercises during the early stages of learning result in longer-lasting memory [1]. With support from the National Science Foundation (NSF), we sought to address: Does spacing decrease performance on retrieval practice exercises in an engineering mathematics course? Results showed that student performance was significantly lower for questions in the spaced condition than questions in the massed condition, indicating that we successfully increased the difficulty of the questions by spacing them out over time. Future work will assess final quiz performance to determine whether spacing improved long-term course performance, i.e., whether the difficulty imposed by spacing was desirable. 

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2. Using SQL to query the difficulty imposed by spaced retrieval in engineering mathematics

   https://doi.org/10.1109/FIE49875.2021.9637171  

   Boyd, Jeremy; Bego, Campbell; Garcia, Osvaldo; Ralston, Patricia; Immekus, Jason; Lyle Keith (January 2021, 2021 IEEE Frontiers in Education Conference (FIE))

   This WIP Research Paper investigates the temporal nature of the difficulty imposed by spacing. Spaced retrieval practice is an evidence-based strategy for improving memory and consists of asking multiple questions on a topic with intermittent delays. Spacing is often thought to impose difficulty by making questions harder to answer. However, this difficulty may be desirable, since spacing ultimately improves memory. In this paper, we (a) outline an implementation of spaced retrieval practice in an engineering mathematics classroom, (b) describe the development of an SQL database to organize and manage the large and complex dataset, and (c) discuss a brief but interesting dive into the rich data we have collected. Statistical analyses revealed that, when three questions targeting the same topic are spaced over multiple quizzes, versus being massed on a single quiz, students are less likely to answer the first and second questions correctly. Spacing does not affect students' ability to answer the third questions. This suggests that spacing may impose difficulty when students are first learning to perform mathematical operations, rather than when they are trying to retrieve memories of how to perform those operations. 

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3. Spaced Retrieval Practice in Undergraduate Engineering Courses: Psychometric Considerations

   Immekus, Jason; Bego, Campbell; Ralston, Patricia; Lyle Keith (January 2022, ASEE 2022 Annual Conference and Exposition)

   The focus of our current National Science Foundation (NSF: IUSE Award #1912253) project is the degree to which spaced retrieval practice, as compared to massed, increases performance on an end-of-semester exam in 10 different STEM courses (e.g., engineering, chemistry, biology). This paper presents an intermediate analysis examining the psychometric properties of the retrieval-practice exercises in two specific courses. The critical question is whether the psychometric properties of the exercises differ depending on whether they are spaced or massed. Preliminary results indicate that spacing does not consistently affect reliability of items, but can impact item difficulty. 

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4. Optimizing Learning Efficiency: Balancing Spacing and Repetition Under Time Constraints

   Cao, M; Yan, V X; Sana, F; Carvalho, P F (July 2025, Proceedings of the 47th Annual Conference of the Cognitive Science Society; eScholarship Open Access Publications from the University of California)

   Barner, D; Bramley, NR; Ruggeri, A; Walker, CM (Ed.)

   Spaced retrieval practice has been repeatedly demonstrated to improve learning, but its implementation is often constrained by real-world time limitations. This study investigated whether, under fixed study durations, learners should prioritize spacing or repetition. Across two experiments (total N = 1589), participants practiced Indonesian vocabulary under four conditions that varied in spacing and repetition. Item difficulty was also manipulated. Results showed that increasing repetitions at the cost of spacing enhanced immediate test performance, particularly for harder items. These findings suggest that spaced retrieval practice is effective only when learners have sufficient prior repetitions to retrieve information successfully. This study highlights the trade-offs between spacing and repetition under time constraints and offers practice insights for optimizing learning strategies. 

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5. Spaced Retrieval Practice Imposes Desirable Difficulty in Calculus Learning

   https://doi.org/10.1007/s10648-022-09677-2  

   Lyle, Keith_B; Bego, Campbell_R; Ralston, Patricia_A_S; Immekus, Jason_C (April 2022, Educational Psychology Review)

   Abstract 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 spacingdecreasesperformance 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. 

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