Unbounded Human Learning: Optimal Scheduling for Spaced Repetition

In the study of human learning, there is broad evidence that our ability to retain information improves with repeated exposure and decays with delay since last exposure. This plays a crucial role in the design of educational software, leading to a trade-off between teaching new material and reviewing what has already been taught. A common way to balance this trade-off is spaced repetition, which uses periodic review of content to improve long-term retention. Though spaced repetition is widely used in practice, e.g., in electronic flashcard software, there is little formal understanding of the design of these systems. Our paper addresses this gap in three ways. First, we mine log data from spaced repetition software to establish the functional dependence of retention on reinforcement and delay. Second, we use this memory model to develop a stochastic model for spaced repetition systems. We propose a queueing network model of the Leitner system for reviewing flashcards, along with a heuristic approximation that admits a tractable optimization problem for review scheduling. Finally, we empirically evaluate our queueing model through a Mechanical Turk experiment, verifying a key qualitative prediction of our model: the existence of a sharp phase transition in learning outcomes upon increasing the rate of new item introductions.

[1]  D. Kendall.  Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain , 1953 .

[2]  Paul Pimsleur.  A MEMORY SCHEDULE , 1967 .

[3]  Anthony Unwin,et al.  Reversibility and Stochastic Networks , 1980 .

[4]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

[5]  F. N. Dempster,et al.  Spacing effects and their implications for theory and practice , 1989 .

[6]  E. Gorzelańczyk,et al.  Optimization of repetition spacing in the practice of learning. , 1994, Acta neurobiologiae experimentalis.

[7]  R. J. Mokken,et al.  Handbook of modern item response theory , 1997 .

[8]  Michael Pinedo,et al.  Queueing networks - customers, signals and product form solutions , 1999, Wiley-Interscience series in systems and optimization.

[9]  Hans C. van Houwelingen,et al.  The Elements of Statistical Learning, Data Mining, Inference, and Prediction. Trevor Hastie, Robert Tibshirani and Jerome Friedman, Springer, New York, 2001. No. of pages: xvi+533. ISBN 0‐387‐95284‐5 , 2004 .

[10]  John R. Anderson,et al.  Knowledge tracing: Modeling the acquisition of procedural knowledge , 2005, User Modeling and User-Adapted Interaction.

[11]  J. Franklin,et al.  The elements of statistical learning: data mining, inference and prediction , 2005 .

[12]  H. Pashler,et al.  Distributed practice in verbal recall tasks: A review and quantitative synthesis. , 2006, Psychological bulletin.

[13]  Ed Vul,et al.  Predicting the Optimal Spacing of Study: A Multiscale Context Model of Memory , 2009, NIPS.

[14]  Robert Tibshirani,et al.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition , 2001, Springer Series in Statistics.

[15]  Steven H Strogatz,et al.  Education of a model student , 2012, Proceedings of the National Academy of Sciences.

[16]  Hermann Ebbinghaus,et al.  Memory: a contribution to experimental psychology. , 1987, Annals of neurosciences.

[17]  Hermann Ebbinghaus (1885).  Memory: A Contribution to Experimental Psychology , 2013, Annals of Neurosciences.

[18]  Robert V. Lindsey,et al.  Improving Students’ Long-Term Knowledge Retention Through Personalized Review , 2014, Psychological science.