A probabilistic system helps understand the transition between models of learning

In learning, the trade-off between reviewing old content and introducing new content poses interesting theoretical questions. Practically speaking, designers of educational software need to build algorithms with optimal sequences of review and new learning. The slow flashcard system (SFS) explores the trade-off dilemma at the critical transition between infinite rapid learning (unbounded) and a limited learning regime encumbered by constant reviewing (bounded). Joel Nishimura reports on findings using methods borrowed from statistical mechanics to study SFS in more detail.

In the SFS, card viewings are tallied and reinserted into the pack at strictly increased depths. This sequence slowly introduces new cards but has combinatorial complexity, making the derivation of a formula to determine a familiarity curve, which describes the proportion of familiar to new content, a daunting task. Nishimura formulated a nearby discrete-time probabilistic system that shares similar states and overarching principles with SFS.

Probabilistic SFS eliminated the permutation-based complexity that is numerically unimportant to the overall statistics but captured the macroscale familiarity curve. “If you just want to look at the familiarity curve, low level details can be blurred away probabilistically,” said Nishimura.

The author developed a partial differential equation to describe the behavior of the familiarity curve, which enabled him to examine other SFS reinsertion schemes. He showed that the invariance of the familiarity curve is a universal feature of slow regime models and argues that the shape of the familiarity curve reveals clues about a student’s learning process. The universality of the curve also suggests that other archetypal learning schemes exist.

Nishimura encourages others to use probabilistic techniques to bypass intrinsic complexity within a system and is now keen to explicitly model the complete phase transition between unbounded and bounded learning.

Source: “Critically slow learning in flashcard learning models,” by Joel Nishimura, Chaos: An Interdisciplinary Journal of Nonlinear Science (2018). The article can be accessed at https://doi.org/10.1063/1.5038027 .