M. Riopel

Université du Québec à Montréal (CANADA)
Even if practice plays an important role in e-learning, it is still difficult to find a consensual and useable mathematical model to describe the shapes of the learning and forgetting curves. As pointed out by [1] “a fundamental understanding of the distributed practice effect is lacking; many qualitative theories have been proposed, but no consensus has emerged” (p. 236). This paper presents a first outlook on some of the reasoning steps used in deducing the mathematical properties of the observed curves for the practice and forgetting effects. The proposed model is based on three scale invariant hypotheses that were shown to be empirically valid in another paper [2] when learning to identify images of constellations with a php/html web application.

As pointed out by [3], search for the shape of the forgetting curve is an old problem going back to Ebbinghaus (1885) and for which there is still no consensual solution. It is proposed that the temporal invariance can be used with the first consequence of the composition invariance to deduce the shape of this curve. This shows that logarithm of correctness should follows an inverse Weibull cumulative function that, for large t, behave as a three-parameter power function. This new three-parameter forgetting law is compatible with the results of [3] with the advantage of being scale invariant. When fitted to their data by minimizing the chi-squared statistic, the two curves are almost equivalent, with a slightly better fit for the new function.

The proposition of a power law of practice concerns the relationship between response time and number of practice trials. However, according to [4] “the evidence for a power law is flawed, because it is based on averaged data”. Their analysis shows that the exponential function is probably a better choice for individual data. It is proposed that the ordinal invariance can be used with the consequences of the composition invariance to deduce another shape for this curve: a four-parameter stretched exponential function. This is compatible with the analysis from [4] that concluded that the three-parameter exponential function was almost as good as the more flexible four-parameter power-exponential function. Adding a fourth parameter to the exponential to stretch it can only improve the fit and hopefully make this new proposition, deduced from the invariance hypotheses, the best choice in all cases. However, this empirical verification still has to be done.

It is interesting to note that scale invariance can be used to deduce de properties of two very different and very strong effects related to learning. The proposed laws are somewhat different but still compatible with the most precise empirical data so far. This is an important achievement in the field that could ultimately lead to a complete and coherent proposition for a uniquely defined scale invariant theory of learning.

[1] Cepeda et al. (2009). Optimizing distributed practice: Theoretical analysis and practical implications. Exp. Psych., 56, 236-246
[2] Riopel et al. (2017). Using invariance to model practice, forgetting, and spacing effects: the constellations’ case. ESERA 2017 21st-25th August, Dublin, Ireland.
[3] Averell, L. & Heathcote, A. (2011). The form of forgetting curve and the fate of memories. J. of Math. Psych., 55(1), 25–35
[4] Heathcote et al. (2000). The power law repealed: The case for an exponential law of practice. Psych. Bull. & Rev. 7(2), 185-207.