Abstract:

Even if practice plays an important role in e-learning, it is still difficult to find a consensual and usable mathematical model to describe the shapes of the learning, forgetting, and spacing curves. Studies on distributed practice go back to Ebbinghaus (1885) and, since then, hundreds more have been conducted on the related effects [2]. One of these effects is the law of practice [3], which relates the higher success rate and shorter response time to the number of practice trials. Another effect is the forgetting curve [4], which relates lower observed probability of retention to longer elapsed time before retrieval. A third and more abstract phenomenon is the spacing effect, which proposes that adequate distribution of repetitions over time usually produces more learning than massing them.

For the purpose of this study, three hypotheses have been chosen for use in deducing the mathematical properties of the observed empirical curves. The first hypothesis, the composition invariance, states that, for each subject, simultaneous success for two equivalent and valid tasks in a certain context should always produce a new valid, but possibly rescaled, task for the same context. The second hypothesis, the temporal invariance, states that slowing a sequence of tasks cannot change the mathematical properties of the observed empirical curve for correctness, except for one scaling parameter. The third hypothesis, the ordinal invariance, similarly states that, when counting successive occurrences of identical tasks, one should be able to choose freely what can be counted as unity (a single task or group of tasks) without changing the mathematical properties of the observed empirical curve for correctness.

An online php/html application was developed for this study. An invitation to participate was sent via social media to professional and amateur astronomy association members. Subjects were first asked to choose the correct name (out of 3) for the image of one of the 88 constellations. They were then given three seconds to review the correct answer, and questions were repeated several times in a within-session randomized block design with different scales. Correctness of the given answers and response time were recorded. The complete sequence consisted of 3 blocks of 136 identifications, each block taking approximately 15 minutes to complete. The three scale invariance hypotheses were tested in a two-step procedure. A scale parameter was first chosen to make each pair of related curves as similar as possible by minimizing the chi-squared statistic. Then, the same statistic was used to test if the first curve could serve as a model for the second. Preliminary results show that an appropriate scale parameter can be used to make all related curves statistically equivalent. The hypotheses are then used to deduce some properties of the observed curves in this context.

References:
[1] Küpper-Tetzel, C. E. (2014), Understanding the Distributed Practice Effect : Strong Effects on Weak Theoretical Grounds, Zeitschrift fr Psychologie, 222(2),71–81.
[2] Cepeda et al. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psych. Bull., 132, 354–380.
[3] Heathcote et al. (2000). The power law repealed: The case for an exponential law of practice. Psych. Bull. & Rev. 7(2), 185-207.
[4] Averell, L. & Heathcote, A. (2011). The form of forgetting curve and the fate of memories. J. of Math. Psych. 55(1), 25–35

Keywords:

Astronomy, Practice, Forgetting, Learning Theory.