A mathematical model to describe the learning process to write Arabic numerals

Abstract

We use results from cognitive psychology and statistical mechanics to describe the ordering process of children when learning to write numerical. Children of the same elementary school grade were asked to write in Arabic format dictated numerals in three sessions and the production of all children in each session was classified. The classification contains one class for the correct production and four classes for syntactic errors. Each class is characterized by the value of a quantity that is assumed to be discrete and valued from the order of appearance of syntactic errors in children's production. We call this quantity distance. The class containing the correct production has the minimum value for distance and the other classes have larger values. To model the system from a statistical mechanics perspective, we consider each class as a subsystem of a theoretical ensemble and define the spectrum of distance from three models, two of them with degenerate states and one using the quantum harmonic oscillator. We calculate the entropy and the temperature of the system and use them to describe the ordering process for children in the first three elementary school grades. We found that both temperature and entropy values reduce with time and that they can be used to describe the learning process of Arabic numerals.