Exponential distributions in the syntactic structure of written numerals: A linguistic model

Abstract

The linguistic structure of numbers supports the grammar that children have to learn to write Arabic numerals The main linguistic components for the base-10 number of primitive number concepts and sum and product relations. Any com numerical expression can be reduced to primitive concepts by applying in successive times both relations. The grammar contains the set of rules that children have to master to produce well-formed numerical expressions. These rules or procedures are the operations that the subject has to perform and their acquisition corresponds to a learning process. Children production when learning numerals contain both well-formed and ill-formed numerical expressions with syntactic structures coming from the sequence of rules that used to write each numeral We analyze the data from two experiments with children from 6 to 8 years old transcoding oral verbal numerals to the Arabic format under dictation. The first experiment makes measurements in short period of time between successive tests, one week, and the second uses periods of two months. The analysis relates to the syntactic structure of written Arabic numerals with the linguistic structure of the dictated number and measures of their frequency distributions in children's production. The results indicate that these distributions fit exponential functions and we calculate the scale parameter associated to temperature in the description of some economic and social systems. Temperature and Shannon's entropy decrease as children learn describing an ordering process. Finally we find that heterogeneous groups of children produce neat distributions with exponential behavior.