Notations from children's mathematical thinking: equalizing and dividing quantities at the roots of multiplicative structures / Nota es da matemática infantil: igualar e repartir grandezas na origem das estruturas multiplicativas
The paper concerns the learning of additive structures on its way to the multiplicative ones according to the equilibration model and Vergnaud's propositions about conceptual fields. It describes the nature and the changes of children's notations produced on tasks of equalization and partition of quantities, and verifies the meaning of those notations in the study of the psychogenetic relationship between those structures. The 12 participants (6,4 to 9,5 years old), attending two State Elementary Schools from suburbs of two different metropolitan areas, performed in triads the tasks, which alternated moments of practical compositions, notations production and their interpretation. The qualitative microgenetic analysis of videotaped data resulted in notation types and in tendencies of the conceptual transformations shown by the notations. The discussion underlines the nature and the progression of the described notations, and the relevant role of equalization and repartition as inherent schemata to the psychogenetic relationship between the focused structures.