Educação matemática e cálculo mental: Uma análise de invariantes operatórios a partir da teoria dos campos conceituais de Gérard Vergnaud

Abstract

The study proposes to identify operatory invariants used in mathematical problem-solving requiring mental computation. The theoretical approach adopted was based on Gérard Vergnaud s theory of conceptual fields, especially the notions of conceptual field, concepts, situations, schemas and operatory invariants. Research procedures aimed at: analysing operatory invariants - namely concepts-in-action and theorems-in-action at a microgenetic level; characterizing the uses of mathematical strategies in everyday activities; identifying speech and actions/gestures which indicate the use of mathematical knowledge. The study took place in a large school in Juiz de Fora, Minas Gerais, where three pairs of 11 years-old pupils, enrolled in the 4th year of primary schooling, were interviewed and carried out a group activity. All data was recorded in videotapes and analysed according to a microgenetic approach, which allowed for detailed analysis of relevant episodes involving mental computation. The problems presented to the pupils required money-based calculations involved in public transport, such as taking a bus, paying for the fare and getting change. Findings concerning pupils use of strategies show that: two strategies used by the pupils, composition and decomposition, require sound knowledge of the decimal system; on many occasions, it is noticeable that strategies are used in combination, which present personal variations; most of the strategies used rely on school-based knowledge, such as the decimal number system and the basic properties of arithmetic operations, although the actual procedures developed by the pupils were far from the typical problem-solving taught in school. Analysis of operatory invariants shows similarities and differences in the strategies used by pupils in school and out-of-school situations. The study offers contributions for primary school maths teaching: first, through the identification of operatory invariants used in different problem-solving situations; second, by way of primary school teachers increased knowledge of problem-solving strategies, leading to further understanding of everyday uses of mathematics.