Construction of Knowledge about Fractional Number

Abstract

Studies, experimentation, and reflections over the course of twenty years have led our investigations in the direction of students’ construction of the concepts of fraction and fractional number. Distinct trends can be identified in our investigations over the years, each resulting from and reviewed following observations of students’ thinking and reactions as well as the depth and reach of the knowledge constructed. In the first of these phases, we used manipulative material; in the second, reflexive abstractions regarding divided geometric figures; in the third, objects that showed themselves divided in the real world, which led us to situations involving these objects and their associated fractional numbers. This revealed to us an interface with the ideas of Vergnaud regarding the formation of concepts, presented in the Theory of Conceptual Fields. We began to investigate and experiment with the development of a proposal for learning fractional numbers according to the elements of this conception, which includes initial situations, schemes of the students in action, and representations of the concept. In this context, we present the trajectory of our work, a re-elaboration of the proposal in the final phase, and our findings. The proposal we developed centers on the learning of fractions in the construction of a number, making explicit its aim and what it quantifies, as well as its relations with natural numbers. Key-words: Fraction as number. Vergnaud – Formation of Concepts. Counting, Division, Measures prioritized in relation to Divided Geometric Figures.