Parts, measures and equivalent fractions: the movement of theoretical thought of teachers who teach mathematics

Abstract

Research on the concept of fraction is faced with the report of teachers and students about difficulties with this concept in the teaching and learning process. It is possible that this difficulty stems from a didactic approach based on purely technical and mechanized processes. In an attempt to overcome this mechanization, understanding the dialectical relationship between teaching activity and study activity, we organized a research that investigated the process of developing the theoretical thought of teachers about the concept of fraction. In this article, we present an excerpt of this research, presenting the movement of approximation of the theoretical thought of teachers mediated by the concept of equivalent fractions. Based on the contributions of the Developmental Didactic System, based on the Historical-Cultural Theory, we organized a formative experiment for teachers, in which Triggering Learning Situations were developed to enable teachers to encounter the logical-historical movement of the concept, revealing its essence in the comparison of continuous quantities. The analysis of the data revealed that the concrete-abstract-concrete movement at work with situations involving equivalent fractions, provided teachers with thinking mediated by the concept, typical of theoretical thought. In this process, teachers demonstrated to overcome empirical actions or crystallized methods of transforming fractions into decimal representations, understanding the importance of the arithmetic, algebraic and geometric interrelationship of the concept of fraction. The results reinforce the importance of teacher training promoting, through the development of theoretical thought, the overcoming of crystallized ways of using algorithms, with a view to the production of theoretical and meaningful knowledge for teachers and students.