Interpretive fields in mathematics education: the Pythagorean theorem case

Abstract

In this work, we expose and explain the elements of a theoretical construct called the interpretive field (IF), which is generated from a mathematical object called the IF core. To exemplify and demonstrate the construction process of an IF, we chose the Pythagorean Theorem (PT) as the core, given its relevance within mathematics and as a learning object in didactics. An IF is made up of conceptual networks that structure the mathematical objects immersed in the core, which are made explicit through hermeneutic tools, particularly analogy. During the construction process of the IF associated with the Pythagorean Theorem (IF-PT), the relevance of the tension between opposites (for example, particularization-generalization) is highlighted as a generator of relationships that articulate meanings and interpretations between mathematical objects. The purpose of generating an IF is to obtain a global vision of the relationships and articulations that emerge from the nucleus, in which the equivocal interpretations in different contexts are highlighted and made transparent. These relationships are useful to identify and design didactic trajectories, which are evidence of mathematical understanding, which can be achieved through a process of oriented reconstruction of knowledge. As a result of the work, we present a graphic representation of the partial network of constituent relationships of the IF-PT. This network is partial and incomplete, because the IC is a dynamic construct in permanent construction, whose structure is dialectically related to the knowledge of each person and to scientific advances at a certain historical moment. The previous ideas exemplify the inherent complexity of the mathematical objects that are placed in the didactic scenario, and which consideration is fundamental for the development of understanding.

Keywords:
Interpretive field; Articulations; Interpretations; Understanding; Tension