A notion of semantic distance between terms or expressions and its application in the case of mathematical terms used in the classroom

Abstract

In this paper a model of semantic difference between various mathematical terms used in the classroom is built. As a basic tool in the construction of this model the concept of codification, which is a mapping σ of a space M of meaning into a set Λ of terms or expressions, is used. Given a set of codifications (which may belong to the teacher or to the pupils), a semantic distance μΣ is defined in Λ by the formula: μΣ(s, t) = sup {d(x, y)|x, y∈M with σ1(x) = s, σ2(y) = t for some σ1, σ2∈Σ} where d is a distance function in M. In particular the 'semantic width' of a term t, WΣ(t) = μΣ(t, t), is generally different from zero. The model is applied in various situations of teaching, but the main application is in the construction of a semantic distance suitable for analysing students' difficulties with quadrilaterals. In this respect a 'Boolean' representation of types of quadrilaterals is combined with a 'tree-like' representation related to Aristotle's γέvη, and as a result, a larger semantic space is constructed which offers a panoramic view of classical and modern conceptions. By using this semantic space it was found, for example, that the term 'rhombus' has in Greek classrooms a considerable 'semantic width'; i.e. if Σ1 (resp. Σ2) denotes the set of codifications of the teachers (resp. 16-year-old students) then WΣ1υΣ2(rhombus) = d (Inclined Rhombus, Square) = 2 This and other similar situations are responsible for a lot of confusions and misunderstandings in geometry lessons. © 1994 Taylor & Francis Ltd.