A notion of semantic distance between terms or expressions and its application in the case of mathematical terms used in the classroom
Date
1994Source
International Journal of Mathematical Education in Science and TechnologyVolume
25Issue
1Pages
31-43Google Scholar check
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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.