On squares, rectangles, rhombuses, ... and the influence of culture and language on students’ conceptions

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

“…Patronis and Spanos [3] analysed from a cultural and linguistic viewpoint the difficulties that students have with the conceptual structure of usual types of quadrilaterals encountered at school, namely the square, the rectangle, the rhombus, the parallelogram and the trapezium. The authors constructed a concept lattice in the sense of Wille [6], which represents the logical/conceptual structure of these types of quadrilaterals (figure 5).…”

“…Patronis and Spanos [3] explained this case as a result of a social and cultural process, in which an identification takes place, of a symbolic form with a geometrical object. More specifically, the symbolic form O (whose shape is in fact a square) has been used for several cultural reasons and has been habitually confused, under the name of 'rhombus', with the geometrical figure which bears the same name.…”

“…Spanos [1], Gagatsis and Patronis [2], and Patronis and Spanos [3] have considered various situations, in which there was a lack of understanding of the meaning of algebraic expressions and of terms used to denote classes of geometrical figures. These situations are described and analysed, in our model, as a result of the difference between the codifications of the pupils and those of the teacher: because of this difference, some expressions have a large semantic width-a technical term introduced in our model-which means that these expressions are difficult to understand because they are interpreted in many ways.…”

“…(ii) 16-year-oldpupiVs codifications: According to the results of [3], the set Z 2 of these codifications contains an 'irregular' case-which holds in fact for many pupils of this age-by which the Square is codified in two possible ways (for the same subjects): (a) as 'square', when the subject sees or imagines a square in its standard position (figure 9 (a)), and (b) as 'rhombus', when the subject sees or imagines a square rotated 45° from its standard position (figure 9 (6)).…”