ABSTRACT
In the past few years, we have conducted a series of studies investigating the development of the number concept from a conceptual change perspective (Vamvakoussi & Vosniadou, 2004, 2007, 2010). A key assumption of our theoretical framework is that students, before they are exposed to instruction about non-natural numbers, have already formed complex and relative coherent explanatory frameworks for numbers; these are tied around their knowledge and experience with natural numbers, which are shaped initially in the context of their socio-cultural environment and further confirmed and strengthened during the first years of instruction which focuses on natural number arithmetic. Among the background assumptions underlying students’ initial number concept is the idea that numbers are discrete, i.e. they obey the successor principle (see also Gelman, 2000; Smith, Solomon, & Carey, 2005). It is amply documented that the idea of discreteness constrains students’ understanding of the density property of rational and real numbers not only at elementary and secondary education (Hartnett & Gelman, 1998; Merenluoto & Lehtinen, 2002), but at tertiary education level as well (Giannakoulias, Souyoul & Zahariades, 2007; Tirosh et al., 1999). However, it is not the only constraint on students’ understanding of the
density of numbers. Consider the following examples, coming from individual interviews with ninth-graders, who were asked how many numbers there are between 3/8 and 5/8 (Vamvakoussi & Vosniadou, 2004).
