“Quaderni di Ricerca in Didattica (Mathematics)”, n. 21, 2011 G.R.I.M. (Department of Mathematics and Informatics, University of Palermo, Italy) 127 A study on the comprehension of irrational numbers Michael Gr. Voskoglou Professor of Mathematical Sciences Graduate Technological Educational Institute Meg. Alexandrou 1, 263 34 Patras, Greece e-mail: voskoglou@teipat.gr ; mvosk@hol.gr URL: http://eclass.teipat.gr/RESE-STE101/document Dr. Georgios Kosyvas Varvakeio Pilot Lyceum Mouson & Papadiamanti, 15452 Palaio Psychico, Athens, Greece e-mail: gkosyvas@yahoo.com Abstract Following a theoretical introduction concerning the difficulties that people face for understanding the structure of the basic sets of numbers, we present a classroom experiment on the comprehension of the irra- tional numbers by students that took place at the 1 st Pilot High School of lower level (Gymnasium) of Athens and at the Graduate Technological Educational Institute of Patras, Greece. The outcomes of our experiment seem to validate our basic hypothesis that the main intuitive difficulty for students towards the understanding of irrational numbers has to do with their semiotic representations (i.e. the ways in which we describe and we write them down). Other conclusions include the degree of affect of age, of the width of mathematical knowledge, of geometric representations, etc, for the comprehension of the irrational numbers. Key words: Irrational and real numbers, learning mathematics 1. Introduction The empiric comprehension of numbers by children is taking place during the pre-school age and it is based on their practical needs to distinguish the one among many similar objects and to count these objects (Gelman 2003). This initial approach of the concept of number helps children in understanding the structure of natural numbers. For example, it supports them to ‘build” the “principle of the next of a given number” and therefore to conclude the infinity of natural numbers (Hartnett & Gelman 1998). It also supports the de- velopment of strategies for addition and subtraction based on counting (Smith et al. 2005), the comparison and order among the naturals, etc. The above approach is strengthened during the first two years of school education, where the natural numbers constitute a basic didactic target. The decimals and fractions are introduced later, after the second year of primary school, while the nega- tive numbers are usually introduced at the first year of high-school. Mathematically speaking, the set Q of ra- tional is an extension of the set N of natural numbers that could be attributed to the necessity for subtraction and division to be closed operations. However Q is not simply a bigger set than N, but it actually has a com- pletely different structure. In fact, while between any two natural numbers there exist at most finitely many other natural numbers (i.e. N is a discrete set), between any two rational numbers there always exists an infi- nite number of other ones (i.e. Q is an everywhere dense set). It is widely known and well indicated by researchers that students face many difficulties for the compre- hension of rational numbers (Smith et al. 2005). It seems that most of these difficulties have to do with a false transfer of properties of natural numbers to the set of rational numbers (Yujing & Yong-Di 2005, Vam- vakousi & Vosniadou 2004 and 2007). For example, some students believe that “the more digits a number has, the bigger it is” (Moskal & Magone 2000), or that “multiplication increases, while division decreases numbers” (Fischbein et al. 1985). The idea of “the discrete” restricts also the understanding of the structure of rational numbers. In fact, many students believe that, as it happens for the natural numbers, the “principle of the next number” holds for the rational numbers as well (Malara 2001, Merenluoto & Lehtinen 2002).