Infinity has played an important role in the history of mathematics. It has been a source of fascination but at the same time a challenging issue for mathematicians. Infinity is also an interesting and a problematic topic for children. There are many reasons in the children's difficulties of understanding of infinity. Dealing with infinity requires reasoning beyond concrete representations. As Falk stresses “it requires the intellect to extend beyond the imaginable and to reach much further than intuition.” (p.35). Kolar and Cadez (2012) attribute the difficulty in childrens’ understanding of infinity to its abstract nature. Infinity cannot be conceretely represented and visualized because it is difficult to relate to our real life experiences. Therefore we have to visualize it mentally. According to Fischbein, Tirosh and Hess (1979), the main source of difficulties related to the concept of infinity is the fundamental contradiction between this concept and our intellectual schemes, which are naturally adapted to finite realities.

Infinity is a very complex and abstract concept which has various aspects. Dubinsky, Weller, McDonald and Brown (2005) state that "key aspect of the concept of infinity has been the distinction between potential infinity, an ongoing activity that never ends, and actual infinity, a definite entity encompassing what was potential "(p. 341).  Students first acquire the understanding of potential infinity when they realize the endlessness of the counting process (Kolar and Cadez, 2012). On the other hand, in order to understand the actual infinity, one has to realize that infinity defines the process that creates infinite sets. In other words, never ending process of counting produces infinitely many numbers. In addition, Pehkonen, Hannula, Maijala & Soro  (2006) distinguish three levels of students understanding of infinity. The lowest level is when they do not understand infinity, but use only finite numbers. In the intermediate level, the students understand potential infinity, and use processes that have no end. Those students, who have reached the third level, are able to conceptualize actual infinity and the final resultant state of the infinite process. After a study with extensive number of middle school students, Pehkonen, Hannula, Maijala and Soro (2006) concluded that infinity of natural numbers is understood earlier than infinity of other subsets of rational numbers. Potential infinity is understood earlier than actual infinity. Despite these findings, they argue that infinity remains mysterious for most students throughout school years.

Although the concept of infinity is neither be concretely represented nor visualized, non-existence of a largest natural number is said to be intuitive.  Studies show that before any formal instruction on the concept of infinity, the majority of the children from age 8 on develop an understanding of the endlessness of numbers (Evans, 1983; Falk et al, 1986). Moreover, children as young as 11 years old are able to explain that there is not a biggest natural number and this is why the set of natural numbers is infinite (Singer and Voica, 2003). In light of these findings, this study is designed to determine whether Turkish middle school students develop an intuitive understanding of non-existence of the largest natural number and in turn, an understanding of potential infinity before any formal instruction on the topic. More specifically, the purpose of this study is to investigate the middle school students’ understanding of the distinction between very large numbers and infinity within the context of finite quantities. 

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