How to Act? A Question about Encapsulating Infinity

The focus of this paper is to outline briefly the key ideas of my presentation at the 12th SIGMAA on RUME conference. This paper addresses aspects of an on-going study of mine which attends to learners ’ conceptions of infinity as manifested in their engagement with variations of a well-known paradox: The Ping-Pong Ball Conundrum. The APOS Theory (Dubinsky & McDonald, 2001) postulates a framework for interpreting learners ’ understanding of two distinct ideas of mathematical infinity: potential and actual infinity. According to Fischbein (2001), potential infinity can be thought of as a process which at every moment in time is finite, but which goes on forever. Actual infinity is described as a completed entity that envelops what was previously potential. Through the mechanisms of internalisation and encapsulation, Dubinsky et al. (2005a,b) propose that learners construct meaning for the concept of infinity as a process (potential infinity) and infinity as an object (actual infinity). This paper takes a closer look at the specific features connected to infinity the object. Theoretical Background In the terminology of the APOS Theory (Dubinsky & McDonald, 2001), an understanding of a mathematical entity begins with an action conception, which is recognised by the need for a