Infinite Iterative Processes: The Tennis Ball Problem

Kirk Weller, Cindy Stenger, Ed Dubinsky, Draga Vidakovic

Abstract

Iterative processes are central to the undergraduate mathematics curriculum. In [4], Brown et al. used a problem situation calling for the coordination of two infinite processes to analyze student difficulties in understanding the difference between the union over k of sets P({1,2,…,k}) (P is the power set operator) and the set P(N) of all subsets of the set of natural numbers.  In this paper we study students’ thinking about the Tennis Ball Problem which involves movement of an infinite number of tennis balls among three bins.  Here, there are three coordinations of infinite processes. 

As in [4], our analysis uses APOS Theory to posit a description of mental constructions needed to solve this problem.  We then interviewed 15 students working in groups on the problem and we detail the responses of five of them, which represent the full range of comments of all the students.  We found that only one student was able to give a mathematically correct solution to the problem.  The responses of the successful student indicated that he had made the mental constructions called for in our APOS analysis whereas the others had not.

The paper ends with pedagogical suggestions and avenues for future research.

Keywords

infinite iterative processes (26A18), APOS theory (97C50), countable sets (11B99), natural numbers (11B99), student learning and thinking (97C20)

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