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Consistency and Representations: The Case of Actual Infinity

Pessia Tsamir, Dina Tirosh 

March 1999, Volume 30, Issue 2, Page 213

Abstract:
In this article we demonstrate how research-based knowledge about students' incompatible solutions to various representations of the same problem could be used to raise their awareness of inconsistencies in their reasoning. In the first part of the article we report that students' decisions as to whether 2 given infinite sets have the same number of elements depend on the specific representation of the infinite sets in the problem. We used these findings to construct an infinite-set activity with the aim of encouraging students to reflect on their own thinking about infinity. The findings indicate that taking part in this activity led a number of the participating students to realize that producing 2 contradictory reactions to the same mathematical problem is problematic; yet, few chose to avoid these contradictions by using 1-to-1 correspondence as a criterion for comparing infinite sets.

Keywords:
Research Summaries
Algebraic Patterns
Article
Conceptual Development
Higher Education
Grades 9-12
Teacher
Problems
Problem Solving / Problem Posing
Representation
Number Patterns
Number Theory