When `The Same' is not perceived as such: The case of infinite sets

Abstract

This paper demonstrates how research-based knowledge about students’ incompatible answers to different representations of the same task could be used in mathematics instruction. The `It's the Same Task' research-based activity is described; this activity encourages students to reflect on their thinking about infinite quantities and to avoid contradictions by using only one criterion, one-to-one correspondence, for comparing infinite quantities. This activity led the vast majority of participants to the realization that producing contradictory reactions to the same mathematical task is problematic and to avoid this contradiction by using the one-to-one correspondence as the unique criterion for the comparison-of-infinite-sets tasks.