Children’s conceptualizations of infinity: the association of mathematical context and middle-grade students’ responses to tasks involving infinity

Abstract

The purpose of this study was to characterize students’ knowledge of infinity by investigating the following questions: 1. Do students’ responses to infinity tasks vary according to the numerical/geometric context of the task situation? 2. Do students’ responses to infinity tasks vary according to the aggregate/serial context of the task situation? 3. Do students’ responses to infinity tasks vary according to the convergent/divergent context of the task situation? A test of infinity conceptualizations, whose items assessed understanding of infinity in numerical and geometric situations, was administered to 31 6th grade students. Responses to items were rated as finitist, transitional, or infinitist. Students’ responses to geometric tasks were rated finitist more often than students’ responses to numerical tasks; students’ responses to aggregate tasks were rated finitist as often as students’ responses to serial tasks; and students’ responses to convergent tasks were rated finitist much more often than students’ responses to divergent tasks. No student gave infinitist responses to all tasks; several students gave finitist responses to all tasks. Many 6th grade students create viable concepts of infinity without the benefit of instruction. It was argued that the range of mathematics experienced by students accounts for the viability of conceptualizations of infinity found in this group.