Envisioning the infinite by projecting finite properties

We analyze interviews with 24 post-secondary students as they reason about infinite processes in the context of the tricky Tennis Ball Problem. By metaphorically projecting various properties from the finite states such as counting and indexing, participants envisioned widely varying final states for the infinite process. Depending on which properties they attended to, some participants recognized contradictions arising from these infinite projections and sought to resolve the contradictions in non-dismissive ways. The findings indicate that increasing their meta-level awareness of the features of their mental constructions may help students overcome common difficulties with the infinite.

Research highlights▶ Details of mental acts involved in reasoning about infinite processes. ▶ Attending to labeling or counting leads to varied envisioned final states. ▶ Properties projected to final state influence subsequent cognitive conflict. ▶ Meta-level awareness of projections influences modes of contradiction resolution.