Prospective Mathematics Teachers’ Perceptions of Infinity and Their Academic Knowledge About the Arithmetic of Infinity

Abstract

This study examines prospective mathematics teachers’ perceptions of infinity and their academic knowledge of the arithmetic of infinity. A concurrent nested mixed-methods design was employed. The participants consisted of 100 volunteer third- and fourth-year prospective teachers enrolled in a mathematics teacher education program at a state university in the Aegean Region of Türkiye during the 2024–2025 academic year. Data were collected in a classroom setting via an online form comprising one open-ended question and 24 multiple-choice items. The open-ended question aimed to elicit participants’ conceptual perceptions of infinity, while the multiple-choice items assessed their academic knowledge of arithmetic operations involving infinity. In particular, participants’ understandings of operations involving  with , 1, 0, and π were examined separately. Qualitative data were analyzed using descriptive analysis, whereas quantitative data were analyzed using descriptive statistics. The findings indicate that prospective teachers’ perceptions of infinity are predominantly grounded in potential infinity rather than actual infinity. Although their overall performance on arithmetic operations involving infinity was above average, notable difficulties emerged, particularly in operations involving  and 0. The results further reveal that intuition-based perceptions of infinity negatively reflect prospective teachers’ procedural performance in operations involving infinity. When interpreted through the combined lenses of conceptual change theory and APOS theory, the findings suggest that prospective teachers have not successfully undergone a conceptual shift from potential to actual infinity and that their understanding of infinity has largely not been constructed at the object level.