Taming Fantastic Beasts of Mathematics: Struggling with Incommensurability

According to commognitive conceptualization, development of mathematical thinking, whether historical or ontogenetic, requires periodic transitions to mathematical discourse incommensurable with the one that has been practiced so far. In this new discourse, some familiar mathematical words will be used in a new way. Historically, such discursive transformations were usually greeted with resistance and led to heated, centuries-long debates. Also for today’s learners, a discourse incommensurable with the one currently in use constitutes a source of much trouble. This is particularly true in those cases in which the change in the use of words occurs tacitly, leading to apparent paradoxes. In this paper, I argue that discourses of finite and infinite sets are mutually incommensurable, and thus the case of students grappling with Sierpiński triangle (ST) may lead to insights about ways in which learners act in the face of incommensurability. Here, possible sources of the confusion reported by the participants are identified with the help of specially designed discourse-analytic tools. It is shown that the students, imperceptibly to themselves, oscillate between the discourses of area-as-a-segment-of-a-plane and of area-as-a-number. The analysis is followed with discussion on theoretical, methodological and practical implications of this study.