Accommodation in the formal world of mathematical thinking

Abstract

In this study, we examined a mathematician and one of his students' teaching journals and thought processes concurrently as the class was moving towards the proof of the Fundamental Theorem of Galois Theory. We employed Tall's framework of three worlds of mathematical thinking as well as Piaget's notion of accommodation to theoretically study the narratives. This paper reveals the pedagogical challenges of proving an elegant theory as the events unfolded. Although the mathematician was conscious of the students'abilities as he carefully made the path accessible, the disparity between the mind of the mathematician and the student became apparent.

Fostering Inquiry and Creativity in Abstract Algebra: The Theory of Banquets and Its Reflexive Stance on the Structuralist Methodology

“…If research results may be expressed as above using theoretical constructs from Fischbein and Tall, the framework of the objects-structures dialectic allowed a finer-grained analysis of the interplay between the formal, symbolic and embodied components of mathematical activity than previous accounts (e.g. Stewart and Schmidt, 2017).…”

“…The conceptual aspects include concept formation but also the structural horizon of the structuralist theorem of decomposition of objects into simpler ones: in the case of banquets, the decomposition of a banquet in a disjoint union of tables. This is a clear illustration of the role that conceptual embodiment, in the sense of Stewart and Schmidt (2017), may play on the journey to the so-called formal world (in other words, structuralist thinking).…”