“I thought points were just... there”: A point-free path from hands-on regions to points fostering spatial reasoning and axiomatic thinking

In this study we investigate how a point-free geometry approach, grounding geometric learning in solid regions rather than abstract points, can support the development of spatial reasoning and promote what we call figural-conceptual covariation (Miranda, 2026) across different geometric axiomatisations. Drawing on a curricular path designed for both primary teacher education students and secondary school pre-service teachers, we explore how students transition from informal, hands-on manipulation of physical objects (solids, paper folding, stacking) to a formal, axiomatic understanding of geometry that starts from regions rather than points. The research is situated within point-free geometry, an emerging tradition which founds Euclidean geometry on notions of region, inclusion, and connectedness, following the philosophical tradition of Whitehead (1919, 1920) and the mathematical formalizations of Tarski (1929), Gerla and Miranda (2004, 2008, 2020). From a didactic standpoint, this approach addresses a well-documented gap between students' spontaneous, spatial intuitions and the classical axiomatic presentation of geometry that begins with the dimensionless point, a concept epistemologically distant from everyday experience (D'Amore et al., 2009; Sbaragli et al., 2004).