Mario Pieri’s View of the Symbiotic Relationship between the Foundations and the Teaching of Elementary Geometry in the Context of the Early Twentieth Century Proposals for Pedagogical Reform

In this paper, we discuss a proposal for reform in the teaching of Euclidean geometry that exposes the symbiotic relationship between axiomatics and pedagogy. We examine the role of intuition in such reform, as expressed by Mario Pieri, a prominent member of the Schools of Peano and Segre at the University of Turin. We are mindful of centuries of attention paid to the notion of intuition by mathematicians, mathematics educators, philosophers, psychologists, historians, and others. To set a context for Pieri’s proposal, we seek only to open a small window on views of its pedagogical role, from primary education to university study that may have informed early 20th century efforts to improve the teaching of geometry at the secondary school level. Pieri addressed the topic of intuition in many of his axiomatizations, including those in projective geometry that was his main area of concentration in foundations. We focus here primarily on his axiom systems for elementary geometry, which embraced the transformational approach of Felix Klein’s vision for the subject. Our goal is to convey Pieri’s thoughts on how to integrate two types of intuition, denoted as sensible and rational, in endeavors to improve the teaching of the geometry of Euclid. We show how Pieri’s views on geometric intuition and pedagogical reform were either ignored or misrepresented in several notable publications at the turn to the 20th century. In particular, we give Pieri a voice in response to specific comments made in the early 1900s by Federigo Enriques, Ugo Amaldi, and Florian Cajori in widely circulated publications inspired by Klein.