On the epistemological interplay among three realities: The Geometric, the Physical and the Algebraic (GPA). - Moreno-Armella, Luis (2026)
Our interests in cognitive and epistemological issues were born from our lived experiences in the classroom as observers of the student's encounter with mathematics. We attend to two essential categories of reality: the mathematical reality, this assumed-to-be-shared among the mathematics community at large and the realities inferred from the students’ actions by an observer. We offer a theoretical model accompanied with episodic illustrations of a particular way of thinking, that of the habit of constructing the three mathematical realities, the Geometric, the Physical and the Algebraic. This way of thinking is largely absent from students’ repertoire of reasoning thus presenting an instructional challenge. Even more challenging is the construction of a suitable Semanticreality, that which coordinates the relationship among the three realities. The environments where these experiential realities can develop and mature are called local organizations, where students can begin to unveil the mathematical meaning of their tasks and gradually refine their understanding. Local organizations work as early versions of ways of thinking where deduction play a central role. Before concluding the paper, we offer examples from Calculus, cardinality of infinite sets, and linear algebra of the pedagogical manifestations of the GPA model.