Exploring Irrational Number Learning Using the Plithogenic Hypothesis Method: Integrating Incommensurability, Decimal Representations, and Cognitive Conflict

Abstract

This dissertation aimed to formulate a theory regarding high school students' learning of irrational numbers with the plithogenic hypothesis method as a modeling instructional approach for incommensurability, decimal expansions, and cognitive conflict. Using a concurrent Design-Based Research (DBR) and Grounded Theory (GT) method with twelve eleventh-grade students during 2025 in Bogotá, Colombia, four didactic interventions occurred addressing incommensurability, geometric representations, approximated sequences, and interdisciplinary uses; data were analyzed through systematic coding and plithogenic matrices, with results presented through truth, uncertainty, and falsity. Conclusions revealed that incommensurability (60% consensus) serves as a conceptual anchor to justify subsequent understandings, decimal expansions (50% dissent) cause misconceptions when not applied to real-life situations, and cognitive conflict (90% consensus) promotes conceptual change; additionally, five categories emerged from the consequent treatment of data including incommensurability, geometric representation, numerical continuum, discursive expansion, and interdisciplinary use each defined along four levels of conceptual understanding. Ultimately, the plithogenic hypothesis method serves as an appropriate modeling transfer for uncertainties in learning irrational numbers, and this study provides a clear guide for implementation without the compartmentalized plithogenic hypothesis method transfers found in current literature with the potential for application in mathematics curriculum as well as teaching and formative assessments.