From Students’ Ways of Thinking to Meaning Construction: Hermeneutics Perspective of the ε-δ Definition

Abstract

The  definition is widely recognized as one of the most conceptually demanding topics in differential calculus and introductory real analysis, as students often rely on intuitive notions of closeness without fully grasping its formal logical structure. Building on prior findings on students’ ways of thinking, this study investigates how such ways of thinking contribute to the construction of mathematical meaning related to the  definition. This study employed a qualitative hermeneutic design involving written responses, task-based interviews, and reflective memos from 28 undergraduate students enrolled in calculus and introductory real analysis courses. Data were analyzed through a hermeneutic cycle consisting of naïve reading, selective highlighting, thematic interpretation, and the integration of students’ interpretations with the formal mathematical meaning of the  definition. The findings reveal three dominant interpretive themes: distance meaning, where  and  are understood as measures of closeness; responsive dependency, where δ is interpreted as a response to a chosen ; and control meaning, where  functions as a mechanism for regulating input proximity. Persistent challenges include equality, dependency and confusion in quantifier sequencing. This study contributes by extending students’ ways of thinking into a hermeneutic account of meaning construction, providing an epistemic basis for future didactic transposition studies. The findings also suggest instructional approaches that explicitly address distance, dependency, and logical sequencing in teaching the  definition.