Epistemological Analysis of Multiple Representations in Secondary Mathematics: The Case of the Function Concept

Abstract

Mathematical concepts are inherently inaccessible to learners except through their representations. Yet what does it mean to "represent" a mathematical object, and why does the movement between different forms of representation constitute such a persistent site of difficulty? This paper presents a sustained theoretical analysis that integrates Duval's theory of semiotic representation registers with Bachelard's notion of epistemological obstacles, using Sfard's reification framework as a bridging construct. Taking the function concept in secondary mathematics as a case study, we argue that representation conversion is not a technical skill superimposed upon conceptual understanding but is constitutive of mathematical thinking itself. We analyze how the function is expressed across five semiotic registers (natural language, algebraic-symbolic, graphical, tabular, and dynamic-technological) and identify four epistemological obstacles that emerge during conversion: the variable obstacle, the graph-as-picture obstacle, the formula-function conflation, and the correspondence-dependence tension. We further ground this analysis in specific examples drawn from the Chinese People's Education Press (PEP) textbook series and the 2022 Chinese mathematics curriculum standards, examining how curricular choices generate particular epistemological conditions. The paper contributes an integrative theoretical framework and a series of philosophical reflections on the nature of mathematical representation, concept formation, and the irreducibility of semiotic mediation to mathematical knowing.