Pre-service mathematics teachers’ intra-mathematical connections: An onto-semiotic analysis of explanations of exponential and logarithmic functions

Abstract

This study analyzed the intra-mathematical connections activated by pre-service mathematics teachers when explaining exponential and logarithmic functions in a Colombian university context. The research was theoretically grounded in the Extended Theory of Connections and the Onto-semiotic perspective, which allow the analysis of relationships established between mathematical concepts, procedures, meanings, and representations during instructional processes. Using an exploratory qualitative comparative case-study design involving two purposively selected pre-service mathematics teachers, the study was conducted in four stages: lesson design, case selection, participant observation in a Didactics of Calculus course, and thematic and onto-semiotic analysis through the networking between ETC and OSA. The findings revealed an interconnected network of mathematical connections, including instruction-oriented, meaning, feature, part–whole, procedural, implication, reversibility, and representational connections. These connections showed how the PMTs coordinated mathematical objects, processes, and representations when explaining exponential and logarithmic functions. Particularly, algebraic, graphical, tabular, and verbal representations supported the construction of inverse relationships. Reversibility emerged as the most structurally complex connection and was manifested through recognition, procedural, and structural forms. Rather than confirming a meta-connection, the findings suggest that reversibility may function as an integrative connection, coordinating meanings, procedures, properties, representations, and logical reasoning. The study concludes that the analysis of mathematical connections provides a valuable framework for strengthening didactic-mathematical knowledge in initial teacher education and understanding how pre-service teachers communicate mathematical meanings during teaching processes.