Mathematical connections from a networking of theories between extended theory of mathematical connections and onto-semiotic approach

This article presents a networking of two theories, the Extended Theory of Mathematical Connections (ETC) and the Onto-semiotic Approach (OSA). In particular, concordances and complementarities are identified in the respective conceptions of mathematical connections, as a result of applying these theoretical frameworks to a student's response protocol when solving a task on the derivative. First the mathematical connections were identified from the perspective of the ETC and, second, an onto-semiotic analysis was carried out where the mathematical practices, the primary mathematical objects activated in them and the semiotic functions established between these objects were analysed. The main conclusion is that both theories complement each other to make a more detailed analysis of the mathematical connections. In particular, the more detailed analysis carried out with the OSA tools, visualizes a mathematical connection like the tip of an iceberg of a conglomerate of practices, processes, primary objects activated in these practices and semiotic functions that relate them, which enables a thorough analysis that details the structure and function of the connection.