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The goal of this research is to make a theoretical reflection about connections that contributes to the development of the model for mathematical connections originally proposed by Businskas, and extended with the contributions of other researchers. This model is used in the most relevant investigations in this field. As a consequence of the reflection, the introduction of a new type of connection is proposed. The context for the reflection was the transcription of several episodes of a lesson on derivative. We first analysed the connections given in this transcription grounded in a qualitative methodology based on a-priori categories given by the theoretical model. We then selected some paragraphs where the model proved to be ambiguous, and finally, argued the need to clarify or extend the categories in the model to deepen the study of the connections in the transcription. In particular, we propose metaphorical connections as a new type of connection.

A new view about connections: the mathematical connections established by a teacher when teaching the derivative