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We report a characterization of the reflections made by high school mathematics teachers when discussing their resolutions of three numerical tasks related to the concept of function understood as a mathematical procept. Thematic analysis was used to code and categorize the reflections. Six teachers with at least 5 years of professional experience and an average age of 30 years participated. The results show that the teachers reflected on their actions in the tasks; their content knowledge; their teaching practice; and their professional training. As a result of these reflections, they recognized the need to improve their knowledge and practice of teaching functions in a more proceptual sense. The results contribute to deepen in how to make teachers reflect and incorporate a much more structural vision of functions in their teaching practices and, at the same time, incorporate the importance and role of their professional training in their reflections.

A theoretical-didactic proposal based on variational ideas and the use of GeoGebra is described as a response to a local problem on the understanding of the concept of function and quadratic equation in pre-university students. The theoretical elements of this work are based on the understanding of concepts and the theory of semiotic registers and representations, while the methodological references were based on hypothetical learning trajectories and the use of dynamic software, considered as a heuristic resource. The proposal was validated by means of expert criteria. This elaboration contributes with a proposal that breaks with the classical schemes of presentation and treatment of the content, which favors through the incorporation of variational ideas generated with the dynamic software GeoGebra the understanding of the articulated concepts of function and quadratic equation in pre-university students.