In this article we share the story of a proposal for permanent training for mathematics teachers who are in charge of the training of students between 12 and 18 years old. We present a task and the fundamentals that lead us to develop it in this training scenario, for the collective construction of specialized knowledge of those who teach topics related to metric geometry and its relationship with optical illusions. We comment on the planning of the scenario and the results achieved from the perspective of the Mathematics Teacher?s Specialized Knowledge Model (MTSK Model),which we relate to studies on the use of technologies in mathematics classrooms. We conclude with an analysis of the strengths and weaknesses, opportunities and threats that may arise in scenarios such as the one described for the permanent training of teachers considered teaching professionals.