16. Opening an Exercise

This text presents two possibilities for transforming exercises of the school mathematics tradition into investigative activities related to landscapes of investigation located in the context of critical mathematics education (CME). This process of transformation was based on what Skovsmose calls “opening an exercise”. In order to open an exercise with reference to pure mathematics, one can explore the theme proposed in the exercise by going in different directions. In order to open an exercise with reference to semi-reality and reality, I have proposed the exploration of comments produced by students as they work through the exercise. A task related to such transformation was developed for prospective mathematics teachers in order to reflect on mathemacy, help students to make mathematical discoveries, and potentially cause changes in traditional teaching practices. The result of the transformation process carried out by the prospective teachers in the course indicates that the investigative activity should have a strong appeal to reality and semi-reality, besides mobilising other actions that are not developed when one solves an exercise. Due to the characteristics of the task, a new reference emerged: an imagined reality. The task and reflections represent the first step in the process of these prospective teachers entering CME.