Emerging models for the slope of a curve: a reinvention activity for upper secondary school

Would it be useful to invert the standard teaching trend and start designing lessons and activities able to push students to "reinvent" a mathematical notion? The research described in this thesis partially answers to this question, focusing on the concept of "slope of a curve". The whole research investigates if an already planned and progressively improved task- whose design is based on the Theory of Didactical Situations (TDS) and Realistic Mathematics Education (RME)- has the potential to make relevant informal models of the concept of slope of a curve in one point emerge from the students' solution; the consequent question inquired is whether the teacher could effectively build a rigorous lesson and present the formal knowledge about the topic, starting from these models. The whole research, conducted in secondary schools in the Netherlands, was conceived in the context of the Erasmus+ project “MERIA” (Mathematics Education Relevant, Interesting and Applicable).