The use of interactive visualizations to foster the understanding of concepts of calculus: design principles and empirical results

Calculus and functional thinking are closely related. Functional thinking includes thinking in variations and functional dependencies with a strong emphasis on the aspect of change. Calculus is a climax within school mathematics and the education to functional thinking can be seen as propaedeutics to it. Many authors describe that functions at school are mainly treated in a static way, by regarding them as pointwise relations. This often leads to the underrepresentation of the aspect of change at school. Moreover, calculus at school is mainly procedure-oriented and structural understanding is lacking. In this work, two specially designed interactive activities for the teaching and learning of concepts of calculus based on dynamic geometry software are presented. They accentuate the aspect of change and the object aspect of functions using a double stage visualization. Moreover, the activities allow the discovery and exploration of some concepts of calculus in a qualitative-structural way without knowing or using curve-sketching routines. The activities were used in a qualitative study with 10th grade students of age 15–16 in secondary schools in Berlin, Germany. Some pairs of students were videotaped while working with the activities. After transcribing, the interactions of the students were interpreted and analyzed focusing to the use of the computer. The results show how the students mobilize their knowledge about functions working on the given tasks, and using the activities to formulate important concepts of calculus in a qualitative way. Also, some important epistemological obstacles can be detected.