Constructing the integral concept on the basis of the idea of accumulation: suggestion for a high school curriculum

Students have a tendency to see integral calculus as a series of procedures with associated algorithms and many do not develop a conceptual grasp giving them the desirable versatility of thought. Thus, instead of a proceptual view of the symbols in integration, they have, at best, a process-oriented view. On the other hand, it is not surprising that many students find concepts such as the integral difficult when they are unable to experience these processes directly in the classroom. With a view towards improving this situation, constructing the integral concept on the basis of the idea of accumulation has been proposed (Educ Stud Math. 1994;26:229-274; Integral as accumulation: a didactical perspective for school mathematics; Thessaloniki: PME; 2009. p. 417-424). In this paper, we discuss a curriculum that is based on this idea and a design for curriculum materials that are intended to develop an improved cognitive base for a flexible proceptual understanding of the integral and integration in high school. The main focus is on how we (mathematics teachers and mathematics educators) might teach the integral concept in order to help high school students to construct meaningful knowledge alongside acquiring technical abilities.