The main focus of this paper is on the study of students' conceptual understanding of two major concepts of Set Theory--the concepts of inclusion and belonging. To do so, we analyze two experimental classroom episodes. Our analysis rests on the theoretical idea that, from an ontogenetic viewpoint, the cognitive activity of representation of mathematical objects draws its meaning from different semiotic systems framed by their own cultural context. Our results suggest that the successful accomplishment of knowledge attainment seems to be linked to the students' ability to suitably distinguish and coordinate the meanings and symbols of the various semiotic systems (e.g. verbal, diagrammatic and symbolic) that encompass their mathematical experience.