The premise of this article is that cognitive development involves both conceptual and semiotic achievements. From this perspective, the authors emphasize the distinctness of the semiotic issues and develop a differentiated appreciation of the semiotic aspects of cognition, particularly in the field of elementary mathematical cognition. The authors provide semiotic analyses of the differences between counting, adding, and multiplying and of the conventional place-value system. The authors introduce the concept of the field of reference of a sign, the differentiation of (...) the field into foreground and background, and the dynamics within the field of reference. Finally, the authors relate these ideas to the dynamics between two dimensions of semiotic relations: the sign-referent dimension and the sign-sign dimension. (shrink)