The aim of this paper is to build up an argument about the importance of a mathematical analysis of young children's activity in relevant for the age educational tasks. Most of current approaches (psychological, social, and pedagogical) are limited to the study of the development of children's thinking, paying less attention to the involved mathematical concepts. In the paper these approaches are briefly presented and an attempt is then made to analyse the mathematical activity within and beyond them. Finally, implications and some examples from a program of early mathematics aimed at developing authentic mathematical activity are provided. [For the complete proceedings, see ED597799.]