Radford, L. (2020). Play and the production of subjectivities in preschool. In M. Carlsen, I. Erfjord, & P. S. Hundeland (Eds.), Mathematics education in the early years. Results from the POEM4 conference 2018 (pp. 43-60). Cham: Springer. Chapter 3 Play and the Production of Subjectivities in Preschool Luis Radford 3.1 Introduction Play has always been a popular topic in early childhood education. And, one way or another, it has been associated with the more general question of children’s develop- ment. Indeed, despite the impressive variety of conceptions of play (see, e.g., Elkonin, 2005), play has usually been considered either as a source of development or as a window through which one can grasp the current state of the child’s development. In the latter view, play appears as a kind of methodological tool. This is the case of Piaget’s conception of play. In observing children play, the children’s under- standing of rules can be made apparent. Reasoning along this line, Piaget (1948) suggested a series of successive stages which children undergo in play: children travel from a motor or individual understanding of rules where the driven force is the child’s desires, to an egocentric stage where although playing together each child plays “on his own” (p. 16), to incipient and, later on, developed stages charac- terized by social forms of collaboration. In the former view, by contrast, play appears as something that can potentially influence the child’s development. For instance, Smirnova and Gudareva argue that “Play is of special importance for the formation of the child’s motivational sphere and voluntariness” (Smirnova & Gudareva, 2017, p. 252). This chapter is about children playing mathematical games in a preschool set- ting. However, it goes in a different direction. Indeed, in this chapter, I am not inter- ested in exploring how play allows children to develop mathematical ideas (the play-as-a-source view mentioned above that confines play to a mere facilitator of knowledge construction and intellectual growth). Nor am I interested in what we can learn about development in observing children play (the play-as a-window view 43