This paper aims to argue how education might be considered and practised if not under the logic of the formation of childhood. As such, it puts into question the traditional way of considering children as representing adults' opportunity to impose their own ideals, and considering education to be an appropriate instrument for such an end. More specifically, it considers how the purposes of practising philosophy with children might be affirmed as other than in the service of the social and political education of childhood. This complex issue calls for a redefinition, not only of philosophy and education, but also of childhood itself. Several ancient (Heraclitus) and contemporary (Deleuze, Lyotard) philosophical contributions are offered in order to reflect on new concepts and vocabularies for childhood. What they have in common is a non-chronological concept of childhood--one that considers the child under the sign of aion rather than chronos, and therefore as something inherently constitutive of human life, which therefore could never be abandoned, forgotten or overcome. As an example of this deterritorialisation of the relation between childhood and education, a practical project undertaken in a couple of public schools in the environs of Rio de Janeiro and its environs is presented, in which a strong emphasis is placed on the concept of the "experience of philosophical thinking". The paper unpacks each of these three terms--experience, philosophy, and thinking--appealing to Foucault, Deleuze and Hadot for conceptual reconstruction. In addition, some basic pedagogical assumptions that informed this project are presented in the context of two philosophers who inspired it--Socrates and Jacques Ranciere. The last section of the paper reflects on how the practice of mainstream schooling seems actually hostile to the experience of philosophical thinking, thus challenging the practitioners to encounter the pedagogical space of the mainstream as if it were possible to establish a new educational relationship to childhood there, and to work fully expecting what cannot be predicted.