In the attempt to improve mathematical thinking for safeguarding our future societal needs, there is a worldwide tendency in schools to start training mathematical and arithmetical operations at an earlier age in children's development. Recent theoretical developments and empirical research have pointed to alternative ways of approaching early mathematical thinking. In these latter approaches, mathematical development in the early developmental stages is seen as an emerging process in the context of children's own activities that contributes to meaningful learning and stimulation of children's cultural identity (Bildung approach). The discussion between the training approach versus the "Bildung" approach is still intemperately going on. In this article, some outcomes of a research programme (based at the Free University Amsterdam) are discussed that present empirical studies and their theoretical background (cultural-historical theory, elaborated in an educational concept called "Developmental Education") that demonstrates the promising potentials of promoting mathematical thinking through supporting young children's appropriation of schematic representations and notations in the context of play. (Contains 2 footnotes.)