Teaching is one of nature’s remarkable inventions. Despite over two millennia of academic discussion about teaching, it remains in need of a broad scientific footing. The cognitive and developmental sciences, writ large, are candidates for such a footing. The purpose of this article is to go beyond teaching as knowledge transmission and present an expanded cognitive developmental account of teaching’s remarkable complexity. A minimalist and maximalist approach for describing teaching are presented, with a preference offered for the latter. It can serve as a map for backward engineering so as to get to the origins and developmental trajectories of the map of teaching’s components. An expanded mind-to-mind coupling version of the complexity of normative adult teaching is presented which includes stage-setting with emotion-, motivation- and mind-reading, organizing teaching sessions, scaffolding, detecting knowledge gaps and intervening by knowledge transmission strategies intended to reduce the gaps. In a backward engineering move, research is presented on infants’ abilities for some of what is presented for adults. Teaching is also prosocial, with heart-to-heart coupling. It is inextricably intertwined with teaching, and is another part of teaching’s complexity. Teaching’s prosociality is analysed via the economic concept of non-rivalry. It is shown that teaching is a form of helping behaviour but is not altruistic. Directions for further work in this maximalist approach to the scientific study of teaching are presented.