My argument for an empirical and normative model of children's development applicable to reasoning in mathematics is in three parts. Part I is a review of the evidence from a recent study of the development of young children's (aged 5-7 years) reasoning by mathematical induction. Part II outlines an interpretation in terms of an inclusive unit of analysis, which is the act of judgement combining causal facts and normative facts. Norms are manifest as rules, obligatory commands, and technical directives about "what has to be done" and "what has to be". Normativity has been marginalised, even omitted, from most other accounts of children's development and education. Causal regularity is not the same as normative regulation. The final part is a review of five educational implications, covering reasoning by induction in the mathematics curriculum for children, diagnostic assessment of both adults' norms and children's norms in children's reasoning, formative assessment through normative regulation, and critical methods in investigating children's norms. Underlying all the three parts are key principles of Piaget's constructivism about norms at work in human minds in action.