This paper aims to shed light on an overlooked but essential aspect of informal reasoning and its radical implication to mathematics education research: Decentralising mathematics. We start to problematise that previous studies on informal reasoning implicitly overfocus on what students infer. Based on Walton's distinction between reasoning and…

The author lists and discusses each of six stages of mathematical learning. The stages are: (1) interaction, (2) rule construction and manipulation, (3) isomorphisms, (4) representations, (5) symbolization, and (6) formalization. Examples from geometry are used to demonstrate the learning processes. (JG)

The theoretical and reference frameworks developed in a 1982 study proved effective for developing a constructivist approach leading children to build a meaningful and efficient symbolism of number. Results of a three-year longitudinal study are described. (MNS)