Dialogue between theories interpreted as research praxeologies: the case of APOS and the ATD

The notion of “praxeology” from the anthropological theory of the didactic (ATD) can be used as a framework to approach what has recently been called the networking of theories in mathematics education. Theories are interpreted as research praxeologies, and different modalities of “dialogues” between research praxeologies are proposed, based on alternatively considering the main features and proposals of one theory from the perspective of the other. To illustrate this networking methodology, we initiate a dialogue between APOS (action-process-object-schema) and the ATD itself. It starts from the theoretical component of both research praxeologies followed by the technological and technical ones. Both dialogue modalities and the resulting insights are illustrated, and the elements of APOS and the ATD that the dialogue can promote and develop are underlined. The results found indicate that a complete dialogue taking into account all components of research praxeologies appears as an unavoidable step in the networking of research praxeologies.