Abstract
In this paper, we illustrate how we use the theory of conceptual fields in our research on the example of transitivity. Our aim is to highlight the interest of adopting a developmental point of view to study the epistemological and didactical questions linked to the emergence of a concept or a mathematical field. We illustrate this by studying the emergence of the notion of order in history, and by putting the results of the epistemological study in perspective with associated didactical issues. The central role of transitivity and the importance of considering the preorder relation, both from an epistemological and a didactical point of view, emerge from our study. After a quick presentation of the main concepts of the theory of conceptual fields that we use in this research, we present the main milestones of the epistemological study. We then show on the one hand that the notion of transitivity is encountered as early as primary school and plays a central role in the development of the enumeration scheme, and on the other hand that the properties of relations taught at the beginning of university find their roots in these first learnings and in return allow to enlighten them, this illustrating the dialectic between operative knowledge and predicative knowledge. The results of this exploratory research show the relevance of continuing this research by including secondary education (students aged 11 to 18).