Learning trajectory as a complex cluster of activity

International audience ; This paper presents a study of students' learning mathematics by modelling. In the study learning was conceptualized in terms of emergent modelling following the four-layer model by Gravemeijer et al. (Gravemeijer, 2020). Aim of the study was to inquire learning trajectories in the context of modelling with differential equations in 21 students' reports. Textual analysis of the reports implied the marking of notions and terms related to progressive, horizontal and vertical mathematising, followed by the linking of them with the four layers. The paper focuses on one case of students' modelling in one report. It demonstrates how the students' learning trajectory was identified and determined, for interaction between populations as an emergent sub-model of a differential equations model. Thereby it contributes to the discussion of i) conceptualizations of modelling and its relations to learning, and ii) methods for research in students' learning.