The notion of proof is a pivotal issue in mathematical teaching. In France, school curricula have been associating this notion with the term Demonstration. The term Preuve is lesser used. In French, the words démontrer, justifier, prouver are considered equivalents, but some researchers emphasize differences between them. For indeed, Balacheff and Soury-Lavergne (1996) consider that a proof is an accepted explanation, whereas the word Démonstration designates an explanation accepted by mathematicians because of its particular structure. This polysemy could have an impact on mathematical learning, especially in ultramarine contexts, where learners are exposed to multicultural and multilingual environments but are asked to follow the same curricula as in mainland. Following our hypothesis, we conduct a longitudinal survey in a French ultramarine context (the French West Indies): 168 students are questioned about their conception of the meaning of the terms Demonstration and Justification at three steps of their course. They were also questioned about the evolution of their ability to argue, reason, and write a proof. We observed the evolution of their abilities. We also established causal links between the various observed conceptions of students on the meaning of these terms and the evolution of their capacity. The results we obtained suggest that there is some adequacy between these conceptions and perception of how their ability to demonstrate a result in mathematics evolve.