Ambiguity as a cognitive and didactic resource

International audienceMathematics is often perceived and described as a domain of perfection, where polysemy is seen as dangerous and ambiguity is banned. But if that is true for the " final " dressing of official mathematics, in doing mathematics and, as we claim, in understanding it, things run the opposite way. Contrasting usual didactic practices, suitable forms of ambiguity can be seen and used as powerful cognitive resources. We present reflections on a long lasting didactic activity with students of different grades (from primary to university level), and show how to exploit some ambiguities (in particular linguistic ones) related to the notion of consecutive numbers