In this paper, we focus on the inquiry-based approach at the elementary school when students try to resolve mathematics problems. After a brief survey about problem solving, problem posing and inquiry-based learning, we analyze two case studies in the French context with the framework of the learning by problematization (Fabre & Orange in ASTER 24:37–57, 1997) and the notion of potential of inquiry of a problem that we introduce. This way, we identify some conditions of possibilities of learning mathematics with inquiry at elementary French schools. The design of the situation chosen by the teacher, that must conduct students to doubt and enroll them on inquiry, plays a crucial role. The teacher must also be able to support students’ inquiry activity, which may require developing didactical skills through teachers’ training.