Argumentation is an important aspect in the field of education because of its impact on learning processes. At the same time, argumentation is a complex activity in terms of cognitive, relational, emotional and social dynamics. In this paper, I investigate and I describe possible difficulties encountered by children during the argumentative process. The study involves 25 preschool children at a kindergarten engaged in three building tasks. The tasks were video-recorded and the argumentative discussions transcribed. For the aim of this paper, I analyze how argumentation are distributed among participants. I select interactions in which participants apparently do not argue or there are differences in the degree of argumentative participation between participants of the same group. I analyze these interactions and moments of impasse in the argumentative steps. The findings show how the simplicity of solving the task (e.g., when children do not encounter any problem in completing the activity) and the children’s self-perception of their competences in solving the task may have an impact on argumentation activities. Moreover, this perception is co-constructed by children within the interaction. The study contributes to the line of research on designing argumentation and highlights the role of the adult in managing children’s interactions.