In this work we present the design of a computer-based environment aimed at engaging primary school children in argumentative practices while solving a story problem. Argumentation has been regarded from two points of view, both as a process and as a product. Students, using the collaborative tools offered by the platform, co-constructed shared reasoned answers. The focus of the activity is not only to reach the solution but above of to justify their thinking. We analyze the script with respect to the argumentative dimension and dimension of the social modes of co-construction, that is, with respect to the argumentative product and the process of co-constructing the argument that leads to the shared answer. So, we investigate the impact of the designed script on the evolution of students' productions of mathematical arguments, looking at the protocols of their interaction both in chat and in the forum. We discuss the outcomes and draw some conclusions with respect to the didactical potential for the emergence of germs of meanings and mathematical concepts.