Handling a language to think together in the classroom: The case of the notion of equivalence

This paper presents a research design carried out with fourth-grade class children. The study explores both an educational question and some theoretical considerations about the emergence of logical-mathematical modalities of thinking during activities involving collective linguistic handling. The activities are framed in an educational device whose core is the construction and handling by children of a procedural-algebraic language with a constant attention to the rela- tionship between syntax and semantics of that language. The activities focus on appropriation of the equivalence notion, considered by many scholars an important prerequisite for understanding high-level mathematics. The main results show how children, to find solutions to the problematic situation, in interaction and within a safe dialogic space for thinking, handle the linguistic ob- jects. It has been observed that the latter act as psychological tools allowing, in a continuous back and forth between syntax and semantics, children to arrive at an appropriation of the equivalence