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 relationship 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 objects. 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.