According to Vergnaud's and Piaget's proposals, the study aims to describe children's conceptions of partitive division expressed on learning tasks of sharing numerical collections, and to identify the grasp of consciousness levels of the typical relations which define that kind of division. This process is supposed to be enhanced by tasks, which alternate moments of practical sharing of numerical collections, notation production and their interpretation. The microgenetic analysis of videotaped data describes the nature and the changes of six students' solutions (7 to 8 years old) to those tasks. These students were attending a State Elementary School and they performed the tasks in triads, during two sessions. Results show children's conceptions of partitive division disposed on two hierarchies, one for the practical executions and other for the notations. Both hierarchies give support to the identification of the grasp of consciousness levels of schemata and relations inherent to that type of division. The status of the sharing activities on the comprehension of partitive division and the role of the grasp of consciousness of actions in this conceptual learning are discussed.
construction of division; multiplicative structures; children's conceptions of division
