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The construction of a numerical sequence requires an ordinal model of conceptual support that can be defined by the given relationships in a system of progressions. Taking into account the ordinal model of conceptual support leads us to a coherent conceptual/interpretative system which takes into consideration conceptions and beliefs regarding numerical sequencing that immediately takes us back to epistemological and didactic considerations. Epistemological approaches are confined to looking at the nature, origin and mode of existence of the natural number and elementary arithmetic, so that construction of the numerical sequence relies, at this point, on the conclusions that are drawn in regard to the said problem. [ABSTRACT FROM AUTHOR]

The object of this presentation consists of analyzing the knowledge that is put into play when the students need to give some validation of their productions. For it, we look for classes fragments on numerical interpretations in the first year of the basic education corresponding to data of an research project on the learning and the education of the numeration system directed by D. Lerner. It interests to us to reflect approaches of the impact on the infantile numerical learning that can get to have the validation processes, in which the students introduce themselves in the criteria search to settle down if their productions have been correct and about some educational interventions directed to install and to maintain such processes. [ABSTRACT FROM AUTHOR]