"Long term" refers necessarily to a developmental perspective: between the first competencies acquired by young children, at the age of five or six, and those acquired by fifteen year-old adolescents (only part of them), there are several steps and processes: continuities and discontinuities can be observed along this development. "Short term" concerns situations that are likely to be offered to students, for them to learn, at some crucial moment of this development. It is therefore concerned by the discovery, even partially, of new properties by students, owing to the teacher's help in conducting and facilitating the acquisitions course. The additive structures, for example, provide many different situations in which a particular reasoning and the choice of an addition or subtraction operation are clearly more delicate than in other situations: a case that requires a specific scenario, the assistance from the adult and, eventually, an original symbolic representation. The same kind of phenomena takes place with multiplicative structures, geometry, algebra, and other domains, far from mathematics, like morals and text comprehension. The present article limits this discussion to the case of mathematics learning.
development of mathematical competencies; mathematics teaching and learning; additive and multiplicative structures; mathematical representations; conceptual fields
