ABSTRACT
This chapter argues that school learning is different to some extent from the spontaneous learning of the child in natural environment. The reader may find some resemblance between the term learning system (LS) and the terms microworlds or learning environments, which are uses freely in artificial intelligence and more recently in education. The ontological status of the mathematical knowledge that concerns requires discussion. Number words are usually used in ordinary language as concept signs, quantitative qualifiers of objects that are specifies by the class term. The educational enterprise strives to teach young children the rules of formation of mathematical language sentences; the sense of different sentences in mathematics, including new symbols that do not appear in ordinary language. The LS is bases on two components: an articulation of the unit of knowledge to be taught, bases on the expert's knowledge, referred to as the knowledge component of the system, and an illustrative domain, homomorphic to the knowledge component.
