A principle of concretization permanence for the formation of arithmetical concepts

The celebrated principle of the permanence of the rules of calculation is contrasted here with another principle, called concretization permanence principle (c.p.p.). Both apply to situations where some arithmetical operation known to children for numbers of a certain kind is to be extended so as to include further numbers (these numbers are already known to the children). In such a case, c.p.p. advises that a suitable concretization schema and a suitable paradigmatic example be chosen and then extended so as to include numbers of the broader domain; students should explore what the given concept becomes in the new situation and in this way find values of the extended operation for a sequence of examples with fixed concretization (or fixed concretization schema) and varying numbers.