Generalization as a basis for algebraic thinking: observations with 11-12 year old pupils

Generalization, intended as the ability to pass from the particular to the general or also the ability to see the general in the particular, represents a fundamental element of algebraic thinking that would otherwise be simply working with symbols. Moreover it is a common experience that often the pure technical aspect prevails in didactic practice and the operative ability sometimes hides the not understanding of the general significance of that which is being done. The bases of algebraic thinking are laid when the properties of the operations between numbers are learnt and one starts to work with symbols in various contexts (arithmetical, geometrical, data processing), but the acquisitions in this field must be considered as a gradual achievement and must be continually consolidated. Our research study proposes to determine the moments of origin of some of the difficulties that consequently cause errors and misunderstandings in working algebraically. This report refers to the level of generalization in 11-12 year old pupils, showing how the ability to grasp the generality of the result may be apparent and trying to understand when it corresponds to real awareness of general relationships