It is one of our basic theoretical

It is one of our basic theoretical assumptions that the assortment of perspectives available to the student grows gradually, roughly following the logical-historical path presented in Table II. The hierarchical structure of algebra and of its different interpretations makes this conjecture fairly plausible. A direct jump, say, over the wide gap separating functional algebra from operationally interpreted algebra may end in broken bones. In the discussion that follows, we shall focus our attention on two critical junctions in school algebra: first, we shall consider the transition from purely operational conception of a symbolic formula to the dual process-product interpretation (from cell 1.1.1. to 1.2.1. in Table II); second, we will investigate the passage from here to the functional approach (to cell 1.2.2.). We shall make an effort to find out how difficult these two steps are for the pupil and what phenomena may be regarded as symptoms of such difficulties. In the last section, the looking glass will be turned at the final outcome of schooling: the question of versatility and adaptability of the conceptions with which the students typically leave the school will be addressed