The construction of algebraic knowledge: towards a socio-cultural theory and practice Ferdinando Arzarello, Dept. Math., Univ. of Turin - Italy, Luciana Bazzini, Dept. Math., Univ. of Pavia-Italy, Giampaolo Chiappini, IMA-CNR, Genova, Italy Abstract. A theoretical model is sketched, which has been elaborated by the authors for analysing pupils’ activities of production and manipulation of algebraic formulas. The model is based on the distinction between sense and denotation of an algebraic expression and on the notion of conceptual frame: these allow describing the way pupils attach a meaning to the algebraic formulas as well as their major misconceptions. Successively, the relationships between the model and the learning environments are examined and a detailed description is given of those features able to support school activities which produce a meaningful learning of algebra. In the end, a few examples are discussed Introduction. Recent research in algebraic problem solving has identified as a key problem the relationship that students create between formulas and their meaning. The inadequacy of such a relationship often induces a void manipulation with symbols or an inefficient use of surrogates (see Sfard, 1991; Kieran, 1992 and 1994). There is general consensus that many students do not master the sense" of those symbols which they have learned to handle formally. Sometimes they do not only ignore the meaning of formulas and concepts, but even arrive to invent meanings which surrogate the authentic ones. Other students, even if good algebraic computers, do use algebra only as a computational machine and not as a tool apt to understand generalisations, to grasp structural connections and to argue in mathematics. From a didactic point of view, it is very hard to overcome such misconceptions and difficulties, mainly because the invented meaning often has its own justification, frequently rooted in previously learned models. It may happen that the teacher and the student use the same words which correspond to very different meanings in their heads; a genuine comedy of errors is thus generated: the pupil and the teacher enter into a vicious circle, which is very difficult to break. Existing literature has shown the possibility of taking instant pictures of students' difficulties but such a microanalysis focuses on short term phenomena and may be inadequate for studying long term cognitive processes of pupils engaged in the learning of elementary algebra. In fact, the unbalance of time scales between pupils and teachers (or school) is fundamental for featuring the dialectic between learning and teaching the symbolic language of mathematics, especially the algebraic one: without such an approach, it is also difficult to elaborate suitable suggestions for teaching. It is our concern to analyse algebraic thinking in the framework of a theoretical model we have elaborated on the ground of observed students' behaviours while solving problems. Starting from the consideration of students' difficulties, their typical