Acta Didactica Universitatis Comenianae Mathematics, Issue 5, 2005 ARITHMETICAL AND ALGEBRAICAL APPROACH TO THE SECOND DEGREE EQUATIONS 1 (AGE: 15 − 19 YEARS) DANIELE BENFANTI, BENEDETTO DI PAOLA, SALVATORE RAIMONDI Abstract. The principal aim of this work is trying to observe/describe in which way the slovak students approach the study of algebra and in particular the second degree equa- tions, underlining some differences and similitude to the italian students` approach to the same subject. This study is inserted in a more big research started two years ago in Italy with a the the- sis of Benedetto Di Paola on Didactic of Mathematics in Palermo (Supervisors: Prof. T. Marino, Prof. F. Spagnolo) (http://math.unipa.it/~grim/). In his thesis, according to the idea that the algebraic thought developed in the History of the Mathematics only after the construction of the arithmetic thought (as for the Western culture), Benedetto Di Paola analyzed a particular didactic experience on the students of secondary school, according to the theory of situations of Guy Brousseau (Brousseau, 1997, 1998) with a semiotic re- vision in this connection with interpretations of phenomena of learning/teaching (Spag- nolo, 1998) as for the methodology of the study: analysis a priori and final results (quanti- tative analysis) of the presented subject. In this paper we want to analyze some of the results observed realizing the same didacti- cal experience with the Slovak students, during our permanence in Slovakia, and to com- pare these with some of the most interesting results of the research in Palermo. This work follows two essential lines: first of all we will briefly report some historical notes about the History of Algebra and therefore the development of the algebraic thought underling the importance of the historical point of view in an experimental research like this, secondly, in the second part of the paper, the most important one, we will analyse the didactical experience (problematic situation) on the italian 2 and Slovak 3 students, report- 1 This paper was created also with support of project Socrates Comenius 2.1 n.106663−CP− 1−2002−1−IT−COMENIUS−C21 2 Supervisor: T. Marino, F. Spagnolo in collaboration with the teachers Prof. Della Ratta, Prof. Messina, Prof. Briguglia and Prof. Mocciaro of the schools Liceo Classico Garibaldi, Liceo Scientifico Galileo Galilei, Istituto Tecnico Commerciale Pareto of Palermo, 2002