Proceedings฀of฀the฀28th฀Conference฀of฀the฀International฀฀ Group฀for฀the฀Psychology฀of฀Mathematics฀Education,฀฀2004฀ Vol฀4฀pp฀393–400 FROM FUNCTIONS TO EQUATIONS: INTRODUCTION OF ALGEBRAIC THINKING TO 13 YEAR-OLD STUDENTS Vasiliki Farmaki, Nikos Klaoudatos, Petros Verikios Department of Mathematics, University of Athens The various difficulties and cognitive obstacles that students face when they are introduced to algebra are well documented and described in the relevant bibliography. If it is possible, in order to avoid these difficulties, we have adopted the functional approach widening the meaning of algebraic thinking. In this paper, which is part of wider research, we concentrate on problems that are modelled by linear equation with the unknown on both sides. We investigate the advantages and disadvantages of the functional approach in the solutions of these kinds of problems. The findings from our research suggest that the functional approach indeed gives the beginners a satisfactory way of answering, while the typical solution by equation demands maturity on the part of the students and could be postponed for a later time. INTRODUCTION Research in teaching and learning algebra has detected a number of serious cognitive difficulties and obstacles especially to novice students, see for example Tall and Thomas (1991). One of the important themes that research has focused on is the solution of linear equations and problems related to them. Kieran (1997) and Sfard & Linchevsky (1994) have indicated difficulties related with the use and meaning of the symbol of equality, while Kieran (1985) and Kuchemann (1981) found misunderstandings about the use and meaning of letters, to name only a few. In the transition from arithmetic to algebra one of the important steps seems to be the solution of ax+b=cx+d and its variation ax+b=cx, Filloy and Rojano (1989), Herscovics and Linchevski (1994). This particular form of equation has been a subject of dispute in bibliography. For example, Filloy and Rojiano suggest that this equation demands teacher’s intervention ‘the didactic cut’, while Herscovics and Linchevski locate their argument in the student’s cognitive development. We accept the position of Pirie and Martin (1997) that rather than an inherent difficulty in the solution of linear equations, the cognitive obstacle is created by the very method, which purports to provide a logical introduction to equation solution. Our classroom experiences say that, indeed, this kind of equation puts a very heavy burden on the students. In this paper we adopted the functional approach to algebra which widens the meaning of algebraic thinking. Then, through problems which are expressed by equations of the form ax+b=cx+d or ax+b=cx, we examine the students’ solution processes by the two approaches, functional and letter-symbolic. Our goal is the Vol 4–26