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The present study explores pupils' constructed definitions of the concept of function in relation to their abilities in dealing with tasks of functions involving different forms of representations and problem solving tasks. A major concern is also to examine the interrelations between these three ways of thinking about or dealing with the concept of function. The sample of the study consisted of secondary school pupils in Cyprus. A test was developed which involved seven items: one item requested pupils to provide a definition of what function is and the other six items were developed in order to investigate pupils' ability to transfer information from one representation to another and to solve problems on function. Findings revealed pupils' difficulties in giving a proper definition for the concept of function and resolving problems on functions involving conversions between diverse modes of representation. Several inconsistencies among pupils' constructed definitions, their competence to use different representations of functions and their problem solving ability, were also uncovered, indicating lack of flexibility between different ways of approaching functions.

Uso del lenguaje natural y algebraico -Estudiantes E15 y E16……………………136 Uso del lenguaje numérico Estudiantes E23 y E24…………………………………136 Uso del lenguaje algebraico -Estudiantes E 11 y E12………………………………136 Uso del lenguaje algebraico -Estudiantes E9 y E10………………………………...136 Respuesta -Estudiantes E7 y E8…………………………………………………...137 Respuesta -Estudiantes E5 y E6…………………………………………………...137 Respuesta -Estudiantes E27 y 28…………………………………………..………138 Respuesta -Estudiantes E17 y E18………………………………………………...139 Respuesta -Estudiantes E3 y E4 …………………………………………………..139 Respuesta -Estudiantes E13 y E14……….

The Mathematics education community has long recognized the importance of diagrams in the solution of mathematical problems. Particularly, it is stated that diagrams facilitate the solution of mathematical problems because they represent problems' structure and information (Novick & Hurley, 2001;Diezmann, 2005). Novick and Hurley were the first to introduce three well-defined types of diagrams, that is, network, hierarchy, and matrix, which represent different problematic situations. In the present study, we investigated the effects of these types of diagrams in non-routine mathematical problem solving by contrasting students' abilities to solve problems with and without the presence of diagrams. Structural equation modeling affirmed the existence of two first-order factors indicating the differential effects of the problems' representation, i.e., text with diagrams and without diagrams, and a second-order factor representing general non-routine problem solving ability in mathematics. Implicative analysis showed the influence of the presence of diagrams in the problems' hierarchical ordering. Furthermore, results provided support for other studies (e.g. Diezman & English, 2001) which documented some students' difficulties to use diagrams efficiently for the solution of problems. We discuss the findings and provide suggestions for the efficient use of diagrams in the problem solving situation.