Semiotic representations have been an important topic of study in mathematics education. Previous research implicitly placed more importance on the development of institutional representations of mathematical concepts in students rather than other types of representations. In the context of an extensive research project, in progress since 2005, related to modelling mathematical situations in Québec secondary schools (grades 8 and 9), we have addressed the problem of constructing a specific mathematical concept: covariation between variables as a prerequisite for the concept of function and its graphical representation. However, our research differs from previous studies as we attempt to take into consideration, in a cultural semiotic perspective, the spontaneous non-institutional representations that students produce when solving a problem situation in mathematics. We report our results with a group of students in grade 9, discussing the evolution of the representations the students produced to solve a problem situation, and the key role that the concept of covariation seems to play in helping students grasp the graphical representation of functions. We also discuss the different stages of the teaching method used, based upon collaborative learning, scientific debate and self-reflection (the ACODESA method of teaching) which aims to help the students acquire a cultural semiotic system.