In a previous paper, we proposed a characterization of algebraic reasoning in primary education, based on the onto-semiotic approach to mathematical knowledge and instruction, where we distinguish three levels of algebraization. In defining these levels we took into account the types of representations used, the generalization processes involved and the analytical calculation performed in mathematical activity. In this paper, we extend this previous model by including three more advanced levels of algebraic reasoning in order to analyze the mathematical activity carried out in secondary education. These new levels are based on the following considerations: 1) using and processing parameters to represent families of equations and functions; 2) the study of algebraic structures themselves, their definitions and properties.