An Analysis on the Fundamental Concept of Levels and Sublevels in the History of Algebra

An Analysis on the Levels and Sublevels of Algebraic Reasoning in Education

Abstract

Based on the onto-semiotic approach to mathematical knowledge and instruction a characterization of algebraic reasoning in education has been proposed, distinguishing three levels of algebraization. These levels are defined taking into account the types of representations used, generalization processes involved and the analytical calculation at stakes in mathematical activity. In this paper we extend this previous model by including three more advanced levels of algebraic reasoning that allow to analyze mathematical activity carried out in education. These new levels are based on the consideration of 1) using and processing parameters to represent families of equations and functions 2) the study of algebraic structures themselves, their definitions and properties.