Developing elementary algebraic reasoning through counting with patterns activities

The counting tasks with patterns are a privileged context for the development of algebraic reasoning. The onto-semiotic complexity of tasks allows the identification of study processes to improve performance in successive algebrization levels, from purely arithmetic manipulation to a practice with a consolidated algebraic level. The objective is twofold: on the one hand, to elaborate indicators that allow evaluating the performance of an operative and discursive practice in the task resolution of counting patterns. On the other hand, to propose a sequence that takes into account the epistemological, cognitive, and teaching restrictions common to these practices. A pre-test and post-test design is proposed using an open-ended questionnaire. A qualitative analysis is carried out through a case study, which provides relevant information on the development of the subjects' algebraic reasoning in the successive algebrization levels. It is concluded that Flexible Mathematical Thinking is a key indicator of the subjects' algebraic competence; we also provided a teaching sequence that start from a basic strategy based on concrete manipulations and close generalizations to the analysis of different analytical expressions.