Operative invariants of algebraic pattern present in the strategies of students of the 3rd grade of elementary school

Background: One of the great challenges for mathematics education in the 21st century is to alleviate the difficulties of students in the transition from arithmetic to algebra. There is already a consensus that there should not be a transition, as several experts have indicated algebraic approaches since the early years of schooling. Objectives: This study aims to describe and analyze the operative invariants of algebraic patterns present in the strategies of students of the 3rd grade of an elementary public school in the countryside of the state of Rio Grande do Sul. Design: The methodology used in this research was the clinical method of manipulation-formalization, created by Jean Piaget and applied in several of his studies. Setting and Participants: Students of the 3rd grade. Data collection and analysis: Clinical interviews. Results: We start from the assumptions of Gerard Vergnaud’s theory of conceptual fields to analyze the strategies used by the research participants. Conclusions: We identified four operative invariants: the theorems-in-action “count the places each time a table is introduced” and “add two places each time a table is introduced”, respectively linked with the concepts-in-action “putting the tables together” and “place at the ends of the tables”.