Abstract

This study explores the process through which students learn to articulate generalized functional rules using both natural and symbolic language. We employed a microgenetic analysis to characterize and describe cognitive changes in four Chilean third-grade students over six intervention sessions, during which they engaged in tasks involving functional relationships. Additionally, a test and interview were administered after the intervention to assess individual performance. Our analysis revealed how students progressively internalized signs in both natural and symbolic language to represent variability. In particular, the expression any number played a crucial role in enabling students to internalize the concept of an indeterminate quantity, which they later transferred to letters. Although students succeeded in internalizing these signs, they did not always use them correctly, especially in tasks with more complex patterns; nonetheless, understanding variability represents a fundamental initial learning milestone in the algebra learning trajectory. Our findings provide a deeper understanding of the learning process associated with these notations and offer valuable insights for the formulation of instructional strategies.