Visualizing the Matrix Product as a Transformation: A Task Design Using GeoGebra in Secondary Mathematics Education

TL;DR

This study demonstrates that integrating manual activities with GeoGebra visualization enhances secondary students' understanding of matrix multiplication as a geometric transformation, fostering conceptual and semiotic development.

Contribution

It introduces a GeoGebra-based instructional sequence grounded in the MWS framework, promoting a transition from algorithmic to conceptual understanding of matrices.

Findings

Students improved in semiotic coordination of representations

GeoGebra was effectively used as a cognitive tool

Students developed explanations of geometric effects of matrices

Abstract

The teaching of matrix multiplication in secondary education is often limited to the mechanical application of the row-by-column algorithm, leaving aside its interpretation as a geometric transformation. This study analyzes the impact of a GeoGebra-mediated instructional sequence, grounded in the Mathematical Working Space (MWS) framework, on students learning of the matrix product. Ten fifth-year secondary students from a school in Lima (Peru) participated in the study. The intervention was carried out over four sessions, combining manual activities with digital exploration using GeoGebra. The results show notable progress in students semiotic genesis, reflected in the coordination of algebraic, graphical, and numerical representations; in instrumental genesis, through the increasingly meaningful use of GeoGebra as a cognitive tool; and in discursive genesis, as students developed explanations of the geometric effects of matrices. A transition is observed from an algorithmic execution of matrix multiplication toward a conceptual understanding based on linear transformations. These findings suggest that task designs integrating manual work, dynamic visualization, and mathematical argumentation support deeper understanding of matrix multiplication and provide criteria for the reflective use of digital technologies in secondary linear algebra instruction.