Fractal geometry in secondary education: characterization of mathematical and didactic-pedagogical knowledge through continuing teacher training

Abstract

This study examines the teaching of fractal geometry in secondary education and its integration into the professional development of mathematics teachers in Uruguay. A sequence of activities was designed to develop the mathematical and pedagogical knowledge required to teach fractals, including applications such as multiband fractal antennas based on self-similarity. The sequence was implemented in two online professional development courses in 2023 with 83 teachers, with revisions in the second edition emphasizing the formalization of intuitively taught concepts. The activities received positive evaluations and were effective even for teachers with no prior knowledge of fractal geometry. Teacher responses were analyzed using the Mathematical Knowledge for Teaching (MKT) framework and its extension MTSK, focusing on levels of formalization and mathematical richness. Teachers showed significant improvement in addressing key concepts such as self-similarity and fractal dimension. A Goodman–Kruskal bivariate analysis revealed a strong association between common and curricular knowledge, indicating that teachers successfully connected prior knowledge with official curricula. These findings support the inclusion of fractal geometry in secondary education.