Rebuilding Number Sense: Teaching Euclid’s Algorithm and LCM through Guided Practice in Prospective Lower Secondary Mathematics Teachers

Abstract

This study analyses the impact of two guided-practice-based teaching strategies —rediscovering the Euclidean Algorithm and Adding Fractions with Meaning— on the conceptual understanding and development of didactic-mathematical competencies in 30 future mathematics teachers in Panama. The intervention was grounded in the Didactic-Mathematical Knowledge and Competencies model (DMKC) and implemented over five sessions within a university-level Mathematics Didactics course linked to the broader formative framework of the EMAS program. The study aimed to address the gap between procedural repetition and structural understanding, particularly in topics such as greatest common divisor (GCD) and least common multiple (LCM), which are central to the teaching of fractions, divisibility, and divisibility, and numerical reasoning in lower secondary mathematics. Initial results showed that over 70% of the participants were unable to justify concepts such as multiple or common divisor, revealing relevant conceptual weaknesses. Throughout the intervention, participants used pseudocode and solved contextualized problems. By the end of the third session, 666% had constructed a functional pseudocode for the Euclidean Algorithm, and in the post-test, 73% correctly solved fraction problems using the lcm, WITH 64% clearly justifying their procedures. Qualitative findings indicate strong pedagogical appropriation: 93% of participants proposed meaningful didactic applications for lower secondary mathematics classrooms. The data suggest progress in multiple dimensions, particularly in supporting the integration of computational thinking and historical algorithms as formative resources in mathematics prospective mathematics teachers. This intervention forms part of a broader doctoral research project aimed at strengthening mathematics teacher education in Panama. Its findings contribute to the redesign of formative cycles within EMAS and to the development of a blended specialization in mathematics didactics that integrates inclusive principles (CID-DUA), computational thinking, and frameworks such as TPACK-XK and the Theory of Instrumental Genesis.