Design and implementation of a pedagogical layer for large language models for mathematics education

Abstract

This paper presents the design and theoretical validation of an intelligent agent aimed at improving mathematics learning in secondary education in Colombia. Despite the new models and opportunities introduced with the use of digital technologies, student performance in mathematics remains critical, particularly in public institutions in Cartagena, as evidenced by Saber 11 and PISA results. Large Language Models (LLMs) offer scalable tutoring capabilities, but off-the-shelf generalist models often lack pedagogical context, risking “overhelp” by providing direct solutions rather than instructional guidance. We introduce a novel intelligent agent that integrates a “Pedagogical Value Layer” (PVL) on top of a general-purpose LLM that constrains its generative behavior through an explicit pedagogical control layer designed to guide the learning process. The proposed system utilizes a four-level prompt architecture (general principles, local curricular context, instructor-defined strategies, and student preferences) to transform the model into an adaptive tutor without the need for expensive and complex fine-tuning. We implemented a prototype platform and conducted a comparative benchmark against a baseline model to evaluate this proposal. Results demonstrate that the PVL successfully suppresses direct answer-giving behaviors, ensuring the agent adopts a questioning approach to encourage deliberate reasoning and conceptual development, while incorporating local and content information for better contextualized interaction.