A problem-solving method for advanced mathematics based on large language models and knowledge graphs

Advanced mathematics is a foundational course for first-year students in science and engineering majors, playing a crucial role in their future academic development. However, existing intelligent problem-solving tools often rely on character recognition or database matching and struggle with generalization beyond known question types. Although Large Language Models (LLMs) exhibit promising zero-shot capabilities, their answers to mathematics problems frequently suffer from hallucinations, making them unreliable for tasks requiring strict logical reasoning. To address this, we propose a hybrid method that combines an advanced mathematics knowledge graph with LLMs to improve both accuracy and generalizability. A fine-grained knowledge graph was constructed based on textbooks, exam questions, and key concepts, encompassing definitions, theorems, formulas, and their interrelations. We then designed a Chain-of-Thought (CoT)-based framework to guide LLMs in structured reasoning using the knowledge graph. We tested our approach on a set of 26 advanced mathematics questions, covering derivatives, integrals, mean value theorems, multivariable calculus, and series. Our method achieved an accuracy of 81.48%, significantly outperforming the baseline LLM performance of 59.26%. These results demonstrate that integrating structured domain knowledge and reasoning pathways can effectively enhance LLM performance in advanced mathematics tasks and holds promise for educational and intelligent tutoring applications.