Multi-Stage Knowledge Injection for Mathematical Problem-Solving in Large Language Models

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Large Language Models (LLMs) have recently demonstrated remarkable capabilities across a wide range of domains, including education, programming, and scientific problem-solving. However, mathematical problem-solving remains challenging because LLMs often generate logically inconsistent solutions and are susceptible to error propagation across reasoning stages. To address this, this study introduces the Multi-Stage Knowledge Injection (MSKI) framework, a unified closed-loop system that synergizes structural decomposition with recursive knowledge injection across four distinct stages. The framework utilizes an invoke–verify–inject mechanism and Progressive Back-Translation (PBT) to dynamically refine intermediate reasoning steps and mitigate error propagation. Evaluated on public benchmark datasets using Gemini 2.5 Flash and GPT-4o Mini models, MSKI consistently outperformed baseline prompting methods. The framework achieved a peak accuracy of 89.71% under the MSKI-Iter3/4 configuration using Gemini 2.5 Flash, demonstrating that the structured contextual knowledge injection significantly improved mathematical correctness.