Knowledge distillation allows smaller neural networks to emulate the performance of larger, teacher models with reduced computational demands. Traditional methods for Large Language Models (LLMs) often necessitate extensive fine-tuning, which limits their accessibility. To address this, we introduce Trace-of-Thought Prompting, a novel framework designed to distill critical reasoning capabilities from high-resource teacher models (over 8 billion parameters) to low-resource student models (up to 8 billion parameters). This approach leverages problem decomposition to enhance interpretability and facilitate human-in-the-loop interventions. Empirical evaluations on the GSM8K and MATH datasets show that student models achieve accuracy gains of up to 113\% on GSM8K and 20\% on MATH, with significant improvements particularly notable in smaller models like Llama 2 and Zephyr. Our results suggest a promising pathway for open-source, low-resource models to eventually serve as both students and teachers, potentially reducing our reliance on high-resource, proprietary models. Our code, featuring data analytics and testing scripts, is provided here: https://github.com/traceofthought/trace-of-thought-prompting/tree/main.