As Large Language Models (LLMs) scale up and gain powerful Chain-of-Thoughts (CoTs) reasoning abilities, practical resource constraints drive efforts to distill these capabilities into more compact Smaller Language Models (SLMs). We find that CoTs consist mainly of simple reasoning forms, with a small proportion ($\approx 4.7\%$) of key reasoning steps that truly impact conclusions. However, previous distillation methods typically involve supervised fine-tuning student SLMs only on correct CoTs data produced by teacher LLMs, resulting in students struggling to learn the key reasoning steps, instead imitating the teacher's reasoning forms and making errors or omissions on these steps. To address these issues, drawing an analogy to human learning, where analyzing mistakes according to correct solutions often reveals the crucial steps leading to successes or failures, we propose mistak\textbf{E}-\textbf{D}riven key reason\textbf{I}ng step distilla\textbf{T}ion (\textbf{EDIT}), a novel method that further aids SLMs learning key reasoning steps rather than mere simple fine-tuning. Firstly, to expose these crucial steps in CoTs, we design specific prompts to generate dual CoTs data with similar reasoning paths but divergent conclusions. Then, we apply the minimum edit distance algorithm on the dual CoTs data to locate these key steps and optimize the likelihood of these steps. Extensive experiments validate the effectiveness of EDIT across both in-domain and out-of-domain benchmark reasoning datasets. Further analysis shows that EDIT can generate high-quality CoTs with more correct key reasoning steps. Notably, we also explore how different mistake patterns affect performance and find that EDIT benefits more from logical errors than from knowledge or mathematical calculation errors in dual CoTs\footnote{Code can be found at \url{https://github.com/C-W-D/EDIT}}.

使用基于错误的驱动关键推理步骤蒸馏（EDIT）方法，可以更有效地帮助小型语言模型学习重要的推理步骤，而不仅仅是简单的微调，验证了其在基准推理数据集上的有效性。

超越模仿：从推理涤纶中学习关键推理步骤的双重思维链