HintMR: Eliciting Stronger Mathematical Reasoning in Small Language Models

Reinforcement learning has become a powerful approach for enhancing large language model reasoning, but faces a fundamental dilemma: training on easy problems can cause overfitting and pass@k degradation, while training on hard problems often results in sparse rewards. Recent question augmentation methods address this by prepending partial solutions as hints. However, uniform hint provision may introduce redundant information while missing critical reasoning bottlenecks, and excessive hints can reduce reasoning diversity, causing pass@k degradation. We propose PieceHint, a hint injection framework that strategically identifies and provides critical reasoning steps during training. By scoring the importance of different reasoning steps, selectively allocating hints based on problem difficulty, and progressively withdrawing scaffolding, PieceHint enables models to transition from guided learning to independent reasoning. Experiments on six mathematical reasoning benchmarks show that our 1.5B model achieves comparable average performance to 32B baselines while preserving pass@k diversity across all $k$ values.

Multi-step Chain-of-Thought (CoT) has significantly advanced the mathematical reasoning capabilities of LLMs by leveraging explicit reasoning steps. However, the widespread adoption of Long CoT often results in sequence lengths that exceed manageable computational limits. While existing approaches attempt to alleviate this by reducing KV Cache redundancy via Markov chain-like structures, they introduce two critical limitations: inherent memorylessness (loss of context) and limited backward reasoning capability. To address these limitations, we propose a novel Chain-of-Thought framework based on Reversible Hierarchical Markov Chain, termed Cognitive Loop of Thought (CLoT), and a backward reasoning dataset CLoT-Instruct. In CLoT, problems are decomposed into sub-problems with hierarchical dependencies. Inspired by human cognitive processes, we introduce a backward verification mechanism at each hierarchical layer. Furthermore, we implement a pruning strategy: once higher-level sub-problems are verified, redundant lower-level sub-problems are pruned to maximize efficiency. This approach effectively mitigates error propagation and enhances reasoning robustness. Experiments on four mathematical benchmarks demonstrate the effectiveness of our method. Notably, on the AddSub dataset using GPT-4o-mini, CLoT achieves 99.0% accuracy, outperforming traditional CoT and CoT-SC by 4.1% and 2.9%, respectively.

Recent advancements in large language models (LLMs) have catalyzed the rise of reasoning-intensive inference paradigms, where models perform explicit step-by-step reasoning before generating final answers. While such approaches improve answer quality and interpretability, they incur substantial computational overhead due to the prolonged generation sequences. In this paper, we propose Tandem, a novel collaborative framework that synergizes large and small language models (LLMs and SLMs) to achieve high-quality reasoning with significantly reduced computational cost. Specifically, the LLM serves as a strategic coordinator, efficiently generating a compact set of critical reasoning insights. These insights are then used to guide a smaller, more efficient SLM in executing the full reasoning process and delivering the final response. To balance efficiency and reliability, Tandem introduces a cost-aware termination mechanism that adaptively determines when sufficient reasoning guidance has been accumulated, enabling early stopping of the LLM's generation. Experiments on mathematical reasoning and code generation benchmarks demonstrate that Tandem reduces computational costs by approximately 40% compared to standalone LLM reasoning, while achieving superior or competitive performance. Furthermore, the sufficiency classifier trained on one domain transfers effectively to others without retraining. The code is available at: https://github.com/Applied-Machine-Learning-Lab/ACL2026_Tandem.