# SHAPE of Chain-of-Thought in Math Reasoning

arXiv: https://arxiv.org/abs/2608.28600 · HTML: https://mlsanity.com/paper/2608.28600/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 22 of its 48 references that are in the map.
- First posted on arXiv 28 Jun 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 22 of 22)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Measuring Faithfulness in Chain-of-Thought Reasoning](https://mlsanity.com/paper/2307.13702/) (2023)
- [Omni-MATH: A Universal Olympiad Level Mathematic Benchmark For Large Language Models](https://mlsanity.com/paper/2410.07985/) (2024)
- [Dynamic Early Exit in Reasoning Models](https://mlsanity.com/paper/2504.15895/) (2025)
- [When More is Less: Understanding Chain-of-Thought Length in LLMs](https://mlsanity.com/paper/2502.07266/) (2025)
- [Chain-of-Thought Reasoning In The Wild Is Not Always Faithful](https://mlsanity.com/paper/2503.08679/) (2025)
- [Thought Anchors: Which LLM Reasoning Steps Matter?](https://mlsanity.com/paper/2506.19143/) (2025)
- [Olmo 3](https://mlsanity.com/paper/2512.13961/) (2025)
- [Between Underthinking and Overthinking: An Empirical Study of Reasoning Length and correctness in LLMs](https://mlsanity.com/paper/2505.00127/) (2025)
- [The Invisible Leash: Why RLVR May or May Not Escape Its Origin](https://mlsanity.com/paper/2507.14843/) (2025)
- [MATH-Perturb: Benchmarking LLMs' Math Reasoning Abilities against Hard Perturbations](https://mlsanity.com/paper/2502.06453/) (2025)
- [Nemotron-Cascade: Scaling Cascaded Reinforcement Learning for General-Purpose Reasoning Models](https://mlsanity.com/paper/2512.13607/) (2025)
- [On the Hardness of Faithful Chain-of-Thought Reasoning in Large Language Models](https://mlsanity.com/paper/2406.10625/) (2024)
- [RL Squeezes, SFT Expands: A Comparative Study of Reasoning LLMs](https://mlsanity.com/paper/2509.21128/) (2025)
- [Do LLMs Really Need 10+ Thoughts for "Find the Time 1000 Days Later"? Towards Structural Understanding of LLM Overthinking](https://mlsanity.com/paper/2510.07880/) (2025)
- [Schoenfeld's Anatomy of Mathematical Reasoning by Language Models](https://mlsanity.com/paper/2512.19995/) (2025)

## Abstract (arXiv)

Large language models (LLMs) achieve strong performance on mathematical reasoning benchmarks, yet final-answer accuracy reveals little about the strategies they use. We introduce textttSHAPE (Structured Heuristic Analysis for Problem-solving Evolution), a framework grounded in mathematics-education research that characterizes local mathematical actions in reasoning trajectories using a problem-independent heuristic taxonomy. Using textttSHAPE, we examine two previously hypothesized phenomena in LLM reasoning. First, we revisit the hypothesis that LLMs reuse learned solution techniques under problem perturbations. Across six models on MATH-Perturb, perturbations requiring a different solution approach yield lower accuracy but greater heuristic shifts than those preserving it. Second, we characterize mode-seeking in RLVR through mathematical heuristic profiles. Across four base--RL pairs, successful RL solutions overlap substantially with base-model profiles. We further introduce Heuristic-Augmented Plan-GRPO (HA-Plan-GRPO), which guides RLVR planning through heuristic selection and applicability reasoning. Across three models and five evaluation sets, it improves mean Avg@16 by up to 7.0% absolute and mean Pass@16 by up to 2.4% absolute over planning-only RLVR. Our code is available at hrefhttps://github.com/holi-lab/shape-of-CoThttps://github.com/holi-lab/shape-of-CoT.