SHAPE of Chain-of-Thought in Math Reasoning

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.