Large language models (LLMs) achieve strong results on math reasoning benchmarks, yet the underlying mathematically meaningful skills remain under‑explored. We introduce SHAPE, a framework that examines Chain‑of‑Thought (CoT) trajectories through two lenses from mathematics education:
- Semantic spaces – the model’s evolving mathematical interpretation of a problem (e.g., algebraic, geometric);
- Heuristics – the concrete mathematical actions taken within those spaces (e.g., simplification, working backward).
Applying SHAPE to various models reveals that the heuristics a model employs explain answer correctness better than traditional CoT features. Moreover, models tend to reach correct solutions by focusing reasoning effort within a few semantic spaces rather than scattering across many unrelated ones, a pattern consistent with human behavior.
We then use the SHAPE perspective to assess whether post‑training truly enhances mathematical proficiency. Findings indicate that reinforcement learning induces mode‑seeking behavior in heuristic usage. Consequently, we post‑train LLMs by encouraging heuristic diversity and demonstrate notable accuracy gains.
The code is publicly available at https://github.com/holi-lab/SHAPE-of-CoT.
Review: SHAPE offers a theoretically grounded diagnostic tool for decoding LLM reasoning, highlighting the importance of semantic‑space and heuristic analysis, and provides a promising route for post‑training improvements in math reasoning.