# Atomic Thinking of LLMs: Decoupling and Exploring Mathematical Reasoning Abilities

arXiv: https://arxiv.org/abs/2509.25725 · HTML: https://mlsanity.com/paper/2509.25725/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 19 of its 75 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Placed in

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

## Builds on (most cited first, 19 of 19)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [MathQA: Towards Interpretable Math Word Problem Solving with Operation-Based Formalisms](https://mlsanity.com/paper/1905.13319/) (2019)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning](https://mlsanity.com/paper/2504.11456/) (2025)
- [Evaluating the Performance of Large Language Models on GAOKAO Benchmark](https://mlsanity.com/paper/2305.12474/) (2023)
- [InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning](https://mlsanity.com/paper/2402.06332/) (2024)
- [Common 7B Language Models Already Possess Strong Math Capabilities](https://mlsanity.com/paper/2403.04706/) (2024)
- [Lean-STaR: Learning to Interleave Thinking and Proving](https://mlsanity.com/paper/2407.10040/) (2024)
- [Atom of Thoughts for Markov LLM Test-Time Scaling](https://mlsanity.com/paper/2502.12018/) (2025)
- [AURORA:Automated Training Framework of Universal Process Reward Models via Ensemble Prompting and Reverse Verification](https://mlsanity.com/paper/2502.11520/) (2025)
- [One Example Shown, Many Concepts Known! Counterexample-Driven Conceptual Reasoning in Mathematical LLMs](https://mlsanity.com/paper/2502.10454/) (2025)
- [Diversity as a Reward: Fine-Tuning LLMs on a Mixture of Domain-Undetermined Data](https://mlsanity.com/paper/2502.04380/) (2025)
- [ReasonAgain: Using Extractable Symbolic Programs to Evaluate Mathematical Reasoning](https://mlsanity.com/paper/2410.19056/) (2024)
- [Attention Basin: Why Contextual Position Matters in Large Language Models](https://mlsanity.com/paper/2508.05128/) (2025)

## Built on by (most cited first, 6 of 6)

- [AdmTree: Compressing Lengthy Context with Adaptive Semantic Trees](https://mlsanity.com/paper/2512.04550/) (2025)
- [Cognitive Mismatch in Multimodal Large Language Models for Discrete Symbol Understanding](https://mlsanity.com/paper/2603.18472/) (2026)
- [Geometry of Reason: Spectral Signatures of Valid Mathematical Reasoning](https://mlsanity.com/paper/2601.00791/) (2026)
- [Aligning Perception, Reasoning, Modeling and Interaction: A Survey on Physical AI](https://mlsanity.com/paper/2510.04978/) (2025)
- [Evaluating Medical LLMs by Levels of Autonomy: A Survey Moving from Benchmarks to Applications](https://mlsanity.com/paper/2510.17764/) (2025)
- [From Atomic to Agentic: Towards Interpretable Evaluation of LLMs' Agentic Mathematical Capabilities](https://mlsanity.com/paper/2608.26950/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) have demonstrated outstanding performance in mathematical reasoning capabilities. However, we argue that current large-scale reasoning models primarily rely on scaling up training datasets with diverse mathematical problems and long thinking chains, which raises questions about whether LLMs genuinely acquire mathematical concepts and reasoning principles or merely remember the training data. In contrast, humans tend to break down complex problems into multiple fundamental atomic capabilities. Inspired by this, we propose a new paradigm for evaluating mathematical atomic capabilities. Our work categorizes atomic abilities into two dimensions: (1) field-specific abilities across four major mathematical fields, algebra, geometry, analysis, and topology, and (2) logical abilities at different levels, including conceptual understanding, forward multi-step reasoning with formal math language, and counterexample-driven backward reasoning. We propose corresponding training and evaluation datasets for each atomic capability unit, and conduct extensive experiments about how different atomic capabilities influence others, to explore the strategies to elicit the required specific atomic capability. Evaluation and experimental results on advanced models show many interesting discoveries and inspirations about the different performances of models on various atomic capabilities and the interactions between atomic capabilities. Our findings highlight the importance of decoupling mathematical intelligence into atomic components, providing new insights into model cognition and guiding the development of training strategies toward a more efficient, transferable, and cognitively grounded paradigm of "atomic thinking".