A Survey on Large Language Models for Mathematical Reasoning
Paper · 2025 · Peng-Yuan Wang, Tian-Shuo Liu, Chenyang Wang et al.
Summary
This survey, by Peng-Yuan Wang et al. (mainly Nanjing University), maps how large language models became capable of solving mathematical problems. It contrasts this with older rule-based and semantic-parsing systems that relied heavily on hand engineering. The authors split the capability into two parts:
- Comprehension: learned during pre-training on mathematical corpora such as textbooks, papers and problem sets, increasingly supplemented with filtered and synthetic data.
- Answer generation: has moved from predicting an answer directly to producing step-by-step Chain-of-Thought (CoT) solutions.
The review then groups enhancement techniques into prompting (In-Context Learning, CoT), supervised fine-tuning with distillation, reinforcement learning, search at inference time, self-improvement, and use of external tools or retrieval. Long CoT and "test-time scaling" receive particular attention. The paper offers no new data or benchmark. Its contribution is a unified taxonomy plus a diagnosis of open problems: exploration efficiency, wasteful reasoning paths, and weak transfer beyond mathematics. This makes it a useful entry point for anyone adapting reasoning methods to other fields.
Key points
- Figure 1 tracks rapid gains in mathematical ability from 2024/3 to 2025/3. It covers open models (DeepSeek-Math 7B, Qwen2.5-Math 7B, QwQ-32B-preview, DeepSeek R1 671B, Qwen3) and closed ones (O1, O3, Claude 3.7 Sonnet, Gemini 2.0 Flash Thinking).
- Grok 3 Beta is cited at 83.9% on AIME 2024. Against MAA human data, that score would rank among the top 2.5% of contestants.
- The authors name three central challenges:
- exploring efficiently under limited sampling budgets during RL;
- producing lengthy or sub-optimal reasoning chains, since outcome-only rewards ignore step quality;
- generalizing poorly from math to open-ended domains.
- Proposed remedies include:
- verification-aware optimization, using formal proof systems such as Lean to check intermediate steps;
- tool and retrieval integration, organized through the learnware paradigm;
- scaling up reward models;
- generative reward models trained via long CoT RL.
- The survey positions itself against earlier reviews that cover traditional methods, datasets and fine-tuning, multi-modal math, System 2 thinking, and RL for reasoning.
Institutions
Fields, topics and concepts
📚 Shelves 6The field of science this work belongs to (OpenAlex fields).Computer Science · 19,503 worksArtificial Intelligence · 18,805 worksArts and Humanities · 17,959 worksLanguage and Linguistics · 17,789 worksComputer Vision and Pattern Recognition · 3,621 worksComputational Theory and Mathematics · 496 works
🧭 Topics 2Research subjects and languages the work is about.Language models & pretraining · 4,145 worksReasoning · 1,948 works
Cite
Cite as Peng-Yuan Wang, Tian-Shuo Liu, Chenyang Wang, Yi-Di Wang, Shu Yan, Cheng-Xing Jia, Xu-Hui Liu, Xin-Wei Chen, Jia-Cheng Xu, Ziniu Li and Yang Yu. 2025. A Survey on Large Language Models for Mathematical Reasoning. arXiv:2506.08446.
@misc{wang2025survey,
title = {{A Survey on Large Language Models for Mathematical Reasoning}},
author = {Peng-Yuan Wang, Tian-Shuo Liu, Chenyang Wang, Yi-Di Wang, Shu Yan, Cheng-Xing Jia, Xu-Hui Liu, Xin-Wei Chen, Jia-Cheng Xu, Ziniu Li and Yang Yu},
year = {2025},
doi = {10.48550/arxiv.2506.08446},
eprint = {2506.08446},
archivePrefix = {arXiv},
url = {https://doi.org/10.48550/arxiv.2506.08446}
}References
- American invitational mathematics examination (aime). https://www.maa.org/math-competitions/aime, 2024. Mathematical Association of America. ↗
- S. Adarsh, K. Shridhar, C. Gulcehre, N. Monath, and M. Sachan. SIKeD: Self-guided iterative knowledge distillation for mathematical reasoning. CoRR, abs/2410.18574, 2024. find
- A. Ahmadian, C. Cremer, M. Gallé, M. Fadaee, J. Kreutzer, O. Pietquin, A. Üstün, and S. Hooker. Back to basics: Revisiting REINFORCE-style optimization for learning from human feedback in LLMs. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024. find
- J. Ahn, R. Verma, R. Lou, D. Liu, R. Zhang, and W. Yin. Large language models for mathematical reasoning: Progresses and challenges. In Proceedings of the 18th Conference of the European Chapter of the Association for Computational Linguistics, Student Research Workshop, 2024.
- Z. Allen-Zhu and Y. Li. Physics of language models: Part 3.1, knowledge storage and extraction. In Proceedings of the 41th International Conference on Machine Learning, 2024.
- A. Amini, S. Gabriel, P. Lin, R. Koncel-Kedziorski, Y. Choi, and H. Hajishirzi. MathQA: Towards interpretable math word problem solving with operation-based formalisms. CoRR, abs/1905.13319, 2019.
- S. An, Z. Ma, Z. Lin, N. Zheng, J.-G. Lou, and W. Chen. Learning from mistakes makes LLM better reasoner. CoRR, abs/2310.20689, 2023.
- Z. Ankner, M. Paul, B. Cui, J. D. Chang, and P. Ammanabrolu. Critique-out-loud reward models. CoRR, abs/2408.11791, 2024.
- T. Anthony, Z. Tian, and D. Barber. Thinking fast and slow with deep learning and tree search. In Advances in Neural Information Processing Systems 31, 2017.
- I. Arcuschin, J. Janiak, R. Krzyzanowski, S. Rajamanoharan, N. Nanda, and A. Conmy. Chain-of-Thought reasoning in the wild is not always faithful. CoRR, abs/2503.08679, 2025.
- S. Arora and B. Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. find
- S. Arora, Y. Li, Y. Liang, T. Ma, and A. Risteski. A latent variable model approach to PMI-based word embeddings. CoRR, abs/1502.03520, 2019. A. Asai, Z. Wu, Y. Wang, A. Sil, and H. Hajishirzi. Self-RAG: Learning to retrieve, generate, and critique through self-reflection. In Proceedings of the 12th… find
- E. Chern, H. Zou, X. Li, J. Hu, K. Feng, J. Li, and P. Liu. Generative AI for math: Abel. — Training Verifiers to Solve Math Word Problems https://github.com/GAIR-NLP/abel, 2023. P. F. Christiano, J. Leike, T. B. Brown, M. Martic, S. Legg, and D. Amodei. Deep reinforcement learning from human preferences. In Advances in Neural Information Processi… ↗
- McGraw-Hill, 1963. D. Feng, B. Qin, C. Huang, Z. Zhang, and W. Lei. Towards analyzing and understanding the limitations of DPO: A theoretical perspective. CoRR, abs/2404.04626, 2024a. G. Feng, B. Zhang, Y. Gu, H. Ye, D. He, and L. Wang. Towards revealing the mystery behind Chain of Thought: A theore… — [arXiv:2404.04626](https://arxiv.org/abs/2404.04626) find
- J. Gehring, K. Zheng, J. Copet, V. Mella, T. Cohen, and G. Synnaeve. RLEF: Grounding code LLMs in execution feedback with reinforcement learning. CoRR, abs/2410.02089, 2024.
- S. Ghosh, C. K. R. Evuru, S. Kumar, U. Tyagi, O. Nieto, Z. Jin, and D. Manocha. Visual description grounding reduces hallucinations and boosts reasoning in LVLMs. CoRR, abs/2405.15683, 2024. — [arXiv:2405.15683](https://arxiv.org/abs/2405.15683) find
- D. Goldwasser and D. Roth. Learning from natural instructions. In Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence, 2014. find
- X. Guan, L. L. Zhang, Y. Liu, N. Shang, Y. Sun, Y. Zhu, F. Yang, and M. Yang. rStar-Math: Small LLMs can master math reasoning with self-evolved deep thinking. CoRR, abs/2501.04519, 2025.
- D. Guo, D. Yang, H. Zhang, J. Song, R. Zhang, R. Xu, Q. Zhu, S. Ma, P. Wang, X. Bi, et al. DeepSeek-R1: Incentivizing reasoning capability in LLMs via reinforcement learning. CoRR, abs/2501.12948, 2025.
- S. Guo, A. Didolkar, N. R. Ke, A. Goyal, F. Huszár, and B. Schölkopf. Learning beyond pattern matching? assaying mathematical understanding in LLMs. CoRR, abs/2405.15485, 2024. — [arXiv:2405.15485](https://arxiv.org/abs/2405.15485) find
- S. Hao, Y. Gu, H. Ma, J. J. Hong, Z. Wang, D. Z. Wang, and Z. Hu. Reasoning with language model is planning with world model. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, 2023. find
- C. He, R. Luo, Y. Bai, S. Hu, Z. L. Thai, J. Shen, J. Hu, X. Han, Y. Huang, Y. Zhang, et al. OlympiadBench: A challenging benchmark for promoting AGI with olympiad-level bilingual multimodal scientific problems. CoRR, abs/2402.14008, 2024a.
- M. He, Y. Shen, W. Zhang, Z. Tan, and W. Lu. Advancing process verification for large language models via tree-based preference learning. In Proceedings of Conference on Empirical Methods in Natural Language Processing, 2024b. find
- N. He, H. Lai, C. Zhao, Z. Cheng, J. Pan, R. Qin, R. Lu, R. Lu, Y. Zhang, G. Zhao, et al. TeacherLM: Teaching to fish rather than giving the fish, language modeling likewise. CoRR, abs/2310.19019, 2023. — [arXiv:2310.19019](https://arxiv.org/abs/2310.19019) find
- D. Hendrycks, C. Burns, S. Basart, A. Zou, M. Mazeika, D. Song, and J. Steinhardt. Measuring massive multitask language understanding. CoRR, abs/2009.03300, 2020.
- D. Hendrycks, C. Burns, S. Kadavath, A. Arora, S. Basart, E. Tang, D. Song, and J. Steinhardt. Measuring mathematical problem solving with the MATH dataset. In The 35th Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2021.
- G. Hinton. Distilling the knowledge in a neural network. CoRR, abs/1503.02531, 2015.
- N. Ho, L. Schmid, and S. Yun. Large language models are reasoning teachers. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics, 2023.
- A. Hosseini, X. Yuan, N. Malkin, A. Courville, A. Sordoni, and R. Agarwal. V-STaR: Training verifiers for self-taught reasoners. CoRR, abs/2402.06457, 2024.
- J. Hu. REINFORCE++: A simple and efficient approach for aligning large language models. CoRR, abs/2501.03262, 2025.
- J. Huang, S. S. Gu, L. Hou, Y. Wu, X. Wang, H. Yu, and J. Han. Large language models can self-improve. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, 2023.
- Y. Huang, X. Liu, Y. Gong, Z. Gou, Y. Shen, N. Duan, and W. Chen. Key-point-driven data synthesis with its enhancement on mathematical reasoning. CoRR, abs/2403.02333, 2024.
- S. Imani, L. Du, and H. Shrivastava. MathPrompter: Mathematical reasoning using large language models. In Proceedings of the 61th Annual Meeting of the Association for Computational Linguistics, pages 37–42, 2023.
- C. Jia, P. Wang, Z. Li, Y.-C. Li, Z. Zhang, N. Tang, and Y. Yu. BWArea model: Learning world model, inverse dynamics, and policy for controllable language generation. CoRR, abs/2405.17039, 2024. — [arXiv:2405.17039](https://arxiv.org/abs/2405.17039) find
- C. Jia, Z. Li, P. Wang, Y.-C. Li, Z. Hou, Y. Dong, and Y. Yu. Controlling large language model with latent actions. CoRR, abs/2503.21383, 2025. — [arXiv:2503.21383](https://arxiv.org/abs/2503.21383) find
- W. Jiang, H. Shi, L. Yu, Z. Liu, Y. Zhang, Z. Li, and J. Kwok. Forward-backward reasoning in large language models for mathematical verification. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024.
- Z. Jie and W. Lu. Leveraging training data in few-shot prompting for numerical reasoning. CoRR, abs/2305.18170, 2023. find
- B. Jin, H. Zeng, Z. Yue, D. Wang, H. Zamani, and J. Han. Search-R1: Training LLMs to reason and leverage search engines with reinforcement learning. CoRR, abs/2503.09516, 2025.
- M. Jin, Q. Yu, D. Shu, H. Zhao, W. Hua, Y. Meng, Y. Zhang, and M. Du. The impact of reasoning step length on large language models. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024.
- A. Joulin. Fasttext.zip: Compressing text classification models. CoRR, abs/1612.03651, 2016.
- A. Kazemnejad, M. Aghajohari, E. Portelance, A. Sordoni, S. Reddy, A. Courville, and N. L. Roux. VinePPO: Unlocking RL potential for LLM reasoning through refined credit assignment. CoRR, abs/2410.01679, 2024.
- T. Kojima, S. S. Gu, M. Reid, Y. Matsuo, and Y. Iwasawa. Large language models are zero-shot reasoners. In Advances in Neural Information Processing Systems 36, 2022.
- R. Koncel-Kedziorski, S. Roy, A. Amini, N. Kushman, and H. Hajishirzi. MAWPS: A math word problem repository. In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics, 2016. find
- T. Kwiatkowski, E. Choi, Y. Artzi, and L. Zettlemoyer. Scaling semantic parsers with on-the-fly ontology matching. In Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing, 2013.
- X. Lai, Z. Tian, Y. Chen, S. Yang, X. Peng, and J. Jia. Step-DPO: Step-wise preference optimization for long-chain reasoning of LLMs. CoRR, abs/2406.18629, 2024.
- L. Lehnert, S. Sukhbaatar, D. Su, Q. Zheng, P. Mcvay, M. Rabbat, and Y. Tian. Beyond A*: Better planning with Transformers via search dynamics bootstrapping. CoRR, abs/2402.14083, 2024. find
- Z. Levonian, C. Li, W. Zhu, A. Gade, O. Henkel, M.-E. Postle, and W. Xing. Retrieval-augmented generation to improve math question-answering: Trade-offs between groundedness and human preference. CoRR, abs/2310.03184, 2023. — [arXiv:2310.03184](https://arxiv.org/abs/2310.03184) find
- A. Lewkowycz, A. Andreassen, D. Dohan, E. Dyer, H. Michalewski, V. V. Ramasesh, A. Slone, C. Anil, I. Schlag, T. Gutman-Solo, Y. Wu, B. Neyshabur, G. Gur-Ari, and V. Misra. Solving quantitative reasoning problems with language models. CoRR, abs/2206.14858, 2022.
- C. Li, W. Wang, J. Hu, Y. Wei, N. Zheng, H. Hu, Z. Zhang, and H. Peng. Common 7b language models already possess strong math capabilities. CoRR, abs/2403.04706, 2024a.
- C. Li, Z. Yuan, H. Yuan, G. Dong, K. Lu, J. Wu, C. Tan, X. Wang, and C. Zhou. MuggleMath: Assessing the impact of query and response augmentation on math reasoning. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024b.
- J. Li, E. Beeching, L. Tunstall, B. Lipkin, R. Soletskyi, S. Huang, K. Rasul, L. Yu, A. Q. Jiang, Z. Shen, et al. NuminaMath: The largest public dataset in AI4Maths with 860k pairs of competition math problems and solutions. Hugging Face repository, page 9, 2024c. find
- Q. Li, L. Cui, X. Zhao, L. Kong, and W. Bi. GSM-Plus: A comprehensive benchmark for evaluating the robustness of LLMs as mathematical problem solvers. CoRR, abs/2402.19255, 2024d.
- X. Li. A survey on LLM test-time compute via search: Tasks, LLM profiling, search algorithms, and relevant frameworks. CoRR, abs/2501.10069, 2025. — [arXiv:2501.10069](https://arxiv.org/abs/2501.10069) find
- X. Li, H. Zou, and P. Liu. ToRL: Scaling tool-integrated RL. CoRR, abs/2503.23383, 2025a.
- Y. Li, Z. Lin, S. Zhang, Q. Fu, B. Chen, J.-G. Lou, and W. Chen. Making large language models better reasoners with step-aware verifier. CoRR, abs/2206.02336, 2022. — [arXiv:2206.02336](https://arxiv.org/abs/2206.02336) find
- Y. Li, K. Sreenivasan, A. Giannou, D. Papailiopoulos, and S. Oymak. Dissecting Chain-of-Thought: Compositionality through in-context filtering and learning. In Advances in Neural Information Processing Systems 37, 2023. find
- Z. Li, T. Xu, and Y. Yu. When is RL better than DPO in RLHF? a representation and optimization perspective. In The Second Tiny Papers Track at International Conference on Learning Representations, 2024e. find
- Z. Li, T. Xu, Y. Zhang, Z. Lin, Y. Yu, R. Sun, and Z.-Q. Luo. ReMax: A simple, effective, and efficient reinforcement learning method for aligning large language models. In The 41st International Conference on Machine Learning, 2024f. find
- Z. Li, C. Chen, T. Xu, Z. Qin, J. Xiao, Z.-Q. Luo, and R. Sun. Preserving diversity in supervised fine-tuning of large language models. In Proceedings of the 13th International Conference on Learning Representations, 2025b. find
- Z.-Z. Li, D. Zhang, M.-L. Zhang, J. Zhang, Z. Liu, Y. Yao, H. Xu, J. Zheng, P.-J. Wang, X. Chen, et al. From system 1 to system 2: A survey on reasoning large language models. CoRR, abs/2502.17419, 2025c.
- P. P. Liang, A. Zadeh, and L.-P. Morency. Foundations & trends in multimodal machine learning: Principles, challenges, and open questions. ACM Computing Surveys, 56(10):1–42, 2024. find
- Z. Liang, W. Yu, T. Rajpurohit, P. Clark, X. Zhang, and A. Kalyan. Let GPT be a math tutor: Teaching math word problem solvers with customized exercise generation. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, 2023. find
- H. Lightman, V. Kosaraju, Y. Burda, H. Edwards, B. Baker, T. Lee, J. Leike, J. Schulman, I. Sutskever, and K. Cobbe. Let’s verify step by step. CoRR, abs/2305.20050, 2023.
- Y. Lin, S. Seto, M. ter Hoeve, K. Metcalf, B.-J. Theobald, X. Wang, Y. Zhang, C. Huang, and T. Zhang. On the limited generalization capability of the implicit reward model induced by direct preference optimization. CoRR, abs/2409.03650, 2024a. find
- Y.-T. Lin, D. Jin, T. Xu, T. Wu, S. Sukhbaatar, C. Zhu, Y. He, Y.-N. Chen, J. Weston, Y. Tian, A. Rahnama, S. Wang, H. Ma, and H. Fang. Step-KTO: Optimizing mathematical reasoning through stepwise binary feedback. CoRR, abs/2501.10799, 2025. find
- Z. Lin, Z. Gou, Y. Gong, X. Liu, Y. Shen, R. Xu, C. Lin, Y. Yang, J. Jiao, N. Duan, et al. Rho-1: Not all tokens are what you need. CoRR, abs/2404.07965, 2024b.
- J. Lindsey, W. Gurnee, E. Ameisen, B. Chen, A. Pearce, N. L. Turner, C. Citro, D. Abrahams, S. Carter, B. Hosmer, J. Marcus, M. Sklar, A. Templeton, T. Bricken, C. McDougall, H. Cunningham, T. Henighan, A. Jermyn, A. Jones, A. Persic, Z. Qi, T. B. Thompson, S. Zimmerman, K. Rivoire, T. Conerly, C. O… find
- W. Ling, D. Yogatama, C. Dyer, and P. Blunsom. Program induction by rationale generation: Learning to solve and explain algebraic word problems. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, 2017. find
- B. Liu, S. Bubeck, R. Eldan, J. Kulkarni, Y. Li, A. Nguyen, R. Ward, and Y. Zhang. TinyGSM: achieving\> 80% on GSM8k with small language models. CoRR, abs/2312.09241, 2023.
- H. Liu, Y. Zhang, Y. Luo, and A. C. Yao. Augmenting math word problems via iterative question composing. CoRR, abs/2401.09003, 2024a.
- Z. Liu, H. Liu, D. Zhou, and T. Ma. Chain of thought empowers transformers to solve inherently serial problems. In Proceedings of the 12th International Conference on Learning Representations, 2024b. find
- P. Lu, L. Qiu, W. Yu, S. Welleck, and K.-W. Chang. A survey of deep learning for mathematical reasoning. In Proceedings of the 61th Annual Meeting of the Association for Computational Linguistics, 2023. find
- L. Luo, Y. Liu, R. Liu, S. Phatale, H. Lara, Y. Li, L. Shu, Y. Zhu, L. Meng, J. Sun, et al. Improve mathematical reasoning in language models by automated process supervision. CoRR, abs/2406.06592, 2024.
- M. Luo, S. Tan, J. Wong, X. Shi, W. Y. Tang, M. Roongta, C. Cai, J. Luo, T. Zhang, L. E. Li, R. A. Popa, and I. Stoica. DeepScaleR: Surpassing O1-Preview with a 1.5b model by scaling RL. https://pretty-radio-b75.notion.site/DeepScaleR-Surpassing-O1-Preview-with-a-1-5B-Model-by-Scaling-RL-19681902c14… ↗
- A. Madaan, N. Tandon, P. Gupta, S. Hallinan, L. Gao, S. Wiegreffe, U. Alon, N. Dziri, S. Prabhumoye, Y. Yang, S. Gupta, B. P. Majumder, K. Hermann, S. Welleck, A. Yazdanbakhsh, and P. Clark. Self-refine: Iterative refinement with self-feedback. In Advances in Neural Information Processing Systems 36… find
- L. C. Magister, J. Mallinson, J. D. Adamek, E. Malmi, and A. Severyn. Teaching small language models to reason. In Proceedings of the 61st Annual Meeting Of The Association For Computational Linguistics, 2023.
- J. Meadows and A. Freitas. A survey in mathematical language processing. CoRR, abs/2205.15231, 2022. — [arXiv:2205.15231](https://arxiv.org/abs/2205.15231) find
- Y. Meng, M. Xia, and D. Chen. SimPO: Simple preference optimization with a reference-free reward. CoRR, abs/2405.14734, 2024. find
- J. Menick, M. Trebacz, M. Mikulak-Krupiński, E. Elsen, A. Valbusa, A. Raichuk, R. Kadlec, G. Irving, J. Kopeckỳ, M. Ring, A. Glaese, N. Elhage, T. Hennigan, S. Saci, T. Cai, M. Fritz, A. Jones, D. Pfau, T. Pohlen, and O. Vinyals. Teaching language models to support answers with verified quotes. Co…
- W. Merrill and A. Sabharwal. The expressive power of Transformers with Chain of Thought. In Proceedings of the 12th International Conference on Learning Representations, 2024. find
- S.-Y. Miao, C.-C. Liang, and K.-Y. Su. A diverse corpus for evaluating and developing english math word problem solvers. CoRR, abs/2106.15772, 2021. find
- S. Mirzadeh, K. Alizadeh, H. Shahrokhi, O. Tuzel, S. Bengio, and M. Farajtabar. GSM-Symbolic: Understanding the limitations of mathematical reasoning in large language models. CoRR, abs/2410.05229, 2024.
- S. Mishra, M. Finlayson, P. Lu, L. Tang, S. Welleck, C. Baral, T. Rajpurohit, O. Tafjord, A. Sabharwal, P. Clark, and A. Kalyan. LILA: A unified benchmark for mathematical reasoning. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, 2022.
- N. Muennighoff, Z. Yang, W. Shi, X. L. Li, L. Fei-Fei, H. Hajishirzi, L. Zettlemoyer, P. Liang, E. Candès, and T. Hashimoto. s1: Simple test-time scaling. CoRR, abs/2501.19393, 2025.
- OpenAI. GPT-4 system card. https://cdn.openai.com/papers/gpt-4-system-card.pdf, March 2023. ↗
- OpenAI. Learning to reason with LLMs. https://openai.com/index/learning-to-reason-with-llms/, 2024. ↗
- L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. L. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, J. Schulman, J. Hilton, F. Kelton, L. Miller, M. Simens, A. Askell, P. Welinder, P. F. Christiano, J. Leike, and R. Lowe. Training language models to follow instructions with human feedback.…
- J.-C. Pang, P. Wang, K. Li, X.-H. Chen, J. Xu, Z. Zhang, and Y. Yang. Language model self-improvement by reinforcement learning contemplation. In International Conference on Learning Representations, 2023.
- R. Y. Pang, W. Yuan, H. He, K. Cho, S. Sukhbaatar, and J. Weston. Iterative reasoning preference optimization. In Advances in Neural Information Processing Systems 38, 2024.
- K. Paster, M. Dos Santos, Z. Azerbayev, and J. Ba. OpenWebMath: An open dataset of high-quality mathematical web text. In Proceedings of the 12th International Conference on Learning Representations, 2024.
- A. Patel, S. Bhattamishra, and N. Goyal. Are NLP models really able to solve simple math word problems? CoRR, abs/2103.07191, 2021.
- D. Paul, M. Ismayilzada, M. Peyrard, B. Borges, A. Bosselut, R. West, and B. Faltings. REFINER: reasoning feedback on intermediate representations. In Proceedings of the 18th Conference of the European Chapter of the Association for Computational Linguistics, 2024.
- X. Peng, C. Xia, X. Yang, C. Xiong, C.-S. Wu, and C. Xing. ReGenesis: LLMs can grow into reasoning generalists via self-improvement. CoRR, abs/2410.02108, 2024. — [arXiv:2410.02108](https://arxiv.org/abs/2410.02108) find
- B. Prystawski, M. Li, and N. D. Goodman. Why think step by step? Reasoning emerges from the locality of experience. In Advances in Neural Information Processing Systems 36, 2023.
- X. Qu, Y. Li, Z. Su, W. Sun, J. Yan, D. Liu, G. Cui, D. Liu, S. Liang, J. He, P. Li, W. Wei, J. Shao, C. Lu, Y. Zhang, X. Hua, B. Zhou, and Y. Cheng. A survey of efficient reasoning for large reasoning models: Language, multimodality, and beyond. CoRR, abs/2503.21614, 2025.
- Y. Qu, T. Zhang, N. Garg, and A. Kumar. Recursive introspection: Teaching language model agents how to self-improve. In Advances in Neural Information Processing Systems, 2024.
- A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019. find
- R. Rafailov, A. Sharma, E. Mitchell, C. D. Manning, S. Ermon, and C. Finn. Direct preference optimization: Your language model is secretly a reward model. In Advances in Neural Information Processing Systems 37, 2023.
- J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization algorithms. CoRR, abs/1707.06347, 2017. find
- A. Setlur, C. Nagpal, A. Fisch, X. Geng, J. Eisenstein, R. Agarwal, A. Agarwal, J. Berant, and A. Kumar. Rewarding progress: Scaling automated process verifiers for LLM reasoning. CoRR, abs/2410.08146, 2024.
- Z. Shao, P. Wang, Q. Zhu, R. Xu, J. Song, X. Bi, H. Zhang, M. Zhang, Y. Li, Y. Wu, et al. DeepSeekMath: Pushing the limits of mathematical reasoning in open language models. CoRR, abs/2402.03300, 2024.
- M. Shen, G. Zeng, Z. Qi, Z. Hong, Z. Chen, W. Lu, G. W. Wornell, S. Das, D. Cox, and C. Gan. Satori: Reinforcement learning with chain-of-action-thought enhances LLM reasoning via autoregressive search. CoRR, abs/2502.02508, 2025.
- Z. Shen. LLM with tools: A survey. CoRR, abs/2409.18807, 2024. — [arXiv:2409.18807](https://arxiv.org/abs/2409.18807) find
- F. Shi, M. Suzgun, M. Freitag, X. Wang, S. Srivats, S. Vosoughi, H. W. Chung, Y. Tay, S. Ruder, D. Zhou, D. Das, and J. Wei. Language models are multilingual chain-of-thought reasoners. CoRR, abs/2210.03057, 2022.
- F. Shi, X. Chen, K. Misra, N. Scales, D. Dohan, E. H. Chi, N. Schärli, and D. Zhou. Large language models can be easily distracted by irrelevant context. In International Conference on Machine Learning, 2023.
- A. Singh, J. D. Co-Reyes, R. Agarwal, A. Anand, P. Patil, X. Garcia, P. J. Liu, J. Harrison, J. Lee, K. Xu, A. T. Parisi, A. Kumar, A. A. Alemi, A. Rizkowsky, A. Nova, B. Adlam, B. Bohnet, G. F. Elsayed, H. Sedghi, I. Mordatch, I. Simpson, I. Gur, J. Snoek, J. Pennington, J. Hron, K. Kenealy, K. Swe… — Beyond Human Data: Scaling Self-Training for Problem-Solving with Language Models
- J. R. Slagle. Experiments with a deductive question-answering program. Communications of the Association for Computing Machinery, 8(12):792–798, 1965. find
- C. Snell, J. Lee, K. Xu, and A. Kumar. Scaling LLM test-time compute optimally can be more effective than scaling model parameters. CoRR, abs/2408.03314, 2024.
- Z. Sprague, F. Yin, J. D. Rodriguez, D. Jiang, M. Wadhwa, P. Singhal, X. Zhao, X. Ye, K. Mahowald, and G. Durrett. To CoT or not to CoT? chain-of-thought helps mainly on math and symbolic reasoning. CoRR, abs/2409.12183, 2024a.
- Z. Sprague, F. Yin, J. D. Rodriguez, D. Jiang, M. Wadhwa, P. Singhal, X. Zhao, X. Ye, K. Mahowald, and G. Durrett. To CoT or not to CoT? chain-of-thought helps mainly on math and symbolic reasoning. CoRR, abs/2409.12183, 2024b.
- P. Srivastava, M. Malik, V. Gupta, T. Ganu, and D. Roth. Evaluating LLMs’ mathematical reasoning in financial document question answering. In Findings of the Association for Computational Linguistics, 2024. find
- N. Stiennon, L. Ouyang, J. Wu, D. M. Ziegler, R. Lowe, C. Voss, A. Radford, D. Amodei, and P. F. Christiano. Learning to summarize from human feedback. CoRR, abs/2009.01325, 2020. find
- Y. Sui, Y.-N. Chuang, G. Wang, J. Zhang, T. Zhang, J. Yuan, H. Liu, A. Wen, H. Chen, X. Hu, et al. Stop overthinking: A survey on efficient reasoning for large language models. CoRR, abs/2503.16419, 2025.
- Q. Tang, Z. Deng, H. Lin, X. Han, Q. Liang, B. Cao, and L. Sun. ToolAlpaca: Generalized tool learning for language models with 3000 simulated cases. CoRR, abs/2306.05301, 2023.
- Z. Tang, X. Zhang, B. Wang, and F. Wei. MathScale: Scaling instruction tuning for mathematical reasoning. In Proceedings of the 41st International Conference on Machine Learning, 2024.
- G. Team and Google. Gemini: A family of highly capable multimodal models. https://storage.googleapis.com/deepmind-media/gemini/gemini_1_report.pdf, December 2023. ↗
- K. Team, A. Du, B. Gao, B. Xing, C. Jiang, C. Chen, C. Li, C. Xiao, C. Du, C. Liao, et al. Kimi k1.5: Scaling reinforcement learning with LLMs. CoRR, abs/2501.12599, 2025.
- A. Templeton, T. Conerly, J. Marcus, J. Lindsey, T. Bricken, B. Chen, A. Pearce, C. Citro, E. Ameisen, A. Jones, H. Cunningham, N. L. Turner, C. McDougall, M. MacDiarmid, C. D. Freeman, T. R. Sumers, E. Rees, J. Batson, A. Jermyn, S. Carter, C. Olah, and T. Henighan. Scaling monosemanticity: Extract… find
- S. Toshniwal, W. Du, I. Moshkov, B. Kisacanin, A. Ayrapetyan, and I. Gitman. OpenMathInstruct-2: Accelerating AI for math with massive open-source instruction data. CoRR, abs/2410.01560, 2024a.
- S. Toshniwal, I. Moshkov, S. Narenthiran, D. Gitman, F. Jia, and I. Gitman. OpenMathInstruct-1: A 1.8 million math instruction tuning dataset. In Advances in Neural Information Processing Systems 38, 2024b.
- H. Touvron, T. Lavril, G. Izacard, X. Martinet, M.-A. Lachaux, T. Lacroix, B. Rozière, N. Goyal, E. Hambro, F. Azhar, et al. Llama: Open and efficient foundation language models. CoRR, abs/2302.13971, 2023.
- L. Q. Trung, X. Zhang, Z. Jie, P. Sun, X. Jin, and H. Li. ReFT: Reasoning with reinforced fine-tuning. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024.
- A. Turing. Computing machinery and intelligence. 1950. find
- R. Tutunov, A. Grosnit, J. Ziomek, J. Wang, and H. Bou-Ammar. Why can large language models generate correct chain-of-thoughts? CoRR, abs/2310.13571, 2023. — [arXiv:2310.13571](https://arxiv.org/abs/2310.13571) find
- Z. Wan, X. Feng, M. Wen, S. M. McAleer, Y. Wen, W. Zhang, and J. Wang. AlphaZero-like tree-search can guide large language model decoding and training. In Proceedings of the 41th International Conference on Machine Learning, 2024. find
- B. Wang, S. Min, X. Deng, J. Shen, Y. Wu, L. Zettlemoyer, and H. Sun. Towards understanding chain-of-thought prompting: An empirical study of what matters. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics, 2023a.
- E. Z. Wang, F. Cassano, C. Wu, Y. Bai, W. Song, V. Nath, Z. Han, S. M. Hendryx, S. Yue, and H. Zhang. Planning in natural language improves LLM search for code generation. CoRR, abs/2409.03733, 2024a. find
- J. Wang, Q. Sun, X. Li, and M. Gao. Boosting language models reasoning with chain-of-knowledge prompting. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024b.
- K. Wang, J. Pan, W. Shi, Z. Lu, H. Ren, A. Zhou, M. Zhan, and H. Li. Measuring multimodal mathematical reasoning with MATH-Vision dataset. In Advances in Neural Information Processing Systems 38, 2024c.
- L. Wang, W. Xu, Y. Lan, Z. Hu, Y. Lan, R. K. Lee, and E. Lim. Plan-and-solve prompting: Improving zero-shot chain-of-thought reasoning by large language models. In Proceedings of the 61th Annual Meeting of the Association for Computational Linguistics, 2023b. find
- P. Wang, L. Li, Z. Shao, R. Xu, D. Dai, Y. Li, D. Chen, Y. Wu, and Z. Sui. Math-Shepherd: Verify and reinforce LLMs step-by-step without human annotations. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024d.
- T. Wang and W. Lu. Learning multi-step reasoning by solving arithmetic tasks. In Proceedings of the 61th Annual Meeting of the Association for Computational Linguistics, 2023. find
- X. Wang and D. Zhou. Chain-of-thought reasoning without prompting. CoRR, abs/2402.10200, 2024. find
- X. Wang, J. Wei, D. Schuurmans, Q. V. Le, E. H. Chi, S. Narang, A. Chowdhery, and D. Zhou. Self-consistency improves chain of thought reasoning in language models. In Proceedings of the 11th International Conference on Learning Representations, 2023c.
- Y. Wang, X. Liu, and S. Shi. Deep neural solver for math word problems. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, 2017. find
- Y. Wang, Q. Liu, J. Xu, T. Liang, X. Chen, Z. He, L. Song, D. Yu, J. Li, Z. Zhang, et al. Thoughts are all over the place: On the underthinking of o1-like LLMs. CoRR, abs/2501.18585, 2025.
- Z. Wang, X. Li, R. Xia, and P. Liu. MathPile: A billion-token-scale pretraining corpus for math. In Advances in Neural Information Processing Systems 37, 2024e.
- S. Warshall. A theorem on boolean matrices. Journal of the Association for Computing Machinery, 1962. find
- J. Wei, Y. Tay, R. Bommasani, C. Raffel, B. Zoph, S. Borgeaud, D. Yogatama, M. Bosma, D. Zhou, D. Metzler, et al. Emergent abilities of large language models. Transactions on Machine Learning Research, 2022a.
- J. Wei, X. Wang, D. Schuurmans, M. Bosma, B. Ichter, F. Xia, E. H. Chi, Q. V. Le, and D. Zhou. Chain-of-thought prompting elicits reasoning in large language models. In Advances in Neural Information Processing Systems 36, 2022b.
- T. Wei, J. Luan, W. Liu, S. Dong, and B. Wang. CMATH: can your language model pass chinese elementary school math test? CoRR, abs/2306.16636, 2023.
- S. Welleck, J. Liu, R. Le Bras, H. Hajishirzi, Y. Choi, and K. Cho. NaturalProofs: Mathematical theorem proving in natural language. In Proceedings of the 1th Neural Information Processing Systems Track on Datasets and Benchmarks, 2021.
- S. Welleck, X. Lu, P. West, F. Brahman, T. Shen, D. Khashabi, and Y. Choi. Generating sequences by learning to self-correct. In Proceedings of the 11th International Conference on Learning Representations, 2023.
- Y. Weng, M. Zhu, F. Xia, B. Li, S. He, S. Liu, B. Sun, K. Liu, and J. Zhao. Large language models are better reasoners with self-verification. In Findings of the Association for Computational Linguistics, 2023.
- C. Wu, Z. Lin, W. Fang, and Y. Huang. A medical diagnostic assistant based on LLM. In Health Information Processing Evaluation Track Papers, 2024a. find
- S. Wu, E. M. Shen, C. Badrinath, J. Ma, and H. Lakkaraju. Analyzing chain-of-thought prompting in large language models via gradient-based feature attributions. CoRR, abs/2307.13339, 2023a.
- Y. Wu, Z. Sun, S. Li, S. Welleck, and Y. Yang. Inference scaling laws: An empirical analysis of compute-optimal inference for problem-solving with language models. CoRR, abs/2408.00724, 2024b.
- Z. Wu, Y. Hu, W. Shi, N. Dziri, A. Suhr, P. Ammanabrolu, N. A. Smith, M. Ostendorf, and H. Hajishirzi. Fine-grained human feedback gives better rewards for language model training. In Advances in Neural Information Processing Systems 37, 2023b.
- W. Xiong, J. Yao, Y. Xu, B. Pang, L. Wang, D. Sahoo, J. Li, N. Jiang, T. Zhang, C. Xiong, et al. A minimalist approach to LLM reasoning: from Rejection Sampling to REINFORCE. CoRR, abs/2504.11343, 2025a. — [arXiv:2504.11343](https://arxiv.org/abs/2504.11343) find
- W. Xiong, H. Zhang, C. Ye, L. Chen, N. Jiang, and T. Zhang. Self-rewarding correction for mathematical reasoning. CoRR, abs/2502.19613, 2025b. — [arXiv:2502.19613](https://arxiv.org/abs/2502.19613) find
- F. Xu, Q. Hao, Z. Zong, J. Wang, Y. Zhang, J. Wang, X. Lan, J. Gong, T. Ouyang, F. Meng, et al. Towards large reasoning models: A survey of reinforced reasoning with large language models. CoRR, abs/2501.09686, 2025.
- S. Xu, W. Fu, J. Gao, W. Ye, W. Liu, Z. Mei, G. Wang, C. Yu, and Y. Wu. Is DPO superior to PPO for LLM alignment? a comprehensive study. CoRR, abs/2404.10719, 2024. — [arXiv:2404.10719](https://arxiv.org/abs/2404.10719) find
- Y. Yan, J. Su, J. He, F. Fu, X. Zheng, Y. Lyu, K. Wang, S. Wang, Q. Wen, and X. Hu. A survey of mathematical reasoning in the era of multimodal large language model: Benchmark, method & challenges. CoRR, abs/2412.11936, 2024. find
- A. Yang, B. Yang, B. Hui, B. Zheng, B. Yu, C. Zhou, C. Li, C. Li, D. Liu, F. Huang, G. Dong, H. Wei, H. Lin, J. Tang, J. Wang, J. Yang, J. Tu, J. Zhang, J. Ma, J. Yang, J. Xu, J. Zhou, J. Bai, J. He, J. Lin, K. Dang, K. Lu, K. Chen, K. Yang, M. Li, M. Xue, N. Ni, P. Zhang, P. Wang, R. Peng, R. Men,… find
- A. Yang, B. Zhang, B. Hui, B. Gao, B. Yu, C. Li, D. Liu, J. Tu, J. Zhou, J. Lin, et al. Qwen2.5-Math technical report: Toward mathematical expert model via self-improvement. CoRR, abs/2409.12122, 2024b.
- K. Yang, A. M. Swope, A. Gu, R. Chalamala, P. Song, S. Yu, S. Godil, R. J. Prenger, and A. Anandkumar. LeanDojo: Theorem proving with retrieval-augmented language models. In Advances in Neural Information Processing Systems 36, 2023.
- L. Yang, K. Lee, R. Nowak, and D. Papailiopoulos. Looped Transformers are better at learning learning algorithms. CoRR, abs/2311.12424, 2024c. find
- S. Yang, D. Schuurmans, P. Abbeel, and O. Nachum. Chain of thought imitation with procedure cloning. In Advances in Neural Information Processing Systems, 2022. find
- T. Yang, M. Yan, H. Zhao, and T. Yang. LemmaHead: RAG assisted proof generation using large language models. CoRR, abs/2501.15797, 2025. — [arXiv:2501.15797](https://arxiv.org/abs/2501.15797) find
- S. Yao, D. Yu, J. Zhao, I. Shafran, T. Griffiths, Y. Cao, and K. Narasimhan. Tree of thoughts: Deliberate problem solving with large language models. In Advances in Neural Information Processing Systems 37, 2023. find
- T. Ye, Z. Xu, Y. Li, and Z. Allen-Zhu. Physics of language models: Part 2.2, how to learn from mistakes on grade-school math problems. CoRR, abs/2408.16293, 2024.
- Y. Ye, Z. Huang, Y. Xiao, E. Chern, S. Xia, and P. Liu. LIMO: Less is more for reasoning. CoRR, abs/2502.03387, 2025.
- Z. Yin, Q. Sun, Q. Guo, Z. Zeng, X. Li, J. Dai, Q. Cheng, X. Huang, and X. Qiu. Reasoning in flux: Enhancing large language models reasoning through uncertainty-aware adaptive guidance. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics, 2024. find
- H. Ying, Z. Wu, Y. Geng, J. Wang, D. Lin, and K. Chen. Lean workbook: A large-scale Lean problem set formalized from natural language math problems. CoRR, abs/2406.03847, 2024a.
- H. Ying, S. Zhang, L. Li, Z. Zhou, Y. Shao, Z. Fei, Y. Ma, J. Hong, K. Liu, Z. Wang, et al. InternLM-Math: Open math large language models toward verifiable reasoning. CoRR, abs/2402.06332, 2024b.
- L. Yu, W. Jiang, H. Shi, Y. Jincheng, Z. Liu, Y. Zhang, J. Kwok, Z. Li, A. Weller, and W. Liu. MetaMath: Bootstrap your own mathematical questions for large language models. In Proceedings of the 12th International Conference on Learning Representations, 2024.
- Q. Yu, Z. Zhang, R. Zhu, Y. Yuan, X. Zuo, Y. Yue, T. Fan, G. Liu, L. Liu, X. Liu, et al. DAPO: An open-source LLM reinforcement learning system at scale. CoRR, abs/2503.14476, 2025.
- L. Yuan, W. Li, H. Chen, G. Cui, N. Ding, K. Zhang, B. Zhou, Z. Liu, and H. Peng. Free process rewards without process labels. CoRR, abs/2412.01981, 2024.
- X. Yue, X. Qu, G. Zhang, Y. Fu, W. Huang, H. Sun, Y. Su, and W. Chen. MAmmoTH: Building math generalist models through hybrid instruction tuning. In Proceedings of the 12th International Conference on Learning Representations, 2024a. X. Yue, T. Zheng, G. Zhang, and W. Chen. Mammoth2: Scaling instruc… — MAmmoTH2: Scaling Instructions from the Web
- H. Zhou, A. Nova, H. Larochelle, A. C. Courville, B. Neyshabur, and H. Sedghi. Teaching algorithmic reasoning via in-context learning. CoRR, abs/2211.09066, 2022. K. Zhou, B. Zhang, J. Wang, Z. Chen, W. X. Zhao, J. Sha, Z. Sheng, S. Wang, and J.-R. Wen. JiuZhang3.0: Efficiently improving mathematica…
Updated
This page in Markdown → https://md.research.elamur.ai/free/en/papers/wang-2025-survey-large-language-models-mathematical/