ChatGLM-Math: Improving Math Problem-Solving in Large Language Models with a Self-Critique Pipeline

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1. Very close Another approach to the problem Not relevant

   Improving Large Language Model Fine-tuning for Solving Math Problems

   Yixin Liu, Avi Singh, C. D. Freeman et al. · 2023 · arXiv.org · preprint, not peer reviewed · cited 78 times · Not summarized yet

   Improving Large Language Model Fine-tuning for Solving Math Problems tests solution fine-tuning, verifier reranking, and related strategies on MATH rather than a self-critique alignment pipeline.

   Abstract

   Despite their success in many natural language tasks, solving math problems remains a significant challenge for large language models (LLMs). A large gap exists between LLMs' pass-at-one and pass-at-N performance in solving math problems, suggesting LLMs might be close to finding correct solutions, motivating our exploration of fine-tuning methods to unlock LLMs' performance. Using the challenging MATH dataset, we investigate three fine-tuning strategies: (1) solution fine-tuning, where we fine-tune to generate a detailed solution for a given math problem; (2) solution-cluster re-ranking, where the LLM is fine-tuned as a solution verifier/evaluator to choose among generated candidate solutio…

2. Very close Same question Cites this paper Not relevant

   SIaM: Self-Improving Code-Assisted Mathematical Reasoning of Large Language Models

   Dian Yu, Baolin Peng, Ye Tian et al. · 2024 · arXiv.org · preprint, not peer reviewed · cited 5 times · Not summarized yet

   SIaM improves LLM mathematical reasoning through self-improvement, using code-assisted solutions and expert-written question-answer data.

   Abstract

   There is a growing trend of teaching large language models (LLMs) to solve mathematical problems through coding. Existing studies primarily focus on prompting powerful, closed-source models to generate seed training data followed by in-domain data augmentation, equipping LLMs with considerable capabilities for code-aided mathematical reasoning. However, continually training these models on augmented data derived from a few datasets such as GSM8K may impair their generalization abilities and restrict their effectiveness to a narrow range of question types. Conversely, the potential of improving such LLMs by leveraging large-scale, expert-written, diverse math question-answer pairs remains une…

3. Very close Another approach to the problem Cited by this paper Used in depth by this paper Not relevant

   MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models

   L. Yu, Weisen Jiang, Han Shi et al. · 2023 · International Conference on Learning Representations · cited 790 times · Not summarized yet

   MetaMath fine-tunes LLaMA-2 on questions rewritten from multiple perspectives, while this paper uses model-generated critique and preference training.

   Abstract

   Large language models (LLMs) have pushed the limits of natural language understanding and exhibited excellent problem-solving ability. Despite the great success, most existing open-source LLMs (e.g., LLaMA-2) are still far away from satisfactory for solving mathematical problem due to the complex reasoning procedures. To bridge this gap, we propose MetaMath, a fine-tuned language model that specializes in mathematical reasoning. Specifically, we start by bootstrapping mathematical questions by rewriting the question from multiple perspectives without extra knowledge, which results in a new dataset called MetaMathQA. Then we fine-tune the LLaMA-2 models on MetaMathQA. Experimental results on …

4. Very close Same method, another problem Not relevant

   DeepCritic: Deliberate Critique with Large Language Models

   Wenkai Yang, Jingwen Chen, Yankai Lin et al. · 2025 · arXiv.org · preprint, not peer reviewed · cited 16 times · Not summarized yet

   DeepCritic develops LLMs’ mathematical critique ability, whereas this paper trains a Math-Critique model to provide feedback for improving problem-solving.

   Abstract

   As Large Language Models (LLMs) are rapidly evolving, providing accurate feedback and scalable oversight on their outputs becomes an urgent and critical problem. Leveraging LLMs as critique models to achieve automated supervision is a promising solution. In this work, we focus on studying and enhancing the math critique ability of LLMs. Current LLM critics provide critiques that are too shallow and superficial on each step, leading to low judgment accuracy and struggling to offer sufficient feedback for the LLM generator to correct mistakes. To tackle this issue, we propose a novel and effective two-stage framework to develop LLM critics that are capable of deliberately critiquing on each re…

5. Very close Same question Not relevant

   Can LLMs Master Math? Investigating Large Language Models on Math Stack Exchange

   Ankit Satpute, Noah Gießing, André Greiner-Petter et al. · 2024 · Annual International ACM SIGIR Conference on Research and Development in Information Retrieval · cited 57 times · Not summarized yet

   Can LLMs Master Math? tests how well LLMs answer real Math Stack Exchange questions, while this paper improves their mathematical problem-solving.

   Abstract

   Large Language Models (LLMs) have demonstrated exceptional capabilities in various natural language tasks, often achieving performances that surpass those of humans. Despite these advancements, the domain of mathematics presents a distinctive challenge, primarily due to its specialized structure and the precision it demands. In this work, we follow a two-step approach to investigating the proficiency of LLMs in answering mathematical questions. First, we employ the most effective LLMs, as identified by their performance on math question-answer benchmarks, to generate answers to 78 questions from the Math Stack Exchange (MSE). Second, a case analysis is conducted on the LLM that showed the hi…

6. Very close Shared data, tool or method Not relevant

   FineMath: A Fine-Grained Mathematical Evaluation Benchmark for Chinese Large Language Models

   Yan Liu, Ren-Ren Jin, Ling Shi et al. · 2024 · ACM Trans. Asian Low Resour. Lang. Inf. Process. · cited 9 times · Not summarized yet

   FineMath provides a manually annotated Chinese math evaluation benchmark, while this paper releases MathUserEval and evaluation scripts for testing mathematical problem-solving.

   Abstract

   To thoroughly assess the mathematical reasoning abilities of Large Language Models (LLMs), we need to carefully curate evaluation datasets covering diverse mathematical concepts and mathematical problems at different difficulty levels. In pursuit of this objective, we propose FineMath in this article, a fine-grained mathematical evaluation benchmark dataset for assessing Chinese LLMs. FineMath is created to cover the major key mathematical concepts taught in elementary school math, which are further divided into 17 categories of math word problems, enabling in-depth analysis of mathematical reasoning abilities of LLMs. All the 17 categories of math word problems are manually annotated with t…

7. Very close Same method, another problem Not relevant

   Enhancing LLM Reasoning via Critique Models with Test-Time and Training-Time Supervision

   Zhiheng Xi, Dingwen Yang, Jixuan Huang et al. · 2024 · Computation and Language · preprint, not peer reviewed · cited 55 times · AI summary available

   Enhancing LLM Reasoning uses critique models to supervise reasoning during training and testing, as this paper uses critique feedback in training to improve math solving.

   Abstract

   Training large language models (LLMs) to spend more time thinking and reflection before responding is crucial for effectively solving complex reasoning tasks in fields such as science, coding, and mathematics. However, the effectiveness of mechanisms like self-reflection and self-correction depends on the model's capacity to accurately assess its own performance, which can be limited by factors such as initial accuracy, question difficulty, and the lack of external feedback. In this paper, we delve into a two-player paradigm that separates the roles of reasoning and critique models, where the critique model provides step-level feedback to supervise the reasoning (actor) model during both tes…

8. Very close Another approach to the problem Not relevant

   WarriorMath: Enhancing the Mathematical Ability of Large Language Models with a Defect-aware Framework

   Yue Chen, Ming-Hua He, Fangkai Yang et al. · 2025 · arXiv.org · preprint, not peer reviewed · cited 5 times · Not summarized yet

   WarriorMath targets model-specific failures with defect-aware data synthesis and progressive training, rather than this paper’s self-critique and preference-training pipeline.

   Abstract

   Large Language Models (LLMs) excel in solving mathematical problems, yet their performance is often limited by the availability of high-quality, diverse training data. Existing methods focus on augmenting datasets through rephrasing or difficulty progression but overlook the specific failure modes of LLMs. This results in synthetic questions that the model can already solve, providing minimal performance gains. To address this, we propose WarriorMath, a defect-aware framework for mathematical problem solving that integrates both targeted data synthesis and progressive training. In the synthesis stage, we employ multiple expert LLMs in a collaborative process to generate, critique, and refine…

Prior work it builds on

1. Close Another approach to the problem Used in depth by this paper Not relevant

   MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning

   Xiang Yue, Xing-Wei Qu, Ge Zhang et al. · 2023 · International Conference on Learning Representations · cited 641 times · Not summarized yet

   MAmmoTH trains math problem-solving models on hybrid chain-of-thought and program-of-thought instruction data rather than self-generated critique feedback.

   Abstract

   We introduce MAmmoTH, a series of open-source large language models (LLMs) specifically tailored for general math problem-solving. The MAmmoTH models are trained on MathInstruct, our meticulously curated instruction tuning dataset. MathInstruct is compiled from 13 math datasets with intermediate rationales, six of which have rationales newly curated by us. It presents a unique hybrid of chain-of-thought (CoT) and program-of-thought (PoT) rationales, and also ensures extensive coverage of diverse fields in math. The hybrid of CoT and PoT not only unleashes the potential of tool use but also allows different thought processes for different math problems. As a result, the MAmmoTH series substan…

2. Close Broader context Cites this paper Used in depth by this paper A method it uses Not relevant

   Black-Box Prompt Optimization: Aligning Large Language Models without Model Training

   Jiale Cheng, Xiao Liu, Ke-Han Zheng et al. · 2023 · Annual Meeting of the Association for Computational Linguistics · cited 178 times · Not summarized yet

   Black-Box Prompt Optimization aligns LLMs by optimizing prompts without training, addressing instruction alignment rather than mathematical problem-solving.

   Abstract

   Large language models (LLMs) have shown impressive success in various applications. However, these models are often not well aligned with human intents, which calls for additional treatments on them; that is, the alignment problem. To make LLMs better follow user instructions, existing alignment methods primarily focus on further training them. However, the extra training of LLMs is usually expensive in terms of GPU computing; even worse, some LLMs are not accessible for user-demanded training, such as GPTs. In this work, we take a different perspective -- Black-Box Prompt Optimization (BPO) -- to perform alignments. The idea is to optimize user prompts to suit LLMs' input understanding, so …

3. Close* Same method, another problem Not relevant

   CritiqueLLM: Scaling LLM-as-Critic for Effective and Explainable Evaluation of Large Language Model Generation

   Pei Ke, Bo-Si Wen, Zhuo-Er Feng et al. · 2023 · arXiv.org · preprint, not peer reviewed · cited 61 times · Not summarized yet

   CritiqueLLM scales LLMs as critics for evaluating generated responses, while this paper trains a critique model to guide mathematical self-improvement.

4. Close Same question Not relevant

   How well do Large Language Models perform in Arithmetic tasks?

   Zheng Yuan, Hongyi Yuan, Chuan-Qi Tan et al. · 2023 · arXiv.org · preprint, not peer reviewed · cited 188 times · Not summarized yet

   How well do Large Language Models perform in Arithmetic tasks? benchmarks arithmetic abilities, while this paper evaluates and improves broader mathematical problem-solving.

   Abstract

   Large language models have emerged abilities including chain-of-thought to answer math word problems step by step. Solving math word problems not only requires abilities to disassemble problems via chain-of-thought but also needs to calculate arithmetic expressions correctly for each step. To the best of our knowledge, there is no work to focus on evaluating the arithmetic ability of large language models. In this work, we propose an arithmetic dataset MATH 401 to test the latest large language models including GPT-4, ChatGPT, InstrctGPT, Galactica, and LLaMA with various arithmetic expressions and provide a detailed analysis of the ability of large language models. MATH 401 and evaluation c…

5. Close Broader context Not relevant

   MathAttack: Attacking Large Language Models Towards Math Solving Ability

   Zihao Zhou, Qiufeng Wang, Mingyu Jin et al. · 2023 · AAAI Conference on Artificial Intelligence · cited 44 times · Not summarized yet

   MathAttack probes robustness by perturbing math word problems while preserving their logic, whereas this paper focuses on improving mathematical solving ability.

   Abstract

   With the boom of Large Language Models (LLMs), the research of solving Math Word Problem (MWP) has recently made great progress. However, there are few studies to examine the robustness of LLMs in math solving ability. Instead of attacking prompts in the use of LLMs, we propose a MathAttack model to attack MWP samples which are closer to the essence of robustness in solving math problems. Compared to traditional text adversarial attack, it is essential to preserve the mathematical logic of original MWPs during the attacking. To this end, we propose logical entity recognition to identify logical entries which are then frozen. Subsequently, the remaining text are attacked by adopting a word-le…

Papers that cite it

1. Close Builds on this paper Not relevant

   LEMMA: Learning from Errors for MatheMatical Advancement in LLMs

   Zhuo-Shi Pan, Yu Li, Hong-Lin Lin et al. · 2025 · Annual Meeting of the Association for Computational Linguistics · cited 36 times · Not summarized yet

   LEMMA uses error data to improve mathematical reasoning, extending the paper’s related use of critique and feedback for math problem-solving.

   Abstract

   Large language models (LLMs) have demonstrated remarkable reasoning capability in solving mathematical problems. However, existing approaches primarily focus on improving the quality of correct training data, e.g., distilling high-quality correct solutions from advanced models, neglecting the value contained in error data, potentially hindering the model's reflective ability. Though some studies attempt to leverage error data, they often involve complex mechanisms, such as Monte Carlo Tree Search (MCTS) to explore error nodes. In this work, we propose to enhance LLMs' reasoning ability by Learning from Errors for Mathematical Advancement (LEMMA). LEMMA constructs data consisting of an incorr…

2. Close Review Not relevant

   A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges

   Yibo Yan, Jiamin Su, Jianxiang He et al. · 2024 · Annual Meeting of the Association for Computational Linguistics · cited 82 times · Not summarized yet

   The survey reviews benchmarks, methods, and challenges in mathematical reasoning for multimodal LLMs, including work on math-LLM improvement.

   Abstract

   Mathematical reasoning, a core aspect of human cognition, is vital across many domains, from educational problem-solving to scientific advancements. As artificial general intelligence (AGI) progresses, integrating large language models (LLMs) with mathematical reasoning tasks is becoming increasingly significant. This survey provides the first comprehensive analysis of mathematical reasoning in the era of multimodal large language models (MLLMs). We review over 200 studies published since 2021, and examine the state-of-the-art developments in Math-LLMs, with a focus on multimodal settings. We categorize the field into three dimensions: benchmarks, methodologies, and challenges. In particular…

3. Close Same question Not relevant

   Skywork-Math: Data Scaling Laws for Mathematical Reasoning in Large Language Models - The Story Goes On

   Liang Zeng, Liangjun Zhong, Liang Zhao et al. · 2024 · arXiv.org · preprint, not peer reviewed · cited 25 times · Not summarized yet

   Skywork-Math studies how training-data quantity affects LLM mathematical reasoning and introduces a large math question-answer dataset.

   Abstract

   In this paper, we investigate the underlying factors that potentially enhance the mathematical reasoning capabilities of large language models (LLMs). We argue that the data scaling law for math reasoning capabilities in modern LLMs is far from being saturated, highlighting how the model's quality improves with increases in data quantity. To support this claim, we introduce the Skywork-Math model series, supervised fine-tuned (SFT) on common 7B LLMs using our proposed 2.5M-instance Skywork-MathQA dataset. Skywork-Math 7B has achieved impressive accuracies of 51.2% on the competition-level MATH benchmark and 83.9% on the GSM8K benchmark using only SFT data, outperforming an early version of G…

4. Close* Shared data, tool or method Not relevant

   MathScape: Evaluating MLLMs in multimodal Math Scenarios through a Hierarchical Benchmark

   Minxuan Zhou, Hao Liang, Tian-Peng Li et al. · 2024 · arXiv.org · preprint, not peer reviewed · cited 20 times · Not summarized yet

   MathScape provides a hierarchical benchmark for evaluating multimodal LLMs in mathematical scenarios, complementing this paper’s mathematical evaluation resources.

5. Close Shared data, tool or method Not relevant

   CriticEval: Evaluating Large Language Model as Critic

   Tian Lan, Wen-Wei Zhang, Chen Xu et al. · 2024 · cited 12 times · Not summarized yet

   CriticEval introduces a benchmark for evaluating LLM critique abilities, a capability this paper uses to generate mathematical feedback.

   Abstract

   Critique ability, i.e., the capability of Large Language Models (LLMs) to identify and rectify flaws in responses, is crucial for their applications in self-improvement and scalable oversight. While numerous studies have been proposed to evaluate critique ability of LLMs, their comprehensiveness and reliability are still limited. To overcome this problem, we introduce CriticEval, a novel benchmark designed to comprehensively and reliably evaluate critique ability of LLMs. Specifically, to ensure the comprehensiveness, CriticEval evaluates critique ability from four dimensions across nine diverse task scenarios. It evaluates both scalar-valued and textual critiques, targeting responses of var…