Title: Atomic Thinking of LLMs: Decoupling and Exploring Mathematical Reasoning Abilities
Venue: NeurIPS 2025
Accepted: yes
Keywords: Large Language Models

Mathematical Reasoning

Atomic Thinking
Abstract: 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".


Peer reviews:
comments: SUMMARY:
This study creates a benchmark that investigates LLMs' atomic-level math reasoning performance by dividing the capability with fields (algebra, geometry, analysis, and topology) and logical reasoning abilities (Conceptual understanding, Forward reasoning with formal language, Backward reasoning with counterexamples). The evaluation with a wide range of math LLMs reveals several insights such as strength and weakness of each LLM, knowledge transfer characteristics between fields and reasoning types.

STRENGTHS AND WEAKNESSES:
It is a novel approach to categorize the math capability into sub-types, create the benchmark, and evaluate each LLMs performance.
In addition, the evaluation results show us several insights which will provide important clues for improving the performance of mathematical LLMs in the future.

However, there are some weaknesses and points to be improved.
- While the findings are interesting, they are not surprising. For example, we often find knowledge transfers between close fields (For example, https://neurips.cc/virtual/2024/poster/93733). In addition, LLMs are not good at Geometry because it cannot treat vision.
- At training for atomic capability interaction, only one model (Qwen2.5-Math-Instruct-7B) is used for the experiment. No less than two models are needed to insist the universality of the findings.
- VLMs are not included within the scope of the analysis. In particular, geometry needs vision information to completely grasp the problems.
- The Supplementary zip file could not be unzipped due to some errors.

QUESTIONS:
See "Strengths And Weaknesses".

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-16

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents an approach that focuses on fundamental atomic capability units to enhance the mathematical reasoning in LLMs.  The authors study field specific abilities vis-a-vis four mathematical fields and logical abilities at different  reasoning levels. The authors also develop training and evaluation datasets for each of the atomic mathematical units. The paper presents an extensive comparative analysis of different types math reasoning LLMs and their own developed supervised models taking various factors into consideration.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper proposes the paradigm of atomic thinking which is intertwined with the cognitive abilities of humans while solving mathematical problems. Although current LLMs owe their mathematical reasoning capabilities to large scale of training data and implicit step-by-step supervision, this paper  breaks the reasoning task at two levels of field atomic units and logical reasoning based on three core reasoning capabilities. The coverage of experiments is vast and the paper analyzes the performance of the models in terms of fields and level of difficulty. The paper also discusses the transfer capabilities of fields and provides empirical evidence for it. Training and evaluation datasets for each of the atomic capability units have been developed which is a significant contribution to the field.\n\nThe authors do not mention the criteria of selection of the baseline LLMs. The construction of the dataset is not fully explained which makes things unclear. It is not clear what the authors mean by Level 1 and Level 2 problems. Although the paper relies on automatic evaluation metrics, a thorough human evaluation is required as a math problem can be solved in multiple ways using different kinds of mathematical axioms and concepts. It should be mentioned how the difficulty level is determined for a problem.\n\nQUESTIONS:\n1. The authors should mention the criteria of selection of the baseline LLMs both the open-source and commercial ones.\n2. The details of the data construction should be detailed with examples in the appendix.\n3. Human evaluation should be conducted for this task.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces \"atomic thinking,\" a new paradigm for analyzing the mathematical reasoning abilities of Large Language Models (LLMs). The authors argue that current models, often trained on long and complex problems, may be memorizing reasoning chains rather than truly understanding fundamental concepts. To address this, they propose decoupling mathematical ability into two dimensions: field-specific capabilities (Algebra, Geometry, Analysis, Topology) and logical capabilities (conceptual understanding, forward reasoning with formal language, and backward reasoning with counterexamples. The study finds that models generally excel in algebra and analysis but struggle with geometry and topology. Furthermore, it highlights that strong conceptual understanding significantly boosts other reasoning abilities, suggesting a shift in training strategy from \"question drilling\" towards a more foundational, concept-based approach.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- Clarity: The paper is well-written and organized. The core idea of \"atomic thinking\" is introduced and motivated clearly, and Figure 1 provides an excellent visual summary of the paper's entire scope and key findings.  The research questions are explicitly stated in the introduction and systematically answered in the analysis sections, making the paper easy to follow and understand.\n\n- Significance & Originality: In my opinion, the paper's primary strength lies in its novel and insightful framing of LLM mathematical reasoning. As we see models achieve near-superhuman scores on some benchmarks, it's becoming crucial to ask how they are solving these problems. This paper provides a structured methodology for doing just that. Moving the focus from \"can it solve the problem?\" to \"what fundamental skills does it possess?\" is a significant step forward.\n\n- Quality: The empirical evaluation is comprehensive and of high quality.  The authors benchmark a diverse and modern set of LLMs, including powerful proprietary models like OpenAI o1, Deepseek-R1, and Gemini 2.5-pro, which lends credibility and relevance to their findings. The methodology for creating the datasets is systematic, and the analysis of the results is thorough, supported by clear tables and illustrative case studies. \n\n## Weaknesses\n\n- While the proposed decoupling is a major strength, the choice of what constitutes an \"atom\" is debatable. Fields like \"Algebra\" or \"Geometry\" are vast and could be considered composites of many smaller, more fundamental skills (e.g., symbolic manipulation, spatial transformation, etc.). The authors briefly justify their choice to avoid over-complicating the interaction analysis, but the framework's claims to \"atomicity\" could be softened.\n\nQUESTIONS:\n1. The paper defines \"field capabilities\" like Algebra and Geometry as atomic units.  However, these are themselves complex domains. Could you elaborate on the process for selecting this level of granularity? Were finer-grained decompositions (e.g., breaking algebra into \"equation solving\" and \"proof by induction\") considered, and what were the primary trade-offs that led to the current four-field structure?\n\n2. The interaction experiments (Tables 3-5) offer great insights but are based on fine-tuning a single model, Qwen2.5-Math-7B.  How confident are you that these interaction effects (e.g., the foundational role of algebra and conceptual understanding) are fundamental principles of mathematical reasoning in LLMs, rather than artifacts of the specific model's architecture or pre-training? Would you expect similar results if you fine-tuned a model from a different family, such as Llama?\n\nLIMITATIONS:\nIn Section 6, \"Conclusion, limitation, and future directions,\" the authors explicitly state that they \"have not explored more advanced strategies to stimulate a specific atomic capability, such as curriculum learning or reinforcement learning.\"  This is a fair and important acknowledgment.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel paradigm termed \"Atomic Thinking\" to evaluate and understand the mathematical reasoning abilities of Large Language Models (LLMs) at a more granular level. The authors argue that current end-to-end evaluation methods fall short of revealing whether LLMs genuinely master mathematical concepts or merely memorize problem-solving patterns. Inspired by human cognition, which decomposes complex problems into fundamental components, this work decouples mathematical intelligence into two primary dimensions: (1) field-specific atomic abilities, covering algebra, geometry, analysis, and topology, and (2) logical atomic abilities, which include conceptual understanding, formal forward reasoning, and counterexample-driven backward reasoning.\n\nTo support this framework, the paper contributes specialized training and evaluation datasets for each atomic capability. Through extensive experiments on a wide array of state-of-the-art open-source and proprietary LLMs, the study provides a detailed-performance landscape. A key contribution lies in the interaction experiments, where the authors fine-tune models on one atomic ability to measure its impact on others.\n\nThe main findings are insightful: LLMs generally exhibit stronger performance in algebra and analysis but struggle with geometry and topology. Crucially, the paper reveals that even top-tier models have significant deficiencies in backward reasoning (i.e., constructing counterexamples). It also establishes that foundational skills, particularly conceptual understanding and algebraic abilities, play a pivotal role in boosting performance across other, more abstract domains. These contributions offer a new lens for analyzing LLM cognition and provide actionable insights for developing more efficient and generalizable training strategies for mathematical reasoning.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths:\nThe \"Atomic Thinking\" framework is a clever and highly original contribution. It moves beyond standard end-to-end benchmarks to offer a more insightful, fine-grained analysis of what LLMs actually learn in mathematics. The experimental design is robust, covering a wide range of relevant models and thoughtfully decomposing skills. The paper's most significant contribution comes from its interaction experiments, which reveal fascinating, non-obvious relationships between different mathematical abilities—for example, showing that training on algebra can boost geometry skills more effectively than training on geometry itself. These findings are not only interesting but also provide practical guidance for developing better training strategies.\n\n\n## Weaknesses: \n1）The paper's core idea is to \"decouple\" skills, but this is practically impossible to do perfectly. An algebra task might still rely on conceptual understanding, for example. The authors should acknowledge this limitation more directly and discuss how the inherent overlap between skills might affect their conclusions. It doesn't invalidate the work, but it's an important nuance to address.\n\n2）The claims about ability transfer (Tables 3, 4, 5) are based on a single training run. Given the stochastic nature of fine-tuning, these results could be due to chance. The conclusions would be far more convincing if the authors ran each experiment multiple times (e.g., with different seeds) and reported means and standard deviations. Without this, it's hard to be certain that the observed performance gains are statistically significant.\n\n3）The finding that models do better on hard topology problems than easy ones is intriguing but left underdeveloped. The authors suggest a data distribution mismatch, which is a reasonable hypothesis, but they don't provide any evidence to back it up. A few concrete examples of the problems in question or a quick analysis of relevant training data would make this point much stronger and less speculative.\n\nQUESTIONS:\n1. The \"decoupling\" of skills is central to your work, but it's hard to do perfectly. How did you control for skill overlap when creating your datasets? For example, how did you isolate \"forward reasoning\" from \"conceptual understanding\"?\n2. Your interaction experiments (Table 3, etc.) are based on single training runs. Can you run these experiments multiple times with different seeds and report the mean and standard deviation? Without this, it's difficult to know if the results are just due to chance. This is my main concern.\n3. The result where models are better at hard topology problems than easy ones is interesting. Can you provide some examples of these problems? Or, even better, show some evidence from training data that might explain this strange result? Right now, it feels too speculative.\n\nLIMITATIONS:\nYes, the authors have included a \"Conclusion, limitation, and future directions\" section (Section 6) that acknowledges some limitations of their work, such as not exploring more advanced training strategies like curriculum or reinforcement learning. This is a good start.\nHowever, the discussion could be improved by also explicitly addressing the key limitations raised in my review, namely:\n\n1. The inherent difficulty of achieving \"pure\" decoupling of atomic abilities and its potential impact on the experimental conclusions.\n2. The lack of statistical validation for the interaction experiments due to single-run evaluations, which affects the robustness of the transfer learning claims.\n\nAdding these points would provide a more complete picture of the work's current scope and strengthen the paper by demonstrating a thorough self-assessment.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}