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A Survey on Large Language Models for Mathematical Reasoning

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This survey argues that mathematical reasoning in LLMs advances through two phases—comprehension, built by pretraining, and answer generation, led by chain-of-thought reasoning—and that this split organizes the field's methods and open…

desk verdict A useful, up-to-date survey of LLM math reasoning that is let down by a framework that doesn't actually organize the content and some fixable production errors. read the letter →

arxiv 2506.08446 v1 pith:VUOVRYXS submitted 2025-06-10 cs.AI cs.CL

Peng-Yuan Wang , Tian-Shuo Liu , Chenyang Wang , Yi-Di Wang , Shu Yan , Cheng-Xing Jia , Xu-Hui Liu , Xin-Wei Chen
show 3 more authors
Jia-Cheng Xu Ziniu Li Yang Yu
This is my paper
classification cs.AIcs.CL
keywords mathematicalreasoninglargelanguagemodelschain-of-thoughtreinforcementlearningtest-timescalingsurvey
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This survey argues that progress in LLM mathematical reasoning is best organized into two high-level phases: comprehension, built by pretraining on diverse mathematical corpora, and answer generation, which has moved from direct prediction to step-by-step chain-of-thought (CoT) reasoning. The survey uses this split to organize the field's methods—prompting, supervised fine-tuning, reinforcement learning, test-time scaling, self-improvement, and external tools—and to identify the open problems. The synthesis matters because it gives researchers a map of where current techniques act and where future progress is bottlenecked.

What carries the argument

The organizing device is the two-phase division of comprehension and answer generation. Comprehension names the mathematical understanding internalized during pretraining; answer generation names the production of solutions, from direct prediction to structured step-by-step reasoning. Within the generation phase, CoT prompting is the key mechanism: intermediate tokens give transformers greater computational reach, so CoT is the load-bearing object connecting the two phases.

What would settle it

A controlled comparison of models matched on pretraining but differing only in CoT training, and vice versa, would show whether one phase alone drives the accuracy gains on benchmarks like MATH or AIME.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that LLMs acquire mathematical competence through comprehension first and answer generation second. Comprehension comes from pretraining on large, curated mathematical corpora; answer generation is the reasoning behavior elicited at inference or fine-tuning time, whose keystone is CoT prompting. The survey reads recent advances as a sequence of ways to strengthen the generation phase, and it gathers evidence that these methods largely unlock latent capabilities rather than create new ones.

Load-bearing premise

The survey's map rests on the assumption that LLM mathematical ability cleanly splits into a comprehension phase and an answer-generation phase.

Editorial extensions

If this is right

  • CoT is more than a prompting trick; it lets models emulate structured procedures, so extending CoT with long reasoning or structural search should keep improving accuracy on multi-step mathematics problems.
  • Reinforcement learning with rule-based rewards improves Pass@1 and, with long CoT, Pass@k, but mostly by activating capabilities already present in the base model.
  • Test-time scaling offers a practical alternative to scaling model parameters for mathematics benchmarks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If reinforcement learning mainly unlocks latent capability, then measuring a base model's Pass@k before training could predict how much RL can help, which the survey does not explicitly propose.
  • The comprehension-versus-generation framing could transfer to non-mathematical reasoning domains, predicting that CoT and search help most when the domain's knowledge is already in pretraining.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This survey reviews research on mathematical reasoning in large language models (LLMs). It organizes the field around two proposed cognitive phases: comprehension (acquiring mathematical understanding through pretraining on diverse corpora) and answer generation (progressing from direct prediction to Chain-of-Thought reasoning). The paper covers background on pretraining, SFT, RL, and prompting; methods for boosting reasoning (SFT, RL, DPO, test-time structural search, self-improvement, external knowledge/tools); discussion of limitations; and future research directions. It also provides tables summarizing pretraining datasets, SFT datasets, and evaluation benchmarks.

Significance. The paper's compilation of recent work on long CoT, rule-based RL, test-time scaling, and self-improvement is timely, and its dataset/benchmark tables (Tables 1–3) are a useful reference. The discussion in Section 5 of RL as unlocking latent capabilities rather than creating new ones is a fair and well-supported observation. However, the central claim—that the paper consolidates fragmented developments into a coherent framework based on comprehension and generation—is not actually realized: the methods in Section 4 are never mapped onto the proposed phases. As a synthesis, the contribution is therefore currently incomplete, though the underlying survey content is largely accurate and potentially valuable after restructuring.

major comments (2)
  1. [Section 3 / Figure 2] The comprehension/generation dichotomy is asserted in the abstract and Section 3, and Figure 2 presents it as the organizing structure, but Section 4 never uses these categories. Section 4.1 (SFT), Section 4.2 (RL), Section 4.3 (test-time search), Section 4.4 (self-improvement), and Section 4.5 (external knowledge) are not classified as either comprehension or answer generation, and the text provides no explanation of how, say, RL with rule-based rewards or RAG relates to the two phases. Because the claimed 'coherent framework' (Section 1) is the paper's stated contribution, this omission is load-bearing: the framework is decorative rather than organizing.
  2. [Section 6] The roadmap at the start of Section 6 promises three directions but cites 'Sections 6 and 6', '(6)', and 'Section 6'—all self-references. The actual subsections are not numbered, so the reader cannot map the three promises (extending performance boundaries, enhancing efficiency, cross-domain generalization) onto the content that follows. This structural error undermines the paper's forward-looking contribution and needs to be fixed, either with proper subsection numbers and cross-references or by removing the broken pointers.
minor comments (5)
  1. [Section 4.1] The first sentence is grammatically incomplete: 'align a pretrained language model with high-quality, human-crafted by supervised learning' appears to be missing a noun such as 'data'.
  2. [Tables 1–3] The 'Synth.' column uses '%' to indicate non-synthetic datasets, which is confusing; use 'No' or a dash instead, since '%' reads as a percentage or a formatting artifact.
  3. [Figure 2] Figure 2 renders 'Comprehension and Generation (§3)' and 'Methods for Boosting Reasoning (§4)' as sibling boxes, visually contradicting the text's claim that Section 4 methods fit within the comprehension/generation framework.
  4. [References] Besta et al. 2024a and 2024b are the same paper (Graph of Thoughts, AAAI 2024) listed twice; similarly, Sprague et al. 2024a and 2024b are duplicates. These should be merged into single entries with consistent in-text citations.
  5. [Throughout] Model naming is inconsistent: 'DeepSeek R1' appears both as 'DeepSeek' and 'Deepseek', and 'Kimi K1.5' appears also as 'Kimi k1.5'. Please standardize.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the survey's two-phase framework and method review are organizational, with no derivation or prediction resting on its own inputs.

full rationale

This is a survey paper; it does not derive or predict results from fitted parameters or from a uniqueness theorem. Its central organizing claim—comprehension via pretraining/data and answer generation via CoT—is introduced as a high-level cognitive framing (Section 3) and used to structure the literature review; it is not obtained from the surveyed methods by construction. The self-citations in Section 6 (BWArea [Jia et al., 2024], CoLA [Jia et al., 2025]) and Section 4.4 (SIRLC [Pang et al., 2023]) are presented as promising research directions or as examples of self-improvement methods, but none of these citations carries the load of the survey's synthesis or of any factual claim about the field; removing them would not change the framework or the survey's conclusions. The potential weakness that Section 4 methods (SFT, RL, DPO, tree search, external tools) are not explicitly classified into the two phases is an organizational/coherence concern, not a circular one: no equation, definition, or fitted quantity reduces to its own input. The paper's own caveat in Section 5 that Pass@k may be an inadequate capability measure is a self-acknowledged limitation, not a circularity. No concrete circular step can be quoted, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim of this survey is the usefulness of its synthesis. That claim rests on the reliability of the cited literature and on the validity of the comprehension/generation taxonomy. No free parameters or invented entities are introduced.

assumptions (3)
  • domain assumption The cited references accurately summarize the state of the art in LLM mathematical reasoning.
    The survey's value rests on accurate representation of the surveyed literature; misquotes or misattributions would undermine its synthesis.
  • ad hoc to paper The comprehension/generation dichotomy is a valid organizing principle for the field.
    The paper introduces this framework in Section 3 as a lens for organizing innovations, but does not justify it against alternative taxonomies.
  • domain assumption Benchmark scores cited (e.g., Grok 3 Beta on AIME 2024) are accurate and comparable.
    The narrative of rapid progress in Section 1 and Figure 1 relies on external benchmark numbers that are not independently verified in the survey.

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Cite this review

Pith. "Pith review of A Survey on Large Language Models for Mathematical Reasoning." pith.science (2026). https://pith.science/paper/VUOVRYXS

@misc{pith2026250608446,
  author       = {Pith},
  title        = {Pith review of: A Survey on Large Language Models for Mathematical Reasoning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VUOVRYXS}},
  note         = {Machine review of arXiv:2506.08446}
}
read the original abstract

Mathematical reasoning has long represented one of the most fundamental and challenging frontiers in artificial intelligence research. In recent years, large language models (LLMs) have achieved significant advances in this area. This survey examines the development of mathematical reasoning abilities in LLMs through two high-level cognitive phases: comprehension, where models gain mathematical understanding via diverse pretraining strategies, and answer generation, which has progressed from direct prediction to step-by-step Chain-of-Thought (CoT) reasoning. We review methods for enhancing mathematical reasoning, ranging from training-free prompting to fine-tuning approaches such as supervised fine-tuning and reinforcement learning, and discuss recent work on extended CoT and "test-time scaling". Despite notable progress, fundamental challenges remain in terms of capacity, efficiency, and generalization. To address these issues, we highlight promising research directions, including advanced pretraining and knowledge augmentation techniques, formal reasoning frameworks, and meta-generalization through principled learning paradigms. This survey tries to provide some insights for researchers interested in enhancing reasoning capabilities of LLMs and for those seeking to apply these techniques to other domains.

Figures

Figures reproduced from arXiv: 2506.08446 by the authors.

Figure 1
Figure 1. The rapid advancements in mathematical capabilities in recent years. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The organization of this survey. Despite their impressive language understanding abilities, pre-trained LLMs often struggle to produce contextually appropriate and relevant responses when applied directly to downstream tasks, frequently resulting in repetitive or irrelevant output. To mitigate these shortcomings, prompt engineering, particularly approaches like In-Context Learning (ICL) [Jie and Lu, 2023, Zhou et al… view at source ↗
Figure 3
Figure 3. Scaling up LLMs leads to surpassing the top 2.5% of human participants on AIME 2024 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: An illustration of short and long CoT. pattern matching; they internalize conceptual structures and flexibly integrate them across diverse reasoning scenarios. 3.2 Chain-of-Thought: A Key Driver of LLM Generation LLMs encode task instructions into latent representation…
Figure 5
Figure 5. Figure 5: Schematic illustrating of different method. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

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Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    Post-training LLMs first on abstract number-free reasoning plans (CoMT), then with confidence-weighted rewards (CCRL), raises math accuracy by ~2-5 points over standard CoT-SFT+RL across four models.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.