Enterprise AI Analysis
A Survey on Large Language Models for Mathematical Reasoning
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 and answer generation. It reviews methods for enhancing mathematical reasoning, discusses extended Chain-of-Thought and test-time scaling, and highlights promising research directions.
Authored by PENGYUAN WANG, TIANSHUO LIU, CHENYANG WANG et al. | Published: 04 February 2026
Executive Impact Summary
Large Language Models are transforming mathematical reasoning, achieving human-competitive performance and unlocking new efficiencies for enterprise applications.
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Reasoning Efficiency Gain (Est.)
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Generalization Improvement (Est.)
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Pre-Training Foundation
Pre-training equips LLMs with foundational mathematical knowledge by exposing them to extensive corpora including textbooks and problem datasets (Section 2.1). This data-driven approach allows models to internalize domain-specific knowledge, terminology, and contextual reasoning patterns, moving beyond traditional rule-based systems. Larger scale pre-training significantly enhances comprehension and generalization capabilities, with examples like OpenWebMath (14.7B tokens) and MathPile (9.5B tokens) improving performance on complex mathematical tasks.
Supervised Fine-Tuning
Supervised Fine-Tuning (SFT) adapts pre-trained LLMs to specific mathematical tasks using high-quality human-crafted demonstrations (Section 2.2). This process aligns models with instruction-following objectives, helping them generate accurate and structured solutions. Constructing diverse, correct, and relevant training data is crucial. Techniques include using strong LLMs to generate synthetic data (e.g., OpenMathInstruct-1 with GPT-4), question bootstrapping, and incorporating error-correction data to develop error detection capabilities.
Reinforcement Learning Approaches
Reinforcement Learning (RL) methods further enhance LLMs' problem-solving via trial-and-error exploration, especially for Chain-of-Thought (CoT) reasoning (Section 2.3, 4.2). RL helps optimize long CoT generation, guiding models towards coherent reasoning trajectories using various reward models: Outcome Reward Models (ORM), Process Reward Models (PRM), and Rule-Based Rewards. PRMs, which provide detailed step-by-step feedback, generally outperform ORMs. RLHF-inspired approaches like Proximal Policy Optimization (PPO) and its variants (ReMax, RLOO) are prominent, with DeepSeek R1 demonstrating breakthrough performance with rule-based rewards for long CoT.
Effective Prompting Strategies
Prompting has emerged as a simple yet effective way to elicit and enhance reasoning capabilities (Section 2.4). Zero-shot prompting provides task instructions, while few-shot prompting includes examples to infer reasoning strategies. Chain-of-Thought (CoT) prompting, particularly with phrases like "Let's think step by step," guides models to generate intermediate reasoning steps, significantly improving accuracy for complex tasks. Advanced strategies include Tree-of-Thoughts (ToT) and Graph-of-Thoughts (GoT) for structured reasoning and exploration of solution paths.
External Knowledge Integration
Integrating external knowledge is crucial for tasks requiring precise computational accuracy or real-time information, mitigating hallucinations (Section 4.5). Two main approaches are external tool integration (e.g., calculators, code interpreters, symbolic computation systems) and Retrieval-Augmented Generation (RAG). External tools extend model capabilities beyond static parameters, enabling dynamic program execution and real-time information retrieval. RAG retrieves relevant documents (theorems, definitions) to enhance accuracy and verifiability, especially for complex mathematical problems.
Test Time Inference Optimization
Test-time scaling, through increased generation burden, enhances mathematical reasoning, especially for complex problems with high token budgets (Section 4.3). While majority voting helps with answer derivation, tree search methods are crucial for exhaustive solution space exploration. Structured tree search, as seen in Tree-of-Thoughts (ToT) and Graph-of-Thoughts (GoT), enables systematic exploration and backtracking. Monte Carlo Tree Search (MCTS) further optimizes search efficiency by constructing structural trees for optimal path exploration.
83.9%
AIME 2024 Score Achieved by Grok 3 Beta
LLM Mathematical Reasoning Process
Comprehension (Problem Understanding)
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Chain-of-Thought (Step-by-Step Reasoning)
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Generation (Solution Synthesis)
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Refinement & Verification (Self-Correction)
| Feature | Supervised Fine-Tuning (SFT) | Reinforcement Learning (RL) |
|---|---|---|
| Primary Goal | Align with human instructions | Optimize problem-solving via exploration |
| Data Dependence | High-quality, human-annotated datasets | Reward signals (ORM, PRM, Rule-Based) |
| Strengths |
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| Limitations |
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| Key Techniques | Data augmentation, error correction | PPO, ReMax, Rule-based rewards, DPO |
Case Study: DeepSeek-R1's Breakthrough in Long CoT RL
DeepSeek R1 demonstrated transformative potential by achieving comparable or superior performance to OpenAI's o1 through a purely RL-based approach for long Chain-of-Thought reasoning. This breakthrough, detailed in Section 4.2.2, emulated System 2-style reasoning by focusing on three critical factors:
- Golden Reward: Employing rule-based rewards derived from ground truth answers to ensure reliable correctness evaluation and stable exploration dynamics, mitigating reward model inaccuracies and hacking behaviors.
- Scaling CoT Length: Extending reasoning sequences from short to long significantly enhanced the model's exploration capabilities and strengthened intrinsic reasoning, promoting deeper analytical thinking and diverse response spaces.
- Pure RL Training: Bypassing the Supervised Fine-Tuning (SFT) phase entirely and directly applying reinforcement learning to the base model. This maximized exploration potential by allowing models to discover novel high-quality responses beyond SFT dataset constraints.
This strategic combination advanced LLMs' mathematical reasoning abilities, particularly for multi-step problems and competition-level mathematics.
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Your AI Implementation Roadmap
A strategic five-phase approach to integrating advanced LLMs for mathematical reasoning into your enterprise workflow.
Phase 1: Foundation & Pre-Training Optimization
Strategically curate diverse, high-quality mathematical corpora (textbooks, proofs, problem sets) to enhance LLM comprehension and generalization. Focus on data distribution and error-correction integration during pre-training to build robust foundational reasoning.
Phase 2: Supervised Fine-Tuning & CoT Integration
Develop high-quality SFT datasets with structured Chain-of-Thought (CoT) demonstrations. Emphasize generating logically coherent, step-by-step reasoning paths. Utilize strong LLMs for synthetic data generation and incorporate error-correction data to train for error detection and self-correction.
Phase 3: Advanced Reinforcement Learning & Reward Modeling
Implement RL with verifiable rewards (rule-based or process reward models) to optimize long CoT generation. Focus on maintaining exploration capabilities while ensuring training stability. Explore efficient RL algorithms (e.g., ReMax, RLOO) and direct preference optimization (DPO) for refining reasoning trajectories.
Phase 4: Test-Time Structural Search & Self-Improvement
Integrate structured tree search methods (ToT, GoT, MCTS) for systematic exploration of solution spaces and strategic backtracking during inference. Develop self-improvement methodologies, including iterative refinement with feedback loops and self-verification mechanisms, to enhance response quality and address inconsistencies.
Phase 5: External Knowledge & Cross-Domain Generalization
Incorporate external tools (calculators, code interpreters, symbolic systems) and Retrieval-Augmented Generation (RAG) for real-time information and precise computation. Research compact representation spaces for efficient RL exploration and develop verification-aware optimization strategies to improve logical consistency and cross-domain reasoning capabilities.
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