# Common 7B Language Models Already Possess Strong Math Capabilities

arXiv: https://arxiv.org/abs/2403.04706 · HTML: https://mlsanity.com/paper/2403.04706/

## Key facts

- Cited by 109 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 7 Mar 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 21 of 21)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [WizardLM: Empowering large pre-trained language models to follow complex instructions](https://mlsanity.com/paper/2304.12244/) (2023)
- [LIMA: Less Is More for Alignment](https://mlsanity.com/paper/2305.11206/) (2023)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [Complexity-Based Prompting for Multi-Step Reasoning](https://mlsanity.com/paper/2210.00720/) (2022)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [A Diverse Corpus for Evaluating and Developing English Math Word Problem Solvers](https://mlsanity.com/paper/2106.15772/) (2021)
- [ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving](https://mlsanity.com/paper/2309.17452/) (2023)
- [How Abilities in Large Language Models are Affected by Supervised Fine-tuning Data Composition](https://mlsanity.com/paper/2310.05492/) (2023)
- [MuggleMath: Assessing the Impact of Query and Response Augmentation on Math Reasoning](https://mlsanity.com/paper/2310.05506/) (2023)

## Built on by (most cited first, 30 of 109)

- [Do NOT Think That Much for 2+3=? On the Overthinking of o1-Like LLMs](https://mlsanity.com/paper/2412.21187/) (2024)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Scaling Synthetic Data Creation with 1,000,000,000 Personas](https://mlsanity.com/paper/2406.20094/) (2024)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking](https://mlsanity.com/paper/2501.04519/) (2025)
- [Step-DPO: Step-wise Preference Optimization for Long-chain Reasoning of LLMs](https://mlsanity.com/paper/2406.18629/) (2024)
- [OpenMathInstruct-2: Accelerating AI for Math with Massive Open-Source Instruction Data](https://mlsanity.com/paper/2410.01560/) (2024)
- [Recursive Introspection: Teaching Language Model Agents How to Self-Improve](https://mlsanity.com/paper/2407.18219/) (2024)
- [MAmmoTH2: Scaling Instructions from the Web](https://mlsanity.com/paper/2405.03548/) (2024)
- [Mutual Reasoning Makes Smaller LLMs Stronger Problem-Solvers](https://mlsanity.com/paper/2408.06195/) (2024)
- [The Good, The Bad, and The Greedy: Evaluation of LLMs Should Not Ignore Non-Determinism](https://mlsanity.com/paper/2407.10457/) (2024)
- [Best Practices and Lessons Learned on Synthetic Data](https://mlsanity.com/paper/2404.07503/) (2024)
- [RL on Incorrect Synthetic Data Scales the Efficiency of LLM Math Reasoning by Eight-Fold](https://mlsanity.com/paper/2406.14532/) (2024)
- [Q*: Improving Multi-step Reasoning for LLMs with Deliberative Planning](https://mlsanity.com/paper/2406.14283/) (2024)
- [DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving](https://mlsanity.com/paper/2407.13690/) (2024)
- [ZipCache: Accurate and Efficient KV Cache Quantization with Salient Token Identification](https://mlsanity.com/paper/2405.14256/) (2024)
- [Scaling Test-Time Compute Without Verification or RL is Suboptimal](https://mlsanity.com/paper/2502.12118/) (2025)
- [NaturalReasoning: Reasoning in the Wild with 2.8M Challenging Questions](https://mlsanity.com/paper/2502.13124/) (2025)
- [T1: Advancing Language Model Reasoning through Reinforcement Learning and Inference Scaling](https://mlsanity.com/paper/2501.11651/) (2025)
- [Critique Fine-Tuning: Learning to Critique is More Effective than Learning to Imitate](https://mlsanity.com/paper/2501.17703/) (2025)
- [Sampling-Efficient Test-Time Scaling: Self-Estimating the Best-of-N Sampling in Early Decoding](https://mlsanity.com/paper/2503.01422/) (2025)
- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [The Limits of Inference Scaling Through Resampling](https://mlsanity.com/paper/2411.17501/) (2024)
- [A Survey on Large Language Models for Mathematical Reasoning](https://mlsanity.com/paper/2506.08446/) (2025)
- [Is Your Model Really A Good Math Reasoner? Evaluating Mathematical Reasoning with Checklist](https://mlsanity.com/paper/2407.08733/) (2024)
- [Scaling Laws of Synthetic Data for Language Models](https://mlsanity.com/paper/2503.19551/) (2025)
- [JiuZhang3.0: Efficiently Improving Mathematical Reasoning by Training Small Data Synthesis Models](https://mlsanity.com/paper/2405.14365/) (2024)
- [The Larger the Better? Improved LLM Code-Generation via Budget Reallocation](https://mlsanity.com/paper/2404.00725/) (2024)
- [SwS: Self-aware Weakness-driven Problem Synthesis in Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.08989/) (2025)

## Abstract (arXiv)

Mathematical capabilities were previously believed to emerge in common language models only at a very large scale or require extensive math-related pre-training. This paper shows that the LLaMA-2 7B model with common pre-training already exhibits strong mathematical abilities, as evidenced by its impressive accuracy of 97.7% and 72.0% on the GSM8K and MATH benchmarks, respectively, when selecting the best response from 256 random generations. The primary issue with the current base model is the difficulty in consistently eliciting its inherent mathematical capabilities. Notably, the accuracy for the first answer drops to 49.5% and 7.9% on the GSM8K and MATH benchmarks, respectively. We find that simply scaling up the SFT data can significantly enhance the reliability of generating correct answers. However, the potential for extensive scaling is constrained by the scarcity of publicly available math questions. To overcome this limitation, we employ synthetic data, which proves to be nearly as effective as real data and shows no clear saturation when scaled up to approximately one million samples. This straightforward approach achieves an accuracy of 82.6% on GSM8K and 40.6% on MATH using LLaMA-2 7B models, surpassing previous models by 14.2% and 20.8%, respectively. We also provide insights into scaling behaviors across different reasoning complexities and error types.