# JiuZhang3.0: Efficiently Improving Mathematical Reasoning by Training Small Data Synthesis Models

arXiv: https://arxiv.org/abs/2405.14365 · HTML: https://mlsanity.com/paper/2405.14365/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 72 of its 76 references that are in the map.
- First posted on arXiv 23 May 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)
- Synthetic data and self-improvement for LLMs / [Synthetic reasoning data and bootstrapping](https://mlsanity.com/synth/synthetic-reasoning-data-and-bootstrapping/)

## Builds on (most cited first, 30 of 72)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [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)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Visual Instruction Tuning](https://mlsanity.com/paper/2304.08485/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [Finetuned Language Models Are Zero-Shot Learners](https://mlsanity.com/paper/2109.01652/) (2021)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context](https://mlsanity.com/paper/2403.05530/) (2024)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [Scaling Instruction-Finetuned Language Models](https://mlsanity.com/paper/2210.11416/) (2022)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [Code Llama: Open Foundation Models for Code](https://mlsanity.com/paper/2308.12950/) (2023)
- [Large Language Models Encode Clinical Knowledge](https://mlsanity.com/paper/2212.13138/) (2022)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (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)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [CRITIC: Large Language Models Can Self-Correct with Tool-Interactive Critiquing](https://mlsanity.com/paper/2305.11738/) (2023)
- [Galactica: A Large Language Model for Science](https://mlsanity.com/paper/2211.09085/) (2022)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)

## Built on by (most cited first, 30 of 45)

- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Step-DPO: Step-wise Preference Optimization for Long-chain Reasoning of LLMs](https://mlsanity.com/paper/2406.18629/) (2024)
- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [A Survey on Large Language Models for Mathematical Reasoning](https://mlsanity.com/paper/2506.08446/) (2025)
- [Scaling Laws of Synthetic Data for Language Models](https://mlsanity.com/paper/2503.19551/) (2025)
- [Position: Multimodal Large Language Models Can Significantly Advance Scientific Reasoning](https://mlsanity.com/paper/2502.02871/) (2025)
- [Unleashing LLM Reasoning Capability via Scalable Question Synthesis from Scratch](https://mlsanity.com/paper/2410.18693/) (2024)
- [Rethinking Data Selection at Scale: Random Selection is Almost All You Need](https://mlsanity.com/paper/2410.09335/) (2024)
- [SWE-Synth: Synthesizing Verifiable Bug-Fix Data to Enable Large Language Models in Resolving Real-World Bugs](https://mlsanity.com/paper/2504.14757/) (2025)
- [ProVision: Programmatically Scaling Vision-centric Instruction Data for Multimodal Language Models](https://mlsanity.com/paper/2412.07012/) (2024)
- [A Survey on LLM Mid-Training](https://mlsanity.com/paper/2510.23081/) (2025)
- [S^3cMath: Spontaneous Step-level Self-correction Makes Large Language Models Better Mathematical Reasoners](https://mlsanity.com/paper/2409.01524/) (2024)
- [Towards Effective and Efficient Continual Pre-training of Large Language Models](https://mlsanity.com/paper/2407.18743/) (2024)
- [LogicPro: Improving Complex Logical Reasoning via Program-Guided Learning](https://mlsanity.com/paper/2409.12929/) (2024)
- [AgentFrontier: Expanding the Capability Frontier of LLM Agents with ZPD-Guided Data Synthesis](https://mlsanity.com/paper/2510.24695/) (2025)
- [Montessori-Instruct: Generate Influential Training Data Tailored for Student Learning](https://mlsanity.com/paper/2410.14208/) (2024)
- [Activation Control for Efficiently Eliciting Long Chain-of-thought Ability of Language Models](https://mlsanity.com/paper/2505.17697/) (2025)
- [Little Giants: Synthesizing High-Quality Embedding Data at Scale](https://mlsanity.com/paper/2410.18634/) (2024)
- [Learning to Pose Problems: Reasoning-Driven and Solver-Adaptive Data Synthesis](https://mlsanity.com/paper/2511.09907/) (2025)
- [Not Everything is All You Need: Toward Low-Redundant Optimization for Large Language Model Alignment](https://mlsanity.com/paper/2406.12606/) (2024)
- [Mix-CPT: A Domain Adaptation Framework via Decoupling Knowledge Learning and Format Alignment](https://mlsanity.com/paper/2407.10804/) (2024)
- [YuLan-Mini: An Open Data-efficient Language Model](https://mlsanity.com/paper/2412.17743/) (2024)
- [ClinicalGPT-R1: Pushing reasoning capability of generalist disease diagnosis with large language model](https://mlsanity.com/paper/2504.09421/) (2025)
- [MathDebugger: Detecting and Diagnosing Errors in Synthetic Mathematical Data](https://mlsanity.com/paper/2502.19058/) (2025)
- [Leveraging Web-Crawled Data for High-Quality Fine-Tuning](https://mlsanity.com/paper/2408.08003/) (2024)
- [MathFimer: Enhancing Mathematical Reasoning by Expanding Reasoning Steps through Fill-in-the-Middle Task](https://mlsanity.com/paper/2502.11684/) (2025)
- [MASS: Mathematical Data Selection via Skill Graphs for Pretraining Large Language Models](https://mlsanity.com/paper/2503.14917/) (2025)
- [A Survey of LLM × DATA](https://mlsanity.com/paper/2505.18458/) (2025)
- [Large-Scale Diverse Synthesis for Mid-Training](https://mlsanity.com/paper/2508.01326/) (2025)
- [Demystifying Data Organization for Enhanced LLM Training](https://mlsanity.com/paper/2605.30334/) (2026)

## Abstract (arXiv)

Mathematical reasoning is an important capability of large language models~(LLMs) for real-world applications. To enhance this capability, existing work either collects large-scale math-related texts for pre-training, or relies on stronger LLMs (eg GPT-4) to synthesize massive math problems. Both types of work generally lead to large costs in training or synthesis. To reduce the cost, based on open-source available texts, we propose an efficient way that trains a small LLM for math problem synthesis, to efficiently generate sufficient high-quality pre-training data. To achieve it, we create a dataset using GPT-4 to distill its data synthesis capability into the small LLM. Concretely, we craft a set of prompts based on human education stages to guide GPT-4, to synthesize problems covering diverse math knowledge and difficulty levels. Besides, we adopt the gradient-based influence estimation method to select the most valuable math-related texts. The both are fed into GPT-4 for creating the knowledge distillation dataset to train the small LLM. We leverage it to synthesize 6 million math problems for pre-training our JiuZhang3.0 model, which only needs to invoke GPT-4 API 9.3k times and pre-train on 4.6B data. Experimental results have shown that JiuZhang3.0 achieves state-of-the-art performance on several mathematical reasoning datasets, under both natural language reasoning and tool manipulation settings. Our code and data will be publicly released in urlhttps://github.com/RUCAIBox/JiuZhang3.0.