---
title: "DART-Math — paper by Tong et al. (2024) — summary and ideas"
description: "The authors show this skews datasets toward easy problems. 68 references, cited by 81 papers here."
authors: ["Yuxuan Tong", "Xiwen Zhang", "Rui Wang", "Ruidong Wu", "Junxian He"]
year: 2024
venue: "arXiv:2407.13690"
source: "https://arxiv.org/abs/2407.13690"
doi: "10.48550/arxiv.2407.13690"
arxiv: "2407.13690"
licence: "arXiv — non-exclusive licence to distribute"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/tong-2024-dart-math-difficulty-aware-rejection/"
---
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# DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving

**Authors:** [Yuxuan Tong](https://md.research.elamur.ai/free/en/authors/yuxuan-tong/), [Xiwen Zhang](https://md.research.elamur.ai/free/en/authors/xiwen-zhang/), [Rui Wang](https://md.research.elamur.ai/free/en/authors/rui-wang/), Ruidong Wu and [Junxian He](https://md.research.elamur.ai/free/en/authors/junxian-he/) · **Year:** 2024 · **Venue:** arXiv:2407.13690 · **Licence:** [arXiv — non-exclusive licence to distribute](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/2407.13690) · [PDF](https://arxiv.org/pdf/2407.13690) · [DOI 10.48550/arxiv.2407.13690](https://doi.org/10.48550/arxiv.2407.13690) · [arXiv 2407.13690](https://arxiv.org/abs/2407.13690) · [OpenAlex](https://openalex.org/W4405792472)

**Citations:** 0 citations (OpenAlex, 2026-10-14) · [81](https://md.research.elamur.ai/free/en/papers/tong-2024-dart-math-difficulty-aware-rejection/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
Instruction tuning on synthetic solutions is a common way to improve LLM math reasoning. Such data usually comes from rejection sampling: a strong model generates several answers per problem, and wrong ones are discarded. The authors show this skews datasets toward easy problems. In MetaMathQA's MATH-derived AnsAug portion, more than half of the hardest (Level 5) training queries get no correct solution at all. DART (Difficulty-Aware Rejection Tuning) fixes this by spending more sampling attempts on harder queries. Difficulty is estimated from how often sampled answers are wrong. There are two variants:
- **Uniform** gathers an equal number of correct solutions per query.
- **Prop2Diff** gives harder queries proportionally more solutions.

All data was generated with the open DeepSeekMath-7B-RL model, without GPT-4. The resulting DART-Math models, fine-tuned from 7B–70B base models, beat vanilla rejection tuning across six in- and out-of-domain benchmarks. They match or exceed models trained on larger public datasets. This makes DART-Math datasets a strong, cheap, fully open resource for math instruction tuning.

## Key points
- **Bias diagnosis:** In MetaMathQA-MATH-AnsAug, 51.1% of Level 5 MATH queries are lost, and the hardest queries supply only 10.5% of responses.
- **Datasets and comparisons:** DART-Math (Uniform) and DART-Math (Prop2Diff) each contain about 590k examples. They were compared against MetaMath (395k), MMIQC (2.3M) and VRT (590k).
- **Base models:** Mistral-7B, DeepSeekMath-7B, Llama3-8B and Llama3-70B. For Llama3-8B, MATH accuracy rose from 21.2% to 46.6% and GSM8K from 51.0% to 82.5%.
- **Synthesis cost:** About 150M raw samples (~160 A100 GPU days) were needed for a >95% query achieving ratio. The authors treat this as a one-time cost.
- **Limitations and availability:** Difficulty is measured only by fail rate, and solutions use natural-language reasoning without code. Data, models and code are released at github.com/hkust-nlp/dart-math.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [Tsinghua University](https://md.research.elamur.ai/free/en/universities/tsinghua-university/) — University, Beijing, China · 415 works · 94,028 citations
- [Hong Kong University of Science and Technology](https://md.research.elamur.ai/free/en/universities/hong-kong-university-of-science-and-technology/) — University, Hong Kong, China · 170 works · 26,341 citations

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## Cite

Cite as: Yuxuan Tong, Xiwen Zhang, Rui Wang, Ruidong Wu and Junxian He. 2024. DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving. arXiv:2407.13690.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{tong2024dart,
  title = {{DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving}},
  author = {Yuxuan Tong, Xiwen Zhang, Rui Wang, Ruidong Wu and Junxian He},
  year = {2024},
  doi = {10.48550/arxiv.2407.13690},
  eprint = {2407.13690},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2407.13690}
}
```

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*Updated 2 November 2026*