# TinyGSM: achieving >80% on GSM8k with small language models

arXiv: https://arxiv.org/abs/2312.09241 · HTML: https://mlsanity.com/paper/2312.09241/

## Key facts

- Cited by 70 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 43 of its 48 references that are in the map.
- First posted on arXiv 14 Dec 2023.

## Placed in

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

## Builds on (most cited first, 30 of 43)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [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)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Finetuned Language Models Are Zero-Shot Learners](https://mlsanity.com/paper/2109.01652/) (2021)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Temporal Ensembling for Semi-Supervised Learning](https://mlsanity.com/paper/1610.02242/) (2016)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [CodeGen: An Open Large Language Model for Code with Multi-Turn Program Synthesis](https://mlsanity.com/paper/2203.13474/) (2022)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Large Language Models Can Be Easily Distracted by Irrelevant Context](https://mlsanity.com/paper/2302.00093/) (2023)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [Improved Knowledge Distillation via Teacher Assistant](https://mlsanity.com/paper/1902.03393/) (2019)
- [Textbooks Are All You Need](https://mlsanity.com/paper/2306.11644/) (2023)
- [Textbooks Are All You Need II: phi-1.5 technical report](https://mlsanity.com/paper/2309.05463/) (2023)
- [Beyond neural scaling laws: beating power law scaling via data pruning](https://mlsanity.com/paper/2206.14486/) (2022)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [TinyStories: How Small Can Language Models Be and Still Speak Coherent English?](https://mlsanity.com/paper/2305.07759/) (2023)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [Detecting Pretraining Data from Large Language Models](https://mlsanity.com/paper/2310.16789/) (2023)
- [The Curse of Recursion: Training on Generated Data Makes Models Forget](https://mlsanity.com/paper/2305.17493/) (2023)
- [Snapshot Ensembles: Train 1, get M for free](https://mlsanity.com/paper/1704.00109/) (2017)
- [CodeT5+: Open Code Large Language Models for Code Understanding and Generation](https://mlsanity.com/paper/2305.07922/) (2023)
- [A Diverse Corpus for Evaluating and Developing English Math Word Problem Solvers](https://mlsanity.com/paper/2106.15772/) (2021)

## Built on by (most cited first, 30 of 70)

- [Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models](https://mlsanity.com/paper/2401.01335/) (2024)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [ReFT: Reasoning with Reinforced Fine-Tuning](https://mlsanity.com/paper/2401.08967/) (2024)
- [Rephrasing the Web: A Recipe for Compute and Data-Efficient Language Modeling](https://mlsanity.com/paper/2401.16380/) (2024)
- [Orca-Math: Unlocking the potential of SLMs in Grade School Math](https://mlsanity.com/paper/2402.14830/) (2024)
- [OpenMathInstruct-1: A 1.8 Million Math Instruction Tuning Dataset](https://mlsanity.com/paper/2402.10176/) (2024)
- [LLaVA-Phi: Efficient Multi-Modal Assistant with Small Language Model](https://mlsanity.com/paper/2401.02330/) (2024)
- [Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining](https://mlsanity.com/paper/2504.07912/) (2025)
- [Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process](https://mlsanity.com/paper/2407.20311/) (2024)
- [Easy-to-Hard Generalization: Scalable Alignment Beyond Human Supervision](https://mlsanity.com/paper/2403.09472/) (2024)
- [MAPoRL: Multi-Agent Post-Co-Training for Collaborative Large Language Models with Reinforcement Learning](https://mlsanity.com/paper/2502.18439/) (2025)
- [Prover-Verifier Games improve legibility of LLM outputs](https://mlsanity.com/paper/2407.13692/) (2024)
- [LLM2LLM: Boosting LLMs with Novel Iterative Data Enhancement](https://mlsanity.com/paper/2403.15042/) (2024)
- [LLM Discussion: Enhancing the Creativity of Large Language Models via Discussion Framework and Role-Play](https://mlsanity.com/paper/2405.06373/) (2024)
- [Augmenting Math Word Problems via Iterative Question Composing](https://mlsanity.com/paper/2401.09003/) (2024)
- [On the Diversity of Synthetic Data and its Impact on Training Large Language Models](https://mlsanity.com/paper/2410.15226/) (2024)
- [TinyAgent: Function Calling at the Edge](https://mlsanity.com/paper/2409.00608/) (2024)
- [The Limits of Inference Scaling Through Resampling](https://mlsanity.com/paper/2411.17501/) (2024)
- [Multi-step Problem Solving Through a Verifier: An Empirical Analysis on Model-induced Process Supervision](https://mlsanity.com/paper/2402.02658/) (2024)
- [A Survey on Large Language Models for Mathematical Reasoning](https://mlsanity.com/paper/2506.08446/) (2025)
- [CultureLLM: Incorporating Cultural Differences into Large Language Models](https://mlsanity.com/paper/2402.10946/) (2024)
- [Physics of Language Models: Part 2.2, How to Learn From Mistakes on Grade-School Math Problems](https://mlsanity.com/paper/2408.16293/) (2024)
- [Reasoning to Learn from Latent Thoughts](https://mlsanity.com/paper/2503.18866/) (2025)
- [SmallToLarge (S2L): Scalable Data Selection for Fine-tuning Large Language Models by Summarizing Training Trajectories of Small Models](https://mlsanity.com/paper/2403.07384/) (2024)
- [MAGDi: Structured Distillation of Multi-Agent Interaction Graphs Improves Reasoning in Smaller Language Models](https://mlsanity.com/paper/2402.01620/) (2024)
- [A Survey on Data Synthesis and Augmentation for Large Language Models](https://mlsanity.com/paper/2410.12896/) (2024)
- [Surveying the Effects of Quality, Diversity, and Complexity in Synthetic Data From Large Language Models](https://mlsanity.com/paper/2412.02980/) (2024)
- [Self-Explore: Enhancing Mathematical Reasoning in Language Models with Fine-grained Rewards](https://mlsanity.com/paper/2404.10346/) (2024)
- [Achieving >97% on GSM8K: Deeply Understanding the Problems Makes LLMs Better Solvers for Math Word Problems](https://mlsanity.com/paper/2404.14963/) (2024)
- [Automatic Instruction Evolving for Large Language Models](https://mlsanity.com/paper/2406.00770/) (2024)

## Abstract (arXiv)

Small-scale models offer various computational advantages, and yet to which extent size is critical for problem-solving abilities remains an open question. Specifically for solving grade school math, the smallest model size so far required to break the 80% barrier on the GSM8K benchmark remains to be 34B. Our work studies how high-quality datasets may be the key for small language models to acquire mathematical reasoning. We introduce textttTinyGSM, a synthetic dataset of 12.3M grade school math problems paired with Python solutions, generated fully by GPT-3.5. After finetuning on textttTinyGSM, we find that a duo of a 1.3B generation model and a 1.3B verifier model can achieve 81.5% accuracy, outperforming existing models that are orders of magnitude larger. This also rivals the performance of the GPT-3.5 ``teacher'' model (77.4%), from which our model's training data is generated. Our approach is simple and has two key components: 1) the high-quality dataset textttTinyGSM, 2) the use of a verifier, which selects the final outputs from multiple candidate generations.