# Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process

arXiv: https://arxiv.org/abs/2407.20311 · HTML: https://mlsanity.com/paper/2407.20311/

## Key facts

- Cited by 108 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 16 of its 23 references that are in the map.
- First posted on arXiv 29 Jul 2024.

## Placed in

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

## Builds on (most cited first, 16 of 16)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [GPT-NeoX-20B: An Open-Source Autoregressive Language Model](https://mlsanity.com/paper/2204.06745/) (2022)
- [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)
- [Physics of Language Models: Part 3.1, Knowledge Storage and Extraction](https://mlsanity.com/paper/2309.14316/) (2023)
- [What Algorithms can Transformers Learn? A Study in Length Generalization](https://mlsanity.com/paper/2310.16028/) (2023)
- [A Careful Examination of Large Language Model Performance on Grade School Arithmetic](https://mlsanity.com/paper/2405.00332/) (2024)
- [Physics of Language Models: Part 3.3, Knowledge Capacity Scaling Laws](https://mlsanity.com/paper/2404.05405/) (2024)
- [Physics of Language Models: Part 3.2, Knowledge Manipulation](https://mlsanity.com/paper/2309.14402/) (2023)
- [Physics of Language Models: Part 1, Learning Hierarchical Language Structures](https://mlsanity.com/paper/2305.13673/) (2023)
- [TinyGSM: achieving >80% on GSM8k with small language models](https://mlsanity.com/paper/2312.09241/) (2023)
- [Length Generalization in Arithmetic Transformers](https://mlsanity.com/paper/2306.15400/) (2023)

## Built on by (most cited first, 30 of 108)

- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Training Language Models to Self-Correct via Reinforcement Learning](https://mlsanity.com/paper/2409.12917/) (2024)
- [Reasoning with Latent Thoughts: On the Power of Looped Transformers](https://mlsanity.com/paper/2502.17416/) (2025)
- [MetaMorph: Multimodal Understanding and Generation via Instruction Tuning](https://mlsanity.com/paper/2412.14164/) (2024)
- [When More is Less: Understanding Chain-of-Thought Length in LLMs](https://mlsanity.com/paper/2502.07266/) (2025)
- [Scaling Latent Reasoning via Looped Language Models](https://mlsanity.com/paper/2510.25741/) (2025)
- [On Memorization of Large Language Models in Logical Reasoning](https://mlsanity.com/paper/2410.23123/) (2024)
- [Towards System 2 Reasoning in LLMs: Learning How to Think With Meta Chain-of-Thought](https://mlsanity.com/paper/2501.04682/) (2025)
- [MLLM can see? Dynamic Correction Decoding for Hallucination Mitigation](https://mlsanity.com/paper/2410.11779/) (2024)
- [Reasoning by Superposition: A Theoretical Perspective on Chain of Continuous Thought](https://mlsanity.com/paper/2505.12514/) (2025)
- [On the Interplay of Pre-Training, Mid-Training, and RL on Reasoning Language Models](https://mlsanity.com/paper/2512.07783/) (2025)
- [LoopFormer: Elastic-Depth Looped Transformers for Latent Reasoning via Shortcut Modulation](https://mlsanity.com/paper/2602.11451/) (2026)
- [On the Diversity of Synthetic Data and its Impact on Training Large Language Models](https://mlsanity.com/paper/2410.15226/) (2024)
- [Latent Space Chain-of-Embedding Enables Output-free LLM Self-Evaluation](https://mlsanity.com/paper/2410.13640/) (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)
- [Reconstruction Alignment Improves Unified Multimodal Models](https://mlsanity.com/paper/2509.07295/) (2025)
- [Theoretical limitations of multi-layer Transformer](https://mlsanity.com/paper/2412.02975/) (2024)
- [Prismatic Synthesis: Gradient-based Data Diversification Boosts Generalization in LLM Reasoning](https://mlsanity.com/paper/2505.20161/) (2025)
- [Causal Language Modeling Can Elicit Search and Reasoning Capabilities on Logic Puzzles](https://mlsanity.com/paper/2409.10502/) (2024)
- [What Makes a Good Diffusion Planner for Decision Making?](https://mlsanity.com/paper/2503.00535/) (2025)
- [Physics of Language Models: Part 4.1, Architecture Design and the Magic of Canon Layers](https://mlsanity.com/paper/2512.17351/) (2025)
- [GameArena: Evaluating LLM Reasoning through Live Computer Games](https://mlsanity.com/paper/2412.06394/) (2024)
- [Why Prompt Design Matters and Works: A Complexity Analysis of Prompt Search Space in LLMs](https://mlsanity.com/paper/2503.10084/) (2025)
- [Energy-Based Transformers are Scalable Learners and Thinkers](https://mlsanity.com/paper/2507.02092/) (2025)
- [RL Fine-Tuning Heals OOD Forgetting in SFT](https://mlsanity.com/paper/2509.12235/) (2025)
- [What Happened in LLMs Layers when Trained for Fast vs. Slow Thinking: A Gradient Perspective](https://mlsanity.com/paper/2410.23743/) (2024)
- [Locate, Steer, and Improve: A Practical Survey of Actionable Mechanistic Interpretability in Large Language Models](https://mlsanity.com/paper/2601.14004/) (2026)
- [Encode, Think, Decode: Scaling test-time reasoning with recursive latent thoughts](https://mlsanity.com/paper/2510.07358/) (2025)

## Abstract (arXiv)

Recent advances in language models have demonstrated their capability to solve mathematical reasoning problems, achieving near-perfect accuracy on grade-school level math benchmarks like GSM8K. In this paper, we formally study how language models solve these problems. We design a series of controlled experiments to address several fundamental questions: (1) Can language models truly develop reasoning skills, or do they simply memorize templates? (2) What is the model's hidden (mental) reasoning process? (3) Do models solve math questions using skills similar to or different from humans? (4) Do models trained on GSM8K-like datasets develop reasoning skills beyond those necessary for solving GSM8K problems? (5) What mental process causes models to make reasoning mistakes? (6) How large or deep must a model be to effectively solve GSM8K-level math questions? Our study uncovers many hidden mechanisms by which language models solve mathematical questions, providing insights that extend beyond current understandings of LLMs.