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Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process
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Builds a controlled synthetic grade-school math dataset (iGSM) and pretrains GPT2-style models from scratch to probe how language models solve math problems internally.
Problem
Whether language models truly learn reasoning skills on grade-school math or merely memorize solution templates, and what hidden mental reasoning process they use; existing internet-pretrained models and GSM8K-style data contamination make these questions hard to answer.
Approach
Generate an effectively unlimited, contamination-free synthetic grade-school math dataset from controlled dependency graphs, pretrain GPT2-style models from scratch on it, and use a V(ariable)-probing method (frozen model plus rank-8 embedding update and linear head) to reveal internal planning states such as necessity and dependency of parameters.
Key results
- GPT2 pretrained on iGSM reaches about 99% accuracy in-distribution and generalizes out-of-distribution to operation counts beyond training (up to op=32 on iGSM-hard), per the paper's Result 2.
- Pretrained models produce on average about 0.5 unnecessary parameters per solution even at op=32, and mostly generate shortest solutions.
- A 16-layer 576-dim GPT2 outperforms a 4-layer 1920-dim model on reasoning, indicating depth matters for math reasoning.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Pretrain GPT2-RoPE on synthetic iGSM data | GPT2-RoPE · iGSM same distribution | accuracy: 99% | Helped |
| Pretrain GPT2-RoPE on synthetic iGSM data | GPT2-RoPE · iGSM OOD longer reasoning lengths | accuracy | Helped |
| Use a deeper, narrower model rather than a shallower, wider model | GPT2 · iGSM math reasoning | reasoning length | Helped |
| Use a 4-layer, 1920-dimension model instead of a 20-layer, 576-dimension model | GPT2 · iGSM math reasoning | reasoning length | Hurt |
| Replace GPT2 architecture with Llama architecture | GPT2-RoPE · iGSM math reasoning | accuracy | No clear effect |
| Use CoT solutions that define the parameter before giving intermediate steps | GPT2-RoPE · iGSM | accuracy | Helped |
| Generate only necessary solution steps rather than computing unnecessary parameters | GPT2-RoPE · iGSM including OOD | unnecessary computations | Helped |
| Probe model states after pretraining on iGSM | GPT2-RoPE · iGSM parameter dependencies, including unnecessary parameters | dependency identification | Helped |
| Use V-probing to identify incorrect necessity predictions | GPT2-RoPE · iGSM reasoning errors | prediction of unnecessary parameters as necessary | Hurt |
| Use V-probing to identify readiness errors | GPT2-RoPE · iGSM reasoning errors | ready-to-compute predictions | Hurt |
Surprising findings
- The model learns all-pair parameter dependency and can_next skills that are not needed to fit its pretraining data, before any question is asked.
- Most model errors trace to incorrectly predicting that a parameter is ready to compute, visible in internal states even before the first wrong solution step.
Limitations
- Uses a simplified synthetic setting (arithmetic mod 23, templated English) that may not transfer to natural math or foundation models.
- GPT-2 with rotary embeddings is a small custom model; results on larger LLMs are not directly tested.
- Probing relies on a custom V-probing method whose interpretation depends on the chosen rank-8 update.
Experimental setup
Datasets: iGSM-med iGSM-hard GSM8K
Benchmarks: iGSM-med iGSM-hard
Metrics: accuracy probing accuracy unnecessary parameters per solution
Baselines: GPT-4 GPT-4o Llama randomly-initialized transformer probing
Builds on: GPT-2 RoPE GSM8K
Closest papers by meaning
- Physics of Language Models: Part 2.2, How to Learn From Mistakes on Grade-School Math Problems (ICLR 2025 · similarity 0.93)
- A Careful Examination of Large Language Model Performance on Grade School Arithmetic (NeurIPS 2024 · similarity 0.92)
- Advancing Mathematical Reasoning in Language Models: The Impact of Problem-Solving Data, Data Synthesis Methods, and Training Stages (ICLR 2025 · similarity 0.91)
- Do Language Models Exhibit the Same Cognitive Biases in Problem Solving as Human Learners? (ICML 2024 · similarity 0.90)
- CogMath: Assessing LLMs' Authentic Mathematical Ability from a Human Cognitive Perspective (ICML 2025 · similarity 0.90)
- GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models (ICLR 2025 · similarity 0.90)
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