# A Causal Framework to Quantify the Robustness of Mathematical Reasoning with Language Models

arXiv: https://arxiv.org/abs/2210.12023 · HTML: https://mlsanity.com/paper/2210.12023/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 38 of its 78 references that are in the map.
- First posted on arXiv 21 Oct 2022.

## Placed in

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

## Builds on (most cited first, 30 of 38)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [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)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter](https://mlsanity.com/paper/1910.01108/) (2019)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Beyond Accuracy: Behavioral Testing of NLP models with CheckList](https://mlsanity.com/paper/2005.04118/) (2020)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [GPT-NeoX-20B: An Open-Source Autoregressive Language Model](https://mlsanity.com/paper/2204.06745/) (2022)
- [Learning the Difference that Makes a Difference with Counterfactually-Augmented Data](https://mlsanity.com/paper/1909.12434/) (2019)
- [A Diverse Corpus for Evaluating and Developing English Math Word Problem Solvers](https://mlsanity.com/paper/2106.15772/) (2021)
- [Direct and Indirect Effects](https://mlsanity.com/paper/1301.2300/) (2013)
- [Causal Mediation Analysis for Interpreting Neural NLP: The Case of Gender Bias](https://mlsanity.com/paper/2004.12265/) (2020)
- [Causal Inference in Natural Language Processing: Estimation, Prediction, Interpretation and Beyond](https://mlsanity.com/paper/2109.00725/) (2021)
- [Do NLP Models Know Numbers? Probing Numeracy in Embeddings](https://mlsanity.com/paper/1909.07940/) (2019)
- [Impact of Pretraining Term Frequencies on Few-Shot Reasoning](https://mlsanity.com/paper/2202.07206/) (2022)
- [Injecting Numerical Reasoning Skills into Language Models](https://mlsanity.com/paper/2004.04487/) (2020)
- [Causal Analysis of Syntactic Agreement Mechanisms in Neural Language Models](https://mlsanity.com/paper/2106.06087/) (2021)
- [CausaLM: Causal Model Explanation Through Counterfactual Language Models](https://mlsanity.com/paper/2005.13407/) (2020)
- [NumGLUE: A Suite of Fundamental yet Challenging Mathematical Reasoning Tasks](https://mlsanity.com/paper/2204.05660/) (2022)
- [Representing Numbers in NLP: a Survey and a Vision](https://mlsanity.com/paper/2103.13136/) (2021)
- [Generate & Rank: A Multi-task Framework for Math Word Problems](https://mlsanity.com/paper/2109.03034/) (2021)
- [Do Language Embeddings Capture Scales?](https://mlsanity.com/paper/2010.05345/) (2020)

## Built on by (most cited first, 30 of 68)

- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://mlsanity.com/paper/2402.00157/) (2024)
- [Distilling Reasoning Capabilities into Smaller Language Models](https://mlsanity.com/paper/2212.00193/) (2022)
- [DynaMath: A Dynamic Visual Benchmark for Evaluating Mathematical Reasoning Robustness of Vision Language Models](https://mlsanity.com/paper/2411.00836/) (2024)
- [CLadder: Assessing Causal Reasoning in Language Models](https://mlsanity.com/paper/2312.04350/) (2023)
- [Language Model Behavior: A Comprehensive Survey](https://mlsanity.com/paper/2303.11504/) (2023)
- [A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis](https://mlsanity.com/paper/2305.15054/) (2023)
- [Beyond Accuracy: Evaluating the Reasoning Behavior of Large Language Models -- A Survey](https://mlsanity.com/paper/2404.01869/) (2024)
- [Autonomous GIS: the next-generation AI-powered GIS](https://mlsanity.com/paper/2305.06453/) (2023)
- [Large Language Models Are Not Strong Abstract Reasoners](https://mlsanity.com/paper/2305.19555/) (2023)
- [Is Your Model Really A Good Math Reasoner? Evaluating Mathematical Reasoning with Checklist](https://mlsanity.com/paper/2407.08733/) (2024)
- [Automatic Generation of Socratic Subquestions for Teaching Math Word Problems](https://mlsanity.com/paper/2211.12835/) (2022)
- [Number Cookbook: Number Understanding of Language Models and How to Improve It](https://mlsanity.com/paper/2411.03766/) (2024)
- [Unveiling the Key Factors for Distilling Chain-of-Thought Reasoning](https://mlsanity.com/paper/2502.18001/) (2025)
- [Quantifying and Mitigating Unimodal Biases in Multimodal Large Language Models: A Causal Perspective](https://mlsanity.com/paper/2403.18346/) (2024)
- [Unveiling Causal Reasoning in Large Language Models: Reality or Mirage?](https://mlsanity.com/paper/2506.21215/) (2025)
- [Causal Inference with Large Language Model: A Survey](https://mlsanity.com/paper/2409.09822/) (2024)
- [Assessing Hidden Risks of LLMs: An Empirical Study on Robustness, Consistency, and Credibility](https://mlsanity.com/paper/2305.10235/) (2023)
- [Causal Prompting: Debiasing Large Language Model Prompting based on Front-Door Adjustment](https://mlsanity.com/paper/2403.02738/) (2024)
- [A Survey of Useful LLM Evaluation](https://mlsanity.com/paper/2406.00936/) (2024)
- [Causally Testing Gender Bias in LLMs: A Case Study on Occupational Bias](https://mlsanity.com/paper/2212.10678/) (2022)
- [AutoPRM: Automating Procedural Supervision for Multi-Step Reasoning via Controllable Question Decomposition](https://mlsanity.com/paper/2402.11452/) (2024)
- [Mathematical Reasoning in Large Language Models: Assessing Logical and Arithmetic Errors across Wide Numerical Ranges](https://mlsanity.com/paper/2502.08680/) (2025)
- [Testing and Understanding Erroneous Planning in LLM Agents through Synthesized User Inputs](https://mlsanity.com/paper/2404.17833/) (2024)
- [WorldSense: A Synthetic Benchmark for Grounded Reasoning in Large Language Models](https://mlsanity.com/paper/2311.15930/) (2023)
- [SemEval-2024 Task 2: Safe Biomedical Natural Language Inference for Clinical Trials](https://mlsanity.com/paper/2404.04963/) (2024)
- [Robust Prompt Optimization for Large Language Models Against Distribution Shifts](https://mlsanity.com/paper/2305.13954/) (2023)
- [Do Language Models Exhibit the Same Cognitive Biases in Problem Solving as Human Learners?](https://mlsanity.com/paper/2401.18070/) (2024)
- [NLI4CT: Multi-Evidence Natural Language Inference for Clinical Trial Reports](https://mlsanity.com/paper/2305.03598/) (2023)
- [Text Generation: A Systematic Literature Review of Tasks, Evaluation, and Challenges](https://mlsanity.com/paper/2405.15604/) (2024)

## Abstract (arXiv)

We have recently witnessed a number of impressive results on hard mathematical reasoning problems with language models. At the same time, the robustness of these models has also been called into question; recent works have shown that models can rely on shallow patterns in the problem description when generating a solution. Building on the idea of behavioral testing, we propose a novel framework, which pins down the causal effect of various factors in the input, e.g., the surface form of the problem text, the operands, and math operators on the output solution. By grounding the behavioral analysis in a causal graph describing an intuitive reasoning process, we study the behavior of language models in terms of robustness and sensitivity to direct interventions in the input space. We apply our framework on a test bed of math word problems. Our analysis shows that robustness does not appear to continuously improve as a function of size, but the GPT-3 Davinci models (175B) achieve a dramatic improvement in both robustness and sensitivity compared to all other GPT variants.