# Reflection of Thought: Inversely Eliciting Numerical Reasoning in Language Models via Solving Linear Systems

arXiv: https://arxiv.org/abs/2210.05075 · HTML: https://mlsanity.com/paper/2210.05075/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 31 of its 36 references that are in the map.
- First posted on arXiv 11 Oct 2022.

## Placed in

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

## Builds on (most cited first, 30 of 31)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [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)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension](https://mlsanity.com/paper/1910.13461/) (2019)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [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)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [DROP: A Reading Comprehension Benchmark Requiring Discrete Reasoning Over Paragraphs](https://mlsanity.com/paper/1903.00161/) (2019)
- [Evaluating the visualization of what a Deep Neural Network has learned](https://mlsanity.com/paper/1509.06321/) (2015)
- [Solving General Arithmetic Word Problems](https://mlsanity.com/paper/1608.01413/) (2016)
- [TAPEX: Table Pre-training via Learning a Neural SQL Executor](https://mlsanity.com/paper/2107.07653/) (2021)
- [UnifiedSKG: Unifying and Multi-Tasking Structured Knowledge Grounding with Text-to-Text Language Models](https://mlsanity.com/paper/2201.05966/) (2022)
- [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)
- [Language Models are General-Purpose Interfaces](https://mlsanity.com/paper/2206.06336/) (2022)
- [Cognitive Graph for Multi-Hop Reading Comprehension at Scale](https://mlsanity.com/paper/1905.05460/) (2019)
- [NumNet: Machine Reading Comprehension with Numerical Reasoning](https://mlsanity.com/paper/1910.06701/) (2019)
- [Giving BERT a Calculator: Finding Operations and Arguments with Reading Comprehension](https://mlsanity.com/paper/1909.00109/) (2019)
- [Reasoning Like Program Executors](https://mlsanity.com/paper/2201.11473/) (2022)
- [A Multi-Type Multi-Span Network for Reading Comprehension that Requires Discrete Reasoning](https://mlsanity.com/paper/1908.05514/) (2019)
- [Turning Tables: Generating Examples from Semi-structured Tables for Endowing Language Models with Reasoning Skills](https://mlsanity.com/paper/2107.07261/) (2021)
- [Question Directed Graph Attention Network for Numerical Reasoning over Text](https://mlsanity.com/paper/2009.07448/) (2020)
- [Awakening Latent Grounding from Pretrained Language Models for Semantic Parsing](https://mlsanity.com/paper/2109.10540/) (2021)

## Built on by (most cited first, 7 of 7)

- [Towards Reasoning in Large Language Models: A Survey](https://mlsanity.com/paper/2212.10403/) (2022)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [ReviewerGPT? An Exploratory Study on Using Large Language Models for Paper Reviewing](https://mlsanity.com/paper/2306.00622/) (2023)
- [Towards Better Chain-of-Thought Prompting Strategies: A Survey](https://mlsanity.com/paper/2310.04959/) (2023)
- [VLM-R3: Region Recognition, Reasoning, and Refinement for Enhanced Multimodal Chain-of-Thought](https://mlsanity.com/paper/2505.16192/) (2025)
- [Large Reasoning Embedding Models: Towards Next-Generation Dense Retrieval Paradigm](https://mlsanity.com/paper/2510.14321/) (2025)
- [RATIONALYST: Mining Implicit Rationales for Process Supervision of Reasoning](https://mlsanity.com/paper/2410.01044/) (2024)

## Abstract (arXiv)

Numerical reasoning over natural language has been a long-standing goal for the research community. However, cutting-edge language models have proven difficult to reliably generalize to a broad range of numbers, although they have shown proficiency in reasoning over common and simple numbers. In this paper, we propose a novel method to elicit and exploit the numerical reasoning knowledge hidden in pre-trained language models using simple anchor numbers. Concretely, we first leverage simple numbers as anchors to probe the implicitly inferred arithmetic expressions from language models, and then explicitly apply the expressions on complex numbers to get corresponding answers. To inversely elicit arithmetic expressions, we transform and formulate the task as an analytically solvable linear system. Experimental results on several numerical reasoning benchmarks demonstrate that our approach significantly improves numerical reasoning capabilities of existing LMs. More importantly, our approach is training-free and simply works in the inference phase, making it highly portable and achieving consistent performance benefits across a variety of language models (GPT-3, T5, BART, etc) in all zero-shot, few-shot, and fine-tuning scenarios.