# Calc-X and Calcformers: Empowering Arithmetical Chain-of-Thought through Interaction with Symbolic Systems

arXiv: https://arxiv.org/abs/2305.15017 · HTML: https://mlsanity.com/paper/2305.15017/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 13 of its 22 references that are in the map.
- First posted on arXiv 24 May 2023.

## Placed in

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

## Builds on (most cited first, 13 of 13)

- [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)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [Scaling Instruction-Finetuned Language Models](https://mlsanity.com/paper/2210.11416/) (2022)
- [WebGPT: Browser-assisted question-answering with human feedback](https://mlsanity.com/paper/2112.09332/) (2021)
- [LaMDA: Language Models for Dialog Applications](https://mlsanity.com/paper/2201.08239/) (2022)
- [MathQA: Towards Interpretable Math Word Problem Solving with Operation-Based Formalisms](https://mlsanity.com/paper/1905.13319/) (2019)
- [PAL: Program-aided Language Models](https://mlsanity.com/paper/2211.10435/) (2022)
- [Program Induction by Rationale Generation : Learning to Solve and Explain Algebraic Word Problems](https://mlsanity.com/paper/1705.04146/) (2017)
- [Analysing Mathematical Reasoning Abilities of Neural Models](https://mlsanity.com/paper/1904.01557/) (2019)
- [TALM: Tool Augmented Language Models](https://mlsanity.com/paper/2205.12255/) (2022)
- [Ape210K: A Large-Scale and Template-Rich Dataset of Math Word Problems](https://mlsanity.com/paper/2009.11506/) (2020)

## Built on by (most cited first, 13 of 13)

- [Tool Learning with Large Language Models: A Survey](https://mlsanity.com/paper/2405.17935/) (2024)
- [Agent Lightning: Train ANY AI Agents with Reinforcement Learning](https://mlsanity.com/paper/2508.03680/) (2025)
- [Self-Training with Direct Preference Optimization Improves Chain-of-Thought Reasoning](https://mlsanity.com/paper/2407.18248/) (2024)
- [Skip a Layer or Loop It? Learning Program-of-Layers in LLMs](https://mlsanity.com/paper/2606.06574/) (2026)
- [Learning by Analogy: Enhancing Few-Shot Prompting for Math Word Problem Solving with Computational Graph-Based Retrieval](https://mlsanity.com/paper/2411.16454/) (2024)
- [Improving Arithmetic Reasoning Ability of Large Language Models through Relation Tuples, Verification and Dynamic Feedback](https://mlsanity.com/paper/2406.17873/) (2024)
- [Improving the OOD Performance of Closed-Source LLMs on NLI Through Strategic Data Selection](https://mlsanity.com/paper/2505.20209/) (2025)
- [Pre-trained Language Models Learn Remarkably Accurate Representations of Numbers](https://mlsanity.com/paper/2506.08966/) (2025)
- [Concept-aware Data Construction Improves In-context Learning of Language Models](https://mlsanity.com/paper/2403.09703/) (2024)
- [Programs-of-Layers in LLMs through the Lens of Cortical Areas](https://mlsanity.com/paper/2609.31360/) (2026)
- [Pre-Calc: Learning to Use the Calculator Improves Numeracy in Language Models](https://mlsanity.com/paper/2404.14355/) (2024)
- [When Efficiency Backfires: Cascading LLMs Trigger Cascade Failure under Adversarial Attack](https://mlsanity.com/paper/2605.17288/) (2026)
- [Fusion Training for Mathematical Generalization in Large Language Models](https://mlsanity.com/paper/2608.09893/) (2026)

## Abstract (arXiv)

Despite outstanding performance in many tasks, language models are notoriously inclined to make factual errors in tasks requiring arithmetic computation. We address this deficiency by creating Calc-X, a collection of datasets that demonstrates the appropriate use of a calculator in reasoning chains. Calc-X is suitable for teaching language models to offload computations to a symbolic system. We survey and unify several existing chain-of-thought datasets into a proposed format, resulting in a standard collection of over 300,000 samples requiring arithmetic reasoning. Finally, we use the new Calc-X collection to train open-source calculator-using models we call Calcformers and show that these models approximately double the accuracy of generating correct results compared to vanilla language model baselines. We make all Calc-X datasets, source code and Calcformers models publicly available.