# MathCanvas: Intrinsic Visual Chain-of-Thought for Multimodal Mathematical Reasoning

arXiv: https://arxiv.org/abs/2510.14958 · HTML: https://mlsanity.com/paper/2510.14958/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Placed in Multimodal reasoning, one of 17 directions in LLM reasoning.
- Builds on 38 of its 41 references that are in the map.
- First posted on arXiv 16 Oct 2025.

## Placed in

- LLM reasoning / [Multimodal reasoning](https://mlsanity.com/lr/multimodal-reasoning/)

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

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [MMMU: A Massive Multi-discipline Multimodal Understanding and Reasoning Benchmark for Expert AGI](https://mlsanity.com/paper/2311.16502/) (2023)
- [Learn to Explain: Multimodal Reasoning via Thought Chains for Science Question Answering](https://mlsanity.com/paper/2209.09513/) (2022)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Emerging Properties in Unified Multimodal Pretraining](https://mlsanity.com/paper/2505.14683/) (2025)
- [Measuring Multimodal Mathematical Reasoning with MATH-Vision Dataset](https://mlsanity.com/paper/2402.14804/) (2024)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [VLMEvalKit: An Open-Source Toolkit for Evaluating Large Multi-Modality Models](https://mlsanity.com/paper/2407.11691/) (2024)
- [Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning](https://mlsanity.com/paper/2105.04165/) (2021)
- [We-Math: Does Your Large Multimodal Model Achieve Human-like Mathematical Reasoning?](https://mlsanity.com/paper/2407.01284/) (2024)
- [Visual Sketchpad: Sketching as a Visual Chain of Thought for Multimodal Language Models](https://mlsanity.com/paper/2406.09403/) (2024)
- [Imagine while Reasoning in Space: Multimodal Visualization-of-Thought](https://mlsanity.com/paper/2501.07542/) (2025)
- [Seed1.5-Thinking: Advancing Superb Reasoning Models with Reinforcement Learning](https://mlsanity.com/paper/2504.13914/) (2025)
- [MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning](https://mlsanity.com/paper/2310.03731/) (2023)
- [MAmmoTH-VL: Eliciting Multimodal Reasoning with Instruction Tuning at Scale](https://mlsanity.com/paper/2412.05237/) (2024)
- [MAVIS: Mathematical Visual Instruction Tuning with an Automatic Data Engine](https://mlsanity.com/paper/2407.08739/) (2024)
- [Zebra-CoT: A Dataset for Interleaved Vision Language Reasoning](https://mlsanity.com/paper/2507.16746/) (2025)
- [Thinking with Generated Images](https://mlsanity.com/paper/2505.22525/) (2025)
- [Kwai Keye-VL 1.5 Technical Report](https://mlsanity.com/paper/2509.01563/) (2025)
- [Math-PUMA: Progressive Upward Multimodal Alignment to Enhance Mathematical Reasoning](https://mlsanity.com/paper/2408.08640/) (2024)
- [GoT-R1: Unleashing Reasoning Capability of MLLM for Visual Generation with Reinforcement Learning](https://mlsanity.com/paper/2505.17022/) (2025)
- [MINT-CoT: Enabling Interleaved Visual Tokens in Mathematical Chain-of-Thought Reasoning](https://mlsanity.com/paper/2506.05331/) (2025)
- [MathCoder-VL: Bridging Vision and Code for Enhanced Multimodal Mathematical Reasoning](https://mlsanity.com/paper/2505.10557/) (2025)
- [Advancing Multimodal Reasoning via Reinforcement Learning with Cold Start](https://mlsanity.com/paper/2505.22334/) (2025)
- [FLUX-Reason-6M & PRISM-Bench: A Million-Scale Text-to-Image Reasoning Dataset and Comprehensive Benchmark](https://mlsanity.com/paper/2509.09680/) (2025)
- [VisuoThink: Empowering LVLM Reasoning with Multimodal Tree Search](https://mlsanity.com/paper/2504.09130/) (2025)

## Built on by (most cited first, 30 of 35)

- [V-Thinker: Interactive Thinking with Images](https://mlsanity.com/paper/2511.04460/) (2025)
- [Visual Generation Unlocks Human-Like Reasoning through Multimodal World Models](https://mlsanity.com/paper/2601.19834/) (2026)
- [Demystifying Video Reasoning](https://mlsanity.com/paper/2603.16870/) (2026)
- [DraCo: Draft as CoT for Text-to-Image Preview and Rare Concept Generation](https://mlsanity.com/paper/2512.05112/) (2025)
- [COOPER: A Unified Model for Cooperative Perception and Reasoning in Spatial Intelligence](https://mlsanity.com/paper/2512.04563/) (2025)
- [GRADE: Benchmarking Discipline-Informed Reasoning in Image Editing](https://mlsanity.com/paper/2603.12264/) (2026)
- [GGBench: A Geometric Generative Reasoning Benchmark for Unified Multimodal Models](https://mlsanity.com/paper/2511.11134/) (2025)
- [SwimBird: Eliciting Switchable Reasoning Mode in Hybrid Autoregressive MLLMs](https://mlsanity.com/paper/2602.06040/) (2026)
- [Unveiling Fine-Grained Visual Traces: Evaluating Multimodal Interleaved Reasoning Chains in Multimodal STEM Tasks](https://mlsanity.com/paper/2604.19697/) (2026)
- [PerceptionBench: Evaluating Atomic Visual Perception in Multimodal Large Language Models](https://mlsanity.com/paper/2607.24957/) (2026)
- [When to Think and When to Look: Uncertainty-Guided Lookback](https://mlsanity.com/paper/2511.15613/) (2025)
- [DuoGen: Towards General Purpose Interleaved Multimodal Generation](https://mlsanity.com/paper/2602.00508/) (2026)
- [TwiFF (Think With Future Frames): A Large-Scale Dataset for Dynamic Visual Reasoning](https://mlsanity.com/paper/2602.10675/) (2026)
- [Generative Visual Chain-of-Thought for Image Editing](https://mlsanity.com/paper/2603.01893/) (2026)
- [CodePercept: Code-Grounded Visual STEM Perception for MLLMs](https://mlsanity.com/paper/2603.10757/) (2026)
- [Uncovering Understanding-Generation Synergy in Native Unified Multimodal Models: From Representation, Task to System](https://mlsanity.com/paper/2609.01607/) (2026)
- [Thinking with Constructions: A Benchmark and Policy Optimization for Visual-Text Interleaved Geometric Reasoning](https://mlsanity.com/paper/2603.18662/) (2026)
- [DeltaV: Thinking with Visual State Updates in Unified Large Multimodal Models](https://mlsanity.com/paper/2607.08434/) (2026)
- [FaithEyes: Towards Faithful Tool Use via Multi-Agent Process-Image Verification](https://mlsanity.com/paper/2607.28225/) (2026)
- [Failure-Informed Image Self-Augmentation for Multimodal Large Language Model Self-Improvement](https://mlsanity.com/paper/2608.03733/) (2026)
- [MultiTalk: Scaling Full-Duplex Speech Models to Long, Multi-Party, Bilingual Conversation](https://mlsanity.com/paper/2609.36903/) (2026)
- [R1-SyntheticVL: Is Synthetic Data from Generative Models Ready for Multimodal Large Language Model?](https://mlsanity.com/paper/2602.03300/) (2026)
- [InterCoG: Towards Spatially Precise Image Editing with Interleaved Chain-of-Grounding Reasoning](https://mlsanity.com/paper/2603.01586/) (2026)
- [LatentGeo: Learnable Auxiliary Constructions in Latent Space for Multimodal Geometric Reasoning](https://mlsanity.com/paper/2603.12166/) (2026)
- [Draw2Think: Harnessing Geometry Reasoning through Constraint Engine Interaction](https://mlsanity.com/paper/2605.20743/) (2026)
- [GeoMathCode: Understanding Interleaved Math-Code Reasoning for Geometry Problem Solving](https://mlsanity.com/paper/2605.25384/) (2026)
- [Model Guides You How to Draw: Adaptive Visual Gating for Unified Multimodal Reasoning](https://mlsanity.com/paper/2607.10004/) (2026)
- [See2Think: Do Multimodal Models Really Use Intermediate Visual States?](https://mlsanity.com/paper/2607.26769/) (2026)
- [MetaReason: Precise Interleaved Multimodal Reasoning via Editing Meta Information for Solving Geometry Problems](https://mlsanity.com/paper/2608.15006/) (2026)
- [Aphanta: Diagnosing Task-Aligned Image-Edited Intermediates for Multimodal Reasoning](https://mlsanity.com/paper/2608.26993/) (2026)

## Abstract (arXiv)

While Large Language Models (LLMs) have excelled in textual reasoning, they struggle with mathematical domains like geometry that intrinsically rely on visual aids. Existing approaches to Visual Chain-of-Thought (VCoT) are often limited by rigid external tools or fail to generate the high-fidelity, strategically-timed diagrams necessary for complex problem-solving. To bridge this gap, we introduce MathCanvas, a comprehensive framework designed to endow unified Large Multimodal Models (LMMs) with intrinsic VCoT capabilities for mathematics. Our approach consists of two phases. First, a Visual Manipulation stage pre-trains the model on a novel 15.2M-pair corpus, comprising 10M caption-to-diagram pairs (MathCanvas-Imagen) and 5.2M step-by-step editing trajectories (MathCanvas-Edit), to master diagram generation and editing. Second, a Strategic Visual-Aided Reasoning stage fine-tunes the model on MathCanvas-Instruct, a new 219K-example dataset of interleaved visual-textual reasoning paths, teaching it when and how to leverage visual aids. To facilitate rigorous evaluation, we introduce MathCanvas-Bench, a challenging benchmark with 3K problems that require models to produce interleaved visual-textual solutions. Our model, BAGEL-Canvas, trained under this framework, achieves an 86% relative improvement over strong LMM baselines on MathCanvas-Bench, demonstrating excellent generalization to other public math benchmarks. Our work provides a complete toolkit-framework, datasets, and benchmark-to unlock complex, human-like visual-aided reasoning in LMMs. Project Page: https://mathcanvas.github.io/