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

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.