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We-Math 2.0: A Versatile MathBook System for Incentivizing Visual Mathematical Reasoning

Runqi Qiao, Qiuna Tan, Peiqing Yang, Yanzi Wang, Xiaowan Wang, Enhui Wan, Sitong Zhou, Guanting Dong, Yuchen Zeng, Yida Xu, Jie Wang, Chong Sun, Chen Li, Honggang Zhang

arXiv:2508.10433v1cs.AIcs.CVcs.LG

TL;DR

MLLMs remain challenged by complex mathematical reasoning, while prior work has left gaps in comprehensive knowledge design and model-centric difficulty modeling. We-Math 2.0 combines structured knowledge, progressive datasets, and two-stage RL training; the authors report competitive performance across four benchmarks and strong MathBookEval results.

  • Problem

    MLLMs struggle with complex visual mathematical reasoning, while existing datasets have limited knowledge coverage and difficulty labels based on human learning stages may not match MLLM learning patterns.

    Source: §1 · p. 2

  • Method

    We-Math 2.0 combines a hierarchical mathematical knowledge system, progressively expanded datasets, and two-stage RL training with knowledge-oriented fine-tuning and dynamic scheduling.

    Source: §1 · pp. 2–3

  • Results

    MathBook-RL performs competitively with existing baselines on four widely used benchmarks and achieves strong results on MathBookEval.

    Source: §1 · p. 3, §6 · p. 10

  • Takeaways & Limitations

    MathBookEval assesses reasoning across all 491 knowledge points with diverse step distributions, alongside the paper's training resources.

    Source: §1 · p. 3, §6 · p. 10

Abstract

from arXiv · show

Multimodal Large Language Models (MLLMs) have demonstrated impressive capabilities across various tasks, but still struggle with complex mathematical reasoning. Existing research primarily focuses on dataset construction and method optimization, often overlooking two critical aspects: comprehensive knowledge-driven design and model-centric data space modeling. In this paper, we introduce We-Math 2.0, a unified system that integrates a structured mathematical knowledge system, model-centric data space modeling, and a reinforcement learning (RL)-based training paradigm to comprehensively enhance the mathematical reasoning abilities of MLLMs. The key contributions of We-Math 2.0 are fourfold: (1) MathBook Knowledge System: We construct a five-level hierarchical system encompassing 491 knowledge points and 1,819 fundamental principles. (2) MathBook-Standard & Pro: We develop MathBook-Standard, a dataset that ensures broad conceptual coverage and flexibility through dual expansion. Additionally, we define a three-dimensional difficulty space and generate 7 progressive variants per problem to build MathBook-Pro, a challenging dataset for robust training. (3) MathBook-RL: We propose a two-stage RL framework comprising: (i) Cold-Start Fine-tuning, which aligns the model with knowledge-oriented chain-of-thought reasoning; and (ii) Progressive Alignment RL, leveraging average-reward learning and dynamic data scheduling to achieve progressive alignment across difficulty levels. (4) MathBookEval: We introduce a comprehensive benchmark covering all 491 knowledge points with diverse reasoning step distributions. Experimental results show that MathBook-RL performs competitively with existing baselines on four widely-used benchmarks and achieves strong results on MathBookEval, suggesting promising generalization in mathematical reasoning.

1 Introduction

MLLMs still struggle with complex visual mathematical reasoning, amid gaps in knowledge coverage, model-centric difficulty modeling, and generalization. We-Math 2.0 combines a structured knowledge system, progressive datasets, and RL training, and reports competitive benchmark performance.

1 Introduction

  • Existing datasets have limited knowledge-point coverage and domain diversity, human-defined difficulty levels may not match MLLM learning patterns, and models struggle to generalize to related problems.The paper frames these as gaps in comprehensive knowledge design, model-centric difficulty modeling, and reasoning generalization.Source: §1 · p. 2
  • We-Math 2.0 combines a five-level hierarchy of 491 knowledge points and 1,819 principles with model-centric data modeling and RL-based training.The hierarchy was refined through clustering and human review.Source: §1 · pp. 2–3
  • MathBook-Standard provides principle-level annotations and dual expansions, while MathBook-Pro uses step, visual, and contextual complexity to create seven difficulty levels per problem.The dual expansions are multiple images per question and multiple questions per image.Source: §1 · pp. 2–3
  • MathBook-RL first uses knowledge-oriented CoT fine-tuning, then progressively aligns through average rewards and dynamic scheduling across difficulty levels.Scheduling redirects errors toward incremental-step samples or single-modality problems, depending on their source.Source: §1 · p. 3
  • MathBook-RL performs competitively with existing baselines on four widely used benchmarks and shows substantial improvements in generalization and robustness.MathBookEval covers all 491 knowledge points with diverse step distributions.Source: §1 · p. 3

2 Related Work

Prior work spans visual-math benchmarks, modality alignment, step-wise reasoning, and RL for visual reasoning. We-Math 2.0 adds a systematic, model-centric knowledge framework with RL alignment and a new dataset.

Visual Mathematical Reasoning

  • Visual-math benchmarks assess overall performance or analyze reasoning mechanisms, with MathVerse examining richer textual information and Dynamath evaluating near-duplicate robustness.The cited benchmarks include MathVista, MathVision, MathVerse, We-Math, and Dynamath.Source: §2 · p. 3
  • Prior methods align visual and textual modalities, introduce step-wise reasoning, or optimize rewards through RL for complex visual reasoning tasks.Examples include Math-LLaVA, MAVIS, MathCoder-VL, Math-PUMA, and URSA.Source: §2 · p. 3
  • We-Math 2.0 offers an alternative through a systematic, model-centric knowledge framework combined with RL-based alignment training and a new dataset.The authors hope these resources support ongoing work in visual mathematical reasoning.Source: §2 · pp. 3–4

3 WE-MATH 2.0

WE-MATH 2.0 combines a structured mathematical knowledge system, multidimensional data construction, and a benchmark aligned with that system. Its components organize mathematical knowledge, expand visual problems, model difficulty, and evaluate reasoning across knowledge points and reasoning depths.

3 WE-MATH 2.0

  • WE-MATH 2.0 brings together a five-level knowledge system, multidimensional data construction, and MathBookEval for systematic evaluation.The system organizes 491 knowledge points and 1,819 principles; its data pipeline includes seed construction, variant expansion, and three-dimensional difficulty modeling.Source: §3 · p. 4

3.1 MathBook Knowledge System

  • The MathBook Knowledge System organizes 491 knowledge points and 1,819 fundamental principles in a five-level hierarchy using a Definition–Theorem–Application paradigm.Experts integrate and review a structure developed with AI assistance; GPT-4o maps solution steps to knowledge points and summarizes associated principles.Source: §3.1 · p. 4

3.2 Multi-Dimensional Data Construction

  • MathBook-Standard problems are built through a model-assisted, expert-led workflow that seeks broad coverage, mathematical fidelity, and semantic and visual diversity.Draft problems and diagrams are generated with language-model and GeoGebra assistance, then revised or reworked by experts; almost all drafts were revised, and only 1.2% were directly adopted.Source: §3.2.1 · pp. 4–5, §3.2.2 · p. 5

3.2.1 MathBook-Standard: Seed and Variant Problem Construction

  • MathBook-Standard expands seed problems through multiple images for one problem and multiple problems for one image, preserving geometric construction or reusing diagrams across questions.The image variants change geometric parameters while retaining the underlying construction; question variants target different knowledge points.Source: §3.2.1 · p. 5

3.2.2 MathBook-Pro: Three-Dimensional Difficulty Modeling

  • MathBook-Pro models difficulty along knowledge depth, visual complexity, and contextual abstraction, creating controlled variants that can be composed across dimensions.Step-complexity variants add knowledge points, with the most complex involving at least six; visual and contextual changes preserve the core mathematical structure while increasing demands.Source: §3.2.2 · pp. 5–6

3.3 MathBookEval

  • MathBookEval covers all 491 knowledge points and problems involving 1–10 knowledge points, with reasoning depth grouped into three levels.The benchmark contains 1,000 annotated problems, including 600 collected from existing benchmarks and 400 newly curated.Source: §3.3 · pp. 6–7

4 Methodology

MathBook-RL progressively trains MLLMs from knowledge-guided reasoning toward harder mathematical tasks. It combines cold-start fine-tuning with reinforcement learning that uses average rewards and dynamic difficulty scheduling.

4 Methodology

  • MathBook-RL has a cold-start fine-tuning stage for knowledge-driven reasoning followed by dynamic reinforcement learning intended to enhance generalization.The framework progressively guides models from easier to harder tasks.Source: §4 · p. 7

4.1 Cold-Start Fine-tuning

  • Cold-start fine-tuning uses MathBook-Standard samples covering all 491 knowledge points, with GPT-4o-rewritten explanations that explicitly reference relevant knowledge.Standard supervised fine-tuning trains the model to follow knowledge-guided reasoning chains.Source: §4.1 · p. 7

4.2 Progressive Alignment Reinforcement Learning

  • Progressive Alignment RL first averages rewards across variants for the same knowledge principle, then schedules increasingly difficult MathBook-Pro problems with incremental practice when a transition is missed.The mean-based reward assigns 0.9 for a correct answer, 0.1 for correct format only, and 0 otherwise; incremental samples isolate newly introduced knowledge or modality before the model retries the harder variant.Source: §4.2 · pp. 7–8

5 Experiments

The experiments evaluate MathBook-7B against baselines on four mathematical reasoning benchmarks and examine performance across MathBookEval’s knowledge and reasoning dimensions. Ablations and SFT analyses investigate the roles of training stages, data format, and data scale.

5.1 Experimental Setup

  • The evaluation uses 1K SFT, 5.8K pre-aligned RL, and 4K dynamic-scheduling RL samples, with comparisons on four standard benchmarks.The benchmarks are MathVista, MathVision, MathVerse, and We-Math; evaluation uses VLMEvalKit with modifications to API-calling components due to network constraints.Source: §5.1 · p. 8

5.2 Main Results

  • On We-Math, which tests complex questions and corresponding subproblems, MathBook-7B outperforms strong RL-based baselines.The authors describe this as evidence for the effectiveness of progressive alignment RL in improving knowledge generalization.Source: §5.2 · p. 8
  • Using 9.8K training samples, MathBook-7B achieves consistently strong performance.The authors attribute this to their structured mathematical knowledge system, which they say supports efficient alignment and generalization with limited data.Source: §5.2 · pp. 8–9
  • MathBook-7B achieves remarkable performance and demonstrates superiority across all four benchmarks, using accuracy for MathVista and MathVision, strict score for We-Math, and vision-only accuracy for MathVerse.Source: §5.2 · p. 8, Table 2 · p. 9

5.3 Results on MathBookEval

  • MathBookEval accuracy declines as more knowledge points are required, falling below 50% for most MLLMs on problems requiring 7–10 points.The paper identifies knowledge-point count as an indicator of problem difficulty.Source: §5.3 · p. 9, Table 5 · p. 10
  • On MathBookEval, most MLLMs exceed 50% accuracy in algebra but struggle with geometry, while larger models improve across dimensions and overall scores.The scale comparison is reported within the InternVL2.5 and Qwen2.5-VL families.Source: §5.3 · p. 9, Table 5 · p. 10

5.4 Quantitative Analysis

  • Both RL stages contribute to progressive gains over the SFT-only model; pre-aligned RL yields strong MathVista and We-Math results, while SFT alone gives marginal gains but boosts RL when combined.The ablation compares MathBook-7B with models at different training stages.Source: §5.4 · pp. 9–10, Table 3 · p. 9
  • Natural-language CoT SFT outperforms structured step-wise SFT in subsequent RL, and scaling SFT data from 1K to 10K does not consistently improve results.Models trained on minimal, curated SFT data perform comparably or better than those trained on larger datasets.Source: §5.4 · p. 10, Table 4 · p. 10

6 Conclusion

WE-MATH 2.0 combines a structured knowledge system, annotated datasets with conceptual expansion and difficulty modeling, a two-stage RL framework, and MathBookEval. The paper reports that experiments validate MathBook’s effectiveness in enhancing MLLM generalization.

6 Conclusion

  • WE-MATH 2.0 integrates a five-level knowledge hierarchy, MathBook-Standard and MathBook-Pro, two-stage MathBook-RL, and the MathBookEval benchmark.The hierarchy covers 491 knowledge points and 1,819 fundamental principles; the datasets use conceptual expansions and principled difficulty modeling.Source: §6 · p. 10
  • MathBook-RL uses knowledge-guided supervision and dynamic data scheduling in a two-stage reinforcement learning framework.Source: §6 · p. 10
  • The experiments validate MathBook’s effectiveness in enhancing MLLM generalization.Source: §6 · p. 10

A Broaden Impact

The broader-impact discussion presents WE-MATH 2.0 as a structured, model-centric resource for mathematical training, evaluation, and education. The appendices describe how its hierarchy and principles are built and how GeoGebra supports diagram generation.

A Broaden Impact

  • The datasets pair problems with GeoGebra files, while hierarchical knowledge and step-wise annotations support personalized learning paths and targeted exercises.The authors describe potential uses in adaptive learning, formative assessment, and intelligent tutoring systems.Source: A Broaden Impact · p. 18
  • The model-centric curriculum organizes problems from easy to hard, and combines multi-image and multi-question strategies with dynamic scheduling.The authors present these choices as a perspective for RL curricula and multimodal augmentation.Source: A Broaden Impact · p. 18

B Ethical Considerations

  • Team-curated data are to be released under CC BY 4.0; the dataset uses public or expert-generated data without personal user information, and expert compensation meets or exceeds local minimum wage.Source: B Ethical Considerations · pp. 18–19

C.1.1 Hierarchical Structure of Knowledge Points

  • The knowledge-point hierarchy is built by combining a human-curated structure with an AI-assisted structure derived through topic tagging and semantic-similarity clustering.The initial structure is sourced from references including textbooks and curriculum standards; the AI-assisted structure informs its revision.Source: §C.1.1 · p. 19

C.1.2 Knowledge Principles

  • For each knowledge point, core principles from authoritative sources are supplemented, refined, and validated using aggregated CoT steps mapped to that point.The examples include definitions, theorems, and other foundational statements associated with knowledge points.Source: §C.1.2 · p. 24

C.2.1 GeoGebra-based Diagram Generation.

  • GeoGebra scripts support scalable, reproducible diagram generation with parameter variation, and its geometric primitives offer more precise mathematical relationships than general-purpose plotting libraries.The paper describes GeoGebra diagrams as high-fidelity and aligned with mathematical conventions.Source: §C.2.1 · p. 26

C.2.2 Dataset Diversity and Variant Construction.

MathBook-Standard expands seed problems through multiple images per question and multiple questions per image. MathBook-Pro varies difficulty across three dimensions, while MathBookEval uses knowledge-point annotations to represent reasoning depth and mathematical coverage.

C.2.2 Dataset Diversity and Variant Construction.

  • MathBook-Standard creates multiple diagram instances for a problem and multiple questions for one diagram, expanding seed problems across knowledge points and principles.GeoGebra parameter variation supports the image variants; questions for a shared diagram can derive from different knowledge points or principles.Source: §C.2.2 · pp. 26–27

C.3 MathBook-Pro

  • MathBook-Pro varies step, visual, and contextual complexity and combines them into seven progressive variants per seed problem.Step complexity adds intermediate conclusions, visual complexity adds shaded regions, and contextual complexity reframes the problem as a real-world scenario.Source: §C.3 · p. 34

C.4.1 Dataset Construction and Annotation Protocol.

  • MathBookEval combines filtered open-benchmark samples with newly constructed problems, retaining items whose experts agree exactly on knowledge points at each solution step.At least two experts independently annotate complete step-by-step solutions, with each step decomposed by underlying knowledge points.Source: §C.4.1 · p. 35

C.4.2 Task Dimensions.

  • MathBookEval classifies reasoning depth as 1–3, 4–6, or 7–10 steps and covers all 491 knowledge points across four top-level domains and 13 subdomains.Each step maps to a knowledge point, and each problem is annotated with all knowledge points involved in its solution.Source: §C.4.2 · p. 35

C.4.3 Dataset Statistics.

MathBookEval contains 1,000 annotated problems covering all 491 knowledge points, including 400 newly constructed problems. Compared with existing benchmarks, it adds substantial coverage of intermediate and complex multi-step reasoning.

C.4.3 Dataset Statistics.

  • MathBookEval has 1,000 annotated problems covering all 491 knowledge points, with 600 sourced from open benchmarks and 400 newly constructed.Its problems include multiple-choice and fill-in-the-blank formats across reasoning depths and knowledge domains.Source: §C.4.3 · p. 35, Table 6 · p. 36
  • Existing benchmarks have less than 3% Level 2 problems and none at Level 3, while MathBookEval substantially supplements both categories.The comparison excludes problems experts classify as physics, chemistry, biology, or other subjects; some benchmark totals therefore fall below 100%.Source: §C.4.3 · p. 35, Figure 24 · p. 36

C.4.4 Evaluation Protocol and Metrics.

MathBookEval uses GPT-4o as an LLM judge and reports overall accuracy. The supplementary setup documents evaluated models, their architectures, and inference settings.

C.4.4 Evaluation Protocol and Metrics.

  • MathBookEval adopts GPT-4o as its LLM judge and reports overall accuracy, following the evaluation protocol used by MathVista and MathVerse.The prompt templates are provided in Table 7.Source: §C.4.4 · p. 36, Table 7 · p. 36

C.4.5 Experiment Setup

  • The benchmark setup lists model release dates and official sources, and summarizes the architectures of the evaluated models.Tables 8 and 9 provide these model details.Source: §C.4.5 · p. 37, Table 8 · p. 37, Table 9 · p. 37
  • Closed-source models use standard API generation settings, while open-source models run on 8 NVIDIA A800-SXM4-80GB GPUs with official-example or default hyperparameters.Generation parameters are summarized in Tables 10 and 11.Source: §C.4.5 · p. 37

C.4.6 Additional Results on MathBookEval

Across MathBookEval subdomains, models generally struggle more with geometry than with algebra and fundamental skills. The appendix also specifies the two-stage training implementation.

C.4.6 Additional Results on MathBookEval

  • Models generally perform worse on geometry-related problems, especially solid and analytic geometry, and better on algebra and fundamental skills.The geometry subdomains involve higher visual complexity and require more advanced reasoning; computational-method subdomains are among the stronger areas.Source: §C.4.6 · p. 37

D.1 Implementation details

  • Training uses Qwen2.5-VL-7B-Instruct as the base model and proceeds through cold-start SFT followed by dynamic RL.Both stages are run on 8×A800 GPUs.Source: §D.1 · p. 38
  • Cold-start SFT trains for 1 epoch at a 1.0 × 10−5 learning rate with a 0.1 warmup ratio to establish knowledge-system awareness and knowledge-driven reasoning.The stage is intended to develop explicit awareness of the knowledge system and a knowledge-driven reasoning paradigm.Source: §D.1 · p. 38
  • Dynamic RL uses 8 rollouts per sample and combines answer accuracy (weight 0.9) with format compliance (weight 0.1) in its reward.MathVerify extracts and compares answers; format compliance checks the required output structure.Source: §D.1 · p. 38, Table 12 · p. 38

D.2 Case Study

The case studies compare MathBook-7B with Qwen2.5-VL-7B on We-Math, MathVision, and MathVista. The appendix also specifies benchmark metrics and the categories of models used for comparison.

D.2 Case Study

  • On We-Math, MathBook-7B gives more concise responses while retaining necessary formulas and reasoning steps, addressing overthinking without sacrificing completeness or rigor.Figure 25 compares the two models’ response patterns.Source: §D.2 · pp. 38–39, Figure 25 · p. 39

D.3.1 Details of the Evaluation.

  • Evaluation uses testmini splits for MathVista, MathVerse, MathVision, and We-Math; it reports average accuracy except for We-Math’s primary “Score (Strict)” metric.The experiments follow each benchmark’s official scoring protocol.Source: §D.3.1 · pp. 39–40

D.2 Case Study

  • On MathVision, MathBook-7B shows improved spatial reasoning over the baseline, especially for geometric relationships and positional logic.Figure 26 illustrates the comparison.Source: §D.2 · p. 39, Figure 26 · p. 40

D.3.2 Details of the Baselines

  • Baselines span closed-source models, open-source general models, and open-source reasoning models, enabling comparisons across these three groups.Examples include GPT-4o and Gemini-1.5-Pro; Qwen2.5-VL-7B and InternVL2.5-8B; and Math-PUMA-7B, URSA-8B, R1-Onevision-7B, R1-VL-7B, and MM-Eureka-7B.Source: §D.3.2 · p. 40

D.2 Case Study

  • On MathVista, MathBook-7B applies mathematical concepts correctly and better integrates mathematical knowledge with real-world contexts than the baseline.Figure 27 compares the models’ responses.Source: §D.2 · p. 39, Figure 27 · p. 41
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