We-Math 2.0: A Versatile MathBook System for Incentivizing Visual Mathematical Reasoning
Authors: Runqi Qiao
Qiuna Tan
Peiqing Yang
Yanzi Wang
Xiaowan Wang
Enhui Wan
Guanting Dong
Shiqiang Lang
Sitong Zhou
Yida Xu
Yuchen Zeng
Jie Wang
Chong Sun
Chen Li
Honggang Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Mathematical Reasoning
Multimodal Large Language Models
Primary area: datasets and benchmarks

Abstract: Multimodal large language models (MLLMs) have demonstrated impressive capabilities across various tasks but still struggle with complex mathematical reasoning. Prior work has mainly focused on dataset construction and method optimization, while often overlooking two critical aspects: comprehensive knowledge-driven design and model-centric data space modeling. We introduce WE-MATH 2.0, a unified system that integrates a structured mathematical knowledge hierarchy, model-centric data space modeling, and a reinforcement learning (RL)-based training paradigm to enhance the mathematical reasoning abilities of MLLMs. Our contributions are fourfold: (1) MathBook Knowledge System: a five-level hierarchy covering 491 knowledge points and 1,819 fundamental principles; (2) MathBook-Standard and MathBook-Pro: datasets that ensure broad conceptual coverage and robust training through dual expansion, a three-dimensional difficulty space, and seven progressive variants per problem; (3) MathBook-RL: a two-stage RL framework including Cold-Start Fine-Tuning to align models with knowledge-oriented chain-of-thought reasoning, and Progressive Alignment RL leveraging average-reward learning with dynamic data scheduling for progressive difficulty alignment; (4) MathBookEval: a benchmark covering all 491 knowledge points with diverse reasoning step distributions. Experimental results show that MathBook-RL achieves competitive performance on four widely used benchmarks and demonstrates strong results on MathBookEval, suggesting promising generalization in mathematical reasoning.

Published as a conference paper at ICLR 2026
WE-MATH2.0: A VERSATILE MATHBOOK SYSTEM FOR INCENTIVIZING VISUAL MATHEMATICAL REASONING
Runqi Qiao! Qiuna Tan '1 Peiqing Yang', Yanzi Wang?, Xiaowan Wang', Enhui Wan', Guanting Dong', Shiqiang Lang', Sitong Zhou', Yida Xu',
Yuchen Zeng', Jie Wang', Chong Sun", Chen Li Honggang Zhang":
BBUT WeChat Vision, Tencent Inc. TTinggu University qrq@bapt.edu.com cn qiunatan&bupt. edu.
chaselli wewelllli@eeeeet.coo
https: //we petps://we-math2/ heirth..gituu.....
ABSTRACT
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 structured mathematical knowl- edge system, model-centric data space modeling, and a reinforcement learning (RL.)-based training paradigm to comprehensively enhance the mathematical rea- soning abilities of MLLMs. The key contributions of We-Math 2.0 are fourfold: (1) MathBook Knowledge System: We construct a five-level hierarchical sys- tem encompassing 491 knowledge points and 1,819 fundamental principles. (2) MathBook-SSandardd & Pro: We develop MathBook -oook--tdddrdd a dataset that ensures broad conceptual coverage and flexibility through dual expansion. Addi- tionally, we define a three-dimensional difficulty space and generate 7 progressive variants per problem to build Math
challenging dataset for robust training. (3) MathBook-RL: We propose a two-stage RL framework compris- ing: (i) Cold-Start Fine-tuning, which aligns the model with knowledge-oriented thain-of-thougee reasoning; and (ii) Progressive Alignment RL, leveraging average- reward learning and dynamic data scheduling achieve progressive alignment across difficulty levels. (4) MathBookEval: We introduce comprehensive bench- mark covering all 491 knowledge points with diverse reasoning step distributions. Experimental results show that MathBook- performs competitively with ex- isting baselines on four widely-used benchmarks and achieves strong results on MathBookEvaal, suggesting promising generalization in mathematical reasoning.
INTRODUCTION
Large Language models (LLMs) have demonstrated remarkable capabilities across a wide range of
tasks (Achiam et al. 2023 DeepSeek-A]] 2025) Jaech et al. 2024 Wan et al. 2024 Trinh et al. [024]]
Xin et al. 2024). Building on this foundation, Multimodal Large Language Models (MLLMs) have
shown impressive performance visual question answering (VQA) (Bai al. 2025b Zhu et al.
2025: Guo et al.
2025) Qiao et al.) [024cc 2025a Zhang et al. 2024c). optical character recognition (OCR) (Ye al.
2023b al Wei et al.) 2024| Tan al. (025), and object detection (Liu et al. 2025b
Ren et al. 2024). However, MLLMS still face difficulties with complex reasoning tasks, particularly
visual mathematical moblem--ollving. where generalization remains fundamental challenge (Lu
ct al.
2023 Zhang et al. 2024a Wang et al. 2024
Recent efforts enhance mathematical reasoning in MILMs have primarily focused on three
directions: dataset construction (Lu et al. 2021a Zhang et al. 2024b Shi et al. 2024a Wang et al.
'Equal contribution.
I Work done as interns at WeChat, Tencent Inc. Corresponning author. zhhg hupt.edu.en

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Published as a conference paper at ICLR 2026
Table 1: Comparison of We-Math 2.0 with several representative multimodal mathematical datasets.
Dataset
Usage
Duta
Knowledge-evee
Principle-tevel Annotation
Difficulty
Annotation
Annotation
Levels
Geometry AK Lu al. 2021a
Training Data & Benchmark
Manual
Votk Shi et al:
X024n
Training Data
Collection
We-Math Iao et al 2024a
Benchmark
Manual
GeoSense
Xu ci al
2025
Benchmark
Manual
2.0 (Ours)
Training Data & Benchmurk
Manual
2023),
preference optimization (Zhuang et al.
2024 Luo et al. 2025), ) and reinforcement learning Foundation approaches aggregated datasets from 2024b). Subsequent efforts introduce structured supervision
(RL) (uung et al., 2025; Meng et al. 2025).
diverse mathematical domains (Shi et al.,
formats (e.g., Chain-of-Thought (CoT)) combined with preference optimization to guide step-by-step reasoning (Zhuang et al.) (025). More recently, RL-based studies with curriculum-ban based training have been employed to further improve model performance on complex reasoning tasks (Huang et al. 2025
Wan et al. 2025). Despite this progress, several fundamental challenges remain:
(1) Lack of a comprehensive knowledge system: Existing MLLMs show uneven performance
across different subfields of math reasoning. (Lu et al. 2023 Wang et al. 2024) Unfortunately, current datasets suffer from limited coverage of knowledge points and domain diversity, underscoring the necessity of establishing more systematic knowledge system.
(2) Lack of model-centric difficulty modeling: Existing multimodal training datasets primarily perform difficulty annotation based human learning stages (Meng al. 2025). However, recent studies (Lei et al., 2024) Qiao et al., 2024b] Lu et al., (2023) reveal that MLLMs do not exhibit learning patterns that align well with these human-defined levels. This highlights the need for more model-centric approach to modeling data difficulty.
reasoning generalization: MLLMs are capable of solving complex problems, but perform poorly on corresponding subproblems (Qiao et al. 2024b) as well as on similar, same-type tasks (Zou al. (024). This underscores the current training methods' focus on problem memorization rather than fostering reasoning generalization.
(3) Lack of emphasis
To address these limitations, we introduce WE-MATH 2.0, versatile framework that combines a structured mathematical knowledge system, model-centric data space modeling, and a reinforcement learning-basee training paradigm to comprehensively improve MILM's s reasoning capabilities (see Table In detail, we begin by establishing the MathBook Knowledge System, five-level hierarchy comprising 491 knowledge points and 1,819 fundamental principles (see Figure This structure systematically derived from sources such as Wikipedia and open- textbooks, refined through hicrarchical clustering, and further revised by human experts.
Building on this foundation, we introduce MathBook-Standard dataset featuring comprehensive annotations at the level of 1,819 knowledge principles, along with carefully curated problems to ensure broad, balanced coverage, particularly in underrepresented mathematical domains. To foster deeper conceptual understanding, MathBook-Standanl employs dual expansions: "multi-images per question" and "multi-questtonn per image". ", enabling diverse problem that achieve conceptual flexibility. Crucially, we propose a pivotal three- faree-dimensional difficulty modeling framework that redefines mathematical problem construction. By explicitly modeling "step complexity", "visual complexity" and "contextual complexity". each problem is systematically expanded into seven difficulty levels to form MathBook-Pro. This design enables structured, progressive learning for MLLMs, laying strong foundation for improved reasoning across difficulty levels.
To further enhance MLLMs' general mathematical reasoning ability, we propose MathBook-RL, a two-stage reinforcement learning framework for progressive and robust training:
(1) Cold-Start Fine-tuning: We first adopt supervised fine-tuning that guides the MILM to learn knowledge-orienter CoT reasoning, internalizing it to acquire conceptual understanding and structured problem-solving paradigms. (2) Progressive Alignment RL: We propose a curriculum- based RL paradigm. Leveraging the MathhBookSttndarrd we first align the model's analogical reasoning by introducing an average reward mechanism. Building this foundation, we progressively train the MLLM on MathBook- Pro and further introduce two dynamic scheduling strategies: i) Knowledge Increment Scheduling: When errors occur due to complex reasoning steps, the model is adaptively redirected to relevant
and knowledge-point features in
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incremental-stey samples in MathBook-Standas ii) Modality Increment Scheduling: When emrors stem from increased modality complexity, the model is guided through ungle-modality incremental problems. This targeted curriculum enables effective knowledge transfer actoss difficulty levels.
To comprchensively evaluate MLLMs' reasoning capability. we introduce EathBo a bench- mark covering all 491 knowledge points with diverse step distributions. Experimental results show that MathBook-RL. performs competitively with existing baselines on four widely used benchmarks and substantially improves generalization and robustness. In summary, our contributions are:
We propose the Math Knowledge System, five-level hierarchical framework with 491 knowledge points and 1,819 fundamental principles, enabling systematic and comprehensive mathematical knowledge supervision.
We develop NathBook-Standard and MathBook-Pro, two novel datasets that combine com- prehensive step-wise annotation, dual expansions for conceptual flexibility, and a principled three-dimensional difficulty modeling framework for structured and progressive learning.
We introduce MathBook-RL a two-stage RL framework that integrates structured knowledge supervision and dynamic data scheduling, improving the reasoning capabilities of MLLMs.
We present MathBookEval, benchmark designed to comprehensively evaluate model reasoning across diverse knowledge points and step distributions. Extensive experiments demonstrate that our approach achieves remarkable performance in both generalization and robustness.
RELATED WORK
Visual Mathematical Reasoning, Recently, visual mathematical reasoning has advanced rapidly (Shi
et al. 2024a| /Zhang et al. 2024b) /Zuuag et al.] 2024 Han et al.) 2024 [ Luo et al.) (025). Bench-
marks such Math Vista (Lu et al. 2023) and Math Vision (Wang al. 2024) assess overall
performance, while Math Verse (Zhang et al. 2024a) and Dynamath (Zou et al. 2024)) examine rea-
ooggcially progress has been made through visual-textual alignment (Shi et al. 2024a ZZhagg et al.
soning mechanisms and robustness. Methodolog
2024b Wang et al. 2025), , step- wise reasoning (Zhuang
al. 2024: Luo et al. | (025): and RL-based optimization (Huang et al. 2025| Zhang et al. | 2025
Meng et al. 2025| Chen et al. 2025 Liu et al. 2025a Al et al. 2025 Wan et al. 2025
Qiao et al.
2025b
Zheng et al. 2025 Yang et al. 2025a; Team al. 2025a Hong et al.) 2025 Team et al.
2025b
which show promising gains complex tasks. However, robust and generalizable visual
reasoning remains an open challenge. Therefore, we propose a systematic, model-centrie knowledge system, integrate it with RL-based alignment and a new dataset, aiming to provide fresh insights for the community.
WE-MATH 2.0
Overview. In this section, introduce WE-MATH 2.0, unified system designed to advance visual mathematical reasoning in MLLMs, developed from three key aspects: (1) We construct a five- level MathBook Knowledge System (3.1)) systematically organizing 491 knowledge points and 1,819 fundamental principles for comprehensive mathematical supervision. (2) We propose a Multi- Dimensional data construction pipeline ($3.2), incorporating seed problem construction, variant expansion, and principled diree-dimensional difficulty modeling. (3) We introduce MathBookEval (3..), benchmark aligned with our knowledge system for systematic evaluation.
3.1 MATHBOOK KNOWLEDGE SYSTEM
System Overview. We construct five-level hierarchical MathBook Knowledge System orga- nized by the Segsiiiionn-Theermm--pppiiiiiiii
paradigm (Fitzpatrick) 2008). The core is set of kw }. N = 491, spanning primary to university mathematics. Each k; is associated with set of fundamental principles P. {Pı1.... Pam,) }, my [L, 7]. where
knowledge points K. {K1, .2....,
P = UN P; and |P| = 1,819. (Figure 3 illustrate examples of principles within the MathBook Knowledge System, while Figure shows how problems aligned with the system.)
Hierarchical Construction via Human-AI Collaboration. We construct through a hybrid process. Human experts first design initial structure KChuman based authoritative sources,
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We-Math 2.0
Demppeer
Areinhiitt
491 Abanledge Ponnte
pronety.222
geoety
1,819 Principar
Alyetro
Figure 1: Overview of MathBook, including knowledge points, principles, and sample problems.
including textbooks, Wikipedia, and national curriculum standards. In parallel, we sample 30K problems from the existing math dataset (Lu et al. 2021b Johnson et al. 2017, Gao et al. 2023 Shi
ct al. 2024a] Peng et al. [024] [uo al.] [025]] Zhuang et al.) [2024]] Zhang et al.]] 2024b) merging them into unified dataset. We then use GPT-4o (OpenAll 2024) to assign multi-level topic tags
T = (l1,-.., ,',,,,,,,,, ). followed by hierarchical clustering on the semantic similarity matrix E [R=xn to
obtain an Al-generated structure K:auto The final knowledge point set IC is produced by expert-guided
integration of X-human and Xi with independent review for quality assurance.
Fine-Grained Principle Annotation. Given the constructed K,, we employ GPT-4o to annotate the step-level knowledge points for each problem qj E Q = {41...., .f4 } by mapping each step in its chain- of-thought solution to the corresponding k; K. This yields mapping M1 : qj + (ki, .......). ). forming a set of step-level solution paths for each knowledge point. Next, for each ka GPT-4o summarizes the set of theorems and principles used across all associated solution paths, resulting in a mapping M2 : K; + (pri,..., Pim. J. Finally, these Al-extracted principles are consolidated and cross-checked with those written by human experts, with iterative refinement to ensure completeness and accuracy of P. Detailed guideline of our system are listed in Appendix B.]
3.2 MULTI-DIMENSIONL DATA CONSTRUCTION
In this section, we introduce our data construction pipeline: MathBook-Standare & Math
3.2.1
SIUUBBOOKSSANDARR SEED AND VARIANT PROBLEM CONSTRUCTION
Seed Problem Construction. To ensure rich coverage and high-quality design, we construct problems based on the knowledge system following 3 guidelines:
(1) All diagrams are rendered with GeoGebra for precise geometric representation; (2) Problems focus on math essence, avoiding reliance on superficial visual cues; (3) Each problem strictly corresponds to its designated principle set P,. To achieve these, we adopt assdel-assisted, expert-led" workflow. Given knowledge point k, E IC and its associated principle set P., an LLM first generates a draft problem, including the question, answer, and XML script. We then GeoGibra, software that renders diagrams from XMsed scripts, to automate the generation of draft images: (k,
- (Ą a afraft x: draft). The resulting visual drafts serve as references to guide human experts in constructing problems and diagrams via GeoGebra scripting. In practice, almost all drafts were revised or reworked by experts ! in order avoid reliance on superficial visual cues and ensure proper alignment with the underlying mathematical principles. The final seed problem set is Dzeed {ki, Pij, qs a,, I, mnm)) }} covering
Only 1.2% of the drafts were directly adopted hy experts.
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(a) Pipeline
SFT
Pre allined RL
Dynamic Scheduling Rd
(b) WethBoakk-Stnnard
Math
00
Seed
(LOK)
Difficulty
[4K]
argle 2BOC.
Compleiity
Mutt--rrzllem
(55
Find the measure angle cOBC.
What
<B00
cehell Hater Priiede Ltfte
Find the messure of angle cbOB.
de
One- tonpFrobien--kii--Zmee
Conterted
Find the neature of angle <BOC
Visua
and hegll comp
Complecity
Figure 2: Overview of MathBook dataset and the corresponding training phase.
all knowledge points and principles. The detailed GeoGebra-based diagram generation guidelines can be found in the Appendix [.2.1
Variant Problem Expansion. To further enhance the diversity and generalization ability of the dataset, we systematically construct two types of variants based on each seed problem:
(1) One-PProlem-Muutti-mmee
Variants (DingVar): Given seed problem (gi, di; I;) DDseed we fix the problem statement and knowledge annotation Piy), and generate set of im- ages (1(")) _1(2) ....., (.....,,,))) 1('))} by varying the parameters GeoGebra while maintaining the under- lying geometric construction. Each image corresponds to different geometric instantiation (e.g..
acute/obtuse/right triangle), resulting in distinct answers a!") (q:, (1")) PimgVar>
m). m.. This approach enriches the visual data diversity while preserving semantic consistency.
(2) One-Image-Multi-PProan Problemm Variants (Dosvar): Given seed image 4, we construct multiple new
problems q:" targeting different knowledge points k(()) and principles Pij curated by experts with (+) II)) E DouVar
(s)
1, n). This strategy leverages the reusability of high-quality diagrams to generate diverse problem variants.
language model assistance: {(gg.))
By systematically applying these variant construction methods to each seed problem, we build the Mathhook-SSandarr dataset with rich semantic and visual diversity.
3.2.2 MATHBOOK- THREE-DIMENSEE DIMENSIONAL DIFFICULTY MODELING
To systematically characterize problem complexity from model-centric perspective, define three-dimensional difficulty space for each seed problem along three orthogonal axes (in Figure :
(1) Step Complexity (o,): Knowledge-ond oriented reasoning depth is quantified by the number of involved knowledge points, which from the MathBook knowledge system. Given a seed problem with process-oriented knowledge points, we construct variants requiring I' > (with least six).
-orienete
(2) Visual Complexity (d.)): We increase complexity by adding auxiliary elements (e.g., lines) to the original image via GeoGebra, while preserving the core structure.
(3) Contextual Complexity (d.): Captures the contextualization of the problem statement. We vary the textual context from concise mathematical descriptions complex linguistic scenarios.
Each seed problem (go, Ag, Tp) E " Dsoed serves as the origin in a structured difficulty space. To enable controlled and interpretable expansion, we generate derived problems by varying single dimension E {Ø., o.. }.} at a time, yielding variants (q (d) (! Through multiple rounds of such single- axis transformations, we progressively construct complex problems by composing changes
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across multiple dimensions. Formally, the most advanced variant takes the form: (q', a. 1") Φg a ov o (e(9 do 1o). In MathBook-Proo, the expansion along each dimension is implemented as:
(1) Along the o, dimension, introduce intermediate conclusions as new conditions, enabling knowledge-drivee progressive deepening of reasoning, expressed as Ki+1 = K; + 1, where K; denotes the number of knowledge points involved step i. In MathBook-Proe the most complex step variants involve at least 6 knowledge points.
(2) Along the 6.., increase visual complexity by adding auxiliary lines, altering geometric configurations or introducing new spatial constructs via GeoGebra, while preserving the core structure.
(3) Along &e. embed the mathematical core into real-world contexts linguistically abstract scenarios, increasing the semantic and contextual demands of the problem statement.
By expanding along the defined dimensions, generate set of difficulty-controlleed problem variants for each knowledge point, forming the difficulty modeling subset Daifficuity of MathBook-Pro..
3.3 MATHBOOKEVAL
Design Principles. To ensure the quality and interpretability of annotations visual math reasoning tasks, MathBookEval is designed based on the following principles: (1) Comprehensive Knowledge Coverage: Problems involve 491 knowledge points, spanning primary to university level, demonstrat- ing broad coverage.
(2) Multi-level Reasoning Depth: Each problem integrates 1-10 knowledge points, compared to 1-3 (Level 1) in existing process- -oiente- benchmarks. as illustrated in Figure 27 Notably, our annotation adheres to three principles: (1) integrating public and newly constructed problems under unified guideline; (2) expert step-by-step annotation with explicit knowledge-poing mapping; and (3) independent cross-validation, retaining only consistently annotated items.
Data Statistics and Evaluation Protocol. MathBookEval contains 1,000 fully annotated problems. covering all 491 knowledge points in the unified knowledge system K., with 600 problems collected from existing benchmarks and 400 newly curated (Detailed statistics are presented in Table 8). We provide detailed statistics and splits along two key dimensions: (1) Reasoning Dimension: Problems divided by reasoning steps into three levels: 1-3 (Level 1), 4-6 (Level 2), and 7-10 (Level 3), reflecting different reasoning depths. (2) Knowledge Dimension: The 491 knowledge points grouped into 4 domains and 13 subdomains, covering primary to university level. Figure 27 demonstrates superior coverage of knowledge points and reasoning depth. All problems are in multiple-choice or fill-in-hhe-blank format.
METHODOLOGY
In this section, we introduce RathBook-RI,, a two-stage framework that progressively guides MLLMs to develop reasoning capabilities from easy to hard. The first stage is cold-start fine-tuning phase that establishes a knowledge-driven reasoning paradigm ($4.I); I) the second is a dynamic reinforcement learning phase that enhances the model's generalization ability ($4.2)
4.1 COLD-START FINE-TUNING
The cold-start supervised fine-tuning (SFT) stage aims to instill explicit awareness of knowledge system and a knowledge-driven reasoning paradigm, avoiding rote memorization. The initial training set Dinit is built from MathBook-Standard, which fully covers all 491 knowledge points. To improve rationale interpretability, we use GPT-4o (OpenAl 2024) to rewrite each sample with natural language explanations that explicitly reference the relevant knowledge. The model is then trained using standard supervised fine-tune -hn-ttnnn:: LSFT (0) = E(r,y)>Da. [olog Pa(y x)]. This stage enhances the model's s ability to internalize the knowledge system and follow knowledge-guidde reasoning chains.
4.2 PROGRESSIVE ALIGNMENT REINFORCEMENT LEARNING
(1) Pre-aligned RL. Prior the dynamic scheduling stage, perform initial RL training dataset to ensure that the model develops genuine understanding of mathematical knowledge. Specifically, we utilize the DingVar subset, where each group contains multiple variants

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of the same knowledge principle: (y;, dj (0) 1100) DingVar, ( 1,..., m. To encourage consistent and robust performance across different formulations, we adopt a mean-based reward function:
r = 1
m Ei= r(0) where r(t) = 0.9 the answer is correct, 0.1 if only the format is correct, and O otherwise. Specifically, for problems corresponding to the same knowledge principle, rollout rewards are first sorted within each problem. Next, the mean reward at each sorted position is calculated across these problems and subsequently employed in the calculation of A, Instead of focusing on individual problems, this design integrates rewards across all problems corresponding to the same knowledge principle, thereby providing a more comprehensive critic.
(2) Dynamic Scheduling RL. In this section, we introduce a dynamic RL algorithm based on MathBook- The training process is organized as a curriculum along a main trajectory increasing difficulty, primarily centered on the knowledge dimension. For each base problem (4o. do, To), denoted as To. we construct a sequence of increasingly challenging variants as follows:
Zo + (20) o, (x + o, (2o) Φ, @ (e(Fp)
(I)
where o. denotes increasing the number of knowledge points, de and de denotes increasing visual complexity and contextual abstraction. This forms progressive path from basic advanced reasoning for each knowledge anchor.
Incremental Learning Mechanism. At each curriculum transition x - 6(x), if the model fails on ((x) after succeeding on z, we introduce an incremental learning step. Specifically, we define the incremental set (z, σ) as collection of samples that isolate the new knowledge or modality introduced by db. The model is first trained on A(x, Φ) to address the incremental challenge, then reattempts (xx). Concretely:
Knowledge Increment Scheduling: For zo - o. (Io). if the model fails on (x()),, we construct (zo. o.). comprising auxiliary problems z/ that target the new knowledge point(s) from &..
Modality Increment Scheduling: For (x0) - oe (1()) (or o, (e(x))), if the model fails on the more complex sample, we construct A (zo)), @h)) (or ((b.(T)) 6c)), which contains samples isolating the visual or contextual complexity.
This incremental adaptation, denoted by A(x, 6) at each step, ensures that the model can efficiently bridge the gap between curriculum stages. Notably, our overall RL objective is optimized using Group Relative Policy Optimization (GRPO) (Shao et al. (202)), which extends PPO by estimating the baseline from group scores instead of separate critic. The GRPO objective is:
(0) = Blg ((,), ~ Toois (Ola)
00.
(2)
Tont (or.llg o+) Ar.., To(Or, on,c))
no(on,l9,,i,4
1 +e) ALE xTx [Tol|tres]
min
To (0t.el oi,et)"
where and 3 are hyperparameters, denotes the input, (o: {o:){_ are sampled outputs, and is the corresponding reward. Å., is the normalized advantage value for the i-th trajectory in the group. This curriculum-driver RL process, augmented with explicit incremental adaptation at each stage, enables the MILM progressively master complex, multi-dimennional reasoning tasks while preserving stability and generalization across knowledge, visual, and contextual variations.
5
EXPERIMENTS
5.1 EXPERIMENTAL SETUP
Datasets. All training data are sourced from WE-MATH 2.0 in compliance with copyright and licensing requirements, and all expert-construuted problems will be released under appropriate CC licenses. We use IK, 5.8K and 4K samples for SFT, pre-aligned RL, and dynamic scheduling RL stages, respectively. Experiments are conducted on four standard mathematical reasoning benchmarks:
Vista (Lu et al. 2023), Math (Wang et al. 2024), Math Verse (Zhang al. 2024a), and
We-Math (Qiao et al. 2024a). Detailed evaluation protocols are provided in Appendix C.5.1
Baselines. We conduct training based both Qwen2.5-VL-7B and Owen2.5-VL-33 and compare our method with three categories of baselines: (1) Closed-source models (c.g., GPT.
4o (OpenAl 2024)); (2) Open-source general models (e.g., Intern VL.2.5 series (Chen al. 2024a
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Table 2: Performance comparison across four widely-used mathematical reasoning benchmarks. Each benchmark follows its standard evaluation metric: Math Vista and Math Vision use accuracy. We-Math reports the strict score, and Math Verse evaluates on the vision-only subset with accuracy. Data sizes used for SFT and RL are annotated in blue and red, respectively.
Model
#Data
Avg.
Math Vista
Math Vision
We-Math
Math Verse
Closed-source
GPT-4o-latest
54.0
71.6
43.8
50.6
49.9
Gemini-1.5-Pro
53.6
67.9
50.5
54.8
Open-source (General)
Qwen2.5-VL-7B
42.6
68.2
25.1
36.0
41.1
LntcmWL22.58B-BoN-
41.7
68.2
25.6
38.6
34.5
Open-source (Reasoning)
Math-PUMA-7B
1.88M
47.9
19.2
26.0
URSA-8B
2.96M
37.8
58.8
28.7
32.8
31.0
Vision-7B
155K+I0K
64.1
29.9
30.1
R1-VL-7B
260K+10K
63.5
24.7
22.7
MM-Eureka-7B
15K
45.2
73.0
26.9
34.5
46.2
WeThink-7B
120K+20K
47.5
71.6
26.0
48.0
44.2
VLAA-Thhinke=-7B
25K
46.0
68.0
26.4
41.5
48.2
OpenVL.Thinker-78
35K+15K
72.3
25.9
MathBook-7B (Ours)
1K+9.8K
48.7
73.0
28.0
48.4
45,2
(vs Qwen2.5-VL-7B)
+6.1
+4.8
+2.9
+12.4
+4.1
Qwen2.5- 2.5- VL series (Bai al. 2025b)); (3) Open-source reasoning models (e.g.. R1-VL Zhang ct al. ). Our evaluation is based on VLMEvalKit (Duan et al. 2024). Detailed descriptions of baselines are provided in Appendix C.5.2
5.2
MAIN RESULTS
Table
displays the performance of our MathBook-7B across various benchmarks. Overall, our method achieves remarkable performance, clearly demonstrating its superiority. Further analysis reveals the following observations. (Results for the 3B setting are provided in Appendix C.2)
2
(1) Overall superiority of MathBook. Compared to the backbone Qwen2.5-VL- MathBook-7B achieves over a 6% improvement across benchmarks, validating the effectiveness our approach.
(2) Effectiveness of progressive alignment reinforcement learning knowledge generalization. Focusing on the We-Math benchmark, which requires solving both complex multi-step questions and their corresponding subproblems, MathBook- 7B outperforms strong RL baselines. This demonstrates the effectiveness of progressive alignment reinforcement learning in knowledge generalization.
(3) Less is More: Efficiency with limited training data. MathBook-7B achieves strong performance using only 9.8K training samples. We attribute this to the high-quality, structured mathematical knowledge system we constructed, enabling efficient alignment and generalization with limited data.
5.3 RESULTS ON MATHBOOKEVAL.
To investigate MLLM abilities in reasoning depth and knowledge coverage breadth, we conduct Math- BookEval and observe the following (see Table Complete results are provided in Appendix B.4.6
(1) MILMs performance negatively correlates with the number of required knowledge points. As reasoning steps increase, model accuracy declines. In particular, problems requiring 7-10 knowledge points yield the lowest accuracy (below 50%). These results highlight the challenge of multi-step reasoning and validate knowledge points key measure for modeling problem difficulty.
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Table 3: The performance of different MLLMs on MathBookEval for reasoning evaluation. Ace.:
Accuracy; FS., Foundational skills; PS.: Probability and statistics; Geo.; Geometry; Alg. Algbra
Reasoning
Knowledge
Models
Acc.
Levell
Level2
Level3
FS.
PS.
Geo.
Alg.
Closed-source MILMs
GPT-4o GPT-4V
50.8
52.8
48.9
41.7
33.8
57.6
44.2
67.2
42.8
44.0
43.0
31.9
36.8
56.6
33.5
59.4
Open yource MIL.Ms
VL2.5722
51.8
52.5
51.8
45.8
50.0
64.2
42.6
67.6
Qwen2.5 -
57.1
58.3
56.4
50.0
52.9
58.5
52.1
68.8
LLaVA-One Vision-72B
43.0
44.8
42.0
31.9
38.2
52.8
37.0
53.5
Intern VI.2m/I...55888 V.2.5-8B
37.9
40.7
34.5
27.8
33.8
46.2
31.4
50.0
Qwen2.5-VL-78
46.7
50.1
43.0
33.3
44.1
58.5
38.8
60.2
LLeVA-One -i
31.6
34.3
28.0
23.6
36.8
41.5
24.9
41.0
RI-VL-7B
38.0
41.9
32.6
27.8
38.2
38.7
32.3
50.4
MM-Eureka-7B
50.0
50.6
51.5
38.9
47.1
52.8
44.0
62.9
MathBook-7B 50.4
52.0
48.2
45.8
57.4
67.9
40.5
63.3
(2) MLLMs perform well on algebra but struggle with geometry. Along the knowledge dimension, most MLLMs demonstrate strong performance in algebra, achieving accuracies above 50%. However, their consistently poor performance geometry highlights ongoing challenges in spatial reasoning.
(3) Larger models yield more consistent improvements across all dimensions. Within the InternVL.2.5 and Qwen2.5-VL. families, increasing model size leads to consistent gains across all dimensions and in overall scores, emphasizing the role of scale in enhancing reasoning capabilities.
5.4 QUANTITATIVE ANALYSIS
Ablation Study. As shown in Table we conduct ablation studies on the training stages. MO denotes MathBook-7B, while M1-M4 represent models different training stages (RL-Pre: Pre- aligned RL; RL-Dyn: Dynamic Scheduling RL). We lead to following two key findings:
(i) Both RL stages contribute significantly. Each RL stage (MO-M3) yields progressive improve- ments over M4. In particular, pre-aligned rein- forcement learning (RL) in the first stage yields im- pressive results on Math Vista and We-Math bench- marks, highlighting the crucial role of knowledge learning enhancing mathematical reasoning abili- ties. (ii) SFT alone offers limited gains, but is cru- cial for unlocking RL potential. Comparing MO, M3, M4, we find that SFT alone yields marginal im- provements over the Qwen2.5- VL backbone. How-
Table 4: Results of the ablation study. MVC Math Vista; MVs: Math Vision; WM: We-Math)
Method SFT RL-Pre RL-Dyn MVt MVs WM
Mo
73.0 28.0 48.4
Mi
/
1
72.4
27.0
47.2
Mr
/
72.0
26.3 43.3
Ma
71.5 26,3 46.7
M
65.8 25.7
38.3
ever, when combined with RI, SFT version substantially boosts overall performance, highlighting its critical role in shifting the model's reasoning paradigm and enabling more effective RI. optimization.
Analysis of SFT Data Paradigm and Scale. We explore the impact of data paradigm and scale during the SFT stage. Based on the MO setting, we consider two variants: (1) Replacing the natural language CoT format with a structured, step- wise CoT format (Zhuang et al. 2024) aligned with K: (2) Increasing the SFT data scale with larger datasets (from IK to 15K).
Table 5: SFT Data Analysis. SFT (Str.) and SFT (Lar.) denotes struc- tured and large-scale SFT training.
Setting
MVt
MVs
WM
(i Natural language CoT outperforms the structured step- wise format in SFT. As shown in Table 5) natural language CoT outperforms the structured format in RL. This highlights the advantage of natural language prompts in cultivating flex- ible reasoning, which in turn strengthens visual mathematical
Me
73.0
28.0
48.4
SFT(Str.) SFT(Lar))
71.9
26.0
46.7
72.8
27.0
49.0
reasoning skills. (ii) Minimal SFT suffices to unlock RL potential. Scaling up SFT data does

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Published as a conference paper at ICLR 2026
improve performance. Models trained minimal, well-curated data perform comparably better than those trained on larger datasets, suggesting that a small, high-quality SFT set suffices to establish the reasoning paradigm for effective RL.
6 CONCLUSION
In this work, we present WE-MATH 2.0, a unified framework for multi-modal mathematical reason- ing. It comprises: (1) MathBook Knowledge System, five-level hierarchy covering 491 knowledge points and 1,819 fundamental principles for comprehensive supervision; (2) MathBook-SSandann and MathBook-Pro, two richly annotated datasets with conceptual expansions and principled dif- ficulty modeling for structured learning; (3) MathhBoo-RRl a two-stage reinforcement learning framework that leverages knowledge-guine guided supervision and dynamic data scheduling; and (4) Math- BookEval, benchmark for evaluating reasoning across diverse knowledge and step distributions. Extensive experiments validate MathBook's effectiveness enhancing generalization of MLLMs.
REPRODUCIBILITY STATEMENT
Data. We ensure that all datasets developed in this work, including MathBook Knowledge System, KaahBook-Standarke MathBook-Pro, and MathBookEval, will be fully released upon publication. To support reproducibility during the review phase, we provide representative data samples (up to the maximum size allowed by the submission system, 100MB) the supplementary material.
Experimental Setup. The experimental protocol, including dataset sizes, training stages, and evaluation benchmarks, is described in Section and and Appendix C.I In addition, our code contains complete environment specifications (e.g., zequi i rements. s txt), along with scripts for dataset preparation. training, and evaluation. These files ensure that the reported results can reproduced on standard hardware with minimal configuration.
Code and Model Checkpoints. We include the full code in the supplementary material, together with scripts for data preparation, training. and evaluation. Due to file size limitations, pretrained model checkpoints cannot be submitted at this stage. Upon acceptance, we will release all checkpoints in the camera-ready version to facilitate rapid validation of our results.
Methodology. We detail the proposed framework in Section 3 and Section Specifically, we (i) specify the MathBook Knowledge System (five-level hierarchy; 491 knowledge points; 1,819 principles), (ii) describe the SaahhBook-Sanddun pipeline with two orthogonal expansions (one- cnestion-multi-ineg one-image ees-iaagemuut--eee nulti-questionn, (iii) formalize the three-axis difficulty space for MathBook-PP Pro (step, visual, contextual) yielding seven progressive variants per seed, and (iv) present MathBook-RL, two-stage training paradigm.
ETHICS STATEMENT
Licensing and Open Access. For all referenced or incorporated data in WE-MATH 2.0, we only existing datasets with clear and appropriate licenses. All data curated by our team will be released under the CC BY 4.0 license, ensuring open access for the research community. The entire MathBook dataset, including both external and newly constructed components, will be made publicly available to facilitate further research and development.
Data Sources and Privacy. All data in WF--MATH 2.0 are either sourced from publicly available datasets or generated by our expert team, and do not contain any personal user information. Therefore, there are no privacy concerns related to our dataset.
Expert Compensation. Experts involved in annotation compensated on a per-task basis, with payment issued only after cross-validation and quality assurance. All compensation meets or exceeds the local minimum wage standards.
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Jinguo Zhu, Weiyun Wang. Zhe Chen, Zhaoyang Liu, Shenglong Ye, Lixin Gu, Yuchen Duan, Hao Tian, Weijie Su, Jie Shao, et al. Internv13: Exploring advanced training and test-time recipes for open-source multimodal models. arXiv preprint arXiv:2504. 70.55444477, 10479, 2025.
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Appendix
CONTENTS
Introduction
2
Related Work
3
WE-MATH 2.0
3.1
MathBook Knowledge System
Multi-Dimensional Data Construction
3.2.1
MathBook Standard: Seed Variant Problem Construction
3.2.2
MathBook-Pros Pro: Phree-Dimensional Difficulty Modeling
MathBookEval
Methodology
Cold-Start Fine-tuning
4.2
Progressive Alignment Keinforcement Learning
5
Experiments
Experimental Setup
5.2
Main Results
Results on MathBookFxall
5.4
Quantitative Analysis
6
Conclusion
10
Broaden Impact
18
Details of WE-MATH 2.0
18
B.1 MathBook Knowledge System
18
B.I.I
Hicrarchical Structure of Knowledge Points
18
B.2.2
Knowledge Principles'
19
B.2
MathBook-Standarl
27
B.2.1
GeoGiebra-based Diagram Generation,
27
B.2.2
Dataset Diversity and Variant Construction.
................................
27
3
MathBook-Pri
35
B.4
4
MathBookEval
35
B.4.1
Dataset Construction and Annotation Protocol.
35
B.4.2
Task Dimensions.
37
B B.4.3
Dataset Statistics
37
B.4.4
Evaluation Protocol and Metrics.
38
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B.4.5
Experiment Setup
38
B. B.4.6
Additional Results on Maok
39
More Details MathBook-RL
40
Implementation details
40
C.2
Additional Experimental Results for MathBook-33 3B
41
Generalization to Text-only Mathematical Reasoning Tasks
C.4
Case Study
41
Experiment Setup
42
C.5, Details of the Evaluation.
42
C.5.2 Details of the Baselines
42
D The Use of Large Language Models (LLMs)
46
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A BROADEN IMPACT
Towards principled and generalizable mathematical model training. WE-MATU 2.0 provides comprehensive and structured mathematical knowledge system, which fills the gap left by previous works that lack a complete and systematic framework. By introducing a five-level hierarchy with 491 knowledge points and 1,819 fundamental principles, MathBook enables more principled and interpretable mathematical learning for MI..Ms. The dual expansion strategy ("multi-images per question" and "multi-questions per image") and the three-dimensional difficulty modeling not only enrich the diversity of training data but also facilitate robust and progressive learning. This system- atic approach can inspire future research to adopt knowledge-driven and model-centric data space modeling, leading to more reliable and generalizable mathematical reasoning models. Furthermore, the fine-graine annotations and progressive difficulty levels provide valuable resource for bench- marking and analyzing the strengths and weaknesses of different MLLMs, promoting transparency and interpretability in model development.
Bridging AI and Education: high-quality datasets for teaching and learning. WE-MATH 2.0's datasets not only designed for model training but also have significant educational value. Each problem is accompanied by GeoGebra (GGB) file, which can serve as high-quality teaching material for educators and students. The hierarchical knowledge system and step-wise annotations make easier to design personalized learning paths and targeted exercises, supporting adaptive learning and formative assessment. The multi- modli and nulli-pprspecctvve problem sets encourage students to develop flexible thinking and deepen their conceptual understanding. By bridging the gap between Al research and educational practice, MathBook has the potential to enhance mathematics education, facilitate interactive and engaging learning experiences, and support the development of intelligent tutoring systems.
Enhancing RL generalization through progressive and dynamic training. WE-MATH 2.0 intro- duces novel, model-centric curriculum for RL-based training. where problems are systematically organized from easy hard based explicit difficulty modeling. This approach provides new perspective for designing RL curricula, enabling more effective and efficient learning. The "one- question-multi-mmagee and sane-image-muliiquuesee mechanisms, enhance the robustness and generalization of RL-trained models. These innovations can inspire the broader RL community explore curriculum learning, dynamic data scheduling, and modal data augmentation for complex reasoning tasks. Moreover, these hierarchical knowledge approaches also offers a new solution for tool learning. MathBook thus serves as a valuable testbed for advancing RI. methods in the context of mathematical reasoning and beyond.
strategies, together with dynamic scheduling
B DETAILS OF WE-MATH 2.0
B.1 MATHBOOK KNOWLEDGE SYSTEM
B.1.1 HIERARCHICAL STRUCTURE OF KNOWLEDGE POINTS
As illustrated in Figure 3 we provide an overall view of the hierarchical structure of knowledge points in the MathBook Knowledge System. Figures 7 further present partial examples of different substructures at varying depths of the hierarchy. The system is organized as a five-level hierarchical structure of knowledge points = (ki, kz.... kN), where N = 491 denotes the total number of knowledge points at the lowest level of the hierarchy. The first level consists of four categories: Geometry, Fundamental Skills, Algebra, and Probability and Statistics.
The construction of K. follows two-track, human-Al collaborative process. First, initial version K'human is constructed by collecting and merging knowledge point lists from authoritative sources, including Wikipedia, open-source mathematics textbooks, and national curriculum standards. This initial structure is deduplicated, reorganized, and refined for logical consistency and comprehensive coverage.
In parallel, a large-scale problem set Q is collected, including 30,000 sampled from existing math datasets. GPT-440 is used to assign multi-level topic tags T {l,,... . (x) to each problem, and
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Fundammntal Mothematical Skills
Algebra
Geometry
Probabiitty and Starisiics
Til: harrow the soope by sclectigneddss
Englfih
Chapter
KnowledgePoint Principles
Definition of Isoscel and Equilitteral Trieng les
Plane Geometry
Trinngle
Basic Plane Figures
Casssiicttio isoeceles. Triangle
Segment, Straight Ray
Angle
Description:
type of angle,
triangles classifled
isoscales triangle
Trianggl
angle e 90'). right ixcscelles triang Oun. bicscosles right triangle, vertex angle = 90", hypotenuse Ve #tegrtt221)) and obnuue isosceles triangle Ivertew angle 90', such as ver tex angle 120". angles
Definition Triangie
Defintion of lscaceles and Equilataral Triangles
Properties of lsosceles and Equilateral Triengles
Definition Triangie
Classiflcation of Triongles
Reletionahip between Eegulatennl and tsosoelee Triangle
Figure 3: Overall view of the hierarchical structure of knowledge points in MathBook.
semantic similarity matrix S E R=** is computed. Hierarchical clustering is then applied to S to generate an Al-assisted hierarchical structure of knowledge points ACauto
Finally, the Al-assisted structure 10 is integrated with the initial structure Kchiman through systematic comparison, merging, and revision. The Ka serves as reference for revising and refining the manually constructed hierarchical structure of knowledge points, resulting in the final knowledge point set K.
1.1.2 KNOWLEDGE PRINCIPLES
we provide several examples knowledge principles, which include definitions, theorems, and other foundational statements associated with each knowledge point.
As shown in Figures 9 12
The annotation of principles P U. P, (PI 1,819) also follows two-track, human-AI
collaborative approach.
Based the constructed knowledge hierarchy K., of principles for each knowledge point k; is first drafted, referencing authoritative sources such as Wikipedia, textbooks, and international curriculum standards.
In parallel, for each k, a set of representative problems from Q is selected and their chain-of-thoughh (CoT) solutions are annotated. Each step in the CoT is mapped to the corresponding knowledge point using GPT-4o, and the relevant CoT steps for each k; are extracted. The CoT steps associated with each knowledge point are then aggregated and reviewed supplement, refine, and validate the set of principles for k..
This process repeated iteratively, consolidating both the expert-written and data-driven principles, cross-checking against original sources and annotated solution paths, until the set of principles P, for each knowledge point comprehensive and precise.
This process is repeated iteratively, consolidating both the expert- written and data-driven principles, cross-checking against original sources and annotated solution paths, until the set of principles P. for each knowledge point is comprehensive and precise. As illustrated in Figure we further provide an example from We-Math 2.0, where each problem is explicitly aligned with its corresponding knowledge point and associated principle.
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Delaaies auck
Aedee
Tagey
Bubiblere prodide
Jonan Aigerss
Prokiive
Traritan
Tou
Enpirto ofowka
Freechel Pollie
Sepeusting Peckia line Uhiord fas
Caea
Falr sale
Ecallagi
Aed Gominn
ifportira
Datans brieus
Ukita Biorion
Projetter
manh
prgplak
Priarti pla
Phonki
Flnx
Delaate pf saedesk gune
Apare
Facked
MarhBook
fiels
Milyaar fireds
Dias
Buliaa d'a dly
Infalesela kyathe
Reemrin debpp
Ajricc
Figure 4: Overview of the hierarchical structure of knowledge points in MathBook (1).
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Goometry
Probability
Algebea
Observe aatural al imgges
Meacure
Find patterns
Fundamental Skilk
Compare images
MathBook
Observa graphs
Observe function images
Observe charls
Delinition Integers
Definitann fractinss
U'dderstnding the number axis
brtwee precentages
Simplfication of complex numbers
Number Quantitivs
Understanding time units
Comparboo tumberr
F Fundintocttal Stäb
Redurtion and common dumominatur of fractions
Defiaition of rational and irrational sumbers
Delinitien of additiin
Four arithmetic operation
Four operations on inleger
Delinition square
Square rewit
Syuare reot and arithmutic square root
Delinition radi
Ratio and propartion
Solving proportion
calculation method
Knowledge of exponents
Exponcacial logarittmii opuration
Calculation of exponents
The concapt of
Operation on sets
The inclusisa relationship betwees sets
The operation rules derivatives
Derivation
Figure 5: Overview of the hierarchical structure of knowledge points in MathBook (2).
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Fundamental Skills
Goumetry
Algebra
Definition of random events
Events
Holationships raadom events
Caleubation rundums
Definitiea randemu varsables
Definition of distribution functions
MathBook
Randeas Variables
Nameiial churactrrstiie of one-timeasiomal varisales
Probability
Prohobility and frequency
Formabs for calculating probability
Prebability Caleniation
Estimating probubility using frequency
Calculation classical probubility models
Probabiliiy and Statheties
Differences different sampling methods
Data Collection
Choose appropriate survey and sampling
Binsmial exgansien
Binomial Theorem
Definition permutation
Principles of Counting
Permutations
Calcalation of permotations
Statistics
Measurss central lemdency data
Amalysis
Mersures dispersion data sel
Application of statstiica tables
Application of line charts
Applications
Application of heat maps
Figure 6: Overview of the hierarchical structure of knowledge points in MathBook (3).
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Protbiiity Statidix
Prodenscte/Sbb
Fredamatta Sequesee
Dufuatio repronzzaiv
Detabion afthr Has Baxtke
Trigeveseri Frectiess
Delaeike al peeer faxikes
Budr
Ekeestary
Panor Pacties
Dosss and rieg peverr fraction
Exuation
Delasikee strt
Defrathe
Dettaiion srrppiegg
MathBeok
Alghtrlc Exprcstan:
Deflskker dipteaa exprosiess
Exskation of ebrlbak eupronses
Delisiins suttic
Defistion ow mours nestike
Dedaisus of ofnennaary and efficct couittion
Neewery and Sufliton Caaditias
prepertio
Dekemtastee prepettio sccoveryy cenfiion:
Legi
Dessisis clss and jenge fins of secmary and Mfficiaa couitiion
The Propesities
propestian
Duflutko crevanse propecttions
Canupt of vicler
Madila tutter
Decorpattion vocters
Coedititee eepronditie of parslld vecters
Csarclankc iuprocataiio perpeddiahe votlers
Therre of cepliner vockes
Nirerias Carter of lime la space
Delinkine
Vetters
Bosi car aliin Vesters
Veaters Masikis
Defeeliee reation
Marrics
Lisor Alscir
Cooples operation.oo
Syoten Linee Exprrions
Delalton of tystne of Baor oquation
Schrnng tysees hane opation
pmpplo
Grople
Graph trancrel
Phatialluppiaateeeoo gr grag
Griph Theary
Tow
Troe
Prantied applications of tfeees
Figure 7: Overview of the hierarchical structure of knowledge points in MathBook (4).
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Problem Retrieval
English
Swarch problems / knowtedig patnneelggggreiiiiiagggg prancipee
insodedge
Agpl
Geamery>
Derstoo:: Whatis moitmam velure the
L
Solld Geometry
Krondadgs Vilurre Cabe
Priscppe Compuration Analysi Calbe and kestannuuaa Cakaid valuree Farmudaa
Basio Solid Pigures
Desction: Fisd the volure of rectangu primn based cuble.
Level 4
Rectangular
trandsdge Vakare Cake
Princgle: Comparativn Arabyb of Cabe and Ractenguiar Cabold Volume Formulee Ganolion thaeekkeer
Knowledge
Valume Oube
Annwert1o
Principle
Formula for Volume of Cube
velure of the cobe based rectargater pram.
Compa oaive Analyaa of Cube and Rectangula Oubcid Volume Formulos
Figure 8: An example from We-Math 2.0 showing how each problem is explicitly aligned with a specific knowledge point and its associated principle.
Geometry
Geametry-Plane Geometry Basic Plane Figures- Triangles Classification triongles
Geametry-Relctinshhip Geometry Plane Geometry Relationships Between Figures
Intersection and Perpendicukeity-Addiie Adjjacet supplementary angies
Princijol::
Princiole:
Difanmos of offooon explewsttry axglr odjere angle fohuner COEENNT ril vesoee spe to secolll olisom applimmtter inghin fra, finr Chroctointss
with bestal angle orthocotv resie well mable drnntaa durger for, Eafil munne Digissioon of rightt viange: mongle with wight sike J sollgf 33, olher empler complemntryy is
te
(Ii kehwen the eapler (2) de ofter ar monoser of anc other in oppostr devctives (3) relr sinigh angle (estionaiccl expestio do + -at Dud aosmatic proad complic axple relatoenkuu.
J'skaponece
essugt posticsing
Delinton
Asigese of aupileeettuy rike ferma shruthh foce des angoo saplewerry axplo ('2) and shars odd D4 and (8l hes onsigh fov All sppleatior contreettg toonie axglo of afaiiiiinsss compainy finge sparn ungle in todors Fecter expressoo afjoon apulenveeryy oxgle angle Semoeet coctors o'
2867 aegle e lngur or cooopeeliigg striude hes rodde misgle bur the other hen shitaks scnd beeioonell oabsai the inangle Sea. for strcnnal mev dhariiative is engineeing Mapproyy
ihan
Geometry-Solid Geometry- Besic Solid Figures-Cones -Section of cone
Geometry-Solid Geometry-Bost Solid Figures
VVolume of irregular solids
Principle:
Principle:
cirdle
Ofjec'
wold Spares onfore
flor
in sppeed rolose statiin
A vericaa rntoocctionaln siing beggtt bourler trangle share butgh -.A. arve wy by uouf is spewevnn catting wooe coseasigguer
coatooatcaaionn elewoee calnalaee spueesly anodlly the volens
smagsior hove cas be abalani aabing rice
Appasing or acarifi section eay paggle cngle gowasaor bvigtt panhecks gronasss bpprball
chinas nedive
Ckpeez auppurettyy Reaew a par solove beowe sdlid Rpure
enliaaa rodooee the sbact Gaatty, the volane wigiina
lieght
Suave ana
Aecalstato reaaradfy and feully roluner
astolike
rellcatio
inopeler faww
padte
Geometry Analytic Geometry-Andyy Geometry -Defiiition the slope of line
Geometry-Analyle Geometry- -rcces
-Problem of circle passing through fixted point
Principle:
Principle:
Conittioe hao howr be paratiei
interpp cfffrest riopor of tan Ban ore spual
woly 203 porallti fins ponulie)
Porpuatre apuation prowmiing equative sof paranatoon indeundiss poad watiger
confiiiest
Panalll abanaisation gesttion fanifiing
or loe perpedicaarr idope no uso toutlnng orloogonttiy of hraay and gossenii consiraction cxesives Reperpesdeaaerr Zoe is ledf: sundon ggoelng dee epecid opeiid/ neniool and bochoosal thrs fooch and yod peppodlesior)
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Gustisse nwo Stee pespaskintt:: Whe prodicc the rkger of tan liss J.
tosake Fairre complodd olonisnt ponseshey redepaadov coalitou Quetios opc. Fiad all Geal poics dat the parn Bhosggh the
Gussairic condinon nataet Coe cosatver of the drek motes (hach in the innorction low ho ctder
muderrate the food portt Roy valued
ensamsss of
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Figure 9: Examples of knowledge principles corresponding to specific knowledge points in MathBook (1).
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Fundamental Skills
Pundomental Skolle-Compore sizne in Images
Pundemantal SHll--Understandeg Nambers Quariities
Imags Defaitian integers
Principli::
Principlc:
Coeparioe fooeee Compare wiker of ofdffooe
eopats anfoms Pouthe lagger, Dreate Desvetive purthr esoper, nogathy
saanal
tysinlyy sed for poovlyy Zere Nether posite aee oegatii, expesssss "voe"" "ona
Caggerbar of anbeese Compare susiease seviiaan naler of didirn foection
refoneer poott Negativ nagee Leu dice reprensing sysurr pavetio
Dasperbor digte sungrat Compars the arca exstored octreos carre
vuatton: The exaqoo decuted the outura cuttory Jennaca
Funiemental Skils- Cdlcultiion Methods-Se Squaring and Square Rocting- befinition
Fandomental Skill Calculation Metheds Faur Arthhmeii Operationa-
diviaion
Princippte:
Principle:
Detlantos of spuare Sqpare refor to noot waatiiii cuuther dualf a Nas
The stontia dethitio of drotie the Iosee operttoo of ofesttiiionnss thd ansiee giveege prodec and feow dadaaa (badeetodd to methonation Ohr coscer of masiderr shan canner ahrite enects &e ronsstilo colld the cossisio oor 7o107 nessiser 1)
sunder eudlyded by wnef coomple the oppone 2-2-4 wathomalte
squirs the
socier evergggate, tlat
coneutatte lae x252 -b)
now door not kave rxpastr volation rrac or (-2proddd)
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esenge Foy
Dytunion
viich
eriii for d'epedeettpare (nck 2- (5
Recomoca conciation w/ily Aibioe connertal eulipiiatio by nuppreat
Figure 10: Examples of knowledge principles corresponding to specific knowledge points in Math- Book (2).
Probability and Statistics
www.-a-rrtyyyaaaae -Oeflettin randen events
Probabiltty Statistics Probabiity Probabiitty Colculation
-Estinating probability using frequency
Principler
Principler
Pasion eeest As erpoissee sstigfor dee conditin dis be repeordd unskr yowe costhoo and al pountle rusts are kxoes ronslo expreeee
premaii o'fequace odieatoo cnobabilty apencinaal mar pecpeaalllll by
Piyin Assts dhe
expeissee rastion
law large caeere wire ((AA) - 7(A)
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Sumph pare pootll nusthef casabe cxperient sample yaast conger quace
paker auseat rover
compordday
Pormal mample Pressenc aucter
aoo awainng dieaa,
Prebabiiiy Statistics Stutishics of Counting Additian Multipliaation Principles Step by-s multipliattion counting principle
Probobility Shatistics Princppess of Counting Comblaations
-Cmb Combination based count ing problems
Principle:
Principle:
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cvlvee indgosder 'Voose slve" 'chooer
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protice
Figure 11: Examples of knowledge principles corresponding to specific knowledge points in Math- Book (3).
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Algebra
Algebro-Panctinn Faniro--Fnnttonsseeoo
Equations- Undiiissndnnddddddddnnnnnn Functions
Algebro- Equetions Fundomentals Sequences -Dfinition and representation sequences
-nalyffical expression of functions
Principle:
Principle:
Gooapt costiiici opresive my cxpess edecciion exatiorstioal
Piylattion ofas
is collenio areeiin orooged aodh conder trcalid dewend
Types exgrrsion Lhase (eg. y Iy-2 gushali (eg. vlalr-a. exposeniil
Kot. lnpordenn (ut, y rxgaromen (63,
replict of yf
number spueees J.2
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y
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gen Ie gespuses the saster reguece 1.3, Pasesing firmnhs Mivinng peeesioo
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proft
wher
Algebro-Logic- Laaerr-LoogcNNeeesarry Necessory and Sufficient Conditions
Algebro-Vectorse Veccoors--eoocceeppppp
Definition of true and folse propositions
Principler
Principlc:
Frur propuctiooe. Sutveeer comtser val foct light (ue. "In vnsedls
Daplettie of wear guenn nier Aar Aesh mepyttur (ossisda) and deventos frosh is selucty Senot upsoal sodber (ach so topooatarr moot The Inveleeents a reter sugestdee
triangl, tagiengl are opea"" mas "343 2 vuc. The Swith optase" propusiiee: Sestenaee that contrddrr
folre Truek of ooudiosd pespocttion to mossossss ohe 'I' thon A (4 it, le fo wabr when d brat and fine se other ouus it the fua. "a the groand propistines Forwad loghtal
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and ous tao nox ext of commraangio snppeoonnoo digrons univered shatonal reas. "Al bart canty digessed by do setick
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Algebra-Lineae Algebro- Vectors and Matrices- Matrrics -Definition of motrices
Algebra- Graph Theary- Tree -Basic concepts of frees
Principle:
Priuciple
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parot disoee tic
worb sagstr cprenel didatoe shore namier pusitos Wlecnt of suste regessees the taber and) iepereenr cel bukeer
witha
Figure 12: Examples of knowledge principles corresponding to specific knowledge points in Math- Book (4).
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B.2
MATHBOOK-STANDAAN
B.2.1
GEOGEBRA-BASED DIAGRAM GENERATION.
As described in the main text, all diagrams in MathBook-Standaan are rendered using GeoGebra, dynamic mathematics software that enables precise and reproducible geometric constructions. GeoGebra supports both interactive design and programmatic generation of diagrams, making it highly suitable for large-scale dataset creation. In our workflow, each problem paired with high-quality diagram constructed in GeoGebra, ensuring mathematical rigor and visual clarity. For automated and scalable generation, we leverage GeoGebra's ' ability to encode diagrams as scripts (e.g., XML, shown in Figure 13]. which allows for efficient parameter variation and reproducibility, but the core advantage lies in GeoGebra's expressive power and accuracy for mathematical figures.
Compared to general- peneral-purpose purpose plotting libraries such as Python's matpl ot 1ib, GeoGebra offers richer geometric primitives and more precise control over mathematical relationships, supporting broader range of problem types and visual styles (Table 6). As shown in Figures [4 15 and 16 diagrams generated by GeoGebra exhibit higher fidelity and better alignment with mathematical conventions, which is essential for both algorithmic evaluation and educational use. It evident that the complexity and precision of these diagrams would be difficult to achieve using Python-bases plotting tools alone.
Table 6: Comparison between GeoGebra and Python Plotting tools.
Command Line Plotting
Interactive Graphic Editing
Image Re-editing
Precise Parameter
Tool
Control
GeoGebra
Python Plotting
X
X
Limited
B.2.2 DATASET DIVERSITY AND VARIANT CONSTRUCTION.
Building on the GeoGebra-based pipeline, we systematically construct a diverse dataset as detailed in the main text. For each knowledge point and principle, seed problem is designed with corresponding diagram. To further enhance diversity, we introduce two types of variants: one- problem-multi-image (generating multiple diagram instances for the same problem by varying parameters in GeoGebra) and one-image-multi- multt-proollpeee (curating multiple questions for a single diagram, each derived from different knowledge points mathematical principles). Representative examples of seed problems and their variants are shown in Figure 17 24 and extensibility of our approach.
demonstrating the flexibility
By leveraging GeoGebra's 's capabilities, SuthBook-Standaale achieves both high-fidelip geometric representation and systematic dataset expansion, ensuring rich semantic and visual diversity for mathematical reasoning tasks.
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Algebma
3D Graphies
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Figure 13: Overview the GeoGebra interface and part of the corresponding XML script, showing core commands for defining geometric objects.
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Examples (1)
Figure 14: Overview of a group of GeoGebra-generated images in MathBook (1).
Examples (2)
R2-41
XC-4 2
Q141A3H
457
Ar4-3)
B44-40
Bi-L.5
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Figure 15: Overview of a group of GeoGeehra-generated images in MathBook (2).
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Examples (3)
912.3)
(LL.BAM
0 1'
2 40 m1
Figure 16: Overview of group of GeoGebra-generateee images in MathBook (3).
Example 1
Seed
How many triangles are there in the picture?
One OieeImage--Mee Multi--b-r-
1. How many triangles are there the picture?
2. How many line segments are there in the picture?
3. How many angles are there the picture?
Tie-Problem-MMutii--mee
ASA
How many triangles are there in the picture?
Figure 17: An example of MathBook-Standant data instance (1).
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Example 2
Seed
Compare the magnitudes of the fractions represented by the shaded parts circles A and B.
Pite-Image-Mane Multti-Poblemm
What is the difference between the fractions indicated by the shaded areas of Circle B and Circle A?
2. What is the sum of the fractions indicated by the shaded areas of Circle B and Circle A?
3. Compare the size of the fractions represented by the blank parts circle and circle B.
One-Problem--o Saii--MMltiiimmn
Compare the magnitudes of the fractions represented by the shaded parts in circles A and B.
A
B
Figure 18: An example of MathBook-SSandard data instance (2).
Example
Seed
What three-dimensiona. geometric solid is generated by rotating the figure about the dotted line?
One-Image-Moe Moltee-Mutt--Pooblem
What three-dimensional geometric solid is generated by rotating the figure about the dotted line?
2. What type of figure represents the side profile of solid generated by rotation?
What the shape of the base of the solid generated by rotation?
Problem-Multi-Eee What three-dimensional geometric solid is generated by rotating the figure about the dotted line?
AR
Figure 19; An example of SathBooo--Stadarrt data instance (3).
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Example 4
Seed
What the Angle between the two straight lines in the figure?
a v8g 220
One-Image-MMr Moine-MMuti--Pooblem
1. What is the slope of the straight line I1?
2. What type of triangle is formed by the two lines and the y-axis in the figure?
3. What is the intercept of the straight line !1?
4:
Mie-Problem-Muun -wl--Muulliiaeeee
What is the Angle between the two straight lines in the figure?
byz Vig 2-0
6:2 Vi2-- 2-0
Figure 20: An example of Mataaaro-Standard data instance (4).
Example
Seed
What the in-degree and out-degree of vertex B respectively?
One- One-Image-Minn Mannge-Muiti-Problen
1. What are the in-degree and out-degree of vertex A respectively?
2. What is the length of the shortest path from to B?
3. Does a cycle exist in the figure?
One-Problem- Boabblee--ulti--nee Multi-Image
What are the in-degree and out-degree of vertex respectively?
D
Figure 21: An example of SarhBook-Sttndarie data instance (5).
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Example
Seed
ain(s)
Determine the monotonically increasing interval of the function in the graph.
One- Ohee-mmag--ae www.immmaatiiiaeeeeeee Multi-Problem
Determine the monotonically decreasing interval of the function in the graph.
(x))
Determine the zero point of the function in the graph.
3. Determine the symmetry points of the functions in the graph.
Cie-Problem--Me -ulti-Mulaiieeeee
AS
Determine the monotonically increasing interval of the function in the graph.
cos(e)
Sante!
oofixl
Figure 22: An example of SathBook-Standarte data instance (6).
Example
Seed
Divide the circles in the following picture into 3 equal parts. How many are there each part?
1. Divide the circles in the following picture into equal parts. How many are there in each part? rectangular grid
2. If the circles are arranged
with 3 rows, how many columns are there?
3. If all cincles tightly packed into square shape, how many circles are on the outermost layer?
Cice-Proble--Mutii-nee
Divide the circles in the following picture into equal parts. How many are there each part?
Figure 23: An example of MathBook-Staaniee data instance (7).
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Example
Seed
The vector OA can be decomposed into the vectors and shown the figure. What are the coordinates vector OA?
100
Beee-Tmgge-Multi-Proole
What is the abscissa of vector OA?
2. What is the ordinate of vector OA?
3. What is the modulus of vector OA?
11401
One-Problem-- Saablem-Mutti-Imagee vector can be decomposed into the vectors and j shown the figure. What are the coordinates of vector OA?
1000
Figure 24: An example of MathBook- -tandddd data instance (8).
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B.3 MATHBOOK-PRO
As shown in Figure 25 we present concrete example from MathBook- to illustrate the con- struction and expansion of problem variants within the diree-dimensioaal difficulty space. The seed problem, positioned at the origin, focuses on the arc length formula in plane geometry. and involves knowledge points such as definition of angle, definition of circles, four arithmetic operations of integers and four arithmetic operations of fractions.
We first demonstrate how the seed problem is expanded along each individual dimension:
The number of required knowledge points is increased by introducing new intermediate conclusions as conditions. In this example, the extended variant requires not only the are length formula, but also incorporates circumference of circle and area of circle as additional knowledge points. The solution to the new problem depends on the answer to the seed problem. reflecting progressive deepening of reasoning.
Step Complexity:
Visual Complexity: The original diagram enhanced by adding shaded regions, which increases the visual and interpretive demands while keeping the core mathematical focus unchanged.
Contextual Complexity: The problem statement is recontextualized from a direct geometric descrip- tion to a real-world application scenario. Although the narrative becomes more complex, the essential assessment of the are length formula remains at the core.
By systematically combining these single-dimension expansions, we further generate multi- dimensional variants that integrate increased step, visual, and contextual complexity. In total, starting from the seed problem, construct 7 variants corresponding all possible combinations of the three dimensions. Each new variant is constructed through progressive modifications to both the problem statement and the accompanying image, resulting in diverse and interpretable set of conttolle- problems. The full set of variants and their corresponding dimensions are summarized in Table 7
As shown in Figure 26 MathBook-Pro supports the Dynamic Scheduling RL framework by pro- viding difficulty-control.. problem variants. These structured training samples enable progressive adaptation across different difficulty levels in a dynamic training process.
Table 7: Difficulty-contronl variants constructed from the seed problem in MathBook-Pro Each variant corresponds to a unique combination of step, visual, and contextual complexity.
Variant
Step
Visual
Contextual
Seed
Variant
Variant
Variant
Variant
Variant
Variant
Variant 7
B.4 MATHBOOKEVAL
B.4.1 DATASET CONSTRUCTION AND ANNOTATION PROTOCOL.
To ensure both comprehensive knowledge coverage and rigorous, interpretable annotations for visual mathematical reasoning, MathBookEval constructed through a multi-stage, process-oriented pipeline. We begin by integrating representative samples collected from five open-source benchmarks: Math Vista (Lu et al. 2023). Math Verse (Zhang et al., (024a). MathVision (Wang et al. 2024). We-Math (Qiao et al. 2024a) and DynaMath (Zou et al. 2024). systematically filtering out redundant or highly similar items to maximize diversity in knowledd ledge point combinations and reasoning patterns. All problems re-annotated under unified guidelines, ensuring consistency in annotation style and
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Step Complexity
Find dircumference the longth
figure, circulor farm. New fence enclese regular hexagonal flower What the bed?
Find of the shoded
If shaded For neade be fully with tiles, how much tile should purchated in advance?
or
AC4
I
Contextual Complexity
What length
Find the length of correspondlng the shodled sector in the
circulor figure, length fencing purchated?
the figure, fance circular form. Find length the fence corvesponding shaded
Visual Complexity
Figure 25: An example from MathBook-PRi Pro in the difficulty space.
Step?
Stept+ +Visualt
Step *Contxtttt
Step Contextt
Case Demonstration
Difficulty Space
Problem 1
The base area the priam 4 and the haight is What of Know belaee/ of Phome Answer:
1 +Visua
Step 1 +Context
AlT
Click one the buttors below to start the step-by-step demonstration.
Visua
Contaxt
Visuel " Conbext
Vinsel
Model Correct
Model Wroong
Contentiull
Figure 26: Overview of the Dynamic Scheduling RL training stage with MathBook-Pris
granularity. For each problem, at least two human experts independently provide complete, step- by-step solution, where each step is explicitly decomposed according the underlying knowledge point(s) from the unified knowledge system K. This knowledge-pons based decomposition is fundamental: it enables precise and systematic quantification of reasoning depth, each reasoning
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Knowledge (Points)
100
Nathitoonhhal
MathBookEval
KartiVista
Natthvenn
DynaMart
60
Mathvisia
DynaMath
20
Reasoning (Steps)
Figure 27: Comparison of MathBookEval and open-source benchmarks
step directly corresponds specific knowledge point. Only those problems for which the set of annotated knowledge points at each step is exactly consistent across expert annotations are retained in the final dataset, ensuring high reliability and objectivity. To address knowledge points and reasoning depths insufficiently covered by existing benchmarks, additional samples are newly constructed by human experts following the same process-oriented,
wowwlege-ponntleeee annotation protocol. result, MathBookEval comprises both collected samples from open-souree benchmarks and newly constructed samples, achieving balanced and comprehensive coverage across mathematical domains and reasoning complexities, with every annotation step tightly aligned to the relevant knowledge points for robust reasoning depth modeling.
B.4.2 TASK DIMENSIONS.
Bath is organized along two dimensions, capturing both the diversity of mathematical knowledge and the depth of reasoning required for problem solving.
(1) Reasoning Dimension: Problems categorized by the number of reasoning steps required to reach the solution. Each step is explicitly defined and directly mapped to a specific knowledge point in the unified knowledge system K.. We define three levels: Level (1-3 steps, basic reasoning), Level (4-6 steps, intermediate reasoning), and Level 3 (7-10 steps, complex reasoning). This step-by-step annotation based knowledge points allows for clear and objective quantification of reasoning depth, enabling detailed analysis of model performance across different levels of reasoning complexity.
(2) Knowledge Dimension: The 491 knowledge points in X. are distributed across 4 top-level domains and 13 subdomains, all of which are covered the benchmark. Each problem is annotated with all the knowledge points involved in its solution, and every reasoning step is aligned with specific knowledge point. This enables fine- mna-irri evaluation model capabilities across various mathematical topics and educational stages.
B.4.3 DATASET STATISTICS.
Table summarizes the key statisties of Evva The benchmark contains 1,000 fully annotated problems, covering all 491 knowledge points in the unified knowledge system. Of these, 600 are sourced from open-source benchmarks and 400 are newly constructed to address coverage gaps and increase diversity. Problems are distributed across multiple formats, including multiple- choice and fill-n-thheblank, and span a wide range of reasoning depths and knowledge domains.
As shown in Figure 27 we compare MathBookEval and existing benchmarks. In the right panel, the y-axis represents the percentage of problems at each reasoning level. Note that for some benchmarks, the total does not reach 100% because we exclude problems that experts annotate belonging to other subjects such as physics, chemistry, or biology, in order to ensure rigorous comparison. is evident from the figure that existing benchmarks contain less than 3% of problems at Level 2 (4-6 steps), and none at Level (7-10 steps). In contrast, MathBookEval substantially supplements these two categories, providing
comprehensive evaluation of multi-step reasoning abilities.
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Table 8: Key statistics of MathBookFyyal.
Statistics
Number
Total Problems
1,000
Open-source Benchmarks
600
Math Vista
150
Math Verse
150
Math Vision
100
We-Math
100
-ynnMathh
100
Newly Constructed
400
Knowledge Points Covered
491
Domains
4
Subdomains
13
Reasoning Depth
Levell 1 (1-3 steps)
62.0%
Level 2 (4-6 steps)
30.2%
Level 3 (7-10 steps)
7.8%
B.4.4 EVALUATION PROTOCOL AND METRICS.
EathBookEval incorporates existing evaluation protocols, including both LLM-aas--jjdde and rule- based approaches. To ensure consistency, MathBookEval adopts the LLM-as-a-judde protocol as the evaluation rule. Specifically, we adopt the LLM-as-a-judge protocol, following Math Vista and Math Verse by employing GPT-4o as the judge model and reporting overall accuracy the evaluation metric. The detailed prompt used for evaluation is shown in Table
Table Prompt templates for evaluation on MathBookEval.
Type
Prompt Template
Now, we require you solve math question. Please briefly describe your thought process and provide the final answer.
For multiple-choice questions, return the selected option and its content. For direct answer selection, return only the chosen result. For fillinnthho-blank questions, answer directly. Question: <Question>
Evaluation Prompt
Regarding the format, please answer following the template below, and be sure to include symbols:
<Thought process>: <<your thought process>>
<Answer>: <<your answer>>
B.4.5 EXPERIMENT SETUP
Details of the Evaluated Models. To evaluate the performance of various multimodal large models on mathematical tasks, we include a diverse set of recent models in our benchmark. Table lists the release dates and official sources of all evaluated models. Additionally, Table I] provides an overview of their architectural designs to support a comprehensive comparison.
Details of the Model Hyperparameters. For all closed-source models accessed via API, we adopt the standard generation settings and perform inference on CPUs, with the process typically completing within a day. For open-source models, inference is conducted on a cluster equipped with 8 NVIDIA 8000SXXM4-80GB GPUs, using the hyperparameter configurations provided in the official inference examples. If no specific instructions are available, default settings are applied. The detailed generation parameters summarized in Table 12 and Table 13
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Table 10: The release time and model source of MILLMs used in MathBookEval
GPT-4a OpenAl 2024
2024-05
httpa::/ggtto....
GPT-4V OpenAA
2025
2024-04
hilpps://ppead: wwwp///////////////o/e//e//eeeee
2.5-78
2024h
2024-12
ps httppe//hoggginggiiie http://wwggggg.oooooooooo///////
5-788
Chen
2024b
2024-12
0.en25.VLL722
Bai
2025a
2025-01
Qwen2.5VVL:22 Bai ct.
2025a
2025-01
Qwv-2.5-VL-3B Bui 2025a
2025-01
LLaVA-OreViskke-770 72B ei
2024
2024-08
LLaVA-OreVisine-7 Lieral
2024
2024-08
quen? Tb
GLM-4V:98
GLM cr 2024
2024-06
bt bLips: httpps://uuggiingaaas..ooo
Table 11: Model architecture of 10 MILMs evaluated on MathBookEval.
Models
LLM
Vision Encoder
GPT-4o
GPT-4V
Intern VL2,5-88
Qwen2.5-72B-Instuer
Inten ViT6B-448pp-22
IL2.5-
1rormlm2_557b-chh
Intem\VIT-BB-48px-2225
Owen2.5-VL-72B
Qwen2..-77BBIIntrruc
CLIP ViT-bigG-P14
Qwen2.5-VL-7B
Qwen2.5-7B-Instruet
CLIP ViT-bigG-P14
Qwen2.5-VL-3B
Qwen2..-3BB-Intrruct
CLIP ViT-bigG-P14
LLaVA-OneViiioon-222
Qwen2-72B
SigLip-so4000mP1443384
ILaVAA-OneVision-77
Qwen2-7B
Siglippso440m--144338
GLM-4V-9B
GLM-9B
EVA_02_CLIIPEEPPI
B.4.6 ADDITIONAL RESULTS ON MATHIBOOEEAAL
Table 14 reports the complete results of evaluated models on MookEvol providing a more comprehensive perspective model behavior across both reasoning depth and knowledge-coverage dimensions. Overall, general- purpose open-source MLLMs exhibit stable performance trends, with larger models such Qwen2.5-VL--72. and Intern VI.2.5-78B demonstrating strong accuracy multiple reasoning levels and knowledge categories. In contrast, crasoning-orientee systems (for example, MM-Eurcka-7BB show performance patterns influenced by their reasoning-foo focused training pipelines, excelling in certain multi-step inference settings while presenting varied outcomes across different knowledge domains. These results collectively illustrate how different training paradigms and objectives shape mathematical reasoning capabilities in complementary ways. Within this broader landscape, MathBook-77 7B achieves competitive performance across both reasoning and knowledge dimensions, reflecting the effectiveness of the structured knowledge design and progressive alignment strategies introduced in We-Math 2.0.
Furthermore, as shown in Figure 31 to Figure 40 we present the performance of different models on various subdomains in Eval,, where subdomains belonging to the same domain are indicated with the same color. It can be observed that models generally perform worse geometry- problems, especially in subdomains such as solid geometry and analytic geometry, which involve higher visual complexity and require more advanced reasoning. In contrast, models tend achieve better results on algebra and fundamental skills, particularly in subdomains related to computational methods.
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Table 12: Generating parameters for Open-Source MILMS.
Model
Generation Setup
Intern VLB.5-78B
do go_sample False, max new tokens 1024
Intern VL2.5-8B
so_sample False, new tokens 1024
Qwen2.5-VL-72B
do_ vo_sample False, max_new______am wow__okees- 1024
Qwcn2.5-VL-7B
do 10__ample- False, max new tokens 1024
Qwen2.5-VL-3B
do_sample False, max_new_tokent 1024
LLaVA-OneeVision-722
do_sample False, maax_new_ookens 1024
LLaVA-OneVision-7B
do_sample False, max moox_new_ooken 1024
GL.M-4V99B
do_sample False
Table 13: Generating parameters for Closed-Source MLLMs.
Model
Generation Setup
GPT-4o
"model" "gpt-4o", "temperature" 0, "max soax_ttkens" 1024
GPT-4V
"model": "gpt-44turboo,, "temperature" 0, "max__tokeee tokens": 1024
Table 14: Full results of different MLLMs on MathBookEval. Acc.: Accuracy; FS.; Foundational
skills; PS.: Probability and statistics; Geo.: Geometry; Alg. Algebra.
Reasoning
Knowledge
Models
Acc.
Levell
Level2
Level3
FS.
PS.
Geo.
Alg.
flosed-soure MILMs
GPT-4o
50.8
52.8
48.9
41.7
33.8
57.6
44.2
67.2
GPT-4V
42.8
44.0
43.0
31.9
36.8
56.6
33.5
59.4
Open-source (General)
IntemmL22.5-78B
51.8
$2.5
51.8
45.8
50.0
64.2
42.6
67.6
Qwen2.5-VLL72B
57.1
58.3
56.4
50.0
52.9
58.5
52.1
68.8
LLaVA-One Vision-72B
43.0
44.8
42.0
31.9
38.2
52.8
37.0
53.5
V.termVV.2.5-8B
37.9
40.7
34.5
27.8
33.8
46.2
31.4
50.0
Dwen2.5-VL-77
46.7
50.1
43.0
33.3
44.1
58.5
38.8
60.2
MadVA-OneViie 7B
28.0
36.8
24.9
Qwen2.5-VL--B
36.9
38.7
34.2
33.3
35.3
49.1
29.1
49.6
GLM-4V-9B
23.7
20.5
16.7
26.5
23.6
18.4
28.9
Open-source (Reasoning)
MM-Eureka-7B
50.0
50.6
51.5
38.9
47.1
52.8
44.0
62.9
RI-VL-7B
38.0
41.9
32.6
38.7
32.3
Open V.
32.6
33.0
32.6
29.2
45.6
38.7
24.9
43.4
VI.A--Thinke--B
35.7
38.3
31.9
29.2
46.6
42.5
30.2
41.6
MathBook-7B
50.4
52.0
48.2
45.8
57.4
67.9
40.5
63.3
C MORE DETAILS ON AATHBOOK-RL.
C.I IMPLEMENTATION DETAILS
We use Owen2.5-VL- L-L----LLBB------ as the base model and conduct all experiments on 8 x A800 GPUs. The training process consists of two stages.
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In the first stage, we perform cold-start supervised fine-tuning (SFT) to help the model develop explicit awareness of knowledge system and a knowledge-driven reasoning paradigm. The SFT stage uses learning rate of 1.0 10 -,,5,,, is trained for 1 epoch, and adopts a warmup ratio of 0.1.
In the second stage, we apply dynamic reinforcement learning (RL) to further improve the model's generalization ability. For RL, set the rollout temperature 1.0 and generate 8 rollouts per sample. The learning rate is set to 1 x 10 10-6, and the maximum completion length is 1024 tokens. The system prompt used in this stage is illustrated in Table [5] The reward function combines answer accuracy (weight 0.9) and response format compliance (weight O. 0). Specifically, we use Math Verify to extract and compare answers for accuracy, while format compliance ensures the output follows the required structure.
Table 15: The system prompt template for response generation in the RL stage.
Type
Prompt Template
A conversation between User and Assistant. The user asks question, and the Assistant solves it. The assistant first outputs the thinking process in <think> </think> and then provides the final answer(number, option, phrase, LaTeX expression as appropriate) <answer> </answer> tags. User: (question) Assistant:
System Prompt
C.2 ADDITIONAL EXPERIMENTAL RESULTS FOR MATHBOOK-3B
Table 16 summarizes the performance of MathBook-3B across multiple benchmarks. Our analysis yields the following observations.
Table 16: Performance comparison between Qwen2.5-VL-3B and MathBook-3B across benchmarks.
Model
Math Vista
Vision
WeMath
Math Verse
MathBookEval
Qwen2.5-VL-3B
60.8
21.1
22.9
29.1
36.9
MathBook-3B 63.2
24.3
28.1
32.6
40.2
(1) Consistent improvements across benchmarks. MathBook-31 -3 surpasses the Qwen2.5-V2 VL-BB backbone on all evaluated datasets, with notable gains on WeMath (+5.2), MathBookEval (43.3), and VathVerse (+3.5). These results indicate that the structured data design and training methodology introduced in We-Math 2.0 remain effective even at smaller model scales, enhancing both multimodal mathematical reasoning and generalization to diverse problem formats.
(2) Performance gains independent of model size. Despite using a significantly smaller backbone, MathBook-333 -B achieves improvements comparable in trend to those observed with MathBook-77B This suggests that the benefits of We-Math 2.0 arise from its principled knowledge structure and training pipeline, rather than relying large model capacity.
C.3 GENERALIZATION TO TEXT-ONLY MATHEMATICAL REASONING TASKS
To further examine whether the structured knowledge design in We-Math 2.0 transfers beyond the multimodal setting, we additionally evaluate MathBook- 7B on two widely used text-only mathemati- cal reasoning benchmarks: GSM8K (Cobbe al., 2021) and MATH-500 (Lightman al. 2023). As shown in Table 17] MathBook-785 7B achieves consistent improvements over the Qwen2.5-VL VL-7B
backbone. Although these datasets differ substantially from the multimodal training distribution, the gains observed here suggest that the structured knowledge supervision introduced in We-Math 2.0 can support more generalizable reasoning behavior across modalities.
C.4
CASE STUDY
To further illustrate the strengths of our approach, we present several representative case studies comparing model with the Qwen2.5-VI--7 baseline across different benchmarks.
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Table 17: Results text-only reasoning benchmarks.
Model
MATH-500
GSM8K
Qwen2.5-VI.-7B
66.0
86.2
MathBook-7B
67.2
87.6
Case 1: Conciseness and Reasoning Process. Figure 28 compares the response patterns of our model and the Qwen2.5-VL-7B baseline on We-Math. Our model produces more concise answers, with reduced average response length, while still retaining all necessary formulas and reasoning steps. This demonstrates that our approach effectively mitigates the issue of overthinking. resulting in more focused and efficient solutions without sacrificing completeness mathematical rigor.
Case 2: Spatial Reasoning Enhancement. Figure 29 highhights the performance of our model on Math MathVision. Compared to the baseline, which often fails to correctly interpret solve such problems, our model (MathBook-7B)) shows clear improvements in spatial reasoning. This is particularly evident in questions requiring the understanding of geometric relationships positional logic, indicating that our training strategy significantly enhances the model's 's ability to handle spatially complex scenarios.
Case 3: Knowledge-Oriented and Context-Aware Mathematical Reasoning. In Figure 30 compare the responses of our model and the Qwen2.5- (wwn2.55VL--7B VL-7B baseline on Math Vista. Our model not only applies the relevant mathematical concepts correctly but also demonstrates improved integration of mathematical knowledge with real-world problem contexts. This case exemplifies the model's strengthened ability bridge abstract mathematical reasoning and practical application, reflecting the benefits of our knowledge-oricet oriented and context-aware training paradigm.
Overall, these case studies provide qualitative evidence that our model achieves more concise, accu- rate, and contextually appropriate reasoning compared to strong baselines, particularly in scenarios demanding spatial understanding and knowledge-oriented problem solving.
C.5 EXPERIMENT SETUP
C.5.1 DETAILS OF THE EVALUATION.
We evaluate our model four representative benchmarks: Math (Lu et al. 2023) Math- Verse (Zhang et al. 2024a), Math Vision (Wang et al.] 2024), and We-Math (Qiao et al. 2024b). During evaluation, we strictly follow the official scoring protocols provided in the respective bench- mark GitHub repositories to ensure fair and consistent comparison. Specifically, we report results on the testmini split for Math Vista, Math Verse, Math Vision, and We-Math. For We-Math, we adopt the main evaluation metric as defined in the original paper, reporting "Score (Strict)" as the primary metric. For the other benchmarks, we report the average accuracy as the main evaluation result.
C.5.2 DETAILS OF THE BASELINES
We compare our method with a comprehensive set of baselines from three perspectives: closed-source
models (GPT-40 ApenAA 2024), Gemini-1.5-Pro (Team et al. 2023) ), open-source general models
(Qwen2.5-VL--7) (Bai et al. (025a),
Intern VL.2. V122.588B (Chen et al. 2023 ) GLM-4V-9B (GLM
et al. 2024). LLa VA- One Vision-78 7B Li et al. 2024). and open-souree reasoning models (Math-
PUMA-7B (Zhuang al. 2024h. URSA-8B Luo al. 2025). RL--OevisionnT7 (Yang al.] 2025b)
RI-VL-7B (Zhang et al. 2025), MM-Eureka- 7B (Meng et al. 2025), WcThink-7B ( (Yag et al.
2025a), VLAA-Thinker- 7B (Chen et al. 2025). Open DhennLThinner-78 (Deng al., 2025)). which
enables thorough and multi-faceted comparison to highlight the advantages of our approach.
GPT-4o (OpenAl 2024): GPT-4o ("o"" for omni) OpenAl's 2024 flagship multilingual, multi-
modal large language model that accepts text, images, and audio input for unified cross-modas understanding. is designed for broad adaptability and seamless integration visual and linguistic information, supporting complex reasoning across modalities and languages.
(Team et al. 2023); Gemini 1.5 Pro, developed by Google DeepMind, a multimodal model capable of processing text, images, audio, and video inputs, with an extremely
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Comparison of MathBook-7B and Qwen2.3-VL-7B We-Math
Question:
Now, we require you to solve a multiple-choice math question. Please briefly describe your thought process and provide the final
4 answer(option). Question: As shown in the diagram, what is the are length of this sector?
90
Options:
A. 2x; 4x:C. 8x; D. No correct answer
Regarding the format, please answer following the template below, and be sure to include two <> symbols: Thhoght process?: <<your thought process>> <Answer>> <<your option>>
Response from Qwen2.5-VL-7B:
<Thought process?: The diagram shows a quarter circle (90 degrees) with a radius of units. The formula for the are length of a sector is (0/360) 2nr, where 0 the angle in degrees and r the radius. For a 90-degree sector, the are length is (90/360) 2x * (1/4)* 2x x 4 4x, Therefore, the are length of this sector
<Answer>: B.4x
X
Response from MathBook-78
<Thought process>: The are length of a sector is calculated using the formula W (tfrae/)thhtaa33666 Vaimes 20pp r N), where W1 Mahheta Vi) the angle degrees and Mr r 1)) is the radius. Here, M lltheta 900 Vi and lig ((r== 1). Plugging in the values, we get li( (fra6/90)/360} litimes 2'ppi ittimss 4 [lfrac/][44) Mimess 81pi 2lipi 11) Thus, the are length is M/ 21lpp it. <Answer>: A. 2x
Figure 28: Comparison of MathBook-77 7B and Qwen2.5-VL-7B We-Math.
long context window (up to 1-2 million tokens). It is optimized for multi-task proficiency and structured tool integration, enabling analysis of lengthy and diverse content.
Qwen2.5-VL-7B (Bai et al. 2025a): Qwen2.5-VL-7B is open-source vision-language model with 7 billion parameters, designed to generate both free-form text and structured outputs such bounding boxes and JSON. It emphasizes fine-grained visual understanding and multi-modal alignment, supporting tasks like document analysis and event detection.
Intern VL2.5-8B (Chen et al. 2023): VL2.5-88 is an 8B-parameter open-source multimodal model that employs progressive scaling and co-training strategies to align vision and language components. incorporates training optimizations and curated dataset to enhance cross-modal reasoning and reduce hallucinations.
GLM-4V-9B (GLM et al. 2024): GLM-4V-9B is BBpaaameterr multimodal model built on the GLM series architecture. It integrates visual understanding and language processing capabilities, and is optimized using large-scale multimodal data. With specialized architectural design and training strategies, it demonstrates strong performance in multimodal reasoning tasks and is capable of effectively handling variety of vision-centric tasks.
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Comparison of MathBook-7B and Qwen2.5-V1-77B on MathVision
<imagel>:
Question:
What do you see if you look at the tower, which is made up of two building blocks, exactly from above?
<imagel> <image?>
<image2>:
(8)
[C]
Response from Qween25-VV--BB
To determine what you would see you looked at the tower from above, we need to analyze the shapes of the building blocks that make up the tower.
1. The tower consists of two building blocks: cylinder and a cone.
2. When viewed from above, the cylinder and the cone will appear as their respective two- dimensional shapes.
The cylinder, when viewed from above, appears as a circle.
The cone, when viewed from above, appears as triangle. Since the question asks for the shape you would see from above, we need to identify the shape that corresponds to the top view of the cone.
The options provided are:
(A) Circle (B) Triangle (C) Square (D) Triangle (repeated) (E) Square
The correct answer the shape that represents the top view of the cone, which is a triangle. Therefore, the answer is (B).
X
Response from MathBook-77
<think The top view of the tower, which consists of a cone and a cylinder, is circle. This is because the top surface of the cone is circle, and the top surface of the cylinder is also circle. Therefore, the top view is a circle. </thin>> <answer>: A</answer>
Figure 29: Comparison of MathBook-77 and Qwen2.5-VL-7B on Math Vision.
LLaVA-OneVision is an open-source model designed for unified single-image, multi-image, and video understanding. It adopts an AnyRes visual repre- sentation strategy to enable pross-scenario capability transfer. Trained via three-stage curriculum, it performs well across various benchmarks and exhibits emerging capabilities like video-to- to-ro-ttoviddo difference analysis and multi-camera self-driving video understanding via task transfer from images.
LLaVA-One Vision-7B et 2024):
(Zhuang et al., 2024): Math-PUMA-78 -B is vision vision-language model focused on mathematical reasoning with visual inputs, introducing a three-stage curriculum for aligning textual and visual modalities. The model is optimized for visual math benchmarks and aims to ensure consistent problem-solving across formats such as text and diagrams.
URSA-8B (Luo al., 2025): URSA-8B is 8B-parameter multimodal model targeting chain-of- thought reasoning in visual mathematical problems, trained on large-scale multimodal CoT datasets. employs reward model for stepwise verification, emphasizing both the generation and validation of reasoning chains for reliable solutions.
RI-Onevision-771 (Yang et al. 2025b): RI-Onevision-7B a 1B-parameter multimodal reasoning model that converts images into structured textual representations for symbolic reasoning. It is trained
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Published as a conference paper at ICLR 2026
Comparison of MathBook-7B and Qwen2.3-VL-7B on MathViste
Question:
Average waiting time at the DMV
Please answer the question requiring an integer answer and provide the final value, e.g.. 1, 3, at the end.
Month
Waiting time (minutes)
August
17
Question: An administrator at the Department of Motor Vehicles (DMV) tracked the average wait time from month to month. According to the table, what was the rate of change between August and September? (Unit: minutes per month)
September
14
October
26
November
17
December
25
Response from Qwen2.5-VL-7B:
<Thought process>: To find the rate of change between August and September, we need to calculate the difference waiting time and divide by the number of months between August and September.
The waiting time in August is 17 minutes.
The waiting time in September is minutes.
The difference waiting time
17 minutes (August) 14 minutes (September) = minutes
Since there is only one month between August and September, the rate of change is: minutes / I month minutes per month Therefore, the rate of change between August and September is minutes per month.
X
Response from Math Book-7B:
<think> To find the rate of change between August and September, we calculate the difference in waiting times and divide by the number of months between them. The waiting time in August is minutes, and in September is 14 minutes. The change is M 17 14 3 19 mimues. Since the time decreased, the rate of change is 11(311) minutes per month. </think> <answer>-3</answen
Figure 30: Comparison MathBook-75 and Qwen2.5-VL-7B on Math Vista.
on step-by-step multimodal reasoning annotations and refined with reinforcement learning, enabling precise multi-hop visual-textual inference.
R1-VL-7B (Zhang et al. 2025): RI-VL-7B is open-source 7B vision-language model designed for stepwise reasoning, applying Step-wise Group Relative Policy Optimization (StepGRPO) for dense intermediate rewards. This approach improves logical coherence and multi-step problem- solving, especially in mathematical and logical tasks.
MM-Eureka-7B (Meng et al., 2025): MM-Eureka-7B a vision-language model based on Qwen2.5-VL-7B, fine-tuned with the MMK12 dataset and rule-based reinforcement learning strategy. It is designed for multidisciplinary visual reasoning in math and science, using rule-based rewards guide the learning of complex reasoning steps.
WeThink-VL-78 (Yang et al. 2025at: WeThink-Ve VL-7B is 7B-parameter general-purpose vision- language reasoning model innedt on Qwen2.5- Dwen2.5-VL-77B via reinforcement learning. It is trained on the WhThi dataset and adopts hybrid reward mechanism combining rule- allebasedd verification. The model enhances performance across both mathematical reasoning and general multimodal tasks by leveraging scalable multimodal QA synthesis pipeline for diverse data generation and GRPO.
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Open' WTThnkee-- (Deng et al. 2025): Open
is open-source model tailored for complex vision-language reasoning. built by iterating between lightweight supervised fine-tuning and curriculum reinforcement learning. SFT initially distills chain-of-thoure traces from text- based reasoning models, while RI. refines these behaviors via a two-stage curriculum. achieves improvements across six benchmarks, outperforming concurrent models with smaller training data.
an open-source large vision- language model optimized for multimodal reasoning, trained via Group Relative Policy Optimization with novel mixed reward module. The mixed reward integrates 4 types of rule-based rewards and open-ended reward, avoiding "pseudo reasoning paths" induced by supervised fine-tuning, It achieves good performance on the open LMM reasoning leaderboard.
VLAA-Thinker-7B (Chen et al. 2025): VLAA-Thinker--e
D THE USE OF LARGE LANGUAGE MODELS (LLMs)
LLMs were used solely for grammar correction and minor language polishing. The conception, methodology, experiments, and analysis presented in this paper were entirely designed and imple- mented by the authors without relying on LLMs.
Figure 31: Detailed performance of GPT-4o across 13 subdomains.
100
Figure 32: Detailed performance of GPT-4V across 13 subdomains.
100
Figure 33: Detailed performance of V125577 across 13 subdomains.
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100
Figure 34: Detailed performance of Qwen2.5-VL.-72B across 13 subdomains.
100
Tran
Tome
Figure 35: Detailed performance of LLaVA-One Wis.onneeiiii2222 across subdomains.
100
Tome
Figure 36: Detailed performance of VLte5MVL2..58B across subdomains.
100
Erb
Tomee
Figure 37: Detailed performance of Qwen2.5-VL-7B across 13 subdomains.
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Published as a conference paper at ICLR 2026
100
Figure 38: Detailed performance of LLLVA---nee Vision-7B across 13 subdomains.
Figure 39: Detailed performance of GLM-4V-9B across 13 subdomains.
100
Figure Detailed performance of Qwen2.5-VL-3BB across 13 subdomains.
48


Reviews:
[{"review_id": "73Tz6bEwTt", "reviewer": "Reviewer_FXnB", "summary": "The authors proposed We-Math 2.0, a comprehensive framework for visual mathematical reasoning benchmarks. The major contribution of this paper is providing a detailed way to curate the dataset alongside with itself, paving the path for future progress. The authors also include a two-stage RL framework for MLLMs, and a extensive evaluation.", "questions": "1. I see in the teaser figure the definition has been emphasized a lot. I wonder if the authors want to state that the solution of the question is related to the definition, or the question is about the definition itself?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well structured and easy to follow. The teaser figure is particularly well structured.\n2. The intuition of the dataset is detailed and inspiring, which could be even more helpful than the dataset iteself.\n3. The evaluation results show the improvement brought by We-Math 2.0.", "weaknesses": "No significant weakness within the dataset scope.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vcHAqoISWc", "reviewer": "Reviewer_ZPo5", "summary": "The paper proposes We-Math 2.0, a unified framework centered around a structured MathBook knowledge system with 491 knowledge points. It introduces associated datasets and the MathBookEval benchmark, and designs a reinforcement learning training strategy based on curriculum learning. The authors claim their trained model achieves a marginal performance advantage on several existing mathematical multimodal benchmarks", "questions": "- How can the evaluation in Table 1 be made fair and consistent, especially regarding the granularity of category definitions?\n- Could the evaluation on MathBookEval include more open-source reasoning models of comparable scale? This would help disentangle whether the observed performance gains primarily stem from the richness and structure of the training data or from the effectiveness of the MathBook-RL training pipeline itself.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper presents a unified framework (We-Math 2.0) that integrates a structured mathematical knowledge system, a model-centric data space, and an RL-based training paradigm.\n\n- The trained model reportedly demonstrates a marginal advantage on some established mathematical multimodal benchmarks.", "weaknesses": "- The categorization and comparison in Table 1 appear inconsistent. For instance, comparing the granularity of the proposed 491-point system with datasets like MathV360k (which contains diverse content like charts and general QA) is not an apples-to-apples comparison and may be misleading.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rjgRwszIUL", "reviewer": "Reviewer_TdME", "summary": "This paper introduces We-Math 2.0, a system involving a knowledge system for mathematical knowledge, a pair of dataset for training, an RL framework, and a comprehensive benchmark for multimodal mathematical reasoning.", "questions": "Do we have a mechanism to use the knowledge system at inference?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This work represents substantial engineering effort in an important area and should be applauded for this.\n2. MathBook Knowledge System (MKS) is comprehensive with 491 knowledge points + 1819 fundamental principles\n3. MathBook-Standard/Pro is Built on MKS with annotated problems which are shown later in experiments to be strong together with the proposed MathBook-RL.\n4. MathBook-RL is presented well with ablation studies.\n5. This paper offers a few insights including the observation that MLLMs performance in geometry is subpar compared to that in algebra, and that performance in general correlates negatively with knowledge points.", "weaknesses": "Maybe some more experiments on a few more model scale are warranted, but considering the amount of work put into the entire system, I do not see this as much of defect.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kkqv53iipJ", "reviewer": "Reviewer_K8f9", "summary": "The paper proposes a unified system to enhance the mathematical reasoning abilities of MLLMs. WE-MATH 2.0 combines a structured math knowledge base, a model-driven data design framework, and an RL-based learning curriculum. Results on various benchmarks suggest the effectiveness of the proposed framework for MLLM reasoning.", "questions": "- How well does MathBook-7B generalize to language-only mathematical reasoning tasks, such as GSM8K or MATH, given that the system is trained and evaluated only on multimodal data?\n- Have the authors conducted human validation or cross-checking to verify the correctness of automatically assigned knowledge labels and reasoning steps?\n- Although MathBook-7B achieved strong results on We-Math and MathBookEval, how do the authors ensure that evaluation items were not seen or paraphrased during training, given the shared data construction pipeline?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- WE-MATH2.0 Integrates data generation, training, and evaluation within a closed-loop system, ensuring consistent metrics between learning objectives and assessment benchmarks.\n\n- The paper provides a large-scale, well-balanced benchmark covering both knowledge breadth and reasoning depth, enabling systematic evaluation of multimodal reasoning models.", "weaknesses": "- The performance improvements of MathBook-7B are mainly observed on its self-constructed benchmarks like We-Math, while gains on external benchmarks such as MathVista and MathVerse are relatively minor.\n- Although MathBookEval is designed to test reasoning depth and knowledge coverage independently from the training data, MathBook-7B does not show significant advantages on this benchmark. \n- The system focuses only on multimodal reasoning. It has not been evaluated on standard textual datasets such as GSM8K or MATH, leaving its language-only mathematical reasoning ability unverified. It would be great to also discuss the capability on language-only but still challenging math reasoning tasks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)