MathBook Knowledge System

• MathBook Knowledge System is a hierarchically organized framework featuring 491 math points and 1,819 fundamental principles that underpin curriculum learning and model training.
• It integrates human-curated and AI-generated ontology, employing spatial, visual, and contextual expansions to create diverse, annotated training datasets.
• The system drives reinforcement learning-based training and dynamic curriculum alignment to enhance multimodal mathematical reasoning and generalization.

The MathBook Knowledge System is a hierarchically structured, explicitly principled framework at the core of We‑Math 2.0—a multimodal system dedicated to enhancing complex mathematical reasoning in LLMs. It provides a rigorously organized mathematical ontology comprising hundreds of knowledge points and fundamental principles, acting as both a conceptual backbone for dataset construction and as a curriculum for model-centric reinforcement learning.

1. Hierarchical Structure and Ontological Organization

The MathBook Knowledge System organizes mathematical content into a five-level hierarchy consistent with the “Definition–Theorem–Application” methodological paradigm. At the top level, domains such as Geometry, Algebra, Fundamental Skills, and Probability & Statistics serve as major categories. The hierarchy comprises 491 knowledge points, denoted in set-theoretic notation as $\mathcal{K} = \{k_1, k_2, \ldots, k_n\}$ with $n = 491$, covering educational content from elementary through university mathematics.

Each knowledge point $k_i$ is associated with a set of fundamental principles $\mathcal{P}_i = \{p_{i1}, p_{i2}, \ldots, p_{im}\}$. Collectively, the system explicitly catalogues 1,819 fundamental principles ($|\mathcal{P}| = 1,\!819$) across all knowledge points, encoding definitions, essential theorems, and canonical applications that are mapped to mathematical problem-solving steps.

The initial construction of the hierarchy is “human-AI collaborative”:

• A human-curated structure $\mathcal{K}^{(\mathrm{human})}$ is compiled from textbooks, Wikipedia, and official curriculum standards.
• In parallel, GPT-4o processes approximately 30,000 mathematics problems, extracting multi-level topic labels and computing a semantic similarity matrix $S \in \mathbb{R}^{n \times n}$. Hierarchical clustering of $S$ yields an AI-generated structure $\mathcal{K}^{(\mathrm{auto})}$.
• Final integration merges both, enforcing logical consistency and exhaustive coverage.

2. Integration with Model-Centric Data Space and Dataset Construction

MathBook’s ontological framework directly supervises the construction of systematically diverse training sets:

• MathBook-Standard: Each problem is authored under deep annotation, being explicitly tagged with knowledge points from $\mathcal{K}$ and relevant principles from $\mathcal{P}$. This guarantees complete and explicit conceptual supervision for every item.
• Dual Expansion: Problems are expanded in two orthogonal manners:
  • One-problem-multi-image: Each problem statement and knowledge tag is paired with multiple unique, semantically consistent diagrams (generated via GeoGebra).
  • One-image-multi-problem: A fixed diagram is used as the basis for generating several distinct problem variants pointing to different underlying knowledge aspects.
• MathBook-Pro: Problems are further developed in a three-dimensional difficulty space:
  • Step Complexity $(\varphi_s)$: Sequentially adding knowledge points or requiring additional intermediate inferences, e.g., $K_{i+1} = K_i + 1$.
  • Visual Complexity $(\varphi_v)$: Modifying diagrams via new elements (auxiliary lines, shaded areas) to increase spatial challenge.
  • Contextual Complexity $(\varphi_c)$: Transforming textual contexts, ranging from abstract to real-world or applied scenarios.

Mathematically, progressive variant generation is formalized as:

$$(q^*, a^*, I^*) = \varphi_s \circ \varphi_v \circ \varphi_c (q_0, a_0, I_0)$$

where $(q_0, a_0, I_0)$ is the seed problem and $*$ denotes a fully transformed, higher-difficulty variant. Seven such progressive variants per seed are produced, enabling curriculum learning and robust generalization capacity.

3. RL-Based Training: Knowledge-Driven Progressive Alignment

Integration with reinforcement learning (MathBook-RL) leverages the knowledge system in a two-stage paradigm:

1. Cold-Start Supervised Fine-Tuning: The model is initially trained on MathBook-Standard using knowledge-oriented chain-of-thought reasoning. The SFT loss is given by:

$$\mathcal{L}_\mathrm{SFT}(\theta) = \mathbb{E}_{(x, y) \sim \mathcal{D}_\mathrm{init}} \left[ -\log P_\theta(y|x) \right]$$

enforcing stepwise conceptual awareness and grounded logical progression.

1. Progressive Alignment RL: Fine-tuning proceeds on MathBook-Pro with dynamic curriculum:
   • Knowledge Increment Scheduling and Modality Increment Scheduling adjust sampling when the model struggles with new knowledge points or visual complexity, respectively.
   • Group Relative Policy Optimization (GRPO) is employed, with the objective:

   $$\mathcal{J}(\theta) = \mathbb{E}\left[ \min\left( \frac{\pi_\theta(o_{i, t} | q, o_{i, < t})}{\pi_{\theta_\mathrm{old}}(o_{i, t} | q, o_{i, < t})} \cdot \hat{A}_{i, t},\ \mathrm{clip}(\cdot) \right) - \beta \mathcal{D}_\mathrm{KL}(\pi_\theta \| \pi_\mathrm{ref}) \right]$$

This systematically aligns the model's policy with progressively challenging subregions of the data space, guided by explicit knowledge annotation.

4. Formal Representation of Knowledge and Reasoning

Mathematical knowledge objects and reasoning steps are grounded in explicit mathematical notation and LaTeX-formulated descriptors. For example:

• Individual knowledge points and their fundamental principles are represented as $\mathcal{K} = \{k_1, \ldots, k_{491}\}$, $\mathcal{P} = \bigcup_{i=1}^{491} \mathcal{P}_i$.
• Expansion steps—for instance, adding an inference requirement—are coded as $K_{i+1} = K_i + 1$.
• Variant generation is notated concisely, as above.

This explicit approach allows direct mapping from problem statements to underlying knowledge, supporting both detailed explainability and fine-grained curriculum design.

5. Impact on Multimodal Mathematical Reasoning and Benchmarking

The MathBook Knowledge System’s rigorous structuration is directly linked to empirical performance advantages:

• Each reasoning step in training and evaluation is explicitly mapped to one or more knowledge points, ensuring supervision is conceptually meaningful (rather than relying on surface features or memorization).
• The multi-dimensional expansion in MathBook-Pro offers unmatched diversity and coverage in the training set, facilitating robust learning across modalities (text, diagrams).
• Curriculum learning and dynamic RL policy adaptation, guided by MathBook, result in improved generalization and spatial reasoning, as evidenced by strong outcomes on benchmarks including MathVista, MathVision, and We‑Math 2.0’s internal MathBookEval (which encompasses all 491 knowledge points with variable reasoning step depth).

6. Human-AI Collaborative Curation and Ongoing Development

The system’s hybrid construction method—integrating authoritative human curation with large-scale AI-driven topic extraction and clustering—ensures both logical rigor and practical breadth. This results in a knowledge framework well-aligned with human curricular standards yet flexible and extensible as more data and mathematical domains are incorporated.

A plausible implication is that this hybrid methodology could be adapted for continuous evolution, whereby feedback from model performance and external benchmarks iteratively refines both the ontology and the dataset annotation.

7. Significance and Future Directions

The MathBook Knowledge System represents a comprehensive paradigm for mathematical knowledge management in AI, offering:

• Explicit, granular mapping between concepts, principles, and problem statements.
• A foundation for curriculum-based, knowledge-oriented training.
• Support for systematic expansion across semantic, visual, and contextual dimensions of mathematical tasks.
• Demonstrated empirical advantages in both accuracy and generalization capacity, particularly in model-centric, RL-augmented training schemes.

Continued development may further emphasize dynamic knowledge updating, automatic knowledge point induction from new data, and integration with broader mathematical ontologies and assessment platforms.

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In summary, the MathBook Knowledge System underpins We‑Math 2.0 with a five-level, 491-point, 1,819-principle mathematical ontology that is tightly integrated with dataset annotation, curriculum learning, and reinforcement-based model optimization, yielding substantial advances in explicit, interpretable, and robust multimodal mathematical reasoning (Qiao et al., 14 Aug 2025).