The Missing Primitive: Diagnosing and Repairing Structural Mathematical Reasoning in LLMs
LLM 數學推理的「失落基元」：如何診斷並修復模型缺乏的結構化理解？
While LLMs perform well on math problems, their structural understanding remains questionable. This paper introduces 'Mathematical Primitives' and a new benchmark to evaluate reasoning across four dimensions: Discovery, Generation, Digestion, and Execution. Diagnostics show that raw accuracy masks specific capability gaps, and 'Discovery' is the primary bottleneck. To address this, the authors develop a primitive-privileged self-distillation framework that successfully transfers structured reasoning capabilities to student models, consistently boosting performance across benchmarks.
Key points
Introducing Mathematical Primitives
Proposes the concept of Mathematical Primitives to systematically probe whether LLMs possess underlying structural understanding of mathematical problems.
Four-Dimensional Diagnostic Framework
Dissects math reasoning into Discovery, Generation, Digestion, and Execution, revealing that surface accuracy often hides internal capability gaps.
Discovery as the Main Bottleneck
Diagnostics reveal that 'Discovery' is the dominant bottleneck in LLM reasoning, yet discovery-limited failures are the most amenable to post-training repair.
Primitive-Guided Self-Distillation
Introduces a primitive-privileged self-distillation framework to selectively transfer guided reasoning capabilities to student models, boosting performance across scales.
Why it matters
This research exposes the gap between surface-level math accuracy and deep structural comprehension in LLMs. By dissecting reasoning into four distinct capabilities, developers can pinpoint exact model weaknesses. Crucially, proving that the dominant 'discovery' bottleneck is highly repairable via self-distillation offers a practical, high-impact post-training pathway for enhancing mathematical and logical reasoning in next-generation models.
Who it affects
- AI Developer
- AI Researcher
- Product Manager
How to use it
- 1LLM mathematical diagnostics: Using the four-dimension framework to pinpoint model vulnerabilities in structural math understanding.
- 2Post-training & distillation: Employing primitive-guided self-distillation to repair and enhance open-source LLMs.
Limitations & caveats
- The effectiveness of the self-distillation framework still heavily depends on the quality of the mathematical primitives generated by the teacher model.
- Currently restricted to mathematics; whether the concept of primitives can generalize to other reasoning tasks like code generation remains to be tested.
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