INSPIRE: An Internalize-Then-Improve Approach for Example-Driven Mathematical Reasoning

• URL: http://arxiv.org/abs/2608.27501v1
• Date: Thu, 27 Aug 2026 03:24:03 GMT
• Title: INSPIRE: An Internalize-Then-Improve Approach for Example-Driven Mathematical Reasoning
• Abstract summary: In human mathematics education, example-based reasoning such as constructing counterexamples to test theorem boundaries reflects deep conceptual understanding.<n>We propose INSPIRE, an Internalize-Then-Improve approach combining Reference-Guided Student Internalization (RGSI) with a stage-wise preference training strategy.<n>Experiments across multiple model scales and families demonstrate consistent improvements, even surpassing larger open-source models.
• Score: 61.60016491769943
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Mathematical reasoning has seen rapid progress in large language models (LLMs), yet existing methods optimize predominantly for final-answer correctness, raising the question whether models truly internalize mathematical concepts or merely memorize solution patterns. In human mathematics education, example-based reasoning such as constructing counterexamples to test theorem boundaries reflects deep conceptual understanding, but remains underdeveloped in current LLMs. Enhancing this capability through preference optimization presents two key challenges: (1) the model's limited example-based reasoning ability makes constructing effective preference pairs inherently difficult; and (2) capability acquisition is progressive, as the model must first learn to adopt this strategy before learning to apply it correctly. Therefore we propose INSPIRE, an Internalize-Then-Improve approach combining Reference-Guided Student Internalization (RGSI), which produces high-quality preference candidates under the policy model's own distribution, with a stage-wise rubric preference training strategy that decomposes learning into method-oriented and correctness-oriented stages. Experiments across multiple model scales and families demonstrate consistent improvements, even surpassing larger open-source models, while evaluations on out-of-distribution benchmarks confirm no degradation in general mathematical reasoning ability.

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