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

Researchers propose INSPIRE, an Internalize-Then-Improve approach for example-driven mathematical reasoning in large language models, combining Reference-Guided Student Internalization (RGSI) with stage-wise rubric preference training. Experiments across multiple model scales and families show consistent improvements, even surpassing larger open-source models, with no degradation on out-of-distribution benchmarks.

arXiv:2608.27501v1 Announce Type: new 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.