INSPIRE: An Internalize-Then-Improve Approach for Example-Driven Mathematical Reasoning
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.
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[Submitted on 27 Aug 2026]
Title:INSPIRE: An Internalize-Then-Improve Approach for Example-Driven Mathematical Reasoning
View a PDF of the paper titled INSPIRE: An Internalize-Then-Improve Approach for Example-Driven Mathematical Reasoning, by Shuai Wang and 6 other authors
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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.
Comments: EMNLP 2026
Subjects:
Computation and Language (cs.CL)
Cite as: arXiv:2608.27501 [cs.CL]
(or arXiv:2608.27501v1 [cs.CL] for this version)
https://doi.org/10.48550/arXiv.2608.27501
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Yinghui Li [view email] [v1] Thu, 27 Aug 2026 03:24:03 UTC (2,123 KB)
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