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

arXiv: https://arxiv.org/abs/2608.27501 · HTML: https://mlsanity.com/paper/2608.27501/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 29 of its 56 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 29 of 29)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [OpenAI GPT-5 System Card](https://mlsanity.com/paper/2601.03267/) (2025)
- [DeepSeek-V3 Technical Report](https://mlsanity.com/paper/2412.19437/) (2024)
- [Phi-4-Mini Technical Report: Compact yet Powerful Multimodal Language Models via Mixture-of-LoRAs](https://mlsanity.com/paper/2503.01743/) (2025)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [Kimi K2.5: Visual Agentic Intelligence](https://mlsanity.com/paper/2602.02276/) (2026)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [GLM-4.5: Agentic, Reasoning, and Coding (ARC) Foundation Models](https://mlsanity.com/paper/2508.06471/) (2025)
- [Rubrics as Rewards: Reinforcement Learning Beyond Verifiable Domains](https://mlsanity.com/paper/2507.17746/) (2025)
- [Step-DPO: Step-wise Preference Optimization for Long-chain Reasoning of LLMs](https://mlsanity.com/paper/2406.18629/) (2024)
- [InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning](https://mlsanity.com/paper/2402.06332/) (2024)
- [Reinforcement Learning with Rubric Anchors](https://mlsanity.com/paper/2508.12790/) (2025)
- [ChatGLM-Math: Improving Math Problem-Solving in Large Language Models with a Self-Critique Pipeline](https://mlsanity.com/paper/2404.02893/) (2024)
- [BrokenMath: A Benchmark for Sycophancy in Theorem Proving with LLMs](https://mlsanity.com/paper/2510.04721/) (2025)
- [Refine Knowledge of Large Language Models via Adaptive Contrastive Learning](https://mlsanity.com/paper/2502.07184/) (2025)
- [Youtu-LLM: Unlocking the Native Agentic Potential for Lightweight Large Language Models](https://mlsanity.com/paper/2512.24618/) (2025)
- [Cognitive Mismatch in Multimodal Large Language Models for Discrete Symbol Understanding](https://mlsanity.com/paper/2603.18472/) (2026)
- [Toward Native Multimodal Modeling: A Roadmap](https://mlsanity.com/paper/2605.25343/) (2026)
- [From Chatbot to Digital Colleague: The Paradigm Shift Toward Persistent Autonomous AI](https://mlsanity.com/paper/2606.14502/) (2026)
- [Learning to Disprove: Formal Counterexample Generation with Large Language Models](https://mlsanity.com/paper/2603.19514/) (2026)
- [TangramPuzzle: Evaluating Multimodal Large Language Models with Compositional Spatial Reasoning](https://mlsanity.com/paper/2601.16520/) (2026)
- [TopoAgent: A Self-Evolving Topological Agent for Multimodal Scientific Reasoning](https://mlsanity.com/paper/2607.14658/) (2026)
- [Deep Thought Alignment: Trajectory-Level Latent Distillation for Video Reasoning](https://mlsanity.com/paper/2608.16316/) (2026)

## Abstract (arXiv)

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.