# LLMs Are Already Good Tutors: Training-Free Prompt Optimization for Pedagogical Math Tutoring

arXiv: https://arxiv.org/abs/2605.27088 · HTML: https://mlsanity.com/paper/2605.27088/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 21 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 11 of 11)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Large Language Models Are Human-Level Prompt Engineers](https://mlsanity.com/paper/2211.01910/) (2022)
- [DSPy: Compiling Declarative Language Model Calls into Self-Improving Pipelines](https://mlsanity.com/paper/2310.03714/) (2023)
- [Automatic Prompt Optimization with "Gradient Descent" and Beam Search](https://mlsanity.com/paper/2305.03495/) (2023)
- [Agentic Context Engineering: Evolving Contexts for Self-Improving Language Models](https://mlsanity.com/paper/2510.04618/) (2025)
- [Optimizing Instructions and Demonstrations for Multi-Stage Language Model Programs](https://mlsanity.com/paper/2406.11695/) (2024)
- [MathDial: A Dialogue Tutoring Dataset with Rich Pedagogical Properties Grounded in Math Reasoning Problems](https://mlsanity.com/paper/2305.14536/) (2023)
- [Big-Math: A Large-Scale, High-Quality Math Dataset for Reinforcement Learning in Language Models](https://mlsanity.com/paper/2502.17387/) (2025)
- [The AI Teacher Test: Measuring the Pedagogical Ability of Blender and GPT-3 in Educational Dialogues](https://mlsanity.com/paper/2205.07540/) (2022)
- [MathTutorBench: A Benchmark for Measuring Open-ended Pedagogical Capabilities of LLM Tutors](https://mlsanity.com/paper/2502.18940/) (2025)

## Abstract (arXiv)

Aligning LLMs for math tutoring typically requires RL-based training with multi-GPU infrastructure. We investigate whether training-free prompt optimization-evolving only the system prompt via API calls-can serve as a practical alternative. We adapt 7 published methods and propose 5 education-specialized methods, evaluating these 12 methods under 5 conditions on 2 OOD benchmark suites. All 12 best-per-method configurations surpass the strongest RL-trained baseline (R_total = 0.633), and our ParetoGrad achieves the best Pareto balance across post-test solve rate, leak control, and helpfulness, rather than dominating any single component. Behavioral analysis with an 82-code educational codebook reveals that training-free methods rely on teaching-knowledge patterns at 2-3x the rate of RL-trained models, with a compensating ~10 percentage-point reduction in intent-level scaffolding. We also find a task-dependent reasoning mode effect consistent across training-free and RL-based paradigms. Our approach enables efficient development of pedagogically aligned LLM tutors with prompts alone and minimal compute.