# Are We Measuring Strategy or Phrasing? The Gap Between Surface- and Approach-Level Diversity in LLM Math Reasoning

arXiv: https://arxiv.org/abs/2606.29985 · HTML: https://mlsanity.com/paper/2606.29985/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 18 of its 46 references that are in the map.
- First posted on arXiv 29 Jun 2026.

## Placed in

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

## Builds on (most cited first, 18 of 18)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [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)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Qwen3 Embedding: Advancing Text Embedding and Reranking Through Foundation Models](https://mlsanity.com/paper/2506.05176/) (2025)
- [Artificial Hivemind: The Open-Ended Homogeneity of Language Models (and Beyond)](https://mlsanity.com/paper/2510.22954/) (2025)
- [Olmo 3](https://mlsanity.com/paper/2512.13961/) (2025)
- [Jina Embeddings 2: 8192-Token General-Purpose Text Embeddings for Long Documents](https://mlsanity.com/paper/2310.19923/) (2023)
- [Jointly Reinforcing Diversity and Quality in Language Model Generations](https://mlsanity.com/paper/2509.02534/) (2025)
- [Diverse Preference Optimization](https://mlsanity.com/paper/2501.18101/) (2025)
- [CoRNStack: High-Quality Contrastive Data for Better Code Retrieval and Reranking](https://mlsanity.com/paper/2412.01007/) (2024)
- [DRA-GRPO: Your GRPO Needs to Know Diverse Reasoning Paths for Mathematical Reasoning](https://mlsanity.com/paper/2505.09655/) (2025)
- [Reasoning Path Divergence: A New Metric and Curation Strategy to Unlock LLM Diverse Thinking](https://mlsanity.com/paper/2510.26122/) (2025)
- [Reaching Beyond the Mode: RL for Distributional Reasoning in Language Models](https://mlsanity.com/paper/2603.24844/) (2026)
- [GuidedSampling: Steering LLMs Towards Diverse Candidate Solutions at Inference-Time](https://mlsanity.com/paper/2510.03777/) (2025)

## Built on by (most cited first, 2 of 2)

- [Strategically Diverse Sampling for Self-Training](https://mlsanity.com/paper/2609.31571/) (2026)
- [Mathematical Transfer in LLMs Follows Reasoning Approach More Than Topic](https://mlsanity.com/paper/2610.00331/) (2026)

## Abstract (arXiv)

Diversity in LLM mathematical reasoning is critical for exploration, but common diversity metrics mostly capture surface-level variation rather than differences in how a problem is solved. We address this gap by introducing approach-level diversity: variation in strategies across correct solutions to the same problem. Using a human-calibrated LLM judge framework, we show that prior diversity measures are unreliable proxies for approach-level diversity, and this mismatch carries over to diversity-aware RLVR, where target metrics are preserved while approach-level diversity declines. Investigating when approach-level diversity helps and whether it can be directly induced, we find that approach-diverse candidate sets improve test-time scaling. However, optimizing an LLM judge diversity reward during training causes the policy to exploit judge-specific preferences rather than broaden its approaches, leaving direct optimization of approach-level diversity as an open problem. Together, our work introduces the notion of approach-level diversity and uncovers a systematic divergence between surface- and approach-level signals, marking a step toward LLMs that reason in genuinely diverse, human-like ways.