Improving Mathematical Reasoning Capabilities in Large Language Models via Reasoning Process Error Classification

Large Language Models (LLMs) achieve impressive accuracy on mathematical reasoning benchmarks, yet their performance drops when problems are modified with simple changes like different names or numbers. Code execution methods, which let models generate and run Python code instead of reasoning in natural language, have been proposed as a solution, but their effect on reasoning robustness (the ability to maintain accuracy across problem variations) has not been systematically tested. This study evaluates three approaches on 1,000 problems from the GSM-Symbolic dataset: pure reasoning using chain-of-thought (CoT) prompting, single-shot code execution using Program-Aided Language models (PAL), and iterative code execution using Step-by-Step Coding (SBSC). All three were run on paired original and modified problems using Claude Haiku 4.5. CoT was the most robust method, with an accuracy drop of 1.3 percentage points and 1.8% of problems breaking under perturbation. PAL was the least robust at 1.7 percentage points and 3.1% broke, with SBSC falling in between. Although these differences were not statistically significant ($p = .096$), the directional trend was consistent across all measures, suggesting that code execution, whether single-shot or iterative, does not improve reasoning robustness on grade-school-level problem variations.

Mathematical reasoning is essential for problem-solving in education, science, and industry, serving as a crucial benchmark for evaluating artificial intelligence systems. As Large Language Models (LLMs) improve their reasoning capabilities, understanding how well they perform mathematical reasoning has become increasingly important. This survey synthesizes recent advancements in mathematical reasoning with LLMs through a structured analysis of datasets, architectures, training strategies, and evaluation protocols. Our systematic review encompasses approximately 120 peer-reviewed studies and preprints, examining the evolution of this research area and providing a unified analytical framework to understand current progress and limitations. Our study particularly introduces a unified taxonomy of mathematical datasets, distinguishing between pretraining corpora, supervised fine-tuning resources, and evaluation benchmarks across varying levels of reasoning complexity. A systematic analysis of reasoning architectures and training strategies, including tool integration, verifier-guided reasoning, and parameter-efficient adaptation, is presented to assess their effects on reasoning robustness and generalization. Moreover, a comparative evaluation of existing metrics highlights the gap between final-answer accuracy and process-level reasoning verification. By synthesizing insights across these areas, our analysis identifies recurring failure modes, such as reasoning faithfulness issues, benchmark biases, and generalization limitations, and outlines key research directions toward improving symbolic grounding, evaluation reliability, and the development of more robust and trustworthy LLM-based reasoning systems.

Despite the pivotal role of numerical reasoning as the cornerstone of mathematical capabilities in large language models (LLMs) across applications, few benchmarks evaluate LLMs by integrating numerical processing and mathematical reasoning, hindering the interpretability of failures in math tasks. We introduce PyraMathBench, a comprehensive hierarchical benchmark with 32,505 questions derived from 7,404 math word problems, spanning 4 key cognitive aspects, 14 subcategories, and 2 modalities. Experiments reveal that LLMs' performance is severely compromised by inadequate numerical computation and weak handling of abstract numerical questions. To address this, we propose the Smart Optimization & Learning-based VErsatile module (SOLVE) and Interactive Relative Policy Optimization (IRPO), which enhance LLMs' numerical-mathematical synergy via efficient tool calls (fuzzy matching and low-quality call rejection). Comparative experiments show Qwen-2.5 achieves a 5.0 score improvement with SOLVE and IRPO training.