Artificial Intelligence for Mathematical Reasoning: An Integrated Survey of Language Models, Neuro-symbolic Systems, and Verified Discovery

Mathematical reasoning has long served as a stringent test of machine intelligence; over the past decade, it has moved from a niche problem within NLP to one of the most consequential AI frontiers. This survey provides a unified account of the field's evolution, from early rule-based math word problem (MWP) solvers and template-driven geometry systems, through neural expression generation and LLM prompting, to contemporary reasoning models, multi-agent systems, neuro-symbolic theorem provers, and verified discovery workflows. We organize the landscape along four axes: (i) informal reasoning over text and diagrams, spanning MWP solving, multimodal geometry, and VLMs; (ii) formal reasoning in proof assistants, including autoformalization, tactic prediction, compiler-guided repair, and proof search; (iii) mathematical discovery, where systems propose constructions, improve bounds, or assist attacks on open problems; and (iv) the inference and training-time techniques, including CoT prompting, tool use, process reward models, and RLVR, that increasingly connect generation with verification. We catalog major benchmarks across grade-school arithmetic, competition mathematics, geometry, formal proving, multimodal and multilingual reasoning, and expert evaluation, and we examine benchmark saturation, contamination, reporting mismatches, and the distinction between pass@1, majority voting, and verifier-assisted pass@$k$. We critically assess failure modes: brittleness under perturbation, reward hacking, multimodal grounding failures, fragile formalization, and the energy cost of reasoning-scale inference. Drawing on recent perspectives from working mathematicians, we identify future directions centered on verified-discovery workflows, reasoning efficiency, and infrastructure to make AI-assisted formalization broadly usable. Companion materials: https://github.com/Starscream-11813/awesome-AI4Math.

PyraMathBench: Evaluating and Improving Mathematical Capability in Large Language Models

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.

NeuroProlog: Multi-Task Fine-Tuning for Neurosymbolic Mathematical Reasoning via the Cocktail Effect

Large Language Models (LLMs) achieve strong performance on natural language tasks but remain unreliable in mathematical reasoning, frequently generating fluent yet logically inconsistent solutions. We present \textbf{NeuroProlog}, a neurosymbolic framework that ensures verifiable reasoning by compiling math word problems into executable Prolog programs with formal verification guarantees. We propose a multi-task Cocktail training strategy that jointly optimizes three synergistic objectives in a unified symbolic representation space: (i) mathematical formula-to-rule translation (KB), (ii) natural language-to-program synthesis (SOLVE), and (iii) program-answer alignment. This joint supervision enables positive transfer, where symbolic grounding in formula translation directly improves compositional reasoning capabilities. At inference, we introduce an execution-guided decoding pipeline with fine-grained error taxonomy that enables iterative program repair and quantifies model self-debugging capacity. Comprehensive evaluation on GSM8K across four model scales (3B--32B parameters) demonstrates consistent improvements: cocktail training achieves significant accuracy gains of +5.23\% (Qwen-32B, $p < 0.01$), +3.43\% (GPT-OSS-20B, $p < 0.01$), and +5.54\% (Llama-3B, $p < 0.05$) over single-task baselines. Systematic error analysis reveals scale-dependent learning dynamics: at 32B scale, cocktail training transforms unfixable type errors (12\% repair rate) into correctable domain errors (96\% repair rate), achieving 92.7\% overall correction; at 8B scale, the same training eliminates syntactic errors but introduces semantic failures, revealing a critical capacity threshold for type-safe symbolic reasoning.

Adaptive Problem Generation via Symbolic Representations

We present a method for generating training data for reinforcement learning with verifiable rewards to improve small open-weights language models on mathematical tasks. Existing data generation approaches rely on open-loop pipelines and fixed modifications that do not adapt to the model's capabilities. Furthermore, they typically operate directly on word problems, limiting control over problem structure. To address this, we perform modifications in a symbolic problem space, representing each problem as a set of symbolic variables and constraints (e.g., via algebraic frameworks such as SymPy or SMT formulations). This representation enables precise control over problem structure, automatic generation of ground-truth solutions, and decouples mathematical reasoning from linguistic realization. We also show that this results in more diverse generations. To adapt the problem difficulty to the model, we introduce a closed-loop framework that learns modification strategies through prompt optimization in symbolic space. Experimental results demonstrate that both adaptive problem generation and symbolic representation modifications contribute to improving the model's math solving ability.

Progress over Points: Reframing LM Benchmarks Around Scientific Objectives

Current benchmarks that test LLMs on static, already-solved problems (e.g., math word problems) effectively demonstrated basic capability acquisition. The natural progression has been toward larger, more comprehensive and challenging collections of static problems, an approach that inadvertently constrains the kinds of advances we can measure and incentivize. To address this limitation, we argue for progress-oriented benchmarks, problem environments whose objectives are themselves the core targets of scientific progress, so that achieving state of the art on the benchmark advances the field. As a introductory step, we instantiate an environment based on the NanoGPT speedrun. The environment standardizes a dataset slice, a reference model and training harness, and rich telemetry, with run-time verification and anti-gaming checks. Evaluation centers on the scientific delta achieved: best-attained loss and the efficiency frontier. Using this environment, we achieve a new state-of-the-art training time, improving upon the previous record by 3 seconds, and qualitatively observe the emergence of novel algorithmic ideas. Moreover, comparisons between models and agents remain possible, but they are a means, not the end; the benchmark's purpose is to catalyze reusable improvements to the language modeling stack. With this release, the overarching goal is to seed a community shift from static problem leaderboards to test-time research on open-ended yet measurable scientific problems. In this new paradigm, progress on the benchmark is progress on the science, thus reframing "benchmarking" as a vehicle for scientific advancement.

Leveraging Large Language Models for Bengali Math Word Problem Solving with Chain of Thought Reasoning

Solving Bengali Math Word Problems (MWPs) remains a major challenge in natural language processing (NLP) due to the language's low-resource status and the multi-step reasoning required. Existing models struggle with complex Bengali MWPs, largely because no human-annotated Bengali dataset has previously addressed this task. This gap has limited progress in Bengali mathematical reasoning. To address this, we created SOMADHAN, a dataset of 8792 complex Bengali MWPs with manually written, step-by-step solutions. We designed this dataset to support reasoning-focused evaluation and model development in a linguistically underrepresented context. Using SOMADHAN, we evaluated a range of large language models (LLMs) - including GPT-4o, GPT-3.5 Turbo, LLaMA series models, Deepseek, and Qwen - through both zero-shot and few-shot prompting with and without Chain of Thought (CoT) reasoning. CoT prompting consistently improved performance over standard prompting, especially in tasks requiring multi-step logic. LLaMA-3.3 70B achieved the highest accuracy of 88% with few-shot CoT prompting. We also applied Low-Rank Adaptation (LoRA) to fine-tune models efficiently, enabling them to adapt to Bengali MWPs with minimal computational cost. Our work fills a critical gap in Bengali NLP by providing a high-quality reasoning dataset and a scalable framework for solving complex MWPs. We aim to advance equitable research in low-resource languages and enhance reasoning capabilities in educational and language technologies.

Elementary Math Word Problem Generation using Large Language Models

Mathematics is often perceived as a complex subject by students, leading to high failure rates in exams. To improve Mathematics skills, it is important to provide sample questions for students to practice problem-solving. Manually creating Math Word Problems (MWPs) is time consuming for tutors, because they have to type in natural language while adhering to grammar and spelling rules of the language. Existing Deep Learning techniques for MWP generation either require a tutor to provide the initial portion of the MWP, and/or additional information such as an equation. In this paper, we present an MWP generation system based on Large Language Models (LLMs) that overcome the need for additional input - the only input to our system is the number of MWPs needed, the grade and the type of question (e.g. addition, subtraction). Unlike the existing LLM-based solutions for MWP generation, we carried out an extensive set of experiments involving different LLMs, prompting strategies, techniques to improve the diversity of questions, as well as techniques that employ human feedback to improve LLM performance. Human and automated evaluations confirmed that the generated MWPs are high in quality, with minimal spelling and grammar issues. However, LLMs still struggle to generate questions that adhere to the specified grade and question type requirements.

A Diversity-Enhanced Knowledge Distillation Model for Practical Math Word Problem Solving

Math Word Problem (MWP) solving is a critical task in natural language processing, has garnered significant research interest in recent years. Various recent studies heavily rely on Seq2Seq models and their extensions (e.g., Seq2Tree and Graph2Tree) to generate mathematical equations. While effective, these models struggle to generate diverse but counterpart solution equations, limiting their generalization across various math problem scenarios. In this paper, we introduce a novel Diversity-enhanced Knowledge Distillation (DivKD) model for practical MWP solving. Our approach proposes an adaptive diversity distillation method, in which a student model learns diverse equations by selectively transferring high-quality knowledge from a teacher model. Additionally, we design a diversity prior-enhanced student model to better capture the diversity distribution of equations by incorporating a conditional variational auto-encoder. Extensive experiments on {four} MWP benchmark datasets demonstrate that our approach achieves higher answer accuracy than strong baselines while maintaining high efficiency for practical applications.

Exploring the Hidden Reasoning Process of Large Language Models by Misleading Them

Large language models (LLMs) and Vision language models (VLMs) have been able to perform various forms of reasoning tasks in a wide range of scenarios, but are they truly engaging in task abstraction and rule-based reasoning beyond mere memorization and pattern matching? To answer this question, we propose a novel experimental approach, Misleading Fine-Tuning (MisFT), to examine whether LLMs/VLMs perform abstract reasoning by altering their original understanding of fundamental rules. In particular, by constructing a dataset with math expressions that contradict correct operation principles, we fine-tune the model to learn those contradictory rules and assess its generalization ability on different test domains. Through a series of experiments, we find that current LLMs/VLMs are capable of effectively applying contradictory rules to solve practical math word problems and math expressions represented by images, implying the presence of an internal mechanism that abstracts before reasoning.

DiffSampling: Enhancing Diversity and Accuracy in Neural Text Generation

Despite their increasing performance, large language models still tend to reproduce training data, generate several repetitions, and focus on the most common grammatical structures and words. A possible cause is the decoding strategy adopted: the most common ones either consider only the most probable tokens, reducing output diversity, or increase the likelihood of unlikely tokens at the cost of output accuracy and correctness. In this paper, we propose a family of three new decoding methods by leveraging a mathematical analysis of the token probability distribution. In particular, the difference between consecutive, sorted probabilities can be used to avoid incorrect tokens and increase the chance of low-probable but accurate words. Experiments concerning math problem solving, extreme summarization, and the divergent association task show that our approach consistently performs at least as well as current alternatives in terms of quality and diversity.