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Large Language Models for Mathematical Reasoning: Progresses and Challenges

Janice Ahn, Rishu Verma, Renze Lou, Di Liu, Rui Zhang, Wenpeng Yin

arXiv:2402.00157v4cs.CL

TL;DR

LLM mathematical-reasoning research is difficult to assess because it spans diverse problems, datasets, metrics, and settings. This survey synthesizes the field’s problems, techniques, influencing factors, and challenges, presenting one of the first comprehensive examinations tailored to mathematics. It highlights progress alongside limited generalization, brittleness, adversarial vulnerability, and the need for human-centric design.

  • Problem

    Diverse mathematical problems, datasets, metrics, and evaluation settings make it difficult to determine progress and understand persistent challenges in LLM-based mathematical reasoning.

  • Method

    The survey reviews mathematical problems and datasets, LLM problem-solving techniques, factors affecting performance, and persistent challenges.

  • Results

    The survey provides one of the first comprehensive examinations of LLMs specifically tailored for mathematics.

  • Takeaways & Limitations

    The survey offers a holistic view of achievements, challenges, and unexplored areas in LLM-driven mathematical reasoning.

  • Takeaways & Limitations

    Current research shows limited robust generalization across datasets, grade levels, and types of mathematical problems.

Abstract

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Mathematical reasoning serves as a cornerstone for assessing the fundamental cognitive capabilities of human intelligence. In recent times, there has been a notable surge in the development of Large Language Models (LLMs) geared towards the automated resolution of mathematical problems. However, the landscape of mathematical problem types is vast and varied, with LLM-oriented techniques undergoing evaluation across diverse datasets and settings. This diversity makes it challenging to discern the true advancements and obstacles within this burgeoning field. This survey endeavors to address four pivotal dimensions: i) a comprehensive exploration of the various mathematical problems and their corresponding datasets that have been investigated; ii) an examination of the spectrum of LLM-oriented techniques that have been proposed for mathematical problem-solving; iii) an overview of factors and concerns affecting LLMs in solving math; and iv) an elucidation of the persisting challenges within this domain. To the best of our knowledge, this survey stands as one of the first extensive examinations of the landscape of LLMs in the realm of mathematics, providing a holistic perspective on the current state, accomplishments, and future challenges in this rapidly evolving field.

1 Introduction

LLMs have become important tools for automating mathematical problem-solving, but the field spans diverse problem types, datasets, metrics, and settings. This survey organizes that landscape across problems and datasets, techniques, influencing factors, and persistent challenges.

  • Mathematical reasoning challenges AI systems to combine textual comprehension, image interpretation, tabular analysis, symbolic manipulation, operational logic, and world knowledge.
  • LLMs have emerged as potent tools for automating intricate tasks, including mathematical problem-solving.
  • Diverse mathematical problems, evaluation metrics, datasets, and settings make it difficult to gauge progress and understand persistent challenges coherently.
  • The survey examines mathematical problem types and datasets, LLM techniques, factors affecting performance, and persisting challenges.
  • The survey presents one of the first comprehensive examinations of LLMs tailored specifically to mathematics and aims to synthesize achievements, challenges, and unexplored areas.

2 Related Work

Prior literature has summarized how LLMs can support mathematical work, while this survey broadens the scope to model-performance factors and educational perspectives. It emphasizes that human factors should not be neglected in pursuit of model performance.

  • Earlier work compared ChatGPT versions and GPT-4 across proof production, proof completion, mathematical search, and computation, while advocating human–LLM collaboration.
  • This survey differs by analyzing factors affecting model performance and considering LLM progress in mathematics from both AI and education perspectives.
  • The survey warns that pursuing model performance while neglecting human factors requires attention.

3 Math Problems & Datasets

The survey spans arithmetic, math word problems, geometry, automated theorem proving, and mathematics in vision contexts. These categories differ in inputs, reasoning demands, representations, and datasets.

  • 3.1 Arithmetic: Arithmetic problems involve pure mathematical operations and numerical manipulation without requiring textual, visual, or contextual interpretation.
  • 3.2 Math Word Problems: Math word problems present scenarios in written or verbal descriptions that require identifying relevant information and formulating equations or expressions.
  • 3.2 Math Word Problems: MWP instances may contain only a question and answer, or additionally provide an equation or step-by-step rationale.
  • 3.2 Math Word Problems: TABMWP studies open-domain word problems over tabular contexts represented as images, semi-structured text, or structured tables.
  • 3.2 Math Word Problems: MWP generation reverses the usual task by generating questions rather than deriving answers for given mathematical problems.
  • 3.3 Geometry: Geometry problems require spatial understanding and application of geometric principles, while current approaches mainly rely on symbolic methods and predefined search heuristics.
  • 3.4 Automated Theorem Proving: Automated theorem proving constructs proofs for conjectures using logical analysis, formal languages, and knowledge bases; MINIF2F, HOList, and COQGYM provide distinct evaluation settings.
  • 3.5 Math in Vision Context: Vision-context benchmarks include CHARTQA, with 9.6K human-written and 23.1K model-generated questions, and MATHVISTA, with 6K items spanning seven reasoning types.

4 Methodologies

The survey groups LLM mathematical-solving methods into prompting frozen models, strategies that enhance frozen models, and fine-tuning. These methods vary from prompt design and external tools to intermediate-step training, verification, distillation, and dataset construction.

  • Prior methods are organized into prompting frozen LLMs, strategies enhancing frozen LLMs, and fine-tuning LLMs.
  • Prompting frozen LLMs: Prompting studies include GPT-3 evaluation for classifying, extracting equations from, and generating math word problems.
  • Prompting frozen LLMs: ChatGPT and GPT-4 evaluations examine mathematical performance across arithmetic, logic, brainstorming, prompting methods, and visual mathematical reasoning.
  • Strategies enhancing frozen LLMs: Preprocessing, chain-of-thought, self-consistency, and explicit code-based self-verification modify inputs or reasoning procedures to improve mathematical solving.
  • Strategies enhancing frozen LLMs: External tools such as Python REPLs and symbolic solvers separate or verify computation, while Program-of-Thought expresses reasoning as executable programs.
  • Strategies enhancing frozen LLMs: Other enhancement strategies use conversational interaction, confidence and verifiable-explanation evaluation, and learned in-context-example selection.
  • Fine-tuning LLMs: Fine-tuning methods train models to generate intermediate steps, learn answer verifiers, use error-correction data, and learn from diverse annotation styles.
  • Fine-tuning LLMs: Further approaches use teacher–student distillation, multi-dataset instruction tuning, and solver ensembles combining complementary reasoning systems.

5 Analysis

LLM performance in mathematics depends on robustness, model design, prompts, training data, scale, and educational context. The survey also highlights risks when LLMs are deployed to support mathematics learning.

  • 5.1 LLMs’s robustness in math: GPT-4 maintains robustness to distracting information in CMATH, whereas other evaluated LLMs fail under the same perturbations.
  • 5.1 LLMs’s robustness in math: Adversarial samples generated by higher-accuracy LLMs can attack lower-accuracy models, while complex MWPs are more vulnerable to such attacks.
  • 5.1 LLMs’s robustness in math: Few-shot prompting with adversarial samples can improve LLM robustness on mathematical problems.
  • 5.2 Factors in influencing LLMs in math: Tokenization and pre-training token frequency influence arithmetic proficiency; T5 performs less effectively than Galactica and LLaMA on arithmetic tasks.
  • 5.2 Factors in influencing LLMs in math: Including code and LATEX in pre-training data correlates with enhanced arithmetic skills, while stronger reasoning ability does not necessarily imply stronger arithmetic performance.
  • 5.2 Factors in influencing LLMs in math: Prompt type and instruction tuning affect arithmetic performance, and larger models generally perform better but may plateau or be outperformed by smaller models.
  • 5.3 Perspectives of mathematics pedagogy: In mathematics education, LLMs can support critical thinking, sequential hints, error analysis, and conversational problem-solving.
  • 5.3 Perspectives of mathematics pedagogy: Educational deployment risks confusion from misinterpreted queries or erroneous explanations, while inaccurate responses may reinforce misconceptions.

6 Challenges

The survey identifies limited generalization, brittle reasoning, and insufficient attention to human interpretation as persistent challenges for LLMs solving mathematics.

  • Current research is strongly data-driven but lacks robust generalization across datasets, grade levels, and types of math problems.The survey suggests continual learning may improve machines’ math-solving capabilities.
  • LLMs show brittle mathematical reasoning across textual variations, alternative reasoning paths, and adversarial inputs.The same question can produce different final answers across trials, while manipulated inputs can exploit pre-trained math-oriented models.
  • Chain-of-thought reasoning often overlooks students’ needs and comprehension abilities, limiting its usefulness for human-oriented math interpretation.GPT-3.5 may misinterpret students’ questions and fail to provide adaptive feedback.
  • GPT-4 frequently overlooks younger students’ practical comprehension abilities and can generate hints that are overly intricate and confusing.The survey therefore calls for greater incorporation of human factors into AI design.

7 Conclusion

The survey reviews LLM capabilities and limitations in mathematical reasoning, covering problem types, datasets, applications in education, and persistent challenges. It emphasizes a human-centric approach and aims to guide future research and practical applications.

  • The survey examines LLM capabilities and limitations, mathematical problem types, datasets, educational applications, and persistent challenges.
  • The paper highlights the need for a human-centric approach in mathematics education and aims to inspire future research and practical applications.
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