Large Language Models for Mathematical Reasoning: Progresses and Challenges
(2402.00157)Abstract
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 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.
Overview
-
The paper surveys the current landscape of LLMs in mathematical problem-solving, analyzing their performance and capabilities.
-
A variety of mathematical domains including Arithmetic, Math Word Problems (MWP), Geometry, Automated Theorem Proving (ATP), and Math in the Vision-Language Context are explored, with specific datasets mentioned.
-
Methodologies to enhance LLMs for mathematical reasoning are discussed, covering prompting, fine-tuning, external verification tools, and knowledge distillation.
-
The paper assesses challenges facing LLMs such as brittleness, limited generalization, and the need for human-centered design, while suggesting their potential in educational contexts.
Introduction
The landscape of mathematical reasoning has been substantially impacted by the rise of LLMs, which have demonstrated impressive capabilities in solving a range of mathematical problems. This paper provides a comprehensive survey of the current state of LLMs in mathematical problem-solving, laying out the diverse problem types and datasets that have been explored, as well as the techniques put in place for this purpose.
Mathematical Problem Types and Datasets
The survey categorizes mathematical problems tackled by LLMs into several domains: Arithmetic, Math Word Problems (MWP), Geometry, Automated Theorem Proving (ATP), and Math in the Vision-Language Context. Each domain presents its unique challenges and datasets. The paper details the characteristics of these problems, from the straightforward arithmetic operations to the intricate MWPs requiring textual comprehension and step-by-step reasoning. Moreover, it outlines how MWPs can vary widely, offering examples and listing key datasets, such as SVAMP and MAWPS, which aid in training and benchmarking LLMs’ mathematical abilities.
Methodologies for Enhancing LLMs’ Capabilities
The paper delineates the various methodologies deployed to augment LLMs for mathematical reasoning. These range from mere prompting of pre-trained models to more intricate techniques like fine-tuning on specialized datasets. Among the methodologies discussed is the use of external tools to verify answers, advanced prompting methods like Chain-of-Thought, which improves models’ reasoning steps, and fine-tuning strategies that entail improving intermediate step generation and learning from enhanced datasets. Consideration is also given to teacher-student knowledge distillation, emphasizing the potential for making smaller models with high proficiency in solving math problems.
Analysis and Challenges
The robustness of LLMs in mathematics is particularly scrutinized, revealing a disparity in models' abilities to maintain performance under the variation of inputs. Factors influencing LLMs in math are also examined, such as prompt efficiency, tokenization methods, and model scale, contributing to a comprehensive understanding of LLMs' arithmetic capabilities. Despite notable advancements, challenges persist in the form of LLMs' brittleness in mathematical reasoning and their limited generalization beyond data-driven approaches. Furthermore, there is a salient need for a human-centered design in LLMs to ensure usability in educational settings, addressing aspects of user comprehension and adaptive feedback.
Educational Impact and Outlook
The implications of utilizing LLMs for mathematics within educational contexts are multifaceted, with LLMs having the potential to serve as powerful tools for aiding in learning and instruction. However, the current approaches often do not address the uniqueness of individual student needs or learning styles, nor do they consider the complexity or practicality of responses in line with students’ cognitive abilities. This paper calls for a delicate balance between machine efficiency and human-centric design, to ensure that LLMs serve as effective educational supplements.
In conclusion, the survey presents an intricate tapestry of achievements and challenges in the interplay between LLMs and mathematical reasoning. LLMs have proven their worth in various mathematical domains, yet the quest for more robust, adaptive, and human-oriented solutions continues to be a dynamic area of research and development.