CogMath: Evaluating LLMs' Authentic Mathematical Ability from a Cognitive Perspective

Jiayu Liu, Zhenya Huang, Wei Dai, Cheng Cheng, Jinze Wu, Jing Sha, Qi Liu, Shijin Wang, Enhong Chen

Keywords: Mathematical Reasoning, Large Language Models

Abstract: As large language models (LLMs) exhibit potential in solving complex mathematical tasks, increasing attention has been directed toward constructing benchmarks to evaluate their mathematical capabilities. However, existing benchmarks are either limited to specific task types (e.g., long-text problem understanding) or rely solely on a coarse measure of answer accuracy, making them insufficient for assessing a model's authentic mathematical proficiency. In this paper, we propose CogMath, which provides a comprehensive assessment of LLMs' mathematical abilities based on human cognitive processes. Specifically, inspired by cognitive theories, CogMath formalizes the reasoning process into 3 stages that align with human cognition: problem comprehension, problem solving, and solution summarization, and encompasses 9 fine-grained evaluation dimensions from perspectives such as numerical calculation, knowledge, and counterfactuals. In each dimension, to carry out a scientific evaluation, we develop an ``Inquiry-Judge-Reference'' multi-agent system, where the Inquiry agent generates inquiries that assess LLMs' mastery from this dimension, the Judge agent ensures the inquiry quality, and the Reference agent provides correct responses for comparison with the LLMs' actual performances. A LLM is considered to truly master a problem only when excelling in all inquiries from the 9 dimensions. In experiments, we evaluate 7 mainstream LLMs by applying CogMath to three benchmarks, which cover the full K-12 mathematical curriculum. The results reveal that the authentic mathematical capabilities of current LLMs are overestimated by 30-40%. Moreover, we locate their strengths and weaknesses across different stages/dimensions, offering constructive insights to further enhance their reasoning abilities.

Primary Area: foundation or frontier models, including LLMs

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Submission Number: 2743