Linear algebra is one of the first abstract mathematics courses that students encounter at university. Research shows that many students find the dense presentation of definitions, theorems and proofs difficult to comprehend. Using a case study approach, we report on a teaching intervention based on Tall’s three worlds (embodied, symbolic and formal) of mathematical thinking, and use a framework combining these with Dubinsky’s Action, Process, Object and Schema (APOS) theory to analyse students’ resulting levels of understanding. Through interviews and analysis of test and examination scripts, we investigate students’ understanding of the basic concepts of linear algebra, their ability to use and explain these concepts and their relationship to definitional clarity. The results show that, while students tend not to learn definitions by rote and can be imprecise when expressing them in words, they seem to understand the concepts, can talk sensibly about them and are able to use their essential features in solving problems.

中文翻译：

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通过数学思维的三个世界发展线性代数的概念理解和定义清晰度

线性代数是学生在大学中遇到的第一批抽象数学课程之一。研究表明，许多学生发现很难理解定义，定理和证明的密集表述。我们使用案例研究的方法，报告了基于塔尔数学思维的三个世界（体现，象征和形式）的教学干预，并使用结合了杜宾斯基的动作，过程，对象和图式（APOS）理论的框架来分析学生由此产生的理解水平。通过对测试和考试脚本的访谈和分析，我们调查了学生对线性代数基本概念的理解，他们使用和解释这些概念的能力以及它们与定义清晰度的关系。结果表明，