Abstract Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R3.

中文翻译：

---

大学生对三维（欧几里得）空间和二元函数概念的理解分析

摘要 数学教育正在强调概念理解。学生通常难以理解多变量函数这一关键概念。本研究基于分析个体在学习数学概念时形成的认知结构并产生与该学习相关的成分的APOS理论，分析大学生对三维空间和二元函数的概念理解。还考虑了 Trigueros 和 Martinez-Planell 提出的这些概念的遗传分解。分析结果表明，只有一名学生在遗传分解的框架内构建了三维空间的概念作为对象。一些学生无法将二维函数和三维空间的概念联系起来。能够进行代数运算的学生在几何表示方面存在问题。这项研究建议改进遗传分解以包括，例如，编写特殊曲面的代数方程的心理构造步骤，其图形在 R3 中给出。