This study is concerned with the reasoning that undergraduates apply when deciding whether a prompt is an example or non-example of the subspace concept. A qualitative analysis of written responses of 438 students revealed five unconventional tacit models that govern their reasoning. The models account for whether a prompt is a subset of a vector space, whether the zero vector is included, the structure of vectors, their number in the formula for the general solution to the system of linear equations, and the corresponding coefficient matrix. Furthermore, a conception was identified in students’ responses, according to which the algebraic structure of a vector space passes from a ‘parent’ space to its subset, turning automatically it into a subspace. For many students this conception of an inheriting structure was instrumental for identifying and reasoning around subspaces. Polysemy of the prefix ‘sub’ and students’ prior experiences in identifying concept examples are used for offering explanations for the emergence of the conception.

中文翻译：

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管理本科生关于子空间的推理的默认模型

这项研究与大学生在决定提示是子空间概念的示例还是非示例时应用的推理有关。对438名学生的书面回答进行定性分析后，发现有五个非常规的默认模型可以控制他们的推理。这些模型说明了提示是否是向量空间的子集，是否包括零向量，向量的结构，它们在线性方程组的一般解公式中的编号以及相应的系数矩阵。此外，在学生的回答中确定了一个概念，根据该概念向量空间的代数结构从“父”空间传递到其子集，并自动将其转变为子空间。对于许多学生而言，这种继承结构的概念有助于识别和推理子空间。前缀“ sub”的多义性和学生在识别概念示例中的先验经验可用于为概念的出现提供解释。