In this paper, we identify various forms of geometric work carried out by student teachers who were asked to perform a geometric task for the estimation of a land area. The theory of Mathematical Working Spaces is used to analyze and characterize the work produced. This study provides evidence that students developed forms of geometric work that are compliant with at least two distinct geometric paradigms, one characterized by the utilization of measuring and drawing tools and the other by a property-based discourse on proof. Significantly, a sizable number of students also developed work forms that do not correspond to any geometrical paradigm. A broader purpose of this paper is to highlight three criteria born by the theory and shown to be useful for the description and evaluation of geometric work: compliance, completeness, and correctness.

中文翻译：

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几何作业的形式：基于数学工作空间理论的职前教师研究

在本文中，我们确定了被要求执行几何任务以估计土地面积的学生教师进行的各种形式的几何工作。数学工作空间理论用于分析和表征所产生的工作。这项研究提供的证据表明，学生开发的几何作品形式至少符合两种不同的几何范式，一种以使用测量和绘图工具为特征，另一种以基于属性的论证论述为特征。值得注意的是，相当多的学生还开发了不符合任何几何范式的工作形式。本文的一个更广泛的目的是强调该理论产生的三个标准，并证明对几何工作的描述和评估很有用：合规性、完整性和正确性。