Abstract The purpose of this study was to characterize the growth of one student’s mathematical understanding and use of different representations about a geometric transformation, dilation. We accomplished this purpose by using the Pirie-Kieren model jointly with the Semiotic Representation Theory as a lens. Elif, a 10th- grade student, was purposefully chosen as the case for this study because of the growth of mathematical understanding about dilation she exhibited over time. Elif participated in task-based interviews before, during and after participating in a variety of transformation lessons where she used multiple representations, including physical and virtual manipulatives. The results revealed that Elif was able to progress in her mathematical understanding from informal levels to the formal levels in the Pirie-Kieren model as she performed treatments and conversions, movements involving different registers of representations. The results also showed numerous examples of Elif’s mathematical understanding based on folding back activities, complementary aspects of acting and expressing, and interventions. Using the two theories together provides a powerful and holistic approach to a deeper understanding of mathematical learning by characterizing and articulating the growth of mathematical understanding and the way of mathematical thinking.

中文翻译：

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表征一个学生在多表现学习环境中数学理解的增长

摘要 本研究的目的是描述一个学生对几何变换、膨胀的不同表示的数学理解和使用的增长。我们通过使用 Pirie-Kieren 模型和符号表示理论作为镜头来实现这一目的。Elif 是一名 10 年级学生，她被有意选择作为本研究的案例，因为她对扩张的数学理解随着时间的推移而增长。在参加各种转换课程之前、期间和之后，Elif 参加了基于任务的访谈，在这些课程中她使用了多种表示，包括物理和虚拟操作。结果表明，当 Elif 进行处理和转换，以及涉及不同表征记录的运动时，她的数学理解能够从非正式级别提升到 Pirie-Kieren 模型中的正式级别。结果还显示了 Elif 基于折叠活动、表演和表达的互补方面以及干预的数学理解的许多例子。将这两种理论结合使用，通过表征和阐明数学理解的增长和数学思维方式，为更深入地理解数学学习提供了一种强大而全面的方法。结果还显示了 Elif 基于折叠活动、表演和表达的互补方面以及干预的数学理解的许多例子。将这两种理论结合使用，通过表征和阐明数学理解的增长和数学思维方式，为更深入地理解数学学习提供了一种强大而全面的方法。结果还显示了 Elif 基于折叠活动、表演和表达的互补方面以及干预的数学理解的许多例子。将这两种理论结合使用，通过表征和阐明数学理解的增长和数学思维方式，为更深入地理解数学学习提供了一种强大而全面的方法。