This study reports the classroom mathematical practices proposed in an instructional sequence grounding on the principles of inquiry-based learning (IBL) through the use of the dynamic geometry software (DGS) in learning to teach geometric transformations. A 5-week instructional sequence consisting of IBL activities that were guided by the hypothetical learning trajectory (HLT) was designed. The participant 23 preservice middle-school mathematics teachers (PMSMTs) used the geometer’s sketchpad in the sequence. The qualitative data obtained from whole-class discussions, classroom artifacts, and teacher candidates’ products were analyzed by the three-phase methodology of Rasmussen and Stephan based on the Toulmin argumentation model. The findings revealed that three mathematical practices emerged at the end of the 5-week instructional sequence: (1) an in-depth understanding of geometric transformations, (2) operational definitions of geometric transformations and the identification of the relationship among them, and (3) the properties and attributes of geometric transformations. The development of the PMSMTs’ understanding of geometric transformations in the instructional sequence was explained by the systematic nature of classroom mathematical practices. Hence, it was concluded that the PMSMTs’ knowledge and understanding of transformation geometry were enhanced through the instructional sequence based on the specific HLT.

中文翻译：

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基于查询的设计研究，通过在动态几何环境中开发数学实践来教授几何变换

这项研究报告了通过基于动态探究学习（IBL）原理的教学顺序，通过使用动态几何软件（DGS）进行几何变换教学，提出了课堂数学实践。设计了一个为期5周的教学序列，其中包括由假设学习轨迹（HLT）指导的IBL活动。参与者23名职前中学数学老师（PMSMT）按顺序使用了几何图形的画板。基于Toulmin论证模型，通过Rasmussen和Stephan的三相方法，分析了从全班讨论，课堂文物和教师应聘者的产品中获得的定性数据。研究结果表明，在为期5周的教学序列结束时出现了三种数学实践：（1）对几何变换的深入理解；（2）几何变换的操作定义以及它们之间的关系的识别；（3）几何变换的属性和属性。课堂数学实践的系统性解释了PMSMT对教学顺序中的几何变换的理解的发展。因此，可以得出结论，通过基于特定HLT的教学顺序，可以增强PMSMT对变换几何的知识和理解。课堂数学实践的系统性解释了PMSMT对教学顺序中的几何变换的理解的发展。因此，可以得出结论，通过基于特定HLT的教学顺序，可以增强PMSMT对变换几何的知识和理解。课堂数学实践的系统性解释了PMSMT对教学顺序中的几何变换的理解的发展。因此，可以得出结论，通过基于特定HLT的教学顺序，可以增强PMSMT对变换几何的知识和理解。