将矩阵乘法视为变换：一种使用GeoGebra的中学数学教育任务设计
Visualizing the Matrix Product as a Transformation: A Task Design Using GeoGebra in Secondary Mathematics Education
AI总结：
本研究通过GeoGebra结合手动活动和动态可视化，帮助学生从算法执行转向理解矩阵乘法作为线性变换的概念。
摘要：
在中学教育中，矩阵乘法的教学往往局限于行乘列算法的机械应用，忽略了其作为几何变换的解释。本研究分析了基于数学工作空间（MWS）框架的GeoGebra介导的教学序列对学习矩阵乘法的影响。十名来自秘鲁利马中学的五年级学生参与了本研究。干预过程持续四次会话，结合了手动活动和使用GeoGebra的数字探索。结果显示，学生在符号生成方面有显著进步，体现在代数、图形和数值表示的协调；在工具生成方面，通过GeoGebra作为认知工具的日益有意义的使用；在言语生成方面，学生发展了对矩阵几何效应的解释。观察到从矩阵乘法的算法执行向基于线性变换的概念理解的转变。这些发现表明，结合手动工作、动态可视化和数学论证的任务设计支持对矩阵乘法的深入理解，并为反思使用数字技术进行中学线性代数教学提供了标准。
英文摘要：
The teaching of matrix multiplication in secondary education is often limited to the mechanical application of the row-by-column algorithm, leaving aside its interpretation as a geometric transformation. This study analyzes the impact of a GeoGebra-mediated instructional sequence, grounded in the Mathematical Working Space (MWS) framework, on students learning of the matrix product. Ten fifth-year secondary students from a school in Lima (Peru) participated in the study. The intervention was carried out over four sessions, combining manual activities with digital exploration using GeoGebra. The results show notable progress in students semiotic genesis, reflected in the coordination of algebraic, graphical, and numerical representations; in instrumental genesis, through the increasingly meaningful use of GeoGebra as a cognitive tool; and in discursive genesis, as students developed explanations of the geometric effects of matrices. A transition is observed from an algorithmic execution of matrix multiplication toward a conceptual understanding based on linear transformations. These findings suggest that task designs integrating manual work, dynamic visualization, and mathematical argumentation support deeper understanding of matrix multiplication and provide criteria for the reflective use of digital technologies in secondary linear algebra instruction.
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