ABSTRACT

Design research was conducted to investigate the development of dynamic conceptions of area. Students used simple physical instruments, squeegees, and paint, to generate rectangular areas continuously as a sweeping motion of one length through another. Then, they dissected the resulting swept space into discrete units of measure by coordinating measures of the lengths of the sides, including side lengths measured in different units, such as “squeegee” and inch. Analysis of classroom discourse, formative assessments, and flexible interviews conducted 2 months after instruction illustrated that students developed dynamic images and meanings of area and its measure. These expanded their understanding of multiplicative product to include referent-transforming quantity and provided means for re-envisioning the meaning of the commutative and distributive properties of multiplication. Students employed these properties to relate fractional and whole number area products. Dynamic generation and dissection in tandem helped students bridge discrete and continuous conceptions of area and its measure.

摘要

进行了设计研究以调查区域动态概念的发展。学生使用简单的物理仪器，橡皮刮刀和油漆，以一种长度的长度连续掠过另一区域的方式连续产生矩形区域。然后，他们通过协调侧面长度的度量（包括以不同单位（例如“吸水扒”和英寸）测量的侧面长度）将生成的扫掠空间分解为离散的度量单位。在授课后两个月对课堂话语，形成性评估和灵活的访谈进行的分析表明，学生发展出了动态图像以及区域及其度量的含义。这些扩展了他们对乘法乘积的理解，包括了指代转换量，并提供了重新设想乘法的交换性和分布性含义的手段。学生利用这些属性来关联小数和整数区域的乘积。串联动态生成和分解有助于学生桥接离散的和连续的面积及其度量概念。