Reasoning Dynamically About the Area of a Rectangle: The Case of Lora and Isaac
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Reasoning Dynamically About the Area of a Rectangle: The Case of Lora and Isaac Nicole Panorkou 1 # Springer Nature Switzerland AG 2020
Abstract This article reports on an exploratory study that engaged students in dynamic experiences of generating the area of a rectangle as a sweep of a line segment over a distance. A case study from a design experiment with one pair of third-grade students is presented to initiate a discussion around the forms of reasoning that students may exhibit as a result of their engagement with these dynamic motion tasks and the characteristics of the design that supported these particular forms of reasoning. The findings of this study show that engaging students in dynamic experiences of area may help them develop a conceptual understanding of the area of a rectangle as a continuous structure that can dynamically change based on the two linear measures that generate it: the length of the line segment swept and the distance of the sweep. These experiences can also help students from an early age develop a flexible understanding of a unit and reason covariationally about the continuous change of the quantities in measurement. Keywords Area measurement . Technology . Student reasoning . Design experiment .
Dynamic measurement . Dynamic geometry software . Multiplicative coordination . Covariational reasoning . Mathematics of motion . Mathematics of variation . Area as sweeping Measurement is usually defined as assigning a number to a continuous quantity (Clements & Stephan, 2004). In terms of area measurement, several studies have focused on using square units to cover surfaces and quantifying that covering based on the number of square units needed to cover the surface (e.g. Barrett & Clements, 2003; Barrett, Clements et al., 2017; Barrett, Cullen et al., 2017; Battista et al., 1998; Clements & Sarama, 2009; Izsák, 2005; Kamii & Kysh, 2006). For instance, one can
* Nicole Panorkou [email protected]
1
Department of Mathematics, Montclair State University, 1 Normal Avenue, Montclair, NJ 07030, USA
Digital Experiences in Mathematics Education
use the fact that 20 1-in.2 tiles cover the 5-in. by 4-in. rectangle that appears in Fig. 1a to make the claim that its area is 20 in.2 Several studies have organized the strategies young students use to solve rectangular covering tasks into developmental levels (e.g. Barrett, Cullen et al., 2017; Battista et al., 1998; Clements & Sarama, 2009; Clements & Stephan, 2004; Outhred & Mitchelmore, 2000). These researchers found that early concepts of area measurement begin with identifying area as the space occupied by a two-dimensional shape, then move to direct comparison of areas by placing two shapes next to each other or on top of each other, and then to indirect comparisons of areas using external or internal units (Fig. 1a & b). Students incrementally progress from a primitive covering to a spatial structuring of area, in which they demonstrate an understanding of how a surface can be tiled with squares arranged in an array of rows and col
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