Rethinking the Teaching and Learning of Area Measurement

This study focused on exploring an innovative way of teaching and learning measurement, what we refer to as Dynamic Measurement or DYME. Without relying on the common approach of counting square units, our goal was to engage students in contextually rich digital dynamic tasks to visualize area as a continuous quantity and evaluate the area of a rectangular region as a multiplicative relationship between the two lengths of the sides. In this paper, we briefly describe the iterative process of designing, testing and refining the tasks for DYME pointing to the significance of the design for developing students’ thinking of area as length times width. Geometric measurement: What we know and pushing forward Extensive research on measurement has described the importance of using square units to cover rectangular surfaces and quantify that covering by counting the square units (e.g. Barrett & Clements, 2003; Battista, Clements, Arnoff, Battista & Borrow, 1998; Izsak, 2005; Kamii & Kysh, 2006). For instance, we can use twenty 1 sq. inch tiles to cover a 5 by 4 inches rectangle as in Figure 1a and claim that its area is 20 sq. inches. Similar to these studies, the Common Core State Standards for Mathematics (CCSSO, 2010) in the United States introduce third grade students to area measurement first by counting unit squares in a rectangular surface, thus forming an array (Content standards 3.MD.C.5 and 3.MD.C.6). Next, the standards assume that students will use this tiling experience to “show that the area is the same as would be found by multiplying the side lengths” (3.MD.C.7.A). However, the standards do not provide information on how students will transition from counting individual units to constructing the multiplicative area formula. Figure 1. Progression of structuring area based on a synthesis of the measurement literature. Although the measurement research studies mentioned above suggest a progression of structuring area (Figure 1 a-d), to understand how area is generated by multiplying lengths is a different notion conceptually from the construction of a matrix like shown in Figure 1d. As Piaget et al. (1960) argued, “the difference between the two operational mechanisms is the difference between a matrix which is made up of a limited number of elements and one which is thought of as a continuous structure with an infinite number of elements” (p. 350). Indeed, area, length and width are continuous quantities (Kamii & Kysh, 2006) that are related multiplicatively while covering a surface with discrete unit squares is one-dimensional and additive in nature (e.g. Outhred & Mitchelmore, 2000; Reynolds & Wheatley, 1996). As a result, this study aimed to go beyond the static perspective of understanding area as the counting of discrete square units and find a more intuitive and accessible approach of illustrating area as a continuous quantity that involves a multiplicative relationship between length and width. To do that, we built on the work of Confrey et al. (2012) and Lehrer, Slovin, Dougherty, & Zbiek (2014) on visualizing area as a ‘sweep’ of a line segment of length a over a distance of b to produce a rectangle of area ab. Figure 2. Visualizing area as a continuous structure through ‘sweeping. ’ ICLS 2018 Proceedings 863 © ISLS For instance, imagine a paint roller with length 5 inches sweeping for a distance of 6 inches and generating a surface of 30 square inches (Figure 2). In this approach, which we refer to as Dynamic Measurement or DYME, area can be visualized as a continuous dynamic quantity which depends on both the length of the roller (length) and the distance of the swipe (width). DYME involves engaging students in dynamic experiences of generating 2D surfaces and 3D shapes by iteratively (and multiplicatively) composing lowerdimensional objects (linear measures).