Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring, Enumeration and Errors in Concretely-Presented 3D Array Tasks

Abstract

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Exploring units-locating in enumerating units of 3D arrays: linking units-locating to units-representation

“…Double and triple counting is a common erroneous enumeration strategy that is well documented in the research literature (Finesilver, 2015(Finesilver, , 2017Kara et al, 2012;Vasilyeva et al, 2013). In the present study, we not only identify double and triple counting, but we also observe how double and triple counting depends on students' units-locating processes and, particularly, their difficulties in locating basic units when they are made up of cubes.…”

“…They found that students who experienced difficulties in enumerating cubes in the array were most often using one of the four following erroneous counting strategies: counting the number of faces visible, counting the number of faces visible and multiplying the result by two, counting the number of visible cubes or counting the number of visible cubes and multiplying the counted number by two. More recent studies have confirmed these findings (Finesilver, 2015(Finesilver, , 2017Kara et al, 2012;Vasilyeva et al, 2013). Ben-Haim et al (1985) suggested that students who counted only the visible faces interpreted the representation strictly as 2D, while students who counted the visible cubes showed more awareness of the three-dimensionality of the array since they were able to discern the unit as a 3D object.…”

“…Several studies have found that students might have difficulties enumerating the cubical units of the arrays both when the arrays are represented as 2D and when working with a physical 3D representation of an array (Battista, 1999;Battista & Clements, 1996;Ben-Haim Fig. 1 Student task from Ben-Haim et al (1985) et al., 1985;Finesilver, 2015Finesilver, , 2017Vasilyeva et al, 2013). Finesilver (2015Finesilver ( , 2017 found that students who tended to count only faces in enumeration tasks, i.e.…”

“…1 Student task from Ben-Haim et al (1985) et al., 1985;Finesilver, 2015Finesilver, , 2017Vasilyeva et al, 2013). Finesilver (2015Finesilver ( , 2017 found that students who tended to count only faces in enumeration tasks, i.e. ignored cubes on the inside and back of the array, moved toward a more layer-based thinking when the arrays came in different coloured layers.…”

“…The ability to enumerate unit-cubes in 3D arrays is crucial for understanding volume and the volume formula (Battista & Clements, 1996;Smith & Barrett, 2017). However, enumerating unit-cubes in a 3D array has shown not to be an easy task for many students (Battista, 1999;Battista & Clements, 1996;Ben-Haim et al, 1985;Finesilver, 2015Finesilver, , 2017Vasilyeva et al, 2013). The students need to interpret the array as a space-filling object, including units inside the array (Finesilver, 2015).…”