Abstract
The aim of the present study is to explore strategies in enumerating units of three dimensional (3D) arrays. We analyse enumeration strategies of students in grade 3 (ages 8 to 9) in situations of cubical and spherical representations of units of 3D arrays. By exploring students’ strategies in these two situations, we find that difficulties in enumerating units in 3D arrays can be traced to difficulties in units-locating, with the consequence of applying double and triple counting. Our results also indicate that spherical units can serve as perceptual clues in units-locating and in assembling units into relevant composites. With input from our findings, we suggest research to investigate the following three hypotheses: (i) spherical units can turn students away from double and triple counting, (ii) spherical units can support students’ units-locating process and their ability to assemble units into relevant composites and (iii) teaching of enumerating 3D arrays should start with spherical units before cubical units.
Highlights
The ability to enumerate unit-cubes in 3D arrays is crucial for understanding volume and the volume formula (Battista & Clements, 1996; Smith & Barrett, 2017)
By exploring students’ strategies in these two situations, we find that difficulties in enumerating units in 3D arrays can be traced to difficulties in units-locating, with the consequence of applying double and triple counting
With input from our findings, we suggest research to investigate the following three hypotheses: (i) spherical units can turn students away from double and triple counting, (ii) spherical units can support students’ units-locating process and their ability to assemble units into relevant composites and (iii) teaching of enumerating 3D arrays should start with spherical units before cubical units
Summary
The ability to enumerate unit-cubes in 3D arrays is crucial for understanding volume and the volume formula (Battista & Clements, 1996; Smith & Barrett, 2017). Limited spatial reasoning skills can lead students to count only what is visible from the drawer’s point of view, not taking the non-visible cubes into account (Battista & Clements, 1996; Ben-Haim et al, 1985; Vasilyeva et al, 2013). This may result in a double and triple counting of unitcubes located at the edges and corners of the array. The discussion will provide a background for our analytical approach
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