Students' understanding of three-dimensional rectangular arrays of cubes

The present study extends previous research in this area by providing a more elaborate and theoretical description of students' solution strategies and errors in dealing with 3-D cube arrays. It describes several cognitive constructions and operations that seem to be required for students to conceptualize and enumerate the cubes in such arrays, exploring in depth general cognitive operations such as coordination, integration, and "structuring" as they are manifested in a spatial context. It describes how, in dealing with 3-D rectangular arrays, students' spatial thinking is related to their enumeration strategies. The findings suggest that students' initial conception of a 3-D rectangular array of cubes is as an uncoordinated set of faces. Eventually, as students become capable of coordinating views, they see the array as space filling and strive to restructure it as such. Those who complete a global restructuring of the array use layering strategies. Those in transition use strategies that indicate that their restructuring is local rather than global. Finally, the data suggest that many students are unable to enumerate the cubes in a 3-D array because they cannot coordinate the separate views of the array and integrate them to construct one coherent mental model of the array.