In dealing with tasks that make use of an array-structure, such as determining the number of square units needed to cover a rectangular area, students need to use spatial understanding combined with multiplicative strategies. The required spatial understanding for area multiplication tasks involves mentally organising or constructing the units (in this case squares) in order to cover the rectangle. Initially, little attention is given to where the squares should be positioned on the rectangle and the student simply follows a meandering path until it appears the area might be covered. No attention seems to be directed to gaps or overlaps that occur while reconstructing the units.
At a more sophisticated level, the student is able to structure the square units in specific groups but may be unable to coordinate the structure of groups in two directions. At a higher level of understanding, the student is able to structure the units in rows, however the rows would be iterated incorrectly when covering the rectangle. At the highest level of understanding, the student follows a grid structure and is able to visualise a column and row structure (Battista, Clements, Arnoff, Battista, & Borrow, 1998).
Coordinating multiplication with an array
In determining the number of squares needed to cover the area, students’ multiplicative strategies move from counting by ones, to skip counting (often supported by the use of finger strategies) and finally to coordinating groups of groups, e.g. 7 rows of 3 (Outhred & McPhail, 2000). However, the link between spatial structure and multiplication is not automatic and needs to be established.