the row with 6 matches was longer because it had more matches. Thus, in measurement, there are situations that differ from the discrete cardinal situations. For example, when measuring with a ruler, the order-irrelevance principle does not apply and every element (e.g., each unit on a ruler) should not necessarily be counted (Fuson and Hall, 1982).
Understanding of area measurement involves learning and coordinating many ideas (Clements and Stephan, 2004). Most of these ideas, such as transitivity, the relation between number and measurement, and unit iteration, operate in area measurement in a manner similar to length measurement. Two additional foundational concepts will be briefly described.
Understanding of the attribute of area involves giving a quantitative meaning to the amount of bounded two-dimensional surfaces.
Equal partitioning is the mental act of cutting two-dimensional space into parts, with equal partitioning requiring parts of equal area (usually congruent).
Spatial structuring. Children need to structure an array to understand area as truly two-dimensional. Spatial structuring is the mental operation of constructing an organization or form for an object or set of objects in space, a form of abstraction, the process of selecting, coordinating, unifying, and registering in memory a set of mental objects and actions. Based on Piaget and Inhelder’s (1967) original formulation of coordinating dimensions, spatial structuring takes previously abstracted items as content and integrates them to form new structures. It creates stable patterns of mental actions that an individual uses to link sensory experiences, rather than the sensory input of the experiences themselves. Such spatial structuring precedes meaningful mathematical use of the structures, such as determining area or volume (Battista and Clements, 1996; Battista et al., 1998; Outhred and Mitchelmore, 1992). That is, children can be taught to multiply linear dimensions, but conceptual development demands this build on multiplicative thinking, which can develop first based on, for example, their thinking about a number of square units in a row times the number of rows (Nunes, Light, and Mason, 1993; note that children were less successful using rulers than square tiles).