From spatial structure to numerical units
The introduction of structure into spatial patterns can assist in interpreting groups of groups, such as the column and row structure of arrays. This starts with developing a sense of the structure of how units are repeated in one direction. Then aligning the structure of units presented in rows can be built upon to develop into an awareness of structure in two directions: rows and columns. This in turn assists in seeing the rows and columns of 2-dimensional arrays as composite units. A composite unit is formed when a group of items is treated spatially or numerically as a unit rather than emphasising the individual items in the group. Simply recognising units is not sufficient and students need to develop ways of coordinating the groups that are formed. (See Level 5, Multiplication and division as operations in Aspect 5).
Units of units in one dimension can also be seen in additive part-whole recognition of number (Figure 1).
Figure 1. A unit of five with a unit of three making eight in total
That is, interpreting number in terms of part-whole additive relationships makes it possible for children to think about a number as being composed of other numbers. The idea of forming a unit made up of smaller pieces is also of fundamental importance in measurement and in multiplication (Figure 2). For both multiplication and measurement the use of “structure” with the units is essential.
Figure 2. Increasing by groups of three