Level 2: Perceptual multiples

The student develops basic methods of counting multiples before being able to construct composite units as counting units (iterable units). Equal groups are modelled and counted using rhythmic, skip or perceptual double counting. The student counts by ones the numberof equal groups and the number of items in each group at the same time, only if the itemsare visible.

Rhythmic counting and skip counting are two procedures that are common at this stage. Perceptual rhythmic counting occurs when a student models equal-sized groups and counts perceived items by ones, following a pattern with emphasis on rhythm e.g. “1, 2; 3, 4; 5, 6;” without any obvious reference to the equal groups. Perceptual skip counting occurs when a student models equal-sized groups and counts groups of items following a pattern of multiples e.g. “3, 6, 9, 12” but this may be simply a short-cut method of counting by ones. A student enacts a skip counting pattern but does not recognise the overall picture of the pattern made up of composite units. The student cannot operate with concealed collections or patterns.

Perceptual multiples

Somewhat less common is the procedure of perceptual double counting. Perceptual double counting occurs when a student counts perceptually or by rhythm by ones with a simultaneous second count of the number of groups used e.g. “1, 2, 3, (1); 4, 5, 6, (2); 7, 8, 9 (3)“ (numbers in brackets emphasise the second count of the number of groups). At this stage, it is individual items, rather than composites, that are counted. The groups are individually and repeatedly constructed prior to counting.

When items are presented in a way that supports the composite structure, students can use the structure to assist in determining the total. The student cannot operate with concealed collections or patterns.