Counting sequences and grouping

Students' arithmetical learning develops through two complementary processes: counting and grouping. Just as counting describes more than generating a sequence of words, grouping captures a range of procedures. The formation of units as well as combining and separating are examples of grouping. Grouping utilises conceptual subitising and recognition of part-part-whole.

In answering 3 + 9, a student who forms 10 saying, "1 plus 9 is 10 and I have 2 more making 12", uses grouping to answer the question. In particular, the student:

• regroups (partitions) the 3 into 1 and 2
• combines the 1 and 9 to form a group (unit) of 10
• uses the unit structure of place value to combine 10 and 2.
  

Although it is possible to answer this question by counting, the use of regrouping is more efficient.

Counting sequences and grouping develop together. Grouping, as a process of "chunking" information, increases in importance as arithmetical strategies increase in sophistication. The counting sequences themselves become nested within numbers.

A student who counts down to 8 from 11 recognises that the counting sequence to 8 is grouped within 11.

Forming and coordinating groups is the basis of multiplication and division as well as place-value. The analogy of using "twin dials" has been used (Janet Bowers) to describe this process with base ten questions. For example, when asked how many groups of ten are contained in a collection of 360, some students use one process of counting by tens (one dial), and a second of keeping track of the number of counting acts (a second dial).

The analogy can be extended to be equally applicable to other group sizes and double counting in general. Forming the counting units on one dial (click on the "count" dial), sequencing the count on the second dial (click on the "groups" dial) and coordinating the two dials (click on the "coordinates" dial) describes the processes underpinning a range of strategies beyond counting by ones.

Viewing number and number patterns as units of units also relies on imagery. These processes are not restricted to number but apply also to measurement. Forming units and units of units describes students' increasing competence in measuring length, area and volume.