The connectionist nature of the teaching taking place is underlined by Carl's comment: "We're teaching arrays as representations of the

commutative property and it needs to have a whole focus to understand the connections".

While the

Commutative Property provides the context for this activity, the activity is about far more than having students understand the property, since students probably already understand it intuitively.

The children had already learned the

commutative property of multiplication (e.

Finally, explanations that were based on the

commutative property of multiplication were considered MB explanations.

Doubles (CGI, KR) * Doubles plus 1 (CGI, KR) *

Commutative property of addition * Different forms of numbers (Ma) * (Y1) Count collections to 100 by partitioning numbers using place value (ACMNA014) * (Y2) Group, partition, and rearrange collections .

These two equations suggested that Sandy had already learned the

commutative property of multiplication, a x b = b x a.

This conversation gets to the heart of the

commutative property as something to consider rather than a definition to memorise (Warren, 2003).

Soon another teacher announced that the same generalization applied to even-numbered rows; the pattern emerged when the

commutative property of multiplication was applied.

The authors explain how arrays can be used to illustrate both the

commutative property (e.

Learners apply the

commutative property when they compute 3 + 14 by counting on from 14 rather than 3, and 23 x 2 by doubling 23 rather than adding 2 twenty-three times.

It is widely noted that the array can simply be rotated by ninety degrees to represent the

commutative property (e.

75; the potential insight of a student who is thinking about the

commutative property will be frustrated by this approach to error correction.