The Learning Framework In Number:

Multiplication & Division

1

Chris Francis

State Numeracy Programs Conference 20102

The Multiplication and division aspect of the learning framework underpins:

Count Me In TooCounting OnTaking Off With Numeracy

State Numeracy Programs Conference 20103

Structure• Early multiplication and division strategies focus on

the structure and use of groups of things.

• A composite unit is formed when a student takes a group of items and treats it as a unit.

• Recognising units is not sufficient. It is necessary to develop ways of coordinating the groups that are formed.

3

State Numeracy Programs Conference 20104

Typical progression

4

State Numeracy Programs Conference 20105

Level 1: Forming equal groups

The student:• uses perceptual counting and sharing to form equal

groups• finds the total of the groups, counting by ones• does not take notice of the structure of the groups

when finding the total.NES1.3PAES1.1

5

State Numeracy Programs Conference 20106

Level 2: Perceptual multiples

The student :• uses items to form equal groups• finds the total of the groups through rhythmic, skip

or double counting, only when the items are visible• cannot deal with concealed items.NS1.3PAS1.1

6

State Numeracy Programs Conference 20107

Level 3: Figurative units

The student:• does not rely on items being visible• counts forwards or backwards using multiples (or a

combination of multiples and rhythmic counting )• uses perceptual markers (e.g. fingers) to keep track

of the groups• reconstructs the groups before counting.NS1.3PAS1.1

7

State Numeracy Programs Conference 20108

Level 4: Repeated abstract composite unitThe student:• uses repeated addition or subtraction to find the

total• does not need perceptual markers to represent the

groups• might use fingers to keep track as they count

• might use skip counting or a double count.NS1.3, NS2.3PAS1.1, PAS2.1

8

State Numeracy Programs Conference 20109

Level 5: Multiplication & division as operations

The student:• recalls a wide range of multiplication and division

facts• uses multiplication and division as inverse operations

to solve problems• explains the composite structure in a range of

contexts; not simply recalling facts.PAS2.1

NS2.3

9

State Numeracy Programs Conference 201011

Observations

• Students are usually confident with some multiples over others.

• Strategy use is strongly dependent on knowledge of multiples.

• Students may use a combination of strategies.• Students typically appear more confident with

multiplication than division.• Modeling of strategies is needed.• The use of arrays is useful in developing visualisation.

11