# Background Information on the Learning Sequence for Number

The Learning Sequence for Number describes key steps in students’ thinking as they develop and use increasingly sophisticated number concepts. We believe that there are five key steps in the learning sequence, which we have summarized below.

**Counting
from one**

Students at this stage
are able to count a set of objects by counting from one.
To do this they need to understand that each object has a single
count. They also need to know
the number names in sequence.

Students at this stage typically solve addition problems by physically representing each quantity (set) in a problem, then combining the sets and finally counting the combined set from one.

**Counting-on**

Students at this more
advanced counting stage essentially recognise that it is not necessary to
actually construct and count sets. They
now use the counting sequence itself to solve problems.
Students at the counting-on stage now appreciate the total count of
a set and use this to solve addition or subtraction problems.
Instead of counting the objects from one, the students begin their
count from the cardinal number of one of the sets.
They can then count-on or count-back from this number.

The students at this stage are still essentially seeing and using numbers as collections of “ones” but they also know that a set of objects can be represented by a single count.

**Additive Strategies
**Students at this stage
have moved from seeing and using numbers as collections of “ones” to
treating numbers as abstract ideas or units.
When they have an “abstract” idea of a number they can treat it
as a “whole” or can partition it and then recombine it to solve
addition or subtraction problems.

Students are able to use addition or subtraction number facts that they know to derive answers to unknown problems. Described below are the three additive strategies that students use:

- Compensation

For example, if they know that 6 + 6 = 12 they may use this to derive 6 + 7 = (6 + 6) + 1 = 13. This same strategy underpins the renaming of 74 – 19 as 74 – 20 + 1 to find the answer to 74 – 19. - Place value partitioning

Breaking or partitioning numbers so that they can be recombined to form “tens” is another additive strategy.

For example, 18 + 6 = (18 + 2) + 4 = 20 + 4. - Inverse Operations

This involves using known addition/subtraction facts to derive the opposite subtraction/addition fact.

For example, 62 – 34 = ?? can be reworked as 34 + ?? = 62 and 34 + (30 – 2) = 62.

Students at this stage solve multiplication problems through repeated addition. For example, 4 x 5 = 5 + 5 + 5 + 5.

**Multiplicative
Strategies
**Students who are
multiplicative in their number strategies can solve problems by choosing
appropriately from a wide range of strategies that involve multiplication
and division facts. This
requires that the student can treat multiplicative units as a whole or as
able to be divided into smaller multiplicative units.
Essentially they use their known multiplication and division facts
to derive answers to unknown multiplication or division problems.

There are four main multiplicative strategies:

- Compensation

For example, 14 x 7 = (15 – 1) x 7 = 15 x 7 – 7. - The associative property

For example, 22 x 4 = 22 x 2 x 2. - The distributive property

For example, 45 x 6 = (40 + 5) x 6 = (40 x 6) + (5 x 6) - Inverse Operations

For example, 66 ÷ 3 can be found by first realising that 20 x 3 = 60 and 2 x 3 = 6 so 66 ÷ 3 = 22.

**Proportional Strategies
**Students at this stage
use their knowledge of multiplication and division to derive answers to
problems that involve proportions, ratios, percentages and advanced
fractional ideas.