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.

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.