Numeracy Stages Stage 0: Emergent
Description of Stage
The student has no reliable strategy to one-to-one count an unstructured collection of
items up to ten.
Observable Behaviours
Students are Emergent if they exhibit any of these behaviours:
•
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fail to start the count at one
count in the wrong order
•
leave out any numbers in the count
•
fail to count one-to-one
Stage 1: One-to-One Counting
Description of Stage
The students can reliably count one-to-one an unstructured collection of items up to
twenty.
Observable Behaviours
Students are at One-to-One Counting if they exhibit both of these behaviours:
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can count one-to-one reliably up to 20
•
cannot solve addition and subtraction problems even with the use of material
Stage 2: Counting from One on Materials
Description of Stage
The students' most advanced addition strategy is counting from one on materials.
Observable Behaviours
Students are at Counting from One on Materials if they exhibit both of these
behaviours:
•
•
choose to model addition and subtraction problems on materials
count the materials one-to-one
Stage 3: Counting from One by Imaging
Description of Stage
The students' most advanced addition strategy is counting from one without the use of
materials.
Observable Behaviours
Students are at Counting from One by Imaging if they exhibit this behaviour:
•
solve addition and subtraction problems by counting mentally one-to-one.
St Cuthbert’s College 2011 1 Numeracy Stages Stage 4: Advanced Counting
Description of Stage
The students’ most advanced addition or subtraction strategy is counting-on or
counting-down.
Observable Behaviours
Students are at Advanced Counting if they exhibit both of these behaviours for
addition or subtraction problems:
•
•
begin with the larger of the two numbers
for addition
counts-on or counts down
Stage 5: Early Part-Whole Addition and Subtraction
Description of Stage
The students use simple part-whole strategies to solve addition or subtraction
problems without counting. Solutions involve splitting numbers into parts, and
recombining the parts to form new wholes. These are the main problem types:
Type 1
One number involved in the problem or the answer is a single digit. And, at some
time in the
calculation a tidy number - a number ending in a zero - is used. For example, 45 - 8
might involve:
Step 1: 45 - 5 = 40,
Step 2: 40 - 3 = 37.
Here the single digit involved is 8 and the tidy number is 40.
Type 2
One number is a "super-tidy number" - a number ending in two or more zeroes. The
calculation must be simple. For example,
998 + 404 might involve:
Step 1: 998 + 2 = 1000. Step 2: 404 - 2 = 402 Step 3: 1000 + 402 = 1402
Type 3
Addition or subtraction problems with multi-digit numbers that do not require
renaming "ten for one". For example,
567 - 401 or 23 + 643 are early part-whole examples, but 56 - 37 or 567 + 78 are not.
Type 4
Multiplication problems are solved by addition. For example, 4 x 5 might be solved
by (5 + 5) + (5 + 5) = 10 + 10 = 20
Observable Behaviours
Students are at Early Part-Whole Addition or Subtraction if they exhibit all of
these behaviours to solve early part-whole addition or subtraction problems:
•
do not count
•
use instantly-recalled known basic addition or subtraction facts - list which ones
St Cuthbert’s College 2011 2 Numeracy Stages have been used
•
mentally split wholes into parts and/or recombine parts and wholes
Stage 6:Advanced Part-Whole Addition and Subtraction
Description of Stage
The students use part-whole strategies to solve harder addition or subtraction
problems; they require more steps than early part-whole problems. All numbers are
two or more digits. A problem like 81 - ? = 38 that is solvable by a variety of
methods is Advanced Additive because, in addition to needing all the thinking in
early additive, needs more steps, thereby that placing a significantly heavier load on
students’ thinking. Whereas, a multi-digit problem like 653 – 241 is not considered
Advanced Additive since it is simple – it does not require any regrouping.
Observable Behaviours
The difference between Early and Advanced Addition or Subtraction stages is the size
of the numbers and the extra steps needed. So the first three steps below are identical.
Students are at Advanced Part-Whole Addition or Subtraction if they exhibit all of
these behaviours to harder part-whole addition and subtraction problems.
•
do not count
•
use instantly-recalled known basic addition or subtraction facts - list which ones
have been
used
•
mentally split wholes into parts
•
use more mental steps than needed in early part – whole thinking with large
numbers
Stage 7: Advanced Part-Whole Multiplication and Division
Description of Stage
The students solve multi-digit multiplication and division problems with whole numbers
mentally. There are two main problem types:
Type 1
These involve the distributive law. For example, 76 ÷ 4 might be solved by:
Step 1: 80 ÷ 4 = 20
Step 2: 4 ÷ 4 = 1
Step 3: 20 – 1 = 19
This uses the distributive law:
76 ÷ 4 = (80 = 4) ÷ 4 = 80 ÷ 4 – 4 ÷ 4 = 20 – 1
Type 2
These involve multiplication or division without using any addition or subtraction. For
example, 86 x 50 might be solved by:
Step 1: 86 ÷ 2 = 43
Step 2: 50 x 2 = 100
Step 3: 43 x 100 = 4300
St Cuthbert’s College 2011 3 Numeracy Stages Another example (intermediate or secondary) 33 1/3 ÷ 3 1/3 might be solved by:
Step 1: 3 1/3 x 3=10
Step 2: 33 1/3 x 3=100
Step 3: 100 ÷ 10 = 10
Observable Behaviours
Students are at Advanced Multiplicative Part-Whole if they exhibit all of these behaviours
to solve multi-digit multiplication or division problems
•
Solve multiplication or division problems without using repeated addition or
subtraction
•
Use instantly recalled multiplication facts – list which ones have been used
•
Use multiple mental steps
Stage 8: Advanced Equivalent Fractions and Ratios
Description of Stage
The students solve problems involving whole numbers with simple percentages, equivalent
fractions, and rations.
There are two main problem types:
Type 1
These involve solving equivalent fraction problems. For example 2/3 – 12
Type 2
These involve solving equivalent ratio problems. For example, 4 apples and 6 bananas are
used in a fruit salad. 20 apples and 30 bananas make the same flavoured fruit salad.
Type 3
These are whole number percentage problems using simple equivalent fractions .For example:
15% of $440 equals $66 because 10% = 1/10 of 440 = 44, 5% = ½ of 10% and 1.2 of $44 =
$22
Type 4 (Intermediate/Secondary only)
These are percentage problems involving a calculator, and the rounding is sensible in
the context of the problem. For example:
15.8% of $446.50 = 0.158 x $446.50 ~ $71 to the nearest dollar.
NB: This type can be deleted for the few primary students who are operating at this
stage.
Observable Behaviours
Students are at Advanced Fractions and Ratios if they exhibit all of these
behaviours to solve proportional problems.
•
Use multiplication or division thinking
•
Solve mentally equivalent fractions, equivalent ratios, or percentage problems
Use multiple mental steps
St Cuthbert’s College 2011 4