Key Ideas for Teaching
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Misunderstandings
Mathematical
Language
Concrete
Resources
Counting
Foundation
Key Ideas
When students are first learning to count, they need repeating practise and exposure to counting through eight counting principles. These eight counting principles are:
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One-to-one
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Stable order
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Cardinality
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Order irrelevance
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Ordinal
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Abstraction
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Conservation of number
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Subitising
Each of the videos demonstrates how to use the counting principle and possible misunderstandings that students may have when practising their counting.
The Four Counting Stages
The four counting stages are methods used to help students learn how to count and understand basic mathematical concepts.
1. Count All
This method involves counting each object in a set one by one, starting from the first object and working your way through the set until you reach the last object. For example, if there are 5 apples in a basket, you would count "1, 2, 3, 4, 5" as you point to each apple.
2. Count On
This method involves counting a set of objects by starting at a certain number and counting on from there. For example, if you are counting a set of 5 apples and you already counted 3 apples, you would start with the number 3 and count "4, 5."
3. Count On from Larger
This method is similar to the 'count on' method but instead of starting at a certain number, you are starting with a larger number and counting on from there. For example, if you have a set of 5 apples and another 3 apples, you would start with the number 5 and count on three more "6, 7, 8".
4. Trust the Count
This method involves recognising the number of a set of objects without counting them. For example, if a child is shown a set of 5 apples and is asked how many apples there are, they should be able to say "five" without counting the apples.
Activity Ideas
*Videos
Counting principles addressed:
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Learning to write numbers
Content descriptions addressed:
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
Counting principles addressed:
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Subitising numbers up to 5
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Part-part-whole
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Numbers to ten
Content descriptions addressed:
Counting principles adressed:
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Part-part-whole
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Trusting the count
Content descriptions addressed:
name, represent and order numbers including zero to at least 20, using physical and virtual materials and numerals
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Counting principles addressed:
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Trusting the count
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Counting on backwards/forwards
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Skip counting
Content descriptions addressed:
name, represent and order numbers including zero to at least 20, using physical and virtual materials and numerals
recognise, copy and continue repeating patterns represented in different ways
Subitising
Foundation
Key Ideas
There are several stages to subitising that need to be consolidated as students are establishing the concept. Following this subitising sequence will help students develop their number sense in this area.
Perceptual Subitising
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Recognising small amounts (1 - 3) immediately with no counting or thinking.
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This is an innate ability and is the subitising starting point as it doesn’t need to be taught.
Conceptual Subitising
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Recognising numbers up to 5 as composite pieces that can be identified through perceptual subitising (e.g., 5 is made up of 3 and 2).
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Recognise numbers up to 10 as composite pieces that can be identified through part-part-whole relationships.
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Conceptual subitising is not innate; it needs to be learned in sequence.
Activity Ideas
*Videos
Subitising principles addressed:
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Perceptual subitising
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Conceptual subitising
Content descriptions addressed:
recognise and name the number of objects within a collection up to 5 using subitising
Subitising principles addressed:
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Perceptual subitising
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Conceptual subitising
Content descriptions addressed:
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
Subitising principles addressed:
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Conceptual subitising
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Visual-spatial awareness
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Pattern recognition
Content descriptions addressed:
recognise and name the number of objects within a collection up to 5 using subitising
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Subitising principles addressed:
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Subitising numbers to 5
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Part-part-whole
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Numbers to ten
*Content descriptions addressed:
Key Ideas
Number lines should be introduced to students as soon as they start working with numbers and counting.
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Students need to be given opportunities to construct and count along a number line by ones.
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Using number lines with early counting allows students to see the consecutive nature of numbers, how they increase when moving to the right and decrease when moving to the left.
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Use the term ‘jump’ to represent moving one unit to the left or right when counting by ones.
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Prompt students to draw their jumps and record their numbers underneath the ticks on the number line.
Continuous exposure to number lines will help to consolidate student understanding of counting and allow them build toward the successful use of addition and subtraction strategies.
Counting on a Number Line
(F - 1)
Key Ideas
1. Partitioning into equal groups
1.1 Students practise sharing from a whole collection into smaller groups first.
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This allows them to physically create the group to get a better understanding of what a group is.
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When introducing groups to students, always use the correct language divide or share equally so that they understand that there needs to be the same amount in each group.
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Once students have their group or groups, ensure that they understand that the one group is the whole quantity, not just one object from that group.
1.2 Begin introducing groups using group markers.
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Initially, use group markers that represent people, animals, or objects that they relate to. For example, a student may represent one group, or if they are dividing a collection into two equal groups, one group might be identified by a fish and the other group by a bird.
1.3 Ensure that students understand that both groups must be equal by asking them to compare the quantity in each group.
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Ask them how they know that both groups are equal. Always ask the students to communicate their thinking to encourage reasoning skills.
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Prompt students to use subitising to identify smaller quantities and counting to identify larger quantities.
1.4 Once practise has determined proficiency in the above areas, ask students to create groups without group markers.
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This is the next level of understanding because students will need to perceive a group as one whole without an identifying marker.
2. Partitioning into different-sized groups
Partitioning into different-sized groups is the lead-up to part-part-whole understanding, as well as addition and subtraction.
2.1 Begin by asking students to make groups that have different quantities.
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Encourage them to place each quantity in an organised group like a row or an array.
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This allows them to recognise the quantity in each group, as well as the combined total.
2.2 Use a number triad or think board to ensure that students practise describing the groups in different ways.
2.3 It is important to lead students in making the conceptual connection to addition and subtraction by describing the groups and how much they make altogether.
3. Partitioning into groups - part-part-whole
3.1 The numbers 0 – 9 make up every number, so learning how to combine these digits to make different numbers to 10 and to 20 is an important part of developing number sense.
3.2 As students begin to get more comfortable with adding groups of different sizes, get them to practise making different groups with numbers to 10.
3.3 Once this has been established, introduce facts to 20.
Practising grouping: Things to remember
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When students are first learning to divide into smaller groups or combine groups to make larger groups, avoid the explicit use of the operating symbols (+, -, =).
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Students need to consolidate their understanding of numbers and groups before formal number sentences are introduced.
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Use a number sentence in words instead of as an equation. Get them to describe what’s happening in words.
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Use more than 2 groups and get students to practise equal sharing among these groups.
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Eventually, give them practise at sharing where there is a remainder.
Partitioning and Grouping
(F - 1)
Activity Ideas
Multiplicative principles addressed:
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Sharing as partition
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Sharing as quotition
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Sharing with remainders
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Multiplication as ‘groups of’
*Content descriptions addressed:
*See Instructions for full details
Additional Activities
Additional Activities
Links to external websites:
Counting principles addressed:
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All
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
recognise and name the number of objects within a collection up to 5 using subitising
quantify and compare collections to at least 20 using counting and explain or demonstrate reasoning
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Year 1
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
use mathematical modelling to solve practical problems involving additive situations including simple money transactions; represent the situations with diagrams, physical and virtual materials, and use calculation strategies to solve the problem
use mathematical modelling to solve practical problems involving equal sharing and grouping; represent the situations with diagrams, physical and virtual materials, and use calculation strategies to solve the problem
Content descriptions addressed:
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one-to-one
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stable order
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cardinality
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ordinal
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conservation of number
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subitising
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
quantify and compare collections to at least 20 using counting and explain or demonstrate reasoning
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Counting principles addressed:
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Trusting the count
Content descriptions addressed:
Year 1
recognise, represent and order numbers to at least 120 using physical and virtual materials, numerals, number lines and charts
Counting principles addressed:
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Number sequences
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Skip counting
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Odd and even numbers
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Operations
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
Year 1*
recognise, copy and continue repeating patterns represented in different ways
*NOTE: This activity can be differentiated to include similar Number and Algebra content descriptors from
F - 6
Counting principles addressed:
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Ordinal numbers
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Sequences
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Operations
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
Year 1*
recognise, copy and continue repeating patterns represented in different ways
recognise, continue and create repeating patterns with numbers, symbols, shapes and objects, identifying the repeating unit
*NOTE: This activity can be differentiated to include similar Number and Algebra content descriptors from
F - 6.
Counting principles addressed:
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Part-part-whole
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Formal addition
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Formal subtraction
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Year 1
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
References
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Dianne Siemon (2011). Teaching Mathematics: Foundation to Middle Years. Oxford University Press.
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Reys et.al. (2022). Helping Children Learn Mathematics p.228. John Wiley and Sons Australia.
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Van de Walle (2019). Primary and Middle Years Mathematics. Pearson Australia.
