Number and Counting
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Sequence Contents
Early Counting Concepts
Counting on Number Lines
Subitising
Partitioning and Grouping
Free Resources
The Eight Counting Principles
When students are first learning to count, they need repeating practise and exposure to counting through eight counting principles. These eight counting principles are:
One-to-one
Stable order
Cardinality
Order irrelevance
Ordinal
Abstraction
Conservation of number
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
Key Ideas
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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.
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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."
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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".
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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.
Subitising
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
Recognising small amounts (1 - 3) immediately with no counting or thinking.
This is an innate ability and is the subitising starting point as it doesn’t need to be taught.
Conceptual Subitising
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).
Recognise numbers up to 10 as composite pieces that can be identified through part-part-whole relationships.
Conceptual subitising is not innate; it needs to be learned in sequence
Key Ideas
Dot Plates: A Subitising Activity
Subitising is one of the earliest maths skills that kids learn. It is exeptionally important because if students can recognise collections, they don't waste precious cognitive load having to count them each time. In this video, I demonstrate a cute little subitising activity that unpacks the complex nature of subitising and how to deliver it successfully in your classrooms.
Counting on a Number Line
Number lines should be introduced to students as soon as they start working with numbers and counting.
Students need to be given opportunities to construct and count along a number line by ones.
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.
Use the term ‘jump’ to represent moving one unit to the left or right when counting by ones.
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.
Key Ideas
Partitioning and Grouping
1. Partitioning into equal groups
1.1 Students practise sharing from a whole collection into smaller groups first.
This allows them to physically create the group to get a better understanding of what a group is.
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.
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.
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.
Ask them how they know that both groups are equal. Always ask the students to communicate their thinking to encourage reasoning skills.
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.
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.
Encourage them to place each quantity in an organised group like a row or an array.
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
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 (+, -, =).
Students need to consolidate their understanding of numbers and groups before formal number sentences are introduced.
Use a number sentence in words instead of as an equation. Get them to describe what’s happening in words.
Use more than 2 groups and get students to practise equal sharing among these groups.
Eventually, give them practise at sharing where there is a remainder.
Key Ideas
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Videos
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Lessons and Activities
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External Resources
