Big Ideas in Number
Big ideas or 'key ideas' in number are statements or definitions that encompass the main mathematical understandings necessary to build a solid foundation in mathematical and numerical understanding. The definitions outlined on this page have been developed around the learning sequences and provide support to the information and activities contained in each area.
Counting is one of the very first skills that children learn in mathematics. In many cases, it begins before students even enter the classroom but needs to be carefully taught in their first year using eight explicit methods. When learning to count, students will need exposure and practise in the following principles:
One-to-one
"Matching an object to its number in a collection."
What is it?
One-to-one correspondence is an important concept for students to learn when they are first learning to count. It is a way of matching each object in a set with a unique number.
Example of use
If you have three apples, you can match the first apple with the number "1", the second apple with the number "2", and the third apple with the number "3". This way, each apple has a corresponding number and each number corresponds to exactly one apple. This helps children understand the concept of quantity and how to count objects accurately.
What you can use
To make it more tangible for kids, teachers can use different manipulatives, such as counting bears or blocks, to help them visualise and understand the correspondence.
Stable order
“Three means a collection of three no matter what it looks like."
What is it?
The stable order principle is a method used to help students understand the process of counting. The stable order principle states that when counting, the numbers should be assigned to the objects in a consistent and stable order.
Example of use
If you are counting a group of apples, you should always start with the first apple, then move on to the second apple, then the third apple, and so on. This ensures that each apple is only counted once and that the final count is accurate.
How you can apply it in your classroom
Teachers can use this principle by using different strategies such as counting from left to right, from top to bottom, or from one end to another end of the set of objects. This helps children to develop their counting skills and to understand the concept of ordering, which is fundamental for mathematical operations. Using different manipulatives like counting bears, blocks or even fingers helps students to visualise and understand the stable order principle.
Cardinality
"The last number counted tells ‘how many’."
What is it?
The cardinality principle is a method of counting that helps students understand the concept of quantity. The cardinality principle states that the last number used when counting a set of objects represents the total number of objects in that set.
Example of use
If a child is counting a group of apples, and they say "1, 2, 3, 4, 5", the number "5" represents the total number of apples in the group.
How you can apply it in your classroom
Teachers can use this principle by providing students with different sets of objects and asking them to count them and tell how many objects are in each set. This helps children to understand that the last number they say when counting represents the total number of objects in the set. Using different manipulatives like counting bears, blocks or even fingers helps students to visualise and understand the cardinality principle.
The cardinality principle is an important concept for children to understand as it lays the foundation for understanding mathematical operations such as addition and subtraction.
Order irrelevance
"The order counted does not matter."
What is it?
The order irrelevance principle is a method of counting that helps students understand that the order of the objects being counted doesn't affect the final count. This principle states that it doesn't matter in what order the objects are counted, the final number will always be the same.
Example of use
If a child is counting a group of apples, and they say "1, 2, 3, 4, 5" or "5, 4, 3, 2, 1" the final number will be the same, 5.
How you can apply it in your classroom
Teachers can use this principle by providing students with different sets of objects and asking them to count them in different orders and compare the final count. This helps children to understand that the order of the objects being counted doesn't affect the final count. Using different manipulatives like counting bears, blocks or even fingers helps students to visualise and understand the order irrelevance principle.
Understanding the order irrelevance principle is an important step for children to understand the concept of commutativity in mathematical operations such as addition and multiplication.
Ordinal
"Each new number is one more than the previous number."
What is it?
The ordinal principle is a method of counting that helps students understand the concept of position or order of objects. The ordinal principle states that each object in a set has a specific position or order and that this order can be represented by a number.
Example of use
If a child is counting a group of apples, and they say "first apple, second apple, third apple, fourth apple, fifth apple", the ordinal principle is being applied.
How you can apply it in your classroom
Teachers can use this principle by providing students with different sets of objects and asking them to order them and then assigning numbers to them in the order they are arranged. This helps children to understand that each object in a set has a specific position and that this position can be represented by a number. Using different manipulatives like counting bears, blocks or even fingers helps students to visualise and understand the ordinal principle.
Understanding the ordinal principle is an important step for children to understand the concept of ordering, which is fundamental for mathematical operations such as sequence, pattern and function.
Abstraction
"Different sized or unrelated objects can be counted the same numerically."
What is it?
The abstraction principle is a method of counting that helps students understand the concept of abstraction, which is the ability to identify and understand common properties or characteristics of objects. The abstraction principle states that when counting, children can group objects together based on their similarities, rather than counting each object individually.
Example of use
If a child is counting a group of apples, and they group them together by color, they can count the number of red apples and the number of green apples and add those numbers together to get the total number of apples.
How you can apply it in your classroom
Teachers can use this principle by providing students with different sets of objects and asking them to group them together based on their similarities and then count them. This helps children to understand that counting can be done in a more efficient way by grouping similar objects together, and also helps them to understand the concept of abstraction. Using different manipulatives like counting bears, blocks or even fingers helps students to visualise and understand the abstraction principle.
Understanding the abstraction principle is an important step for children to understand the concept of grouping, which is fundamental for mathematical operations such as addition and subtraction.
Conservation of number
"The number of objects does not change even when they are arranged differently."
What is it?
The conservation of number principle is a method of counting that helps students understand the concept of quantity remaining the same regardless of how it is arranged or grouped. The conservation of number principle states that when objects are grouped or rearranged, the total number of objects remains the same.
Example of use
If a child is counting a group of 10 apples and they group 5 of them into one pile and 5 of them into another pile, the total number of apples remains the same, which is 10.
How you can apply it in your classroom
Teachers can use this principle by providing students with different sets of objects and asking them to group or rearrange them in different ways and then count them. This helps children to understand that the total number of objects remains the same regardless of how it is arranged or grouped. Using different manipulatives like counting bears, blocks or even fingers helps students to visualise and understand the conservation of number principle.
Understanding the conservation of number principle is an important step for children to understand the concept of quantity, which is fundamental for mathematical operations such as addition, subtraction, and multiplication.
Subitising
"The instant recognition of a small quantity without counting."
What is it?
The subitising principle is a method of counting that helps students understand the concept of quickly recognising the quantity of a small number of objects without having to count them one by one. The subitising principle states that the human brain has the ability to quickly and accurately recognise the quantity of a small number of objects without counting.
Example of use
If a child is presented with three dots, they will be able to recognise the quantity as "3" without counting each dot individually.
How you can apply it in your classroom
Teachers can use this principle by providing students with small sets of objects, such as dot cards, and asking them to identify the quantity without counting. This helps children to develop their ability to quickly recognise the quantity of small numbers of objects, which is a fundamental skill for mathematical operations such as addition, subtraction, and multiplication. Additionally, teachers can use games, activities, and real-life examples that focus on the subitising principle to make this concept more interactive and engaging for students.
Trusting the count is a teaching approach that emphasises the importance of children understanding the meaning of numbers, rather than just memorising them. It involves encouraging children to count objects and group them in different ways, to understand that numbers represent a quantity and that numbers can be broken down into smaller parts. This approach helps children develop a deeper understanding of number relationships and the ability to apply mathematical concepts in a flexible way. It also helps them to build a strong foundation for more advanced mathematical concepts they will encounter later in their education. For students to be successful in working with numbers beyond 20, they need to:
1.
Successfully recite and name the number sequences to beyond 20.
2.
Recognise number numerals to 10.
3.
Read and write number numerals to 10.
4.
Count and model small collections to 20.
5.
Use a successful strategy to identify ‘how many’ in a collection.
6.
Identify which single digit number is larger/smaller when presented, orally or in written form.
Subitising is the ability to recognise quantities without counting. In order for students to successfully subitise, they need to:
1.
Recognise collections up to five without counting.
2.
Name numbers in terms of their parts (part-part-whole relationships)
An early understanding of place value involves the idea that whole numbers can be seen as countable units. This means that any number can be seen as how many in a set (e.g., 6 ones) or as a countable unit (e.g., 1 six) (Siemon, 2011).
Teaching children to understand the meaning of numbers, rather than just memorising them, is crucial in helping them develop a deeper understanding of number relationships and the ability to apply mathematical concepts in a flexible way. Using a hands-on approach, such as counting objects and grouping them in different ways, can help children understand that numbers can be broken down into smaller parts and built up from smaller numbers. This understanding of numbers as countable units lays a strong foundation for more advanced mathematical concepts children will encounter not just in place value but also in later mathematical concepts.
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Unitising Big idea A: 1 of these is 10 of those
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Unitising Big idea B: 1000 of these is 1 of those
Unitising is a concept that involves breaking numbers down into smaller units, such as ones, tens, and hundreds. One way to teach unitising is through the use of the phrase "one of these is ten of these". This phrase helps children understand the relationship between different place values, such as how one group of ten units is equivalent to ten individual units. After counting, this is the next step in building a strong grounding in place value and requires repeated exposure to the new idea that for example one group of ten blocks can be shown to be equivalent to ten individual blocks and this new group can be used as a countable unit.
Our base-10 number system has developed around the ten digits 0 – 9. The term 'base' means a collection and in our number system, 10 is the value that determines a new collection (Reys, 2022). From a multiplicative perspective, our place value numeration system increases and decreases by a factor of 10.
The five aspects of the place value framework listed below have been adapted from the work of Dr. Angela Rogers (Rogers, A. (2014)), and are interchangeable when teaching place value. For students to be successful in place value and develop a deep conceptual understanding, they need repeated, interleaved practice in all five areas.
Make, Name, Represent
When working with place value, provide opportunities for students to make their number using materials, write their number in symbols and words and record their number on their place value charts. Using the number triad board can help students to document their work in an organised way and show it clearly in all of the different ways
Count
Whenever students are working with their place value, it is important to provide opportunities for them to practise counting forward and backwards in place value parts. For example, if working with two-digit numbers, counting forwards and backwards in groups of 10 both on and off the decade while using language such as more, less, greater, before, and after.
Compare, Order
Another important component of learning place value is comparing numbers to determine which is larger and which is smaller. Students can compare numbers by placing them in ascending or descending order, using the symbols <, > and =, or by placing them on an open number line. Number lines are particularly important because the skills learned through number lines can be transferred to many different strands of mathematics. Eventually, students’ comparison of numbers will move from an additive to a multiplicative model. For example, while they may start by comparing 45 and 54 by stating that 54 is 9 more than 45, they will eventually move to comparing 45 to 450 by stating that 450 is ten times greater than 45.
Rename
Renaming numbers, also called regrouping, is one of the most important skills that students need to practise when learning place value. It involves partitioning numbers into standard and non-standard place value parts as well as non-place value parts. For example the number 235 can be renamed in standard place value parts as 2 hundreds, 3 tens and 5 ones, non-standard place value parts as 23 tens and 5 ones, or non-place value parts as 1 hundred, 13 tens and 5 ones. Practise in partitioning into non-place value parts is particularly important for developing strong number sense in students.
Calculate
When any type of operation is applied to numbers, students need to use their knowledge of place value to complete the calculation successfully. The understanding of the place value system and students’ ability to rename, is crucial for developing the conceptual understanding necessary to complete calculations flexibly. Understanding the place value system also underpins students’ development of number sense which allows for efficiency when working with numbers.
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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Rogers, A. (2014). Investigating whole number place value assessment in Years 3-6: Creating an evidence-based Developmental Progression, RMIT University.
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Angela Rogers (2017). Teaching Place Value: A Framework, PRIME NUMBER: Volume 32, Number 1., Mathematical Association of Victoria.
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Van de Walle (2019). Primary and Middle Years Mathematics. Pearson Australia.
