Numeracy

Counting Principles

In my last numeracy article, I mentioned that there is a lot more to counting than first meets the ear, so let’s unpack that a little more today. Before I get going I want to assure you all that we are not sharing this information with an expectation on any of you becoming teachers, but to help you understand the developmental process your children are going through when they learn mathematical concepts.

Counting is one of the first mathematical concepts children learn and they often come to school with some understanding of the counting sequence before they have started Foundation. Children will proudly recite the numbers they know - sometimes even extending beyond one hundred! When we teach children counting, we are looking for evidence of ten different skills and understandings that go beyond knowing the names of numbers. These are listed below and are roughly in developmental order, but it is not unusual for children to skip one skill or understanding, while fully developing the others.

Subitising

Subitising is the ability to ‘see’ how many objects are in a collection without counting - an example of this is knowing that you have scored a ‘three’ when you roll a dice. As an approximate rule, people can typically only subitise up to their age, but this stops at seven; once a collection reaches more than seven objects, we tend to engage in what is called ‘Perceptual Subitising’ - where you break the collection into smaller groups that are more easily seen.

Stable Order

Stable Order refers to the idea that our counting sequence doesn’t change; 3 always comes after 2 and 7 always comes before 8. Most people can count forwards and backwards as well as skip count (such as counting by twos) - these skills are an important part of children’s counting development. If we were to survey who could recite the alphabet forwards and backwards and skip-recite the alphabet by twos or fives, it’s likely that most people would find that exceptionally challenging. The reason for this is that as we learn to count, we build mental images in our head about what ‘four’ looks like. This is the reason why we don’t like to rush children into using symbols for numbers and spend lots of time looking at collections of objects and dots in random and organised arrangements (such as in a ten frame).

Stable Order gives our number system a distinct pattern and once you understand that pattern, you can count indefinitely. Not only do we know that fives comes after four, we also know that our numbers are read in hundreds, tens and ones through each ‘place value house’ - though you may need to be told the name of the next house! The exception to our pattern is the tricky teens - perhaps that’s why teenagers are sometimes tricky too?

One-to-One Correspondence

This understanding involves knowing that when we are counting, every object should be counted and only counted once. I’m sure you have all seen a child count without one-to-one correspondence; while pointing to and counting objects, these children might start reciting numbers quicker than they are touching the objects or they might double (or triple!) count objects without knowing where to stop. As children get older and start counting things that are higher in number or more abstract, they need to develop different strategies to track what has and has not been counted.

Order Irrelevance

Order irrelevance refers to the idea that it doesn’t matter which object is counted first or last, as long as all the objects are counted. While this might seem obvious to us grown ups that have developed our understanding of counting, children who are still developing their understanding of this may believe that the number of objects in a collection will be different if we start with the blue pencil instead of the red one, or if we start counting the objects from the middle of the line instead of the end.

Cardinality

Cardinality is understanding that the last number name used to count a group of objects represents how many are in the group - I finish counting on the seventh object so there are seven objects in the set. A child who does not have a solid understanding of cardinality might recount a set they have just counted in order to answer the question of ‘How many are there?’ This can sometimes appear as though it relates to the child’s memory, but is actually most often due to them developing their cardinality.

Conservation

Conservation is also known as Trusting the Count and involves understanding that the number of objects in a collection does not change if the collection is squished together, spread apart or popped into a cup that the child can’t see through. 

As children move through working with physical materials to more abstract mathematics, this concept becomes even more important as it asks children to hold numbers in their mind’s eye.

Abstraction

Abstraction requires an understanding that we can count any collection of objects, whether tangible or not - we can count pencils, claps or days. It also refers to children being able to compare sets of objects - understanding that five large items is the same count as five small items or a mixed group of five small and large items. 

Hierarchical Inclusion

Hierarchical Inclusion involves understanding that all numbers preceding a number are systematically included in the value of another number. For example, knowing that within a group of 5 items, there is also a group of 4 items within that group; 3 items within that group; 2 items… and so on.

Hierarchical inclusion is an important landmark that children must reach in order to fully understand cardinality and to begin composing numbers (i.e. composing a set of 5 items by combining a set of 2 items with a set of 3 items) and decomposing numbers (i.e. breaking apart a set of 6 items by separating into a set of 4 items and a set of 2 items).

Hierarchical Inclusion can also be referred to as Part-Part-Whole or ‘The Eightness of Eight’ and is an important skill that children will depend on throughout the remainder of their time at primary school and beyond.

Movement is Magnitude

Magnitude requires children to understand that as you move up the counting sequence (or forwards), the quantity increases by one and as you move down (or backwards), the quantity decreases by one - unless you are skip counting! 

Children develop an understanding of magnitude up to ten quite readily - they often see collections of objects up to ten and can comprehend how much ‘bigger’ nine is than one. As they start working with larger numbers, this sense of magnitude becomes more challenging purely because their experience with those numbers is more limited. Consider your concept of the numbers one hundred and one billion; can you visualise one hundred pencils? How about one billion pencils?

Unitising

Unitising is the ability to look at a collection of objects or numbers and group them as one set - an essential understanding of our place value system. Our number system is base-10 which means our numbers increase in sets of 10. We start with ones, but once we get to ten we ‘rename’ them as one-ten. Then we count in one-ten and ones - here are those tricky teens again! Many educators talk about how much easier it would be if our teen numbers were renamed as onety-one, onety-two, onety-three etc. If I bypass the teens, we can see that 25 is made up of two-tens and five more. Moving along the place value columns, once we get to ten-tens, we rename that number as one-hundred and so on.

It is not just for place value that children are required to unitise numbers, however. We unitise seconds into minutes and minutes into hours, for example. As children develop their multiplicative and fractional thinking, we want them to be confident with renaming numbers flexibly - children counting three groups of four (3 x 4) are actually seeing each group of four as one unit, children finding out how many people are in a quarter of their class will need to see the class of 20 students as one-whole.

As I said at the start - there is a lot more to counting that one might first consider! These skills are foundational to future mathematical understandings; by rushing children through counting without ensuring each understanding is well developed can lead to difficulties with maths in later primary school and beyond. 

As always, if you have any questions, feel free to send me an email (hannah.reid@education.vic.gov.au) - I welcome questions about one of the topics I have written about so far, or one you are keen to understand more about!