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The 5 counting principles explained

The science of counting: A visual explainer of the 5 Counting Principles

3 min read
  • The science of learning

What are the key principles to helping young students learn how to count?

Numbers can be quite an abstract concept, and until children learn the principles that underpin how to count correctly, numbers can be challenging for them to understand. Fortunately, with just a few key principles, we can create a foundation and platform to help students learn how to do this quicker. This is what the science of counting suggests…

The 5 Counting Principles

Generally, research agrees that there are 5 principles that children follow when learning how to count. These principles can be divided into two parts:

  • How to count – These principles explain the rules and methods that children should follow when they count. These include the One-To-One Principle, the Stable Order Principle, and the Cardinal Principle.
  • What to count – These principles explain what children can count. They include the Abstraction Principle and the Order Irrelevance Principle.

1. The One-To-One Principle

This describes the necessity to count each item in a group only once. When we count, we assign one distinct name to each number. Children need a lot of practice doing this, because they have a tendency to: 

  • Skim – Children may count too quickly and miss out an object.
  • Flurry – Children may count an object more than once. 
the science of counting the one to one principle

2. The Stable Order Principle

This is the idea that the names of numbers should be said in the correct order, and that the order of those numbers will not change. 

The difficulty with this is that the names of numbers in the English language don’t always have a pattern. This is the case specifically for the numbers between 10 and 19, which don’t have the same pattern as the numbers between 1 and 9, 20 and 29 or 30 and 39, and so on. 

Children often learn how to recite names of numbers from quite a young age, without really understanding the quantities that numbers represent until they are around 4 years old. 

Children need to learn the correct names of numbers so that they can then count correctly. Essentially, by understanding that the names of numbers have a correct order, children can understand that numbers also have a progressive value.

the science of counting the stable order principle

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3. The Cardinal Principle

This is the understanding that the final number said when counting tells you how many objects are in that group. 

Research suggests that children develop a sense of cardinality for different numbers at different stages. They first understand the cardinal meaning of the number 1 and believe that all other numbers are simply “bigger than one”.

the science of counting the cardinal principle

4. The Abstraction Principle

This is idea that we count everything in the same way, no matter what it is. Anything can be counted, from physical things to things that can’t be touched.

Research has shown that children find it easier to count tangible objects compared to abstract concepts.

the science of counting the abstraction principle

5. The Order Irrelevance Principle

This is the understanding that the order in which objects in a group are counted is not important. The number order does not change just because the counting order does. However, this principle only rings true if all the other principles are followed.

the science of counting the order irrelevance principle


Understanding the 5 counting principles is key to understanding numerosity, which is the ability to perceive the number of items in a group. Sometimes, numerosity happens consciously, such as with counting. But other times, numerosity happens unconsciously, such as with subitising. 

Subitising is the ability to accurately determine numerosity without having to consciously count. For example, we are able to determine the numbers on one face of a dice without having to actively count the dots. 

Subitising itself is not a counting principle. Instead, a child’s ability to subitise is a measure of how well they understand the 5 counting principles, and numerosity as a concept.

the science of counting subitising

Final thoughts

Counting is an essential life skill that children sometimes struggle to grasp. Using these five principles can help provide a range of activities that can help them learn more effectively and more efficiently. A child’s ability to subitise is a great indicator of how much they understand the 5 counting principles, numbers and their meanings.

About the editor

Bradley Busch

Bradley Busch

Bradley Busch is a Chartered Psychologist and a leading expert on illuminating Cognitive Science research in education. As Director at InnerDrive, his work focuses on translating complex psychological research in a way that is accessible and helpful. He has delivered thousands of workshops for educators and students, helping improve how they think, learn and perform. Bradley is also a prolific writer: he co-authored four books including Teaching & Learning Illuminated and The Science of Learning, as well as regularly featuring in publications such as The Guardian and The Telegraph.

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