The Complexity of Counting

Counting is one of the most common forms of mathematics with which young children engage. They love to enter the world of counting as they see and hear the adults around them count.

Counting is often considered a simple process. It is commonly misunderstood in terms of the complex range of underpinning processes that are required to "count" and how these contribute to young children's number sense.

Let's think of this first in terms of reading and writing. These are considered complex processes. Young children of three or four-year-old might be able to sing or recite the alphabet but this doesn't mean they can read and write. They have, probably quite excitedly, remembered the order of the letters of the alphabet. They may also be able to read but this isn't necessarily the case.

Similarly, we may hear a child sing or recite the numbers to 5 or 10 and hear parents or educators state that the child can count. Maybe they can, or maybe, like the alphabet, they have remembered a series of words by rote in singing order.

Counting is actually a very complex process and takes more than reciting names of numbers in order - just like reciting the names of letters for the alphabet does not mean a child can automatically read, write, spell etc.

Mastering counting requires the development of the following concepts (sometimes referred to as counting principles).

One-to-one correspondence
Stable order principle
Cardinality
Order Irrelevance principle
Abstraction

One-to-one Correspondence

When we count, we assign one distinct number name (one, two, three…etc) to one object only, in the set of things being counted. Children need a lot of practice doing this, and some difficulties or evidence of this concept needing further development could be indicated by

Counting the set too quickly and missing an object, or
Continuing to count an object that has already been assigned a ‘count’.

Ways to promote this concept:

Physically touching objects/ moving objects as they are counted are a great strategy. Bringing children’s attention to “have we counted all?” “has each object only been counted once?” are also good questions and prompts for children to check their understanding.

Stable Order Principle:

The stable order principle is founded on the idea that the names of numbers have one, correct order in which we say them to count. It means 4 can ONLY come after 3 and before 5 when we are counting a set of objects and naming each count.

Difficulties of this concept are evident when children skip a number when they are counting, like counting 1,2,3,5,6… and also become challenging when children move into the teens, where the pattern of the numbers behave differently to others.

Ways to promote this concept:

Children need to see and practice counting in sequence with physical objects, by making and and naming the quantities of each number name. That is making a group of 1, group of 2, group of 3 etc and comparing them as they name them (not necessarily with symbols but these can be introduced as appropriate). This type of physical experience with quantity will help build an understanding that the names of numbers not only have a correct order, but also have a progressive value.

Order Irrelevance principle

This principle focuses on the understanding that although the names of the count I assign to each object must only occur once (one to one) and that the order I say each count is a fixed order (stable order principle), the order of the objects I count does not affect the total quantity.

For example, I could have three blocks, a red, blue and green block. No matter what order I count them in blue first, red then green; green, blue, red etc, I will still always have three blocks.

To check for this understanding, observe the order a child counts a set of objects, and then deliberately move them and ask them how many they have. If they re-count, this principle may not be established yet.

Cardinal Principle

“The last number I say is the total number of items I have”. This phrase sums up the cardinal principle whereby the last count of the collection represents the total quantity.

This starts with the understanding or recognition of one versus many. Difficulties of this principle may be evident when you ask a child how many d do you have, and rather than answer with the total of the collection, they recount the set. To develop this idea, children need to experience making groups of items to count, so they emphasizing that when I say and count my final object, that tells me how many I have altogether.

Abstraction

This is idea that we count everything in the same way – that is, utilizing the same principles above, no matter what it is. Anything can be counted, from physical things such as how many children are in our room, to things that can’t be seen or touched, like the stars in the sky or the number of times a clock chimes.