Counting: There is more to it than you know!

Most children first learn to count when they are still very young, repeating numbers their parents say as they count fingers, toys, crackers, and such. But just like being able to sing the ABC song does not mean a child has learned all there is to know about letters, being able to count aloud to 10 or 20 or higher may not mean a child has learned all there is to know about counting.

Research¹ shows that there are five principles of counting that children must master to have the best success with learning numbers and mathematics. Generally, these counting principles examples are in the order that children master them.

First Counting Principle: Counting is always done in the same order.

Otherwise known as the Stable Order Principle, this early skill is knowing that when we count, numbers always come in the same order: one, two, three, four, five, six, and so on. If you have ever heard a young preschooler count “One, two, three, five, eight,” you’ve experienced a child who has not yet mastered this principle.

To help your child learn to count in order, simply count with her throughout your day. Count the snaps on her jacket as you snap her up, count the apple slices on her plate as you serve her a snack, and count the steps you walk up to your front door. Encourage her to count with you and provide gentle corrections when she gets the order wrong.

Second Counting Principle: We count each item once and only once.

This counting skill is more challenging than just knowing which number comes next. The One-to-One Principle requires the knowledge that each item in a group needs to be counted without any item being skipped or counted more than once. Each number goes with one object; each object gets one number.

When counting with your child, touch each item and, if possible, move it aside. Children will learn the one-to-one correspondence of item and number by seeing you do it. Have your child practice it with small groups of objects that are easily movable.

Third Counting Principle: The last number we say when counting tells how many objects were counted.

This concept of understanding that the final number we counted is the number of items in the group is often referred to as the Cardinal Principle. Knowing the last number we say is how many there are is a part of this, but research shows that children are often able to answer “how many” correctly before they fully understand other aspects of the principle. Children have full mastery of the Cardinal Principle when they understand that the next number when counting means adding one item, and the numbers beyond the next number mean adding more than one.

You can help your child develop an understanding of the last number telling us how many there are by modeling the concept for her. When you count fingers, say, “One, two, three, four, five. Five. I have five fingers. How many fingers do I have? I have five!” When she is ready, start giving your child the chance to answer “how many” after you count.

Once your child is doing well with “how many,” you can move on to the more complex aspects of the Cardinal Principle. Count out a group of two stuffed animals, “One, two.” Then, while your child watches, add one more to the group. Ask, “Are there one or three stuffed animals now?” After your child answers, whether she is correct or not, count out the group. “One, two, three. There are three stuffed animals.”

After a while, switch to taking one of the items away. Set out a group of three cups. Say, “One, two, three. There are three cups.” As the child watches, take one away. Then ask, “Are there two or four cups now?” Again, count the group, or have your child count it, regardless of her answer.

Fourth Counting Principle: The order we count groups in does not change the number of items in the group.

If we count a row of buttons from left to right and then from right to left, the number of buttons remains the same. This concept seems simple or obvious, but the Order Irrelevance Principle may not be fully mastered even beyond preschool. Children may not understand that calling a different button “one” will not change how many buttons there are in all. Note: a child can develop the Order Irrelevance Principle after having a good understanding of “how many,” even if she is still working on fully mastering the Cardinal Principle.

Count groups with your child in more than one way. Count a pile of blocks, and then line them up and have your child count them. Talk with your child about how the number of blocks didn’t change. This can take time to develop, so continue to practice it with your child periodically. You’ll know she mastered the concept when you rearrange the group and she tells you how many there are without needing to recount them.

Fifth Counting Principle: We can count anything, even if we can’t touch it, and every item we count has the same value.

This is known as the Abstraction Principle, and children need to have a good handle on the preceding counting principles before they can fully understand this one. Young learners find it much easier to count objects they can touch, especially objects they can move. But non-tangible things can be counted as well, like hops, sounds, or even imaginary castles. Even counting words can be counted, which is what we do when we add by counting on. Interestingly, when a child is asked to count how many syllables in a word, it may not be her inability to hear the syllables that keeps her from being able to answer. It may be that she needs more practice with the Abstraction Principle to be comfortable counting things she can’t touch.

Also, this principle covers the concept that things of different sizes, colors, and shapes are counted the same. A very big beachball is not of more value than a tiny marble. If counted together, they are “one, two,” just two items.

Working on this principle is the time to start counting mixed groups. Put rubber duckies, washcloths, and shampoo bottles together for a group to be counted. Start counting non-tangible things as well. Ask your child to count while you clap and count with her. Count hugs and kisses, count apples and oranges, count songs and dances.

Development of the Counting Principles

Children master the basics of counting slowly throughout their preschool years, with most children having a solid grasp of them by the time they are 5 years old. However, some children need extra support. If your child is 4 or 5 and still having trouble with counting out of order, counting objects more than once, having to recount when objects are rearranged or when one more is added or removed, and so on, it is time to give some daily focus to practicing counting. The tips above will help you, and the All About Math Level 1 program makes it easy to work on these skills with your student.

Beyond the 5 Principles of Counting

Once your child has conquered the five basic principles of counting, there is still more to learn. Here are some of the other counting skills that children need to master for a full understanding of our number system:

• Counting backward – “Five, four, three, two, one.”
• Counting by 10s – “Ten, twenty, thirty, forty…” Counting by other numbers, often called skipped counting, will need to be learned as well in time.
• Counting on – “This group has five, and we are adding this other group of three to it. So, six, seven, eight. There are eight items altogether.”

Lastly, I want to mention a skill that isn’t technically counting but still answers the question, “How many are there?” It is what allows us to look at a stack of books and just know that there are three books in the stack. This skill is called subitizing. When children learn to notice groups of one, two, three, and four objects and know how many there are without counting, they are developing an understanding of what it means to be “two” or the fourness of four. This is foundational for higher mathematical thinking.

All About Math Level 1 ensures that your child learns each of these fundamental principles of counting and skills as a part of its “no gaps” approach to learning math. This means that Level 1 may seem easy at first, spending time noticing small groups and learning to count, but ensuring these foundational skills are mastered will ensure your child succeeds in the more complex concepts found in later lessons and levels.

What is your favorite way to work on these counting skills with your children?

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¹Gelman, R., & Gallistel, C. R. (1978). The Child’s Understanding of Number. Harvard University Press.

InnerDrive Team. (2021, December 2). The Science of Counting: A Visual Explainer of the 5 Counting Principles (B. Busch, Ed.). InnerDrive. https://www.innerdrive.co.uk/blog/counting-principles-explained/

Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108(3), 662–674. https://doi.org/10.1016/j.cognition.2008.05.007