The Cardinality Principle

A Milestone on the Learning Trajectory for Counting

Although children have an innate visual appreciation for which collection of objects is larger,  the skill of accurate counting needs to be taught.    After learning the number sequence and beginning to develop some coordination and rhythm in the counting process, a milestone signaling a move to a higher level of thought is understanding the cardinality principle.

Cardinality Principle: The last word said in a count represents the total number in the collection.

This idea is significant because it shifts the child’s thinking from the mechanics of counting to the result of the process.  Knowledge of cardinality leads to the use of more sophisticated and efficient counting strategies, as well as to the ability to compare numbers and understand addition and subtraction.  I propose to you that with abundant experiences counting objects, as well as adult guidance, most children can understand the cardinality principle before their fourth birthdays.      You will know that the “cardinality light-bulb” has not yet been illuminated if the child repeats the count (“one, two, three,…”) or just guesses when you ask questions about how many objects are in the collection. (Guessing can also come from mental laziness.)   This is a time to say with enthusiasm and gestures, “That means there are five things all together here!”  I encourage you to begin early asking “How many?”  Ask the question often.  Sometimes, particularly with cardinalities of 5 or less,  children know how many there are without counting.   When counting objects with children, remind them that collections of things in counted groups can be alike or different but still have the same cardinality.  The idea of one-to-one correspondence can also be related to cardinality.   Here are some examples of ways you can talk about one-to-one correspondence.

• Look here! We need 5 cups because there are five children at this table
• In this carton for a dozen eggs there are 12 spaces, one for each egg.
• In a game of Musical Chairs, the one-to-one correspondence between chairs and participants is continually removed.

Cardinality and Conservation of Number

Furthermore, the arrangement of objects — in a long line, a shorter line, a circle, an array, a pile, or a cluster — does not affect the collection’s cardinality.   This second principle was named “conservation of number” by the famous learning theorist Jean Piaget (1896-1980),  who studied children’s ways of knowing and thinking throughout his career.   In educational psychology courses, most college students preparing to be teachers will read from the  many volumes of Piaget’s  interviews with children about tasks he had created, and he found conservation tasks especially intriguing.  He designed tasks related to all kinds of measurements to show that changes in physical arrangement of the quantity did not change its measurement — the total length of two sections of a ribbon would be the same as before it was cut apart and the total area of four quarters of a sheet of paper would be the same as the whole piece.   Some of Piaget’s favorite questions were challenging questions about conservation of a liquid quantity that had been poured from one container into one that was taller and skinnier.    Later researchers have found these conservation of length, area, and liquid capacity tasks to be more difficult for young children than conservation of number.  Perhaps children’s early experiences with the counting and cardinality help to make number conservation believable and understandable.    Here is a photo of a father-son conversation about cardinality and conservation. 

CHILD: Look! I have 5 red circles in a line. WithLearning Trajectories the blue circles, I see 3 and 2 and that’s 5.

DAD: Yes, and there are 5 here, too. No matter whether things are arranged in a short line, a long line, or other ways, there are still the same number of them.

A Learning Trajectory for Counting

Whether you have a collection of 3 objects or 300,  you can help a child increase counting skills by connecting with a child’s thinking and guiding new thoughts hierarchically up the steps of a trajectory.   Here are some typical milestones along the path.

1. Memorize the count sequence (1,2,3, 4, 5,…)
2. Counting accuracy increases as children coordinate saying the number’s name with pointing to the number — one point, one count.  In other words, objects should not be skipped or counted twice.
3. Gain an understanding of the Cardinality Principle (the last number named in a count is the number in the collection).  This is an important milestone for preschoolers.
4. For small collections, determine how many there are without counting.
5. Skip count by 2s (2,4,6,8,…); by 3s (3,6,9,12,…); by 5s (5,10,15,20, …): and by 10s (10, 20, 30, 40, 50, …)
6. Count forward from a certain number and stop at a higher number number.
7. Count backward from a certain number and stop at a lower number.
8. Compose 2 collections and find the cardinality for the total.
9. For collections between 10 and 100, introduce a grouping-in-tens strategy.
10.  Use row-and-column arrangements, skip counting, and fitting 1-inch square tiles into a rectangle to introduce multiplication concepts (3 groups of 8 objects is 24 objects in all).