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In the previous post, At age four, the idea of number, we examined how math is learned through a spiral curriculum and explored the types of content—both social and logical-mathematical—that form the foundation for learning about numbers. In this post, we will focus on specific learning related to the “concept of number,” which begins or expands at age four.
Content
What will they learn about number at age 4?
As we have seen, children around the age of 4 gradually develop their understanding of numbers. They begin to connect various ideas and processes associated with this concept. The goal is to enhance their number sense, which is defined as the ability to intuitively and flexibly understand numbers and quantities and to apply them in real-life situations.
However, this learning process is not straightforward; it requires time and diverse experiences for children to form a solid understanding of numbers. Let’s consider what some researchers say on the topic:
“For most adults, the knowledge and use of the first nine natural numbers (one, two, three…) seems very simple and obvious. However, a child needs around five years—approximately from ages two to seven—to learn how to use these numbers coherently and apply them in a variety of everyday situations. This period is even longer when the use of arithmetic operations is included.”
Dickson, Brown and Gibson
Why does this learning take so long? The concept of number is likely one of the first significant abstract notions that children grasp. Additionally, this idea encompasses many other notions and processes that must be interconnected within a single concept.
These include subitizing, rote counting, counting backward, identifying the previous and next numbers, cardinal counting, transforming quantities, addition and subtraction, decomposing numbers, understanding the concept of zero, and using arithmetic symbols such as +, −, and =.
Mastering all these aspects and integrating them into a cohesive approach to solve real-life situations requires considerable time and practice.
So how do four-year-olds absorb all of this information? Do they learn one concept after another? Is it essential to fully master one aspect before moving on to the next? The answer is no. This brings us back to Bruner’s concept of spiral learning: in the classroom, these notions and processes will be explored in parallel and often simultaneously, without waiting for complete mastery of one concept before starting another. This approach is generally applied to real-life or contextualized situations.
To delve deeper into the idea of numbers in early childhood education, we can say that children at this age learn to quantify groups of objects, meaning they can determine “how many there are” when dealing with small quantities. They achieve this through two primary methods: subitizing and cardinal counting. Let’s explore these two concepts further.
Subitizing
Subitizing is the ability to instantly recognize the number of items in a group without counting them individually. There are two types of subitizing: perceptual and conceptual.
Perceptual subitizing occurs when we recognize a quantity simply by looking at it, without needing to think deeply or make connections. Piaget referred to these as perceptual numbers because the quantity is immediately apparent. This type of recognition works very well with small quantities, such as 1, 2, 3, or 4, especially when objects are arranged in a familiar way, such as the dots on dice up to 5 or 6.
Perceptual subitizing: if we ask “How many are there?”, they will probably answer “three” right away, because they see it instantly.
When quantities become larger or arranged in different ways, conceptual subitizing comes into play. In this case, children recognize the total quantity by connecting small groups to each other. It involves not just looking but also recognizing different small quantities and relating them mentally. They begin to understand that quantities can be transformed by joining and separating groups of items. This understanding is linked to concepts like decomposition, addition, and subtraction, helping them develop number sense. Thus, we consider it a more cognitive form of subitizing.
Conceptual subitizing: if we ask “How many are there?” and they answer “four,” we can ask “How did you see it?” Often, they might explain their thought process by saying things like, “one on top and three on the bottom” or “two in the middle and one on each side.” This means they recognized small groups and combined them to arrive at the total.
Both perceptual and conceptual subitizing are very important for developing number sense. They help prepare the brain for counting and calculating later on. While these skills naturally develop in children, they can be strengthened with regular practice. At the age of 4, we primarily focus on quickly recognizing small quantities at a glance; that is, perceptual subitizing. However, we can also start to introduce conceptual subitizing with small quantities.
Subitizing activities are typically conducted as a whole class exercise, involving group discussions and lasting a short duration. You will find some suggestions that work on this skill in some Workshops, seasonal content, and a special gift mentioned in Judith’s last letter ;).
Cardinal counting
Cardinal counting is one way to quantify a group of objects, but it is a process that children take quite a while to master. Furthermore, children may not naturally choose counting as a useful tool to solve real-life situations. Why is this the case?
Research indicates that counting does not become a fully reliable tool until around the age of 6 (Dickson et al., Kamii, Piaget). This is largely because, for students to use cardinal counting efficiently, they must master a series of important skills:
- Being able to recite number words in order; that is, mastering rote counting.
- Having eye-hand-verbal coordination to point to one, and only one, object at the same time as each number word is said.
- Keeping track of which objects have been counted and which have not, without repeating any or leaving any uncounted.
- Knowing that the last word said determines the cardinal number; that is, the number that tells the total in the group and answers the question “How many are there?”.
As we can see, counting effectively requires mastering many different skills simultaneously. Since these develop gradually, it is normal that at early ages children still struggle with them. This explains why cardinal counting is often not a fully effective tool in solving real situations.
So, the fact that many children under age 7 do not choose counting, even if they know how to count, makes sense: research shows that they still do not perceive it as a safe or necessary strategy.
To address these challenges in school, it is important to work in two directions. First, educators should help children develop each of the skills needed for counting, allowing ample time for them to try, make mistakes, and gradually integrate their learning.
Second, educators should create situations where children can choose counting as a strategy to solve problems, without feeling pressured to do so. A child can be said to have successfully integrated cardinal counting as an effective tool when they use it spontaneously, as they recognize its usefulness for addressing a specific challenge. As Kamii points out, children need time and real-life experiences to gradually become comfortable with the counting process. The use of counting cannot be forced; each child will adopt it as their own tool when they understand that it is an effective strategy for solving specific challenges.
How Jean-Pierre discovers counting
Next is an example described by Constance Kamii of how a specific child learns to use cardinal counting by observing a real-life situation.
A mother asked her five-year-old son to set a napkin on each person’s plate every day at lunchtime. Usually, four people sat at the table. Jean-Pierre knew how to count to 30 or more. However, he went to the cabinet to get the first napkin and placed it on one plate, went back to the cabinet to get a second napkin and placed it on the second plate, and so on, making a total of four trips.
At five years, three months, and sixteen days, he spontaneously thought to count the plates, counted four napkins he needed to take from the cabinet, and distributed them on the table. He did this for six days. On the seventh day, a guest arrived, which meant there was one more plate than usual. Jean-Pierre took four napkins as usual, distributed them, and realized that one plate was still empty. Instead of getting one more napkin, he picked up the four that were already on the plates and took them back to the cabinet. Then he started again and made five trips to complete the task.
The next day the guest was not there, but Jean-Pierre continued making four trips for five more days, until he discovered again how to count. After using this method for ten days, they told Jean-Pierre that there was again a guest. The child distributed four napkins as usual, but this time, when he saw the empty plate, he simply went to get the missing napkin. The next day, when there were only four people again, he counted the number of plates before going to get the same number of napkins. The arrival of a new guest was never a problem again.
“In the previous example, we saw the difference between counting mechanically and counting when the child chooses to do so to solve a real problem. Knowing how to count is one thing. Knowing how to use it when we face a question we want to answer is something quite different.”
Kamii
If Jean-Pierre had been instructed to count the plates and napkins, he would have learned to depend on others. Since he was not given a specific instruction, he had the opportunity to develop his intellectual autonomy and self-confidence.
To close
In this piece, we have focused mainly on increasing number sense, focusing on subitizing and cardinal counting, but while we talked about them, other notions involved in this concept of number have also appeared. In fact, At the age of 4, more new ideas about the concept of number appear. However, since we are just beginning to explore these ideas, we will revisit them in detail as we continue this fascinating journey of building a deep understanding of numbers.
Recognizing that knowledge of number has different layers of depth (Bruner), what can we expect for the upcoming year, as children transition from age five to six? Discovering and building ideas such as, for example, knowing that this number we found through counting:
- is always smaller than all the numbers that come after it and bigger than all the ones before it,
- includes inside it all the numbers smaller than it,
- can be transformed: if we add, it gets bigger; if we take away, it gets smaller,
- can be decomposed; that is, it can be split into different parts and, when put back together, the total always stays the same,
- can be split in different ways into two parts, and we can find them all,
- there is a very strange number that contains no elements and is called zero,
- there are new symbols (+, −, =) for writing these quantity transformations…
The exploration continues, as numbers are infinite and capacity to learn is boundless.
See you soon,
Mequè
References
Dickson, L., Brown, M., Gibson, O. (1991). Learning mathematics. Labor.
Kamii, C. (1984). Number in preschool education. Visor
Piaget, J., Inhelder, B. (1977). The psychology of the child. Ediciones Morata
Smith, S.S. (2013). Early Childhood Mathematics. Pearson.
