*This post is part of the Little Brains, Big Concepts series. Read the introduction to this series here.*

Define the number 4 by following these rules: You can’t define it by its surrounding numbers, and you can’t use the words “number” or “four.”

Harder than you think, right? I mean, what *is *a number, and how on earth do children come to understand a concept that is so obvious yet so abstract?

## The Big Concept

Numbers answer a fundamental question: *how many?* The concept of number integrates three essential understandings: the number-word sequence, one-to-one correspondence, and cardinality.

#### Understanding 1: Number-word Sequence

The first step to understanding *number* starts with memorizing the number-word sequence: “one, two, three, four…” From the alphabet to “Head, Shoulders, Knees and Toes,” children learn to memorize sequences early on.

In the beginning, children can recite the numbers in order, but the words themselves have no real meaning. The challenge, then, is “*to transform the number sequence from a mere sequence of words into a conceptual tool for doing mathematics*” (Dougherty et al., 2010).

#### Understanding 2: One-to-One Correspondence

The next step is to use the memorized number-word sequence to count items in a set. A child counts with **one-to-one correspondence** when they count one number-word for each object. Many times children will physically touch each object or move objects to one side to remember which objects have been counted and which haven’t. Children who count the same item multiple times or skip over items have not yet mastered one-to-one correspondence.

#### Understanding 3: Cardinality

Counting one item at a time is an exciting achievement, but it still doesn’t answer *how many? *This is where **cardinality** comes along. Cardinality refers to the understanding that the last number counted represents the total number of items in the set. And this, ladies and gentlemen, is where children begin to form a full understanding of *number*: the word “seven” in the number-word sequence *is* the group of seven items that they just counted.

As children count a variety of sets, they’ll come to understand the concept of **conservation**. This is the understanding that the total number of items remains the same, even if you count the items in a different order or arrange them differently.

## Experiences that Build Understanding

#### Build One-to-One Correspondence

Give your child a set of physical objects to count – this could be coins, buttons, beans, paperclips, you name it. Fingers are also a great option – and they’re with you all of the time! Encourage your child to touch each object as they count and move it to one side to symbolize that it has been counted.

If your child struggles with this, that’s ok! In fact, it means that their brain is growing and learning from each mistake. Children learn from watching and from repeated practice. Start by modeling how to touch each object as you count. Then, have your child try it while you count together.

#### Build Cardinality

Once your child is routinely counting with one-to-one correspondence, start to focus on cardinality. Remember, counting one item per number-word does not necessarily mean that your child can answer *how many*! Explicitly ask, *How many are there?* to ensure your child is developing the understanding that the last number counted is how many objects there are.

If your child doesn’t answer correctly, gently correct them by modeling. Count the set out loud, touching each object as you count. Then, say how many there are in total: *“One, two, three, four, five, six. There are six acorns! You try!”*

#### Build Conservation

Once your child has started to grasp cardinality, you can deepen their understanding to include the concept of conservation.

Have your child count two sets of the same objects but put them in different arrangements. Take a look at some of the ways you can arrange a set of 6 items:

Children might initially think that each set has a different amount simply because they look different. I love this about children’s brains – of course it makes sense to think that arrangement B has more objects because it’s longer than arrangement A!

This *same but different* approach is perfect for helping children correct their initial assumptions and develop strong understandings of *number*. As children come to see that different arrangements can have the same number of items, their brains build connections and pathways around the concept of conservation: “six” can look a variety of ways, but “six” is always the same amount of items.

Similar to changing the arrangement, challenge your child to count the objects in different orders. Instead of counting left to right, count right to left. Instead of counting the top row and then the bottom row, count by columns.

## Take Action

- Determine where your child is in their understanding of
*number*by progressing through these levels. If they get stuck, that’s where they are in their understanding!- Level 1: Ask your child to recite the number-word sequence to 10.
- Level 2: Give your child a set of 5-10 physical objects to count. As they count, pay close attention to
*how*they count. Do they count one number-word per object? Do they touch each object or move it aside? If so, they have successfully mastered one-to-one correspondence. - Level 3: Give your child a set of physical objects to count and afterwards ask,
*how many*? If they can answer correctly, they’ve mastered cardinality.

- Spend 15 minutes this week counting with your child. You can do this all at once or spread it out across multiple days. Use the strategies in the “Experiences that Build Understanding” section to help your child build a strong understanding of
*number*.

## References

Feikes, David., Schwingendorf, Keith. and Gregg, Jeff. (2018) *Children’s Mathematical Learning*. Retrieved from this website.

Dougherty, et al. (2010). *Developing Essential Understanding of Number and Numeration: Pre-K-Grade 2*. National Council of Teachers of Mathematics.

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