This post is part of the Little Brains, Big Concepts series. Check out the previous posts: Introduction, Number.
Counting plays a critical role in understanding number. Once children begin to understand the concept of number, they start to develop counting strategies that utilize deeper understandings of the number-word sequence.
Counting strategies are the foundation for addition and subtraction and are an essential building block for future math understanding. All children go through the same stages of counting strategies, and each stage provides a fascinating view into how children’s little brains learn big concepts.
The Big Concept
The order of numbers is one of inclusion. Let’s use these cute penguins to explore this concept.
The set of three penguins, outlined in yellow, are included in the set of four penguins, outlined in orange. This highlights an important relationship in the number sequence: when we add 1 to a number, the previous number is included in the new number. We can see in the equation 3 + 1 = 4 that both 3 and 1 are included in 4.
You can also think of included in as part of. Inclusion is closely connected to the concept of part and whole (which we’ll dive deeper into next week!). In the above example, 4 is the whole, and 3 and 1 are the parts.
So why does this matter? The understanding of inclusion is a key step in the process of developing efficient counting strategies. Counting strategies are the foundation for addition and subtraction and can be powerful tools for developing mental math strategies:
So what are counting strategies? Counting strategies refer to how a child joins two groups of objects. Due to the amazing nature of the human brain, all children go through the same series of stages as they develop more efficient counting strategies. Each new stage means a deeper understanding of counting, and inclusion is integral to reaching the final stage.
Let’s explore the fascinating journey through the levels of counting strategies with this example: We start with 3 penguins, then we add four more penguins. How many penguins are there?
Level 1: Count All
In this stage, children start by counting the first three penguins: “one, two, three”. Then, they count the added four penguins: “one, two, three four”. Finally, they count all seven penguins starting at 1: “one, two, three, four, five, six, seven.”
Level 2: Count-All from the first number
In this stage, children still count all of the objects, but they don’t need to count the second set of objects twice. They count the first three penguins (“one, two three”) and continue counting the set of four penguins (“four, fix, six, seven”).
Level 3: Count-on
In this stage, children don’t start by counting the number of items in the first set. Instead, they start with the number in the first set, “three” and then count the items in the second set, “four, five, six, seven.” Reaching the count-on phase is a huge step and demonstrates a new level of sophistication in children’s understanding of number.
Reaching this stage means a child has developed an understanding of inclusion! They know that the set of three is included in the combined set of seven, so they don’t need to count it.
Level 4: Count-On from the larger number
The last phase in the counting strategy progression is counting on from the larger number. Children recognize that the second set with four penguins is larger, so they start there (“four”) and then count the three penguins in the second set (“five, six, seven”). This increase in efficiency is an important step that creates space for more complex numbers later on.
Experiences that Build Understanding
Children go through the above stages naturally and tend to begin counting-on around kindergarten or first grade. The best way to help your child develop an understanding of inclusion is to count – a lot! As children repeat the counting process, their understanding of number and the number-word sequence deepens.
When counting, use these general tips:
- Use physical objects that your child can touch and move as they count.
- Have your child touch each object as they count out loud.
- Ask, “how many are there?” after counting a set of items.
Count a Combined Set of Objects
Provide your child with two sets of the same objects (small erasers, beans, cubes, Legos, etc.), and have them find the total. Display one set and count how many together. Then, display the second set and ask, “If we add three more, how many do we have now?” By counting the first set before you display the second set, your child will start to realize that they recently counted that group and don’t need to count it again.
Number-word Sequence Practice
To count-on, children need to be able to start counting from any number in the number-word sequence. For example, given six, a child needs to know that seven is the next number without having to start counting at one. Develop this by counting up to a number, then pause and ask your child what number comes next. You say, “one, two, three, four, five…what comes next?” Then your child says “six!”
- Show your child two sets of the same item, one set of 3 and one set of 2. Ask, “If we have three and we add two more, how many do we have?” Then, pay attention to how they count the total to see where they are in the counting strategy stages.
- Spend intentional time counting with your child this week. Count toys as you pick them up, count the number of carrots you need for dinner, count how many crackers they get for a snack – whatever it takes to get your child counting!
- Get excited for next week’s Little Brains, Big Concepts post on Part-Whole Relationships!
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.