How Many?

When was the last time you counted something? What did you count? Why did you count it?

"How many?" It’s a simple question, but a complex set of learned skills goes into answering it. We’re able to quickly recognize small numbers of objects like  1, 2, or 3 dots, but for larger amounts, counting strategies come into play.

How many dots do you see?

How did you determine the number of dots?

• Count by 1s
• Count by 2s
• Use a 5

Maybe you used an entirely different strategy. Counting is the first mathematical skill we learn, and over time we develop different strategies to count more efficiently.

Names for Number

Before we can count a set of objects, we have to learn the words for numbers. Much like learning the names of letters in the alphabet before knowing their sounds, young children learn and practice number names before knowing the quantity they represent.

We learn our number names in stages, first learning to count to 10, then to 20, then to 100, and so forth.

One-to-One Correspondence

After memorizing number names we learn to connect them to a sense of quantity, giving us our first mathematical tool, counting one by one.  

One-to-one correspondence is a skill that involves pairing a number name with each object in a set once and only once, identifying each object’s number in the set as it is counted. Practicing this one-to-one counting by physically moving or touching each object as it’s counted helps young learners begin to develop their sense of number.   

Soon, this child will be able to count with one-to-one correspondence without moving the cubes but by pointing to them and mentally tracking which cubes have been counted.

Cardinality

As we continue to count numbers of objects, something else begins to develop—cardinality. Cardinality refers to the total number of objects in a group.

The last number we say when counting matches the last item in our set and is the total number of items being counted; it’s the cardinal number. 

To determine the cardinal number after a block is added, the child recounts from 1. What might she do next to determine the total more efficiently?

Ordinality

As our number sense develops, we learn that numbers have order, and that each number we count is one more than the previous number.

We begin to identify numbers by their order and position in a sequence of numbers. These numbers—first, second, third, and so on—are called ordinal numbers. 

Numerals

In early childhood we also learn the symbols for numbers, like the symbols "3" and "8" for the numbers “three” and “eight.”

These symbols are called numerals, and they enable  us to efficiently identify any number in our number system. This is an important foundational skill needed for mathematical work.

Have you ever seen a child write numerals backwards or upside down? Numerals, 6 and 9, can be particularly confusing for children.

Number Paths

By labeling each successive dot with its numeral, we create a number path, a basic visual model that helps us see numbers in sequence, growing by “one” with each dot.

In this number path, what is the  number that is 2 more than 5?  

Once children have a strong sense of the sequence of numbers, they begin to identify larger or smaller numbers without recounting.

What About Zero?

What’s 1 less than 1 in this number path?

How do we show zero dots? Do we add a dot to the left and put the numeral "0" on it?

Because zero has no quantity, it’s not considered to be a counting number. Zero represents an empty set where there is nothing to count.

The next time you count something, think about the counting skills you use. Notice how your math brain uses subitizing, one-to-one correspondence, cardinality, and words and numerals to identify, represent, and communicate quantity.

Simple counting skills provide the foundation for counting much larger quantities, like the number of pennies in a piggy bank. Is there a more efficient strategy for counting large groups of pennies than counting them one by one? In the next chapter, we’ll explore ways to count large quantities by grouping and arranging objects.