Learning to count is important for all children in developing their math skills. The idea of number sense is the basis for all later math learning. Counting is a companion skill to numerosity which is the ability to discriminate sets of objects on the basis of quantity of items. Learning to count is a first step into mathematics and forms the basis for the many important fundamental concept in mathematics.
Children around the age of two years old can identify small numbers of objects usually 1, 2, or 3 objects. Around the age of 4 years old, children can recognize groups of objects up to 4 without counting. The ability to recognize the number of objects in a small set without counting is called subitizing. Even into adult life we continue to use our ability to subitize. However even adults can’t seem to recognize groups of larger than 5 objects unless they are arranged in a particular way that aids in memorization. Subitizing refers to the mind’s ability to recognize specific patterns and associate them with a fixed number. You can think of patterns on dice or a deck of cards as an example of how we use subitizing instead of counting.
Researchers believe we are born with a certain sense of numbers. Some infants as early as 18 months can recognize number patterns. In mathematics, number sense refers to number meaning, relationships, magnitude, operations, and referents for numbers and quantities.
Counting is the companion skill to numerosity. Counting requires language using number words. The objective of counting is to count a set of objects. We learn counting through verbal exchanges, songs, and games parents play with their children. Although young children can develop the ability to count. The skill of counting is objects is quite involved.
There are typically 5 concepts associated with counting. Firstly there is a one-to-one correspondence between one and only one number word and one and only one object to be counted. Number words have a specific order that must be followed in order. Cardinality is the idea that the last number word counted in the sequence of number words represents the numerosity, or amount, of the set of objects counted. When we count a set of objects the order in which we count those objects is irrelevant. So we can count a set of objects in any order we want. Lastly we have the idea of abstraction in counting which means we can count any set of objects, even completely dissimilar objects and still arrive at the amount of objects through counting.
Children learn to follow and understand the principles of counting early on and mastery of counting becomes an important foundational skill to acquire higher level math skills. The principles of counting show us that counting is an ordered sequence of number names specific to each language and culture. When we extend the counting sequence beyond 0 through 9, the counting sequence repeats certain aspects of previously used number names. Unfortunately for cultural reasons, western languages do not strictly follow this principle when counting from 11 to 19 by introducing additional new counting words like eleven, twelve, etc… Asian languages more closely follow the naming principle by repeating the number words instead of introducing new words. For example in Japanese the number name for eleven is “ten – one”. This reuse of the number names also begins to reinforce the notion of place value. So children simply counting are learning the fundamental number concept of place value and repeating the same 10 digit names over and over. As children learn to count assigning exactly one number name from the counting sequence to each object in the set, they learn that the quantity or amount of objects is the last number name used in the sequence, cardinality. To understand quantity and later how to manipulate quantities, children need a lot of counting experience. In order to perform more sophisticated mathematics like addition and subtraction, counting needs to become automatic. Just like when children are first learning to multiply multi-digit numbers they need to have first memorized their multiplication facts through 9×9. Counting has the same importance for children when they are first learning addition and subtraction. Children need lots of practice even well beyond counting to 20 or 30.
Let’s take a moment to explore the importance of this idea of cardinality in mental math. When counting children learn they need to keep track of cardinality while counting so at the end of the counting sequence they have the answer. In the abacus world students do something similar to cardinality by only tracking the last state of their mental image of an abacus. So when the calculation is done, the brain simply translates the abacus image into its numerical representation which is the cardinal value.
In counting, we learn there are patterns. Our base 10 numbering system is design to keep reusing the same digits, 0 – 9, over and over again as we shift in place value. As we count we begin to develop the ability to discriminate the magnitude between two numbers. Simply saying the counting numbers alone does not necessarily help children understand magnitude. But by counting sets of objects children gain the experience to connect numbers with different magnitudes.
What do we learn from counting? Children learn the concept of more or less in their introduction to magnitude. Through counting, children can see there are more objects in one set over another. Counting shows children that numbers are ordered. Order means the next item is more than the previous but less than the one following it in counting a set of objects.
Fluency in learning the counting words helps children learn new math skills later on. As children memorize the number words they understand order is important and numbers are different from letters since they can not skip or repeat words.
Using words like same, more, or less are meaningful but lack precision. The process of counting give us precision in determining the exact number of a set. The cardinal number tells us how many objects there are. Enumeration gives children a process to say the number words in correct order, match one number word with only one object, and count each object only once. Counting is the most important mechanism used by children in estimating numbers of all sizes.