Number Sense involves important aspects of counting, number relationships, and operations.
Preschool children first understand the concept of quantity through their experiences with small groups of items without counting the number of items in the group. For example, a child might see two groups of chocolate chips and perceive that one group has more chips in it than the other without knowing how many chips are in each group. These experiences allow young children to develop the ability to compare groups and know if they are the same or if one is larger or smaller than the other.
Eventually, preschool children will understand the concept of cardinality. That is, children will know that the last number named when counting a group of object is the quantity of objects in that group.
Building on a child’s capacity to recognize quantitative differences between small groups of objects, a child will develop a basic understanding of mathematical operations as they . By “adding” two small groups of objects together, the child now has one larger group of objects. They will “put together”, “take apart”, and “take away” small groups of objects. Through these experiences, children develop an informal understanding of mathematical operations.
Preschool-aged children learn number names and recite numbers in order from memory. As they begin to count they also begin to understand one-to-one number correspondence, the ability to match an object to its corresponding number. You can check to see if your child understands this concept by asking them to count a set of objects. Children who understand one-to-one correspondence will be able to coordinate number names with the corresponding number of items. If you hear more number names than objects or more objects than number names this concept is not yet fully developed.
Further Reading: http://www.cde.ca.gov/sp/cd/re/documents/preschoollf.pdf, page 145 and following.