The most fundamental concept in elementary school mathematics is that of number, specifically whole number. To get a sense of both the difficulty of the concept and how much of it is taken for granted, try to define what a whole number is.
One common conception of whole number says that two sets have the same numerosity (same number of members) if and only if each member of one set can be paired with exactly one member of the other (with no members left over from either set). If one set has members left over after this pairing, then that set has a greater numerosity (more items in it) than the other does.
This definition allows one to decide whether two sets have the same number of items without knowing how many there are in either set. The Swiss psychologist Jean Piaget developed a task based in part on this definition that has been widely used to assess whether children understand the critical importance of this one-to-one correspondence in defining numerosity.1 In this task, children are shown an array like the one below, which might represent candies. They are then asked a question like the following: Are there more light candies, the same number of dark and light candies, or more dark candies?
Most preschoolers recognize that the sets have the same amount of candy, based on the one-to-one alignment of the individual pieces. Next, the child watches the experimenter spread out the items in one set, which alters the spatial alignment of the pieces:
Shown this diagram, many children younger than 5 years assert that there are more of whichever kind of candy is in the longer row (the light candies in this example). Piaget argued that a true understanding of number requires an ability to reason about the effects of transformations that is beyond the capacity of preschool children. It was not uncommon several decades ago for educators aware of Piaget’s findings and his claims to make assertions such as