For example, preschool children know the difference between larger and smaller objects or amounts. They know that drinking milk from a glass reduces the amount inside the glass and that pouring milk into the glass increases its contents. They know that a quantity remains constant if nothing is added or subtracted. They know that if an object is cut into pieces, the pieces taken together contain the same amount of material as the uncut object. And most of them know both how to count and how to use counting to quantify sets of objects.
Children regard such knowledge as self-evident.
What preschool children do not know, and what formal instruction has traditionally been assigned to teach them, are the rules, symbols, and language of mathematics and how to use them in calculations. In this case, the gap exists between mathematical intuitions that develop easily and almost universally and the difficulties that arise in the learning of formal mathematical notations and procedures in school. She further suggests that the reason for the discontinuity is that children's intuitions arise directly from their real-world experiences, whereas formal instruction in math requires them to reason about abstractions such as numbers and operators that they cannot experience directly. The student writes 9 correctly but fails to continue borrowing from the next column to the left. Thus, the mechanical operations appear to break down at the point where their connection to the meaning of the problem is lost an error that reflects the gap between intuitive and formal processes of learning.