In addition to using counting to solve simple arithmetic problems, preschool children show understanding at an early age that written marks on paper can preserve and communicate information about quantity.^{35} For example, 3- and 4-year-olds can invent informal marks on paper, such as tally marks and diagrams, to show how many objects are in a set. But they are less able to represent changes in sets or relationships between sets, in part because they fail to realize that the order of their actions is not automatically preserved on paper.

Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations and to justify and ultimately prove the correctness of a mathematical procedure or assertion. Adaptive reasoning also includes reasoning based on pattern, analogy, or metaphor. Research suggests that young children are able to display reasoning ability if they have a sufficient knowledge base, if the task is understandable and motivating, and if the context is familiar and comfortable.^{36} In particular, preschool children can generate solutions to problems and can explain their thinking.

Situations that require preschoolers to use their mathematical concepts and procedures in unconventional ways often cause them difficulty. For example, when preschool children are asked to count features of objects (e.g., the tines of forks) or subsets of objects (e.g., just the red buttons in a mixed set), they often cannot overcome their tendency to count all the separate objects.^{37}

Another example of the limitations on preschoolers’ ability to generalize their mathematics is that they perform better in situations that require them to think about adding or subtracting actual objects (even if those objects are hidden from view in a box) than they do when simply asked an equivalent question (e.g., “What’s 3 and 5?”).^{38} Four- and 5-year-olds do begin to use their knowledge to answer correctly the Piagetian number task presented above involving equivalent sets of candies, and later they recognize without counting that the sets have the same number of candies.^{39}

A major challenge of formal education is to build on the initial and often fragile understanding that children bring to school and to make it more reliable, flexible, and general.

Most preschool children enter school with an initial understanding of procedures (e.g., counting, addition, subtraction) that forms the basis for much of their later mathematics learning, although they have limited ability to generalize that knowledge and to understand its importance. A major challenge of formal education is to build on the initial and often fragile understanding that children bring to school and to make it more reliable, flexible, and general.^{40}