From an early age, children begin to develop an informal understanding of quantity and number. Careful research conducted by developmental and cognitive psychologists has mapped the progression of children’s conceptual understanding of number through the preschool years. Just as healthy children who live in language-rich environments will develop the ability to speak according to a fairly typical trajectory (from single sound utterances to grammatically correct explanations of why a parent should not turn out the light and leave at bedtime), children follow a fairly typical trajectory from differentiating more from less, to possessing the facility to add and subtract accurately with small numbers. But just as a child’s environment influences language development, it influences the rate of acquisition of number concepts. For many children whose early years are characterized by disadvantage, there is a substantial lag in the development of the number concepts that are prerequisite to first grade mathematics.
Between the ages of 4 and 6, most children develop what Case and Sandieson (1987) refer to as the “central conceptual structure” for whole number mathematics. The concepts are central in the sense that they are vital to successful performance on a broad array of tasks, and their absence constitutes the major barrier to learning. That structure involves four steps (pictured in Figure 3.1) that are developed in sequence:
FIGURE 3.1 “Mental counting line” central conceptual structure.
The bottom row of Figure 3.1 indicates that children recognize the written numerals. This information is “grafted on” to the conceptual structure above.