With careful examination of infants, toddlers, preschoolers, and children in primary grades, one can see evidence of all these criteria.
Young children may not be able to add or subtract, but the relationships they are forming and their interactions with a stimulating environment encourage them to construct a foundation and framework for what will eventually become mathematical concepts (Powell & Butterworth, 1971; Butterworth, 1999a, 1999b, 1999c). If you watch long enough you will see some amazing mathematical thought going on in even the youngest children (Butterworth, 1999). Consider the following example:
An 18-month-old child playing in a large pit filled with different colored balls drops one ball, then a second ball, and then a third ball over the side of the pit. The child then goes to the opposite side of the pit and drops two balls. He then goes back to the first side, reexamines the grouping of balls, moves to the second side and drops another ball over the side to make a grouping of three.
The coordination and comparison of “threes” on opposite sides of a structure is clear evidence of this 18-month-old child making a mathematical relationship and putting order into his world. It is not yet a numerical relationship because the child is solely using visual perception—what he sees—to make the judgment of “same” or “different.” However, the coordination of dropping three balls each time is evidence of an early understanding of “more,” “less,” and basic equality. This child may not be developmentally ready for number concepts like counting and quantification, which we will discuss in Part II, but this simple task shows that children as young as 18 months are making relationships and exercising their logical thought processes. Teachers of infants and toddlers need to be aware of these actions and abilities and provide activities to encourage construction of these mathematical concepts. Emergent mathematics continues throughout early childhood. Encouraging children to construct many different relationships between and among objects, to interact with other children and adults, and to mentally and physically act on the objects around them promote the concept of building we refer to in discussions of emergent mathematics.
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