arithmetic operations of adding and subtracting. Through their surprise or search reactions, young children are able to tell us when an item is added or subtracted from what they expected (Wynn, 1990, 1992a, b; Starkey, 1992). For example, 5-month-old infants first saw two objects repeatedly; then a screen covered the objects and they watched as an experimenter proceeded to add another object or remove one from the hidden display. The screen was then removed, revealing one more or one less item than before. In both the less and more conditions, infants looked longer at the numerically “incorrect” display—that is, the unexpected value that did not correspond to their initial training; if they saw one added, they expected three, not one, and vice versa (Wynn, 1992a, b).
Experimental evidence of this kind implies a psychological process that relates the effect of adding or removing items to a numerical representation of the initial display. A similar line of evidence with preschool children indicates that very young children are actively engaged in using their implicit knowledge of number to attend to and make sense of novel examples of numerical data in their environments; see Box 4.2.
There are many other demonstrations of young children’s interpreting sets of objects in terms of number. Together, the findings indicate that even young children can actively participate in their own learning and problem solving about number. This ability is why children often deal with novel conditions rather well, as when they tell puppets who are “just learning to count” if they are correct and if they are wrong or even invent counting solutions (Groen and Resnick, 1977; Siegler and Robinson, 1982; Starkey and Gelman, 1982; Sophian, 1994).
But just because children have some knowledge of numbers before they enter school is not to say that there is little need for careful learning later. Early understanding of numbers can guide their entry into school-based learning about number concepts. Successful programs based on developmental psychology already exist, notably the Right Start Program (Griffin and Case, 1997). Although making the entry levels easier, these early number concepts can also be problematic when it comes to the transitions to higher-level mathematics. Rational numbers (fractions) do not behave like whole numbers, and attempting to treat them as such leads to serious problems. It is therefore noteworthy that many children experience just these sorts of problems in mathematics when they encounter “fractions”: They believe the larger number always represents a bigger quantity or larger unit.
We introduced the idea that children come equipped with the means necessary for understanding their worlds when considering physical and biological concepts. It should not be surprising that infants also possess