The concrete operations stage begins at age six or seven, when the schoolage child becomes capable of keeping in mind and logically manipulating several concrete objects at the same time. The child is no longer the prisoner of the momentary appearance of things. In no case is the change more evident than in the sort of problem in which a number of objects (such as twelve black checkers) are spread out into four groups of three. While the four-year-old, preoperational child would be likely to say that now there are more checkers because they take up a larger area, to the eight-year-old it is obvious that this transformation could easily be reversed by regrouping the checkers. Piaget describes the capacity to visualize the reversibility of such transformations as "conservation." This understanding is fundamental to the comprehension of simple arithmetical manipulations. It is also fundamental to a second operational skill: categorization. To the concrete-operational child, it seems obvious that while Rover the dog can for other purposes be classified as a household pet, an animal, or a living organism, it will still be a "dog" and still be "Rover." A related skill is seriation: keeping in mind that an entire series of objects can be arranged along a single dimension, such as size (from smallest to largest). The child now is also capable of role-taking, of understanding the different perspective of a parent or teacher. No longer egocentric (assuming that everyone shares one's own perspective and cognitively unable to understand the different perspective of another), the child becomes able to see himself or herself as others see him or her and to temper the harshness of absolute rules with a comprehension of the viewpoints of others.
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