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piaget's ideas about the formal operations stage are often misunderstood. teachers who use them often speak of children going from concrete operations to formal operations, as if there is an abrupt change. piaget, however, was always cautious about judging and had clear ideas about how children move from one stage to the next. he was unimpressed with the idea that children go from concrete operations to formal operations without any idea at all of what they were doing. piaget argued that the 'formal operations' stage refers to the period of life when children begin to make logical inferences and think from symbols to meaning.
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in addition, the logical operations of the "formal operations" stage are always governed by an evaluative criterion, which determines whether the operation is sufficient to produce a final product of some kind. a child who does a grade five algebraic equation of the form 2x3 = 4 may see that the operation produces a final product of interest to him, but he will not know the mathematical reasons why he arrived at that answer. such reasons are always available (as are similar reasons in the other domains of life), but if they are not present, the child will not be able to arrive at that answer.
since it is only the formal operations that have such an "informal" evaluative criterion, the operational stages are not more and more concrete, but simply reflect the growth of the ability to take account of the perspectives of others. this makes the changing into the formal operations a gradual, reflective process, which requires that the child be able to understand situations from the perspective of someone else. the possibility of thinking about an event in terms of its consequences for oneself and others depends on a concrete, practical sense of time (this is what piaget calls the "present" or "hypothetical" sense of time). the child's "hindsight," to put it in current psychological jargon, is thus formed in an iterative way, as he reflects on his prior experiences: "if i had been in a similar situation before, then i must be able to account for such situations as this" (e. piaget, the moral development of the child, new york: basic books, 1955. piaget's more recent book on creativity, vers une biologie generative, is published in france as la connaissance de l'invention. paris: plon, 1992).
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