How Do Children Learn to "Think Logically"?

In early education, we design many activities that teach children to classify, order, and perceive quantity, and we have them carry out hands-on manipulations throughout. Why do we do this? We have, at most, a vague sense that it cultivates children's mathematical literacy and familiarizes them with simple forms of formal (logical) thinking, in preparation for their future capacity to learn more difficult and complex mathematical reasoning. Yet when we view the matter this way, our attention is usually fixed on the content of learning: young children can understand "1+1=2," while high school students can understand second derivatives. Learning is thus regarded as the continual acquisition of ever "more complex" and "more difficult" knowledge, so that the differences between children of different ages are reflected merely in differences in learning content. This conception is indeed widespread in practice. Early childhood educators are seen as teaching knowledge that is "very simple," and hence the professional threshold is held to be low, whereas higher education is seen as teaching knowledge that is "very difficult," and hence the professional threshold is held to be high. This has become a commonly held view, resulting in an "inequitable" hierarchy within the education system itself. Few educators are able to approach education itself from a more "complete" and "holistic" perspective; that is, they cannot fully perceive the continuity of individual cognitive development. In other words: what, in fact, changes cognitively as a child moves from operations that can only be carried out through counting, ordering, and classifying, to the capacity to employ formulas in formal operations?

As educators, we should detach ourselves from specific knowledge in order to understand children's learning as such. That is, what we should attend to first is not how to enable children to gradually master more complex reasoning, such as how to understand a particular formula; nor do we need to become logicians in order to prove which forms of reasoning are valid. Rather, we should attend to how children's modes of cognition are progressively changing, that is, to how the manner in which the brain processes information undergoes transformation.

Drawing on psychological experiments, Piaget offers an explanation of this process of change. First, let us consider how we think. In daily life, we engage in roughly two forms of thinking. One is "natural thought," such as "What shall I eat for lunch today?" The other is "logical thinking," namely formal thinking: proceeding from premises, according to rules, to the derivation of conclusions. Piaget holds that "natural thought" manifests itself in the large number of operational actions that children perform. In the course of these, children carry out concrete actions, such as taking an apple out of a basket and putting it back, and discover that "putting in" and "taking out" can cancel each other. From this they abstract the "law of identity" (A is A; it cannot be both A and not A at the same time). Likewise, in the action of "taking out one apple and then taking out another apple, yielding two apples," the child discovers the abstract form "1+1=2." Such "natural thought," which takes direct life experience as its vehicle, accumulates abundant material for the later emergence of "logical thinking." Piaget employs a key concept, "reflective abstraction," to define this process: the brain, from actions or operations it has already performed, "reflects" out higher-order rules. The more numerous and varied such operations are, the richer the "rules of formal operation" the child reflects out. The process by which children logically formalize concrete action-based experience is, in essence, the gradual formation in the brain, as humans solve problems, of the co-ordination of schemes of action ( unions, inclusions, order, correspondences, etc.). Therefore, for educators seeking to understand the development of children's logical thinking, attention should be directed to whether children can continually enrich the internal "action-operation" schemes of the brain from direct experience, attending to the development of cognitive capacity rather than to cognitive outcomes. That is, teachers need not, like logicians, concern themselves with particular inferences grounded in axioms or hypotheses, such as the reasoning of Euclidean or non-Euclidean geometry.

Piaget's central claim is that what is logically formalized is neither the physical world nor language per se, but the human subject's "structure of action-operations." The formulas that mathematicians write on paper (formalized systems) are not inventions but discoveries. What they discover is the "skeleton of thought" that has been deposited in the human brain through millions of years of evolution and through development from infancy to adulthood, by way of countless "action-operations" (classifying, ordering, cancelling, combining).

In addition, we can understand this arguement more specifically through Piaget's classic conservation experiment. In children's intellectual development, there is a highly advanced cognitive capacity called "conservation." For example, when a glass of water is poured into a tall, thin glass, the water appears to have increased (the water level is higher). Yet a child at the concrete operational stage or the formal operational stage will say: the total quantity of water has not changed. Why can the child reach this conclusion? It is not because he has seen water molecules (the physical world), nor because he has memorized a formula (language). Rather, it is because a set of "action-operations" has been carried out in his brain: "the water has become taller," but at the same time "the water has become narrower." His brain has "co-ordinated" these two actions, so that they cancel each other out, yielding the logical conclusion that "the total quantity is unchanged." By contrast, a child at the preoperational stage can attend to change in only one dimension and cannot co-ordinate actions so as to "operate" on them together.

Understanding, from the perspective of "action-operations," how children's cognitive capacity transitions from "natural thought" to "logical thinking" helps us, in early education practice, to emphasize the cultivation of children's cognitive capacity, that is, the cultivation of learning potential, rather than narrowly focusing on the mastery of specific knowledge at any given stage.