Reasoning

Indeed, this is the reasoning that school children acquire and that largely replaces transductive reasoning. As we shall see, however, another still higher level of reasoning emerges in adolescence.

Piaget called the Aristotelian mode of reasoning concrete operations. He argued that these operations form an interconnected group much like the operations of arithmetic. Addition, for example, can be reversed by the operation of subtraction, and multiplication can be reversed by the operation of division. From this perspective, once the child has concrete operations, we can predict from his performance on one task how he will be able to perform on others.

Concrete operations enable the child to do a number of things that he was unable to do at a younger age. First, he can now follow rules and play games with rules. Moreover, because much of formal education involves the inculcation of rules, he is able to profit from such instruction. Second, he is now able to grasp units, whether these be in reading, math, science, or social studies. The unit concept presupposes the understanding that a single object can possess two properties or relationships at the same time. Third, the child is able to nest classes and relationships.

With respect to rules, consider first children's new ability to play games. In a simple board game such as Candy Mountain, the basic rule is: "Move your token as many spaces as the spinner says" (major premise). The child flicks the spinner with his finger. "The spinner says three" (minor premise). "I move my token three places" (conclusion). Young children who cannot yet engage in such reasoning are not really able to play simple board games. To be sure, they play according to a rule, but it is the rule, "I win, you lose!" Not surprisingly, older children complain that when younger children play the game, they "cheat."

Learning rules is a critical component in children's academic achievement. For example, a common spelling rule (not a very good one!) is "i before e except after c." Likewise, a common pronunciation rule is "when two vowels go walking, the first one does the talking." Both of these rules require syllogistic reasoning to be applied. In a particular instance, sayan encounter with the word speak, the child must say to himself:

When two vowels go walking! the first one does the talking [major premise].
In this word, speak, the first vowel is e [minor premise].
In this word, the e does the talking so I say "speak" [conclusion].

Children's understanding of rules at this age has far-reaching consequences. For example, it enables children to play with other children in organized games with rules.

Moreover, children get obsessed with rules and are unhappy when they are broken or when they are bypassed. The child's sense of morality is very much tied up with his sense of rules. During the early elementary school years, the child believes that breaking the rules is bad absolutely, but later, toward the age of nine or ten, he begins to take intention into account as a mitigating circumstance.

The attainment of the unit concept, the understanding that one and the same thing can be two things at once, enables the child with concrete operations to understand both elementary mathematics and basic English phonics. For example, a true understanding of number requires the child to grasp the fact that one number, say the number 3, is like every other number in the sense that it is a number but is different from every other number in its order of enumeration. It is the only number that comes after 2 and before 4. Once children understand a number in this unit sense, they can perform all of the elementary operations of arithmetic.

Something similar is required to comprehend English phonics. What is difficult about English phonics is that one vowel or consonant sound can be represented in different ways. For example, the ay vowel sound can be represented by the long a as in ate, by the vowel combination ai as in vain, or by the vowel-consonant combination ay as in say. In the same way, the consonant sound k can be represented by the letter k; by the letter c; or even by the first sound of the combination of the letters qu as in quote. To comprehend phonics, then; the child has to be able to understand that one and the same sound can be represented in different ways, depending upon the context. Once the child understands the letter as a unit that is the same and different from other units, he can understand that the letter can represent both the same and different sounds as other letters.

Concrete operations also enable children to nest classes and relationships. Nesting simply means being able to group smaller classes within larger superordinate classes and to subordinate single relationships to multiple relationships. With respect to nesting classes, imagine that you are in a kindergarten with ten boys and eight girls. The children take roll every morning and know how many children there are in the class as well as the number of boys and girls. If you ask a five-year-old child in this class whether there are more boys or girls in the class, he will instantly respond, "More boys than girls, eight girls and ten boys." If you then ask, "Are there more boys or more children in the class?" the child appears perplexed and answers, "There are more boys than girls." At this stage, the child does not have a general concept of children that subsumes the class of boys and the class of girls. By the age of eight or nine, children can nest all sort of concepts. They know that eagles and robins are birds and that tulips and roses are flowers.

Nesting family relations follows a similar pattern. For example, if you ask a five-year-old child whether he has a brother, he may answer, "Yes, his name is Robert." If you now ask the child, "Does your brother Robert have a brother?" he is likely to answer, "No." At this stage, the child cannot nest the brother relation within a pattern of relationships and grasp that if he has a brother, his brother must have a brother as well, namely, himself. Children of seven or eight, however, readily acknowledge that their brothers have brothers. Similarly, if you show a child of five or so three objects and ask whether the one in the middle is on the right or the left of the object on the right, the child will say that it is "in the middle." He is as yet unable to nest relations and see that one and the same object can be both on the right of one object and on the left of another. Older children handle this question with ease.