As children continue into formal schooling, they become able to represent ideas and events more flexibly and logically. Their rules of thinking still seem very basic by adult standards and usually operate unconsciously. These rules allow children to solve problems more systematically than before, and children at this stage can be successful with many academic tasks.
In the concrete operational stage, for example, a child may unconsciously follow the rule: “If nothing is added or taken away, then the amount of something stays the same.” This simple principle helps children to understand certain arithmetic tasks, such as in adding or subtracting zero from a number, as well as to do certain classroom science experiments, such as ones involving judgments of the amounts of liquids when mixed.
Piaget called this period the concrete operational stage because children mentally “operate” on concrete objects and events. They are not yet able, however, to operate (or think) systematically about representations of objects or events. Manipulating representations is a more abstract skill that develops later, during adolescence.
Concrete operational thinking differs from preoperational thinking in two ways, each of which renders children more skilled as students.
One difference is reversibility, or the ability to think about the steps of a process in any order.
Example 1 of reversibility:
Imagine a simple science experiment, such as one that explores why objects sink or float by having a child place an assortment of objects in a basin of water. Both the preoperational and concrete operational child can recall and describe the steps in this experiment, but only the concrete operational child can recall them in any order.
This skill is very helpful on any task involving multiple steps—a common feature of tasks in the classroom.
Example 2 of reversibility:
In teaching new vocabulary from a story, a teacher might tell students: “First make a list of words in the story that you do not know, then find and write down their definitions, and finally get a friend to test you on your list”.
These directions involve repeatedly remembering to move back and forth between a second step and a first—a task that concrete operational students—and most adults—find easy, but that preoperational children often forget to do or find confusing. If the younger children are to do this task reliably, they may need external prompts, such as having the teacher remind them periodically to go back to the story to look for more unknown words.
The other new feature of thinking during the concrete operational stage is the child’s ability to decentre, or focus on more than one feature of a problem at a time. There are hints of decentration in preschool children’s dramatic play, which requires being aware on two levels at once—knowing that a banana can be both a banana and a “telephone”.
But the decentration of the concrete operational stage is more deliberate and conscious than pre-schoolers’ make-believe. Now the child can attend to two things at once quite purposely.
Example of decentration:
Suppose a teacher gives students a sheet with an assortment of subtraction problems on it, and ask them to do this: “Find all of the problems that involve two-digit subtraction and that involve borrowing’ from the next column. Circle and solve only those problems.” Following these instructions is quite possible for a concrete operational student because the student can attend to the two subtasks simultaneously—finding the two-digit problems and identifying which actually involve borrowing.
In real classroom tasks, reversibility and decentration often happen together. A well-known example of joint presence is Piaget’s experiments with conservation, the belief that an amount or quantity stays the same even if it changes apparent size or shape (Piaget, 2001; Matthews, 1998).
Imagine two identical balls made of clay. Any child, whether preoperational or concrete operational, will agree that the two indeed have the same amount of clay in them simply because they look the same. But if one ball is squished into a long, thin “hot dog”, the preoperational child is likely to say that the amount of that ball has changed—either because it is longer or because it is thinner, but at any rate because it now looks different. The concrete operational child will not make this mistake, thanks to new cognitive skills of reversibility and decentration: for him or her, the amount is the same because “you could squish it back into a ball again” (reversibility) and because “it may be longer, but it is also thinner” (decentration). Piaget would say the concrete operational child “has conservation of quantity”.
The classroom examples described above also involve reversibility and decentration. As already mentioned, the vocabulary activity described earlier requires reversibility (going back and forth between identifying words and looking up their meanings); but it can also be construed as an example of decentration (keeping in mind two tasks at once—word identification and dictionary search).
And as mentioned, the arithmetic activity requires decentration (looking for problems that meet two criteria and also solving them), but it can also be construed as an example of reversibility (going back and forth between subtasks, as with the vocabulary activity).
Either way, the development of concrete operational skills support students in doing many basic academic tasks; in a sense they make ordinary schoolwork possible.
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