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When kindergarten teacher Ms. Taylor asks Janie to count five randomly arranged chips, Janie counts some objects more than once, overlooks others, and announces that there are 12. Ms. Taylor then models counting the chips by moving them one by one while saying "One, two, three..."
MANY 5-YEAR-OLDS CAN IMITATE THE teacher but then return to their own way of counting the same objects more than once and overlooking the others. Why is it so hard for young children to count five objects? The answer lies in the nature of logico-mathematical knowledge that Piaget studied scientifically for more than 50 years.
Three kinds of knowledge
Piaget made fundamental distinctions among three kinds of knowledge, according to their sources: physical knowledge, social-conventional knowledge, and logicomathematical knowledge (Piaget [1945] 1951; [1967] 1971).
Physical knowledge
This is knowledge of objects in the observable world. Knowing that marbles roll but counters don't is an example of physical knowledge. Knowing that paper tears easily but cloth does not is also an example of physical knowledge. Physical knowledge is acquired through actions on objects and observation.
Social-conventional knowledge
An example of social-conventional knowledge is languagefor example, understanding the words one, two, three or uno, dos, tres. Knowing about the Thanksgiving holiday is also an example of social knowledge. Social-conventional knowledge's ultimate source is conventions that people create over time.
Logico-mathematical knowledge
While physical and social-conventional knowledge have sources in the external world, the source of logicomathematical knowledge is inside each person's mind. If I show you a red counter and a blue counter, you will probably agree that the two counters are different. In this situation, the difference between the counters might erroneously be believed to be observable. But the difference does not exist in the observable world and is therefore not observable. The same two counters can also be said to be similar-because they have the same shape, are made of the same material, and are the same size. However, similarity and difference are mental relationships-they exist only in the minds of the people who think about the objects as being similar or different. A third mental relationship we can create between the red and blue counters is the numerical relationship "two." When we think about the counters as "two,"...