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teenager or adult could -- with a little time and effort. Consider this rule about a set of cards that have letters on one side and numbers on the other: “If a card has a vowel on one side, then it has an even number on the other side.” Take a look at the cards below and tell me, which cards do I need to turn over to tell if this rule is actually true? You’ll find the answer at the end of this chapter.
It is the formal operations stage that allows one to investigate a problem in a careful and systematic fashion. Ask a 16 year old to tell you the rules for making pendulums swing quickly or slowly, and he may proceed like this:
A long string with a light weight -- let’s see how fast that swings. A long string with a heavy weight -- let’s try that. Now, a short string with a light weight. And finally, a short string with a heavy weight.
His experiment -- and it is an experiment -- would tell him that a short string leads to a fast swing, and a long string to a slow swing, and that the weight of the pendulum means nothing at all!
The teenager has learned to group possibilities in four different ways:
By conjunction: “Both A and B make a difference” (e.g. both the string’s length and the pendulum’s weight).
By disjunction: “It’s either this or that” (e.g. it’s either the length or the weight).
By implication: “If it’s this, then that will happen” (the formation of a hypothesis).
By incompatibility: “When this happens, that doesn’t” (the elimination of a hypothesis).
On top of that, he can operate on the operations -- a higher level of grouping. If you have a proposition, such as “it could be the string or the weight,” you can do four things with it:
Identity: Leave it alone. “It could be the string or the weight.”
Negation: Negate the components and replace or’s with and’s (and vice versa). “It might not be the string and not the weight, either.”
Reciprocity: Negate the components but keep the and’s and or’s as they are. “Either it is not the weight or it is not the string.”
Correlativity: Keep the components as they are, but replace or’s with and’s, etc. “It’s the weight and the string.”
Someone who has developed his or her formal operations will understand that the correlate of a reciprocal is a negation, that a reciprocal of a negation is a correlate, that the negation of a correlate is a reciprocal, and that the negation of a reciprocal of a correlate is an identity (phew!!!).
Maybe it has already occured to you: It doesn’t seem that the formal operations stage is something everyone actually gets to. Even those of us who do don’t operate in it at all times. Even some cultures, it seems, don’t develop it or value it like ours does. Abstract reasoning is simply not universal.
(Reference. Beilen, H. (Mar 1992). Piaget's enduring contribution to developmental psychology. Developmental Psychology, 28, 191-204.)
(Reference. Papert, S. (1992). The Children's Machine: Rethinking School in the Age of the Computer. New York: Basic. 137-156.)
It is the formal operations stage that allows one to investigate a problem in a careful and systematic fashion. Ask a 16 year old to tell you the rules for making pendulums swing quickly or slowly, and he may proceed like this:
A long string with a light weight -- let’s see how fast that swings. A long string with a heavy weight -- let’s try that. Now, a short string with a light weight. And finally, a short string with a heavy weight.
His experiment -- and it is an experiment -- would tell him that a short string leads to a fast swing, and a long string to a slow swing, and that the weight of the pendulum means nothing at all!
The teenager has learned to group possibilities in four different ways:
By conjunction: “Both A and B make a difference” (e.g. both the string’s length and the pendulum’s weight).
By disjunction: “It’s either this or that” (e.g. it’s either the length or the weight).
By implication: “If it’s this, then that will happen” (the formation of a hypothesis).
By incompatibility: “When this happens, that doesn’t” (the elimination of a hypothesis).
On top of that, he can operate on the operations -- a higher level of grouping. If you have a proposition, such as “it could be the string or the weight,” you can do four things with it:
Identity: Leave it alone. “It could be the string or the weight.”
Negation: Negate the components and replace or’s with and’s (and vice versa). “It might not be the string and not the weight, either.”
Reciprocity: Negate the components but keep the and’s and or’s as they are. “Either it is not the weight or it is not the string.”
Correlativity: Keep the components as they are, but replace or’s with and’s, etc. “It’s the weight and the string.”
Someone who has developed his or her formal operations will understand that the correlate of a reciprocal is a negation, that a reciprocal of a negation is a correlate, that the negation of a correlate is a reciprocal, and that the negation of a reciprocal of a correlate is an identity (phew!!!).
Maybe it has already occured to you: It doesn’t seem that the formal operations stage is something everyone actually gets to. Even those of us who do don’t operate in it at all times. Even some cultures, it seems, don’t develop it or value it like ours does. Abstract reasoning is simply not universal.
(Reference. Beilen, H. (Mar 1992). Piaget's enduring contribution to developmental psychology. Developmental Psychology, 28, 191-204.)
(Reference. Papert, S. (1992). The Children's Machine: Rethinking School in the Age of the Computer. New York: Basic. 137-156.)
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