So if a man's wit be wandering, let him study the mathematics; for in demonstrations, if his wit be called away never so little, he must begin again. - Francis Bacon, "Of Studies"
Bruner's CONCRETE-PICTORIAL-ABSTRACT approach is the basis for all Early Childhood educator. The notion of using concrete materials and giving concrete experiences is a long standing method of working with children.
To "teach" or present a concept to a child, we must first present a basis for the child.
Instead of merely presenting a rule, as many schoolbooks do, a better way is to teach children to visualize fractions, and perform some simple operations with these visual images or pictures, without knowingly applying any given 'rule'.
If a child is able to visualize fractions in his mind, they become more concrete - not just a number on top of other number without meaning. Then the child can estimate the answer before calculating, and evaluate the reasonableness of the final answer, and perform many of the simplest operations in his head.
Of course textbooks DO show fractions with pictures, and they DO show one or two examples of how a certain rule connects with a picture. But that is not enough! A better way is to make kids do lots of problems with fraction manipulatives - and DRAW fraction pictures for problems. That way they will form a mental visual model and can think through the pictures for simple problems.
As an educator, we would have to be aware of the end goal and work backwards to see the gradual steps that each child have to tale before attaining these goals.
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