6 Helping the Older Child with the Rods
The first looks difficult. The second is easy, and the third is ridiculously easy. So why does Johnny need to learn any notational tricks to do such subtractions? Now, if we combine the two examples we get:
These are equivalent differences again and, by breaking down the subtraction into two parts and finding easy equivalent differences, Johnny can readily solve the apparently difficult subtraction–and he enjoys doing it because it is an interesting challenge, and he can see just what is happening. He is using his intelligence instead of blindly following a complicated rule. Even if Johnny knew no more than this, he would be able to tackle the subtractions that come his way. He would break them up and add or subtract rapidly in his head as required to find the answer. He would learn to look on such operations as a whole, using his understanding of equivalent differences to meet every case. But let us suppose the teacher gives him a very long subtraction with all the difficulties he can devise to defeat Johnny to show that he must know one of the methods usually
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