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• #### Index 441-454

ematics is an important reason for its usefulness: A single idea can apply in many circumstances. On the other hand, it is difficult to learn an idea in a purely abstract setting; one or another concrete interpretation must usually be used to make the idea real. But having been introduced to a mathematical concept by means of one interpretation, children then need to pry it away from only that interpretation and take a more expansive view of the abstract idea. That kind of learning often takes time and can be quite difficult. Sometimes the way in which a concept is first learned creates obstacles to learning it in a more abstract way. At other times, overcoming such obstacles seems to be a necessary part of the learning process.

### Properties of the Operations

When three numbers are to be added, there are several options. To add 1 and 2 and 3, I can add 1 and 2, giving 3, and then add the original 3 to this, to get 6. Or I can add 1 to the result of adding the 2 and the 3. This process again gives 6. These two ways of adding give the same final answer, although the intermediate steps look quite different:

(1+2)+3=3+3=6=1+5=1+(2+3).

This statement of equality uses what is known as the associative law. Again, it holds for any three numbers. I know that

(83,449+173,248,191)+417=83,449+(173,248,191+417)

without doing either sum.

The commutative and associative laws in combination allow tremendous freedom in doing arithmetic. If I want to add three numbers, such as 1, 2, and 3, there are potentially 12 ways to do it:

 (1+2)+3 (2+1)+3 (1+3)+2 (3+1)+2 (2+3)+1 (3+2)+1 1+(2+3) 2+(1+3) 1+(3+2) 3+(1+2) 2+(3+1) 3+(2+1)

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