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.

Experience with the operations of addition and multiplication leads to the observation of certain regularities in their behavior. For example, it does not matter in what order two numbers are added. If I dump a basket of three apples into a basket with five apples already in it, there will be eight apples in the basket; and if I dump the basket of five apples into the basket with three, I will also have eight. Thus 5+3=8=3+5. The similar fact is true for any two numbers. Thus, I know that 83,449+173,248,191=173,248,191+83,449 without actually doing either addition. I have used what is known as the *commutative law* of addition.

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) |