>I am smart, but.... I have struggled with a good metaphor
>for subtracting integers. When it is equated to money, the
>metaphor doesn't make sense. For example, if I say that I
>have 7 cents and I owe Johnny 4 cents, I could express this
>as:
>
>+7 + (-4)
>
>Resulting in a net of 3 cents.

Just do not forget about that.

>However, let's assume that I take away that four cent debt:
>
>+7 - (-4) = ?

If you take away the four cent debt, you get

3 - (-4) = 7,

which is OK.

>In the real world, I should have 7 cents. But if the
>subtraction algorithm is applied to the problem (turn it
>into an addition problem and add the opposite) I get:
>
>+7 - (-4) = ? becomes +7 + (+4) = 11

No. I think the right way to describe the action is

7 + (-4) - (-4) = 7

This is what truly describes the real world situation.

>This is the part that confuses me, because I don't have
>eleven cents, only seven. I'm sure that there an English
>language explanation of this, but I don't get it.
>
>Instead, I have used a different metaphor passed along to me
>by another teacher. It involves an island where the cooks
>cook using special cubes: Hot cubes (+1) and Cold cubes
>(-1). Each cube raises or lowers the temperature of the
>stew by one degree. The mathematical notation in the
>expressions given (as an arithmetic problem) are the recipes
>the cooks use.

This is an excellent metaphor. But of course all brains work differently. It's always a good idea to have several approaches up your sleeve. No one will work with all children.