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.
However, let's assume that I take away that four cent debt:
+7 - (-4) = ?
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
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.
So, adding a hot cube and a cold cube changes the temperature of the stew by zero: (+1) + (-1) equals zero change in temperature. This "zero pair" of cubes is a significnt notion.
Say the recipe says: +2 + (-3)
This indicates that two hot cubes are added and three cold cubes, for a net fall in temperature of -1.
+2 + (-3) = -1
This is clear.
But what happens in subtraction?
+2 - (-3)
Consider:
If I start the pot with two hot cubes:
+
+
How can I remove three cold cubes that don't exist?
The solution is in the notion of zero pairs.
I can add three zero pairs to get
+
+
+ -
+ -
+ -
(Imagine the plusses and minuses as cubes.)
This does not change the temperature of the stew. Now I can remove three cold cubes from the stew. The result is:
+
+
+
+
+
Written as a recipe, it looks like this:
+2 - (-3) = +5
It is difficult with number lines to develop a reasonable method for subtractive movement by operation. But this metaphor holds for addition and subtraction of integers (there is an extension for multiplication that involves bunches of cubes).
Is it useful? Well, some children learn the algorithms rotely: "Turn subtraction problems into addition problems and add the opposite...."
The same children often generalize the rule and turn addition problems into subtraction problems with the opposite and the whole thing gets very messy.
I think the value could be in establishing a simple way of constructing the algorithm. If a child is presented with a problem and they are not sure of the correct algorithm, they could, with the metaphor, complete a simple problem to verify the procedure, estimate the sign and magnitude of the number, and pull the whole problem solving process into the realm of common sense and mental math.
As for the money metaphor, maybe it has something to do with double entry bookkeeping....I don't know. It just hasn't worked for me.
I welcome any further comments on this: jmeyer@cusd.net