I was asked for some ideas on teaching operations (+, -, x, /) on integers the other day. I have tried lots of metaphors (see here for some thoughts about the use of metaphor when adding fractions) when teaching this in the past, but have found that learners have found some aspects of working with negative numbers difficult, possibly due to said metaphors not holding in the whole range of cases.
What metaphors are commonly used when talking about operations on integers? Examples include subtraction as taking away, division as how many a in b, or as sharing, and multiplication as repeated addition, or 'lots of'. When we get into the domain of negatives, these metaphors become less useful.
The question is then: what are addition, subtraction, multiplication and addition? Addition is the an operation on two elements of the same set, counting them up together to create another element of that set. Defining multiplication is a bit more tricky, but a useful metaphor for multiplication that might hold throughout work on negatives is scaling (see this article). Subtraction and division are then (only) the inverse operations of addition and multiplication.
I have in the past taught addition and multiplication of integers using the field axioms to year 7s (in a simplified form), and it is this which I wish to explain here. It might all sound a bit complicated, but it is worth noting that the children seemed to find it useful.
Combing the fact that addition is the counting up of two quantities of the same thing, with the axiom a + (-a) = 0, can simplify any addition calculation. I would go further and say it draws attention to important structure.
Adding 7 + 4, or (-7) + (-4) presents few problems, as we are counting up two quantities of the same thing. Now consider the calculation 7 + (-4). This is more problematic. But given the fact that 4 + (-4) = 0, we can then transform 7 + (-4) into 3 + 4 + (-4), which is 3 + 0. This seems abstract at first glance, but further reflection might reveal that this mirrors a fundamental way of thinking about adding numbers with opposing signs.
I suggest moving onto multiplication next, as it is the other fundamental operation. I am going to consider multiplication as scaling: a x b means 'a scaled (stretched) by factor b'. Here are two possible diagrams that could represent the scaling of 3, I'm not sure which is preferable (click to view):
Scaling of negative values of a can also be dealt with in a similar way.
Metaphors for a divided by b, such as 'how many b in a', or 'a shared between b people', don't hold for a calculation such as 12 / (-4). Considering division as the inverse operation to multiplication, 12 / (-4) = ? becomes (-4) x ? = 12.
An alternative to this approach might be to consider division as multiplying by the multiplicative inverse (i.e. the reciprocal), which might be most useful in the long run, but could be difficult for some learners.
Subtraction may be considered in a similar way, so a - b = ? becomes a = b + ?. As an example, (-7) - (-4) = ? becomes (-7) = (-4) + ?, which is relatively simple for someone already fluent with addition of integers.
A better alternative in this case could be to considered subtraction as addition of the (additive) inverse. So any subtraction calculation, a - b can be transformed into a + (-b). Under this approach, (-7) - (-4) becomes (-7) + 4, which is straightforward for a learner fluent in addition. I have used this approach and it was successful.
The important thing about all of this is that there is little need for metaphor. The only metaphor that is used (scaling) holds for all integers. Beyond that, calculations are performed through the use of the notion of equivalence, based on the definitions of the various operations.