@MrMathsTeacher tweeted (5/6/2012) “Is 5x-3=7 easy, but -3+5x=7 hard due to the way neg numbers have been taught or because of algebra difficulties, or something else?”

A lot of the difficulty for kids in learning mathematics, is knowing what models to use to think about things.

For example, in thinking about functions it helps to have a model of a function as a machine with an input and an output:

This may not be strictly correct from a rigorous point of view, yet it serves as a model that is accurate, that can be later modified as students require more rigor, and does not need to be thrown overboard to do that. It is an instance of a cognitive root.

A useful cognitive root for signed numbers, such as -3 and 5, is as instructions to move from a given place on the number line.

For example, if we imagine ourselves as being at 0 on the number line, then the number -2 instructs us to move 2 places to the left:

If we are at the point 1 on the number line the number -2 still instructs us to move 2 places to the left – it just lands as at a different spot:

Similarly, a positive number such as 4 tells us to move 4 places to the right from where we are:

Now a number such as 3 has an ambiguity of interpretation: does it mean the place 3 to the right of 0 on the number line, or does it mean to move 3 places right from where we are?

The answer, of course, depends on the context. For a number to mean a place on the number line we move that many spots from 0 (left for negative numbers, right for positive numbers). Otherwise, we keep the interpretation of a number open, as meaning move to the left or right of where we currently are on the number line.

Now with addition interpreted as following on (concatenation) we have our cognitive root of signed number addition:

-3 + 5 means move 3 places to the left of where we are and then 5 places to the right. This lands us 2 places to the right of where we started.

On the other hand, 5-3 means move 5 places to the right of where we are and then move 3 places to the left. This again lands us 2 places to the right of where we started.

In both cases, if we started at 0, then we would end at 2.

So -3+5= 2 = 5-3

At least, that’s one way to think about it.