In my first full-time teaching post, it is very likely that I'll need to be teaching a small amount of GCSE Mathematics to students retaking it. One thing that has been bugging me is that I can't seem to find any sort of "proof" or explanation of why the BODMAS (PEMDAS, for Americans) rule for doing calculations works. I learned this rule in school, and applying it is second nature. It's obvious to me that multiplication precedes addition, and brackets proceeds powers, via elementary properties of real numbers.
What would be a good way of explaining this rule to students who hadn't seen it before, and does anyone have any understanding-focused ways of explaining why calculations are ordered this way?