Indeed, but as I noted, I now see in primary school math textbooks that it stands for "of." Which is problematic, because I tutor once a week at a home schooling centre where they use one of these textbooks.

Of course, "of" really means multiplication, so "half of 6" is pretty much the same thing as "half x 6", so if you use your BODMAS rule like that you'll still get correct answers, but only if you treat "of", x and divide as equals (and therefore evaluate them from left to right). And I don't think that is the way they do it in schools.

I get the impression that at school level (or at least in many schools), the BODMAS rule (or PEMDAS, or whatever else they use as mnemonic) is taken too literally, as a sort of rigid rule that must be followed precisely as stated, i.e. always doing division before multiplication, rather than treating them as equals. And as we can see, in some cases that will result in wrong answers.

And why does this happen? Because the teachers and the authors of the textbooks have "qualifications" in education rather than in math. That's my theory, anyway. :-)

The essential problem with BODMAS/PEMDAS et al. is that such mnemonics fail to indicate that multiplication and division are two sides of the same coin, as are addition and subtraction. Without this insight, BODMAS wrongly indicates that division precedes multiplication, while PEMDAS wrongly indicates the opposite, and both wrongly indicate that addition precedes subtraction.

One can revise the mnemonics to BODA/BOMA/BODS/BOMS and PEMA/PEMS/PEDA/PEDS on the understanding that “M” (or “D”) specifies both multiplication and division, and “A” (or “S”) specifies both addition and subtraction. Alternatively, one could be explicit about it with BODaMAaS or PEMaDAaS where the lower case “a” means “and”.

In any case, once a person properly understands the precedence rules and the reasons for them (mostly notational compactness and non-ambiguity), such mnemonics are entirely superfluous. In fact, it’s my considered view that they are actually an impediment to such proper understanding because it’s easier memorising a mnemonic than penetrating to the proverbial grass roots of the matter.

As to why the confusion happens, it’s very likely as you say, namely that understanding of the subject matter is far less important than the hazy, imprecise wafflings of pedagogic theory.

'Luthon64