Yesterday, I received an email from Stacey, a teacher and good friend and former student. Stacey was asking for my opinion of “order of operations”, having been encouraged to contact me by Dave, also a teacher and good friend and former student. Apparently, Dave had suggested that I had “strong opinions” on the matter. I dashed off a response which, in slightly tidied and toned form, follows.
1) The general principle is that if mathematicians don’t worry about something then there is good reason to doubt that students or teachers should. It’s not an axiom, but it’s a very good principle.
a) No mathematician would ever, ever write that.
b) I don’t know what the Hell the expression means. Honestly.
c) If I don’t know what it means, why should I expect anybody else to know?
The fact that schools don’t instruct this first and foremost, that demonstrates that BODMAS or whatever has almost nothing to do with learning or understanding. It is overwhelmingly a meaningless ritual to see which students best follow mindless rules and instruction. It is not in any sense mathematics. In fact, I think this all suggests a very worthwhile and catchy reform: don’t teach BODMAS, teach USBB.
[Note: the original acronym, which is to be preferred, was USFB]
4) It is a little more complicated than that, because mathematicians also write arguably ambiguous expressions, such ab + c and ab2 and a/bc. BUT, the concatenation/proximity and fractioning is much, much less ambiguous in practice. (a/bc is not great, and I would always look to write that with a horizontal fraction line or as a/(bc).)
5) Extending that, brackets can also be overdone, if people jump to overinterpret every real or imagined ambiguousness. The notation sin(x), for example, is truly idiotic; in this case there is no ambiguity that requires clarification, and so the brackets do nothing but make the mathematics ugly and more difficult to read.
6) The issue is also more complicated because mathematicians seldom if ever use the signs ÷ or x. That’s partially because they’re dealing with algebra rather than arithmetic, and partially because “division” is eventually not its own thing, having been replaced by making the fraction directly, by dealing directly with the result of the division rather than the division.
So, this is a case where it is perfectly reasonable for schools to worry about something that mathematicians don’t. Arithmetic obviously requires a multiplication sign. And, primary students must learn what division means well before fractions, so of course it makes sense to have a sign for division. I doubt, however, that one needs a division sign in secondary school.
7) So, it’s not that the order of operations issues don’t exist. But they don’t exist nearly as much as way too many prissy teachers imagine. It’s not enough of a thing to be a tested thing.
7 Replies to “BODMAS v USBB”
I totally agree that brackets should be used and that BODMAS is a meaningless ritual (unless you get sucked in by the “Only 0.1% of the population get this question right …” click-bait, or you have to sit a Naplan test, or you’re a contestant on Hard Quiz ….)
And what might be more interesting anyway (for junior students) would be questions where you have to put brackets in things like 3 x 5 + 2 x -3 to get particular answers such as 9 or -63 or -90 or …. Or to ask how many different answers you can get by using brackets in 3 x 5 + 2 x -3 etc.
I actually like brackets around my sin(x)’s etc. (I’m quite happy to be called idiotic for this lol). Because I see far too many students become lazy or complacent and start writing sin 2x etc. which for me is the thin of the wedge.
But where I absolutely *loathe* and *detest* brackets is around an integrand!! For some unfathomable reason, there are teachers by the dozen (often writers of trial exams) who insist on using them in this context. Now *that* is TRULY idiotic.
You’d be surprised (or maybe not) where the division sign comes in handy in Secondary school (especially Yr 12 Maths Methods!):
I find it useful when showing some students how to get from (a/b)/(c/d) to (ad)/(bc), by using the intermediate step a/b divided by c/d and then recruiting the ‘flip and multiply’ rote learnt rule for dividing two fractions.
John, your argument for the use of the division sign is only valid to the extent that secondary school mathematics is being taught (perhaps necessarily) in a primary school manner.
As for sin(x), yep, you’re wrong. But, also yes, if sin(x) is idiotic notation, it’s difficult to think of the appropriate expression for the practice of bracketing integrands.
Thanks John and *hear hear* for the damned brackets around integrands – thankfully I haven’t seen VCAA do it yet, but damn! annoying.
Just as a side hobby I actually did some research into the history of the obelus (division sign) and it seems to have been intended to mean something TOTALLY DIFFERENT to what modern teachers (and textbooks) take it to mean.
a+b div c+d in the original texts actually meant (a+b) div (c+d) rather than a+(b div c) +d which I found interesting to say the least… if true (I don’t know the text was in German and copied from something written in the 15th century).
Thanks, RF. After reading your post I took a quick look at Cajori’s book. Cajori suggests that using the symbol ÷ to indicate division dates to the 17th century, though the symbol was previously used for subtraction, and there was a maze of earlier notation for division. Since early mathematics was so sporadically symbolic, and conventions so varied and fleeting, it was probably impossible to have any accepted order of operations.
My (limited) understanding is that order of operations basically comes from the language of polynomials and was somehow extrapolated from there.
However, in the pragmatic sense, what we are really talking about is implied brackets – which in itself suggests using brackets makes more sense than just implying them.
It appears that the acronyms for order of operations are a bit of a modern trend. So they ring a little bit off to those of us (talkin’ ’bout mah generation), who didn’t learn with the mnemonics. See, for example, the nice research by mweiss (halfway down page)
Thanks, A. Very interesting. I did a little hunt myself a few months back, and wound up in similar territory.