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.

Strong opinions? Me? No, just gentle suggestions. I assume they’re the same as Dave’s, but this is it:

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.

2) Specifically, if I see something like

3 x 5 + 2 x -3

my response is

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?

3) The goal in writing mathematics is not to follow God-given rules, but to be clear. Of course clarity can require rules, but it also requires common sense. And in this case common sense dictates

USE

SOME

BLOODY

BRACKETS

For Christ’s sake, why is this so hard for people to understand? Just write (3 x 5) + 2 or 3 x (5 + 2), or whatever. It is almost always trivial to deambiguousize something, so do so.

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.

Our second sabbatical post concerns, well, the reader can decide what it concerns.

Last year, diagnostic quizzes were given to a large class of first year mathematics students at a Victorian tertiary institution. The majority of these students had completed Specialist Mathematics or an equivalent. On average, these would not have been the top Specialist students, nor would they have been the weakest. The results of these quizzes were, let’s say, interesting.

It was notable, for example, that around 2/5 of these students failed to simplify the likes of 81^-3/4. And, around 2/3 of the students failed to solve an inequality such as 2 + 4x ≥ x² + 5. And, around 3/5 of the students failed to correctly evaluate $\boldsymbol {\int_0^{\pi} \sin 5x \,{\rm d}x}\,$ or similar. There were many such notable outcomes.

Most striking for us, however, were questions concerning lists of numbers, such as those displayed above. Students were asked to write the listed numbers in ascending order. And, though a majority of the students answered correctly, about 1/4 of the students did not.

What, then, does it tell us if a quarter of post-Specialist students cannot order a list of common numbers? Is this acceptable? If not, what or whom are we to blame? Will the outcome of the current VCAA review improve things, or will it make matters worse?

Tricky, tricky questions.

Tag: arithmetic

BODMAS v USBB

USE

SOME

BLOODY

BRACKETS

Signs of the Times