BODMAS v USBB

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.

AMSI’s Brain Teaser

Last week, AMSI released yet another paper on the issue of school mathematics being taught by “out of discipline” teachers. It will come as no surprise to readers of this blog that we have many issues with AMSI’s paper. Here, we’ll focus on just one aspect.

The Sydney Morning Herald’s report on AMSI’s paper begins:

Fewer than one in four Australian high school students have a qualified maths teacher …

That statement is, of course, utter nonsense. By any reasonable definition, a much higher percentage of secondary students are taught by formally “qualified” teachers. It is concerning that an “education reporter” would lead with such an implausible claim, but SMH was not alone. The news.com.au report was titled:

Only 1 in 4 high-schoolers are being taught maths by qualified teachers

The Australian’s barely comprehensible sentence, courtesy of another education reporter, appeared to suggest that matters are even worse:

Fewer than one in four students are taught by a qualified maths teacher — one with at least a university minor in the subject — at some stage between Years 7 to 10.

So, what is the source of all these inflated declarations of educational doom? It would appear to be on page 2 of AMSI’s paper. In the first of the paper’s eye-catching Key Points, the authors write:

The extent of the problem [with the supply of qualified teachers] is illustrated by the estimated amount of out‐of‐field teaching occurring with less than one in four students having a qualified mathematics teacher in each of Years 7 to 10.

That reminds us: we must buy AMSI a box of commas for Christmas.

The above sentence, which turned out to be the grabber of AMSI’s paper, is like an optical illusion: you think you’ve got the meaning, and then it slips around to mean something entirely different. It is no wonder if reporters misinterpreted.

What did the AMSI authors intend to convey, and on what basis? It is difficult to tell. A linked endnote in AMSI’s paper refers to a 2017 AMSI publication. The page reference to this second document is clearly incorrect, but it appears that the intention is to refer to page 4, which has its own list of key points, including:

At least 26% of Years 7–10 maths teachers are not fully qualified.

This is an admirably clear statement and, if true, one may (or may not) regard it as a relatively major problem. The statement, however, is not remotely supportive of the educational catastrophe that AMSI’s garbled 2019 statement led gullible reporters to declare.

Also puzzling, it is not clear how AMSI’s 2017 statement, or any other AMSI declaration that we could find, leads reasonably to any natural interpretation of AMSI’s 2019 statement. This is the case even if one ignores that “not fully qualified” does not clearly equate to “not qualified”, and that 26% of teachers does not equate to 26% of classes, nor to 26% of students. Even with the most liberal assumptions and generous interpretations, we still cannot determine the basis, any basis, for the 2019 statement. The reader is invited to give it a go.

There are plenty more serious issues with AMSI’s paper which, though raising some very important issues and suggestions, also connects some distant and very disputable dots. It probably doesn’t matter, however. We worked hard to read AMSI’s clumsily written paper. It seems unlikely that many others will do likewise.

It’s Time to ATAR and Feather the Labor Party

Tanya Plibersek, Australian Labor’s Shadow Minister for Education, has just been reaching out to the media. Plibersek has objected to the low ATAR sufficient for school leavers to gain entry to a teaching degree, and she has threatened that if universities don’t raise the entry standards then Labor may impose a cap on student numbers:

We [should] choose our teaching students from amongst the top 30 per cent …

This raises the obvious question: why the top 30 per cent of students? Why not the top 10 per cent? Or the top 1 per cent? If you’re going to dream an impossible dream, you may as well make it a really good one.

Plibersek is angry at the universities, claiming they are over-enrolling and dumbing down their teaching degrees, and of course she is correct. Universities don’t give a damn whether their students learn anything or whether the students have any hope of getting a job at the end, because for decades the Australian government has paid universities to not give a damn. The universities would enrol carrots if they could figure out a way for the carrots to fill in the paperwork.

The corruption of university teaching enrolment, however, has almost nothing to do with the poor quality of school teachers and school teaching. The true culprits are the neoliberal thugs and the left wing loons who, over decades, have destroyed the very notion of education and thus have reduced teaching to a meaningless, hateful and hated profession, so that with rare exceptions the only people who become teachers are those with either little choice or little sense or a masochistically high devotion to civic duty.

If Plibersek wants “teaching to be as well-respected as medicine” then perhaps Labor could stick their neck out and fight for a decent increase in teachers’ wages. Labor could fight for the proper academic control of educational disciplines so that there might be a coherent and deep Australian curriculum for teachers to teach. Labor could fight against teachers’ Sisyphean reporting requirements and against the swamping over-administration of public schools. Labor could promise to stop, entirely, the insane funding of poisonously wealthy private schools. Labor could admit that for decades they have been led by soulless beancounters and clueless education hacks, so as much as anyone they have lost sight of what education is and how a government can demand it.

But no. Plibersek and Labor choose an easy battle, and a stupid, pointless battle.

None of this is to imply that Labor’s opponents are better. Nothing could be worse for education, or anything, than the sadistic, truth-killing Liberal-National psychopaths currently in power.

But we expect better from Labor. Well, no we don’t. But once upon a time we did.

Update (27/02/19)

Tanya Plibersek has announced a new Labor policy, to offer $40,000 grants for “the best and the brightest” to do teaching degrees, and to go on to teach in public schools. Of course Plibersek’s suggestion that this will attract school duxes and university medal winners into teaching is pure fantasy, but it’s a nanostep in the right direction. 

 

 

 

Eddie Versus the Forces of Woo

No one appears to have a bad word for Eddie Woo. And no, we’re not looking to thump Eddie here; the mathematics videos on Eddie’s WooTube channel are engaging and clear and correct, and his being honoured as Local Australian of the Year and as a Top Ten Teacher is really cool. We do, however, want to comment on Eddie’s celebrity status and what it means.

What do Eddie’s videos exhibit? Simply, Eddie is shown teaching. He is explaining mathematics on a plain old whiteboard, with no gizmos, no techno demos, no classroom flipping, rarely a calculator, none of the familiar crap. There’s nothing at all, except a class of engaged students learning from a knowledgeable and engaging teacher.

Eddie’s classroom is not the slightest bit revolutionary. Indeed, it’s best described as reactionary. Eddie is simply doing what good maths teachers do, and what the majority of maths teachers used to do before they were avalanched with woo, with garbage theories and technological snake oil.

Sure, Eddie tapes his lessons, but Eddie’s charmingly clunky videos are not in any way “changing the face of mathematics teaching“. Eddie’s videos are not examples of teaching, they are evidence of teaching. For actual instruction there are many better videos out there. More importantly, no video will ever compare to having a real-live Eddie to teach you.

There are many real-live Eddies out there, many teachers who know their maths and who are teaching it. And, there would be many, many more real-live Eddies if trainee teachers spent more time learning mathematics properly and much less time in the clutches of  Australia’s maths ed professors. That’s the real message of Eddie’s videos.