MitPY 7: Diophantine Teen Fans

This MitPY is a request from frequent commenter, Red Five:

I’d like to ask what others think of teaching (mostly linear) Diophantine equations in early secondary school. They are nowhere in the curriculum but seem to be everywhere in competitions, including the AMC junior papers on occasion. I don’t see any reason to not teach them (even as an extension idea) but others may have some insights into why it won’t work.

DIY Teaching Degrees

Dan Tehan, the Federal minister for screwing up education, has announced a rescue package for Australia’s universities. This was clearly necessary, since the universities are no longer in a position to fleece international students. The package guarantees funding for the universities, and introduces a range of cheap six-month courses in “areas considered national priorities”.

The government’s package is “unashamedly focused on domestic students”. That was inevitable since:

a) the government, and Tehan in particular, doesn’t give a stuff about international students;*

b) Tehan is a born to rule asshole, entirely unfamiliar with the notion of shame.

And, what of these “priority” courses? According to the ABC,

The Government said prices would be slashed for six-month, remotely delivered diplomas and graduate certificates in nursing, teaching, health, IT and science provided by universities and private tertiary educators.

OK, so ignoring all the other nonsense, we have a few questions about those six-month online teaching diplomas:

  • Will such a diploma entitle the bearer to teach?
  • If not, then what is it good for?
  • If so, then what is a school to do with the mix of 6-month diploma-qualified applicants and the standard 24-month Masters-qualified applicants?
  • And, if so, what does that tell us of the intrinsic worth of those standard 24-month Masters?

To be clear, we have no doubt that six months is plenty sufficient for the initial training of a teacher, and indeed is at least five months too many. We also have no doubt that a diploma-trained teacher has the same chance to be a good teacher as someone who has suffered a Masters. They have a better chance, in fact, since there will have been less time to pervert natural instincts and feelings and techniques with poisonous edu-babble.

But, good or bad, who is going to give these diploma teachers a shot? Then, if the teachers should be and are given a shot, who is going to address the contradiction, the expensive and idiotic orthodoxy of demanding two year post-grad teaching degrees?


*) Or anyone, but international students are near the bottom.

MitPY 4: Motivating Vector Products

A question from frequent commenter, Steve R:

Hi, interested to know how other teachers/tutors/academics …give their students a feel for what the scalar and vector products represent in the physical world of \boldsymbol{R^2} and \boldsymbol{R^3} respectively. One attempt explaining the difference between them is given here. The Australian curriculum gives a couple of geometric examples of the use of scalar product in a plane, around quadrilaterals, parallelograms and their diagonals .

Regards, Steve R

A Question from a Teacher

A few days ago we received an email from Aaron, a primary school teacher in South Australia. Apparently motivated by some of our posts, and our recent thumping of PISA in particular, Aaron wrote on his confusion on what type of mathematics teaching was valuable and asked for our opinion. Though we are less familiar with primary teaching, of course we intend to respond to Aaron. (As readers of this blog should know by now, we’re happy to give our opinion on any topic, at any time, whether or not there has been a request to do so, and whether or not we have a clue about the topic. We’re generous that way.) It seemed to us, however, that some of the commenters on this blog may be better placed to respond, and also that any resulting discussion may be of general interest.

With Aaron’s permission, we have reprinted his email, below, and readers are invited to comment. Note that Aaron’s query is on primary school teaching, and commenters may wish to keep that in mind, but the issues are clearly broader and all relevant discussion is welcome.


Good afternoon, my name is Aaron and I am a primary teacher based in South Australia. I have both suffered at the hands of terrible maths teachers in my life and had to line manage awful maths teachers in the past. I have returned to the classroom and am now responsible for turning students who loathe maths and have big challenges with it, into stimulated, curious and adventure seeking mathematicians.

Upon commencing following your blog some time ago I have become increasingly concerned I may not know what it is students need to do in maths after all!

I am a believer that desperately seeking to make maths “contextual and relevant” is a waste, and that learning maths for the sake of advancing intellectual curiosity and a capacity to analyse and solve problems should be reason enough to do maths. I had not recognised the dumbing-down affect of renaming maths as numeracy, and its attendant repurposing of school as a job-skills training ground (similarly with STEM!) until I started reading your work. Your recent post on PISA crap highlighting how the questions were only testing low level mathematics but disguising that with lots of words was also really important in terms of helping me assess my readiness to teach. I have to admit I thought having students uncover the maths in word problems was important and have done a lot of work around that in the past.

I would like to know what practices you believe constitutes great practice for teaching in the primary classroom. I get the sense it involves not much word-problem work, but rather operating from the gradual release of responsibility (I do – we do – you do) explicit teaching model.

I would really value your thoughts around this.

Warm regards,

Tootering Your Own Horn

Eddie Woo is reportedly concerned about private tutoring. His warning comes courtesy of SMH‘s education editor, Jordan Baker, in an article entitled ‘Be very, very careful’: Experts raise warning on private tutoring. The article begins,

Maths teachers including high-profile mathematician Eddie Woo have sounded an alarm on private tutoring, warning that bad tutors could be “fatal” to students’ future in the subject.

Eddie said it, so it must be true. And, Baker quotes another expert, the chief executive of the Australian Tutoring Association, Mohan Dhall:

I am absolutely dismayed at the lack of creativity and lack of real-world applicability most tutors bring to maths …The main problem stems from this idea that they focus on the outcome – ‘this is what students need to know’, rather than ‘this is what kids need to learn to be interested and engage’.

Finally, Baker quotes expert Katherin Cartwright, a lecturer in mathematics education at The University of Sydney. Cartwright, according to Baker, is concerned that poor tutoring could lead to a lack of confidence:

If it becomes about skill and drill and speed, and it becomes an anxious, emotional issue for students, then they are not going to like it, and they will not want to take it further.

Yep, of course. The most important consideration when framing an education is to be sure to never make a student anxious or emotional. Poor, fragile little petals that they are.

Baker’s fear-mongering is nonsense. Almost every line of her article is contentious and a number contain flat out falsehoods. Beginning with the title. Woo and Dhall and Cartwright are “experts” on the issues of tutoring? According to whom? Based on what? Perhaps they are experts, but Baker provides no evidence.

OK, we could concede Baker’s point that Eddie is a mathematician. Except that he isn’t and we don’t. Not that it matters here, since most mathematicians are unlikely to know much about the role of tutoring in Australian education. But the false and pointless puffery exemplifies Baker’s unjustified appeals to authority.

What of the declared concerns of Baker’s “experts”? Cartwright is supposedly worried about “skill and drill and speed”. This in contrast to school, according to Baker:

Most schools no longer emphasise speed and rote learning when teaching maths, and now focus on students’ understanding of key concepts as part of a concerted effort to improve engagement in maths across the system.

This hilarious half-truth undercuts the whole thrust of Baker’s article. It is true that many schools, particularly primary schools, have drunk the educational Kool-Aid and have turned their maths lessons into constructivist swamplands. But that just means the main and massive job of competent Year 7 maths teachers is to undo the damage inflicted by snake-oilers, and to instil in their students, much too late, an appreciation of the importance of memory and skill and efficient technique. Such technique is critical for formal success in school mathematics and, which is sadly different, for the learning of mathematics. Baker seems entirely unaware, for example, that, for better or worse, Year 12 mathematics is first and foremost a speed test, a succession of sprints.

As for Dhall, does he really expects tutors to be more offering of “creativity” and “real-world applicability”? Dhall seems blissfully unaware that most “real-world” applications that students must suffer through are pedagogically worthless, and are either trivial or infinitely tedious. Dhall seems unaware that some subjects have warped “applicability” into a surrealist nightmare.

And Eddie? What worries Eddie? Not much, as it happens, but too much. Eddie’s quoted comments come from a NSW podcast, which appears to have been the genesis of Baker’s piece; stenographic fluffing is of course the standard for modern reportage, the cheap and easy alternative to proper investigation and considered reflection.

Eddie’s podcast is a happy public chat about teaching mathematics. Eddie is demonstrably a great teacher and he is very engaging. He says a number of smart things, the half-hour podcast only being offensive for its inoffensiveness; Eddie, or his interviewer, was seemingly too scared to venture into a deep public discussion of mathematics and the sense of it. The result is that, except for the occasional genuflection to “pattern”, Eddie may as well have been talking about turtle farming as teaching mathematics.

Eddie’s comments on tutoring are a very minor part of the podcast, a response in the final question time. This is Eddie’s response in full:

When I think about external tuition – again just like before this is a really complex question – there is tuition and then there is ‘tuition’. There is some which is enormously helpful to individual students to come in at a point of need and say “you have got gaps in your knowledge, I can identify that and then help you with those and then you can get back on the horse and off you go, fantastic”. There are other kinds of tuition which are frankly just pumping out an industrial model of education which parents who are very well intentioned and feel like they cannot do anything else, it is like “at least I can throw money at the problem and at least they are spending more time on maths hopefully that will help”. Maybe it does and maybe it is making your child hate maths because they are doing it until 9pm at night after a whole day? That to me is heartbreaking.

I think that students need to be very, very careful and parents need to be very, very careful about how they experience mathematics. Because yes the time is a worthwhile investment, it is a practical subject, but if you are just churning through, often tragically learning things which actually are just machine processes. I have students come to me and they say “I can differentiate, I am really good at that, I am only fifteen years old”. You don’t need to know what differentiation is, but they come to me with this ability to turn a handle on this algorithm this set of steps. Just like me; I don’t know how to bake, but I can follow a recipe. I have no idea what baking powder does or why 180 degrees Celsius is important but I can follow steps. That is okay for a cake because you can still eat it at the end, but that is fatal for mathematics because you don’t know why you are doing any of the things that you are doing. If that is what you are, you are not a mathematician, you are a machine and that is not what we want our children to become. We have to be careful.

Eddie says plenty right here, touching on various forms of and issues with tutoring, and school teaching. The issues do not get fleshed out, but that is the nature of Q & A.

Eddie also gets things smugly wrong. Sure, some tutoring might be characterised as “industrial”. But more so than schools? How can mass education not be industrial? This isn’t necessarily bad: mostly, it just is. Unless, of course, little Tarquin’s parents have the time and the money to arrange for individual or small-group lessons with an, um, tutor.

All the concerns Baker and her experts raise about tutoring apply as much or more so to school education and, as a matter of business necessity, are largely a reflection of school education. And, how do tutors and tutoring companies deal with this? Some well, some poorly. But mostly with industry, which is not a dirty word, and with good and honest intent.

Baker notes the underlying issue, seemingly without even realising it:

However, Australian students’ performance in maths has either stalled or declined on all major indicators over that period, and academics have raised concerns about students arriving at university without the maths skills they need.

Why do parents employ tutors? Having enjoyed and suffered forty years of tutoring, in pretty much all its forms, we can give the obvious answer: there’s a zillion different, individual reasons. Some, as Eddie suggests, are looking for a little damage control, the filling of gaps and a little polishing. Some, as Eddie suggests, think of mathematics, falsely, as a syntactic game, and are looking for lessons in playing that dangerously meaningless game. Some believe, correctly or otherwise, that their teacher/school is responsible for little Johnny’s struggling. Some are trying to get darling Diana into law school. Some are hothousing precious little Perry so he/she can get a scholarship into Polo Grammar or Mildred’s College for Christian Ladies.

But, underlying it all, there is one obvious, central reason why parents employ tutors: parents are unsatisfied with the education their child receives at school.

Why are parents unsatisfied? Are they right to be? Of course, it depends. But, whatever the individual analyses, the massive growth of the tutoring industry indicates a major disconnect, and either a major failing in schools’ performance or a major blindness in parents’ expectations, or both.

That would be a much more worthwhile issue for Baker, and everyone, to consider.


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





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 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.