WitCH 26: Imminent Domain

The following WitCH is pretty old, but it came up in a tutorial yesterday, so what the Hell. (It’s also a good warm-up for another WitCH, to appear in the next day or so.) It comes from the 2011 Mathematical Methods Exam 1:

For part (a), the Examination Report indicates that f(g)(x) =([x+2][x+8]), leading to c = 2 and d = 8, or vice versa. The Report indicates that three quarters of students scored 2/2, “However, many [students] did not state a value for c and d”.

For Part (b), the Report indicates that 84% of students scored 0/2. After indicating the intended answer, (-∞,-8) U (-2,∞) (-∞,-8] U [-2,∞) or R\backslash(-8,-2), the Report goes on to comment:

“This question was very poorly done. Common incorrect responses included [-3,3] (the domain of  f(x); x ≥ -2 (as the ‘intersection’ of  x ≥ -8 with x ≥ -2); or x ≥ -8 (as the ‘union’ of x ≥ -8 with x ≥ -2). Those who attempted to use the properties of composite functions tended to get confused. Students needed to look for a domain that would make the square root function work.”

The Report does not indicate how students got “confused”, although the composition of functions is briefly discussed in the Study Design (page 72).

Obsessive Compulsion

Last week, the New South Wales government came out with the next great plan to Save Mathematics Education: make mathematics compulsory up until the end of high school. Why? According to Premier Gladys Berejiklian, this will “ensure students have the numeracy skills required to succeed in today’s society”.

Yes, of course. In exactly the same way, for example, that compulsory instruction in ethics ensures that lawyers and cops act ethically.

What’s the source for this latest nonsense? Well, it’s kind of, sort of from the Interim Report of the NSW Curriculum Review, which was released a few days earlier, and which is prominent in the Government’s media release. Like all such reports, the NSW Report is barely readable, the predictable mishmosh of pseudoscience, unweighted survey, statistics of undeterminable worth and contradictory motherhoodisms. Thankfully, there’s no reason to read the Report, since the NSW Government hasn’t bothered to read it either; nothing in the Report points to making mathematics compulsory throughout high school.

Still, it was easy enough to find “maths experts” who “applauded the move”. Jordan Baker, the Sydney Morning Herald‘s education reporter, quoted four such “experts”, although the only expert appearing to say much of substance was doing anything but applauding. Greg Ashman, who is always worth reading (especially when he is needling nitwits), pointed to the need for specialist teachers in lower years. He is then quoted:

“You need to move away from the fashion for inquiry learning and problem-based learning and instead focus on high quality, interactive, explicit teaching of mathematics. Do that, and I believe numbers in year 12 would organically grow.”

In other words, if you stop having shit teachers teaching shit maths in a shit manner in lower years then maybe more kids will choose to stick around a little longer. (Ashman is more collegial than this writer.)

The NSW government’s compulsion will undoubtedly push mathematics in the exact opposite direction, into ever more directionless playing and mathematical trivia dressed up as real world saviour. You know the stuff: figuring out credit cards and, God help us, “how to choose cancer treatment“.

To illustrate the point perfectly, Melbourne’s Age has just published one of its fun exam-time pieces. Titled “Are you smarter than a 12th grader?“, the reader was challenged to solve the following problem from yesterday’s Further Mathematics exam:

A shop sells two types of discs: CDs and DVDs. CDs are sold for $7.00 each and DVDs are sold for $13.00 each. Bonnie bought a total of 16 discs for $178.00. How many DVDS did Bonnie buy?

The question this problem raises isn’t are you smarter than a 12th grader. The real question is, are you smart enough to realise that making mathematics compulsory to 12th grade will doom way too many students to doing 7th grade mathematics for six years in a row? For the NSW government and their cheer squad of “maths experts”, the answer appears to be “No”.

Melbourne’s “Toxic” Arts

Having given Monash University a whack, it’s time to take a quick look at the University of Melbourne. A couple of intriguing reports about the University appeared earlier this week in Melbourne’s Age newspaper. The reports are most interesting for what was not written.

The first report, by Kylar Loussikian and which appeared on Monday, detailed allegations apparently raised by Professor Jennifer Milam, Head of the School of Culture and Communications. Professor Milam is reportedly in a legal battle with the University of Melbourne, stemming from accusations of bullying against her.

That battle is notable (and see an earlier report here), but the focus of Loussikian’s report, and his second report with Tom Cowie the following day, was on a more general issue, the supposedly “toxic” environment in Melbourne’s Faculty of Arts. That characterisation appears to be due to the university’s former vice-chancellor, Professor Glyn Davis.

According to Loussikian and Cowie:

  • A legal review conducted by the University of Melbourne “found four heads of school [in the Faculty of Arts] were ‘undermining’ acting dean [of Arts] Denise Varney“.
  • Professor Varney, who had been in the School of Culture and Communications, was promoted to acting dean in February. Academics claimed to The Age that school heads had been discouraged from applying for the acting dean position.
  • Milam has “alleged to have implied that the faculty [of Arts] was hiding profits”.

There’s plenty more detail and colour in Loussikian and Cowie’s reports, including the suggestion that the four heads in question could be investigated for misconduct, which “could lead to dismissal”. Two of the heads are also named: Professor Milam and Professor Trevor Burnard, who was head of the School of Historical and Philosophical Studies up until July and who is reportedly leaving the University.

So, what is really going on? God only knows. But there is one glaring question underlying all this: if those four heads in Arts were indeed “undermining” Dean Varnie then why were they undermining her? What is the underlying substance of the dispute? The secondary but still important question is how Professor Varnie came to be acting dean. If heads in Arts were discouraged from applying, then by whom, and why?

We know nothing about this dispute other than what has been reported. There’s plenty nasty we could say in general about Australian arts and humanities and vice chancellors and heads and deans. We have friends at the University of Melbourne who pretty much loathe everything about the place. But can the systemic awfulness of Australian universities offer any insight into this very specific dispute? God only knows.

All that seems clear is that there’s a larger and, we’ll guess, more important story that, for whatever reason, Loussikian and Cowie aren’t telling.

Monash Extends a Backhander

One of the better offerings for Victoria’s senior students is Extension Studies. Corresponding roughly to America’s Advanced Placement program, ES permits a school student to undertake a university subject as part of VCE, albeit as a lower weighted, fifth or sixth subject.

The extension studies program is not without its flaws. In particular, there are no externally defined curricula or standards, with, rather, each participating university shaping their ES subjects to match their own university subjects. Consequently, there is significant variance in the content, quality and difficulty of the ES subjects offered. This also creates issues for the AP aspect of the program; on occasion, students aligned with one university have had difficulty receiving credit from another; this subject mismatching has also been exacerbated by the arrogance of some university administrators. It can also be a non-trivial task finding keen and competent teachers for ES which, as always, means the wealthier private schools benefit much more than public schools. And, some weirdness from VTAC hasn’t helped matters.

Nonetheless, extension studies functions reasonably well overall and can be of genuine value to a keen or strong student. Apart from the immediate reward of richer study while at school, ES can give a student a jump on their university education and effectively lower their uni fees. (The fees, one is always obliged to mention, which were introduced by this Labor asshole.)

Which is why Monash University’s decision this year to cease offering extension studies is so disappointing, and so annoying. This has created the ridiculous situation where the John Monash Science School, which is, you know, Monash University’s science school, is having to look elsewhere for their extension studies. And of course it is not just future JMSS students that are being screwed around.

What was Monash’s reason? All they wrote to ES subject administrators was, “In recent years, there has been a consistent decline in the number of students taking up this opportunity due to a range of factors.”

Yeah, well, maybe. Maybe numbers have declined, although enrolment in mathematics (with which we’ve been associated) has been healthy and stable. And, Monash might have mentioned that amongst the “factors” in that “range” are Monash’s relatively high cost for a participating student, combined with Monash’s effective discouragement of the participation of smaller schools.

It’s difficult to tell what is really going on, what is the real reason for Monash’s decision. The obvious suspicion is it has to do with money, although the program is not administratively heavy and ought to be pretty cheap to run; indeed, it’s the individual departments that have to pay for the academics to teach and administer and grade the subjects, almost certainly at a loss. The Mathematics Department has always lost money on the deal, and has never whined about it.

The other suspicion is that Monash’s extension program wasn’t attracting sufficient school students to study at Monash, whatever “sufficient” might mean. In contrast, the Mathematics Department has never worried about whether the program attracts more students to do mathematics at Monash; they’ve just accepted that that’s what a principled Department should do.

So, what was it? Was it Monash engaging in particularly obtuse neoliberal bean counting? Or, was it Monash disregarding any notion of community obligation? We’re not sure. But, we’re guessing the answer is “Yes”.

Golden Numbskulls

One of the strangest and most enjoyable presentations we’ve given was at a Brisbane conference on art and design. The conference organiser appeared from nowhere, requesting that we give an address on the golden ratio and art. This was a little puzzling, since our views on such matters were presumably known to them. We replied as such:

“But it’s all bullshit.”

“Yes, we want you to come up and tell us it’s all bullshit.”

Which we dutifully did.

Our talk was well received. The artists seemed relieved that they could stop trying to make sense of something that made no sense. And, the conference was a blast. (It turns out that artists and designers have a significantly better idea of fun than mathematicians. Who knew?)

Golden ratio garbage, linked to both art and nature, has been around for ages, but it has really piled up in the last century. As it was taking off, the art critic Sir Theodore Cook wrote a brilliant, scathing critique. About 25 years ago, mathematician George Markowsky published a careful and thorough debunking. There have been plenty of other critiques, and of course we gave the cult a whack as well. But none of it helps. There will always be another clown waiting in the wings, ready to bring out her nautilus shells or Parthenons or whatnot.

That is all by way of introduction to last month’s bumper crop of nonsense. We try to steer clear of the phi fetish; the systemic perversion of education and (thus) democracy matters a hell of a lot more than some dumb clickbait. On occasion, however, it’s too much to ignore. And some golden nonsense is significantly disturbing.

Most recently we had Bella Hadid declared to be the most beautiful woman in the world because of, you know, ‘science’. This astonishing theorem was announced by that august journal, The Daily Mail. The theorem was repeated all around the world, almost always without irony. Predictably, the source for the theorem was a plastic surgeon, who performed some computery gimcrack eyes divided by chin plus nose ratio thing.

Whatever. Just some clown wanting to sell his dubious wares. Who passes his stuff to the newspapers, run by clowns selling their dubious wares. But then there’s the serious science as well.

Earlier in the month, Dr. Rafael Tamargo and Dr. Jonathan Pindrik, two medical researchers at John Hopkins, published a research paper in The Journal of Craniofacial SurgeryMammalian Skull dimensions and the Golden Ratio (Φ). We heard about their paper because the media clowns grabbed onto John Hopkins’ catchy press release, with its very catchy title: Golden Ratio Observed in Human Skulls.

Tamargo and Pindrik observed nothing of the sort, of course. One simply cannot detect an irrational number in our fuzzy, approximate view of nature. What Tarmago and Pindrik did was imagine the golden ratio in human skulls, and their imaginations were pretty wild.

Tamargo and Pindrik were exploring the accepted idea that the human cranium evolved to accommodate an increasingly large and complex brain. They measured 100 human skulls (and 70 skulls of other mammals). For each skull they calculated the ratio of the ‘nasioiniac arc’ and ‘parieto-occipital arc’ (green divided by red in the diagram below), as well as the ratio of the parieto-occipital arc and the ‘frontal arc’ (red divided blue). With a little algebra one can show that that the two ratios are equal if and only if that common ratio equals the golden ratio, (1 + √5)/2 ≈ 1.618…

(Ian Suk, John Hopkins University)

Tamargo and Pindrik found that the two ratios in their human skulls averaged to about 1.64 and 1.57, which, they write, “are essentially identical and closely approximate Φ”. Well, yeah, sort of. As they admit, however, the ratios also “closely approximate” 1.6. But then the title The Ratio 8/5 observed in Human Skulls isn’t quite as grabby, is it?

Still, maybe Tamargo and Pindrik are correct? After all, the golden ratio sometimes appears in nature, in approximate form. So, why not here? Because there is simply no reason to think so. Tamargo and Pindrik bear the burden of proof, and there is not a hint of a proof in their paper.

Any claim for the appearance of the golden ratio must be supported by a model, an argument why the golden ratio might be appearing. Without that, or without a lot of decimal places, the claim is just number-mongering; the claim is no stronger or weaker than the claim for 1.6 or 1.62, or whatever.

Predictably, the media tended to swallow the press release whole, in the lazy and gullible manner they’ve turned into an art form. We could only find one properly sceptical report, one that was willing to suggest that at least some anatomists considered the skull-phi thing to be “ridiculous”.

Of course Tamargo and Pindrik are by no means the first people to spot phi in the human body and, in proper scholarly fashion, they cite earlier research. They write: “In the clinical sciences, Φ has been found to underlie cardiac anatomy and physiology, gait mechanics, and the aesthetic dimensions of the face.” And no, they didn’t reference The Daily Mirror for that last one. Without checking, however, there’s no particular reason to believe that the literature they cite has any greater validity. It is reasonable and proper to question whether the majority of this research is anything more than silly number games.

Sir Cook’s brilliant critique was titled “A New Disease in Architecture”. Perhaps it’s time for someone to write a follow-up: “A New Disease in Medicine”.

WitCH 24: The Fix is In

We’ve finally found some time to take a look at VCAA’s 2019 NHT exams. They’re generally bad in the predictable ways, and they include some specific and seemingly now standard weirdness that we’ll try to address soon in a more systematic manner. WitCHwise, we were tempted by a number of questions, but we’ve decided to keep it to two or three.

Our first NHT WitCH is from the final question on Exam 2 (CAS) of Mathematical Methods:

As usual, the NHT “Report” indicates nothing of how students went, and little of what was expected. In regard to part f, the Report writes,

p(x) = q(x) = x, p'(x) = q'(x) = 1, k = 1/e

For part g, all that the Report provides is the answer, k = 1.

The VCAA also provides sample Mathematica solutions to schools trialling Methods CBE. For the questions above, these solutions are as follows:

Make of it what you will.

The MAV and a Matter of Opinion

This post is tricky. It is not about us, but there is context, and that context should be kept in mind.

Many readers of this blog will be aware of the long relationship we have had with the Mathematical Association of Victoria. It dates back to 2001, when we first came up with the weird idea that mathematics teachers may be interested in learning some maths beyond the thin gruel they were typically served while at university. That idea morphed into 15+ years of teaming up with the Evil Mathologer, of presenting under the banner of and as a consequence of the MAV, of spreading ideas and rousing the rabble. It was quixotically stupid and exhausting and incredibly rewarding. The prehistory of this blog is an interesting story, which is probably of interest to no one.

Fewer readers of this blog will be aware that our association with the MAV ended a few years ago, when the MAV threatened to (and arguably did) censor the abstract of our (invited) keynote. That story may be of more interest, and we hope to write on it in the near future.

In summary, and notwithstanding our long association with and our gratitude to the MAV, we have no love for the MAV in its current form. That is the context. Now for the post.


A few months ago we heard that an article was rejected for publication in the MAV’s teachers’ journal Vinculum. The manner of and the reason for that rejection sounded very strange, and so we began to ask questions. As indicated below, the MAV has not been particularly forthcoming, but this is our current understanding of the story:

1) An opinion piece was submitted to Vinculum. In the piece, the author argued that all VCE mathematics exams in Year 12 should be calculator-free.

2) Roger Walter, the editor of Vinculum, accepted the piece for publication and included it to be published in the next issue.

3) Peter Saffin, the CEO of the MAV, overruled the editor, instructing Walter to retroactively reject the piece.

4) Saffin’s stated reason for the rejection was that the author’s position was in conflict with the VCAA’s strong advocacy of calculator use.

That is the bare bones of the story. Here is a little flesh (once again, as we understand it):

a) The author of the article is a long-standing member of the MAV, a respected gentleman who has devoted decades to Australian mathematics education generally and to the MAV specifically.

b) The author’s piece was topical, well-written and not flame-throwing.

c) In early September we contacted Michael O’Connor, the President of the MAV, seeking information and clarification. After a back and forth, the President declined to confirm or deny point 3, declaring that as a member of the public we had “no need to know”, and that “even MAV members would have to show sufficient reason”. O’Connor citied his “duty of care towards MAV staff and volunteers”.  Similarly, O’Connor declined to confirm or deny point 4.

d) To our knowledge, no MAV editor has ever previously been overruled in such a manner, by anyone.

e) The author has not contested the rejection.

f) Notwithstanding (d), O’Connor indicated that “proper processes have been followed”.

g) O’Connor indicated that he is “expecting there to be a policy discussion at the next publications meeting”.

h) At this stage, the rejection of the article has not been rescinded.

i) At this stage, no one at the MAV, nor the MAV as a body, has apologised to the author for the rejection of the article or the manner of that rejection,

j) In late September we replied to O’Connor, critiquing various aspects of this incident and his characterisation of it. O’Connor indicated his intention to respond.


That then is the post. O’Connor and Saffin were invited to comment on a close version of the above. O’Connor reiterated his intention to reply and suggested out posting now was “premature”, arguing that the MAV had not had “sufficient time to perform due diligence”. Saffin did not reply as of the time of posting.

We will update the post if and when any new information comes to hand.