VTACKY

It’s been a long, long time. Alas, we’ve been kept way too busy by the Evil Mathologer, as well as some edu-idiots, who shall remain nameless but not unknown. Anyway, with luck normal transmission has now resumed. There’s a big, big backlog of mathematical crap to get through.

To begin, there’s a shocking news story that has just appeared, about schools posting “wrong Year 12 test scores” and being ordered to remove them by the Victorian Tertiary Admissions Centre. Naughty, naughty schools!

Perhaps.

The report tells of how two Victorian private schools had conflated Victoria’s VCE subject scores and International Baccalaureate subject scores. The schools had equated the locally lesser known IB scores of 6 or 7 to the more familiar VCE ATAR of 40+, to then arrive at a combined percentage of such scores. Reportedly, this raised the percentage of “40+” student scores at the one school from around 10% for VCE alone to around 25% for combined VCE-IB, with a comparable raise for the other school. More generally, it was reported that about a third of IB students score a 6 or 7, whereas only about one in eleven VCE scores are 40+.

On the face of it, it seems likely that the local IB organisation that had suggested Victorian schools use the 6+ = 40+ equation got it wrong. That organisation is supposedly reviewing the comparison and the two schools have removed the combined percentages from their websites.

There are, however, a few pertinent observations to be made:

None of the sense or substance of the above is hinted at in the schools-bad/VTAC-good news report.

Of course the underlying issue is tricky. Though the IBO tries very hard to compare IB scores, it is obviously very difficult to compare IB apples to VCE oranges. We have no idea whether or how one could create a fair and useful comparison. We do know, however, that accepting VTAC’s cocky sanctimony as the last word on this subject, or any subject, would be foolish.

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.

Chicken Shit

The ACCC has released guidance on the meaning of “free range eggs”, to come into force in April. There are a number of conditions for hens to be designated free range, but the clear mathematical requirement is that the chickens be subject to “a stocking density of 10,000 hens or less [sic] per hectare.” This compares to the maximum of 1500 hens per hectare recommended by the CSIRO. And by Choice. And by the Humane Society International. And by the RSPCA. And by pretty much everyone except Coles and other industry thugs.

The ACCC is just the messenger here, their guidance mirroring the Australian Consumer Law (Free Range Egg Labelling) Information Standard 2017, passed last April. The legislation was introduced by the Minister for Small Business, Michael McCormack. It was McCormack who took credit for the definition of stocking density:

 … my decision takes into consideration the views of consumers, advocacy groups and industry, and provides a sensible balance with a focus on informing consumers – so they can make the choice that’s right for their needs.

The reader can assess whether McCormack’s “consideration” has resulted in anything remotely resembling “sensible balance”, or in the ability of consumers to make an informed choice. Or, rather, whether Minister McCormack is simply another National Party asshole.

Downwardly Mobile

In response to France’s move to ban mobile phones from schools, now other countries are considering the same.

Well, sort of. Since 2010, France has already banned mobile phones from classrooms; what is controversial is the French proposal to ban mobiles from schools entirely. So, countries like England and Australia are only actively considering what France has accepted without question for years.

Of course, following the consideration to do the blindingly obvious, there is the backlash from the professionals. The ABC quotes NSW Secondary Principals’ Council president Chris Presland as saying

We talk about trying to stimulate STEM education in our schools … it seems quite bizarre that we’re talking about banning the most obvious forms of technology at our disposal. 

Dr Joanne Orlando, an “expert on children and technology” at UWS is also against any such ban. Responding to government comments, Dr. Orlando responds that

 it takes us a few years back from all the work we are doing in education and training … There are so many new ways that mobile devices can add to the classroom.

Thank God for experts.

The Oxford is Slow

Last year, Oxford University extended the length its mathematics exams from 90 to 105 minutes. Why? So that female students would perform better, relative to male students. According to the University, the problem with shorter exams is that “female candidates might be more likely to be adversely affected by time pressure”.

Hmm.

There’s good reason to be unhappy with the low percentage of female mathematics students, particularly at advanced levels. So, Oxford’s decision is in response to a genuine issue and is undoubtedly well-intentioned. Their decision, however also appears to be dumb, and it smells of dishonesty.

There are many suggestions as to why women are underrepresented in mathematics, and there’s plenty of room for thoughtful disagreement. (Of course there is also no shortage of pseudoscientific clowns and feminist nitwits.) Unfortunately, Oxford’s decision appears to be more in the nature of statistical manipulation than meaningful change.

Without more information, and the University has not been particularly forthcoming, it is difficult to know the effects of this decision. Reportedly, the percentage of female first class mathematics degrees awarded by Oxford increased from 21% in 2016 to 39% last year, while male firsts increased marginally to 47%. Oxford is presumably pleased, but without detailed information about score distributions and grade cut-offs it is impossible to understand what is underlying those percentages. Even if otherwise justified, however, Oxford’s decision constitutes deliberate grade inflation, and correspondingly its first class degree has been devalued.

The reported defences of Oxford’s decision tend only to undermine the decision. It seems that when the change was instituted last (Northern) summer, Oxford provided no rationale to the public. It was only last month, after The Times gained access to University documents under FOI, that the true reasons became known publicly. It’s a great way to sell a policy, of course, to be legally hounded into exposing your reasons.

Sarah Hart, a mathematician at the University of London, is quoted by The Times in support of longer exams: “Male students were quicker to answer questions, she said, but were more likely to get the answer wrong”. And, um, so we conclude what, exactly?

John Banzhaf, a prominent public interest lawyer, is reported as doubting Oxford’s decision could be regarded as “sexist”, since the extension of time was identical for male and female candidates. This is hilariously legalistic from such a politically wise fellow (who has some genuine mathematical nous).

The world is full of policies consciously designed to hurt one group or help another, and many of these policies are poorly camouflaged by fatuous “treating all people equally” nonsense. Any such policy can be good or bad, and well-intentioned or otherwise, but such crude attempts at camouflage are never honest or smart. The stated purpose of Oxford’s policy is to disproportionally assist female candidates; there are arguments for Oxford’s decision and one need not agree with the pejorative connotations of the word, but the policy is blatantly sexist.

Finally, there is the fundamental question of whether extending the exams makes them better exams. There is no way that someone unfamiliar with the exams and the students can know for sure, but there’s reasons to be sceptical. It is in the nature of most exams that there is time pressure. That’s not perfect, and there are very good arguments for other forms of assessment in mathematics. But all assessment forms are artificial and/or problematic in some significant way. And an exam is an exam. Presumably the maths exams were previously 90 minutes for some reason, and in the public debate no one has provided any proper consideration or critique of any such reasons.

The Times quotes Oxford’s internal document in support of the policy: “It is thought that this [change in exam length] might mitigate the . . . gender gap that has arisen in recent years, and in any case the exam should be a demonstration of mathematical understanding and not a time trial.” 

This quote pretty much settles the question. No one has ever followed “and in any case” with a sincere argument.

Smoke Gets in Your IQs

There’s not much more revolting than the tobacco industry. Well, OK, there’s racist scum like Trump and Turnbull. And there’s greasy media apologists. And Bill Gates. And Mia Farrow.*

Alright, the world is full of awful people. But you get the point: it is difficult to be on the side of smoking and tobacco-pushing sociopaths.

Difficult, but not impossible.

Recently, the media was full of shock and horror at a new study on smoking. It was widely reported that 2/3 of people who try one cigarette end up as “daily smokers”. This was the conclusion of a meta-analysis, covering over 200,000 respondents from eight surveys. Professor Peter Hajek, one of the study’s authors, noted the meta-analysis constituted documentation of the “remarkable hold that cigarettes can establish after a single experience.”

Which is crap, and obvious crap. The implied suggestion that a single cigarette can turn a person into a helpless addict is nothing but Reefer Madness madness.

How can a respected and sophisticated academic study come to such a conclusion? Well, it doesn’t.

Anyone who has read the great debunking by Susan Traynor‘s son knows to never take a statistical study, much less a one sentence summary of a study, at face value. In this case, and as the authors of the study properly and cautiously note, that “2/3 of people” hides a wide variance in survey quality, response rates and response types.

More fundamentally, and astonishingly, the study (paywalled) never attempts to clarify, much less define, the term “daily smoker”. How many days does that require? The appendix to the study suggests that only three of the eight surveys included in the meta-analysis asked about “daily” smoking with specific reference to a minimal time period, the periods being 30 days, “nearly every day” for two months, and six months.

Of these three studies, the 2013 US NSDUH survey, which used the 30-day period, had around 55,000 respondents and the highest response rate, of around 72%. Amongst those respondents, about 50% of those who had ever smoked had at some time been “daily smokers” (i.e. for 30 days). Hardly insignificant, nor an insignificant time period, but a significant step down from “2/3 daily smokers”. (For some reason, the figures quoted in the meta-analysis, though close, are not identical to the figures in the NSDUH survey; specifically the number of people answering “YES” to the questions “CIGEVER” and “CIGDLYMO” differ.)

Even accepting the meta-analysis as sufficiently accurate, so what? What does it actually indicate? Reasonably enough, the authors suggest that their study has implications for efforts to stop people becoming regular smokers. The authors are tentative, however, rightly leaving the policy analysis for another forum. In particular, in the study the authors never make any claim of the “remarkable hold” that a single cigarette can have, nor do they make any remotely similar claim.

The “remarkable hold” line, which was repeated verbatim in almost every news report, originates from a media release from Hajek’s university. Of course barely any media organisations bothered to look beyond the media release, or to think for half a second before copying and pasting.

There is indeed a remarkable hold here. It is the remarkable hold university media units have on news organisations, which don’t have the time or experience or basic nous to be properly skeptical of the over-egged omelettes routinely handed to them on a platter.

Update: Just a quick addition, for those might doubt that Turnbull is racist scum.

* Yeah, yeah, no one knows, except Mia and Woody. But I believe Moses.

Wenn Will We Ever Learn?

Another day, another person banging the STEM drum.

Today we’re supposedly learning about The case for making maths mandatory in high school.

Except we’re learning nothing of the sort.

What we are learning is that the author is incapable of composing a paragraph containing more than one sentence.

This is very annoying.

We are also learning nothing about mathematics education.

This is also very annoying.

We are learning, however, that a sequence of tendentious and unsupported and unconnected dot points makes for a boring and pointless newspaper column.

We are learning all this from Kim Wenn, the retiring CIO of Tabcorp.

This is perfect.

Nothing to See Here, Folks

Image copyright Bodleian Library, University of Oxford

Some pretty cool mathematical history made the news recently. Researchers at Oxford University investigated the Bakhshali manuscript, an ancient Indian text, and using carbon dating they apparently “pin[ned] the moment” of the “discovery of zero”.

Well, no. Dating one particular manuscript to “the 3rd or 4th century [AD]” is not pinpointing anything. And there are other issues.

The story is genuinely interesting, and much of the media reported the conclusions of the (not yet peer-reviewed) research accurately and engagingly. Others, however, muddled the story, particularly in the headlines. In order to clear things up, we can distinguish four related but distinct ideas to which “zero” might refer:

1)(a) The use of some symbol, say , as a placeholder in positional notation. We can then distinguish, for example, 43 and 43 (i.e. four hundred and three).

1)(b) The use of some symbol, say , to represent the number zero, for example in the equation 5 – 5 = .

2)(a) The use of something resembling the symbol 0 as a placeholder (as in 43 versus 403).

2)(b) The use of something resembling the symbol 0 to represent a number (as in 5 – 5 = 0).

All these ideas are of genuine interest, but 1(a) and, particularly, 1(b) much more so. Famously, from about 2000 BC Babylonian mathematicians employed a form of positional notation, using spacing when required to make the positions clear; so, it would be as if we used 43 and 4 3 to indicate forty-three and four hundred and three, respectively. From around 400 BC Babylonian mathematics began to employ a double-wedge symbol as a placeholder. That’s the earliest such occurrence of symbol for “zero”, in any sense, of which we are aware.

It took much longer for zero to be employed as a genuine number. The first known use was in 628 AD, in a text of the Indian mathematician Brahmagupta. He stated algebraic rules of the integers, though in words rather than symbols: a debt [negative] subtracted from zero is a fortune [positive], and so on. The symbolic arithmetic of zero may have followed soon after, though it is not clear (at least to me) even approximately when. By the end of the ninth century, however, the use of the symbol for the number 0 had appeared in both Indian and Arabic arithmetic.

The interest in the Bakhshali Manuscript is its use of (something resembling) the symbol 0: it is the filled-in dot on the bottom line in the photograph above. As for the Babylonians, this dot was employed as a placeholder rather than to represent a number. It had been thought that the Manuscript dated from the ninth century, and more recent than the (placeholder) 0 appearing on the walls of the famous Gwalia Temple, also from the ninth century. The recent carbon dating, however, determined that portions of the Manuscript, including pages that used the dot as a placeholder zero, were much older, dating to around 300 AD. That’s the big news that hit the headlines.

Now, none of that is as mathematically interesting as the still cloudy origins of the number zero. Combined with our knowledge of Brahmagupta, however, this new dating of the Bakhshali Manuscript suggests the possibility that the use of the number 0 in arithmetic occurred centuries earlier than previously suspected. So, not yet the magnificent historical revelation suggested by some newspaper reports, but still very cool.

 

The “Marriage Theorem” Theorem

The Marriage Theorem is a beautiful piece of mathematics, proved in the 1930s by mathematician Philip Hall. Suppose we have a number of men and the same number of women. Each man is happy to marry some (but perhaps not all) of the women, and similarly for each woman. The question is, can we pair up all the men and women so that everyone is happily married?

Obviously this will be impossible if too many people are too fussy. We’ll definitely require, for example, each woman to be happy to marry at least one man. Similarly, if we take any pair of women then there’s no hope if those two women are both just keen on the one and same man. More generally, we can take any collection W the women, and then we can consider the collection M of men who are acceptable to at least one of those women. The marriage condition states that, no matter the collection W, the corresponding collection M is at least as large as W.

If the marriage condition is not satisfied then there’s definitely no hope of happily marrying everyone off. (If the condition fails for some W then there simply aren’t enough acceptable men for all the women in W.) The Marriage Theorem is the surprising result that the marriage condition is all we need to check; if the marriage condition is satisfied then everyone can be happily married.

That’s all well and good. It’s a beautiful theorem, and you can check out a very nice proof at (no pun intended) cut-the-knot. This, however, is a blog about mathematical crap. So, where’s the crap? For that, we head off to Sydney’s University of New South Wales.

It appears that a lecturer at UNSW who has been teaching the Marriage Theorem has requested that students not refer to the theorem by that name, because of the “homophobic implications”; use of the term in student work was apparently marked as “offensive”. How do we know this? Because one of the affected students went on Sky News to tell the story.

And there’s your crap.

But, at least we have a new theorem:

The “Marriage Theorem” Theorem

a) Any mathematician who whines to her students about the title “Marriage Theorem” is a trouble-making clown with way too much time on her hands.

b) Any student who whines about the mathematician in (a) to a poisonously unprincipled pseudonews network is a troublemaking clown with way too much time on his hands.

Proofs: Trivial.

Going off at a Tangent

So Plimpton 322, the inscrutable Babylonian superstar, has suddenly become scrutable. After a century of mathematics historians puzzling over 322’s strange list of Pythagorean triples, two UNSW mathematics have reportedly solved the mystery. Daniel Mansfield and Norman Wildberger have determined that this 3,800-ish year old clay tablet is most definitely a trigonometry table. Not only that, the media have reported that this amazing table is “more accurate than any today“, and “will make studying mathematics easier“.

Yeah, right.

Evelyn Lamb has provided a refreshingly sober view of all this drunken bravado. For a deeper history and consideration, read Eleanor Robson.

Babylonian mathematics is truly astonishing, containing some great insights. It would be no surprise if (but it is by no means guaranteed that) Plimpton 322 contains.great mathematics. What is definitely not great is to have a university media team encourage lazy journalists to overhype what is probably interesting research to the point of meaninglessness.