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.

The Marriage Equality Theorem

Theorem: Let V be the set of valid arguments against marriage equality. Then is empty.

Proof: Let P be a valid argument. Then, by now, someone would have argued P. This has not occurred. (Proof: by exhaustion.) By contradiction, it follows that P does not exist, and thus V is empty. QED.

An alternative, direct proof of the theorem was provided by the California Supreme Court; their proof applied the definition of equality.

Consideration of the many straight-forward corollaries of this theorem are left to the reader.

Three Apples + Two Oranges = Infinite Nonsense

The key findings of Australia’s 2016 National Drug Strategy Household Survey were released earlier this year, and they made for sobering reading. The NDSHS reported that over 15% of Australians had used illicit drugs in the previous year, including such drugs as cannabis, ice and heroin. Shocking, right?

Wrong. Of course.

We’re being silly in a way that the NDSHS reporting was not. Yes, the NDSHS reported that 15% had used illicit drugs at least once (including the possibility of exactly once) in the previous year, but NDSHS also emphasised the composition of that 15%. By far the most commonly used drug was cannabis, at about 10% of the population. Ice use was around 1%, and heroin didn’t register in the summary.

Illicit drug use is a serious problem, and a problem exacerbated by idiotic drug laws. Nothing can be learned, however, and nothing can be solved if one focuses upon a meaningless 15% multicategory. Whatever the specific threats or the reasonableness of concerns over the broad use of cannabis, such concerns pale in comparison to the problems of ice and heroin. The NDSHS makes no such categorical mistake. Unfortunately, there are plenty of clowns who do.

Last week, the Federal Ministers for Social Services and Human Services announced the location of a drug testing trial for job seekers who receive federal benefits. The ironically named Christian Porter and the perfectly named Alan Tudge announced that receipients would be tested “for illicit substances including ice (methamphetamine), ecstasy (MDMA) and marijuana (THC) … People who test positive to drug tests will continue to receive their welfare payment but 80 per cent of their payment will only be accessible through Income Management.” The plan is deliberately nasty and monumentally stupid, and it has been widely reported as such. For all the critical reporting, however, we could find no instance of the media noting the categorical lunacy of effectively equating the use of ice and ecstasy and THC.

Still, one should be fair to Porter and Tudge. They are undeniably dickheads, but Porter and Tudge are hardly exceptional. They are members of a very large group of thuggish, victim-blaming politicians, which includes Malcolm Turnbull, and Peter Dutton, and Adolf Hitler.

It is also notable that this kind of multicategory crap is only practised by social conservatives. It’s not like a nationwide survey on sexual harrassment and sexual assault in universities would ever couch the results in broadly defined categories in such a clouded and deceptive manner. Nope, not a chance.

NAPLAN’s Numerological Numeracy

This year Australia celebrates ten years of NAPLAN testing, and Australians can ponder the results. Numerous media outlets have reported “a 2.55% increase in numeracy” over the ten years. This is accompanied by a 400% increase in the unintended irony of Australian education journalism.

What is the origin of that 2.55% and precisely what does it mean to have “an increase in numeracy” by that amount? Yes, yes, it clearly means “bugger all”, but bugger all of what? It is a safe bet that no one reporting the percentage has a clue, and it is not easy to determine.

The media appear to have taken the percentage from a media release from Simon Birmingham, the Federal Education and Training Minister. (Birmingham, it should be noted, is one of the better ministers in the loathsome Liberal government; he is merely hopeless rather than malevolent.) Attempting to decipher that 2.55%, it seems to refer to the “% average change in NAPLAN mean scale score [from 2008 to 2017], average for domains across year levels”. Whatever that means.

ACARA, the administrators of NAPLAN, issued their own media release on the 2017 NAPLAN results. This release does not quote any percentages but indicates that the “2107 summary information” can be found at the the NAPLAN reports page. Two weeks after ACARA’s media release, no such information is contained on or linked on that page, nor on the page titled NAPLAN 2017 summary results. Both pages link to a glossary, to explain “mean scale score”, which in turn explains nothing. The 2016 NAPLAN National Report contains the expression 207 times, without once even pretending to explain what it means. The 609-page Technical Report from 2015 (the latest available on ACARA’s website) appears to contain the explanation, though the precise expression is never used and nothing remotely resembling a user-friendly summary is included.

To put it very briefly, each student’s submitted test is given a “scaled score”. One purpose of this is to be able to compare tests and test scores from different years. The statistical process is massively complicated and in particular it includes a weighting for the “difficulty” of each test question. There is plenty that could be queried here, particularly given ACARA’s peculiar habit of including test questions that are so difficult they can’t be answered. But, for now, we’ll accept those scaled scores as a thing. Then, for example, the national average for 2008 Year 3 numeracy scaled scores was 396.9. This increased to 402.0 in 2016, amounting to a percentage increase of 1.29%. The average percentage increases from 2008 to 2017 can then be further averaged over the four year levels, and (we think) this results in that magical 2.55%.

It is anybody’s guess whether that “2.55% increase in numeracy” corresponds to anything real, but the reporting of the figure is simply hilarious. Numeracy, to the very little extent it means anything, refers to the ability to apply mathematics effectively in the real world. To then report on numeracy in such a manner, with a who-the hell-cares free-floating percentage is beyond ironic; it’s perfect.

But of course the stenographic reportage is just a side issue. The main point is that there is no evidence that ten years of NAPLAN testing, and ten years of shoving numeracy down teachers’ and students’ throats, has made one iota of difference.