Occasional reflections on Life, the World, and Mathematics

Posts tagged ‘education’

Slippery slope

At the Kepier School, a secondary school in North England, it is reported that children were being sent home on the first day of school if their trousers were not purchased (at inflated prices) from a particular supplier. The teachers walked around with colour swatches, checking that they were exactly the right shade of grey. Lest you think this was merely an irrelevant distraction from education — if not actually evidence of a corrupt kickback — there was this explanation from the headteacher:

If you have different types of trousers it leads on to different types of shoes, different types of shirts, etc.

“Etc.” indeed. Once they have different shirts, it’s just a short step to different thoughts, and then it’s straight downhill to heroin addiction and human sacrifice in the parking lot.

Perhaps this is why the school inspectorate Ofsted wrote in their report on the school in October 2013

leaders and managers do not always focus their actions where they are most needed and do not check the impact on students’ achievement.

“Getting a bit of extra assistance is never cheating”

If I were a philosophy student with a looming deadline for an essay on casuistry, I know I’d turn to BuyEssay for expert help. The Guardian has reported on government moves to crack down on essay mills, that sell individually crafted essays for students who need “extra help” –anything from a 2-page essay to a PhD dissertation (for just £6750!) The article reprints some of the advertising text that these websites offer to soothe tender consciences.

“Is Buying Essays Online Cheating?” it asks, in bold type. You’d think this would be an easy question, hardly something you could spin a 300-word essay out of. But they start with a counterintuitive answer: “We can assure you it is NOT cheating”. The core of the argument is this:

What is essential when you are in college or university is to focus on scoring high grades and to get ready for your career ahead. In the long run, your success will be all that matters. Trivial things like ordering an essay will seem too distant to even be considered cheating.

Given that high grades are so essential, it seems almost perverse that universities make it so difficult to obtain them. Why do they put all these essays and other hurdles in the way — “unreasonable demands from unrelenting tutors in expecting extensive research in a short time”, as the essay puts it? It’s shitty customer service, that’s what it is.

The only critique I might make is that the essay is a bit generic. I’d worry that when I submitted it for the assignment “Is Buying Essays Online Cheating”, that the marker might notice that someone else bought almost the same essay for the assignment “Is Murder Wrong?” In the long run, your success will be all that matters. Wasn’t this the plot of Woody Allen’s Crimes and Misdemeanors?

Government intervention

It seems Theresa May has found her strategy for rescuing the British economy from the political damage the Tories are planning to inflict:

The prime minister will publish the strategy at a cabinet meeting in the north-west of England, setting out five sectors that could receive special government support: life sciences, low-carbon-emission vehicles, industrial digitalisation, creative industries and nuclear.

She will say the government would be prepared to deregulate, help with trade deals or create new institutions to boost skills or research if any sector can show this would address specific problems.

Great idea! As one of the people working on developing “skills and research”, I’d like to suggest that it might be a good idea to arrange an agreement to share students, workers, and researchers with our neighbours, who are similarly technologically-developed and share common scientific and educational traditions. We could call it the Anglo-European Union, or something like that.

But no, that would help “old institutions” like my own. The Westminster Pharaoh is only interested in boosting skills or research if it can create “new institutions” as a monument to her greatness.

American carnage

This is probably the best title for a reality TV show ever proposed in a presidential inaugural address. And then there’s this.

Mothers and children trapped in poverty in our inner cities, rusted-out factories, scattered like tombstones across the landscape of our nation, an education system flush with cash but which leaves our young and beautiful students deprived of all knowledge.

And you have to admit, most of us really are tired of our education system being flush with cash. Such a burden, and what do you get for it? Early-onset gout from the rich cafeteria food, and gold-plated textbooks that give the children scoliosis from having to lug them around, that’s what. And the rest of the economy starved for talent as the best and the brightest compete for the overpaid classroom posts. We really don’t know what to do with all that money, and our young and beautiful students won’t learn anything anyway. (We don’t care much what happens to the non-beautiful…)

That’s why we need Trump as president: Someone who has learned his whole life to cope with the burden of extreme wealth, and is willing to lift it from us.


Don’t do the maths!

Journalist Simon Jenkins has launched a broadside against the teaching of maths in school, or at least against taking it seriously. He goes further than Andrew Hacker, who argues prominently for a focus on more concrete mathematical skills.

No one would argue that pupils should not be able to add, subtract and multiply. But I studied higher maths, from calculus to number theory, and have forgotten the lot. All the maths I have needed comes from John Allen Paulos’s timeless manual, Innumeracy. It is mostly how to understand proportion and risk, and tell when a statistician is trying to con you.

Presumably, once you know how to count to 1000 you’ve learned enough. (I’m wondering about this claim about his having “studied higher maths”. At least according to Wikipedia his university subjects were philosophy, politics, and economics. Now, I have no doubt that some people can learn very advanced mathematics in their spare time and understand it wonderfully. I wouldn’t even object to them saying they had “studied” the subject. But if your private study of mathematics left you with no memory of what you thought you had learned, that suggests that perhaps the fault was in your mode of study, and not in the subject. It’s rather like someone who says, “There’s no point learning to swim. I spent years on it, and I still can’t cross a pool without drowning.”

And why is it that statisticians are always accused of trying to “con” people? Is it that statisticians are particularly dishonest? Or is it that statisticians make things sufficiently clear that you can see where you might disagree with them. What subject would you study to understand when a journalist is trying to con you? There isn’t one, because the journalist’s con is ambiguous, and for the most part his claims are clouded in rhetorical smog.

Then there’s this:

I agree with the great mathematician GH Hardy, who accepted that higher maths was without practical application. It was rather a matter of intellectual stimulus and beauty.

Now, GH Hardy was indeed a great mathematician. He probably knew more about higher maths, from calculus to number theory, than even Simon Jenkins in his prime (before he forgot everything). But I think we can also agree that the man who wrote in 1940

No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years

did not have the most acute vision of the scope of mathematical application. In any case, Hardy’s goal was not to argue for or against the potential utility of mathematics, but rather to defend mathematics against the charge of uselessness — basically, to defend it against people like Jenkins.

Any league table that has China at the top, Britain at 26th and America at 36th tells me something more important than merely who is good at maths. If the US and Britain – among the most vigorous economies and most successful at science – are so bad at maths, it suggests their young people are applying themselves to something more useful. Chinese students are rushing to British and US universities to join them….

Maths is merely an easy subject to measure, nationally and internationally.


I am reminded of a bumper sticker I saw in Florida, responding to the popular boastful messages that parents would paste on, saying “My kid is an honor roll student at Dingdong Middle School”; the response said “My kid can beat up your honor roll student.” This is that bully-boy bumper sticker expanded to a national scale. Let the inferior races waste their time on mathematics. Our kids will learn how to be “vigorous” and kick their asses.

Jerome Karabel’s wonderful book The Chosen describes how elite universities in the US in the first half of the 20th century, dismayed at how the meritocratic elements of their admissions process were being abused by Jews, who were simply outperforming their gentile compatriots on admissions tests, leading to the freshman class at Harvard in 1922 being more than 20% Jewish. The response, driven by fear that Jews would “drive away the Gentiles” (in the words of Harvard president A. Lawrence Lowell) was to de-emphasise quantitative measures and tests, in favour of the all-important “character” of applicants, a quality husbanded mainly by WASP families in exclusive boarding schools.

There’s kind of a Nietzschean flavor here: Mathematics has replaced Christianity as the intellectual tool used by the weak (nerds) to dominate their natural superiors (men of action and vigor like Jenkins). The soul-breaking catechism has been replaced by the binomial theorem. The priests are statisticians and bureaucrats, obsessed with counting and what can be measured. I am reminded of a remark by CS Lewis (I can’t find the exact quote now), that soft virtues like love and mercy had come to be more discussed than rigid virtues like chastity and courage, because it is easier to persuade yourself that you have been loving than that you have been chaste or courageous.

Cheating at maths

One thing you get used to as a mathematician: You meet someone in a non-professional context, you tell them what you do (“mathematics” coming after they’ve pushed through vague dodges like “teaching”… “at the university”…), and they look away furtively, as though you’d gratuitously inquired after the origin of their scar or their PTSD, and say something like “I could never do maths”; occasionally a more wistful “I always liked maths at school”. I thought of this when reading this article about a recent Christmas chat by Labour leader Jeremy Corbyn and shadow Chancellor of the Exchequer John McDonnell:

Corbyn was followed by McDonnell (“he’s about to spend all our money,” said the Labour leader by way of introduction), who thanked the Eastern Daily Press for publishing a letter from a former classmate who revealed that he used to “whisper the maths answers to me to avoid me being caned”. He joked of the Daily Mail headline he expected: “Chancellor cheats at maths again”.

Clearly, he thinks his creative solution to maths anxiety — backed up by the cane — is something that right-thinking people should, if not admire, at least condone, and possibly chuckle at in self-recognition. But as the Labour Party’s aspirant to helm the Treasury, which does presumably require some sort of numeracy, doesn’t he owe the public some sort of explanation of when, if at all, he did actually learn to do sums?

Politicians debate statisticians and philosophers

I should have known the writing was on the wall for my career in Canada when, at the first federal election debate in 2006, the Bloc Québécois leader Gilles Duceppe said

We don’t need inspectors. We don’t need statisticians. We need doctors and nurses.

The rest of academia kept their heads down, hoping the storm would blow over. But now, not even a decade later, just south of the border, presidential candidates have another academic discipline in their sights. In yesterday’s Republican presidential debate Marco Rubio said

Welders make more money than philosophers. We need more welders and less philosophers.

As is pointed out here, the first statement isn’t actually true. Whether it should be true is another question. We might say, a philosophical question; although, in a serious dispute over the issue between a philosopher and a welder, I would not be surprised if the latter came out the better for it.

First they came for the statisticians…

Confidence game: Growth mindset for the secret police

Powers that give MI5, MI6 and GCHQ a “dizzying” range of electronic surveillance capabilities will be laid out in the investigatory powers bill next month, in a move that will bolster the confidence of the intelligence agencies but pave the way for a row with privacy campaigners.

According to one headline announcing this report in the Times, the security services will get the “legal right” to hack into people’s computers and other electronic devices. Under must circumstances, “legal right” might be seen as redundant, but not here. They already do these things, they have the power to do these things, but what they lack, apparently, is confidence in their abilities.

Cue the Growth Mindset (TM). I suppose it was only a matter of time before education fads started sloshing over into spying: After all, aren’t GCHQ and the others supposed to be “learning things”? What they need is confidence. The standard critiques apply:

Confidence and motivation are crucial, but confidence without competence is simply hot air.


According to a report on The Intercept, a US anti-Muslim group has been pushing back against claims that Texas teenager Ahmed Mohammed, who was recently arrested for bringing a homemade clock to school, was the victim of anti-Muslim prejudice, or, indeed, that he was unfairly treated in any way.

Center for Security Policy vice president Jim Hanson argued on his organization’s podcast that the clock “looks exactly like a number of IED triggers that were produced by the Iranians and used to kill U.S. troops in the war in Iraq.” He said the clock “was half a bomb.”

Rightwing organisations spouting nonsense is nothing worth commenting on, but I find the particular logical construction here fascinating. He’s right, after all. It is indeed half a bomb. It just happens to be the half without explosives. And if any Muslim teens think of bringing homemade telescopes to school, I trust they’ll be arrested for bringing “half a sniper rifle” to school. That may look like an innocent block of wood to you, but it’s actually half a combat knife; no more innocuous for being the part without a blade.

All very logical. I admit, it’s slightly odd to hear this obsession with dangerous components coming from the same side of the political spectrum that inclines to dismiss the dangerousness of firearms because they can’t kill people all on their own.

You can’t have your pocket money and save it too

My 13-year old child received the following maths problem in school:

Paul saves 4/15 of his pocket money and spends 5/12 on topping up his mobile phone. What fraction of his pocket money does he have left?

(The question was part of a sheet from Cambridge Essentials.) With a PhD in mathematics, I usually feel myself adequately qualified to deal with school maths questions, but this one stymied me. I have decided to stop blaming myself, though. This question is

  1. misleadingly formulated;
  2. ambiguous;
  3. exceptionally dependent on hidden cultural assumptions.

Let’s start with number 1. Who counts fractions of pocket money? This makes about as much sense as asking

Paul and Paulina order a pizza together. Paul eats 0.375 pizza. Paulina eats 0.5 pizza. How much pizza do they take home?

It’s like you were trying to teach children about toothbrushes, and showing them how useful they are by having them use the toothbrush to clean the floor. Sure, you can do it, but it’s really not the tool anyone would choose to use, and it doesn’t give them a fair impression of what it could really be good for.

Okay, maybe Paul lives in a socialist country, where “from each according to his ability”, so that prices are stated as fractions of your income. But it gets worse. Point 2: My first thought was that Paul had spent 11/15 of his money on other things — probably drugs — and now had to top up his phone, which cost 5/12 of his pocket money. But he only has 4/15, which is smaller, so he needs to go into debt by 5/12-4/15=3/20. Okay, that didn’t seem likely. So then I figured that the 5/12 was intended to be a proportion of the 4/15 that he has remaining. Then it would at least make a little bit of sense to express it as a fraction. (Extreme socialism: Prices are all formulated as a fraction of the money you have in your pocket. Customer: How much? Merchant: How much you got?) So the amount remaining is 4/15*7/12=7/45.

But on further discussion with my partner I recognized that neither of these versions was what was intended by the people who set the question. I was thinking in terms of a model of sequential spending: The money you “save” is the money you have available to spend the next time an expense arises. The question, though, presumes that money that is “saved” is being saved from yourself. Whereas I would think that the money you “save” is part of — or possibly identical with — the money you “have left”, you were supposed to think of spending and saving as just two different ways of losing money. You add the two together to get a total loss of 4/15+5/12=17/30, leaving Paul with 13/30 pocket money units to spend on non-mobile-phone and non-banking expenses. (Probably drugs.)

Of course, I’m overthinking this. The point is that you’re not supposed to think. You’re just supposed to see two fractions and add them, because that’s what you’ve been learning to do. It’s a kind of pseudo-applied maths problem that is quite common — even at university level — where any actual thought about the issues involved will only penalise you. It’s a puzzle, where you’re supposed to read through the irrelevant verbiage to get to the maths problem that has been concealed there.

I call this “adding up the temperatures”, after the story by Richard Feynman (in Surely You’re Joking, Mr Feynman) about his time evaluating textbooks for the state of California. He describes a problem from one elementary school textbook:

Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees, Green stars have a temperature of seven thousand degrees, blue stars have a temperature of ten thousand degrees, and violet stars have a temperature of … (some big number).

John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?

Feynman points out that the temperatures aren’t really right, and that there is no such thing as green and violet stars, which he is willing to tolerate, but then blows up at the sheer pointlessness of adding up temperatures. Like the above, it only looks like an application of the mathematical tool being presented (in this case addition).

But I’m even more amazed at the absurdity of the story. How is it possible that John sees only 3 stars, his father sees 4, and they see completely different stars? But the point is, in school mathematics you’re supposed to do, not think.

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