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.

More uncertainty confusion

After commenting on the confusion between different clichés about physics and physicists in reporting about Angela Merkel, I feel obliged to note this sentence, from an article in the New Statesman about the fake traveller-tourist dichotomy:

The rush to witness the “authentic” ultimately alters the reality, in a kind of behaviourist butterfly effect.

Once again, physics clichés are being confounded. When you’re looking for an educated-sounding way to make the banal observation that it’s hard to observe things without getting mixed up in them, and so changing them, the cliché you want is “uncertainty”. The “butterfly effect” is what you cite when you’re bloviating about how small actions can have large long-term effects.

It’s slightly depressing for anyone who has hopes for general science education. It suggests that even if you come up with compelling ordinary-language metaphors for scientific concepts, the result will just be a salad of interchangeable expressions gesturing vaguely at an undifferentiated mass of physics woo-woo concepts.

Too many orang-utans?

I recently read Pierre Boulle’s Planète des Singes [Planet of the Apes]. I knew about the novel, of course, but hadn’t read it. It is very much of its time and place — though, as I have commented, the origins of the story have been sufficiently obscured by the various film versions, as to make a French version seem to an American cartoonist a plausible punchline. What I had not anticipated was the extent to which the novel is a satire about scientists, the management of science, and science education. The point is well summarised mid-way through the story, when we are finally given an overview — from the chimpanzee perspective — of the social structure of the planet Soror. I say social structure, but the only apes who are of any interest are scientists of some sort or other, and the only social or political organisation we hear about is scientific, though we do hear about a more brutal past, where the gorillas ruled by force. They have maintained the habit of power.

Ils excellent dans l’art de tracer des directives générales et de manoeuvrer les autres singes. Quand un technicien a fait une découverte interéssante, tube lumineux par exemple ou combustible nouveau, c’est presque toujours un gorille qui se charge de l’exploiter et d’en tirer tout le bénéfice possible. Sans être véritablement intelligents, ils sont beaucoup plus malins que les orang-outans. Ils obtiennent tout ce qu’ils veulent de ceux-ci en jouant de leur orgueil. Ainsi, à la tête de notre Institut… il y a un gorille administrateur, que l’on voit très rarement…
[They excel in composing general instructions and in manipulating other apes. When a technician has made an interesting discovery, for example a luminiferous tube, or a new fuel, it is almost always a gorilla who takes charge of the development and extracting the maximum possible benefit. Without being genuinely intelligent, they are much more clever than the orang-utans. The gorillas get everything they want from them by playing on their pride. Thus, our Institute is headed by a gorilla administrator, who is almost never seen.]

The gorillas also produce, when they do occasionally stoop to research, massive tomes that are expertly structured and organised, even if the content is produced by others, each one by a different subaltern chimpanzee.

The orang-utans are referred to as the “official science”, although

certains se poussent parfois dans la politique, les arts et la littérature. Ils apportent les mêmes caractères dans toutes ces activités. Pompeux, solennels, pédants, dépourvus d’originalité et de sens critique, acharnés à maintenir la tradition, aveugles et sourds à toute nouveauté, adorant les clichés et les formules toutes faites, ils forment le substratum de toutes les academies. Doués d’une grande mémoire, ils apprennent énormément de matières par coeur, dans les livres. Ensuite, ils écrivent eux-mêmes d’autres livres, ou ils répètent ce qu’ils ont lu, ce qui leur attire de la considération de la part de leurs frères orang-outans…. Le malheur c’est qu’ils fabriquent ainsi tous les livres d’enseignement, propageant des erreurs grossières dans la jeunesse simienne.
[some of them do occasionally make their way into politics, art, and literature. They display the same characteristics in all their activities. Pompous, solemn, pedantic, lacking in originality and critical sense, obsessed with preserving traditions, blind and deaf to all novelty, adoring clichés and settled formulas, they form a substratum in all the academies. Gifted with excellent memories, they learn enormous amounts of material by heart from books. Then they write it all down in other books, repeating exactly what they read, thus attracting the approbation of their brother orang-utans… The real tragedy is that they write, in this way, all the textbooks, perpetuating gross errors among the simian youth.]

As for the chimpanzees,

Ceux-ci semblent bien représenter l’élément intellectuel de la planète. Ce n’est pas par forfanterie si Zira soutient que toutes les grandes découvertes ont été faites par eux. C’est tout au plus une généralisation un peu poussée, car il y a quelques exceptions. En tout cas, ils écrivent la plupart des livres intéressants, dans les domaines les plus divers. Ils paraissent animés par un puissant esprit de recherche.
[They appear to be the intellectual element of the planet. It is not mere boastfulness when Zira claims that all the great discoveries have been made by chimpanzees. To be sure, it is a bit exaggerated, as there are some exceptions. In any case, they write most of the interesting books on all subjects. They appear to be motivated by a powerful spirit of research.]

There must be important lessons for us here, in the age of the REF. Also for teaching. The government wants us to produce more gorillas, but our education system is optimised for orang-utans. As for the chimpanzees, they’ve recognised that they’ll muddle through anyway, or enough of them anyway, motivated by this “powerful spirit of research”, willing to work for a few bananas on fixed-term contracts.

Hannah’s sweets

The following problem appeared on one of yesterday’s GCSE maths exams, leading to considerable frustration and media attention:

Hannah has 6 orange sweets and some yellow sweets.

Overall, she has n sweets.

The probability of her taking 2 orange sweets is 1/3.

Prove that: n^2-n-90=0

^ is “to the power of”

Now, I am a professional probabilist, and I wasn’t immediately sure how to do it. Why not? Well, there’s something missing: The problem doesn’t tell us what Hannah’s options are. Did she pick sweets at random from the bag? How many? Are we asked the probability that she took 2 orange sweets rather than 3 yellow, given that she actually prefers the orange?  Did she choose between taking sweets out of the bag and putting it away until after dinner?

There should have been a line that said, “She picks two sweets from the bag, at random, without replacement, with each sweet in the bag equally likely to be taken.”

According to the news reports

Hannah’s was just one of the many supposed “real life” problems that the students were required to tackle.

This is just an example of the ridiculous approach to mathematical “applications” induced by our testing culture. It’s not a “real life” maths problem. It’s a very elementary book problem, decked out with a little story that serves only to confuse the matter. You are supposed to know a standard rule for decoding the chatter. If you try to make use of any actual understanding of the situation being described you will only be misled. (Richard Feynman described this problem, when he was on a commission to examine junior high school maths textbooks in California in the 1960s. His entertaining account is the chapter “Judging Books by their Covers” in Surely You’re Joking, Mr Feynman.)

Sikh and ye shall find

… a primary school place.

Apparently the government’s decision to wipe out new school construction and put huge amounts of the remaining schools budget into independently run “free schools” has led to some unexpected consequences:

More than 20 pupils have been allocated places at a Sikh-ethos free school in Leeds that they did not choose, amid a shortage of school places.

The school is not a faith school, but it is run with a Sikh ethos.

As someone with a child who had little option but to attend a Church of England school, that really was a “faith school” — secular options were much further away, even if they did have places — I wonder what the fuss was about.

On its website, the Khalsa Free School stresses it is not a faith school, adding: “We are firmly committed to developing our pupils’ understanding and appreciation of the diverse world in which they live.” It says: “We welcome all children regardless of their backgrounds or faiths and we aim to help all our children develop a lifelong love of learning, which will support them throughout their academic careers and beyond.” 
“For Sikhs, education not only prepares students for work and life in society but also supports spiritual growth. Education is understood by Sikhs to raise aspirations and personal standards, encourage self-awareness and humility, and inspire all to seek a greater purpose in life.”

Humility! Purpose! No wonder parents are outraged. What impact will self-awareness have on lifetime earnings? How will humility help them get in to Cambridge? Where is the acknowledgement of the divine imperative to maximise test scores?

14th Century NIMBYism

In Juliet Barker’s book on the Great Revolt of 1381 I was struck by this comment on the spread of local grammar schools in England in the second half of the 14th century:

Where there was no dedicated room or building available, classes were held in the local church. In 1373 the Bishop of Norwich prohibited this practice in the schools of King’s Lynn, on the grounds that the cries of beaten children interrupted services and distracted worshippers.

Nowadays the bishop and the local residents would have cited the shortage of parking… Continue reading “14th Century NIMBYism”