Occasional reflections on Life, the World, and Mathematics

Posts tagged ‘math anxiety’

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?

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.

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.)

More scary maths

I was just working through the sheet music for “As Time Goes By”. I don’t remember ever having heard the intro — it was left out when the song appeared in Casablanca, and seems never to have been performed since. It begins with the lines

This day and age we’re living in gives cause for apprehension,

With speed and new invention, and things like the third dimension.

Yet, we grow a trifle weary with Mister Einstein’s theory,

So we must get down to earth, at times relax, relieve the tension.

“Third dimension” — pretty scary! (Those interested in the influence of geometric ideas on early 20th century literature, in particular the work of Franz Kafka, could look at this paper. Though typically the anxiety about dimensions started at four.)

I’ve commented earlier on suggestions that math education should be confined to the rudiments to spare children from math anxiety, and the use of math anxiety by unscrupulous politicians to distract attention from their policies.

Tag Cloud