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.)

The Nobel prize in mathematics

There was an interesting article in Der Spiegel about Angela Merkel’s visit to a Berlin secondary school as part of the the “EU-Projekttag”, a national day for teaching about the EU and its institutions. (No surprise that nothing like this happens in Britain.) This school has mostly Muslim immigrant children, and she found that instead of asking about the functions of the European Parliament the children wanted to tell her about discrimination in Germany.

Fatma, eine 15-jährige Jugendliche mit Kopftuch, klagt über Schwierigkeiten beim Praktikum im Kindergarten, weil die Eltern keine Erzieherinnen mit Kopftuch wollen. Das habe ihr Chef ihr gesagt. Ja ja, sagt Merkel, die inmitten der Schüler auf der Bühne Platz genommen hat, man kenne das Problem von Bewerbungen junger Menschen mit komplizierten, ausländisch klingenden Namen. “Viele glauben da nicht, dass jetzt gleich ein Nobelpreisträger in Mathematik um die Ecke kommt.”

[Fatma, a 15-year-old with head-scarf, complains about her difficulties in an internship in a kindergarten, because the parents don’t want a teacher with head-scarf. Her boss told her that. Yes, yes, says Merkel, who is sitting on the podium with the students, we know these problems, as with job applications from young people with complicated, foreign-sounding names. “Many people don’t think, this is a future Nobel-prize winner in mathematics coming around the corner.”]

Never mind this bizarre and nearly incomprehensible stream-of-consciousness from a major world leader asked an uncomfortable question by a 15-year-old. What is it about the chimeric Nobel prize in mathematics? Alfred Nobel established prizes in subjects that were related to the kind of practical science that he made his fortune with (chemistry and physics) and to the kind of selfless causes (medicine, literature, peace) that he hoped would blur the association of his name with weapons manufacture. There are lots of subjects that he did not create prizes in. Mathematics. Geology. Engineering. Astronomy. History. Cooking. No one thinks it odd that any of these subjects don’t have a Nobel prize, except mathematics. They think it so odd, that they either imagine that there actually is one, as above, or they invent outlandish stories to explain this lacuna, generally involving some mathematician — Gosta Magnus Mittag-Leffler, when he is given a name — running off with Nobel’s wife. (This story has the advantage of Mittag-Leffler actually having been Swedish, but the fact that Nobel never married is usually counted against its credibility.)

The Pope’s Shluchim

I’ve just been reading Amir Alexander’s book Infinitesimal, about the intellectual struggle over the concepts of infinitesimals and the continuum in mathematics and science (and theology) in the 17th century. The early part of the book is a history of the Society of Jesus, presented as a ruthless and intellectually daring force for religious conservatism, strictly hierarchical, devoted to its holy founder, a thoroughly mystical movement that built its reputation and influence on educational outreach. And then it struck me: The Jesuits were just like Chabad-Lubavitch!

Correct me, Lord, but in moderation…

Jeremiah 10:24.

Accounts of error-correcting codes always start with the (3,1)-repetition code — transmit three copies of each bit, and let them vote, choosing the best two out of three when there is disagreement. Apparently this code has been in use for longer than anyone had realised, to judge by this passage from the Jerusalem Talmud:

Three scrolls [of the Torah] did they find in the Temple courtyard. In one of these scrolls they found it written “The eternal God is your dwelling place (maon)“. And in two of the scrolls it was written, “The eternal God is your dwelling place (meonah)”. They confirmed the reading found in the two and abrogated the other.

In one of them they found written “They sent the little ones of the people of Israel”. And in the two it was written, “They sent young men…”. They confirmed the two and abrogated the other.

In one of them they found written “he” [written in the feminine spelling] nine times, and in two they found it written that way eleven times. They confirmed the reading found in the two and abrogated the other. (tractate Ta’anit 4:2, trans. Jacob Neusner)

(h/t Masorti Rabbi Jeremy Gordon, who alluded to this passage in an inter-demominational panel discussion yesterday at the OCHJS. He was making a different point, which for some reason had very little to do with information theory.)

The next war

The BBC reports that education secretary Nicky Morgan “wants England to be in the top five in the world for English and maths by 2020. It is currently 23rd.” They quote her:

Returning us to our rightful place will be a symbol of our success. To achieve this, we will launch a war on illiteracy and innumeracy.

So, I’m thinking about wars that Britain has prosecuted over the past half century or so, often with the goal of “returning us to our rightful place”. Suez. Falkland Islands. Bosnia. Iraq. Yemen. Cyprus. Kenya. Afghanistan. Northern Ireland. Not all disasters, but not an unbroken record of glory either. Not really a set of memories you want to activate if you want your audience to think “overwhelming success” rather than, say “useless drain on national resources”, “antiquated racist ideology”, or “undermining democracy and human rights”.

Putting aside the absurd-sounding ambition for England to be among the top 5 for English, (I’ll just guess this wording reflects the slightly vague British awareness that foreigners tend to speak Foreignish, and so might have literacy skills to be tested that aren’t literally “English”) the battle plan for maths all comes down to tables:
Continue reading “The next war”

Math and science

Corey Robin updates us on l’affaire Salaita. I was struck by his comment

Thirty-four heads of departments and academic units at the University of Illinois at Urbana-Champaign wrote a scorching letter to the University of Illinois’s new president[…] Clearly, far from diminishing, the controversy on campus has only expanded.

What’s even more amazing is where it has expanded: three of the signatories are chairs of the departments of chemistry, math, and statistics. The opposition has spilled beyond the walls of the humanities and social sciences. During the summer, lots of folks dismissed this story because the natural sciences weren’t involved. Well, some of them are now.

Math and statistics aren’t really natural sciences, in the crucial economic sense. The people who dismiss the boycott because it’s just the humanities and social sciences are somewhat expressing a sense that those academics are woolly-headed cultural relativists; but even more, I think it’s about the idea that “serious” academics have big grants and big labs and generally deal with big money. Chemistry is the outlier here. Math and statistics are still much more constructed on the same economic model as the humanities, hence barely one step removed from socialism.

The peer-review fetish: Let’s abolish the gold standard!

I’ve just been reading two books on the climate-change debate, both focusing on the so-called “hockey stick graph”: Michael Mann’s The Hockey Stick and the Climate Wars: Dispatches from the Front Lines, and A. W. Montford’s The Hockey Stick Illusion: Climategate and the Corruption of Science. I’ll comment on these in a later post, but right now I want to comment on the totemic role that the strange ritual of anonymous peer review plays for the gatekeepers of science.

One commonly hears that anonymous peer review (henceforth APR) is the “gold standard” for scientific papers. Now, this is a reasonable description, in that the gold standard was a system that long outlived its usefulness, constraining growth and innovation by attempting to measure something that is inherently fluid and abstract by an arbitrary concrete criterion, and persisting through the vested interests of a few and deficient imagination of the many.

That’s not usually what people mean, though.

An article is submitted to a journal. An editor has read it and decided to include it. It appears in print. What does APR add to this? It means that the editor also solicited the opinion of at least one other person (the “referee(s)”). That’s it. The opinion may have been expressed in three lines or less. She may have ignored the opinion.

Furthermore, to drain away any incentive for the referee(s) to be conscientious about their work,

  • They are unpaid.
  • They are anonymous. We know how well that works for raising the tone of blog comments.
  • Anonymity implies: Their contributions will never be acknowledged. If they contribute important insights to the paper, they may be recognised in the acknowledgement section: “We are grateful for the helpful suggestions of an anonymous referee.” Very occasionally an author will suggest, through the editor, that a referee who has made important contributions be invited to join the paper as a co-author. More commonly, a paper will be sent from journal to journal, collecting useful suggestions until it has actually become worth publishing.*
  • No one will ever take issue with any positive remarks the referee makes, as no one but the authors (and the editor) will ever see them. Negative comments, on the other hand, may get pushback from the author, and thus need to be justified, requiring far more work.
  • Normally, the author will be forced to demonstrate that she has taken the referee’s criticism to heart, no matter how petty or subjective. This encourages the referee to adopt an Olympian stance, passing judgement on what by rights ought to be the author’s prerogative.

Of course, I don’t mean to say that most referees most of the time don’t do a very conscientious job. I take refereeing seriously, and make a good-faith effort to be fair, judicious, and helpful. But I’m sure that I’m not the only one who feels that the incentives are pushing in other directions, and to the extent that I do a careful job, it is mainly out of some abstract sense of duty. I am particularly irritated when I find myself forced to put original insights into my report, to explain why the paper is deficient. I would much rather the paper be published as is, and then I could make my criticism publicly, and then, if I’m right, be recognised for my contribution. Continue reading “The peer-review fetish: Let’s abolish the gold standard!”

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.

Who needs math?

According to a study by sociologist Michael Handel, summarised here by Jordan Weissman, 75% of American workers never use any mathematics more complicated than fractions in their work. (It goes without saying that most partake of recreational calculus, at least on weekends…) Writing in the NY Times last year, Andrew Hacker argued that most schoolchildren are wasting their time learning mathematics: They’ll never understand it, and they won’t be any the worse off for it. As for scientists, the great entomologist E. O. Wilson has recently taken to the pages of the Wall Street Journal to argue that

 exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory.

For that matter, even Albert Einstein famously remarked to a schoolgirl correspondent

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.

But that was after he’d mostly decamped from physics for sagecraft.

Wilson goes on to portray mathematical biologists as technicians, armed with useful tools and useless ideas. And if you need them, you just hire them. (It’s not like they have anything important to do with their time.) So what’s going on? Are mathematicians scamming the public, teaching algebra and other unnecessaries to justify their existence? I would suggest that there are several important issues that these wise men are ignoring or underplaying:

  • It may be true (as Hacker argues) that only a tiny techno-elite actually needs to know how a computer works, or how to compute the trajectory of a spacecraft, or how to program a Bayesian network. But when they’re 11 years old you don’t know who will have the interest or aptitude to join that elite. If you start sieving the children out early because they don’t seem like a likely candidate for that track — and let’s be honest, a lot of the tracking is going to be based on parental status and educational attainment — most of them will have no way to change tracks later on, because of the cumulative nature of mathematical understanding. Worth noting, in this context, is Handel’s observation (cited above) that skilled blue collar jobs are actually slightly more likely to require “advanced maths” (algebra and beyond) than skilled white collar jobs. So you can’t decide who needs the advanced maths based on the kinds of work they’re going into. Those without the education are simply more likely to be stuck at the lower rungs of whatever trade or profession they go into. (On the other hand, a larger fraction of white collar workers are in Handel’s “upper” (skilled) category, so an average blue collar worker probably needs less maths than an average white collar worker.)
  • Mathematics is a language. And what is discussed in that language is, as Hacker recognises, crucial to the fate of everyone in the world. Those who have not learned at least the rudiments of the language are excluded from the conversation. I am reminded of a friend who dismissed the value of learning to speak French, with the argument that “Everyone in France speaks English.” Now, France might have been a bad choice for his claim, but even if it were true, it puts you at a significant disadvantage to be surrounded by people who speak your language, while you can’t decipher their language to understand what it is they’re saying to each other.
  • Think about that Einstein quote: Everyone finds mathematics difficult when they’re pushing beyond their current knowledge. If we’re going to drop mathematics training when it becomes challenging, we might as well stop counting when we run out of fingers and toesies.
  • I would suggest that Wilson may be using more sophisticated mathematics in his work than he is aware. To paraphrase J M Keynes, practical biologists who believe their work to be quite exempt from any need for mathematics, are usually the slaves of some defunct mathematician. Modern biologists of bench and field are often quite attached to some mathematical and statistical machinery that happens to be some years old, and seemed impossibly abstruse when it first seeped in from the pure mathematics or theoretical statistics world. Many of the attempts to apply mathematical techniques in biology (or sociology or economics or whatever) will prove more clever than enlightening, but some will stick, and become part of the basic toolkit that the biologists who think they don’t need any sophisticated math do use. Wilson’s arrogant posture really reflects the fact that there are far more trained mathematicians who are intellectually flexible enough to try and figure out what the biologists are doing, and what the connections might be to their own field, than trained biologists willing to work in the other direction.