*Above: a meeting of Portuguese mathematicians*

Cordelia Fine’s book “Delusions of Gender” finds that a repeated theme in arguments about biologically-determined inequalities is that girls and women are worse at maths, and mathematically-structured fields like physics.

The “compensating” claim is that girls and women are, supposedly, biologically determined to be better at empathy and caring.

The nearest to a “proof” of this claim is the results for “mental rotation performance”, “the largest and most reliable gender difference in cognition”. Males are usually found to do better, on average, than females in telling rotations of three-dimensional shapes apart from mirror-reflections.

Fine reports research with Italian high school students who were told, before being tested, that “women usually perform better than men in this test”. The women then performed just as well as the men.

Even if the difference were real, why would that generate difference in maths more than in, say, visual arts, design, and architecture?

Aside from visual dexterity, the other qualities required for maths are, if you think about it, ones stereotypically identified as “feminine”: precision, attention to detail, neatness, imagination, ability to hold onto many ideas simultaneously, persistence and stamina, self-critical attitudes (most mathematicians get most things wrong most of the time), and sociability and cooperativeness (maths is the most social of sciences).

The quality cited as essential for maths and supposedly “masculine” is logical, analytic, systematic thinking.

In the first place, why would logical, analytic, systematic thinking be less important for law than for maths? Yet the photo which my younger daughter has just put on her Instagram account of herself with the other Judge’s Associates at the Supreme Court of Queensland (the top-end just-graduated law students there) shows 13 women and 5 men.

In the second place, different people are good at maths in many different ways. Some can “see” mathematical relations quickly by visualisation; others can get little from diagrams. Some are brilliant calculators; others are quick at seeing possible approaches, but relatively slow at checking them out by calculation…

And some excel at painstaking logical analysis, while others excel at leaps of intuition and imagination. (See here for an example: Heaviside and Schwartz).

I see no evidence that women are systematically “good at maths” in different ways from men. But even if they were, they would still be “good at maths”.

The “mental rotation” test, as above, should identify women as worse at visual studies as much as at maths. Yet in British universities, 65% of students in creative arts and design subjects are women, 50% of architecture students (and 62% of law students). In all those fields, men do better at the best-paid senior end, but that can hardly be because of aptitudes biologically imprinted from babyhood.

Yet in the UK still only 37% of university maths students are women. (Only 21% of uni maths lecturers, and only 9% of professors).

Things are better elsewhere, and, oddly, not necessarily in countries where women’s equality is generally better.

Possibly best in the world is Portugal, where 46% of university maths and computer science students are women, 68% of maths Ph D students, 50% of maths lecturers, and 32% of maths professors.

Fine reports that in Armenia close to 50% of university students specialising in computer science are women (UK: 17%). The explanation which Fine got from Hasmik Gharibyan, an Armenian woman professor of computer science now working in the USA, is unexpected:

“In Armenia, ‘there is no cultural emphasis of having a job that one loves… The source of happiness for Armenians is their family and friendships, rather than their work… [So] there is a determination to have a profession that will guarantee a good living’.”

Fine explains how “stereotype threat” – the need to use mental energy fending off negative stereotypes – hurts in maths.

“Stereotype threat hits hardest those who actually care about their maths skills… and thus have the most to lose by doing badly, compared with women who don’t much identify with maths. Also, the more difficult and non-routine the work, the more vulnerable its performance will be to the sapping of working memory”.

She also quote findings from Shelley Correll in 1988 that among US high school students, boys tended to rate themselves as better at maths than girls would, at each level of actual achievement. There wasn’t the same male self-overrating, and female self-underrating, on verbal skills.

“Boys do not pursue mathematical activities at a higher rate than girls do because they are better at mathematics. They do so, at least partially, because they *think* they are better”.

Heather Mendick, who did a Ph D study on maths and gender in UK schools and came to talk with one of my classes about it in July 2018, found the same: girls are more reluctant to call themselves “good at maths”.

I think this may have shifted, or at least that more subtle mechanisms are operating.

Ten of my students were in class when Heather came, five girls, five boys. I asked them at the end: “Do you think you are good at maths?” Four of the girls said yes; one said she was just “ok”. Only one of the boys said “yes”; four said they were just “ok”. I’ve had similar responses elsewhere.

When in the course of the discussion Heather said something like: “Of course, maths isn’t easy for anyone”, the whole class immediately responded by laughing and indicating one of the girls: “Except for some!”

At least at some level, I think that girl knows she’s brilliant. In fact, most of my classes at that school have included girls recognised by me and by their classmates, and, at least eventually, by themselves, as stars.

Yet I’ve always found that the boys are more likely to attempt prize problems, outside the syllabus, than the girls. The boys are less worried by the risk of falling flat on their face with such problems.

My reading is that girls are more likely (on average! for socially-conditioned reasons!) to think that they’re good, even very good… but maybe not quite good enough. Boys are more likely to think that they’re maybe not brilliant… but good enough.

Fine’s argument about “stereotype threat” would fit into that picture.

Shortage of confidence (even at the level of thinking “I’m very good… but may still not good enough”) is a bigger difficulty in maths than in other subjects, I think. Asked to write an essay on the rise of Hitler, even the least confident student is likely to write something (and then can be told: “This is good!”) Asked to show, say, that in every right-angled triangle with whole-number sides, the three sides multiplied together must make a number divisible by 60, the less confident student is likely just to freeze and write nothing, while the more confident will make attempts, not be too worried if some attempts are way off the mark, and probably come up with at least some good ones.

The examples of Armenia and Portugal suggest that winning equality for girls and women in maths is not just a straightforward subsection of winning general equality. But it can be done.

]]>“Maths Appeal”, with Bobby Seagull and Johnny Ball.

https://itunes.apple.com/gb/podcast/maths-appeal/id1445702010

or https://open.spotify.com/show/3IbZnnrIgKuKdr7MQTSy3c

The problem they discuss is: if you add the squares of the ages of two grandchildren, then take the square root, you have the age that their grandmother was 50 years ago. If the grandmother is less than 100 years old, what are the ages?

Solution at https://mathsmartinthomas.wordpress.com/2017/12/24/solution-to-susan-okerekes-grandmother-problem/

]]>By considering the possible factorisations of 33,127, show that there are exactly two values of m for which is a perfect square, and find the other value.

]]>The diagram shows that the rectangle of area F_{n+1}F_{n} is the sum of the squares F_{1}^{2}, F_{2}^{2}… F_{n}^{2}.

It proves that F_{n+1}F_{n} = F_{1}^{2} + F_{2} + ^{2} + … + F_{n}^{2} more quickly, neatly, and vividly than by induction.

In the same way, the attached diagram proves

F_{n+1}F_{n-1} – F_{n}^{2} = (-1)^{n}

more quickly, neatly, and vividly than by induction.

]]>Luciano Rila has illuminatingly discussed this STEP problem as an example where the “working” is found easy by most STEP students, but writing the answer clearly and logically is much harder.

]]>In many areas of maths, especially mechanics, drawing clear, neat, big diagrams is useful to organise your information.

Also, in many areas, *imaginative* diagrams, which do not just organise your information, but change the way you see it, are useful.

For example, a problem about a triangle can be transformed into a problem about three points on a circle. A problem about three circles can be transformed into a problem about three right-angled triangles.

]]>Click here for notes from James Jones, a maths teacher in the USA

]]>Convalescing from an operation, I’ve been reading *The Girl Who Played With Fire*, a thriller by Stieg Larsson, who as well as being a novelist and investigative journalist was also a revolutionary socialist.

It’s a good thriller, but also includes what has plausibly been called “the silliest fictional mathematics” ever, the silliest ever version of the myth that mathematics is all about inexplicable, given-by-nature, eccentric flashes of inspiration. As the Polish-Canadian mathematician Izabelle Laba has explained:

The worst thing about the series *[of stories, of which TGWPWF is no.2]* is the mathematical interludes in *The Girl Who Played With Fire*. We’re told that Lisbeth Salander *[the heroine]* is also a puzzle-loving math genius who solves Fermat’s last theorem, or thinks she does, in a passage that *[the Cambridge mathematician]* Tim Gowers singled out for attention some time ago *[as “the silliest piece of fictional mathematics I have ever come across”]*.

Mind you, I’m all for having more novels and movies with strong, resourceful and mathematically talented heroines. I just wish that the math part weren’t so far off the mark…

Salander comes to mathematics by way of puzzles: Rubik’s cube, intelligence tests in magazines, every logical puzzle that she can lay her hands on. She has always been good at solving them, but was not aware of their mathematical side until sometime between the end of TGWTDT and the start of TGWPWF. Mathematics, to her, is “a logical puzzle with endless variations”, a meta-riddle where the goal is to understand the rules for solving numerical or geometric puzzles.

Salander’s primary resource is a book called *Dimensions in Mathematics* by a Dr. L. C. Parnault (Harvard University Press), a 1,200 page book that’s allegedly considered the bible of mathematics. Quite unsurprisingly, neither Dr. Parnault nor the book in question exist in real life, but Larsson tells us that *Dimensions* is a book about “the history of mathematics from the ancient Greeks to modern-day attempts to understand spherical astronomy”. It’s supposed to be pedagogical, entertaining, gorgeously illustrated and full of anecdotes. Salander is fascinated by a theorem on perfect numbers – one can verify it for as many numbers as one wishes, and it never fails! – and then advances through “Archimedes, Newton, Martin Gardner, and a dozen other classical mathematicians”, all the way to Fermat’s last theorem.

Unwilling to look at the “answer key”, she skips the section on Wiles’s proof, and tries to figure it out for herself… *[Andrew Wiles’s proof, with Richard Taylor, of Fermat’s last theorem, completed in 1995, is lengthy and uses 20th-century techniques which require years of postgraduate study to acquire: click here for an idea of it. There is no way that a brilliant amateur could find it in a flash of inspiration, any more than she could design a self-driving truck in a similar flash].*

This is all easy to mock, but unfortunately it seems to be a pretty accurate reflection of what mathematics means to most people… *[Gardner, who died in 2010, was a (brilliant) writer of popular mathematical puzzles, not a working, let alone a “classical”, mathematician]*… With all due respect to Gardner and his work, I have a problem with the image of mathematics as the art of puzzle solving.

Sure, mathematics involves logical arguments and so do mathematical puzzles. An appreciation of that does offer some insight into what we do. Regrettably, it can also lead to the notion that we get paid for playing with Rubik’s cube and solving crossword puzzles and newspaper-style intelligence tests. It’s the equivalent of a blind person touching an elephant’s trunk and concluding that elephants look like snakes.

In case any non-mathematicians are reading this: logical puzzles convey no sense whatsoever of how vast the subject actually is or how much work it takes to learn the craft. They can’t, for the simple reason that they’re created for entertainment. Their target audience can’t be expected to take a calculus class, never mind advanced graduate courses, before they can even understand the statement of the question. This already narrows it down to simple Euclidean geometry, basic combinatorics, possibly some manipulation of numbers, and eliminates most of mathematics as we know it.

You’d never learn for example that analysis, PDE or ergodic theory even exist, let alone how much accumulated knowledge there is in each of these areas. You wouldn’t get any good picture of contemporary geometry or combinatorics, either. The puzzles you’re left with may be tricky and entertaining, but they’re at best peripheral to mainstream mathematics.

The difficulty of math puzzles is usually calibrated so that the readers would have a good shot at solving them within a short time, usually ranging from a few minutes to an hour or two. A really hard puzzle is one that takes more than a few hours. No wonder that Salander was disappointed when she couldn’t solve Fermat’s theorem within a couple of days, or that she would expect a short solution with no background required.

In real-life mathematics, we don’t have a Ceiling Cat to set up problems for us and control their level of difficulty. Advisors can sometimes do that for their graduate students, to a very limited extent, but mostly we’re left stumbling in the dark, not knowing whether there even is a solution or whether we’re asking the right question in the first place. Learning to navigate this is possibly the hardest part of becoming an independent researcher.

Then there’s *Dimensions of Mathematics*. The very idea that mathematics should have a “bible” looks like a continued misunderstanding of the nature and scope of the subject. However, Larsson’s description is more reminiscent of any number of popular math books, except for the length.

If I had to suggest a real-life book for Larsson to use instead, it might be a collection of national or international Math Olympiad problems with solutions. It would not be a bible of anything, but it should present a challenge to someone like Salander at about the right level.

Olympiad problems only require normal high school background, which Salander should be able to catch up on. (Did I mention that she had dropped out of school?) They can be very hard, immeasurably more so than logical puzzles in popular magazines, but not necessarily out of reach for an extremely smart newcomer to mathematics who’s willing to put in the time and effort.

*[Or the Stanford Mathematics Problem Book, of which there is a copy in the CoLA Maths Library. The ingenuity and imagination and persistence required for these problems is something which every research mathematician has to learn, as well as learning the more advanced techniques developed by other mathematicians’ collective ingenuity and imagination over the centuries. It is also useful in many other areas of thought.*

*And sometimes a mathematician does – through knowing a lot of possible ways to approach problems, through being imaginative and persistent about trying them out, and through being quick and precise in checking them – make a simple-once-you-see-it breakthrough which produces a quicker, easier proof of something which previously required complicated and abstruse reasoning – John Conway’s proof of Morley’s Miracle, for example.]*

Click here for pdf. These notes on some beginning ideas of combinatorics cover only:

the Pigeonhole Principle

basic rules about combinations and permutations

a few features of Pascal’s Triangle

the Inclusion-Exclusion principle

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