Problem... The Sophie Germain Identity

#AlgebraicManipulation,  #ExpandingBrackets, #Proof

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• For a blog post on a solution, see this post.

On The Inadvertencies of Teaching (Girls & STEM)

The lack of engagement in @TeachFMaths' female-only Twitter #MathOlympiad2017 — in relative comparison, that is, with the engagement generated by the male-only Olympiad (won by Euler, incidentally) — has spiked my interest, particularly after seeing what I think I can uncontroversially describe as a fairly exasperated tweet from @TeachFMaths pop up in my timeline, and especially since today is the international day of the girl, and yesterday was Ada Lovelace Day, an international day celebrating the achievements of women in science, technology, engineering and mathematics.

Yes, of course, @TeachFMaths' (fantastic) math Olympiads are just the kind of innocent, knockabout fun that Dorsey, Glass, Stone, and Williams could almost have made Twitter for (see @richardosman's glorious World Cup of Crisps, for example, or his World Cup of Biscuits), but as it is likely that a large proportion of the audience engaged with the Olympiads are bothered, in some respect, with the business of education, and the fact that this disparity in engagement between the male and female-only Olympiads so patently exists, inadvertently throws up, I think, some pertinent questions that it's worth schools — and particularly we as teachers — ponder a tad.  So, if you will amuse me for a moment or two:

As of today, with the female-only Olympiad still in progress, if we take a quick look at the number of votes cast in each — as a proxy for engagement (ignoring, that is, the secondary tweets, re-tweets and the like generated by the competition, where, incidentally, the disparity is even greater) — we see:

The mean number of votes cast per match up in the male-only Olympiad was 315, or 233 if we remove the two outliers that were the epic Newton v Euclid semi-final, and the final.  The mean number of votes cast per match up after the first ten rounds of matches for the female-only Olympiad is 61.  In other words, the men’s Olympiad attracted around four times the degree of engagement than the women’s Olympiad is currently attracting.  The match — or should that be mismatch — that attracted the least number of votes in the male-only Olympiad, still attracted more votes than the match that has thus far attracted the most number of votes in the female-only Olympiad.

My first instinctual reaction to this disparity was one of almost resignation, that this was not, in other words, unsurprising nor, indeed, something to read anything into; to blow out of proportion.  Of course, there are an inordinate amount of factors that could and would impact on the relative disparity in the degrees of engagement between the male and female-only math Olympiads — and I won't be going into any eigenvalue-ridden factor analysis about it here.  (It is worth pointing out, nonetheless, that @TeachFMaths has more followers now, and thus potential participators for the female-only math Olympiad, than he did when he ran the male-only math Olympiad, and I’m pretty sure we can rule out math Olympiad fatigue, so to speak.)  But I do wonder, from the perspective of a Dad of a daughter, a one-time Engineer turned maths teacher and Headteacher, from the perspective of someone who has seen Bourdieuian socio-cultural reproduction at play first hand day in day out when it comes to the life 'choices' students' make, whether this neutrality of sorts — or this out of sorts neutrality perhaps — to put it quite simply, matters, and matters more that is than we give it credence.

When voting in the female-only Olympiad (put yourself in the position of voting if you haven't, yet), are we voting more in terms of what mathematician we have heard of, rather than what we perceive to be the relative merits of their mathematics, as I would contend would be more the case in the male-only Olympiad?  Are our female-only votes more a function of the relative celebrity status of the mathematicians, as it were, rather than their achievements, in contrast to our male-only votes?  Are our votes simply an act of reinforcement?  (Take Ada Lovelace, for example, relatively unknown ten years ago, she now has a day named after her, and trounced the opposition in the group stage of the Olympiad.)  What are our choices — to vote or not to vote, rather than who to vote for — a function of?

When it comes to our engagement in a fun, onthefaceofit innocuous social-media competition, it does not — or, moreover, should not — matter whether groundbreaking achievements have been made by men, historically, or not.  It is irrelevant to the act of voting, or to the act of choosing to vote or not.  To put it another way, if I am being asked to vote between Isaac Newton and Euclid of Alexandria, and then to vote between say Phoebe Sarah Hertha Ayrton and Linda Goldway Keen, the very act of me voting is — or should be — independent of the makeup of the group from which I am choosing.  The only thing stopping me from voting for the female-only match up would be the fact that — as I am embarrassed to admit — I know very little of either of Ayrton or Keen, and would need to do some research.  Or, perhaps, in choosing not* to vote in the female-only math Olympiad, I am adopting some kind of aesthetic/intellectual distance (cf. Bourdieu, 1984, p34) to distinguish myself from something, to signify something about myself to others.  Maybe my 'detachment, disinterestedness, indifference' (in that order) should really be read as 'disinvestment [my emphasis], detachment, indifference, in other words, the refusal to invest oneself and take things seriously' (ibid.).  (*I am voting, incidentally.) 

And therein lies the issue that perhaps @TeachFMaths' fantastic Olympiads have, in an unintentional, Milgram-esque sense, exposed (whether the issue needed to be or not).  From the perspective of the girls that I teach, whether I am ignorant of, or ambivalent to the achievements of female mathematicians in comparison to their male counterparts, irrespective of the relative degrees of their achievements, the outcome is the dangerous same, namely that I run the risk of reproducing and reinforcing the conditions whereby girls are inadvertently denied access to the positive gender role models they may need (see, for example, Dennehy and Dasgupta, 2017; Lockwood, 2006; Allen and Eby, 2004).

This is, of course, not, in any sense, to dilute the joyous sense of inspiration that comes from exposure to and appreciation of the amazing achievements and breakthroughs made by the great male mathematicians of history.  We all, boy or girl, need these subject-centred, or intellectual role-models.  And I want my sons and daughter, and the sons and daughters whom I teach, to aspire to be mathematicians, or scientists, or engineers, or the whateveritistheywanttobe's of the future — but we must remember that the daughters we teach may need female role-models as well, in contrast to the sons we teach, by virtue of the simple fact that history is relatively uncluttered by them.  As a teacher and school leader I of course ensure that students are exposed to — and develop an appreciation for — the great achievements of thinkers throughout history, irrespective of gender.  The point that I am labouring to end on is that girls need female role models as well, and that as a teacher it is part of my job to do more than just not deny them this.  I need to think, carefully, differently maybe, about my own, individual, inadvertent behaviours, limitations (of knowledge), and perspectives, that may inconspicuously — but burningly — limit or even curb girls' notions of what they can or cannot do. 

(Thank you to @TeachFMaths for his #MathOlympiad2017 and for making me think again about my own practice, however inadvertent that may have been!)

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On Ada Lovelace Day

Founded in 2009 by Suw Charman-Anderson, a social technologist, journalist and writer (and, I must say, an enviable talent for naming cats), Ada Lovelace Day (ALD) is more than merely a day of observance, if you like; it is a day when schools should unashamedly revel in a pure celebration of scientific and technological achievement, made, as it happens, by relatively un-celebrated women, whose passion for learning, for understanding and discovery — for education — overcame any socio-culturally formed gender barriers that may have otherwise got in the way (read Project Ada's interview with Suw Charman-Anderson here, discussing why she founded the day.)

As a teacher and school leader, marking ALD with and for our students is absolutely about raising the profile of women in science, technology, engineering and mathematics (STEM) — but, importantly, for our boys, as well as our girls.  The day is also, and indeed, unashamedly about raising awareness of contemporary female STEM role models explicitly for the girls under our care: as Lockwood (2006) suggests, 'women... may derive particular benefit from the example of an outstanding woman who illustrates the possibility of overcoming gender barriers to achieve success... in contrast [to] men'.  But to suggest that we put aside one day in the school year to celebrate the achievements of women in STEM is — or should be — anathema to any institution that bothers itself with the business of education.  Marking ALD is just one of the many ways schools use historical and contemporary achievements to motivate and inspire young minds.  The day is an opportunity to boot for schools to unabashedly reinforce the ethos underpinning our organisations, to reaffirm the values that implicitly or otherwise inform the everyday work we do.  Yes, we are celebrating the historic achievements of women who have excelled in their field, but by also framing their achievements against the contemporary work of today's pioneers, we are cultivating a set of cultural conditions for the future where boys as well as girls — or arguably where boys more than girls — understand that it is not just OK for girls to be STEM nerds too, many quite simply are, so get over it. 

This post is not about me retelling Lovelace's story, or discussing the history or controversy over the extent of her work — others have done this with scope and rigour and you will find or may stumble across links to some excellent sources through this post.  Suffice to say, in the words of Ursula Martin, Professor of Computer Science at the University of Oxford,  'Lovelace saw the potential for the computer to do amazing things', but to really 'understand Lovelace’s work, you need to know the context of the ideas of the time.'  (You can read a short, fascinating interview with Professor Martin about Lovelace's papers here (you may also enjoy perusing Oxford's Bodleian library's blog dedicated to Lovelace).  Through this post I am simply encouraging schools to mark the day, hopefully providing some simple suggestions, or spikes for further exploration, for how this may be done, without any huge organisational implications.

For example, ALD also, coincidentally, marks the start of Global Math Week (October 10-17), and the end of the United Nations' World Space Week (October 4-10).  With this in mind, two potential avenues for exploration with or by students on the day could be around the work of Maryam Mirzakhani, the Fields Medallist who tragically passed away only recently, and perhaps around the story of Valentina Tereshkova, the first woman in space and subject of a current science museum  exhibition (read this from Doug Millard at the Science Museum blog). 

You will find a Twitter Moment I created marking Maryam Mirzakhani's death here, with links to a number of sources you may wish to explore and/or share with students, including a brief documentary about her groundbreaking work.  You may also wish to explore with students this fantastic task around topology, Mirzakhani's field of research, devised by Professor Yahya Tabesh, one of Mirzahkani's mentors when she was studying mathematics, and shared by Jo Boaler and the youcubed.org team.  The question is 'can you morph a donut [sic]?'   (You may also want to explore the excellent Women of Mathematics throughout Europe website.)

Whilst, despite the name, ALD is not explicitly about Ada Lovelace, her story, achievements, and visionary insights are more than worth exploring with students.  You will find some other sources of information — and links to other sources — about Lovelace in the Twitter Moment I created here, and the Bodelian library have an excellent short biography of her here.  Rachel Thomas at Plus magazine has written another excellent and accessible article here, and you may want to share this brief video by @YourLifeTeam and narrated by Hannah Fry (@FryRsquared), who also told Lovelace's story in more depth in this superb BBC documentary.  (You may also want to use Hannah's wonderful talk on Benford's Law and 'repeat victimisation', at the Royal Institution for ALD 2014, 'Can Maths Predict the Future?')  If you want to delve a little deeper into the enigma that is Ada Lovelace, consider 'Untangling the Tale of Ada Lovelace', an excellent biographical blog post by Stephen Wolfram.

This link will take you to a digital transcription — by John Walker (@fourmilab) — of Lovelace's translation of Luigi Federico Menabrean's write up of Babbage's lecture series on the Analytical Engine at the Academy of Sciences in Turin, including Lovelace's famous notes, including Note G, the first published computer program: Instructions for the Analytical Engine to calculate Bernoulli numbers.  (The First Edition of Lovelace's translation is currently valued at $28,000.)

You may also wish to explore and share the videos from the 2016 ALD Live event organised by findingada.com, and held at the Institution of Engineering and Technology, featuring design engineer Yewande Akinola, planetary physicist Dr Sheila Kanani, science writer Dr Kat Arney, developer Jenny Duckett, mathematician Dr Sara Santos, computational biologist Dr Bissan Al-Lazikani, and climate scientist Dr Anna Jones.  This year's event is taking place at the Royal Institution, and you can find more information about it here.  In addition, findingada.com have produced an excellent pack of resources for schools, pitched  for use with 11-14 year olds.

Gloucestershire STEM network have produced a worksheet providing an introduction to coding using free online resources that you can access here, and the University of Edinburgh has produced a series of Open Educational Resources to complement their wonderfully hectic ALD schedule.  Adalogical Ænigma produces sets a tricky logical problem every month that are definitely worth a look at; this one published by Alex Bellos in The Guardian is a good place to start.

There are of course an inordinate amount of simple ways that schools and teachers can mark ALD, not least through assemblies, publicising via the school website and/or social media, etc.  Marking ALD most effectively, I believe, however — and as ever — is through our interactions as teachers with students in our classrooms.  Below, for example, is a sheet I affixed to my Y7 class' page in their exercise books on last year's ALD, to invoke their curiosity and questioning — which it certainly did!

Just recently, my little girl succeeded in a making three-strand plait for the first time, alone, after an inordinate amount of time trying, i.e. in her own words, "6 years" (she's 6.)  We will now, of course, be exploring more plait and braid types (see SecureRF's 'Introduction to the mathematics of braids' and NRICH's 'Plaiting and Braiding').  Drenched in the joy at her achievement, with the plait clasped firmly in her hand like a theorem, she ran to me screaming, "Look, I figured it out by myself.  I'm an Ada!"  Marking ALD can make a difference, to someone, even it is just one person.  

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