Wednesday, 15 November 2017

On Proving the Sophie Germain Identity

Marie-Sophie Germain (1776-1831) was, in short, an extraordinary mathematician.  Noted for her work across a range of disciplines, from number theory to physics and astronomy, she is widely recognised as one of the first women to make significant and original contributions to mathematical research.

Germain's story is, in more ways than one, a remarkable one, and it is not hyperbole to describe her achievements, whether framed within the context of time and place or not, as inspirational. Forbidden to attend University and thus unable to make a career from her mathematics, Germain, with the financial and moral support of her father, famously taught herself.  She was inspired by Jean-Étienne Montucla's Histoire des Mathématiques, read Newton and Euler, and, under the nom de plume of a Monsieur LeBlanc, corresponded with preeminent mathematicians of the time, most notably Adrien-Marie LegendreJoseph-Louis Lagrange and Carl Friedrich Gauss.  It was through her persistence, resilience, ingenuity, inquisitiveness and most importantly the sheer, outrageous brilliance of her enduring work, that she overcame — to an arguable exten— the prejudices of 19th-century French society that threatened to impede her way in mathematics.

Germain was inspired by Ernst Chladni's beautiful, awe-inspiring 'figures' to formulate a mathematical theory of elastic surfaces that described Chladni's experimental observations, eventually publishing Récherches sur la théorie des surfaces élastiques in 1821 (see Gardner's and Kowk's fascinating paper), and in doing so winning the prix extraordinaire from the esteemed Académie des sciences.  She was also one of a select number of mathematicians to have worked on Fermat's Last theorem, including 'Euler, Legendre, Gauss, Abel, Dirichlet, Kummer and Cauchy' (Dickson, 1917), resulting in arguably her most notable achievement, what is now known as the Sophie Germain Theorem (for more detail see herethis typically insightful account from Simon Singhand this from Robin Whitty of, who notes that Germain provided 'the first real breakthrough on Fermat's Last Theorem since Fermat’s death in 1665').

Whilst this post is not about Germain's life or accomplishments per se, it is an advocation — it is about sharing Germain's story with our students, but in a way moreover that achieves more than mere reach, or appreciation, or admiration.  This post is about sharing Germain's story with our students in a way that supports their learning and enhances their mathematical experience.  It is about using narrative to personify, if you like, Germain's mathematics — in an effort to make it more accessible, to make the seemingly difficult be more 'learn-able', to undermine in turn the pervasive (and often persuasive) fear of 'hard maths'.  For all of our students, boy or girl, for the sake of their own mathematical maturation and in the perspectivist sense of the messages we inadvertently transmit as teachers, it is the way in which we tell Germain's story that matters more than the story itself [1].

Germain, along with Emmy Noether, Hypatia of AlexandriaSofia Kovaleskaya and Ada Lovelace, are often the women schools refer — or should that be, if you will forgive me a degree of provocativeness here, defer — to when celebrating the women of mathematics, usually on a perfunctory once or twice a year, such as on the albeit superb Ada Lovelace Day[2].  But of course every day is (or should be) Ada Lovelace Day, and of course the stories of these amazing mathematicians should be told, celebrated, and learned from.  However, to encourage the greatest effect (i.e. deep and lasting learning for our students), we must take care to share these stories at appropriate — i.e. not overly contrived — points in the curriculum, and that when doing so, in the face of such overwhelmingly compelling socio-cultural narratives, we do not lose or dilute the underlying story of their astounding mathematics and mathematical achievements. 

As I have elaborated upon in more detail in an earlier post, we must do more as teachers than just not deny (inadvertently or otherwise) the girls we teach access to the positive gender role models they need (see, for example, Dennehy and Dasgupta, 2017Lockwood, 2006Allen and Eby, 2004).  And this is about more than just telling their stories, it is also about considering what it is that makes them positive role models, considering what it was that drove their intellectual achievements.  To this end, we should want all of our students, boy or girl, to be inspired by Germain's mathematics, to be inspired by Germain because of her mathematics and the manner in which she did it, not just by the albeit inspirational fact that her mathematics was done in the face of almost insurmountable odds. 

I am suggesting, therefore, that we look for genuine ways of exposing our students to the accessible aspects of Germain's mathematics (and the mathematics of her ilk) as early as possible in their mathematical careers  wherever that is possible, wherever that is doable.  It is a deliberate effort, in other words, to expose students to what they will perceive to be ‘hard maths’ earlier than is perhaps typical, when they may not ordinarily be considered 'ready', to increase therefore the opportunities for ‘inspiration’ and support our students' mathematical maturation, to sail against the prevailing cultural and pedagogical winds and make students begin to see themselves as mathematicians in the making.  Exploring Sophie Germain primes when working on properties of number, for example, or, indeed, when working on algebraic manipulation, this:

Prove the Sophie Germain Identity
\[{a^4} + 4{b^4} \equiv \left( {{a^2} + 2{b^2} + 2ab} \right)\left( {{a^2} + 2{b^2} - 2ab} \right)\]

The outline presented as follows is a ‘thinking through’ of how this problem could be approached in a way to make it accessible to secondary students yet to have formally learned how to factorise expressions.  The problem is intended, thus, to be shared and explored with students before they typically learn the relatively more advanced techniques that would normally (and more efficiently) be used, i.e. completing the square and factorising by finding the difference between two squares.  It is intended in other words for use and exploration with secondary students who have some knowledge of simplifying and manipulating algebraic expressions by collecting like terms, multiplying a single term over a bracket, expanding products of two binomials, and the multiplication index law.

The idea behind this exercise is to consider with students how they could prove the identity, using their existing knowledge, without the direct verification of the identity that they are likely to suggest — without, in other words, ‘going that way’, multiplying out the right-hand side and simplifying (which should still of course be applauded and reinforced as a sound proof).  This then situates students at the limits of their existing knowledge and frames their approaches within a more ‘problem solving’ context.  Students’ understanding of the techniques they have already learned — or are in the process of learning — should deepen as a result of working with this problem.

The accompanying text is intended purely to help teachers consider how to approach the problem with students, structurally [3] and pedagogically, and consider what are the questions that can/should be asked of their students, should they not yet be able to ask them for themselves. (This from could be used to illustrate how the identity is often used in modern mathematics, and  thus perhaps lead onto other areas of learning.)

Notes & (Select) Links:

[1]  For posts of a more biographical bent see, for example, this from Brain Pickingsthis from Evelyn Lamb in her wonderful 'Roots of Unity' blog, and peruse the select links at the end of this post for other suggested sources. For a more in depth treatise on Germain, consider this authoritative biography from 1980 by Bucciarelli and Dworsky.

[2]  See this Twitter moment celebrating Ada Lovelace Day 2017, and see this, by Evelyn Lamb and her Twitter list of contemporary 'mathy ladies'.

[3]  Working on proving the identity could be the centre piece of a larger, longer body of work on algebraic manipulation.  Students could be introduced to the identity and told that by the end of the body of work they are about to embark on, they — or the class as a whole — will have proven it.  It could be used as a conceit before work on the more advanced techniques mentioned so that students could then gain — over time — an appreciation of the relative elegance of different proofs.  It is also a problem that will encourage depth and maturation, and give us as teachers the opportunity, as I wrote about in a previous post, to 'model for [students] what it is to be mathematically mature, to behave in a mathematically mature manner when we are befuddled by the beautiful sixes and sevens of uncertainty.'  

[4]  If you want to know why I have used the identity sign throughout, see this.

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