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Mathematics Today

Mathematics Today is the membership publication of the Institute of Mathematics and its Applications.

Issued six times a year, this general interest mathematics publication provides articles, reports, reviews and news for mathematicians.

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Current features and articles 

Editorial

If you were asked to summarise your vision for how mathematics education should develop over the next 20 years or so, what would you say? The Royal Society has been thinking hard about this question (in the broader context of science and mathematics) and in June published its Vision for Science and Mathematics Education [1]. As might be expected, the committee that was responsible for the report was made up of numerous extremely distinguished individuals and it is nice to be able to report that the Chair of the group was a mathematician (Sir Martin Taylor). The Vice-Chair, Professor Dame Julia Higgins, spoke about the Vision at the IMA@50 event that took place at the Royal Institution in October. Incidentally, this was a great event made even more special by being the first time that I had ever been in the lecture theatre that is used for the Christmas Lectures. The knowledge that I was in the very room where Faraday enthused audiences in the 19th century more than made up for the lack of leg-room!


Celebrating Mathematics: IMA-Ri joint event

The IMA President Dame Celia Hoyles chaired the third and final major IMA 50th Anniversary Celebration at the Royal Institution (Ri) on 7 October. The event was well attended by over 150 IMA members and invited guests, Ri members, school children and the public. The first speaker, Dame Julia Higgins, spoke about the Royal Society Vision for Science and Mathematics Education. Dame Julia’s talk was based on the Royal Society Report which explains their ambition to make mathematics and science education fit for purpose in 20 years’ time. The full report and summary can be found at: https://royalsociety.org/education/policy/vision/. A detailed article based on Dame Julia’s talk can be found on page 284 of this issue.


Career Profile: Dominic Thorrington MIMA

E ThorringtonJob Title: Research degree student, London School of Hygiene & Tropical Medicine.
Number of years in current position: 2 years.
Qualifications: MSc Modern Epidemiology, BSc Mathematics.
What stimulated your interest in maths, and when? It’s difficult to choose one particular point in time but I certainly enjoyed mathematics more than the other subjects at school. My teacher was incredibly helpful in encouraging me to study the subject at a more advanced level after GCSE and to consider the further applications of the tools we acquired in the classroom. My teachers at A-level were equally encouraging. One additional source of inspiration was the book Zero: the biography of a dangerous idea by Charles Seife – I bought this at the National Air and Space Museum in Washington DC and loved it. This was my first real taste of mathematics outside of textbooks. My collection of mathematics/popular science books has since expanded somewhat.


Transition to STEM Degrees – Further Maths A-level

The rapid pace of change within post-16 mathematics education over the last fifteen years has created both opportunities and challenges for mathematics educators. Since the well-documented damaging outcomes of Curriculum 2000, A-level Mathematics has arguably gained a reputation as a challenging domain, with recent research suggesting that Mathematics and particularly Further Mathematics A-levels are amongst the most demanding A-level subjects [1].
This article gives an overview of topical research and curriculum activity in relation to post-16 mathematics and outlines the utility and incidence of Further Mathematics A-level qualifications in progressing to a wide range of degree courses. Data relating to trends in admissions to STEM courses over the past eight years will be presented, providing evidence of the extent of uptake of the A-level Further Mathematics qualification by those progressing to STEM undergraduate degrees. The role of universities in influencing post-16 mathematics education is also considered.


Pendulum Pattern Perception

Cycle PolishThis paper considers the imagery generated by multiple pendula swinging from a single beam. Although it is not generally detectable, the pendulum bobs lie on a sinusoid of contracting wavelength. Instead, an onlooker sees a sequence of diverse patterns produced by the bobs. Disparity between the undetected and perceived imagery is caused by insufficient signal sampling. The sequence of visual effects is explained mathematically.
Our ability to learn is, in many ways, a measure of resourcefulness. Like others, we sought the answers of our youth in the dusty basements of university libraries as all but a rite of passage. Today, the halls of knowledge are often just a mouse click away. Through video-sharing website YouTube for instance, MIT throw open their illustrious lecture theatres while fellow Americans Harvard have captivated users with truly mesmerising scientific demonstrations (e.g. a coffee mug survives a perilous fall in [1] thanks to a pencil, some string and angular momentum).


Full contents page of the December 2014 printed issue
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Content from the October 2014 issue

Editorial

Simon Jenkins wrote a piece for the Guardian on 7 August [1], which might charitably be described as a diatribe against things that he doesn’t understand. Some of his er­rors were pointed out in a letter from Paul Glendinning [2] but Jenkins’ ignorance regarding science and mathematics does not mean everything in the article is off the mark. There are quite a few ideas put forward in the article which I am entirely comfort­able with; that science graduates should be articulate and able to explain themselves clearly, that modelling our maths educa­tion system on that of the old USSR or of China is a bad idea, for example. But nevertheless the article leaves a sour taste, I think because there are large numbers of influential people in the UK who have virtually no understanding of mathematics but whose ignorance results in them failing to see the importance of our discipline. I do not subscribe to the view that a degree in Politics, Philosophy and Economics from Oxford is the best basis on which to pontificate about the importance or otherwise of science or mathematics, which is somewhat of a worry when you realise that apart from Jenkins this group includes both our current prime minister and the leader of the opposition, as well as a host of other MPs.


How Graph Theory is Changing Marketing

Most people who have studied Mathematics beyond A-Level will be familiar with the bridges in Königsberg. The seven bridges that helped Citizens cross from one part of the city to another are enshrined in mathematical history thanks to Leonard Euler and the Seven Bridges Problem.
Königsberg, which, since 1946 has been called Kaliningrad, is located a few miles inland from the Baltic Sea along the Pregel River. If you were aboard a boat heading through the city towards the sea, you’d first come across the island of Lomse. Once past Lomse, you’d then need to navigate around a small ten hectare island on which the Königsberg cathedral stands proud.


IMA Festival of Mathematics and its Applications

For many years we have had, in the UK, excellent science festivals, science fairs and even a science week. However, despite mathematics being the Queen of the Sciences and underpinning all of STEM, there has, up to now, never been a Festival of Mathematics, and mathematics tends to have a small presence in the major science festivals. This is an unaccepta­ble situation for the future of UK mathematics, as it denies the general public, and young people in particular, the op­portunity to see mathematics and its many applications in all of its glory. To mark its 50th anni­versary year the IMA decided to change this, and launch, what it hopes, will be an exciting annual event: the Festival of Mathematics and its Applications.


Sex Drugs and Sausage Rolls

Professor Sir David Speigelhalter FRS, OBE gave the evening popu­lar lecture at the IMA Festival of Mathematics and its Applications at The University of Manchester. The talk, Sex, Drugs and Sausage Rolls, was aimed primarily at sixth formers, but was enjoyed by all – from primary school children to professors.

David Spiegelhalter is Winton Pro­fessor for the Public Understanding of Risk, University of Cambridge. The talk and reception was generously sponsored by Winton Capital Management. You can read David’s blog at understandinguncertainty.org. He has also written a book about risk and it is accompanied by a promotional website, http://thenormchronicles.com/, which illustrates the story of Norm (a normal guy) and his risk of death from birth onwards.


Full contents page of the October 2014 printed issue
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Content from the August 2014 issue

Editorial

During the early years of the eighteenth century the accurate determination of longitude at sea was a very pressing con­cern, particularly acute for a maritime nation like Great Britain. A number of shipwrecks and the associated loss of life were directly at­tributable to mistakes in the computation of longitude and aside from high-profile major disasters the difficulties in navigation caused by the inability to accurately determine one’s position led to longer voyages which in turn led to greater risk of scurvy and other illnesses which afflicted the sailors of the age. The impact on trade was also significant and many fortunes were ruined for want of a solution to this basic problem of navigation.


The IMA’s 50th Anniversary Celebration

The IMA’s 50th Anniversary Celebration at the Royal Society on 14 May was a great success. The IMA President, Professor Dame Celia Hoyles DBE welcomed The Princess Royal and expressed warm thanks to her for agreeing to be our Patron in our anniversary year and for graciously agreeing to come to our event.
The audience, including IMA members, representatives from government and other STEM societies, enjoyed lectures on the diverse areas of mathemat­ics that the IMA supports.


50 Years of Mathematics in Industry

How does maths affect the lives of ordi­nary people? How does it benefit us all, improving our quality of life? And how best can we bring mathematicians and busi­ness people together, so that maths can continue to contribute to our economy in the UK?
These were some of the issues that we discussed recently, as we celebrated the fiftieth anniversary of the Institute of Mathematics and its Applications. The occasion provided an opportunity to reflect on the contribution of maths to industry over the last half century, and to look to the future.


Mathematics for the Billion

My title pays homage to Lancelot Hogben’s Mathematics for the Million, a book that explained basic mathematics and practical uses in simple terms for a general audience. I think that today we have to be more ambitious.
When I was seeking inspiration for this lecture, my mind was drawn to the phrase πόλλ’ οἶδ’ ἀλώπηξ, ἀλλ’ ἐχῖνος ἓν μέγα. I did Latin at school, not Greek, but mathematicians know their Greek letters. Even I can recognise ‘echinos’ and ‘mega’, and make an educated guess, so it’s something about a big hedgehog. It actually translates as ‘The fox knows many things, but the hedgehog knows one big thing’. It is a fragment attributed to the Greek lyric poet Archilochus (c. 680 – c. 645 bc).


Two Trains and a Crazy Bird

The examples of least-squares applications given in Parts I and II have been used by the author when invited to give lectures about the importance of mathematics to non-specialists, including teenagers still at school with different abilities and interests. The mathematics of the communications example is arguably too technical for such audiences, but the explanation of how we form different speech sounds and then model them for use in mobile telephones ensures that they at least appreciate the value of the mathematics. The underlying principles of the navigation examples are perhaps easier to understand, as is the idea of making a guess and then iterating to converge to a solution.


Full contents page of the August 2014 printed issue
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Content from the June 2014 issue

Editorial

At some point in my school education (in the late 1970s if my memory serves) I was introduced to logarithms as a means of facilitating calculations. I was taught how to use log tables to multiply, divide and take roots at a time when calculators were just becoming commonplace and I doubt many children younger than me were exposed to these techniques. What those who have grown up in the calculator age perhaps don’t realise is that logarithms were the principal means of calculating with decimals for over 300 years. Log tables and their associated techniques arrived from nowhere, were an instant sensation, dominated scientific calculation for centuries, and disappeared just as quickly as they had arisen. It seems appropriate to reflect on this in 2014 as the first set of such tables were published in 1614 in Napier’s Mirifici logarithmorum canonis descriptio (‘A Description of the Wonderful Principle of Logarithms’) which contained tables and a brief explanation of how they were to be used; the theory behind the construction of the tables appeared posthumously in 1619.


Why Does Mathematics Matter in the Digital Arts?

Celia Hoyles, Malcolm Sabin and Jon Macey presented Why Does Mathematics Matter? The Case of the Digital Arts at the British Congress of Mathematics Education in April 2014.
Celia Hoyles introduced the session which was inspired by one of the Mathematics Matters series of mathematics research case studies, ‘Advancing the Digital Arts’ and introduced her co-presenters. Jon Macey is from the University of Bournemouth which is now a world leading institution in the field of computer animation. Jon, Celia explained, would describe the crucial underpinning role that mathematics plays in the truly astonishing field of computer animation. Malcolm Sabin, a pioneer of many techniques that have permitted the development of modern CAD tools, would describe the closely related field of computer aided design.


The Festival of Mathematics and its Applications

A major activity to celebrate the IMA’s 50th anniversary will be the Festival of Mathematics and its Applications, to be hosted in the Alan Turing Building, School of Mathematics at the University of Manchester. The event will bring together all elements of our broad community: academic mathematicians; mathematicians working in the corporate, industrial and business sectors; undergraduate and postgraduate students; teachers and lecturers; members of the general public with an interest in mathematics; school and college students and their families.


How Microinsurance can Help to Reduce Poverty

According to the World Bank around 2.2 billion people, or almost 31% of the global population, live on incomes less than $1.25 a day, with 3.6 billion living on less than $2 per day [1]. People living on such low incomes usually have no access to formal financial services. In particular, they often have little access to formal risk management solutions, including traditional insurance products. As a result, the world's poor are often the most exposed to the impact of many high-risk events that can cause financial difficulties in their lives.

This article gives an introduction to one possible approach to mitigating the impact of high-risk events on low-income groups: microinsurance.


Fifty Visions of Mathematics Solution

The IMA’s book Fifty Visions of Mathematics was launched at the 50th anniversary celebration at the Royal Society on 14 May. The book contains 50 Images – Visions – of Mathematics and 50 articles on many topics, including sport, sports gambling, CAT scans, viruses, space, biographies, to name a few.

Below, there are ten quotes from the book matched to their articles. The unmatched quotes are available on page 159 of the June 2014 issue of Mathematics Today.


Full contents page of the June 2014 printed issue
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Content from the April 2014 issue

Editorial

As many readers will know, EPSRC commissioned in­ternational reviews of UK mathematics research (IRMs) which were published in 2004 and 2010. The reports, available on the EPSRC website, make interesting reading. A considerable part of these two reports addresses the question of the strength or otherwise of the UK mathematics research commu­nity at the time of writing and how this compares with the best world­wide. Both the reports, and the re­view processes that led to them, were valuable but it is arguable that the 2010 report has had limited impact on the research landscape. In contrast, the 2004 report led, more or less directly, to a signifi­cant investment in a number of areas that had been highlighted by the review. Having said that, neither report is as bold in scope, as thought provoking, as challenging or as important as the recently published The Mathematical Sciences in 2025.


The Festival of Mathematics and its Applications

A major activity to celebrate the IMA’s 50th anniversary will be the Festival of Mathematics and its Applications, to be hosted in the Alan Turing Building, School of Math­ematics at the University of Manchester. The event will bring to­gether all elements of our broad community: academic mathema­ticians; mathematicians working in the corporate, industrial and business sectors; undergraduate and postgraduate students; teach­ers and lecturers; members of the general public with an interest in mathematics; school and college students and their families.


Maths Joins the Rolls-Royce Science Prize

The Rolls-Royce Science Prize was first implemented as an annual awards programme in 2004 as an incentive for teachers to enhance science teaching in their schools and colleges. There is a total of £120,000 to be won by schools each year. In 2014 Rolls-Royce, in partnership with the IMA and the NCETM, has extended the Prize to include mathematics teaching in addition to science. The Prize recognises and rewards excel­lence in Science and Mathematics teaching for all children, from those with special education needs to high ability pupils, and cov­ers Primary, Secondary and post-16 age ranges.


Beautiful Music

Beautiful music is not easy to define. Some would claim that it is impossible to define because the terms `beautiful' and to some extent `music' are entirely subjective -- the aural equivalent of `in the eye of the beholder' (presumably `in the ear of the listener', although the phrase doesn't have quite such a ring to it). Such people might contend that anyone who finds the sound of rush-hour traffic more beautiful musically than, say, Mozart's Elvira Madigan Concerto is taking a perfectly reasonable position.


Communication, Navigation and Mathematics: Part I

This article is based on a public lecture given by the author at the University of Reading on 13 November 1997 to celebrate his appointment as a Visiting Industrial Professor. The material has remained in his files and unpublished until now. The primary aim of the lecture was to demonstrate that the basic mathematics which underpins one technology (e.g. digital speech communications) can often be applied to a quite different technology (e.g. navigation systems). This article follows the basic theme of the lecture, but includes more of the mathematics. This is the first part of the article, Speech Communications, the second part on Navigation will appear in the next issue of Mathematics Today.


The Dome that Touches the Heavens

This year is the 50th year of the IMA and my 50th year too. Birthdays are times for celebration and reflection – to identify what really inspires us. Appropriately then, very soon, on my birthday, I will give a talk to a group of architects in Paris about when architecture inspired new mathematics and vice versa. One example is the invention of the catenary curve, which is popularly attributed to Sir Christopher Wren (1632–1723) and his design of St Paul’s Cathedral (built 1675–1720).

Wren’s cathedral was the highest building in London from 1710 to the not so distant 1962. It was built after the Great Fire of London brought destruction to much of the city and changed the landscape not only of London, but of the building trade in England too.


Full contents page of the April 2014 printed issue
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Content from the February 2014 issue

Editorial
Maths Today August 11

What the IMA is for can readily be seen from its website, where the first paragraph is the following: ‘The Institute of Mathematics and its Applications (IMA) is the UK’s learned and professional society for mathematics and its applications. The IMA exists to support the advancement of mathematical knowledge and its applications and to promote and enhance mathematical culture in the United Kingdom and elsewhere, for the public good.’

As usual this mission statement doesn’t tell us much, but a few more clicks reveal a wealth of information.  


A Message from HRH The Princess Royal, the IMA Patron for 2014
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I am delighted to be the Patron of the Institute of Mathematics and its Applications (IMA) in its 50th Anniversary year. This is not the first occasion in which my family has been involved with the Institute: His Royal Highness The Duke of Edinburgh was President from 1976 to 1977, and the Institute was incorporated by Royal Charter in 1990.

In 1964 the founders of the IMA created a learned society to support mathematics and provide professional status for mathematicians, with membership open to those working in academia, commerce, industry and education. Over the ensuing 50 years it has contributed to many developments in mathematics and provided professional support and services to mathematicians.


The Next 50 Years
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In trying to write a vision for where (applied) mathematics, (applied) mathematicians and the IMA will be in the next 50 years, I am ever mindful of John von Neumann’s attempt to do this for the subject of computing, in which he said that any statement about the future was bound to look pretty silly in just a few years time. Given the exponential rate at which mathematics and its applications are developing, and the incredible creativity of mathematicians in thinking up utterly new ideas, I think I will be lucky if I even get a few years grace. But having said this, here goes.


Interview with Celia Hoyles, the New IMA President
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Professor Dame Celia Hoyles, CMath CMathTeach FIMA is a Professor of Mathematics Education at the London Knowledge Lab, Institute of Education, University of London. She took over as President of the IMA at the beginning of the year. Professor Hoyles interests include mathematics policy and research in students’ conceptions of proof, mathematical skills in modern workplaces, computational environments for learning and sharing mathematics and systemic change in teaching mathematics. She was awarded the International Commission of Mathematics Instruction (ICMI) Hans Freudenthal medal in 2004 and the Royal Society Kavli Education Medal 2011. I interviewed Celia Hoyles in May 2013.


Sherlock Holmes and the Three-Body Problem
One of the joys of reading fiction is speculating on how the world created by the author can relate to the real world. No series has been more successful in this respect than Conan Doyle’s Sherlock Holmes stories, which have spawned a considerable literature of interpretation based on the pretence that Sherlock Holmes was a real person, perhaps for the following reasons:

Full contents page of the February 2014 printed issue
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