Mathematical Angles the official magazine of the MA The Mathematics of Tomorrow President Prof Nira Chamberlain What, or who, inspired you to become a maths teacher? Mathematical PiE No. 221 Peter’s French excursion February 2024 Issue 2
3 A word from the Editors Welcome to the second edition of Mathematical Angles! Thank you to those who sent feedback after the first edition, and to those who have sent in articles or letters for this issue. We wish you a happy New Year and all the best for 2024. Did you know that 2024 is a Harshad number? It is divisible by the sum of its digits (2024 ÷ (2 + 0 + 2 + 4) = 2024 ÷ 8 = 253). Whilst this property is not particularly rare, what is rare is that we are currently on a run of four consecutive Harshad number years: 2022, 2023, 2024 and 2025 all hold this property. The next pair of consecutive years is 2119, 2120, and the next triple is not until 2464, 2465, 2466. For the next quadruple you have to wait until 3030, 3031, 3032, 3033! The early pages of this issue feature the usual MA news features. MA President Professor Nira Chamberlain’s article is entitled Mathematics of Tomorrow and is his last Presidential article prior to Charlie Stripp taking over Presidential duties in April. This is followed by the results from the MA’s research project Moving Forwards which was set up with aim of finding out what the mathematical community want from a subject association. News from Committees details news from Teaching Committee, Branches Committee (including) Branch events, and Primary Group. The regular article on Maths Teacher Training Scholarships provides all the key dates for 2024 and includes an article What or who inspired you to become a maths teacher? by maths scholar Telkya Donyai. It is with much sadness that we report on the ‘passing of friends of the MA’, which includes Vicky Neale and Tony Gardiner. There is an obituary for Professor Ian Grant Macdonald FRS, a long-time MA member, which is kindly provided by his son Christopher. We always try to acknowledge fully MA members who have recently deceased and we are keen to hear from readers who would wish to provide us such details to print. The mathematical features then follow. There is a plethora of articles from our regular contributors as well as puzzles from two new authors. Jenny Ramsden writes about Fabrizio Mordente, pioneer of instruments of precision. Paul Stephenson provides four articles: a fractions problem, a revisit to Pascal’s triangle, a piece on the music of the spheres and, later, an explanation of Peter Ransom’s French excursion from the previous issue. Neil Walker adds his unique touch by providing well-crafted school tasks: one an algebraic maze and the other a continuation of his systems of equations from issue 1. Neil Curwen poses a question of age while Erick Gooding once again provides fresh insights and questions, this time on palindromic products and applying the mean to geometry. Our new puzzle contributors are Kjartan Poskitt, who has created two interesting K-Puzz’s, and Suresh G who poses three challenging geometrical problems. The letters page gives opportunity for readers’ mathematical photos – please keep them coming in – and a new section ‘From the Archives’ republishes two articles from Mathematical Digest, a related publication, from nearly forty years ago. Mathematical Pie provides many thought provoking puzzles amidst of all of this. With thanks to Dave Lee, Bill Richardson and Jane Appleton for proof reading. Please send letters, photos and articles to the email address below. We hope you enjoy the magazine. Oli Saunders and Dave Pountney angles@m-a.org.uk Annual General Meetinn Date: Thursday 4th April Time: 18.00 Location: Crowne Plaza, Stratford-upon-Avon Mathematical Angles | February 2024 | Issue 2
4 IF I COULD TELL YOU ONE THING Edited by Ed Southall + eBook download ‘If I could tell you one thing’ is the pocket-sized maths departmental office you always wanted. Wise words and advice from a range of experienced professionals in maths education offering everything from thought provoking teaching ideas to discussions around why we do what we do in the classroom, and whether we should do it differently. Each contributor has navigated their way through years of maths teaching to bring you a slice of advice to learn from, think about, and bring back to your own department. £9.80 Members £14.00 Non Members Mathematical Angles | February 2024 | Issue 2
5 CONTENTS February 2024 | Issue 2 3 A word from the Editors 6 The Mathematics of Tomorrow Professor Nira Chamberlain 8 Moving Forwards 10 N ews from Committees Teaching Committee 10 N ews from Committees Branches Committee 10 Forthcoming Branch Events 11 News from the Joint ATM/MA Primary Group 12 T he Launch of the Loughborough University Mathematics Education Network Curriculum! 12 Oxford Dictionary of National Biography (ODNB) – Michael Atiyah 12 Maths Teacher Training Scholarships 13 What, or who, inspired you to become a maths teacher? 14 Fabrizio Mordente Jenny Ramsden 16 Fractions of infinity? Paul Stephenson 16 Pascal-like arrays from cellular automata 17 Simply solve: just algebraic equations Neil Walker 17 The music of the spheres Paul Stephenson 18 A Question of Age Neil Curwen 18 About Turns Erick Gooding Mathematical PiE 19 Average? Mean? Or something in between? Erick Gooding 20 K-Puzz Kjartan Poskitt 20 Three puzzles Suresh G 21 Peter’s French excursion Paul Stephenson 23 From the Archives Kevin Naidoo 24 Mathematics in Bed Pierre-Olivier Legrand 25 Notes for Mathematical PiE 27 Solutions 31 Passing of Friends of the MA 34 Letters The Mathematical Association Charnwood Building, Holywell Park Loughborough University Science and Enterprise Park Leicestershire LE11 3AQ Tel: 0116 221 0013 Email: office@m-a.org.uk Website: www.m-a.org.uk Please note that the copyright of all material published in Mathematical Angles is held by the Mathematical Association. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the copyright owner. Requests for permission to copy material should be sent to the Editor-in-Chief via email: editor-in-chief@m-a.org.uk or by post to the address to the left. The Mathematical Association produces a number of journals including Primary Mathematics, Mathematics in School and The Mathematical Gazette For more information visit: https://www.m-a.org.uk/ma-journals The views expressed in this publication are not necessarily those of the Mathematical Association. The inclusion of any advertisement in the journal does not imply endorsement by the Mathematical Association. Mathematical Angles is produced by Media Shed on behalf of the Mathematical Association. ISSN 2977-182X Mathematical Angles | February 2024 | Issue 2
The Mathematics of Tomorrow Throughout my MA Presidency, I have run a series of VLOGs called “What is the Point of Mathematics?” I have enjoyed doing this and I hope you have enjoyed hearing the diverse voices of mathematics teachers, educationalists, experts and stakeholders. However, the question “What is the Point of Mathematics?” is an important one. There isn’t one consistent answer that rolls off the tongue – like “What is the point of music?” or “What is the point of sports?”. Nevertheless, we all agree that mathematics is important and vital to the individual, their community, their nation and to the whole world. Before I started my VLOG, most answers to “What is the Point of Mathematics?” looked at how mathematics was used in the past! For example, how mathematics was used in the great industrial revolution, how mathematics helps humanity in agriculture and to understand the dynamics of the universe. Rarely do we hear “how is mathematics going to be used in the future?” For those who are teachers/educationalists, the students and the pupils you are currently teaching, how will they be using mathematics when they are in their 30s, 40s, 50s etc.? It is important that we teach the mathematics that our students will need for today but also the mathematics they will need for tomorrow. Hence the title of this article – The Mathematics of Tomorrow. The Mathematics of Tomorrow? Some may argue that mathematics will stay the same for the next 100 years; 1 + 1 will always equal 2, 2 + 2 will always equal 4. However, technology, which has mathematics as one of its foundations, will change. I shall take myself as an example. When I was in infant school, I remember using the abacus, when I was in secondary school the logarithm book and, later I remember the paper tape and punch card. Most of these are a distant memory. The paper tape and punch card – “what’s that?” I hear you say! It was a way of programming mechanical machines called computers. I remember a maths teacher telling us that in the future we would be using computers that would have screens that you could shuffle like paper! Rubbish, I thought. How wrong was I? As technology develops, new and emergent mathematical fields become alive. One good example is the MonteCarlo simulation. This was founded in the 1940s but, as computers have become more powerful, the MonteCarlo simulation has become a versatile method that can help solve the simplest to the most complex of problems. For example, in schools, teachers could explore using the Monte-Carlo method to approximate the area of a country. Other applications of the Monte-Carlo simulation are shown in a talk of mine called Monte-Carlo Simulation and the Pirates of the Caribbean at the Royal Society Summer Science program 2021. (see Nira Chamberlain on maths and The Pirates of the Caribbean - YouTube). In this talk, aimed at school pupils, I reiterate that these are some calculations too complicated for even established mathematicians to solve. However, I discuss how the technique is used for search and rescue, forest fire and artificial-intelligence take-over. I even showed an example of its use in long multiplication! In the Q&A, I was asked what is the future of MonteCarlo simulation? Carrying on with the same technology-related theme, the world will likely be affected by the rise of artificial-intelligence applications as we become more and more digital and data driven. This will have its 6 Mathematical Angles | February 2024 | Issue 2
pros and cons. The obvious con would be that mathematics will become more about prediction and less about understanding. To the student who doesn’t understand the underlying mathematics, they may become a consumer in a world that is making decisions at a fast pace. We have seen an example of this already when, during the recent pandemic, A level grades were predicted by an algorithm. Before we delve into this further, similar debates occurred with the rise of computer algebra systems (CAS) in the 1990s. Some had the view that this would be the end of mathematics as we knew it. For example, a student wouldn’t need to know how to differentiate, but just press the button on the machine and there is your answer! Despite the fears, to a certain extent this technology has not had the expected impact. According to the online question & answer platform Quora, the question ‘can computer algebra systems such as Mathematica make it easier to learn mathematics’, gave rise to the most popular answer: Computer algebra systems (CAS) such as Mathcad or Mathematica can make it easier to learn math[s] in several ways: 1. Visualization: CAS tools can help you visualize mathematical concepts, such as functions and equations, in a more interactive and dynamic way. This can help you better understand the relationships between different mathematical objects. 2. Automation: CAS tools can automate complex calculations, such as differentiation and integration, allowing you to focus on the concepts rather than the mechanics of the calculations. This can save time and reduce errors. 3. Experimentation: CAS tools allow you to experiment with different mathematical concepts and test hypotheses without having to do all the calculations by hand. This can help you develop a deeper understanding of the underlying principles. 4. Feedback: CAS tools can provide immediate feedback on your work, allowing you to quickly identify and correct errors. This can help you learn from your mistakes and improve your understanding of the material. Overall, CAS tools can be a powerful aid in learning maths, especially for students who struggle with calculations or who benefit from visual aids. However, it is important to note that CAS tools should be used as a complement to traditional learning methods, rather than a replacement. It is still important to develop a strong foundation in mathematical concepts and techniques, and to practise using pencil and paper. From my experience, the rise of CAS has never made me redundant as a professional mathematician, as most of my problems have fallen in the numerical/Monte-Carlo-simulation domain. However, I did enjoy experimenting with CAS, and I agree fully with the Quora output marked above in bold type. For someone who has witnessed the rise of CAS in the 1990s, I would have expected it to be much further on and more widely used in the 2020s. This may not have happened due to the rise of Big Data, AI, machine learning etc. Algorithms that will give you the answer as a number rather than a formula. So, will the future of mathematics be AI black boxes that give you the answer as a number? We may not understand how it has been arrived at. Or is the future already here? For example, those who watch football may have encountered the metric Expected Goals. According to StatsBomb https://statsbomb. com/, Expected Goals (xG) is a metric designed to measure the probability of a shot resulting in a goal. But how is this calculated? Each xG model has its own characteristics, but these are the main factors that have traditionally been fed into the large majority of Expected Goals models: distance to goal, angle to goal, body part with which the shot was taken, and type of assist or previous action (through-ball, cross, set-piece, dribble, etc…). Based on historical information of shots with similar characteristics, the xG model then attributes a value between 0 and 1 to each shot that expresses the probability of it producing a goal. This is the future where more of our life will be faced with these xG types of models, making decisions that may impact us and/or telling us when a scenario is good or bad. As another example, we walk into a restaurant and an algorithm may tell us how large a tip we should pay based on time at the table, number of drinks ordered, expressions on our face, speed of which we consumed the meal, left-overs on the plate etc. However, to go even further, we may not have a choice on how much we pay as a tip, it will be automatically debited from our bank account! Am I exaggerating? Consider this true story; In 2020, an insurance company refused to pay for car damage when it was hit by an outside Venetian blind, as the company had predicted that the wind speed was less than 75 km/h. The car owner calculated, using mathematics, that the wind speed could have exceeded this value, but backed down when threatened 7 Mathematical Angles | February 2024 | Issue 2
by the insurance company’s lawyers. (https://ima.org.uk/20878/ the-estimation-of-wind-speedchallenging-the-insurancecompanys-decision/) In the future, it may be possible that those who know a little bit of maths will argue with numbers in order to make big decisions about our lives. It is already happening. With the rise of AI, will this occur even more? So, our A level Mathematics syllabus may have to change to equip our students with the mathematical skills to challenge the decision(s) made by these algorithms! However, that may require some traditional topics being dropped – but what would you drop? Calculus techniques? Nevertheless, for the mathematics of tomorrow, mathematics will become a way for the student to challenge and validate decisions made about themselves, where these decisions may have a big influence on their lives. To me, this is the point of mathematics. I would be happy to hear views from MA members on ‘mathematics of tomorrow’ and how you see the future developing in your role as mathematician or mathematics teacher. Please email me at the address below. Thank you. Professor Nira Chamberlain MA President 2023-24 nira.chamberlain@yahoo.com Moving Forwards I n 2023 MA launched a research project to shape our future strategy with data-driven insights. Its aim was to understand how members and the wider mathematical educational community perceive us and to identify any internal barriers. The main objective of this research project was to find out what the mathematics community want from a subject association in mathematics and to provide actionable insight needed to improve engagement, increase renewal, and better demonstrate the value of membership. We did this by working with Membership Matters - the membership sector’s leading consultancy in providing strategic research to associations. We asked them to: • Conduct in-depth research to gain valuable insights that will inform our future strategy and ensure that our direction is guided by data. • Build a richer and more useful picture of members and the wider market, including schools, teachers and the wider sector, answering ‘what should membership look like?’ • Provide guidance on how to improve our offer, increase membership, enhance membership retention, foster engagement, and effectively showcase value. • Identify barriers and limitations related to systems, resources, branding, and other relevant areas, with a clear focus on outlining the desired direction of travel. To do this, Membership Matters worked alongside the HQ team and MA trustees conducting workshops, desktop research, online surveys, interviews and focus groups. We were delighted that so many of you engaged with us during this process with 1,841 respondents to the survey alone, 45% of which were MA members and 55% non-members (34% who have never been members of the MA) or members of the other classroom-facing mathematics associations. Initial findings of the research indicated that the MA is well respected and viewed very positively as a strong and successful membership organisation, known predominantly for our production of journals and publications. We have a small team with limited resources and technology constraints creating barriers to membership growth, therefore our challenge is to meet the needs of this fast-changing sector in which we operate. We would like to share with you the findings from this research. The report containing all of the data can be found on the MA website. The following are highlights of some of the key findings: Reasons for joining “The children at my school really enjoy having the opportunity to be creative and competitive.” 8 Mathematical Angles | February 2024 | Issue 2
Awareness and engagement Most valued aspect CPD Resources – to aid teacher growth particularly for early career teachers and non-specialists. Mathematics in School (MiS) Journal – valued for its contributions to effective student learning. Primary Mathematics Journal – provides specific primary resources and relevant content for primary education Priority aims survey participants want You will see from the results that there are clear priorities of what you as members and potential members expect from your association. You are telling us that those key priorities are: • To support professional development through courses, events and resources • Provide resources, journals, publications and research • Make mathematics enjoyable and accessible to all and facilitate challenges • Support advocacy and voice initiatives We will now be analysing all the data produced from this project to develop strategic goals and action plans to deliver on your priorities. Our commitment to you and to supporting those who work within mathematics education is at the forefront of our vision to ensure sustainability and future growth. Thank you for engaging so enthusiastically in this project; your views are what shapes our future. For the full results, visit the MA website and follow the link on the hompage. If you have any questions or further input, we are always here to listen: Sandi Atkinson, Chief Executive Officer sandi.atkinson@m-a.org.uk Amber Richardson, Marketing and Communications Officer amber.richardson@m-a.org.uk “Time-poor teachers benefit from easy resource access, saving hours on Internet searches for worksheets, challenges and activities.” 9 Mathematical Angles | February 2024 | Issue 2
News from Committees Teaching Committee T he latest virtual meeting of Teaching Committee took place on the morning of Saturday 29 September 2023. We discussed the Teaching Committee section of the MA Regulations and agreed a change of wording that was to be approved at the December Council meeting. There was a guest presentation from Liz Woodham and Charlie Gilderdale explaining the process behind the development of new NRICH tasks. We are grateful to Liz and Charlie for giving up their time to brief the Committee. Following a discussion at Teaching Committee, the Joint MA/ATM Primary Group and the 11-18 subcommittee separately replied to the four questions posed in the recent Royal Society ‘Mathematical Futures’ paper ‘A new approach to mathematics and data education’. You can read these responses here: https://www.m-a.org.uk/ news/?id=390 Paul Harris is producing the initial draft of a position paper on the withdrawal of some mathematics degree courses and we intend to publish this document in due course. This is certainly a live issue as, in November 2023, it was announced that the mathematics degree course at Oxford Brookes is to close due to “declining student numbers”. Our 11-18 subcommittee met virtually on Saturday 16th September and the joint MA/ATM Primary Group met virtually on Saturday 30th September. Subsequently, two documents were released by working groups of the Joint MA/ATM Primary Group. The first paper is a response to the Ofsted mathematics subject report ‘Coordinating mathematical success’ and the second paper addresses the end of statutory assessment in KS1. Both documents can be accessed from the MA ‘We Say’ page: https:// www.m-a.org.uk/we-say By the time this article is published, Teaching Commiittee will have met online on Saturday 13th January. We have also requested a slot for an Open Meeting of Teaching Committee at the upcoming April Conference but the details are yet to be confirmed. David Miles Chair of Teaching Committee 11th December 2023 Branches Committee Branches Committee had its half yearly meeting in October by Zoom. Just under half the branches were represented with apologies from three others. It was agreed to have a Branches bursary for the 2024 Annual Conference. The Liverpool Branch and the Yorkshire Branch have had various events and the Sussex Branch has run Masterclasses. In particular, after three years of exile in cyberspace, the YBMA Christmas Quiz returned as an in-person event at the School of Mathematics, University of Leeds. The event was enhanced by offerings of different rounds from a number of different members. There was geometry, algebra and coding as well as a stamp collection containing Mensa questions and also the usual ‘How many numbers can you make from 2024’ using various operations, but this year with an added twist! Although the numbers attending were low, great fun was had by all present and helped by seasonal nibbles and beverages as well as prizes to take home. This event was a great way to start the Christmas festivities. Both Liverpool and Yorkshire Branches have events running in 2024 and these are listed under ‘Forthcoming Branch Events’ within this magazine. The Yorkshire Branch have also announced the retirement of Alan Slomson as Branch Secretary, after many years of invaluable service in that role. The new Secretary is Bill Bardelang (email: rgb43@gmx.com). A few members have been in touch with me with regard to the possibility of starting up branches in areas where there are none at present or they are at present inactive. I hope that this coming year we will hear more about these ventures. The next Branches Committee meeting will be during the upcoming Conference in Stratford upon Avon 3rd -5th April. As yet the time and date are waiting to be confirmed to see if there is a possible slot earlier than after the closing plenary. Cindy Hamill (Chair of Branches) Forthcoming Branch Events Liverpool Mathematical Society (LivMS) Wednesday 6th March 2024 from 5.00pm to 6.00 pm. Venue: to be confirmed Fantastical modes and how to find them by Finn Allison We hunt a two-fanged finite shapeshifter made from masses, pointlike, and springs, linear. At natural frequencies its true nature is revealed - the modes fantastical otherwise concealed. With a radial pulsing it lures its prey and devours them whole so the legends say. Its syncopated beats form a hypnotic dance, sending all who dare look into an endless trance. A creature found lurking in the depths of caves, you say “hello” and to you back, it waves. Suitable for year 11 and upwards, their teachers and all those interested in mathematics. Note: For the above event, registration may be required via the Eventbrite page for the event. Further details about all of the Society’s events, including a direct link to the Eventbrite page, can be found at the Society’s website at http://www.livmathssoc.org.uk/ Yorkshire Branch (YMBA) Thursday, 21st March 2024 Venue: University of Leeds W. P. Milne lecture Title: to be confirmed By Professor Nira Chamberlain OBE President of the Mathematical Association The lecture forms part of a KS5 Maths Day at the University of Leeds. Further details: https://www.stem.leeds.ac.uk/ mathematics/tasterday-2/ 10 Mathematical Angles | February 2024 | Issue 2
News from the Joint ATM/MA Primary Group The Joint Primary Group’s meeting held on Saturday September 30th 2023 from 10.00am – 12.30pm, was a virtual meeting on Zoom, chaired by the vice chair Gemma Parker. The group now comprises 128 members. 31 members were in attendance. KS2 SATs Data Analysis: Rose Keating Rose has collected data on KS2 SATs from 2016 and she gave some background on how she has categorised and organised this data and the lengths she has gone to ensure consistency and comparability in terms that would be familiar to teachers (when the frameworks themselves might seem a little arcane). The percentage of total marks required for expected/higher standard has dropped this year particularly… and only just over 50% for Expected Standard and usually 85-90% for Higher Standard. Rose shared some useful graphics showing the proportion of content from each year group in KS2. We noted that Expected Standard can be achieved without any Y6 content. She also illustrated at what point in the paper (i.e. what number question) content from each KS2 year group came. It is not a simple progression in difficulty…there are plenty of ‘easier’ questions spread throughout the paper. So, some test technique involving ‘moving on to find easier questions’ can be helpful. Around 75% of the content of the papers every year is based on number and calculation: major number content sub-domains: 31% on multiplication and division, 18% on fractions and 9% on place value. Members really appreciated the depth and presentation of the analysis and requested slides. Members who had shared Rose’s analysis with schools shared the positive impact this had, particularly on the confidence and attainment of Y6 learners. Rose ran a Mathematics Association Webinar on 9th November 2023, which some members may have attended. OFSTED Working Group Update: Alison Borthwick, Lorraine Hartley, Liz Woodham Members had the opportunity to work in break out rooms and discuss the response to the Ofsted report: Coordinating Mathematical Success. Alison gave some background about the approach taken and clarified how the work was completed. A great deal was done over the summer and it was a good result getting it published both by the MA and ATM in September. There were deep concerns from members about the quality of the advice from OFSTED on primary maths. The response published was viewed as a measured and helpful one that helps teachers to question and discuss alternative perspectives. Responding to the Ofsted Report: Coordinating Mathematical Success, the mathematics subject report ATM website link: https://www. atm.org.uk/News/responding-tothe-ofsted-report-coordinatingmathematical-success-themathematics-subject-report MA Website link: https:// www.m-a.org.uk/resources/ coordinating%20 mathematics-%20a%20 response%20from%20the%20 joint%20primary%20group.pdf The Chair and everyone present commended the response and the quality of the work done on it by the working party, despite being over the summer holiday. KS2 Assessment Working Group Update: Pip Huyton The focus for this working group is to support teachers with future tests. Pip shared the Padlet that has been created for the group to work from and was keen to include work that Rose shared earlier in the meeting. Pip raised a concern about the ‘compartmentalisation’ of the curriculum, particularly algebra and ratio, when we know that these areas appear far earlier in the curriculum and that many teachers focus on this only in the summer term in year 6. Pip asked any other interested participants to contact her and get involved: pip@accomplisheducation.com . Pip will update us all at the next meeting in January. KS1 Assessment Working Group Update: Gill Knight A major principle for this group is on prioritising formative assessment to inform teaching. There is an emphasis on understanding what learners know from listening and observing to children working together. Gill explained there is an introduction and some activities, based on KS1 NC test questions since 2016, to support teachers to understand how they can assess more holistically and avoiding throwing out the baby (well written questions) with the bath water (standardized tests). Future meeting dates confirmed: 27th January 2024 online 7th June 2024 in person at The Royal Society, London. Sue Lowndes 11 Mathematical Angles | February 2024 | Issue 2
Oxford Dictionary of National Biography (ODNB) – Michael Atiyah I n the Obituary to Professor Ian Macdonald FRS located within this magazine, reference is made to Michael Atiyah, a former President of the MA. Michael Francis Atiyah was born in London on 22nd April 1929 and died in Edinburgh on 11th January 2019. His entry in the ODNB appeared in April 2023. It is possible that you may be able to view it online via your institution or, as I did, via your local library or by some other means. He was one of the greatest mathematicians of his era. It seems quite surprising that he was a part of the MA. The initial connection with the MA was that he was the President from Easter 1981 to Easter 1982. As a consequence of this, his Presidential Address, ‘What is Geometry?’ appeared in the October 1982 issue of the Gazette which, as with all issues of the Gazette, can be accessed online via the MA website. Also he was appointed an Honorary Member in July 2016. He died in January 2019 in Edinburgh. His Obituary, written by Geoffrey Howson, appeared in the July 2019 of the Gazette. As well as the Obituaries in the ODNB and in the Gazette, there is one on the MacTutor website: https:// mathshistory.st-andrews.ac.uk/ Biographies/Atiyah/ Bill Richardson (MA President, 1996-1997) Maths Teacher Training Scholarships The Mathematics Teacher Training Scholarships is now in its twelfth year, poised to bestow 270 prestigious awards upon dedicated individuals with the potential to ignite a passion for mathematics in the next generation. Recipients of the Maths Scholarship not only receive a £30,000 bursary, but also gain complimentary membership to key mathematical societies – including The MA – and access to exciting Continuing Professional Development (CPD) opportunities throughout their teacher training year. If you know someone starting initial teacher training in 2024/25 who aspires to become an outstanding secondary maths teacher, send them to our website; teachingmathscholars.org . Applications are planned to remain open until 09:00 on Monday 8 July 2024; however, in previous years the scheme closed earlier than expected due to the overwhelming number of applications. In the event of a similar circumstance, we will close early. Important Dates To delve deeper into the Maths Scholarship, interested individuals can attend one of our three upcoming webinars or meet us at various Get into Teaching events in the coming months — both excellent The Launch of the Loughborough University Mathematics Education Network Curriculum! The LUMEN Curriculum is a completely free, thoroughly research-informed and entirely editable set of teaching resources for Key Stage 3 mathematics, designed by a team at Loughborough University led by Immediate Past President of the MA, Colin Foster. The Launch Event will be at 6:30 pm on Thursday 21 March at Friends House, 173-177 Euston Road, London NW1 2BJ. Prof Marcus du Sautoy (another former MA President) will be the guest speaker. After he speaks, Colin Foster will present an overview of the design of the Curriculum, and then there will be a hot buffet. Please join us for the Launch – we can reimburse your travel costs for attending if that would enable you to come. Please email Manisha Mistry m.mistry@lboro.ac.uk if you would like to take advantage of that. Please register at: http://lumencurriculum-launch.eventbrite. co.uk/ and do sign up as soon as possible, as places are limited. 12 Mathematical Angles | February 2024 | Issue 2
opportunities to address any lingering questions! Saturday 2 March 2024 London Get into Teaching Event Saturday 9 March 2024 Manchester Get into Teaching Event Saturday 16 March 2024 Birmingham Get into Teaching Event Monday 25 March 2024 at 19:30 – 20:30 Maths Scholarships Webinar Thursday 25 April 2024 National Get into Teaching Event (online) Tuesday 4 June 2024 19:30 – 20:30 Maths Scholarships Webinar For more information on Get into Teaching events, visit their website; getintoteaching.education.gov.uk. To sign up for one of the Maths Scholarships webinars, individuals can email us at scholarships@ima.org.uk with their preferred date. What, or who, inspired you to become a maths teacher? Becoming a maths teacher often starts with something that really inspires you. It could be a teacher who made you love numbers, a special moment in your education that changed how you see maths, or just a natural love for the subject. Everyone has their own unique story that led them to become a maths teacher. Here’s Maths Scholar Telka Donyai on what, or who, inspired her to become a maths teacher. Growing up I was always good at maths and it came fairly easily to me (though this didn’t mean I did not have to study or work hard at it to do well in my exams!). However, I noticed that this was not the same for the majority of students at my school. In fact, it was seen as very acceptable to say “oh I’m just not good at maths, my brain doesn’t work like that”. This is something I still hear all the time and this mindset is carried forwards into adulthood. I always wondered why maths was seen as something you were either innately good at or doomed to never understand. For the latter, the opinion seemed to be that they shouldn’t even bother trying to change this. I love mathematics and I see it as a language in its own right. As a language I think it should be seen to be just as important as learning to speak the language of the country you are born or live in. It is such a crucial life skill and is used in everyday life just as we use our communication skills. I think this has been a big drive for me in becoming a maths teacher. I want everyone to stop just accepting they are ‘not good at maths’ and realise that it is very important and they should not settle for anything less than having a high level of proficiency in maths. Nobody would give up trying to learn to speak or read fluently so why do students give up understanding the language of maths? I also want people to realise that with practice you can actually get better at maths and it’s not just an intrinsic gift only some people have. This leads me onto my next main reason for wanting to become a maths teacher. My younger brother is autistic and really struggled with maths even though he is actually very numerate. His brain works very differently, and it takes him a lot longer to learn things. There is a lot of repetition needed for him to develop good retention or fluency in something, but once he does understand the concept, he applies it well. I helped him with studying for his maths GCSE and he came out with much higher than his predicted grade! Being able to see the difference I could make made me really want to be able to do this for other people. Ultimately, I want to be able to break the stigma around mathematics, make more people see its importance, realise it can be learned and they should want to learn it because it is so useful. I am also passionate, because of my brother, at trying to make mathematics more accessible for those with neurodivergence or other learning differences. Telka Donyai I always wondered why maths was seen as something you were either innately good at or doomed to never understand. 13 Mathematical Angles | February 2024 | Issue 2
14 Fabrizio Mordente Fabrizio Mordente was, in his day, an acclaimed mathematician who invented a proportional compass, which could perform many mathematical operations such as measuring fractions of a degree accurately, to help mariners at that time calculate longitude with precision. The compass could also perform arithmetic and geometric operations such as calculating a square root, or performing area, volume and trigonometrical calculations. Mordente was not alone in inventing such a device, since other mathematicians such as Michel Coignet (during the 1580s), Galileo Galilei (1587) and Thomas Hood (1598) had also invented similar devices, independently, but Mordente’s (1554) was the first. However, if you search for Mordente in books on the history of mathematics, or on the Internet, you will find very few references to him. His compass was eventually superseded by other instruments of measurement and the history of it was only rediscovered relatively recently when scholars were researching Giordano Bruno, a mathematician who had created quite a controversy about Mordente and his invention, within four dialogues that Bruno published in 1586. Fabrizio Mordente was born in 1532 in Salerno, in Campania, south-west Italy. Almost nothing is known about his early education but he attended the University of Naples until 1552, studying philosophy and mathematics. He then embarked on a tour of various countries, which lasted for many years. He firstly visited Crete, Cyprus, Palestine and Egypt, where he studied the pyramids and many places of Christian worship. Thereafter he travelled to Mesopotamia, seeking the ruins of the mythical Tower of Babel. From the Persian Gulf he sailed on a Portuguese ship to India and stayed for three years in Goa, after which he sailed to Lisbon in Portugal. Mordente also travelled to Ireland, to visit the legendary Well of St Patrick, after which he embarked on a trip to various cities in Europe, including London and Paris, before returning to Italy, where he visited Venice, Florence, Rome and Naples. He continued to travel periodically during his working life since he was always looking for a position as a mathematician in one of the European Courts. His scientific career really started in Venice in 1567, when he was 35 years old. He published a one-page treatise describing his newlyinvented proportional compass, which was also known as a sector. He stated in this treatise that the idea for the compass came to him during his travels in 1554. His proportional four-pointed compass had two arms with sliding cursors that moved along a proportional scale engraved into the arms, to enable the calculation of angles to a fraction of a degree. The increase in accuracy of angle measurements helped increase the precision of other astronomical instruments in use at that time, such as the astrolabe. During the years 1568 to 1570, Mordente was in Urbino, at the court of Duke Della Rovere, where he developed a new version of his compass. This instrument is currently held by the Correr Museum in Venice, under the title of “sextus roverinus”. This compass, pictured above, had a series of points along each arm, which indicated the proportional ratios for dividing lines. It also had a groove for a sliding point along each arm, and was accompanied by a ruler for taking measurements and to assist with the proportional calculations. In July 1570, Mordente was hired by the Republic of Lucca to undertake a consultation on the city walls. Two years later, he travelled to Vienna and presented a third version of his compass to Emperor Maximillian II, which was known as the “admirabilis circinus”. This compass now resides in the Adler Planetarium in Chicago, along with others of Mordente’s compasses, some of which are illustrated in the picture below. For the coronation of Maximillian’s successor, Rudolph II, Mordente prepared new instruments: one Mathematical Angles | February 2024 | Issue 2
15 for dividing the circumference of a circle, a precision scale and a revised compass. The new compass incorporated a point located in the joint of the compass’s two arms, which had been suggested by Rudolph II himself. Mordente’s younger brother, Gasparo, wrote a treatise about the newly developed compass, which was published in 1584, entitled “Mr Fabrizio Mordente’s compass”. A fourth incarnation of the compass was described in a publication “Fabrizio Mordente’s compass and figure” in Paris the following year, where Mordente also gave public demonstrations of his compass, in the hope that he could secure a position at the court of Henry III, as well as gain funding from the Queen Mother for a voyage. The fourth compass became known as the eight-point compass, having two slender square-section arms along which four cursors slid with points orthogonal to the arms. Of the other four points, two were on the ends of the arms and two at the hinged joint. The compass was always used with a quadrant, to carry out proportional operations and to measures fractions of degrees. There was much publicity in Paris about Mordente’s compass. The philosopher and mathematician, Giordano Bruno, was intrigued by the invention and attended one of the public demonstrations to learn more about it. Impressed by the invention, he wrote two dialogues (with Mordente’s approval) to reveal the workings of the compass, to bring it to the wider attention of the scientific community. In these dialogues, he called Mordente the “god of geometers”, but unfortunately, the way in which he described Mordente’s work caused some scientists to think that Mordente had not really invented it himself but had plagiarised the work of others. Mordente was furious with the shadow that Bruno had cast on his reputation and accused Bruno publicly of arrogance. Bruno’s response was to publish a further two dialogues in which he described Mordente as a “triumphant idiot”. This did not appease Mordente. Known to be a difficult and temperamental man, Mordente decided to retaliate. He purchased and destroyed all the copies he could find of Bruno’s dialogues and called upon his friend, Henry I, Duke of Guise, in Antwerp, to help him defame Bruno. Meanwhile, Bruno had also upset others in Paris with what many of the Catholic faith viewed as his heretical views on religion, and together with the fallout from the quarrel with Mordente, Bruno was forced to flee the country. In 1587 Mordente joined the Court of Henry I in Antwerp. There he designed war machines and wrote a treatise entitled “Offensive and Defensive Machines”. He remained in Antwerp until 1591, when he joined the court of Alessandro Farnese, Duke of Parma and Piacenza, and the Governor of the Spanish Netherlands. Mordente created a fifth iteration of his compass while he was working for Farnese, describing this version in a treatise entitled “The squaring of the circle, the science of residuals, the compass and rule by Fabrizio and Gasparo Mordente, brothers from Salerno”. The “science of residuals” mentioned in the title of the treatise referred to the measurement of fractions of a degree, potentially into an infinite number of parts, which was at that time a new branch of mathematics. Mordente intended this publication to be a response to Bruno’s accusations of plagiarism and idiocy. In his treatise, the new compass was described, which had an adjustable double point on one of the cursors, and was used in conjunction with the parallel ruler that had been developed by his brother, Gasparo. You can see a representation of a parallel ruler at Poole Harbour, where the ruler is set into the quayside pavement (pictured above). When Farnese died in December 1592, Mordente remained in Flanders for a while, but thereafter returned to Italy, settling in Rome in 1596 where he planned to meet Christopher Clavius, reader of mathematics at the Roman College of the Society of Jesus. Clavius had published many years previously an instrument for drawing sundials, which had included a detailed description of Mordente’s method of measuring fractions of a degree. However, when Mordente arrived in Rome, Clavius was residing in Naples, so Mordente firstly met his successor, Christopher Grienberger, to whom he explained in detail how the compass operated. He finally met Clavius later that year in Naples. Mordente returned to Rome in 1598, where he published his seventh and final mathematical paper on the compass, “The propositions of Fabrizio Mordente of Salerno”. He mostly remained in Rome until his death in 1608. However, his fame and his compass lived on due to Michel Coignet, another Italian mathematician who composed in 1608 a detailed treatise on the now nine-point compass. Coignet’s publication was translated into various languages, including German, French and Latin, and disseminated widely. The proportional compass was used as a calculating tool until the 1800s, when new tools replaced it, such as the slide rule. Parallel rulers are still used by navigators today to transfer an angle accurately from a compass rose printed on nautical chart across to another part of the chart. Jenny Ramsden Mathematical Angles | February 2024 | Issue 2
16 Fractions of infinity? Here is part of an infinite tiling of triangles and hexagons. Every vertex is the same: 4 triangles meet 1 hexagon. (As you see, there are no mirror lines, so there’s a left and a right version, but that doesn’t matter for our problem.) What fraction of the area do the hexagons account for, in other words, how much of the plane is blue? As a check, try to work out the answer by more than one method. Once you’ve finished, look at the notes at the back of the magazine to check your answer and see if you’ve found a way different from the four given. Paul Stephenson Pascal-like arrays from cellular automata Pascal’s triangle appeared frequently in SYMmetryplus and it will continue to do so in Mathematical Angles. It’s the best known example of a number array. Each number is a binomial coefficient. When these numbers are laid out in space, many relations between them become apparent which would otherwise lie concealed in a forest of symbols. In this piece we shall let a cellular automaton generate the triangle and, by changing the rule the automaton applies, generate other, Pascal-like, triangles. We know certain properties Pascal’s triangle itself exhibits. You are challenged to find which are shared by all the others by virtue of the way we generate them. On a sheet of squared paper write a single ‘1’ in the middle of the top row. Set the automaton to scan the cells, sweeping across each row before dropping down and scanning the next. The scanner looks like this: Cut it from card and try the experiment. Pascal’s Triangle should appear in its right-angled form. Call Pascal’s Triangle itself the 2-box array, and do the same with the 3-box scanner to obtain the 3-box array. Try the 4-box scanner and the 5-box scanner to produce the 4-box array and the 5-box array respectively. Here are three properties of Pascal’s Triangle itself. Predict what you think the analogous (= equivalent) property of the 3-box, 4-box, ..., n-box array may be. The hints below may help. 1. To see why summing along the red arrows gives the Fibonacci sequence, examine the sheared version below. 2. To see why powers of 11 give the rows, multiply 11 by 11 and consider how the displaced rows sum. 3. To see why the 2-box automaton causes the sum to double each time, think about the adding rule. Results towards the back of the magazine. Mathematical Angles | February 2024 | Issue 2
17 Simply solve: just algebraic equations Neil Walker The music of the spheres The Pythagoreans considered that concordant sounds were produced by simple ratios of pitch, made visible in the lengths of vibrating strings. This led to the idea that the crystalline spheres, in which the stars and planets were thought to be embedded, made sounds as they rubbed against each other (too beautiful to be apprehended by human ears), ‘the music of the spheres’. Like all the best ideas, this has never really gone away. Those working at the frontiers of theoretical physics say that what we took for elementary particles are really just different modes of vibration of a multidimensional surface, a (mem)brane. But we shall move up in scale from the unimaginably small to the very large but conceivable: the solar system, or rather a solar system, not ours, whose origin is too messy, but HD110067. The idea that unites a vibrating string and an orbiting planet is that of resonance. Depress the sustaining pedal on a piano (so as to raise the dampers from the strings) and strike bass C. You will hear middle C, treble G, upper C and possibly the E above that. If we give bass C unit frequency, those notes have frequencies close to 2, 3, 4, 5, respectively (‘close to’ only because of the way pianos are tuned). Other strings remain silent. Imagine that we have a solar system, all planets in the same plane, circling in the same direction at their own speeds. When two planets come closest (the astronomer’s phrase is ‘into conjunction’), the attraction of one may give the other an extra push, like a parent pushing a child on a swing. If the orbital periods (and therefore frequencies, the reciprocals of the periods) are close to a simple ratio, the effect of the push may serve to bring the orbits closer to that exact value. At the time of writing (November 2023) observations from two orbital satellites put up in the previous five years have just been pooled to reveal a remarkable example of this. Google ‘HD110067’ Mathematical Angles | February 2024 | Issue 2
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