"It's failure to prepare mentally and failure to take practicing penalties really seriously." This is Ken Bray's explanation for England's dismal performance in penalty shootouts. England are successful in only 17% of their encounters, compared to Germany's impressive 80%. Bray is an expert in the science of football, and he has studied the physics as well as the psychology of penalties and analysed the statistics. The result are three steps to ensure a perfect penalty, which he explains in this video. You can find out more about the science and maths of football in Bray's Plus articles on football.
I was lucky enough to see the beautiful Matisse exhibition at the Tate Modern in London last week. A few days later I was asked, by a TV researcher, how do you make maths interesting and understandable to people when so many people, in her experience, had hated and avoided it at school? In response I found myself telling her about the Matisse exhibiton. Not to change the subject or avoid discussing the bad reputation maths has in many people's minds, but to explain what we at Plus, along with many other maths communicators, try to do.
Now I loved maths at school. It made sense, I could express what was in my brain and how I saw the world through maths, and I found it fun. What I found a challenge was art class, I didn't feel I had any artistic ability at all. People who could accurately draw or realistically paint people (or bowls of fruit or jugs of water...) amazed me - it was like they had a magic power. And for a long time I didn't think much of modern art. For example Picasso's pictures seemed ugly and didn't look like the things they apparently depicted, and Matisse's bright art seemed simplistic.
My opinion on art, particularly modern art, has changed, thanks particularly to two fantastic exhibitions. The first was over a decade ago at a light-filled gallery in Spain that showed many of Picasso's preliminary sketches and studies leading up to the Guernica, the full work of which filled the final room of the gallery. Following the journey that Picasso took as he built towards this huge masterpiece, along with the commentary of the social and artistic context of the piece, gave me a new appreciation of the skill, effort and genius of this work. It was awesome. I began to see all his work differently.
The Matisse show at the Tate contains many of what are called the Cut-Outs. Very late in his life, after a serious illness, he initially wasn't able to paint the huge canvases he previously produced. Instead, sitting in a chair or up in bed, he began to cut out coloured paper which he would instruct assistants to pin to the walls of his room and studio, moving it a little to the left or right, a little up or down, experimenting with these coloured shapes to build collages that had enormous power. He was so taken by this new genre he had invented that even when he recovered enough to paint he preferred this new approach.
The exhibition contains video of Matisse at work, photographs of his studio and commentary that explain the personal, artistic and social context for his work. Walking through the chronologically arranged rooms it gave me a sense of his motivation for working this way, and how it changed his perspective, and the significance of this work both artistically but also its wider cultural context. The exhibition included different types of content: recordings of the artist himself (or someone speaking his words) about his motivation, clear descriptions of the works and how they were made, explanations of the significance of this work in art, and how they interpreted Matisse's world. And most importantly, of course, was seeing the works themselves, up close, smooth curves, jagged edges, bright colours, pinned together.
It was the brilliant curation of this exhibition that reminded me and inspired me about the work we do here at Plus. We want to allow anyone a chance to see some of the wonders of mathematics up close. We hope that hearing the words of researchers will give a sense of their motivations in doing their mathematics, and we aim to show the significance of this work both in mathematics, and in a wider, cultural setting. We hope to give people a glimpse into how mathematicians perceive the world and how they use mathematics to express their perception of the world. And we hope this gives our readers a new appreciation of maths, of its power and beauty, that they might not have noticed or enjoyed before.
Now I'll never be an artist. But thanks to clever and passionate curation I have over time developed an appreciation of many different types of art, that makes seeing new art less confronting and more exciting. And sometimes, when the urge arises, I might have a go at capturing something on pencil and paper or in cut-outs of coloured paper, just for my own pleasure. I hope that Plus plays a small part in helping everyone have a similar appreciation of mathematics. We might not all be mathematicians, but I hope we can all enjoy and engage with mathematical ideas, appreciate their beauty and power. And that the next time maths pops up in our lives, it's something to be excited about, rather than avoided.
The main editor of the book is Sam Parc (follow her on Twitter) and we were part of a devoted editorial team that assisted her. If you like to read about maths with all its beauty and many applications, then why not have a look at the book? We're off to practice our curtsies...
Where were your most creative experiences at school? In art class? In music? English? In your maths lesson? That last one might not be the obvious choice for many of us, unless you were lucky enough to have a really inspiring maths teacher. But that is exactly the type of opportunity we are hoping to create for maths students aged 7-16 as part of the project, Developing Mathematical Creativity, with our sister site, NRICH.
One aspect of the project that we are particularly excited about is highlighting the role of creativity in mathematics research. All mathematicians tell us that doing original mathematics is highly creative – but what exactly do they mean by that? We asked some researchers from a range of subjects about the role of creativity in their work.
Working within constraints
We started with David Berman who has a very interesting perspective on creativity. As well as being a theoretical physicist at Queen Mary, University of London, he also has a long standing collaboration with the Turner prize winning artist, Grenville Davey. Deconstructing the artistic idea of creativity, Berman told us that rather than an unbridled release of ideas where anything is possible, beauty comes from creating work withing very tight syntactic constraints. "Think of music: the tight system of key and chord makes music very constrained and yet capable of amazing emotional power," he said. For example Schoenberg's experiments with atonal music, though completely new and boundary breaking, were far from unconstrained. "Maths is like this. There are enormous syntactic constrains but still enough freedom to say something new. The beauty lies in between the constraints of syntax and the freedom of meaning."
David Spiegelhalter, the Winton Professor for the Public Understanding of Risk at the University of Cambridge, said he recognised creativity in research when someone makes a leap between two different areas, making new connections between them, perhaps using a tool from one area in another. And, as is fitting for someone with his expertise, Spiegelhalter highlighted that creativity is about stepping into the unknown and taking risks, something that can be different to our experience of mathematics in school. "Research is different to maths at school because you don't know where it will end, nobody has gone there before to tell you the answer but you have a feeling that something will come out."
Asking good questions
Ben Allanach, a professor of theoretical physics from the University of Cambridge, also mentioned this exploratory nature of creativity and research: "You don't know the answer [to the questions you're asking] before you start… You don't even know if there is an answer." He gave some examples such as inventing new models of particle physics, or new algorithms for analysing data. In this sense, a key aspect of creativity is the ability to ask good questions. "Creativity has a role in research in deciding which projects to work on," said Allanach. And as well as creating or choosing good questions that, as Spiegelhalter said, you have a feeling that something will come out, it's also important to be able to spot when your questions turn out to not be such good choices after all. "Personality and judgement play a large role," said Allanch. "When do you persist? When do you struggle on and when is the time to give up on a line of enquiry?"
Interestingly, our discussions with researchers from chemistry, although quite a different field from mathematics, produced some very similar ideas. Charlotte Williams, Senior Research Scientist at CSIRO in Australia, also emphasised the importance of asking good questions, persevering with them, and constantly reviewing if you were still going in the right direction. "Most research projects come from a position of fundamental direction: you need your research grounded in reason, precedent and understanding of the subject, but then you need to innovate and that is where the creativity comes in," said Williams. But, as well as being able to ask good questions, Williams also emphasised that persistence was an important attribute to research success.
Animation of the small subunit of the Thermus thermopiles ribosome. RNA shown in orange, protein in blue. (Image in public domain)
Although the emphasis is often on the new, Williams made the point that great research also comes from thorough investigation and understanding of current work. "Nobel prize winner Ada Yonath (in Chemistry, 2009), said just this when I heard her speak. She would never have solved the structure of ribosome or got the Nobel prize if she hadn’t continued to do the same work for her whole career. And importantly have the funding bodies and universities support that (and not say, ‘you’ve been trying to do this for too long, do something new’...). She said she thought that tenacity and follow through in research was vastly under-rated." Perhaps this persistence with an idea, following it through, testing it out and really gaining an understanding of what is going on, is also a valuable element of creativity.
Williams felt that it is only once you have learnt the fundamentals of chemistry at school and in undergraduate degrees, that real creativity is possible. This could be seen as in contrast to our aim to encourage creative thinking in maths from the very earliest ages. Perhaps this is core difference between the subjects of maths and chemistry - there are a lot more costs and dangers associated with doing chemistry than playing with maths. "Too much creativity [in reasearch] and you could waste a lot of time and effort," says Williams. "So it's the combination that is key: thinking of a new direction or interesting area to explore that has a reasonable likelihood of being successful." Also, due to the real physical dangers of some chemical reactions, it's not appropriate to just let students follow their own ideas without proper supervision. "I respect and appreciate a student’s thoughtfulness and creativity but given that they are students, they still need guidance. In particular in chemistry, going off being creative without having the appropriate experience or without having consulted one's supervisor, can be really dangerous to be honest."
Collaboration and interaction
The final theme that really emerged from these researchers was the importance of interaction and collaboration in allowing for creativity in research. Williams said the process of attending conferences, listening to seminars and staying in touch with new research stimulates collaboration with people that might lead off into unexpected directions. Allanach too said that creativity often comes from interacting with others.
So what role do you think creativity has in learning and researching maths, both as an academic but also as a student? What have been your favourite experiences of mathematical creativity?
About this article
This article was inspired by content on our sister site Wild Maths, which encourages students to explore maths beyond the classroom and designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.
If you're not sure how maths is relevant to real life, then go to see this year's popular lectures put on by the London Mathematical Society. Kevin Buzzard of Imperial College will explore the digitisation of our lives — the fact that many of the things we enjoy and work with are now stored on computers which reduce them to numbers. This has weird consequences. For example, some numbers are copyrighted, and there are even some illegal prime numbers. But way before this digital revolution, mathematicians realised that numbers could encode all of mathematics (even the parts of it that aren't about numbers), and this has some even stranger consequences.
How do diseases spread?
Julia Gog, of the University of Cambridge, will look at how mathematics has been applied to help understand and control infectious diseases, from the scale of a single virus particle through to a global influenza pandemic, and some mathematical challenges for the future.
The lectures will take place on Wednesday 9th July at the Institute of Education in London (7:00pm) and again on
Wednesday 24th September at the Haworth Lecture Theatre, University of Birmingham (6:30pm). They are free, but if you wish to attend, please write to firstname.lastname@example.org. A form for registration is available on the LMS website.