Teaching Mathematics

Following my talk in Madrid in November, I was asked to answer a few questions about the current status of maths teaching based on my experience as a university lecturer. Here are my answers…

How should mathematics be taught in schools?

Through stories. Teaching through story-telling is an incredibly powerful tool and one that is not used enough in mathematics. For example, when teaching trigonometry, rather than just stating the formulae, why not explain WHY they were needed in the first place – by ancient architects trying to construct monuments, by explorers trying to estimate the height of a distant mountain – these are the reasons that mathematics was developed, and I think that teaching it through these stories will help to engage more students with the subject.

Are teachers prepared to teach this subject correctly?

I don’t believe the teachers are at fault – they are told to follow a particular curriculum and due to their heavy workload have no time to develop lessons with engagement at the heart of their design. There are of course ways that we can help teachers, by providing examples of ways to make maths content more interesting and engaging. This can be through story-telling or applications to topics of interest to students such as sport and video games. This is what I try to do with ‘Tom Rocks Maths’, for example see my video teaching Archimedes Principle by answering the question ‘how many ping-pong balls would it take to raise the Titanic from the ocean floor?’.

In your view, how should a math teacher be?

The most important thing is to have passion for the subject. The level of excitement and interest that the teacher demonstrates when presenting a subject will pass on to the students. Just as enthusiasm is infectious, so too is a lack of it. Beyond passion, there is no typical profile of a maths teacher. Anyone can be a mathematician, and it is very important that people don’t feel that they have to conform to a particular stereotype to teach the subject. I have always just been myself, and hopefully as a public figure in mathematics will inspire others to do the same.

Sometimes, this subject becomes more complicated for some students, not so much because of its difficulty, but because of the way in which they have been taught. What should be done with these students?

The trick is to find a way to explain a topic that resonates with a particular group of students. Let me give you an example from my research: the Navier-Stokes Equations (NSEs). For students who have no real interest in mathematics, I would try to get them to engage by explain the $1-million prize that can be won by solving these equations. For students who have more interest in real-world applications such as in Engineering or Biology, I would tell them about how the aerodynamics of a vehicle or the delivery of a drug in the bloodstream rely on an understanding of Fluid Mechanics and the NSEs. If the students are fans of sport, I can explain how the equations are used to explain the movement of a tennis ball through the air, or for testing the perfect formation in road cycling. Finally, for students who are already keen mathematicians, I would explain how the equations work in almost every situation, except for a few extreme cases where they result in ‘singularities’, which as a mathematician are the ones you are most interested in understanding. Once you know the interests of your audience, you can present a topic in a way that will help them to engage with the material.

Can you get to hate math?

It is certainly possible – though of course alien to mathematician such as myself! I think this feeling of ‘hate’ relates back to either the way that you have been taught the subject, or from a lack of understanding. If you did not enjoy your maths lessons at school and harbour ill feelings towards your teacher, then you will begin to develop negative feelings towards the subject. This is not because you dislike the subject, but more because of the way that it was taught to you. Likewise, if you do not understand mathematics then it is very easy to develop a ‘fear’ of the subject, which can quickly turn into hatred due to feelings of inadequacy or stupidity if not addressed. It all comes back to finding a way to approach the subject that fits with the knowledge and experiences that you already have. If you present a problem in an abstract manner of manipulating random numbers to find a given total, then most people will struggle – regardless of their mathematical ability. But the same problem presented in a relatable situation suddenly becomes understandable. Here’s an example:

(a). Using the following numbers make a total of 314: 1, 1, 2, 5, 10, 10, 20, 20, 50, 100, 100, 500.

(b). You go shopping and the total is €3.14. What coins would you use to pay for your items?

They are the same question, but in (a). the problem looks like a maths question, and in (b). it is an everyday situation that people all over the world are used to. Both require the same maths to solve, but even people who ‘hate’ maths could tell you the correct answer to (b). using their own real-life experience.

Women are at a great disadvantage compared to men when entering a STEM career, why do you think this is happening?

First of all, as a man I am certainly not qualified to answer this question, but I will at least try to provide you with my opinion based on personal experience. At high school level I believe that the difference is less severe (eg. see article here) and even at university there is a slightly higher number of females than males studying science-based subjects. BUT, the issue occurs after this. In graduate degree programmes and beyond there is a definite lack of female researchers, and this is amplified even further at more senior level positions. One explanation could be that academic ‘tenure-track’ positions exist for life, and so many of the men that now hold these positions have done so for the past 30-40 years and were employed when we were doing a much worse job of tackling the gender gap. Now that awareness of these issues has increased, and in general we are doing a much better job at addressing them that we were 30 years ago, hopefully we will begin to see more females in leading positions over the coming years, it will just take a little while for the effect to be seen. I also think that in general there are not enough female role models within many subjects (especially maths) that have reached the pinnacle of their field (through no fault of their own), and as such there is a lack of role models for young female researchers. The achievements of female mathematicians such as Maryam Mirzakhani (2014 Fields Medal) and Karen Uhlenbeck (2019 Abel Prize) should be even more celebrated precisely for this reason.

Do you think that enough importance is given to mathematics in the educational world?

In the past perhaps not, but attitudes are certainly changing. With the increased role that technology and algorithms play in our lives, people are beginning to realise that we need to better understand these processes to be able to make informed decisions – and maths is the key to doing this. Employers are certainly aware of the invaluable skillset possessed by a mathematician and as a result more and more students are choosing to study the subject at degree level and beyond to improve their competitiveness in the job market. Ultimately, attitudes are changing for the better, but there is still more that can be done.

In your opinion, what is the best way to teach this subject?

Exactly as I have described in questions 1 and 4. Storytelling is key to making the material as engaging as possible and knowing the interests of your audience allows you to present the subject in a way that will appeal to them most effectively.

What is the current situation of mathematics research in the university?

I think the main issue facing research mathematics is the relatively recent trend of short-term research outcomes. The majority of funding available to mathematicians requires either continuous publication of new results or outcomes that can readily be used in an applied setting. The issue of continuous publication means that researchers feel the need to publish a new manuscript every few months, which leads to very small advances at each step, and a wealth of time spent writing and formatting an article instead of conducting actual research. In many cases, the work would be much clearer if published as one piece in its entirety after several years of careful work. The drive for short-term research outcomes means that it is now very difficult to study mathematics just for the sake of it – you have to be able to convince your funding body that your work has real-world applications that will be of benefit to society within the next 5-10 years. To show why this is a disaster for maths research, let’s take the example of Einstein and his work on relativity. Now seen as a one of the most fundamental theories of physics, his work had no practical applications until the invention of GPS 60 years later. In today’s short-term outcomes driven market, it is highly unlikely that Einstein’s work would have been funded.

Photo: Residencia de Estudiantes

January 9, 2020 0

El Pais: The Oxford teacher who stays in briefs to teach mathematics

Tom Crawford talks about how mathematics can help win a football league or the real ability of algorithms to manipulate people’s behaviour.

Tom Crawford (Warrington, United Kingdom, 1989) is presented as an atypical math teacher. He teaches mathematics to first and second year students at the University of Oxford (United Kingdom) and carries out an intense dissemination work in which he tries to approach a discipline that is not usually found among the favourites of young students.

In his attempt to popularise science, he does not hesitate to stay in his underpants , using the striptease as a metaphor for his work deepening the meaning of equations such as Navier-Stokes, unveiling them layer by layer, to make something affordable that can result in principle esoteric.

This week, Crawford visited the Student Residence, in Madrid, where, within the Mathematics in Residence cycle organised by the ICMAT, he offered the conference Mathematics of sport . In it, he uses sport as an example of a daily activity that can be better understood and practiced using mathematical equations.

Question. You undress or use sports to make mathematics impose less. Why is it necessary to show that mathematics is fun? I don’t see lawyers or judges, who also deal with very complex issues, trying to present the law as something fun.

Answer. I think it’s because people, for whatever reason, happily admit that they don’t like math, it’s socially acceptable. If you tell someone that you are a lawyer, their default answer is not going to be “I don’t like the law,” and that does happen with math. And it shouldn’t be like that. Everyone should have a basic understanding of math, but many people don’t have it. For me, that is why I want to emphasize that mathematics is fun and accessible. It doesn’t have to be something very hard or something that was taught badly in school.

Q. Do you think mathematics is taught especially badly in school, worse than other subjects?

A. Mathematics has a hard time competing with other subjects in the sense of teaching them through stories. When you learn something, if they can teach you through stories, it is something very powerful, which serves to catch people. And that is easier with literature or history.

A very simple example of how to add stories to mathematics would be trigonometry. The properties of the triangles you learn in high school. If you think about how these functions were discovered or invented, why we invented the sine, the cosine and the tangent, it was the ancient architects who tried to build buildings, churches, pyramids and created those intellectual tools. This is how trigonometry should be taught to me. Imagine they are in ancient Rome and you have to build a concrete building. How would you do it with the technologies available at that time? This prompts you to think about angles and distances and that is where trigonometry is useful and what it was invented for.

Q. A little more than a century ago, in a country like Spain, more than half of the population was illiterate. Do you think it would be possible and desirable to get a large majority of people to be able to handle basic mathematical tools?

A. It is completely possible and I would say that we are already doing it. It depends on what you consider a basic level of mathematics. Most people can, for example, looking at a clock know that the needles return to the same place every 12 hours, it is modular arithmetic, something you don’t study until you get to college. Even being able to calculate changes when they give you a ticket is to do mental arithmetic. Or calculate when you have to leave home if it takes 35 minutes to the station and the train leaves at 12.45. There are many things you do without thinking, but that involve mathematical calculations. So it depends on what you consider a desirable level of mathematics, but a large part of the population already has some capacity to use them.

“You can question whether trying to influence voters is good or bad”

Q. He also talks about the possibilities of mathematics to improve the performance of athletes. There is a movie like Money Ball , which talks about the experience of a baseball coach who uses mathematical analysis to lead a small team to compete against the big ones in the league with much less budget. Do you use math a lot in elite sport?

A. As far as I know, it is an important part of the scout systems of large teams. Today, these scouts, in addition to the classic analysis of a player’s performance, strengths and weaknesses, include teams of mathematicians and data scientists. As in Moneyball , your job is to analyse large amounts of data and detect marginal gains to take advantage of. That works well in baseball, because you have many controllable factors: The pitching of the pitcher, the batter, the race to the base. It is very formulable and they are repetitive behaviours. In football it is more difficult to find those marginal gains because it is less controllable.

The best example I can think of in football is Leicester City, which won the Premiere League in 2016. A big surprise. They had climbed to the first few years before and suddenly they win. In that victory, N’Golo Kanté was very important. He was the star of the season and won the player of the year award. He had been signed by a French second division team because the scout network had identified him among all the defensive midfielders in Europe at any level. As a defensive camper centre, one of your jobs is to stop the attacks of opponents. You can measure this in tickets, but one of the best ways to do this is through interceptions, which has to do with the player’s ability to read a game. It is something very difficult to assess with a number, quite subjective. But interceptions suggest that you are very often in the right place. And from that point of view, their number of interceptions was much higher statistically than the rest of midfielders. If the average of all midfielders in Europe is two, but most of the players are between 1.9 and 2.1 and Kanté is at 3, we see that it is an atypical case. It was not just a statistical analysis, because the human element is valued, but it was a factor to hire him.

elpais — The mathematician Tom Crawford at the Student Residence, in Madrid. ÁLVARO GARCÍA

Q. Can mathematics tell us what is the limit of human performance in sport? There have already been examples in the past, such as Roger Bannister’s, which went down four minutes on the mile when almost everyone said it was impossible, in which the predictions were completely wrong. Can these limits be accurately identified using mathematics?

R.If you look at the men’s marathon record during the last century, the marks descend, but not at a constant pace. You can estimate, for example, that every 10 years, 10 minutes are trimmed at the beginning, but then, in the 1940s and 1950s, the curve begins to flatten out and already in the 1990s it seems completely flat. So if we had sat here 30 years ago, when the record was around two hours and five minutes, we could have thought we would never run below two hours, because even if it keeps going down, the pace is getting slower. But in recent years, there has been much progress in long-distance races, such as new shoes that can provide 4% more energy. In addition, there is a professionalisation that allows you to train all day and not have a job besides running.

“I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes”

So these are new factors that modify our calculations. In the future, in 30 years, new improvements may appear, but it is certain that we will not run a marathon in less than an hour. Given what has happened in the past, I think I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes.

Q. Some people, when talking about the possibilities of mathematics to bring humans to the limit of perfection, may think that sports will become more boring, because there will be less and less space for the unexpected.

A. I think that also has to do with the human psychological trait that is nostalgia. But sport evolves and there is always a human factor. If the study allows you to perfect the place where it is better to throw a penalty, the goalkeepers can also work with that information. And then, there are some players who do not shoot at that supposedly perfect space, such as Eden Hazard, of Real Madrid, who when he threw the penalties for Chelsea waited until the last moment to decide where he threw it, a method that goes against what he says The mathematical model. In the end there are many variables in sports.

Q. Can mathematics help us better understand human groups? Does that technology have the potential to improve living together or to make it worse?

A. With all the data available, there are huge technology companies that can make profiles of people. Knowing that you are white, American, that you earn so much money and live in such a state, they can try to predict what you like or what you do and influence your vote in one direction. But this technology could also be used for good and you can also question whether trying to influence voters is good or bad. I think that ultimately we depend on the big companies that have control over these data so that they assume their moral responsibility and use the data well.

In any case, I think that most of the mathematicians working in this field would say that the idea of using mathematical data, algorithms and models to try to predict people’s behaviour is incredibly new and we don’t know exactly what we are doing. Algorithms may be a part of the decision making process, but not the only criteria for making a decision.

You can read the original article on El Pais here.

November 28, 2019 0

BIG STEM Communicators Network

As a new member of the BIG STEM Communicators Network I was very pleased to be featured in the member spotlight for spring 2019. (The original article is ‘members only’ so I’ve copied the text below.)

“As a new member of the BIG community I would like to introduce myself as the ‘Naked Mathematician’ (yes you did read that correctly). I am a Maths Tutor at the University of Oxford with a goal to reduce fear and anxiety towards maths. One of the ways in which I do this is to take my clothes off – what better way to emphasise that the subject is not as serious and intimidating as many people think than by teaching in my underwear! The concept began as a series of videos on my YouTube channel entitled ‘Equations Stripped’ where I strip back some of the most famous equations in maths (and myself) layer-by-layer so that everyone can understand, and has since evolved into a live performance now touring universities across the UK. My efforts to bring maths to a new audience have been recognised by the University of Oxford, where I was awarded first prize in the Outreach and Widening Participation category at the OxTALENT awards, and I have also been shortlisted for the Institute of Physics Early Career Communicator award.

The ‘Naked Mathematician’ is of course not appropriate for every audience and as such is only a small part of the work that I do to share my love of maths. My ‘Funbers’ series was broadcast throughout 2018 on BBC Radio, where in each episode I look at numbers more closely than anyone really should to bring you the fun facts that you didn’t realise you’ve secretly always wanted to know… I also try to involve my audience in the creative process as much as possible by issuing a call for questions on social media and then hosting a vote to decide the topic of my next video in the ‘I Love Mathematics’ video series. Finally, I combine my love of sport with maths in my popular ‘Maths v Sport’ talk which features a live penalty shootout on stage and an attempt to break a running world record (appropriately scaled of course!).

All of the material that I produce is available for free on my website tomrocksmaths.com and associated social media profiles @tomrocksmaths on Facebook, Twitter, Instagram and YouTube. I am very excited to have joined BIG and look forward to working with the community to help to share STEM subjects with the world!”

May 3, 2019 0

Maths and the Media

Arriving at St John’s in 2008 to begin my study of mathematics, I was certain that within 4 years I would be working in the city as an actuary or an investment banker. Whilst I loved my subject, I saw it as means to obtain a good degree that would set me up for a career in finance. I’m not sure I could have been more wrong…

My current journey began towards the end of my second year, where I found myself enjoying the course so much that I wanted to continue to do so for as long as possible. This led me to research PhD programmes in the UK and the US, and I was fortunate enough to be offered a place to study Applied Maths at the University of Cambridge in 2012. During my time at Oxford, I found myself straying further and further into the territory of applied maths, culminating in a fourth-year course in fluid mechanics – the study of how fluids such as water, air and ice move around. This ultimately led to my PhD topic at Cambridge: where does river water go when it enters the ocean? (If you’re interested to find out more I’ve written a series of articles here explaining my thesis in simple terms.)

As part of my PhD I conducted experiments, worked on equations and even took part in a research cruise to the Southern Ocean. It was on my return from 6 weeks at sea that I had my first taste of the media industry via a 2-month internship with the Naked Scientists. I would spend each day searching out the most interesting breaking science research, before arranging an interview with the author for BBC radio. It was great fun and I learnt so much in so many different fields that I was instantly hooked. Upon completion of my PhD I went to work with the Naked Scientists full time creating a series of maths videos looking at everything from beehives and surfing, to artwork and criminals. You can watch a short trailer for the Naked Maths series below.

My work with the BBC and the media in general ultimately led me to my current position as a Mathematics Tutor at three Oxford colleges: St John’s, St Hugh’s and St Edmund Hall. This may not sound like the media industry, but the flexibility of the position has allowed me to work on several projects, including launching my website and my YouTube channel @tomrocksmaths where I am currently running two ongoing series. In the first, Equations Stripped, I strip back the most important equations in maths layer-by-layer; and for the second series in partnership with the website I Love Mathematics, I answer the questions sent in and voted for by students and maths-enthusiasts across the world.

Alongside my online videos, I am also writing a book discussing the maths of Pokémon – Pokémaths – and have a weekly show with BBC radio called ‘Funbers’ where I tell you the fun facts about numbers that you didn’t realise you’ve secretly always wanted to know. I have also recently presented at conferences in the US and India and hold regular talks at schools and universities, including for the Oxford Invariants and the Maths in Action series at Warwick University where I faced my biggest audience yet of 1200.

Looking back at my time at St John’s, I never would have imagined a career in the media industry lay before me, but the skills, experience and relationships that I formed there have undoubtedly helped to guide me along this path. I think it just goes to show that Maths is possibly the most universal of all subjects and really can lead to a career in any industry.

You can follow Tom on Twitter, Facebook and Instagram @tomrocksmaths for the latest updates.

April 3, 2019 0

Tom Crawford, and Rockin’ Maths Matters

Esther Lafferty meets Dr Tom Crawford in the surprisingly large and leafy grounds of St Hugh’s College Oxford as the leaves begin to fall from the trees. It’s a far cry from the northern town of Warrington where he grew up.

Tom is a lecturer in maths at St Hugh’s, where, defying all ‘mathematics lecturer’ stereotypes with his football fanaticism, piercings, tattoos, and wannabe rock musician attitude, he makes maths understandable, relevant and fun.

‘It was always maths that kept me captivated,’ he explains, ‘ever since I was seven or eight. I remember clearly a moment in school where we’d been taught long multiplication and set a series of questions in the textbook: I did them all and then kept going right to the end of the book because I was enjoying it so much! It was a bit of a surprise to my teacher because I could be naughty in class during other subjects, messing around once I’d finished whatever task we’d been set, but I’ve loved numbers for as long as I can remember and I still find the same satisfaction in them now. There’s such a clarity with numbers – there’s a right or else it’s wrong. In English or History you can write an essay packed with opinion and interpretation and however fascinating it might be, there are lots of grey areas, whereas maths is very black and white. I like that.’

‘My parents both left school at sixteen for various reasons but they appreciated the value of education. My mum worked in a bank so she perhaps had an underlying interest in numbers but it wasn’t something I was aware of. I went to the local school and was lucky enough to be one of the clever children but it wasn’t until I got my GCSE results [10 A*s] that the idea of Oxford or Cambridge was suggested to me. I would never have thought to consider it otherwise.

‘I remember coming down for an interview in Oxford, at St John’s, arriving late on a Sunday night and the following morning I took a stroll around the college grounds – I could feel the history and traditions in the old buildings and it was awesome. I really wanted to be part of everything it represented. I thought it would be so cool to study here so I was very excited when I was offered a place to read maths.

‘Studying in Oxford I found I was most interested in applied maths, the maths that underpins physics and engineering for example. ‘Pure’ maths can be very abstract whereas I prefer to be able to visualise the problems I am trying to solve and then when you work out the answer, there’s a sudden feeling when you just know it’s right.’

In his second year, Tom became interested in outreach work, volunteering to take the excitement of maths into secondary schools under the tutelage of Prof Marcus Du Sautoy OBE as one of Marcus’s Marvellous Mathematicians (or M³), a group who work to increase the public understanding of science.

‘I went to China one summer to teach sixth formers and it was great to have the freedom to talk about so many different topics. I spent another summer in an actuary’s office because I was told that was the way to make real money out of maths – it was a starkly different experience. I realised I was not at all cut out for a suit and a screen!’ Tom smiles. ‘I am a real people-person and get a real buzz from showing everyone and anyone that you can enjoy maths, and that it is interesting and relevant. I love the subject so much and I think numbers get a bad press for being dull and difficult and yet they underpin pretty much everything in the whole universe. They can explain almost everything and you’ll find maths in topics from the weather to the dinosaurs.

Take something like the circus for example – hula-hoops spinning and circles in the ring, and then the trapeze is all about trigonometry: the lengths and angles of the triangle. Those sequinned trapeze artists are working out the distances and directions they need to leap as they traverse between trapezes and its maths that stops them plummeting to the floor!’

Having spent four years in Oxford Tom then spent five years at Cambridge University looking at the flow of river water when it enters the sea, researching the fluid dynamics of air, ice and water, and conducting fieldwork in the Antarctic confined to a boat for six weeks taking various measurements in sub-zero temperatures. You’d never expect a mathematician to be storm-chasing force 11 gales in a furry-hooded parka, but to get the data needed to help to improve our predictions of climate change, that was what had to be done!

Tom also spent a year as part of a production group known as the Naked Scientists, a team of scientists, doctors and communicators whose passion is to help the general public to understand and engage with the worlds of science, technology and medicine. The skills he obtained allowed him to kick-start his own maths communication programme Tom Rocks Maths, where he brings his own enthusiasm and inspiring ideas to a new generation alongside his lectureship in maths at St Hugh’s.

A keen footballer (and a massive Manchester United fan) it’s no surprise Tom has turned his thoughts to football and as part of IF Oxford, the science and ideas festival taking over Oxford city centre in October, Tom is presenting a free interactive talk (recommend for age twelve and over) on Maths versus Sport – covering how do you take the perfect penalty kick? What is the limit of human endurance – can we predict the fastest marathon time that will ever be achieved? And over a 2km race in a rowing eight, does the rotation of the earth really make a difference? Expect to be surprised by the answers.

Esther Lafferty, OX Magazine

The original article can be found here.

January 7, 2019 0