Visiting Students at St Edmund Hall

Calling all US-based students, if you have ever thought you would like to have me as your college professor, now is your chance. I am currently in charge of the visiting student mathematics programme at St Edmund Hall, which means anyone accepted onto the programme will have weekly tutorials with yours truly. Information on the course specifics and how to apply can be found on the St Edmund Hall website here.

Courses available include (but are not limited to):

Michaelmas Term (Autumn)

  • Linear Algebra
  • Geometry
  • Real Analysis: Sequences and Series
  • Probability
  • Introductory Calculus
  • Differential Equations
  • Metric Spaces and Complex Analysis
  • Quantum Theory

Hilary Term (Winter)

  • Linear Algebra
  • Groups and Group Actions (continues next term)
  • Real Analysis: Continuity and Differentiability
  • Dynamics
  • Fourier Series and PDEs
  • Multivariable Calculus
  • Differential Equations
  • Numerical Analysis
  • Statistics
  • Fluid and Waves
  • Integral Transforms

Trinity Term (Spring)

  • Constructive Mathematics
  • Groups and Group Actions (continued)
  • Real Analysis: Integration
  • Statistics and Data Analysis
  • Calculus of Variations
  • Special Relativity
  • Mathematical Biology

The detailed course synopses, as well as some course materials can be found here.

If you have any questions please get in touch with Tom via the contact form, or the admissions office at St Edmund Hall via admissions@seh.ox.ac.uk.

Photo: Flemming, Heidelberg Laureate Forum

La Redada Podcast Interview

I was recently interviewed by Lucia Taboada for La Redada Podcast about my love of maths and how it is used in today’s world to model everything from penalty kicks to the next TV series you watch on Netflix. The interview was translated into Spanish for the actual podcast so I’ve also included the original recording of my answers in English – enjoy!

Podcast version

La Redada T08 E29: Tom Crawford the Naked Mathematician

Questions

  1. On your YouTube channel, you present science in an entertaining way. Why is maths so unpopular sometimes, maybe students are afraid of maths?
  2. How would you define the importance of mathematics in our life?
  3. Tom, I’m a huge supporter of a Spanish team called Celta de Vigo. You explain the possibilities using maths to improve the performance of football players. How can Celta de Vigo use this to improve? (unfortunately, we are now in the last positions)
  4. Penalty kicks are a science? Can you predict them?
  5. Have you been hired by any football team?
  6. Do you think football teams should hire math workers?
  7. You are a tutor in St John’s College at the University of Oxford where you teach maths to the first and second year undergraduates. Oxford is a traditional university – how are your methods received there?
  8. You have some maths tattoos on your body, thats right? Explain them to us?

English (unedited) version

Image: Residencia de Estudiantes

How do Stinging Nettles Inject Poison?

The leaves of stinging nettles are covered in ‘pipette-like’ stingers which penetrate the skin on contact and deposit a small amount of poison. The ‘pipette-like’ design means that almost all of the poison contained in the stinger can be injected at once if sufficient force is applied to bend the stinger to an angle of 90 degrees. This is demonstrated in laboratory experiments conducted by Kaare Jensen at the Technical University of Denmark.

This video is part of a collaboration between FYFD and the Journal of Fluid Mechanics featuring a series of interviews with researchers from the APS DFD 2017 conference.

Sponsored by FYFD, the Journal of Fluid Mechanics, and the UK Fluids Network. Produced by Tom Crawford and Nicole Sharp with assistance from A.J. Fillo.

What’s your type? The Maths behind the ‘Qwerty’ Keyboard

The maths behind the most unshakeable technology of the 20th century

Martha Bozic

In 1867, a newspaper editor in Milwaukee contemplated a new kind of technology. He had previously patented a device which could be used to number the pages of books but, inspired by the suggestions of fellow inventors, he decided to develop it further. The idea itself wasn’t exactly new – it had been echoing around the scientific community for over 100 years. The challenge was to realise it in a way that was commercially viable, and Christopher Latham Sholes was ready.

His first design, in 1868, resembled something like a piano. Two rows of keys were arranged alphabetically in front of a large wooden box. It was not a success. Then, after almost 10 years of trial and error came something much more familiar. It had the foot-pedal of a sewing machine and most of the remaining mechanism was hidden by a bulky casing, but at the front were four rows of numbers, letters and punctuation… and a spacebar.

Surprisingly little is certain about why he chose to lay it out as he did, probably because to Sholes the layout was no more than a side-effect of the machine he was trying to create. But as the most influential component of the typewriter, the qwerty keyboard has attracted debates about its origin, its design and whether it is even fit for purpose. Without historical references, most arguments have centred on statistical evidence, jostling for the best compromise between the statistical properties of our language and the way that we type. More recently, questions have been posed about how generic ‘the way that we type’ actually is. Can it be generalised to design the perfect keyboard, or could it be unique enough to personally identify each and every one of us?

The first typewriter was designed for hunt-and-peck operation as opposed to touch typing.  In other words, the user was expected to search for each letter sequentially, rather than tapping out sentences using muscle-memory. Each of the 44 keys was connected to a long metal typebar which ended with an embossed character corresponding to the one on the key. The typebars themselves were liable to jam, leading to the commonly disputed myth that the qwerty arrangement was an effort to separate frequently used keys.

Throughout the 20th century new inventors claimed to have created better, more efficient keyboards, usually presenting a long list of reasons why their new design was superior. The most long-lasting of these was the Dvorak Simplified Keyboard, but other challengers arrived in a steady stream from 1909, four years after qwerty was established as the international standard.

Is it possible that there was a method behind the original arrangement of the keys? It really depends who you ask. The typebars themselves fell into a circular type-basket in a slightly different order to the one visible on the keyboard. Defining adjacency as two typebars which are immediately next to each other, the problem of separating them so that no two will jam is similar to sitting 44 guests around a circular dinner table randomly and hoping that no one is seated next to someone they actively dislike.

Picture 1
The qwerty keyboard and typebasket (Kay, 2013)

For any number, n, of guests, the number of possible arrangements is (n-1)!. That is, there are n places to seat the first guest, multiplied by (n-1) places left to seat the second guest, multiplied by (n-2) for the third guest and so on. Because the guests are seated round a circular table with n places, there are n ways of rotating each seating plan to give another arrangement that has already been counted. So, there are (n x (n-1) x (n-2) x…x 1)/n = (n-1) x (n-2) x…x 1 arrangements, which is written (n-1)!.

By pairing up two feuding guests and considering them as one, you can find the total number of arrangements where they are sat next to each other by considering a dinner party with one less person. From our calculation above we know the total number of possible arrangements is (n-2)!, but since the feuding pair could be seated together as XY or YX we have to multiply the total number of arrangements by two. From this, the final probability of the two feuding guests being sat together is 2(n-2)!/(n-1)! = 2/(n-1), and so the probability of them not being sat together is 1-(2/(n-1)) = (n-3)/(n-1).

But what if one or more of the guests is so unlikable that they have multiple enemies at the table? Say ‘h’ who has been known before now to clash with both ‘e’ and ‘t’. Assuming the events are independent (one doesn’t influence the other) we just multiply the probabilities together to get the chance of ‘h’ being next to neither of them as [(n-3)/(n-1)]2. And the probability that on the same table ‘e’ is also not next to her ex ‘r’ is [(n-3)/(n-1)]2 x [(n-3)/(n-1)] = [(n-3)/(n-1)]3. So, for any number of pairs of feuding guests, m, the probability of polite conversation between all is [(n-3)/(n-1)]m.

Now, returning to the problem of the typebars, a frequency analysis of the English language suggests there are roughly 12 pairings which occur often enough to be problematic. For n=44 symbols, the dinner party formula gives a probability of [(44-3)/(44-1)]12 = [41/43]12 = 0.56. That is a better than 50% chance that the most frequently occurring letter pairs could have been separated by random allocation. An alternative theory suggests that Sholes may have looked for the most infrequently occurring pairs of letters, numbers and punctuation and arranged these to be adjacent on the typebasket. The statistical evidence for this is much more compelling, but rivals of qwerty had other issues with its design.

August Dvorak and his successors treated keyboard design as an optimisation problem. With the advantage of hindsight now that the typewriter had been established, they were able to focus on factors which they believed would benefit the learning and efficiency of touch typing. Qwerty was regularly criticised as defective and awkward for reasons that competing keyboards were claimed to overcome.

The objectives used by Dvorak, qwerty’s biggest antagonist and inventor of the eponymous Dvorak Standard Keyboard (DSK), were that:

  • the right hand should be given more work than the left hand, at roughly 56%;
  • the amount of typing assigned to each finger should be proportional to its skill and strength;
  • 70% of typing should be carried out on the home row (the natural position of fingers on a keyboard);
  • letters often found together should be assigned positions such that alternate hands are required to strike them, and
  • finger motions from row to row should be reduced as much as possible.
Picture 2
The Dvorak Simplified Keyboard (Wikimedia commons)

To achieve these aims, Dvorak used frequency analysis data for one-, two-, three-, four- and five- letter sequences, and claimed that 35% of all English words could be typed exclusively from the home row. He also conducted multiple experiments on the ease of use of his new design over qwerty, although the specifics were sparsely published.

Of course, however good Dvorak’s new design may have been, there was a problem. Qwerty being pre-established meant that finding subjects who were unfamiliar with both keyboards was difficult. Participants who tested the DSK had to ‘unlearn’ touch typing, in order to relearn it for a different layout, while those using qwerty had the advantage of years of practice. The main metric used to determine the ‘better’ design was typing speed but clearly this was not only a test of the keyboard, it was also a measure of the skill of the typist.

Alone, average typing speed would not be enough to distinguish between individuals – any more than 40 words per minute (wpm) is considered above average and since a lot more than 40 people are average or below average typists, some of them must have the same wpm – but other information is available. Modern computer keyboards send feedback from each letter you type, leading to a wealth of data on the time between each consecutive key press. This can be broken down into the time between any particular letter pairing, building a profile on an individuals specific typing patterns, and combined with typing speed it is surprisingly effective at identifying typists.

In a battle of the keyboards, despite its suboptimal design and uncertain past, qwerty has remained undefeated. Today it is so ubiquitous that for most people to see a different layout would be jarring, yet our interactions with it are still identifiably unique. Nearly 150 years after its conception, the keyboard is embedded in our culture – it’s an old kind of technology, just not the one Scholes thought he was inventing.

Million Millimeter March for MoMath

I went to visit the National Museum of Mathematics (MoMath) in New York City to walk the Million Millimeter March to celebrate the 1 millionth visitor to MoMath. Puzzle enthusiast and mathematician Peter Winkler (Dartmouth) joined me to provide some fun facts about numbers along the way…

The route is shown below.

map

The fun facts about numbers explained by Peter Winkler are copied below for reference:

142,857 – a cyclic number. Try multiplying it by 1, 2, 3, 4, 5 or 6 and see what happens!

219,978 – the only 6-digit number such that if you multiply it by 4 it reverses!

322,560 = 9! – 8!

422,481 – the smallest number whose fourth power is itself the sum of three smaller fourth powers: 422,481^4 = 414,560^4 + 217,519^4 + 95,800^4

548,834 – a six-digit number equal to the sum of the sixth powers of it’s six digits: 548,834 = 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6

604,800 – the number of seconds in a week.

742,900 – the number of different ways to walk from the bottom left to the top right (only moving along grid lines to the right or upward) in a 13×13 grid, always staying below the diagonal.

801,125 – the smallest number that is the sum of two positive squares in at least 2^2^2 ways.

873,613 = 1^1 + 2^2 + 3^3 + 4^4 + 5^5 + 6^6 + 7^7

1,000,000 = the number of visitors to MoMath!

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

Tracking Beetles using the Sound of their Wings

The Cocunut Rhinoceros Beetle is an invasive species that if left alone would decimate citrus crops across California. To prevent this from happening, John Allen and his team at the University of Hawai’i have been working to hunt the insects down before they are able to reach the West Coast of the USA. By identifying the frequency of the beetles wing beat, they are able to track them down by listening out for the unique flapping sound of their wings and alert pest control to their whereabouts.

This video is part of a collaboration between FYFD and the Journal of Fluid Mechanics featuring a series of interviews with researchers from the APS DFD 2017 conference.

Sponsored by FYFD, the Journal of Fluid Mechanics, and the UK Fluids Network. Produced by Tom Crawford and Nicole Sharp with assistance from A.J. Fillo.

What is P versus NP?

TRM intern and University of Oxford student Kai Laddiman speaks to St John’s College Computer Scientist Stefan Kiefer about the infamous million-dollar millennium problem: P versus NP. 

You can read more about P vs NP here.

Numberphile: Pi Million Subscribers

It’s incredible to see a channel dedicated entirely to maths reach this quite frankly ridiculous number of subscribers – congratulations Numberphile!! If you haven’t seen it yet check out the many famous faces, including yours truly at 1:27…

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