TN Argentina: Oxford teacher teaches in underpants to explain math

Translation of my interview with TN Argentina below, and original (in Spanish) here.

Tom Crawford introduces himself as an atypical Math teacher. He teaches first and second year students at the University of Oxford where he carries out an intense work of dissemination in which he tries to bring closer a discipline that is not usually among the favorites of young students. In his attempt to popularize science, he does not hesitate to stay in shorts. He uses striptease as a metaphor for his work, delving into the meaning of equations like Navier Stokes’, revealing them layer by layer, to make something that can be too complex even more level.

This week, Crawford visited the Student Residence in Madrid, where, within the Mathematics in Residence cycle organized by ICMAT , he offered the conference The Mathematics of Sport . In it, she uses sports as an example of an everyday activity that can be better understood and practiced using mathematical equations.

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You undress or use sports to make mathematics better understood. 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.

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

Do you think that mathematics is taught especially poorly in school, worse than other subjects?

Math is difficult to compete 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, it 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 triangles that 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, they were the ancient architects who tried to construct buildings, churches, pyramids, and created those intellectual tools. For me this is how trigonometry should be taught. Imagine that 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 encourages thinking about angles and distances, and that’s where trigonometry is useful and what it was invented for.

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 use basic mathematical tools?

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 hands return to the same place every 12 hours, it is modular arithmetic, something that you do not study until you get to university. Even being able to calculate changes when you get a ticket is doing 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 they involve mathematical calculations. So it depends on what you consider to be a desirable level of mathematics , but much of the population already has some ability to use it.

It also talks about the possibilities of mathematics to improve the performance of athletes. There’s a movie like Moneyball, which talks about the experience of a baseball coach who uses mathematical analysis to lead a small team to compete against the big league players with much less budget. Is math widely used in elite sports?

As far as I know, it is an important part of the selector systems of large teams. Today, these specialists, in addition to the classic analyzes talking about a player’s performance, his strengths and weaknesses, include teams of mathematicians and data scientists. Like Moneyball , your job is to analyze large amounts of data and spot marginal profits to tap. That works well in baseball, because you have so many controllable factors: the pitch, the hitter, the run to the base. It is very formulable and are repetitive behaviors. In soccer it is more difficult to find those marginal gains because it is less controllable.

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Can mathematics tell us what is the limit of human performance in sport? There have already been examples in the past where predictions were completely wrong. Can those boundaries be accurately identified using mathematics?

If you look at the record of the men’s marathon during the last century, the marks fall, but not at a constant rate. You can estimate, for example, that every 10 years, they are cut 10 minutes at the beginning, but then, in the 1940s and 1950s, the curve begins to flatten and 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 that we would never get to run for less than two hours, because although it continues to drop, the pace is every time slower. But in recent years, there has been a lot of progress in long distance running, such as new sneakers that can provide 4% more energy. In addition, there is a professionalization that allows training all day and not having a job besides running.

Some people, when talking about the possibilities of mathematics to push 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.

-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 refine the place to which it is better to shoot a penalty, also the archers can work with that information. And then, there are some footballers who do not shoot into that supposedly perfect space, such as Eden Hazard, from Real Madrid, who waited until the last minute when deciding where to shoot, a method that goes against what he says. the mathematical model. In the end there are many variables in sports.

Can mathematics help us to better understand human groups? Does that technology have the potential to improve coexistence or to worsen it?

With all the data available, there are huge technology companies that can create 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 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 this data to take their moral responsibility and use the data well.

Oxbridge admission question: how many paths are there between opposite corners of a cube?

This is one of my favourite Maths questions. I encountered it among a list of past interview questions for Oxbridge admissions: how can we get from point A to point B using the edges of the cube, without going over the same edge twice?

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I encourage you to try it yourself first, as getting to the solution gave me another glimpse at the beauty of Maths problem solving. And who knows? Maybe you’ll get to a different solution than the one below…

Now, let’s take a look at my process for coming to a solution. I often look at beautiful solutions and think to myself ‘how on Earth did they come up with that?’ So, I’ll show you my exact steps.

First, we need a way of representing the road we take from A to B. Since we can represent an edge by stating its two endpoints, let’s try to label all the vertices of the cube. The natural way to do that is with the standard 3D Cartesian coordinates.

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As you can see, A is chosen to be the origin of the coordinate system. Also, for ease of notation B was chosen to be the point (1, 1, 1), which can be done without changing the problem (all we are doing is re-scaling the cube to have side length 1 which doesn’t affect the number of paths).

It was at this point my friend remarked that the cube was too hard to visualise on paper, which made me realise that we don’t actually have to visualise it in 3D. The only information we need is how the points are connected together – ie. where we can move to from a given point.

So, picking a random point say (1, 0, 1), where can we move? Looking at the diagram you can see that there are three points connected to this point: (1, 1, 1), (1, 0, 0) and (0, 0, 1). From here it is not too hard to deduce that any two points are connected by an edge if their coordinates are different in exactly one place (only in x, y or z from (x, y, z)). And we can use this to show the lines in a 2D way as follows:

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Next, we need some logic for the way we arrange these points in a diagram. In other words, we need more symmetry in the picture. A cube is very symmetric (48 symmetries in total – can you work them all out?) and so our 2D representation should ideally be more symmetric. This was my second attempt:

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…we can do better than that. The central part in the next one is less jumbled.

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Now we’re getting somewhere. Do you notice something in the middle? The path in blue looks like a crumpled hexagon – what would happen if we unravelled it?

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Much better. I’d say we’re almost at the point where we can start counting the ways to get from A to B, we just have a couple of little problems to iron out…

First, B is not symmetrical in the picture. Sure, you could say this is only an aesthetic thing, but symmetry in Maths often helps us to solve things more easily so let’s try to fix it. One idea would be to add 3 ‘B’s into the picture – one for each of the nodes connected to it. But, thinking about this a little deeper, we see that because the path has to end at B – and once we are there we cannot go back – then in fact B does not need to be connected to the rest of the picture. Instead, we can just draw some EXIT signs at the nodes connected to B. Once you reach one of these points, you have two choices: either “exit” the picture through a B (only one way to do that: move to B from that point), or continue along the hexagon. This observation will be very helpful when it comes to counting paths.

Second, we are not quite sure where to go from A. There is no obvious choice for a starting move. From A to… which of the three? Thankfully, due to the symmetry of the picture (and of the problem, of course), we can simply count the ways to get from A to B with the first move being ‘A to (0, 0, 1)’ and then multiply by 3 to get the result. So, let’s do exactly that!

Given our new-found knowledge, we can make very useful simplifications to the picture. Firstly, the line from A to (0, 0, 1) can be replaced simply by an ENTRY sign at (0, 0, 1), since that is our starting move (which we do in exactly one way). Secondly, since we cannot use that first line anymore, we can simply remove it.

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Now things are really starting to take shape. In this form we see that the line through the point A is equivalent to just going from (1, 0, 0) to (0, 1, 0) or the reverse (once we are in A, we cannot go back to where we came from), and so we can simply delete the point A. Also, for neatness, let’s rotate the whole diagram 60 degrees clockwise.

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Perfect! I’d say we are now in a position to start counting paths – remembering that we have to multiply by 3 at the end because of the symmetry of the first move. In fact, looking at our much simplified – and very symmetric – diagram, it becomes apparent that we can also count all of the paths that start by going from the ENTRY POINT to (1, 0, 1) and then multiply by 2 at the end. I told you symmetry was going to be helpful!

The following picture should help with the counting:

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It perhaps looks a little complicated so let’s break it down. Following all the arrows, or doing it on your own, you can hopefully see that by going through an edge once and starting as decided above (ENTRY to (1,0,1)), we have the following:

  • one way to get to the (1, 0, 1) EXIT
  • two ways to get to the (1, 1, 0) EXIT (ah, that pesky purple arrow!)
  • two ways to get to the (0, 1, 1) EXIT (either only on the hexagon, or using the shortcut).

In total, with the chosen start, we have 5 ways to get from A to B. But, don’t forget we need to multiply due to the symmetries that we’ve used to simplify the problem. First, multiply by 2 (we can go to (0, 1, 1) instead of (1, 0, 1) starting from (0, 0, 1)) and get 10 ways. And then multiply by 3, the number of ways to choose an ENTRY point. Thus, the actual total is 30.

ANSWER: 30 Ways to get from A to B!

Whilst we take a well-deserved moment to pat ourselves on the back for getting the right answer, let’s look at some things to take away from this problem.

  1. Even if you are asked about a cube in a problem, you mustn’t get stuck on the idea of a cube. Maths is flexible that way. You can model your cube into any other structure that keeps the important elements in place. In our case, only the ways in which the vertices were connected mattered, not the angles between the lines or anything. Thus, a graph was a fairly good choice.
  2. You can sometimes make use of the symmetry of a problem to make your life easier, as we did with the way we counted the total number of paths. Symmetry plays a very important role in Mathematics, so keep an eye out for it.

I hope you enjoyed my solution and best of luck with your own Maths problems! If you want to discuss something, leave a comment below – this place is always open to interesting solutions or remarks.

The original version of this article can be found here.

Vlad Tuchilus

La Razon: Math with Rock

A translation of an article about my work in Spanish newspaper La Razon. You can read the original article here.

Mathematics was, as for so many classmates with little numerical capacity, the coconut of my adolescence. In a twisting mortal with pedagogy, my teacher came to suspend me with a 4.9. I always stayed 0.1 to understand algebra and today I can’t survive without a calculator. I am not proud. I wonder if everything would have gone better with Tom Crawford. This Brit is a professor at Oxford, but he doesn’t wear a herringbone jacket or bottle-butt glasses nor is he older than the polka. Tom is an AC / DC math, the punk kid in the bunch. Unlike the old masters, he does not use the ruler as a throwing weapon but, at most, to measure the meters of cloth that is removed from each lesson. He is a “naked scientist”, not as a nod to precariousness but as a seduction pedagogical strategy. “I want to take the solemnity off the math, make it entertaining,” he says.

That goes through a “look” of a hangover rocker with a given shirt, sucks, piercing, tattoos and hair dye. He calls himself “Tom Rocks Maths.” His profiles on networks and his informative videos, in which he ends up posing in leopard-print briefs, have legions of followers. Will it be the solution to my problems? Be that as it may, Crawford was in Madrid yesterday, for the first time in Spain, to give a talk in his own way about mathematics applied to sport. The event took place at the Student Residence, where in 1923 another weird boy, with more clothes and more hair, Einstein, summed up his theory of relativity in an act presented and translated by Ortega and Gaset. The list of visits to that leading institution is as interesting as that of its well-known students: Lorca, Dalí, Buñuel …

The Residence has long become part of a memorial of what it was, but its teaching program continues far from the spotlight, without neglecting the field of science, which seems to have been overlapped when speaking of the Residence due to talent. creative of the boys of Letters already mentioned. Tom Crawford is the last visit and, although we may feel like a histrion or a secondary actor in “Trainspotting”, we must not forget that this is purely an eminence from Oxford.

Photo: Jesus G. Feria

Brian – 1st year Maths

First year St John’s Maths student Brian discusses his favourite areas of undergraduate Maths (featuring a famous sequence) and his plans for a future career in Data Science. Produced for the SJC Inspire Programme.

Teddy Rocks Maths Essay Competition

Entries for the first ‘Teddy Rocks Maths’ Essay Competition are now open! This is YOUR chance to write a short article about your favourite mathematical topic which could win you a prize of up to £100. ENTER HERE: https://seh.ac/teddyrocksmaths

All entries will be showcased on tomrocksmaths.com with the winners published on the St Edmund Hall website. St Edmund Hall (or Teddy Hall as it is affectionately known) is a college at the University of Oxford where Tom is based.

Entries should be between 1000-2000 words and must be submitted as Microsoft Word documents or PDF files using the form at https://seh.ac/teddyrocksmaths

The closing date is 12 March 2020 and the winners will be announced in April 2020.

Two prizes of £50 are available for the overall winner and for the best essay from a student under the age of 18. There are no eligibility requirements – all you need is a passion for Maths and a flair for writing to participate!

The winners will be selected by Dr Tom Crawford, Maths Tutor at St Edmund Hall and the creator of the ‘Tom Rocks Maths’ outreach programme. The mathematical topic of your entry can be anything you choose, but if you’re struggling to come up with ideas here are a few examples to get you started:

Where does river water go when it enters the ocean? – Numberphile

Would alien geometry break our brains? – Tom Rocks Maths intern and maths undergraduate Joe Double

How many ping-pong balls would it take to raise the Titanic from the ocean floor?

If you have any questions or would like more information please get in touch with Tom using the contact form here – Good luck!!

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