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The Imitation Game: Part 3

In The Imitation Game: Part 2 we looked at Alan Turing’s work doing codebreaking at Bletchley Park during World War 2. In this final instalment of blog posts about the film The Imitation Game, we look at the work Turing did during the final years of his life, about pattern formation. This is based on a talk given by Professor Jamie Davies from the University of Edinburgh at an event called The Maths of the Imitation Game at the Filmhouse cinema.

You might be fooled into thinking that the film doesn’t touch on Turing’s work after Bletchley at all, but you would be wrong. The reference is subtle, but it is there from the very first scene. The opening part of the film shows Turing in his house, clearing up mess from a burglary. On a table we see a pine cone; on the walls we see pictures of strange spirals of dots and curious pictures of starfish. These pictures are again present in the final, very moving, scenes of the film, when Turing is at home suffering from the effects of his enforced chemical castration. But what do these images tell us about the work that Turing did?

pinecone with spirals

Count the number of spirals on a pinecone (first four shown here). You often find a Fibonacci number.

Count the number of petals in a daisy, or the number of spirals in a pinecone, or the spirals in the head of a sunflower. You will very often get a Fibonacci number: one that appears in the sequence 1,1,2,3,5,8,13,21,34,… In this sequence, you get the next number by adding up the previous two.

Turing claimed that the ubiquity of Fibonacci numbers in plant forms was not a coincidence, and wanted to show that such patterns could arise as the result of chemical reactions in the cells of the plant. In this view he was influenced by the book On Growth and Form by the Scottish mathematical biologist D’Arcy Wentworth Thompson.

In 1952 Turing wrote the paper “The chemical basis of morphogenesis” which was ground-breaking work in explaining how local chemical reactions could result in large-scale patterns. Fibonacci numbers in pine cones, stripes on a tiger, tentacle patterns on a starfish and hexagons in a fly’s eye were all predicted to be manifestations of the same type of chemical reaction. Today Turing’s idea is called a reaction-diffusion equation and is used in even more areas of science than just animal biology.

Haeckel's Kunstformen der Natur

Haeckel‘s picture showing various sea anemones with their tentacles. If you look closely in the film you can spot this picture on the wall of Turing’s house.

The idea is that patterns are formed by the interplay between two types of chemical: one that activates growth (called the activator) and one that inhibits growth (called the inhibitor). Production of the activator stimulates further production of the activator, but also stimulates creation of the inhibitor. Imagine a ring of cells around an embryo. A particular cell might grow ever-so-slightly faster than its neighbours, resulting in more of the activator being produced and thus even further growth. At the same time, the cell is producing the inhibitor chemical, meaning that the growth of cells adjacent to it is stunted. However, cells that are far enough away are not affected by the inhibitor, and so will grow as normal. This results in a sort of wave-like pattern around the embryo, of particular cells growing at regular intervals. And this is exactly what causes the regular arms of a starfish or the tentacles of a sea-anemone.

Turing was not only the first to come up with these ideas, but the first to test them using computer simulations. Working at the University of Manchester in 1951, he had access to the world’s first commercially available general-purpose digital computer, the Ferrati Mark 1, and he immediately started programming it to explore the consequences of his equations. Working by hand his equations would never have been tractable to solve, but using a computer he could see that they were a great model for the physical phenomena he was trying to explain.

Sadly Turing’s work was far ahead of his time, too difficult to understand for biologists (even though he made a conscious effort to explain things well!), and was largely ignored by the scientific community for fifty years. It was only in the late 1990s that the work was picked up again and Turing’s paper rediscovered, and today reaction-diffusion models are considered fundamental to the study of pattern-formation. Prof. Jamie Davies’ current research in Edinburgh is testing Turing’s ideas by ‘programming’ real living cells to make patterns on command, and his lab has been able to replicate all the predictions made by Turing.

To conclude, I think it is a shame that the film portrays Turing as being this incredibly autistic and single-minded person, giving all his love to machines and algorithms instead of real people. In reality he was a warm and friendly individual who had strong friendships and relationships, and, moreover, he did some of his most important work into living things rather than cold machines.

I hope my series of blog posts have highlighted the amazing research that Turing did, which revolutionised computer science, mathematics and biology, not to mention the pivotal role he played in breaking Enigma during World War 2. Many thanks go to John Longley, Tom Leinster and Jamie Davies for providing the material on which these posts have been based, and to the Filmhouse cinema in Edinburgh for hosting our event explaining the Maths of the Imitation Game.

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The Imitation Game: Part 2

In The Imitation Game: Part 1 we looked at the early work of Alan Turing about universal computing machines, the limits of what computers can do, and whether computers could ever successfully imitate human brains. In this post we look at Turing’s work doing codebreaking at Bletchley Park during the Second World War, and the similarities and differences with codebreaking today. The material in this post is based on a talk given by Dr Tom Leinster from the School of Mathematics at the University of Edinburgh, as part of an event called The Maths of the Imitation Game at the Filmhouse cinema.

Bletchley Park in Buckinghamshire was the site of the Allied codebreakers in World War 2.

Bletchley Park in Buckinghamshire was the site of the Allied codebreakers in World War 2.

Bletchley Park, otherwise known as Station X, was a mansion in Milton Keynes that was used during World War 2 as the base for the Government Code and and Cypher School (GC&CS). (Earlier this year it was reopened as a museum, and I’m told it’s very good!) One of its main purposes was to decipher German messages, which were being encrypted using a device called the Enigma machine. It was thought that Enigma was unbreakable, and so it might have been were it not for Turing’s brilliance and subtle mistakes made by the German operators.

In a simple substitution cipher, each letter of the alphabet is replaced by another letter of the alphabet. For example, if we had S->F, H->U, E->R and P->Y, the word SHEEP would be encoded as FURRY. Such a code is easily breakable using frequency analysis: if E is the most common letter in the English language and R is the most common letter in the message, it’s likely that E has been encoded as R. The Enigma machine is much cleverer than that.

A 3-rotor Enigma machine with parts labelled.

A 3-rotor Enigma machine with parts labelled.

In a standard Enigma machine there are 3 rotors. Each rotor has each of the 26 letters of the alphabet inscribed around it, and each is set to an arbitrary position at the beginning of the day. When the operator types a letter on the keyboard, the signal from that key is sent through wiring to the first rotor, which encodes it as a new letter. It is then sent through wiring to the second rotor, which changes it again, and finally it is sent to the third rotor, which changes it again. The signal then goes around some fixed wiring (called a reflector) and then it returns through each of the rotors, finally lighting up a new letter on the lampboard which the operator writes down. Here’s the clever bit: after each new letter is entered, the rotors turn. Therefore, if you had typed S the first time and got an F, you could type S again and get a Z. The same letter is encoded differently every time it is pressed. The message itself is part of the encoding.

There are 3 different rotors, which the operator could choose out of a possible 5. This gives 5x4x3=60 options already for the initial setup. Then, each rotor can start out in any one of 26 different positions. This gives 26x26x26=17,576 options. So far, this is only about a million combinations. It sounds like a lot, but this is at a level where, with a little ingenuity, you could simply brute force all the possibilities. To make Enigma unbreakable a final layer of encryption was added: the plugboard. This used cables to pair up letters. For example, if E was paired with Q, then if E was typed then the machine would interpret this as a Q before transmitting it through the rotors. With this additional scrambling, the number of combinations of an Engima machine was over 159 million million million. There was no way they could ever hope to check all possible settings in a reasonable amount of time. And, to make matters worse, the Germans changed their settings each day, meaning that there was only 24 hours to figure out each code before everything changed.

What Turing did to crack Enigma was to build a machine capable of doing logical calculations that would eliminate a vast number of the possible settings. This was called the bombe and built on earlier work by Polish cryptanalysists. Basically it could try different rotor settings in turn, and look for logical contradictions that would show the settings to be impossible. (For an analogy, think of doing a sudoku puzzle where you might postulate that a 6 goes in a box, but that would result in two 3s in another row which can’t happen, so therefore it can’t be a 6.) Such contradictions might include:

  • Deciding that a letter was encoded as itself. This was impossible due to the way the signals through Enigma were sent round the reflector.
  • Having an asymmetry in the plugboard. If the B is connected to the N, then the N must also be connected to the B.
  • Having a letter in the plugboard connected to its neighbour. Operators were told not to do this.
  • Having plugboard settings that were used the previous day. Operators had to change all the settings every day.

There were many more contradictions like these. There were also sloppy practices among the operators. They would often not set the rotors to a truly random initial position, but would use their names (e.g. “BOB”) or would simply turn the rotors a few places from the day before. There were also common phrases in messages, e.g. the word “ein” appeared in 90% of messages, and “Heil Hitler” also appeared often. Wikipedia has a fascinating description of the methods used to crack Enigma, as well as the methods the Germans used to make Enigma even harder to crack.

Nowadays, encryption of information is done using mathematical algorithms rather than mechanical machines. The most commonly used is called RSA, and relies on the difficulty of factorising large numbers into primes. Just as in Turing’s time, codebreakers rely on having a large amount of information to help them look for patterns, and on computers to do the decryption. The big difference between then and now, according to Tom Leinster, is that during the war the government was spying on the Nazis; today they are spying on us.

Edward Snowden

Edward Snowden, who revealed details about the mass surveillance carried out by the NSA and GCHQ.

Earlier this year it was revealed that the UK’s Government Communications Headquarters (GCHQ) and the USA’s National Security Agency (NSA) had been systematically monitoring all of our emails, phonecalls, texts, web browsing and bank transactions. Their goal was to “collect all of the signals, all of the time”, regardless of whether or not anybody had done anything suspicious. And, just as in the film, the codebreakers often had complete autonomy over the information they collected, with even the highest in command being unaware of what they were doing.

Just as in the film, this codebreaking was only possible because of errors in the encryption of the data. The information leaked by Snowdon showed that the NSA had inserted a secret back door into the world’s most widely used cryptosystem, allowing it to break the encryption.

Alan Turing was a homosexual at a time when homosexuality was illegal, and his conviction and subsequent chemical castration were what led to his suicide in 1954. Today it is legal in the UK for two men to have sex, and even get married. This change in our law has come about because of campaigning and activism, but it is always dangerous to be an activist, speaking out for something that is considered against the law. It is easily argued that such campaigning is even more dangerous today, with the government carrying out mass surveillance of everyone in the population.

I shall end this post with the question asked by Tom Leinster in a piece he wrote for New Scientist in April: is it ethical for mathematicians to work for government intelligence agencies like GCHQ?

I look forward to hearing your comments!

Read The Imitation Game: Part 3 about Turing’s work in biology and pattern-formation.

The Imitation Game: Part 1

Imitation Game posterToday we had a wonderful event at the Filmhouse cinema in Edinburgh talking about The Maths of the Imitation Game. This is the film which tells the story of mathematician Alan Turing and his work codebreaking at Bletchley Park during the Second World War.

I’m going to admit up front: I liked the film. I’ve met other people who’ve been foaming at the mouth with anger over inaccuracies in the film, both with the historical aspects and also with how Turing was portrayed. I agree that the film isn’t perfect – they had to take a lot of liberties with how things were presented in order to make the story appeal to a mass audience. Events certainly did not happen the way the film depicts. And I do take issue with the film’s insinuation that Turing assisted Soviet spying, and also that Turing would have told classified secrets to a police officer. But I also think the film gets a lot of things right, it tries its best to explain key aspects of the story, and is wonderfully acted, especially by Cumberbatch as Turing.

If nothing else, the film is a great platform to start discussions about the important (theoretical) work that Turing did in his life and the repercussions of that work today. This is what our event at the Filmhouse was all about. We had three speakers talking about different aspects of his work, and I’ll summarise those discussions here. I’ll break it down into three blog posts as there is a lot to say about each one.

First up, Dr John Longley from the School of Informatics talked about Turing’s early work on computability. In our lives we are used to having different machines for different tasks: a toaster that warms your bread, a dishwasher to wash your plates and a blender to mash your food. We wouldn’t expect a single machine to perform all these different tasks. Yet, walk through to your study and you’ll find a machine on which you can do many things: play solitaire, manage your accounts, watch videos, search for prime numbers. Nowadays we aren’t at all surprised that a computer can do so much, but in Turing’s day such a machine was inconceivable. In fact, it was Turing who proposed that such a machine could exist. He had the idea of a ‘universal machine‘ which could emulate the output of any other given machine.

Every machine takes some sort of input, performs a certain task and then gives an output. The toaster’s input is your bread, which is heated according to the settings you’ve told it, and its output is some toasted bread. The Enigma machine takes as input the key presses of letters, it performs an encryption, and it outputs letters forming a coded message. For Turing’s universal machine, the input is a sequence of symbols that provide the instructions for what another simulated machine does and the input to this machine. The output is then the answer that the simulated machine would have provided if you had given it this particular input. This idea that the input may itself comprise a set of instructions was groundbreaking. Today we simply call this kind of input ‘software’.

Turing’s invention of the universal machine was designed to answer a long-standing problem in mathematics: can the answer to every mathematical question be determined mechanically? This was posed by David Hilbert in 1928. When Turing was only 23 years old, he showed that the answer to this question was negative, and he did this by coming up with something called the Halting problem.

Turing's bombe

Will Turing’s machine output the correct Enigma settings, or could it keep searching forever? This is called the Halting problem.

This simply asks: given a particular input to a machine, is it possible to determine whether the machine will ever halt and give an output? For example, in the film the machine (called “Christopher”) that Turing has invented is checking through possible settings of Enigma and everyone is waiting for it to tell them what the correct answer is. Will it ever stop, or will it keep whirring away for ever, stuck in some logical loop that prevents it from finishing? If you pull the plug, how can you be sure that in the next minute it wouldn’t have told you the answer? Turing proved that there is no single algorithm that can decide whether a given arbitrary program will halt for a given input. The halting problem is ‘undecidable’.

The conclusion from his work is that there are some questions in mathematics that are unknowable –  there is no computer program that will ever tell us whether they are true or false. However, this begs the question: is the human mind the same as a computer? Or could a brain do computations, do mathematics, that a machine could not? If so, what makes a brain different from a computer?

This question fascinated Turing and led him to invent The Imitation Game, which gives the film its title. Rather than asking the question “Can machines think?” which is very abstract and hard to pin down, he instead asked whether it was possible for a machine to imitate a human so well that another human could not tell that it was really a machine. In a Turing test, person A can interrogate player B by asking a series of questions. Turing said that if the interrogator decides as often as not that player B is a real person, then the machine has passed the test. Despite predicting that computers would be built to pass the test by the year 2000, it was only in June this year (2014) that a computer was said to have passed the test. This was a computer masquerading as a 13 year old Ukrainian boy, who fooled 33% of a panel of judges over the course of a 5-minute conversation.

What do you think? Could computers come to imitate humans one day? Is it just a case of having more computing power, or is it possible that brains can do things that computers fundamentally cannot?

Many thanks to John Longley for the material on which this blog post is based.

Read The Imitation Game: Part 2 about Turing’s role in codebreaking during World War 2 and moral questions about codebreaking today…

Most random masterclass question ever

sheep-tea-towel

Would your DNA be long enough to weave into a tea towel?

Yesterday I was in Stirling giving my usual knots masterclass when I got asked the most unusual question ever! I was explaining how DNA is like a very very long piece of string sitting inside each of our cells. (It’s actually about 2-3m in each cell!) One pupil then put their hand up and said “If you took all the DNA from all the cells in my body, would it be long enough…to weave into a tea towel?” I was so taken aback by this that I just laughed, but actually the answer is probably yes! Depending on the size of tea towel that he wanted of course. And it would be so thin that it wouldn’t be very good at drying dishes, but that is beside the point.

It is always a pleasure to be surprised by children. And I was especially impressed at all the questions I was asked in Stirling. It helped that the masterclass was much smaller than usual – 15 pupils compared to about 70 in Edinburgh and 40-50 in Glasgow. It meant that I was able to talk to everyone while they were doing exercises, and it was much less intimidating for people to stop me and ask questions during the presentations. But even given those things, I was genuinely taken aback by the insightful questions I was asked and the interest that they showed in what I had to say.

My other favourite moment was when another child pondered “I could take a one-dimensional piece of string and weave it into a 2-dimensional object…”. (I assume this was inspired by the previous question about tea towels.) Now, I had not mentioned the word ‘dimension’ once in my whole masterclass. Most people don’t understand the concept of dimension. Even the undergraduates I teach would have trouble understanding why a knot is inherently one dimensional and not three dimensional. And here is a 12-year old child explaining to me how you could use one dimension to fill out 2-dimensional space!

In case you still don’t understand why this bowled me over, you should go and Google “space filling curve”. (Wikipedia is not a bad reference, but is a little technical.) In the mid-19th century, mathematicians had the idea that a curve could be drawn inside a square so that it went through every single point of the square. This is counter-intuitive, as it seems like the square is ‘bigger’ than the curve, so how could the curve fill it all out? Cantor had showed in 1878 that the infinity which is the number of points in a line segment is the same as the infinity which is the number of points in a square, but it was not until 1890 that Peano came up with this geometrical argument that demonstrated it.

Hilbert space-filling curve

How to construct a space-filling curve iteratively. After infinitely many iterations, it will fill out the whole square.

It is still very counterintuitive to mathematicians that this curve is continuous (i.e. can be drawn without taking your pen off the paper) but is nowhere differentiable (i.e. every point is a ‘corner’, so the curve is always changing direction) and is everywhere self-intersecting (every point on the curve touches another point on the curve). Maths is full of these great examples that challenge our assumptions and intuitions and I hope that I can teach this to my undergraduates later in the semester.

So although the masterclass pupil probably didn’t have infinities and deep thoughts in mind when he made the comment about weaving, it’s exactly questions like this which got mathematicians discovering such things over 100 years ago. I hope that his teachers continue to encourage this wonderful imagination and willingness to ask questions, however silly they may seem at first.

MathsJam 2011

This post is a VERY long overdue one… I had meant to write a post after the (very first) MathsJam conference in November 2010, then after the first few Edinburgh MathsJams… We’ve now had the second national MathsJam conference and 5 Edinburgh ‘Jams so it’s about time I told you readers what it’s all about!

HaggisBalloons2010

Balloons at MathsJam 2010

In a nutshell, MathsJam is a place for people to meet to share mathematical puzzles, games, toys, ideas, stories and tricks. It was originally the brainchild of Colin Wright, who organised the first ever MathsJam conference last year, bringing together geeky enthusiasts from all over the country for a weekend of mathematical fun. We had Rubiks cubes of all shapes and sizes, mathematically folded balloons, mirrors, Klein bottles, magic tricks, soma cubes, post-it note dodecahedra, vortex cannons… And that was just the list of physical toys! I also learnt facts like:

  • Round pegs fit into square holes better than square pegs fit into round holes…until you get to 9 dimensions!
  • Almost every integer contains a 3.
  • That given 5 numbers, you can always find 3 of them which add up to a multiple of 3. (But what is the generalisation?)
  • That there is only one number whose spelling is in alphabetical order. (Can you find it?)
  • That a blindfolded person given an even number of coins, placed on a table so that half are facing heads up and half are facing tails up, can separate them into two piles so that the number of heads in each pile is the same.
  • That you can work out the distance to the moon using only a pendulum.

The weekend was such a success that people started asking “Can’t we have a MathsJam every month?”. Pretty soon there were ‘Jams in Manchester, Nottingham and London with Edinburgh, Glasgow, Reading, Liverpool, Newcastle, Dorset, Leeds, Bath, Dublin and Belfast following on their tails.

August Edinburgh MathsJam

Attempting topology at the August Edinburgh MathsJam

Everyone meets on the second to last Tuesday of the month and we have a shared Twitter account, @MathsJam, so that everyone can see the puzzles being worked on around the country. The Edinburgh ‘Jam was set up by myself and Ewan Leeming, and we meet at Spoon Café Bistro on Nicholson Street. Further details are on the MathsJam website, together with an email address and Facebook page, and also contact details for all the other ‘Jams around the UK (and indeed, the world!).

This weekend I travelled down to somewhere near Crewe for the second annual MathsJam conference, together with my buddies Albert, Julia and Michael. I was very excited about all the toys and games I’d get to play with, but at the same time incredulous that the weekend could possibly be better than the first MathsJam weekend. Well, I shouldn’t have had any such thoughts.

Ring on a chain

Albert wearing his ring-on-a-chain

One thing I loved about this year’s conference was the chance to purchase goody bags with exciting toys to take home and show friends. Last year I shot some videos and got photos, but nothing compares to being able to go home and show your friends in person the amazing things you’ve seen. My favourite was the ring-on-a-chain trick (pictured left) where a ring is dropped from a chain with unexpected consequences. Next favourites the falling rings and James Grime’s amazing non-transitive dice.Maths and science is much more cool than sleight-of-hand magic. 🙂

Here are some pencil and paper questions you might like to get your teeth stuck into (metaphorically speaking):

  • A consecutive sum is a sum of consecutive digits. Are there any numbers which are not consecutive sums? How many ways can a number be written as a consecutive sum?
  • Why is 100/81 equal to 1.2345678…?
  • How can you cut any shape out of a piece of paper using only one cut?
  • Does a running sand timer weigh more, less or the same as a finished sand timer?
  • How do you make 2 paperclips link together using a strip of paper?
  • Given that we can make a regular pentagon by tying a knot into a strip of paper, is it possible to make a dodecahedron by folding 12 knots into a piece of paper and then folding it up?
  • How is it possible to randomly play two games, each of which would individually lose you money, and make an overall gain? (This is called Parrondo’s Paradox.)
  • Split the numbers 1,..,16 into two sets X and Y so that the sum of the elements in X equals the sum of the elements in Y; the sum of the squares of X equals the sum of the squares of Y; the sum of the cubes of X equals the sum of the cubes of Y. (I am currently working on a generalisation!)

Plus I learnt  that a 9999-sided polygon is called a nonanonacontanonactanonaliagon. (This seems to be the most popular thing I have ever posted on Twitter.) I encountered Pat Ashforth, one of the founders of Woolly Thoughts, who showed me her dragon-curve blankets and crocheted hexaflexagons. I also saw a magic square that worked upside down and some Platonic solid maps of the world.

Maths knitting by Pat Ashforth

Dragon curves and other mathematical knitting by Pat Ashforth

Cushion magic square

A magic square cushion which works both ways up

Julia found herself on the panel for the Math/Maths podcast, which you can listen to here,with contributions also from Matt Parker, James Grime and Katie Steckles. The laughter on the podcast is a really good reflection of the fun that everyone had at the MathsJam, and once again I have to extend a huge thank you to Colin and all the other people who helped to organise the event this year. There’s no other conference in the world which is this enjoyable and it is wonderful to see so many people enjoying the fun and beauty of mathematics.

If you’ve never been to a MathsJam, I hope this article persuades you to go along to the next one on 22nd November! They are all over the country now so there’s bound to be one nearby. And if there isn’t, start one up yourself! All you need is a pub and a couple of people willing to come sit with you on a Tuesday evening. I look forward to seeing more people MathsJamming in Edinburgh in a week’s time!

Clicking infinity

On Wednesday I was asked to give a talk to a small bunch of S6 (Yr 13 in in England) pupils who were visiting Edinburgh from Fife. It wasn’t any particular special occasion – the enterprising teacher just wanted his students to get out and learn some exciting and different mathematics. It was the perfect opportunity for me to try out a new piece of technology that I’d heard my boss raving about: clickers.

clicker

A clicker

A clicker is like one of those things they have in Who Wants to be a Millionaire where the audience votes for what they think is the right answer. It is an absolutely wonderful teaching aid and we are very lucky to have them at the University of Edinburgh. It means that students can tell the teacher their thoughts without letting anyone else know what they are thinking, so they needn’t worry about the embarrassment of having the wrong answer.

The clickers we use are exactly those pictured on the left. There are 6 buttons which can be used for multiple choice questions, and also a True/False option. The software that comes with the clickers is capable of storing a huge amount of data about your sessions, which really comes into its own when you are monitoring a specific class over many weeks rather than just having impromptu sessions. You can see whether students are improving, how often they change their minds about questions, and even (if you have a strict seating plan) how ideas are spread around the classroom.

I decided to make my class about infinity, using the story of Hilbert’s Hotel to hold the plot together. My first question, just to get people used to the clickers, was a simple true/false question: “Infinity exists only in our imaginations”. There was a fairly strong preference for ‘false’ from the class, which was interesting for me because I would usually vote the other way myself. The students gave examples of ‘real’ infinities that I would argue are purely abstract mathematics, such as the infinity of numbers or infinities in fractals. It’s actually quite nice to meet people who believe that abstract thoughts are as real as anything else in life.

balls in a barrel

Take 2 balls out, put 1 back, repeat. How many are left in the end?

The surprises didn’t stop there, and I really believe it was the clickers which made the session work. One of my favourite infinity questions is the ‘balls in a barrel‘ paradox which I learnt from the ever-wonderful Colin Wright. If balls numbered 1,2,3…etc are in a barrel, and at each time step two are taken out and one replaced, then after infinitely many time steps how many balls are left in the barrel? (a) None (b) All of them (c) One (d) Half of them, or (e) Not enough information to decide? As I’d hoped, people were very split on this question, with a small majority going for either (b) or (d). A lot of the students were very shy and I don’t think they would have volunteered an opinion without being able to do it anonymously. But once they saw that nobody else really knew the answer either, they were more inclined to speak up in favour of the option they had voted for, and we really got a great discussion going. (If you don’t know the answer yourself, have a good think about it before reading Colin’s article!)

My favourite clicker question was at the end, where I was basically proving uncountability. (I got the idea for how to incorporate this into Hilbert’s Hotel from an xkcd chat forum!) We’d got to the point where I’d done the diagonal argument and asserted the existence of an element which was not in the infinite list we had assumed contained every element. I asked them if they agreed with this. Usually when I teach this I just assume that the argument is crystal clear, and students usually nod and smile. This time, using the clickers, I found that the class were exactly split each way! Half agreed that the new element was definitely not in our list because of the way it was constructed, while half asserted that it must be in the list, because that’s what we assumed at the beginning. Once again we got to have a great discussion, examining our implicit assumptions and coming to the mind-bending conclusion that there are different sizes of infinity!

counting four fingers

Get people to vote using fingers, then hold them against their chest so nobody but you can see

I really hope that more schools and universities will start using this method of teaching. You don’t even need the fancy technology – some voting cards or fingers against chests are adequate for the purpose. It comes into its own with the quieter members of the class, giving them a voice they otherwise wouldn’t have had, and pushing weaker students to have opinions about things they’d otherwise not bother thinking about. There’s certainly a skill in asking the right questions and in not being scared to ask things which seem obvious to you. For example, my boss Toby asked his class  “2 ≤ 3, true or false?”. Stupid question, right? But half the class disagreed, asserting it was wrong because two is less than three, not less than or equal to.  Misconceptions occur at the deepest levels and we must work hard to root them out!

I’d love to hear other people’s stories of using clickers or other similar teaching methods. What have been your most surprising results?

New blogs

Braids on a blackboard

Topology on a blackboard, courtesy of Ryan Budney

Haggis the Sheep is back on the blog! And not just this blog, but two others that I’ve started up. The first is a photo-blog about mathematicians’ blackboards: What’s on my blackboard? Every week I want to upload a photo of a blackboard with some interesting or beautiful (or both!) maths on it, along with a short description of the mathematics. I think that there really is something wonderful about seeing the random scribblings of a great mathematician, or seeing the beautiful abstract pictures that we draw. Spread the word and get your local mathematicians to send me photos! (And it can be whiteboards too – I’m not discriminating!)

The other blog is to document a project that my friend Madeleine Shepherd and I have just got funding for. It’s called The Mathematician’s Shirts and is being funded by ASCUS, the Art Science Collaborative in Edinburgh. Madeleine and I beat off a lot of competition to secure the funding and we’re really excited about getting started on the project.

shirt

A humble shirt, but what will it become?

The idea here is that we are going to make a series of soft sculptures out of shirts to represent different mathematical concepts. For example, we could pass a shirt’s sleeve through itself to make a Klein Bottle, or we could sew successively smaller sleeves onto each other to make a fractal object. It was Madeleine’s idea to use the shirt, since it is an iconic piece of clothing, representing the formal and largely male world of mathematics. Perhaps some of the shirts will be donated by mathematicians themselves!

Here’s the timeline for the project. Over the next few weeks we’ll get together with local mathematicians to brainstorm ideas and make a concrete plan for between 5 and 7 sculptures. Then in September and October it’ll be time for the practical work to begin, actually sewing and making the sculptures in Madeleine’s studio. Finally, in November there’ll be an exhibition in a ‘non-standard’ location. That is, not a maths department or a science museum or an art gallery. We thought maybe we could have our exhibition in a shop window to entice passing shoppers.

If you have ideas on either of my two new projects, I’d be very glad to hear from you!