It’s not quantum physics

As we finish the Edinburgh & Lothians maths masterclasses (which have been a great success!) I am still following the efforts of another group of young people – those on The Young Apprentice.

The Young Apprentice candidates

These are apparently the best 12 young entrepreneurs of the country, chosen from many thousands to compete for Alan Sugar’s money to set up a new business of their own. Initially they were divided into two teams (boys and girls) and asked to choose names for their teams. Independently, both teams went for science-sounding names: Atomic and Kinetic. This bodes well, I thought. At the very least, it shows that young people view science as exciting and fast-paced; at best, maybe these future entrepreneurs have a deep appreciation for maths and science.

Ha! Wishful thinking.

Despite the Physics-sounding team names, only 2 of the 12 contestants are studying any kind of science, and in both cases it is Biology. 3 of them are doing an A-Level in Maths and a few more in Economics, but we’ll see soon enough that this implies nothing about their basic arithmetical skills.

Episode 1: The contestants have to make and sell their own brand of ice cream. Problems start early on, when they have to work out how many kilograms of ingredients are necessary to make so many litres of ice cream. Not just that, but they have to work out, given how many scoops they can sell in an hour, how many litres to make in the first place, and how much profit will accrue from various costings. The voice-over dubs this ‘A-Level Maths’. Hmm. Is multiplication and division really not taught before the age of 16 any more?

Sainsburys clearly need to employ someone with an A-Level in Maths in order to do their pricing...

But maybe the voice-over man is right. The boys have a team member with A-level maths, who copes fine with the numbers. The girls, on the other hand, have sent both their A-levellists off to do the design work, leaving chaos behind them. The remaining team think a gram is equivalent in weight to a litre. They think 3 x 4 is 28. “The real surprise here is that the team cannot add up, subtract, divide or multiply”, says a bewildered Nick Hewer, who is keeping an eye on proceedings. In the boardroom, the project leader defends herself saying “I only have a GCSE in maths”, to which Lord Sugar responds with “I don’t care if you only have air miles in maths – this is baby stuff!”

Episode 2: The contestants have to design a new product for the mother & baby market. Thankfully it’s more about market research and good pitching than making and selling this week, so no maths to worry about.

Episode 3: The contestants have to set up their own floristry businesses, designing and selling displays to corporate customers, as well as flogging bouquets on the street. Each team is divided into two, with some people staying behind to learn flower arranging and the others going to win deals with companies. Maths makes an early appearance, with one team leader choosing to keep one of the girls in the flower arranging group, despite her being a proven success at pitching to companies. The reason? Because she’s the only one who can do the maths in pricing the displays.

One of the boys, who had got the highest score in GCSE Economics in Northern Ireland, proclaimed “I’m not very good with numbers.” Prompted further, he said “There’s not a lot of numbers in Economics”, to which Lord Sugar replied “We’re not talking about quantum physics here, are we? We’re talking ‘this rose costs 40p, so 10 roses costs £4′.”

In the end though, the most frustrating thing for me about watching the show is that the team who gets it right every week is almost certain to lose the task! The teams who work hard to correctly do their costings, to make a good product and to provide good customer service always fall down because they haven’t overcharged for their product like the other team have. It is depressing to live in a world where being ruthless and greedy gets you further in life than being honest and intelligent, and even more depressing to think that young people will be watching this programme and learning exactly this lesson.

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!

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.

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.

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.

Dragon curves and other mathematical knitting by Pat Ashforth

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.

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.

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!

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?

Braided knitting

A few months ago a new sheep, Fernilee, appeared in my flat. (You may remember him from the New Year’s Eve party.) Luckily for him I’m not a very territorial sheep (except for the sofa, which is MINE) but I was a bit put out that he had a lovely hat and scarf and I didn’t. I know that it’s currently summer in Edinburgh, but that isn’t the point.

Seeing as Julia was totally bored after having finished her PhD, I asked if she would make me the scarf to rival all scarves. Not only should it be a warm and functional piece of winter knitwear, but it should embody some sort of mathematical principle so that I can continue inspiring my followers wherever I go. We brainstormed a few ideas: having a stripey scarf where the number of rows of each colour followed the digits of pi (e.g. see here), or having a hidden mathematical image knitted into the scarf (so-called illusion knitting), or having a braid pattern using cabling. The first idea seemed too easy and the second one quite hard, so we decided on a braid.

The braid pattern we chose

Julia had never tried cabling before and wasn’t sure how to design a pattern from scratch, so we decided to find a ready-made pattern for our first braiding attempt. Eventually we decided to go with this one modulo some modifications – changing the border to seed stitch to make it easier, and adjusting the cabling pattern to make the braid alternate (over-under-over-under). It’s a 6-stranded braid with no special mathematical properties (that I can see). In particular, it is definitely not the same as this commutative braid which my officemate Patrick and my old supervisor Andrew are working with.

Being mathematicians (and knot theorists!) definitely helped us to figure out how the cabling pattern worked. In a mathematical braid there are a number of strands running parallel to each other, and every now and then two adjacent strands are allowed to cross. If the strands are labelled 1 to n, then the crossings are denoted by a sequence of numbers, where i means that strand i crosses OVER strand i+1, and -i means strand i crosses UNDER strand i+1. For example, the braid below would be denoted by (1,-2,1,-2):

The braid 1,-2,1,-2

This is in some ways very similar to knitting braids, because in a knitted braid only adjacent strands are allowed to cross. The cable pattern denotes whether the crossing is OVER or UNDER by using F (‘front’) and B (‘back’). The first obvious difference between the maths braids and the knitted braids is that knitted strand-crossings are allowed to occur simultaneously between non-interfering strands. E.g. strands 1 and 2 can cross at the same time as strands 3 and 4 do. Mathematically it makes no difference, but aesthetically it is more pleasing to have simultaneous strand crossings.

The next similarity between maths and knitting is how we ‘add’ braids together. It is simply by putting them side-by-side, the second braid following on from the first. Similarly, the knitting pattern only gives the first 16 rows – the first ‘block’ – and then the braid is continued by placing these blocks next to each other along the scarf.

Braid theory can design paths for these robots so they don't crash into each other

Mathematically, braids are interesting because their addition has a lot in common with numbers. There is a ‘zero’ braid which does nothing when added to another braid – it is the braid with n strands running in parallel. There are also ‘inverse’ braids, which are like negative numbers in the sense that when you add a braid to its inverse you get the zero braid back. (Can you figure out the inverse braid to the (1,-2,1,-2) braid drawn above?) This additive structure makes the collection of braids into a group, and the braid group is of great interest to a lot of people in the world right now! Engineers use them for motion planning in robotics, cryptographers use them to design new codes and computer scientists are using them to design quantum computers.

Braids can also be turned into knots by joining the strands at the end of the braid back to the beginning. I think if I make another braided scarf I shall try to encode the braid for the knot 12a₆₃₁, which is the only knot in my thesis where I couldn’t decide if it was slice or not.

I will end this post by showing you some pictures of how Fernilee reacted to my beautiful new scarf. You can see that he was quite jealous!

Fernilee looks on from behind as I try on my new scarf...

We have a little chat...

He inspects the scarf and admires the mathematical braid

We make friends and play a game of Ninja Sheep!

If you’d like any more information or more photos of the scarf, please get in touch!

Bright Club

When I told friends I was planning to spend the evening watching university academics doing stand-up comedy, the response was a look of confusion and a placatory “That should be…interesting.”

Academics and comedians normally form completely non-intersecting parts of a Venn diagram. After all, what is funny about a gamma ray burst, tree conservation or crayfish? When was the last time a seminar on genetics cracked you up with laughter? To be fair, I do often hear mathematicians making jokes, but normally they are so obscure that only the 3 other people in the room would have a clue why they were funny.

So I really didn’t know what to expect when I went along to Edinburgh’s first Bright Club at the City Cafe on Blair Street. I certainly didn’t expect there to be a long queue at the door because all the tickets were sold out! After a tense wait I was relieved to find myself inside, albeit with standing room only. In a stroke of luck, I spotted my colleague and fellow tweeter Karon McBride who squeezed me in on the seat beside her. She explained that she was quite interested at having a go at the comedy herself and was excited to see how the first session went.

Well, I don’t think that any of you readers are going to be surprised when I say that it was a fantastic evening.  Steve Cross, the founder of Bright Club, came all the way from London to start the proceedings, and we had the enthusiastically foul-mouthed Susan Morrison as our compère for the night. The first academic was none other than fellow mathematics PhD student Hari Srithkantha, which I’m very proud of because it was me who encouraged him to sign up for Bright Club! Hari is already making a name for himself in stand-up, taking part in the Chortle Student Comedy Awards and playing gigs around Edinburgh. However, it was great to listen to him making his research (into gamma ray bursts) the butt of his gags, which I think is something he hadn’t tried before. Even more than that, it was great to find out what he was actually researching! He is probably only the second person (after Matt Parker) to use a graph to make the audience laugh.

Evil American crayfish, likened by Zara Gladman to Madonna

Of the other 7 academics, I don’t think any of them had tried stand-up before, so I was really really impressed with their efforts. Highlights for me were Dan Ridley-Ellis, who talked about the stiffness of wood but managed to avoid all the obvious jokes, Zara Gladman, a zoologist studying crayfish who wrote a song about how they are damaging our ecosystems, and Dan Arnold, who talked about uncertainty and ‘unknown unknowns’. As well as laughing for the whole two hours, I also felt like I learnt a lot about all the science topics on offer, and thought it was an unexpectedly brilliant way of doing public engagement with science.

It is also great to see the nationwide news coverage that Bright Club is getting. The BBC covered the story back in February and the Edinburgh Evening News wrote an article earlier this week. BBC Radio Scotland are running a piece on Friday at 13:15 as part of their Comedy Cafe and there is going to be a BBC Fringe show in Edinburgh on 24th August with all the academics doing the show again. So if you missed it then don’t worry, there will be a chance to catch up – but only if you’re quick! Tickets for the BBC show are only available until 8th August (despite what it says on the website) so make sure you sign up pronto!

Bright Club is going to be a monthly event in Edinburgh, so I look forward to seeing Karon and other academics getting on stage and making people laugh with their research. It’s going to take a lot more persuading to get me to think about having a go though!

Graduation

It’s been just over a week since graduation, and with the official documents in my hands I can finally feel confident at calling myself Dr Haggis.

The 28th June was an auspicious day for a graduation. Not only did the skies produce some amazing sunshine, but it was a mathematically interesting day for at least two different reasons.

The first reason, which actually got a decent amount of news coverage on the BBC and elsewhere, is that June 28th is Tau Day. There is a small but dedicated group of people who believe that pi is the ‘wrong’ number. Not that there is anything wrong with the number itself, 3.14159…, but that we should be using the ratio of the radius to the circumference of a circle, not the ratio of the diameter to the circumference.  So they define tau to be equal to 2 times pi (which is 6.28, hence June 28th!), and have even invented their own symbol:

Tau = 2 pi = 6.2831853...

To modern mathematicians, tau is a more natural constant to use than pi because a circle is defined as the set of points which are a fixed distance (the radius) away from a point. For any of you readers who have studied maths beyond AS level (or Highers), you will know that the standard unit of measure of angles is the radian, not the degree. One radian gives an arc on the circle which has the same length as the radius (see picture). Learning to use radians and converting to degrees can be the bane of a young mathematician’s life! How many radians are in a circle? It’s the circumference divided by the radius, which is 2 pi. So 360 degrees is 2 pi radians. That means we get horrible formulas, like a quarter of a circle being pi/2 radians, instead of what you would expect: pi/4. Using tau instead of pi, we do indeed get a much more intuitive measure for angles.

Still, despite all the arguments in favour of tau, I have to say that I don’t think it will ever catch on. Pi is far too engrained in our history and our culture. We made our choice over 2000 years ago and it’s too late to change things now!

A much better reason, I think, that 28th June is interesting is because it is the perfect day of the perfect month. A perfect number is one which is the sum of its divisors (not including itself). So 6 = 1+2+3, while 28 = 1+2+4+7+14. Perfect numbers, as you might imagine, are pretty rare: the next one after 28 is 496, while the next perfect year is not going to be until 8128. The ancient Greeks were the first people to pay attention to perfect numbers, and it was Euclid who noticed that the perfect numbers had the form 2^p−1(2^p−1), where p is a prime number. He proved that if 2^p-1 is prime then the formula 2^p−1(2^p−1) will always be a perfect number. However, it was not until Euler came along in the 18th century that the converse was shown: that every even perfect number comes from this formula. So every time we find a new perfect number, we find a new prime too!

Me and Nagyi

Graduation was also an exciting day because all of Julia’s family were able to come up and see it. This included her Hungarian grandmother Emily, who we all call Nagyi (which is Hungarian for grandma, pronounced nudge-ee). Although Nagyi could hardly speak any English, she loved the historic atmosphere of the graduations (in the beautiful McEwan Hall) and all the lovely people she met.

I don’t know why this is, but the Hungarian nation produces a surprising number of top-class mathematicians. So it wasn’t very hard to find one at the University of Edinburgh who was happy to chat to Nagyi during the afternoon reception. Our victim was the charming Tibor Antal, who works on modelling biological processes (such as cancer growths) and population dynamics. I hope that he got to tell Nagyi a bit about his research, but I suspect that he spent most of the conversation listening to her life story!

Hyperbolic crochet: add a new stitch for every 4

Another thing which surprised Nagyi was how much everyone appreciated her crochet work here. Before she came, I gave her the double challenge of making a Möbius strip and a hyperbolic disc. Although she made two beautiful models, she had absolutely no idea what they represented or why they were important. And yet, when we took them out at the maths reception, everyone was cooing over them in amazement!

To any other mathematicians who have grandmas: I suggest you get them knitting and crocheting and embroidering any mathematical objects you can think of! They really learn something new from it and have fun in the process. One day I’m still hoping to run an interactive maths knitting exhibition where anyone can come along and try it for themselves.

Hyperbolic disc, McHaggis and a Mobius strip

Thank you once again to everyone who sent messages of congratulation for the graduation, and to all my friends and family for being there on a great day. I wouldn’t have been there without you all!

New blogs

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.

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!

Doctor Haggis!

Dr Haggis, the knot surgeon. Not to be confused with real doctors who do surgery on people.

Well, one step closer to Dr Haggis at least! As you may have surmised from the comments and Twitter feed, I did indeed pass the viva yesterday, and have just some minor corrections to complete before I can be officially awarded my title. I promise to use it responsibly!

As expected, the defence was quite enjoyable, and it felt more like an extended seminar than an exam. We got off to a late start because the ‘Non-Examining Chair of the Examination Board’ (thanks Laura!) didn’t show up. Since the internal examiner Mark had never examined a thesis before, we needed a third person to keep an eye on him; eventually a substitute was found, and it was Chris Smyth, who is actually my second supervisor. I really hope there isn’t some rule against this which makes the viva null and void!

So after that brief faff I began proceedings by giving a little presentation summarising the aims, importance and main results of the thesis. Although intended to be only 15-20 minutes long, it was easily 45 minutes with all the interruptions and questions by Brendan and Mark!

I thought it would be a bit pointless summarising results which they already knew, having read carefully through the thesis, but of all the chapters I think it was my main results chapter which they hadn’t read in detail. Disappointingly, they didn’t spot the mistake in one of my big proofs, which I had frantically spent the last week trying (and eventually succeeding) to correct! It does make me wonder how many incorrect results actually manage to pass through vivas and referees, and how many never get noticed. Is it any better in the other science disciplines?

Talking about my own results was the easiest part of the viva. The kinds of questions I failed at answering were the ‘elementary’ ones. Results that are written in so many textbooks that you take them for granted without making the effort to understand the proof. And there were even a couple of questions that none of us, even the examiners, could answer! Hopefully I will sort these out in the coming week.

Brendan Owens, Julia and Mark Grant. I am wearing my lucky Seifert surfaces.

At this point I’d like to say a big thank you to Mark and Brendan for being such great examiners and for helping me to feel relaxed about the whole thing. They caught lots of my mistakes but also gave me much-needed encouragement that my results were important and interesting.

The viva was over 4 hours in the end (fairly long for a maths defence) and poor Andrew (my supervisor) was pacing the corridors ‘like an expectant father’ (in his own words). There were many sighs of relief when the examiners finally delivered the verdict, and much drinking of alcohol afterwards! Thank you also to all those who emailed, texted, tweeted or otherwise conveyed messages of congratulations, especially little sister Suzanne who seemed very emotional about it!

I now have 8 days to complete all corrections, get the thesis printed, bound and signed and handed in, so that I can graduate at the ceremony in June. Of course, if I miss the deadline it’s not the end of the world, as I can graduate in November instead, but all of Julia’s family are coming up to Edinburgh in the hopes of seeing some be-robed students, so I had better make the effort. It’ll be a tight deadline, but since the corrections are all pretty minor I think I can do it if I work hard.

And then I shall be free! I pledge now to

• Do more exercise
• Do more blogging
• Do more mathematical knitting
• Do more exploring of interesting places
• Do more exciting public engagement things
• Do more keeping up with poor neglected friends
• Do more cooking of healthy things
• Do more enjoying of the beautiful Edinburgh summer weather

But for now, I shall simply be doing more sleeping!  G’night all!

P.S. Congratulations are also due to my mathematical brother Mark Powell, who passed his viva on Monday, beating me by 2 days. Grr.

Judgement day!

Well, this is it. Half an hour to go until we begin the 3 hours that will judge the past 4.5 years of my life. Yes folks, it is my thesis defence!

It is pretty nerve-wracking for anyone to have their work closely scrutinised by a group of experts, but I think it is especially so for a mathematician. The standards of rigour and clarity are exceptionally high, and it is easy to work on a problem by yourself for a long time and not see subtle errors creeping in. Myself, I’ve not talked about my thesis since last summer, so I am expecting lots of corrections from my examiners. And to have to explain things in detail that I’ve never had to consider before.

Despite the nerves, I’m pretty excited too. As mentioned in a previous post, this is the one time in my life when I get to talk about my work to people who have spent a month reading it and trying to understand it. Perhaps we will even shed new light on some of the problems I failed to solve.

And on top of the fear and excitement is a sort of melancholy.  It’s the end of an era today; the end of being a student, the end of being a research mathematician (at least for the time being) and the end of thinking about all the problems in my thesis. I may go back and think about some of them again, but realistically this is unlikely to happen unless I am surrounded by other mathematicians interested in the same ideas.

Many thanks to all the people who have wished me (and Julia) luck via email, Twitter and Facebook. Your support means a lot, and I will report back with the result soon enough. Fingers crossed!

A week in the life: Friday

Gosh, I’ve not done very well in keeping up with this series of blog posts, have I? For the past week I’ve been caught up in the Edinburgh International Science Festival, helping to chair some talks and run an exhibition at the National Museum of Scotland about game theory. More on those in another post perhaps.

So, Friday. Although it’s been over two weeks since this particular Friday, I remember it very well. It was on this day that we were lucky enough to have a visit by the distinguished professor Eric Mazur.

Mazur & Me

Although Mazur is distinguished in his field of physics (lasers, semi-conductors, optical fibres), it wasn’t a physics lecture that everyone turned out to hear. Surprisingly (even to himself) he has become most well known for his radical teaching method, known as peer instruction. The talk that he gives about this is really fantastic, and I recommend that everyone watches it on YouTube.

It is ironic that Mazur should be touring universities around the world, giving lectures about peer instruction, when a fundamental tenet of the theory is that we shouldn’t be lecturing to students! The idea is that people don’t take in information when they are forced to sit and listen to something; they have to be doing and discussing the subject matter in order to really engage with it. This became especially clear when Harvard University physics students were given a simple exam which tested their basic understanding of Newtonian physics.

Big truck vs little car

For example, suppose that a heavy truck and a small car crash into each other. At the moment of impact, is the force of the truck on the car (a) larger than, (b) smaller than or (c) equal to the force exerted by the car on the truck? Think about this for a moment…….. Your intuition is probably telling you that the truck exerts a larger force on the car than the car on the truck. Yet anyone who can remember their high-school mechanics should know Newton’s 3rd law: that forces are always equal and opposite.

What was interesting was that Harvard physics students got this question wrong almost as often as the general public did. It’s an extremely strong indication that students are only superficially learning information, memorising things in order to pass an exam but not really internalising the concepts. We see this all the time with our maths undergraduates too. They can compute all manner of difficult integrals and solve complex matrix equations, but in the end very few know what the answers mean or why they are important. What actually is the determinant of a matrix, or what does it mean to have an infinite decimal expansion?

But if lecturing doesn’t work as a means of education, then what else can we do? Eric Mazur’s answer is peer instruction, which works something like this. The students read the material in a textbook before the lecture, submit a list of things they don’t understand to the lecturer, then during class the students work through questions designed to address and correct their misconceptions. Questions are often presented in multiple-choice format and students have the chance to vote on an initial answer before discussing with their peers and then voting again on a new answer. From research into this method, it seems that students teach each other a lot more effectively than a lecturer can. They understand each other’s problems and can more easily get to the heart of the explanation. And nobody can just sit and sleep through the lecture, because there is constant discussion of the material in the class.

At Edinburgh, we would love to try and implement this method with our first year undergraduates in September. The main difficulty in starting out is getting those multiple choice questions which can really change people’s opinions about a subject. What are the common misconceptions in maths? When does our natural intuition override the definitions we are given in lectures, like in the physics example before?

The only university which has really implemented teaching like this in mathematics is Cornell University, and you can take a look online at their list of ‘Good Questions‘. Here’s one to get you talking:

Was there a time in your life when you were exactly pi feet tall?

A good question, to me, is one where you have an instinctive immediate answer, but then when you think more carefully you really get to very deep questions about the subject.

Can a person ever be exactly pi feet tall?

For example, someone might immediately think that the answer to the above question is “no”, because pi is an infinite non-repeating decimal and nothing can be ‘exactly’ pi feet long. But then they might think, is it really because pi is irrational that the answer is “no”? Was there ever a time when they were exactly 3 feet tall? Or they might think, there was a time when they were less than pi feet tall and now they are bigger, so surely there must have been a moment (however brief) when they were exactly pi feet tall. Discussions of this will get to the heart of the real number system and questions of approximation, which are essential for anyone studying analysis to master .

I would be really interested to hear from anyone with opinions about this. Do you remember which concepts in maths you struggled with the most? When do you think you learnt your specialist subject: through listening to lectures, or at home with a textbook, or chatting to friends? Do you believe that peer instruction can work or do you think the system is fine as it is?

Send me your comments!