Travels in a Mathematical World Blog

Barriers to teaching

2012-02-20T09:25:00.000Z

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Reading around the Alan Turing Pardon

2012-02-17T00:08:00.000Z

I have a piece in this week's Pod Delusion episode 123 at 45:00 on the pardon for Alan Turing.

Here are links to some of the bits I talked about in this.

I spoke about concerns of overdoing the Turing celebrations, saying: what Turing did was brilliant, but we should celebrate what Turing actually did, not some imagined feats, and we should not forget others in doing so. You can read more about this and find out about the article which suggested that had Turing lived then Silicon Valley might have been started in the UK at 'Beware the Alan Turing fetish' by John Graham-Cumming.

Turing was convicted under Section 11 of the Criminal Law Amendment Act 1885. In 2009 Gordon Brown issued an official apology for the way Turing was treated. Read about the official Government apology in 'PM's apology to codebreaker Alan Turing: we were inhumane'. Read how the apology came about in 'How Alan Turing Finally Got a Posthumous Apology' by John Graham-Cumming.

Now there is a current e-petition calling for a pardon for Turing. John Leech MP issued an early day motion calling for this pardon. (I also mentioned the current e-petition calling for a pardon for Oscar Wilde.)

Asked a question in House of Lords, a Government Justice Minister said "a posthumous pardon was not considered appropriate". Read the text of Lord McNally's statement.

I've seen the refusal to pardon Turing described as "homophobic" and an "act of malice". Particularly, the complaint is that Turing is still seen as a criminal in the eyes of the law.

John Graham-Cumming on 'Why I'm not supporting the campaign for a pardon for Alan Turing', in which he writes about the Protection of Freedoms Bill, which "specifically allows for the disregarding of convictions under the old law that was used against Turing".

To honour Turing I suggested you might attend events under the Alan Turing Year banner, or donate to Bletchley Park's Action This Day! fundraising campaign.

This piece used audio from episodes 84 and 85 of the Pulse-Project Math/Maths Podcast.

A puzzle from James Grime about abcdef

2012-02-16T15:17:00.001Z

Today James Grime tweeted this question/puzzle:

Is there a six digit number abcdef such that the following all hold?

a+b+c+d+e+f = y
ab+cd+ef=10y
abc+def=100y

If not, show why not.

A little tweeting back and forth verified that "ab" means 10a+b not a×b.

If you want to have a go at this, don't read any further until you have!

First, rewrite the expressions so that both sides use standard arithmetic:

a+b+c+d+e+f = y
10a+b+10c+d+10e+f=10y
100a+10b+c+100d+10e+f=100y

I noticed that since a, b, c, d, e and f are all positive integers, so must y be. Then 10y must end in 0 and 100y must end in 00.

From 3, we see that in 100y only c and f contribute to the units, so c+f=0, or f=-c. See also that only b and e contribute to the 10s, so b+e=, or e=-b.
Since a, b, c, d, e and f are positive digits 0-9, the only values that satisfy these equations are c=f=0 and b=e=0.

From 2, we see that in 10y only b, d and f contribute to the units. Therefore b+d+f=0, or d=b+f. Since b=f=0, we know that d=0.

So solutions to this problem have a being any digit 0-9, b=c=d=e=f=0, and so y=a.

The answer isn't no, nor is it quite yes. These multiples of 100000 are not exactly interesting!

After I emailed Jim I was interested to see this solution, posted at The Wandering Monster, which took a substitution approach. On Twitter Dave Hughes gave his approach: "My solution was inelegant - I threw a C# program at it". How interesting to see how different people approach a problem!

Update (20 mins after posting!): Please check the comments for a caveat I've missed.

Things to do in London on a Tuesday

2012-02-15T08:51:00.000Z

Next Tuesday I will spend a day off in London. I am asking people to offer suggestions for things I could do with my time by adding pins to this Google Map: PR's Day Off. A few people have already added their suggestions but it would be great to hear more.

View PR's Day Off in a larger map

Load the map, then you need to log in with your Google Account then a big red edit button should appear and you can add a pin. The description I put on the map:

Ground rules:
Date is Tuesday 21st Feb.
Free or low cost. It's expensive just to get to London.
I have a zone 1 & 2 tube card.
Train times are non-negotiable (fixed tickets).
I am going on the Maths in the City tour.
All else is up for grabs.
Make suggestions for places to go and things to see by putting pointers on the map. Explain why and give timings and a link if appropriate.
For timed events: Remember to allow enough time for travel (for someone who doesn't know where they are going).
Please don't delete other people's entries. If you disagree, by all means leave a comment to say so but don't delete something you haven't put there.

If you can't get the Google Map to work please leave your suggestion in the comments of this blog post.

George and Julian

2012-02-13T10:15:00.000Z

Yesterday, the @mathshistory Twitter feed tells me, was the anniversary of the birth of Julian Schwinger (1918-1994), one of the great physicists of the 20th century. (Technically I queued this tweet up but there are a lot of days and a lot of mathematicians to remember...)

Schwinger is known to me particularly through his connection to the story of George Green. Green was a Nottingham mathematician who did work on electricity and magnetism (among other things) that, largely unrecognised in his lifetime, was discovered and brought after his death to further attention by William Thomson (later Lord Kelvin). The application of Green's work in 19th century science was impressive but it found a new legacy in the 20th century.

At the 1993 celebration in Nottingham of the bicentenary of Green's birth, Schwinger spoke about his use of Green's work (a talk written up as The Greening of Quantum Field Theory: George and I).

Schwinger's account is worth reading. He describes his use of Green's work first on microwave radar during World War II, then in the development of the microtron and synchrotron particle accelerators, and finally to solve a problem on quantum electrodynamics, work which earned him a share, with Sin-Itiro Tomonaga and Richard Feynman, of the 1965 Nobel Prize for Physics.

In the preface to his most famous work, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), Green had written:

Should the present Essay tend in any way to facilitate the application of analysis to one of the most interesting of the physical sciences, the author will deem himself amply repaid for any labour he may have bestowed upon it.

Schwinger's account helps us to understand how Green not only impacted the physics of his age, but how it continued to have impact beyond anything Green could have imagined.

Mathematicians are people too

2012-02-09T09:06:00.000Z

Last Saturday in the Telegraph there was a feature announcing the start of a numeracy campaign: Make Britain Count. This included an article by Rachel Riley about "the stigma around maths". She writes about the "image problem" of maths and numeracy:

I’m a blonde Essex girl, so I’m well used to being talked down to, but when I tell people I did a degree in mathematics at Oriel College, Oxford, I see their jaws hitting the floor. Mathematicians labour under a negative stereotype – older men in anoraks with beards and glasses. Maths isn’t sexy.

She talks about problems of attitude and relevance to the real world, and the need for creative teaching to

teach children number skills from first principles. They have to know the underlying “why” of maths, not just memorise the formulas.

Let's talk a little about the issue of the image of mathematicians. Last night on Twitter I was approached by user @philhumpo, a teacher from Exeter, with this query: "I need a 'top 5 crazy mathematicians' (duelling Romans, drowning kittens etc)."

This sort of thing concerns me. I wondered in what sense he meant "crazy". Mathematics can seem to have an association with mental illness in popular culture and so I'm naturally concerned if "crazy" is being handled sensitively. Also, many of the interesting historical anecdotes turn out to be false or exaggerated, an issue touched on in my previous post.

Thankfully, it was just an issue of the brevity of messages on Twitter. Phil explained the heart of the problem. It's the last day of term today and Phil has his class of 15 year olds for a shortened lesson. He has discovered many of them think "all mathematicians are grey suit baldies with social problems" and hopes to disabuse them of this view.

With the reference to duelling mathematicians, Phil is clearly aware of Évariste Galois, who clearly has a romantic and stereotype-breaking story. Ramanujan is another good story. You can find online biographies of women mathematicians - Ada Lovelace, Mary Somerville and Sophie Germain are typical examples, though there are many more.

I also wondered about more contemporary sources. Recently I came across a photo blog "This is what a scientist looks like" via the @HESTEM Twitter feed. A quick search reveals just one mathematician featured so far. As Phil put it "hmmm… not a duelling Frenchman but not a grey suited baldy that's for sure".

I recommended Katie Steckles' video Playing Games with Squares. Katie certainly doesn't fit the stereotype and the video shows her having fun with mathematics.

There are a host of careers profiles from a range of different people in the Maths Careers Career profiles, where just scrolling down the page gives an idea of some of the stereotype-breaking people involved with mathematics, and a similar list is available with the Plus Careers Interviews.

I am sure there are countless more examples of mathematicians breaking the mold - mathematicians really are people too! - and I've only had a quick think about it. Perhaps you can suggest your favourites in the comments.

Why do we enjoy maths history misconceptions?

2012-02-02T11:06:00.000Z

I don't think I have come to a conclusion from my previous blog post about historical accuracy and popularisation, though there were some interesting points in the comments (relating less to my comments about the 'errors which may be gently corrected' and more to the 'demands of the narrative').

George Jelliss and Thony C. both read the famously inaccurate Men of Mathematics by E.T. Bell in their youth and were inspired to mathematical lives as a result.

Will Daniels suggests I should hold different standards for different people, so those writing historical research are held to a higher level of accuracy than those writing for a popular audience. I'm not sure this feels right. Thony asks a really interesting question:

is it possible to achieve the inspiration generated by Bell's book and be historically accurate at the same time?

I think this is at the heart of the matter. If it is possible to inspire through popularisation while remaining completely accurate then I can safely hold everyone to this high standard. However, if inspiration requires a little showmanship, if telling a good tale means not getting lost in minor distractions and sub-clauses, then we have our double standard.

This brings me to a final, anonymous comment that includes the following statement:

I am a great believer in the wisdom of stories, regardless of their provenance. If some stories persist despite being disproven, there must be a reason.

My first reaction on reading this is that it is preposterous. If you start presenting stories you know to be disproven you are in the realm of historical fiction. Historical fiction is fine, but these are now just stories and have no place being presented as real accounts of historical mathematics and mathematicians. Then today I was struck by something relevant.

I was listening to Paul Dirac and the religion of mathematical beauty on the Royal Society Library podcast while wrangling with the washing machine. This recording, of a talk given in March 2011 by Graham Farmelo, covers the life of Paul Dirac. Farmelo talks about how Paul Dirac is considered to be the theoreticians' theoretical physicist, yet he had a very practical schooling and took a practically-focused engineering degree. Farmelo says (15:40):

Let's get one thing right, he was a very practically-minded person. Completely different from the image that he has among most theoretical physicists.

This is as so often the case; the established fact isn't just slightly wrong but completely wrong. This is the case in the story that Einstein did poorly at school, a misconception that Thony C. tells me is not as well known as I thought it was when I used it as an example in my previous post.

Do we really want to believe Dirac is a theoretician with no practical sense, that Einstein was a terrible student made good? Is there really some "wisdom" in these stories that causes them to "persist despite being disproven"?

Rather than necessarily being wise, I think we are drawn to certain types of story. The Dirac perception reinforces the view of a flawed genius; a theoretical physicist with no sense of the real world. The Einstein story perhaps speaks to a desire for the plucky underdog to win out in the end.

Aren't these classic Hollywood ideas? Do other common misconceptions fit into the Hollywood-style? (Galois' heroic struggle against the odds to invent Galois theory in a single night before the dual springs to mind. What others?) Do, in fact, stories that deviate from historical record and persist deviate when the story fails to fit a certain sort of narrative?

Perhaps more importantly, are there correct historical stories which fit a classic Hollywood narrative? (I'm thinking, for example, of George Green teaching himself advanced mathematics "in the hours stolen from [his] sleep".) Perhaps stories of this type are the key to achieving Bell-like inspiration while maintaining historical accuracy.

Puzzle from Maths Jam Nottingham: Kathryn's cube of cheese

2012-01-30T16:03:00.000Z

At Maths Jam Nottingham January 2012, Kathryn brought this puzzle.

Kathryn has a cube made of cheese. Her question is simple: What is the smallest number of tetrahedra (not necessarily regular) that you can cut the cube into, leaving no cheese left over?

If you think you've solved this, see the solution page below for a follow on question.

For occasional puzzles from Nottingham Maths Jam meetings are tagged so you can search for "mathsjam" and find them.

It is important in problem solving that you have an honest attempt before reading a solution. Once someone has shown you the solution you are forever robbed of the chance to have that experience (in future you will half-remember the solution rather than reason it out) so it is important that you attempt this puzzle before reading the solution. If you are ready, check out: Kathryn's cube of cheese solution.

N.B. I assume the puzzles written about are old puzzles. They are brought to Maths Jam meetings, or half remembered at the time, by attendees. If I have done something wrong by posting a puzzle here please tell me and I will be happy to correct the mistake.

Apparently Gauss got in this bar fight with Hilbert...

2012-01-27T08:21:00.000Z

The title is silly, of course, but is meant to refer to a problem with historical accuracy. I have had this blog post in draft for a long time and I am struggling to finish it. I would like to talk about an area in which I appear to have cognitive dissonance. I'm intending to ask a bunch of questions to which I do not have answers. I hope you will help me come to some.

I firmly believe that what is published on the history of mathematics should be correct. The history of mathematics is full of misconceptions and apocryphal stories and to propagate these is a terrible sin. Call this Principle A.

Now, from time to time I see someone who has had a good go at producing something on a historical topic which is mostly correct but repeats a few common errors. This work (or person) is then picked apart by those in the know, or the piece of work is roundly dismissed as entirely without merit. I've heard this in the case of very popular books - "it's written well and tells a good story but it has this fact wrong so nobody should ever read it".

I'm not talking about someone who copies wholesale from some website nobody has ever heard of without checking any of the facts. Nor am I talking about a serious academic history of mathematics work. Nor silly errors. I'm talking about cases where an enthusiastic amateur has put in the effort; they've read fourteen sources for a particular piece of information and when they publish it they are picked up for not having read the fifteenth - a recent research paper in a journal they can't access - which debunks the fact.

I believe popularisation is good. Mathematicians would do well to know more of the history of their subject. I value the use of history in teaching as a way to engage students with the curriculum. I also believe history can be useful in outreach, the use of engaging stories to bring in more people to study of mathematics or its history. When I see someone having an honest attempt at telling some historical story, and they have done a reasonable level of research, I think it is bad to tear them apart or dismiss their effort. Instead we should encourage their keeness and perhaps gently steer them towards a better understanding (and they, in turn, should be pleased to learn). Sometimes this might mean you overlook a series of small errors to work, for now, on the major one. Pointing out everything that is wrong with a piece of work in minor detail can be very discouraging and, since popularisation and keeness are good, we hope to encourage this person not put them off from trying again. Call this Principle B.

You see the problem? Principle A tells me nothing should be produced with errors, but Principle B suggests work with minor errors should be taken in good faith. Both cannot hold. This is particularly a problem when I might be the person naively committing the sin (as I will be more often than the expert spotting the error). The fear of what might happen makes me feel very uncomfortable and hesitant to publish content on history.

There is another issue running along with this one. Perhaps the minor errors were not through ignorance but by choice, either due to restrictions of the format (word count or time available for a performance) or out of an attempt to keep the momentum of a story without getting sidetracked. This is like a piece of historical fiction where a character's sister and cousin are amalgamated into one character because it would confuse the main thread to introduce a new minor character for some small interaction with the plot before they disappear. If the main story is basically being told correctly but a few peripheral details are being ignored or muddled to keep the momentum, is that a bad thing? We want an audience for our story, after all; is it possible that too much accuracy (or too many caveats) can make the story uninteresting?

This puts me in mind of a piece of advice I was once given about writing popular mathematics. I was told that nobody should write a popular mathematics book unless they are a researcher in the topic of the book. I don't agree with this at all. Sometimes the researchers are too close to the topic to explain it well, or to make it interesting, or perhaps there isn't a talented writer researching a particular area but it should still be popularised. I wonder if people hold the same view - people should steer clear of history unless they are professional historians of mathematics? Won't this lead to less history being told?

There are also cases where someone learns or remembers something, or builds confidence, as a result of a historical story. I can't think of a better example right now but say for example I meet a twelve year old who was really struggling with mathematics when they were eight until a teacher told them that Einstein had failed mathematics in school and gone on to be a great physicist. A lot of ability in mathematics comes from perseverance which comes from confidence. Was the person who told the eight year old this story to boost their confidence wrong to do so? (There are surely cases where less decidedly wrong misconceptions apply to more nuanced situations but this will do as a placeholder; please don't get too hung up on Einstein or my imagined twelve year old.)

I really don't know the answer to these questions. I am asking them here in the hope that you might share your views. I really am interested to hear arguments either way.

Puzzle from Maths Jam Nottingham: Jon's coloured balls

2012-01-25T08:56:00.000Z

At Maths Jam Nottingham January 2012, Jon brought this puzzle.

You have three pairs of coloured balls - 2 each of red, white and blue. Within each pair one ball is heavy and one is light but you do not know which. All three heavy balls are equally heavy and all three light balls are equal weight too.

Q: What is the minimum number of weighings needed to identify each ball?

A: 1 is fairly clearly impossible. 3 is trivial (weigh each pair separately). So, in order for us to have an interesting puzzle the answer must be 2.

Q: How can it be done in 2?

For a while Christian Perfect has suggested that Maths Jam local organisers write blog posts based on what went on at their meetings. Rather than write a wholesale account of what happened at each meeting once a month I have decided to drop occasional puzzles from Nottingham Maths Jam meetings into this blog. I will tag these posts so you can search for "mathsjam" and find them.

It is important in problem solving that you have an honest attempt before reading a solution. Once someone has shown you the solution you are forever robbed of the chance to have that experience (in future you will half-remember the solution rather than reason it out) so it is important that you attempt this puzzle before reading the solution. For this reason I will post the solution separately. I will post this as a 'Page' since I don't think they appear in the blog stream so you have a reduced chance of inadvertently stumbling upon it. Jon's coloured balls solution.

N.B. I assume the puzzles written about are old puzzles. They are brought to Maths Jam meetings, or half remembered at the time, by attendees. If I have done something wrong by posting a puzzle here please tell me and I will be happy to correct the mistake.

Maths Jam Conference talk write-ups

2012-01-23T08:05:00.000Z

I have attended the two Maths Jam conferences - 2010 near Stone and 2011 near Crewe. At each I gave two talks, one of each I have written up over at Second-Rate Minds, the mathematics writing blog Samuel Hansen and I share over at ACMEScience.com.

Most recently I wrote up my 2011 talk Why the hot light bulb annoys me. In this I describe a puzzle which annoys me and explore why this is the case through a couple of other puzzles:

The light bulb puzzle presents you with three switches, one of which controls a light bulb inside a closed room. You are permitted to flip switches as much as you like, then you must open the door and say which switch controls the light bulb.
You don’t seem to have enough information. You can flip one switch and open the door. If the light is on then you have found your switch. However, if the light is off you can’t tell which of the other two switches controls the bulb.

My first post over at Second-Rate Minds was a write-up of my 2010 Maths Jam conference talk Moving on a strange diagonal. In this, I describe a puzzle I have given to students and why I like what it reveals about their thinking.

Given a 4×4 grid of sixteen dots, draw six straight lines that form a continuous path passing through all of the dots. Here, continuous means you must be able to draw over your six lines in one go without taking your pen off the paper.
This task is easy to complete with seven lines and impossible with five. Six is where the interesting puzzle lies.

The way Second-Rate Minds works each of these posts was written by me with Samuel Hansen providing editorial direction.

Favourite popular mathematics books

2012-01-19T17:06:00.000Z

I consider popular mathematics writing to be a good thing. I even tried a little myself and would be keen to try more. I am not, however, an expert in this genre. I certainly read popular maths and science books as a teenager and I remember fondly, along with a couple of physics books and biographies, the mathematical stories told in James Gleick's Chaos, Ivars Peterson's The Mathematical Tourist and Simon Singh's Fermat's Last Theorem. I'm not sure this is sufficient qualification to have a strong critical opinion. I have a copy of Alex Bellos' Alex's Adventures in Numberland that I was bought last birthday and, although it is on the top of my pile and I feel sure I will enjoy this when I get chance (perhaps someday I'll spend a holiday not worrying about my PhD), I haven't quite got around to reading it.

This week Guardian Books offered Ian Stewart's top 10 popular mathematics books in which, the description promises, "the much-acclaimed author chooses the best guides to 'the Cinderella science' for general readers". Why Cinderella you ask? Stewart means this in the sense at the start of the story, "undervalued, underestimated, and misunderstood", and perhaps intends popular mathematics to take mathematics to the ball, saying:

Popular mathematics provides an entry route for non-specialists. It allows them to appreciate where mathematics came from, who created it, what it's good for, and where it's going, without getting tangled up in the technicalities. It's like listening to music instead of composing it.

It will be no surprise, after the opening paragraph, if I admit that I neither own nor have I read any of Stewart's choices. I've heard of several of them but by no means all. I was surprised by the inclusion of Newton's Principia. In the back of my mind I have collected the 'fact' (citation needed) that Newton is a difficult read and I felt this made it a strange choice against the aim to bring "the best guides to 'the Cinderella science' for general readers" (though I'm aware the description will have been added later, possibly without Stewart's knowledge). Stewart justifies its inclusion as "a great classic" saying that although this is "not popularisation in the strict sense", this "slips in because it communicated to the world one of the very greatest ideas of all time: Nature has laws, and they can be expressed in the language of mathematics" and claims "no mathematical book has had more impact".

On Twitter, Tony Mann confirmed my half-remembered notion that "Principia is hard, very hard. Even in English". As to the claim of impact, Tony suggested Stewart should have chosen the Latin version as having more impact. Thony Christie agreed this is "a very hard book to read and comprehend", though Christian Perfect suggested that he found the scans of Newton's college notebooks which were recently made available online to be "quite readable".

Reading what Stewart wrote about Newton's Principia and its impact in the history of science, I wonder if the book was chosen more to tell the story in the article than out of a serious suggestion that it might be read. Christian Perfect makes this point more generally about the list over on my Google+ page:

I think he's chosen 10 books about his favourite mathematical ideas rather than 10 books which most effectively communicate mathematical ideas to a member of the "populace".

To include a classic, I wondered if something like Euler's Elements of Algebra, which I had heard travels fairly well to a modern reader, might be a more appropriate choice. On my G+ page, Sarah Kavassalis suggested "one of Poincaré's popular books instead though, for readability".

I asked people for their thoughts on the list and what else they would include. It's quite noticeable that several respondents report not having read many on the list (the same is true of the comments under the original article). Alex Bellos, on G+ expands on this:

I guess there are two types of "popular" - 1) something accessible for people who know no maths and 2) something fun for the math literate. I'd say Ian's list is very much the latter. If a lay friend asked me for a maths book suggestion they might understand and enjoy, I would only recommend the first two on his list [Robert Kanigel's The Man Who Knew Infinity and Douglas Hofstadter's Gödel, Escher, Bach].

Given the popular medium and Stewart's introduction to the article, in which he talks about popular mathematics as "an entry route for non-specialists", it is strange to see the list being regarded in this way. There's nothing wrong with a list of fun books for maths folks, with something to surprise us rather than just the obvious choices, but if that was what was intended then this probably should have said so. I worry about someone using this list to build a 'must-read' list and perhaps being put off popular mathematics as a result.

I also asked for your suggestions and these follow. It may not be fair but I have listed these in the order they were suggested. I've included descriptions, except where stated these are those given on Amazon UK.

Thank you to everyone who played along with this little game. We've got more than ten and I can't vouch for which would suit "people who know no maths" or "the math literate", but I've enjoyed looking through the suggestions. Further suggestions are, of course, welcome via the comments.

Alex Bellos' Alex's Adventures in Numberland (US title: Here's Looking at Euclid)
Suggested by Vincent Knight and Singing Hedgehog on G+.

In this richly entertaining and accessible book, Alex Bellos explodes the myth that maths is best left to the geeks. Covering subjects from adding to algebra, from set theory to statistics, and from logarithms to logical paradoxes, he explains how mathematical ideas underpin just about everything in our lives.

Edwin Abbott's Flatland: A Romance of Many Dimensions
Suggested by Sarah Kavassalis ("very different approach to popular mathematics") and Singing Hedgehog ("strange since Ian Stewart wrote the follow up Flatterland!") on G+.

How would a creature limited to two dimensions be able to grasp the possibility of a third? Edwin A. Abbott's droll and delightful 'romance of many dimensions' explores this conundrum in the experiences of his protagonist, A Square, whose linear world is invaded by an emissary Sphere bringing the gospel of the third dimension on the eve of the new millennium. Part geometry lesson, part social satire, this classic work of science fiction brilliantly succeeds in enlarging all readers' imaginations beyond the limits of our 'respective dimensional prejudices'.

Ian Stewart's Cabinet of Mathematical Curiosities and Hoard of Mathematical Treasures
Singing Hedgehog, on G+, recognises that Stewart can't choose his own books for the list but would add Cabinet and Hoard, which he calls "fabulous repositories of interesting stuff".

A book of mathematical oddities: games, puzzles, facts, numbers and delightful mathematical nibbles for the curious and adventurous mind.

A new trove of entrancing numbers and delightful mathematical nibbles for adventurous mind.

Clifford Pickover's The Math Book
Suggested by Singing Hedgehog on G+.

Maths infinite mysteries and beauty unfold in this fascinating book. Beginning millions of years ago with ancient 'ant odometers' and moving through time to our modern-day quest for new dimensions, it covers 250 milestones in mathematical history.

Barry Mazur's Imagining Numbers: (Particularly the Square Root of Minus Fifteen)
Suggested by Singing Hedgehog on G+.

The book shows how the art of mathematical imagining is not as mysterious as it seems. Drawing on a variety of artistic resources the author reveals how anyone can begin to visualize the enigmatic 'imaginary numbers' that first baffled mathematicians in the 16th century.

Florian Cajori's A History of Mathematical Notations
Suggested by Singing Hedgehog on G+, who says this "covers the history of mathematics through the methods of writing it".

Described even today as "unsurpassed," this history of mathematical notation stretching back to the Babylonians and Egyptians is one of the most comprehensive written. In two impressive volumes--first published in 1928-9--distinguished mathematician Florian Cajori shows the origin, evolution, and dissemination of each symbol and the competition it faced in its rise to popularity or fall into obscurity.

Richard Elwes' Maths 1001: Absolutely Everything That Matters in Mathematics
Susan Turnbull insists this mustn't be forgotten over on G+.

Maths 1001 provides clear and concise explanations of the most fascinating and fundamental mathematical concepts. Distilled into 1001 bite-sized mini-essays arranged thematically, this unique reference book moves steadily from the basics through to the most advanced of ideas, making it the ideal guide for novices and mathematics enthusiasts.

William Poundstone's The Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge
Suggested by John Read on G+.

In The Recursive Universe, William Poundstone uses Conway's Life as a vehicle to explore complexity theory and modern physics. Poundstone demonstrates how simple rules can produce complex results when applied recursively and suspects our own universe was created in a similar manner. (Description source)

Ivan Moscovich's Super-games
Suggested by John Read on G+, but of which I cannot find a description.

Benoit Mandelbrot's The Fractal Geometry of Nature
Suggested by John Read on G+.

"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

John Allen Paulos' Innumeracy: Mathematical Illiteracy and Its Consequences
Suggested by John Read on G+.

Why do even well-educated people often understand so little about maths - or take a perverse pride in not being a 'numbers person'?
In his now-classic book Innumeracy, John Allen Paulos answers questions such as: Why is following the stock market exactly like flipping a coin? How big is a trillion? How fast does human hair grow in mph? Can you calculate the chances that a party includes two people who have the same birthday? Paulos shows us that by arming yourself with some simple maths, you don't have to let numbers get the better of you.

Martin Gardner's Mathematical Puzzles and Diversions
Suggested by John Read on G+ who says this is "the first I bought and the one I go back to most" but I can't find a cover blurb description of this.

Marcus Du Sautoy's The Music of the Primes: Why an unsolved problem in mathematics matters
Suggested by John Read on G+.

In this breathtaking book, mathematician Marcus du Sautoy tells the story of the eccentric and brilliant men who have struggled to solve one of the biggest mysteries in science. It is a story of strange journeys, last-minute escapes from death and the unquenchable thirst for knowledge. Above all, it is a moving and awe-inspiring evocation of the mathematician's world and the beauties and mysteries it contains.

Ian Stewart's Game Set and Math: Enigmas and Conundrums
John Read on G+ says "I'd also pick an Ian Stewart - probably Game, Set and Math". Again, I can't find a description.

William Cook's In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation
Mitch Keller on Twitter notes that only one book on Stewart's list focuses on a specific problem and suggests this as another.

What is the shortest possible route for a traveling salesman seeking to visit each city on a list exactly once and return to his city of origin? It sounds simple enough, yet the traveling salesman problem is one of the most intensely studied puzzles in applied mathematics--and it has defied solution to this day. In this book, William Cook takes readers on a mathematical excursion, picking up the salesman's trail in the 1800s when Irish mathematician W. R. Hamilton first defined the problem, and venturing to the furthest limits of today's state-of-the-art attempts to solve it.

On G+ Alex Bellos recommended the following three for an accessible list for "people who know no maths", saying "the challenge when writing a maths book is to find a strong narrative - and these three books do it better than any others".

Simon Singh's Fermat's Last Theorem: The story of a riddle that confounded the world's greatest minds for 358 years
Recommended by Alex Bellos on G+.

The extraordinary story of the solving of a puzzle that has confounded mathematicians since the 17th century... A remarkable story of human endeavour and intellectual brilliance over three centuries, Fermat's Last Theorem will fascinate both specialist and general readers.

Apostolos Doxiadis and Christos Papadimitriou's Logicomix: An Epic Search for Truth
Recommended by Alex Bellos on G+.

This brilliantly illustrated tale of reason, insanity, love and truth recounts the story of Bertrand Russell's life... An insightful and complexly layered narrative, Logicomix reveals both Russell's inner struggle and the quest for the foundations of logic. Narration by an older, wiser Russell, as well as asides from the author himself, make sense of the story's heady and powerful ideas. At its heart, Logicomix is a story about the conflict between pure reason and the persistent flaws of reality, a narrative populated by great and august thinkers, young lovers, ghosts and insanity.

Apostolos Doxiadis' Uncle Petros and Goldbach's Conjecture
Recommended by Alex Bellos on G+.

Uncle Petros and Goldbach's Conjecture is an inspiring novel of intellectual adventure, proud genius, the exhilaration of pure mathematics - and the rivalry and antagonism which torment those who pursue impossible goals.

For a list of "something fun for the math literate", Alex recommended the following three.

Petr Beckmann's A History of Pi
Recommended by Alex Bellos on G+.

The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress -- and also when it did not, because science was being stifled by militarism or religious fanaticism.

Tobias Dantzig's Number: The Language of Numbers
Recommended by Alex Bellos on G+.

A new edition of the classic introduction to mathematics, first published in 1930 and revised in the 1950s, explains the history and tenets of mathematics, including the relationship of mathematics to the other sciences and profiles of the luminaries whose research expanded the human concept of number.

Paul Hoffman's The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth
Recommended by Alex Bellos on G+.

The biography of a mathematical genius. Paul Erdos was the most prolific pure mathematician in history and, arguably, the strangest too.

For this group, Alex also recommends "the complete works of Martin Gardner".

James Gleick's Chaos and The Information
Recommended by Alex Bellos on G+. Alex says these are between the two lists as they are "both utterly brilliant but might lose the casual reader in parts".

Chaos: This book brings together different work in the new field of physics called the chaos theory, an extension of classical mechanics, in which simple and complex causes are seen to interact. Mathematics may only be able to solve simple linear equations which experiment has pushed nature into obeying in a limited way, but now that computers can map the whole plane of solutions of non-linear equations a new vision of nature is revealed. The implications are staggeringly universal in all areas of scientific work and philosophical thought.

The Information: We live in the information age. But every era of history has had its own information revolution: the invention of writing, the composition of dictionaries, the creation of the charts that made navigation possible, the discovery of the electronic signal, the cracking of the genetic code.
In The Information James Gleick tells the story of how human beings use, transmit and keep what they know. From African talking drums to Wikipedia, from Morse code to the ‘bit’, it is a fascinating account of the modern age’s defining idea and a brilliant exploration of how information has revolutionised our lives.

E-Learning in Mathematical Subjects

2012-01-17T11:00:00.000Z

In 2005 I was asked to attend a meeting at Nottingham Trent University with some fellow PhD students. I explained my topic, e-learning in university mathematics, to one student who said, "oh, you should go and talk to Mike in Physics; he's interested in that sort of thing". When I found him, it turned out Mike had a little interest in primary school teaching but he said "oh, you should go and talk to Dave downstairs; he's interested in that sort of thing".

Feeling like I was on a wild goose chase, I went downstairs and knocked on the relevant door. Dave turned out to be Dr. David Fairhurst, a physicist who had recently moved to Nottingham Trent University. Dave was indeed interested in university education and after a quick chat we agreed there are lots of subjects all trying to deliver mathematical content through electronic means who might benefit from getting together and a seminar series might be useful. I was keen on this; as a PhD student I was encouraged to attend departmental seminars but I hadn't even managed to understand the titles of any so far.

We booked a room and sent an email around whoever we could think of as an invitation to a general discussion on setting up a seminar series. Thirty or so people turned up and someone suggest Trevor Pull could give the first talk. Over the next three-and-a-bit years we held a total of 25 meetings which were attended by teachers of mathematicsfrom subjects such as mathematics, statistics, physics, chemistry, biosciences, environmental sciences, engineering, computing, social sciences, business and economics, as well as researchers from both computing and education and university learning technology developers. It was really pleasing to meet all these people and see how mathematics is taught in nearly every academic school in the university. We were joined in the organisation by Pete Bradshaw from the School of Education.

In 2006 we had a talk by Steve Maddox about his work supporting two blind physics students at the University of Nottingham. This was excellent and I felt a great sense of loss at having only made this available to the twenty or so people in the room who saw it live. Steve did write his talk up for MSOR Connections, but this only appeared a whole year later because these things take time. I applied for funding from the Institute of Mathematics and its Applications (I was a member but this is long before I worked for them) and received £600 towards speakers' expenses and refreshments at meetings in exchange for recording the talks and making them available online. The Maths, Stats and OR Network (long before I worked for them too) were kind enough to host the large video files on their web server.

For the first two talks I borrowed video cameras from the computing department and recorded and edited these myself. I wrote an article in MSOR Connections about this experience: A quick and easy (rough and ready) method for online video. From 2007 onwards, much to my relief, these were recorded and edited professionally by Chris Shaw of Nottingham Trent University.

Many of the seminar videos cover teaching, learning, assessment and support using specific technologies – wikis, podcasting, Logo, interactive whiteboards, GeoGebra, VLEs and interactive voting systems – a series of talks related to accessibility, particularly access to mathematics by students with visual impairments, and several related to more general pedagogy such as designing effective online questions and relevance of learning styles to e-learning. In the end we recorded fourteen talks as videos. These have been available for a while on the ELMS website and have been downloaded quite a few times (by unique ip addresses: min 30; max 176; mean 51; SD 38; median 37). Now I have transferred the videos also to a YouTube playlist. The website has downloads relating to talks (such as slides) where these are available.

I wrote updates on the availability of new ELMS talks in Mathematics Today (43(3), p. 85; 43(5), p. 164); 45(1), p. 11) and MSOR Connections (7(2), pp. 49-50; 7(4), p. 43; 9(2), pp. 29-30), and a final grant report in Mathematics Today (46(6), pp. 287-288).

ELMS seminars stopped when I moved to full time employment and couldn't get to Nottingham Trent very often to organise them. I'm really glad for this experience. I met a lot of interesting people doing these seminars and later ran workshops for the MSOR Network's Accessing MSOR group, staff development seminars for the School of Mathematical Sciences at the University of Nottingham when I worked there and in my current job for the MSOR Network we run seminars and workshops. ELMS was the experience that demystified this process for me and that alone was incredibly useful.

What is mathematics?

2012-01-12T15:53:00.000Z

This morning on Twitter Tony Mann asked the question: "This morning's class is "What is Mathematics?" Answers in a tweet please." Answers were collected via the #MATH1103 hashtag.

Some of the answers were what you might expect: patterns, abstraction, order.

Stuart Ravn sent a series of tweets giving his views:

Math is everything you can do with the abilities to count and deduce. There's literally no end to the fun you can have. No joke.
Math is the only thing which is truly universal; it underlies and makes it possible to understand and communicate with everything.
Everything, immediately or ultimately, is mathematical and arises from mathematics.
Ask yourself what isn't mathematics, and try to prove yourself right.
Noam Chomsky said of love, "I can't tell you what it is, but life's empty without it." The same is viscerally true of mathematics.
I don't just have enthusiasm for maths. I love it. It's the closest thing to my heart after my family. I'm emotional about it.

Noel-Ann Bradshaw noted that "What is mathematics?" is the name of "an excellent book by Courant & Robbins revised by Stewart". I have the tenth printing from 1960. Although this has a lot to say on the subject, it opens a discussion of historical development with:

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasise different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

I was interested in a related question: When does mathematics become something else? At some point some topic is clearly applied maths and at some point it is physics, astronomy, engineering, economics, computer science, biology, and so on.

We struggle with this a little on the Math/Maths Podcast, where we try to report news from mathematics and its applications. On Twitter I said that I think I tend to stray a little further from that which is unambiguously mathematics than does Samuel Hansen. We both report applications but I think mine are often more tangential than Samuel's. This was quite noticeable on episode 80 this week when Samuel picked me up on an astrophysics story I was defending as involving statistical models. He said:

This is 'Math/Maths', not 'Stat/Stats', and it's definitely not 'Astronomy/Astronomies'. I'm assuming you put 's' at the end of every single science - you have 'Chemistries' and 'Physicses', right?

and later

Now you've turned us into 'Geology/Geologies'.

Answering my question on Twitter, Samuel said: "if an application has been around long enough to have own name, Physics astronomy or thermodynamics. It's not math". I don't fundamentally disagree with this and some disciplines, notably computer science, were born this way. However, Sharon Evans made a very practical (if teasing) counter-point: "so it's only maths if it hasn't got a name? You're not leaving much to report on in [the podcast]".

Clarissa Wornack replied to say "Well, when you start writing code; it's IT/software eng/comp sci; if you create something that is a material object; it's eng" and Charles Brain said "Applied maths becomes Engineering when it hits the real world and money becomes part of the equation!" I don't particularly agree with these. I know people who use high end computing to do mathematics and just because they are using computing as a tool (and writing bespoke code) this doesn't mean they are doing computer science research. I also don't agree that it stops being applied maths when it creates a material object. Defining mathematics as that which doesn't involve the real world or money seems very self-defeating.

Felipe Pait offered this definition: "Applied math interests mathematicians and non mathematicians. Otherwise it's pure math, or pure engineering. Math stops being applied math and becomes pure physics when it doesn't interest mathematicians. An operational definition."

There's something in this. When I think about "what is mathematics?" I am really thinking "what can mathematicians do?" I am particularly interested in what university mathematics graduates might become and would like this to be a broad as possible. I meet a lot of mathematics researchers working in different application areas. For example, back when I was doing the Travels in a Mathematical World podcast for the IMA I spoke to Paul Shepherd. I am much more naturally inclined to consider Paul a mathematician working in architecture than an architecture researcher who once did a maths degree. By extension, I am happy to include Paul's use of geometry in architecture as part of mathematics than to exclude it.

Daniel Colquitt suggested "a lot depends on the user" and "in many cases, the distinction is arbitrary". I think this may be the wisest view on the subject I have heard. From my point of view I am biased towards including topics on the edge within mathematics rather than excluding them, and maybe even collecting a little of the host subject along with them. I would rather cast the arbitrary net as wide as possible.

Have you used maths in the news in school?

2012-01-09T15:11:00.000Z

Later this year I am to give a session at a teachers conference on using maths in the news for enriching school maths lessons.

In my session, I intend to go over some recent maths news. I would also like to give some real examples of teachers having used some news in class.

Samuel Hansen and I keep track of mathematics news and mathematics in the news for our podcast. I am aware that people have written in from time to time to say they have used some bit or another in class but I haven't recorded these instances.

My plea, then, is this: Whether from the podcast or not, please could you send me your examples of how you've used current events in mathematics class for enrichment? I'd like to know what the news story was, what you did and how it worked.

You can leave a message in the comments of this post or send me a message various ways that are listed on the contact page of my website.

Thank you!

Card trick video from Christian Perfect

2012-01-04T14:26:00.001Z

A while ago Christian Perfect suggested the monthly local Maths Jam organisers might write up what happens at Maths Jams to their blogs so others can get a feel for what goes on. I regard this as a good idea I haven't got around to yet.

Luckily, Christian has just made a video showing a card trick we have played with at the Nottingham Maths Jam, so that makes this an easy post!

I was shown this trick by Matt Parker in a hotel bar in Coventry, who refused to say how it works. I went to the Nottingham Maths Jam in November 2011 having worked out how to do the trick but having spent no time at all considering how it might work, saving this for Maths Jam. I showed John Read, Kathryn Taylor and Sharon Evans and together we worked out the details given in Christian's video.

I made a joke on Twitter based on Gauss' reaction to Bolyai's work on non-Euclidean geometry: "Enjoying video by @christianp. However, 'to praise it would amount to praising myself' ;)". Gauss is reported to have written to Bolyai's father:

To praise [Bolyai's work] would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.

while privately writing to a friend to say:

I regard this young geometer Bolyai as a genius of the first order.

Of course, by invoking the former I meant to imply the latter. Perhaps a more suitable quote might be that of Kelvin, having first read George Green's Essay on electricity and magnetism:

I have just met with Green's memoir, which renders a separate treatise on electricity less necessary... I have, most unwittingly, trodden almost exactly in his steps as far as regards electricity.

I'd say playing around with tricks and working out how they work is a very Maths Jam activity so anyone considering attending one should regard this as very much the sort of thing that happens at a Maths Jam. Find your local one, or set one up!

300 posts later, who is Peter Rowlett?

2012-01-02T11:45:00.002Z

This is the 300th post to this blog. At 100 posts and 200 posts I paused for a recap of my current circumstances. This 300th post coincides with the change of calendar year, which seems to bring out a great deal of reflection from people. Nevertheless, I will try not to get too mushy on you!

When I started this blog in February 2008, I had recently begun work as University Liaison Officer for the Institute of Mathematics and its Applications, and decided to blog about my travels around the UK talking to university student groups about why they should join the IMA. After 100 posts, in March 2009, I had recently started working alongside the IMA job in e-learning for the School of Mathematical Sciences at the University of Nottingham.

By my 200th post, in June 2010, I had reduced my hours with the IMA to continue to work for Nottingham. My role title at Nottingham had changed from e-learning to Technology Enhanced Learning and a change of emphasis, as I saw it, from being a teaching support person acting through technology to a tech support person who dealt with teaching meant that I was feeling much less well placed. Shortly afterwards, in August 2010, I finished at Nottingham and, extremely sadly, with the IMA to move full-time to the University of Birmingham. There I work on the Mathematical Sciences HE Curriculum Innovation Project for the Maths, Stats and OR (MSOR) Network as part of the National HE STEM Programme.

The National HE STEM Programme is a major higher education intervention seeking to enable HE to engage with schools, enhance curricula, support graduates and develop the workforce. My part is focused around curriculum development in the mathematical sciences. A major part of this work had us running the HE Mathematics Curriculum Summit, an event this time last year that brought together those with an interest in mathematics teaching at university whose priority recommendations we are acting on in a series of curriculum innovation projects this academic year.

What of the future? The National HE STEM Programme is a three-year initiative which finishes on 31st July 2012 and the Higher Education Academy has withdrawn funding for Subject Centres like the MSOR Network, so my job will end with no chance of follow on work. Of course this means I am quite preoccupied with worries about income in the latter half of 2012. I have a strong interest in teaching and would love it if someone would employ me as a mathematics lecturer. I think my CV is strong for curriculum development aspects and schools outreach but many lecturing posts are really about serious mathematics research, while my research is in the curriculum development aspects of teaching, learning, assessment and support. Even for those few that aren't, the number of candidates applying for jobs now means that, while I have some relevant teaching experience, my lack of mathematics PhD means I am not at the top of the pile. I believe I would make a good lecturer, strongly interested in pedagogy (as it could improve student learning and the student experience, rather than as a philosophical pursuit), and that I would enjoy such a role. I just need to convince someone else of this, or stop barking up this tree and find something else to aspire to do.

Outside of work, I remain registered for a PhD in e-assessment in mathematics, which I must complete by July 2013. I think this is on track as it moves into a final experimental phase.

At 200 posts, I had recently started a weekly mathematics-based conversation with Samuel Hansen of ACME Science. Well, we've just published the 79th almost-weekly episode of the Math/Maths Podcast, which was a review of the year 1811 (not wanting to merely rehash 2011). Samuel and I have started a shared blog for writing practice over at Second-Rate Minds. My write-up of my 2010 Maths Jam Conference talk about a simple puzzle and what I think it can reveal about student thinking got a lot of attention and I am pleased with a piece I wrote reflecting on Hardy's Apology. I have also been editing posts written by Samuel, which has been an illuminating experience.

I no longer work for the IMA but I remain a member (MIMA) and have a volunteer role on the committees for the East Midlands Branch and the Early Career Mathematicians Group. Having been co-opted to Council of the British Society for the History of Mathematics at my 200th post, I have since been elected to Council and continue to serve in this voluntary role. I remain a STEM Ambassador and contributed a mathematics stall to the East Midlands Big Bang STEM Festival.

Nobel prize for mathematics

2011-12-28T08:45:00.000Z

There's no Nobel Prize for Mathematics

This is a common statement. I've certainly used it myself. Recently it occurred to me to be annoyed with this.

Nobel Prizes are awarded in physics, chemistry, medicine, literature, peace and economics, but not mathematics.

On the other hand, mathematics is widely applicable and I think I could convince you it is certainly used in physics (career), chemistry (career), biology (career), medicine (career) and economics (career). (Links to the excellent Plus Magazine and Maths Careers.) The case for literature and peace might be a bit harder to sell. But even without these two we still have a majority.

So perhaps from now on I will try to remember to say:

Most of the Nobel Prizes are for Mathematics¹

[1. there is a fallacy here: for example, saying that some mathematics can be applied to economics does not mean that all economics involves mathematics. But, shh!]

Christmas presents

2011-12-26T10:58:00.000Z

What did I get for Christmas (mathematically, at least)? My big present was an Acme Klein Bottle, whose website claims it to be one of "the finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions". This is a 3D representation of a 4D Klein bottle; a pale shadow, of course, although the cheerful and entertaining information leaflet that came with the bottle claims this an advantage: "You can actually hold an Acme Klein Bottle in your hand. Those highfalutin' 4-dimensional ones can only be held in your mind". Here is a photo of mine:

I also got a set of physical puzzles which are nice to have a supply of. I find I am sometimes laying puzzles on a table for students to play with and solid, physical puzzles, while perhaps not the most mathematically interesting, are certainly an attractive draw. People can't resist picking up and playing with wooden blocks, it seems!

I got a copy of The Great Mathematicians by Raymond Flood and Robin Wilson, a past and present President of BSHM, which claims to present "mathematics with a human face, celebrating the achievements of the great mathematicians in their historical context". You can watch a lecture given by Raymond and Robin at the launch of the book at Gresham College.

As a bit of Christmas day craft, inspired by the escapades of the Manchester Maths Jam, we made dodecahedron star lanterns. Unlit, these are like this:

And lit they come alive like this:

Merry Christmas everyone! What mathematical presents did you receive and what mathematical activities did you do this year?

Pictorial proofs

2011-12-20T15:59:00.001Z

I received this message from Alan Stevens, Nottingham Maths Jam attendee. I am putting it here so readers of this blog and the other Maths Jams might consider the topic as well.

Although I won't be able to make the next MathsJam at Nottingham I've thought of a theme you might like to consider. I don't know if you have themes, but, if you do, how about "Pictorial proofs and derivations"?

Probably the most obvious pictorial proof is of Pythagoras's theorem (in fact there are probably several such). Do your mathsjammers know of any more?

I thought of this while viewing James Tanton's YouTube channel, where he has a very pictorial way of looking at maths, including a very nice pictorially based derivation of the geometric series 1/3 + 1/3^2 +1/3^3 + ... = 1/2.

If you haven't seen someone in a t-shirt displaying a pictorial proof of Pythagoras, you haven't been going to the right sort of conferences! The James Tanton video reminds me of a pictorial demonstration of summing 1/2 + 1/4 + 1/8 + ... which I used in a lecture, after Zeno's paradox of Achilles and the tortoise, when I was trying to get across the idea of an infinite series summing to a finite amount. Shading half a square, then a quarter, then an eighth, and so on it looks like you will eventually shade the whole square and nothing more, a useful illustration that the series converges to one. In fact, I repeated this in the micro-teaching session of my Postgraduate Certificate in Higher Education course and a screenshot of the slide after I drew on it using the interactive whiteboard is below. I'm not sure if this constitutes a proof, though.

Why I like some bad maths stories

2011-12-19T10:11:00.000Z

My two most recent posts here have been about a story reporting a coincidence as more exceptional that it is and 'bad maths' reported in the media. Both are examples of mathematical stories being reported in a way that is not desirable. Somehow, though, I like the whist story and dislike the PR equations. I have been thinking about why this might be the case.

The PR-driven, media-friendly but meaningless equations from the first article are annoying because they present an incorrect view of mathematics and how mathematics can be applied to the real world. Applications of mathematics are everywhere and compelling, yet the equations in these sorts of equations seem to present little more than vague algebra. The commissioned research with seemingly trivial aims I find more difficult because, as commenters on that article pointed out, it is really difficult to decide what is trivial. Still, reporting that a biscuit company has commissioned research into biscuit dunking is either meaningless PR or else a matter of internal interest, and certainly nothing like what I expect mathematicians do for a living.

Coming back to our Warwickshire whist drive: what do I like about this story? It too presents incorrect information about mathematics and the real world, claiming that the event, four perfect hands of cards dealt, is so unlikely that it is only likely to happen once in human history (and it happened in this village hall!).

I think the difference is that the mathematics used, combinatorics and probability, appear to be correctly applied. The odds quoted, 2,235,197,406,895,366,368,301,559,999 to 1, are widely reported and I see no reason to doubt them.

The problem, then, is one of modelling assumptions. Applying a piece of mathematics to the real world involves describing the scenario, or a simplified version of it, in mathematics, solving that mathematical model and translating the solution back to the real world scenario. In this case, the description of the scenario in mathematics assumes that the cards are randomly distributed in the pack. This modelling assumption, rather than the mathematics, is where the error lies.

The result is still a bad maths news story, presenting a mathematical story as something other than what it is, but while the PR formulae are of little consequence, this incorrect application of a correct combinatorial analysis is something we can learn from.

'Bad maths' news stories

2011-12-13T10:29:00.000Z

On the Math/Maths Podcast, we frequently cover 'bad maths' stories. A recent example was the bobbing apples story we spoke about in episode 71: Halloween Fruit Special. This proposed a "mind-bogglingly complicated equation", provided by a supermarket, for finding the perfect bobbing apple:

D = 3 x (2 + T^2) x M / (10 x T), where D is diameter, T is typical texture of an apple, and M is average mouth size.

Along with another formula from a rival supermarket (I reproduce this as reported; I imagine the second equals is supposed to be a plus):

B = (BU + S) x (C = BI), where B is bobability, BU is buoyancy, S is size, C is colour and BI is biteability.

A headline like "Mathematician finds formula for the perfect bobbing apple" is a tell-tale sign of bad maths in the news. Or is it? Actually there are several types of story that all appear under this same style of headline.

A couple of years ago Simon Singh launched a mini-campaign against 'phoney formulae' being reported in the news. Simon describes being asked by a PR company for an equation to say that the perfect shopping day coincided with the launch of a shopping exhibition, saying to the PR company,

I would engineer the equation so that the graph peaked on the day you require. There would be no real science behind the equation, but it would look sensible and convincing.

and getting the response,

Your ideas and formulas are perfect and exactly what we are looking for and it would be great to confirm you working with us.

As well as calling them "absurd PR equations", Simon also gave a more serious warning, saying this,

demeans mathematics and science by giving the impression that academics waste their time on frivolous topics and are willing to come up with the appropriate answer if someone is prepared to pay them enough money.

A 2004 BBC article, "Formula for the perfect formula", claims the origin of this media fascination with formulae was a piece of work by Len Fisher on how to dunk biscuits, sponsored by a biscuit company. What is described in that article is, I think, something subtly different from what Simon is railing against.

Len seems to be taking corporate sponsorship to do real experiments (the original article describes a "two-month investigation"), just that they are experiments with fairly trivial goals. The article describes a scientist who is aware that newspapers just want "to make a story look scientific" but motivated by a desire to communicate science to the public through any means available. In the case of the biscuit dunking, Len is quoted describing the physical processes and saying: "As with most things in physics, we can write equations which govern this".

I doubt the biscuit manufacturer cared in advance how long a biscuit should be dunked for (though they are presumably interested in getting their brand in the news) and Len claims to be doing real science to answer the question. I have also heard people describe being approached by PR companies with ready-made meaningless formulae to which the approachee is asked to put their name, on behalf of some brand. This seems similar to but slightly different from Simon’s case, where he is apparently asked to manufacture a realistic-looking piece of pseudo-mathematics to back up a pre-determined conclusion, or Len’s, where a brand name is associated with some trivial piece of research.

Simon also speaks about the danger of confusing real research with these PR equations. In Math/Maths 52: World's Smallest Klein Bottle we covered side-by-side the formula for the perfect cup of tea and that for the perfect golf putt. The former appears to be just PR for a milk company but the latter, although you could imagine a headline like "Mathematicians discover the formula for the perfect putt" being dumb PR for a golf equipment company, appears to be based on genuine research. Using mathematics to improve the performance of sports people and manufacturing of sports equipment is big business. The author, a physics professor at Yale, filed a patent in 2005 through the university for a golf training aid, marketed through a spin-off company. This was described by the Dean of the Engineering Faculty at Yale as "a great example of the joy in practical application of basic science and engineering". Over in the UK, I expect this would be an example of 'knowledge-transfer' and much sought-after evidence of the positive impact university research has on wider society.

There is another interesting case, when a real piece of research is explained using a metaphor which then gets reported as its real purpose. This may be what happened in a 2010 story in the Guardian, which is about a paper on Recursive Binary Sequences of Differences published in Complex Systems that we spoke about on Math/Maths 7. Attempting to explain potential applications to blending problems, the researcher seems to have given the trivial example of pouring coffee. The work is reported under the headline "How to pour the perfect cup of coffee" with explanation: "Years of research have resulted in the definitive way to pour the best second cup of coffee". I don't know what happened here but it looks to me like some number theory with potential applications to chemistry is being presented in a way that might lead you to believe it is one of Simon's "absurd PR equations".

So what have we learned? Rail against bad maths in the media when you see it, but be careful to check you've found what you think you've found and be aware there are levels of severity.

Four perfect hands: An event never seen before (right?)

2011-12-06T18:26:00.000Z

A couple of weeks ago there appeared several reports of an astonishing coincidence. Reports in the Daily Express, The Sun and the Daily Mail tell of a game of whist at the village hall at Kineton, Warwickshire. In whist, one deals 52 cards equally between four players. During this particular game, all four players were dealt one entire suit each.

All three reports refer to an analysis by Dr Alexander Mijatovic of Warwick University. It is always difficult to know how much of what is reported is faithful, but the fullest account of his words was given in The Sun:

The chances of this happening are so humongous that it is almost impossible.
The event can only be compared to natural occurrences.
It would be the same as a person having a tiny drop of water and then finding that same drop of water in the Pacific Ocean.
I would question whether the cards were shuffled the correct number of times but if they were, and the people involved are sure they were, then it is probably safe to say this is the first time this hand has ever been dealt in the history of the game.

It is this last sentence, in particular, that caught my eye: "it is probably safe to say this is the first time this hand has ever been dealt in the history of the game". I took a quick look on Google News, which indexes old newspapers. I obeyed the following rules: I ignored results when only one perfect hand was dealt (hardly remarkable at all!); I didn't pick a second result from the same decade (although there were plenty, particularly in the 1920s and 30s) and I didn't spend very long at all on this. Here are a set of articles I found:

I was particularly taken with an account by Catherine Ford in The Calgary Herald of 29th November 1983, which contains,

Every bridge player fantasizes about the perfect hand - being dealt the 13 cards of one suit - and the perfect game, in which each of the four players receives all 13 cards of one suit. The odds of this happening are 2,235,197,406,895,366,368,301,559,999 to 1, which explains why a plain brown envelope, sealed in 1946, is among my mother's prized possessions. It contains the cards which dealt one perfect suit to each player.

William Hartson, commenting in The Independent on one such incident in 1998, said:

There are about six billion people in the world. If they all played one hand of cards every five minutes, 12 hours a day, such a coincidence would happen about once every ten trillion years. On the other hand, there are a good few practical jokers around who would love to sneak a doctored pack of cards to four unsuspecting players to create the perfect whist hands when dealt. I know which possibility my money is on.

It is tempting to suggest that someone made these stories up, or stacked the deck as a joke. However, it turns out these assumptions aren't needed to explain what is happening.

Essentially, Dr Mijatovic was right to question whether the cards were shuffled correctly (so I wonder if this was actually the main thrust of what he said). Basically, whist is a game in which the objective is to stack the deck. A card is played and the other players must follow suit if they can, meaning the cards at the end of the game are particularly well ordered into suits. If the shuffling does not completely randomise the deck (and it often doesn't) then the probability of a perfect game occurring is increased greatly. There is a good summary of this on the MAA website at Ivars Peterson's MathTrek.

Samuel Hansen pointed out on the Math/Maths Podcast when we spoke about this that this is still very unlikely and may even be worthy of note in a local newspaper, so we should let people have their fun. He's right, of course - I am mostly just amused by the claims of just how unlikely this is and the way an event that happens every few years is set up as unlikely to happen during the lifetime of the human race.

Shifting decline of mathematical preparedness?

2011-12-04T11:48:00.000Z

Last year I wrote On the Decline of Mathematical Studies, and ever was it so, which looked at several examples of people complaining that the new generation of mathematics students were not as well prepared as the current one, with quotes from the late 20th C, mid 20th C. and even from the early 19th C. I wondered whether the problem was one of perception, or whether mathematics teaching could really be in constant (or, as Tony Mann pointed out, cyclical) decline.

I have just read 'Mathematics at the Transition to University: A Multi-Stage Problem?', an essay by Michael Grove (of the National HE STEM Programme, which supports my project) which offers an interesting view on this question. Though the complaint, that students are not prepared for university courses, sounds the same, Michael suggests the root cause and manner in which this problem manifests itself has changed. He backs up his argument with findings from several recent reports. His essay is worth a read if you are interested in this issue.

Having identified a possible root cause for the current situation, Michael also makes recommendations for what can be done to address this and points to relevant work the Programme is doing.

My Erdős number?

2011-11-24T09:22:00.001Z

People ask, from time to time, what is your Erdős number? For a long time I've said I haven't got one because I haven't published a mathematical research paper. When I gave this answer to Samuel Hansen last year he told me that any research paper counts, not just those in mathematics. This left me idly wondering and today, having listened to Samuel discussing social network theory on the Big Science FM podcast, I finally decided to have a go. There are a few possibilities, none of which seem, to me, entirely satisfactory.

Short answer: At most 4. Probably.

Long answer:

My list of publications is on my website. I do not appear on MathSciNet.

MathSciNet tells me Edmund Harriss wrote 'Flattening functions on flowers' with Oliver Jenkinson, who wrote 'Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type' with R. Daniel Mauldin , who wrote 'The nonexistence of certain invariant measures' with Paul Erdős. This gives Edmund an Erdős number of 3. Edmund and I are both authors on the paper 'The unplanned impact of mathematics' in Nature, which would make my number at most 4. However, this was a strange piece and although it is listed as one paper on the Nature website it was actually a series of seven short pieces under a common title and introduction. It is difficult to say whether this counts as a collaboration.

From the same root article, MathSciNet tells me Mark McCartney has an Erdős number of 5, Graham Hoare has 4 (although this is via a biography 'Stefan Banach (1892–1945). A commemoration of his life and work' in Mathematics Today which may not count (see below)), Juan Parrondo has 5, Julia Collins has 5 and Chris Linton has 4. These all have the same problem as Edmund above and are all greater numbers than Edmund's anyway.

For another route, MathSciNet tells me Joel Feinstein wrote 'A fixed-point theorem for holomorphic maps' with Richard Timoney, who wrote 'An extremal property of the Bloch space' with Lee Rubel, who wrote 'Tauberian theorems for sum sets' with Paul Erdős. This makes Joel's Erdős number 3. Joel and I have a paper 'Media Enhanced Teaching and Learning', in the new issue of MSOR Connections. This would make my number 4. MSOR Connections is a mathematics education practitioner journal and I am not sure if this counts.

Stephen Hibberd and Cliff Litton, with whom I collaborated on some of my first articles, including 'MELEES - Managing Mayhem?' (Proc. Mathematical Education of Engineers IV), both have an Erdős number of 4. Going this route, mine would be 5. Being a paper in conference proceedings, this seems the most 'real' route.

So what are the rules? The Erdős Number Project says:

Our criterion for inclusion of an edge between vertices u and v is some research collaboration between them resulting in a published work. Any number of additional coauthors is permitted. Not normally included are joint editorships, introductions to books written by others, technical reports, problem sessions, problems posed or solved in problem sections of journals, seminars, very elementary textbooks, books on history, memorial or other tributes, biography, translations, bibliographies, or popular works.

The Nature article perhaps doesn't count then, even if it counts as a collaboration, as it is a history or perhaps even popular article (exposition rather than original research). By this definition, the article with Joel Feinstein does seem to count. We've collaborated for a couple of years on using tablet PCs to deliver mathematics lectures, both while I was at Nottingham and since then, and have run several workshops on this topic. This collaboration led to this paper on Joel's use of this technology in his lectures. So I suppose that would make my Erdős number 4.

Of course, like any social network analysis, the members of a network may not be able to find the shortest path through it. For example, I co-authored with Claire Chambers who has a healthy number of co-authors but as her work now is in education around geography they are not included in MathSciNet. I have other co-authors on my list of publications that don't appear in MathSciNet but may, as I appear to, have a number. Indeed, any of the authors listed above could have a shorter route found by leaving MathSciNet's database. So I should say my Erdős number is "at most 4".

I find you can only go so far down this rabbit hole before it all seems far too preposterous to go on, so I will stop.