Where do old issues of MSOR Connections live online these days? @peterrowlett?

— Christian Perfect (@christianp) November 26, 2015

It’s complicated, but here is what I know.

Volumes 1-12 (actually 0-12, if you include the ‘Maths, Stats and OR’ newsletter published in 2000 as volume 0) were published by the Maths, Stats and OR Network, which I worked for in its dying days. At that point, the website previously at mathstore.ac.uk was archived by the Plymouth International Centre for Statistical Education at icse.xyz/mathstore. It’s still there, so you can still get volumes 1-12 (published 2001-2012) via its Newsletter archive, which acts as a by-issue index of individual PDFs.

MSOR Connections was relaunched as a peer-reviewed journal by the Higher Education Academy in 2013. These were online at journals.heacademy.ac.uk, and indeed that is currently still where the DOI links direct you, but that site was taken down earlier this year in favour of the Knowledge Hub. So if you know the name of an article, you can find it there – though I’m not sure there is a contents listing of issues.

However, there’s a catch. When I spent some time earlier this year comparing the online archives with my printed copies, I found that not every article is available. Volume 13 appears entirely available in the Knowledge Hub. For volumes 1-12, my fairly blunt approach was basically to look at the articles on mathstore and then, if the number of PDFs differs from the number of articles in my print copy of that issue, investigate why. Mostly that happened because articles were combined in the same PDF, but there were a few times (to my surprise) where the mathstore version missed some articles. In such cases, I was able to find most of the missing articles in the HEA Knowledge Hub. (There are also articles not in the Knowledge Hub that do appear on mathstore; it’s a mess.) Most frustratingly, I couldn’t find the following articles in PDF on either archive:

- ‘The False Revival of the Logarithm’ by Colin Steele 7(1):17-19 (I have found an author pre-print);
- ‘PowerPoint Accessibility within MSOR Teaching and Learning’ by Sidney Tyrrell 7(1):26-29;
- ‘Have You Seen This? RExcel – An interface between R and Excel’ by John Marriott 7(1):43;
- ‘Book Review – SPSS for Dummies’ by Arthur Griffin by Sidney Tyrrell 8(4):38-39.

These are not on the mathstore site or the Hub, but appear in my print copies. If you can locate electronic copies of any of these I would be pleased to hear it.

Volume 13 was the only volume published before the HEA finished publishing MSOR Connections and agreed to release the title back to a group coordinated by sigma and the University of Greenwich. I am one of the editors of MSOR Connections in its current form, and you should find volume 14 (published in 2015) onwards indexed on the Greenwich journals website.

]]>The format, wholly original and not in any way ripped off by Colin and Dave from anywhere else, saw two teams compete by giving correct and incorrect definitions of a word for the other team to determine who was telling the truth and who was bluffing. Team members challenged the other team to ‘call my bluff’, as it were.

There were three rounds, in which the teams defined first a mathematician, then a constant, then a theorem. Colin’s team included Dominika Vasilkova along with The Aperiodical’s own Christian Lawson-Perfect, with Elizabeth A. Williams and Nicholas Jackson opposing them on Dave’s team.

**Ways to listen**: Listen online. Download. Get the podcast via RSS or via iTunes.

If you enjoy this, you might like other episodes of Wrong, But Useful. At least that’s what Colin’s WordPress thinks:

]]>@icecolbeveridge insightful stuff from WordPress here. pic.twitter.com/Eq8tQheUeJ

— Peter Rowlett (@peterrowlett) November 23, 2015

Get the Being a Professional Mathematician podcast in RSS format.

Get the Being a Professional Mathematician podcast on iTunes.

The wider project includes resources and suggestions for using this audio in teaching undergraduates, inclunding the booklet Being a Professional Mathematician.

Enjoy!

]]>Last week, I decided I would discuss myths and inaccuracies. Though I am aware of a few well-known examples, I was struggling to find a nice, concise debunking of one. I asked on Twitter for examples, and here are the suggestions I received, followed by what I did.

Jason Dyer, @Brobuntu and Rob Eastaway all three suggested an article, Gauss’s Day of Reckoning, which discusses the tale of Gauss as a boy quickly summing the first 100 integers. @Brobuntu also mentioned the story of Hippasus and the Pythagoreans and “the worn story” of Euler debating Diederot, but without sources debunking them. Dan Wood also mentioned the former of these, of the Pythagoreans ordering the death of a student who proved $\sqrt{2}$ to be irrational.

@haggismaths suggested a blog post, Logic and Madness?, that debunks the idea that thinking about the continuum hypothesis drove Cantor mad.

Nicholas Jackson suggested Galois “frantically scribbling maths the night before his fatal duel”, and the article Genius and Biographers: The Fictionalization of Evariste Galois giving a detailed debunking. @haggismaths also suggested this as a story “much romanticised by Bell”, linking to the same article and also suggesting that “the section on Wikipedia about Galois’ death and final hours is not bad”. Thony Christie made the same suggestion and said the bebunking should be covered by Boyer, by which I guess he means A History of Mathematics.

Rob Eastaway suggested his blog post about the Golden Rectangle and Donald Duck.

@theoremoftheday suggested “something on how Nobel did NOT shun mathematicians cos one cuckolded him”, linking to Why is there no Nobel in mathematics?.

John Read suggested “whether Euclid was Greek, African or made up from a collective of people”, although without a debunking source.

So what did I go with? I had planned to give the students a page from E.T. Bell’s Men of Mathematics and a short article debunking it, as a reading exercise. Lacking a short debunking, I instead made a short lecture giving a typical story of Galois, quoting Bell:

he had spent the fleeting hours feverishly dashing off his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death which he foresaw could overtake him. Time after time he broke off to scribble in the margin ‘I have not time; I have not time,’ and passed on to the next frantically scrawled outline. What he wrote in those desperate last hours before the dawn will keep generations of mathematicians busy for hundreds of years.

Then I gave a brief debunking from Genius and Biographers: The Fictionalization of Evariste Galois, by Tony Rothman (1982) including:

It is unclear how far one can go in forgiving Bell. . . . I believe consciously or unconsciously Bell saw his opportunity to create a legend. The details which are absent in his account . . . are those details which lend a concreteness and a humanness to Galois’s life which a legend must not have. Unfortunately, if this was Bell’s intent, he succeeded.

I also included some discussion from Mathematical Myths by G.A. Miller (1938), who writes that some readers of Bell will think that errors of detail are unimportant, but that

there are others who will be very much annoyed by errors of detail, and whose interest in the book will be greatly diminished when they become convinced that they cannot assume that the author took a reasonable amount of care to avoid misleading remarks even when they are striking.

I also included the story of Archimedes leaping from the bathtub shouting ‘Eureka!’ in order to discuss the place of legends in folklore and the value this can have. For a discussion, I used Life on the Mathematical Frontier: Legendary Figures and Their Adventures by Roger Cooke, who writes:

What is valuable in the story is the picture of the sudden flash of inspiration that mathematicians sometimes experience. Whether true or not, this story will continue to be told because it amuses people and because it expresses some folklore concerning a legendary figure.

To encourage my students to consider such issues when writing their own work, I ended with a quote from Miller suggesting that

]]>the reader should realize that he is in danger of contributing toward the spreading of mathematical myths when he quotes from these writings without verifying the accuracy of statements contained therein.

This journal was published by the Maths, Stats and OR Network 2001-12, then by the Higher Education Academy in 2013. The first new issue for two years, published by a volunteer group coordinated and supported by **sigma** and the Greenwich Maths Centre, is volume 14 issue 1.

This issue includes articles about maths support, active learning of game theory, support for numerical reasoning tests in graduate recruitment, an implementation of the Maths Arcade, and an article about the new maths learning space at Sheffield Hallam University at which I am going to work later this month, written by my new head of department (you can just about see my office-door-to-be in figure 2).

Submissions are encouraged, which could be case studies, opinion pieces, research articles, student-authored or co-authored articles, resource reviews (technology, books, etc.), short update (project, policy, etc.) or workshop reports and should be of interest to those involved in the learning, teaching, assessment and support of mathematics, statistics and operational research in higher education.

]]>There is a second type of DLHE survey, which is longitudinal. This surveys graduates 3.5 years after graduation, and the 2010/11 longitudinal data has just been released. This deserves some investigation and I don’t have time right now, but I did notice a couple of tables that make me proud of my subject.

The first reports the proportions of graduates who are in jobs rated as ‘professional’ and ‘non-professional’. These data are taken from Table 8 of the 2010/11 DLHE longitudinal data set. I’ve chosen all levels (postgrad and undergrad) and ordered the data by percentage in professional jobs (descending). I’ve highlighted mathematical sciences, which includes maths, stats and operational research.

Level of qualification obtained, mode of study and subject area 2010/11 | Total professional | Total non-professional |
---|---|---|

All levels | ||

Medicine & dentistry | 98.8% | 1.2% |

Veterinary science | 92.9% | 7.1% |

Subjects allied to medicine | 92.5% | 7.5% |

Architecture, building & planning | 91.8% | 8.2% |

Education | 87.7% | 12.3% |

Mathematical sciences |
86.5% |
13.5% |

Computer science | 86% | 14% |

Engineering & technology | 84.6% | 15.4% |

Physical sciences | 83.3% | 16.7% |

Law | 81.7% | 18.3% |

Social studies | 79.9% | 20.1% |

Business & administrative studies | 77% | 23% |

Biological sciences | 76.4% | 23.6% |

Combined | 73.5% | 26.5% |

Languages | 72.9% | 27.1% |

Historical & philosophical studies | 72.5% | 27.5% |

Mass communications & documentation | 71.6% | 28.4% |

Creative arts & design | 67.2% | 32.8% |

Agriculture & related subjects | 55.8% | 44.2% |

The second table is this one showing whether graduates felt the subject they studied was a formal requirement, important or helpful in gaining their current job. These data are from Table 15 of the 2010/11 DLHE longitudinal data set. Again, I’ve chosen all levels and I’ve ordered the table by those that felt their subject was not important (ascending). Again, I’ve highlighted maths.

Level of qualification obtained and subject area 2010/11 | Formal requirement’, ‘Important’ or ‘Not very important but helped’ |
Not important |
---|---|---|

All levels | ||

Veterinary science | 97.3% | 2.7% |

Medicine & dentistry | 96.4% | 3.6% |

Subjects allied to medicine | 93.6% | 6.4% |

Education | 91.7% | 8.3% |

Architecture, building & planning | 87.8% | 12.2% |

Engineering & technology | 87.7% | 12.2% |

Mathematical sciences |
87.5% |
12.5% |

Computer science | 84.7% | 15.3% |

Law | 81.5% | 18.5% |

Business & administrative studies | 81.1% | 18.9% |

Physical sciences | 78.5% | 21.5% |

Social studies | 77.1% | 22.9% |

Biological sciences | 76.8% | 23.2% |

Agriculture & related subjects | 75.9% | 24.1% |

Mass communications & documentation | 73.2% | 26.8% |

Creative arts & design | 70.7% | 29.2% |

Combined | 68.7% | 31.3% |

Languages | 68.6% | 31.3% |

Historical & philosophical studies | 57.8% | 42.2% |

Looking at these tables fairly naively, I’d say there are some subjects represented which are really a profession for which you require a degree (medicine, education, architecture, engineering, law). A student might decide before coming to university “I want to be a doctor” and then take medicine. That’s okay, provided you know at that stage what you want to do with your life (I didn’t). Clearly not everyone who takes these subjects goes into the associated profession, but it is reasonable to expect a large number to do so, and therefore a high proportion in professional jobs.

Then there are subjects that I guess are aligned to a job sector, but less closely to a particular job. I’d put Physical sciences, Biological sciences and Computer science into this category. I suppose we’d expect a moderate number to progress from these into the associated job sectors, but many to go into more general employment.

Finally, there are subjects that are extensions of subjects done in school that I imagine are taken out of interest or ability in the subject, but which don’t align to a particular job or job sector. Here is where I’d put maths. We might expect that these students have less of a specific job goal in mind, so may end up further down the tables. And this is why I am proud of maths — as we tend to tell applicants, maths leads to lots of different jobs, and graduates 3.5 years into their career seem to be doing very well. I’d say maths is the top subject not aligned to a particular profession on both proportion in a professional job and proportion saying the subject was helpful or important in gaining their current job.

Well, I think it’s interesting, anyway. Kids: choose maths! ;)

]]>I wrote previously on mathematician stereotypes, and suggested a few ideas for sources of biographies of historical and contemporary people working in maths who would be stereotype-breaking. I’m happy to report that since I wrote that post, This Is What A Scientist Looks Like has attracted more mathematicians.

Of course, in the background to all this is the issue of who counts as a mathematician. This issue has been well-discussed here, including by Katie Steckles, Liz Hind and on at least one previous occasion by me, plus the amusing satirical take by Christian Lawson-Perfect. My view is that if you’re using maths and are happy to think of yourself as a mathematician, that’s good enough for me.

Anyway, back to Kit and the #realfaceofmath, the hashtag on Twitter has attracted some interesting pictures — but there’s always room for more. I’m not keen on pictures of myself, and I don’t think I particularly break the stereotype, but I suppose if I am going to suggest you contribute a picture of yourself, I should play ball myself. So here we go.

]]>For @Kit_Yates_Maths, here I am doing something not mathematical. #realfaceofmath pic.twitter.com/YBr70ilCdg

— Peter Rowlett (@peterrowlett) August 9, 2015

My title is ‘Developing Strategic and Mathematical Thinking via Game Play: Programming to Investigate a Risky Strategy for Quarto‘ and the abstract is below.

The Maths Arcade is an extracurricular club for undergraduate students to play and analyse strategy board games, aimed at building a mathematical community of staff and students as well as improving strategic and mathematical thinking. This educational initiative, used at several universities in the U.K., will be described.

Quarto is an impartial game played at the Maths Arcade, in that there is one set of common pieces used by both players, and one where stalemates are a common outcome. While some students play without apparent direction until a winning opportunity appears, others adopt a more risky strategy of building the board towards a winning position, which could allow either player to win. Whether building towards a win is a sensible strategy, when the other player could equally well benefit, is a topic of debate at the Maths Arcade. Intending to suggest a possible student project, this article will describe a method to represent Quarto as an array of binary numbers, making the game suitable for programming in Python. Then, one strategy is programmed to play at random unless a winning move becomes available, while another is programmed to work towards a winning position. These are calibrated by playing against a completely random strategy and against themselves, then they are played against each other. The more risky strategy is found to win over the more naive player in around two thirds of one million games. Some limitations and possible areas of development are discussed.

Download (free): Developing Strategic and Mathematical Thinking via Game Play: Programming to Investigate a Risky Strategy for Quarto.

]]>In the latest Taking Maths Further podcast (Episode 19: Computer games and mechanics), we had a puzzle that we say could be answered roughly, but the precise answer 23.53 (2 d.p.) required a little calculus. On Twitter, @NickJTaylor said

Not sure the @furthermaths podcast Ep 19 solution "requires calculus" to arrive at 23.5cm Just use v² = u² + 2as and solve for s @stecks

— Nick Taylor (@NickJTaylor) May 11, 2015

The question was: “Susan the Hedgehog runs at 20cm/s across the screen while the run button is held down. Once the run button is released, she slows down with constant deceleration of 8.5cm/s^{2}. Will she stop within 32cm more of screen?”

Taking the position to be $x$, we have constant acceleration $x^{\prime\prime}=-8.5$ and initial speed $x'(0)=20$. Therefore we get, w.r.t. time $t$,

\[ x’ = \int x^{\prime\prime} \mathrm{d}t = -8.5 t + 20\text{.} \]

Setting $x’=0$ gives $t=\frac{20}{8.5}=\frac{40}{17}$ when Susan has stopped.

Now we can integrate again to get position and, since we can decide $x(0)=0$, we can omit the constant:

\[ x = \int x’ \mathrm{d}t = -4.25 t^2 + 20 t\text{.} \]

Putting in $t=\frac{40}{17}$ gives

\[ x = -4.25 \left(\frac{40}{17}\right)^2 + 20 \left(\frac{40}{17}\right) = \frac{400}{17} \approx 23.53\text{.} \]

@NickJTaylor is suggesting that we use the fact that “$v^2 = u^2 + 2as$” or, using the notation above, $(x’)^2 = u^2 + 2ax$, where $x'(0)=u$ and $x^{\prime\prime}=a$ is a constant. This is okay, and it works, but to me it still uses calculus.

To get to this, we start with $x^{\prime\prime}=a$, $x'(0)=u$ and $x(0)=0$, and obtain

\[ \begin{align}

x’ &= \int x^{\prime\prime} \mathrm{d}t = at + u\text{;}\tag{1}\label{1}\\

x &= \int x’ \mathrm{d}t = \frac{1}{2}at^2 + ut\text{.}\tag{2}\label{2}

\end{align} \]

From (1), we rearrange for $t$ to give, for non-zero acceleration,

\[ t = \frac{x’-u}{a}\text{.} \]

Substituting this into (2), we get

\[ \begin{align}

x &= \frac{1}{2}a\left(\frac{x’-u}{a}\right)^2 + u \left(\frac{x’-u}{a}\right)\\

&= \frac{1}{2a} (x’-u)^2 + \frac{1}{a}u(x’-u)\\

&= \frac{1}{2a} ((x’)^2-2x’u+u^2) + \frac{1}{a}(x’u-u^2)\\

&= \frac{1}{2a} ((x’)^2 – u^2)\text{.}

\end{align} \]

So

\[ (x’)^2 = u^2+2ax\text{.}\]

Setting $a=-8.5$, $u=20$ and $x’=0$ gives

\[ 0 = 400-17x\text{,}\]

so we see $x=\frac{400}{17} \approx 23.53$.

If you are happy to accept $v^2 = u^2 + 2as$ as a given, or to work out the area under a graph of the velocity to get displacement, then you could say there’s no calculus needed. I’d say that deriving the formula, or knowing that the area gives the displacement, uses calculus. And if you’re doing a calculus question on my exam, you should expect to have to show me the calculus.

]]>This uses a formula for $\pi$ due to John Machin (1680–1751) (for which a derivation can be found):

\[ \pi = 16 \tan^{-1}\left(\frac{1}{5}\right) – 4 \tan^{-1}\left(\frac{1}{239}\right)\text{.} \]

First, we need a Maclaurin series for $\tan^{-1}$. That would be:

\[ f(x) = f(0) + f'(0)x + \frac{f”(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 + \ldots \]

To find this, we need to know the derivative of $f(x)=\tan^{-1}(x)$, which I claim to be $\frac{1}{x^2+1}$.

(To see this, let $x=\tan(\theta)$ in $\int \frac{1}{x^2+1} \, \mathrm{d}x$, remembering $\frac{\mathrm{d}x}{\mathrm{d}\theta}=\sec^2(\theta)$ and $\tan^2(\theta)+1 = \sec^2(\theta)$.)

So, back to our Maclaurin series, the relevant derivatives are: $f(x)=\tan^{-1}(x)$, $f'(x)=(x^2+1)^{-1}$, $f”(x)=-2x(x^2+1)^{-2}$, and so on (I’m waving my arms here because the quotient rule is involved at this point and it gets messy!).

Then the function values end up as: $f(0)=0$, $f'(0)=1$, $f”(0)=0$, $f^{(3)}(0)=-2!$, $f^{(4)}(0)=0$, $f^{(5)}(0)=4!$, $f^{(6)}(0)=0$, $f^{(7)}(0)=-6!$, etc.

So

\[ \begin{align*}

\tan^{-1}(x)&=0+x+\frac{0}{2!}x^2+\frac{-2!}{3!}x^3+\frac{0}{4!}x^4+\frac{4!}{5!}x^5+\frac{0}{6!}x^6+\frac{-6!}{7!}x^6+\ldots \\

&= x – \frac{x^3}{3} + \frac{x^5}{5} – \frac{x^7}{7} + \ldots

\end{align*}\]

I’m happy, for an approximation, to say $\tan^{-1}(x) \approx x – \frac{x^3}{3} + \frac{x^5}{5} – \frac{x^7}{7}$, so that

\[ \tan^{-1}\left(\frac{1}{5}\right) \approx \left(\frac{1}{5}\right) – \frac{\left(\frac{1}{5}\right)^3}{3} + \frac{\left(\frac{1}{5}\right)^5}{5} – \frac{\left(\frac{1}{5}\right)^7}{7} \approx 0.197395504761905 \]

and

\[ \tan^{-1}\left(\frac{1}{239}\right) \approx \left(\frac{1}{239}\right) – \frac{\left(\frac{1}{239}\right)^3}{3} + \frac{\left(\frac{1}{239}\right)^5}{5} – \frac{\left(\frac{1}{239}\right)^7}{7} \approx 0.004184076002075\text{.}\]

Finally,

\[ \pi \approx 16 \times 0.197395504761905 – 4 \times 0.004184076002075 = 3.141591772182177\text{.} \]

I think it is neat to get agreement with the first five decimal places from only four terms.

The first time I did this example in a lecture, I started by joking “this is a long and complicated example. When I get to the end, I fully expect a round of applause”. When I finished, somewhat embarrassingly, I received one — along with ironic whoops from the back row!

To take this a little further, I wrote this quick Python code.

import decimal import math for loop in range(1,12): pivalue=0 firstterm=0 secondterm=0 for i in range(0, loop): firstterm = firstterm + decimal.Decimal((-1)**i * (1/5**(2*i+1))/(2*i+1)) secondterm = secondterm + decimal.Decimal((-1)**i * (1/239**(2*i+1))/(2*i+1)) pivalue = decimal.Decimal(16 * firstterm - 4 * secondterm) print("Using {} terms: {:.15f}".format(loop,pivalue)) print('math.pi: {:.15f}'.format(math.pi))

This gives the following values, showing that this finds 15 digits of $\pi$ by the time eleven terms of the sequence are computed.

Using 1 terms: 3.183263598326360 Using 2 terms: 3.140597029326060 Using 3 terms: 3.141621029325035 Using 4 terms: 3.141591772182177 Using 5 terms: 3.141592682404400 Using 6 terms: 3.141592652615309 Using 7 terms: 3.141592653623555 Using 8 terms: 3.141592653588602 Using 9 terms: 3.141592653589836 Using 10 terms: 3.141592653589792 Using 11 terms: 3.141592653589793 math.pi: 3.141592653589793

Apparently Machin used his formula to compute 100 digits of $\pi$, but to do that I’d need to get my head around increasing Python’s decimal places. Or get a lot more free time and calculate it by hand!

]]>