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!

]]>My title is: ‘The unplanned impact of mathematics’ and its implications for research funding: a discussion-led educational activity.

Abstract:

‘The unplanned impact of mathematics’ refers to mathematics which has an impact that was not planned by its originator, either as pure maths that finds an application or applied maths that finds an unexpected one. This aspect of mathematics has serious implications when increasingly researchers are asked to predict the impact of their research before it is funded and research quality is measured partly by its short term impact.

A session on this topic has been used in a UK undergraduate mathematics module that aims to consider topics in the history of mathematics and examine how maths interacts with wider society. First, this introduced the ‘unplanned impact’ concept through historical examples. Second, it provoked discussion of the concept through a fictionalized blog comments discussion thread giving different views on the development and utility of mathematics. Finally, a mock research funding activity encouraged a pragmatic view of how research funding is planned and funded.

The unplanned impact concept and the structure and content of the taught session are described.

Rowlett, P., 2015? ‘The unplanned impact of mathematics’ and its implications for research funding: a discussion-led educational activity. *BSHM Bulletin: Journal of the British Society for the History of Mathematics*. DOI: 10.1080/17498430.2014.945136.

Green attended Robert Goodacre’s school in Nottingham 1801-2 and took part in scientific culture in Nottingham, including at Bromley House Library, in the 1810s and 20s, before going to Cambridge in 1833. I speak about each of these aspects and some of the people involved. The audience was mixed public. I was aware I was being recorded and tried quite hard to make audible what was on the slides, so I hope you can follow along just fine.

My title was ‘George Green’s Mathematical Influences’ and the abstract is below:

George Green was an “almost entirely self-taught mathematical genius” (NM Ferrers, 1871) whose work was a major influence on the mathematical physics of the 19th and 20th centuries and shows no signs of stopping in the 21st. But from where or from whom did Green learn his mathematics? Peter Rowlett from Nottingham Trent University surveys Green’s education in Nottingham and Cambridge and those who influenced him.

Get the audio by streaming it from the exhibition page ‘George Green: Nottingham’s Magnificent Mathematician‘ or by direct download (mp4, 28.2MB). The talk is approx. the first 43 minutes, after which are questions, which you might or might not be able to hear but mostly consist of me saying “interesting idea, but I don’t know”.

While there, you can also listen to the previous talk to mine, ‘George Green’s contribution to MRI’ by Prof. Roger Bowley. The George Green exhibition at Nottingham’s Lakeside Arts Centre remains open until Sunday 4 January 2015. I recommend you visit, if you are able.

Related post: George Green: Nottingham’s Magnificent Mathematician.

]]>What particularly caught my ear was this section (around 5:30):

I was looking into going into engineering … I wanted to do something in industry, I didn’t know what … I went to a careers fair that was specifically for scientists and the people they’d sent to those fairs weren’t sure what to do with me — they recommended the accounts department. So I think there’s more to be done between universities and industry to realise what skills — especially for me: mathematicians — have, and working with degrees and universities to make sure that what you’re learning there is then applicable.

I recognise this frustrating situation, and I’d say this describes fairly well part of what I am supposed to do in my new job when I’m not teaching maths.

]]>

Cory Doctorow described himself on boingboing as “a great fan of Relatively Prime” and the Chinook episode as “one of the best technical documentaries I’ve heard“. Tim Harford described it on Twitter as “a great podcast of storytelling about mathematics“.

This series was funded via a successful Kickstarter in 2011. This is where people pledge to support the project, but only have to pay if the project reaches its target, and get funder rewards in return. Maybe you supported it. I certainly did.

You probably also know Samuel is trying to raise funds via Kickstarter for a second series of episodes. Funder rewards include video updates from Samuel, stickers, a ‘zine, your adverts on episodes, the opportunity for Samuel to do voice work for you, right through to the chance to get involved with production of an episode. The deadline for funding is Tuesday October 21st, and Samuel has over 100 backers and is more than one third of the way to his goal. Maybe you’ve already pledged to support it. I certainly have.

Samuel is tweeting about the Kickstarter, and I am occasionally retweeting him. Katie wrote a blog post here at The Aperiodical about the project. However, it gets to the point where we are just telling the same people over and over, many of whom will have already pledged. What the project really needs is for you to help by telling other people about the Kickstarter. Can you tweet about it? Or post it somewhere other than Twitter? Can you write your own blog post about the project and/or why you chose to support it?

You can watch an entertaining animated video giving the pitch and embed the video in your own website or blog at the Kickstarter page.

Note: I have nothing to do with this project, have no inside information and do not benefit as a result. If you want to ask questions about the series or the Kickstarter, contact Samuel Hansen. Samuel has appeared on a couple of podcasts talking about the Kickstarter, and I know he is keen to do interviews to promote the idea.

I’ve used the Kickstarter page to embed the current total below, so people of the future can see whether my words offer a heart-warming story of success, or a tragically unheard cry for help. People of the present: you have the power to influence this outcome.

]]>