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.

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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.

]]>E-assessment in higher education mathematics is explored via a systematic review of literature and a practitioner survey, and compared with other assessment approaches in common use in higher education mathematics in the UK. E-assessment offers certain advantages over other approaches, for example question randomisation allows individualisation of assessment, but it is restricted in the range of what can be assessed due to the limitations of automated marking.

A partially-automated approach is proposed in which e-assessment techniques are used to set an individualised assessment which is taken and marked by hand. This approach is implemented in a higher education mathematics module. The module uses individual coursework assignments alongside group work to attempt to account for individual contribution to learning outcomes. The partially-automated approach is used as a method for reducing the risk of plagiarism in this coursework, rather than replacing it with a written examination or e-assessment.

Evaluation via blind second-marking indicates that the approach was capable of setting a reliable and valid assessment. Evaluation of student views and analysis of assessment marks leads to the conclusion that plagiarism does take place among the undergraduate cohort, was a risk during this assessment, but was not in fact a particular problem.

The partially-automated approach is recommended as an appropriate addition to the repertoire of higher education mathematics assessment methods, particularly in cases where an assessment carries a high risk of plagiarism but the need for open-ended or deeper questions make an examination or automated marking system sub-optimal.

Alright, so you might not want to read the whole 184 pages, but you might be interested to relive the thrill of submission in 26th July 2004–23rd July 2013, or listen to an interview I did about my PhD and viva experience in Peter Rowlett: Viva Survivor.

Rowlett, P.J., 2013. *A Partially-automated Approach to the Assessment of Mathematics in Higher Education*. PhD thesis, Nottingham Trent University.

We don’t regard him as a miller, I’m afraid, we regard him as a very eminent mathematician whose work today is still being used in major industries and concerns.

– George Saunders, descendant of George Green, on being asked a question about bags of flour on the Alan Clifford show on BBC Radio Nottingham of 11th September 2014 (starts approx. 1:16).

The above quote is from a short interview with George Saunders and Kathryn Summerwill on BBC local radio about George Green. Green, of whom you may have heard, was a mill-owner in Nottingham and a genius mathematical physicist. The interview marks the opening of an exhibition, curated by Kathryn, ‘George Green: Nottingham’s Magnificent Mathematician‘ in the Weston Gallery at the Lakeside Arts Centre, University of Nottingham.

I was pleased to attend the opening of the exhibition and meet some of the people involved, including Lawrie Challis. You can’t read much about Green without coming across Lawrie, who has written on Green himself and led a campaign to restore the mill in the 1970s and 80s when a rumour emerged that the City was thinking of demolishing the derelict. The exhibition is fantastic, with many interesting items from the university’s Manuscripts and Special Collections archive relating to various aspects of Green’s life and works.

As well as the exhibition, there are three lunchtime talks on aspects of Green’s life, though I wouldn’t bother with the third one if I were you.

- ‘George Green and his Mill‘ by Tom Huggon, Chairman of the Friends of Green’s Mill, Wednesday 1st October, 1-2pm;
- ‘George Green’s Contribution to MRI‘ by Roger Bowley, Emeritus Professor in the School of Physics, University of Nottingham, Tuesday 21st October, 1-2pm;
- ‘George Green’s Mathematical Influences‘ by Peter Rowlett, some guy, Wednesday 12th November, 1-2pm.

The talks are free and take place in the theatre adjacent to the exhibition, but you must book a ticket in advance by phoning the Box Office on 0115 846 7777.

The wonderful Theorem of the Day website marked the opening of the exhibition on 12th September 2014 by featuring Green’s Theorem. The exhibition is open until Sunday 4th January 2015.

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On puzzles and games, the report says:

The inherent interest of mathematics and the appeal which it can have for many children and adults provide yet another reason for teaching mathematics in schools. The fact that ‘puzzle corners’ of various kinds appear in so many papers and periodicals testifies to the fact that the appeal of relatively elementary problems and puzzles is widespread; attempts to solve them can both provide enjoyment and also, in many cases, lead to increased mathematical understanding. For some people, too, the appeal of mathematics can be even greater and more intense.

…

We do not believe that mathematical activity in schools is to be judged worthwhile only in so far as it has clear practical usefulness. The widespread appeal of mathematical puzzles and problems to which we have already referred shows that the capacity for appreciating mathematics for its own sake is present in many people. It follows that mathematics should be presented as a subject both to use and to enjoy.

…

Whatever the level of attainment of pupils, carefully planned use of mathematical puzzles and ‘games’ can clarify the ideas in a syllabus and assist the development of logical thinking.

Cockcroft, W. (1982), *Mathematics counts: report of the Committee of
Inquiry into the teaching of mathematics in schools*. London: HMSO.

The premier maths documentary podcast, created by Samuel Hansen in 2012 following a successful Kickstarter. Relatively Prime is eight shows of stories behind mathematics, with topics including game-playing computers, music and architecture.

Cobbled together while I was working for the IMA, Travels in a Mathematical World mostly features short interviews with working mathematicians about their work. 2008-2010.

The Plus Podcast mostly features interviews on a diverse range of mathematical topics to support their written articles. Going since 2007 and still releasing new episodes, though infrequently.

All Squared is the Aperiodical’s own Katie Steckles and Christian Perfect in conversation with interesting mathematicians. Started in 2013 and still going, as far as I know, though infrequently.

Proper BBC Radio 4 documentary, A Brief History of Mathematics is ten episodes of maths history by Marcus du Sautoy first broadcast in 2010, each themed around a particular mathematician or group of mathematicians.

Math Mutation is a curious, short podcast in which Erik Seligman aims to explore “fun, interesting, or just plain weird corners of mathematics that you probably didn’t hear in school” (school, presumably, in the American sense). Going since 2007 and still releasing new episodes, though infrequently.

I believe host Samuel Hansen would describe Combinations and Permuations as a jokey panel discussion about mathematics. 2009-2011.

Strongly Connected Components is a series of 51 episodes released between 2009-2012 in which Samuel Hansen interviews an interesting range of mathematicians.

The Math Factor is a podcast on mathematics, logic and puzzles with Chaim Goodman-Strauss and Kyle Kellams in a series of radio episodes released between 2005 and 2010.

Will Davies made four recommendations.

- Wrong, But Useful: Colin Beveridge and Dave Gale discuss mathematical miscellany. I hadn’t mentioned Wrong, But Useful because I feel it is quite teacher-oriented and the query was for undergrads, but perhaps that’s unfair.
- More or Less. Statistical current affairs from Tim Harford and the team, broadcast on BBC Radio 4. Didn’t I recommend More or Less? Oops! I always assume everyone listens to More or Less.
- The Naked Scientists, apparently, comprises “a media-savvy group of physicians and researchers from Cambridge University who use radio, live lectures, and the Internet to strip science down to its bare essentials, and promote it to the general public”.
- Probably Science: “four professional comedians and incompetent scientists take you through this week in science. Incompetently.”

Colin Beveridge suggested science story-fest RadioLab, saying “it’s not often on maths, but it’s rarely dull.”

Samuel Hansen, as well as some of those above, added Math/Maths. I didn’t include this because I think of it as a topical show and it is on extended hiatus or ended (the hosts aren’t clear which), and because a problem with the Pulse-Project site currently means the RSS feed (and therefore iTunes) don’t show any episodes.

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