I’m involved with three sessions – a fun Maths Jam, a ‘how I used history in my teaching’ workshop and a research talk based on half my PhD. Here are the details:

**Monday 14th April 2014**

*A Taste of Maths Jam *- with Katie Steckles and some other MathsJammers.

19:30 – we’re one of the after-dinner entertainment options!

Maths Jam is a monthly opportunity for like-minded self-confessed maths enthusiasts to get together in a pub and share stuff they like. Puzzles, games, problems, or just anything they think is cool or interesting. Attendees range from hobbyists to researchers, with every type of mathematician and maths enthusiast in between. Events happen simultaneously in over thirty locations worldwide (mostly in the UK) listed on the website at www.mathsjam.com. Come to this event to get a taste of what happens at a typical Maths Jam night.

**Wednesday 16th April 2014**

*The unplanned impact of mathematics and how research is funded: a discussion-led activity*

Session F6 – 09:05-10:05

Mathematics is sometimes developed (or discovered) by a mathematician following curiosity with no thought of application. Later, perhaps decades or centuries later, this mathematics fits some application area perfectly. This aspect of mathematics has serious implications as 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 has been used successfully in a UK undergraduate mathematics module on how maths interacts with wider society. This explored the concept of ‘unplanned impact’ and views on the phenomenon, as well as its impact on the way research is funded. This workshop will describe the session and demonstrate some of the activities used.

This session is one of a series on the History of Mathematics in Education coordinated by BSHM.

*Development and evaluation of a partially-automated approach to the assessment of undergraduate mathematics*

Session RI15 – 16:15-17:45

This research explored assessment and e-assessment in undergraduate mathematics and proposed a novel, partially-automated approach, in which assessment is set via computer but completed and marked offline.This potentially offers: reduced efficiency of marking but increased validity compared with examination, via deeper and more open-ended questions; increased reliability compared with coursework, by reduction of plagiarism through individualised questions; increased efficiency for setting questions compared with e-assessment, as there is no need to second-guess the limitations of user input and automated marking. Implementation was in a final year module intended to develop students’ graduate skills, including group work and real-world problem-solving. Individual work alongside a group project aimed to assess individual contribution to learning outcomes. The deeper, open-ended nature of the task did not suit timed examination conditions or automated marking, but the similarity of the individual and group tasks meant the risk of plagiarism was high. Evaluation took three forms: a second-marker experiment, to test reliability and assess validity; student feedback, to examine student views particularly about plagiarism and individualised assessment; and, comparison of marks, to investigate plagiarism. This paper will discuss the development and evaluation of this assessment approach in an undergraduate mathematics context.

Nathan says:

I see this as a great resource for researchers, research staff, academics and those who support researcher training. On the run up to my viva I had great advice from my supervisor, but I didn’t know about other resources that might help me prepare. The Viva Survivors Podcast will provide listeners with stories from people who have been through the viva, and give some insight on what to expect, what to do to prepare, and where to turn for help.

On the run up to my viva, I listened to a lot of the back catalogue and read Nathan’s e-book ‘Fail Your Viva: Twelve Steps To Failing Your PhD (And Fifty-Eight Tips For Passing)‘. Having used this in my preparation, I volunteered to be interviewed (you can offer to be interviewed too). The description for my episode says:

In this episode I’m talking to Dr Peter Rowlett, who recently completed his PhD at Nottingham Trent University. Peter’s research was multidisciplinary and was in the areas of computing and maths education; he did his PhD part time as well, and so we had a lot to talk about for this episode!

Viva Survivors Episode 23: Dr Peter Rowlett. Or subscribe.

Drinking game idea: take a drink every time I say “viva” when I clearly mean ‘thesis’.

Aside: Nathan is conducting a survey of PhD graduates of UK institutions since 2000. He explains why in a blog post. Consider filling it in.

]]>As well as rehashing Why do $0!$ and $a^0$ equal $1$?, and engaging in a little silliness, I think we touched on some interesting topics. Is $6\times 4$ four lots of $6$ or six lots of $4$ (Dave’s blog post on this is here)? Is the Pythagorean theorem part of trigonometry? Why should people be made to study maths? What is ‘mathematics’, and why don’t we have different words for maths at different levels? Why do students hate ‘show that’ questions? Why do people think maths is about getting the correct answer? What are continued fractions good for? And more. I’m not sure we answered any of this very well, so perhaps you will listen and let Colin and Dave know what you think.

Listen, and leave comments, over at Flying Colours Maths: Wrong, But Useful: Episode 13 or find the podcast on iTunes.

]]>My talk is part of the research strand of the conference, organised by the British Society for Research into Learning Mathematics. This society is “for people interested in research in mathematics education”, and I am a member.

I’m presenting the ‘what I did’ portion of my PhD; well, most of it. ~~Anyway, the peer-reviewed proceedings have now been published~~ **(not true! See edit below)**. My article is ‘Development and evaluation of a partially-automated approach to the assessment of undergraduate mathematics’. The abstract is below.

This research explored assessment and e-assessment in undergraduate mathematics and proposed a novel, partially-automated approach, in which assessment is set via computer but completed and marked offline. This potentially offers: reduced efficiency of marking but increased validity compared with examination, via deeper and more open-ended questions; increased reliability compared with coursework, by reduction of plagiarism through individualised questions; increased efficiency for setting questions compared with e-assessment, as there is no need to second-guess the limitations of user input and automated marking. Implementation was in a final year module intended to develop students’ graduate skills, including group work and real-world problem-solving. Individual work alongside a group project aimed to assess individual contribution to learning outcomes. The deeper, open-ended nature of the task did not suit timed examination conditions or automated marking, but the similarity of the individual and group tasks meant the risk of plagiarism was high. Evaluation took three forms: a second-marker experiment, to test reliability and assess validity; student feedback, to examine student views particularly about plagiarism and individualised assessment; and, comparison of marks, to investigate plagiarism. This paper will discuss the development and evaluation of this assessment approach in an undergraduate mathematics context.

Rowlett, P., 2014. Development and evaluation of a partially-automated approach to the assessment of undergraduate mathematics. *In*: S. Pope (ed.). *Proceedings of the 8th British Congress of Mathematics Education*. pp. 325-332. ~~Available via: bsrlm.org.uk/IPs/ip34-2/BSRLM-IP-34-2-38.pdf~~.

**Edit 28/03/2014**: It seems these papers were released early in error, and have now been removed. The page on the BSRLM website says “The proceedings will be available here after 14 Apr 2014″.

That is to say, the university have sent me a degree certificate, and I’ve shown it to the bank. So that’s pretty darn official.

]]>One of my students, Callum Mulligan, tweeted this question.

Why does 0! = 1 better yet, why does a^0 = 1 I must see a proof! #Mission #Unanswered #MathRage

— Callum Mulligan (@Calified) February 1, 2014

Saying “by definition” or “because it makes a bunch of stuff work” won’t cut it. So how to answer this question? To give a somewhat intuitive understanding of why this should be the case to a first year undergraduate. It may be obvious, but it wasn’t immediately obvious to me how to explain this, so I share some thoughts here.

Basically, I think it comes down to $1$ being the multiplicative identity.

Think about adding zero. If I take $x$ and add no things to it, I get $x+0=x$ (zero being the additive identity). Similarly, if I take $x$ and multiply it by no things, I ought to expect to get $x$ back, not $0$. The empty product must therefore be $1$, so that I get $1x=x$.

For $a^0$, think of this as $a$ multiplied by itself zero times. This is an empty product, so $a^0=1$.

$n!$ is $n$ things multiplied together, so $0!$ is zero things multiplied together. This, again, is an empty product, so $0!=1$.

I also came across some interesting ways to try to get a feeling for what is going on.

For $a^0$, I quite like this:

\[1=\frac{a^n}{a^n}=a^{n-n}=a^0\text{.}\]

Alternatively, since

\[a^b=a^{b+0}=a^ba^0\text{,}\]

it follows that $a^0=1$.

For $0!=1$, I quite like the reasoning that uses the definition of each factorial in terms of the previous. So $6!=6\times 5!$, and, in general,

\[ (n+1)!=(n+1)n!\text{.} \]

Rearranging this, we get

\[ n! = \frac{(n+1)!}{(n+1)}\text{.} \]

So

\[ 0! = \frac{1!}{1} = 1\text{.} \]

Also, if you like $n!$ as the number of ways of arranging $n$ objects, then think that there’s only one way to arrange zero objects, so $0!=1$.

I found this useful on ‘Why is x^0 = 1?‘ and this on 0!=1. I’d be interested to hear what you think in the comments. How do you convince someone that $a^0=1$ and $0!=1$ feel *right*?

Katie Steckles pointed out via the latest Carnival of Mathematics that quantum computer scientist Scott Aaronson posted an explanation of his research using only the 1000 most common words in English, inspired by the xkcd comic ‘Up-Goer Five‘, which did the same for a labelled diagram of the Saturn V rocket (the ‘Up-Goer Five’). Scott’s post links to The Up-Goer Five text editor, a fabulous innovation that allows typing in a box and highlights when a word isn’t on the same list of words used in the xkcd diagram. I used this to write a version of my thesis abstract. Beyond what the text editor wanted, I also voluntarily adjusted some terms that are on the list, but presumably not in the way I mean them. Particularly, ‘deep learning’ and ‘open-ended questions’ didn’t get highlighted. I’ve gone for a fairly close, word-by-word translation, though clearly some parts could be rewritten completely to be clearer.

My thesis abstract (the version I handed in) is in a previous blog post, if you want to view it for comparison. Here’s my Up-Goer Five version.

]]>Asking questions on big school numbers and stuff using computers is studied by reading lots of books and papers and asking questions of people who do it, and looked at along with other approaches to asking questions on numbers and stuff usually used in big school in my land. Asking questions using computers offers some good points over other approaches, like changing numbers in questions using chance allows asking different questions for each person, but it means that things can be asked in fewer areas because the computers can not mark everything.

An approach is suggested that uses computers a bit in which approaches like asking questions using computers are used to ask different questions for each person, which are taken and marked by hand. This approach is used in a big school numbers and stuff bit of schooling. The bit of schooling uses sets of questions that are different for each person when doing group work to attempt to give different numbers of marks to students who have done the bit of schooling more or less well than other students. The approach that uses computers a bit is used as a way to lower the chance of one student looking at the work done by another and writing theirs the same, instead of watching students write their answers or asking the questions completely using computers.

Seeing whether it worked by having someone else mark the work as well suggests that the approach could be used to make a set of questions that: are marked the same whether I am doing the marking or someone else is; and, that mean that I can tell whether the students have learned what I wanted them to learn. Seeing whether it worked by asking students what they think and looking at the marks I gave them for the work leads to the idea that sometimes one student does look at the work done by another and writes theirs the same at big school, that this might have happened with this piece of work, but was not in fact a problem in this case.

I suggest you add the approach that uses computers a bit to the other ways you know of asking questions on numbers and stuff at big school, especially in cases where asking questions carries a high chance of one student looking at the work done by another and writing theirs the same but the need for questions which students could take in different directions or where they learn more makes watching students write their answers or asking the questions completely using computers less perfect.

The query was from someone who is training to be a maths teacher and is doing some web development. They had seen mention of my MSc dissertation topic on this blog and asked where they could read more about the underlying web technologies. Here, basically, is what I replied.

My MSc was ten years ago and things have moved on since then, technology-wise. Some of what I did is now so commonplace as to be boring. You can get the basics of what I did for my dissertation from this article: ‘Pseudo-Randomised CAA by “Preprocessing” MathML‘ (*Maths-CAA Series*, HEA MSOR Network, 2004). You can also read a critique I wrote about it, from an education point of view, in: ‘Developing a Healthy Scepticism About Technology in Mathematics Teaching‘ (*Journal of Humanistic Mathematics*, 3(1), pp. 136-149, 2013).

Anyway, back then I used PHP to write raw MathML code, so I could dynamically stick numbers in it. Now, if I had to do the same, I would probably just spit out LaTeX and give it to MathJax to deal with. For example, the (free) Numbas e-assessment system does just this, as do other systems. Numbas makes tests like this sample exam.

During my PhD, I shifted from a home-grown approach to developing this kind of thing, to an approach where free to use software is perfectly capable of doing the sort of thing I’m interested in doing. This is particularly as I’m more interested now in what can be done with such things in teaching than I am with the technical development aspects. I’d be pleased to hear via the comments from anyone who has anything to add on the topic of putting maths on the web, and particularly of dynamically changing that maths.

]]>Anyway, last week I was asked where the quoted figure 2,235,197,406,895,366,368,301,559,999 to 1 actually comes from. Here’s my shot at it.

There are 52 cards in a standard deck, each belonging to one suit of thirteen cards each. The number of ways of dealing 13 cards from a deck is therefore $\binom{52}{13}$. The probability of one player being dealt 13 cards from 52 all of any one of the four available suits (we don’t care which) is then $$\frac{\binom{4}{1}}{\binom{52}{13}}\text{.}$$

Having dealt thirteen cards from one suit to one player, 39 cards remain in three suits. The probability of the second player being dealt 13 cards from 39 all of the the same suit is $$\frac{\binom{3}{1}}{\binom{39}{13}}\text{.}$$

Then 26 cards remain in two suits. The probability of the third player being dealt 13 cards from 26 all of the same suit is $$\frac{\binom{2}{1}}{\binom{26}{13}}\text{.}$$

Finally, the remaining 13 cards are all of one suit and are dealt to the fourth player with probability $1$.

Then the probability of all of this happening together is $$ \frac{\binom{4}{1}}{\binom{52}{13}} \times \frac{\binom{3}{1}}{\binom{39}{13}} \times \frac{\binom{2}{1}}{\binom{26}{13}}\text{.}$$

If you calculate this, you should find the quoted probability. Or, if you prefer, the magic ‘to one’ number we are looking for arises from $$\frac{\binom{52}{13}\binom{39}{13}\binom{26}{13}}{4!}\text{.}$$

]]>Up to a point, it might seem reasonable to explore an issue by finding a bunch of examples and extrapolating a general rule that your examples seem to obey. I realise there’s a little more to it than that, but this is basically what science does. This process is called inductive reasoning, because a general theory is ‘induced’ from the ground up.

Mathematics, on the other hand, follows a deductive process. A set of basic ideas are assumed (we call these axioms), and a series of propositions are ‘deduced’ from these via proof. Of course, in reality there are mathematicians on the applied side who are effectively doing science, but at its heart, mathematics is a process of deductive reasoning.

So science induces from evidence, while mathematics deduces from assumed truths. This is why a mathematical truth (a true statement within a constrained system) remains true throughout time, while scientific truth (an idea based on a lot of evidence) can be overturned by new observations.

My officemate, a forensic scientist, is currently reading some Sherlock Holmes stories. I’ve never read these, and am only aware of the material through various screen incarnations, but I’m aware that what Holmes does is sometimes referred to as “deduction“. Actually, Holmes is proceeding from close observation and works to establish the facts of a case from these — a process of inductive reasoning.

We approached the case, you remember, with an absolutely blank mind, which is always an advantage. We had formed no theories. We were simply there to observe and to draw inferences from our observations.

(The Adventure of the Cardboard Box)

It is interesting to me that Moriarty, nemesis of Holmes, is set up as a Professor of Mathematics. I wonder if this is deliberate, to establish Moriarty as Holmes’ opposite, who reasons completely differently to Holmes (but is almost his match), and whether this distinction features overtly in the stories. This ability to reason appears crucial to Moriarty’s evil nature; he is not just any villain, because his “criminal strain” is “increased and rendered infinitely more dangerous by his extraordinary mental powers” (The Final Problem).

Of course, this makes science the hero and maths its diabolical nemesis. So be aware, dear reader, that we may be on the wrong side of good.

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