# You're reading: Travels in a Mathematical World

### A new place to hang my hat

I have moved my blog Travels in a Mathematical World to The Aperiodical!

### Pale imitations: newcomers in the Math/Maths Podcast hiatus

Since the start of the year, the Math/Maths Podcast has been on hiatus. I’m very much enjoying the extra thesis-writing time but apparently this has left some missing their regular mathematical listen. Not infrequently I get an email from someone wishing me well with my thesis and asking when we’ll be back podcasting. Well, nature abhors a vacuum and here are three offerings that I’m aware are working to fill the void. (Oh, and “pale imitations” – I’m joking, of course!)

My Aperiodical co-conspirators Katie Steckles and Christian Perfect started All Squared, a maths magazine podcast, in February. The description for the first episode (or “number”, as Katie and Christian have it) overtly points out the “unusual paucity of maths podcasts at the moment” and promises “a half-hour podcast featuring maths, guests, puzzles and links from the internet”. The name is designed to be recognisable to mathematicians, who might find themselves reporting that an expression is “all squared”. As someone who named a podcast as overtly as it is possible to be, “Math/Maths”, this obfuscation amuses me. The three episodes so far have been enjoyable with a guest and main topic in each. As far as I’m concerned, this is far more the Aperiodical podcast that should exist than is The Aperiodcast with that third guy.

#### TES Maths Podcast (iTunes)

This one started just before Samuel Hansen and I went on our hiatus, but if you enjoyed the teaching aspects of what we did you can get a lot more on the theme from Craig Barton and his guests on the TES Maths Podcast. Craig promises “to share the latest news, resources and ideas that are relevant to secondary/high-school maths teachers and general number enthusiasts”.

Wrong, but Useful is a new podcast featuring “a mathematical conversation” between Colin Beveridge and Dave Gale that sets out its stall as a response to the lack of Math/Maths episodes. The title is another nod to the mathematically minded without being overt, referring to a quote from George Box and Norman Draper who wrote “essentially, all models are wrong, but some are useful” (Empirical Model-Building and Response Surfaces, 1987). Episode 1 sees Colin and Dave finding their feet in a rambling, wide-ranging mathematically-themed discussion. There were a couple of awkward moments that gave me Math/Maths early episode flashbacks but I’m looking forward to Colin and Dave getting into the swing for the next episode.

Happy listening!

### A simple proof that π is rational

I present a new paper, ‘A simple proof that π is rational‘. The abstract is:

The number pi, written using the symbol π, is a mathematical constant that is the ratio of a circle’s circumference to its diameter, and has been claimed since antiquity to be an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, and that therefore its decimal expansion never ends or settles into a permanent repeating pattern. Here a proof is given that π can indeed be expressed as a ratio of two integers, 4/17, a fact that has unbelievably been overlooked until now. Moreover, this proof is understandable to anyone with a basic knowledge of algebra and calculus and arises from simply considering a standard integral at two values of x, x=1/4 and x=1. Of course I doubted the result at first, given that it has been overlooked for so many years, but I have checked the proof and verified it to be correct. This is a crucial and important revelation that will significantly alter all of mathematics.

### Reading aloud and enlarging mathematics

My collaborators at The Aperiodical, Katie Steckles and Christian Perfect, have launched a new mathematics magazine podcast called All Squared. In the first episode, number one1, Katie and Christian speak to Edmund Harriss about speaking mathematics out loud.

Towards the end of the conversation, they speak a little about some of the ambiguity in spoken mathematics and how this might affect blind mathematicians. Particularly, they speak about timing and ambiguity, and Christian gives the example (where the comma is a pause): ‘x plus, y squared’ and ‘x plus y, squared’. The placement of the pause changes the meaning of the formula substantially.

A few minutes ago the conference ‘Supporting Disabled Students in STEM‘ began at the Royal Society. This is the spring conference of the STEM Disability Committee (an alphabet soup collaboration between IOP, RAEng, RSC, SoB, CaSE and RS). I am not attending this, though I was asked a couple of weeks ago for two examples for a session at this.

For the first example, I was asked for a formula that would be a bit difficult to read aloud. The aim is not for an unrealistic expression or to trick the audience, but just for something that might cause ambiguity and mean that the reader would stop and think. Here is the equation (apologies for the use of images throughout):

The exercise, later today, will use two volunteers. One reads the formula aloud and the other, with back turned to the screen, will write it down. Try it now with the person on the next desk or in the next office. I’ll wait.

This is the Taylor expansion of ln(x) around x=2. It is an example from a problem sheet in my first year methods course a few weeks ago, so it isn’t unrealistic. I chose it because there are two particular points of possible ambiguity. The first is in “minus one to the power i plus one”. The second is to have the top be “all over n times two to the n”. Is the “one” included in the power? Is the “two to the n” included in the bottom half of the fraction? There are also some conventions for non-mathematicians in the room to take note of, such as the pronunciation of “ln” and the way we read out “sum over i from one to infinity”. All of this means that to be read well requires a thoughtful, mathematically-trained reader.

If the person reading the equation in the session today does it badly then the other volunteer will write the equation wrong and the point will be made. If they do it well, it will be interesting to hear them speak to the group about what choices they made when reading the equation to reduce the ambiguity.

The second example was a piece of mathematics that would need to be broken over two lines when enlarged, and that would be difficult to do so. Here the grey is supposed to indicate the edge of the page. Have a think about it: how would you break this equation to fit on the page (you can’t make it smaller!)?

Again this is an example from class, a partial fractions question. A naive solution might be to simply cut the equation at the page edge. You can hopefully see in the following image that this is unacceptable, in particular because the 2 on the next line looks like -2.

A more intelligent cut might be to take whole terms onto the next line. However, this may cause confusion because it looks like the (x+3) is part of a separate fraction. Is it one fraction plus another? One fraction multiplied by another? This perhaps isn’t unusable, provided you explain to the reader what you have done, but you are presenting the reader (your student?) with non-standard and, strictly speaking, incorrect mathematical notation. If part of being a mathematician is learning to speak and write mathematics so that other mathematicians can understand you, you are doing damage here.

Here I try to keep the two halves of the fraction together by including in-line cuts. This is still weird, particularly the bottom half, but I could imagine writing something like this by hand (perhaps with a multiplication symbol on the bottom cut) so maybe it’s okay.

Finally, here is what I think I would do. The result is mathematically correct and using fairly standard notation (we wouldn’t ordinarily use multiplication symbols in g(x) but this is unusual rather than strictly wrong).

This is hopefully a good solution, but there are still two problems. Interpreting the two functions back into the partial fractions form is an extra cognitive load for this student (not related to the intended learning outcomes of the question), and producing this formatting is a giant amount of extra work compared with the original code. Here is the original LaTeX:

Find the following integral by first resolving the integrand into its partial fractions.    $$\int \frac{x^3+x^2+x+2}{(x+1)(x-2)(x+3)} \, dx$$

And here is the adapted LaTeX code that produced the last version.

Find the following integral by first resolving the integrand into its partial fractions.    $$\int \frac{f(x)}{g(x)} \, dx \text{,}$$    where    \begin{align*}      f(x)= \, &x^3+x^2\\      &+x+2    \end{align*}    and    \begin{align*}      &g(x)=(x+1)\times\\      &(x-2)\times(x+3)    \end{align*}

The person who adapts this page must be properly trained in mathematics, so they don’t introduce errors into the notation and adapt it sensibly, and in this case they must know LaTeX to be able to write the code. This is an unusual set of skills for a generalist disability support professional, and the maths department might be unable to commit personnel to do this. Sometimes the problems are that mathematics notation is difficult to adapt, and sometimes they are to do with the practicalities of who is able to do the work.

I will leave you with this quote from Christian in the podcast.

It is very rare for someone who is blind to get through to becoming a research mathematician, isn’t it? … So is that because the culture isn’t accommodating or because maths is a thing that really is much easier to understand visually?

Something for you to think about. Listen to the podcast to hear what Edmund thinks.

If you are interested in these issues, Emma Cliffe is running a workshop ‘Mathematical Study Without Pen and Paper: Experiences, Impacts and Options‘ for the Higher Education Academy on 20th March 2013 at Manchester Metropolitan University. Attendance is highly recommended.

1. It is an open question, as far as I know, whether the second episode will be number two or number four.

### "Developing a Healthy Scepticism About Technology in Mathematics Teaching"

I have an article in the current issue of the Journal of Humanistic Mathematics (Vol 3, Issue 1). The title is Developing a Healthy Scepticism About Technology in Mathematics Teaching. This will be a chapter of my PhD thesis and provides some background context. I am following a model in which teaching draws on a body of theory which is based on scholarship as well as reflective evaluation of previous experience. So as well as a literature survey, I present a reflective account of experiences which have taken place alongside, but outside of, my PhD research that have shaped my thinking.

This journal is an online-only, diamond open-access, peer-reviewed journal with an emphasis on “the aesthetic, cultural, historical, literary, pedagogical, philosophical, psychological, and sociological aspects as we look at mathematics as a human endeavor”. They publish “articles that focus mainly on the doing of mathematics, the teaching of mathematics, and the living of mathematics”. (Quotes from the Journal’s About page.)

My article’s synopsis is:

A reflective account is presented of experiences which took place alongside a research project and caused a change in approach to be more sceptical about implementation of learning technology. A critical evaluation is given of a previous e-assessment research project, undertaken from a position of naive enthusiasm for learning technology. Experiences of teaching classes and designing assessment tasks lead to doubts regarding the extent to which the previous project encouraged deep learning and contributed to graduate skills development. Investigations of the benefits of another technology—in-class response systems—lead to revelations about learning technology: its enthusiastic introduction in isolation cannot be expected to produce educational benefit; instead it must address some pedagogic need and should be evaluated against this. Overall, these experiences contribute to a shift away from a naive enthusiasm to an approach based on careful consideration of educational need before technology implementation.

P.S. Sorry the blog has become rather infrequent and quite education-focused. I am currently splitting my time between teaching and writing my thesis, so I have little time for anything else. My employment contract is only to teach until May and my thesis is due in July.

I write to share and invite discussion of something I presented at a conference at Nottingham Trent University last week.

I have been thinking a lot about assessment methods and their advantages and limitations for a chapter I am writing for my PhD thesis. For example, I could set a paper test and mark it by hand, as indeed I set one last week and will be marking it when I finish this post, and this allows me to give a personal touch and assess students’ written work but one downside is that I can’t return marks to students very quickly. I could return marks immediately if I used automated assessment, but then setting the assessment would be more difficult and I may be limited in the range of what I could assess. And so on.

I have been trying to classify these advantages and their paired limitations. My thinking is that by viewing different assessment methods as balanced sets of advantages and limitations we can justify different approaches in different circumstances and, particularly for my PhD, explore the advantage/limitation space for any untapped opportunities, which I won’t go into now (but ask me).

Here is my current list of potential advantages that assessment could access. These advantages are each something that I think that some assessment method can offer. My question is: what am I missing? I would be pleased to receive your thoughts on this in the comments.

• Immediate feedback. This is linked to learning from mistakes, confidence and motivation. It can also prioritise procedural learning over conceptual understanding.
• Detailed, personalised feedback. Though there is much disagreement in what I have read whether a human, who can respond to individual student work, or a computer, which will tirelessly generate worked examples using the context of the question asked, will in practice provide this.
• Individualised assessment. This is achieved through randomisation of questions and is linked to repeated practice, deterring plagiarism, allowing students to discuss the method of a piece of work without the risk of copying or collusion.
• Assessing across the whole syllabus. For example, computers can’t mark every topic.
• Testing application of technique. Whether students can apply some procedure.
• Assessing deep or conceptual learning. For example, open-ended or project work may require a detailed manual review to mark. This is linked to graduate skills development, etc.
• Easy to write new questions. Assume it is easy for a lecturer to write questions that students can answer (it isn’t, but we’re talking principle here). Difficulty is introduced by having to second guess an automated system, or having to second guess students to program misconceptions.
• Quick to set assessments. Assume that writing a test manually takes time. By quickly, I really mean choosing items from a question bank.
• Quick to mark assessments. Assume that marking by hand is not quick, perhaps unless the assessment is very short and student answer format very prescribed, in which case the assessment is limited. This is perhaps linked to problems of consistency and fairness when using multiple markers.
• Easy to monitor students. Clearly marking individual work from every student by hand will give great insight, but here I refer to the ability to gain a snapshot of how individuals and the cohort are doing as a whole with a concept, perhaps very soon after a lecture that introduced that concept has taken place.
• Perception of anonymity. I’ve read that some students are happier to make their mistakes if only a computer knows. This can reduce stress.
• Testing mathematical writing. Clearly requires hand-written work.
• Testing computer skills. Clearly requires use of a computer.

For example, then it might be possible to offer ‘Easy to write new questions’, ‘Assessing deep or conceptual learning’ and ‘Testing mathematical writing’ through a traditional paper-based, hand-marked assessment, but this would preclude, for example, ‘Immediate feedback’.

Similarly, a multiple-choice question bank might offer ‘Quick to set assessments’ and ‘Quick to mark assessments’ at the expense of ‘Assessing across the whole syllabus’ and ‘Assessing deep or conceptual learning’.

And so on. I have loads of these for different assessment types.

My question really is, is there anything missing from my list that might be delivered by an assessment method?

### Podcasting update

I have a job. This is not the podcasting update, but it does affect it! If you have listened to the latest Math/Maths Podcast you will know that I will be lecturing mathematics from January while trying to finish my PhD thesis, and that we will be putting that podcast on hiatus while I do so. This means no more talking to Samuel Hansen for at least six months.

There is something you can do to fill this mathematical podcasting gap, however. Samuel is trying to raise money through a Kickstarter to allow him upgrade his equipment and improve the quality, to pay for the travel to conduct face-to-face interviews and to make this his full-time job so he can concentrate on a regular release schedule, for his work in maths (math) and science communication over at ACMEScience.com.

At Kickstarter, Samuel says:

ACMEScience.com has spent the last four years trying to do something that very few others have ever attempted, create entertaining, insightful, and interesting content about mathematics and science. Started by Samuel Hansen in the beginning of 2009, ACMEScience has produced a pop-culture joke filled mathematical panel show, Combinations and Permutations, a show that interviews everyone from the CEO of a stats driven dating site to a stand up mathematician to Neil deGrasse Tyson, Strongly Connected Components, a show that tells the stories of the fights that behind DNA, dinosaurs, and the shape of the universe, Science Sparring Society, a video interview show that has featured predatory bacteria and crowdsourced questions, ACMEScience News Now, and a series of hour long journeys into the world of competitive AI checkers computers and stories of the most interesting 20th C mathematician and much more, Relatively Prime.

You may remember that Samuel raised money through a Kickstarter before, for the extremely well-received documentary series Relatively Prime. So you might judge this as evidence that he is capable of delivering this project. However, you may also remember that if he doesn’t raise the whole amount he needs then he gets nothing.

There are various pledge levels, with various rewards. Some of these are aimed at the individual who wants to own a piece of the project. Others are aimed at people who want to sponsor/advertise via the shows and get their message out there. Looking at the level of pledges so far, Samuel could really do with a few companies or individuals who want to get a message out to a mathematics or science audience coming forward and pledging some money. Relatively Prime was very well listened to, and you could get your message to a large, focused, engaged set of listeners.

There is not long to go (only four days at time of writing) and it doesn’t look good. So please pitch in and also tell everyone you know via your own blog/podcast/social networks/etc. so that others will support his effort.

Here is the video in which Samuel makes his case. It’s six minutes so at least watch that! The Kickstarter page is ACMEScience.com by Samuel Hansen. Donating is easy through Amazon payments.