How to Play like a Mathematician

[Originally posted on Edmund Harriss' blog Maxwell's Demon, this is a transcript of a talk at the Twitter Math Camp 2014, a truly energising event, teacher organised peer professional development. Anyone interested in education, whether parent, academic, teacher or administrator should check it out.]

Zero…

Start by clearing your mind.

One…

Now imagine one dot pop into view.

Two…

A second dot joins it. Let the two dots flow around each other, rotating and getting closer and further apart.

Three…

Now a third dot, creating a line or a triangle…

Four…

Keep on adding, with each addition try to see all the dots, find a shape you like…

Five…

Six…

Seven…

Eight…

Nine…

Ten…

Eleven…

Twelve…

Thirteen…

Fourteen…

Fifteen…

Sixteen…

Seventeen…

This is about having fun and playing with math, which often sounds a little:

TMC14_Talk_Images.002This is not the holy grail, it is not even a challenge to bring into the classroom. Teachers have too many challenges, sometimes the challenge is just to get through the day without messing up too badly. It is an encouragement to relax and have fun, yet remember that this fun is part of your teaching prep!

TMC14_Talk_Images.007Playing with maths can often start with going back, returning to something you know well, and trying something new, testing an idea. If it fails try something a little different, or go back to work out how it went wrong. If it works, can you try everything? Mathematicians can say everything and really mean it! Even then do not settle, go back with your new knowledge and try something new. You might notice once you have started you cannot escape! You can always just stop. This is play not work. Though it might not be relaxing, just as playing a sport is exciting, fun and cathartic but you put effort in.

This is why this can build into your teaching, once you have fun you have a chance to help your students have fun. If they have fun they will put far more effort in than if you have to push them. Also I do not feel that mathematics has a huge number of facts, but isolated they are not that useful, going back and playing with ideas helps build the dense web of connections that really drives understanding.

General strategies are great, but it can be hard to know where to start, I will describe two tools:

  • Analogy and the concept of same/different (mathematics is the world’s greatest metaphor!)
  • Breaking rules! (yes mathematics is often about creating them, but also about changing them and seeing what happens).

To get further, we need an example, and not one that will lose half the audience just with its title so…

Counting.

TMC14_Talk_Images.009Three dots, are they the same or different? They are in different positions, but are the same shape. We have to be clear what we mean.

TMC14_Talk_Images.010Now we take pairs of dots, we can spin them around and pull them apart. We could say they were the same if they can be moved on top of each other. Yet to define that precisely we have to use most of plane geometry. We have not even counted past two and we already need that!

TMC14_Talk_Images.011Getting to three the line and the triangle, different in ways that the pair of dots can never be.

Lets change tack, we have been looking at how the same number of dots can be different, what about how different numbers of dots can be the same?TMC14_Talk_Images.012

These patterns for four, six and eight have some similar features. How might we describe those precisely so we can identify other ones? Saying that the numbers are all even is an obvious way to do it, but maybe they also share something with this:TMC14_Talk_Images.013

Like the earlier examples nine dots drawn like this form a rectangle (specifically a square). Following this definition we can define prime numbers (technically composite numbers!).

TMC14_Talk_Images.014Here are another collection of dot patterns that share features, one dimension, two dimension and three dimension, and at this point reality gives up on us. Yet we really went past our page after two, we can use the notions of analogy to push further. We know the next pattern will have sixteen dots. For example we can make this image, with lines to show the structure. Can you find the eight cubes?

TMC14_Talk_Images.016With a little work from here we can work out that an nn-dimensional cube has 2n (n-1)-dimensional faces. So we know very little about 172 dimensional space, but we do know that a hypercube in that space has 344 faces! Playing with some of these tricks we can get this:

TMC14_Talk_Images.017There is a lot more to discover in this image. If you are interested in getting a version send me an email, I am looking into options.

Lets move to the other trick, breaking the rules. Mathematics is made of rules, yet there is not one rule that is not broken somewhere else in mathematics. For example this might make you uncomfortable:

7 + 7 = 2

2 + 1 = 2

If I say instead that seven months after July (the seventh month) is February then the first makes perfect sense. In this case 7+7 is still 14 but 14 is the same as 2, we have modular arithmetic.

That trick will not work for 2 + 1 = 2. Yet in Chemistry two hydrogen molecules combine with an oxygen molecule to create two water molecules. There is an even greater rule, though one that has been enshrined in legend. Yet this image shows what happens when we divide by zero (at the centre)!

TMC14_Talk_Images.021(the mathematical trick is to use what is called the Riemann sphere).

In conclusion playing with math can happen with the simplest structures and lead to a variety of thoughts and adventures. No one should be shy of having a go!

Here is a neat animation from my play:

Notes

I have a list of some other materials to inspire your mathematical play, and there is a whole world of examples in Sue van Hattan’s book Playing with mathematics. That should be available for pre-order soon!

Many other have explored the idea of simple pictorial versions of numbers, often using prime factorisation. With dots and circles, with monsters, or even to make a game. Although my personal favorite are these dots, with their illusion of simplicity.

 

A. E. Kyprianou: The UK financial mathematics M.Sc.

A. E. Kyprianou: The UK financial mathematics M.Sc. arXiv:1405.6739v2 [math.HO]

Abstract:

Postgraduate taught degrees in financial mathematics have been booming in popularity in the UK for the last 20 years. The fees for these courses are considerably higher than other comparable masters-level courses. Why? Vendors stipulate that they offer high-demand, high-level vocational training for future employees of the financial services industry, delivered by academics with an internationally recognised research reputation at world-class universities.

We argue here that, as the UK higher education system moves towards a more commercial environment, the widespread availability of the M.Sc. in financial mathematics exemplifies a practice of following market demand for the sake of income, without due consideration for the broader consequences. Indeed, we claim that, as excellent as such courses can be in intellectual content and delivery, they are mismatching needs and expectations for such education and confusing the true value of what is taught.

The story of the Mathematical Finance MSc serves as a serious case study, highlighting some of the incongruities and future dangers of free-market education.

Why Do Americans Stink at Math?

An article by in the NYT; it is about America, but is very timely in the context of the National Curriculum reform in England. A quote:

It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious “new math,” only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.

The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them. [Emphasis is mine -- AB.]

Read the whole article.

 

 

Tony Gardiner: Teaching mathematics at secondary level

A. D. Gardiner, Teaching mathematics at secondary level. The De Morgan Gazette 6 no. 1 (2014), 1-215.

From the Introduction:

This extended essay started out as a modest attempt to offer some supporting structure for teachers struggling to implement a rather unhelpful National Curriculum.  It then grew into a Mathematical manifesto that offers a broad view of secondary mathematics, which should interest both seasoned practitioners and those at the start of their teaching careers.  This is not a DIY manual on How to teach.  Instead we use the official requirements of the new National Curriculum in England as an opportunity:

  • to clarify certain crucial features of elementary mathematics and how it is learned — features which all teachers need to consider before deciding `How to teach’.

Continue reading

Campaign to change UK immigration practices that deter foreign students

Please sign this petition to the Home Secretary and Minister of
Education https://you.38degrees.org.uk/p/foreignstudents to implement
the House of Lords Science & Technology Report recommendations
http://tinyurl.com/nbqbvca
international_students[1]

When we really need to send the message that international STEM students
will get a warm welcome in the UK, they’re getting the cold shoulder and
they are heading elsewhere. We’ve seen over the last few years how
international student numbers have fallen dramatically. As a result
we’re missing out on the talent and the economic and cultural
contribution that international students bring when they come here to
study, and our competitors are reaping the rewards.

Please sign the petition and ask your friends to sign it. The more
support we can get behind this campaign, the better chance we have of
succeeding.

MY REASON FOR STARTING THIS PETITION
The LMS June newsletter drew my attention to the House of Lords Science
and Technology report (April 2014) http://tinyurl.com/nbqbvca that
recommends a change in immigration practices relating to foreign
students.

This is a campaign of national and international importance, and not just
about a few individuals, but I decided to act because this week some
friends of ours from Chennai met with the unpleasant face of UK
immigration practices, and academic visitors coming to work with us have
had similar experiences. Mandira, age 17, who has permanent residency in
the UK, and was on her way to the UK to do a 2 week course for High
School students in Cambridge that starts this week, was turned back and
put on a plane back to India because she has not been in the UK for 2
years. She was with her mother and younger sister, and they also planned
to visit some other universities because Mandira wants to apply to UK
universities to study medicine starting in September 2015. This was
probably a mistake on the part of some officious individual but
never-the-less it is typical.

Can you also take a moment to share the petition with others? It’s
really easy – all you need to do is forward this email or share this link
on Facebook or Twitter: https://you.38degrees.org.uk/p/foreignstudents

Thank you! Toni

Toni Beardon OBE Retired from University of Cambridge NRICH/MMP
http://mmp.maths.org
& African Institute for Mathematical Sciences Schools Enrichment Centre
(AIMSSEC) http://aimssec.aims.ac.za

Programmes of study for Mathematics at Key Stage 4

Programmes of study for  Mathematics at Key Stage 4, which will be taught in schools from September 2015 alongside the new English and mathematics GCSEs, are published today. This appears to be the final pack of statutory documents:

Can you pass the maths test for 11-year-olds?

A sample KS2 test based on the official publication from Standards and Testing Agency,
2016 key stage 2 mathematics test: sample questions, mark scheme and commentary,
was published in The Telegraph. One question attracts attention. In The Telegraph version, it is

A question as published in The Telegraph.

The answer given is £12,396.

And this is the original question from 2016 key stage 2 mathematics test: sample questions, mark scheme and commentary

The official version of the same question

In my opinion, both versions contain serious didactic errors. Would the readers agree with me?

And here are official marking guidelines:

Official marking guidelines

And the official commentary:

In year 6 pupils are expected to interpret and solve problems using pie charts. In this question pupils can use a number of strategies including using angle facts or using fractions to complete the proportional reasoning required.
Pupils are expected to use known facts and procedures to solve this more complex problem. There are a small number of numeric steps but there is a demand associated with interpretation of data (or using spatial knowledge). The response strategy requires pupils to organise their method.

Mathematics after 16: the state of play, challenges and ways ahead

On Wednesday 02 July the Nuffield Foundation published report Mathematics after 16: the state of play, challenges and ways ahead. It argues that reforms to GCSEs and A levels risk undermining the government’s goal of universal participation in post-16 mathematics education, particularly if new ‘Core Maths’ qualifications are not backed by universities. The report brings together a wide range of evidence and warns that plans to make GCSE Maths more demanding, detach AS from A levels, and replace the modular structure in favour of terminal exams could actually discourage students from continuing to study the subject beyond the age of 16.

The report is available to download from the Nuffield Foundation website.

The Math Myth

D. Edwards,   The Math Myth, The De Morgan Gazette 5 no. 3 (2014), 19-21.

Abstract

I’ve been concerned with what skills those who are working as scientists and engineers actually use. I find that the vast majority of scientists, engineers and actuaries only use Excel and eighth grade level mathematics. This suggests that most jobs that currently require advanced technical degrees are using that requirement simply as a fi lter.

[A version of this text appeared in the August, 2010 issue of The Notices of The American Mathematical Society.]