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.