## Thursday 14 May 2015

### A Maths Game with a Psychological Twist

Today I'm going to tell you about one of my favourite ways to kill a spare fifteen minutes in a maths lesson. It's great because it requires absolutely no preparation, no prior knowledge, and no differentiation. All you need is a class set of mini whiteboards and pens.

Tell students that you're going to ask them to write down a number. It must be a whole number (no decimals, fractions or surds) and it must be zero or greater. They have to write this secretly and not show anyone their number. Tell them you are going to count down from 5. At 2, everyone's pens need to be down. At 1, everyone needs to show their boards. The winner, and this is the important bit, is the person who has the smallest number that no one else has.

This should be simple enough to understand, and if a student doesn't quite get it, they can at least write down a number and participate until it clicks.

When students hold up their boards, first look for zeros. If there's just one, the zero wins. If there's more than one, tell all the zeros to put their boards down and then look for ones. Repeat until you have found a number that only one person has written. They are the winner. This doesn't take very long at all, even with 32 students like my year 7 class.

I have found that this game intrigues students immediately. They have to try to predict what everyone else is thinking. I have found that I can usually predict who in the class will write zero. Zero rarely wins, and I've never seen it win in the first round. I've found that the winning numbers vary massively from class to class. The results are also very different when I've played this with adults. I'm sure there's a lot that can be explored here in terms of psychology (and economics, actually) but I'll leave that for you to think about.

I like to try to encourage strategic thinking and annoying my students at the same time by saying things like "oh, so three won that time. That means you are probably thinking about choosing three this time. But if other people think that too, you won't win, so you'd better choose four instead. But now that I've said that, you can't choose four, it's too obvious, so..." The students usually interrupt me at this point and beg me to shut up because I'm ruining their strategy.

The beauty of this game is it can be done with year 7s, year 13s, and even the Maths faculty as a pre-meeting warm-up. I have never met a class (or group of adults even) that doesn't enjoy this game.

A more mathematical version of this game is to have the same rules but this time the winner is the student whose number is the closest to the mean of all of the numbers. This is too difficult to do as a whole class, so I do this in table groups. The four students each choose a number, they calculate the mean, then the winner is whoever's closest. They keep a tally of how many times each person wins so they can declare a winner at the end of the fifteen minutes.

What's really nice about this variation is that students will be calculating means with much more enthusiasm and motivation than if they were meaningless numbers on a worksheet. Also, by playing this game and trying out different strategies, students begin to appreciate the nature of the mean. There will always be a student who will write down a million, thinking it will skew the mean towards them. However, if the other three numbers are low they still won't win.

You can also play this game with the median instead of the mean. It doesn't quite work with mode! Actually, maybe one of my amazing readers could come up with a way of making a mode variation. Comment below!

Try one of these games out when you have ten minutes to spare. Let me know how it goes by commenting below.

Emma x x x

## Thursday 7 May 2015

### The Mathematics of Voting

I'm going to do something completely out of character and write a TOPICAL post! Yes, I may live underneath a proverbial rock (what, there was an earthquake recently?) but even I am aware that there is a General Election today in the UK. In fact, I was the first person at my polling station to vote this morning!

I've mentioned in a previous post that I love Lewis Carroll. He combines two of my favourite things: books and maths (Lewis Carroll is the pen name of mathematician Charles Dodgson. Please keep up!)

Dodgson became involved in college elections in the early 1870s at Oxford university where he was a professor. He became interested in the theory of voting, of the accuracy and fairness of different voting systems.

## First Past The Post

Dodgson was not a fan of this voting system. He claimed "the extraordinary injustice of this Method may be very easily demonstrated". He then gives an example to show how stupid it is:
Suppose there are 11 electors and 4 candidates a, b, c and d. Each elector ranks the four candidates in order of preference. The 11 columns here show their choices:

 a a a b b b b c c c d c c c a a a a a a a a d d d c c c c d d d c b b b d d d d b b b b

It's easy to see that a is considered best by three of the electors and second best by the rest. But in actual fact, it is b who ends up winning, even though he/she was considered the worst by seven voters.

I don't think Dodgson looked at "Alternative Vote", although he did write about lots of other systems.

## The Method of Elimination

In this method, each voter chooses their favourite, and then the one who gets the fewest votes is eliminated, and the process is repeated (a bit like Big Brother? The TV show, not the Orwellian thing). This method at first seems pretty flawless. However, consider the following situation:

 b b b c c c d d d a a a a a a a a a a a b c d c d b b b c c b d d c d c d d d b b c c b

Notice that a is everybody's first or second choice, and hence appears to be the best candidate. However, he/she will be eliminated first. c will be elected instead.

## The Method of Marks

In this method, each voter is given a specified number of marks that they can divide between the candidates. Then the candidate who gets the most marks wins. Dodgson said that this method would be perfect as long as the voters divided their marks fairly: giving most to their favourite but some to the candidates that they wouldn't mind electing. But Dodgson commented that "since we are not sufficiently unselfish and would assign all our votes to our favourite candidate, the method is liable in practice to conicide with that of the simple majority [first past the post] which has already been shown to be unsound".

I hope you voted today! Let's get rid of the current education-ruining idiots!

Emma x x x

All quotes are from Robin Wilson's "Lewis Carroll in Numberland", a book I highly recommend.