Clicking infinity

On Wednesday I was asked to give a talk to a small bunch of S6 (Yr 13 in in England) pupils who were visiting Edinburgh from Fife. It wasn’t any particular special occasion – the enterprising teacher just wanted his students to get out and learn some exciting and different mathematics. It was the perfect opportunity for me to try out a new piece of technology that I’d heard my boss raving about: clickers.

A clicker

A clicker is like one of those things they have in Who Wants to be a Millionaire where the audience votes for what they think is the right answer. It is an absolutely wonderful teaching aid and we are very lucky to have them at the University of Edinburgh. It means that students can tell the teacher their thoughts without letting anyone else know what they are thinking, so they needn’t worry about the embarrassment of having the wrong answer.

The clickers we use are exactly those pictured on the left. There are 6 buttons which can be used for multiple choice questions, and also a True/False option. The software that comes with the clickers is capable of storing a huge amount of data about your sessions, which really comes into its own when you are monitoring a specific class over many weeks rather than just having impromptu sessions. You can see whether students are improving, how often they change their minds about questions, and even (if you have a strict seating plan) how ideas are spread around the classroom.

I decided to make my class about infinity, using the story of Hilbert’s Hotel to hold the plot together. My first question, just to get people used to the clickers, was a simple true/false question: “Infinity exists only in our imaginations”. There was a fairly strong preference for ‘false’ from the class, which was interesting for me because I would usually vote the other way myself. The students gave examples of ‘real’ infinities that I would argue are purely abstract mathematics, such as the infinity of numbers or infinities in fractals. It’s actually quite nice to meet people who believe that abstract thoughts are as real as anything else in life.

Take 2 balls out, put 1 back, repeat. How many are left in the end?

The surprises didn’t stop there, and I really believe it was the clickers which made the session work. One of my favourite infinity questions is the ‘balls in a barrel‘ paradox which I learnt from the ever-wonderful Colin Wright. If balls numbered 1,2,3…etc are in a barrel, and at each time step two are taken out and one replaced, then after infinitely many time steps how many balls are left in the barrel? (a) None (b) All of them (c) One (d) Half of them, or (e) Not enough information to decide? As I’d hoped, people were very split on this question, with a small majority going for either (b) or (d). A lot of the students were very shy and I don’t think they would have volunteered an opinion without being able to do it anonymously. But once they saw that nobody else really knew the answer either, they were more inclined to speak up in favour of the option they had voted for, and we really got a great discussion going. (If you don’t know the answer yourself, have a good think about it before reading Colin’s article!)

My favourite clicker question was at the end, where I was basically proving uncountability. (I got the idea for how to incorporate this into Hilbert’s Hotel from an xkcd chat forum!) We’d got to the point where I’d done the diagonal argument and asserted the existence of an element which was not in the infinite list we had assumed contained every element. I asked them if they agreed with this. Usually when I teach this I just assume that the argument is crystal clear, and students usually nod and smile. This time, using the clickers, I found that the class were exactly split each way! Half agreed that the new element was definitely not in our list because of the way it was constructed, while half asserted that it must be in the list, because that’s what we assumed at the beginning. Once again we got to have a great discussion, examining our implicit assumptions and coming to the mind-bending conclusion that there are different sizes of infinity!

Get people to vote using fingers, then hold them against their chest so nobody but you can see

I really hope that more schools and universities will start using this method of teaching. You don’t even need the fancy technology – some voting cards or fingers against chests are adequate for the purpose. It comes into its own with the quieter members of the class, giving them a voice they otherwise wouldn’t have had, and pushing weaker students to have opinions about things they’d otherwise not bother thinking about. There’s certainly a skill in asking the right questions and in not being scared to ask things which seem obvious to you. For example, my boss Toby asked his class  “2 ≤ 3, true or false?”. Stupid question, right? But half the class disagreed, asserting it was wrong because two is less than three, not less than or equal to.  Misconceptions occur at the deepest levels and we must work hard to root them out!

I’d love to hear other people’s stories of using clickers or other similar teaching methods. What have been your most surprising results?