Swan, Malcolm, Mathematics Teaching
I love it when students surprise me. I recently worked with a group that appeared to believe that area and perimeter are related (if you increase one, you increase the other). I tried to help them realise that this is incorrect by introducing a counterexample: "Look at this sandwich". I took a bite out of one side. "What has happened to its area?" "It has gone down." "And what about its perimeter?" Some were surprised: "It goes up!" I took another bite. "The area has gone down still more and the perimeter has gone up again!" And now it was my turn to be surprised: "By the time you've finished eating it, the perimeter will be enormous!" I could see we were heading for fractals . . .
How often do your pupils surprise you in a mathematics lesson? And how often are they surprised? As well as describing an emotional response the word is also used to describe an intentional action. Do you deliberately set out to cause surprises in pupils? In my dictionary it defines surprise as 'to encounter or discover unexpectedly or suddenly', and also 'to cause to feel amazement, delight or wonder'. Surely it is every good teacher's job to provoke these emotions?
As I visit classrooms, however, it seems to me that most lessons seem deliberately designed to reduce the possibility of surprises arising. The typical triple X teacher* (explanation, example, exercise) knows exactly what mathematics the lesson will contain before it starts, limits the range of responses possible through rapid closed questioning, and reduces unpredictability and pupil creativity by sticking closely to the textbook or powerpoint presentation. (Examples of this mentality are offered by Barbara Ball, Howard Tanner and Sonia Jones in this issue. Barbara mentions how one teacher was unwilling to use a piece of software simply because it generated examples at random and thus offered no chance to know answers in advance; Howard and Sonia describe how 'interactive' whiteboards are often used in non-interactive ways). Such teachers introduce amazing theorems in matter-of-fact ways without pausing to wonder at how they can possibly be true in every case. Contrast the emotional response of the 40-year-old philosopher Thomas Hobbes (1588-1679) when he first came across Pythagoras theorem: "He read the proposition. By G-, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible!" 1
Reflecting on my own life, I can see how surprises have been vivid learning experiences. I can still remember when I was 10 and my dad bought me the Readers Digest Junior Omnibus. There was one question in it that really caught my imagination:
HOW HIGH IS THE PILE ?
Imagine that you have a very large sheet of tissue paper, uy a thousandth of an inch thick. The exact area and thickness don't matter. Now tear the sheet in half and place one half on top of the other. Then tear the two pieces in half again and stack them together, making a pile four pieces high. Tear these in half, making a pile of eight pieces. If you keep this up until you have done it fifty times, how high will the pile be? Make a few guesses before turning to the answer on page 171.
Reproduced with permission from The Reader's Digest Association Ltd, Reader's Digest Junior Omnibus 1959.
I just couldn't believe the answer on page 171. It said 'over seventeen million miles'. I couldn't believe it. I even tried to tear a newspaper in half fifty times to check the answer! Many years later, as a teacher, I asked my own class this same question and got the same disbelieving response. They also tried to prove me wrong by tearing paper!
Surprises aren't just about the results. I have often been surprised by methods. I was once asked the following question at a job interview for an engineering company. 'If you keep rolling a dice and keep track of the running total (maybe by moving a counter along a board), what is the probability that the total will hit exactly 100 after some number of rolls? …