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?' Many months afterwards, I came across a surprising method for estimating the answer in an instant (Can you guess it?).2
'Surprise', the theme of this special issue, was coincidentally the theme of this year's Institute of Mathematics Pedagogy (IMP06) (see box below) and this issue contains several articles by people that attended. Over three days we experienced many mathematical surprises through working on problems together, we made surprising connections, we looked at ways to create surprise in classrooms and even considered the ethics of engaging pupils' emotions and trying to 'engineer' surprises.
In this issue, Anne Watson and John Mason offer several examples of tasks that we explored together; Jenni Back offers a few more and also raises the ethical issues; Geoff Faux considers how we respond to surprising pupil 'howlers' and Jenny Piggott reflects on how she surprised herself! Mundher Adhami takes a more analytical view and considers the design of activities and the different types of surprise that can occur during different phases of them. The quotes and questions on page 46 -7 are some of the thoughts and questions that we all shared. So this issue should give you a good idea of what you missed! Just one further bit of self-indulgence; I offer a few of my own favourite mathematical surprises from IMP06:
* Is it possible to design a drill that can make square holes?3
* What proportion of all the natural numbers contain the digit 3?4
* You have a good supply of identical bricks. Place a brick at the edge of a table, so that it hangs over the edge. Now place another brick on top of this one so that it hangs over the edge even further. If you continue in this way, what is the largest overhang you can create without the whole lot collapsing?5
* Fold a piece of A4 paper twice as shown below:
What is the resulting shape? What is its perimeter?6
Other articles in this issue also resonate with this theme. Laurinda Brown and AIf Coles remind me that surprises often occur when we suddenly see something in a different way or when we recognise a new connection. This often generates an emotional reaction, such as laughter. Dick Tahta and Helen Williams pick up this theme in their response. My own fascination with teaching began when I tried to 'get alongside' pupils and spend time listening without interrupting. I tried to predict how they would think in particular situations and was often astonished by how wrong I was. Such listening experiences should form part of every teacher education programme. Barabara Jaworski takes this a step further and reminds us of the power of sharing classroom anecdotes and stories to help us reflect on teaching and learning.
I now regularly try to engineer surprises in classrooms. As surprises, by definition, conflict with expectations, I try to magnify the surprise by making expectations explicit at the very start. I use questions that I'm pretty sure will cause alternative ideas and common 'misconceptions' to surface and list these on the board without comment. I try to get some commitment to these wrong answers - often you can provoke more than half the class to give the same wrong answer. A typical example might be: 'If a price is first reduced by 20%, then increased by 25%, how will the final price compare with the original one?'. When commitments have been made, I'll introduce a new way of thinking about this type of problem, maybe through a group activity that uses a range of different representations (in this case percentage, decimal and fraction multipliers printed on cards dial pupils sort). As pupils work with diese you can often hear them returning to the original question with new insights. Towards the end of the lesson we discuss the surprises they have found. There's nothing new in this approach, I know. We have been using it for years at Nottingham and have research results diat show that it can produce more permanent learning compared to traditional expository approaches.7 Interestingly, I recently tried to incorporate this approach into algebra teaching materials, but most of the teachers I trialled it with rejected the notion of giving their low attainers questions at the beginning of the lesson that they 'can't do'. They said it would 'undermine confidence' and 'cause confusion'. Their position was that you must show and tell pupils the correct methods and ideas before asking pupils to use them. I argued that I wanted to surprise pupils with the implications and limitations of their own intuitive ideas before attempting to introduce new ones. I was not trying to cause confusion - I was trying to expose confusion where it already existed. I wanted pupils to realise that they needed to learn something new and where there was already some understanding, we could build on that.
One of the teachers I have recently observed now begins every lesson with questions such as: 'Tell me what you already know about . . ." There have been many surprises and he says that by taking the replies seriously it has revolutionised his teaching.
The Institute of Mathematics Pedagogy (IMP) is organised by John Mason, Malcolm Swan and Anne Watson and is held annually over three days. It normally involves about two dozen mathematics educators (teachers, teacher-educators, researchers, curriculum developers) working together in small groups on promising classroom activities and reflecting on the process. These reflections are carried out from diverse perspectives. Colleagues focus variously on the mathematics itself, such as their own engagement with it, its meanings, or progression in concept formation; on the collaborative process; on the emotions and values involved in the work; on the pedagogic value of the activity; and on how the activity or the process link with other practices and with theory.
John, Malcolm and Anne will be presenting IMP07, the 7th annual Institute of Mathematics Pedagogy, from 30th July to 2nd August next year, email email@example.com to be put on the circulation list.
* I am indebted to Jane Annets for this description.
1 This quote is taken from David Acheson's wonderful little book that is lull of surprises: 1089 and all that, published by OUR 2002.
2 The answer is not 1/6! The method I saw used the fact that the mean roll on a single dice is 3.5.
3 Yes it is possible to drill square holes. see for example the website http://uppcr.us.edu/faculty/smith/reuleaux.htm
4 This problem is taken from Clifford Pickover (1995) Keys to Infinity, Wiley, pp3-7. Consider the set of natural numbers up to 10^sup n^ (n>0). The proportion of these numbers containing the digit 3 is 1- (9/10)^sup n^. As n tends to infinity, this proportion tends to 1. So one might say that the probability of randomly choosing a natural number that contains the digit 3 is 1. And there is nothing special about 3...
5 This well-known problem is discussed in Martin Gardner (1971) The 6th Book of Mathematical Diversions from Scientific American, University of Chicago, page 168. The surprising answer is that the overhang can be as large as one wishes!
6 I am indebted to Corinne Angler for this gem.
7 These research results are summarised in Swan (2006) Collaborative Learning in Mathematics, NRDC/NIACE.
Malcolm Swan teaches at Nottingham University.…