By Swan, Malcolm | Mathematics Teaching, January 2007 | Go to article overview
Save to active project


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:


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?

The rest of this article is only available to active members of Questia

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • Ad-free environment

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
Loading One moment ...
Project items
Cite this article

Cited article

Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited article



Text size Smaller Larger
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

While we understand printed pages are helpful to our users, this limitation is necessary to help protect our publishers' copyrighted material and prevent its unlawful distribution. We are sorry for any inconvenience.
Full screen

matching results for page

Cited passage

Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.

Are you sure you want to delete this highlight?