Reflections

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

Reflections


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? …

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

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.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

Reflections
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
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?

Help
Full screen

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    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.

    Buy instant access to save your work.

    Already a member? Log in now.

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.