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# Secondary Magazine - Issue 59: Focus on

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Created on 26 April 2010 by ncetm_administrator
Updated on 11 May 2010 by ncetm_administrator

# Focus on...short image sequences

“The mind does not spontaneously adopt a logical approach to the study of a subject but rather acts intuitively on the material presented to it.” Caleb Gattegno

Being able to form, hold, modify, manipulate, change and think about our own personal mental images is profoundly useful in doing mathematics. It helps us explore situations and consider possibilities and implications.

How can we help our students cultivate their powers of imagery, explore the ambiguity of words and refine their abilities to describe their mental images?

We can use various resources and different kinds of strategy.

Inviting students to create mind-pictures
Challenge students to conjure up short sequences of mental images from verbal instructions, talk about their images, and possibly draw them. For example:

Picture two points.
Stop them moving.
Are they near to each other, or far apart?
Is one higher than the other?
Bring in a straight-line segment.
Place it somewhere between your two points.
Move it about until it passes through one of the points.
Can you move it so that it also passes through the other point?
What kind of movement?
If you can’t make it pass through the other point, why can’t you?

Students’ images will be personal and idiosyncratic, and in a supportive atmosphere students usually enjoy sharing them.

This way of working with learners is described, with many examples, in ‘Geometric Images’, a handbook for teachers written by Dick Tahta, Roger Beeney, and others in 1982 for the ATM.

Instructions could be more specific. For example:

Visualise a square.
Join diagonally opposite corners with straight lines.
Join opposite mid-points of the sides.
Fit a circle inside the square so that it touches the sides of the square.
Join up the points where the circle cuts the diagonals of the square.
Join the mid-points of neighbouring sides.
How are the two shapes that you have made related to each other?
Join up all neighbouring corners of the two shapes that you have made.
What new shape have you made?

To facilitate more reflection and discussion about this situation, show students Square doodling, pausing it frequently and prompting thought by, for example, asking what they think will appear next. How do they know that the octagon is regular? How do they know that the smaller squares are congruent?

Visualising possibilities
Convince me that the area of this tetromino is halved by the cut:

Picture in your mind other ways of dividing the area in half with one cut.

Challenge students to visualise and describe possibilities before showing this animation, which you can pause and replay as many times as you like. How would you cut other polyominoes?

Can you visualise every possible pentomino?
Use this image sequence to confirm students’ images and develop discussion.

Does this image sequence generate every possible hexomino? What is its system?

Picture in your mind the ‘growing’ of a 7-point mystic rose. How many line segments will there be altogether? Invite students to talk about their ideas and their images, which they can confirm here.

Extend your ideas to a 10-point mystic rose. Use this animation to generate questions and confirm students’ mind-pictures.

Imagine slicing through a cube.
How do you know that the shape that slicing in this way creates is a rectangle?

What other two-dimensional shapes can you make by slicing in different ways through the cube?
Use this Animated Geometry film, designed by Gattegno, to prompt, encourage, support and confirm students’ ideas.

Picturing paths
Imagine rolling along a line: a line segment, an equilateral triangle, a square, a regular hexagon, a circle. What path is traced by:

one end of the line segment,
one corner of the equilateral triangle,
one corner of the square,
one corner of the regular hexagon,
the point of the circle that is originally touching the line?

Watch this sequence and pause it after just a fraction of a second. How do students think the ‘complete’ paths will look? Eventually you can ask: Are the paths as you imagined they would be?

Re-creating together what you saw
Encourage students to ‘replay’ in their minds an image sequence that you all watched, and use their descriptions of their images to stimulate classroom discussion.

You could show this circle doodling sequence, then challenge students to ‘replay it’ in their minds, describing what they are ‘seeing’.
What can you deduce about the centres and radii of the circles?

Invite students to describe what they ‘see’ when they replay in their minds this parallel lines sequence.

Show this sequence, pausing to let students discuss what they think will happen next. Can they explain how they might create a similar tessellation? What if they started with a triangle, or another shape?
Students might re-create aspects of what they see in Journey to the Center of a Triangle, which is a sequence developed in 1976 by Bruce and Katharine Cornwell. At the end of this stimulating five-minute sequence we see where the four centres go as the triangle changes shape! You can read Margaret Jones’ reflections on using this film with a group of 15-year-old pupils in MT206.

Margaret compares working with a moving film to working with software tools.

Using a short image sequence as the starting point for exploration
Image sequences often suggest further explorations and prompt students naturally to look for explanations.

For example, the Triangles in polygons sequence created by the Swiss mathematics teacher Jean Nicolet leads naturally to asking oneself whether a right-angled triangle can be created in a similar way using other regular polygons. And why does it work? Might other polygons lead to other special kinds of triangle?

During the 1940s and 50s Jean Nicolet, Caleb Gattegno and Trevor Fletcher produced short, silent animated films. You can find information about a DVD version of 22 of Nicolet’s films here.

A van shooten’s theorem animation at the ‘Maths Films’ website also prompts students to explain what they see.

Using ‘artistic’ image sequences to encourage creativity, exploration and inquiry
Some image sequences are quite dazzling. For example, think what students might create with the stimulus of Notes sur un triangle by Réné Jodoin. This five-minute film is absolutely fabulous! – packed with arrangements that slide into each other. View it ‘full-screen’! A printable sheet of equilateral triangles to use in explorations generated by watching this film is available from NRICH.

Dance Squared is another sequence of the same kind made in 1961 by Réné Jodoin with help from Trevor Fletcher.

View this issue in PDF format

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