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Using visualisation in maths teaching

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# 3. What does visualisation mean?

‘Imagery is a powerful force for perception and understanding. Being able to “see” something mentally is a common metaphor for understanding it. An image may be of some geometrical shape, or of a graph or diagram, or it may be some set of symbols or some procedure.

Visualising means summoning up a mental image of something – seeing it in your mind. Some people can actually close their eyes and “see” a picture, but for others it has much more to do with imagining, than seeing. Try to picture a cube, the seven-times table, a graph of sin x. Describe what you “see”.

The point of this is that if you really want to grasp a concept or idea, struggling to visualise it is worthwhile. There are many aids to visualisation. Diagrams or symbols on paper often help, or physical apparatus. Trying to “say what you see” (or cannot see!) can be helpful, too. Visualisation and articulation go hand in hand.’

(Open University, 1988, p. 10)

Click on the link below and read ‘Mental mathematics’ by Chris Bills. This text suggests a number of mathematical visualisation activities for you to try out. You are asked to think about the mental strategies and processes you have used (metacognition) in engaging with these activities.

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## Activity 2: Questionnaire

• For each of the questions below write few very brief notes of your own.

1. What do we mean by the term mental imagery?

2. What do we think is the value (for the learner) of working with images?

3. Does mental imagery lend itself more easily to shape and space topics?

4. How can we use it in number work?

5. Can this work have a place within the School Curriculum?

6. Do pupils improve their mental imagery as they are given more opportunities to use it?

7. Is mental imagery part of the problem-solving process?

8. How do you think it improves concepts and skills?

9. Describe any classroom-management issues and any useful hints for the use of this approach for the first time.

10. Can students of different abilities cope with this approach?

• Click on the link below to read the various components of the questionnaire and compare your views with those of other teachers.

(Source: These responses were collected by Brenda Stevens and Aled Williams, of Oxford Brookes University. bjstevens@brookes.ac.uk). The responses here were made by teachers attending a course at Oxford Brookes University.

1. What do we mean by the term mental imagery?

• Learning by example – analogy – except that the analogies are visual pictures. Then instead of having objects in front of us we imagine it.

• Any mathematics done just in your head: for example, imagining shapes/pictures and moving objects/lines/angles around.

• Whenever we use pictures or diagrams, whether moving or still, imagined (straightaway) or initially drawn to aid understanding.

• When any form of problem is tackled using only the imagination.

• Anything which requires a ‘picture’ – mental or drawn(?)

2. What do we think is the value (for the learner) of working with images?

• Because we can say ‘it is similar to this’ and the image in particular can usually be less ambiguous than words and have more meaning.

• It helps to think more about other topics and apply the idea themselves.

• Sometimes it is a lot easier than doing something practically.

• Visual imagery should start with images with which the learner is familiar, thus providing confidence.

• The visual is very powerful and young people are already more comfortable with television and computers than with abstract concepts and procedures.

• It provides analogies for concepts and procedures that can connect a new concept with an old one, or elaborate the existing concepts. It can therefore be more memorable and aids understanding.

• The skill of being able to manipulate mental images improves spatial abilities.

• Once the problem has been visualised, not only does the problem appear easier to comprehend, but any solutions found will tend to be remembered longer.

• It strengthens and/or creates ideas and concepts; variety leads to fun.

3. Does mental imagery lend itself more easily to shape and space topics?

• Not necessarily. We may have to look harder for algebra and number, but worth doing because it makes it more fun. Probably not appropriate in data handling.

• Yes, as this is the main topic that it would be used for if it has been previously taught by another teacher. Also, when pupils (and teachers) are new to mental maths this is generally the easiest topic to grasp.

• It has a different role in other topics in providing analogies and contexts for concepts. There are probably more ways of using it for shape and space.

• Yes, as it is easier to imagine a picture or object.

• I think so although algebra can be spatial

4. How can we use it in number work?

• For example, ratios, equal ratios. I found it useful to use images of cakes being split up etc. I suppose division is always a good one for mental imagery (and division stands in a good relationship to number).

• Fractions, decimals, percentages and percentage change.

• Four operations.

• Ratio; sharing and mixing.

• Adding – climbing up; subtraction – falling down.

• Fractions – imagine an object divided up.

• Imagine a number line.

• Fractions, decimals, percentages, ratio, four rules of number, solving problems.

5. Can this work have a place within the National Curriculum?

• It should because it makes learning fun and memorable and methods can be recalled/replicated in the future.

• Could be mentioned, but is more a method of delivery.

• A teaching method rather than a separate topic.

• Definitely.

• Of course – can teaching have a place in the National Curriculum?

6. Do pupils improve their mental imagery as they are given more opportunities to use it?

• Pupils seem to get better at understanding your way of communicating whatever style you use (within reason) so of course they would get more skilled at using this ‘tool’ in particular.

• Yes, but they find it very difficult.

• I believe that it is like a muscle in the brain that can improve like any other with exercise. So, yes.

• Yes.

7. How do you think it improves concepts and skills?

• It facilitates communication of meaning. A good natural ‘pointer’ at the concepts you want to get across. It is a fairly universal medium of communication. (Sometimes language – words, writing and reading are obstacles, but ‘imagining’ is easy for most of us.)

• It can strengthen existing concepts, or provide a new way of understanding the concept by clarifying and extending it. It provides a justification for procedures and it is easier to remember.

• If they have a mental image it is easier to recall.

• By visualising concepts you can generally better see the whole picture.

• Lots of different ‘pictures’ – overall view – generalise.

• More likely to remember having constructed your own idea of what it is.

• Don’t need to remember – it’s just there.

8. Is mental imagery part of the problem-solving process?

• I don’t know. It isn’t for me, not naturally. Maybe my problem-solving skills would benefit from trying to use it. For those brought up using it, so it comes naturally. Why not?

• To solve fully any problem you need to be able to have an image or at least some of it. In creating this image, you are starting to solve the problem.

• Yes, but often pupils don’t use it and find questions more difficult because of it.

• Yes – finding a picture to fit the situation is an invaluable tool in problem solving. It is a way of understanding the situation.

• Yes.

9. Describe any classroom-management issues and any useful hints for the use of this approach for the first time?

• I strongly believe that complete silence must be achieved.

• I have found it useful to get the class to ‘travel’ to a different place where they can imagine all of this happening and are allow to relax. (I even gave a two-minute meditation lesson on one occasion with some excellent results.)

• Try to empty any images by imagining a blank sheet of paper. Focusing on the picture helps them to concentrate (e.g. ask them what colour their shape is).

• Make it accessible to all by transferring it to paper so everyone knows what they should have seen and then building on the image so it has some kind of purpose – use it as an introduction rather than an end in itself so that they can see it as a tool to be used again. Differentiate so that all can visualise something.

• Little and often. Use to help explanations or setting up problems. Clarifying ideas. Finding out misconceptions. Classes like imagining things so it could be used to settle a class at times, though it depends on what is being imagined.

• Need silence; need to trust each other, not mock. I told them that we were going to try something new and some of them might not feel comfortable, but that didn’t matter. If it didn’t work we’d stop and do something else. If it was going to work they needed to be quiet and mustn’t laugh at each other – needed to support each other. Then settled down by ‘imagine you are walking through a desert … (first time – one off of 15 minutes; bottom set Year 8, 24 pupils).

• It worked well; some wanted to tell what their pictures were. Asked if I would hypnotise them again next week!

10. Can pupils of different abilities cope with this approach?

• Yes, but the images for lower abilities must be kept simple or related to a real-life situation.

• I see absolutely no reason why not; although obviously it should be recognised that not all pupils will be able to follow the activity fully right through to the end.