Copyright © University of Cambridge. All rights reserved.
This article first appeared in
Mathematics Teaching 207, published by the Association of
Teachers of Mathematics .
We rely on visualising when we solve problems. Sometimes we
create an image of the situation that is being discussed in order
to make sense of it; sometimes we need to visualise a model that
can represent the situation mathematically before we can begin to
develop it, and sometimes we visualise to see 'what will happen if
...?'. But are there other ways in which we visualise when solving
mathematical problems and if so how can we encourage, value and
develop visualising in our classrooms?
This article is based on some of the ideas that emerged during
the production of a book (and accompanying CD) which takes
visualising as its focus (Piggott and Pumfrey, 2007). It was while
we were working on this book we began to identify problems which
helped us to take a structured view of the purposes and skills of
visualising that we thought worth sharing with you.
We often associate visualising in mathematics with drawing pictures
or diagrams as an aid to getting started on problems.
But visualising has a much wider role to play in problem solving
including supporting the development of ideas and facilitating
communication of results and understanding. In these senses it is
not just about pictures and diagrams.
For example, sometimes we place particular emphasis on using
visualising to help understand and develop a plan to solve a
problem. In producing such a visualisation, the problem solver is
identifying the key components of the problem and the relationships
between them. This process has two main elements:
- an internal model or visualisation (described as imagery by
Crapo et al. (2000))
- an external representation (described as a visualisation by
Crapo et al. (2000)).
Crapo et al. emphasise the importance of the interplay between
these internal and external representations which support the
development of an effective model. Key to this is both the physical
representation which sparks ideas about how the model can be
improved, and also discussion and communication with others to come
to a shared visualisation.
If we want learners to utilise and improve their capacity to
visualise, we need to identify why visualisation is important (the
purposes of visualising) and what visualising skills we want our
pupils to develop. To do this we need to know what a visualising
opportunity is, make that opportunity available and have a language
with which to talk about it.
Purposes of visualising
We have been able to identify three purposes for visualising:
- to step into a problem,
- to model,
- to plan ahead.
The first of these purposes relates closely to the idea of
'getting started'. The second and third purposes go beyond this and
suggest that visualising occurs in the depth of problem solving,
not just at the beginning.
Visualising to step into the problem:
Here visualisations are used to help with understanding what the
problem is about. The visualisation gives pupils the space to go
deep into the situation to clarify and support their understanding
before any generalisation can happen. For example:
One out some under:
Imagine a stack of twenty cards in numerical order (1 at top
to 20 at bottom).
Imagine the cards being dealt - one out, one under, one out,
one under, one out, one under ...
What would be the last card left in your hand?
Investigate piles of different numbers of cards and justify
any rules you discover. In particular:
- With a stack of one hundred cards what is the last card left in
- A stack of a thousand cards - what would be the last card left
in your hand this time?
- Can you always predict which will be the final card in your
hand no matter how many there are in the pack?
An introduction to the problem
Imagine a stack of 10 cards in order from 1 (Ace) on the top
to 10 on the bottom.
Remove the top card and put it on the bottom.
What is the card on the top now?
Repeat this process, carrying on from where you finished, but
this time move four cards.
What is the card on the top this time?
And again, this time move six cards.
What is the number on the top now?
Repeat with different numbers of cards until the class can
confidently predict the top card.
The visualisation moves on to a pile of 4 then 8 cards using
the rule 'one out - one under'.
The pupils are led through the context with a simple example.
The teacher models the process being investigated with frequent
pauses to check for understanding. Whilst the teacher is
demonstrating, the pupils are asked to visualise and describe what
will happen next. The teacher's actions give a focus, and a
motivation for the visualisation and enable immediate feedback. The
use of language to explain what is seen 'in the mind's eye' also
helps with reinforcing the process. Pupils are being encouraged to
make sense of the situation through the visualisation.
Visualising to model a situation:
This is particularly useful when the situation is physically
unattainable, in other words to try to see the 'unseeable', for
example the inside of a 3D object, or considering a case involving
a very large number. To illustrate this, have a look at the problem
Cubes Within Cubes .
Supporting pupils in modelling
Ask the children to imagine they have an unlimited supply of
interlocking cubes (all the same size) in different colours.
Invite them to imagine starting with one yellow cube.
This is covered all over with a single layer of red
Describe what you see:
- How many red cubes touch the yellow cube face-to face?
- How do the cubes along the edge of the red cube touch the
- How many red cubes touch the yellow cube in this way
- How do the red cubes in the corners of the large red cube touch
the yellow cube?
- How many touch in this way?
- How many red cubes are there altogether?
- How many red and yellow cubes?
- Can you explain why?
In this example it is not possible to 'see inside' a large cube
and, although smaller cubes can be constructed as the layers
increase, it is not possible to see the centre cube at the same
time as the surrounding layer. The modelling in this problem is
two-fold. Initially the pupils are encouraged to think in stages
and later to use the visualising strategies to tackle the main
problem. An outcome of the lesson can be pupil production of story
boards as representations of the visualisations and stages they
used whilst problem solving. These images have two aspects:
- they reflect the imagery the pupils used,
- they represent the route the pupils may have taken to obtain
In this particular problem, there are multiple representations
and there are potentially multiple routes to a solution. For
example in the figure below, pupils show two different ways of
describing the number of surrounding cubes:
Visualising to plan ahead:
This involves using visualising
during the problem-solving process to anticipate. In other words
asking yourself: 'What will be the consequence if I do this?'. This
is related to problem posing 'What would happen if ...?'. It is not
possible to ask the question 'What if?', if you have not thought
ahead and any thinking ahead necessarily includes visualisation.
For example, have a go at the problem
Supporting pupils in thinking
- What would be your first move?
- Your second move?
- Is there more than one option for the next move?
- What if I move the frog to here next?
- Does it matter which move you choose when you do have a choice?
To answer a question like 'What if I move the frog to here
next?', I would need to be aware that this is one of several
possibilities and therefore have a visualisation of the wider
context of the problem (as opposed to a form of tunnel vision -
going for the first thing that comes into your head). A lesson
plenary could focus on a solution to the problem, but there is also
the opportunity to discuss the range of recording systems the
pupils adopt and the role of thinking ahead.
We have spoken about why you might visualise. But, what are
the underpinning skills that support the visualising when problem
solving? That is, the ways in which we visualise as we step into,
model and plan. We have started to think about specific visualising
skills that we should be offering opportunities for our pupils to
practise and hone.
the 'imagery' of Crapo et al (2000) this involves being able to
(close your eyes) and focus on a problem, then pick out salient
features to represent and make sense of the situation. In the
Cubes Within Cubes
it is necessary to spend the time creating
an internal representation that you can draw on as you work through
the problem. Without this internal image any ownership of the
generalisation seems impossible.
Identifying : Being
able to identify a useful image or representation of an idea, which
may be someone else's, that means something to you. This
representation helps you see or describe the structure of a
problem. This is why a teacher might produce diagrams and images to
support their pupils - sharing their visualisation may help
learners to access the problem situation.
Comparing : Being able
to scrutinise different images to identify what is the same or
- Being able to compare other people's representations with our
own. For example in
Cubes Within Cubes , comparing the two visualisations given
above with your own.
- Being able to identify the general and the specific in a
representation and their significance in terms of the problem at
hand. For example in a problem involving polyominoes, you might
start with a domino (from which you are going to build triominoes),
asking learners to visualise putting two squares together.
Recognition that there is an infinite number of orientations
(general) is important but, in preparing to build triominoes, you
only need to consider one orientation and to have that very clearly
in your mind.
- Trying to hold more than one image in your head. For example:
remembering a starting point and being able to 'rewind' when a
sequence of moves does not work; keeping more than one aspect of
the problem in 'view'. For example in the problem
Roundabout the visualisation requires you to be thinking about
both the circle moving round and the locus its centre makes.
able to make connections by remembering the processes or
underpinning structure rather than individual images. For example,
it is possible to build all the hexominoes given all the
pentominoes. The underpinning structure involves the visualisation
of each pentomino (which could involve the visualisation of each
tetromino ...). On top of the pentomino structure is superimposed a
system for systematically generating all the hexominoes. In this
case it might be by visualising the movement of a sixth square
around each pentomino to produce a family of 'related' hexominoes.
: Being able
to describe a personal visualisation to an audience. This may be
necessary when trying to explain or clarify thinking or share an
interpretation. There are times when the visualisation is not
simply an image but has an element of variation. For example in the
the visualisation might include the effect of
enlarging or shrinking the square (or circle).
The skills underpin the processes, which extend beyond using
images and diagrams to help familiarise the problem solver with the
problem situation. Visualising is at the heart of problem solving
There is more to visualising than meets the eye!