A group activity using visualisation of squares and triangles.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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.
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.
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 .
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: