Imagery, Spatial Ability and Problem Solving
McLeay, Heather, Mathematics Teaching
Heather McLeay describes a pilot study to investigate the extent to which learners use imagery in a variety of spatial problems.
'The difference between good and poor problem solvers is often the extent to which they use imagery'
(Wheatley, 1997: 295)
'I preferred visual methods of solving mathematical problems, finding that these frequently suggested shortcuts that the algebra missed.'
(Presmeg, 1997: 300)
Is imagery important in developing problem-solving skills? What is it about spatial ability and solving spatial problems that seems to differ between individuals?
Spatial ability is often regarded as special skill possessed by the few. Moreover, it is often said that girls are less able than boys at spatial tasks and some researchers have even set out to show this. Learners of both genders may grow up with a wide range of experiences which can help or hinder their acquisition of spatial awareness. Their ability to solve spatial tasks may be related to the range of experiences they have had and it is often suggested that since boys play with construction toys more often than girls do then this can develop their spatial awareness. Concrete experiences will give rise to some form of mental images in the brain recording these experiences. Does the recalling of past experiences help develop imagery? In what way do imagery and spatial ability interact?
Learners need concrete experiences of the objects and situations encountered in mathematics so as to become familiar with these objects and to 'know' about them. Later, the learner may be able to recall mental images and abstract from them. Wherever mathematics can be embedded within the learner's experience then this helps form the schemas necessary to deal with abstractions.
An example of abstractions based upon recalling a mental image of a concrete experience occurs when learners work at linking plastic cubes together and view them from different angles, turning and toppling them. The use of isometric paper to represent these objects can be introduced, leading to the use of visualisation to 'imagine' the rotation of a range of shapes. Similarly, pupils could experience making and folding nets for cubes so that later they may be able, not only to fold a net for a cube mentally, but to adapt the idea to four equilateral triangles arranged in a continuous strip and imagine how it may be folded up to make a tetrahedron.
Imagery is useful not only for 3D tasks but in all kinds of problem solving. It is suggested here that one way to improve pupils' problem-solving ability is to encourage pupils to use imagery and visualisation strategies. Indeed, it has been reported by Wheatley and others, that pupils with good problem-solving skills are frequently those who are good visualisers. Ben-Chaim, Lappan and Houang (1989:58) stated that "Visualisation provides the learners with additional strategies potentially enriching their problem solving repertoire." This suggestion, that imagery aids creative problem solving in unfamiliar problems, is also supported in the psychology literature. Kaufmann (1985) reported that whereas linguistic strategies are helpful in familiar situations, for new or unfamiliar situations visual imagery is more effective.
David Fielker (1993:23) provides a very useful book on the topic of mental geometry and one of his imagery tasks concerns visualising and manipulating a cube:
'Start with a cube. Cut a little piece off each corner. What have you got now? How many faces has it? What shape are they? How many of each shape? How many corners?'
These tasks are suitable for beginning to teach learners to visualise. (Note that it is possible to present these tasks to pupils concretely by using cubes of potato!) John Mason has worked in this way with adults over many years (see for example Mason, 1988). More recently, the new Open University geometry course ME627 (Johnston-Wilder and Mason, 2005) encourages students to polish their imagery skills by means of audio and dynamic geometry files. …