Mathematics Developmental Continuum P-10 – Space
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About spatial visualisation: 3.5
Howard Gardner, in his book 'Frames of Mind: The Theory of Multiple Intelligences' (1983) proposed that spatial ability is one of the seven separate human intelligences. It is an ability that is essential in many occupations and it also makes an important contribution to being able to solve mathematical problems. It is also essential in many everyday activities such as sewing, assembling toys and furniture, fixing things, finding your way, reading a map and drawing.
Different modes of thought are used in mathematics. Some people have a strong preference for verbal-logical thinking, others for spatial-visual thinking and most people are in between. Verbal-logical and spatial-visual thinking are believed to be carried out in opposite hemispheres of the brain.
While spatial ability seems to be something that some people are born with, many studies have shown that it can be improved through structured experiences and training. For example, there are studies that show that children who use concrete materials in primary schools tend to have better spatial ability than those who do not, and that geometry lessons where students study transformations (flips, slides and turns, etc.) improve spatial ability more than geometry lessons that only involve learning names for shapes.
One of the most difficult spatial tasks is to mentally rotate an object. An example is to imagine what an object looks like from the back and underneath, or in cross-section. This ability improves as students get older, but can be quite limited for students even in the junior secondary years. Practical experience with real three-dimensional objects, in mathematics and other school subjects, is helpful, as is careful observation of the world around us e.g. for drawing. However, students need more than just hands-on experience. They need activities which encourage them to predict the effect of mental transformations first and then check with real objects, not always just using the real object.
There is evidence to suggest that, on average, girls are not quite as able as boys in three-dimensional visualisation. This may be due to different early experiences and interests, or it may be due to slightly different innate abilities. This does not mean that all boys will have good spatial skills, nor that all girls have poor spatial skills. It is important to include practical visualisation activities for all students.
Some milestones of development of spatial ability
Piaget had a general theory that very young children see the world only from their own point of view and that an important part of cognitive development in spatial reasoning and in many other cognitive abilities was for them to be able to 'decentre'. Part of his evidence for this was the 'three mountains task' which is sketched below. The child has to describe the view that someone on the other side of the three mountains would see. Many children even up to age 9, think the view is the same. The difficulty of this task is partly due to its abstract nature. In Martin Hughes’ ‘naughty boy’ task, a young child has to place a naughty boy doll so that he cannot be seen by the policemen dolls. (The brown lines in the diagram are walls, and the policemen are standing near them). If the task is carefully explained, most students beginning school should perform well. When there are more policemen or more walls, the task gets harder. About half of four year olds can solve a problem with 3 policemen and six wall segments. This scenario helps the child put themselves in the place of the naughty boy.
Piaget's 3 mountains task
Hughes' naughty boy task
Some authors separate spatial abilities into two components. First, spatial orientation is the ability to understand and operate on relationships between objects in space. This ability is needed for finding your way in a building, for example. The second component is spatial visualisation, which enables a person to carry out mental movements of two and three-dimensional objects in space. This is the ability required to do Piaget's three mountains task. Either imagine yourself behind the mountains looking at them, or imagine the mountains themselves rotating around in front of you.
Other authors add the ability to understand visual representations.
Others believe that the essence of true spatial ability is being able to form and transform visual images as an organised whole. Some people are able to deal with the whole image at once, whilst others have to attend to it in parts.
The ability to mentally transform objects increases with age. Mentally transforming an object may include rotating it mentally (e.g. imagining the mountains from the other side), sliding it (e.g. imaging the view if I drew the mountain with the tree on top on the left hand side instead) or flipping it (e.g. reading a notice in a mirror).
Combinations of rotations around different axes are even more difficult. For example, it is easier to visualise what my cup would look like it I gave it two quarter turns around a vertical axis (so that the handle moved from the right to the back to the left), than if I gave it one quarter turn around the vertical axis and then a quarter turn around a horizontal axis (so that the handle went from the right to the back to the top of the sideways cup).
References
Clements, D.H. & Battista, M.T. (1992) Geometry and Spatial Reasoning. In D. Grouws (ed.) Handbook of Research on Mathematics Teaching and Learning. (pp. 420 - 464) New York: Macmillan.
Donaldson, M. (1978) Children’s Minds. Glasgow : Collins.
Gardner, H. (1983) Frames of Mind: The Theory of Multiple Intelligences. Fontana.