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Article by Jenni Way# The Development of Spatial and Geometric Thinking: 5 to 18

### In the beginning . . .

### Piaget and Inhelder

### Sensori-motor Stage 0-2 years

### The four basic topological concepts are as follows:

### Pre-operational Stage 2-7 years

### Concrete Operational Stage 7-12 years

### Formal Operational Stage 12-18 years

### References

Links to the University of Cambridge website
Links to the NRICH website Home page

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Age 5 to 16

Published 2011 Revised 2012

This is the first in a series of articles looking at the development of spatial reasoning which aim to provide some insight into the way spatial thinking develops in children, and draws on a range of reported research. The first article in the series focuses on the work of Piaget and Inhelder. The next two look at coordinating space in drawings and the importance of instruction.

The development of spatial awareness is already underway in new-born babies. To begin with, a baby's world is limited to the space close around him/her. Gradually, the baby explores objects within reach and begins to develop concepts such as 'near', 'far', 'up', 'down', 'big' and 'little'. At first these concepts are relative to oneself - "I reach out but can't touch it", or, "I can touch it but it won't fit in my hand". As the child gains the ability to move around, his/her world expands and new spatial ideas such as 'here', 'there', 'in', 'out', 'around', 'under' and 'over' form. Eventually these concepts are linked with language and refined and clarified to apply to a variety of situations.

Research by Piaget and Inhelder (1956, 1960) suggest that early spatial conceptions are topological in nature. These basic topological ideas are very general and inclusive, and so give an infant a very broad understanding of his/her spatial world that can be refined with more detailed and complex perceptions. These perceptions can be described through the features of various types of geometry.
The observations gathered by Piaget and Inhelder led them to propose four stages of development in spatial thinking.

Babies notice certain elementary features of objects, such as size and some contour, in a topological way. For example, a baby notices mother's eyes are inside the face shape. At first, the location of objects in space is seen in relation to the infant's own body. Later, as the infant begins to move around, the position of each object in space is referenced to its surroundings. For example, a
first perception of a ball might be that 'it is within reach of my hand', and a later view might be 'I see it under the table'. According to Piaget and Inhelder perceptions of surroundings include events, such as mother or father entering the room, and not just objects.

- Proximity - the relative nearness of an object or event to any other object or event.
- Order - the sequence of objects or events (in time) according to size, colour or some other attribute. For example, if three toys are suspended in a line over a crib long enough for an infant to become familiar with them, he/she will notice if the sequence of the toys is changed.
- Separation - an object, event or 'space' coming between other objects or events. It also involves distinguishing between objects and parts of objects.
- Enclosure - an object or event surrounded by other objects or events, which involves the ideas of inside, outside and between.

The idea of a simple closed curve is also important and helps to explain why very young children perceive shapes such as circles, squares and triangles as being essentially the same shape, particularly when they draw their own.

Children begin to represent spatial features through drawing and modelling. Their topological thinking is evident in their drawings. For example, in the drawing of a duck below, done by a five-year-old, the sky and the ground are represented as separate objects - there is no comprehension of the horizon. Both eyes are drawn on one side of the head because, to the child, the important feature
is that they are inside (enclosed within) the head shape (McNally, p.29). As is typical around this age, the child does not yet possess the type of thinking that can be described by Projective Geometry, and which would allow him/her to imagine the other side of the duck.

Gradually, between the ages of about 4 and 9 years, the child begins to perceive and represent objects from different points of view and incorporates ideas of perspective. The placement of features or objects in relation to each other and taking account of vertical and horizontal relationships becomes part of the child's way of viewing the world. These sorts of ideas can be classified as
belonging to the type of geometry called Projective Geometry. In the drawing of 'Dogs playing soccer', done by a 7 year old, evidence of this type of thinking can be found. When asked why the dogs had only one eye she replied, "The other one's on the other side but we can't see it". When questioned about the numbers of legs drawn for each dog, she explained that the dogs on the left were running
so we could see all their legs, but the dog on the right was standing still so two legs were hidden from view (the third appendage is a tail!).

The child also begins to use the ideas associated with Euclidean Geometry such as distinguishing between straight and curved lines, specific shapes (like squares and circles), the length and number of sides and angles. These 'measurement' concepts allow children to bring objects and parts of object into relative proportion in their drawings.

The development of the co-ordination of horizontal and vertical planes is illustrated in the sequence of drawings below (McNally pp.45-46). Children ranging from 4 years to 10 years were asked to draw liquid in a tilted jar on a table, and to draw people or trees on a hillside. The youngest drawer clearly demonstrated Topological thinking, with the liquid simply shown inside the jar, and the
people enclosed by the hill. Gradually, as spatial thinking matures, the co-ordination of vertical and horizontal can be seen.

The understanding of geometric concepts continues to develop until complex formal systems of plane geometry can be used.

While Piaget and Inhelder suggest that the development of perception as described by the types of geometry are sequential (i.e. Topological, Projective, Euclidean), other researchers believe that all types of geometric thinking continue to develop over time and become increasingly integrated.

McNally, D. (1975). Piaget, Education and Teaching. Sydney:Hodder & Stoughton.

Piaget, J. and Inhelder, B. (1967). The Child's Conception of Space. New York: Norton.

Piaget, J., Inhelder, B. and Szeminski, A. (1960). The Child's Concepti