### Why teach Geometry?

There are a number of reasons why geometry holds an important place in the curriculum. We list some of these below.

- We live in an obviously three-dimensional world that we walk through, explore and use every minute of every day. We need to come to grips with this environment in order to carry out even the simplest of tasks. So it is important that we learn both a vocabulary with which we can talk about the space that we occupy and the properties that this space has.
- In particular, we need to know about shapes as part of our everyday vocabulary because many common objects are in the shape of squares, circles, cubes, spheres etc. (or are sufficiently close approximations to these shapes). From time to time we need to construct certain shapes to wrap up presents and so on. Some people, such as carpenters, construct shapes on a daily basis as part of their job.
- Position is an important aspect of our world. Many billions of dollars has been spent in satellites that enable ships to accurately determine their exact location (the Global Positioning System). On a more mundane level we use maps to find our way to a friend’s new house and give directions to people on the street. All such activities are geometric in nature.
- Aspects of geometry are predominant in art and in aesthetics. We like to have symmetrical shapes in our houses. (How many asymmetric doors and windows can you remember seeing?) Wallpaper patterns are prime examples of symmetry in our everyday life. And art, dance, and fine arts generally, are often based on the pleasing nature of symmetry.
- Geometry has been central to the historical development of mathematics itself. Through it were developed concepts such as abstraction, generalisation, deduction and proof. It still provides an avenue through which students can come to a deeper understanding of the nature of mathematics itself. Although such an understanding of mathematics and geometry is beyond the scope of primary students, nevertheless, they can, and should, still acquire spatial intuition and knowledge of various geometric concepts. This should provide a firm foundation on which secondary school and university geometry, and indeed mathematics itself, can be built.
- Spatial thinking is essential to scientific thought. The visual-pictorial mode of thought is one of two (the other is verbal-logical) that some believe occur in different quantities in individuals. For those of us for whom visual-pictorial is the dominant mode, problems that have no apparent geometric properties are often tackled geometrically. Einstein, for instance, said that the elements that he thought with were not words but certain signs or less clear images. These he could reproduce and combine.

### The Skeleton

Despite its power as part of mathematics, much of what geometry is able to convey mathematically is beyond the primary school syllabus. Hence in Levels 1 to 4 no consideration is given to theorems, either proving them or learning them or using them. The major aim of geometry at these levels is to develop geometric awareness in students. By this we mean that we want them to develop three things:

- a feeling for shape and the properties of objects so that they can tell objects apart, classify them and know their basic properties
- an intuition of two- and three-dimensional space so that they are happy to locate themselves, find directions, and navigate using simple maps
- an appreciation of pattern in space so that they can see the symmetry in wallpaper and other common objects, make enlargements and reductions of simple shapes, and understand the underlying ideas behind tessellations.

Though these three aspects of the geometry curriculum are related and can, and at times should, be taught together, they provide a convenient skeleton on which to base the geometry units of Levels 1 to 4.

Shape covers those aspects of the curriculum that deal with both two-and three-dimensional geometrical shapes, their names, properties, construction and relationships to each other.

Position and Orientation develops from simple concepts of beside, under, etc., to following directions, drawing and interpreting simple maps, using simple coordinates, using a compass and so on up to the beginnings of the study of loci.

Transformation involves lines of symmetry, symmetrical patterns, tessellations, translation, reflection, rotation, and the enlargement and reduction of shapes.

### General Background

We now want to consider how students learn geometry. In particular we want to discuss the contributions of Piaget and Inhelder, and the van Hieles. We also give an example from outside of geometry that might illustrate what we are trying to say.

Piaget and Inhelder suggest that students construct their internal representation of space by active manipulation of their environment. This means that this is not just a ‘reading off’ of the spatial world but results from active interaction with it. This concept is generally accepted. The implication for teaching here is that it is not enough to just show students a triangle, for instance, but rather that students will learn more about triangles if they pick one up and explore and play with it. This is very much in line with constructivist theories in other areas of learning.

The van Hieles, and subsequent workers, propose six levels of thought in geometry. (We take the number six for convenience. This is still under debate.) We will refer to them here though, as stages. In this way there will be no confusion with the Levels of the curriculum.

Underpinning these stages are the following characteristics:

- learning is not continuous – it proceeds in jumps that reveal discrete stages of thinking
- stages are ordered and qualitatively different – to get to the next stage, students must essentially have mastered their current stage
- there is a progression from implicit to explicit – knowledge becomes deeper as the students move through the stages
- each stage has its own language – a student at one stage will not be able to converse with one at a higher stage because of the way that the concepts have developed between the two stages.

Here then are the six stages.

Stage 0: Pre-recognition. At this stage students recognise shapes and can even name some. However, they perceive only some of the properties of a shape. Hence they may be able to distinguish between a square and a circle but not between a square and a triangle.

Stage 1: Visual. Here students recognise shapes but recognise them as a whole. That is, they have a mental image of the overall shape rather than checking out its individual properties and then concluding what it is. The students may be able to draw the shape quite accurately but to describe it they will have to use some general expression as a circle being ‘like a wheel’. At this stage it may be difficult for teachers to understand the students’ use of language. They may use words like ‘lop-sided’ and ‘slanting’ to describe shapes.

Stage 2: Descriptive/Analytic. Logic starts to creep in at this stage. Shapes are recognised by carefully checking out their properties. So while students at Stage 1 say ‘this is a square because all shapes like this are squares’, at Stage 2, they say ‘this is a square because it has the properties that I know that a square has’. Such properties usually have to be perceived by the students themselves. They may not be able to understand or apply descriptions imposed by the teacher. Hence it is necessary to give students at this stage, simple yet accurate definitions of shapes.

Stage 3: Abstract/Relational. At this stage students understand formal definitions and can use them in a hierarchical way. So, for instance, a rectangle is a parallelogram with some extra properties (the four right angles). They also understand the idea of necessary and sufficient conditions to test an object. Students may occasionally be able to put together a logical argument but generally any proof that they need to know must be copied and repeatedly practised in order for them to reproduce it.

Stage 4: Formal Deduction. Now students are able to develop logical arguments by themselves and they can appreciate the necessity for such arguments. They also understand the difference between definitions, axioms and theorems. At this point they have grasped the idea that geometry is a logical system that can be used to describe the world.

Stage 5: Rigour/Metamathematical. Here students are able to reason formally. They are able to work geometrically in the absence of models to check their work against. Reasoning involves the formal manipulation of geometric statements.

Which age, which stage? Generally speaking primary age students are able to reach Stage 2. Very few of them get to Stage 3. Stage 4 is likely to be reached by students at the top end of secondary school. Stage 5 is beyond all but the most able secondary school students.

There is still some debate about the precise nature of these stages. For instance, it is not clear whether or not they are actually discrete. It is possible that a student may be at one stage in one area and at another elsewhere. However, the stages do seem to provide a worthwhile basis on which to build a picture of the development of students’ geometric learning.

### New Cities

We want to illustrate what we have said above under the van Hiele stages, by using an ‘everyday’ example that shows that the stages also appear in learning in non-school situations.

When you first get to a new big city you can easily get lost. If you are lucky, a friend will show you around. You are at Stage 0. You have a vague notion of how to get to important places like the city centre, the cinema and the local pub. But you probably go to places using a rote learned route.

Gradually you get to recognise the red house on the corner and the old church further down and the lady who is often outside weeding her garden. You start to fill in the gaps that you didn’t really know existed and your route then is less rote-learned. You have much more of a ‘feel’ of where you are going and how you get there. This is much like van Hiele’s Stage 1.

At the next stage, you realise that when you were first shown the route, you were shown the ‘simple’ way of getting to where you wanted to go. Your friend had taken you by a longer route that was easier to follow than the more direct route. When you think about it, going for several blocks and turning right and then going for several more blocks, can be replaced by a shorter route that requires several turns and a short walk across a park. You begin to think about the way that you are going, especially if you are late one day. And you think that there ought to be a better way.

At Stage 4, you have a good internal ‘map’ of your surroundings. Now you can tell visitors to the city how to get places, even places that you haven’t been to. You can even say that you’ll need to turn at the red Victorian house, No. 355, and go past the grey Uniting Church, and you might even see Betty Martin out weeding her garden. You also know why your friend initially told you to go a certain way. The shorter route often has heavy traffic so it takes a while to cross the road.

Finally you might get a job in the City Engineers Office. Then you might suggest that certain roads should be turned into one-way routes, that parking might be better at an angle than parallel, that maybe another bridge is needed and where it should go. Or you may end up as a taxi driver and have an intimate knowledge of the streets throughout the whole city.

Of course, because of your needs, you can’t be expected to have a strong overview of the city as a whole. So you might be particularly strong in one or two suburbs and perhaps even the central business district. It is possible for you, therefore, to have developed to different Stages in different areas of the city.

So while van Hiele suggested the stages as stages for the learning of geometry, there is nothing special about the geometry. Much of what he is saying applies to other things as well. This probably means that the stages are not so surprising and so not so strange and hard to remember or learn.

### References

Bobis, J., Muligan, J., Lowrie, T and Taplin, M., 1999, Chapter 4: Finding connections in space: shape, structure, location and transformations, in *Mathematics for Children*, Prentice Hall.

Booker, G., Bond, D., Brigs, J. and Davey, G.,1998, Chapter 5: Space and Geometry in *Teaching Primary Mathematics*, South Melbourne: Addison Wesley Longman.

Clements, D.H. and Battista, M.T., 1992, Geometry and Spatial Reasoning, in D. Grouws (Ed.), *Handbook of Research in Mathematics Learning and Understanding*, pp 420-464, New York: Macmillan.

For more information on the progression of mathematical learning in the Geometry strand you may want to try the Geometry Curriculum Exemplar. Note that while the progressions shown in the exemplars are still valid, their links to curriculum levels and AOs relate to MiNZC (1992) rather than to NZC (2007).