Teaching geometry

There is a considerable amount of research in mathematics education that concerns the teaching and learning of geometry. It is neither sensible nor feasible to attempt to summarize it all (for a comprehensive review, see Clements 2001). Instead, a selection of issues is addressed below covering theories of geometric thinking, learning, and teaching.

In order to teach geometry most effectively, and give some coherence to classroom tasks, it is helpful if, in your preparation and teaching you keep in mind, and highlight where appropriate, key ideas in geometry. These include:

Invariance: In 1872, the mathematician Felix Klein revolutionized geometry by defining it as the study of the properties of a configuration that are invariant under a set of transformations. Examples of invariance proposition are all the plane angle theorems (such as Thales’ theorem in task 8.5), and the theorems involving triangles (such as the sum of the angles of a plane triangle is 1800). Pupils do not always find it straightforward to determine which particular properties are invariant. The use of dynamic geometry software (see task 8.6) can be very useful in this respect.

Symmetry: Symmetry, of course, is not only a key idea in geometry but throughout mathematics, yet it is geometry that it achieves its most immediacy. Technically, a symmetry can be thought of as a transformation of a mathematical object which leaves some property invariant. Symmetry is frequently used to make arguments simpler, and usually more powerful. An example from plane geometry is that all of the essential properties of a parallelogram can be derived from the fact that a parallelogram has half-turn symmetry around the point of intersection of the diagonals. Symmetry is also a key organizing principle in mathematics. For example, probably the best way of defining quadrilaterals (except for the general trapezium, which is not an essential quadrilateral in any case, since there are no interesting theorems involving the trapezium that do not also hold for general quadrilaterals), is via their symmetries.

Transformation: Transformation permits students to develop broad concepts of congruence and similarity and apply them to all figures. For example, congruent figures are always related either by a reflection, rotation, slide, or glide reflection. Studying transformations can enable students to realize that photographs are geometric objects, that all parabolas are similar because they can be mapped onto each other, that the graphs of y = cos x and y = sin x are congruent, that matrices have powerful geometric applications, and so on. Transformations also play a major role in artwork of many cultures - for example, they appear in pottery patterns, tiling's, and friezes.

The teaching and learning of proof in geometry

While the deductive method is central to mathematics and intimately involved in the development of geometry, providing a meaningful experience of deductive reasoning for students at school appears to be difficult. Research invariably shows that students fail to see a need for proof and are unable to distinguish between different forms of mathematical reasoning such as explanation, argument, verification and proof. For example, a large-scale survey in the US found that only about 30% of students completing full-year geometry courses that taught proof reached a 75% mastery level in proof writing. Even high-achieving students have been found to get little meaningful mathematics out of the traditional, proof-oriented high school geometry course.

Corresponding difficulties with proof have also been found with mathematics graduates. A number of reasons have been put forward for these student difficulties with proof. Amongst these reasons are that learning to prove requires the co-ordination of a range of competencies each of which is, individually, far from trivial, that teaching approaches tend to concentrate on verification and devalue or omit exploration and explanation, and that learning to prove involves students making the difficult transition from a computational view of mathematics to a view that conceives of mathematics as a field of intricately related structures. Further reasons are that students are asked to prove using concepts to which they have just been introduced and to prove things that appear to be so obvious that they cannot distinguish by intuition the given from what is to be proved.

Nevertheless, despite the sheer complexity of learning to prove and the wealth of evidence suggesting how difficult it can be for students, there are a few studies that show that students can learn to argue mathematically. One promising approach is that being developed by de Villiers (see de Villiers 1999). De Villiers points out that, in addition to explanation, proof has a range of functions, including communication, discovery, intellectual challenge, verification, systematisation, and so on. These various functions, de Villiers argues, have to be communicated to students in an effective way if proof and proving are to be meaningful activities for them. In fact, de Villiers suggests that it is likely to be meaningful to introduce the various functions of proof to students more or less in the sequence . Focusing on explanation, de Villiers argues, should counteract students becoming accustomed to seeing geometry as just an accumulation of empirically discovered facts in which explanation plays no role.

There can be a tendency to teach geometry by informing students of the properties associated with plane or solid shapes, requiring them to learn the properties and then to complete exercises which show that they have learned the facts. Such an approach can mean that little attempt is made to encourage students to make logical connections and explain their reasoning. Whilst it is important that students have a good knowledge of geometrical facts, if they are to develop their spatial thinking and geometrical intuition, a variety of approaches are beneficial. For example, some facts can be introduced informally, others developed deductively or found through exploration.

To teach geometry effectively to students of any age or ability, it is important to ensure that students understand the concepts they are learning and the steps that are involved in particular processes rather than the students solely learning rules. More effective teaching approaches encourage students to recognize connections between  different ways of representing geometric ideas and between geometry and other areas of mathematics. The evidence suggests that this is likely to help students to retain knowledge and skills and enable them to approach new geometrical problems with some confidence.

When planning approaches to teaching and learning geometry, it is important to ensure that the provision in the early years of secondary school encourages students to develop an enthusiasm for the subject by providing opportunities to investigate spatial ideas and solve real life problems. There is also a need to ensure that there is a good understanding of the basic concepts and language of geometry in order to provide foundations for future work and to enable students to consider geometrical problems and communicate ideas. Students should be encouraged to use descriptions, demonstrations and justifications in order to develop the reasoning skills and confidence needed to underpin the development of an ability to follow and construct geometrical proofs.