"Chapter 1
The Role of Geometry
WHY LEARN GEOMETRY?
Geometry has an immediate intuitive appeal at a simple level. From an early age children play
with shapes, noting their obvious properties and seeing how they relate to each other. They also
observe a wide variety of both simple and sophisticated geometrical configurations in their
surroundings and acquire some of the language associated with them. They learn informally to
recognize shapes such as circles, squares and triangles and they begin to understand words like
horizontal, vertical and parallel. The first steps in the study of geometry are concerned with
naming, describing, classifying and making links to measurement, position and movement.
Describing and classifying are not necessarily trivial tasks, as becomes very evident when, for
example, we look at the properties of all the different quadrilaterals and the links between
them. Moreover they lead on to the more demanding requirements of defining and deducing.
Measurement is clearly of immense practical importance and establishes students' understanding of the ideas of length and angle and associated ideas like area and volume, but the
essence of geometry and the source of its endless fascination is the way in which deduction plays
a central role in proving results that are often both simple and surprising.
An excellent classroom example which encapsulates this element of surprise is illustrated in
Figure 1.1. Students are asked to draw any quadrilateral, to mark the midpoints of the edges
and then to join those four points to form another quadrilateral. Comparing the many examples
that are produced suggests that the result is always a parallelogram. This can be demonstrated
very dramatically by constructing the configuration on a computer screen using dynamic
geometry software and observing what happens as the shape of the quadrilateral is changed as
its vertices are dragged to different positions.

Figure 1.1 Why is it a parallelogram?

The result itself has an immediate appeal, but the proof which explains why it is true also has
its own distinctive appeal by virtue of the simplicity of the argument. In the right-hand diagram
of Figure 1.1, one of the diagonals of the initial quadrilateral has been constructed. The
midpoint theorem, or an equivalent argument based on enlargement, tells us that the diagonal
is parallel to and double the length of the two edges that lie on either side of it. This pair of

2

Teaching and Learning Geometry

opposite sides are therefore both parallel and equal in length, thus proving that the quadrilateral
formed by joining the midpoints is a parallelogram. This result, known as Varignon's theorem,
features as one of the 'desert island theorem' in The Changing Shape of Geometry, edited by
Pritchard (2003), a fascinating collection of articles which celebrates a century of geometry and
geometry teaching.
Geometrical problems can be approached in a variety of ways: it can be an experimental,
practical subject where problems are solved by measurement and calculation, but to a mathematician geometry is essentially a deductive subject which either uses purely geometrical
reasoning or embraces algebraic procedures as well. Algebra had no place in geometry until
the seventeenth century, when Descartes established the notion of coordinates as a way of
describing the position of a point in space leading to the idea of equations to describe lines and
curves. Since that time geometry has been subdivided into two broad strands. Pure or synthetic
geometry employs geometrical reasoning alone in a style which was formalized two thousand
years ago in Euclid's Elements (Heath 1967), but which has subsequently been simplified and
made more accessible. Analytic or coordinate geometry harnesses the power of algebra to
geometrical problems by using equations for curves and by employing vectors and matrices.
At school level there is an obvious progression from experimental to pure to analytic geometry,
but there are difficulties about deciding the relative importance of each and the balance between
pure and analytic in the later years of secondary schooling.
The arguments for including geometry in the mathematics curriculum are closely linked to the
reasons why mathematics as a whole is studied, but it is interesting to note that geometry, in
the form of a close study of Euclid's Elements, had a dominant place in the curriculum in the
United Kingdom and was a prerequisite for university entrance, until the end of the nineteenth
century. During the twentieth century it has been increasingly recognized that mathematics
should have a central place in the education of all school students and that geometry in some
form has a vital role in the wider mathematics curriculum. Three broad reasons for including
geometry can be identified:
• to extend spatial awareness
• to develop the skills of reasoning
• to stimulate, challenge and inform
Spatial awareness is a rather vague, but nonetheless important, idea concerned with our
ability to perceive and manipulate geometrical objects. We recognize the utilitarian role of
having a sense of shape and space in a whole range of situations in simple everyday tasks like
erecting shelves in the home or reading a map and in more sophisticated activities like those of
the builder, architect, navigator and graphic designer. Also within mathematics itself geometrical representations are significant tools for dealing with numerical and algebraic problems.
We only have to look at the sequence of triangle numbers, shown in Figure 1.2, to see how a
geometrical representation can provide both insight and appeal.
It is commonly argued that mathematics, and geometry in particular, are valuable components of the school curriculum because they provide a context for developing students'
reasoning skills. While there may be some truth in this it does not provide a sufficient justification for the dominant place that mathematics is given at school because these skills in a
general form can be learnt through other subjects. However, if it is accepted that mathematics is
important for other reasons related to its utility and the intellectual challenge and stimulation
it can provide, then geometry can be seen as offering an important way of establishing what

The Role of Geometry

1

3

6

1 0

3

1 5

Figure 1.2 The triangle numbers

Goldenburg, Cuoco and Mark (1998) refer to as mathematical 'habits of mind'. These 'habits'
relate to the ways of thinking that become habitual to the mathematician: the constant urge to
pose and solve problems, to seek patterns, to note links and connections and above all to prove
conjectures. Geometry, particularly in its pure or synthetic form, is particularly appropriate as a
way of developing these qualities because it is rich in problems and theorems which both have
an intuitive appeal and require arguments that may often be simple, but are not as routine
or procedural as is often the case with a purely algebraic approach. That is not to underrate
the importance of algebra, because it has an important role in geometry and there is much to be
gained from the interplay between different perspectives on a problem.
The third of the reasons for learning geometry is that it can stimulate, challenge and inform.
This embraces the idea that geometry is an intrinsically interesting subject in its own right which
is worth engaging with for its aesthetic and intellectual appeal and because it is part of developing an informed background to so many aspects of our world: the symmetry of so many objects
around us, the ubiquity and utility of the circle and other geometric figures, the parabolic path
taken by a projectile, the elliptical orbits of planets and so on. Moreover there are geometrical
results, of which the theorem of Pythagoras is a towering example, which are of vital practical
significance and yet also a source of great intrinsic interest. Geometry is worth doing because it
is enjoyable, but that enjoyment is often lost because the subject is so often presented in school
merely as facts to remember and procedures to be followed to solve standard problems, rather
than as something to be explored and puzzled over in an endeavour to understand and make
sense of its riches.
An interesting example which illustrates many of the key characteristics of geometry is
provided by Figure 1.3. The problem is to construct three mutually touching circles when their
centres are given. This is used in Old know and Taylor (2000) as an example of the use of
dynamic geometry software in mathematics teaching and the problem is particularly effective
when it is posed in that context, but it can equally well be presented as a problem to be solved
on paper with ruler and compasses. The configuration that is required is shown in the left-hand

Figure 1.3 Constructing three touching circles

4

Teaching and Learning Geometry

diagram of Figure 1.3, where the centres of the circles are taken as the vertices of a given triangle.
The right-hand diagram shows an abortive attempt to draw such a configuration by trial and
error. It is easy to construct two touching circles: the problem is to find the right place for them
to touch so that the third circle precisely touches the other two. This can certainly be achieved
by experimentation for a particular triangle using dynamic geometry, but as soon as one of the
vertices is moved the construction fails. It is necessary to analyse the problem to find a general
method and that provides a good exercise in geometrical thinking.
A first step is to make a rough diagram of the result we are seeking to obtain and to explore its
key features. These are indicated by the additional lines in Figure 1.4. The requirement that the
circles touch means that there is a common tangent to two circles at each touching point and
we can then see that determining their point of intersection would be a useful step. Joining that
point to the vertices of the triangle tells us that the point of intersection of the tangents is also
the point of intersection of the angle bisectors of the triangle. Thinking about this in different
terms, we see that the intersection point is the centre of the inscribed circle of the triangle and

Figure 1.4 Finding where the circles touch

that the three touching points correspond to the points at which that circle touches the triangle.
We now have a procedure for determining the position of the three points where the circles touch:
•
•
•
•

Construct the bisectors of two of the angles of the triangle.
From their point of intersection, drop a perpendicular to one of the sides of the triangle.
Draw the two circles that touch at that point.
Construct the third circle.

If this is done using dynamic geometry software the triangle can immediately be varied to see
that the construction always works. If it is done on paper using ruler and compasses, students
can draw several diagrams using different triangles. Solving the problem requires a carefully
reasoned approach which is clearly dependent on a background of geometrical knowledge
about tangent properties. It has the considerable virtue that the thinking results in an exact
procedure whose effectiveness can be demonstrated.
Many people, particularly if they do not have a strong background in geometrical thinking,
prefer to approach geometrical problems algebraically. With the touching circles problem an
algebraic approach does provide a simple procedure which leads intriguingly to an alternative
geometrical construction.
If we denote the lengths of the sides of the triangle by a, b and c and the radii of the circles by
x, y and z, we have three equations:

The Role of Geometry 5
Adding these three equations gives

where s is the semi-perimeter of the triangle. From this last equation we can express the three
radii in terms of the semi-perimeter and the sides of the triangle, as follows:

To take a particular example, consider a triangle with sides of 5, 7 and 8 units. The semiperimeter is then 10 units and the values of the three radii are found to be 5, 3 and 2. It is not
difficult then to determine which vertex to use for the centre of each circle. This point is clarified
in Figure 1.5 which shows an alternative geometrical construction based on these algebraic
results.

Figure 1.5 An alternative construct ion for the touching circles problem

To find the position of one of the touching points, we need to construct a length corresponding to the perimeter. In Figure 1.5 this has been done by producing the side A B of the triangle
in both directions, so that the lengths AD and BE are equal to the other two sides, EC and AC
respectively. The midpoint, P, of the line DE is then the required point on AB. The diagram
shows clearly how AP and BP correspond to the required radii, s - a and s-b.
While this alternative construction to the touching circle problem is not as straightforward as
the first method using the centre of the inscribed circle, it does illustrate neatly the interplay
between algebra and geometry, with an algebraic result suggesting a geometric procedure.
It should be clear that geometry has an important place in the mathematics curriculum,
because it has many characteristics that can contribute to students' general intellectual
development as well as to their mathematical development. In addition it provides practical
skills and knowledge that have useful applications in everyday situations and in a wide variety of
occupations. This leads on to the next two sections of this chapter which examine what should
be taught - the geometry curriculum - and the all important question of how the subject should
be taught and learnt.
THE GEOMETRY CURRICULUM
While there is much common ground between different countries, there are greater contrasts
in content and emphasis with geometry curricula than are to found with arithmetic or algebra.
Hoyles, Foxman and Kuchemann (2002) in a comparative study of geometry curricula in a
representative group of countries found a considerable contrast between, for example, the
Netherlands where geometry is linked strongly to the real world and France, Japan and
Singapore where there is little emphasis on the real world. There are also marked differences

6

Teaching and Learning Geometry

in how procedural and formal the approaches are and the extent to which the idea of proof is
given emphasis. There are also significant differences between countries in the relative emphasis
given to transformations, vectors and coordinate geometry as compared to synthetic geometry
in the style of Euclid. The extent to which the use of computers is encouraged or expected varies
very widely. There is clearly much interest and concern in many countries about the geometry
curriculum so the situation is far from static. The ICMI study, Mammana and Villani (1998),
reflects this widespread concern as does the report on geometry produced jointly by the Royal
Society and the Joint Mathematical Council in the United Kingdom, RS/JMC (2001).
In England the mathematics curriculum is determined by the National Curriculum, DfEE/
QCA (1999). In response to growing concerns the current version has specified in a much clearer
way than earlier versions the precise geometrical content that is required. The key headings
below indicate the range of topics that are expected for students aged between 11 and 16 years:
properties of triangles and other rectilinear shapes, properties of circles, specifying transformations,
properties of transformations, coordinates, vectors, measures, construction, mensuration, loci.

The curriculum does specify particular definitions, theorems and constructions which students
are expected to know. For example, three typical detailed statements are as follows:
recall the definitions of special types of quadrilateral, including square, rectangle, parallelogram,
trapezium and rhombus;
prove and use the facts that the angle subtended by an arc at the centre of a circle is twice the angle
subtended at the circumference, the angle subtended at the circumference by a semicircle is a right
angle, that angles in the same segment are equal, and that opposite angle of a cyclic quadrilateral
sum to 180°;
use straight-edge and compasses to do standard constructions including an equilateral triangle with
a given side, the midpoint and perpendicular bisector of a line segment, the perpendicular from a
point to a line, and the bisector of an angle.

Interestingly we find the statement 'understand, recall and use Pythagoras' theorem' without
any suggestion that students should encounter any proof of this important result. This is
surprising in the light of the expectation, as indicated above, that proofs are expected of the
circle theorems which, although of obvious interest and value, are much less fundamental and
lack such a rich range of applications and alternative possibilities for proof.
Turning to the United States, the geometry curriculum has been strongly influenced by the
Curriculum and Evaluation Standards for School Mathematics, NCTM (1989) and the more
recent revised document Principles and Standards for School Mathematics, NCTM (2000b).
These documents are not so prescriptive as the English National Curriculum, but the range of
topics and the underlying ideas have many similarities. They do make very specific references
to the particular value of geometry as 'a rich context for the development of mathematical
reasoning', NCTM (2000b, p. 232), whereas the English National Curriculum does not
explicitly give geometry a special place as a vehicle for developing reasoning. However, the
statements in DfEE/QCA (1999) are clearly aimed at giving greater emphasis to proof within
the geometry curriculum as compared to earlier versions of the curriculum.
The other noteworthy contrast between England and the United States is the greater
emphasis that is given in the latter to approaching geometry through real world contexts. This
is evident in the two standards documents, NCTM (1989, 2000b) and is a very striking feature
in many of the curriculum projects, described in Senk and Thompson (2003), that have been set

The Role of Geometry

1

up since 1989 in response to those standards. There is little attempt in English textbooks to
relate geometry to the real world in other than trivial ways, so that geometry appears for the
most part as an abstract study in the same way as is evident in France, Japan and Singapore,
referred to previously. This is clearly an issue where there is little consensus. There does not
appear to be any research about the relative effects on attainment and attitudes of presenting
geometry in terms of real-world contexts, as compared to presenting it as a relatively abstract
study of configurations with an intrinsic interest of its own. The stance adopted in this book
is that geometry is worth studying for its own sake and that it is particularly attractive both
because it has an immediate visual appeal and because it offers a variety of challenges at
different levels. Links should be made on occasion to simple real-world applications, but not in
the false belief that students' motivation is necessarily or only increased by appeals to utility.
There is much common ground between geometry syllabuses across the world in terms of
topics. Variations occur in the emphasis and timing of different elements and in the relative
importance accorded to practical approaches, proofs and applications, resulting in wide differences in the formality of style of presentation that is adopted. Similarly there are variations
in the relative importance accorded to synthetic and analytical geometry and the extent to which
algebra and trigonometry, and symmetry, transformations, matrices and vectors are involved.
The objects upon which the study of geometry is based are listed below, but even at this
simple level there is a lack of consensus about the relative importance to be accorded to each
item. The properties of polygons and some aspects of circles are a common feature of all
geometry curricula, but the role of the third and fourth items in the list varies greatly.
• polygons and their properties, giving particular emphasis to triangles, quadrilaterals and
regular polygons
• circles and their properties related to chords, tangents and angles
• three-dimensional figures such as polyhedra and the sphere, cylinder and cone
• other curves, such as the parabola and ellipse, and their properties
In conjunction with these objects there are the ideas that are used in looking at their properties
and relationships with similar variations in the emphasis that is given to each:
measurement and calculation of length, angle, area and volume
symmetry and transformations
the use of algebra
the theorem of Pythagoras and trigonometry
congruence and similarity
coordinates and equations of straight lines, curves and surfaces
matrices and vectors
Finally we have the place of reasoning and the importance to be accorded to proof. Linked to
that there are questions as to what should be accepted as 'self-evident' and what it is reasonable
to expect students to be able to deduce or prove. This issue is discussed by Barnard (2002)
who contrasts a deductive system based on a 'minimal set of axioms' with a collection of
isolated facts that lack 'the logical connections that are intrinsic to geometry'. The sense that
'one thing follows from another' and that geometrical results are linked together in an
essentially hierarchical way is an important one to develop, but it is necessary to be clear what
are to be taken as starting points. For example, we can deduce that vertically opposite angles

8

Teaching and Learning Geometry

are equal on the assumption that the angles on a straight line sum to 180°. We are then able to
deduce that alternate angles are equal by taking the equality of corresponding angles as the
defining property of parallel lines.
Developing the reasoning skills of students has been discussed in the first section of this
chapter as one of the major reasons for including geometry in the mathematics curriculum. The
urge to prove what has been suggested by intuition or experiment is one of the characteristic
activities of a mathematician. Finding a proof is not just a matter of verifying that a conjecture
is true, although that can certainly be a powerful motivation. Additional reasons for learning
about proof are advocated by de Villiers (1998, 1999), including explanation and discovery, and
the element of challenge with the fulfilment that comes from successful resolution of a problem.
A proof can add greatly to our understanding of a theorem, whose truth we already accept,
by providing a deeper or alternative explanation. Furthermore the process of proof finding
can lead to the discovery of new results and, certainly at a higher level, it has been a major
motivation for developing new mathematical ideas. The proof of Varignon's theorem discussed
at the beginning of this chapter leads readily to the discovery of the further result that the area
of the parallelogram formed by joining the midpoints of any quadrilateral is half the area of the
quadrilateral. That is not something which is instantly obvious from the diagrams illustrating
the theorem.
The problem illustrated in Figure 1.6, referred to by Duval (1998), is a good example of a
geometrical result that is not intuitively obvious. It can be proved in several contrasting ways
which show how different approaches provide different insights. The point P is a variable point
on the diagonal AC of rectangle A BCD and the line segments EG and FH through P are drawn
parallel to the sides of the rectangle. The surprising fact is that the two shaded rectangles, BEPH
and DFPG, have the same area and the problem is to explain why. It is necessary to identify pairs
of triangles which form subconfigurations of the diagram which then lead to various ways of
proving the area property. Three alternative approaches are considered here.

Figure 1.6 Why do the rectangles have the same area?

• There are three pairs of congruent triangles: AHP and AGP, PEC and PFC and the large
pair ABC and ADC. In each case the three pairs of corresponding sides are equal because
each pair of triangles is created by drawing the diagonal of a rectangle. The rectangles are
obtained by subtracting two smaller triangles form a larger triangle as follows:

Since the corresponding pairs of triangles are congruent, and therefore equal in area, the two
rectangles have the same area.

The Role of Geometry

9

• The six triangles mentioned above are all similar. Focusing on triangles AGP and PFC, we
have a pair of equal ratios leading to a pair of equal products that correspond to the areas of
the two rectangles:

Since AG=HP and FC = PE, the product AG. FC is the area of rectangle BEPH and that has
been shown above to be equal to the product PF.GP, the area of the rectangle DFPG.
Using pairs of letters to denote lengths of sides can make the algebraic steps of geometrical
arguments look unnecessarily daunting. In this case it looks much simpler if the lengths AG,
PF, GP and FC are denoted on the diagram by single letter variables a, b, c and d, so that the
argument can be presented as:

Alternatively the argument can be given a trigonometrical form by using the tangent to
determine equal ratios. Denoting the equal angles APG and PCFby 0, that gives:

• An alternative trigonometrical argument uses the same pair of similar triangles as above.
With the angles APG and PCF denoted by 0, as above, and letting the lengths AP and PC
be p and q, we have:
Sides of rectangle BEPH: p sin 6 and q cos 6
Sides of rectangle DFPG: q sin 9 and p cos 6
It follows then that the area of both rectangles is given by the same expression/^ sin 6 cos 6
These varied ways of approaching the same problem draw on different geometrical ideas,
together with some algebra and trigonometry, and provide a good example of the rich range
of possibilities that arise with even a simple problem. Specifying the topics and ideas of a
curriculum is only part of ensuring that students have a fruitful encounter with geometry.
The type of problems to which the ideas are applied and the spirit and style in which they are
approached are also a crucial element.
TEACHING AND LEARNING GEOMETRY
Many examples in school geometry involve calculating angles. These are a valuable exercise in
mathematical thinking because their solution requires a knowledge of the properties of the
figures involved together with an ability to find and communicate a chain of reasoning that
takes you from what you know to what you are asked to find. Figure 1.7 shows two regular
polygons, a hexagon and an octagon, which share an edge. The problem, which appeared
in an examination paper for 16 year old students, is to determine the angle between the two
polygons as shown in the left hand diagram. The question could be posed as 'find the angle' or

10

Teaching and Learning Geometry

Figure 1.7 What is the angle between the hexagon and the octagon?

alternatively as 'show that the angle is 105°'. The latter has the advantage that the focus of the
student's response has to be on the explanation rather than the numerical result.
There are two obvious strategies for solving the problem. One is to subtract the respective
interior angles from 360°: 360° - (120° + 135°) = 105°. Another method, suggested by the
right-hand diagram, is to add the two exterior angles to give the required angle directly:
60°-(-45° =105°.
The essence of the problem for the student lies in realizing that either the interior or the
exterior angles of the two polygons have to be found. The problem for the teacher is to help a
student who lacks a way forward to think what is required and then to find ways of calculating
the necessary angles. There are two questions to ask: 'What do you need to find?' and then
'What do you know about finding the angles of polygons?' That involves remembering, or
reinventing, the usual methods for finding such angles, which are discussed in detail in Chapter
4. Another approach to the problem of helping the student to see a strategy, assuming an earlier
familiarity with the software, would be to ask: 'How could you draw this diagram with LOGO?'
That would focus attention on the exterior angle property and shows how initial exploration of
geometrical ideas with computer software can be drawn upon at a later stage to help students
solve problems. It is not necessary to actually use the software to create a diagram, but thinking
about how it could be done with LOGO provides one possible key to a solution.
To learn geometry successfully requires a knowledge of facts, an ability to reason and a less
definable ability of seeing the essential features of a configuration that provide clues to solving a
problem or proving a theorem. Skemp (1976) identified two broad types of understanding:
instrumental and relational. Instrumental understanding is knowing how to carry out procedures, whereas relational understanding involves knowing why a procedure works as well as
how to carry it out. The report of the RAND Mathematics Study Panel in the United States,
RAND (2003), speaks of 'mathematical proficiency' as that which is needed in order to be
competent in mathematics. The first two of five strands of this proficiency are 'procedural
fluency' and 'conceptual understanding' which are closely allied to Skemp's two types of understanding, but in addition 'strategic competence' and 'adaptive reasoning' are identified. These
embrace the ability to reason and how to search for ways of approaching problems. The fifth
strand is to have a 'productive disposition' which is concerned with motivation, which comes
through seeing mathematics as a worthwhile study, confidence in one's ability to make sense of
the subject and a willingness to be persistent.
It is, of course, relatively easy to state what is required to learn geometry successfully, but far
less clear how this is to be put into practice when a teacher is faced with a class of students.
Perhaps there are some clues in the notion of mathematical proficiency if we take the five elements
in reverse order by recognizing first that little is achieved if students do not have a productive
disposition towards geometry and that requires careful thought about presenting geometry in
ways which engage students' attention. Strategic competence and adaptive reasoning cannot be

The Role of Geometry

11

developed independently of conceptual understanding and procedural fluency, but it seems
to be a common assumption that these should precede rather than be developed alongside the
all-important thinking skills. In fact, as the RAND study RAND (2003) suggests, all the strands
need to be intertwined, mutually reinforcing each other:
Because the five strands are interdependent, the question is not which ones are most critical
but rather when and how they are interactively engaged. The core issue is one of balance and
completeness, which suggests that school mathematics requires approaches that address all of the
strands.

This description of mathematical proficiency can usefully be coupled with observations of
good mathematics teaching in countries, like Hungary and Japan, which come out well in international comparisons. Andrews and Sinkinson (2000) have listed some of the characteristics
of successful mathematics teaching in such countries and, in particular note that:
Mathematics is presented as a problem-solving activity.
Relatively few problems are completed each lesson.

They characterize the problems that are used as ones that are chosen 'to exemplify generality',
'to generate discussion, proof and justification' and 'to help learners make links between
topics'. One of the secrets of success would seem to lie in choosing good problems which offer
some intrinsic motivation and using them as the focus for lessons where understanding, fluency
and thinking skills can be developed together. Geometry is particularly rich in good problems at
all levels as I hope the few examples in this chapter have demonstrated. The purpose of this
book is to look at different aspects of geometry, always with problems at the centre, but making
constant reference to the essential factual knowledge, conceptual understanding and thinking
skills and strategies that are required to develop a good mathematician's 'habits of mind'.
" give me summary and main ideas of this text