USER
" Chapter 2
Learning Geometry
THE INFLUENCE OF EUCLID
Until the early years of the twentieth century in the United Kingdom, geometry only featured as
part of the school curriculum for the small minority, largely male, who had the good fortune to
be able to continue with their education beyond an elementary level. The geometry curriculum
was determined exclusively by Euclid's Elements which was commonly seen as a text to be learnt
by rote, so that the theorems and their proofs could be produced verbatim in the order and the
form prescribed by the book. This inevitably led to failure on the part of many with success only
for those few students who were able to appreciate the purpose and meaning of Euclid's work in
spite of such a narrow approach to learning. Even the talented minority who were successful
within these constraints acquired a very restricted view of geometry. Students had little
opportunity to exercise their creative talents through solving problems or relating the ideas to
wider perspectives involving a wider range of mathematical techniques, particularly algebraic
methods, or relating the ideas to applications drawn from the world outside mathematics.
Concern among teachers about the effects of this rigid adherence to Euclid on the teaching of
geometry led to the founding of the Association for the Improvement of Geometrical Teaching
in 1871 whose name was later changed to become the Mathematical Association in 1897, as
interests extended beyond geometry. Little progress could be made until the entry requirements
of the universities of Oxford and Cambridge, and bodies such as the civil service and the army,
were relaxed so that a verbatim knowledge of Euclid ceased to be expected. The later years of
the nineteenth century saw a mounting campaign to encourage a more practical introduction to
geometry in schools and a less rigid adherence to the precise form and the prescribed order of
the proofs in Euclid. These developments are described in detail in Price (1994) and Howson
(1973, 1982). Two names that are particularly associated with this pressure for reform are
Professor John Perry (1850-1920) and Charles Godfrey (1873-1924), a public school master.
They came from two very different perspectives to become major influences on the way in which
school geometry developed over the fifty years or so that followed the relaxation by the universities in 1903 of the requirement to adhere rigidly to Euclid. Perry advocated a much more
practically based school geometry course with far less emphasis on the deductive proofs of
Euclid whereas Godfrey advocated a practical stage as a prelude to developing many of the
theorems that feature in Euclid, but allowing a variety of methods of proof and giving greater
emphasis to 'riders' - problems linked to the theorems.
The first few years of the new century produced a flurry of activity with many new geometry
textbooks being published of which Elementary Geometry by Godfrey and Siddons (1903)
was perhaps the most significant, remaining in print for many years and offering a much richer
view of geometry than that presented by Euclid. The deductive aspects followed on from more
experimental activities involving measuring, cutting and folding and informal demonstrations
of results which are considered as well as formal proofs.
Learning Geometry
13
The proposal that deductive geometry should be preceded by informal experimental work
appears in the first report on geometry teaching produced by the Mathematical Association
(1923) which introduced the idea that there should be three stages in learning geometry: Stages
A, B and C. The idea of three stages were further developed in the Association's second report,
Mathematical Association (1938), and this was a major influence on geometry teaching and
textbook writers until at least the 1960s, when other more radical changes were proposed and
began to be adopted.
Stage A is an experimental stage where students engage in a range of practical work involving
measurement, cutting and folding to investigate geometrical properties and begin to acquire an
intuitive feel for geometrical objects and relationships. For example, the fact that the angle sum
of a triangle is 180° is established experimentally by measuring the angles with a protractor or
by tearing the three corners from a paper triangle and placing them together to see that the three
angles appear to lie on a straight line. This contrasts with earlier teaching based on Euclid where
measurement was not used and students did not learn to use a protractor or compasses in their
geometry lessons. The rationale for stressing the importance of Stage A approaches is summed
up in the second geometry report, Mathematical Association (1938), in a message which is still
valid today:
One of the great mistakes in the teaching of mathematics, and one to which we are always liable, is
that of presenting abstractions familiar to ourselves to minds unprepared for them.
Stage B is a deductive stage where ideas are defined precisely and results are presented as
theorems and proved in a formal way. However, unlike the presentation of geometry based
rigidly on Euclid, there is flexibility about the precise order in which theorems are presented and
about the mathematical tools that can be used in proofs, in particular allowing the use of
algebra. The move to Stage B did not preclude a return to a Stage A approach initially when a
new topic such as the circle theorems was encountered. The final Stage C is a systematizing stage
in which a more global view of the subject is taken bringing out the links between theorems,
considering a logical order of presentation of results and clarifying initial assumptions by
making reference to axioms. This last stage always had a rather shadowy existence and was, and
is, perhaps only relevant to a small minority of students.
The early years of the twentieth century also saw many other changes in school mathematics
together with an increase in the proportion of both boys and girls having access to secondary
education through increased state provision of grammar schools. The introduction of graphs
and trigonometry into school mathematics were two curriculum changes at this time that were
significant in providing additional tools that could be applied to geometrical proofs and
problems. Familiarity with graphs led to coordinate geometry, particularly linked to a study
of the conic sections, becoming a common feature of mathematics courses in the final years of
secondary education.
CURRICULUM CHANGE FROM THE 1960s
The 1960s were characterized by a period of intense curricular development in many countries:
an informative account will be found in Cooper (1985) of some aspects of these changes in
England. Although this movement for change was international, the form it took varied
between different countries. It was variously referred to as 'new' or 'modern' mathematics and
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Teaching and Learning Geometry
influenced both the content of the curriculum and the way it was taught. The changes to
teaching methods that were encouraged were in the direction of more experimental and exploratory approaches building to some extent on the Stage A approach to geometry that had its
origins about fifty years before. Content changes included the introduction of sets and matrices,
probability and statistics, some reference to computers, and transformation geometry as
an alternative to the traditional deductive geometry with its strong Euclidean influence. The
emphasis on sets was controversial at the time and was given much greater emphasis in
the United States than in the United Kingdom, but has had little long-term effect on the
curriculum. Probability and statistics have become a ubiquitous, although sometimes controversial, feature of school mathematics such that it seems surprising today that they were only
introduced into the curriculum fifty years ago. Early attempts to use computers in school
mathematics were very limited and it was not until the 1990s that their use began to become
widespread, although there is considerable variation both across schools and between different
countries. The impact of computers in mathematics teaching is very varied and there is limited
consensus about the precise role they should play, although there is widespread recognition of
the potential value of software such as LOGO and dynamic geometry.
The changes to the geometry curriculum in the United Kingdom that arose in the 1960s have
been very significant, particularly through the influence of the School Mathematics Project
(SMP) founded in 1961. The publication of the first book in the project's main school course,
SMP (1965), and the subsequent books in the series provided a substantial contrast to the then
current mathematics textbooks, where geometry was very much in the style of Godfrey and
Siddons' Elementary Geometry with Stage A activities leading to a formal deductive approach
to a range of standard theorems and their application to 'riders'. SMP retained and developed
Stage A approaches to the basic properties of geometrical figures, but included the informal
study of transformations such as reflection, rotation, translation and enlargement. Standard
results such as the angle sum of a triangle, the angle properties of polygons and the theorem
of Pythagoras all featured in the course, but with much less emphasis on the formal theorem
followed by proof approach hallowed by Euclid. Congruent triangles were barely mentioned,
but similarity was given an important place using the idea of a scale factor and the transformation of enlargement initially rather than the conceptually harder idea of equal ratios.
By the 1980s a widespread common curriculum had emerged where theorems and their
proofs in the style of Euclid had very little place and the important Cockcroft report of 1982
on school mathematics could say: 'the differences between "modern" and "traditional" mathematics have become less marked', Cockcroft (1982). It is perhaps significant of thinking at that
time that the words 'geometry' and 'proof do not feature in the index of the report! Although
elements of a transformation approach to geometry did feature in the curriculum elsewhere in
the world, countries such as the United States, France and Japan have retained a geometry
curriculum which is much more in the spirit of Euclid than has been the case in the United
Kingdom. However, recent thinking, as reflected in the most recent version of the National
Curriculum for England, DfEE/QCA (1999), and in the documents of the Key Stage 3 National
Strategy, DfEE (2001), is moving the English curriculum firmly back in the direction of a more
deductive approach to geometry. In the United States, where school geometry was not so
strongly influenced by the changes of the 1960s and has retained a traditional Euclidean
form, the standards produced by the National Council of Teachers of Mathematics, NCTM
(1989, 2000b), seem to be moving curricular thinking in directions which give more emphasis
to informal approaches and greater links between areas of mathematics, while not losing the
essential deductive element.
Learning Geometry
15
STAGES IN LEARNING GEOMETRY
The three-stage model for teaching geometry from the geometry reports of the Mathematical
Association (1923, 1938) can be seen as a precursor to more sophisticated models of how
students learn geometry developed in more recent times. Inhelder and Piaget (1958) describes
the shift in students' thinking from concrete operations to formal operations whereby the focus
moves from the consideration of particular examples to the ability to reason in more general
and abstract terms. This should not be interpreted to mean that a transition from concrete to
formal operations takes place at a particular time or that the transition is in any way a smooth
one. Neither should it be assumed that the move cannot be influenced by the nature of
the classroom tasks that the teacher sets up. Indeed the evidence from the work on cognitive
acceleration described in Adey and Shayer (1994), which is linked to Piaget's ideas, suggests
that students' ability to think at higher levels can be strongly influenced by classroom tasks
specifically structured to develop their general thinking skills.
More specific to geometry, the five levels proposed by van Hiele (1986) are a significant
attempt to describe a hierarchy of stages through which students go in learning geometry and as
such provide a guide to how a geometry course might be structured. Van Hiele (1986) proposed
five levels as follows:
• Level 1. Shapes are recognized as 'wholes' without any strong perception or recognition
of their parts or properties. Thus a young child will recognize a circle or a square without any
strong sense of their particular features or any ability to distinguish them from ellipses or
rectangles that do not diverge too much from the special cases.
• Level 2. At this stage students become aware of the individual properties of shapes and are
able to describe them in terms of those properties. Thus a square is seen as having four equal
sides and right angles at each of its corners and, if attention is drawn to them, there is
recognition of the equal diagonals intersecting at right angles. However, the relationship of
the square to the rhombus or rectangle will not be seen or accepted readily.
• Level 3. Here there is an increasing appreciation of the definitions of shapes and their
relationship to other shapes. The distinction between a definition and a listing of properties
is becoming clear and a sense of the relationship between shapes is evident so that the student
recognizes, for instance, that the square is a special case of either the rhombus or rectangle
and that these in turn are special cases of the parallelogram, as shown in Figure 2.1.
• Level 4. This is the stage at which deductive competence is established so that students can
make sense of local chains of reasoning to prove geometrical results with some awareness of
their initial assumptions. They are thus able to distinguish between properties that define a
shape and properties that can be deduced. For example, the equality of the opposite angles
of a rhombus is a property that can be derived from the fact that a rhombus is defined as a
quadrilateral with four equal sides.
• Level 5. The final stage is linked to an understanding of the role of axioms in a systematic
development of a geometry characterized by an interlinked system of theorems derived from
a minimal set of initial assumptions.
Like all attempts to characterize learning the van Hiele levels are of necessity a very simplified
version of the reality they are seeking to describe by seeming to imply a smooth progression
between levels and not taking account of the varying conceptual difficulty of different ideas. In
practice any learner moves backwards and forwards between different levels according to their
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Teaching and Learning Geometry
Figure 2.1 The relationship between some quadrilaterals
depth of knowledge of the particular ideas being considered. Thus a student who is generally
quite happy operating at level 4 with familiar material may revert back to earlier levels when
faced with making sense of new ideas. The same sort of thing happens whenever the solution of
a problem is causing difficulty when it is frequently a good strategy to explore the ideas in an
experimental manner initially reflecting the approach of a student operating at level 2.
While van Hiele's levels do provide a description of the sort of progression that a teacher may
expect as students develop their geometrical understanding, they are perhaps more useful in
thinking about how a curriculum should be structured than in offering advice to the teacher
planning a particular lesson or dealing with a student's specific difficulties with a topic. They
do not offer a model of how to develop student's thinking and understanding beyond the
rather obvious fact that the ability to make sense of ideas is dependent on the extent to which
particular modes of thinking have been developed at an earlier level.
SPATIAL AWARENESS AND GEOMETRICAL INTUITION
As noted in Chapter 1 one of the common aspirations of school geometry is to develop
students' spatial awareness. This is perceived to be relevant to many aspects of the real world
in terms of employment where, for example, builders, architects, surveyors and navigators
obviously require some kind of geometrical sense. There is a similar perceived relevance to
everyday tasks in the home such as constructing shelves or making curtains and making sense
of maps when planning a journey or finding a place in a town. However, any link between these
practical applications and what happens in the classroom is far from clear. While we may
reasonably surmise that activities involving spatial skills might improve performance with such
tasks, it is inevitably difficult to separate the influence of out of school learning from what
happens in the classroom. Thus, for example, childhood experiences with constructional toys
may be more influential than school geometry, and they are likely to contribute in some measure
to success with school geometry as well.
The essence of a Stage A and B approach to geometry and of van Hiele's levels is that success
in using deduction in conjunction with geometry is dependent on a variety of appropriate
practical experiences giving a feel for the elements involved. Fischbein (1982), in discussing
Learning Geometry
17
intuition in relation to proof, speaks of 'the quality of self-evidence, which is the basic characteristic of an intuition'. He contrasts the self-evident equality of vertically opposite angles
with mathematical truths that are not self-evident like the theorem of Pythagoras, where
the result is surprising and not at all intuitively obvious. Learning geometry involves both
intuitive and analytical elements. Intuition is involved not only in seeing obvious features of
geometrical figures, but also in the creative task of spotting critical hidden features of figures,
making productive conjectures and identifying useful constructions when solving geometrical
problems.
While analytical skills - the ability to reason - can be to some extent learnt systematically, it is
much less clear how intuitive skills can be developed, although they are certainly a product of
our experiences of geometrical configurations of all kinds arising in many different contexts. It
is certainly clear that school geometry should aim to develop geometrical intuition by providing
a rich experience through observing, handling, manipulating, linking, discussing and describing
shapes as well as seeking to develop analytical skills.
MISCONCEPTIONS IN GEOMETRY
The idea of angle is a fundamental geometrical concept. Understanding angles is an essential
requirement from an early stage in learning geometry. There is much evidence concerning
students' misconceptions about angles and all teachers are aware of the difficulties students
encounter in mastering how to measure angles with a protractor. Figure 2.2 shows an example
taken from APU (1987) where a large sample of 11 year old students were asked to indicate
which statement they thought was correct. The percentages show the proportion of responses
for each statement. The two angles are in fact the same size, but the second most popular
response suggests that many respondents were seduced by the length of the arms or the 'space'
or 'distance' between the arms into deciding that one angle was larger than the other. Other
evidence from the same source confirms the influence of these extraneous factors.
Figure 2.2 Which is (he bigger angle?
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Teaching and Learning Geometry
The two angles have been drawn on a square grid as a simple means of making the gradient,
and hence the angle, the same for each, but with the arms approximately doubled in length in
one case. Since over half the respondents made a wrong response this sophisticated way of
seeing that the angles are equal clearly had no influence on many of them, although we have no
evidence as to how the correct responses were derived. However, it is likely that many made their
judgement on the basis of the angles looking about the same rather than through any reasoned
approach. A lot of evidence suggests that many students do not fully appreciate the nature of
angles as a measure of turning or rotation, something that is no doubt hindered by the frequent
occurrence of angles in static diagrams rather than in contexts where a rotation can be observed.
Practical approaches in the early stages of learning geometry over many years, reinforced in
recent times by a variety of opportunities provided by computer images, have sought to link the
idea of angle to the idea of turning, but the difficulties persist.
Returning to the example of Figure 2.2, one way to follow up such errors is to ask students
to measure the pair of angles, assuming that they have learnt how to use a protractor correctly.
For those who responded incorrectly to the original question the equality of the measurements produces an element of conflict, because they have to reconcile the evidence of their
measurements with their incorrect intuitive sense that one angle is bigger than the other.
Creating this sense of conflict between an intuitive sense and the evidence provided by
measurement is one step in helping students to modify their ideas by drawing attention to the
inadequacy of their current thinking. Adey and Shayer (1994) emphasize the importance of
cognitive conflict as a means of helping students to develop their cognitive skills. They describe
cognitive conflict as 'an event or observation which the student finds puzzling and discordant
with previous experience or understanding'. Errors and misconceptions can be a valuable
source of situations involving conflict between the students' perception of a situation and the
accepted interpretation.
Another frequent source of difficulty is the confusion between perimeter and area and, at a
later stage, between surface area and volume. At one level there may simply be a confusion of
language - the meaning of the words - but there is certainly very often a deeper misconception
at work which makes students think that perimeter and area are linked in a simple way so that
an increase in one is thought to lead to an increase in the other. It is valuable for students
to consider examples which conflict with this notion, like the various rectangles in Figure 2.3
which show that constant perimeter does not imply constant area and vice versa. However,
like many misconceptions, this confusion about area and perimeter is very persistent and will
not necessarily be eliminated by making reference to a single counter-example or through a
single lesson task investigating the perimeter of rectangles of constant area. It requires frequent
Figure 2.3 Misconceptions about perimeter and area
Learning Geometry 19
encounters through a variety of examples which draw out the conflict between faulty intuition
and geometrical truths.
At a simple level conflicts may arise through misunderstanding the meaning of words. Fielker
(1973) gives an interesting example based on a diagram with three parallel lines labelled a, b
and c as in Figure 2.4. Eleven year old students happily told him that 'a is parallel to 6, and b is
parallel to c\ but when Fielker said 'then a is parallel to c\ they said 'no, because b is in the
way!' Clearly the students had a restricted view of the meaning of parallel which required
the lines to be adjacent to each other. Perhaps this is induced by the usual examples that
teachers draw upon, invariably using pairs of lines such as railway track or the opposite sides
of rectangles. Railway track may be a deceptive example in another way because the effect of
perspective makes the lines appear to meet at a distant point. Another misconception was noted
by Kerslake (1979), who observed that young children were less likely to see a pair of lines as
parallel if the two segments differed significantly in length. These examples make very clear
that, even with a seemingly simple idea like parallel lines, surprising misconceptions can occur.
Similar difficulties arise with the idea of perpendicularity, discussed by Gal and Vinner (1997).
In contrast to parallel lines, if a is perpendicular to b and b is perpendicular to c, it is not true
in two dimensions that a is perpendicular to c. With both these concepts students need to
experience a wide variety of examples over time and teachers need to be alert to the possibility
of unexpected and surprising misinterpretations.
Figure 2.4 Three parallel lines
Many words have mathematical meanings which are different to their everyday meanings:
the word 'similar' is an obvious example. A striking example concerning the word diagonal is
given by Pimm (1987). A student was asked to find the number of diagonals in various polygons
- two of the responses are shown in Figure 2.5. The student thinks that the rectangle has no
diagonals, but that the triangle has three, so there is clearly a strange misconception at work
here. Eventually light dawns and we see that the student is counting the number of sloping edges
using the more everyday meaning of diagonal to describe a line that is sloping rather than
horizontal and vertical.
My own favourite example of this sort of misunderstanding of mathematical words, which is
Figure 2.5 A misconception about diagonals
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Teaching and Learning Geometry
also referred to in Pimm (1987), arose in a lesson I was observing where the teacher had asked
the twelve year old students to illustrate and name all the different types of triangles and other
polygons that they could think of. One boy, acting in a supremely logical way, had decided as
shown in Figure 2.6, that right should be seen in opposition to left rather than as an indication
of perpendicularity.
Figure 2.6 What does 'right'-angled mean?
Mathematical words often acquire an erroneous 'default' meaning which excludes general
cases. Figure 2.7 shows a variety of different pentagons. Typically, when the word pentagon has
become familiar, it becomes attached to a regular pentagon. Moreover a request to draw a
pentagon will invariably result in a regular pentagon being drawn with a horizontal edge at the
base. However, pentagons do not have to be regular, their sides do not have to be of the same
length, one edge does not have to be horizontal and the pentagon does not even have to be
convex. The selection of pentagons illustrated in the figure all have equal edges, but only the first
is regular and that has been drawn in the usual 'default' position. Examples like these are also
useful in drawing attention to the meaning of 'regular' where the polygon not only has all its
sides equal but also all its angles. Constructing a pentagon with five equal plastic rods freely
jointed at the ends, or an equivalent construction using dynamic geometry software, is a good
way of emphasizing that a regular polygon has equal angles.
Figure 2.7 Some pentagons with equal sides
Difficulties often arise because assumptions are made about relationships on the basis of a
particular diagram. Figure 2.8 shows two related examples. In the first case it is easy to assume
that the median bisects the angle of the triangle, but that is incorrect except when the triangle is
isosceles with the median positioned between the equal sides. Conversely in the second diagram
it is easy to assume wrongly that an angle bisector bisects the opposite side if the triangle is not
too far from being isosceles. In each of these cases investigating a few cases by drawing or using
Figure 2.8 Misleading conclusions from inappropriate diagrams
Learning Geometry 21
dynamic geometry is sufficient to highlight the error and warn of the dangers of arguing from
one particular diagram which may have special features.
Figure 2.9 shows the two diagonals from one vertex of a regular pentagon, which appear to
trisect the angle. Here we have a problem where intuition does lead to a correct result, but
intuition is not sufficient to be sure that the result is correct. The fact that the angle is trisected
can be proved readily by calculating angles in the three isosceles triangles in the figure or,
interestingly, by seeing the pentagon as inscribed in a circle and recognizing that the three angles
at a vertex are angles subtended by arcs of equal length.
Figure 2.9 Why is the angle in a regular polygon trisected by the diagonals?
A further area of misconception arises through the difficulty of interpreting two-dimensional
diagrams of three-dimensional objects. A simple example arises with the typical depiction of a
cube shown in Figure 2.10, which can equally well be seen as a two-dimensional picture of three
rhombuses forming a hexagon. It is clear that many people have considerable difficulty 'seeing'
three-dimensional objects through conventional mathematical diagrams. Whether the ability to
visualize three-dimensional objects from such representations can be improved by using suitable
classroom tasks is an open question. However, it is certainly important for the teacher to be
aware of the difficulty that many students have and to give them frequent access to actual objects
and models to help their appreciation of the shapes that are being depicted and discussed.
Figure 2.10 Is it a cube or three rhombuses in a hexagon?
Similar difficulties arise when diagrams of the nets of solids are depicted or students are
asked to design such nets. Figure 2.11 shows two configurations of six squares: the first will be
recognized as the conventional net for a cube, but the second is not a net for a cube because
when folded up two of the squares will overlap. For many people it is not an easy matter to
visualize how the net will fold up in order to determine which creates a cube and which does
not. Practical experience through the use of suitably designed tasks should improve powers of
visualization.
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Teaching and Learning Geometry
Figure 2.11 Possible nets for a cube?
LEARNING ABOUT PROOF
Proof in mathematics is commonly presented as a means of verifying the truth of conjectures,
but it is usually something that is pursued after evidence has established that a conjecture is at
least plausible, if not so convincing that the possibility of it being wrong seems very improbable.
Mathematical proof involves a certainty, subject to agreement on initial premises, that does not
apply to proofs in a scientific or legal context, where the argument hinges upon the evidence that
is available and is always subject to revision if new evidence emerges.
For students mathematical proofs often seem to serve little purpose. In their own terms they
feel that they already know that a result is true either because it is in some way self-evident, as
with the equality of the base angles of an isosceles triangle, or on the basis of experimental
evidence, as with measuring the angles of a number of triangles to verify that the angle sum is
180°. This comment indicates two significant areas of difficulty in motivating students to learn
about proof.
The first concerns what is taken to be self-evident. Deduction has to take place from agreed
assumptions which may be some formal minimal set of axioms, but, much more acceptably for
beginners, starting points which are accepted as true either because they have been deduced
previously or because they will be widely accepted as self-evident. School geometry should give
students an understanding of the geometrical ideas and language of these starting points
together with an ability to follow chains of reasoning and to develop their own deductive
arguments.
One of the difficulties arising from the traditional adherence to Euclid was that a result
like the equality of the base angles of an isosceles triangle was deduced rather than taken as
self-evident. The proof shown in Figure 2.12 uses the three-sides case of congruence, a result
which itself might also be seen as self-evident although a proof is given in Euclid. Proving
In triangle ABC, AB and AC are of equal length.
Let D be the midpoint of BC.
Then triangles ABD and ACD are congruent,
because AB = AC, BD = CD and AD is common.
Hence, angles ABD and ACD are equal.
Figure 2.12 Proving that the base angles of an isosceles triangle are equal
Learning Geometry 23
such facts in a formal way is demotivating for most students because there seems little purpose
in showing why something that is so obvious is true. The two examples are obvious results that
should be taken as true on the basis of a combination of intuition and experiment. That does
not preclude helping students see that the equal side property defines an isosceles triangle
whereas the equal angle property is a derived property, but that distinction is best made by
considering the information needed to construct an isosceles triangle accurately. At a later stage,
it may be instructive for some students to see how congruence can be used to prove the result
in the style of Figure 2.12. The congruence of the triangles also implies that the median bisects
the angle at the vertex A and is perpendicular to the base BC. However, for most students such
properties should be taken as self-evident features which arise from the symmetry of isosceles
triangles.
The second difficulty is making the distinction between a deductive proof and acceptance of
a result on the evidence of appearance or measurement, referred to as 'naive empiricism' by
Balacheff (1988). The fact that the angle sum of a triangle is 180° is not a self-evident fact,
but making measurements does provide strong, although not conclusive, evidence that it is true,
so that there is a more obvious place for deductive argument in such a case. However, since
students will only meet such an argument after they have come to accept that the result is true,
they may still fail to see purpose in a proof. Hoyles (1997) speaks of developing in students 'an
inner compulsion to understand why a conjecture is true if they have first engaged in experimental activity where they have "seen" it to be true', because a proof is more than an argument
to demonstrate the truth of a proposition for, as suggested by de Villiers (1998), it offers
explanation and insight into the ideas involved. Moreover, the ingenuity or neatness of the
argument can often be intriguing in its own right. When proofs are discussed with students we
are seeking to encourage them to think as mathematicians, to extend their conceptual understanding and to respond to the insights and stimulation provided by mathematical arguments
and results.
As with solving problems, appreciating and developing proofs requires more than an understanding of the geometrical ideas involved and an ability to reason. Having failed to find a
proof of a proposition or to solve a geometrical problem, it is a common subsequent experience
to follow somebody else's argument with no difficulty and then either to be surprised that you
missed some obvious feature or to wonder how the solver came to think of their solution
strategy. Students need to be equipped with a range of strategies which enable them to explore
different avenues, to view geometrical figures in different ways and to make links to other ideas.
THE ROLE OF COMPUTERS
The continuing rapid development of a wide range of powerful computer hardware and software has led to a dramatic extension of ways of exploring and presenting geometry. Two broad
styles can be identified and each is linked to two styles of teaching: these are indicated by the
two-way table of Figure 2.13.
Computers can be used, typically in a dedicated room, with students working individually
or in small groups or a single computer can be used as focus for work with a whole class. Each
type of use has possibilities and limitations, both practical and pedagogical. Dedicated rooms
are expensive and are commonly in high demand, but they do provide individual experience,
a feature that may increasingly become more widely available in ordinary classrooms as powerful portable computers for educational use become cheaper. When an individual mathematics
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Teaching and Learning Geometry
WHOLE
CLASS
INDIVIDUAL
OR SMALL GROUP
OPEN
CLASED
Figure 2.13 Two dimensions for classroom computer use
classroom is equipped with a single computer with projection facilities, alone or in conjunction
with an interactive whiteboard, there is abundant scope for frequent use which may often only
be for short periods of time to illustrate a point quickly or to try out an idea.
The other dimension of Figure 2.13 relates to the style of use, specified in broad terms as
open or closed, which refer to the nature of the tasks that students work on. Closed tasks
are based on precise instructions and routine exercises whereas open tasks involve more
exploratory problem-oriented activities which require initiative and creative skills. The same
characterization can be applied to whole-class activity, depending on the extent to which it
involves presentation and practice of skills with closed questions or more open-ended discussion drawing on students' ideas. The two categories are extremes of a spectrum, because
many lessons will have elements of both. Software and accompanying resources are commonly
designed with particular styles of use in mind.
Software can be placed in a number of broad categories:
• Small software is a term used to refer to items designed to develop a particular skill or to
reinforce ideas about a specific process or topic. This often allows students to practise skills
in interesting ways which are self-checking and may involve puzzles and games incorporating
several levels of difficulty.
• Generic software can be used for a wide range of different tasks. Widely used examples are
LOGO, dynamic software packages like Cabri Geometry and Geometer's Sketchpad and
graph plotters, such as Omnigraph and Autograph, which, besides plotting graphs, allow
transformation geometry to be explored. Increasingly these features are becoming available
on graphical calculators, although the small screen and poor resolution are much less clear
and attractive than a computer screen.
• Teaching and resource packages. Typically these are available on DVD or CD ROM and vary
from material presented like a textbook with varying degrees of interactivity to collections
of resources of a more imaginative and flexible nature.
In addition the internet is a vast, varied and continually growing source of information and
ideas, which includes examples of all the previous types of resources.
As an example of small software, Figure 2.14 shows a sample screen taken from one of the
components of a software package on symmetry and transformations produced by SMILE.
This displays half of a figure with a line of symmetry on a square grid on the screen. Students
are asked to complete the picture by inserting line segments in appropriate places. Other items
in the same package involve inserting lines of symmetry and finding centres of rotation, with
each offering tasks at different levels of difficulty. Details of software produced by SMILE will
be found at www.smilemathematics.co.uk.
Learning Geometry
25
Figure 2.14 Symmetry and transformations: a SMILE program
MSW LOGO, a version of the programming language LOGO, is available free from
www.softronix.com. The software provides a simple and versatile means of enabling students to
draw geometrical figures using a small, but powerful set of commands that are very easy to
learn. The example in Figure 2.15 shows an eighteen-pointed star created by drawing a segment
200 units in length, turning through an exterior angle of 140° and then repeating those two
commands 18 times to give two lines at each of the vertices. This is an extension of one of the
simplest uses of LOGO to explore the properties of regular polygons, a topic that is discussed in
detail in Chapter 4.
Dynamic geometry packages allow geometrical figures to be drawn on the screen and then
modified by dragging points in the figure. Figure 2.16 shows a Cabri Geometry screen where the
three medians of a triangle have been constructed. If any of the vertices of the triangle are
dragged the triangle changes its shape, but the medians are still seen to intersect in a common
point. As shown the software can also display lengths of segments so that the division of each
median in the ratio 2:1 can be demonstrated as a prelude to proving the result, which is discussed in Chapter 7. Cabri Geometry is published by Texas Instruments and further details will
be found at education.ti.com. The software originated at the University of Grenoble in France
and some interactive examples will be found at www-cabri.imag.fr. The Geometer's Sketchpad,
published by Key Curriculum Press, is another widely used dynamic geometry package. Further
details can be found at www.keypress.com/sketchpad.
As well as their main function of plotting graphs, most graph plotters have a facility for
plotting shapes and applying standard transformations to them. Figure 2.17 shows a screen
26
Teaching and Learning Geometry
Figure 2.15 Drawing an eighteen-pointed star with LOGO
produced using Autograph. The triangle in the first quadrant has been reflected in the y axis and
then the image has been reflected in the line y = x. It is easy to see that the pair of reflections
is equivalent to a single rotation of 90° clockwise about the origin. Carrying out the two
transformations in the reverse order does not produce the same result. The single equivalent
transformation is a 90° rotation about the origin as before, but it is anti-clockwise. The same
result as previously can be obtained by reflecting the triangle in the first quadrant in the line
y = x first and then by reflecting the image in the x axis rather than the y axis.
Further details of the graph plotting software, Autograph, will be found at
www.autograph-maths.com. Omnigraph, published by Spa Software, is another widely used
graph plotter and details of that will be found at www.spasoft.co.uk.
As an example of the vast range of material available on the internet Figure 2.18 shows an
interactive geometry page taken from the website www.mathsnet.net. A point, A, with its mirror
image, B, in a line are displayed together with a point, X, which can be moved on the line. The
points A and B can also be moved and the distances AX and XB are displayed. The student
is asked to choose between three assertions about the situation, the correct one stating that
X moves along the perpendicular bisector of A B. This website presents other geometrical
situations in a similar style with a range of other mathematical resources. It is but one example
of this ever-growing source of ideas and resources for learning mathematics readily accessible
to both teachers and students. The author's website at ces.huU.ac.uk/people/DougFrench
includes classroom resources and links to other sites of interest.
Learning Geometry
Figure 2.16 Properties of a triangle on a Cabri Geometry screen
Figure 2.17 Transformations on an Autograph screen
27
28
Teaching and Learning Geometry
Figure 2.18 Mathsnet: Interactive geometry from the internet
CONCLUSION
Geometry in nineteenth-century schools, through slavish adherence to Euclid, was characterized by an exclusive emphasis on deduction with little attention to providing any underlying
intuitive feel for the geometrical elements involved. The first half of the twentieth century saw
an increased emphasis in the United Kingdom on introductory practical experiences, but it was
still the case until at least the 1960s that many students experienced geometry as a dull routine
of mysterious theorem-proving and impossible problems. In the latter half of the twentieth
century there has been a strong swing against the deductive side of geometry with the result that
the practical and experimental aspects, and to a lesser extent algebraic approaches, have tended
to become more dominant. To a varying extent the same has been true in many other countries
with moves towards greater emphasis on the practical. There is widespread debate about the
future of geometry and in the United Kingdom there has been a move back towards giving
greater emphasis to reasoning and proof.
The use of computers is increasingly becoming a feature of geometry teaching, but there
is little consensus about the precise role that they should play or their effectiveness beyond
the experimental stage of learning geometry. However, there is a strong sense that they are a
potentially valuable resource that can significantly enhance students' motivation and develop
understanding when used thoughtfully.
Success in learning and doing geometry requires a blend of intuition and deduction. Intuition
is founded on a wide range of experience of geometric figures and their properties developed
Learning Geometry
29
both by practical activities and by discussion of their features. The art of deduction is learnt
by seeing it in action through a wide variety of examples presented by teacher and text,
accompanied by commentary and discussion of what is being stated and why. Alongside this
students need abundant opportunities to develop their own arguments by working on tasks
tackled individually or in collaboration with peers with guidance from sympathetic teachers.
Teachers need to be clear about the respective roles of experiment and deduction, to be aware
of the many sources of conceptual difficulty and to be alert to issues of misunderstanding
linked to the use of technical language. Links between exploratory activities and deductive
approaches should be strong so that each reinforces the other. Helping students to see the
richness of possibilities in a geometrical figure and providing them with ways of thinking about
novel situations requires more than practice in solving routine problems or involvement in
unregulated investigation. The rest of this book looks at many of the key aspects of school
geometry and offers advice to the teacher about appropriate introductory tasks, common
sources of difficulty and ways of extending students' ability to solve problems and generate
proofs.
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