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The development of dynamic geometry software in recent
years is certainly the most exciting development in geometry
since Euclid. Besides rekindling interest in some basic
research in geometry, it has revitalized the teaching of
geometry in many countries where Euclidean geometry was in
danger of being thrown into the trashcan of history. For
example, someone recently made the claim at the
International Congress on Mathematical Education (ICME) in
Spain (July 1996) that dynamic geometry had saved the
geometry curriculum in the United States.
As we have seen earlier, one of the main
reasons for the poor performance of pupils in geometry can
be found in terms of the Van Hiele theory. For example, many
pupils have undeveloped visualisation skills which are an
important prerequisite for success in geometry. Furthermore,
pupils are introduced too early to formal geometry without
allowing sufficient experimental exploration of the
properties of figures and the gradual introduction of
appropriate formal terminology.
In the past, many teachers have simply
avoided the informal exploration of geometric relationships
by construction and measurement with paper-and-pencil, since
they are so time-consuming (and relatively inaccurate). (Of
course, there are also those teachers who from an extreme
formalist philosohical position, disregard any form of
experimental work in mathematics). Another problem is that
such constructed figures are "static ", one
either has to redraw the figure or be able to visualise how
it might chance shape.
This however has now all changed with the
development of some sophisticated software packages for
geometry. One of the first such "state of the
art " packages to be produced was Cabri-Geometry,
a French program that was first introduced to the
international Mathematics Education community at a
conference in Budapest in 1988. Since then other similar
packages have been developed, for example, Geometer's
Sketchpad by an American company and with
assistance from the National Science Foundation and the
Visual Geometry Project at Swarthmore College, USA.
These geometric software packages were
designed with the specific intention of putting at the
disposal of the pupil or student a microworld type
environment for the experimental exploration of elementary
plane geometry. In the past one either had to draw the
geometric configurations on a sheet of paper, thereby
obtaining a more or less exact, but fixed representation,
thus severely limiting exploration. In these software
packages the geometric figures can be constructed through
actions and in a language which are very close to those in
use in the familiar "paper-and-pencil "
universe. In contrast to paper-and-pencil construction,
dynamic geometry is accurate and is it extremely quick and
easy to carry out complex constructions, and to vary them
afterwards.
Once created, these figures can be redrawn
by "grasping " their basic elements directly on
the screen and moving them, while keeping the properties
which had been explicitly given to them. In this way one can
"continuously " change a triangle, and for
instance notice that its altitudes always stay concurrent
during the transformation. The software therefore allows one
to easily repeat experiments in many different orientations
and thereby checking which geometric properties stay
invariant. In fact, Cabri has a property checking facility
(only Macintosh version) that can check whether certain
properties (eg. parallelism, concurrency, collinearity,
orthogonality, etc) are true in general, and if they're not,
it can construct counter-examples.
Probably the most welcome facility of
dynamic geometry is its potential to encourage
(re-introduce) experimentation and the kind of pupil
oriented "research " in geometry described by
Luthuli (1996) and others. In such a research-type approach,
students are inducted early into the art of problem posing
and allowed sufficient opportunity for exploration,
conjecturing, refuting, reformulating and explaining as
outlined in Figure 14 (compare Chazan, 1990). Dynamic
geometry software strongly encourages this kind of thinking
as they are not only powerful means of verifying true
conjectures, but also extremely valuable in constructing
counter-examples for false conjectures.
Figure 14: Pupil research in geometry
However, the development of dynamic geometry has also
necessitated a radical change to the teaching of proof.
Traditionally, the typical approach to geometry has always
been to try and create doubts in the minds of pupils about
the validity of their empirical observations, and thereby
attempting to motivate a need for deductive proof. From
experience, these strategies of attempting to raise doubts
in order to create a need for proof are simply not
successful when geometric conjectures have been thoroughly
investigated through their continuous variation with dynamic
software like Cabri or Sketchpad.
When pupils are able to produce numerous corresponding
configurations easily and rapidly, they then simply have no
(or very little) need for further
conviction/verification.
Although pupils may exhibit no further
need for conviction in such situations, the author has found
it relatively easy to solicit further curiosity by asking
them why they think a
particular result is true; i.e. to challenge them to try and
explain it (also see De Villiers, 1990; 1991;
Schumann & De Villiers, 1993). Pupils quickly admit that
inductive verification merely confirms; it gives no
satisfactory sense of illumination ; i.e. an
insight or understanding into how it is a consequence of
other familiar results. Pupils therefore find it quite
satisfactory to then view a deductive argument as an attempt
at explanation, rather than verification.
Particularly effective appears to be to
present pupils early on with results where the provision of
explanations (proofs) enable surprising further
generalizations (using proof as a means of discovery).
Rather than one-sidedly focussing only on proof as a means
of verification in geometry, it therefore appears that other
functions of proof such as explanation and discovery should
be effectively utilized to introduce proof as a meaningful
activity to pupils.
The following is an example of a possible
worksheet in this regard from De Villiers (1995a):
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WORKSHEET
(a) Construct a dynamic kite using
the properties of kites explored and discussed in
our previous lessons.
(b) Check to ensure that you have a dynamic kite,
i.e. does it always remain a kite no matter how you
transform the figure? Compare your construction(s)
with those of your neighbours - is it the same or
different?
(c) Next construct the midpoints of the sides and
connect the midpoints of adjacent sides to form an
inscribed quadrilateral.
(d) What do you notice about the inscribed
quadrilateral formed in this way? (Make some
measurements to check your observation).
(e) State your conjecture.
(f) Grab any vertex of your kite and drag it to a
new position. Does it confirm your conjecture? If
not, can you modify your conjecture?
(g) Repeat the previous step a number of times.
(h) Is your conjecture also true when your kite is
concave ?
(i) Use the property checker of Cabri to
check whether your conjecture is true in
general.
(j) State your final conclusion. Compare with your
neighbours - is it the same or different?
(k) Can you explain
why it is true? (Try
to explain it in terms of other well-known
geometric results. Hint : construct
the diagonals of your kite. What do you
notice?)
(l) Compare your explanation(s) with those of your
neighbours. Do you agree or disagree with their
explanations? Why? Which explanations are the most
satisfactory? Why?
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Figure 15: Explanation & discovery
Formulation
The line segments consecutively connecting the
midpoints of the adjacent sides of a kite form a
rectangle.
Deductive
explanation
A deductive analysis shows that the inscribed
quadrilateral is always a rectangle, because of the
perpendicularity of the diagonals of a kite.
For example, according to an earlier discussed property
of triangles, we have EF//AC in triangle ABC and HG//AC
in triangle ADC (see Figure 15a). Therefore EF//HG.
Similarly, EH//BD//FG and therefore EFGH is a
parallelogram. Since BD^ AC
(property of kite) we also have for instance
EF^EH, but this implies that
EFGH is a rectangle (a parallelogram with a right angle
is a rectangle).
Looking
back
Notice that the property of equal adjacent sides
(or an axis of symmetry through one pair of opposite
angles) was not used at all. In other words, we can
immediately generalize the
result to a perpendicular quad as shown in
Figure 15b. (Note that it is also true for concave and
crossed cases). This shows the value of understanding
why
something is true. Furthermore, note that the
general result was not suggested by the purely empirical
verification of the original conjecture. Even a
systematic empirical investigation of various types of
quadrilaterals would probably not have helped to discover
the general case, since most people would probably have
restricted their investigation to the more familiar
quadrilaterals such as parallelograms, rectangles,
rhombi, squares and rectangles. (Note that from the above
explanation we can also immediately see that EFGH will
always be a parallelogram in any
quadrilateral. Check on Cabri or
Sketchpad
if you like!).
The teacher's language is particularly crucial in this
introductory phase to proof. Instead of saying the
usual:
"We cannot be sure that this result is
true for all possible variations, and we therefore have
to (deductively) prove it to make absolutely sure
",
pupils (and students) find it much more meaningful if the
teacher says:
"We now know this result to be true from
our extensive experimental investigation. Let us however
now see if we can EXPLAIN WHY it is true in terms of
other well-known geometric results . In other words, how
it is a logical consequence of these other
results. "
It is usually necessary to discuss in some detail what is
meant by an "explanation ". For example, the
regular observation that the sun rises every morning clearly
does not constitute an explanation; it only reconfirms the
validity of the observation. To explain something, one
therefore has to explain it in terms of something else, e.g.
the rotation of the earth around the polar axis. Similarly,
the regular observation that say the sum of the angles of a
triangle is 180 does not constitute any explanation; in
order to explain it, we need to show how (why) it is a
logical consequence of some other results that we know.
Of course, proof has many other functions,
eg. verification, systematization, communication, discovery,
intellectual challenge, etc. which also have to be
communicated to pupils to make proof a meaningful activity
for them. In fact, it seems meaningful to introduce the
various functions of proof more or less in the sequence
given in Figure 16. It is important not to delay the first
introduction to proof as a means of explanation unduly, as
pupils might become accustomed to seeing geometry as just an
accumulation of empirically discovered facts, and in which
explanation plays no role. For example, even pupils at Van
Hiele Level 1 could easily use symmetry to explain why
certain results are true (eg. why base angles of isosceles
triangle are equal). Although the other functions can be
introduced gradually as pupils progress through the levels
from Level 1 to 3, the function of systematization should
however be delayed until pupils have reached at least Van
Hiele Level 3 or 4. (Examples of activities aimed at some
different functions are given in De Villiers (1995b)). The
function of communication is of course present all the time
as the teacher needs to continuously negotiate with pupils
the criteria for what constitutes an explanation, proof,
etc.
Figure 16: Teaching functions of proof
The dynamic nature of geometric figures constructed in
Sketchpad or
Cabri may
also make the acceptance of a hierachical classification of
the quadrilaterals far less problematic than it is at the
moment. For example, if pupils construct a quadrilateral
with opposite sides parallel, then they will notice that
they could easily drag it into the shape of a rectangle,
rhombus or square as shown in Figure 17. Further research
into this particular area would be of great value.
Dynamic transformation of parallelogram
Editor note: learn on the right,
check
on the left with the CabriJava
figure
Figure 17
The ability to quickly and efficiently transform
geometric configurations with dynamic geometry software also
allows one to effectively
model
real world situations and problems by dynamic scale
drawings. It therefore becomes possible to give much more
complicated real world problems to pupils to solve than is
currently the case. Some examples are given in De Villiers
(1994b). These software programs also have facilities for
tracing the loci of certain objects, eg. points. This
facility could easily be used, not only in many real world
contexts, but also makes it feasible to introduce and study
the conics as loci (in the classical Greek way) instead of
treating it purely algebraically as in the present
syllabus.
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