School of Education

Murdoch University

**Introduction**

In a rather short space of time, computers
have changed in character from being large numerical devices that
could only be communicated with obliquely to small visual devices
that allow for much more direct forms of person-machine communication.
We have gone from the roomfull to the pocketfull, from paper tape
and punched cards to keyboards, mice and touch screens and from
strings of binary digits to visual images. All of this has taken
not much more than one (human) generation. The IBM Corporation
confidently predicted in 1945 that there would never be a market
for more than two or three computers in the *world*, and
yet in affluent countries like Australia, there are already many
*households* with more computers than that, depending a bit
on how one defines 'computer'.

Such dramatic technological changes have many consequences,
and one of them is the possibility that computers may be of value
to children studying geometry with some access to technology.
The main purpose of this paper is to describe one development
of this kind, the so-called 'dynamic geometry' software that has
recently begun to appear in educational settings. The first early
experiments in this field involved the *Geometric Supposer*
series of software, which allowed secondary school students to
explore geometric situations efficiently. However, recent refinements
of this idea are much more powerful, and include *Cabri Géomètre*
and *The Geometer's Sketchpad*. A version of *Cabri*
is now also available on personal technology, the Texas Instruments'
astonishingly powerful TI-92 graphics calculator, reinforcing
its status as an important new mathematical tool.

Geometry has certainly fallen from its position of prominence in western countries over the past generation. Indeed, even the word 'geometry' is in danger of disappearing from the vocabulary of many students who may undergo a secondary education without ever reading a book with 'geometry' in the title or taking a course described as a course in 'geometry'. While changes of language are natural and not necessarily problematic, and there is much of great value in the space strands of various Australian curricula, there have also been voices of concern raised about the rapid demise of what used to be a significant part of the curriculum for all students.

All too often, in fact, the 'geometry' of years gone by referred
to the rather formal synthetic geometry typical of school geometry
until recently, and quite similar to the original formulation
in Euclid's *Elements* (the most successful textbook in history,
with a print run at least two millennia long!). The focus was
on the development of formal geometric results concerned with
the plane, particularly congruence and similarity, as well as
some of the geometry of circles. We have now recognised that the
emphasis on formal proof was misplaced for most young students,
and that many important spatial ideas were neglected while students
were trying to come to terms with proofs of geometric results.
Consequently, many changes have occurred to the geometry curriculum
in the past two decades, and most of them have been for the better.

In this vein, in a recent superb publication associated with the NCTM Curriculum and Evaluation Standards for School Mathematics, Art Coxford (1991) suggested that a continuation of the broadening of geometry is to be welcomed:

Geometry, today and tomorrow, must be approached from multiple perspectives to permit the user to make the most of the content as its uses broaden and expand into heretofore unknown regions of science and nature. Fractals, which are founded on the concept of similarity, which are represented graphically (visually), and which are a creation at least partially dependent on powerful computers for their existence, are the new geometric tool of the near future. But what of the more distant future? What will the new tool be? What geometric content will it build on? No one knows, but you can be sure that it will demand an awareness of geometry from multiple perspectives for its comprehension. (p. 4)

This indicates a significant change in direction for the US mathematics geometry curriculum , which has long fixed solely on the geometry of Euclid.

The development of computers and calculators has affected our views of what is important in arithmetic, algebra and statistics, and have affected what sorts of realistic applications of mathematics we regard as appropriate for schools. So it is perhaps surprising that there has been almost no simultaneous impact of the computer on geometry in schools so far, especially in view of the earlier observation that computers have become more visual in nature.

Despite the broadening and the enriching of geometry into 'space',
there have persisted some lingering doubts that some important
babies may well have been inadvertently discarded along with the
Euclidean bath water. It is perhaps ironic that it may be the
computer that will generate fresh interest in Euclidean geometry.
Indeed, in a rich recent text, Heinz Schumann & David Green
(1994) even describe *Cabri* as 'Euclid's revenge':

Geometry used to have pride
of place in mathematical education but in the last fifty years
its role has diminished and formal geometry has disappeared.
There are three major reasons for this: firstly the difficulty
of actually performing the necessary constructions accurately,
secondly the considerable time consumed in repeating drawings,
and thirdly the realisation that the proof concept fundamental
to traditional Euclidean geometry is inherently difficult for
most students and that parrot learning of proofs has no merit.
However, new IT tools are now arriving, one of which is called
*Cabri-géomètre*, which can address these
issues and make traditional geometry live again. the cry of the
modern mathematics movement of the early Sixties was "Euclid
must go!"; the cry of the Nineties could be "Come back
Euclid!" (p 9)

Whether dynamic software realises this ambitious description remains to be seen. However, it is timely now that we turn attention to what the computer may have to offer the students and the teacher as far as geometry is concerned.

**Description**

The *Geometric Supposer* software
allowed students to construct geometric objects and make measurements
on them. For example, they could construct a triangle, measure
the angles and add them. What the computer added that normal geometric
construction tools could not was efficiency: once a construction
was made, a single command allowed it to be repeated. In this
way, having constructed a triangle and found that the sum of its
angle sizes was 180o, students could quickly construct some other
triangles and check that the property was common to all of them,
rather tan being unique to the first one.

What dynamic geometry software adds to this process, as its
name suggests, is *dynamism*. Rather than repeat the construction,
the object constructed can be *moved* in some sense, so that
what is general can be distinguished from what is particular.
In the case of the angles of a triangle, a dynamic geometry program
such as *Cabri* allows the user to move the vertices of a
triangle at will, all the while repeating the measurement of the
angles and the determination of their sum. (It is easier to do
this than it is to describe it in words.)

So, an important aspect of *Cabri* is that it allows users
to construct a geometric object that can be readily manipulated
to observe what changes and what stays the same. The example below
shows an elementary instance of this, allowing students to see
that alternate angles formed by a transversal across a pair of
parallel lines are congruent. As point B, constrained to stay
on the line AB, is dragged left and right with the mouse, the
two angle measurements at B and C are continually updated and
are always the same.

As another example concerning angles, the two angles subtended by the arc below are the same, readily seen by moving point Q around the circle to different places. Furthermore, if the arc is changed (by moving point B, for example), the same congruence is evident. Together, these movements suggest the generalisation that the angles subtended by the same arc are congruent.

As a third example of dynamic geometry , the series of perspective drawings below start with a rectangular box with a square end located in the plane of this page. One vertex of the square is A, and there is a vertex B at the back. The horizon and the vanishing point P are also shown.

A change of perspective is accomplished by dragging suitable parts of the object, using the mouse. The object looks different if the horizon is 'lowered':

If the vanishing point is moved to the right, a new view appears automatically:

If the box is changed - made 'longer' by dragging point B - its perspective drawing changes:

Similarly, dragging point A to make a smaller box also changes the perspective view.

The static page does not do justice to the dynamic aspect of
*Cabri* and similar software. An impression of movement is
readily created by actions such as those represented in the snapshots
above. It is interesting to speculate what the effect on the history
of art would have been if the pre-Renaissance painters (who did
not understand perspective) had had access to a device permitting
these kinds of explorations!

**Capabilities**

So what can be constructed in *Cabri*?
Essentially anything that can be constructed with the traditional
Euclidean tools of compass and straightedge. The difference is
that manual dexterity with the instruments is not necessary, and
arguably of course was never really as important as knowing which
constructions to make and in which particular order.

The two screens above show some of the menu choices available
to a *Cabri* user. Most of these are familiar and collectively
they comprise a powerful new kind of mathematical tool. Later
versions of *Cabri* (all the screen shots in this paper are
from version 1) and the *Geometer's Sketchpad* include even
more constructions, including the elementary transformations of
reflection, rotation and translation.

For example, the screens below, taken from a Texas Instruments TI-92 graphics calculator, show some of these transformational constructions as well as some of the measurements that can be automated with software of this kind. In the first screen, for example, a triangle has been reflected about a line segment. It is possible to change the triangle (by dragging vertices of the triangle) and seeing the effect on the image, or changing the line segment (again by dragging) and seeing the effect on the reflection.

It is indeed interesting that the owner and distributors of
microcomputer versions of more recent versions of *Cabri*
in the USA is the Texas Instruments Company, well known for its
development of fine graphics calculators. This new partnership
reflects both the significance of dynamic geometry software for
mathematics and the significance of personal technology for education.

Other powerful features can be built into software of this
kind. For instance, locus can be studied directly, by instructing
the computer to trace the path of a point as it moves in a certain
way. Measurements of length, angle and area allow for quantitative
geometric relationships to be explored. Macros, particular sequences
of constructions, can be constructed by sophisticated users or
used by others; an example is the construction of the circumcircle
of a triangle, using a single menu entry. *Cabri* incorporates
a 'property-checker', which allows the user to ask the software
whether or not a particular (apparent) property, such as the concurrency
of the perpendicular bisectors of the sides of a triangle, holds
generally. The screens below, snapshots from a dynamic range,
suggest quite persuasively that the property holds in general.

Reviewing the constructions might help students to see *why*
this extraordinary property should hold, but this will not always
be the case. Of course, providing a formal proof of properties
that hold generally involves a great deal more than recognizing
the generality, and dynamic geometry software will not do this
for us. But the computer at least implicitly raises the issue
of *why* a particular relationship holds generally or doesn't.

**Implications**

Although experience is being accumulated, it is still a little early to tell what are the best ways for this sort of software to be used in schools, and it seems likely that its importance will depend quite critically on students having enough experience with it to begin to use it independently - which makes the idea of building it into a calculator especially interesting, of course.

As with spreadsheets, a range of levels of student involvement is possible. Teachers may draw an object (such as those shown above) and provide students with the chance to explore some of its geometric features by manipulating the object. Students might be given instructions, at various levels of detail and specificity, regarding how to draw particular kinds of objects to examine. Students may even be given license to explore geometric objects in whatever way they wish, once they have learned how to get started with the software. There have already been a number of instances of young students discovering new geometric generalisations using software of these kinds! Euclid and his colleagues did not record all the possibilities, and did not have the benefits of access to computers to help their thinking.

Chris Little and Rosamund Sutherland (1995, pp 5-6) describe
a generic approach in a set of student activities for *Cabri*:

Geometry has in the past often
been taught as a set of facts to learn. *Cabri* provides
an ideal medium for pupils to construct elementary theorems for
themselves. In *Cabri* pupils can work on a variety of examples
of the same geometrical figure. They do this by dragging the
basic points of their constructed figure. Pupils can then use
the evidence from their constructed figure to make conjectures,
and check under what conditions their conjectures hold for other
drawings.

*Cabri* helps pupils understand what changes and what
stays the same in a geometric figure. In this sense pupils are
learning about geometric invariants.

Doing geometry requires pupils to use perception, experimentation
but also logic. An equally important aspect of geometry is to
seek *reasons* for results and theorems, using previous
knowledge and results.

All the computer work in this unit follows the same basic pattern.

*Construct
*First construct a figure in Cabri.

*Question and conjecture*

By moving the basic points and observing the size of the angles,
make a simple *conjecture*. For example:

the angles add up to 360 degrees

the two angles ... are equal

*Cabri check*

Once the conjecture is written down clearly, more drawings
can be produced by dragging basic objects. Is the conjecture
*always* true? Can you find a counter-example?

*Communicate and explain*

Can you *explain why*?

How do you *know* that the angles of a triangle add up
to *exactly* 180 degrees?

How do you *know* the sum isn't 179.9 degrees,
or 180.0000001 degrees?

Can you *use* what you know already?

*Communicate and explain on paper*

Make a note of what you have found. This should include copies of diagrams or printouts.

The merits of these kinds of activities are not yet clear, although they appear on the surface to offer important new ways of coming to grips with geometry. Thorough exploration of the classroom and curriculum implications of this kind of software is still to be completed. The development in the long term of new partnerships between geometry and computers will take a good deal of work, especially within classrooms. But it certainly has a great intuitive appeal for mathematics education.

The question of visualisation is important in many parts of mathematics, not only in geometry, of course. In their introduction to a stunning recent publication, Walter Zimmermann and Steve Cunningham, referred to the words of David Hilbert, arguably the greatest mathematician of the twentieth century:

"In mathematics we find
two tendencies present. On the one hand, the tendency toward
abstraction seeks to crystallize the *logical* relations
inherent in the maze of material that is being studied, and to
correlate the material in a systematic and orderly manner. On
the other hand, the tendency toward *intuitive understanding*
fosters a more immediate grasp of the objects one studies, a
live rapport with them, so to speak, which stresses the concrete
meanings of their relations. ... With the aid of visual imagination
[*Anschauung*] we can illuminate the manifold facts and
problems of geometry, and beyond this, it is possible in many
cases to depict the geometric outline of the methods of investigation
and proof."

Paraphrasing Hilbert, it is our goal to explore how "with
the aid of visual imagination" one can "illuminate
the manifold facts and problems of mathematics." We take
the term *visualization* to describe the process of producing
or using geometrical or graphical representations of mathematical
concepts, principles or problems, whether hand drawn or computer
generated. (p 1)

It certainly seems as if dynamic graphing software such as Cabri has the potential to enhance visualisation.

The possibilities opened by the development of dynamic geometry
software reflect the new liaison between the computer and geometry,
and it is hard not to believe that the consequences for mathematics
education are positive. Exploring these consequences is made relatively
accessible, since the AAMT offers each of the excellent references
below for sale in its catalogue of publications, and also distributes
the *Cabri Géomètre* software, all at member
discounts. Much more is to be gained from manipulating objects
and constructing new ones then from reading about others doing
so, of course. Schools, students and teachers need to have at
least access to some software for this to be possible.

After the hand calculator was invented, arithmetic could never be the same again. Following the invention of data analysis software, statistics could never be the same again. Now that algebra is available not only on large computer systems, but also on graphics calculators and personal technologies like the TI-92, algebra and calculus can never be the same again. It now seems, too, that geometry can never be the same again.

**References**

Coxford, Art (ed.) 1991, *Geometry
from multiple perspectives*, Reston VA, National Council of
Teachers of Mathematics.

Little, C. & Sutherland, R. 1995, *Geometry with Cabri:
Taking a new angle* , Chartwell-Bratt.

Schumann, H. & Green, D. 1994, *Discovering geometry
with a computer - using Cabri Géomètre* , London,
Chartwell-Bratt.

Zimmermann, W. & Cunningham, S. (eds) 1991, *Visualization
in Teaching and Learning Mathematics*, Mathematical Association
of America.

Please cite as:

Kissane, B. 1996. Geometry meets the computer.

Cross Section8 (1): 3-8.

[http://wwwstaff.murdoch.edu.au/~kissane/geom/CabriPaper.htm]