How does ‘dragging’ affect the learning of geometry

REINHARD HOLZL
HOW DOES 'DRAGGING' AFFECT THE LEARNING OF
GEOMETRY
ABSTRACT. This paper analyses two components of the epistemological domain of
validity of the Dynamic Geometry Environment (DGE) Cabri-g6om6tre: first, the nature
of its phenomenological interface, and second, the possible implication on the resulting
pupils' conceptions. Particularly, it is asked what effect dragging has on familiar geometric
problems whose nature could be described as 'static'. How do pupils apply Cabri's dynamic
tools to such static problems and are there any specific approaches with their problem solv-
ing behaviour? Trying to answer such questions leads to reconstructing pupils' conceptions
of 'Cabri geometry' and to the discussion of situated descriptions and generalisations of
geometric experience in the case of Cabri.
1. INTRODUCTION
The end of the last decade saw the advent of Dynamic Geometry Envi-
ronments (DGEs) - software specifically designed for the teaching and
learning of plane geometry and endowed with tools that enable users
to manipulate figures directly and dynamically on the computer screen.
Representatives are for instance Cabri-g~omdtre (Baulac, Bellemain &
Laborde 1990), Euklid (Mechling 1994), GEOLOG (Holland 1993), Thales
(Kadunz & Kautschitsch 1994), The Geometer's Sketchpad (Jackiw 1992)
and The Geometric superSupposer (Yerushalmi & Schwartz 1993).
Despite their different appearances, menu options, icons and buttons,
they all have in common that they
• simulate ruler and compass constructions as laid down in Euclid's
Elements some 2000 years ago
• support those constructions by macros that can be defined by the user
• and - most strikingly - allow certain parts of a figure to be moved
without changing its underlying geometric relationships.
As a result DGEs considerably extend opportunities for investigations
and problem solving in a school subject. This is in contrast to its traditional
Euclidean form, which many pupils dislike for its tricky problems and
seemingly superfluous proofs; many teachers also dislike it for its absence
of reliable and easily executable algorithms either.
International Journal of Computersfor Mathematical Learning 1: 169-187, 1996.
~) 1996Kluwer Academic Publishers. Printed in the Netherlands.

170 P~INHAm~HOLZt,
This paper takes a closer look at one representative for DGEs: Cabri.1
In doing so it focuses on Cabri's dynamic feature, the so-called drag-mode
It is the drag-mode that gives Cabri (and the other representatives) its
aesthetic and mathematical power but, as will become clear, it is also the
drag-mode that enhances the complexity of a specific learning situation.
Besides identifying one example of increased complexity, I pinpoint some
of its elements.
My general theoretical framework for this paper is partly based on
D6rfler's work on the computer as a cognitive tool (D0rfler 1993) as
well as on work by Hoyles & Noss (1992) regarding the situativeness of
pupils' mathematical experience in computational environments. The goal
of the paper is to analyse two components of Cabri's 'epistemological
domain of validity' (Balacheff & Suthefland 1994): first, the nature of its
phenomenological interface, and second, the possible implication on the
resulting pupils' conceptions. As for the latter, I draw upon qualitative
empirical data that have accumulated during more and less extended and
systematic Cabri projects in the past (H01zl 1994, Noss, Hoyles, Healy &
HSlzl 1994).
2. THE ROLE OF THE SOFTWARE MODEL
Before developing software for geomelry one must decide what style
of geometry is to be represented. Logo's Turtle geometry, for instance,
embodies an 'intrinsic' style: whatever the Turtle does next, it does only
with respect to its current state (position and orientation). Thus, Turtle
movements in their purest form are relative and without reference to an
'external' or global frame (e.g. coordinate system). By contrast, Cabri
represents an 'extrinsic' style: Cabri constructionsusually refer to external
elements. An example might be Cabri's circle-by-centre-and-radial-point,
in which the centre serves as a reference frame.
But even after one has decided which style of geometry to implement,
a great number of further questions, basic as well as detailed ones, remain.
For instance, we know that due to the global design of computerhardware,
programs like Cabri are forced to model a synthetic view of geometry
with discrete analytical means. This leads to questions about efficient data
structures and operations. Such data structures may be important so that
the resulting program can handle geometric structures, but they essentially
1 Throughout the paper the term 'Cabri' refers to the 'old' Cabri. This is the version I
have worked with for the last 4 years but there is now a more powerful Cabri available, the
so-called Cabri II (Bellemain & Laborde 1994).

HOW DOES 'DRAGGING' AFFECT THE LEARNING OF GEOMETRY 171
carry more than purely geometric information. Therefore that we can ask
whether the geometry that results - when the software is used - is still the
kind of geometry that we wanted to implement. This question is particularly
relevant if the software is to reinforce (though not necessarily extend) the
former tools.
A typical case in point is Cabri's drag-mode: Basically, it is designed
to overcome the 'inertia' of the traditional media: paper and pencil, ruler
and compass. At first sight, the drag-mode is not a new construction tool,
and so it should not alter Euclidean geometry. However, as is shown
below, implementing a drag-mode demands decisions about the behavior of
geometrical objects when they are moved; and some of those characteristics
may not at all lie in the realm of geometry. In short, what type of geometry
evolves out of the computer code? A different geometry?
The question about a different geometry has been controversially
discussed during specific workshops in Germany (e.g. Striisser 1991a). It
has much to do with the question of what we view as the characteristics of
geometry: only the body of stated axioms, permitted operations and proven
theorems? Or should the available tools together with the actions of those
who use them be taken into account? Viewed purely mathematically, Cabri
should not go beyond the realm of Euclidean geometry. But it does, since
there are examples where Cabri constructs geometrical objects which,
according to theory, should not exist (Str~isser 1991b). The question then
is, is this because of a programming error or because the model within the
software is based upon discrete analytical calculations.
Besides those difficulties that arise from this 'computational trans-
position' (Balacheff 1993), there are also didactic reasons which suggest
that a different geometry evolves out of Cabri and its successors - a
Cabri geometry so to say. What are those reasons? I think that for didactic
purposes in particular it is useful not only to look at the various geometric
axioms and their deductive consequences, but also to take into account
the geometric tools and the behaviour of those who use them. Put another
way, Cabri's drag-mode may be axiomatically neutral but certainly not
heuristically neutral. Thus, dragging suggest new styles of consideration
and reasoning which are in a way characteristic of Cabri geometry. But,
as mentioned above, not in an axiomatic sense but in a didactic one.
The problem is furthermore highlighted by another feature of Cabri's
drag-mode. Apparently, it was supposed to be just a tool for exploring the
various invariant relationships inherent in a geometric construction. How-
ever, the drag-mode alters the relational character of geometric objects. If
one constructs, for example, an equilateral triangle ABC where points A
and B are given, then C cannot be dragged whereas A and B can. From

172 REINHARDHOLZL
a relational viewpoint there is no need to distinguish the points A, B and
C, as each pair of them determines the original equilateral triangle.2 From
a functional viewpoint (essentially Cabri's) the situation looks different:
A and B determine the position of C but in return C does not determine
the position of A and B. Thus Cabri does not permit one to drag con-
structed (i.e. intersection) points; a distinction arises between 'dragable'
and 'non-dragable' points.3 This distinction may be 'ungeometrical' and
totally unknown (because unnecessary) in a paper-and-pencil environment
but is nevertheless important for pupils working in a DGE (see section
3.2).
Implementing the drag-mode requires yet other decisions. Balacheff
(1993) as well as Goldenberg (1995) and Goldenberg & Cuoco (in press),
for instance, point to the behaviour of 'points-on-objects' when the object
is dragged. How is a point P, placed on a segment AB, supposed to behave
if A or B is dragged? Two options are left open to the designer: change
the original position of P or do not. If the designer opts for changing the
position the next question to decide is how to change it? In Cabri and
some other DGEs the decision was taken that original (though accidental)
ratios are preserved. P still divides the segment AB in the same ratio after
dragging as it did when it was first placed on the segment. Dragging has
the effect of dilating, although the user has not explicitly called for such
a dilation (and may not expect it). In Euklid, however, a point remains at
a fixed distance from the 'first' endpoint of the segment on which it was
placed.
I illustrate this with an example from my Cabri work with pupils:
following a construction task, Peter and Chris (both 14-year-olds) are
expected to investigate when triangle ABC is an equilateral triangle (see
Figure 1).
By accident Peter and Chris chose the points-on-object Q and P in
such a way that the triangle ABC looked equilateral (see Figure 2a). While
Peter was first dragging the radial point of the circle, the shape of triangle
remained as if it were equilateral, due to fact that the points-on-object Q
and P, on which the triangle depended, were dilated (see Figure 2b).
Watching the drawing varying on the screen Peter arrived at the conjec-
ture: it's always an equilateral triangle. It was only when the boys dragged
P or Q that they realised that the triangle was equilateral in only two cases.
2 Suitable orientation assumed.
3 At first sight this may be different with Sketchpad, for intersection points can be
dragged there. But when an intersection point is dragged the whole figure shifts and the
underlying relationships remain untouched. Psychologically, it may make a difference in
so far as pupils do not come across non-dragable points.

Q A
HOWDOES 'DRAGGING' AFFECTTHE LEARNINGOFGEOMETRY 173
Figure 1. When is the tangent triangle ABC equilateral?
fa) ~x
c
Figure 2. Dragging the radial point does not reveal that ABC only looks equilateral.
Thus.... dynamic geometry shouM not be treated as if it is merely a
new interface to Euclidean construction. Line segments that stretch and
points that move relative to each other are not trivially the same objects
that one treats in the familiar synthetic geometry, and this suggests new
styles of reasoning (Goldenberg 1995, p. 220).
3. INTERACTING WITH DRAG-MODE
'Drag-mode' and 'macro constructions' are arguably the most powerful
extensions that distinguish Cabri from the traditional paper-and-pencil
setting. Macros, which can be defined by the user, serve as modules for
more complex constructions and in this respect they are similar to proce-
dures or functions in programming languages. The main difference is that,
unlike functions or procedures, macros have as yet no control structures

174 REII,mARDHO~.~.
like loops or selections. As a result functions or procedures have more
expressive power (but are conceptually more difficult).
The benefit of Cabri macros above all lies in the simplification of the
construction process; certain building blocks frequently needed in geo-
metric constructions can be used with ease. Drawing becomes essentially
simplified and more complex constructions are possible. Beyond that,
however, it is an open question whether macros act more like amplifiers -
by simplifying the construction process- than reorganisers by establishing
functional bridges, e.g. between a triangle and its circumcircle (see Pea
1985, 1987 and Dt~rfler 1993 for a more general discussion of the amplifi-
er/reorganiser metaphor). Empirical investigations addressing the issue of
modular constructions would be helpful, yet are not known to me.
My own studies have been directed at Cabri's drag-mode because it
seems to be the most salient point with respect to the learner's conceptual
thinking if one compares paper-and-pencil work to Cabri's. Thus, the fol-
lowing subsections address pupil Cabri interactions under three different
aspects:
1. What effect does dragging have on familiar geometric problems whose
nature could be described as 'static'. How do pupils apply Cabri's
dynamic tools to such static problems? Are there any specific ap-
proaches with pupils' problem solving behaviour?
2. The drag-mode is viewed as a mediator between the concepts 'drawing'
and 'figure' (Laborde 1993, Str~sser 1992). I try to take a closer look
at this by reconstructing pupils' subjective views of Cabri geometry.
3. Research on pupils' behaviour in computational learning environments
has showed that learners express their descriptions and generalisations
of observations context-bound, specific to the computational media at
hand (Hoyles & Noss 1992). This subsection discusses such situated
descriptions and generalisations in the case of Cabri.
3.1. Dynamic Problem Solving Strategies
Geometry at the secondary level of the German Gymnasium4 - like corre-
sponding school types in some other European countries - is strongly
influenced by the Euclidean tradition. Though its teaching underwent
some changes since New Math, including opening up to more than just
the Euclidean perspective (Neubrand 1995), it has retained its classical
4 Germany's schoolsystemis basicallya tripartitesystemwiththe Gymnasiumon top
(providingqualificationfor the university).Next comes the Realschule(preparingpupils
forbusinessandtrade)andfinallytheHauptschule(nowadaysrealisticallya schoolforthe
'remainder', thatis thosepupilswhohavenot been successfulat theprimaryschool).

Each triangle below is divided into two isoceles triangles:
C C
A P B
A B
Can any triangle be divided in such a way or do the triangles above
have any particular characteristics ?
Figure3. The TRIANGLE task.
contents. Therefore, we give our pupils the following problem (see
Figure 3).
Notice that the problem is in a way 'static' as it does not involve any
transformations of the triangle nor does the solution require watching any
movements of a point with the aid of the menu item 'locus of points'.
In fact we could arrive at an expert solution by resorting to some basic
properties of isosceles triangles in regard to angles.5 But the problem is by
no means easy for pupils, nor even for college students who are preparing
to be teachers.
We take a look at the work of Marc, a 14-year-old, attending a Realschule.
His construction was based on (see Figure 4)
• a perpendicular bisector of the segment BC, and
• the intersection P of the segment AB and the circle with centre A and
radial point C.
The line CP is the cut, as it were, which divides the triangle ABC
into the smaller triangles APC and PBC. By construction, APC remains
isosceles if any of the vertices of the large triangle is dragged (see Figure
4b, c). The other sub-triangle becomes isosceles if m goes through P (see
Figure 4a, d).
It is remarkable that Marc was able to obtain precisely working con-
structive indicators (namely P and m) to display triangles which can be
divided in tune with the task. Though Marc did not solve the problem in the
5 There are three types of such separable triangles: their characteristic properties are (1)
a right angle, or (2) one angle is three times the size of another one, or (3) one angle is
twice the size of another one (with the smaller angle being less than 45°).

176 REINHARDHOLZL
C
A
J
f
n
c)
in
C
A
/
-, f
B in
J
in C
Figure4. Displayingseparabletrianglesthroughdragging.
end, he found a result of his own: a procedure for constructing separable
triangles, which would have been a very good heuristic starting point for
the actual solution.
Another example. This time an explicitly stated construction task, the
SQUARE task: Given a line g and a point A. Construct a square ABCD
such that B and D lie on g (see Figure 5).
Solution I: A solution, completely in line with the Euclidean tradition,
could look like the following: Analyse the target figure and recognise that
the diagonal BD of the square must lie on g. Hence, g is a mirror line
of the square. Now reflect A through g and get C. With A given and C
constructed, the square can be easily completed.
Being completely based on reflection symmetry, this solution could be
called 'static' in that it implies no further movement of parts of the figure.
Whether or not the learner succeeds in solving the task depends solely on
his or her insight into the symmetry of the problem.
In contrast to this some of our project pupils attempted the following
dynamic strategy. Chris for instance choseB as a point on g and constructed
the square ABCD (see Figure 6a). Then he dragged B along g until D lay
on g and finally tried to 'link' the intersection point D to the object g (see
Figure 6b).

C B A
J
Figure5. The SQUAREtask.
B
C A
C A
Figure6. Chris'drag & linksa'ategy.
At this stage some extra information about Cabri is necessary: There
is a menu item 'link a point to an object'. This option links a so-called
basic point to a geometrical object, for instance a line or a circle. It is still
possible to drag the point along the object, but it cannot be removed from
the object. Mathematically speaking, linking points is a way of establishing
member of relationships. If it had been possible to link D to the line g -
which the pupil desired to do - it would have resulted in a solution to the
task. But Cabri refused to link D for functional reasons (see section 2).
Obviously, the pupil did not solve the construction problem as he could
not construct point D, but we can clearly see how the solution he attempted
was based on the means offered by the software:
1. Conditions B andD on g cannot be satisfied at the same time, hence only
one condition is satisfied while the second is provisionally dropped.
2. To satisfy the second condition too, dragging is used.
3. Being aware that dragging alone does not yield a drag-mode-proof
solution, the pupil wanted to use Cabri's link option.

178 REINHARDH01.7.I,
C
A
J
Figure7. ImprovingChris' method.
Both Marc's and Chris' approaches are clearly dynamic and equivalent
to each other with respect to 1. and 2. Before we discuss 3. in more detail
(see next sub-section) let us first be concemed with 1. and 2., and develop
a fully-fledged solution for the SQUARE task.
Solution 2: Create B as a 'point on g' and construct the square ABCD
by macro. Next watch the locus of D while dragging B (see Figure 7).
The locus appears to be a segment. Is it? If so, the target point D must
be the intersection of g and the segment. How can we be sure about the
geometrical nature of the locus of D? To this, observe that a rotation at
A with angle 90 degrees maps B onto D. Thus, as B is dragged along g
D traces the image of g subject to this rotation. Construct the image of g
and get D at its correct position. With A given and D finally constructed,
complete the square.
Clearly, solution 2 is 'dynamic', too, but seems, at first sight, to be
rather complicated and somewhat awkward given the nature of the task.
While I would go along with this critical appraisal, I would also point to
some features of solution 2 that develop its full force in a broader view of
construction activities:
• the dynamic solution (2) employs two major heuristic principles,
namely drop one of the conditions and vary the data (Polya 1973),
whereas solution 1 is only specific to the given task. Both principles,
above all the second, are natural in computational environments, where
data can be manipulated with ease. Thus, the strategy inherent in solu-
tion 2 can help the learner to get his or her solution process going even
when faced with a task such as this: 'Given three lines that intersect in

one point. A is a point on one of the lines. Construct a triangle ABC
where the lines are angle bisectors.'
• Carefully handling this solution strategy reveals some transformational
aspects of known figures, for instance the close relationship of a square
and a rotation with 90 degrees; similarly, the relationship of an equi-
lateral triangle and a rotation with 60 degrees, that of a parallelogram
and pointwise symmetry etc.
As dynamic solution strategies gain importance, so does the weight of
the problems shift. In the case of the SQUARE task, it is not the construction
of the target square that is difficult but the interpretation of the locus is
crucial to finally producing a solution. Though Cabri is oriented towards
the traditional way of Euclidean construction, its tools favour new styles
of tackling known problems.
3.2. Drag & Link or: Between Drawing and Figure
It is worthwhile viewing Chris' approach from a different angle (not only
because it is dynamic). For how should we interpret his attempt to link
intersection point D to line g? Which understanding of 'construction' does
his attempt indicate? On the one hand the pupil fabricates a solution purely
visually, but on the other hand he is fully aware that dragging B along g
until D intersects with g is not sufficient; thus the attempt to link D to g.
Notwithstanding the failures with such drag & link strategies - they
remained a favourite option for some of the project pupils. This could be
clearly seen with Igor, Marc and Wolfgang, three 14-year-olds with whom
we could work for half a year. At the end (!) of the project we asked the
group whether they would recommend Cabri, supposing that they were
mathematics teachers. Marc replied:
018 Marc:
019
025
026
• . °
029
Recommend it? I would say there is still
a lot to be developed with this program.
[The others are smiling and laughing]
First, the constructed, the constructed points,
that they could be linked, that would be great•
That would be terrific, I think
There it was again: the demand for the link option for intersection (=
'constructed') points, after I had thought that this issue had been settled
for good• The boys' problem solving behavior did not indicate anymore

180 REn~rHARDHOI:ZL
that they wanted to employ 'drag & link.' But it seemed that for Marc the
desired link option was a computational deficiencyof Cabri; the transcript
lines 18 - 29 do not indicate any logical misgivings on Marc's part.
I did not contradict Marc in that episode, saying for instance, that
his link option could not be realised that way. Instead, I took up Marc's
suggestion and applied it to the SQUARE task above. I showed the pupils
Chris' solution strategy and asked them what Cabri's reaction would be
look like, if the software could indeed link intersection points. In posing
this hypothetical question to the pupils and having them discuss it I hoped
to gain insight into their understanding of the (hidden) functional aspects
of Cabri geometry.
More precisely, my question was: supposing Cabri could link con-
structedpoints and you would link D to g, what should then happen ifyou
dragged B? Luckily, a lively discussion arose between the boys, in which
different viewpoints were articulated. As an observer, I acted merely as
someone who tried to encourage them to clarify their respective position,
if I had the feeling a contribution was not appreciated by the other par-
ticipants. The whole of the discussion was transcribed and subsequently
analysed by means of interpretative case studies (Maier & Voigt 1991).
Although it would be desirable here to present key episodes from the
transcript, I confine myself to my interpretation of the discussion because
of language translation difficulties.6 I reiterate for the reader that we are
in a hypothetical, only imagined situation: We assume Cabri could link
intersection points and ask what should happen if we dragged B. Wolfgang
and Marc agree that one could not drag B anymore (but they provide no
reasons for their statement). However, Wolfgang throws the point A into
the discussion, stating that A could be dragged! Igor contradicts because -
such is his argument - if one could drag A then B would have to move as
well (see Figure 8).
But why does this point [B] know how far it should go this way [up
on the line] if it is not constructed? Igor asks. Thus, if B cannot move
- because it is only a point on object, therefore in no dependency on A
- so cannot A. Igor's argument sounds compelling, nevertheless it does
not convince Wolfgang, because Wolfgang has recognised that there are
indeed positions for A, where B need not move at all! (see Figure 9)
6 Myanalysesdependondataprovidedbytranscriptsofpupil-Cabriinteractionsaswell
as pupil-pupilinteractions.The analysesare doneby meansof a methodicallycontrolled
style of interpretionof the transcriptsdevelopedby Banersfeld,Krummheuer& Voigt
(1988) at IDM in Bielefeld.It is demandingworkto comeeventuallyto a consistent
interpretationofpupils' conceptionsevenif one has the advantageofdoingsuchworkin
onesmothertongue- it is impossibleto do it ina foreignlanguageunlessoneis perfectly
bilingual.

ra~
I=
J
Figure 8. IfA could be moved...
B
t2
A
~ r
J
c if"
A-
Figure 9. A couldmoveon a lineandB couldstayinits position.
And as for Marc: he even thinks one could drag A completely freely,
provided that Cabri has calculated backwards. What Marc meant by
'calculating backwards' was not completely clear, but as we talked I got
the sense that he meant that Cabri should restructure the geometric rela-
tionships of the figure so that dragging B becomes possible. If this is an
adequate interpretation, then Marc's term 'calculating backwards' might
be a slight indication that he sees that geometrical relationships have to
be taken into account. It also potentially suggests a kind of computational
insight.
New tools favour new means of solving problems. The drag & link
approach was liked by many pupils because it allowed working on a
task without simultaneously concentrating on all conditions. And it is a
valuable heuristic means, as I have tried to show, because it helps to tackle
and solve even challenging construction tasks. It also necessitates a shift
of perspective from the construction of certain points to the interpretation
of certain loci.

182 Pa~IN~ HOLZL
:
Figure10. Igor'sconstruction.
In Wolfgang's case it is not only a solution that is wanted but new knowl-
edge. Cabri's dynamic possibilities are supposed to create that knowledge.
In the above example Wolfgang's question is: where can A go such that the
square ABCD is still a solution and B remains fixed? A few months before
this discussion took place Igor worked on the TRIANGLE task (see 3.1).
He, too, tackled the task with drag & link at the time (see HOlzl 1995 for
details).
The perpendicular bisector of CB, m intersects the segment AB at the
point M and the line l (perpendicular to the angle bisector w through C)
intersects m at the point P.
Igor divided the large triangle ABC with the aid of C and M into the
smaller triangles AMC and MBC. As a result of this construction MBC
remains isosceles even if one of the basic points A, B or C is dragged
whereas AMC happens to be isosceles only in certain cases. Figure 10b
represents such a particular configuration: The triangle AMC becomes
isosceles if P and M come together. In this situation Igor wanted to link P
to the segment AB.
Despite the fact that Cabri obviously was not able to link the intersection
point Igor had a clear understanding of Cabri's 'duties': Cabri should have
been able to let him drag C while keeping P on AB. This is possible
when C is only allowed to move so that the equality <ACB = 3 <ABC
holds. Consequently, this equality could have been discovered by simply
dragging C and measuring the respective angles at the same time.
Similar to Wolfgang, Igor did not simply want to fix a solution but to
create new knowledge; and this was what distinguished some participants
from the others: those who were mathematically more successful and used
drag & link had heuristic purposes in mind, those who were mathematically
weaker simply wanted to 'freeze in' the target figure.

f
Figure11. Findthemirrorline.
3.3. Situated Mathematical Experience
It has been observed that pupils working in computational environments
develop mathematical language that reflects the nature of the interaction
with the environment. A 'situated abstraction', as is proposed by Hoyles &
Noss (1992) is thefirst step in constructing a mathematical generalisation.
It is situated in that the knowledge is defined by the actions within a context
but it is an abstraction in that the description is not a routinised report of
action but exemplifies the pupil's reflections on their actions as they strive
to communicate with each other and with the computer (Hoyles 1992,
her emphasis). Examples of situated abstractions within Excel and Logo
environments can be found in Hoyles (1993) and Hoyles & Noss (1992)
respectively.
Pupils' language of mathematical experience in Cabri reflects the
dynamics of the drag-mode. Situated descriptions, abstractions or theo-
rems tend to be expressed with active verbs, in particular ones of move-
ment. For example, Bindya and Noora, two Year 8 pupils at an English
comprehensive school, were asked to construct a mirror line in a Cabri
figure.
The pair solved the task by setting up Cabri midpoints between sym-
metrical vertices of the given flags and joining them by a line. Asked why
they thought their line was the mirror line they described a basic property
of mirror lines this way: If it's a mirror line, you can squash all the basic
points onto it and make the figure a line. The pair's characterisation of an
axis of symmetry makes perfectly sense in a Cabri environment. The fact
that the axis of symmetry is the set of all fixed points is implicit in their
statement (but the second property of perpendicularity is not captured).

184 REINHARDHOLZL
C -- dieser Puakt
B
Figure 12. Account for the locus of Z.
Another example is Marc, whose construction was discussed in section
3.1. I asked him about his 'result'. Marc answered (to the effect) that he
thought it was not possible to separate every triangle into two isosceles
triangles but it did not matter what the (interior) angles of the triangle were.
Obviously a surprising answer because- from a purely logical viewpoint
- every triangle could indeed be separated if the angles did not matter.
An analysis of the whole episode shows that Marc did not give static
statements about his triangles at any time but thought of varying triangles:
start with an arbitrary triangle (= angles do not matter) and you can always
drag it in such a way that it becomes separable. The observer's arbitrary but
fixedtriangle was not at Marc's disposal; his reasoning applied to a varying
triangle. This dynamic view culminates in his words: Not always [can a
triangle be separated] but again and again. The observations which he
carded out on a single but varying triangle and the language he developed
reflected his Cabri-specific experience with the drag-mode.
A third example: Miriam and Stefanie, two 17-year-olds, were asked to
account for the locus of Z that is drawn by Cabri when C moves around
the circle (see Figure 12).
A traditional explanation could take the route via Thales' theorem but
the pair did not succeed in finding it.7 Miriam, however, suddenly reasoned
in a Cabri-specific style:
7 First, show that the angle <AZM is a right angle, then apply Thales' theorem which
implies that Z lies on a circle with diameter MA.

475 Mid:
476 RH:
477 Mid:
478
479
but that's always the same when I move Z on the
circle
(corrects) when you move C on the circle
move C on the circle .. then, it's always a dilatation,
and the circle .., I mean, the locus of Z comes from
the dilatation .. doesn't it?
Miriam had already observed that Z could be interpreted as the image
of C with a dilatation at centre A and factor 1/2. In the transcript lines
475-479 she makes the transition from a point-oriented viewpoint to an
object-oriented viewpoint: If C scans, as it were, the large circle, so Z prints
the image of this circle through dilatation. Miriam's idea is particularly
instructive because it connects closely to the solution idea mentioned in
section 3.1. Both ideas can be put into the conceptual framework of trans-
formation geometry: static constructions can be interpreted as dynamic
transformations; and it is not unreasonable to suggest that DGEs could
give a new lease of life to this perspective.
4. CONCLUSION
My expositions were supposed to exemplify how, under the influence of
DGEs, geometric objects change their traditional status (e.g. points) and
new approaches to known problem situations arise.
DGEs emphasise aspects like the 'hierarchy of geometric objects' and
the relationship of 'drawing and figure'. Dependencies of geometrical
objects play an important role in such computational media and a program
like Thales, for instance, shows respective dependencies (and not only
hints at them) each time the user wants to delete parts of a figure. The
relationship of drawing and figure comes to the fore because of the nature
of the drag-mode. A program like GEOLOG for instance emphasises this
aspect by interactively linking a graphics window to a text window.
Furthermore, the paper intended to sketch out features of the meta-level
that emerges as the cognitive technology DGE becomes readily available
for the learning and teaching of geometry. A salient feature of this meta-
level is a shift of importance from constructing (i.e. creating figures) to
varying (i.e. investigating) them. Besides being able to handle the software,
new abilities are demanded from the pupils; above all abilities in relation
to 'meaningful experiments', for instance the control of parameters in
an experiment and the interpretation of its outcomes. Varying different
parameters can result in qualitatively completely different outcomes and

186 REn,ata,RDHOLZL
the challenge for the pupils is to make sense of these outcomes (see section
2). It is hard to see how teaching with DGEs can be effective without the
explicit inclusion of the meta-activity of 'controlled variation'.
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Didaktik der Mathematik
Universitiit Augsburg
86135 Augsburg, GERMANY

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