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 solving 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 Computers for Mathematical Learning 1: 169-187, 1996. ~) 1996 Kluwer 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 constructions usually 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 computer hardware, 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 constructed (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 conjecture: 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


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 i f 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 o f 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 procedures or functions in programming languages. The main difference is that, unlike functions or procedures, macros have as yet no control structures

174 REII, mARD HO~.~.

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 geometric 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 following 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 approaches 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 Gymnasium 4 - 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 school system is basically a tripartite system with the Gymnasium on top (providing qualification for the university). Next comes the Realschule (preparing pupils for business and trade) and finally the Hauptschule (nowadays realistically a school for the 'remainder', that is those pupils who have not been successful at the primary school).


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 ?

Figure 3. 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)

i n

C

A

\ /

-, f

B i n

J

in C

Figure 4. Displaying separable triangles through dragging.

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 chose B 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

Figure 5. The SQUARE task.

B

C A C A

Figure 6. Chris' drag & link sa'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. I f 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 REINHARD H01.7.I,

C

A

J

Figure 7. Improving Chris' 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 solution 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 equilateral 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~rHARD HOI:ZL

that they wanted to employ 'drag & link.' But it seemed that for Marc the desired link option was a computational deficiency of 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 constructed points and you would link D to g, what should then happen i f you 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 participants. 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] i f 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 My analyses depend on data provided by transcripts of pupil-Cabri interactions as well as pupil-pupil interactions. The analyses are done by means of a methodically controlled style of interpretion of the transcripts developed by Banersfeld, Krummheuer & Voigt (1988) at IDM in Bielefeld. It is demanding work to come eventually to a consistent interpretation of pupils' conceptions even if one has the advantage of doing such work in ones mother tongue - it is impossible to do it in a foreign language unless one is perfectly bilingual.

ra~


I=

J

Figure 8. IfA could be m o v e d . . .

B

t2

A

~ r

J

c if"

A-

Figure 9. A could move on a line and B could stay in its 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 relationships 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 P a ~ I N ~ HOLZL

:

Figure 10. Igor's construction.

In Wolfgang's case it is not only a solution that is wanted but new knowledge. 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


Figure 11. Find the mirror line.

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 the first 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 o f 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 theorems tend to be expressed with active verbs, in particular ones of movement. 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: I f 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 fixed triangle 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 transformation 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, RD HOLZL

the challenge for the pupils is to make sense of these ou tcomes (see section 2). It is hard to see how teaching with DGEs can be effective without the

explicit inclusion o f the meta-act ivi ty of 'control led var ia t ion ' .

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