CABRI AS A COGNITIVE TOOL FOR THE EVOLUTION OF A SENSE OF PROVING

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CABRI AS A COGNITIVE TOOLFOR THE EVOLUTION OF ASENSE OF PROVINGCatia Mogetta aa Graduate School of Education, University ofBristol,Version of record first published: 14 Apr 2008.

To cite this article: Catia Mogetta (2000): CABRI AS A COGNITIVE TOOL FOR THEEVOLUTION OF A SENSE OF PROVING, Research in Mathematics Education, 2:1,107-124

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8 CABRI AS A COGNITIVE TOOL FOR THE EVOLUTION OF A SENSE OF PROVING

Catia Mogetta

Graduate School of Education, University of Bristol

Cognitive technologies have been described in the literature as reorganisers of thinking processes, especially where problem solving is concerned. This paper aims to analyse the possible use of Cabri-GkomBtre as a cognitive tool in the elaboration of mathematical justifications in the context of problem-based mathematics. Some empirical examples are given to illustrate the significance of the specific learning situation. The complexity of learning environments incorporating computer-based activities is stressed as a condition for them to be eflective in the introduction of the idea of mathematical justification and its evolution towards a sense ofproving.

A COGNITIVE TOOL?

The main aim of this paper is to analyse the role of Cabri-GkomBtre [I] (Baulac, Bellemain and Laborde, 1988) as a cognitive tool in the teaching and learning of mathematics, with a particular focus on the issue of proof.

In the recent literature on the role of ICT in education computers are seen as particular cognitive technologies, i.e., media that help

transcend the limitations of the mind (e.g. attention to goals, short-term memory span) in thinking, learning and problem-solving activities. (Pea, 1987, p. 91)

In the specific case of mathematics, they act not only as amplifiers of the intellect; they are reorganisers of thinking processes (Pea, ibid.). Following this approach to the use of technology in the mathematics classroom, Cabri might be considered as a support for learners to transcend cognitive limitations and construct a new relation to knowledge.

This paper sets out some theoretical ideas concerning the affordances provided by the software to the student as learnerluser with respect to the exploration and justification of geometrical statements. In the first part, ideas related to informal and formal reasoning in geometry are discussed with a particular focus on problem solving activities incorporating the use of Cabri. In the second part, empirical examples from a study conducted with GCE Advanced Level students are presented, with the aim of illustrating these ideas. Particular attention will be paid to the production of arguments in order to justify constructions and conjectures within problem solving activities.

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Research in Mathematics Education Volume 2

PROOF AS A PROCESS

This paper is centred around the issue of the appropriation by students of mathematical proving practice, and aims to analyse the role of Cabri, as an integrated part of a complex learning environment [2], in the process of proving statements within the context of problem solving. Mathematical proof is a particular type of discourse, which has different functions in different settings. School mathematics has peculiarities related both to the aims and the content which has to be taught. In this particular context a proof needs to be convincing and explaining (Hanna and Jahnke, 1993). The convincing function is related to the structure and the rules governing the mathematical discourse and aims at establishing that a given proposition is valid beyond any possible doubt; the explaining function is, to a certain extent, related to the content of the proposition and aims at understanding why it is valid.

Taking a more general perspective on the issue, it is worth highlighting the fact that proof has specific aims in the construction of mathematical knowledge. The introduction of the idea of validation, as a necessary aspect of mathematical activity, needs to make use of proofs of statements or, at least, rigorous justifications. This idea is a basic component of a collective and long term process of construction of a sense of theory, whereby theorems are seen and introduced as a cognitive unity of three elements: statement, proof and theory (Mariotti, Bartolini Bussi, Boero, Ferri and Garuti 1997). In this paper a proof is not seen just as the final product of a reasoning conducted according to specific rules, but as a whole process of conjecturing (i.e., producing a statement through an exploration of the problem situation), justifying and formalising (i.e., elaborating a proof and including it in a theoretical system) within the context of a problem-based approach to mathematics.

ARGUING VS. PROVING?

The original elaboration of a mathematical proof, especially when located within the context of problem-based mathematics, involves different reasoning modes, which can be classified into two main categories: argumentative and deductive reasoning. Argumentation [3] privileges the semantic content of the propositions involved and aims to establish the truth of the statements, based on the plausibility of the arguments produced, in order to support the reasoning itself. The idea of truth in mathematics is a tricky one because it is similar to the one of validity but, at the same time, different in its nature. While validity is established in relation to the consistency of the argument within a theoretical system, truth is an absolute claim and, since it is not attainable, it is usually attached to a perceptual judgement.

Informal arguments, supported by heuristics and intuitions, often draw on empirical observations and trial and error strategies. They can mislead the learners in the process of justifying statements, sincethey might be convincing on a semantic level and, given the lack of a theoretical control, might highlight some irrelevant aspects

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Cabri as a Cognitive Tool

of the problem situation under analysis, diverting the attention from the necessary elements. The structure of an argumentation is such that the deductive steps undertaken do not always lead to a unique conclusion and sometimes add information to the initial statement (Duval, 1992-93).

On the other hand, a deductive mode of reasoning directed towards proof needs to take into account the syntactic aspects of the propositions and aims to provide validity, based on mathematical reasons and on the acceptability of both arguments and links among them in terms of the conventions established by the mathematical community. Analytical arguments, according to the definition given by Toulmin (see Krurnmheuer, 1 999 , characterise this mode of reasoning; their conjunction into a deductive chain follows well-established rules, leading to statements which are equivalent to the initial one and do not add any information. The elaboration of a deductive chain of arguments is neither straightforward nor spontaneous; it requires a specific knowledge of the rules structuring this particular form of discourse. In this paper I shall argue that the disjunction between argumentative and deductive practices might be overcome in terms of an evolution from the informal, intuitive and empirical approach to the formal, passing through a pre-formal stage [4] which can provide a ground for merging meaningful practices and syntactically correct operations.

BETWEEN INFORMAL AND FORMAL REASONING IN GEOMETRY

The focus of this paper is on geometrical reasoning, given that this particular mathematical domain provides a ground where the proving practice can be appropriated by students in an effective way. This is partly due to the deductive structure of the subject matter, as it is traditionally taught, and partly to the inherent coexistence of visual, intuitive and theoretical aspects. Students are likely to face some difficulties in solving geometrical problems, due to intrinsic epistemological obstacles (Brousseau, 1986) related to specific reasoning processes involved in geometrical thinking.

Visualisation processes (Duval, 1998) fulfil specific epistemological functions, such as the identification of configurations, the illustration of a statement by spatial representation and the heuristic exploration of a complex situation. The choice of a particular configuration for the problem being tackled is often based on visual perceptions of relations between geometrical objects involved and empirical arguments and reflects only one of the possible ways of representing the problem statement. One obstacle intrinsic to visualisation processes is the translation from the verbal to the graphical register, which requires a conscious use of well- conceptualised mathematical objects and facts. Other epistemological obstacles are strictly connected to representation processes: the choice of a representation for the problem situation involves the use of different registers of the mathematical discourse and therefore a flexibility in connecting different aspects of the same

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concept. The pictorial representation of statements through figures, which is prevalent in tackling geometrical problems, can create some difficulties and, in a sense, misconceptions, due to the intuitive self-evidence of the represented statements. In the case of geometrical figures, the self-evidence of properties is particularly highlighted and can heavily influence the process of conceptualisation and, at the same time, the identification of properties and relations linking the different elements constituting the figure. Learners have to face difficulties of a different nature in both visualisation and representation processes and the intermingled presence of informal and formal arguments supporting these processes needs to be achowledged explicitly in order to enable students to overcome these obstacles.

BRIDGING THE GAP: HOW DOES CABRI HELP?

It is generally accepted that there is a cognitive, epistemological and structural gap between argumentative and proving practices. I would argue that it is possible to bridge these 'gaps' provided that suitable learning environments are designed with respect to the needs of the learners to come to h o w the structure of the mathematical discourse they are expected to produce and its relation to the wider mathematical (theoretical) domain of reference. This claim may sound far too ambitious, and therefore it requires some clarification. The differences between the two practices being examined and the associated forms of reasoning are certainly deep. Nevertheless, students need to evolve from an empirical and heuristic approach to a formal one and to be able to perform proving oriented tasks in a meaningful way. The learning environment becomes crucial in this respect, since it can provide a pre- formal space whereby the activities performed have a theoretical ground, but at the same time keep the student close to the meaning of objects and operations performed with them. The issue of language, both in oral and written form, is a core one. The request for formal justifications or proofs can highlight the break between the modes of reasoning involved, since the phase of discovery of the solution is usually carried out through argumentation and makes use of semantically based propositions, while the formalisation phase requires a detachment from the context of the problem and a theoretical control of the sequencing and cohesion of sets of propositions. A possible evolution from the ideas involved in the solution to their translation into mathematical terms can be supported through the setting-up of a suitable learning environment, providing a balanced mixture of informal and formal elements.

Cabri-Gkorndtre is a microworld (Hoyles, 1991) which provides tools to explore mathematical situations in an 'informal' environment and gives space for activities which might connect students' initial conceptions with more structured mathematical ideas. Following Balacheff and Sutherland (1994), a computer microworld can be characterised by two main features:

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its internal structure, consisting of a set of primitive objects, elementary operations and rules expressing the way the operations can be performed;

the constitution of a domain of phenomenology, which relates objects and operations to phenomena on the surface of the screen.

The internal structure of a microworld, therefore, reflects that of a mathematical formal system, while the actual affordances provided through the interface (for instance the dynamically manipulable drawinglfigure on the screen) reflect figural and visual aspects not strictly connected to the underlying mathematics.

Some specific internal characteristics of Cabri, such as the correspondence between menu commands and axioms of the theory, and the need to construct objects under certain geometrical constraints are relevant (though implicit) features which might support the elaboration of geometrical reasoning. Productive thinking might be enhanced within this environment since Cabri provides a ground where figural and conceptual aspects of geometrical figures coexist (Fischbein, 1993, Mariotti, 1996) and may be harmonised and linked to each other. Conceptual aspects of a Cabri- figure, which may differ fiom those of a geometrical figure on paper, may constrain and direct the reasoning within geometrical 'borders'. The visual perception of the figure and an exploration of its characteristics and properties on the screen could be the first step towards a (systematic) production of mathematical arguments. The way a figure is perceived in a Cabri environment could influence its conceptualisation and the construction of links between properties and their geometrical setting.

But Cabri, because of the distance between the internal mathematical structure and the phenomenological space at the interface level, cannot by itself be enough to help learners make the links explicit and move from a figural level to a conceptual one. The complexity of situations where Cabri is just one of the components still needs to be explored and defined. I would argue that a learning environment constituted by this software, a specifically designed set of problems and well-designed situations for the devolution of these problems (Brousseau 1986), can provide a sort of pre- formal space whereby the transition from informal to formal arguments can be carried out within the constraints of a theoretically ruled world.

Features of the software.

I will now outline some features of Cabri, which might be relevant in terms of the transition from informal to systematic arguments. The environment itself is neither formal nor constraint-free, but the need to respect the 'rules of the game' might support this transition process.

The dynamic representation and manipulation of geometrical objects allows a particular configuration of elements to be modified in order to span the whole range of possibilities. This feature could help the learner:

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visualise the multiplicity of configurations included in a single geometrical figure, and therefore overcome the difficulties related to the conflict drawinglfigure (Laborde, 1993).

understand the idea of invariant, through the use of the dragging mode, under which the properties and relations linking the objects to one another resist the change of configuration.

The use of commands and functions in order to construct and then explore a figure requires an implicit reference to the underlying mathematical (theoretical) structure. I would argue that the correspondence between Cabri commands1 constructions and the axioms1 theorems of Euclidean geometry [5] can help students substantiate their arguments with mathematically based reasons and thereby support a process of step by step formulation of the solution and of the related justificatiodproof.

The features of Cabri as a microworld can be important in providing conceptual feedback to the learner. After carrying out a construction and exploring the properties of the figure represented, the dragging function, under which the inherent properties are kept invariant, and the property checker are useful means of understanding the nature of the relations and the operations involved in the solution process. While in a traditional paper and pencil environment the provided feedback is merely of a visual and perceptual nature, Cabri offers a potential means of analysing the mathematical features of the problem under consideration and, at the same time, a visual demonstration of the properties and relations characterising the figure.

Characteristics of problems and situations for devolution

Although the features of the software are important in the evaluation of its role within the learning environment, it is also crucial to consider the types of problems which can be effective in fostering an idea of mathematical justification. Exploration and construction problems can both be suitable for producing conjectures and then testing them. There are some differences between the nature of justifications given in these two types of tasks. In exploration problems, what needs to be justified is a conjecture produced through the manipulation of the Cabri figure, while, in construction type tasks, what needs to be justified is the procedure leading to the construction, so that the output cannot be 'messed up'. Within these two broad classes of problems, mainly presented in an open form, a more specific characterisation might be helpful in order to address the issue of bridging the gap between informal and formal practices [6]. In particular, problem-based activities should be developed that highlight those aspects of the mathematical ideas involved that deal with issues of evidence and the perceived truth of the statements to be justified.

In order to give a complete 'definition' of the pre-formal space, attention should be paid to the situations in which problems are presented and tackled by students.

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Isolated episodes of problem solving sessions are not likely to bring about a real evolution of a sense of proving. A careful a priori analysis of the activities and a systematic use of discussion sessions aiming at the systematisation of the attained results should be part of the whole process (Brousseau, 1986).

Given all the above considerations, the main hypothesis underpinning these ideas can be expressed as follows:

The integration of Cabri-Gbom2tre within a complex learning environment including speczfxally designed problem-based activities can provide a pre-formal space whereby the students ' pre-existing rationality can evolve towards a sense of proving mathematically.

HOW TO IMPLEMENT SUITABLE PROBLEM SOLVING ACTIVITIES?

One of the central points related to this issue consists of finding a suitable and effective implementation of problem solving activities meeting the aims pursued. On the basis of an earlier study (Mogetta, 1996) the role of the characteristics of problems [7] seems to be relevant. In order to provide a fertile ground for the evolution of a sense of justification in mathematics, problems should be open-ended, requiring a conjecturing phase and a consequent exploration of the given situation, and not expressing intuitively evident properties.

One of the main issues is the construction of a link between mathematical properties and the explanations provided in order to justify their validity. The affordances of Cabri seem to provide the opportunity to build up this link and to reflect on the nature of the geometrical objects and their manipulation. In order to analyse the processes involved in the transition from the conjecture to the formalisation of the solution, and the role played by Cabri within these processes, I will report some examples from a short term experiment [8], involving five 17-year-old students, who had no previous experience with Cabri and who had not been introduced to geometry in a traditional deductive way. A short learning path was designed to introduce the students to the idea of justification in mathematics within a 'creative' problem solving process. Three sessions of one hour were held outside the classroom context, in a computer laboratory, and the researcher had the role of participant observer, introducing activities as well as interacting with students asking questions. Data were collected in written form, through completion of specially designed worksheets, and through videotapes and audiotapes.

Given the nature of the students' background in geometry and their lack of experience with the software, some time was spent to get them started with Cabri and to introduce them to geometrical properties and the idea of justification. The whole set of activities proposed was designed as a path aimed at the exploration of geometrical configurations, enunciation of properties and relations found and justifications expressed in verbal form. In the following, I bill focus on some results

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from the students' work to illustrate issues related to the features of the software and the production of arguments in order to justify constructions and conjectures.

JUSTIFYING CONSTRUCTIONS

One of the tasks had been set following exploratory work on the properties of parallel lines and a short collective discussion. Students were asked to tackle the following task:

Construct a line parallel to a given line through a given point without using the command parallel line.

Write a message describing your construction and pass it to your fiends. They will have to replay the construction following your instructions and check whether they obtain the same figure.

The students elaborated two substantially different constructions: the first one (Figure 1) using perpendicular lines and the second one (Figure 2) using symmetry with respect to a point.

1. Construct a line

2. choose two points on the line

3. construct two perpendicular lines from these points

4. select a point on one of the perpendicular lines and construct another perpendicular line

5. Measure the distance between the first line and the line last constructed. Do the same on the other side measuring between the point on the original line and the point of intersection of the last line constructed.

6. Measure the angles where

. the original line is crossed by the perpendicular line [angle x in thefigure]

. where the perpendicular line passes through the last line constructed [angle y in the

figure1

Figure 1: Zara's construction

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Zara's construction in Figure 1 clearly uses perpendicular lines in order to obtain parallel lines; there is an implicit use of the relation between these two concepts. The argument provided in order to support the validity of the construction and to convince the interlocutors of its correctness draws on measurement. This is not merely use of a Cabri function, but is related to the idea that the distance between two parallel lines is always the same. What is interesting is that the measurement in Cabri needs to be done from point to point and this needs the definition of all intersection points which would not need to be considered in a paper-based construction. As a further condition for the construction to be checked Zara proposes the measurement of the angles formed by the lines and their perpendicular transversal. In this case the property being used is the fact that two lines are parallel if the corresponding angles formed by their intercepts on a transversal line are equal. I would argue that measurement is not used in an entirely empirical way, but rather draws 011 ge01ne~ca2 properties of the f i p e constructed. Nevertheless, it is not yet a rigorous argument.

I Draw a line

I . draw a point directly below this line

I . use the symmetry option to reflect the line with respect to the point

use the option on the computer with a ? [referring to the icon corresponding to the property checker] and choose parallel. Click on the first line, then the second line. When a dotted box comes up click in any blank part of the screen and the message 'Objects are parallel' should come up.

Figure 2: Amy and Martha's construction

The construction in Figure 2 presents different features. A first remark concerns the use of the word 'draw' instead of 'construct', which might reflect a prevailing figural aspect in the perception of the figure on the screen. The mathematical transformation is performed by the software without any explicit construction by the students. The choice of this construction derives fkom an exploration of the Cabri commands and stresses the outcome of the construction, i.e. the parallel lines, rather than the procedure leading to this product. The validating function is then delegated to the software, whch is in charge of the mathematically based argument. What the students seem to need is a convincing argument for their interlocutors and this is obtained by checking parallelism through the property checking command. This tool simply reinforces the idea that the construction is correct, stressing the truth value of the statement.

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These two types of construction are substantially different and they could be perceived as providing more or less convincing arguments. Another pair of students first performed a construction similar to that in Figure 1, commenting that the command 'perpendicular line', used twice consecutively could automatically give parallel lines. This argument, drawing on the (implicit) conceptual equivalence between parallelism and perpendicularity, lacked a strong convincing value for the students themselves and led them to move to a construction using symmetry.

The lack of a formalised and well-structured theoretical background in geometry, together with the time constraints of the experiment, have highlighted the fact that Cabri itself might lead students to focus on the output of the construction, that is the objects constructed. The justification is therefore deprived of its mathematical basis, which is embodied in the Cabri commands used for the construction, and the focus is shifted to the elaboration of convincing arguments, describing the features of the objects constructed.

JUSTIFYING CONJECTURES

The final activity in the experiment was the solution of an open problem, concerning properties of the angle bisectors of a parallelogram.

PROBLEM OF THE PARALLELOGRAM 7 1 Construct a parallelogram ABCD and draw its angle bisectors. I Give a characterisation of the quadrilateral obtained through the intersection of the four lines.

I Discuss any particular configuration.

The conjecture of this property is not a difficult one after the construction of the figure and its dynamic manipulation. The empirical exploration of the configuration on the screen should quickly suggest a conjecture, namely that the requested quadrilateral is a rectangle. The difficult part consists of proving this conjecture on the basis of the properties emerging fiom the exploration itself.

In a second part of the activity students were asked to show whether and under which conditions the quadrilateral obtained is a square. Most of the students conducted the discussion of the particular configuration whereby EFGH is a square on the basis of the visual perception of the figure. In the case of Zara, the initial statement was just a description of the figure and only afterwards did she refer to the symmetry of the figure itself (see Figure 3).

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I In which case is the obtained quadrilateral EFGH a square? Justfi your answer. I When the angle bisectors cross the angle at 45' from all of the separate points they will all intersect at right angles to each other forming a square when the triangles made by the sides of the parallelogram and the points form a right angled triangle The middle quadrilateral is a square when it is symmetrical fiom all four comers.

Figure 3: Zara's conditions for a square

The question is not explicitly given an answer: Zara seems to be describing what she 'sees' on the screen, without perceiving the parallelogram as a rectangle in this specific situation. The Cabri figure in this case highlights some sub-configurations which might be relevant for a further justification, such as the right angled triangles formed by the sides of the parallelogram and the bisectors. The initial statement, suggested by the use of measurement and dragging, is not made explicit and Zara does not seem to link the properties of the internal quadrilateral to those of the external one. She seems to provide a number of different descriptions of the configuration remaining on a figural level. The final reference to the symmetry of the figure is also based on perceptual judgement, without any explicit mathematical justification.

In some cases students constructed the parallelogram using the symmetry with respect to the centre and then used this property implicitly in order to justify their statements about the quadrilateral formed by the intersection of the four angle bisectors. The assumption underlying this construction was then used as a proving argument. This is an example of the confision that can arise between premises and consequences within the proving process. This problem is a general one (Duval, 1992-93; Mogetta, 1998) but in this case it is reinforced by the use of Cabri because the construction used suggests some property related to transformations which had not been mentioned in the previous introductory activities. The construction method is not recognised as an implicit premise.

The conceptual aspects of the figure constructed are sometimes hidden by the figural ones, as shown by the following excerpt:

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Even though, when dragged, the rectangle often changed to be bigger than the parallelogram, the opposite lengths were the same and the angles were always 90" also the parallel lines on the parallelogram are always parallel.

The latter claim refers to the parallelism of angle bisectors, which is visually evident but would need a justification in terms of the hypothesis of the problem. Again in this case the dragging function allows a general claim based on a perceived configuration and no justification of the fact that the bisectors are parallel is provided.

The further construction required by the task, i.e. to draw the angle bisectors of the quadnlateral EFGH, leads to a figure which should be recognised as a square or even predicted to be a square, since this particular configuration can be inferred from the previous step in the sequence of questions presented. Zara persisted in her empirical approach, measuring both sides and angles of the obtained quadrilateral. Her written arguments are shown in Figure 4.

The premise fi-om which Zara drew her conclusion about the quadnlateral being a square appears confused: the fact that the triangles formed by the quadrilateral with the intersecting points are right-angled does not necessarily imply that the quadrilateral is a square. The only correct inference which can be drawn from this statement is about the measure of the internal angles of the quadrilateral, but this is not enough to claim it is a square. The statement made by Zara here seems to follow fi-om a perceptual and visual conviction about the properties holding in this particular case and, at the same time, from an attempt to find in the figure properties which are intuitively perceived or known as valid.

the quadrilateral obtained has four sides of equal length and four angles of 90". The triangles formed by the quadrilateral and the intersecting points form right-angled triangles. Therefore the 'new' quadrilateral is a square.

Figure 4: Zara's argument for a square

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In most cases, the students came up with a conjecture and then stopped. One pair of students conjectured that EFGH is a square only when ABCD is a rectangle and their justification was:

We measured the size of angles and distance between points of intersection. Sides are all equal, angles are all 90".

The measurement provides a description of the features characterising the figure, and it may be seen as a justification, but on an empirical level. Nevertheless, the argument relates to a correct characterisation of the square and the fact that the measures are invariant under dragging provided a convincing argument for the students.

Quadrilateral EFGH is a rectangle. All angles are 90"- angles remain same when dragged.

There are pairs of sides of equal length.

The invariance under the dragging function is here used as one of the intrinsic properties of the software, with convincing power because of its potential for the exploration of all possible cases. This seems to discourage students from looking for other mathematical justification, though this effect cannot be completely attributed to Cabri. These students' lack of familiarity with geometry and the idea of justifying statements mathematically and rigorously is certainly a contributing factor.

The use of linguistic markers of general statements, such as 'always', or 'in all cases' was quite frequent as a comment on the results of the investigations carried out with the dragging function. The justification was usually a descriptive one and the validating function was transferred to the software, which acquired an authority in mathematical terms.

Which quadrilateral do you obtain when you draw the angle bisectors of the quadrilateral EFGH? Justifi your answer.

It is a square no matter where you drag it because all the sides are equal all the time and the angles are 90".

Summarising the elements emerging from the students' work, it is clear that although there is a prevailing consideration of figural aspects of the figures on the screen, the arguments provided as justifications are empirical and of a perceptual nature. At the same time, however, they focus on the properties characterising the figures themselves. A validating function seems to be attributed to the software and, while this brings about the production of general statements or arguments expressing generality (at least in a verbal form) these predominately have the function of convincing. Measurement and the property checker are used as means to support the arguments provided; the properties are usually 'seen' or intuited from the dynamic exploration and the justification is just an additional explanation attached to something already perceived as 'true7'.

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SOME CONCLUDING REMARKS

The initial aim of this paper was to outline some features of a learning environment incorporating Cabri as an integrated part and as a cognitive tool which could possibly support the evolution of a sense of proving. The coexistence of figural and conceptual aspects in any geometrical figure, regardless of its representation on paper or on a computer screen, raises issues related to the role of the mediational tools in the conceptualisation and the management of geometrical configurations. Drawing on the theoretical discussion and on the empirical results of the short term experiment carried out, it is possible to claim that the use of Cabri as part of a complex learning environment offers a support as well as raising problems for proving.

Many of the difficulties met by students in providing justifications for their conjectures are brought about by the fact that the geometrical relations among the objects constructed are implicit, while the conceptualisation of figures in terms of properties and relations requires that those relations be made explicit.

The internal features of Cabri (in particular the fact that geometrical properties and relations are kept invariant under perturbation of the configuration) are potentially helpful in this respect because they help the learner find invariants and relate them to the properties of the configuration under examination. But these same features may also hinder the process of elaborating mathematical justifications, because they provide evidence of the actual 'truth' of the observed properties by means of precise and sophisticated instruments for empirical verification.

The fact that the figure constructed on the screen and its dynamic manipulation show the invariance of the geometrical properties conjectured inhibits students from giving linther justifications in mathematical terms. This does not mean that their arguments are not mathematical - they have a basis in the geometric domain - but they are not explicitly stated as such. The main difficulty met by students in the correct use of arguments lies in being aware of the status and the function of these arguments. This cannot be attained spontaneously.

Cabri-based activities need to take into account these elements and offer some complementary ones, which may provoke a sense of mathematical uncertainty and a consequent need for a justification. Therefore it seems feasible to suggest the use of dynamic geometry software in conjunction with open problems and a simultaneous introduction of the related mathematical theory, in the context of collaborative work within the classroom.

The examples presented suggest that the use of Cabri without an appropriate path aimed at understanding the links between software features and mathematical concepts, might hinder the process of justification, since the software might be considered (and employed) simply as a sophisticated tool allowing more precise operations and verifications. As a consequence, the idea of verifLing statements

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could possibly be confused with the idea of validating them with respect to a theory. Teachers and researchers should pay attention to this aspect in the use of software for educational purposes.

Other issues can be raised in relation to the use of Cabri in classroom contexts. The study reported here, although limited by time constraints and the overall length of the experiment, has shown the need for a long term path, implementing activities which allow a fruitful use of the potentialities of the environment and taking into consideration the rationality students have previously developed.

The exploratory power of Cabri and its dynamic features make it a tool for thinking and conceptualising figures in terms of relations and properties. The availability of a (geometrical) theory supporting problem-based activities to be carried out in this environment seems to be a necessary condition for the development of the idea of validation within this dynamic geometry context.

Even though this study has been conducted on a small scale, nevertheless it seems that the lack of 'formal' mathematical background of English students [9] does not lead them to approach problems with a 'theoretical' attitude; the use of intuitive and empirical results prevails over any attempt to justifir mathematically. Nevertheless, Cabri seems to offer space for a focus on mathematical aspects of the geometrical configurations. Making explicit what is an acceptable justification is a necessary step in order to build the links between the figural and the conceptual in geometry. The design of the learning environment should, as a consequence, take into account all the elements considered above, but also the baggage learners carry from their previous mathematical experience, in terms of notions, methods and general approaches to mathematical problems. This (cultural) element affects the choice and design of tasks, the use of the software features in the management of those tasks and, finally, the way (and the extent to which) the geometric theory is built up throughout the process [ 101.

Further research is needed in order to clarifir the characteristics of a learning environment which might support the evolution of a sense of proving. The characteristics of Cabri are not necessarily positive and fruitful. A careful definition and characterisation of problems and situations for the devolution of these problems is needed, in order to foster the idea of mathematical justification, beyond intuitions and empirical arguments. Within the outlined complex environment Cabri might act as a cognitive tool and enhance students' appropriation of the basic mathematical ideas underlying the proving practice.

NOTES

1. The experiment described later in this paper used the second version of the software, Cabri 11, but for the general description of the functions and role it plays in the process of teaching and learning it is not essential to choose one of the two versions.

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2. The learning environment I refer to is characterised by the coexistence of a microworld, a set of problems and activities to be carried out with both paper-based and computer-based materials, and whole class discussion sessions, orchestrated by the teacher (Bartolini Bussi and Boni, 1995) and aiming at students' negotiation of their own meanings.

3. In this context I refer to mathematical argumentation, as distinct f?om other forms of argumentation within different discourses. For a detailed analysis of this distinction see Duval (1992-93).

4. This pre-formal stage relates to some features of the learning environment, which will be discussed in the following section of the paper.

5. A simple example of this correspondence is given by the command parallel line. In order to construct a line which is parallel to a given one it is necessary to select the given line and a point through which the parallel line is going to pass. One cannot construct a parallel line without considering these two initial objects. This is exactly what is required by the fifth Euclidean postulate: given a straight line and a point not lying on it, there exists one and only one straight line which is parallel to the given line and goes through the given point.

6. This characterisation goes beyond the purposes of this paper and will be object of a wider research study, currently being carried out at the University of Bristol.

7. In this context the distinction is drawn between open and closed (or proof) problems.

8. The experiment was designed as a pilot study for the research study mentioned above. The main aims of the study were to investigate and evaluate the appropriateness of specific setting and problem-based activities for English GCE Advanced Level students, lacking a 'formal' education in geometry, the overall aim being the analysis of the evolution of the sense of proving mathematically.

9. This point has already been stressed by other studies: see Healy and Hoyles (1998) and Coe and Ruthven (1 994).

10. The specific requirements of curricula, time constraints and availability of computer facilities are also important elements which pose conditions on the flexibility of the learning environments in order to be adaptable to the needs of different cultural contexts.

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CABRI AS A COGNITIVE TOOL FOR THE EVOLUTION OF A SENSE OF PROVING