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Mariotti M.
A., Bartolini
Bussi M. G., Boero
P., Franca Ferri F., Rossella Garuti M. R.
(1997) PME XXI, Lahti, Finland. pp.180-195 |
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Abstract |
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1. IntroductionThis paper is based on research work regarding the approach to geometry theorems in schools carried out over the last five years by teams in Genoa, Modena and Pisa. These studies have involved students of different age groups (from grade 5 to grade 10) and different thematic contexts. Although the specific goals of these projects differed to some extent, they did share some common features such as general goals, research methodology, epistemological analysis and cultural, cognitive and educational hypotheses. A common framework is emerging as a result of a longstanding dialectic discussion dating back to the design of our teaching experiments: this framework has brought to light some of the deep yet implicit common motives and theoretical perspectives of our independent research designs. This paper provides a unified framework and an original survey of the research studies and their findings, which have been partially reported in other papers (Bartolini Bussi, 1996; Bartolini Bussi & Bergamini, 1996; Costa et al., 1994; Boero et al., 1996; Garuti et al., 1996; Mariotti, 1995b; 1996). We will focus on the following general points: the function of the different contexts in approaching geometry theorems; the role of the teacher in classroom interaction; and the idea of theorems as unities of statement, proof and theory. In this framework we shall analyse the function of dynamic exploration for approaching geometry theorems in those contexts at different school levels. Our studies take into account the following issues in current research into the school approach to theorems: the present-day value of proof in mathematics and mathematics education (Hanna, 1996), even for very young pupils (Maher, 1995); the social dimensions of the approach to proof (Balacheff, 1991) and the distinction between argumentative reasoning and deductive reasoning (Balacheff, 1988; Duval 1991); the classification of student proof schemes (Harel 1996) and the relevance of 'transformational reasoning' in the production of statements and the construction of proofs (Simon, 1996); the study of the potentialities of geometrical software (Goldenberg & Cuoco 1996; Laborde, 1992; 1993). Inspired by the seminal work of Balacheff (1988) and new studies (e.g. De Villiers 1991; Hanna & Jahnke 1993) on the pragmatic of proof, we focused on the link between epistemological, cognitive and didactic analysis. 2. Theoretical FrameworkThe general theoretical framework of our research studies is based on the construct of 'field of experience' and the construct of 'mathematical discussion'. As reported in Boero et al. (1995), a field of experience can be metaphorically defined as a system of three evolutive components (external context, the student's internal context and the teacher's internal context) referred to a sector of human culture which the teacher and students can recognize and consider as unitary and homogeneous. Classroom activities within any field of experience can have different goals: in this paper we shall limit ourselves to those related to approaching geometry theorems. In this perspective, the features of a field of experience that are meaningful for us can be described as follows: - the presence of 'concrete' and semantically pregnant referents (external context) for performing concrete actions that allow the internalisation of the visual field where dynamic mental experiments are carried out; this feature is consistent with Vygotskij's general theory on mental processes and with specific findings on the function of dynamic processes both in the production of conjectures and in the construction of proofs (see Polya's idea of variational strategies, as well as the recent consideration of 'transformational reasoning' by Simon, 1996); These points are consistent with general ideas about the production of geometry statements and the construction of proofs relying on the one hand on 'reified' (Sfard, 1991) pieces of knowledge produced by the historical evolution of mathematics and, on the other, on figural (Fischbein, 1993) referents, which may be either static or dynamic (Dreyfus, 1991). As concerns mathematical discussion, we refer to the metaphorical definition given by Bartolini Bussi (1996): mathematical discussion is a polyphony of articulated voices on a mathematical object, which is one of the motives of the teaching - learning activity. In this case the motive of the discussion is a specific theorem together with the idea of theorem itself (see below). Therefore the complex of conjecturing, arguing, proving and systematizing proofs related to a specific problem situation is taken into account by the teacher by means of mathematical discussion. The continuity between argumentation and proof is naturally emphasized by argumentative behaviours, but at the same time the distance between argumentation and proof (Balacheff, 1988; Duval, 1991; Moore, 1994) is taken into account by the teacher's careful management of discussion with the specific aim of the social construction of the sense and value of a theorem (Bartolini Bussi & Bergamini, 1996). Concerning this issue, we believe that two crucial points emerge from current literature: on the one hand, the problem of the motivation to proof; and on the other, the distinction between argumentation and mathematical proof. These two aspects are linked to each other in a complex way. Motivation to proof can be expressed at different levels. At the first level the truth of the fact is central: Is a fact true? At the second level, truth may no longer be in question, but a foundation of truth is needed: Why is a fact true? Hence the sense and the need for this grounding process (validation) is detached from the truth of the fact. In the first question, the truth of the fact is uncertain whilst in the second the truth of the fact may be certain. In our opinion, the uncertainty status of the truth of a statement is crucial for the initial construction of the meaning of theorems and calls for the careful selection of problem-solving situations, where the production of a conjecture is required. A third level, which is not considered in the research studies reviewed in this paper, concerns the release of theorems from the issue of truth search. In other words, we do not deal with formal proofs and their release from semantics. Within this general framework, we introduce two specific theoretical constructs, the 'cognitive unity' and the 'mathematical theorem', which may help the management of class work on geometry theorems, the functioning of problem-solving situations and the interpretations of student behaviour. These constructs may also represent instruments for analysing some difficulties students meet when following the traditional school approach to geometry theorems. 2.1. Cognitive UnityAnalysis of work done by past and present geometers highlights the continuity that exists between the process of statement production and the construction of its proof, as well as providing meaningful examples. This continuity is not evident at all in the theoretical systematization of ancient classical geometers such as Euclid and Apollonius, but is emphasised as from the 17th century, in documents that reveal the process by which a result has been obtained (Barbin, 1988). What is in play is the relationship between conjecturing and looking for a proof, in particular specifying the objects of the conjecture and determining stricter hypotheses or stating a new weaker conjecture (Alibert & Thomas, 1991; Lakatos, 1976; Thurston, 1994). More generally, the development of the relationship between conjecturing and proving witnesses the longstanding process of elaboration of the idea of rigour. Does a cognitive counterpart of this analysis exist? A metaphorical definition may be useful in analysing student processes. The continuity between the processes of conjecture production and proof construction, recognizable in the close correspondence between the nature and the objects of the mental activities involved, expresses a cognitive phenomenon, which will henceforth be referred to as 'cognitive unity'. Some hints about 'cognitive unity' are given in Harel's investigation into student behaviour (Harel, 1996). 2.2. Mathematical TheoremHowever, in mathematicians' mathematics the aforementioned continuity between statement and proof is always considered in a theoretical context, even if the context can change over time; the existence of a reference theory as a system of shared principles and deduction rules is needed if we are to speak of proof in a mathematical sense. Principles and deduction rules are intimately interrelated so that what characterises a mathematical theorem is the system of statement, proof and theory. Historical-epistemological analysis highlights important aspects of this complex link and shows how it has evolved over the centuries. 3. Towards Teaching ExperimentsAccording to the theoretical framework presented in the previous section, two crucial elements characterize the approach to geometry theorems in our teaching experiments: the function of a particular field of experience, and the role of the teacher as a cultural and cognitive mediator. Every field of experience has to be analysed in terms of limits and potentialities in fostering cognitive unity and a systemic approach to geometry theorems. Historical and epistemological analysis has allowed us to identify the following criteria which, in the presence of a culturally relevant piece of mathematical knowledge, make it possible to choose a field of experience and particular activities within it: the need for concrete and semantically pregnant referents that promote dynamic processes; and the availability of tasks, meaningful to the field of experience, that foster cognitive unity. Dynamic exploration of the problem situation can determine the production of conditional statements and the construction of proofs, with strong functional relationships between these processes (Boero et al., 1996; Garuti et al., 1996; Bartolini Bussi & Bergamini, 1996). The conditional form of statements, from Euclid to the present day, represents the functional connection between statement and proof: actually, a proof develops, in the form of a deductive chain, the link (which is implicit in the statement) between facts that are assumed as starting points in the frame theory and the 'thesis' of the theorem, under some conditions that are given as 'hypotheses'. As far as the role of the teacher is concerned, we assume that the process of construction of the meaning of theorems, although rooted in the field of experience, requires cultural and cognitive mediation. Actually, the teacher is responsible for introducing pupils to a theoretical perspective, which, although not spontaneous, is needed for a systemic view of mathematical theorems. In our teaching experiments, the construction of a theory is pursued in the form of accepted principles: invariants in perspective representation; the evident properties of shadows produced by vertical nails; and the underlying logic of Cabri. The research methodology is typical of long term teaching experiments: classrooms are observed for several months (or even years), by collecting individual texts and transcripts of collective discussions, together with teacher's reports. The length of the process determines evolution in the general assumptions, until specific hypotheses are reached. The specific aim of these studies is on the one hand to single out the conditions under which students can approach geometry theorems, and on the other to study the mental processes involved in such an approach. For these reasons, the direct and productive involvement of teachers in all the phases of research is called for in each of the three experiments. In spite of the common features they share, the studies deal with different didactic problems, and actually concern different school levels (5th, 8th and 10th grades). This requires completely different approaches to geometry theorems for two reasons: the different levels of cognitive development and geometrical knowledge that pupils have reached (geometry is taught in Italian schools from the 1st grade); and the general attitude towards mathematics and its methods derived from their past experiences. At the outset of work on geometry theorems, younger students do not yet have a sufficient grasp of geometry notions. For them, the approach to geometry theorems is a fundamental step in the process by which geometry gradually becomes a 'field of experience' (Boero et al., 1995) and a corpus of mathematical knowledge as well. At high-school level, where students have a grounding in geometry, the problem is how to manage the delicate relationship between their geometrical background and a new approach to this knowledge from a deductive point of view. 3.1. An Historical Digression: the Birth of a TheoryThe history of geometry gives meaningful examples of the development of fully-fledged theories from a long standing tradition of spatial practices. In this section we shall explore a paradigmatic example: the birth of projective geometry from the long-standing process of assuming properties of space and vision as axioms and modelling definitions, and of proving practical rules of painting as theorems. Natural perspective was developed from the classical age (Euclid's Optics) onward with the aim of representing objects with illusionistic effects. Practical rules for painting were transmitted in artist's workshops and collected in treatises of practical geometry. In the 15th century natural perspective gradually gave way to artificial perspective. This was based first of all on the idea of the (central) vanishing point or point of flight: if we consider the picture plane as a vertical window the spectator stands in front of, the central vanishing point is the point of the picture plane where a line from the spectator's eye, ortogonal to the picture plane, cuts it. This definition is taken from a more recent treatise (by Brook Taylor, 1715) where the genesis from practice was already somewhat hidden. The genesis is more evident if we consider that in early treatises, which contain also a theory of vision in space (e.g. Piero della Francesca, 1464) the central vanishing point was named 'eye'. The history of the theoretical development of artificial perspective up to projective geometry is actually the history of its progressive independence from painting practices, from Desargues' first introduction of invariants (Field & Gray, 1987) to the 18th century treatises of linear perspectives of Brook Taylor and Lambert (Bessot & Le Goff, 1992): the incidence axioms listed by Brook Taylor gave birth to a projective approach to problems, and Lambert's use of perspective to prove properties of plane configurations stated definite autonomy from painting. Within the theory of projective geometry, based on incidence axioms, practical rules of painting assumed the status of theorems. A similar analysis could be made for the genesis of other theories in the history of geometry (see the analysis of sunshadows in Serres, 1993 and the analysis of geometrical construction in Lebesgue, 1950). Actually, the above perspective has guided our experimental research studies into the school approach to geometry theorems. 4. A Survey of the Teaching ExperimentsThe teaching experiments that have been carried out by our research groups concern the approach to geometry theorems (production of statements and construction of proofs inside a frame theory) in three different fields of experience: the representation of visible space by means of geometrical perspective; sunshadows; and geometrical constructions in the Cabri - environment. 4.1. The Representation of Visible Space by Means of Geometrical PerspectiveThis experiment concerns the field of experience of the representation of the visible world by means of geometrical perspective. The early approach to theorems takes place in the educational context of 'mathematical discussion' as a fundamental tool for the social construction of knowledge. The most meaningful problem-solving situation, used in several 5th and 6th grade classrooms, concerns the drawing of a small ball in the centre of a table, drawn in perspective. In particular we shall refer to data from an experiment carried out with the same class over four years, from grade 2 to grade 5. Yet before explaining this problem, some information about previous school activities is needed (a more detailed account of which is contained in Bartolini Bussi, 1996). At the beginning of the whole experiment most of pupils (2nd graders) were able to draw simple three dimensional objects, such as a set of boxes, tables and chairs. The level of mastery was variable: a teaching experiment on the coordination of points of view (Bartolini Bussi,1996) had contributed to solve some of the early problems of real life drawing ( the drawing of hidden lines; the drawing of a base line for all the elements lying on the ground plane). Anyway no explicit reference to techniques of perspective drawing had previously been made in the classroom. This is the school context where the crucial problem situation was given. A table, drawn in perspective, was given together with the task: 'Draw a small ball in the centre of the table. You can use instruments. Explain your reasoning'.  The strategies from the classroom can be roughly divided into three categories: a) ROUGH ESTIMATE BY SIGHT: the ball is put directly on the table, without comment. Solution a) refers to everyday practice; solution b) is an attempt to rationalise by means of a scientific tool (measuring); while the correct approach, solution c), depends on an invariant of perspective representation, i.e. alignment of points. The solution based on rough estimation is predominant among young pupils such as 2nd graders, while older pupils prefer measure-based solutions as more precise (when compared to rough estimate) and therefore true. The alignment-dependent solution is the least popular even with older students up to university level. In the 2nd grade class of the experiment, there was no discussion about this problem or the different solutions proposed by pupils. Later a tool of semiotic mediation was built to direct perspective drawing as well as to evaluate perspective representations (Ferri 1993). The following table shows the scheme built by the same class one year later (3rd grade): the geometrical properties of three-dimensional figures (still alluding to real objects because of the age of the pupils) and the corresponding properties of two-dimensional representations were compared.
The scheme was built collectively under the teacher's guidance and reflects the random order of students' observations: each student recorded it in his/her copybook. The crucial row is the following: straight lines in the first column correspond to straight lines in the second one. This property was stated on the basis of empirical observation of both objects and images. Later it was used systematically and autonomously by pupils both to check drawing output and to direct the drawing process itself. Yet this early scheme of invariants (and of variants as well) functioned also as a germ of a whole theory of perspective representation. In the same classroom, the problem of the table and small ball was posed again at 5th grade level, three years after the first attempt: in the meantime different activities about perspective drawing had been carried out in the classroom (Bartolini Bussi 1996) but the 'table and ball problem' itself had not been discussed any more. The second formulation of the problem was a bit different. With the same drawing, the following task was assigned: 'Draw the small ball in the centre of the table. Prove the method you have used'. Actually the (correct) solution to the 'table and ball problem' happens to produce a statement and a proof framed by the germ - theory of perspective embodied in the table of invariants *). In the classroom, all the pupils (19) gave the correct solution, but not everybody succeeded in proving the method. Of those who succeeded, we distinguish between those who gave statement and proof (6 pupils); and those who gave statement, proof and the genesis of the theorem (5 pupils). The latter term refers to the more or less detailed reconstruction of the meaning of the theorem within the activity on perspective drawing carried out over three years in the classroom. Some pupils proved to be aware that the problem had already been set three years before, even though it was not exactly the same problem: in this case historical reconstruction or metacognitive explanations of one's own long- term processes were offered. Of the other pupils (8), 5 'proved' the method by stating that the other methods were wrong, while the last 3 simply gave the correct solution. Of course, due to the age of the pupils involved, we accepted diversions from a rigorous formulation provided the intuition of generalisation, the intuition of abstractness and the deductive structure were conserved. As an example, we report excerpts from the work of two pupils: Federico: 'The right method is to draw diagonals because in any figure, in perspective or in plan, the intersection of diagonals gives the centre, because straight lines retain their characteristic and remain straight'. To interpret the data correctly, it should be noted that the words 'proof' and 'proving' ('dimostrazione' and 'dimostrare' in Italian) had not yet any institutional meaning in the classroom: the words had been used freely by pupils in collective discussions to mean the proposal of very safe arguments. The above data show that most of the pupils related the statement to the shared theory in order to produce the safest deduction: i. e., they were confident that, as the theory was true, the logical consequences of the theory were also true. The previous summary shows that the field of experience of visible space representation using geometrical perspective, through the approach of mathematical discussion, offers a suitable context for introducing to very young pupils an early idea of theorem, namely a statement with a justification that refers to an accepted theory. While measurement shifts reasoning to the empirical level, a geometry without measurement can open up different possibilities of exploration. Hence, perspective provides a field of experience in which measurement is no longer the empirical instrument for the validation of the results. Even if some of the students' products are expressed in a bare language that refers to logical rules and not to control in reality (see Laura above), their idea of theorem is undoubtedly that of a true statement whose justification lies in the theory: justification is thought to be needed mainly because it is an astonishing statement, if compared with the more readily acceptable measure - dependent solution. 4.2. SunshadowsThis experiment concerns the field of experience of sunshadows. The most meaningful problem solving situation, used with several 8th grade classes, concerns the study of the parallelism of shadows of two non-parallel sticks, leading to the proof that the conjectured condition for parallelism is sufficient and necessary. In two classes, activities were organised according to the following stages (for further details, see Boero et al., 1996; Garuti et al.,1996): a)
Setting the problem : 'In the past years we
observed that the shadows of two vertical sticks on
horizontal ground are always parallel. What can be said of
the parallelism of shadows in the case of a vertical stick
and an oblique stick? Can the shadows be parallel?
Sometimes? When? Always? Never? Formulate your conjecture as
a general statement.' b) Producing conjectures (individually, or in pairs). c) Discussing conjectures: the conjectures were discussed with the help of the teacher until statements of correct conjectures were collectively obtained that reflected the students' different approaches to the problem. d) Arranging statements: through different discussions orchestrated by the teacher, the following statements,'cleared' of metaphors and more linguistically precise than those produced by students at the beginning, were collectively attained: 'If the sun rays belong to the vertical plane of the oblique stick, the shadows are parallel. The shadows are parallel only if sun rays belong to the vertical plane of the oblique stick '; 'If the oblique stick is on a vertical plane containing sun rays, the shadows are parallel. The shadows are parallel only if the oblique stick is on a vertical plane containing sun rays' e) Preparing proof; the following activities were performed: - individual search for analogies and differences between one's own initial conjecture and the 'cleared' statements; f) Proving that the condition is sufficient. g) Proving that the condition is necessary. h) Final discussion, followed by an individual report about the whole activity. The most interesting results concern: - Analysis of the sunshadows field of experience, as the basis of pieces of knowledge and empirical evidence allowing the production of general statements and their justification. The sunshadows field of experience is a context in which students can naturally explore problem situations in different dynamic ways. In order to study the relationships between sun, shadow and the object which produces the shadow, one can imagine (and, if necessary, perform a concrete simulation of) the movement of the sun, of the observer and of the objects which produce the shadows. The sunshadows field of experience also offers the possibility of producing, in open problem - solving situations, conjectures which are: meaningful from a space geometry point of view; not easy to prove; and lacking the possibility of substituting proof with the creation of drawings. As to conjecture production: A) the conditionality of the statement can be the product of a dynamic exploration of the problem situation during which the identification of a special regularity leads to a temporal section of the exploration process, which will be subsequently detached from it and then 'crystallized' from a logic point of view ('if.., then..'). As to proof construction: B) for a statement expressing a sufficient condition, proof can be the product of the dynamic exploration of the particular situation identified by the hypothesis; It seems to us that the students' collected texts clearly reveal that dynamic exploration of the situation singled out by the hypothesis fulfils an important function in promoting the logical connection between the property accepted as true (parallel sticks produce parallel shadows) and the property to be confirmed (shadows are parallel): movement of the stick keeps the direction of its shadow (since it happens in the vertical plane containing sun rays) and, therefore, opens up the possibility to reason in a transitive way (e.g.: the real vertical stick produces a shadow parallel to the one of the imaginary vertical stick; the oblique stick produces a shadow aligned with that of the imaginary, vertical stick; so the oblique stick produces a shadow parallel to that of the real vertical stick). - Experimental evidence of cognitive unity between the phases of conjecture production and proof construction through the link revealed between the dynamic exploration in conjecture production and the construction of proofs. We detected a process with the following characteristics: during production of the conjecture, the student progressively works out his/her statement through an intense argumentative activity functionally intermingling with the justification of the plausibility of his/her choices. During the subsequent statement proving stage, the student links up with this process in a coherent way, organising some of the justifications ('arguments') produced during the construction of the statement according to a logical chain. Actually, as concerns the production of the statement, argumentative reasoning fulfilled a crucial function: it allowed students to consciously explore different alternatives, to progressively specify the statement and to justify the plausibility of the produced conjecture. On the other hand, students who produced wrong conjectures later showed the need to reconstruct the valid conjecture in order to produce the proof. The fact that poor argumentation during production of the statement always corresponded to lack of arguments during construction of the proof seems to confirm the close connection that exists between production of the conjecture and construction of the proof. Moreover, consistency between personal arguments provided during the production of statements and the ways of reasoning developed during the proof seems to be confirmed by two facts: I) the type of argumentative reasoning made during the production of the statement by one student was resumed by him/her (often with similar linguistic expressions) in the justification of the statement subject to proof; and II) the kind of dynamic process (movement of the sun or the stick) recorded at the conjecture stage was almost always the same as the one used at the proof stage. Yet, the dynamic exploration implemented during the construction of the proof, though remarkably similar to that implemented during the production of the conjecture in respect to the type of movement, differs deeply as to the function assumed in the thinking process: from a support to the selection and specification of the conjecture, to a support for the implementation of a logical connection between the property assumed as true ('vertical sticks produce parallel shadows' ) and the property to be validated. 4.3. Geometrical Construction in the Cabri EnvironmentThis experiment concerns the field of experience of geometrical construction. The main point was to introduce pupils to the deductive approch to geometry, i. e. to the construction of a system of geometrical facts, coherently related according to the choice of primitives (axioms) and the method of deduction. The history of the classic impossible problems (which puzzled the Greek geometers) tells us about the fundamental theoretical importance of the notion of construction (Henry, 1993). Despite the fact that there is a concrete counterpart to geometrical construction which can be accomplished on a sheet of paper, geometrical constructions have a theoretical meaning that overcomes the apparent practical objective. The tools and their rules of use correspond to axioms and theorems of a theoretical system; a construction given there is a theorem validating it, i.e. it states the relationships between the elements of the geometrical figure, which is represented by the drawing produced. The complex relationship (Heath, 1956, pp. 124-31) between constructions and theorems is not immediate and is difficult for students to grasp. A drawing is a material product of concrete operations and its correctness is definitely controlled by empirical evaluation; theoretical control is not spontaneously achieved, but can result from the activities that pupils perform within the chosen field of experience (Mariotti, 1996). The main motive is the evolution of the idea of construction from the empirical to the theoretical level and the evolution of the justifying process from generical argumentaion to proving. The main elements characterizing the project are: the semiotic mediation (Vygotskij, 1978) offered by the Cabri environment (the primitive commands and macros force students to make explicit geometrical properties hidden in free-hand drawing); the dragging function which starts as a control tool to check the correctness of the construction, then becomes the external sign of the theoretical control; mathematical discussion as a basic element in the social construction of the meaning of theorem. 4.3.1. The Cabri Enviroment In the Cabri environment, the construction activity, i. e. drawing figures through the available commands on the menu, is integrated with the dragging function: in other words, the construction of a figure can be associated with a control by dragging. Thus a construction task is solved if the figure on the screen passes the dragging test. In this case, the necessity of a justification for the solution comes from the need to explain why a certain construction works (i. e., it passes the dragging test); thus, a justification comes from the need to validate one's own construction, in order to explain why it works and/or foresee that it will do so.The key point is that what must be validated is the correctness of the construction; i. e., it is not the product of a procedure that must be validated, but the procedure itself; the necessity of this validation is mediated by the necessity of explaining why the Cabri-figure will not be messed up. When dragging is used, why do some constructions work and not others? The dragging function is accepted as a validating tool, but the problem must be shifted from validating by dragging to explaining the 'proof by dragging' itself. According to the theory of figural concepts (Fischbein, 1993; Mariotti, 1992, 1995a, 1995b), pupils must achieve the conceptual control over what they see on the screen. It is this very control that we want to promote, together with the idea of a coherent system. The richness of the environment may emphasize the ambiguity about intuitive facts and theorems and may constitute an obstacle to the choice of correct hypotheses. Thus, the basic idea of working inside a microworld was adapted to our objective. At the beginning, an empty menu is presented and the choice of commands discussed, according to specific statements selected as axioms, then the other elements of the microworld are added, according to new constructions and in parallel with corresponding theorems. In this way the system is slowly built up; step by step the complexity increases. The analysis of pupils' texts shows the slow evolution in the meaning of construction. At first, a construction is conceived as a concrete process towards production of a drawing, which has its justification in the acceptability of the product; then, a construction is conceived as a theoretical procedure which has its justification in a theorem. On the one hand, the descriptions of the procedure change, improving clarity through increasing mastery of correct terms; on the other hand, the argumentations approach the status of theorems, i. e. the justifications provided by the pupils assume the form of a statement and a proof. 4.3.2. Mathematical Discussion According to the general hypothesis about social construction of knowledge, mathematical discussion (see Section 2) plays an essential part in classroom activities. In this case, the motive concerns the evolution of the personal senses of justification, related to the problem of construction. The cognitive dialectics between personal senses and the general meaning (which is constructed and promoted by the teacher) concerns the senses of justification and the general meaning of mathematical proof. Different senses of justification correspond to different goals of discussion, so that there is an evolution from a first goal: to determine the epistemic value (Duval, 1991) of a certain fact or statement (Is the figure a square? Is the figure always a square? Why?), to a second goal: The second goal is related to a general motive of activity, i.e. awareness of a new theoretical status of certain statements and of their relationships within a theory. The development from the first to the second goal can be described as follows:
At the beginning, the focus of the discussion is on the comparison of the different drawings produced; it then moves on to the comparison of different procedures. This is the first crucial change required to finally approach the heart of the problem: validation in terms of a system of definite rules. In each phase the role of the teacher is fundamental in order to move the goal of the discussion and to guide the evolution of the personal senses towards the geometrical meaning of a construction problem. Moreover, in each phase the reference to Cabri is fundamental; passing from empirical control of the drawing to validation of the procedure, reference to the Cabri-figure and the dragging function is vital. The specificity of the Cabri environment clarifies the question about the procedure, rather than about the product. Without the mediation of the Cabri environment this sense may be unintelligible. When the drawing is done on a sheet of paper, it is very difficult to avoid validation of the construction focusing on the drawing itself and on direct control by observation. Actually, the question is about the drawing, but its sense concerns the geometrical figure that it represents, thus the ambiguity between drawings and figures constitutes an obstacle. On the contrary, in order to grasp the theoretical status of the question, a crucial factor is the reference to a 'microworld' which embodies a theory, that has its own independence (the machine has its own logic); and which is external to the subject (but is accessible to the pupil via the dragging function). In addition, the microworld is also independent from the teacher's authority and promotes a personal relation of the learning subject with geometrical activity (changing the didactic contract, Brousseau, 1986). Looking for a justification within the system of rules of the machine (in this case, the Cabri environment) introduces pupils to a theoretical status of justification; thus it is not the discussion in itself, but the discussion guided by the teacher according a specific goal which determines the meaning of the construction problem to evolve from the empirical to the theoretical level . 5. ConclusionsFirst of all, we consider in what sense during our experiments students have performed a mathematical activity concerning theorems. This problem is particularly relevant for the first and the second experiment. The objects of these experiments were hypotheses concerning the rationalisation of drawing practice and the physical phenomenon of sunshadows; they had as geometric counterparts, at the model level, a statement of central projection and a statement of parallel projection geometry. In the first experiment, pupils produced conjectures about drawing, collected them in the table of invariants and proved to be able to use those conjectures (assumed as postulates) to justify a process with solid arguments. In the second experiment, students produced their conjecture as a hypothesis concerning the phenomenon of sunshadows; when they verified their conjecture most of them seemed to be aware of the fact that they had to get to the truth of the statement by reasoning, starting from true facts. Most of them produced a validation realized through a deductive reasoning. Actually their reasoning started from properties considered as true, (such as 'two vertical sticks produce parallel shadows') and got the truth of the statement in the 'scenario' determined by the hypothesis. In this way, students produced neither statement of geometry in the strict sense, nor a formal proof: objects were not yet geometric entities, deduction was not yet formal derivation. But their deductive reasoning shared some crucial aspects with the construction of a mathematical proof. In the third experiment, in spite of the presence of concrete referents (drawings in the Cabri environment), axioms and modes of deduction were in the foreground. Moreover, the whole activity performed by students in all the experiments shared many aspects with mathematicians' work when they produce conjectures and proofs in some mathematics fields (e.g. differential geometry): mental images of concrete models are frequently used during those activities. As to proof, mathematicians frequently come close to realising the ideal of the formal proof only during the final stage of proof writing. During the stage of proof construction, the search for 'arguments' to be 'set in chain' in a deductive way is frequently performed through heuristics, reference to analogical models and taking into account the semantics of considered propositions (cf Alibert & Thomas, 1991; Thurston, 1994). For these reasons we think that the activities performed during our teaching experiments may represent an approach to mathematics theorems which is correct and meaningful from the cultural point of view. In this way, our experiments also show that the contradiction highlighted by Duval between everyday argumentation and deductive reasoning, between empirical and geometrical knowledge, can be managed in dialectic terms within the evolution of classroom culture. This can be achieved through the distinction between the nature of the productive process and the formal features of the final product, on the one hand and between the 'concreteness' of the referents in the productive process and the conventionality and abstraction of geometrical objects on the other. In all three experiments classroom culture is strongly determined by recourse to mathematical discussion orchestrated by the teacher to change the spontaneous attitude of students towards theoretical validation. In our research studies, this general conclusion is related to the study of the conditions and processes that allow students' constructive approach to geometry theorems. Features of the contexts were analysed as sources of 'concrete' referents and dynamic processes for the production of statements and the construction of proofs. Each context allowed and encouraged students to develop specific processes of dynamic exploration and promoted work with spatial metaphors, which are essential in the production of geometry theorems. Features of dynamic exploration were analysed to detect different functions. In all three fields of experience, the initial outcome is the generation of a space of possible configurations to be explored with different goals. In the field of experience of representation of visible space, dynamic exploration contributes to the collective construction of the theory (table of invariants) and to solution of the problem. In the sunshadows field of experience, dynamic exploration allows production of the conjecture and construction of the proof in the subspace defined by the hypotheses. In the field of experience of geometrical constructions in the Cabri environment, the need to produce figures that resist dynamic exploration (dragging) is the driving force behind the production of acceptable constructions and proof of their validity. The hypothesis of 'cognitive unity' on which we worked seems to have important didactic implications, since it calls into question the traditional school approach to theorems. In fact, teachers in Italy, as well as in other countries, usually ask students to understand and repeat proofs of statements that they supply; this appears one of the most difficult and selective tasks for grade IX-X students. Only as a possible final stage (often reserved to top-level students or students choosing an advanced mathematical course) does the teacher ask the students to prove statements, generally not produced by students but suggested by the teacher. Even more seldom are students involved in the process of producing conjectures. If our hypothesis is valid, during this traditional path student difficulties may at least partly depend on the fact that they have to reconstruct the cognitive complexity of a process in which mental acts of different nature functionally intermingle, beginning with tasks that by their nature lead towards partial activities that are difficult to reassemble in a single whole. 6. Some Emerging ProblemsIn this paper we discuss the problem of approaching geometry theorems at different age levels, within the same theoretical framework. Our findings confirm the possibility of early introduction to theorems in suitable fields of experience, provided a suitable epistemological analysis is performed. The problem of transferring the capabilities developed in our fields of experience to other "purely mathematical" contexts remains to be tackled. With regard to this problem, later observations made while the two classes involved in the sunshadows experiment (Section 4.2.) worked on traditional geometry theorems produced some evidence of transfert. In particular, many students (of both high and average level) could imagine the dynamic exploration of the geometric figures proposed for the formulation of conjectures and proofs. Another research problem concerns the epistemological analysis of statements from the point of view of their conditional form, and its relation to the production process. As conditional form is crucial in the production of theorems, for a given statement we need to look for contexts and tasks (within contexts) that can induce this form. We are well aware that dynamic exploration is not the only possible source of this form. For an example, see Boero & Garuti (1994): during a teaching experiment concerning production of statements, the following type of reasoning was identified in 3 students out of 34: "The length of the shadows is proportional to the height of the sticks due to the parallelism of the sun shadows .... If the lines are parallel, the lengths of the segments cut on another two lines will be proportional". In this case the student passes from a recognition of causal dependency between parallelism and proportionality in the physical phenomenon, to the conditional statement that takes into account the possibility that lines cannot be parallel. Different fields of experience are related in the literature to the issue of geometry proofs: sunshadows (4.2.); geometric construction in Cabri (4.3.) or other software (e.g. Geometric Supposer or Geometer's Sketchpad in the perspective of the 'dynamic geometry' indicated by Goldenberg & Cuoco, 1995); the 'mathematical machines' (Bartolini Bussi, 1993), gears and mechanisms (Bartolini Bussi et al., submitted) and the 'representation of the visible space' (4.1). Further analysis and specific comparison needs to be done concerning the potentialities and limits of those contexts in the genesis of conditionality and the transition from justification to proof. Note*)The complex of the
statement (which expresses a construction process) and the
proof of the theorem can be given in the following
terms: ReferencesAlibert, D. & Thomas, M.: 1991,
'Research on mathematical proof'. In D. Tall (Ed.), Advanced
Mathematical Thinking, Kluwer Ac. Pub. |
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