1. Introduction
1.1. Let us start with a very short commentary on the title and the content of the chapter. The term "rational", as opposite to intuitive, is meant to refer to any aspect of the logical and theoretical organization of the knowledge of Geometry, of its presentation, and of course, since our principal concern is the educational process, to all the implications that the theoretical structure has on teaching activities and choices. This distinction, between the intuitive part and the rational part of geometry, loosely corresponds to the points of view adopted by the official programs for the teaching of geometry at two different school levels (primary and secondary), in Italian school organization. Therefore in this chapter papers shall be taken into account which deal essentially with problems and experiences of the secondary school: this will be the rule, but many exceptions will occur.
On the other hand, a very important distinction has to be made in the meaning of the general term "mathematics education", since it obviously refers not only to the process of becoming acquainted with various kinds of mathematical notions and tools, but also on the acquiring of a correct mathematical way of thinking. Therefore we shall distinguish between the training in using mathematics and the gradual appropriation of mathematical mentality.
We recall also that geometry has played a very important role in the Italian mathematical tradition of the last centuries, also for what concerns the domain of mathematics education. The inheritance of this is in particular a large influence of geometry scholars in the organization of the universities curricula for future school teachers, and also the presence in Italy of a plurality of research groups and branches in the borderline area between theoretical studies of geometry and educational applications.
We also recognize that new fields of research that link the mathematical contents to problems of a psychological nature or to the general theory of the learning processes are becoming more and more important in recent years.
To do research in mathematics education may consist in many different activities. On the one hand one can study efficient teaching techniques using experimental methods or results from general psychology. On the other hand any educational process requires a deep knowledge of the particular subject to be taught, a long experience of teaching and the strong confidence in having important values to pass on. For this reason we can say that many outstanding mathematicians have made important improvements to mathematical teaching and education just reflecting on their experience as researchers and teachers.
For the above reasons, it is very difficult to delineate a good map of all the main and secondary research streams; moreover, since the strict pertinence and the relative importance of them is often judged in different ways by different people, not all shall agree with our personal choices. So we apologize in advance for the possible discrepancies between our personal taste and the preferences of other members of the scientific community.
We shall divide the Italian research in mathematics education, concerning the rational aspects of geometry, into the following main themes:
1.2. Now we give a short outline of the contents of the three themes.
We also observe that some of the papers quoted in this section could equally well be referred to in the chapters devoted to the intuitive aspect of geometry or to Logic.
Many of the papers quoted in this section refer to particular tools employed in teaching strategies, in particular consider the aid provided by computers in many classroom activities. Therefore some of the material examined here is on the border between this chapter and the chapter on Mathematics and Technology.
Other authors stress the relevance of some knowledge of history of mathematics for the teacher. As a matter of fact, at least some minor remarks about the importance of history for didactics is present in a large number of the works collected in this chapter.
In this section we shall also consider studies that suggest how to improve the interest and the motivation of the pupils in learning geometry, and their ability in the usage of a correct mathematical language.
Of course, when a contribution is mostly oriented towards problems of psychological nature of the teaching/learning process, then some superposition is possible with the content of the report on Teaching and Learning Processes, in this volume.
2. The disciplinary theme
2.1. A research group which is active mainly at the University of Catania has been studying for a long time various problems concerning geometric transformations of the Euclidean plane and space. Many papers contain their results. Their main idea is to study the different transformations, isometries, similarities and affinities, giving a classification and some characterizations of their groups and main subgroups and considering various examples of invariant properties. The algebraic point of view is stressed and the analytical method is used extensively in all these papers.
Going into detail, in (Mammana and Micale, 1991a; 1991b; 1991c) the group of similarities is studied respectively in the real line, in the real Euclidean plane, in the real Euclidean space. The transformations are defined by means of their equations and then are studied with regards to their fixed points or to some particular subgroups like those of the isometries and of the dilatations and their generators. In (Mammana and Micale, 1994) a characterization of the plane similarities in the group of the plane affinities is given: an affinity is a positive similarity if and only if the angle of corresponding lines is constant. The theorem is proved using instruments of analytic geometry.
In (Mammana and Micale, 1992a; 1992b) a theoretical problem concerning proper isometries is studied, and some possible teaching implications of the obtained results are deduced.
In (Mammana and Micale, 1995a; 1995b; 1995c; 1995d) the case of affinities in the real Euclidean plane is exhaustively studied. Using analytic geometry, the equations of such transformations are given and many of their properties are deduced.
Finally, in (Micale and Pennisi, to appear) some properties of the geometry of triangles are considered and their congruence theorems are studied.
2.2. The introduction of geometric transformations for the study of Euclidean geometry in the secondary school in Italy has given rise to many deep discussions and to a large debate in the Italian research community. More generally, the question has been put forward of comparing the different approachs to the logical foundations of Euclidean geometry from many points of view. An interesting survey of several aspects of this problem can be found for example in (Villani, 1995), where a comparison is made between the "spirits" of the different axiomatizations of geometry due to Euclid and Hilbert, to Choquet and to Peano.
It is well known that geometry is at the same time a rational description of the physical world, that means a science having a precise content, and a purely theoretical and formal language characterized by a hypothetical-deductive structure. Then it is obvious that the main didactical task is to reach a good balance between the necessary intuitive approach and a proper theoretical experience of the formal structure of the discipline. The problem exists both for the teacher and for the students. That means that the teacher must be able to distinguish at any moment of his educational task the intuitive side from the scientific aspect of the subject. To this purpose, it is necessary that (s)he has sure and clear ideas about the logical basis and the structure of geometry and at the same time is able to recognize in the real world the roots from which the geometrical ideas come. For this reason it is important and useful for the teachers to know about the different logical foundations of geometry and moreover to be able to compare the different logical approaches that have been proposed and studied by various scholars.
Many papers recently published give in depth contributions in this important area. For example in (Maraschini and Menghini, 1992) an interesting discussion can be found about the validity of the Euclidean method in teaching geometry at the secondary school and, furthermore, an analysis of different solutions to this problem proposed in Italy in the last century . In the Authors' opinion this problem has three aspects: i) the relationship between the hypothetical-deductive structure of geometry and the role of geometrical intuition; ii) the possible ways of introducing the fundamental notion of equality for geometric figures; iii) the comparison between a purely synthetic introduction of geometry and a mixture of geometry with arithmetical and algebraic methods. The first aspect is not discussed in the paper. For the second one, many different answers are presented: for example from Hilbert, Klein and, in Italy, from Veronese and Enriques and from Amaldi and De Paolis. The third aspect is illustrated by the work of Legendre and by the different reactions it caused in the Italian school.
In (Menghini, 1992) a notion of geometric constructibility is introduced by means of the ruler and the square. In this way, adding further axioms, e.g. some order properties and Desargues proposition, some interesting classical constructions in the affine plane may be considered.
Another set of axioms for Euclidean geometry together with some suggestions of teaching strategies is contained in (Cruciani, 1988). In this paper the primitive notions of monoid of lengths and areas are introduced and then, as a consequence, by means of the notion of perpendicularity, the Author deduces the definition of parallel lines and the order properties of the plane.
Among the different approaches to the foundations of geometry, Hilbert's plays a peculiar role. His point of view originates in a very rich phase of the history (of mathematics, but not only). In (Marchi, 1992) a brief survey is offered of the deep changes occurring in the beliefs of the science community of that period, and a commentary about the influence that this scientific environment had on Hilbert's ideas.
In the study of the affine planes and spaces, a very important problem is their representation (when it is possible!) as vector spaces over a suitable skew field. In (Fiori and Pellegrino, 1995b) and in (Fiori and Pellegrino, 1996) a classical construction is described of the skew field associated to a Desarguesian affine plane. In this case the affinities of the plane are, as it is well known, the linear and semi-linear transformations defined on the associated vector space.
As we know, Hilbert's system of axioms is nothing else but the formal logic foundation of Euclid's geometry. Nevertheless Hilbert's notion of straight line is really very far away from any intuitive idea. For this reason the Italian logician G. Peano, stating his system of axioms for Euclidean geometry, preferred to stress the notion of segment instead of that of line. This is just an example of the different mentalities of the two great authors and, as a consequence, of the differences in their sets of axioms for the absolute plane. This situation is extensively discussed in (Marchi, 1994), where a comparison between the different axiomatic choices of Hilbert and Peano for the Euclidean plane are examined.
The point of view of G. Peano is extensively examined also in another contibution to this debate, (Manara, 1994a), where it is compared with the ideas of A. C. Clairaut, A. M. Legendre, H. Helmholtz, M. Chasles, F. Klein and H. Grassmann.
Finally, in (Lucchini, 1992) a comparison is made among different editions of the original works of Hilbert, including much information about discrepancies in the respective contents. The interest is due to the possible utilization of the Hilbert axiomatization of geometry in the Italian Scientific Lyceum.
A very different approach to the foundations of geometry inspires some recent research, involving the possibility of building an axiomatic system in which the notion of point is not primitive. Starting from some ideas contained in a classical philosophical work by A. N. Whitehead, and translating them into a mathematical system of axioms, (Gerla and Tortora, 1992) and (Gerla and Tortora, to appear) offer an unusual approach which could be of some usefulness as a teaching strategy.
2.3. The notion of order plays an important role in Euclidean geometry but, nevertheless, seldom appears in the teaching literature. The paper (Cruciani, 1990) is a praiseworthy exception to this general situation. In this work one of the classical sets of axioms for the ordering of the line and the plane is recalled. Starting from there, a careful analysis of possible classroom presentations of the order concept is made and some consequent notions such as segments, angles and so on, are considered and discussed.
Two main concepts which arise in the study of geometry are those of continuity and of equality. The paper (Manara, 1988a) deals with the notion of continuity which is first considered from an historical point of view and then is studied in the modern conceptual frameworks of Dedekind, Cantor and Hilbert. The construction of the real field and the problem of measuring continuous quantities is then discussed. In (Manara, 1988b and 1988c) an interesting discussion on the possible meanings of the concept of equality in geometry is developed. To this aim many statements by outstanding scholars, from Euclid to Peano, are quoted in order to convince the reader of the great difficulties which are hidden in this seemingly simple and intuitive concept. It is very important that the teacher be conscious of these difficulties in order to avoid misunderstandings and misconceptions in the students' minds.
In a more general setting, the evolution of geometrical ideas in the last century and the related teaching consequences are studied and presented in (Manara, 1994b) and (Manara, 1994c). The first paper is a deep survey of the main ideas of analytical, projective, differential and algebraic geometry. The second paper deals with the educational aspects of teaching geometry: as a matter of fact geometry can give good opportunities to improve the inspection and description of sensorial perceptions or to educate creativeness and planning abilities. Many illuminating examples and problems are also exhibited.
One of the most challenging points in the teaching of geometry is to find interesting and meaningful problems which could stimulate the students' curiosity and enthusiasm. Contributions to this purpose can be found for example in (Cruciani, 1993), (Pennisi, 1994) and (Campedelli, 1990-91). In the former the notion of diameter of a plane figure is introduced and then many interesting properties connected with this notion are studied. In (Pennisi, 1994) the family of triangles is studied for which the square of the length of one side is equal to the product of the lengths of the other two sides: in this family one can find an interesting example of two triangles which have five elements (sides or angles) congruent without being themselves congruent. Pairs of triangles with five congruent elements are also considered in (Campedelli, 1990-1991), as particular examples of unusual properties of similar figures in the plane. The Author makes several considerations and suggests, as a result of a classroom experience, that the subject offers the opportunity of improving intuition as well as logical ability.
A very useful task is also that of providing new and stimulating applications of known results, in order to give direct suggestions to the teachers for their classroom activity or else for improving their knowledge of different subjects which might be connected to geometry. In this direction, in (Fiori and Pellegrino, 1995a), a famous combinatorial theorem due to Pólya is presented and many interesting applications of it are discussed concerning several groups of motions of polyhedra.
3. Didactical transposition and school curricula
3.1. As we have already stressed, the teaching of geometry is a continuous search for a balance between intuitive motivations and suggestions and formal reasoning. Then an important area of didactical research includes proposals or projects of new strategies fulfilling this important request, and all the considerations that arise from actual experiments devised and performed on the basis of such projects. Many of the following papers are in fact based on specific activities and experiences.
To begin with, in (Bosco et al., 1995a; 1995b), reporting the activities developed in a course addressed to school teachers, the Authors suggest a various and interesting set of logical and methodological questions about statements and problems of geometry, which a teacher could have the opportunity to deal with. In the second part of the paper, the questions are extensively discussed and some teaching suggestions are made.
In (Bagni and D'Amore, 1992) many criteria for classifying plane quadrangles are suggested (among them, whether there are - and how many - congruent sides, or pairs of parallel opposite sides, congruent angles, right angles, and so on), and many equivalences are proved. The Authors derive from several experiences with 9th and 10th grade students, the opinion that juxtaposing and comparing various criteria, those here suggested and possibly other ones, results in an unusual and very stimulating didactical activity. In this way the pupils are induced to feel free from rigid schemata, that are often responsible for a superficial learning, and look for autonomous definitions and proofs to submit to general discussion in the classroom.
The paper (Gallo and Goldin, 1994) contains the report of an experience concerning the introduction, in a secondary school where the classical Euclidean axiomatic system is already known, of the new set of axioms due to Birkhoff. The aim is to evaluate the students' capability to understand a new axiomatic system.
In (Pacini et al., 1992) and in (Banchi et al., 1995) non-euclidean geometry is proposed as the argument of didactical sequences in two kinds of secondary school, Liceo Scientifico and Istituto Tecnico, respectively. The general aim of the Authors' proposal is to discourage the pupils, particularly those of the technical school, to think of mathematics as a mere auxiliary tool for practical purposes, and to stress its formative role. At the end of the presentation the pupils should a) understand the role and the importance of the postulates in an axiomatic theory, and in general the significance of mathematics as a science with a precise content, and not as a repertory of formulas, and b) attain a clear vision of the historical evolution of geometry, particularly in XIX century. The Authors report their experience in some detail, and express their argumented opinion that the results are satisfactory enough.
In (Berni and D'Angelo, 1995) a classroom experience is presented, whose principal aim is to justify the introduction of an axiomatic system, and to clarify its meaning and its structure. The starting point is a "naive proof" of Pythagora's theorem, then the necessity arises of settling hypotheses and providing rigorous proof procedures, up to the final formal structure of Euclidean geometry.
In (Lucchini, 1995) several examples of symmetries observable in various situations are given, and considerations are made concerning how they are generated.
The book (Bazzini, et al., 1988) presents an exhibition of 181 posters dealing with plane isometries. Three sets of isometries are considered: axial symmetries, translations and rotations. Their reciprocal relationships are taken into account. The role of the axial symmetries as generators of the whole group of isometries is pointed out. A chapter is devoted to the theoretical framework which inspired the exhibition, that is metric axiomatics due to G. Choquet. Didactical suggestions are also provided.
Also in (Baistrocchi, et al., 1992) the plane transformations are the instrument of teaching strategies, to be applied in the first two years of high secondary school (9th and 10th grade). The starting-points are many graphic works of C. Escher, where a quantity of transformations can be observed and drawn out. The booklet is articulated in didactic units: each of them contains the particular subject but also many remarks about the pupils' reactions, their conquests and their difficulties.
3.2. A large number of didactical contributions emphasize the role of the history of mathematics in planning a class activity. Some authors simply recover ancient problems or procedures and suggest that these could be used as subjects of lectures; others propose to directly read, perhaps with the guidance of the teacher, ancient texts of classical authors, and finally in other authors there is a more extensive discussion about the characteristics, the finalities (and also the difficulties) of the use of history in the teaching of mathematics (particularly of geometry): in this direction see for example (Pergola, 1992), to quote just one paper, but see also, for a more extensive and deep discussion, the chapter devoted explicitly to History in this book.
In (Bianchini and Velardi, 1995) a classroom activity is proposed, concerning plane figures inscribed in other figures. The problem is suggested from a famous dispute occurred in 1547 between L. Ferrari and N. Tartaglia, namely how to inscribe a square in a regular pentagon. The solution is achieved after a long preparatory study, that offers the possibility of recalling several geometrical as well as algebraic notions.
In (Lucchini, 1991) the regular polyhedra are studied from the cultural, historical and didactical point of view. Not only the convex or Platonic, but also the starred Keplero-Poinsot polyhedra are taken into account.
(Campedelli, 1991-1992) consists of four parts in which many questions are examined concerning Pythagora's theorem, namely: a) historical notes; b) a survey of interesting proofs, other than the famous one due to Euclid, and some extensions of the theorem; c) a number of applications; d) some properties of Pythagorean triples and of arithmetic right-angled triangles, with particular emphasis on didactical applications.
In (Campedelli, 1994), some pre-euclidean geometric questions are examined, such as Hippocrates' problem of squaring the lunes, with the purpose of adding liveliness to the class activity and also of illustrating some human aspects of the research enterprise. Some other ancient themes of geometry (from Archimedes and Apollonius) are taken into account in (Campedelli, 1995a), where a particular affinity is employed to yield a deep and unifying insight into different geometric situations. The arguments are presented to secondary school students who are also urged to deall with non routine exercises. The proposal in (Campedelli, 1995b) is an activity addressed to students of the 9th grade. Its content is inspired by the representation of numbers by means of geometric figures, dating back to the Pythagorean school.
3.3. The notion of conic section is surely one of the most valuable in geometry, because of its many different approachs, and besides, for the connections of this concept with many other questions. In (Menghini, 1991) an extensive survey can be found, in which the different ways of introducing this notion and all its meanings (either purely geometrical or analytical) are collected. Connections between different definitions are discussed in a way that may be understandable at a secondary school level.
The same theme of conics is conceived as a part of a more comprehensive project, Mathematical machines in the classroom, undertaken by a research group (Pergola, 1992), (Pergola and Zanoli, 1994), (Bartolini Bussi and Pergola, 1994) and (Bartolini Bussi, 1993). The project is addressed to students of secondary school, and assigns a particular role to some artifacts and machines for learning geometry. (Pergola, 1992) reports the various phases of the classroom activity. The students are requested, at first, to design and, if possible, to build some of these instruments, otherwise available in a laboratory. Then they examine the machines and analyse their structures and ways of working, produce written answers to appropriate questions, make conjectures and try to prove properties. The teacher has the role of a guide in this process. Finally much attention is paid to ancient texts, where the evolution of the different mathematical theories embodied in the machines can be retraced.
In (Pergola and Zanoli, 1994) the history of geometry and mechanics is divided, for educational purposes, in three periods, from the ancient Greece to the beginning of scientific revolution, from 1400 to 1700, and from 1700 to the end of 1800. Each of these periods is characterized by particular relationships between practical activity and abstract science. Finally, a detailed account is given of some devices by Descartes, De l'Hospital and De la Hire, designed for tracing curves corresponding to conic sections or for generating projectivities.
General remarks are also contained in (Bartolini Bussi and Pergola, 1994), where in particular the Authors emphasize the role of history of mathematics, considered as a fundamental constituent of mathematical knowledge as well as of the image of mathematics to be built in the classroom. An experiment is described concerning the use of a special device, the orthotome, to trace a parabola in a way that gives a concrete meaning to its canonical equation. A discussion follows about some problems to be faced in doing such activities, problems often due to the general school organization, and about the transferability of the teaching experiment.
The use of machines, or more generally of concrete objects and devices, is often considered as a precious resource for the teacher: for example, in (Lucchini, 1994a) three cases are presented of tools for treating geometrical subjects: optical illusions; models of regular polyhedra; and various images, obtained from drawings by C. Escher or from pictures of some objects.
A learning path outside the classroom: this is the proposal in (Zuccheri, 1992), where the permanent exposition "Oltre lo Specchio" (Beyond the Mirror) is presented. In this laboratory the students can manage many instruments based on semitransparent mirrors, in particular a device called a symmetroscope, allowing them to see at the same time the reflected image and any object which is beyond the glass. The students, through guided exercises, "discover" the most important concepts of plane and space geometry, and the properties of the various transformations, in an amusing and stimulating environment.
3.4. It is well known that the introduction of computers in the school has determined deep changes either in the teachers' strategies or in the students' interest. In (Pellegrino, 1994) one can find a discussion about the possibility of improving the interest for teaching and learning geometry by the use of the specific software Cabri-géomètre. In a recent book, (Pellegrino and Zagabrio, 1996), the Authors extend the previous note and provide a very comprehensive guide for using Cabri in the school.
We have already noticed that a large number of people agree on the importance of the geometric transformations as the principal tool for introducing geometry in secondary schools. In (Scimemi, 1994) this point of view is deeply motivated. It allows a "soft" approach to geometry: formal definitions and proofs are avoided, while intuition and physical evidence suggest conjectures about geometrical properties. For these reasons it is convenient to limit the study to plane isometries and similitudes, to ignore the analytical representation and to make systematic use of an appropriate software (for example, Cabri-géomètre). With this euristic procedure, the students become able to understand geometrical concepts but also general and algebraic notions (mapping, composition of mappings, group), develop intuition rather than deduction, and make an active use of computer tools.
Before the introduction of sophisticated computer programs, like Cabri, the projective transformations, in particular the plane affinities, were already examined, studied and classified, as suggested in (Avanzini et al., 1988), by means of a program written in BASIC, based on simple algorithms of matrix calculus. The "images" of the transformations appear on the video display.
The computer tools have a very wide range of applications in the school: for example they provide a precious aid for reading a text. The thirteen books of Euclid's Elements are readable by means of a computer program presented in (Ferro and Lucchini, 1989): the software consists of several files containing the text and various utilities allowing many research activities on definitions, common notions, postulates, propositions, lemmas and corollaries. Also in (Lucchini, 1994b) the computer is used as a tool to analyze in details the work of Euclid: distinctions are made between mathematical and lexical questions, and between the original text and its Italian translations and commentaries.
4. Epistemological problems and educational value
4.1. One of the possible meanings of this theme in the context of geometrical education is well illustrated by a series of papers due to F. Speranza. In (Speranza, 1988) an outline is given of the traditional roots of geometrical knowledge and of the difficulties that the teaching of geometry encounters nowadays in Italy at every school level (from primary school to the university). It is important to conclude that geometry must be studied as a complex system, by means of the different methods which contribute to its characterization.
In (Speranza, 1989) some contributions to the task of building a curriculum of geometry in high school are given. To this end the Author provides a list of indispensable contents, and discusses the opportunity and the limits of a logical systematization and the necessity of forming a critical sense: in particular he gives: i) a commentary on fundamental geometrical notions and properties (such as transformations, logical tools for their study, etc.); ii) an analysis of Euclid's Elements, in order to emphasize the differences between the "classical" and the present way of facing problems; iii) a comparison among the coordinate method, the synthetic method and the group organization of geometry; iv) an "exemplary" axiomatic treatment, together with some basic ideas about the question of non-euclidean geometry.
In the same line of thought, in (Speranza, 1990) some acute observations are made about the purposes, the value and the meaning of the teaching of geometry, in the light of a comparison between old and new syllabuses. From such observations some possible teaching suggestions are deduced. These considerations are carried out discussing the notions of axiomatic structure and of a model for a given theory, either in Euclidean or in non-euclidean geometry. Non-euclidean geometry is also the argument of (Speranza, 1992). After a survey of the basic ideas of that geometry and of their relationships with the real world, a number of consequent teaching considerations are extensively discussed.
The formation of a critical attitude is often pointed out as a primary objective of mathematics teaching. Nevertheless, as in (Speranza, 1994a) is stressed, the traditional presentations of geometry, either from Euclid's Elements, or from Bourbaki texts, because of the abundance of their technical contents, often do not allow to reach these goals. On the contrary, geometry can be an invaluable subject for giving to mathematics teaching a critical content; to this end, together with a more careful study of Euclidean geometry, other arguments like Klein's programme for the study of non-euclidean geometries and of finite planes could be very helpful.
Finally, (Speranza, 1994b) presents a survey of the different epistemological points of view concerning geometry, known as "conventionalism" and "neo-empiricism". Starting from these notions, the Author gives some suggestions to improve the transition from the experimental stage of geometry to its hypothetical-deductive structure.
A recent resolution of the official Committee for the Piano Nazionale dell'Informatica is the proposal of new Mathematics programs for the first two years of the secondary school (grades 9th and 10th). In (Lucchini, 1989) some reflections are made about the theme "Plane and Space Geometry" of those programs, with particular emphasis on the cultural and educational value of mathematics.
The problem of the educational aspects of mathematics teaching (with particular attention to geometry) is considered also in (Marchi, 1988a). Mathematics, as an educational chance, may be considered either as an opportunity and a model of rigour or as a stimulus to improve intuition and imagination. The thesis upholded by the Author is that teaching axiomatic approaches to mathematics, at the right level, can combine rigour and imagination giving a proper experience of mathematical thought.
The discussion started in the previous work is carried on in (Marchi, 1988b), which deals with the problem of the relationship between the rigour of a rational treatment and the levels of truth we are able to reach. In order to answer to this question, it is necessary first of all to stress the need of a good definition of the concept of truth. We know that in mathematics (and particularly in geometry) truth may be ensured by two different ways: referring to an external reality (therefore assuming truth as synonymous of good correspondence between a statement and the actual state of affairs) or by means of a self-referring concept of non-contradiction or logical consistency. The different psychological and educational implications of these points of view are then discussed.
4.2. The question of evaluating various methods for mathematics teaching is faced in (Bartolini Bussi, to appear). After a rapid survey of several experiences at different school levels, elsewhere detailed, the thesis is put forward that the comprehension of (mathematical and) geometrical contents is highly improved by the employment in the classroom activity of a concrete working environment, and by the manipulation of instruments and artifacts that have had a role in the history of mankind.
In (Bartolini Bussi, 1993) an experience is reported concerning the activity developed in a class of the 11th grade. The students are invited to work with a particular machine or linkage, the pantograph of Sylvester, by means of which it is possible to trace curves and to determine correspondences of points in the plane. A single part of the experiment is described in full details, in order to emphasize the shift from the level of expérience mentale to the level of démonstration as defined by Balacheff in his study of the process of proof. The Author also claims that what happens in the class can be interpreted using concepts from the Russian school of Vygotskij, better than with reference to a Piagetian point of view.
A pioneer investigation about the process of constructing geometrical statements, is the content of (Boero and Garuti, 1994). The experiment there reported was conducted in a class of middle school (7th grade) and concerns the Theorem of Thales. By means of a gradual process guided by the teacher, and pointing on some previous experiences, the students are induced to formulate a statement expressing the well known proportionality assertion. Then a comparison is made between the statements produced by the students and the official formulations contained in the text-books. The students' papers are recorded by the Authors according to some relevant distinctions such as particular/general, concrete/abstract, procedural/relational. The analysis of the data is encouraging with respect to the possibility that the students can reach to some extent the objective of approaching rational geometry constructively. However many questions are still unresolved and deserve further investigation.
The question faced in (Barberini and Bartolini Bussi, 1993a; 1993b) is the relationship between the common sense and the mathematical formalization of the concept of geometrical transformation, at the level of primary school. So the principal concern of the paper is the intuitive rather than the rational aspect of geometry, but nevertheless we are here interested in the problem since at least one of the terms of the question is of pertinence of the rational geometry. Now, what appears in the first analysis is that from the primitive notions of metamorphosis and motion, to the idea of relation between figures and of plane and space tranformations until the final and most abstract way of considering the geometric transformations as elements of a group, the process is very long and difficult. Moreover, there is no evidence that the intuitive notions lead naturally to the subsequent levels of abstraction. Therefore an accurate analysis is made of the principal obstacles to be faced in this process, such as the object-image and the image-figure relationships, or the asymmetry between the elements to be transformed and the tranformed one, and so on. Finally two solutions are suggested, namely an appeal to the resources of history of mathematics and the study of transformations by means of instruments.
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| Mario Marchi | Aldo Morelli | Roberto Tortora |
| Dipartimento di | Dipartimento di | Dipartimento di |
| Matematica | Matematica | Matematica |
| Università Cattolica | Università "Federico II" | Università "Federico II" |
| del Sacro Cuore | Complesso Monte | Complesso Monte |
| Via Trieste 17, | S. Angelo, via Cintia, | S. Angelo, via Cintia, |
| 25121 Brescia, Italy | 80126 Napoli, Italy | 80126 Napoli, Italy |