1. Introduction
Mathematics today is in a different situation when compared to the other sciences. Traditionally it is considered to be the most important subject throughout the school curriculum: of the sciences it is that which figures most strongly in the various types of school and it is the one to which most time is dedicated. However, if we consider the degree of popularization, mathematics is considerably worse off than the other sciences. As regards the situation in Italy, we refer to the inquiry conducted by COASSI (Co-ordinating Committee for Italian Scientific Associations) (cf. Bianca et al[1] 1986) which, though dated, provides some interesting pointers. Consequently, the popular image of mathematics, that is, as it is perceived by the public at large, depends almost exclusively on the schools and on the quality of the level of learning attained by the pupils, and is therefore a result of the training, the skills and the ideas of those that teach it. This, on the whole, is true everywhere, but in Italy - considering the current situation of the schools and universities and given the absence of a proper teacher training system - the situation is all the more worrying.
Mathematics today is in fact going through a difficult period. This, not so much as regards research (currently there is a considerable amount of research activity producing important and often long awaited results), but as regards the role it plays in culture in general and in science in particular. This state of affairs has become so acute that the international mathematics community, normally largely unconcerned with the image it projects of itself and of its discipline, has begun to examine the question of the promotion and the image of mathematics. Indeed, in the last few years, conferences have been organised in countries such as France, the United States, the United Kingdom and Italy to debate this problem and to try to find suitable remedies. One such conference was the "ICMI Study", held in 1989 in Leeds, UK, whose proceedings (cf. Howson and Kahane[2] 1990) constitute an interesting point of reference on this subject.
In this article, as well as mentioning some of the more notable Italian initiatives undertaken in this field, we will outline the contributions of Italian researchers during the period from 1988 to 1995. Most of these researchers are operating within the didactic research groups of the CNR (National Research Council) or the National 40% Project (Didactic Research Projects for Mathematics and Computer Studies) of the MURST (Ministry for University and Scientific Research). In part 2 we will be looking at images, beliefs and notions associated with mathematics, in part 3 we report on investigations carried out to assess the mathematical knowledge of the pupils, in part 4 we deal with the popularization of mathematics with particular reference to some of the aspects involved and the problems posed and, finally, in part 5 we briefly draw some conclusions.
2. Images, Beliefs, Notions
Mathematics is a discipline that can boast the singular distinction of arousing extremely contrasting opinions: it is either loved or hated, considered to be easy or difficult to understand, alive and stimulating or arid and unattractive. Everyone feels strongly about it (cf. Furinghetti 1993), though, unfortunately, the majority are hostile. The image of mathematics is one which is difficult to alter, especially in adults, and has its origins in school (ibid.). The mechanisms involved during the formation of this image are being studied and examined by all researchers in mathematical education. For example, Burton[3] (1994) has found that the negative image that most people have of mathematics is strongly connected with a poor performance at school, in general correlated with a disadvantaged social status.
It also has to be said that mathematics today is found to be increasingly discriminating as regards gaining access to and progressing in many careers. The extensive changes that have occurred in industrialised societies in the last few years have also made themselves felt in the world of employment and in every day life. As shown by Malara (1995), this imposes new educational demands on schools in general and on the teaching of mathematics in particular.
As mentioned previously, the public image of mathematics is for the most part formed at school (cf. Furinghetti 1993 and Bernardi 1994c), so it is at school that one has to intervene if this image is to be altered in order to make it more representative of reality and more useful in the education of each individual. Teacher training therefore takes on an important role right from the university stage (Bernardi and Arzarello[4] 1996).
On the other hand, the image of mathematics, a source of controversy even amongst mathematicians, is not a clearly defined concept for teachers either. Each teacher, then, will unconsciously transmit their (unconscious) understanding of the discipline. In any case, the discussions on the notions of how mathematics teachers involves mathematics, its teaching, educational theories, the philosophies of mathematics, classroom practice and the renewal of the curricula. Many of these aspects have been and still are being studied by various Italian researchers whose works we will now briefly describe.
Bottino et al (1991) are studying the way in which teachers approach the problems of teaching in schools and have found that there is a conflict between that which the teachers think should be done to make the teaching effective and that which it actually achieves. Furinghetti (1995), one of the authors of the previous work, concentrates on the notions of the teachers with a view to highlighting elements that characterise mathematics teaching. Bottino and Furinghetti (1996), with a case study, concentrate instead on the how mathematics teachers conceive the role of informatics and its teaching; in this way they define a set of types of conceptions based on the various attitudes that emerged (note that in the proposals for the new syllabuses for the upper secondary school, the teaching of informatics is paired with that of mathematics). Cannizzaro (1989 and 1994) in a general investigation in the conceptions relating to mathematics conducted amongst university students of the second two-year course in mathematics, studies the connection between "spontaneous visions" of mathematics and misconceptions that persist even after concerted educational efforts.
Bazzini (1993 and 1994) contributes to the debate on the cultural dimension of mathematics in primary education. The implications arising for teaching are examined based on many years' experience in the development of syllabuses, matured with close working relationships with the researchers of the University of Pavia and teachers. In particular, the importance in teaching of reconciling the aspects of mathematics as a discipline unto itself and that of mathematics as a means of describing the real world is highlighted.
A series of researches address the study of the image that the pupils have of mathematics or to ways for promoting one which is more appropriate and representative of its actual characteristics and peculiarities. Malara and Pellegrino (1990) delineate working styles and classroom techniques that are able to engender a more dynamic and appealing image of the discipline. They stress the importance of games, viewed as an intellectual challenge, as a catalyst for the didactic activities and highlight the connections between the development of important branches of mathematics with classical problems or with that which was once termed amusing and curious mathematics. D'Amore and Sandri (1994) analyse, by means of an interview, the image that adults and students of secondary upper schools have of mathematics. They find a subject which is entirely closed in on itself, which does not have any connection with reality and with history and which can not offer any source of interest or amusement. Malara (1995) shows how it is possible, using appropriate didactics, to overcome the traditional image of mathematics which, on a social level, is the cause of widespread prejudice and severe limitations. In addition to strategies for the teaching of mathematics that may be employed to give it a more positive image, Furinghetti (1993) suggests activities such as lectures given by university lecturers for pupils of pre-university schools or the suggestion of suitable reading of a mathematical nature (not school texts). To this end, Furinghetti (1988) had already published a small volume containing a collection of proposals for suitable further reading.
Speranza's three works (1990a, 1990b, 1994) stress how the cultural role of mathematics in school and in society should not be ignored. In particular, the first work suggests that the current role of mathematics in culture is unsatisfactory and suggests a few topics which, if developed and suitably introduced into teaching, may help the pupils to understand the cultural dimension of mathematics. Amongst these topics are symbolism, the problem of applications, the connection between mathematics and art and epistemological thinking. The second work deals with the relationship between mathematics and the other sciences; from those with which there is a relationship dating centuries, such as physics and philosophy, to the newer ones, such as psychology and sociology, and also taking in the natural sciences; the third work stresses the central role of geometry in knowledge and therefore in teaching.
3. The Mathematical Knowledge of Pupils
The manner in which the pupils learn and the results thereby obtained at any given level of schooling are influenced by the type of teaching received up to that stage, in all its aspects, not only the cognitive but also metacognitive and affective aspects. Despite there being interesting branches of research at an international level concentrating on these last aspects (cf., for example, Schoenfeld[5] 1992 and McLeod[6] 1992 ), the research directed specifically at analysing the overall mathematical content of what has been learned at any given age level should not be overlooked, especially as regards the transition from one schooling phase to another. It is well known that these transitions are amongst the most delicate and traumatic phases for the pupils. For a number of decades now there has been a progressive increase, both at an institutional level and amongst researchers, in the level of awareness of the problem of the assessment and analysis of performance at school, which has engendered a greater attention to detail in the preparation of suitable reference and assessment guides (syllabuses, tests, etc.). In this regard, as an example of investigations undertaken for the institutional assessment of classroom performance, we note the APU (The Assessment of Performance Unit) investigations carried out in the United Kingdom from 1978 with the objective of assessing the mathematical knowledge of pupils from the ages of 11 to 16. As regards research work, on the other hand, we will simply note, as a typical example, the CSMS (Concepts in Secondary Mathematics and Science) (cf. Hart[7] 1981) directed at investigating - by means of closed answer testing of a large sample of pupils from the ages of 11 to 16 - the evolution of abilities, the presence of difficulties and the identification of typical errors.
In Italy, during the period in question, we note a single investigation of an institutional nature and a number of research projects. The first (cf. Bolletta 1992a and 1992b), commissioned by the MPI (Ministry of Education) and carried out by the CENSIS (Social Investments Study Centre), is an investigation concerned with monitoring the first five year period of application of the "New Syllabuses for Primary Schools" introduced in 1987. The other research projects are concerned with the changeover from middle school to upper school and from the upper school to university[8] and were carried out with the preparation of tests for the pupils and sometimes combined with forecast tests for the teachers.
As regards the transition from middle school to upper school we will first note that Bolletta (1988a and 1988b) in a large scale investigation, the only one conducted on a national level, commissioned by CEDE (European Centre for Education) supplies a realistic assessment of the situation as regards the application of the middle school syllabuses introduced in 1979.
Again, as regards the transition from middle school to upper school, there are two independent studies conducted at a provincial level in Emilia Romagna. The first (cf. Vené et al. 1989) deals with the results that emerged from an investigation based on a closed answer test, set by the authors, concerning all the topics of the ministerial syllabuses. This investigation, conducted in 5 schools in the province of Parma (covering 12 classes and 260 pupils), comprising the analysis of the mistakes found, highlighted large gaps in the understanding of numerical sets, both in terms of calculation and conceptualisation. The second study consisted of two parts. In the first (cf. Ferretti et al. 1988) a closed answer test was devised following a study conducted on the general criteria for the construction, analysis and evaluation of this type of test; the second (cf. Fiori 1991) contains the results obtained from the test set in 14 schools in the province of Modena (covering 71 classes and 1792 pupils). In this investigation it was found that approximately one third of the pupils had an inadequate level of mathematical knowledge for satisfactory placement in the upper secondary school. On a more macroscopic level, the gaps concerned above all the comprehension of texts, geometry and relative numbers (both in terms of calculation and in terms of their use).
In order to assist the teachers in their need for an objective means for assessing the pupils' level of mathematical knowledge at this point in their schooling, researchers of the University of Pavia provided a wide ranging test divided into a number of parts that could also be used individually. The first of these (cf. Pesci and Reggiani 1989, Reggiani 1989) enables the assessment of mathematical knowledge as regards: (i) the ability to interpret and elaborate the information contained in a text, a graph or a formula; (ii) the ability to mathematise a real life situation; (iii) mastery of calculation procedures. The second (cf. Amoretti et al. 1993) consists of a wide range of tests on knowledge obtained at school with psychological criteria which were standardised on the basis of large samples.
Studies concerning the transition from upper school to university are more recent. Firstly we note two articles, one by Lucchini (1992) and the other by Bencivelli and Villani (1994), concerning admission tests for restricted entry degree courses (only recently instituted in Italy). These articles discuss the quality and effectiveness of pre-university teaching on the basis of the results of these tests and the negative influence that instrumental use of this type of test can have on the image of mathematics.
A research carried out by Rinaldi and Vené (1994) is also concerned with the passage from upper school to university and is directed at identifying the minimum requirements that can be expected of students coming from different types of schools entering into the scientific faculties. The research is based on an investigation carried out for two consecutive years by means of a test and with a sample of approximately 900 students each year. The "basic" questions are those that registered the highest percentages of correct answers, even though these were lower than expected, whereas those of a logical nature registered the lowest percentages.
We note that research projects are under way - conducted by Research Groups in the Didactics of Mathematics of the Universities of Rome, Turin and Trieste - concerned with the problem of mathematical knowledge for entry into the scientific faculties, which, making use of vast samples, set out to compare: (i) knowledge secondary school teachers presume the students to posses; (ii) knowledge university teachers expect the students posses; (iii) knowledge the students actually posses, as determined by suitable tests devised for this purpose.
4. Popularization
The popularization of mathematics, in the literal sense of the term, is an arduous task; for some, it is almost impossible (cf., for example, Emmer 1992) as J. Dieudonné[9] (1987) himself highlights in his book of reflections on mathematics and translated in many languages.
But why is it so difficult to popularise mathematics? Why should it be more difficult than with other sciences? Above all, in the specific case of mathematics, it is not easy to find effective metaphors, and formulas and specialist terms are prohibited. The intended public is not homogeneous and one can not expect a reader to follow complex reasoning. At the same time one has to be able to communicate difficult or highly abstract concepts even to those who had difficulty with mathematics at school. This, without loosing rigour or conceding scientifically unacceptable "discounts".
It has to be said, though, that the attitude of indifference, if not hostility, towards those who write about mathematics has only recently begun to change. In this respect, we recall the contrasting assertions of G.H. Hardy and G.C. Rota, where Hardy[10] (1940) wrote: << It is a melancholy experience for a professional mathematician writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done >> (p. 1). Rota[11] (1986), on the other hand, wrote: << Gifted expositors of mathematics are rare, indeed rarer successful researchers. It is unfotunate that they are not rewarded as they deserve, ... >> (p. 1). Paraphrasing Rota (ibid., p. 2) we can assert that making mathematics accessible to the uninitiated is like navigating between Scylla and Charybdis, that is, between professional contempt and public incomprehension.
However, despite the difficulties, there is an increasing awareness, as noted by Bernardi (1994c), of the importance of popularising mathematics with a view to: (i) "faithfully" conveying its mentality; (ii) highlighting its formative and cultural aspects; (iii) contributing in exploding the myth, believed even by some educated people, that it is a subject which has been consolidated, definitively settled, and in which there is no longer anything to be discovered. In this way it will be possible to contribute in creating a more representative image of mathematics, which, it being also a method, a language, a way of thinking - is an integral part of the cultural heritage of humanity and as such is to be appropriately safeguarded. All this by taking the trouble to examine the relationship between mathematics and society in such a way as to highlight the effective role that it has had and continues to have, leaving aside the image that each individual and the mathematicians themselves currently see.
But how may this be made to happen? How can one talk about research and of the results obtained by mathematics to persons who have not studied mathematics to a level above the first two years of a degree course? How may the Chevallard[12] (1988) paradox be overcome, which sees an increasing "demathematisation" of the individuals of a society which is becoming increasingly mathematics-dependent? Otherwise, how can the usefulness and the beauty of mathematics be shown to those who, even without realising it, are able to ignore it because every day they use tools or objects, such as calculators, computers, electronic systems for topographic measurement, cash point cards, compact discs, that incorporate an increasing number of old and new mathematical ideas and methods?
Certainly, popularization needs to be managed in a differentiated manner with communications at various levels and in various directions that take into account both the level of knowledge and the attitude towards mathematics: we should not forget that despite a majority that is prejudiced or hostile, there are also those, even amongst the young, who love mathematics. These persons enjoy expanding their knowledge of the subject or being posed mathematical questions, but the occasions that properly take into account these aspirations are few and far between.
One solution would be that of speaking only of the great mathematicians of the past and of classical problems, or that of concentrating on the aspects concerning the application of mathematics. But this would in the end have the effect of consolidating the idea that mathematics is static and lacking in cultural content. Another solution could be that of making use of mass media[13]. In this case, though, it would be important to avoid broadcasting information of little or dubious value, as occurs unfortunately in the daily newspapers that try to make an impression on the reader by using sensationalistic headlines. This, in fact, would have the effect of worsening the situation by making people believe that there is nothing of value to be divulged. To avoid this inconvenience it would be better to use the mass media to promote initiatives connected with mathematics such as reviews of old and new books that have been well written or that have been widely published, interviews with prominent mathematicians, communiqués reporting the awards of the main mathematical prizes (unfortunately there is no Nobel prize for mathematics), news of the results of the mathematics Olympiads at various levels and games and problems features in newspapers. From this point of view a certain amount has been done even if these have appeared in magazines that do not have a high circulation.
Amongst these, we will begin by mentioning a series of articles by M. Menghini published in a magazine intended mainly for secondary school teachers (not only of mathematics). In the first one, Menghini (1988a), starting from a comparison of two classic books by M. Kline[14] (1953, and 1980) retraces the various perspectives from which man has viewed mathematics: from the (optimistic) conviction of being able to predict and simulate nature, to the (pessimistic) idea of a human creation, lacking in absolute points of reference, that functions only because it reflects certain of our mental patterns. In subsequent articles, Menghini (1988b, 1988c, 1989, 1990, 1993a, and 1993b), prompted by ideas from books or from the comparisons with mathematics "dictionaries" or from interviews, deals with subjects ranging from the fundamentals of geometry to the history of mathematics and of how it may be included in teaching. Along the same lines, Menghini (1994) talks of the classic connection between mathematics and art (as seen by the painter C. Bouleau); Menghini (1993c) wrote about the Fermat theorem at the same time as even the newspapers were talking about what was then the incomplete proof of the now famous theorem; Menghini together with Cannizzaro (1992) stresses the influence of research, in the fields of mathematics and psychology, on the world of research in the didactics of mathematics.
Another interesting initiative is the periodical column Archimedes replies (cf. Bernardi 1991-1995) in a magazine for teachers and scholars of mathematics, an increasingly successful feature that has been running now for five years, in which queries sent in by the readers are discussed. The reference school level is generally that of the upper schools. The questions are both about didactic matters and more strictly mathematical ones. The subject matter varies widely: it is strictly disciplinary (prime numbers, polyhedra, Pythagorean triples, the last Fermat theorem, derivatives and integrals, numerical and function series, algebraic structures, ruler and compasses constructions, combinatorial analysis, statistics, etc.) without, however, overlooking more general topics (questions of language, philosophy of mathematics, history, small "discoveries" of the students, mathematics competitions, the teacher's profession, etc.).
The PRISTEM (Historical and Methodological Research Project) group of the Bocconi University of Milan, again with a view to improving the image of mathematics and with the idea that its best "ambassadors" will be those who know it best, have been publishing a quarterly magazine (PRISTEM Letter) for the last six years and organising a yearly conference. Both initiatives are intended for teachers, researchers and anyone else involved in mathematics or who use it in their work. The first of the said conferences was dedicated to the role of mathematics in society in the `90s (cf. Di Sieno 1990).
Another sector that may be able to provide useful means for popularising mathematics is that of mathematical games in its widest sense, from brain-teasers to paradoxes via the so-called amusing and curious mathematics. Scimemi (1990), concluding an article on the solution of problems of a higher order than two using of folded paper, notes that << historically, many chapters in mathematics developed specifically because its creators did not treat it as a definitive science, but as a game whose rules one can change: traditional geometry, for example, is a game that follows the rules, or postulates, established by Euclid; remove a few of the rules or invent some new ones and you end up with a different geometry, new problems, new solutions! >> (p. 86).
Malara and Pellegrino (1990) note that games can provide a metaphor for mathematical activities. As an example, they describe an experience in which, starting with problems based on the antique Chinese game of Tangram, pupils aged 13-14 had the opportunity of gaining an insight into the work carried out by researchers. D'Amore (1992) presents an interesting collection of logical, linguistic and mathemagical games in a book. Of this genre there is also an article by Bernardi and Bindi (1989) which, starting from a linguistic brain-teaser, examines paradoxes (some well known, some less so) connected with self-referencing. The last section of this work, dedicated to self-referencing in mathematical logic, deals with the famous incompleteness theorems of Gödel and Tarski.
A good way of involving people who are well disposed towards mathematics is that of suggesting problems: "resolving a mathematical problem requires more concentration than reading an item of news, but enables the mathematical experience to be felt directly" (Bernardi 1994, p. 8-9). With this idea in mind Bagni (1990), Pellegrino (1992), Pellegrino and Marchini (1993) deal with questions of arithmetic from the point of view of geometry problems, which helps to understand that mathematics is not divided up into watertight compartments. The last of the three works cited is presented in the form of a dialogue and appeared in a widely published popular scientific magazine. Scimemi (1992a) proposes an interesting collection of basic problems of indeterminate analysis as being a good tool for introducing arithmetic to junior high-school pupils.
To convey a lively and stimulating image of mathematics and to cultivate the passions of the more interested youngsters, taking the lead from an initiative of B. de Finetti of more than twenty years ago, "Mathematics Clubs" have been set up during the last few years in some cities (at the moment still only a few), that organise lectures and mathematics competitions for secondary school students. As regards mathematics competitions, Italy has been participating in the International Mathematics Olympiads for over ten years with the backing of the MPI by means of an agreement with the Normal Upper School of Pisa, and is guided by, also during the preparatory stages, a specially nominated commission selected by the UMI (Italian Mathematical Union). In addition to these initiatives, publications are being set up which collect and resolve the problems appearing in the above selection (cf., for example, Conti et al. 1994, AA. VV. 1995). << It is worth noting [...] that the importance of all these competitions, as has always been maintained by the UMI (which supports it but does not control it directly), lies in the considerable effect they have during the preparatory phases: the impression that the many youngsters (and their 200.000 families! [figure relative to 1995, ed.]) form of mathematics during the selection process is very different to the traditional one. The questions require the competitors to have abilities that are different to those typically associated with the "top of the class" pupil: as well as accuracy in carrying out calculations (the aspect public opinion most commonly associates with our discipline) the questions require imagination, intuition, speed and concentration. For many youngsters this is more agreeable than the traditional school activities; the teachers are also stimulated by this, using the occasion to improve their teaching methods and widening their range of problems >> (Scimemi 1996, p. 66-67).
Another way of breaking free of the restricted environment of mathematics teachers and scholars is that of identifying the mathematical aspects of general topics, of highlighting lesser known mathematical contributions to the other sciences or connections between mathematics and art. Bernardi and Menghini (1990) provide a good example of the first approach with an article, unfortunately only published in a magazine intended for mathematicians, in which they analyse the characteristics of electoral systems and discuss the non contradictoriness of a system of axioms. Pellegrino and Fiori (1995) provide an example of the second with an article illustrating connections between the Pólya's enumeration theorem, the set of movements of polyhedrons, graphs and chemistry.
As well as in the articles by Speranza (1990a) and by Menghini (1994) cited previously, the connections between mathematics, sciences and art are amply illustrated in three volumes. In the first, Emmer (1991a) provides an historical outline of the subject. Amongst the topics discussed, there are the works of Escher, platonic solids and the role of computer graphics in mathematics and art. In the second, which includes many quality colour photographs, Emmer (1991b) takes a detailed look at the history of soap bubbles and films in art, in mathematics and in the other sciences. In the third, Bagni and D'Amore (1994) examine the origins of perspective. Furthermore, Scimemi (1992b; 1994) in two papers, similar in content, describes the basic laws of sound consonance and illustrates the solution of the problem of the equal temperament scales using the approximations of rational numbers.
The institution of museums and the organisation of mathematical exhibitions appear to be an effective way of overcoming the Chevallard paradox and of achieving a wider audience for what may be termed "mathematical culture". The mathematical communities of a number of countries have, in fact, been working in this direction with a number of initiatives. A outline of the state of the art as regards museums and mathematics exhibitions around the world, together with a detailed overview of the various initiatives is given in an article by Emmer (1993). In Italy, the institution of a museum entirely dedicated to mathematics is only now beginning to become a reality (cf. Conti[15] 1995), whereas in the last few years various exhibitions have been organised. Of these we will cite the permanent exhibition, Beyond the Mirror, in Trieste on the premises of the Laboratorio dell'Immaginario Scientifico (the Laboratory of Scientific Imaginary Reality) (cf. Zuccheri 1992), and the travelling exhibitions Horus's Eye (cf. Emmer 1989a), Beyond the Compasses (cf. Giusti and Conti, 1992), Mathematical Machines and other Objects (cf. NRSDM Modena 1992a), Forms and Figures (NRD-Mathesis Florence, to appear), all of which contain specially prepared materials, including videocassettes and interactive materials.
Finally, we note that in the last few years, as a result of the new possibilities of computer graphics, mathematicians have created a large number of new forms that have generated a sort of mathematical imaginary reality. Emmer (1989c) examines the role and influence of this new happening on our culture. He (1989b) also discusses the role and influence on teaching of the various technologies, such as computers with graphic capabilities, videocassettes and films. Didactic videocassettes on various subjects have been produced by the NRSDM of Modena (1992b).
5. Conclusion
Italy is currently going through an extremely delicate period. After many years during which many people gave little thought to the future, certainties can no longer be counted on. Society in general, with its increasingly pressing and ever greater problems, is in search of a new direction and of new equilibrium. This could be the appropriate moment to recommence with pride and renewed vigour the construction of a healthier, fairer and more "beautiful" society. We believe that mathematics and its teaching can and must play a part. One of the priorities of a country that purports to be civilised and technologically advanced, or which is aspiring to become so, is the greater popularization of scientific and mathematical culture. It has to be said though that the good will of individuals is not enough to confer consistency to truly incisive initiatives with a lasting effect. We therefore hope that all mathematicians (and not only those involved in Mathematical Education), spurred on by the fact that the year 2000 has been declared a world mathematics year by UNESCO, will ask themselves what role they envisage for mathematics in shaping society in the third millennium and, taking a look around themselves for once, will be able to provide appropriate answers and their own contribution of ideas and initiatives: not only will Humanity benefit from this, but also Mathematics!
References
AA. VV. : 1995, XXXV Olimpiadi di Matematica (Selezione Italiana, Cesenatico 1994), Proceedings of the Meeting, publication supported by AgipPetroli; Roma.
Amoretti, G., Bazzini, L., Pesci, A. and Reggiani, M.: 1993, Test di matematica per la scuola dell'obbligo, Ed. OS, Firenze, 2 vols
Bagni, G.T.: 1990, Il piano di Pick e i numeri primi, Periodico di Matematiche, VI, 65, 3, 82- 85
Bagni, G. T. and D'Amore, B.: 1994, Alle radici storiche della prospettiva, Franco Angeli, Milano
Bazzini, L.: 1993, Mathematics as intellectual venture and as instrument to approach reality: implications for teaching, in GDM (ed.), Beiträge zum Mathematikunterricht, 4 Franzbacker, Hildesheim, 6-13
Bazzini, L.: 1994, Cultural Choices and teaching implications in primary mathematics education, in L. Bazzini and H.G. Steiner (eds), Proc. Second Italian-German Bilateral Symposium on Didactics of Mathematicsi, (Osnabrück-Germany, 1992) IDM Materialien und Studien Band 39, Bielefeld, 17-29
Bencivelli, W. and Villani, V.: 1994, Su un test per l'ammissione ad un corso di laurea, La Matematica e la sua Didattica, n. 2, 157-167
Bernardi, C.: 1991-1995, << Archimede risponde>>, Archimede, vol. 43, 176-183; vol. 44, 175- 185; vol. 45, 66-75; vol. 46, 65-77 and 163-172; vol. 47, 51-61
Bernardi, C.: 1994, La Matematica pura nella divulgazione: un fascino discreto, in B. D'Amore (ed.), L'apprendimento della Matematica: dalla ricerca teorica alla pratica didattica, Pitagora Editrice, Bologna, 3-12
Bernardi, C. and Bindi, R.: 1989, Questo è il titolo di un articolo sull'autoreferenza, La Matematica e la sua Didattica, vol. 3, n. 1, 6-12
Bernardi, C. and Menghini, M.: 1990, Sistemi elettorali proporzionali. La "soluzione" italiana, Bollettino U.M.I., 7, 4-A, 271-293
Bolletta, R.: 1988a, Preparazione matematica in Italia al termine della Scuola Media: Rapporto dell'indagine VAMIO, CEDE, Frascati
Bolletta, R.: 1988b, The curriculum of the Scuola Media since 1979, in D.F. Robitaille (ed.), Evaluation and Assessment, Unesco, Paris, 81-87
Bolletta, R.: 1992a, Le competenze matematiche, in CENSIS (ed.), Monitoraggio della Riforma della Scuola Elementare: Rapporto finale, vol. 1, Roma, 174-255
Bolletta, R.: 1992b, La rilevazione delle competenze matematiche, in CENSIS (ed.), Monitoraggio della Riforma della Scuola Elementare: Rapporto finale, vol. 2, Roma, 47-67 e 216-227
Bottino, M. R. and Furinghetti, F.: 1996, The emerging of teachers conceptions of new subject inserted in mathematics programs: the case of Informatics, Educational Studies in Mathematics, vol. 30, 109-134
Bottino, M. R., Ghiarugi, I. and Furinghetti, F.: 1991, Teachers opinions on maths teaching at ages 14-16, in M. Ciosek (ed.), Proc. CIEAEM 42 (Szczyrk-Poland, 1990), 278- 290
Cannizzaro, L.: 1989, Mathematical Models and Modelling: Different Views of University Students and Mathematics Teachers, in W. Blum et al. (eds), Applications and Modelling in Learning and Teaching Mathematics, Ellis Horwood, Chichester, 55- 59
Cannizzaro L.: 1994, Mathematical models of real world phenomena: some historical developments as a guidline for identifying cultural aims, in L. Bazzini and H.G. Steiner (eds), Proc. Second Italian-German Bilateral Symposium on Didactics of Mathematics (Osnabrück-Germany, 1992), IDM Materialien und Studien Band 39, Bielefeld, 115-126
Conti, F., Barsanti, M. and Franzoni T.: 1994, Le Olimpiadi di Matematica: Problemi delle Gare Italiane (with an introduction by G. Prodi), Zanichelli, Bologna
D'Amore, B.: 1992, Giochi logici, linguistici e matemagici, Franco Angeli, Milano
D'Amore, B. and P., Sandri: 1994, L'immagine della Matematica negli studenti. Come vedono gli studenti e gli ex studenti la Matematica ed il suo insegnamento?, La Didattica, n. 2, 83-86
Di Sieno, S. (ed.): 1990, Il Pensiero Matematico nella Cultura e nella Società Italiana degli Anni '90, Quadeni PRISTEM, Università Bocconi, Milano
Emmer, M. (ed.): 1989a, L'occhio di Horus: Itinerari nell'immaginario matematico, Ist. Enciclopedia Italiana, Roma
Emmer, M.: 1989b, Art and Mathematics: an interdisciplinary model for Math education, in W. Blum et al. (eds), Applications and Modelling in Learning and Teaching Mathematics, Ellis Horwood, Chichester, 213-218
Emmer, M.: 1989c, Il nuovo immaginario matematico, in B. Vertecchi (ed.), Una Scuola per tutta la Vita, La Nuova Italia Ed., Firenze, 119-134
Emmer, M.: 1991a, La perfezione visibile: Matematica e Arte, Ed. Theoria, Roma
Emmer, M.: 1991b, Le bolle di sapone: Viaggio tra Arte, Scienza e Fantasia, La Nuova Italia Ed., Firenze
Emmer, M.: 1992, Scrivere sulla Matematica, La Matematica e la sua Didattica, n. 3, 8-12
Emmer, M.: 1993, Il museo di Matematica, La Matematica e la sua Didattica, n. 2, 131- 147
Ferretti, G., Fiori, C. and Quattrocchi, P.: 1988, Dalla Scuola Media al Biennio: Prove di verifica e di ingresso di Matematica, CDE, Collana Materiali di Studio e di Lavoro, Modena
Fiori, C.: 1991, Dalla Scuola Media al Biennio: Prove di verifica e di ingresso di Matematica. Risultati di una sperimentazione condotta all'inizio degli anni scolastici 1988-89 e 1989- 90, Epsilon, IV, n. 1, 56-58
Furinghetti, F. (ed.): 1988, Ipotesi per una biblioteca di area matematica per studenti della scuola secondaria superiore, IRRSAE Liguria, ECIG, Genova
Furinghetti, F.: 1993, Images of Mathematics outside the community of mathematicians: evidence and explanations, For the Learning of Mathematics, vol. 13, n. 2, 33-38
Furinghetti, F.: 1994, Ghosts in the classroom: beliefs, prejudices and fears, in L. Bazzini (ed.), Proc. SCTP (Grado-Italy, 1994), 81-91
Giusti, E. and Conti, F.: 1992, Oltre il Compasso: La Geometria delle Curve, Scuola Normale Superiore, (book-catalogue of a exhibition), Pisa
Lucchini, G.: 1992, La matematica in prove di selezione per l'ammissione a corsi di laurea: un'occasione per riflettere, La Matematica e la sua Didattica, n. 4, 18-22
Malara, N. A.: 1995, È possibile limitare le difficoltà della Matematica e farla apprezzare agli allievi?, L'Insegnamento della Matematica e delle Scienze Integrate, vol. 18A-18B, n. 5, 551-570
Malara, N. A. and Pellegrino, C.: 1990, Il gioco come mezzo per promuovere una corretta immagine della Matematica, in B. D'amore (ed.), Gioco e Matematica, Apeiron Ed., Bologna, 53-62
Menghini, M.: 1988a, Matematica, cultura, incertezza, Epsilon, n. 1, 58-59
Menghini, M.: 1988b, Le voci della Matematica, Epsilon, n. 2, 55-58
Menghini, M.: 1988c, Quale Geometria per lo spazio?, Epsilon, n. 3, 61-62
Menghini, M.: 1989, Dopo una conversazione con H.G. Steiner, Epsilon, n. 4, 20-23
Menghini, M.: 1990, Divulgare Matematica: al cittadino non far sapere..., Epsilon, n. 8, 57- 59
Menghini, M.: 1993a, Libri di Storia della Matematica, Epsilon, n. 14, 52-53
Menghini, M.: 1993b, Quella dimensione in più..., Epsilon, n. 15, 49-51
Menghini, M.: 1993c, Il teorema di Fermat, Epsilon, n. 15, 60-61
Menghini, M.: 1994, C'è Geometria e Geometria, Epsilon, n. 17, 50-52
Menghini, M. and Cannizzaro L.: 1992, La didattica della Matematica, Epsilon, n. 11, 49- 51
NRD-Mathesis Firenze: to appear, Tra Forme e Figure (book-catalogue of a exhibition)
NRSDM Modena: 1992a, Macchine matematiche e altri oggetti (book-catalogue of a exhibition), Comune di Modena, 2 vols
NRSDM Modena: 1992b, Videocassette didattiche (20 didactic videocassettes)
Pellegrino, C. and Fiori, C.: 1995, Il teorema di Enumerazione di Pólya: Escursione tra movimenti, gruppi, rappresentazioni, conteggi ed altro ancora, Didattica delle Scienze e dell'Informatica, n. 175, 40-47
Pellegrino, C.: La tela di Arithmós, La Matematica e la sua Didattica, VI, n. 3, 25-32
Pellegrino, C. and Marchini, C.: 1993, Come giocando al calcio senza pallone si possono incontrare insospettati personaggi matematici, Le Scienze (italian edition of Scientific American), n. 244, 90- 93
Pesci, A. and Reggiani M.: 1989, Verifica di conoscenze e abilità matematiche allo snodo scuola media-scuola superiore, L'Insegnamento della Matematica e delle Scienze Integrate, vol. 12, n. 2, 320-325
Reggiani, M.: 1991, Test of the learning of Mathematics at the and of the obligatory school (pupils age 14), in A. Warbecq (ed.), Proc. CIEAEM 41 (Bruxelles, 1989), 457-462
Rinaldi, M. G., Vené M.: 1994, Un test d'ingresso per le facoltà scientifiche, La Matematica e la sua Didattica, n. 2, 142-154
Scimemi, B.: 1990, Algebra e Geometria piegando la carta, in B. D'amore (ed.), Gioco e Matematica, Apeiron Ed., Bologna, 79-87
Scimemi, B.: 1992a, Aritmetica come scoperta, Scuola e Didattica, 17, 42-47
Scimemi, B.: 1992b, Aritmetica e Musica, Atti Conv. Il Pensiero Matematico nella Ricerca Storica Italiana, IRRSAE Marche, Ancona, 68-76
Scimemi, B.: 1994, Musica e Matematica, in XXXIV Olimpiadi di Matematica (Selezione Italiana, Cesenatico 1994), Proceedings of the Meeting, publication supported by AgipPetroli, Roma, 98-107
Scimemi, B.: 1996, Olimpiadi di Matematica. Attività tradizionali e novità organizzative, Notiziario UMI, n. 1-2, 64-68
Speranza, F.: 1990a, La Matematica e la cultura oggi, in A. Barlotti (ed.), Atti Conv. Cultura Matematica e Insegnamento (Firenze, 1988), 155-169
Speranza, F.: 1990b, Matematica e Scienze: quale distinzione, quale integrazione?, L'Educazione Matematica, III, 1, suppl. 2, 47-54
Speranza, F.: 1994, Il valore conoscitivo della Geometria, Periodico di Matematiche, VII, 1, n. 4, 5-18
Vené, M. and Vighi, P. et al.: 1989, Un test d'ingresso per la prima superiore, Periodico di Matematiche, VI, vol. 65, n. 2, 33-77
Zuccheri, L.: 1992, Oltre lo specchio: storia e motivazioni di una esposizione didattica, in LIS (ed.), Atti Conv. La Matematica tra Didattica e Cultura (Trieste, 1992), 143-150
| Carla Fiori | Consolato Pellegrino |
| Dipartimento di Economia Politica | Dipartimento di Matematica |
| Università di Modena | Università di Modena |
| Via Berengario, 51 | Via Campi, 213/B |
| I - 41100 Modena, Italy | I - 41100 Modena, Italy |