1. Introduction

The premises of the present situation of mathematics education in Italy date back to the post-war II, when the echo of innovative movements in various countries produced a stimulus towards renewing the contents of teaching and studying its methods and effects. International contacts were developed through privileged channels such as CIEAEM, but prior to the Sixties only a few university professors participated actively.

Nevertheless we have to underline, as reported in Barra et al. (1992), that since the Unification of Italy (1861) prestigious mathematicians, such as Cremona, Betti, Veronese and later Peano, Vailati, and Enriques were actively involved in the improvement of the teaching of mathematics, which however underwent a considerable involution when, at the beginning of the Fascist age, the Gentile Reform was introduced (1923).

A particularly significant moment for the involvement of the academic world was the reaction to a study of UNESCO, appearing in 1975, which had pointed out a serious deficiency in the scientific teaching in Italy. In order to provide for this situation, the CNR (Consiglio Nazionale delle Ricerche - National Council for Research), together with UMI (Unione Matematica Italiana - Italian Mathematical Association), immediately promoted the institution of NRD (Nuclei di Ricerca Didattica - Didactics Research Groups) in several universities. These groups were composed by university professors operating in departments of mathematics and teachers of different school levels, with the aim of renewing the contents and methods of teaching, starting from the requirements and difficulties pointed out by the teachers themselves through systematic and joint comparisons. The first results of NRD concerned the setting-up of innovative projects of teaching for the various school cycles and their contribution to the drafting of the new syllabuses for middle school in 1979, primary school in 1985 and for the first two years of secondary school in 1987 [1].

With time, other research groups were set up in several other universities. About ten years ago these groups founded the Seminario Nazionale di Ricerca in Didattica della Matematica (National Seminar of Research in Didactics of Mathematics), with the aim of promoting common studies among various groups and of realising a confrontation on contents and styles of work.

The teachers belonging to the NRD contribute in various ways to the research: from partecipating to the planning and classroom-experimenting of didactical innovations, to writing papers on the realized experimentations. Some of the more involved teachers have acquired the professional status of researchers (see Navarra and De Plano, 1992, 1993), which in the classroom work allows them to assume the double role of participant and observer (Eisenhart[2] 1988). We need to point out that in Italy the classes are organized in such a way that a teacher follows the same students for an entire school cycle (from a minimum of two years to a maximum of five years). This has promoted research focused on the observation of the same students over many years.

From 1975 to today the activity of the NRD has gradually been refined and specialized, thanks to the active and more numerous partecipation of the researchers in international conferences such as ICME, PME, CIEAEM, ICMI studies etc, and to the realization of bilateral meetings between Italy and neighbouring countries such as Germany, Spain and France (see Bazzini and Steiner[3] 1988, 1994; Malara and Rico[4] 1994; Druhard and Maurel[5] 1995 ).

We present here the studies on such matters which, because of their transversal nature, are not directly related to any of the themes discussed in the other reports contained in this book.

2. Studies on the Italian Research in mathematics education

We now concentrate on some works concerning the character and the trends of Italian research in mathematics education, written between 1988-1995 on the occasion of national and international meetings on such theme. To begin with, let us consider the contributions of Arzarello, Boero and Malara at the session of the National Seminar devoted to Ricerca didattica ed Insegnamento (Didactic Research and Teaching ), Pisa 1991.

Arzarello (1992) scrutinizes the research in mathematics education of the last thirty years in Italy, pointing out its different roots and presenting the most relevant differences and similarities when comparing it to the reearch developed in other countries. In particular the author highlights three components in Italian research, which he often finds intermingled within the same work: the first concerning the conceptual organization of the discipline, the second the concrete innovation in the classroom, and the third the observation of the processes of teaching and learning in the classroom (this last component appears only in the works from the second half of the Eighties on). These components reflect general trends which have emerged within the international scientific community; they can be referred, for instance, by the research trends delineated by Bishop[6] (1992) and respectively called "scholastic philosopher tradition", "pedagogue tradition" and "empirical scientist tradition". Moreover the author asserts that from the analysis of the more recent works, there appears a fourth component concerning the study of the mutual relationships among the previous three and that it belongs to a level superior to the others.

Boero (1992) deals with the problem of the specificity of the research in mathematics education, underlining how this research is related to the cultural traditions and to the social situations of the various countries and how it sharply depends on the cultural background of the researchers involved (specialists in the discipline itself or in the science of education). The author asserts that the research in mathematics education should, on one hand, resort to competencies and cognitions typical of the discipline, of its epistemology and history, and on the other hand to methods and frames of reference proper to psychology of learning, sociology of education, docimology, etc. He underlines that the specificity and the value of the disciplinary didactical research consists in the ability to select, to integrate, to unify linguistically, to finalize, to make precise, to deepen methodological contributions and results of various sciences around significant questions concerning the processes of the teaching and learning of the discipline in the school environment. (A distinction between general didactics and disciplinary didactics is traced by D'Amore, 1994a.)

Malara (1992a), on the other hand, deals with questions regarding the influence of the research in mathematics education on the didactics in school, examining in particular the problem of the reproducibility of the experiments of innovation. She describes the motivation, cultural choices and style of work done with and for the teachers in her research group and develops some general considerations on what should be done in order to favour the impact of the didactical research on the school practice, or to strengthen the latter.

Malara (1995) resumes these considerations on the occasion of the fifth international meeeting Systematic Cooperation Between Theory and Practice in Mathematics Education (Grado-Italy, 1994) framing them in the context of the situation of research in mathematics education in Italy and underlining how the work of NRD is developed at a double level: the cultural teacher-training (either on the disciplinary or the psycho-pedagogical side) on one hand and the planning and joint control of classroom-experimentations on the other. She underlines how this method produces in time a high professional qualification of the teachers and she spends time on describing the correlations among the methodology of work in the classroom, the behaviour of the teacher and the results reached by the students.

On the occasion of the ICMI Study What is Research in Mathematics Education and What are its Results (Washington-US, 1994) Arzarello, in cooperation with Bartolini Bussi (to appear), broadens and deepens his analysis of recent Italian research. The scholars cope with a twofold problem: on one hand, to convey to the international professional community of researchers in mathematics education a meaningful set of information about the Italian national context; on the other, to detect the aspects of local research studies that should be relevant to the international community, and vice versa, the aspects of the international research that should be relevant for local research studies. First, they trace the history of Italian research in mathematics education from the sixties, by pointing out two internal trends, namely, concept-based didactics and innovation in the classroom, alongside the external trend of classroom observation of processes. In the second part of the paper, Arzarello and Bartolini Bussi analyse the more recent developments of Italian research in mathematics education and call it research for innovation. They claim that a systemic approach characterises many recent research studies that produce paradigmatic examples of research for innovation, where mutual relationships between components inherited from the different trends are studied and tools for systemic analysis are developed.

On the occasion of the same ICMI Study, Boero -in collaboration with Szendrei (to appear), analyses difficulties and contradictions in the field of mathematics education on the basis of the results of the research. The authors classify the studies into: "innovative patterns" (regarding innovative materials for teaching, didactical proposals or reports on innovations or projects which have been experimented); "quantitative information" (regarding studies based on quantitative data collected and analysed according to statistical methods); "qualitative information" (regarding studies based on a careful analysis of students' written works, of the teacher-student interactions, of recorded group or classroom discussions, etc.); "theoretical perspectives" (regarding descriptions, classifications or interpretations of "phenomena", "models", historical or epistemological analyses of specific contents of teaching). They point out that many papers referable to the first point have pragmatic aims and belong to the "energizer practice" category, whereas many of those referable to the other points have fundamental scientific aims and belong to the "demolishers of illusions" category or to the "economizers of thought" category. They describe some contradictions underlying the results of the "innovative patterns" and "quantitative information" type, showing how these contradictions may be overcome with suitable results of the "qualitative information" and "theoretical perspectives" type. In particular, they underline how the latter are needed to keep under control, in the field of mathematics education, the usage of experimental and statistical methodologies borrowed from pedagogy and for this reason these should gradually enter the teachers' preparation.

Moreover, on the occasion of the meeting Vingt Ans de Didactique des Mathématiques en France: Hommage à Guy Brousseau and Gerard Vergnaud, Boero (1994) deals with the problem of connecting the results of the Italian research in mathematics education to the French theoretical framework and he spends time in particular on some of them highlighting their compatibility or divergence as to the standards of the French research.

Reflections on the differences between the character of the research in didactics of mathematics in France and innovational research in other countries are discussed by Bartolini Bussi (1994), who also underlines the different perspectives induced by the Piagetian and Vygostkian conceptions about the learning and the role of social interaction in this process. In the end she assumes, on the basis of her own research and in accordance with Steiner[7] (1985), that the complementarity among the various theoretic elaborations is a necessary characteristic of the research in didactics of mathematics.

3. Studies on the curriculum

An aspect which has characterized the research in mathematics education, especially in the past, is the attention devoted to the curriculum of the various levels of schooling. Such studies were also favoured by the fact that in the second half of the Seventies and during the Eighties the syllabuses of primary and middle school were meaningfully modified, and many suggestions presented for the renewal of the syllabuses and of the structure of secondary school were tested on a wide scale. The research groups therefore faced the problem of creating and testing didactic proposals for the application of the syllabuses themselves.

To begin with, let us see some studies concerning childhood education (age 3-6). The activities of kindergardens are not guided by proper syllabuses in Italy, but by ministerial indications, in the more recent of which (1991) six fields of educational experience are identified. Among these, the one titled "Space, order, measure" deals with the mathematical activities for this age. Italy has never produced wide disciplinary research on the kind of work possible at this age; general pedagogical studies are more frequent. Still, some projects for the innovation of pre-school mathematics education have appeared. One was created by the research group in the history and didactics of mathematics of the university of Modena, wich has operated in this area since 1983 and has published a collection of papers (Bartolini Bussi, 1992) on language, relations, numbers, measure, time, space, games, problem situations, school routines, representation, which reports the experiments carried out by kindergarden teachers in Modena. Bartolini Bussi, the scientific director of the project, describes its methodological features in Bartolini Bussi (1990).

The NRD of Bologna, also, was concerned with kindergardens and worked out game-like proposals focusing on the above-mentioned themes (space, order and measure, including measurement of time) (Aglì and D'Amore B., 1995; Sandri, 1995) on the approach to chance as precursor of the creation of a probabilistic mentality (Aglì and Martini, 1989).

Recently, studies on the period of compulsory schooling in Italy (age 6-14) have been more frequent. The syllabuses for the middle school were renewed thanks to the introduction of new topics such as Elements of Logic, Probability and Statistics and, above all, through a new way of dealing with traditional themes. Nevertheless, since we are reporting here the studies published since 1988, it is clear that by that date most of the curricular studies on the age range 11-14 had already been carried out. In these years, studies of a cognitive/methodological nature have been developed, or on specific themes for that period of schooling, which are presented in other reports contained in this volume.

Different is the situation for primary school, whose new syllabuses date back only to 1985.

Two main projects for the teaching of mathematics at this level appeared: one created by the NDR of Pavia, the other by the NDR of Genoa. The curricular proposal by the group of Pavia (Ferrari, 1989) aims at helping the teachers to turn the ministerial syllabuses into didactical practice and focuses on three main themes: environment, time, economy. This project underlines two aspects of mathematics: cognitive/operative on one hand (mathematics to know the world, to describe it, to master it) and cognitive/contemplative (mathematics reflecting on itself in order to produce new mathematical knowledge and new conceptual instruments not immediately finalized) on the other. The basic choices of this curricular research are: the problem solving methodology and the notion of plurality. This means a plurality of approaches, languages, strategies and didactic materials. The project is the result of a tight cooperation between university researchers and primary school teachers and has been experimented in several classes.

The other important project for primary school is the one realized by the Genoa group, called "Children, teachers, reality" and used by more than 200 teachers in more than 140 classes (25-50 for each level) each year since 1980. Among these teachers, 20 are actively involved in the project, in which special attention is paid to "situated" teaching and learning of mathematics, to the linguistic aspects of mathematics learning (especially for verbal competence and verbalization processes), to the teacher's role in the classroom, and to management problems of various kinds of activities (Boero, 1990). This project has coherently attempted to bring the concepts of "semantic fields" and "experience fields" to bear on the teaching of mathematics and other subjects, integrating suitable experience fields concerning natural and social reality into teaching activities. Some of the experience fields used in the project are for instance "money and prices", "sun shadows", "Italian emigration in the last century". The theorical arrangement of the problem of the context in the teaching of mathematics and the concepts of "experience field" and "semantic field" are introduced and justified on the basis of cognitive, epistemological and anthropological "evidence" in Boero (1989). The use of experience fields within the project is explained in Boero (1994a), where the author underlines the importance of distinguishing the external context of the field from the internal context of the pupil and the internal context of the teacher when referring to a given experience field.

The same principles appear again in the Genoan project for the group age 11-14. Boero et al. (1995) examines different aspects of the relations between mathematics and culture in the teaching-learning of mathematics within the larger range 6-15. It shows in particular how everyday culture can be used at school for constructing mathematical concepts, and it tries to understand whether mathematics and the way in which it is taught at school, can help interpret natural or social phenomena. It considers finally the teaching of mathematics as part of the scientific culture and it tries to clarify some of the potentialities and limits of the teaching of mathematics in context.

Many other curricular projects have been developed in the last few years in Italy. Among them we mention the one examined by the Cagliari research group, the main feature of which is its being inter-disciplinary. An example of the work carried out by this group can be found in Grugnetti et al. (1989), which illustrates the didactic conduction of an inter-disciplinary activity involving history, language and mathematics.

Despite these broad projects and several isolated activities which obtained considerable feedback on the general functioning of school, the problem of bringing about the 1979 reform of the syllabuses for middle school has not yet been solved, with respect to the mutual integration of the themes, or the choice of programming an activity adequate to the different classes are concerned. A survey on teachers' planning in the Bologna area, carried out in 1991, is reported and commented on by D'Amore (1993): the conclusions drawn underline the need of teacher training on contents as well as methodology and assessment.

The survey realized within the project VAMIO ("Verifica Abilità Matematiche Istruzione dell'Obbligo" - "Verification of Mathematical Ability in Compulsory schooling") concerns the whole national territory. This research has accurately depicted the actual level of the presentation of the national syllabuses and the actual knowledge achieved by pupils at the end of compulsory schooling (Bolletta, 1989) on the basis of a national sample of 1300 teachers and 2800 pupils. The former were interviewed through a questionnaire about the topics actually covered, the latter were given a multiple choice test about their knowledge. High school, is non compulsory and offeris several options, each one of which has its own academic program. One of the problems reported by teachers and in the recent years by the researchers in didactics of mathematics, is the connection between the different types of high schools in contents as well as methodology. The problem is particularly serious in the transition from compulsory education to high school. On the matter, Speranza (1990) states that one of the obstacles to the continuity among the levels of schooling is the low prescriptiveness of the syllabuses and their detatchment from many teachers' didactical practice. It would be more productive if the teachers working at the beginning of a new level relied on skills and conceptual nets, rather than on their knowledge of notions and if they were more familiar with the syllabuses and the main features of the previous level of schooling. D'Amore (1993) also believes that the entry tests from one level to another often concentrate too much on knowledge of contents and point out what the pupil doesn't know rather than what he/she knows. He offers some examples of possible ways that might favour continuity between the middle school and high school.

The comparisons among mathematics syllabuses from different nations are very interesting. In particular, IRRSAE Emilia Romagna has promoted a comparative study of the syllabuses of scientific subjects for 6- to 16-year-old pupils of different European countries with the aim of highlighting the trends and delineating their common features. In this work Malara (1994a and 1994b) has examined the Spanish syllabuses and has compared what the geometry syllabuses in France, United Kingdom, Spain and Hungary. The analysis is not limited to examining the contents but also underlines the didactical-methodological differences, with reference to the political educational choices and to the socio-political situation of the nations considered.

4. On assessment

Italian researchers usually deal with the problem of assessment when working on general studies (curriculum studies, or about learning processes, general or specific), so that rarely does assessment constitutes a theme for research in itself.

Grugnetti (1994) takes into consideration the difference between 'assessment', which focuses on the skills acquired by the pupil and 'evaluation', meant as appraisal of the syllabus and of its concrete realization; moreover she discusses the question of assessment in problem solving and how to train teachers in assessment.

Furinghetti and Somaglia (1994) describe how a group of five teachers carried out a transversal assessment of students between 14 and 19 years old in order to analyse the development of their skills and the factors which influence them.

The main tools used for evaluating and assessing the results obtained by the curricular project of the Genoa research group described in the previous paragraph, are examined by Guala (1990). This study deals with diagnostic worksheets, summative evaluation worksheets, argumentative worksheets, which are presented through examples which highlight the reasons for their choice and construction, their features and effectiveness either as means of testing the pupils' results or for giving information about the periodical revision of the project.

According to Boero and Bondesan (1994) the assessment of the pupils' potentialities is one of the most important fields in which the teachers can become aware of the complexity of the problem of assessment itself, of the relative value of the tools and methods of assessment, and of the close relationship existing between assessment and theories of learning. The authors suggest some experiments of interactive assessments of "potentialities" which involve future teachers and pupils of primary school.

5. Future teachers training and in-service teachers training

5.1. General problems concerning teachers training

The training of future mathematics teachers for high school is influenced by a strong Italian belief (cf. note 1): that a good mathematician (as well as a good biologist or a good man of letters) automatically becomes, when sitting at his teaching desk, a good teacher. Prodi (1995) believes that this conviction is also supported by a political design (to keep teaching as a safety-valve against intellectual unemployment). He wonders if it is the university education itself that conditions teachers in a negative sense; and if it is wrong to train, as the Italians do, mathematicians first, and then to transform some of them into teachers. And Prodi further states that his experience allows him to draw exactly the opposite conclusion. In his opinion if one ever tried to train good mathematics teachers, one would automatically produce good researchers.

Indeed some difficulties that frequently arise in teachers' behaviour (difficulty in accepting didactical innovation, lack of mathematical "affectivity", i. e. little enthusiasm towards "beautiful" mathematical facts, difficulties in reasoning authentically on one's work in class) appear somehow connected to university teaching.

The prevalent kind of teaching in Italian universities still shows traces of the Bourbaki reform, by which the learning of abstract notions, internal or external to mathematics, must precede examples and applications. Some aspects which belong to didactics broadly speaking are therefore ignored. For instance, quoting Prodi again, the internal process of simplification of mathematics, the institutional moment of which is the teaching itself, is fundamental; and it is still didactics that leads to the need of classifying problems according to their priority, which too often pure researchers forget. Our university courses should teach and stress more the awareness of reflecting even on the operations and reasonings we do. Thus new subjects such as Probability and Statistics, Logic, Computer Science would not be considered additional specialistic subjects, but rather seen as ways of thinking of great value both in applications and within mathematics. It is moreover necessary to avoid fragmentation in didactics.

The same problems in the praxis of teaching (fragmentation, excess of theory, lack of applications) are underlined by D'Amore (1992). He tells a series of anecdotes collected by observers in various schools were teaching becomes the "triumph" of an empty language, merely made of sensations. Again, these problems are due to the way in which the university forms future teachers.

In Italian universities, in fact, students who want to take a degree ("laurea") in mathematics, have three options ("indirizzi"): Pure Mathematics, Didactics of Mathematics, Applied Mathematics. Students who choose the second option have to attend two or three special courses, the most common of which are "Matematiche Complementari" ("Complementary Mathematics") and "Matematiche elementari da un punto di vista superiore" ("Elementary Mathematics from a higher point of view"). After the degree but before taking the qualifying exam to become a teacher, a student will have to attend a "School of Specialisation" (to be established).

Bernardi (1995) points out that by the end of the degree courses the neo-graduate entering the world of school already needs updating, not only on methodology, but also on contents. Furthermore, not only the courses of the second option (didactics) should be concerned with teachers training. The problems to which we refer regard the whole degree course from its very beginning. As Bernardi acknowledges, there is no didactic continuity in the transition from secondary school to university. A gap in the educational process can be opportune, but the situation should be kept under control by the teaching staff. Some of the difficulties in the first year of university are, for instance: the relationship between symbolic manipolation and spatial interpretation in Geometry, the reference to Arithmetic in Algebra and the theory of measure in Analysis.

Even if the university formation of the teachers is not satisfactory, the university should play an important role in in-service teachers. Nevertheless, as stated by Fasano e Polo (1992), the role of the university, in particular as regards continuous training on specific disciplinary contents, is unfortunately based only on the engagement that many lecturers undertake at personal level, rather than on concrete initiatives institutionally promoted by the university and therefore their work has no generalized impact on the didactical praxis. According to the authors, in-service teachers training cannot avoid coping with choices concerning the theories of learning and with a reflection on the role of the teacher. Vital are precise methodological choices for the teaching in class, but also for the development of professional skills. This paper suggests a model (already experimented) of training which provokes the acquisition by the teachers of the awareness of the intellectual and formative process they encompass. This happens through the "setting-up into situation", which consists in letting the teachers work on the didactical situation through the "explicitation" of the context, the suggestion of an activity, the theoric-didactical analysis on what has been done.

On the other hand the authors warn about the tendency to overlap training and research, which in the end might confuse the different professional levels and areas.

Pesci e Reggiani have a different opinion. In their paper (1994) they describe an experience of future teachers training, in-service teachers training and didactic research carried out in collaboration with university researchers and in-service teachers. The aim of the group was to study, by means of classroom experience and analysis of students' protocols, the problems that might crop up in the relative teaching-learning process. In this case the three aspects (teachers training, classroom activity, didactic research) are not separate but they are usually part of the same experience (cf. also Malara 1994a, Arzarello and Bartolini Bussi, to appear).

Also Ferrari (1994) underlines that the role of the university in on-going teacher training is important: because it is one of its tasks, because the university has the mathematical and bibliographical facilities that the teachers do not always have, and also because university researchers need continuous training, above all of a didactic nature. Besides this, on-going teacher training must have a didactic and a cultural nature, it must be voluntary, it must be recognised, and must be... continuous. Ferrari reports moreover a brief historical introduction about the continuous in-service teachers training in Italy, the current situation, with its positive and negative sides, and the university contribution.

Although the debates on teachers training usually concern the future teachers of high school, when it comes to updating training we should think about all levels of schooling and for all possible pupils.

Therefore Grugnetti and Speranza (1994) present a synthetic outline of the problems and the reality of today's situation of teachers training in Italy for primary through to high school. Michelotti et al. (1993) illustrate an experiment of two-year courses, promoted by MPI, concentrating on the training of support teachers for disabled and learning deficient pupils.

The problem of the attitude of mathematics teachers towards curriculum changes and their role in this process, about which we previously hinted, is reconsidered in (Furinghetti 1994). As shown by the history of mathematics education, certain problems periodically arise in mathematics teaching (such as the natural conservatism of the school world). Furinghetti discusses a possible philosophy of work in retraining teachers aimed at preparing them to update their teaching in an active and critical way ("any project of curriculum innovation has to envisage a project of inservice teacher retraining centred on reshaping teachers' beliefs"). This aim in retraining tries to bridge the gap often observed between the proposals by curriculum developers and their realization.

5.2. Materials for updating teachers

In addition to the various initiatives arranged in order to update teachers, it is necessary to provide them with materials which give them some concrete ideas for improving, testing or practicing their knowledge. It goes without saying that such materials are quite varied in nature, and in particular all studies of didactics of mathematics do certainly help. We shall recall here only some materials produced by the Italian research groups with the precise aim of service, which though not belonging to specific branches of research, can contribute to the updating of teachers.

Let us begin with the materials for primary school teachers, who most of all need updating, particularly in the discipline, because today their formation derives uniquely from attending a special high school, even if the introduction of new syllabuses (1985) took place alongside an in-service-teachers-training "Pluriannual Plan" which involved all primary school teachers. Materials for the training of formators within such plan have been produced by various research groups, see for instance Arzarello et al. (1988), Chini Artusi et al. (1988), Bernardi et al. (1990/91).

Usually primary school teachers have no specific competence in mathematics, little for teaching it, as for mastering the whole subject and therefore seeing in its wide outline what is fundamental and what isn't. In order to allow the primary teachers to do more mathematics, but chosen with precise aims, not at random, Bagni et al. (1993) suggest exercises for self-testing about Arithmetic, Geometry, Logic, Probability, Statistics and Computer Science, as well as games and historical curiosities which help them test their skills and knowledge. Again Martelli et al. (1994) suggest problem solving activities. The problems suggested are inserted in a general theoretical context, described in the first part of the paper, in order to give productive ideas and spur to the teachers. Giovannoni (1989) suggests lessons of mathematics for childhood educators, too, aimed at developing specific skills (on themes analogous to the above-mentioned) for correct didactics, or in order to help them read critically other's proposals.

We already spoke about primary school teachers, but middle school teachers don't have any sufficient preparation in mathematics either: in most cases they have a degree in Biology rather than in mathematics. D'Amore and Picotti (1991) offer a critical analysis of the importance of inviting the pupils to learn, and present a collection of ideas and suggestions which trace a rigorous but operative didactics, created along the stream of the middle school syllabuses in force. Besides, because of the above-mentioned fear for innovation, and owing to the problems concerning the updating of teachers, the syllabuses for middle school can still be considered new. D'Amore (1988) describes to middle school teachers how to use a mathematics laboratory to stimulatee the cleverer pupils, and to help and support the weaker ones.

A different tool offered to the teachers is the analysis of textbooks. Malara (1992b) shows the results of research aimed at studying the features of mathematics textbooks for middle school, trying to investigate whether, and to what extent, they contribute to a real educational process. On the bases of a complex scheme promoted by UNESCO, the author deals with the global analysis of 13 text books. As far as the mathematical contents are concerned, the study detects their inadequacy to deal on what the syllabuses prescribe (either on new themes such as Statistics and Probability, or on traditional themes, such as: rational numbers and the use of letters) and even degenerations (e.g. many recent textbooks deal with the theme 'structural analogies' only by quoting with highly abstract language, the formal notions of monoid, group, ring and field). The survey revealed scarce attention to language, mostly in the recent textbooks, and a lack of a gradual approach to the mathematical language. The amount of mistakes, too many and of various nature, is pointed out.

Another survey on textbooks, this time for high school, is edited by Quattrocchi and Fiori (1988). A reading card was prepared in order to obtain a sort of objectivity during the examination of the books' contents and also to help the comparison of the gathered data. Trying to avoid mistakes and to save characteristics of the discipline, slight changes in the card were allowed. Fifty selected books were tested (at least 50% of the books adopted in various secondary schools).

The aim of these works is to help teachers, authors and schools in general. The purpose is not to point out mistakes, but to give a contribution to the improvement and to the proper exploitation of mathematical texts which have such a great importance in the scientific development of students.

Moreover, another didactical tool deserves mentioning: two recently published indexes of reviews of didactics of mathematics, including an index of the issues, and an index by authors and by topics. The first (cf. Pellegrino and Borrelli, 1994) was realized on the occasion of the 25th anniversary of the review L'Insegnamento della Matematica e delle Scienze Integrate. The second (cf. Mathesis 1995), realized by L. Citrini, concerns Periodico di Matematiche, one of the oldest reviews published for teachers of mathematics, and appeared in a special issue including also some notes on its history and on the history of the Mathesis Society. In order to help the diffusion of analogous initiatives and to give the teachers a basis for an interesting classroom work devoted to classification, with or without the help of a word processor, Pellegrino and Borrelli (1996) explain the criteria on which the index they edited has been realized.

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Nicolina A. Malara	Marta Menghini	Maria Reggiani
Dipartimento di Matematica	Dipartimento di Matematica	Dipartimento di Matematica
Università	Università La Sapienza	Università
Via Campi 213/B	Piazzale A. Moro 2	Via Abbiategrasso 209
41100 Modena - Italy	00185 Roma - Italy	27100 Pavia - Italy
E-mail: malara@dipmat.unimo.it	E-mail: menghini@mat.uniroma1.it	E-mail: reggiani@dragon.ian.pv.cnr.it