• Preface
  • 1. Introduction
  • 2. Research for Innovation: A New Paradigm?
  • 2.1. The Roots of Research for Innovation
  • 2.2. The Need for a New Paradigm
  • A) The design of classroom experiments
  • B) The analysis of classroom experiments
  • C) Design and analysis were not independent part of the work
  • 2.3. The Birth of the New Paradigm
  • 2.4. An Example. Theorems in School: from History and Epistemology to Cognition
  • 3. Some Open Problems
  • 4. References
  •  


     

    Preface

    This short essay aims at giving landmarks for understanding a recent trend in Italian research in the didactics of mathematics, which, without exhausting the whole of the research studies produced in Italy, is more and more widely represented in the proceedings of the international conferences and in the main international journals and books. The genesis of this original model of educational research is a matter of historical reconstruction rather than a collection of objective facts (if they exist at all). As such it suffers from the bias of the author's scientific and professional interests. This short paper summarises information, opinions and discussions that have been developed elsewhere. The interested reader is referred to [1], [2] and [4]. The last-mentioned book was prepared on the occasion of the international congress ICME 8 (Seville 1996). It is composed of a wide introduction and of series of essays on specific themes, each of which offers an outline of studies carried out between 1988-1995 on the topic examined, and tries to frame them within the international literature. The volume was not for sale. Some copies are still available. (To obtain one, please contact Marta Menghini ( menghini@axrma.uniroma1.it ), Dipartimento di Matematica, Università, Piazzale Aldo Moro, 1, I-00185 Roma, ITALY)

    1. Introduction

    The tradition of an involvement of professional mathematicians in the debate about mathematics education from the very beginning of primary school is very ancient in Italy (see [2] for an historical reconstruction). A crucial episode in this long history is represented by the proposal in 1966-7 of the so-called 'Frascati programs', prepared for secondary school by a mixed group of mathematicians, teachers and government officials, appointed by the Italian Commission for the Teaching of Mathematics, that had been, since 1908, the Italian reference body to ICMI. These programs were not put into practice but influenced all the further development in the field [2].

    The desire to realise something in the spirit of these programs, was one of the cultural mainsprings that lead some years later to the institution of the NRDs (Nuclei di Ricerca Didattica - Didactics Research Teams) in several universities. These groups were composed of university professors, operating in Departments of Mathematics, and school teachers, with the aim of renewing the contents and methods of teaching. They had the scientific support of the UMI (Unione Matematica Italiana - the Italian Mathematical Union) and the financial support of the CNR (Consiglio Nazionale delle Ricerche - National Council for Research). In the early years the major effort was directed towards the production of innovative projects to be tested and disseminated, first of all for secondary school and later for all the school grades. The experience that matured in this way was precious and later transposed into the drafting of the new national syllabuses for middle school (grades 6-8) in 1979, primary school (grades 1-5) in 1985, and the first two years of secondary school (grades 9-10) in 1987, thanks to some leading mathematicians who had directed the activity of the NRDs themselves and had been appointed by the Ministry of Education to serve on the national committees. Besides, the tradition of peer co-operation between university professors and teachers was firmly established and influenced very deeply the further developments, leading to an original model of educational research, whose presentation is the main focus of this paper.

    2. Research for Innovation: A New Paradigm?

    2.1. The Roots of Research for Innovation

    In a paper coauthored with Ferdinando Arzarello [1] we have reconstructed the genesis of this model by means of a conceptual structure that allowed us to identify three distinct research traditions: concept-based didactics; innovation in the classroom; laboratory observation of processes. The arguments of the quoted paper are briefly summed up in what follows.

    The first two trends, internal to the Italian tradition, were born in two different places (the university / the school) and carried out by different communities (the mathematicians / the teachers of mathematics); addressed different objects (the teaching of Maths in 'generic' / 'specific' situations); answered different needs (to produce ideas for improving Maths teaching / to produce improvement in Maths teaching); adopted different methodologies (top-down model / action-research); focused on different problems (products / short and long term processes in Maths teaching); and offered different products (textbooks, tests, syllabuses / projects for curricular innovation). Both trends had a long tradition rooted, in the universities, in the activity of many leading mathematicians, from the seminal work of Cremona, Betti, Veronese, Peano, Vailati, Enriques and, in the schools, in the work promoted by spontaneous groups of teachers and teacher associations; some of these teachers (like Emma Castelnuovo) became well known also outside Italy.

    In other countries, in spite of the presence of similar traditions, the pressure of academic institutionalisation of didactics as a scientific discipline led to a complete separation between forms of theoretical research (closer to 'concept based research') and forms of action-research (closer to 'innovation in the classroom'). In Italy a different pathway was followed, namely the progressive integration of both traditions. As we have said in the introduction, the small but active groups of university mathematicians interested in didactics and of innovative teachers of mathematics came in touch with each other and started to work together, leading to the constitution of the early NRDs. On both sides, participants were volunteers, driven by cultural and social needs, which placed this cooperative work as an additional activity in their respective institutions.

    In the late seventies, several NRDs were working in the Departments of Mathematics at various universities on projects for curricular innovation, with the financial support of the CNR. The grants were used partly to cover the costs of classroom experiments, but the biggest investment was made for researchers (from both university and schools - the latter being named teacher - researchers) to allow them to take part in important seminars, conferences, summer schools, symposia, held in Italy or abroad. Some important international conferences were organised in Italy and new ideas started to circulate among Italian researchers. They came in contact with different research traditions, e. g. with the methods of what may be called laboratory observation of processes (e. g. in the PME Conferences). It consisted in basic research studies about short term processes, developed around laboratory experiments (where the classroom itself could be used as a laboratory), with analytical tools borrowed from psychology, sociology, pedagogy and so on; the methodology was based on experimental induction, as it is conceived in natural science.

    A complementary influence was exerted by the original French paradigm. Some French researchers were invited to Italy and several Italian researchers took part in the French summer schools. One of the points of disagreement was the limited role (if any) that the teacher had in the early elaboration of the theory; that clashed against the system of beliefs matured within the activity of the NRDs. However, the influence can hardly overestimated that the French paradigm had in identifying some points of weakness in the earlier work developed in Italy, on the one side, and, on the other side, in offering a model of a way to transform facts coming from the design and the implementation of classroom activity into phenomena of a didactic theory.

    2.2. The Need for a New Paradigm

    A need emerged of the necessity to make the didactic dimension of the projects more precise, in order to understand, for instance, why some innovations would work in some classrooms and not in others. For this problem, no tool was available either in 'concept based didactics' or in 'innovation in the classroom'. Researchers became aware that 'laboratory observation of processes' could have furnished suitable tools, but they encountered a lot of problems.

    A) The design of classroom experiments

    Teachers insistently claimed that long term teaching experiments were the privileged setting within which to study deep changes in the development of mathematical thinking. The habit of planning classroom activities on a long term basis was surely influenced by the institutional constraints of the Italian school system, where a teacher teaches the same group of students for several years. The teachers felt ill at ease with the short term detailed observations of most educational researchers (typical of the 'laboratory observation of processes') that were supposed not to grasp the very important things in the teaching-learning process. Yet, no long term experiment could have been implemented in any classroom without the strong involvement of the teacher: as it was the teacher who had the institutional responsibility for teaching, s/he could have 'lent' her/his classroom for a session (short term experiment), but not for a month or several months. So the teachers had to be active members of the research team already in the design phase.

    B) The analysis of classroom experiments

    'Laboratory observation of processes' was usually carried out by detached external observers on the basis of carefully designed protocols. Also in the French paradigm, the role of the detached observer was emphasised. No contradiction was found when the object of observation were processes with the individual pupil, but when the object of observation were classroom processes, the teacher was to be observed together with pupils. Several analytical studies had been developed outside Italy and had highlighted the presence of hidden patterns and routines in classroom interaction which sometimes had the effect of being in contrast with the teacher's aims, against his/her will. These results were really important, as they contributed to demolish the illusion that 'teacher - proof' projects could be produced and disseminated. But to put the teachers under a lens directed by the university researchers clashed against the tradition of a peer cooperation. Hence, if some results or analytical methods were to be borrowed from 'laboratory observation of processes' or whatever else, they should have been reconceptualised by means of a deep epistemological analysis of the way to gain knowledge concerning human activities. This was the reason why literature on participant observation, inherited from the anthropological tradition, entered the NRDs' libraries.

    C) Design and analysis were not independent part of the work

    In the design phase, a problem immediately came to the fore: For how long and how deeply was it necessary or timely to analyse classroom data? Surely, at the very beginning only a coarse grain planning of the experiment was done on the basis of previous epistemological and didactic analysis. Yet data from the classroom could upset this analysis and suggest a change of the strategy. When the experiment contained also teacher directed sessions (and this was usually the case in long term experiments) the quality of teachers' management could have been very different with different reactions from the classroom.

    The designers had different choices:

    to compress the experiment in a short period making sketchy analysis between sessions and emphasising the continuous and global feature of experience to dilate the experiment over a long period making detailed analysis between sessions, and re-establishing the continuous and global feature of experience by a careful recourse to contract

    In the first case, the detailed analysis, which was postponed, could have pointed at inappropriate choices; but in the second case, the effort in re-establishing the continuous and global feature could have resulted in something that is very time consuming (long teacher introductions; replication of problems in putting students into the situation; reading of previous documents, and so on) and, anyway, not always successful.

    No best choice existed. Every choice had to be discussed by the research team (which included the teacher) and carefully examined on the basis of available information about the classroom. Some classrooms might already have been accustomed to having long intervals between two sessions on the same problems; others might not. In this case the teacher offered an invaluable expertise to settle the design.

    The above were only three of the 'perturbations' introduced into the universe of didactic research, when the two internal traditions of 'concept based didactics' and of 'innovation in the classroom' came together and were confronted with external traditions. The challenge was to find a solution that could have made the most of the contributions coming from any of the traditions.

    The discussion went on for years in the due places: the yearly meeting of NRDs for each school level (primary, middle and secondary) and, from 1986, in the National Seminar, held every year to discuss in depth some research projects, with the presence of official reactors from inside the field of didactics of mathematics or from neighbouring disciplines and, sometimes, from the traditions of other countries.

    2.3. The Birth of the New Paradigm

    The early results of this reflective process were stated officially in 1992, in the 8th session of the National Seminar, where it was pointed out that most of the Italian studies in the didactics of mathematics addressed the phenomenon of teaching and learning mathematics and were based on the tradition of 'innovation in the classroom' with a deep mixing with the essential elements taken from 'concept-based didactics' and 'laboratory observation of processes'. The main aims of research were stated as follows:

    1. to produce paradigmatic examples of improvement in mathematics teaching and learning (in the form of projects for curricular innovations concerning either the whole mathematics curriculum or some special parts of it);

    2. to study the conditions for the realisation of such improvements as well as the possible factors underlying their effectiveness.

    The term research for innovation has been introduced [1] to characterise this trend. From then on, several studies on 'research for innovation' have been produced, and the findings havebeen presented at international conferences and published in international journals or books. A strong experimental component (i. e. teaching experiments in the classroom) is characteristic of the studies, which, in this sense, are different from the ones produced earlier in 'concept based didactics'. A strong cognitive component (i. e. analysis of mental processes) is present as an influence of the participation in the PME group. The differences between'laboratory observation of processes' and 'research for innovation' have been discussed already. Hence what is left is to point out the main differences between 'innovation in the classroom' and 'research for innovation'. These could be summed up by answering the question: in what sense is 'research for innovation' different from the well known action-research, practised by a lot of teachers in their own classrooms?

    Action-research is oriented towards practice and, often, ideologically a-theoretical. The emphasis is put on the 'art' of teaching and on the individual sensitivity of the teacher. The products are examples of good functioning in the classroom, to be used by other teachers as sources of ideas. Unfortunately, what happens very often is that a project that has worked quite well in one classroom, does not work in another one. In practice this failure is ascribed to the different background situations or the different personalities of the teachers.

    In research for innovation, the design and the implementation of classroom experiments is linked to the development of models of the teaching-learning processes, which are basic results in themselves (hence used to advance the knowledge about classroom processes), yet might be used also to transfer experiments to new situations. In one sentence, action research produces didactic facts, while research for innovation produces and interprets didactic phenomena.

    'Research for innovation' tries to overcome the distinction between theoretical and pragmatical relevance, by means of developing the relations between the two from the very beginning. This means the assumption of a diverse epistemological attitude towards the inquiring activity and implies the attribution of a new theoretical status to teachers. Actually, it is possible to distinguish three modes of relationships between observer and observed, the last being represented in our case by the whole of classroom processes, where teaching and learning cannot be separated from each other (this model is adapted from Raeithel, see [1] for details).

    In what is usually called action-research (which shapes 'innovation in the classroom'), there is a na´ve problem solver (usually the teacher) who considers the meaning of the observed to be inherent, and who is not able to (or not interested in) building a symbolic structure inseparable from the perceived reality.

    In what is known as the classical science approach (which not only inspired 'laboratory observation of processes' but also influenced by the French paradigm) there is a detached observer (usually different from the teacher) who aims at understanding the flow of activity by means of modelling the process in order to cope with its complexity.
    In what we call 'research for innovation', there is a participant observer (the teacher-researcher), who develops a split between observer and observed in a dialogical relation. This is not an easy task, and it can be studied only over time, by analysing (self-analysing from the teacher's perspective) the development of the relationship between teaching and research activity during classroom work.

    In the research studies that have been carried out until now, the legacies of the different traditions are evident. From 'concept based didactics', the interest in non-trivial pieces of mathematical knowledge and in their epistemological analysis is taken; from 'innovation in the classroom' the interest in the design and the implementation of teaching experiments is taken; from 'laboratory observation of processes' the interest in borrowing or inventing analytical tools for the study of classroom processes is taken. The influence of the French paradigm is visible in the tendency to produce a set of didactic theories, connected to each other, concerning different aspects of classroom process. A further elaboration of this point can be found in [1].

    2.4. An Example. Theorems in School: from History and Epistemology to Cognition

    To give a more specific idea of the type of 'research for innovation' outlined above, an example is presented. It is an ambitious project that involves four different teams (each including university researchers - namely F. Arzarello, M. Bartolini Bussi, P. Boero and M. A. Mariotti - with their co-workers and several teacher-researchers from all school levels) and which has already got a place in the international literature (the list of publications is in included in [3]). The project is still in progress and will lead to an articulated exposition expected for 1999-2000. By reconstructing some elements of the historical genesis of this project, it is possible to have an idea of the collaborative way of working developed not only within a team but also between different teams.

    In the early nineties research teams at the universities of Genoa (P. Boero), Modena (M. Bartolini Bussi), Pisa (M. A. Mariotti) and Turin (F. Arzarello) started to work independently on the problem of proof. There was a shared need to counteract the documented international trend of cancelling theorems and proofs from mathematics curricula as a reaction to the formal approach. According to a legacy of the Italian tradition, attention was paid mainly (yet not exclusively) to geometry theorems. Some theoretical constructs developed earlier were assumed by all the teams, implicitly or explicitly, to help in the design, the implementation and the analysis of classroom experiments, whose aim was to create the conditions for most pupils to become able to produce proofs. The theoretical constructs concerned the setting of students' activity (see the construct of field of experience developed by Boero, [5]) and the quality of classroom interaction (see the construct of mathematical discussion developed by Bartolini Bussi [5] ). Exploratory studies were produced at different school levels (from primary to secondary school). The presence of the teachers was decisive in each phase (design, implementation, collection of data and analysis). Their sensitivity and competence proved to be essential not only in the careful management of classroom activity but also in the elaboration of analytical tools and of the theoretical framework. Last but not least, while taking part in the design of experiments, the teachers were put in the condition of deepening some issues concerning the theoretical dimension of mathematics and its relationship with experiential reality. in other words, the theoretical dimension of mathematics became part of the intellectual life of teachers.

    Generally speaking, most of the teaching experiments developed in the project shared (and continue to share) some common features, from the design phase to the implementation in the classroom:

    1. the selection, on the basis of historico-epistemological analysis, of fields of experience, rich in concrete and semantically pregnant referents (e. g. perspective drawing; sunshadows; Cabri-constructions; gears; linkages and drawing instruments);

    2. the design of tasks, within each field of experience, which require the students to take part in the whole process of production of conjectures, of construction of proofs and of generation of a theoretical organisation of the subject matter;

    3. the use of a variety of forms of classroom organisation (e. g. individual problem solving, small group work, classroom discussion orchestrated by the teacher, lectures);

    4. the explicit introduction of primary sources from the history of mathematics into the classroom at any school level.

    The outcomes of the teaching experiments were astonishing when compared with the general insistence on the difficulty (or the impossibility) of coping with the theoretical dimension of mathematics. Most of students even in compulsory education (e. g. grades 5-8) succeeded in producing conjectures and constructing proofs. Were these studies action research based, the process could have stopped here with the production and documentation of facts, i. e. paradigmatic examples of improvement in mathematics teaching. But the second aim of 'research for innovation' concerned the study of the conditions for realisation of such innovation, as well as the possible factors underlying effectiveness; in other words, this success had to be treated as a didactic phenomenon.

    This created the need to framing, in an explicit way, the existing studies within a theoretical framework that allowed for the interpretation of them in a unitary way and for the suggestion of issues for a research agenda (see [5]). Two exemplary elements of the theoretical framework are described below.

    On the basis of historico-epistemological analysis, a mathematical theorem is conceived as the system constituted by three interrelated elements: a statement (i.e. the conjecture produced through experiments and argumentations), a proof (i.e. the special case of argumentation that is accepted by the mathematical community) and a reference theory (including postulates and deduction rules - i.e. meta-theory). This conception emphasises the importance that students are confronted with the entirety of this complexity rather than with the mechanical repetition of given proofs. The cognitive unity is meant as the continuity between the processes of conjecture production and proof construction, recognisable in the close correspondence between the nature and the objects of the mental activities involved. This theoretical construct is, on the one side, a formidable tool for designing activities within the reach of students and, on the other side, a pointer of the difficulty, for analysing activities and for understanding some of the reasons for success and failure.

    3. Some Open Problems

    The different traditions in which 'research for innovation' is rooted represented different attitudes towards the problems of the impact on the educational system. Basic research usually does not address this issue, leaving the problem of wider applications to others. In the two internal traditions, in contrast, the issue was addressed, but only with an optimistic faith in teachers: in 'concept-based didactics', the teachers, provided with better pre- and in-service education were expected to be able to realise change; in 'innovation in the classroom', the teachers, provided with good opportunities for collaborative work, were expected to be able to passprofessional competence on to each other.

    In 'research for innovation' both traditions coexist: the strong involvement of the members (both academic researchers and teacher-researchers) of the NRDs in the design of pre-service and in-service teacher education is rooted in the former tradition, whilst the involvement of new teachers is obtained by passing professional competence on to each other. However, with the increasing theorisation of the field, the attitude towards the teaching experiments has changed: from projects to be disseminated as such to research-type settings for the production of results to be disseminated.

    The constitution of the NRDs was a social and cultural phenomenon that gradually expanded to the creation of a research network throughout the country. It must be said that this phenomenon was neither demanded nor initiated or controlled by the institutional agencies (like the universities, the schools or the Ministry of Education). This relative independence, on the one hand, gave, for years, an immense freedom to involve enthusiastic persons as volunteers, to generate new ideas and to cooperate intellectually with experts from outside the standard agencies, but created, on the other hand, the conditions for a scarce acknowledgedment (if any) of the image and the role of this new generation of academic researchers and teacher-researchers which had consequences for years.

    Whilst the academic researchers are supposed to have found their own place within the community of mathematicians, the most problematic issue seems to concern teachers-researchers. They are now overbusy with their school activity (with no reduction of the schedule), with research activity, with teacher training activity. In the absence of institutional acknowledgement of what they do in the research field, their social status highly depends still on the individual attitude in playing a fundamental role within their own schools.

    Some deep changes are in progress or expected soon in the Italian school system: a restructuration of cycles, and, as far as pre-service teacher education is concerned, an undergraduate university course for pre-primary and primary teachers and a postgraduate course for secondary teachers. In these courses an institutional role will be played also by expert teachers, who will be involved in practical lessons and training. The research activity of the NRDs has prepared not only a substantial corpus of research results on the teaching and learning of mathematics but also a lot of expert teachers. This creates, in a sense, a privileged situation for mathematics, because there no other research network in the country with the same diffusion and international acknowledgement for the other school subjects.

    However optimism is untimely. The true impact on the education system of either the dissemination of research results or the activity of teacher-researchers will depend on additional variables that are neither under the control of academic researchers in didactics nor of teacher-researchers (and not even of mathematicians). Just to quote one of those involved: teaching is often (at least in Italy) conceived as a 'part time' job and not as a complex, important and demanding intellectual profession, where the teacher represents the role of science and knowledge with respect to human existence.

    4. References

    [1] Arzarello F. & Bartolini Bussi M., 'Italian Trends in Research in Mathematical education. A National Case Study from an International Perspective', in Sierpinska A. & Kilpatrick J. (eds.), Mathematics Education as a Research Domain: A Search for Identity, book 2 (241-262), Kluwer Academic Publishers: 1998.

    [2] Barra M., Ferrari M., Furinghetti F., Malara N. A., Speranza F. (eds.), The Italian Research in Mathematics Education : Common Roots and Present Trends, Rome : Consiglio Nazionale delle Ricerche : 1992.

    [3] Bartolini Bussi M., 'Drawing Instruments: Theories and Practices from History to Didactics', Documenta Mathematica - Extra Volume ICM 1998 - III - 735-746.

    [4] Malara N. A. , Menghini M. & Reggiani M. (eds.), Italian Research in Mathematics Education 1988 - 1995, Rome: Consiglio Nazionale delle Ricerche: 1996.

    [5] Mariotti M. A., Bartolini Bussi M., Boero P., Ferri F. & Garuti R. (1997), 'Approaching Geometry Theorems in Contexts: From History and Epistemology to Cognition', in Proc. XXI PME International Conference, vol. 1, 180-195, Lahti (Finland).

     

    Maria Bartolini Bussi
    Dept. ofMathematics
    via G. Campi 213/b
    I 41100 Modena
    ITALY
    e-mail: bartolini@unimo.it